Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 895784 11 pageshttpdxdoiorg1011552013895784

Research ArticleQuantity Discount Supply Chain Models with FashionProducts and Uncertain Yields

Hongjun Peng and Meihua Zhou

School of Management China University of Mining and Technology Xuzhou Jiangsu 221116 China

Correspondence should be addressed to Meihua Zhou mhzhou63126com

Received 11 October 2012 Revised 3 December 2012 Accepted 7 December 2012

Academic Editor Tsan-Ming Choi

Copyright copy 2013 H Peng and M Zhou This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper explores the quantity discount coordination models in the fashion supply chain with uncertain yields and randomdemand The paper proves that under the independent and noncoordinated decision patterns there exists a Nash equilibriumbetween the supplier and the manufacturer which reduces the supply chainrsquos profit margin In order to achieve the ldquooptimalrdquocentralized supply chain expected profit margin new quantity discount models have been established Both the supplier-orientedand the manufacturer-oriented Stackelberg supply chain gaming models are investigated Our analytical and numerical analysesshow that the quantity discount contract proposed in this paper can largely reduce the negative influence brought by the uncertaintyof yields and demand Therefore the profit margin of supply chains based on quantity discount can reach the optimal level of thesupply chain under the centralized setting

1 Introduction

Supply chain coordination has been the central theme for themajority of supply chainmanagement researches Efforts havebeen made in the development of coordination mechanismsaiming to allow a decentralized supply chain to perform aseffectively as a centralized one by aligning chain membersrsquoobjectives with a chain-wide objective [1] Efforts have alsobeenmade in exploring sourcing strategies in a decentralizedsupply chain as multiple sourcing is not uncommon in vari-ous industrial sectors [2ndash5] and in analyzing the dynamics ofinteraction among chain members [6ndash8]

This paper adds to these growing efforts of improvingsupply chain effectiveness by analyzing the quantity discountcontract in a fashion supply chain characterized by theuncertainty in two-echelon yields and random demandnamely the uncertainty in the production of raw materialsand in the production of finished goods (as Figure 1)

The coordination of a supply chain with themanufactureroutput uncertain especially with uncertainity in two-echelonyields and demand has not researched The optimal plannedproduction and the ordering quantity have a close bearingon the uncertainty in two-echelon yields and demand [9]

It is of important significance to research the coordinationmechanism of supply chain with uncertainity in two-echelonyields and demandThis papermainly researches the quantitydiscount contract andmodel in the fashion supply chain withuncertainty in two-echelon yields and demand It analyseshow to determine the qualification point of quantity discountand the discount rate so as to reach the profit level of thesupply chains based on centralized decision making

2 Literature Review

Quantity discounts have received significant attention in theliteratures and numerous inventory models have consideredquantity discounts [10ndash14] Some literatures showed thatthe application of all-unit discounts contributes to reducingthe buyerrsquos inventory cost and improving the supplierrsquosprofit simultaneously [15ndash20] Amixed integer programmingapproachwas presented for solving themultiperiod inventoryproblem with price quantity discount [17 20ndash23] Rezaei andDavoodi [24] considered inventory lot sizing and imperfectitems in a supply chain and proposed a MIP for supplierselection under an environment with multiple suppliersand multiple products with quantity discounts Duan et al

2 Mathematical Problems in Engineering

Supply chain

Finished goods Raw materials

Uncertainty in production Uncertainty in demand Uncertainty in production

Supplier Manufacturer Demand

Figure 1 Supply chain with uncertainty in two-echelon yields and demand

[25] considered a single-vendor single-buyer supply chainfor a fixed lifetime product and proposed decision makingmodels to analyze the benefit of coordinating the supplychain through a quantity discount strategy Munson andHu [14] presented methodologies to calculate the optimalorder quantities and computed the total purchasing andthe inventory costs when products have either all-unit orincremental quantity discounts Chen and Ho [26] presentedan analysis method for the single-period inventory problemwith fuzzy demands and incremental quantity discounts withthe objective of minimizing the total cost per unit time Linand Ho [27] aimed to find the optimal pricing and orderingstrategies for an integrated inventory system when a quantitydiscount policy is present and the market demand is pricesensitive Krichen et al [28] constructed a single suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in payments A decisionrule that helps each retailer to enumerate the preferredcoalitions was proposed and an algorithm that generatedcoalition structures in the core was presented Shi et al[29] aimed to maximize the expected profit of the retailerthrough jointly determining the ordering quantities and sell-ing prices for the products with the consideration of all-unitsquantity discounts and newsvendor pricing The problemwas formulated as a generalized disjunctive programmingmodel and a Lagrangian heuristic approach was developedTaleizadeh et al [23] studied a multiproduct multiconstraintjoint-replenishment inventory control problem in which anincremental discount policy was used and transportationclearance fixed order holding and shortage costs wereconsidered

Almost all the papers researched the quantity discountcontract in the assumption that the supplier was in thedominant position In fact to a supply chain made up ofsuppliers and manufacturers the manufacturers may pre-dominate in the negotiations of quantity discount by decidingthe qualification point and price discount rate Secondlymost quantity discount models consider deterministic yieldsand demand assuming that the buyer faces no risk whenaccepting quantity discounts from the supplier [30]

Uncertainty on the supply side was researched by severalpapers [31ndash42] Tomlin [41] for instance studied a firmrsquosdisruption-management strategy in a dual-sourcing settingin which its two independent suppliers have heterogeneouscapacities reliabilities flexibilities and cost structures Hsiehand Wu [35] studied coordination mechanisms in a supply

chain which consists of two suppliers with capacity uncer-tainties selling differential yet substitutable products througha common retailer who faces price-sensitive random demandof these two products Keren [38] explored a special form ofthe single-period inventory problem (newsvendor problem)with a known demand and stochastic supply (yield) in thesupply chain with a supplier and a distributor He and Zhang[32 33] studied a simple supply chain with one supplier andone retailer where there was random-yield production anduncertain demand Yan et al [40] studied the coordinationof a decentralized assembly system in which the demand ofthe assembler is deterministic and the component yields arerandom Kelle et al [37] focused on the inventory-relatedcosts that can be influenced by adjusting the ordering setupand delivery policy to the random yield The yield model ofhaving a random proportion of defective items is assumedwith known mean and variance

The papers above mainly touched upon the issue ofoptimized operation and coordination under one supplierand one distributor In the supply chain the supply bearsuncertainty The papers above did not research the operationor coordination of supply chainwith themanufacturer outputuncertain especially with uncertain two-echelon yields anddemand To deal with the decision-making problems ofproducing and ordering in the supply chain which are char-acterized by the uncertainty two-echelon yields and randomdemand Peng and Zhou [9] established inventory models ofsupply chains including both centralized and decentralizeddecision makingThe study showed that the optimal plannedproduction and ordering quantity have a close bearing on therisks in two-echelon yields and demand It is thus worthwhileto further research the coordination mechanism and themodel of supply chainwith uncertainity in two-echelon yieldsand demand

This paper mainly deals with the quantity discountcontract in a fashion supply chain when two-echelon yieldsand demand are nondetermined at the same time Amodel ofquantity discount coordination has been established accord-ingly intended to figure out the optimal planned productionby the supplier and the optimal ordering quantity by themanufacturer under the quantity discount contract It alsoanalyses how to determine the qualification point of quantitydiscount and the discount rate so as to reach the same profitmargin as the centralized decision making

This paper contributes to the current literature in thefollowing aspects

Mathematical Problems in Engineering 3

Firstly available literatures mainly cover supply chainsconsisting of suppliers and distributors with the productionfrom the former bearing uncertainty This paper deals witha supply chain consisting of one supplier and one manu-facturer among which the production of two-echelon yieldsand the demand of the products bear uncertainty Finishedgoods are produced after the manufacturer has ordered theraw materials and the uncertainty appears in the processof production In supply chains as such the supplier andthe manufacturer are faced up with a more complex anduncertain environment in operation and decision makingwith its operational processes and coordination mechanismslargely differentiated from supply chains consisting of onesupplier and one distributor

Secondly a quantity discount model has been establishedand two sets of the Stackelberg game of quantity discounthave been put forward on the basis of a supplier-orientedpattern and a manufacturer-oriented pattern respectively

Thirdly the result shows that quantity discount contractproposed by this paper can largely reduce the negativeinfluence imposed by the uncertainty of two-echelon yieldsand demand upon the market benefits of the supply chainsTherefore the profit margin of supply chains based onquantity discount can reach the profit level of the supplychains based on centralized decision making

3 Presumptions and Markings of Model

In the analysis the major presumptions are as follows

(1) The research design is based on a two-level fashionsupply chain systemconsisting of a single supplier anda single manufacturer The supplier produces the rawmaterial which is to be sold to the manufacturer andthe latter then produces the finished products

(2) There exist uncertainty and independent attribute inall the three elementsmdashthe output of the supplier theoutput of the manufacturer and the market demandof the finished products In the meanwhile the pre-condition is set that the expected mean value of theactual output by the supplier and the manufactureramounts to the planned output

(3) JIT is adopted as the mode of supply and the deliveryquantity by the supplier does not exceed the orderingquantity by the manufacturer

(4) The output ratio from the rawmaterial to the finishedproducts is 1 1 specifically one unit of finishedproducts requires one unit of raw material If theoutput ratio is not as designed it can be realizedthrough adaptation of the model Therefore theratio hypothesized would not affect the result of theexperiment

(5) The supplier and the manufacturer would have tooperate abiding by the Stackelberg game This paperrespectively analyzes the quantity discount contractson the condition either when the supplier is set as thedominant side or when the manufacturer is set as thedominant side

Table 1 Instruction for variables

Variable Definitionℎ0 Inventory cost for raw material per unit

ℎ1 Inventory cost for finished products perunit

1199030 Shortage cost for raw material per unit1199031 Shortage cost for finished product per unit1198880 Cost of production of raw material per unit

1198881 Cost of production of finished product perunit

1199010 Price of the raw material per unit1199011 Price of finished product per unit1205870 Total profit of the supply chain1205871 Profit for the supplier1205872 Profit for the manufacturer1199020 Planned output of the supplier1199021 Ordering quantity by the manufacturerX Random variables by the supplierY Random variable by the manufacturerZ Random variables of market demandD Expectation of the market demand1198911(119909) Probability density function of X1198912(119910) Probability density function of Y119892(119911) Probability density function of Z119902lowast Qualification point of quantity discount120572 Discount rate of the price

U Independent and non-coordinateddecision

C Centralized decisionQD Quantity discount decision

Major variables are shown in Table 1

4 Analysis of the Necessity ofthe Coordination of the SupplyChain When Two-Echelon Yieldsand Demand Are Uncertain

41 Production and Ordering Decision Making by an Indepen-dent and Noncoordinated Decision-Making Pattern Whenthe supplier and the manufacturer work in an independentand noncoordinated fashion the decision for output andordering is made by the standard of maximizing their profitsThe manufacturer first orders a certain amount of rawmaterial 1199021 from the supplier and then the latter preparesthe production quantity 1199020

Two-echelon yields and demand in the supply chain areuncertain at the same time Suppose that the actual output is1199020119883 according to hypothesis (3) the amount of raw materialreceived by the manufacturer is no more than the orderingquantity so the final production output is min(1199021 1199020 sdot 119883) sdot 119884The market demand of the finished products is 119863 sdot 119885 then

4 Mathematical Problems in Engineering

the expected value of 119885 is 1 According to hypothesis (2) theexpected values of119883 119884 are 1

Under the independent and noncoordinated decision-making pattern the profit gained by the supplier is

1205871198801 (1199020) = 1199010min [1199021 1199020 sdot 119883] minus 11988801199020 minus ℎ0max [1199020 sdot 119883 minus 1199021 0]minus 1199030max [1199021 minus 1199020 sdot 119883 0]

(1)

The first term is its sales revenue the second term is theproduction cost of the supplier the third term is the inventorycost of the raw material by the supplier and the fourth termis the shortage cost of the raw material The expected profitby the supplier is

119864 [1205871198801 (1199020)] = 1199010 [int119902111990200

(1199020 sdot 119909) 1198911 (119909) 119889119909+int+infin11990211199020

11990211198911 (119909) 119889119909]minus 11988801199020minusℎ0 int+infin

11990211199020

(1199020 sdot 119909 minus 1199021) 1198911 (119909) 119889119909minus 1199030 int119902111990200

(1199021 minus 1199020 sdot 119909) 1198911 (119909) 119889119909

(2)

Proposition 1 (1) Under the independent and noncoordi-nated decision-making pattern the expected profit by thesupplier 119864(1205871198801 ) is concave in 1199020 and the optimal plannedoutput of the raw material 1199021198800 conforms to the followingformula

int119902111990211988000

1199091198911 (119909) 119889119909 = 1198880 + ℎ01199010 + ℎ0 + 1199030 = 1 minus1199010 + 1199030 minus 11988801199010 + ℎ0 + 1199030 (3)

(2) 1199021198800 = 1205821199021 where 120582 is determined by 1199010 ℎ0 1199030 1198880 and1198911(sdot)Proof (1)119889119864 [1205871198801 (1199020)]1198891199020 = (1199010 + ℎ0 + 1199030) int11990211199020

01199091198911 (119909) 119889119909 minus 1198880 minus ℎ0

1198892119864 [1205871198801 (1199020)]11988911990220 = minus (1199010 + ℎ0 + 1199030) 11990221119902301198911 (

11990211199020) lt 0(4)

So 119864(1205871198801 ) is concave in 1199020 and setting (119889119864(1205871198801 ))1198891199020 = 0 wecan derive (3)(2) All we need to show is that function 1198791(119906) =int1199060119909119891(119909)119889119909 is a monotone function of 119906 which is obvious

from 1198891198791119889119906 = 119906119891(119906) gt 0 Therefore (2) has a uniquesolution and 11990211199020 is denoted as 1120582 here

Under the independent and noncoordinated decisionmaking pattern the expected profits by the manufacturer are

1205871198802 (1199021) = 1199011min [119863 sdot 119885 119896 sdotmin (1199021 1199020 sdot 119883) sdot 119884]minus 1199010min [1199021 1199020 sdot 119883] minus 1198881min (1199021 1199020 sdot 119883)minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minusmin (1199021 1199020 sdot 119883) sdot 119884 0]

(5)

The first term is the manufacturerrsquos sales revenue thesecond term is the ordering cost for the rawmaterial the thirdterm is the production cost the fourth term is the inventorycost and the fifth term is the shortage cost

The manufacturer finds out the optimal order quan-tity where the supplierrsquos problem of finding the optimalproduction quantity is a constraint in the mathematicalprogramming problem

max1199021

119864 [1205871198802 (1199021)]= 1199011 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881) [int11990211199020

011990201199091198911 (119909) 119889119909

+int+infin11990211199020

11990211198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909times int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909timesint+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]

Mathematical Problems in Engineering 5

minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909times int11986311991111990201199090

(119863119911 minus 1199020119909119910)1198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909timesint11986311991111990210

(119863119911minus1199021119910)1198911 (119909) 1198912 (119910) 119889119910]st 1199021198800 = 1205821199021

(6)

Proposition 2 Under the independent and noncoordinateddecision-making pattern the manufacturerrsquos expected profitfunction 119864(1205871198801 ) is concave in 1199021 and the optimal orderquantity 1199021198801 satisfies the following equation

int+infin0

119892 (119911) 119889119911 [int11205820

119889119909int1198631199111205821199021198801 1199090

1205821199091199101198911 (119909) 1198912 (119910) 119889119910

+int+infin1120582

119889119909int11986311991111990211988010

1199101198911 (119909) 1198912 (119910) 119889119910]

= (1199010 + 1198881 + ℎ1) [int11205820 1205821199091198911 (119909) 119889119909 + int+infin1120582 1198911 (119909) 119889119909]1199011 + 1199031 + ℎ1 (7)

Proof Since 1199021198800 (1199021) = 1205821199021

119889119864 [1205872]1198891199021 = (1199011 + 1199031 + ℎ1) int+infin0

119892 (119911) 119889119911times [int11205820

119889119909int11986311991112058211990211199090

1205821199091199101198911 (119909) 1198912 (119910) 119889119910+int+infin1120582

119889119909int11986311991111990210

1199101198911 (119909) 1198912 (119910) 119889119910]minus 120582 (1199010 + 1198881 + ℎ1)+ (1199010 + 1198881) int+infin

1120582(120582119909 minus 1) 1198911 (119909) 119889119909

(8)

1198892119864 [1205872]11988911990221 = minus (1199011 + ℎ1 + 1199031) int+infin0

119892 (119911) 119889119911

times [int11205820

(119863119911)212058211990231119909 1198911 (119909) 1198912 (1198631199111205821199021119909)119889119909

+int+infin1120582

(119863119911)211990231 1198911 (119909) 1198912 (1198631199111199021 )119889119909] lt 0(9)

So 119864(1205871198802 ) is concave in 1199021 and setting (119889119864(1205871198802 ))1198891199021 = 0 wehave (7)

42 Production and Ordering Decision Making Guided bya Centralized Decision-Making Pattern In the centralizedsupply chain the goal of the supply chain operation is tomaximize the total profit The manufacturer directly conveysthe demand information to the supplier In the centralizedcase the shortage cost of the supplier can be seen as amatter of an internal revenue transfer which barely affectsthe decision making in production and ordering The profitgenerated by the supply chain dependent on the centralizeddecision making is

1205871198620 = 1199011min [119863 sdot 119885min (1199021 1199020 sdot 119883) sdot 119884]minus 1198881min (1199021 1199020 sdot 119883) minus 11988801199020 minus ℎ0max [1199020 sdot 119883 minus 1199021 0]minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minusmin (1199021 1199020 sdot 119883) sdot 119884 0]

(10)

The operational goal is to realize the maximized profit inthis supply chain and the operational model is

max 119864 [1205871198620 ]= 1199011 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910

+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]

minus 11988801199020 minus ℎ0 int+infin11990211199020

(1199020 sdot 119909 minus 1199021) 1198911 (119909) 119889119909

6 Mathematical Problems in Engineering

minus 1198881 [int119902111990200

11990201199091198911 (119909) 119889119909 + int+infin11990211199020

11990211198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910] (11)

Proposition 3 Consider 119864[1205871198620 ] ge 119864[1205871198801 ] + 119864[1205871198802 ]Proof From (1) (5) and (10) we can obtain

119864 [1205871198620 ] ge 119864 [1205871 (1199021198620 )] + 119864 [1205872 (1199021198621 )] = max1199020 1199021

119864 [1205870 (1199020 1199021)]ge 119864 [1205871 (1199021198800 )] + 119864 [1205872 (1199021198801 )] = 119864 [1205871198801 ] + 119864 [1205871198802 ]

(12)

Proposition 3 illustrates that under the independent andnoncoordinated decision making pattern the Nash equilib-rium existing between the supplier and the manufacturerreduces the profit of the supply chain In order to fulfillthe profit margin in the centralized supply chains under theindependent decision-making pattern there is a necessity todesign an efficient coordinating mechanism

5 Production and Ordering Decision underQuantity Discount Coordination

According to (7) the ordering quantity by the manufacturerdecreases with the price of the raw material In this sense aquantity discount contract amounts to the reduction of theordering price of the raw material through which the order-ing quantity by themanufacturer can be promotedThe studyis based on the quantity discount contract conforming towhich qualification point of quantity discount 119902lowast and pricediscount rate 120572 can be established When the amount of theraw material provided by the supplier to the manufacturer isless than 119902lowast the price is set at 1199010 when the amount of theraw material exceeds 119902lowast the price of the excess part is set at(1 minus 120572)1199011 where 0 lt 120572 lt 1

Under the quantity discount contract the profit of thesupplier is given as follows

120587QD1 (1199020) = 1199010min [1199021 1199020119883]

minus 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 11988801199020 minus ℎ0max [1199020119883 minus 1199021 0]minus 1199030max [1199021 minus 1199020119883 0]

(13)

The first term is the sales revenue of the supplier at theprice of 1199010 the second term is the price discount given tomanufacturer on the basis of quantity discount the thirdterm is the production cost of the supplier the fourth termis the inventory cost of the supplier and the fifth term is theshortage cost of the raw material and the sixth term is theshortage cost of the raw material

Under the quantity discount contract the expected profitof the supplier is

119864 [120587QD1 (1199020)]= 1199010 [int11990211199020

0(1199020119909)1198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909] minus 1205721199010times [int11990211199020119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus 11988801199020 minus ℎ0 int+infin

11990211199020

(1199020119909 minus 1199021) 1198911 (119909) 119889119909minus 1199030 int119902111990200

(1199021 minus 1199020119909)1198911 (119909) 119889119909(14)

Proposition 4 Under the quantity discount contract thesupplierrsquos expected profit function 119864(120587QD1 ) is concave in 1199020and the optimal production quantity of the raw material 119902QD0 satisfies the following equation

int119902111990200

1199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(15)

Mathematical Problems in Engineering 7

Proof Since

119889119864 [120587QD1 (1199020)]1198891199020 = (1199010 + ℎ0 + 1199030) int11990211199020

01199091198911 (119909) 119889119909

minus 1205721199010 int11990211199020119902lowast1199020

1199091198911 (119909) 119889119909 minus 1198880 minus ℎ01198892119864 [120587QD

1 (1199020)]11988911990220 = minus (1199010 + ℎ0 + 1199030 minus 1205721199010) 11990221119902301198911 (

11990211199020)

minus 1205721199010 119902lowast211990230 1198911 (119902lowast1199020 ) lt 0

(16)

So 119864(120587QD1 ) is concave in 1199020 and setting (120597119864(120587QD

1 ))1205971199020 = 0we have (15)

Under the quantity discount contract the profit of themanufacturer is

120587QD1 (1199021) = 1199011min [119863 sdot 119885min (1199021 1199020 sdot 119883) sdot 119884]

minus 1199010min [1199021 1199020 sdot 119883]+ 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 1198881 [min (1199021 1199020 sdot 119883)]minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minus (min (1199021 1199020 sdot 119883) sdot 119884) 0]

(17)

The first term is the sales revenue of the manufacturerthe second term is the ordering cost of the rawmaterial at theprice of 1199010 the third term is the price discount based on thequantity discount contract the fourth term is the productioncost of the manufacturer the fifth term is the inventory costof the manufacturer and the sixth term is the shortage cost ofthe manufacturer

The manufacturer operates under the constraint restric-tions of the supplier and it precedes the decision making ofproduction and ordering by the standard of maximizing theprofits

max1199021

119864 [120587QD2 (1199021)]

= 1199011 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910

+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881) [int11990211199020

011990201199091198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909]+ 1205721199010 [int11990211199020

119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910) 1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910]st int11990211199020

01199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int

11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(18)

Setting 119889119864[120587QD2 (1199021)]1198891199021 = 0 we obtain the optimal

order quantity 119902QD1 which satisfies the following equation

(1199011 + ℎ1 + 1199031) int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

119902101584001199091199101198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

1199101198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881 + ℎ1 minus 1205721199010)times [int119902111990200

119902101584001199091198911 (119909) 119889119909 + int+infin11990211199020

1198911 (119909) 119889119909]

minus 1205721199010 int119902lowast1199020

0119902101584001199091198911 (119909) 119889119909 = 0

(19)

The 1199020 is a function of 1199021 from (15) and 11990210158400 is 1199020rsquos derivativeby 1199021

8 Mathematical Problems in Engineering

Table 2 Numerical results for all models (119898 = 015 119899 = 01 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4256 4197 1014 74585 146830 221415

C centralized decision mdash mdash 4351 4351 1 mdash mdash 222105

QD quantity discount withthe supplier as a dominantparty

01 3853 4415 4387 1006 75275 146830 22210502 4221 4307 4251 1013 75275 146830 22210503 4231 4307 4251 1013 75275 146830 22210504 4236 4307 4251 1013 75275 146830 222105

QD quantity discount withthe manufacturer as adominant party

01 4004 4310 4267 101 74585 147520 22210502 4033 4397 4381 1004 74585 147520 22210503 4173 4310 4270 1009 74585 147520 22210504 4197 4392 4379 1003 74585 147520 222105

6 The Design of Quantity Discount Contracts

The actual producing activities can be boiled down to theissue of the Stackelberg game problem Quantity discountcontracts under both the supplier as the dominant side andthe manufacturer as the dominant side are to be discussedseparately The principle of designing the pact is as follows

(1) In the Stackelberg game the dominant side fulfills thequantity discount coordination of the supply chainthrough the determination of the quantity discountqualification point and the price discount rate

(2) Considering the rationality of the follower the dom-inant side for the acceptance of the strategy by thefollower should ensure that the interest of the followershould be no less than the interest gained under thenoncoordinated pattern

(3) Based on the fulfillment of the above conditionsthe expected profit of the dominant side should bemaximized

61 The Quantity Discount Contract with the Supplier as theDominant Side Under the uncertainty of the two-echelonyields and demand provided that the supplier holds thedominant position with themanufacturer being the followerthe supplier is in the position to determine the parameterpackage of the quantity discount including the quantitydiscount qualification point 119902lowast and the price discount rate120572 so as to realize the coordination in the supply chain Themodel is as follows

max 119864 (120587QD1 )

st Equation (19)119864 (120587QD2 ) ge 119864 (1205871198802 )

(20)

For the first item as the supplier holds the dominantposition the goal is to achieve the maximized profit of thesupplier in the decision making The second item represents

the ordering strategy of the manufacturer the third itemguarantees the profits of the manufacturer should not be lessthan that under independent and noncoordinated decisionmaking pattern

62 The Quantity Discount Contract with the Manufactureras the Dominant Party Under the uncertainty of the two-echelon yields and demand in supply chains given thatthe manufacturer is in the dominant position in quantitydiscount and coordination with the supplier in a secondaryposition the manufacturer determines the parameter pack-age of quantity discount including the quantity discountqualification point and the price discount rate so as to realizecoordination in supply chains The model is shown as below

max 119864 (120587QD2 )

st Equation (15)119864 (120587QD1 ) ge 119864 (1205871198801 )

(21)

The first item shows that the goal to be achieved is tomaximize the profit of the manufacturer in the decisionmaking the second item shows that the ordering strategyis provided by the supplier the third item shows that thesupplierrsquos profit level should be kept no less than the levelunder the independent andnoncoordinated decision-makingpattern

7 Numerical Examples and Discussions

The parameters of a certain supply chain are set as below1198880 = 20 ℎ0 = 5 1199030 = 15 1198881 = 10 ℎ1 = 7 1199031 = 301199010 = 40 1199011 = 95 119863 = 400 The output of the supplier andthe output of the manufacturer random variables for marketdemand of the finished products are uniformly distributedover [1minus119898 1+119898] [1minus119899 1+119899] and [1minus 119897 1+ 119897] respectively

Tables 2ndash4 present the decision making for productionand ordering and the expected profits under an independent

Mathematical Problems in Engineering 9

Table 3 Numerical results for all models (119898 = 02 119899 = 02 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4345 4293 1012 71085 141007 212093

C centralized decision mdash mdash 4537 4537 1 mdash mdash 216542

QD quantity discount withthe supplier as a dominantparty

01 4125 4592 4558 1007 75535 141007 21654202 4330 4590 4558 1007 75535 141007 21654203 4433 4515 4460 1012 75535 141007 21654204 4440 4515 4460 1012 75535 141007 216542

QD quantity discount withthe manufacturer as adominant party

01 3517 4534 4534 1 71869 144673 21654202 4058 4533 4532 1 72759 143783 21654203 4222 4532 4531 1 73007 143535 21654204 4302 4532 4531 1 73123 143419 216542

Table 4 Numerical results for all models (119898 = 015 119899 = 01 119897 = 03)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4291 4232 1014 75196 125231 200427

C centralized decision mdash mdash 4387 4387 1 mdash mdash 201123

QD quantity discount withthe supplier as a dominantparty

01 4227 4342 4286 1013 75892 125231 20112302 4256 4342 4286 1013 75892 125231 20112303 4266 4342 4286 1013 75892 125231 20112304 4271 4342 4286 1013 75892 125231 201123

QD quantity discount withthe manufacturer as adominant party

01 4036 4346 4305 101 75196 125927 20112302 4162 4346 4305 101 75196 125927 20112303 4207 4346 4305 101 75196 125927 20112304 4231 4346 4305 101 75196 125927 201123

and noncoordinated decision-making pattern and the cen-tralized decision making pattern under the the uncertaintyof the two-echelon yields and demand they also respectivelypresent quantity discount coordination decisionmaking withthe supplier as the dominant role solving by model (20) andthe quantity discount coordination decision making with themanufacturer as the dominant role solving by model (21)

With Tables 2ndash4 the following conclusions can be drawn

(1) Under an independent decision-making patternthere exists the Nash equilibrium between the sup-plier and the manufacturer and the profit under theNash equilibrium is less than that gained under thecentralized decision-making pattern under the quan-tity discount pact themanufacturer has increased theordering quantity of the raw material the ratios ofthe planned output of the supplier to the orderingquantity of the manufacturer have decreased

(2) Under the quantity discount pact with the supplierin the dominant position the supplier sets different

combinations of the quantity discount qualificationpoint and the price discount rate under the situationwhere the profit of the manufacturer under the quan-tity discount coordination pact exceeds that underindependent and noncoordinated decision-makingpattern In this sense the profit of the supply chainbased on quantity discount reaches the profit level ofthe chain under a centralized pattern The supplierand the manufacturer can select a certain combina-tion and set forth a quantity discount coordinationpact

(3) Under the quantity discount coordination with themanufacturer in the dominant position the manu-facturer sets different combinations as well With theprofit of the supply chain under the quantity discountcoordination pact exceeding the profit under theindependent and noncoordinated decision-makingpattern the chain profit based on quantity discountreaches the profit level gained in a centralized patternThe supplier and themanufacturer can select a certain

10 Mathematical Problems in Engineering

combination and set forth a quantity discount coordi-nation pact

(4) With a comparison of Tables 2 3 and 4 it can befound that the larger the uncertainty of two-echelonyields and demand the larger the contribution of thequantity discount coordination pact to the profit ofthe supply chains With 119898 = 015 119899 = 01 and 119897 =02 the quantity discount coordination pact helpsincrease the total profit by 03 With 119898 = 015119899 = 01 and 119897 = 03 the quantity discountcoordination pact helps increase the total profit by04 With 119898 = 02 119899 = 02 and 119897 = 02 thequantity discount coordination pact helps increasethe total profit by 21 This shows that the quantitydiscount coordination contract can effectively reducethe negative effect by the uncertainty of two-echelonyields and demand upon the efficiency and benefit ofthe supply chains

(5) We also find out that undermost scenarios the supplychain benefits from the yields and demand risksreduction and generates a higher profit ContrastingTables 2 and 3 we find out that sometimes in thequantity discount contract the supplierrsquos profit canincrease as yieldsrsquo randomness increases And con-trasting Tables 2 and 4 we find out that in the unco-ordinated case and quantity discount contract thesupplierrsquos profit can increase as demand randomnessincreases

8 Conclusions

This paper explores the quantity discount coordinationmod-els in the fashion supply chain with uncertain yields andrandom demand The planned production outputs of thesupplier and the ordering quantity of the manufacturer havebeen set as the decision-making parameters The indepen-dent and noncoordinated and centralized decision-makingpatterns have been established under the uncertainty of two-echelon yields and demandThrough the analysis it has beenfound that the profit of supply chains under independentand noncoordinated decision-making pattern is less than thatunder the centralized pattern which proves the necessityof the coordination of supply chains Based on this twoparametersmdashquantity discount qualification point and pricediscount ratiomdashhave been introduced and quantity discountcoordination models have been established The quantitydiscount coordination pacts with both the supplier and themanufacturer in the dominant position have been set forth

The result reveals that the quantity discount coordinationpact proposed by this paper can promote the manufacturerrsquosordering quantity of the raw material which contributesto the result that the profit of the supply chain based onquantity discount reaches the profit level of the supply chainunder the centralized decision-making pattern The largerthe uncertainty of two-echelon yields and demand the largercontribution of the pact to the benefit of the supply chainwill be The quantity discount coordination pact under theuncertainty of two-echelon yields and demand can reduce the

negative influence brought by the uncertainty and thereforecan contribute to the purpose of the coordination of thesupply chain

Acknowledgments

The authors would like to thank the referees for theirconstructive comments and suggestions that have greatlyimproved the quality of the paper The research was sup-ported by the National Natural Science Foundation of China(Grant no 71201164)

References

[1] D J Thomas and P M Griffin ldquoCoordinated supply chainmanagementrdquo European Journal of Operational Research vol94 no 1 pp 1ndash15 1996

[2] N Agrawal S A Smith andA A Tsay ldquoMulti-vendor sourcingin a retail supply chainrdquo Production and Operations Manage-ment vol 11 no 2 pp 157ndash181 2002

[3] C Chiang and W C Benton ldquoSole sourcing versus dualsourcing under stochastic demands and lead timesrdquo NavalResearch Logistics vol 41 no 5 pp 609ndash624 1994

[4] J D Hong and J C Hayya ldquoJust-in-time purchasing singleor multiple sourcingrdquo International Journal of ProductionEconomics vol 27 no 2 pp 175ndash181 1992

[5] M Treleven and S B Schweikhart ldquoA riskbenefit analysisof sourcing strategies single vs multiple sourcingrdquo Journal ofOperations Management vol 7 no 3-4 pp 93ndash114 1988

[6] G P Cachon and P H Zipkin ldquoCompetitive and cooperativeinventory policies in a two-stage supply chainrdquo ManagementScience vol 45 no 7 pp 936ndash953 1999

[7] M Leng and M Parlar ldquoGame theoretic applications in supplychainmanagement a reviewrdquo INFOR vol 43 no 3 pp 187ndash2202005

[8] MMoses and S Seshadri ldquoPolicy mechanisms for supply chaincoordinationrdquo IIETransactions vol 32 no 3 pp 245ndash262 2000

[9] H J Peng and M H Zhou ldquoProduction and ordering decisionin supply chain with uncertainty in two-echelon yields anddemandrdquo Journal of Systems Engineering vol 25 no 5 pp 622ndash628 2010 (Chinese)

[10] C T Chang ldquoAn acquisition policy for a single item multi-supplier system with real-world constraintsrdquo Applied Mathe-matical Modelling vol 30 no 1 pp 1ndash9 2006

[11] C T Chang C L Chin and M F Lin ldquoOn the single itemmulti-supplier system with variable lead-time price-quantitydiscount and resource constraintsrdquo Applied Mathematics andComputation vol 182 no 1 pp 89ndash97 2006

[12] J Li and L Liu ldquoSupply chain coordination with quantitydiscount policyrdquo International Journal of Production Economicsvol 101 no 1 pp 89ndash98 2006

[13] C L Munson and M J Rosenblatt ldquoTheories and realitiesof quantity discounts an exploratory studyrdquo Production andOperations Management vol 7 no 4 pp 352ndash369 1998

[14] C L Munson and J Hu ldquoIncorporating quantity discountsand their inventory impacts into the centralized purchasingdecisionrdquoEuropean Journal ofOperational Research vol 201 no2 pp 581ndash592 2010

[15] H Y Kang and A H I Lee ldquoInventory replenishment modelusing fuzzy multiple objective programming a case study

Mathematical Problems in Engineering 11

of a high-tech company in Taiwanrdquo Applied Soft ComputingJournal vol 10 no 4 pp 1108ndash1118 2010

[16] A Mendoza and J A Ventura ldquoIncorporating quantity dis-counts to the EOQ model with transportation costsrdquo Interna-tional Journal of Production Economics vol 113 no 2 pp 754ndash765 2008

[17] J Moussourakis and C Haksever ldquoA practical model forordering in multi-product multi-constraint inventory systemswith all-units quantity discountsrdquo International Journal ofInformation and Management Sciences vol 19 no 2 pp 263ndash283 2008

[18] P A Rubin and W C Benton ldquoA generalized framework forquantity discount pricing schedulesrdquo Decision Sciences vol 34no 1 pp 173ndash188 2003

[19] G Zhang and L Ma ldquoOptimal acquisition policy with quantitydiscounts and uncertain demandsrdquo International Journal ofProduction Research vol 47 no 9 pp 2409ndash2425 2009

[20] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[21] A H I Lee and H Y Kang ldquoA mixed 0-1 integer programmingfor inventory model a case study of TFT-LCD manufacturingcompany in Taiwanrdquo Kybernetes vol 37 no 1 pp 66ndash82 2008

[22] J Shi and G Zhang ldquoMulti-product budget-constrained acqui-sition and pricing with uncertain demand and supplier quantitydiscountsrdquo International Journal of Production Economics vol128 no 1 pp 322ndash331 2010

[23] A A Taleizadeh S T A Niaki and R Nikousokhan ldquoCon-straint multiproduct joint-replenishment inventory controlproblem using uncertain programmingrdquo Applied Soft Comput-ing vol 11 pp 5143ndash5154 2011

[24] J Rezaei and M Davoodi ldquoA deterministic multi-item inven-tory model with supplier selection and imperfect qualityrdquoApplied Mathematical Modelling vol 32 no 10 pp 2106ndash21162008

[25] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[26] S P Chen and Y H Ho ldquoAnalysis of the newsboy problemwith fuzzy demands and incremental discountsrdquo InternationalJournal of Production Economics vol 129 no 1 pp 169ndash177 2011

[27] Y J Lin and C H Ho ldquoIntegrated inventory model withquantity discount and price-sensitive demandrdquo Top vol 19 no1 pp 177ndash188 2011

[28] S Krichen A Laabidi and F B Abdelaziz ldquoSingle suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in paymentsrdquo Computers andIndustrial Engineering vol 60 no 1 pp 164ndash172 2011

[29] J Shi G Zhang and J Sha ldquoJointly pricing and ordering fora multi-product multi-constraint newsvendor problem withsupplier quantity discountsrdquo Applied Mathematical Modellingvol 35 no 6 pp 3001ndash3011 2011

[30] H Shin and W C Benton ldquoA quantity discount approach tosupply chain coordinationrdquo European Journal of OperationalResearch vol 180 no 2 pp 601ndash616 2007

[31] H Gurnani and Y Gerchak ldquoCoordination in decentralizedassembly systems with uncertain component yieldsrdquo EuropeanJournal of Operational Research vol 176 no 3 pp 1559ndash15762007

[32] Y He and J Zhang ldquoRandom yield risk sharing in a two-levelsupply chainrdquo International Journal of Production Economicsvol 112 no 2 pp 769ndash781 2008

[33] Y He and J Zhang ldquoRandom yield supply chain with a yielddependent secondary marketrdquo European Journal of OperationalResearch vol 206 no 1 pp 221ndash230 2010

[34] C C Hsieh and C H Wu ldquoCapacity allocation ordering andpricing decisions in a supply chain with demand and supplyuncertaintiesrdquo European Journal of Operational Research vol184 no 2 pp 667ndash684 2008

[35] C C Hsieh and C H Wu ldquoCoordinated decisions for substi-tutable products in a common retailer supply chainrdquo EuropeanJournal of Operational Research vol 196 no 1 pp 273ndash2882009

[36] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[37] P Kelle S Transchel and S Minner ldquoBuyer-supplier coopera-tion and negotiation support with random yield considerationrdquoInternational Journal of Production Economics vol 118 no 1 pp152ndash159 2009

[38] B Keren ldquoThe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmegavol 37 no 4 pp 801ndash810 2009

[39] P Kouvelis and J M Milner ldquoSupply chain capacity andoutsourcing decisions the dynamic interplay of demand andsupply uncertaintyrdquo IIE Transactions vol 34 no 8 pp 717ndash7282002

[40] X Yan M Zhang and K Liu ldquoA note on coordination indecentralized assembly systems with uncertain componentyieldsrdquo European Journal of Operational Research vol 205 no2 pp 469ndash478 2010

[41] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[42] K Zimmer ldquoSupply chain coordination with uncertain just-in-time deliveryrdquo International Journal of Production Economicsvol 77 no 1 pp 1ndash15 2002

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Supply chain

Finished goods Raw materials

Uncertainty in production Uncertainty in demand Uncertainty in production

Supplier Manufacturer Demand

Figure 1 Supply chain with uncertainty in two-echelon yields and demand

[25] considered a single-vendor single-buyer supply chainfor a fixed lifetime product and proposed decision makingmodels to analyze the benefit of coordinating the supplychain through a quantity discount strategy Munson andHu [14] presented methodologies to calculate the optimalorder quantities and computed the total purchasing andthe inventory costs when products have either all-unit orincremental quantity discounts Chen and Ho [26] presentedan analysis method for the single-period inventory problemwith fuzzy demands and incremental quantity discounts withthe objective of minimizing the total cost per unit time Linand Ho [27] aimed to find the optimal pricing and orderingstrategies for an integrated inventory system when a quantitydiscount policy is present and the market demand is pricesensitive Krichen et al [28] constructed a single suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in payments A decisionrule that helps each retailer to enumerate the preferredcoalitions was proposed and an algorithm that generatedcoalition structures in the core was presented Shi et al[29] aimed to maximize the expected profit of the retailerthrough jointly determining the ordering quantities and sell-ing prices for the products with the consideration of all-unitsquantity discounts and newsvendor pricing The problemwas formulated as a generalized disjunctive programmingmodel and a Lagrangian heuristic approach was developedTaleizadeh et al [23] studied a multiproduct multiconstraintjoint-replenishment inventory control problem in which anincremental discount policy was used and transportationclearance fixed order holding and shortage costs wereconsidered

Almost all the papers researched the quantity discountcontract in the assumption that the supplier was in thedominant position In fact to a supply chain made up ofsuppliers and manufacturers the manufacturers may pre-dominate in the negotiations of quantity discount by decidingthe qualification point and price discount rate Secondlymost quantity discount models consider deterministic yieldsand demand assuming that the buyer faces no risk whenaccepting quantity discounts from the supplier [30]

Uncertainty on the supply side was researched by severalpapers [31ndash42] Tomlin [41] for instance studied a firmrsquosdisruption-management strategy in a dual-sourcing settingin which its two independent suppliers have heterogeneouscapacities reliabilities flexibilities and cost structures Hsiehand Wu [35] studied coordination mechanisms in a supply

chain which consists of two suppliers with capacity uncer-tainties selling differential yet substitutable products througha common retailer who faces price-sensitive random demandof these two products Keren [38] explored a special form ofthe single-period inventory problem (newsvendor problem)with a known demand and stochastic supply (yield) in thesupply chain with a supplier and a distributor He and Zhang[32 33] studied a simple supply chain with one supplier andone retailer where there was random-yield production anduncertain demand Yan et al [40] studied the coordinationof a decentralized assembly system in which the demand ofthe assembler is deterministic and the component yields arerandom Kelle et al [37] focused on the inventory-relatedcosts that can be influenced by adjusting the ordering setupand delivery policy to the random yield The yield model ofhaving a random proportion of defective items is assumedwith known mean and variance

The papers above mainly touched upon the issue ofoptimized operation and coordination under one supplierand one distributor In the supply chain the supply bearsuncertainty The papers above did not research the operationor coordination of supply chainwith themanufacturer outputuncertain especially with uncertain two-echelon yields anddemand To deal with the decision-making problems ofproducing and ordering in the supply chain which are char-acterized by the uncertainty two-echelon yields and randomdemand Peng and Zhou [9] established inventory models ofsupply chains including both centralized and decentralizeddecision makingThe study showed that the optimal plannedproduction and ordering quantity have a close bearing on therisks in two-echelon yields and demand It is thus worthwhileto further research the coordination mechanism and themodel of supply chainwith uncertainity in two-echelon yieldsand demand

This paper mainly deals with the quantity discountcontract in a fashion supply chain when two-echelon yieldsand demand are nondetermined at the same time Amodel ofquantity discount coordination has been established accord-ingly intended to figure out the optimal planned productionby the supplier and the optimal ordering quantity by themanufacturer under the quantity discount contract It alsoanalyses how to determine the qualification point of quantitydiscount and the discount rate so as to reach the same profitmargin as the centralized decision making

This paper contributes to the current literature in thefollowing aspects

Mathematical Problems in Engineering 3

Firstly available literatures mainly cover supply chainsconsisting of suppliers and distributors with the productionfrom the former bearing uncertainty This paper deals witha supply chain consisting of one supplier and one manu-facturer among which the production of two-echelon yieldsand the demand of the products bear uncertainty Finishedgoods are produced after the manufacturer has ordered theraw materials and the uncertainty appears in the processof production In supply chains as such the supplier andthe manufacturer are faced up with a more complex anduncertain environment in operation and decision makingwith its operational processes and coordination mechanismslargely differentiated from supply chains consisting of onesupplier and one distributor

Secondly a quantity discount model has been establishedand two sets of the Stackelberg game of quantity discounthave been put forward on the basis of a supplier-orientedpattern and a manufacturer-oriented pattern respectively

Thirdly the result shows that quantity discount contractproposed by this paper can largely reduce the negativeinfluence imposed by the uncertainty of two-echelon yieldsand demand upon the market benefits of the supply chainsTherefore the profit margin of supply chains based onquantity discount can reach the profit level of the supplychains based on centralized decision making

3 Presumptions and Markings of Model

In the analysis the major presumptions are as follows

(1) The research design is based on a two-level fashionsupply chain systemconsisting of a single supplier anda single manufacturer The supplier produces the rawmaterial which is to be sold to the manufacturer andthe latter then produces the finished products

(2) There exist uncertainty and independent attribute inall the three elementsmdashthe output of the supplier theoutput of the manufacturer and the market demandof the finished products In the meanwhile the pre-condition is set that the expected mean value of theactual output by the supplier and the manufactureramounts to the planned output

(3) JIT is adopted as the mode of supply and the deliveryquantity by the supplier does not exceed the orderingquantity by the manufacturer

(4) The output ratio from the rawmaterial to the finishedproducts is 1 1 specifically one unit of finishedproducts requires one unit of raw material If theoutput ratio is not as designed it can be realizedthrough adaptation of the model Therefore theratio hypothesized would not affect the result of theexperiment

(5) The supplier and the manufacturer would have tooperate abiding by the Stackelberg game This paperrespectively analyzes the quantity discount contractson the condition either when the supplier is set as thedominant side or when the manufacturer is set as thedominant side

Table 1 Instruction for variables

Variable Definitionℎ0 Inventory cost for raw material per unit

ℎ1 Inventory cost for finished products perunit

1199030 Shortage cost for raw material per unit1199031 Shortage cost for finished product per unit1198880 Cost of production of raw material per unit

1198881 Cost of production of finished product perunit

1199010 Price of the raw material per unit1199011 Price of finished product per unit1205870 Total profit of the supply chain1205871 Profit for the supplier1205872 Profit for the manufacturer1199020 Planned output of the supplier1199021 Ordering quantity by the manufacturerX Random variables by the supplierY Random variable by the manufacturerZ Random variables of market demandD Expectation of the market demand1198911(119909) Probability density function of X1198912(119910) Probability density function of Y119892(119911) Probability density function of Z119902lowast Qualification point of quantity discount120572 Discount rate of the price

U Independent and non-coordinateddecision

C Centralized decisionQD Quantity discount decision

Major variables are shown in Table 1

4 Analysis of the Necessity ofthe Coordination of the SupplyChain When Two-Echelon Yieldsand Demand Are Uncertain

41 Production and Ordering Decision Making by an Indepen-dent and Noncoordinated Decision-Making Pattern Whenthe supplier and the manufacturer work in an independentand noncoordinated fashion the decision for output andordering is made by the standard of maximizing their profitsThe manufacturer first orders a certain amount of rawmaterial 1199021 from the supplier and then the latter preparesthe production quantity 1199020

Two-echelon yields and demand in the supply chain areuncertain at the same time Suppose that the actual output is1199020119883 according to hypothesis (3) the amount of raw materialreceived by the manufacturer is no more than the orderingquantity so the final production output is min(1199021 1199020 sdot 119883) sdot 119884The market demand of the finished products is 119863 sdot 119885 then

4 Mathematical Problems in Engineering

the expected value of 119885 is 1 According to hypothesis (2) theexpected values of119883 119884 are 1

Under the independent and noncoordinated decision-making pattern the profit gained by the supplier is

1205871198801 (1199020) = 1199010min [1199021 1199020 sdot 119883] minus 11988801199020 minus ℎ0max [1199020 sdot 119883 minus 1199021 0]minus 1199030max [1199021 minus 1199020 sdot 119883 0]

(1)

The first term is its sales revenue the second term is theproduction cost of the supplier the third term is the inventorycost of the raw material by the supplier and the fourth termis the shortage cost of the raw material The expected profitby the supplier is

119864 [1205871198801 (1199020)] = 1199010 [int119902111990200

(1199020 sdot 119909) 1198911 (119909) 119889119909+int+infin11990211199020

11990211198911 (119909) 119889119909]minus 11988801199020minusℎ0 int+infin

11990211199020

(1199020 sdot 119909 minus 1199021) 1198911 (119909) 119889119909minus 1199030 int119902111990200

(1199021 minus 1199020 sdot 119909) 1198911 (119909) 119889119909

(2)

Proposition 1 (1) Under the independent and noncoordi-nated decision-making pattern the expected profit by thesupplier 119864(1205871198801 ) is concave in 1199020 and the optimal plannedoutput of the raw material 1199021198800 conforms to the followingformula

int119902111990211988000

1199091198911 (119909) 119889119909 = 1198880 + ℎ01199010 + ℎ0 + 1199030 = 1 minus1199010 + 1199030 minus 11988801199010 + ℎ0 + 1199030 (3)

(2) 1199021198800 = 1205821199021 where 120582 is determined by 1199010 ℎ0 1199030 1198880 and1198911(sdot)Proof (1)119889119864 [1205871198801 (1199020)]1198891199020 = (1199010 + ℎ0 + 1199030) int11990211199020

01199091198911 (119909) 119889119909 minus 1198880 minus ℎ0

1198892119864 [1205871198801 (1199020)]11988911990220 = minus (1199010 + ℎ0 + 1199030) 11990221119902301198911 (

11990211199020) lt 0(4)

So 119864(1205871198801 ) is concave in 1199020 and setting (119889119864(1205871198801 ))1198891199020 = 0 wecan derive (3)(2) All we need to show is that function 1198791(119906) =int1199060119909119891(119909)119889119909 is a monotone function of 119906 which is obvious

from 1198891198791119889119906 = 119906119891(119906) gt 0 Therefore (2) has a uniquesolution and 11990211199020 is denoted as 1120582 here

Under the independent and noncoordinated decisionmaking pattern the expected profits by the manufacturer are

1205871198802 (1199021) = 1199011min [119863 sdot 119885 119896 sdotmin (1199021 1199020 sdot 119883) sdot 119884]minus 1199010min [1199021 1199020 sdot 119883] minus 1198881min (1199021 1199020 sdot 119883)minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minusmin (1199021 1199020 sdot 119883) sdot 119884 0]

(5)

The first term is the manufacturerrsquos sales revenue thesecond term is the ordering cost for the rawmaterial the thirdterm is the production cost the fourth term is the inventorycost and the fifth term is the shortage cost

The manufacturer finds out the optimal order quan-tity where the supplierrsquos problem of finding the optimalproduction quantity is a constraint in the mathematicalprogramming problem

max1199021

119864 [1205871198802 (1199021)]= 1199011 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881) [int11990211199020

011990201199091198911 (119909) 119889119909

+int+infin11990211199020

11990211198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909times int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909timesint+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]

Mathematical Problems in Engineering 5

minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909times int11986311991111990201199090

(119863119911 minus 1199020119909119910)1198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909timesint11986311991111990210

(119863119911minus1199021119910)1198911 (119909) 1198912 (119910) 119889119910]st 1199021198800 = 1205821199021

(6)

Proposition 2 Under the independent and noncoordinateddecision-making pattern the manufacturerrsquos expected profitfunction 119864(1205871198801 ) is concave in 1199021 and the optimal orderquantity 1199021198801 satisfies the following equation

int+infin0

119892 (119911) 119889119911 [int11205820

119889119909int1198631199111205821199021198801 1199090

1205821199091199101198911 (119909) 1198912 (119910) 119889119910

+int+infin1120582

119889119909int11986311991111990211988010

1199101198911 (119909) 1198912 (119910) 119889119910]

= (1199010 + 1198881 + ℎ1) [int11205820 1205821199091198911 (119909) 119889119909 + int+infin1120582 1198911 (119909) 119889119909]1199011 + 1199031 + ℎ1 (7)

Proof Since 1199021198800 (1199021) = 1205821199021

119889119864 [1205872]1198891199021 = (1199011 + 1199031 + ℎ1) int+infin0

119892 (119911) 119889119911times [int11205820

119889119909int11986311991112058211990211199090

1205821199091199101198911 (119909) 1198912 (119910) 119889119910+int+infin1120582

119889119909int11986311991111990210

1199101198911 (119909) 1198912 (119910) 119889119910]minus 120582 (1199010 + 1198881 + ℎ1)+ (1199010 + 1198881) int+infin

1120582(120582119909 minus 1) 1198911 (119909) 119889119909

(8)

1198892119864 [1205872]11988911990221 = minus (1199011 + ℎ1 + 1199031) int+infin0

119892 (119911) 119889119911

times [int11205820

(119863119911)212058211990231119909 1198911 (119909) 1198912 (1198631199111205821199021119909)119889119909

+int+infin1120582

(119863119911)211990231 1198911 (119909) 1198912 (1198631199111199021 )119889119909] lt 0(9)

So 119864(1205871198802 ) is concave in 1199021 and setting (119889119864(1205871198802 ))1198891199021 = 0 wehave (7)

42 Production and Ordering Decision Making Guided bya Centralized Decision-Making Pattern In the centralizedsupply chain the goal of the supply chain operation is tomaximize the total profit The manufacturer directly conveysthe demand information to the supplier In the centralizedcase the shortage cost of the supplier can be seen as amatter of an internal revenue transfer which barely affectsthe decision making in production and ordering The profitgenerated by the supply chain dependent on the centralizeddecision making is

1205871198620 = 1199011min [119863 sdot 119885min (1199021 1199020 sdot 119883) sdot 119884]minus 1198881min (1199021 1199020 sdot 119883) minus 11988801199020 minus ℎ0max [1199020 sdot 119883 minus 1199021 0]minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minusmin (1199021 1199020 sdot 119883) sdot 119884 0]

(10)

The operational goal is to realize the maximized profit inthis supply chain and the operational model is

max 119864 [1205871198620 ]= 1199011 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910

+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]

minus 11988801199020 minus ℎ0 int+infin11990211199020

(1199020 sdot 119909 minus 1199021) 1198911 (119909) 119889119909

6 Mathematical Problems in Engineering

minus 1198881 [int119902111990200

11990201199091198911 (119909) 119889119909 + int+infin11990211199020

11990211198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910] (11)

Proposition 3 Consider 119864[1205871198620 ] ge 119864[1205871198801 ] + 119864[1205871198802 ]Proof From (1) (5) and (10) we can obtain

119864 [1205871198620 ] ge 119864 [1205871 (1199021198620 )] + 119864 [1205872 (1199021198621 )] = max1199020 1199021

119864 [1205870 (1199020 1199021)]ge 119864 [1205871 (1199021198800 )] + 119864 [1205872 (1199021198801 )] = 119864 [1205871198801 ] + 119864 [1205871198802 ]

(12)

Proposition 3 illustrates that under the independent andnoncoordinated decision making pattern the Nash equilib-rium existing between the supplier and the manufacturerreduces the profit of the supply chain In order to fulfillthe profit margin in the centralized supply chains under theindependent decision-making pattern there is a necessity todesign an efficient coordinating mechanism

5 Production and Ordering Decision underQuantity Discount Coordination

According to (7) the ordering quantity by the manufacturerdecreases with the price of the raw material In this sense aquantity discount contract amounts to the reduction of theordering price of the raw material through which the order-ing quantity by themanufacturer can be promotedThe studyis based on the quantity discount contract conforming towhich qualification point of quantity discount 119902lowast and pricediscount rate 120572 can be established When the amount of theraw material provided by the supplier to the manufacturer isless than 119902lowast the price is set at 1199010 when the amount of theraw material exceeds 119902lowast the price of the excess part is set at(1 minus 120572)1199011 where 0 lt 120572 lt 1

Under the quantity discount contract the profit of thesupplier is given as follows

120587QD1 (1199020) = 1199010min [1199021 1199020119883]

minus 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 11988801199020 minus ℎ0max [1199020119883 minus 1199021 0]minus 1199030max [1199021 minus 1199020119883 0]

(13)

The first term is the sales revenue of the supplier at theprice of 1199010 the second term is the price discount given tomanufacturer on the basis of quantity discount the thirdterm is the production cost of the supplier the fourth termis the inventory cost of the supplier and the fifth term is theshortage cost of the raw material and the sixth term is theshortage cost of the raw material

Under the quantity discount contract the expected profitof the supplier is

119864 [120587QD1 (1199020)]= 1199010 [int11990211199020

0(1199020119909)1198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909] minus 1205721199010times [int11990211199020119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus 11988801199020 minus ℎ0 int+infin

11990211199020

(1199020119909 minus 1199021) 1198911 (119909) 119889119909minus 1199030 int119902111990200

(1199021 minus 1199020119909)1198911 (119909) 119889119909(14)

Proposition 4 Under the quantity discount contract thesupplierrsquos expected profit function 119864(120587QD1 ) is concave in 1199020and the optimal production quantity of the raw material 119902QD0 satisfies the following equation

int119902111990200

1199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(15)

Mathematical Problems in Engineering 7

Proof Since

119889119864 [120587QD1 (1199020)]1198891199020 = (1199010 + ℎ0 + 1199030) int11990211199020

01199091198911 (119909) 119889119909

minus 1205721199010 int11990211199020119902lowast1199020

1199091198911 (119909) 119889119909 minus 1198880 minus ℎ01198892119864 [120587QD

1 (1199020)]11988911990220 = minus (1199010 + ℎ0 + 1199030 minus 1205721199010) 11990221119902301198911 (

11990211199020)

minus 1205721199010 119902lowast211990230 1198911 (119902lowast1199020 ) lt 0

(16)

So 119864(120587QD1 ) is concave in 1199020 and setting (120597119864(120587QD

1 ))1205971199020 = 0we have (15)

Under the quantity discount contract the profit of themanufacturer is

120587QD1 (1199021) = 1199011min [119863 sdot 119885min (1199021 1199020 sdot 119883) sdot 119884]

minus 1199010min [1199021 1199020 sdot 119883]+ 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 1198881 [min (1199021 1199020 sdot 119883)]minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minus (min (1199021 1199020 sdot 119883) sdot 119884) 0]

(17)

The first term is the sales revenue of the manufacturerthe second term is the ordering cost of the rawmaterial at theprice of 1199010 the third term is the price discount based on thequantity discount contract the fourth term is the productioncost of the manufacturer the fifth term is the inventory costof the manufacturer and the sixth term is the shortage cost ofthe manufacturer

The manufacturer operates under the constraint restric-tions of the supplier and it precedes the decision making ofproduction and ordering by the standard of maximizing theprofits

max1199021

119864 [120587QD2 (1199021)]

= 1199011 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910

+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881) [int11990211199020

011990201199091198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909]+ 1205721199010 [int11990211199020

119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910) 1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910]st int11990211199020

01199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int

11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(18)

Setting 119889119864[120587QD2 (1199021)]1198891199021 = 0 we obtain the optimal

order quantity 119902QD1 which satisfies the following equation

(1199011 + ℎ1 + 1199031) int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

119902101584001199091199101198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

1199101198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881 + ℎ1 minus 1205721199010)times [int119902111990200

119902101584001199091198911 (119909) 119889119909 + int+infin11990211199020

1198911 (119909) 119889119909]

minus 1205721199010 int119902lowast1199020

0119902101584001199091198911 (119909) 119889119909 = 0

(19)

The 1199020 is a function of 1199021 from (15) and 11990210158400 is 1199020rsquos derivativeby 1199021

8 Mathematical Problems in Engineering

Table 2 Numerical results for all models (119898 = 015 119899 = 01 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4256 4197 1014 74585 146830 221415

C centralized decision mdash mdash 4351 4351 1 mdash mdash 222105

QD quantity discount withthe supplier as a dominantparty

01 3853 4415 4387 1006 75275 146830 22210502 4221 4307 4251 1013 75275 146830 22210503 4231 4307 4251 1013 75275 146830 22210504 4236 4307 4251 1013 75275 146830 222105

QD quantity discount withthe manufacturer as adominant party

01 4004 4310 4267 101 74585 147520 22210502 4033 4397 4381 1004 74585 147520 22210503 4173 4310 4270 1009 74585 147520 22210504 4197 4392 4379 1003 74585 147520 222105

6 The Design of Quantity Discount Contracts

The actual producing activities can be boiled down to theissue of the Stackelberg game problem Quantity discountcontracts under both the supplier as the dominant side andthe manufacturer as the dominant side are to be discussedseparately The principle of designing the pact is as follows

(1) In the Stackelberg game the dominant side fulfills thequantity discount coordination of the supply chainthrough the determination of the quantity discountqualification point and the price discount rate

(2) Considering the rationality of the follower the dom-inant side for the acceptance of the strategy by thefollower should ensure that the interest of the followershould be no less than the interest gained under thenoncoordinated pattern

(3) Based on the fulfillment of the above conditionsthe expected profit of the dominant side should bemaximized

61 The Quantity Discount Contract with the Supplier as theDominant Side Under the uncertainty of the two-echelonyields and demand provided that the supplier holds thedominant position with themanufacturer being the followerthe supplier is in the position to determine the parameterpackage of the quantity discount including the quantitydiscount qualification point 119902lowast and the price discount rate120572 so as to realize the coordination in the supply chain Themodel is as follows

max 119864 (120587QD1 )

st Equation (19)119864 (120587QD2 ) ge 119864 (1205871198802 )

(20)

For the first item as the supplier holds the dominantposition the goal is to achieve the maximized profit of thesupplier in the decision making The second item represents

the ordering strategy of the manufacturer the third itemguarantees the profits of the manufacturer should not be lessthan that under independent and noncoordinated decisionmaking pattern

62 The Quantity Discount Contract with the Manufactureras the Dominant Party Under the uncertainty of the two-echelon yields and demand in supply chains given thatthe manufacturer is in the dominant position in quantitydiscount and coordination with the supplier in a secondaryposition the manufacturer determines the parameter pack-age of quantity discount including the quantity discountqualification point and the price discount rate so as to realizecoordination in supply chains The model is shown as below

max 119864 (120587QD2 )

st Equation (15)119864 (120587QD1 ) ge 119864 (1205871198801 )

(21)

The first item shows that the goal to be achieved is tomaximize the profit of the manufacturer in the decisionmaking the second item shows that the ordering strategyis provided by the supplier the third item shows that thesupplierrsquos profit level should be kept no less than the levelunder the independent andnoncoordinated decision-makingpattern

7 Numerical Examples and Discussions

The parameters of a certain supply chain are set as below1198880 = 20 ℎ0 = 5 1199030 = 15 1198881 = 10 ℎ1 = 7 1199031 = 301199010 = 40 1199011 = 95 119863 = 400 The output of the supplier andthe output of the manufacturer random variables for marketdemand of the finished products are uniformly distributedover [1minus119898 1+119898] [1minus119899 1+119899] and [1minus 119897 1+ 119897] respectively

Tables 2ndash4 present the decision making for productionand ordering and the expected profits under an independent

Mathematical Problems in Engineering 9

Table 3 Numerical results for all models (119898 = 02 119899 = 02 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4345 4293 1012 71085 141007 212093

C centralized decision mdash mdash 4537 4537 1 mdash mdash 216542

QD quantity discount withthe supplier as a dominantparty

01 4125 4592 4558 1007 75535 141007 21654202 4330 4590 4558 1007 75535 141007 21654203 4433 4515 4460 1012 75535 141007 21654204 4440 4515 4460 1012 75535 141007 216542

QD quantity discount withthe manufacturer as adominant party

01 3517 4534 4534 1 71869 144673 21654202 4058 4533 4532 1 72759 143783 21654203 4222 4532 4531 1 73007 143535 21654204 4302 4532 4531 1 73123 143419 216542

Table 4 Numerical results for all models (119898 = 015 119899 = 01 119897 = 03)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4291 4232 1014 75196 125231 200427

C centralized decision mdash mdash 4387 4387 1 mdash mdash 201123

QD quantity discount withthe supplier as a dominantparty

01 4227 4342 4286 1013 75892 125231 20112302 4256 4342 4286 1013 75892 125231 20112303 4266 4342 4286 1013 75892 125231 20112304 4271 4342 4286 1013 75892 125231 201123

QD quantity discount withthe manufacturer as adominant party

01 4036 4346 4305 101 75196 125927 20112302 4162 4346 4305 101 75196 125927 20112303 4207 4346 4305 101 75196 125927 20112304 4231 4346 4305 101 75196 125927 201123

and noncoordinated decision-making pattern and the cen-tralized decision making pattern under the the uncertaintyof the two-echelon yields and demand they also respectivelypresent quantity discount coordination decisionmaking withthe supplier as the dominant role solving by model (20) andthe quantity discount coordination decision making with themanufacturer as the dominant role solving by model (21)

With Tables 2ndash4 the following conclusions can be drawn

(1) Under an independent decision-making patternthere exists the Nash equilibrium between the sup-plier and the manufacturer and the profit under theNash equilibrium is less than that gained under thecentralized decision-making pattern under the quan-tity discount pact themanufacturer has increased theordering quantity of the raw material the ratios ofthe planned output of the supplier to the orderingquantity of the manufacturer have decreased

(2) Under the quantity discount pact with the supplierin the dominant position the supplier sets different

combinations of the quantity discount qualificationpoint and the price discount rate under the situationwhere the profit of the manufacturer under the quan-tity discount coordination pact exceeds that underindependent and noncoordinated decision-makingpattern In this sense the profit of the supply chainbased on quantity discount reaches the profit level ofthe chain under a centralized pattern The supplierand the manufacturer can select a certain combina-tion and set forth a quantity discount coordinationpact

(3) Under the quantity discount coordination with themanufacturer in the dominant position the manu-facturer sets different combinations as well With theprofit of the supply chain under the quantity discountcoordination pact exceeding the profit under theindependent and noncoordinated decision-makingpattern the chain profit based on quantity discountreaches the profit level gained in a centralized patternThe supplier and themanufacturer can select a certain

10 Mathematical Problems in Engineering

combination and set forth a quantity discount coordi-nation pact

(4) With a comparison of Tables 2 3 and 4 it can befound that the larger the uncertainty of two-echelonyields and demand the larger the contribution of thequantity discount coordination pact to the profit ofthe supply chains With 119898 = 015 119899 = 01 and 119897 =02 the quantity discount coordination pact helpsincrease the total profit by 03 With 119898 = 015119899 = 01 and 119897 = 03 the quantity discountcoordination pact helps increase the total profit by04 With 119898 = 02 119899 = 02 and 119897 = 02 thequantity discount coordination pact helps increasethe total profit by 21 This shows that the quantitydiscount coordination contract can effectively reducethe negative effect by the uncertainty of two-echelonyields and demand upon the efficiency and benefit ofthe supply chains

(5) We also find out that undermost scenarios the supplychain benefits from the yields and demand risksreduction and generates a higher profit ContrastingTables 2 and 3 we find out that sometimes in thequantity discount contract the supplierrsquos profit canincrease as yieldsrsquo randomness increases And con-trasting Tables 2 and 4 we find out that in the unco-ordinated case and quantity discount contract thesupplierrsquos profit can increase as demand randomnessincreases

8 Conclusions

This paper explores the quantity discount coordinationmod-els in the fashion supply chain with uncertain yields andrandom demand The planned production outputs of thesupplier and the ordering quantity of the manufacturer havebeen set as the decision-making parameters The indepen-dent and noncoordinated and centralized decision-makingpatterns have been established under the uncertainty of two-echelon yields and demandThrough the analysis it has beenfound that the profit of supply chains under independentand noncoordinated decision-making pattern is less than thatunder the centralized pattern which proves the necessityof the coordination of supply chains Based on this twoparametersmdashquantity discount qualification point and pricediscount ratiomdashhave been introduced and quantity discountcoordination models have been established The quantitydiscount coordination pacts with both the supplier and themanufacturer in the dominant position have been set forth

The result reveals that the quantity discount coordinationpact proposed by this paper can promote the manufacturerrsquosordering quantity of the raw material which contributesto the result that the profit of the supply chain based onquantity discount reaches the profit level of the supply chainunder the centralized decision-making pattern The largerthe uncertainty of two-echelon yields and demand the largercontribution of the pact to the benefit of the supply chainwill be The quantity discount coordination pact under theuncertainty of two-echelon yields and demand can reduce the

negative influence brought by the uncertainty and thereforecan contribute to the purpose of the coordination of thesupply chain

Acknowledgments

The authors would like to thank the referees for theirconstructive comments and suggestions that have greatlyimproved the quality of the paper The research was sup-ported by the National Natural Science Foundation of China(Grant no 71201164)

References

[1] D J Thomas and P M Griffin ldquoCoordinated supply chainmanagementrdquo European Journal of Operational Research vol94 no 1 pp 1ndash15 1996

[2] N Agrawal S A Smith andA A Tsay ldquoMulti-vendor sourcingin a retail supply chainrdquo Production and Operations Manage-ment vol 11 no 2 pp 157ndash181 2002

[3] C Chiang and W C Benton ldquoSole sourcing versus dualsourcing under stochastic demands and lead timesrdquo NavalResearch Logistics vol 41 no 5 pp 609ndash624 1994

[4] J D Hong and J C Hayya ldquoJust-in-time purchasing singleor multiple sourcingrdquo International Journal of ProductionEconomics vol 27 no 2 pp 175ndash181 1992

[5] M Treleven and S B Schweikhart ldquoA riskbenefit analysisof sourcing strategies single vs multiple sourcingrdquo Journal ofOperations Management vol 7 no 3-4 pp 93ndash114 1988

[6] G P Cachon and P H Zipkin ldquoCompetitive and cooperativeinventory policies in a two-stage supply chainrdquo ManagementScience vol 45 no 7 pp 936ndash953 1999

[7] M Leng and M Parlar ldquoGame theoretic applications in supplychainmanagement a reviewrdquo INFOR vol 43 no 3 pp 187ndash2202005

[8] MMoses and S Seshadri ldquoPolicy mechanisms for supply chaincoordinationrdquo IIETransactions vol 32 no 3 pp 245ndash262 2000

[9] H J Peng and M H Zhou ldquoProduction and ordering decisionin supply chain with uncertainty in two-echelon yields anddemandrdquo Journal of Systems Engineering vol 25 no 5 pp 622ndash628 2010 (Chinese)

[10] C T Chang ldquoAn acquisition policy for a single item multi-supplier system with real-world constraintsrdquo Applied Mathe-matical Modelling vol 30 no 1 pp 1ndash9 2006

[11] C T Chang C L Chin and M F Lin ldquoOn the single itemmulti-supplier system with variable lead-time price-quantitydiscount and resource constraintsrdquo Applied Mathematics andComputation vol 182 no 1 pp 89ndash97 2006

[12] J Li and L Liu ldquoSupply chain coordination with quantitydiscount policyrdquo International Journal of Production Economicsvol 101 no 1 pp 89ndash98 2006

[13] C L Munson and M J Rosenblatt ldquoTheories and realitiesof quantity discounts an exploratory studyrdquo Production andOperations Management vol 7 no 4 pp 352ndash369 1998

[14] C L Munson and J Hu ldquoIncorporating quantity discountsand their inventory impacts into the centralized purchasingdecisionrdquoEuropean Journal ofOperational Research vol 201 no2 pp 581ndash592 2010

[15] H Y Kang and A H I Lee ldquoInventory replenishment modelusing fuzzy multiple objective programming a case study

Mathematical Problems in Engineering 11

of a high-tech company in Taiwanrdquo Applied Soft ComputingJournal vol 10 no 4 pp 1108ndash1118 2010

[16] A Mendoza and J A Ventura ldquoIncorporating quantity dis-counts to the EOQ model with transportation costsrdquo Interna-tional Journal of Production Economics vol 113 no 2 pp 754ndash765 2008

[17] J Moussourakis and C Haksever ldquoA practical model forordering in multi-product multi-constraint inventory systemswith all-units quantity discountsrdquo International Journal ofInformation and Management Sciences vol 19 no 2 pp 263ndash283 2008

[18] P A Rubin and W C Benton ldquoA generalized framework forquantity discount pricing schedulesrdquo Decision Sciences vol 34no 1 pp 173ndash188 2003

[19] G Zhang and L Ma ldquoOptimal acquisition policy with quantitydiscounts and uncertain demandsrdquo International Journal ofProduction Research vol 47 no 9 pp 2409ndash2425 2009

[20] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[21] A H I Lee and H Y Kang ldquoA mixed 0-1 integer programmingfor inventory model a case study of TFT-LCD manufacturingcompany in Taiwanrdquo Kybernetes vol 37 no 1 pp 66ndash82 2008

[22] J Shi and G Zhang ldquoMulti-product budget-constrained acqui-sition and pricing with uncertain demand and supplier quantitydiscountsrdquo International Journal of Production Economics vol128 no 1 pp 322ndash331 2010

[23] A A Taleizadeh S T A Niaki and R Nikousokhan ldquoCon-straint multiproduct joint-replenishment inventory controlproblem using uncertain programmingrdquo Applied Soft Comput-ing vol 11 pp 5143ndash5154 2011

[24] J Rezaei and M Davoodi ldquoA deterministic multi-item inven-tory model with supplier selection and imperfect qualityrdquoApplied Mathematical Modelling vol 32 no 10 pp 2106ndash21162008

[25] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[26] S P Chen and Y H Ho ldquoAnalysis of the newsboy problemwith fuzzy demands and incremental discountsrdquo InternationalJournal of Production Economics vol 129 no 1 pp 169ndash177 2011

[27] Y J Lin and C H Ho ldquoIntegrated inventory model withquantity discount and price-sensitive demandrdquo Top vol 19 no1 pp 177ndash188 2011

[28] S Krichen A Laabidi and F B Abdelaziz ldquoSingle suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in paymentsrdquo Computers andIndustrial Engineering vol 60 no 1 pp 164ndash172 2011

[29] J Shi G Zhang and J Sha ldquoJointly pricing and ordering fora multi-product multi-constraint newsvendor problem withsupplier quantity discountsrdquo Applied Mathematical Modellingvol 35 no 6 pp 3001ndash3011 2011

[30] H Shin and W C Benton ldquoA quantity discount approach tosupply chain coordinationrdquo European Journal of OperationalResearch vol 180 no 2 pp 601ndash616 2007

[31] H Gurnani and Y Gerchak ldquoCoordination in decentralizedassembly systems with uncertain component yieldsrdquo EuropeanJournal of Operational Research vol 176 no 3 pp 1559ndash15762007

[32] Y He and J Zhang ldquoRandom yield risk sharing in a two-levelsupply chainrdquo International Journal of Production Economicsvol 112 no 2 pp 769ndash781 2008

[33] Y He and J Zhang ldquoRandom yield supply chain with a yielddependent secondary marketrdquo European Journal of OperationalResearch vol 206 no 1 pp 221ndash230 2010

[34] C C Hsieh and C H Wu ldquoCapacity allocation ordering andpricing decisions in a supply chain with demand and supplyuncertaintiesrdquo European Journal of Operational Research vol184 no 2 pp 667ndash684 2008

[35] C C Hsieh and C H Wu ldquoCoordinated decisions for substi-tutable products in a common retailer supply chainrdquo EuropeanJournal of Operational Research vol 196 no 1 pp 273ndash2882009

[36] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[37] P Kelle S Transchel and S Minner ldquoBuyer-supplier coopera-tion and negotiation support with random yield considerationrdquoInternational Journal of Production Economics vol 118 no 1 pp152ndash159 2009

[38] B Keren ldquoThe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmegavol 37 no 4 pp 801ndash810 2009

[39] P Kouvelis and J M Milner ldquoSupply chain capacity andoutsourcing decisions the dynamic interplay of demand andsupply uncertaintyrdquo IIE Transactions vol 34 no 8 pp 717ndash7282002

[40] X Yan M Zhang and K Liu ldquoA note on coordination indecentralized assembly systems with uncertain componentyieldsrdquo European Journal of Operational Research vol 205 no2 pp 469ndash478 2010

[41] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[42] K Zimmer ldquoSupply chain coordination with uncertain just-in-time deliveryrdquo International Journal of Production Economicsvol 77 no 1 pp 1ndash15 2002

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Firstly available literatures mainly cover supply chainsconsisting of suppliers and distributors with the productionfrom the former bearing uncertainty This paper deals witha supply chain consisting of one supplier and one manu-facturer among which the production of two-echelon yieldsand the demand of the products bear uncertainty Finishedgoods are produced after the manufacturer has ordered theraw materials and the uncertainty appears in the processof production In supply chains as such the supplier andthe manufacturer are faced up with a more complex anduncertain environment in operation and decision makingwith its operational processes and coordination mechanismslargely differentiated from supply chains consisting of onesupplier and one distributor

Secondly a quantity discount model has been establishedand two sets of the Stackelberg game of quantity discounthave been put forward on the basis of a supplier-orientedpattern and a manufacturer-oriented pattern respectively

Thirdly the result shows that quantity discount contractproposed by this paper can largely reduce the negativeinfluence imposed by the uncertainty of two-echelon yieldsand demand upon the market benefits of the supply chainsTherefore the profit margin of supply chains based onquantity discount can reach the profit level of the supplychains based on centralized decision making

3 Presumptions and Markings of Model

In the analysis the major presumptions are as follows

(1) The research design is based on a two-level fashionsupply chain systemconsisting of a single supplier anda single manufacturer The supplier produces the rawmaterial which is to be sold to the manufacturer andthe latter then produces the finished products

(2) There exist uncertainty and independent attribute inall the three elementsmdashthe output of the supplier theoutput of the manufacturer and the market demandof the finished products In the meanwhile the pre-condition is set that the expected mean value of theactual output by the supplier and the manufactureramounts to the planned output

(3) JIT is adopted as the mode of supply and the deliveryquantity by the supplier does not exceed the orderingquantity by the manufacturer

(4) The output ratio from the rawmaterial to the finishedproducts is 1 1 specifically one unit of finishedproducts requires one unit of raw material If theoutput ratio is not as designed it can be realizedthrough adaptation of the model Therefore theratio hypothesized would not affect the result of theexperiment

(5) The supplier and the manufacturer would have tooperate abiding by the Stackelberg game This paperrespectively analyzes the quantity discount contractson the condition either when the supplier is set as thedominant side or when the manufacturer is set as thedominant side

Table 1 Instruction for variables

Variable Definitionℎ0 Inventory cost for raw material per unit

ℎ1 Inventory cost for finished products perunit

1199030 Shortage cost for raw material per unit1199031 Shortage cost for finished product per unit1198880 Cost of production of raw material per unit

1198881 Cost of production of finished product perunit

1199010 Price of the raw material per unit1199011 Price of finished product per unit1205870 Total profit of the supply chain1205871 Profit for the supplier1205872 Profit for the manufacturer1199020 Planned output of the supplier1199021 Ordering quantity by the manufacturerX Random variables by the supplierY Random variable by the manufacturerZ Random variables of market demandD Expectation of the market demand1198911(119909) Probability density function of X1198912(119910) Probability density function of Y119892(119911) Probability density function of Z119902lowast Qualification point of quantity discount120572 Discount rate of the price

U Independent and non-coordinateddecision

C Centralized decisionQD Quantity discount decision

Major variables are shown in Table 1

4 Analysis of the Necessity ofthe Coordination of the SupplyChain When Two-Echelon Yieldsand Demand Are Uncertain

41 Production and Ordering Decision Making by an Indepen-dent and Noncoordinated Decision-Making Pattern Whenthe supplier and the manufacturer work in an independentand noncoordinated fashion the decision for output andordering is made by the standard of maximizing their profitsThe manufacturer first orders a certain amount of rawmaterial 1199021 from the supplier and then the latter preparesthe production quantity 1199020

Two-echelon yields and demand in the supply chain areuncertain at the same time Suppose that the actual output is1199020119883 according to hypothesis (3) the amount of raw materialreceived by the manufacturer is no more than the orderingquantity so the final production output is min(1199021 1199020 sdot 119883) sdot 119884The market demand of the finished products is 119863 sdot 119885 then

4 Mathematical Problems in Engineering

the expected value of 119885 is 1 According to hypothesis (2) theexpected values of119883 119884 are 1

Under the independent and noncoordinated decision-making pattern the profit gained by the supplier is

1205871198801 (1199020) = 1199010min [1199021 1199020 sdot 119883] minus 11988801199020 minus ℎ0max [1199020 sdot 119883 minus 1199021 0]minus 1199030max [1199021 minus 1199020 sdot 119883 0]

(1)

The first term is its sales revenue the second term is theproduction cost of the supplier the third term is the inventorycost of the raw material by the supplier and the fourth termis the shortage cost of the raw material The expected profitby the supplier is

119864 [1205871198801 (1199020)] = 1199010 [int119902111990200

(1199020 sdot 119909) 1198911 (119909) 119889119909+int+infin11990211199020

11990211198911 (119909) 119889119909]minus 11988801199020minusℎ0 int+infin

11990211199020

(1199020 sdot 119909 minus 1199021) 1198911 (119909) 119889119909minus 1199030 int119902111990200

(1199021 minus 1199020 sdot 119909) 1198911 (119909) 119889119909

(2)

Proposition 1 (1) Under the independent and noncoordi-nated decision-making pattern the expected profit by thesupplier 119864(1205871198801 ) is concave in 1199020 and the optimal plannedoutput of the raw material 1199021198800 conforms to the followingformula

int119902111990211988000

1199091198911 (119909) 119889119909 = 1198880 + ℎ01199010 + ℎ0 + 1199030 = 1 minus1199010 + 1199030 minus 11988801199010 + ℎ0 + 1199030 (3)

(2) 1199021198800 = 1205821199021 where 120582 is determined by 1199010 ℎ0 1199030 1198880 and1198911(sdot)Proof (1)119889119864 [1205871198801 (1199020)]1198891199020 = (1199010 + ℎ0 + 1199030) int11990211199020

01199091198911 (119909) 119889119909 minus 1198880 minus ℎ0

1198892119864 [1205871198801 (1199020)]11988911990220 = minus (1199010 + ℎ0 + 1199030) 11990221119902301198911 (

11990211199020) lt 0(4)

So 119864(1205871198801 ) is concave in 1199020 and setting (119889119864(1205871198801 ))1198891199020 = 0 wecan derive (3)(2) All we need to show is that function 1198791(119906) =int1199060119909119891(119909)119889119909 is a monotone function of 119906 which is obvious

from 1198891198791119889119906 = 119906119891(119906) gt 0 Therefore (2) has a uniquesolution and 11990211199020 is denoted as 1120582 here

Under the independent and noncoordinated decisionmaking pattern the expected profits by the manufacturer are

1205871198802 (1199021) = 1199011min [119863 sdot 119885 119896 sdotmin (1199021 1199020 sdot 119883) sdot 119884]minus 1199010min [1199021 1199020 sdot 119883] minus 1198881min (1199021 1199020 sdot 119883)minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minusmin (1199021 1199020 sdot 119883) sdot 119884 0]

(5)

The first term is the manufacturerrsquos sales revenue thesecond term is the ordering cost for the rawmaterial the thirdterm is the production cost the fourth term is the inventorycost and the fifth term is the shortage cost

The manufacturer finds out the optimal order quan-tity where the supplierrsquos problem of finding the optimalproduction quantity is a constraint in the mathematicalprogramming problem

max1199021

119864 [1205871198802 (1199021)]= 1199011 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881) [int11990211199020

011990201199091198911 (119909) 119889119909

+int+infin11990211199020

11990211198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909times int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909timesint+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]

Mathematical Problems in Engineering 5

minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909times int11986311991111990201199090

(119863119911 minus 1199020119909119910)1198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909timesint11986311991111990210

(119863119911minus1199021119910)1198911 (119909) 1198912 (119910) 119889119910]st 1199021198800 = 1205821199021

(6)

Proposition 2 Under the independent and noncoordinateddecision-making pattern the manufacturerrsquos expected profitfunction 119864(1205871198801 ) is concave in 1199021 and the optimal orderquantity 1199021198801 satisfies the following equation

int+infin0

119892 (119911) 119889119911 [int11205820

119889119909int1198631199111205821199021198801 1199090

1205821199091199101198911 (119909) 1198912 (119910) 119889119910

+int+infin1120582

119889119909int11986311991111990211988010

1199101198911 (119909) 1198912 (119910) 119889119910]

= (1199010 + 1198881 + ℎ1) [int11205820 1205821199091198911 (119909) 119889119909 + int+infin1120582 1198911 (119909) 119889119909]1199011 + 1199031 + ℎ1 (7)

Proof Since 1199021198800 (1199021) = 1205821199021

119889119864 [1205872]1198891199021 = (1199011 + 1199031 + ℎ1) int+infin0

119892 (119911) 119889119911times [int11205820

119889119909int11986311991112058211990211199090

1205821199091199101198911 (119909) 1198912 (119910) 119889119910+int+infin1120582

119889119909int11986311991111990210

1199101198911 (119909) 1198912 (119910) 119889119910]minus 120582 (1199010 + 1198881 + ℎ1)+ (1199010 + 1198881) int+infin

1120582(120582119909 minus 1) 1198911 (119909) 119889119909

(8)

1198892119864 [1205872]11988911990221 = minus (1199011 + ℎ1 + 1199031) int+infin0

119892 (119911) 119889119911

times [int11205820

(119863119911)212058211990231119909 1198911 (119909) 1198912 (1198631199111205821199021119909)119889119909

+int+infin1120582

(119863119911)211990231 1198911 (119909) 1198912 (1198631199111199021 )119889119909] lt 0(9)

So 119864(1205871198802 ) is concave in 1199021 and setting (119889119864(1205871198802 ))1198891199021 = 0 wehave (7)

42 Production and Ordering Decision Making Guided bya Centralized Decision-Making Pattern In the centralizedsupply chain the goal of the supply chain operation is tomaximize the total profit The manufacturer directly conveysthe demand information to the supplier In the centralizedcase the shortage cost of the supplier can be seen as amatter of an internal revenue transfer which barely affectsthe decision making in production and ordering The profitgenerated by the supply chain dependent on the centralizeddecision making is

1205871198620 = 1199011min [119863 sdot 119885min (1199021 1199020 sdot 119883) sdot 119884]minus 1198881min (1199021 1199020 sdot 119883) minus 11988801199020 minus ℎ0max [1199020 sdot 119883 minus 1199021 0]minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minusmin (1199021 1199020 sdot 119883) sdot 119884 0]

(10)

The operational goal is to realize the maximized profit inthis supply chain and the operational model is

max 119864 [1205871198620 ]= 1199011 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910

+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]

minus 11988801199020 minus ℎ0 int+infin11990211199020

(1199020 sdot 119909 minus 1199021) 1198911 (119909) 119889119909

6 Mathematical Problems in Engineering

minus 1198881 [int119902111990200

11990201199091198911 (119909) 119889119909 + int+infin11990211199020

11990211198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910] (11)

Proposition 3 Consider 119864[1205871198620 ] ge 119864[1205871198801 ] + 119864[1205871198802 ]Proof From (1) (5) and (10) we can obtain

119864 [1205871198620 ] ge 119864 [1205871 (1199021198620 )] + 119864 [1205872 (1199021198621 )] = max1199020 1199021

119864 [1205870 (1199020 1199021)]ge 119864 [1205871 (1199021198800 )] + 119864 [1205872 (1199021198801 )] = 119864 [1205871198801 ] + 119864 [1205871198802 ]

(12)

Proposition 3 illustrates that under the independent andnoncoordinated decision making pattern the Nash equilib-rium existing between the supplier and the manufacturerreduces the profit of the supply chain In order to fulfillthe profit margin in the centralized supply chains under theindependent decision-making pattern there is a necessity todesign an efficient coordinating mechanism

5 Production and Ordering Decision underQuantity Discount Coordination

According to (7) the ordering quantity by the manufacturerdecreases with the price of the raw material In this sense aquantity discount contract amounts to the reduction of theordering price of the raw material through which the order-ing quantity by themanufacturer can be promotedThe studyis based on the quantity discount contract conforming towhich qualification point of quantity discount 119902lowast and pricediscount rate 120572 can be established When the amount of theraw material provided by the supplier to the manufacturer isless than 119902lowast the price is set at 1199010 when the amount of theraw material exceeds 119902lowast the price of the excess part is set at(1 minus 120572)1199011 where 0 lt 120572 lt 1

Under the quantity discount contract the profit of thesupplier is given as follows

120587QD1 (1199020) = 1199010min [1199021 1199020119883]

minus 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 11988801199020 minus ℎ0max [1199020119883 minus 1199021 0]minus 1199030max [1199021 minus 1199020119883 0]

(13)

The first term is the sales revenue of the supplier at theprice of 1199010 the second term is the price discount given tomanufacturer on the basis of quantity discount the thirdterm is the production cost of the supplier the fourth termis the inventory cost of the supplier and the fifth term is theshortage cost of the raw material and the sixth term is theshortage cost of the raw material

Under the quantity discount contract the expected profitof the supplier is

119864 [120587QD1 (1199020)]= 1199010 [int11990211199020

0(1199020119909)1198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909] minus 1205721199010times [int11990211199020119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus 11988801199020 minus ℎ0 int+infin

11990211199020

(1199020119909 minus 1199021) 1198911 (119909) 119889119909minus 1199030 int119902111990200

(1199021 minus 1199020119909)1198911 (119909) 119889119909(14)

Proposition 4 Under the quantity discount contract thesupplierrsquos expected profit function 119864(120587QD1 ) is concave in 1199020and the optimal production quantity of the raw material 119902QD0 satisfies the following equation

int119902111990200

1199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(15)

Mathematical Problems in Engineering 7

Proof Since

119889119864 [120587QD1 (1199020)]1198891199020 = (1199010 + ℎ0 + 1199030) int11990211199020

01199091198911 (119909) 119889119909

minus 1205721199010 int11990211199020119902lowast1199020

1199091198911 (119909) 119889119909 minus 1198880 minus ℎ01198892119864 [120587QD

1 (1199020)]11988911990220 = minus (1199010 + ℎ0 + 1199030 minus 1205721199010) 11990221119902301198911 (

11990211199020)

minus 1205721199010 119902lowast211990230 1198911 (119902lowast1199020 ) lt 0

(16)

So 119864(120587QD1 ) is concave in 1199020 and setting (120597119864(120587QD

1 ))1205971199020 = 0we have (15)

Under the quantity discount contract the profit of themanufacturer is

120587QD1 (1199021) = 1199011min [119863 sdot 119885min (1199021 1199020 sdot 119883) sdot 119884]

minus 1199010min [1199021 1199020 sdot 119883]+ 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 1198881 [min (1199021 1199020 sdot 119883)]minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minus (min (1199021 1199020 sdot 119883) sdot 119884) 0]

(17)

The first term is the sales revenue of the manufacturerthe second term is the ordering cost of the rawmaterial at theprice of 1199010 the third term is the price discount based on thequantity discount contract the fourth term is the productioncost of the manufacturer the fifth term is the inventory costof the manufacturer and the sixth term is the shortage cost ofthe manufacturer

The manufacturer operates under the constraint restric-tions of the supplier and it precedes the decision making ofproduction and ordering by the standard of maximizing theprofits

max1199021

119864 [120587QD2 (1199021)]

= 1199011 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910

+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881) [int11990211199020

011990201199091198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909]+ 1205721199010 [int11990211199020

119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910) 1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910]st int11990211199020

01199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int

11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(18)

Setting 119889119864[120587QD2 (1199021)]1198891199021 = 0 we obtain the optimal

order quantity 119902QD1 which satisfies the following equation

(1199011 + ℎ1 + 1199031) int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

119902101584001199091199101198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

1199101198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881 + ℎ1 minus 1205721199010)times [int119902111990200

119902101584001199091198911 (119909) 119889119909 + int+infin11990211199020

1198911 (119909) 119889119909]

minus 1205721199010 int119902lowast1199020

0119902101584001199091198911 (119909) 119889119909 = 0

(19)

The 1199020 is a function of 1199021 from (15) and 11990210158400 is 1199020rsquos derivativeby 1199021

8 Mathematical Problems in Engineering

Table 2 Numerical results for all models (119898 = 015 119899 = 01 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4256 4197 1014 74585 146830 221415

C centralized decision mdash mdash 4351 4351 1 mdash mdash 222105

QD quantity discount withthe supplier as a dominantparty

01 3853 4415 4387 1006 75275 146830 22210502 4221 4307 4251 1013 75275 146830 22210503 4231 4307 4251 1013 75275 146830 22210504 4236 4307 4251 1013 75275 146830 222105

QD quantity discount withthe manufacturer as adominant party

01 4004 4310 4267 101 74585 147520 22210502 4033 4397 4381 1004 74585 147520 22210503 4173 4310 4270 1009 74585 147520 22210504 4197 4392 4379 1003 74585 147520 222105

6 The Design of Quantity Discount Contracts

The actual producing activities can be boiled down to theissue of the Stackelberg game problem Quantity discountcontracts under both the supplier as the dominant side andthe manufacturer as the dominant side are to be discussedseparately The principle of designing the pact is as follows

(1) In the Stackelberg game the dominant side fulfills thequantity discount coordination of the supply chainthrough the determination of the quantity discountqualification point and the price discount rate

(2) Considering the rationality of the follower the dom-inant side for the acceptance of the strategy by thefollower should ensure that the interest of the followershould be no less than the interest gained under thenoncoordinated pattern

(3) Based on the fulfillment of the above conditionsthe expected profit of the dominant side should bemaximized

61 The Quantity Discount Contract with the Supplier as theDominant Side Under the uncertainty of the two-echelonyields and demand provided that the supplier holds thedominant position with themanufacturer being the followerthe supplier is in the position to determine the parameterpackage of the quantity discount including the quantitydiscount qualification point 119902lowast and the price discount rate120572 so as to realize the coordination in the supply chain Themodel is as follows

max 119864 (120587QD1 )

st Equation (19)119864 (120587QD2 ) ge 119864 (1205871198802 )

(20)

For the first item as the supplier holds the dominantposition the goal is to achieve the maximized profit of thesupplier in the decision making The second item represents

the ordering strategy of the manufacturer the third itemguarantees the profits of the manufacturer should not be lessthan that under independent and noncoordinated decisionmaking pattern

62 The Quantity Discount Contract with the Manufactureras the Dominant Party Under the uncertainty of the two-echelon yields and demand in supply chains given thatthe manufacturer is in the dominant position in quantitydiscount and coordination with the supplier in a secondaryposition the manufacturer determines the parameter pack-age of quantity discount including the quantity discountqualification point and the price discount rate so as to realizecoordination in supply chains The model is shown as below

max 119864 (120587QD2 )

st Equation (15)119864 (120587QD1 ) ge 119864 (1205871198801 )

(21)

The first item shows that the goal to be achieved is tomaximize the profit of the manufacturer in the decisionmaking the second item shows that the ordering strategyis provided by the supplier the third item shows that thesupplierrsquos profit level should be kept no less than the levelunder the independent andnoncoordinated decision-makingpattern

7 Numerical Examples and Discussions

The parameters of a certain supply chain are set as below1198880 = 20 ℎ0 = 5 1199030 = 15 1198881 = 10 ℎ1 = 7 1199031 = 301199010 = 40 1199011 = 95 119863 = 400 The output of the supplier andthe output of the manufacturer random variables for marketdemand of the finished products are uniformly distributedover [1minus119898 1+119898] [1minus119899 1+119899] and [1minus 119897 1+ 119897] respectively

Tables 2ndash4 present the decision making for productionand ordering and the expected profits under an independent

Mathematical Problems in Engineering 9

Table 3 Numerical results for all models (119898 = 02 119899 = 02 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4345 4293 1012 71085 141007 212093

C centralized decision mdash mdash 4537 4537 1 mdash mdash 216542

QD quantity discount withthe supplier as a dominantparty

01 4125 4592 4558 1007 75535 141007 21654202 4330 4590 4558 1007 75535 141007 21654203 4433 4515 4460 1012 75535 141007 21654204 4440 4515 4460 1012 75535 141007 216542

QD quantity discount withthe manufacturer as adominant party

01 3517 4534 4534 1 71869 144673 21654202 4058 4533 4532 1 72759 143783 21654203 4222 4532 4531 1 73007 143535 21654204 4302 4532 4531 1 73123 143419 216542

Table 4 Numerical results for all models (119898 = 015 119899 = 01 119897 = 03)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4291 4232 1014 75196 125231 200427

C centralized decision mdash mdash 4387 4387 1 mdash mdash 201123

QD quantity discount withthe supplier as a dominantparty

01 4227 4342 4286 1013 75892 125231 20112302 4256 4342 4286 1013 75892 125231 20112303 4266 4342 4286 1013 75892 125231 20112304 4271 4342 4286 1013 75892 125231 201123

QD quantity discount withthe manufacturer as adominant party

01 4036 4346 4305 101 75196 125927 20112302 4162 4346 4305 101 75196 125927 20112303 4207 4346 4305 101 75196 125927 20112304 4231 4346 4305 101 75196 125927 201123

and noncoordinated decision-making pattern and the cen-tralized decision making pattern under the the uncertaintyof the two-echelon yields and demand they also respectivelypresent quantity discount coordination decisionmaking withthe supplier as the dominant role solving by model (20) andthe quantity discount coordination decision making with themanufacturer as the dominant role solving by model (21)

With Tables 2ndash4 the following conclusions can be drawn

(1) Under an independent decision-making patternthere exists the Nash equilibrium between the sup-plier and the manufacturer and the profit under theNash equilibrium is less than that gained under thecentralized decision-making pattern under the quan-tity discount pact themanufacturer has increased theordering quantity of the raw material the ratios ofthe planned output of the supplier to the orderingquantity of the manufacturer have decreased

(2) Under the quantity discount pact with the supplierin the dominant position the supplier sets different

combinations of the quantity discount qualificationpoint and the price discount rate under the situationwhere the profit of the manufacturer under the quan-tity discount coordination pact exceeds that underindependent and noncoordinated decision-makingpattern In this sense the profit of the supply chainbased on quantity discount reaches the profit level ofthe chain under a centralized pattern The supplierand the manufacturer can select a certain combina-tion and set forth a quantity discount coordinationpact

(3) Under the quantity discount coordination with themanufacturer in the dominant position the manu-facturer sets different combinations as well With theprofit of the supply chain under the quantity discountcoordination pact exceeding the profit under theindependent and noncoordinated decision-makingpattern the chain profit based on quantity discountreaches the profit level gained in a centralized patternThe supplier and themanufacturer can select a certain

10 Mathematical Problems in Engineering

combination and set forth a quantity discount coordi-nation pact

(4) With a comparison of Tables 2 3 and 4 it can befound that the larger the uncertainty of two-echelonyields and demand the larger the contribution of thequantity discount coordination pact to the profit ofthe supply chains With 119898 = 015 119899 = 01 and 119897 =02 the quantity discount coordination pact helpsincrease the total profit by 03 With 119898 = 015119899 = 01 and 119897 = 03 the quantity discountcoordination pact helps increase the total profit by04 With 119898 = 02 119899 = 02 and 119897 = 02 thequantity discount coordination pact helps increasethe total profit by 21 This shows that the quantitydiscount coordination contract can effectively reducethe negative effect by the uncertainty of two-echelonyields and demand upon the efficiency and benefit ofthe supply chains

(5) We also find out that undermost scenarios the supplychain benefits from the yields and demand risksreduction and generates a higher profit ContrastingTables 2 and 3 we find out that sometimes in thequantity discount contract the supplierrsquos profit canincrease as yieldsrsquo randomness increases And con-trasting Tables 2 and 4 we find out that in the unco-ordinated case and quantity discount contract thesupplierrsquos profit can increase as demand randomnessincreases

8 Conclusions

This paper explores the quantity discount coordinationmod-els in the fashion supply chain with uncertain yields andrandom demand The planned production outputs of thesupplier and the ordering quantity of the manufacturer havebeen set as the decision-making parameters The indepen-dent and noncoordinated and centralized decision-makingpatterns have been established under the uncertainty of two-echelon yields and demandThrough the analysis it has beenfound that the profit of supply chains under independentand noncoordinated decision-making pattern is less than thatunder the centralized pattern which proves the necessityof the coordination of supply chains Based on this twoparametersmdashquantity discount qualification point and pricediscount ratiomdashhave been introduced and quantity discountcoordination models have been established The quantitydiscount coordination pacts with both the supplier and themanufacturer in the dominant position have been set forth

The result reveals that the quantity discount coordinationpact proposed by this paper can promote the manufacturerrsquosordering quantity of the raw material which contributesto the result that the profit of the supply chain based onquantity discount reaches the profit level of the supply chainunder the centralized decision-making pattern The largerthe uncertainty of two-echelon yields and demand the largercontribution of the pact to the benefit of the supply chainwill be The quantity discount coordination pact under theuncertainty of two-echelon yields and demand can reduce the

negative influence brought by the uncertainty and thereforecan contribute to the purpose of the coordination of thesupply chain

Acknowledgments

The authors would like to thank the referees for theirconstructive comments and suggestions that have greatlyimproved the quality of the paper The research was sup-ported by the National Natural Science Foundation of China(Grant no 71201164)

References

[1] D J Thomas and P M Griffin ldquoCoordinated supply chainmanagementrdquo European Journal of Operational Research vol94 no 1 pp 1ndash15 1996

[2] N Agrawal S A Smith andA A Tsay ldquoMulti-vendor sourcingin a retail supply chainrdquo Production and Operations Manage-ment vol 11 no 2 pp 157ndash181 2002

[3] C Chiang and W C Benton ldquoSole sourcing versus dualsourcing under stochastic demands and lead timesrdquo NavalResearch Logistics vol 41 no 5 pp 609ndash624 1994

[4] J D Hong and J C Hayya ldquoJust-in-time purchasing singleor multiple sourcingrdquo International Journal of ProductionEconomics vol 27 no 2 pp 175ndash181 1992

[5] M Treleven and S B Schweikhart ldquoA riskbenefit analysisof sourcing strategies single vs multiple sourcingrdquo Journal ofOperations Management vol 7 no 3-4 pp 93ndash114 1988

[6] G P Cachon and P H Zipkin ldquoCompetitive and cooperativeinventory policies in a two-stage supply chainrdquo ManagementScience vol 45 no 7 pp 936ndash953 1999

[7] M Leng and M Parlar ldquoGame theoretic applications in supplychainmanagement a reviewrdquo INFOR vol 43 no 3 pp 187ndash2202005

[8] MMoses and S Seshadri ldquoPolicy mechanisms for supply chaincoordinationrdquo IIETransactions vol 32 no 3 pp 245ndash262 2000

[9] H J Peng and M H Zhou ldquoProduction and ordering decisionin supply chain with uncertainty in two-echelon yields anddemandrdquo Journal of Systems Engineering vol 25 no 5 pp 622ndash628 2010 (Chinese)

[10] C T Chang ldquoAn acquisition policy for a single item multi-supplier system with real-world constraintsrdquo Applied Mathe-matical Modelling vol 30 no 1 pp 1ndash9 2006

[11] C T Chang C L Chin and M F Lin ldquoOn the single itemmulti-supplier system with variable lead-time price-quantitydiscount and resource constraintsrdquo Applied Mathematics andComputation vol 182 no 1 pp 89ndash97 2006

[12] J Li and L Liu ldquoSupply chain coordination with quantitydiscount policyrdquo International Journal of Production Economicsvol 101 no 1 pp 89ndash98 2006

[13] C L Munson and M J Rosenblatt ldquoTheories and realitiesof quantity discounts an exploratory studyrdquo Production andOperations Management vol 7 no 4 pp 352ndash369 1998

[14] C L Munson and J Hu ldquoIncorporating quantity discountsand their inventory impacts into the centralized purchasingdecisionrdquoEuropean Journal ofOperational Research vol 201 no2 pp 581ndash592 2010

[15] H Y Kang and A H I Lee ldquoInventory replenishment modelusing fuzzy multiple objective programming a case study

Mathematical Problems in Engineering 11

of a high-tech company in Taiwanrdquo Applied Soft ComputingJournal vol 10 no 4 pp 1108ndash1118 2010

[16] A Mendoza and J A Ventura ldquoIncorporating quantity dis-counts to the EOQ model with transportation costsrdquo Interna-tional Journal of Production Economics vol 113 no 2 pp 754ndash765 2008

[17] J Moussourakis and C Haksever ldquoA practical model forordering in multi-product multi-constraint inventory systemswith all-units quantity discountsrdquo International Journal ofInformation and Management Sciences vol 19 no 2 pp 263ndash283 2008

[18] P A Rubin and W C Benton ldquoA generalized framework forquantity discount pricing schedulesrdquo Decision Sciences vol 34no 1 pp 173ndash188 2003

[19] G Zhang and L Ma ldquoOptimal acquisition policy with quantitydiscounts and uncertain demandsrdquo International Journal ofProduction Research vol 47 no 9 pp 2409ndash2425 2009

[20] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[21] A H I Lee and H Y Kang ldquoA mixed 0-1 integer programmingfor inventory model a case study of TFT-LCD manufacturingcompany in Taiwanrdquo Kybernetes vol 37 no 1 pp 66ndash82 2008

[22] J Shi and G Zhang ldquoMulti-product budget-constrained acqui-sition and pricing with uncertain demand and supplier quantitydiscountsrdquo International Journal of Production Economics vol128 no 1 pp 322ndash331 2010

[23] A A Taleizadeh S T A Niaki and R Nikousokhan ldquoCon-straint multiproduct joint-replenishment inventory controlproblem using uncertain programmingrdquo Applied Soft Comput-ing vol 11 pp 5143ndash5154 2011

[24] J Rezaei and M Davoodi ldquoA deterministic multi-item inven-tory model with supplier selection and imperfect qualityrdquoApplied Mathematical Modelling vol 32 no 10 pp 2106ndash21162008

[25] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[26] S P Chen and Y H Ho ldquoAnalysis of the newsboy problemwith fuzzy demands and incremental discountsrdquo InternationalJournal of Production Economics vol 129 no 1 pp 169ndash177 2011

[27] Y J Lin and C H Ho ldquoIntegrated inventory model withquantity discount and price-sensitive demandrdquo Top vol 19 no1 pp 177ndash188 2011

[28] S Krichen A Laabidi and F B Abdelaziz ldquoSingle suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in paymentsrdquo Computers andIndustrial Engineering vol 60 no 1 pp 164ndash172 2011

[29] J Shi G Zhang and J Sha ldquoJointly pricing and ordering fora multi-product multi-constraint newsvendor problem withsupplier quantity discountsrdquo Applied Mathematical Modellingvol 35 no 6 pp 3001ndash3011 2011

[30] H Shin and W C Benton ldquoA quantity discount approach tosupply chain coordinationrdquo European Journal of OperationalResearch vol 180 no 2 pp 601ndash616 2007

[31] H Gurnani and Y Gerchak ldquoCoordination in decentralizedassembly systems with uncertain component yieldsrdquo EuropeanJournal of Operational Research vol 176 no 3 pp 1559ndash15762007

[32] Y He and J Zhang ldquoRandom yield risk sharing in a two-levelsupply chainrdquo International Journal of Production Economicsvol 112 no 2 pp 769ndash781 2008

[33] Y He and J Zhang ldquoRandom yield supply chain with a yielddependent secondary marketrdquo European Journal of OperationalResearch vol 206 no 1 pp 221ndash230 2010

[34] C C Hsieh and C H Wu ldquoCapacity allocation ordering andpricing decisions in a supply chain with demand and supplyuncertaintiesrdquo European Journal of Operational Research vol184 no 2 pp 667ndash684 2008

[35] C C Hsieh and C H Wu ldquoCoordinated decisions for substi-tutable products in a common retailer supply chainrdquo EuropeanJournal of Operational Research vol 196 no 1 pp 273ndash2882009

[36] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[37] P Kelle S Transchel and S Minner ldquoBuyer-supplier coopera-tion and negotiation support with random yield considerationrdquoInternational Journal of Production Economics vol 118 no 1 pp152ndash159 2009

[38] B Keren ldquoThe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmegavol 37 no 4 pp 801ndash810 2009

[39] P Kouvelis and J M Milner ldquoSupply chain capacity andoutsourcing decisions the dynamic interplay of demand andsupply uncertaintyrdquo IIE Transactions vol 34 no 8 pp 717ndash7282002

[40] X Yan M Zhang and K Liu ldquoA note on coordination indecentralized assembly systems with uncertain componentyieldsrdquo European Journal of Operational Research vol 205 no2 pp 469ndash478 2010

[41] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[42] K Zimmer ldquoSupply chain coordination with uncertain just-in-time deliveryrdquo International Journal of Production Economicsvol 77 no 1 pp 1ndash15 2002

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

the expected value of 119885 is 1 According to hypothesis (2) theexpected values of119883 119884 are 1

Under the independent and noncoordinated decision-making pattern the profit gained by the supplier is

1205871198801 (1199020) = 1199010min [1199021 1199020 sdot 119883] minus 11988801199020 minus ℎ0max [1199020 sdot 119883 minus 1199021 0]minus 1199030max [1199021 minus 1199020 sdot 119883 0]

(1)

The first term is its sales revenue the second term is theproduction cost of the supplier the third term is the inventorycost of the raw material by the supplier and the fourth termis the shortage cost of the raw material The expected profitby the supplier is

119864 [1205871198801 (1199020)] = 1199010 [int119902111990200

(1199020 sdot 119909) 1198911 (119909) 119889119909+int+infin11990211199020

11990211198911 (119909) 119889119909]minus 11988801199020minusℎ0 int+infin

11990211199020

(1199020 sdot 119909 minus 1199021) 1198911 (119909) 119889119909minus 1199030 int119902111990200

(1199021 minus 1199020 sdot 119909) 1198911 (119909) 119889119909

(2)

Proposition 1 (1) Under the independent and noncoordi-nated decision-making pattern the expected profit by thesupplier 119864(1205871198801 ) is concave in 1199020 and the optimal plannedoutput of the raw material 1199021198800 conforms to the followingformula

int119902111990211988000

1199091198911 (119909) 119889119909 = 1198880 + ℎ01199010 + ℎ0 + 1199030 = 1 minus1199010 + 1199030 minus 11988801199010 + ℎ0 + 1199030 (3)

(2) 1199021198800 = 1205821199021 where 120582 is determined by 1199010 ℎ0 1199030 1198880 and1198911(sdot)Proof (1)119889119864 [1205871198801 (1199020)]1198891199020 = (1199010 + ℎ0 + 1199030) int11990211199020

01199091198911 (119909) 119889119909 minus 1198880 minus ℎ0

1198892119864 [1205871198801 (1199020)]11988911990220 = minus (1199010 + ℎ0 + 1199030) 11990221119902301198911 (

11990211199020) lt 0(4)

So 119864(1205871198801 ) is concave in 1199020 and setting (119889119864(1205871198801 ))1198891199020 = 0 wecan derive (3)(2) All we need to show is that function 1198791(119906) =int1199060119909119891(119909)119889119909 is a monotone function of 119906 which is obvious

from 1198891198791119889119906 = 119906119891(119906) gt 0 Therefore (2) has a uniquesolution and 11990211199020 is denoted as 1120582 here

Under the independent and noncoordinated decisionmaking pattern the expected profits by the manufacturer are

1205871198802 (1199021) = 1199011min [119863 sdot 119885 119896 sdotmin (1199021 1199020 sdot 119883) sdot 119884]minus 1199010min [1199021 1199020 sdot 119883] minus 1198881min (1199021 1199020 sdot 119883)minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minusmin (1199021 1199020 sdot 119883) sdot 119884 0]

(5)

The first term is the manufacturerrsquos sales revenue thesecond term is the ordering cost for the rawmaterial the thirdterm is the production cost the fourth term is the inventorycost and the fifth term is the shortage cost

The manufacturer finds out the optimal order quan-tity where the supplierrsquos problem of finding the optimalproduction quantity is a constraint in the mathematicalprogramming problem

max1199021

119864 [1205871198802 (1199021)]= 1199011 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881) [int11990211199020

011990201199091198911 (119909) 119889119909

+int+infin11990211199020

11990211198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909times int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909timesint+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]

Mathematical Problems in Engineering 5

minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909times int11986311991111990201199090

(119863119911 minus 1199020119909119910)1198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909timesint11986311991111990210

(119863119911minus1199021119910)1198911 (119909) 1198912 (119910) 119889119910]st 1199021198800 = 1205821199021

(6)

Proposition 2 Under the independent and noncoordinateddecision-making pattern the manufacturerrsquos expected profitfunction 119864(1205871198801 ) is concave in 1199021 and the optimal orderquantity 1199021198801 satisfies the following equation

int+infin0

119892 (119911) 119889119911 [int11205820

119889119909int1198631199111205821199021198801 1199090

1205821199091199101198911 (119909) 1198912 (119910) 119889119910

+int+infin1120582

119889119909int11986311991111990211988010

1199101198911 (119909) 1198912 (119910) 119889119910]

= (1199010 + 1198881 + ℎ1) [int11205820 1205821199091198911 (119909) 119889119909 + int+infin1120582 1198911 (119909) 119889119909]1199011 + 1199031 + ℎ1 (7)

Proof Since 1199021198800 (1199021) = 1205821199021

119889119864 [1205872]1198891199021 = (1199011 + 1199031 + ℎ1) int+infin0

119892 (119911) 119889119911times [int11205820

119889119909int11986311991112058211990211199090

1205821199091199101198911 (119909) 1198912 (119910) 119889119910+int+infin1120582

119889119909int11986311991111990210

1199101198911 (119909) 1198912 (119910) 119889119910]minus 120582 (1199010 + 1198881 + ℎ1)+ (1199010 + 1198881) int+infin

1120582(120582119909 minus 1) 1198911 (119909) 119889119909

(8)

1198892119864 [1205872]11988911990221 = minus (1199011 + ℎ1 + 1199031) int+infin0

119892 (119911) 119889119911

times [int11205820

(119863119911)212058211990231119909 1198911 (119909) 1198912 (1198631199111205821199021119909)119889119909

+int+infin1120582

(119863119911)211990231 1198911 (119909) 1198912 (1198631199111199021 )119889119909] lt 0(9)

So 119864(1205871198802 ) is concave in 1199021 and setting (119889119864(1205871198802 ))1198891199021 = 0 wehave (7)

42 Production and Ordering Decision Making Guided bya Centralized Decision-Making Pattern In the centralizedsupply chain the goal of the supply chain operation is tomaximize the total profit The manufacturer directly conveysthe demand information to the supplier In the centralizedcase the shortage cost of the supplier can be seen as amatter of an internal revenue transfer which barely affectsthe decision making in production and ordering The profitgenerated by the supply chain dependent on the centralizeddecision making is

1205871198620 = 1199011min [119863 sdot 119885min (1199021 1199020 sdot 119883) sdot 119884]minus 1198881min (1199021 1199020 sdot 119883) minus 11988801199020 minus ℎ0max [1199020 sdot 119883 minus 1199021 0]minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minusmin (1199021 1199020 sdot 119883) sdot 119884 0]

(10)

The operational goal is to realize the maximized profit inthis supply chain and the operational model is

max 119864 [1205871198620 ]= 1199011 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910

+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]

minus 11988801199020 minus ℎ0 int+infin11990211199020

(1199020 sdot 119909 minus 1199021) 1198911 (119909) 119889119909

6 Mathematical Problems in Engineering

minus 1198881 [int119902111990200

11990201199091198911 (119909) 119889119909 + int+infin11990211199020

11990211198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910] (11)

Proposition 3 Consider 119864[1205871198620 ] ge 119864[1205871198801 ] + 119864[1205871198802 ]Proof From (1) (5) and (10) we can obtain

119864 [1205871198620 ] ge 119864 [1205871 (1199021198620 )] + 119864 [1205872 (1199021198621 )] = max1199020 1199021

119864 [1205870 (1199020 1199021)]ge 119864 [1205871 (1199021198800 )] + 119864 [1205872 (1199021198801 )] = 119864 [1205871198801 ] + 119864 [1205871198802 ]

(12)

Proposition 3 illustrates that under the independent andnoncoordinated decision making pattern the Nash equilib-rium existing between the supplier and the manufacturerreduces the profit of the supply chain In order to fulfillthe profit margin in the centralized supply chains under theindependent decision-making pattern there is a necessity todesign an efficient coordinating mechanism

5 Production and Ordering Decision underQuantity Discount Coordination

According to (7) the ordering quantity by the manufacturerdecreases with the price of the raw material In this sense aquantity discount contract amounts to the reduction of theordering price of the raw material through which the order-ing quantity by themanufacturer can be promotedThe studyis based on the quantity discount contract conforming towhich qualification point of quantity discount 119902lowast and pricediscount rate 120572 can be established When the amount of theraw material provided by the supplier to the manufacturer isless than 119902lowast the price is set at 1199010 when the amount of theraw material exceeds 119902lowast the price of the excess part is set at(1 minus 120572)1199011 where 0 lt 120572 lt 1

Under the quantity discount contract the profit of thesupplier is given as follows

120587QD1 (1199020) = 1199010min [1199021 1199020119883]

minus 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 11988801199020 minus ℎ0max [1199020119883 minus 1199021 0]minus 1199030max [1199021 minus 1199020119883 0]

(13)

The first term is the sales revenue of the supplier at theprice of 1199010 the second term is the price discount given tomanufacturer on the basis of quantity discount the thirdterm is the production cost of the supplier the fourth termis the inventory cost of the supplier and the fifth term is theshortage cost of the raw material and the sixth term is theshortage cost of the raw material

Under the quantity discount contract the expected profitof the supplier is

119864 [120587QD1 (1199020)]= 1199010 [int11990211199020

0(1199020119909)1198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909] minus 1205721199010times [int11990211199020119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus 11988801199020 minus ℎ0 int+infin

11990211199020

(1199020119909 minus 1199021) 1198911 (119909) 119889119909minus 1199030 int119902111990200

(1199021 minus 1199020119909)1198911 (119909) 119889119909(14)

Proposition 4 Under the quantity discount contract thesupplierrsquos expected profit function 119864(120587QD1 ) is concave in 1199020and the optimal production quantity of the raw material 119902QD0 satisfies the following equation

int119902111990200

1199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(15)

Mathematical Problems in Engineering 7

Proof Since

119889119864 [120587QD1 (1199020)]1198891199020 = (1199010 + ℎ0 + 1199030) int11990211199020

01199091198911 (119909) 119889119909

minus 1205721199010 int11990211199020119902lowast1199020

1199091198911 (119909) 119889119909 minus 1198880 minus ℎ01198892119864 [120587QD

1 (1199020)]11988911990220 = minus (1199010 + ℎ0 + 1199030 minus 1205721199010) 11990221119902301198911 (

11990211199020)

minus 1205721199010 119902lowast211990230 1198911 (119902lowast1199020 ) lt 0

(16)

So 119864(120587QD1 ) is concave in 1199020 and setting (120597119864(120587QD

1 ))1205971199020 = 0we have (15)

Under the quantity discount contract the profit of themanufacturer is

120587QD1 (1199021) = 1199011min [119863 sdot 119885min (1199021 1199020 sdot 119883) sdot 119884]

minus 1199010min [1199021 1199020 sdot 119883]+ 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 1198881 [min (1199021 1199020 sdot 119883)]minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minus (min (1199021 1199020 sdot 119883) sdot 119884) 0]

(17)

The first term is the sales revenue of the manufacturerthe second term is the ordering cost of the rawmaterial at theprice of 1199010 the third term is the price discount based on thequantity discount contract the fourth term is the productioncost of the manufacturer the fifth term is the inventory costof the manufacturer and the sixth term is the shortage cost ofthe manufacturer

The manufacturer operates under the constraint restric-tions of the supplier and it precedes the decision making ofproduction and ordering by the standard of maximizing theprofits

max1199021

119864 [120587QD2 (1199021)]

= 1199011 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910

+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881) [int11990211199020

011990201199091198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909]+ 1205721199010 [int11990211199020

119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910) 1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910]st int11990211199020

01199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int

11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(18)

Setting 119889119864[120587QD2 (1199021)]1198891199021 = 0 we obtain the optimal

order quantity 119902QD1 which satisfies the following equation

(1199011 + ℎ1 + 1199031) int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

119902101584001199091199101198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

1199101198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881 + ℎ1 minus 1205721199010)times [int119902111990200

119902101584001199091198911 (119909) 119889119909 + int+infin11990211199020

1198911 (119909) 119889119909]

minus 1205721199010 int119902lowast1199020

0119902101584001199091198911 (119909) 119889119909 = 0

(19)

The 1199020 is a function of 1199021 from (15) and 11990210158400 is 1199020rsquos derivativeby 1199021

8 Mathematical Problems in Engineering

Table 2 Numerical results for all models (119898 = 015 119899 = 01 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4256 4197 1014 74585 146830 221415

C centralized decision mdash mdash 4351 4351 1 mdash mdash 222105

QD quantity discount withthe supplier as a dominantparty

01 3853 4415 4387 1006 75275 146830 22210502 4221 4307 4251 1013 75275 146830 22210503 4231 4307 4251 1013 75275 146830 22210504 4236 4307 4251 1013 75275 146830 222105

QD quantity discount withthe manufacturer as adominant party

01 4004 4310 4267 101 74585 147520 22210502 4033 4397 4381 1004 74585 147520 22210503 4173 4310 4270 1009 74585 147520 22210504 4197 4392 4379 1003 74585 147520 222105

6 The Design of Quantity Discount Contracts

The actual producing activities can be boiled down to theissue of the Stackelberg game problem Quantity discountcontracts under both the supplier as the dominant side andthe manufacturer as the dominant side are to be discussedseparately The principle of designing the pact is as follows

(1) In the Stackelberg game the dominant side fulfills thequantity discount coordination of the supply chainthrough the determination of the quantity discountqualification point and the price discount rate

(2) Considering the rationality of the follower the dom-inant side for the acceptance of the strategy by thefollower should ensure that the interest of the followershould be no less than the interest gained under thenoncoordinated pattern

(3) Based on the fulfillment of the above conditionsthe expected profit of the dominant side should bemaximized

61 The Quantity Discount Contract with the Supplier as theDominant Side Under the uncertainty of the two-echelonyields and demand provided that the supplier holds thedominant position with themanufacturer being the followerthe supplier is in the position to determine the parameterpackage of the quantity discount including the quantitydiscount qualification point 119902lowast and the price discount rate120572 so as to realize the coordination in the supply chain Themodel is as follows

max 119864 (120587QD1 )

st Equation (19)119864 (120587QD2 ) ge 119864 (1205871198802 )

(20)

For the first item as the supplier holds the dominantposition the goal is to achieve the maximized profit of thesupplier in the decision making The second item represents

the ordering strategy of the manufacturer the third itemguarantees the profits of the manufacturer should not be lessthan that under independent and noncoordinated decisionmaking pattern

62 The Quantity Discount Contract with the Manufactureras the Dominant Party Under the uncertainty of the two-echelon yields and demand in supply chains given thatthe manufacturer is in the dominant position in quantitydiscount and coordination with the supplier in a secondaryposition the manufacturer determines the parameter pack-age of quantity discount including the quantity discountqualification point and the price discount rate so as to realizecoordination in supply chains The model is shown as below

max 119864 (120587QD2 )

st Equation (15)119864 (120587QD1 ) ge 119864 (1205871198801 )

(21)

The first item shows that the goal to be achieved is tomaximize the profit of the manufacturer in the decisionmaking the second item shows that the ordering strategyis provided by the supplier the third item shows that thesupplierrsquos profit level should be kept no less than the levelunder the independent andnoncoordinated decision-makingpattern

7 Numerical Examples and Discussions

The parameters of a certain supply chain are set as below1198880 = 20 ℎ0 = 5 1199030 = 15 1198881 = 10 ℎ1 = 7 1199031 = 301199010 = 40 1199011 = 95 119863 = 400 The output of the supplier andthe output of the manufacturer random variables for marketdemand of the finished products are uniformly distributedover [1minus119898 1+119898] [1minus119899 1+119899] and [1minus 119897 1+ 119897] respectively

Tables 2ndash4 present the decision making for productionand ordering and the expected profits under an independent

Mathematical Problems in Engineering 9

Table 3 Numerical results for all models (119898 = 02 119899 = 02 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4345 4293 1012 71085 141007 212093

C centralized decision mdash mdash 4537 4537 1 mdash mdash 216542

QD quantity discount withthe supplier as a dominantparty

01 4125 4592 4558 1007 75535 141007 21654202 4330 4590 4558 1007 75535 141007 21654203 4433 4515 4460 1012 75535 141007 21654204 4440 4515 4460 1012 75535 141007 216542

QD quantity discount withthe manufacturer as adominant party

01 3517 4534 4534 1 71869 144673 21654202 4058 4533 4532 1 72759 143783 21654203 4222 4532 4531 1 73007 143535 21654204 4302 4532 4531 1 73123 143419 216542

Table 4 Numerical results for all models (119898 = 015 119899 = 01 119897 = 03)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4291 4232 1014 75196 125231 200427

C centralized decision mdash mdash 4387 4387 1 mdash mdash 201123

QD quantity discount withthe supplier as a dominantparty

01 4227 4342 4286 1013 75892 125231 20112302 4256 4342 4286 1013 75892 125231 20112303 4266 4342 4286 1013 75892 125231 20112304 4271 4342 4286 1013 75892 125231 201123

QD quantity discount withthe manufacturer as adominant party

01 4036 4346 4305 101 75196 125927 20112302 4162 4346 4305 101 75196 125927 20112303 4207 4346 4305 101 75196 125927 20112304 4231 4346 4305 101 75196 125927 201123

and noncoordinated decision-making pattern and the cen-tralized decision making pattern under the the uncertaintyof the two-echelon yields and demand they also respectivelypresent quantity discount coordination decisionmaking withthe supplier as the dominant role solving by model (20) andthe quantity discount coordination decision making with themanufacturer as the dominant role solving by model (21)

With Tables 2ndash4 the following conclusions can be drawn

(1) Under an independent decision-making patternthere exists the Nash equilibrium between the sup-plier and the manufacturer and the profit under theNash equilibrium is less than that gained under thecentralized decision-making pattern under the quan-tity discount pact themanufacturer has increased theordering quantity of the raw material the ratios ofthe planned output of the supplier to the orderingquantity of the manufacturer have decreased

(2) Under the quantity discount pact with the supplierin the dominant position the supplier sets different

combinations of the quantity discount qualificationpoint and the price discount rate under the situationwhere the profit of the manufacturer under the quan-tity discount coordination pact exceeds that underindependent and noncoordinated decision-makingpattern In this sense the profit of the supply chainbased on quantity discount reaches the profit level ofthe chain under a centralized pattern The supplierand the manufacturer can select a certain combina-tion and set forth a quantity discount coordinationpact

(3) Under the quantity discount coordination with themanufacturer in the dominant position the manu-facturer sets different combinations as well With theprofit of the supply chain under the quantity discountcoordination pact exceeding the profit under theindependent and noncoordinated decision-makingpattern the chain profit based on quantity discountreaches the profit level gained in a centralized patternThe supplier and themanufacturer can select a certain

10 Mathematical Problems in Engineering

combination and set forth a quantity discount coordi-nation pact

(4) With a comparison of Tables 2 3 and 4 it can befound that the larger the uncertainty of two-echelonyields and demand the larger the contribution of thequantity discount coordination pact to the profit ofthe supply chains With 119898 = 015 119899 = 01 and 119897 =02 the quantity discount coordination pact helpsincrease the total profit by 03 With 119898 = 015119899 = 01 and 119897 = 03 the quantity discountcoordination pact helps increase the total profit by04 With 119898 = 02 119899 = 02 and 119897 = 02 thequantity discount coordination pact helps increasethe total profit by 21 This shows that the quantitydiscount coordination contract can effectively reducethe negative effect by the uncertainty of two-echelonyields and demand upon the efficiency and benefit ofthe supply chains

(5) We also find out that undermost scenarios the supplychain benefits from the yields and demand risksreduction and generates a higher profit ContrastingTables 2 and 3 we find out that sometimes in thequantity discount contract the supplierrsquos profit canincrease as yieldsrsquo randomness increases And con-trasting Tables 2 and 4 we find out that in the unco-ordinated case and quantity discount contract thesupplierrsquos profit can increase as demand randomnessincreases

8 Conclusions

This paper explores the quantity discount coordinationmod-els in the fashion supply chain with uncertain yields andrandom demand The planned production outputs of thesupplier and the ordering quantity of the manufacturer havebeen set as the decision-making parameters The indepen-dent and noncoordinated and centralized decision-makingpatterns have been established under the uncertainty of two-echelon yields and demandThrough the analysis it has beenfound that the profit of supply chains under independentand noncoordinated decision-making pattern is less than thatunder the centralized pattern which proves the necessityof the coordination of supply chains Based on this twoparametersmdashquantity discount qualification point and pricediscount ratiomdashhave been introduced and quantity discountcoordination models have been established The quantitydiscount coordination pacts with both the supplier and themanufacturer in the dominant position have been set forth

The result reveals that the quantity discount coordinationpact proposed by this paper can promote the manufacturerrsquosordering quantity of the raw material which contributesto the result that the profit of the supply chain based onquantity discount reaches the profit level of the supply chainunder the centralized decision-making pattern The largerthe uncertainty of two-echelon yields and demand the largercontribution of the pact to the benefit of the supply chainwill be The quantity discount coordination pact under theuncertainty of two-echelon yields and demand can reduce the

negative influence brought by the uncertainty and thereforecan contribute to the purpose of the coordination of thesupply chain

Acknowledgments

The authors would like to thank the referees for theirconstructive comments and suggestions that have greatlyimproved the quality of the paper The research was sup-ported by the National Natural Science Foundation of China(Grant no 71201164)

References

[1] D J Thomas and P M Griffin ldquoCoordinated supply chainmanagementrdquo European Journal of Operational Research vol94 no 1 pp 1ndash15 1996

[2] N Agrawal S A Smith andA A Tsay ldquoMulti-vendor sourcingin a retail supply chainrdquo Production and Operations Manage-ment vol 11 no 2 pp 157ndash181 2002

[3] C Chiang and W C Benton ldquoSole sourcing versus dualsourcing under stochastic demands and lead timesrdquo NavalResearch Logistics vol 41 no 5 pp 609ndash624 1994

[4] J D Hong and J C Hayya ldquoJust-in-time purchasing singleor multiple sourcingrdquo International Journal of ProductionEconomics vol 27 no 2 pp 175ndash181 1992

[5] M Treleven and S B Schweikhart ldquoA riskbenefit analysisof sourcing strategies single vs multiple sourcingrdquo Journal ofOperations Management vol 7 no 3-4 pp 93ndash114 1988

[6] G P Cachon and P H Zipkin ldquoCompetitive and cooperativeinventory policies in a two-stage supply chainrdquo ManagementScience vol 45 no 7 pp 936ndash953 1999

[7] M Leng and M Parlar ldquoGame theoretic applications in supplychainmanagement a reviewrdquo INFOR vol 43 no 3 pp 187ndash2202005

[8] MMoses and S Seshadri ldquoPolicy mechanisms for supply chaincoordinationrdquo IIETransactions vol 32 no 3 pp 245ndash262 2000

[9] H J Peng and M H Zhou ldquoProduction and ordering decisionin supply chain with uncertainty in two-echelon yields anddemandrdquo Journal of Systems Engineering vol 25 no 5 pp 622ndash628 2010 (Chinese)

[10] C T Chang ldquoAn acquisition policy for a single item multi-supplier system with real-world constraintsrdquo Applied Mathe-matical Modelling vol 30 no 1 pp 1ndash9 2006

[11] C T Chang C L Chin and M F Lin ldquoOn the single itemmulti-supplier system with variable lead-time price-quantitydiscount and resource constraintsrdquo Applied Mathematics andComputation vol 182 no 1 pp 89ndash97 2006

[12] J Li and L Liu ldquoSupply chain coordination with quantitydiscount policyrdquo International Journal of Production Economicsvol 101 no 1 pp 89ndash98 2006

[13] C L Munson and M J Rosenblatt ldquoTheories and realitiesof quantity discounts an exploratory studyrdquo Production andOperations Management vol 7 no 4 pp 352ndash369 1998

[14] C L Munson and J Hu ldquoIncorporating quantity discountsand their inventory impacts into the centralized purchasingdecisionrdquoEuropean Journal ofOperational Research vol 201 no2 pp 581ndash592 2010

[15] H Y Kang and A H I Lee ldquoInventory replenishment modelusing fuzzy multiple objective programming a case study

Mathematical Problems in Engineering 11

of a high-tech company in Taiwanrdquo Applied Soft ComputingJournal vol 10 no 4 pp 1108ndash1118 2010

[16] A Mendoza and J A Ventura ldquoIncorporating quantity dis-counts to the EOQ model with transportation costsrdquo Interna-tional Journal of Production Economics vol 113 no 2 pp 754ndash765 2008

[17] J Moussourakis and C Haksever ldquoA practical model forordering in multi-product multi-constraint inventory systemswith all-units quantity discountsrdquo International Journal ofInformation and Management Sciences vol 19 no 2 pp 263ndash283 2008

[18] P A Rubin and W C Benton ldquoA generalized framework forquantity discount pricing schedulesrdquo Decision Sciences vol 34no 1 pp 173ndash188 2003

[19] G Zhang and L Ma ldquoOptimal acquisition policy with quantitydiscounts and uncertain demandsrdquo International Journal ofProduction Research vol 47 no 9 pp 2409ndash2425 2009

[20] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[21] A H I Lee and H Y Kang ldquoA mixed 0-1 integer programmingfor inventory model a case study of TFT-LCD manufacturingcompany in Taiwanrdquo Kybernetes vol 37 no 1 pp 66ndash82 2008

[22] J Shi and G Zhang ldquoMulti-product budget-constrained acqui-sition and pricing with uncertain demand and supplier quantitydiscountsrdquo International Journal of Production Economics vol128 no 1 pp 322ndash331 2010

[23] A A Taleizadeh S T A Niaki and R Nikousokhan ldquoCon-straint multiproduct joint-replenishment inventory controlproblem using uncertain programmingrdquo Applied Soft Comput-ing vol 11 pp 5143ndash5154 2011

[24] J Rezaei and M Davoodi ldquoA deterministic multi-item inven-tory model with supplier selection and imperfect qualityrdquoApplied Mathematical Modelling vol 32 no 10 pp 2106ndash21162008

[25] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[26] S P Chen and Y H Ho ldquoAnalysis of the newsboy problemwith fuzzy demands and incremental discountsrdquo InternationalJournal of Production Economics vol 129 no 1 pp 169ndash177 2011

[27] Y J Lin and C H Ho ldquoIntegrated inventory model withquantity discount and price-sensitive demandrdquo Top vol 19 no1 pp 177ndash188 2011

[28] S Krichen A Laabidi and F B Abdelaziz ldquoSingle suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in paymentsrdquo Computers andIndustrial Engineering vol 60 no 1 pp 164ndash172 2011

[29] J Shi G Zhang and J Sha ldquoJointly pricing and ordering fora multi-product multi-constraint newsvendor problem withsupplier quantity discountsrdquo Applied Mathematical Modellingvol 35 no 6 pp 3001ndash3011 2011

[30] H Shin and W C Benton ldquoA quantity discount approach tosupply chain coordinationrdquo European Journal of OperationalResearch vol 180 no 2 pp 601ndash616 2007

[31] H Gurnani and Y Gerchak ldquoCoordination in decentralizedassembly systems with uncertain component yieldsrdquo EuropeanJournal of Operational Research vol 176 no 3 pp 1559ndash15762007

[32] Y He and J Zhang ldquoRandom yield risk sharing in a two-levelsupply chainrdquo International Journal of Production Economicsvol 112 no 2 pp 769ndash781 2008

[33] Y He and J Zhang ldquoRandom yield supply chain with a yielddependent secondary marketrdquo European Journal of OperationalResearch vol 206 no 1 pp 221ndash230 2010

[34] C C Hsieh and C H Wu ldquoCapacity allocation ordering andpricing decisions in a supply chain with demand and supplyuncertaintiesrdquo European Journal of Operational Research vol184 no 2 pp 667ndash684 2008

[35] C C Hsieh and C H Wu ldquoCoordinated decisions for substi-tutable products in a common retailer supply chainrdquo EuropeanJournal of Operational Research vol 196 no 1 pp 273ndash2882009

[36] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[37] P Kelle S Transchel and S Minner ldquoBuyer-supplier coopera-tion and negotiation support with random yield considerationrdquoInternational Journal of Production Economics vol 118 no 1 pp152ndash159 2009

[38] B Keren ldquoThe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmegavol 37 no 4 pp 801ndash810 2009

[39] P Kouvelis and J M Milner ldquoSupply chain capacity andoutsourcing decisions the dynamic interplay of demand andsupply uncertaintyrdquo IIE Transactions vol 34 no 8 pp 717ndash7282002

[40] X Yan M Zhang and K Liu ldquoA note on coordination indecentralized assembly systems with uncertain componentyieldsrdquo European Journal of Operational Research vol 205 no2 pp 469ndash478 2010

[41] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[42] K Zimmer ldquoSupply chain coordination with uncertain just-in-time deliveryrdquo International Journal of Production Economicsvol 77 no 1 pp 1ndash15 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909times int11986311991111990201199090

(119863119911 minus 1199020119909119910)1198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909timesint11986311991111990210

(119863119911minus1199021119910)1198911 (119909) 1198912 (119910) 119889119910]st 1199021198800 = 1205821199021

(6)

Proposition 2 Under the independent and noncoordinateddecision-making pattern the manufacturerrsquos expected profitfunction 119864(1205871198801 ) is concave in 1199021 and the optimal orderquantity 1199021198801 satisfies the following equation

int+infin0

119892 (119911) 119889119911 [int11205820

119889119909int1198631199111205821199021198801 1199090

1205821199091199101198911 (119909) 1198912 (119910) 119889119910

+int+infin1120582

119889119909int11986311991111990211988010

1199101198911 (119909) 1198912 (119910) 119889119910]

= (1199010 + 1198881 + ℎ1) [int11205820 1205821199091198911 (119909) 119889119909 + int+infin1120582 1198911 (119909) 119889119909]1199011 + 1199031 + ℎ1 (7)

Proof Since 1199021198800 (1199021) = 1205821199021

119889119864 [1205872]1198891199021 = (1199011 + 1199031 + ℎ1) int+infin0

119892 (119911) 119889119911times [int11205820

119889119909int11986311991112058211990211199090

1205821199091199101198911 (119909) 1198912 (119910) 119889119910+int+infin1120582

119889119909int11986311991111990210

1199101198911 (119909) 1198912 (119910) 119889119910]minus 120582 (1199010 + 1198881 + ℎ1)+ (1199010 + 1198881) int+infin

1120582(120582119909 minus 1) 1198911 (119909) 119889119909

(8)

1198892119864 [1205872]11988911990221 = minus (1199011 + ℎ1 + 1199031) int+infin0

119892 (119911) 119889119911

times [int11205820

(119863119911)212058211990231119909 1198911 (119909) 1198912 (1198631199111205821199021119909)119889119909

+int+infin1120582

(119863119911)211990231 1198911 (119909) 1198912 (1198631199111199021 )119889119909] lt 0(9)

So 119864(1205871198802 ) is concave in 1199021 and setting (119889119864(1205871198802 ))1198891199021 = 0 wehave (7)

42 Production and Ordering Decision Making Guided bya Centralized Decision-Making Pattern In the centralizedsupply chain the goal of the supply chain operation is tomaximize the total profit The manufacturer directly conveysthe demand information to the supplier In the centralizedcase the shortage cost of the supplier can be seen as amatter of an internal revenue transfer which barely affectsthe decision making in production and ordering The profitgenerated by the supply chain dependent on the centralizeddecision making is

1205871198620 = 1199011min [119863 sdot 119885min (1199021 1199020 sdot 119883) sdot 119884]minus 1198881min (1199021 1199020 sdot 119883) minus 11988801199020 minus ℎ0max [1199020 sdot 119883 minus 1199021 0]minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minusmin (1199021 1199020 sdot 119883) sdot 119884 0]

(10)

The operational goal is to realize the maximized profit inthis supply chain and the operational model is

max 119864 [1205871198620 ]= 1199011 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910

+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]

minus 11988801199020 minus ℎ0 int+infin11990211199020

(1199020 sdot 119909 minus 1199021) 1198911 (119909) 119889119909

6 Mathematical Problems in Engineering

minus 1198881 [int119902111990200

11990201199091198911 (119909) 119889119909 + int+infin11990211199020

11990211198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910] (11)

Proposition 3 Consider 119864[1205871198620 ] ge 119864[1205871198801 ] + 119864[1205871198802 ]Proof From (1) (5) and (10) we can obtain

119864 [1205871198620 ] ge 119864 [1205871 (1199021198620 )] + 119864 [1205872 (1199021198621 )] = max1199020 1199021

119864 [1205870 (1199020 1199021)]ge 119864 [1205871 (1199021198800 )] + 119864 [1205872 (1199021198801 )] = 119864 [1205871198801 ] + 119864 [1205871198802 ]

(12)

Proposition 3 illustrates that under the independent andnoncoordinated decision making pattern the Nash equilib-rium existing between the supplier and the manufacturerreduces the profit of the supply chain In order to fulfillthe profit margin in the centralized supply chains under theindependent decision-making pattern there is a necessity todesign an efficient coordinating mechanism

5 Production and Ordering Decision underQuantity Discount Coordination

According to (7) the ordering quantity by the manufacturerdecreases with the price of the raw material In this sense aquantity discount contract amounts to the reduction of theordering price of the raw material through which the order-ing quantity by themanufacturer can be promotedThe studyis based on the quantity discount contract conforming towhich qualification point of quantity discount 119902lowast and pricediscount rate 120572 can be established When the amount of theraw material provided by the supplier to the manufacturer isless than 119902lowast the price is set at 1199010 when the amount of theraw material exceeds 119902lowast the price of the excess part is set at(1 minus 120572)1199011 where 0 lt 120572 lt 1

Under the quantity discount contract the profit of thesupplier is given as follows

120587QD1 (1199020) = 1199010min [1199021 1199020119883]

minus 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 11988801199020 minus ℎ0max [1199020119883 minus 1199021 0]minus 1199030max [1199021 minus 1199020119883 0]

(13)

The first term is the sales revenue of the supplier at theprice of 1199010 the second term is the price discount given tomanufacturer on the basis of quantity discount the thirdterm is the production cost of the supplier the fourth termis the inventory cost of the supplier and the fifth term is theshortage cost of the raw material and the sixth term is theshortage cost of the raw material

Under the quantity discount contract the expected profitof the supplier is

119864 [120587QD1 (1199020)]= 1199010 [int11990211199020

0(1199020119909)1198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909] minus 1205721199010times [int11990211199020119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus 11988801199020 minus ℎ0 int+infin

11990211199020

(1199020119909 minus 1199021) 1198911 (119909) 119889119909minus 1199030 int119902111990200

(1199021 minus 1199020119909)1198911 (119909) 119889119909(14)

Proposition 4 Under the quantity discount contract thesupplierrsquos expected profit function 119864(120587QD1 ) is concave in 1199020and the optimal production quantity of the raw material 119902QD0 satisfies the following equation

int119902111990200

1199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(15)

Mathematical Problems in Engineering 7

Proof Since

119889119864 [120587QD1 (1199020)]1198891199020 = (1199010 + ℎ0 + 1199030) int11990211199020

01199091198911 (119909) 119889119909

minus 1205721199010 int11990211199020119902lowast1199020

1199091198911 (119909) 119889119909 minus 1198880 minus ℎ01198892119864 [120587QD

1 (1199020)]11988911990220 = minus (1199010 + ℎ0 + 1199030 minus 1205721199010) 11990221119902301198911 (

11990211199020)

minus 1205721199010 119902lowast211990230 1198911 (119902lowast1199020 ) lt 0

(16)

So 119864(120587QD1 ) is concave in 1199020 and setting (120597119864(120587QD

1 ))1205971199020 = 0we have (15)

Under the quantity discount contract the profit of themanufacturer is

120587QD1 (1199021) = 1199011min [119863 sdot 119885min (1199021 1199020 sdot 119883) sdot 119884]

minus 1199010min [1199021 1199020 sdot 119883]+ 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 1198881 [min (1199021 1199020 sdot 119883)]minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minus (min (1199021 1199020 sdot 119883) sdot 119884) 0]

(17)

The first term is the sales revenue of the manufacturerthe second term is the ordering cost of the rawmaterial at theprice of 1199010 the third term is the price discount based on thequantity discount contract the fourth term is the productioncost of the manufacturer the fifth term is the inventory costof the manufacturer and the sixth term is the shortage cost ofthe manufacturer

The manufacturer operates under the constraint restric-tions of the supplier and it precedes the decision making ofproduction and ordering by the standard of maximizing theprofits

max1199021

119864 [120587QD2 (1199021)]

= 1199011 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910

+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881) [int11990211199020

011990201199091198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909]+ 1205721199010 [int11990211199020

119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910) 1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910]st int11990211199020

01199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int

11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(18)

Setting 119889119864[120587QD2 (1199021)]1198891199021 = 0 we obtain the optimal

order quantity 119902QD1 which satisfies the following equation

(1199011 + ℎ1 + 1199031) int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

119902101584001199091199101198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

1199101198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881 + ℎ1 minus 1205721199010)times [int119902111990200

119902101584001199091198911 (119909) 119889119909 + int+infin11990211199020

1198911 (119909) 119889119909]

minus 1205721199010 int119902lowast1199020

0119902101584001199091198911 (119909) 119889119909 = 0

(19)

The 1199020 is a function of 1199021 from (15) and 11990210158400 is 1199020rsquos derivativeby 1199021

8 Mathematical Problems in Engineering

Table 2 Numerical results for all models (119898 = 015 119899 = 01 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4256 4197 1014 74585 146830 221415

C centralized decision mdash mdash 4351 4351 1 mdash mdash 222105

QD quantity discount withthe supplier as a dominantparty

01 3853 4415 4387 1006 75275 146830 22210502 4221 4307 4251 1013 75275 146830 22210503 4231 4307 4251 1013 75275 146830 22210504 4236 4307 4251 1013 75275 146830 222105

QD quantity discount withthe manufacturer as adominant party

01 4004 4310 4267 101 74585 147520 22210502 4033 4397 4381 1004 74585 147520 22210503 4173 4310 4270 1009 74585 147520 22210504 4197 4392 4379 1003 74585 147520 222105

6 The Design of Quantity Discount Contracts

The actual producing activities can be boiled down to theissue of the Stackelberg game problem Quantity discountcontracts under both the supplier as the dominant side andthe manufacturer as the dominant side are to be discussedseparately The principle of designing the pact is as follows

(1) In the Stackelberg game the dominant side fulfills thequantity discount coordination of the supply chainthrough the determination of the quantity discountqualification point and the price discount rate

(2) Considering the rationality of the follower the dom-inant side for the acceptance of the strategy by thefollower should ensure that the interest of the followershould be no less than the interest gained under thenoncoordinated pattern

(3) Based on the fulfillment of the above conditionsthe expected profit of the dominant side should bemaximized

61 The Quantity Discount Contract with the Supplier as theDominant Side Under the uncertainty of the two-echelonyields and demand provided that the supplier holds thedominant position with themanufacturer being the followerthe supplier is in the position to determine the parameterpackage of the quantity discount including the quantitydiscount qualification point 119902lowast and the price discount rate120572 so as to realize the coordination in the supply chain Themodel is as follows

max 119864 (120587QD1 )

st Equation (19)119864 (120587QD2 ) ge 119864 (1205871198802 )

(20)

For the first item as the supplier holds the dominantposition the goal is to achieve the maximized profit of thesupplier in the decision making The second item represents

the ordering strategy of the manufacturer the third itemguarantees the profits of the manufacturer should not be lessthan that under independent and noncoordinated decisionmaking pattern

62 The Quantity Discount Contract with the Manufactureras the Dominant Party Under the uncertainty of the two-echelon yields and demand in supply chains given thatthe manufacturer is in the dominant position in quantitydiscount and coordination with the supplier in a secondaryposition the manufacturer determines the parameter pack-age of quantity discount including the quantity discountqualification point and the price discount rate so as to realizecoordination in supply chains The model is shown as below

max 119864 (120587QD2 )

st Equation (15)119864 (120587QD1 ) ge 119864 (1205871198801 )

(21)

The first item shows that the goal to be achieved is tomaximize the profit of the manufacturer in the decisionmaking the second item shows that the ordering strategyis provided by the supplier the third item shows that thesupplierrsquos profit level should be kept no less than the levelunder the independent andnoncoordinated decision-makingpattern

7 Numerical Examples and Discussions

The parameters of a certain supply chain are set as below1198880 = 20 ℎ0 = 5 1199030 = 15 1198881 = 10 ℎ1 = 7 1199031 = 301199010 = 40 1199011 = 95 119863 = 400 The output of the supplier andthe output of the manufacturer random variables for marketdemand of the finished products are uniformly distributedover [1minus119898 1+119898] [1minus119899 1+119899] and [1minus 119897 1+ 119897] respectively

Tables 2ndash4 present the decision making for productionand ordering and the expected profits under an independent

Mathematical Problems in Engineering 9

Table 3 Numerical results for all models (119898 = 02 119899 = 02 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4345 4293 1012 71085 141007 212093

C centralized decision mdash mdash 4537 4537 1 mdash mdash 216542

QD quantity discount withthe supplier as a dominantparty

01 4125 4592 4558 1007 75535 141007 21654202 4330 4590 4558 1007 75535 141007 21654203 4433 4515 4460 1012 75535 141007 21654204 4440 4515 4460 1012 75535 141007 216542

QD quantity discount withthe manufacturer as adominant party

01 3517 4534 4534 1 71869 144673 21654202 4058 4533 4532 1 72759 143783 21654203 4222 4532 4531 1 73007 143535 21654204 4302 4532 4531 1 73123 143419 216542

Table 4 Numerical results for all models (119898 = 015 119899 = 01 119897 = 03)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4291 4232 1014 75196 125231 200427

C centralized decision mdash mdash 4387 4387 1 mdash mdash 201123

QD quantity discount withthe supplier as a dominantparty

01 4227 4342 4286 1013 75892 125231 20112302 4256 4342 4286 1013 75892 125231 20112303 4266 4342 4286 1013 75892 125231 20112304 4271 4342 4286 1013 75892 125231 201123

QD quantity discount withthe manufacturer as adominant party

01 4036 4346 4305 101 75196 125927 20112302 4162 4346 4305 101 75196 125927 20112303 4207 4346 4305 101 75196 125927 20112304 4231 4346 4305 101 75196 125927 201123

and noncoordinated decision-making pattern and the cen-tralized decision making pattern under the the uncertaintyof the two-echelon yields and demand they also respectivelypresent quantity discount coordination decisionmaking withthe supplier as the dominant role solving by model (20) andthe quantity discount coordination decision making with themanufacturer as the dominant role solving by model (21)

With Tables 2ndash4 the following conclusions can be drawn

(1) Under an independent decision-making patternthere exists the Nash equilibrium between the sup-plier and the manufacturer and the profit under theNash equilibrium is less than that gained under thecentralized decision-making pattern under the quan-tity discount pact themanufacturer has increased theordering quantity of the raw material the ratios ofthe planned output of the supplier to the orderingquantity of the manufacturer have decreased

(2) Under the quantity discount pact with the supplierin the dominant position the supplier sets different

combinations of the quantity discount qualificationpoint and the price discount rate under the situationwhere the profit of the manufacturer under the quan-tity discount coordination pact exceeds that underindependent and noncoordinated decision-makingpattern In this sense the profit of the supply chainbased on quantity discount reaches the profit level ofthe chain under a centralized pattern The supplierand the manufacturer can select a certain combina-tion and set forth a quantity discount coordinationpact

(3) Under the quantity discount coordination with themanufacturer in the dominant position the manu-facturer sets different combinations as well With theprofit of the supply chain under the quantity discountcoordination pact exceeding the profit under theindependent and noncoordinated decision-makingpattern the chain profit based on quantity discountreaches the profit level gained in a centralized patternThe supplier and themanufacturer can select a certain

10 Mathematical Problems in Engineering

combination and set forth a quantity discount coordi-nation pact

(4) With a comparison of Tables 2 3 and 4 it can befound that the larger the uncertainty of two-echelonyields and demand the larger the contribution of thequantity discount coordination pact to the profit ofthe supply chains With 119898 = 015 119899 = 01 and 119897 =02 the quantity discount coordination pact helpsincrease the total profit by 03 With 119898 = 015119899 = 01 and 119897 = 03 the quantity discountcoordination pact helps increase the total profit by04 With 119898 = 02 119899 = 02 and 119897 = 02 thequantity discount coordination pact helps increasethe total profit by 21 This shows that the quantitydiscount coordination contract can effectively reducethe negative effect by the uncertainty of two-echelonyields and demand upon the efficiency and benefit ofthe supply chains

(5) We also find out that undermost scenarios the supplychain benefits from the yields and demand risksreduction and generates a higher profit ContrastingTables 2 and 3 we find out that sometimes in thequantity discount contract the supplierrsquos profit canincrease as yieldsrsquo randomness increases And con-trasting Tables 2 and 4 we find out that in the unco-ordinated case and quantity discount contract thesupplierrsquos profit can increase as demand randomnessincreases

8 Conclusions

This paper explores the quantity discount coordinationmod-els in the fashion supply chain with uncertain yields andrandom demand The planned production outputs of thesupplier and the ordering quantity of the manufacturer havebeen set as the decision-making parameters The indepen-dent and noncoordinated and centralized decision-makingpatterns have been established under the uncertainty of two-echelon yields and demandThrough the analysis it has beenfound that the profit of supply chains under independentand noncoordinated decision-making pattern is less than thatunder the centralized pattern which proves the necessityof the coordination of supply chains Based on this twoparametersmdashquantity discount qualification point and pricediscount ratiomdashhave been introduced and quantity discountcoordination models have been established The quantitydiscount coordination pacts with both the supplier and themanufacturer in the dominant position have been set forth

The result reveals that the quantity discount coordinationpact proposed by this paper can promote the manufacturerrsquosordering quantity of the raw material which contributesto the result that the profit of the supply chain based onquantity discount reaches the profit level of the supply chainunder the centralized decision-making pattern The largerthe uncertainty of two-echelon yields and demand the largercontribution of the pact to the benefit of the supply chainwill be The quantity discount coordination pact under theuncertainty of two-echelon yields and demand can reduce the

negative influence brought by the uncertainty and thereforecan contribute to the purpose of the coordination of thesupply chain

Acknowledgments

The authors would like to thank the referees for theirconstructive comments and suggestions that have greatlyimproved the quality of the paper The research was sup-ported by the National Natural Science Foundation of China(Grant no 71201164)

References

[1] D J Thomas and P M Griffin ldquoCoordinated supply chainmanagementrdquo European Journal of Operational Research vol94 no 1 pp 1ndash15 1996

[2] N Agrawal S A Smith andA A Tsay ldquoMulti-vendor sourcingin a retail supply chainrdquo Production and Operations Manage-ment vol 11 no 2 pp 157ndash181 2002

[3] C Chiang and W C Benton ldquoSole sourcing versus dualsourcing under stochastic demands and lead timesrdquo NavalResearch Logistics vol 41 no 5 pp 609ndash624 1994

[4] J D Hong and J C Hayya ldquoJust-in-time purchasing singleor multiple sourcingrdquo International Journal of ProductionEconomics vol 27 no 2 pp 175ndash181 1992

[5] M Treleven and S B Schweikhart ldquoA riskbenefit analysisof sourcing strategies single vs multiple sourcingrdquo Journal ofOperations Management vol 7 no 3-4 pp 93ndash114 1988

[6] G P Cachon and P H Zipkin ldquoCompetitive and cooperativeinventory policies in a two-stage supply chainrdquo ManagementScience vol 45 no 7 pp 936ndash953 1999

[7] M Leng and M Parlar ldquoGame theoretic applications in supplychainmanagement a reviewrdquo INFOR vol 43 no 3 pp 187ndash2202005

[8] MMoses and S Seshadri ldquoPolicy mechanisms for supply chaincoordinationrdquo IIETransactions vol 32 no 3 pp 245ndash262 2000

[9] H J Peng and M H Zhou ldquoProduction and ordering decisionin supply chain with uncertainty in two-echelon yields anddemandrdquo Journal of Systems Engineering vol 25 no 5 pp 622ndash628 2010 (Chinese)

[10] C T Chang ldquoAn acquisition policy for a single item multi-supplier system with real-world constraintsrdquo Applied Mathe-matical Modelling vol 30 no 1 pp 1ndash9 2006

[11] C T Chang C L Chin and M F Lin ldquoOn the single itemmulti-supplier system with variable lead-time price-quantitydiscount and resource constraintsrdquo Applied Mathematics andComputation vol 182 no 1 pp 89ndash97 2006

[12] J Li and L Liu ldquoSupply chain coordination with quantitydiscount policyrdquo International Journal of Production Economicsvol 101 no 1 pp 89ndash98 2006

[13] C L Munson and M J Rosenblatt ldquoTheories and realitiesof quantity discounts an exploratory studyrdquo Production andOperations Management vol 7 no 4 pp 352ndash369 1998

[14] C L Munson and J Hu ldquoIncorporating quantity discountsand their inventory impacts into the centralized purchasingdecisionrdquoEuropean Journal ofOperational Research vol 201 no2 pp 581ndash592 2010

[15] H Y Kang and A H I Lee ldquoInventory replenishment modelusing fuzzy multiple objective programming a case study

Mathematical Problems in Engineering 11

of a high-tech company in Taiwanrdquo Applied Soft ComputingJournal vol 10 no 4 pp 1108ndash1118 2010

[16] A Mendoza and J A Ventura ldquoIncorporating quantity dis-counts to the EOQ model with transportation costsrdquo Interna-tional Journal of Production Economics vol 113 no 2 pp 754ndash765 2008

[17] J Moussourakis and C Haksever ldquoA practical model forordering in multi-product multi-constraint inventory systemswith all-units quantity discountsrdquo International Journal ofInformation and Management Sciences vol 19 no 2 pp 263ndash283 2008

[18] P A Rubin and W C Benton ldquoA generalized framework forquantity discount pricing schedulesrdquo Decision Sciences vol 34no 1 pp 173ndash188 2003

[19] G Zhang and L Ma ldquoOptimal acquisition policy with quantitydiscounts and uncertain demandsrdquo International Journal ofProduction Research vol 47 no 9 pp 2409ndash2425 2009

[20] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[21] A H I Lee and H Y Kang ldquoA mixed 0-1 integer programmingfor inventory model a case study of TFT-LCD manufacturingcompany in Taiwanrdquo Kybernetes vol 37 no 1 pp 66ndash82 2008

[22] J Shi and G Zhang ldquoMulti-product budget-constrained acqui-sition and pricing with uncertain demand and supplier quantitydiscountsrdquo International Journal of Production Economics vol128 no 1 pp 322ndash331 2010

[23] A A Taleizadeh S T A Niaki and R Nikousokhan ldquoCon-straint multiproduct joint-replenishment inventory controlproblem using uncertain programmingrdquo Applied Soft Comput-ing vol 11 pp 5143ndash5154 2011

[24] J Rezaei and M Davoodi ldquoA deterministic multi-item inven-tory model with supplier selection and imperfect qualityrdquoApplied Mathematical Modelling vol 32 no 10 pp 2106ndash21162008

[25] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[26] S P Chen and Y H Ho ldquoAnalysis of the newsboy problemwith fuzzy demands and incremental discountsrdquo InternationalJournal of Production Economics vol 129 no 1 pp 169ndash177 2011

[27] Y J Lin and C H Ho ldquoIntegrated inventory model withquantity discount and price-sensitive demandrdquo Top vol 19 no1 pp 177ndash188 2011

[28] S Krichen A Laabidi and F B Abdelaziz ldquoSingle suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in paymentsrdquo Computers andIndustrial Engineering vol 60 no 1 pp 164ndash172 2011

[29] J Shi G Zhang and J Sha ldquoJointly pricing and ordering fora multi-product multi-constraint newsvendor problem withsupplier quantity discountsrdquo Applied Mathematical Modellingvol 35 no 6 pp 3001ndash3011 2011

[30] H Shin and W C Benton ldquoA quantity discount approach tosupply chain coordinationrdquo European Journal of OperationalResearch vol 180 no 2 pp 601ndash616 2007

[31] H Gurnani and Y Gerchak ldquoCoordination in decentralizedassembly systems with uncertain component yieldsrdquo EuropeanJournal of Operational Research vol 176 no 3 pp 1559ndash15762007

[32] Y He and J Zhang ldquoRandom yield risk sharing in a two-levelsupply chainrdquo International Journal of Production Economicsvol 112 no 2 pp 769ndash781 2008

[33] Y He and J Zhang ldquoRandom yield supply chain with a yielddependent secondary marketrdquo European Journal of OperationalResearch vol 206 no 1 pp 221ndash230 2010

[34] C C Hsieh and C H Wu ldquoCapacity allocation ordering andpricing decisions in a supply chain with demand and supplyuncertaintiesrdquo European Journal of Operational Research vol184 no 2 pp 667ndash684 2008

[35] C C Hsieh and C H Wu ldquoCoordinated decisions for substi-tutable products in a common retailer supply chainrdquo EuropeanJournal of Operational Research vol 196 no 1 pp 273ndash2882009

[36] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[37] P Kelle S Transchel and S Minner ldquoBuyer-supplier coopera-tion and negotiation support with random yield considerationrdquoInternational Journal of Production Economics vol 118 no 1 pp152ndash159 2009

[38] B Keren ldquoThe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmegavol 37 no 4 pp 801ndash810 2009

[39] P Kouvelis and J M Milner ldquoSupply chain capacity andoutsourcing decisions the dynamic interplay of demand andsupply uncertaintyrdquo IIE Transactions vol 34 no 8 pp 717ndash7282002

[40] X Yan M Zhang and K Liu ldquoA note on coordination indecentralized assembly systems with uncertain componentyieldsrdquo European Journal of Operational Research vol 205 no2 pp 469ndash478 2010

[41] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[42] K Zimmer ldquoSupply chain coordination with uncertain just-in-time deliveryrdquo International Journal of Production Economicsvol 77 no 1 pp 1ndash15 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

minus 1198881 [int119902111990200

11990201199091198911 (119909) 119889119909 + int+infin11990211199020

11990211198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910] (11)

Proposition 3 Consider 119864[1205871198620 ] ge 119864[1205871198801 ] + 119864[1205871198802 ]Proof From (1) (5) and (10) we can obtain

119864 [1205871198620 ] ge 119864 [1205871 (1199021198620 )] + 119864 [1205872 (1199021198621 )] = max1199020 1199021

119864 [1205870 (1199020 1199021)]ge 119864 [1205871 (1199021198800 )] + 119864 [1205872 (1199021198801 )] = 119864 [1205871198801 ] + 119864 [1205871198802 ]

(12)

Proposition 3 illustrates that under the independent andnoncoordinated decision making pattern the Nash equilib-rium existing between the supplier and the manufacturerreduces the profit of the supply chain In order to fulfillthe profit margin in the centralized supply chains under theindependent decision-making pattern there is a necessity todesign an efficient coordinating mechanism

5 Production and Ordering Decision underQuantity Discount Coordination

According to (7) the ordering quantity by the manufacturerdecreases with the price of the raw material In this sense aquantity discount contract amounts to the reduction of theordering price of the raw material through which the order-ing quantity by themanufacturer can be promotedThe studyis based on the quantity discount contract conforming towhich qualification point of quantity discount 119902lowast and pricediscount rate 120572 can be established When the amount of theraw material provided by the supplier to the manufacturer isless than 119902lowast the price is set at 1199010 when the amount of theraw material exceeds 119902lowast the price of the excess part is set at(1 minus 120572)1199011 where 0 lt 120572 lt 1

Under the quantity discount contract the profit of thesupplier is given as follows

120587QD1 (1199020) = 1199010min [1199021 1199020119883]

minus 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 11988801199020 minus ℎ0max [1199020119883 minus 1199021 0]minus 1199030max [1199021 minus 1199020119883 0]

(13)

The first term is the sales revenue of the supplier at theprice of 1199010 the second term is the price discount given tomanufacturer on the basis of quantity discount the thirdterm is the production cost of the supplier the fourth termis the inventory cost of the supplier and the fifth term is theshortage cost of the raw material and the sixth term is theshortage cost of the raw material

Under the quantity discount contract the expected profitof the supplier is

119864 [120587QD1 (1199020)]= 1199010 [int11990211199020

0(1199020119909)1198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909] minus 1205721199010times [int11990211199020119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus 11988801199020 minus ℎ0 int+infin

11990211199020

(1199020119909 minus 1199021) 1198911 (119909) 119889119909minus 1199030 int119902111990200

(1199021 minus 1199020119909)1198911 (119909) 119889119909(14)

Proposition 4 Under the quantity discount contract thesupplierrsquos expected profit function 119864(120587QD1 ) is concave in 1199020and the optimal production quantity of the raw material 119902QD0 satisfies the following equation

int119902111990200

1199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(15)

Mathematical Problems in Engineering 7

Proof Since

119889119864 [120587QD1 (1199020)]1198891199020 = (1199010 + ℎ0 + 1199030) int11990211199020

01199091198911 (119909) 119889119909

minus 1205721199010 int11990211199020119902lowast1199020

1199091198911 (119909) 119889119909 minus 1198880 minus ℎ01198892119864 [120587QD

1 (1199020)]11988911990220 = minus (1199010 + ℎ0 + 1199030 minus 1205721199010) 11990221119902301198911 (

11990211199020)

minus 1205721199010 119902lowast211990230 1198911 (119902lowast1199020 ) lt 0

(16)

So 119864(120587QD1 ) is concave in 1199020 and setting (120597119864(120587QD

1 ))1205971199020 = 0we have (15)

Under the quantity discount contract the profit of themanufacturer is

120587QD1 (1199021) = 1199011min [119863 sdot 119885min (1199021 1199020 sdot 119883) sdot 119884]

minus 1199010min [1199021 1199020 sdot 119883]+ 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 1198881 [min (1199021 1199020 sdot 119883)]minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minus (min (1199021 1199020 sdot 119883) sdot 119884) 0]

(17)

The first term is the sales revenue of the manufacturerthe second term is the ordering cost of the rawmaterial at theprice of 1199010 the third term is the price discount based on thequantity discount contract the fourth term is the productioncost of the manufacturer the fifth term is the inventory costof the manufacturer and the sixth term is the shortage cost ofthe manufacturer

The manufacturer operates under the constraint restric-tions of the supplier and it precedes the decision making ofproduction and ordering by the standard of maximizing theprofits

max1199021

119864 [120587QD2 (1199021)]

= 1199011 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910

+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881) [int11990211199020

011990201199091198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909]+ 1205721199010 [int11990211199020

119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910) 1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910]st int11990211199020

01199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int

11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(18)

Setting 119889119864[120587QD2 (1199021)]1198891199021 = 0 we obtain the optimal

order quantity 119902QD1 which satisfies the following equation

(1199011 + ℎ1 + 1199031) int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

119902101584001199091199101198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

1199101198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881 + ℎ1 minus 1205721199010)times [int119902111990200

119902101584001199091198911 (119909) 119889119909 + int+infin11990211199020

1198911 (119909) 119889119909]

minus 1205721199010 int119902lowast1199020

0119902101584001199091198911 (119909) 119889119909 = 0

(19)

The 1199020 is a function of 1199021 from (15) and 11990210158400 is 1199020rsquos derivativeby 1199021

8 Mathematical Problems in Engineering

Table 2 Numerical results for all models (119898 = 015 119899 = 01 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4256 4197 1014 74585 146830 221415

C centralized decision mdash mdash 4351 4351 1 mdash mdash 222105

QD quantity discount withthe supplier as a dominantparty

01 3853 4415 4387 1006 75275 146830 22210502 4221 4307 4251 1013 75275 146830 22210503 4231 4307 4251 1013 75275 146830 22210504 4236 4307 4251 1013 75275 146830 222105

QD quantity discount withthe manufacturer as adominant party

01 4004 4310 4267 101 74585 147520 22210502 4033 4397 4381 1004 74585 147520 22210503 4173 4310 4270 1009 74585 147520 22210504 4197 4392 4379 1003 74585 147520 222105

6 The Design of Quantity Discount Contracts

The actual producing activities can be boiled down to theissue of the Stackelberg game problem Quantity discountcontracts under both the supplier as the dominant side andthe manufacturer as the dominant side are to be discussedseparately The principle of designing the pact is as follows

(1) In the Stackelberg game the dominant side fulfills thequantity discount coordination of the supply chainthrough the determination of the quantity discountqualification point and the price discount rate

(2) Considering the rationality of the follower the dom-inant side for the acceptance of the strategy by thefollower should ensure that the interest of the followershould be no less than the interest gained under thenoncoordinated pattern

(3) Based on the fulfillment of the above conditionsthe expected profit of the dominant side should bemaximized

61 The Quantity Discount Contract with the Supplier as theDominant Side Under the uncertainty of the two-echelonyields and demand provided that the supplier holds thedominant position with themanufacturer being the followerthe supplier is in the position to determine the parameterpackage of the quantity discount including the quantitydiscount qualification point 119902lowast and the price discount rate120572 so as to realize the coordination in the supply chain Themodel is as follows

max 119864 (120587QD1 )

st Equation (19)119864 (120587QD2 ) ge 119864 (1205871198802 )

(20)

For the first item as the supplier holds the dominantposition the goal is to achieve the maximized profit of thesupplier in the decision making The second item represents

the ordering strategy of the manufacturer the third itemguarantees the profits of the manufacturer should not be lessthan that under independent and noncoordinated decisionmaking pattern

62 The Quantity Discount Contract with the Manufactureras the Dominant Party Under the uncertainty of the two-echelon yields and demand in supply chains given thatthe manufacturer is in the dominant position in quantitydiscount and coordination with the supplier in a secondaryposition the manufacturer determines the parameter pack-age of quantity discount including the quantity discountqualification point and the price discount rate so as to realizecoordination in supply chains The model is shown as below

max 119864 (120587QD2 )

st Equation (15)119864 (120587QD1 ) ge 119864 (1205871198801 )

(21)

The first item shows that the goal to be achieved is tomaximize the profit of the manufacturer in the decisionmaking the second item shows that the ordering strategyis provided by the supplier the third item shows that thesupplierrsquos profit level should be kept no less than the levelunder the independent andnoncoordinated decision-makingpattern

7 Numerical Examples and Discussions

The parameters of a certain supply chain are set as below1198880 = 20 ℎ0 = 5 1199030 = 15 1198881 = 10 ℎ1 = 7 1199031 = 301199010 = 40 1199011 = 95 119863 = 400 The output of the supplier andthe output of the manufacturer random variables for marketdemand of the finished products are uniformly distributedover [1minus119898 1+119898] [1minus119899 1+119899] and [1minus 119897 1+ 119897] respectively

Tables 2ndash4 present the decision making for productionand ordering and the expected profits under an independent

Mathematical Problems in Engineering 9

Table 3 Numerical results for all models (119898 = 02 119899 = 02 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4345 4293 1012 71085 141007 212093

C centralized decision mdash mdash 4537 4537 1 mdash mdash 216542

QD quantity discount withthe supplier as a dominantparty

01 4125 4592 4558 1007 75535 141007 21654202 4330 4590 4558 1007 75535 141007 21654203 4433 4515 4460 1012 75535 141007 21654204 4440 4515 4460 1012 75535 141007 216542

QD quantity discount withthe manufacturer as adominant party

01 3517 4534 4534 1 71869 144673 21654202 4058 4533 4532 1 72759 143783 21654203 4222 4532 4531 1 73007 143535 21654204 4302 4532 4531 1 73123 143419 216542

Table 4 Numerical results for all models (119898 = 015 119899 = 01 119897 = 03)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4291 4232 1014 75196 125231 200427

C centralized decision mdash mdash 4387 4387 1 mdash mdash 201123

QD quantity discount withthe supplier as a dominantparty

01 4227 4342 4286 1013 75892 125231 20112302 4256 4342 4286 1013 75892 125231 20112303 4266 4342 4286 1013 75892 125231 20112304 4271 4342 4286 1013 75892 125231 201123

QD quantity discount withthe manufacturer as adominant party

01 4036 4346 4305 101 75196 125927 20112302 4162 4346 4305 101 75196 125927 20112303 4207 4346 4305 101 75196 125927 20112304 4231 4346 4305 101 75196 125927 201123

and noncoordinated decision-making pattern and the cen-tralized decision making pattern under the the uncertaintyof the two-echelon yields and demand they also respectivelypresent quantity discount coordination decisionmaking withthe supplier as the dominant role solving by model (20) andthe quantity discount coordination decision making with themanufacturer as the dominant role solving by model (21)

With Tables 2ndash4 the following conclusions can be drawn

(1) Under an independent decision-making patternthere exists the Nash equilibrium between the sup-plier and the manufacturer and the profit under theNash equilibrium is less than that gained under thecentralized decision-making pattern under the quan-tity discount pact themanufacturer has increased theordering quantity of the raw material the ratios ofthe planned output of the supplier to the orderingquantity of the manufacturer have decreased

(2) Under the quantity discount pact with the supplierin the dominant position the supplier sets different

combinations of the quantity discount qualificationpoint and the price discount rate under the situationwhere the profit of the manufacturer under the quan-tity discount coordination pact exceeds that underindependent and noncoordinated decision-makingpattern In this sense the profit of the supply chainbased on quantity discount reaches the profit level ofthe chain under a centralized pattern The supplierand the manufacturer can select a certain combina-tion and set forth a quantity discount coordinationpact

(3) Under the quantity discount coordination with themanufacturer in the dominant position the manu-facturer sets different combinations as well With theprofit of the supply chain under the quantity discountcoordination pact exceeding the profit under theindependent and noncoordinated decision-makingpattern the chain profit based on quantity discountreaches the profit level gained in a centralized patternThe supplier and themanufacturer can select a certain

10 Mathematical Problems in Engineering

combination and set forth a quantity discount coordi-nation pact

(4) With a comparison of Tables 2 3 and 4 it can befound that the larger the uncertainty of two-echelonyields and demand the larger the contribution of thequantity discount coordination pact to the profit ofthe supply chains With 119898 = 015 119899 = 01 and 119897 =02 the quantity discount coordination pact helpsincrease the total profit by 03 With 119898 = 015119899 = 01 and 119897 = 03 the quantity discountcoordination pact helps increase the total profit by04 With 119898 = 02 119899 = 02 and 119897 = 02 thequantity discount coordination pact helps increasethe total profit by 21 This shows that the quantitydiscount coordination contract can effectively reducethe negative effect by the uncertainty of two-echelonyields and demand upon the efficiency and benefit ofthe supply chains

(5) We also find out that undermost scenarios the supplychain benefits from the yields and demand risksreduction and generates a higher profit ContrastingTables 2 and 3 we find out that sometimes in thequantity discount contract the supplierrsquos profit canincrease as yieldsrsquo randomness increases And con-trasting Tables 2 and 4 we find out that in the unco-ordinated case and quantity discount contract thesupplierrsquos profit can increase as demand randomnessincreases

8 Conclusions

This paper explores the quantity discount coordinationmod-els in the fashion supply chain with uncertain yields andrandom demand The planned production outputs of thesupplier and the ordering quantity of the manufacturer havebeen set as the decision-making parameters The indepen-dent and noncoordinated and centralized decision-makingpatterns have been established under the uncertainty of two-echelon yields and demandThrough the analysis it has beenfound that the profit of supply chains under independentand noncoordinated decision-making pattern is less than thatunder the centralized pattern which proves the necessityof the coordination of supply chains Based on this twoparametersmdashquantity discount qualification point and pricediscount ratiomdashhave been introduced and quantity discountcoordination models have been established The quantitydiscount coordination pacts with both the supplier and themanufacturer in the dominant position have been set forth

The result reveals that the quantity discount coordinationpact proposed by this paper can promote the manufacturerrsquosordering quantity of the raw material which contributesto the result that the profit of the supply chain based onquantity discount reaches the profit level of the supply chainunder the centralized decision-making pattern The largerthe uncertainty of two-echelon yields and demand the largercontribution of the pact to the benefit of the supply chainwill be The quantity discount coordination pact under theuncertainty of two-echelon yields and demand can reduce the

negative influence brought by the uncertainty and thereforecan contribute to the purpose of the coordination of thesupply chain

Acknowledgments

The authors would like to thank the referees for theirconstructive comments and suggestions that have greatlyimproved the quality of the paper The research was sup-ported by the National Natural Science Foundation of China(Grant no 71201164)

References

[1] D J Thomas and P M Griffin ldquoCoordinated supply chainmanagementrdquo European Journal of Operational Research vol94 no 1 pp 1ndash15 1996

[2] N Agrawal S A Smith andA A Tsay ldquoMulti-vendor sourcingin a retail supply chainrdquo Production and Operations Manage-ment vol 11 no 2 pp 157ndash181 2002

[3] C Chiang and W C Benton ldquoSole sourcing versus dualsourcing under stochastic demands and lead timesrdquo NavalResearch Logistics vol 41 no 5 pp 609ndash624 1994

[4] J D Hong and J C Hayya ldquoJust-in-time purchasing singleor multiple sourcingrdquo International Journal of ProductionEconomics vol 27 no 2 pp 175ndash181 1992

[5] M Treleven and S B Schweikhart ldquoA riskbenefit analysisof sourcing strategies single vs multiple sourcingrdquo Journal ofOperations Management vol 7 no 3-4 pp 93ndash114 1988

[6] G P Cachon and P H Zipkin ldquoCompetitive and cooperativeinventory policies in a two-stage supply chainrdquo ManagementScience vol 45 no 7 pp 936ndash953 1999

[7] M Leng and M Parlar ldquoGame theoretic applications in supplychainmanagement a reviewrdquo INFOR vol 43 no 3 pp 187ndash2202005

[8] MMoses and S Seshadri ldquoPolicy mechanisms for supply chaincoordinationrdquo IIETransactions vol 32 no 3 pp 245ndash262 2000

[9] H J Peng and M H Zhou ldquoProduction and ordering decisionin supply chain with uncertainty in two-echelon yields anddemandrdquo Journal of Systems Engineering vol 25 no 5 pp 622ndash628 2010 (Chinese)

[10] C T Chang ldquoAn acquisition policy for a single item multi-supplier system with real-world constraintsrdquo Applied Mathe-matical Modelling vol 30 no 1 pp 1ndash9 2006

[11] C T Chang C L Chin and M F Lin ldquoOn the single itemmulti-supplier system with variable lead-time price-quantitydiscount and resource constraintsrdquo Applied Mathematics andComputation vol 182 no 1 pp 89ndash97 2006

[12] J Li and L Liu ldquoSupply chain coordination with quantitydiscount policyrdquo International Journal of Production Economicsvol 101 no 1 pp 89ndash98 2006

[13] C L Munson and M J Rosenblatt ldquoTheories and realitiesof quantity discounts an exploratory studyrdquo Production andOperations Management vol 7 no 4 pp 352ndash369 1998

[14] C L Munson and J Hu ldquoIncorporating quantity discountsand their inventory impacts into the centralized purchasingdecisionrdquoEuropean Journal ofOperational Research vol 201 no2 pp 581ndash592 2010

[15] H Y Kang and A H I Lee ldquoInventory replenishment modelusing fuzzy multiple objective programming a case study

Mathematical Problems in Engineering 11

of a high-tech company in Taiwanrdquo Applied Soft ComputingJournal vol 10 no 4 pp 1108ndash1118 2010

[16] A Mendoza and J A Ventura ldquoIncorporating quantity dis-counts to the EOQ model with transportation costsrdquo Interna-tional Journal of Production Economics vol 113 no 2 pp 754ndash765 2008

[17] J Moussourakis and C Haksever ldquoA practical model forordering in multi-product multi-constraint inventory systemswith all-units quantity discountsrdquo International Journal ofInformation and Management Sciences vol 19 no 2 pp 263ndash283 2008

[18] P A Rubin and W C Benton ldquoA generalized framework forquantity discount pricing schedulesrdquo Decision Sciences vol 34no 1 pp 173ndash188 2003

[19] G Zhang and L Ma ldquoOptimal acquisition policy with quantitydiscounts and uncertain demandsrdquo International Journal ofProduction Research vol 47 no 9 pp 2409ndash2425 2009

[20] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[21] A H I Lee and H Y Kang ldquoA mixed 0-1 integer programmingfor inventory model a case study of TFT-LCD manufacturingcompany in Taiwanrdquo Kybernetes vol 37 no 1 pp 66ndash82 2008

[22] J Shi and G Zhang ldquoMulti-product budget-constrained acqui-sition and pricing with uncertain demand and supplier quantitydiscountsrdquo International Journal of Production Economics vol128 no 1 pp 322ndash331 2010

[23] A A Taleizadeh S T A Niaki and R Nikousokhan ldquoCon-straint multiproduct joint-replenishment inventory controlproblem using uncertain programmingrdquo Applied Soft Comput-ing vol 11 pp 5143ndash5154 2011

[24] J Rezaei and M Davoodi ldquoA deterministic multi-item inven-tory model with supplier selection and imperfect qualityrdquoApplied Mathematical Modelling vol 32 no 10 pp 2106ndash21162008

[25] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[26] S P Chen and Y H Ho ldquoAnalysis of the newsboy problemwith fuzzy demands and incremental discountsrdquo InternationalJournal of Production Economics vol 129 no 1 pp 169ndash177 2011

[27] Y J Lin and C H Ho ldquoIntegrated inventory model withquantity discount and price-sensitive demandrdquo Top vol 19 no1 pp 177ndash188 2011

[28] S Krichen A Laabidi and F B Abdelaziz ldquoSingle suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in paymentsrdquo Computers andIndustrial Engineering vol 60 no 1 pp 164ndash172 2011

[29] J Shi G Zhang and J Sha ldquoJointly pricing and ordering fora multi-product multi-constraint newsvendor problem withsupplier quantity discountsrdquo Applied Mathematical Modellingvol 35 no 6 pp 3001ndash3011 2011

[30] H Shin and W C Benton ldquoA quantity discount approach tosupply chain coordinationrdquo European Journal of OperationalResearch vol 180 no 2 pp 601ndash616 2007

[31] H Gurnani and Y Gerchak ldquoCoordination in decentralizedassembly systems with uncertain component yieldsrdquo EuropeanJournal of Operational Research vol 176 no 3 pp 1559ndash15762007

[32] Y He and J Zhang ldquoRandom yield risk sharing in a two-levelsupply chainrdquo International Journal of Production Economicsvol 112 no 2 pp 769ndash781 2008

[33] Y He and J Zhang ldquoRandom yield supply chain with a yielddependent secondary marketrdquo European Journal of OperationalResearch vol 206 no 1 pp 221ndash230 2010

[34] C C Hsieh and C H Wu ldquoCapacity allocation ordering andpricing decisions in a supply chain with demand and supplyuncertaintiesrdquo European Journal of Operational Research vol184 no 2 pp 667ndash684 2008

[35] C C Hsieh and C H Wu ldquoCoordinated decisions for substi-tutable products in a common retailer supply chainrdquo EuropeanJournal of Operational Research vol 196 no 1 pp 273ndash2882009

[36] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[37] P Kelle S Transchel and S Minner ldquoBuyer-supplier coopera-tion and negotiation support with random yield considerationrdquoInternational Journal of Production Economics vol 118 no 1 pp152ndash159 2009

[38] B Keren ldquoThe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmegavol 37 no 4 pp 801ndash810 2009

[39] P Kouvelis and J M Milner ldquoSupply chain capacity andoutsourcing decisions the dynamic interplay of demand andsupply uncertaintyrdquo IIE Transactions vol 34 no 8 pp 717ndash7282002

[40] X Yan M Zhang and K Liu ldquoA note on coordination indecentralized assembly systems with uncertain componentyieldsrdquo European Journal of Operational Research vol 205 no2 pp 469ndash478 2010

[41] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[42] K Zimmer ldquoSupply chain coordination with uncertain just-in-time deliveryrdquo International Journal of Production Economicsvol 77 no 1 pp 1ndash15 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Proof Since

119889119864 [120587QD1 (1199020)]1198891199020 = (1199010 + ℎ0 + 1199030) int11990211199020

01199091198911 (119909) 119889119909

minus 1205721199010 int11990211199020119902lowast1199020

1199091198911 (119909) 119889119909 minus 1198880 minus ℎ01198892119864 [120587QD

1 (1199020)]11988911990220 = minus (1199010 + ℎ0 + 1199030 minus 1205721199010) 11990221119902301198911 (

11990211199020)

minus 1205721199010 119902lowast211990230 1198911 (119902lowast1199020 ) lt 0

(16)

So 119864(120587QD1 ) is concave in 1199020 and setting (120597119864(120587QD

1 ))1205971199020 = 0we have (15)

Under the quantity discount contract the profit of themanufacturer is

120587QD1 (1199021) = 1199011min [119863 sdot 119885min (1199021 1199020 sdot 119883) sdot 119884]

minus 1199010min [1199021 1199020 sdot 119883]+ 1205721199010max [min (1199021 1199020119883) minus 119902lowast 0]minus 1198881 [min (1199021 1199020 sdot 119883)]minus ℎ1max [min (1199021 1199020 sdot 119883) sdot 119884 minus 119863 sdot 119885 0]minus 1199031max [119863 sdot 119885 minus (min (1199021 1199020 sdot 119883) sdot 119884) 0]

(17)

The first term is the sales revenue of the manufacturerthe second term is the ordering cost of the rawmaterial at theprice of 1199010 the third term is the price discount based on thequantity discount contract the fourth term is the productioncost of the manufacturer the fifth term is the inventory costof the manufacturer and the sixth term is the shortage cost ofthe manufacturer

The manufacturer operates under the constraint restric-tions of the supplier and it precedes the decision making ofproduction and ordering by the standard of maximizing theprofits

max1199021

119864 [120587QD2 (1199021)]

= 1199011 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

11990201199091199101198911 (119909) 1198912 (119910) 119889119910+ int119902111990200

119889119909int+infin1198631199111199020119909

1198631199111198911 (119909) 1198912 (119910) 119889119910+ int+infin11990211199020

119889119909int11986311991111990210

11990211199101198911 (119909) 1198912 (119910) 119889119910

+int+infin11990211199020

119889119909int+infin1198631199111199021

1198631199111198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881) [int11990211199020

011990201199091198911 (119909) 119889119909 + int+infin

11990211199020

11990211198911 (119909) 119889119909]+ 1205721199010 [int11990211199020

119902lowast1199020

(1199020119909 minus 119902lowast) 1198911 (119909) 119889119909+int+infin11990211199020

(1199021 minus 119902lowast) 1198911 (119909) 119889119909]minus ℎ1 int+infin

0119892 (119911) 119889119911

times [int119902111990200

119889119909int+infin1198631199111199020119909

(1199020119909119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int+infin1198631199111199021

(1199021119910 minus 119863119911)1198911 (119909) 1198912 (119910) 119889119910]minus 1199031 int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

(119863119911 minus 1199020119909119910) 1198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

(119863119911 minus 1199021119910)1198911 (119909) 1198912 (119910) 119889119910]st int11990211199020

01199091198911 (119909) 119889119909 minus 12057211990101199010 + ℎ0 + 1199030 int

11990211199020

119902lowast1199020

1199091198911 (119909) 119889119909= 1198880 + ℎ01199010 + ℎ0 + 1199030

(18)

Setting 119889119864[120587QD2 (1199021)]1198891199021 = 0 we obtain the optimal

order quantity 119902QD1 which satisfies the following equation

(1199011 + ℎ1 + 1199031) int+infin0

119892 (119911) 119889119911times [int119902111990200

119889119909int11986311991111990201199090

119902101584001199091199101198911 (119909) 1198912 (119910) 119889119910+int+infin11990211199020

119889119909int11986311991111990210

1199101198911 (119909) 1198912 (119910) 119889119910]minus (1199010 + 1198881 + ℎ1 minus 1205721199010)times [int119902111990200

119902101584001199091198911 (119909) 119889119909 + int+infin11990211199020

1198911 (119909) 119889119909]

minus 1205721199010 int119902lowast1199020

0119902101584001199091198911 (119909) 119889119909 = 0

(19)

The 1199020 is a function of 1199021 from (15) and 11990210158400 is 1199020rsquos derivativeby 1199021

8 Mathematical Problems in Engineering

Table 2 Numerical results for all models (119898 = 015 119899 = 01 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4256 4197 1014 74585 146830 221415

C centralized decision mdash mdash 4351 4351 1 mdash mdash 222105

QD quantity discount withthe supplier as a dominantparty

01 3853 4415 4387 1006 75275 146830 22210502 4221 4307 4251 1013 75275 146830 22210503 4231 4307 4251 1013 75275 146830 22210504 4236 4307 4251 1013 75275 146830 222105

QD quantity discount withthe manufacturer as adominant party

01 4004 4310 4267 101 74585 147520 22210502 4033 4397 4381 1004 74585 147520 22210503 4173 4310 4270 1009 74585 147520 22210504 4197 4392 4379 1003 74585 147520 222105

6 The Design of Quantity Discount Contracts

The actual producing activities can be boiled down to theissue of the Stackelberg game problem Quantity discountcontracts under both the supplier as the dominant side andthe manufacturer as the dominant side are to be discussedseparately The principle of designing the pact is as follows

(1) In the Stackelberg game the dominant side fulfills thequantity discount coordination of the supply chainthrough the determination of the quantity discountqualification point and the price discount rate

(2) Considering the rationality of the follower the dom-inant side for the acceptance of the strategy by thefollower should ensure that the interest of the followershould be no less than the interest gained under thenoncoordinated pattern

(3) Based on the fulfillment of the above conditionsthe expected profit of the dominant side should bemaximized

61 The Quantity Discount Contract with the Supplier as theDominant Side Under the uncertainty of the two-echelonyields and demand provided that the supplier holds thedominant position with themanufacturer being the followerthe supplier is in the position to determine the parameterpackage of the quantity discount including the quantitydiscount qualification point 119902lowast and the price discount rate120572 so as to realize the coordination in the supply chain Themodel is as follows

max 119864 (120587QD1 )

st Equation (19)119864 (120587QD2 ) ge 119864 (1205871198802 )

(20)

For the first item as the supplier holds the dominantposition the goal is to achieve the maximized profit of thesupplier in the decision making The second item represents

the ordering strategy of the manufacturer the third itemguarantees the profits of the manufacturer should not be lessthan that under independent and noncoordinated decisionmaking pattern

62 The Quantity Discount Contract with the Manufactureras the Dominant Party Under the uncertainty of the two-echelon yields and demand in supply chains given thatthe manufacturer is in the dominant position in quantitydiscount and coordination with the supplier in a secondaryposition the manufacturer determines the parameter pack-age of quantity discount including the quantity discountqualification point and the price discount rate so as to realizecoordination in supply chains The model is shown as below

max 119864 (120587QD2 )

st Equation (15)119864 (120587QD1 ) ge 119864 (1205871198801 )

(21)

The first item shows that the goal to be achieved is tomaximize the profit of the manufacturer in the decisionmaking the second item shows that the ordering strategyis provided by the supplier the third item shows that thesupplierrsquos profit level should be kept no less than the levelunder the independent andnoncoordinated decision-makingpattern

7 Numerical Examples and Discussions

The parameters of a certain supply chain are set as below1198880 = 20 ℎ0 = 5 1199030 = 15 1198881 = 10 ℎ1 = 7 1199031 = 301199010 = 40 1199011 = 95 119863 = 400 The output of the supplier andthe output of the manufacturer random variables for marketdemand of the finished products are uniformly distributedover [1minus119898 1+119898] [1minus119899 1+119899] and [1minus 119897 1+ 119897] respectively

Tables 2ndash4 present the decision making for productionand ordering and the expected profits under an independent

Mathematical Problems in Engineering 9

Table 3 Numerical results for all models (119898 = 02 119899 = 02 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4345 4293 1012 71085 141007 212093

C centralized decision mdash mdash 4537 4537 1 mdash mdash 216542

QD quantity discount withthe supplier as a dominantparty

01 4125 4592 4558 1007 75535 141007 21654202 4330 4590 4558 1007 75535 141007 21654203 4433 4515 4460 1012 75535 141007 21654204 4440 4515 4460 1012 75535 141007 216542

QD quantity discount withthe manufacturer as adominant party

01 3517 4534 4534 1 71869 144673 21654202 4058 4533 4532 1 72759 143783 21654203 4222 4532 4531 1 73007 143535 21654204 4302 4532 4531 1 73123 143419 216542

Table 4 Numerical results for all models (119898 = 015 119899 = 01 119897 = 03)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4291 4232 1014 75196 125231 200427

C centralized decision mdash mdash 4387 4387 1 mdash mdash 201123

QD quantity discount withthe supplier as a dominantparty

01 4227 4342 4286 1013 75892 125231 20112302 4256 4342 4286 1013 75892 125231 20112303 4266 4342 4286 1013 75892 125231 20112304 4271 4342 4286 1013 75892 125231 201123

QD quantity discount withthe manufacturer as adominant party

01 4036 4346 4305 101 75196 125927 20112302 4162 4346 4305 101 75196 125927 20112303 4207 4346 4305 101 75196 125927 20112304 4231 4346 4305 101 75196 125927 201123

and noncoordinated decision-making pattern and the cen-tralized decision making pattern under the the uncertaintyof the two-echelon yields and demand they also respectivelypresent quantity discount coordination decisionmaking withthe supplier as the dominant role solving by model (20) andthe quantity discount coordination decision making with themanufacturer as the dominant role solving by model (21)

With Tables 2ndash4 the following conclusions can be drawn

(1) Under an independent decision-making patternthere exists the Nash equilibrium between the sup-plier and the manufacturer and the profit under theNash equilibrium is less than that gained under thecentralized decision-making pattern under the quan-tity discount pact themanufacturer has increased theordering quantity of the raw material the ratios ofthe planned output of the supplier to the orderingquantity of the manufacturer have decreased

(2) Under the quantity discount pact with the supplierin the dominant position the supplier sets different

combinations of the quantity discount qualificationpoint and the price discount rate under the situationwhere the profit of the manufacturer under the quan-tity discount coordination pact exceeds that underindependent and noncoordinated decision-makingpattern In this sense the profit of the supply chainbased on quantity discount reaches the profit level ofthe chain under a centralized pattern The supplierand the manufacturer can select a certain combina-tion and set forth a quantity discount coordinationpact

(3) Under the quantity discount coordination with themanufacturer in the dominant position the manu-facturer sets different combinations as well With theprofit of the supply chain under the quantity discountcoordination pact exceeding the profit under theindependent and noncoordinated decision-makingpattern the chain profit based on quantity discountreaches the profit level gained in a centralized patternThe supplier and themanufacturer can select a certain

10 Mathematical Problems in Engineering

combination and set forth a quantity discount coordi-nation pact

(4) With a comparison of Tables 2 3 and 4 it can befound that the larger the uncertainty of two-echelonyields and demand the larger the contribution of thequantity discount coordination pact to the profit ofthe supply chains With 119898 = 015 119899 = 01 and 119897 =02 the quantity discount coordination pact helpsincrease the total profit by 03 With 119898 = 015119899 = 01 and 119897 = 03 the quantity discountcoordination pact helps increase the total profit by04 With 119898 = 02 119899 = 02 and 119897 = 02 thequantity discount coordination pact helps increasethe total profit by 21 This shows that the quantitydiscount coordination contract can effectively reducethe negative effect by the uncertainty of two-echelonyields and demand upon the efficiency and benefit ofthe supply chains

(5) We also find out that undermost scenarios the supplychain benefits from the yields and demand risksreduction and generates a higher profit ContrastingTables 2 and 3 we find out that sometimes in thequantity discount contract the supplierrsquos profit canincrease as yieldsrsquo randomness increases And con-trasting Tables 2 and 4 we find out that in the unco-ordinated case and quantity discount contract thesupplierrsquos profit can increase as demand randomnessincreases

8 Conclusions

This paper explores the quantity discount coordinationmod-els in the fashion supply chain with uncertain yields andrandom demand The planned production outputs of thesupplier and the ordering quantity of the manufacturer havebeen set as the decision-making parameters The indepen-dent and noncoordinated and centralized decision-makingpatterns have been established under the uncertainty of two-echelon yields and demandThrough the analysis it has beenfound that the profit of supply chains under independentand noncoordinated decision-making pattern is less than thatunder the centralized pattern which proves the necessityof the coordination of supply chains Based on this twoparametersmdashquantity discount qualification point and pricediscount ratiomdashhave been introduced and quantity discountcoordination models have been established The quantitydiscount coordination pacts with both the supplier and themanufacturer in the dominant position have been set forth

The result reveals that the quantity discount coordinationpact proposed by this paper can promote the manufacturerrsquosordering quantity of the raw material which contributesto the result that the profit of the supply chain based onquantity discount reaches the profit level of the supply chainunder the centralized decision-making pattern The largerthe uncertainty of two-echelon yields and demand the largercontribution of the pact to the benefit of the supply chainwill be The quantity discount coordination pact under theuncertainty of two-echelon yields and demand can reduce the

negative influence brought by the uncertainty and thereforecan contribute to the purpose of the coordination of thesupply chain

Acknowledgments

The authors would like to thank the referees for theirconstructive comments and suggestions that have greatlyimproved the quality of the paper The research was sup-ported by the National Natural Science Foundation of China(Grant no 71201164)

References

[1] D J Thomas and P M Griffin ldquoCoordinated supply chainmanagementrdquo European Journal of Operational Research vol94 no 1 pp 1ndash15 1996

[2] N Agrawal S A Smith andA A Tsay ldquoMulti-vendor sourcingin a retail supply chainrdquo Production and Operations Manage-ment vol 11 no 2 pp 157ndash181 2002

[3] C Chiang and W C Benton ldquoSole sourcing versus dualsourcing under stochastic demands and lead timesrdquo NavalResearch Logistics vol 41 no 5 pp 609ndash624 1994

[4] J D Hong and J C Hayya ldquoJust-in-time purchasing singleor multiple sourcingrdquo International Journal of ProductionEconomics vol 27 no 2 pp 175ndash181 1992

[5] M Treleven and S B Schweikhart ldquoA riskbenefit analysisof sourcing strategies single vs multiple sourcingrdquo Journal ofOperations Management vol 7 no 3-4 pp 93ndash114 1988

[6] G P Cachon and P H Zipkin ldquoCompetitive and cooperativeinventory policies in a two-stage supply chainrdquo ManagementScience vol 45 no 7 pp 936ndash953 1999

[7] M Leng and M Parlar ldquoGame theoretic applications in supplychainmanagement a reviewrdquo INFOR vol 43 no 3 pp 187ndash2202005

[8] MMoses and S Seshadri ldquoPolicy mechanisms for supply chaincoordinationrdquo IIETransactions vol 32 no 3 pp 245ndash262 2000

[9] H J Peng and M H Zhou ldquoProduction and ordering decisionin supply chain with uncertainty in two-echelon yields anddemandrdquo Journal of Systems Engineering vol 25 no 5 pp 622ndash628 2010 (Chinese)

[10] C T Chang ldquoAn acquisition policy for a single item multi-supplier system with real-world constraintsrdquo Applied Mathe-matical Modelling vol 30 no 1 pp 1ndash9 2006

[11] C T Chang C L Chin and M F Lin ldquoOn the single itemmulti-supplier system with variable lead-time price-quantitydiscount and resource constraintsrdquo Applied Mathematics andComputation vol 182 no 1 pp 89ndash97 2006

[12] J Li and L Liu ldquoSupply chain coordination with quantitydiscount policyrdquo International Journal of Production Economicsvol 101 no 1 pp 89ndash98 2006

[13] C L Munson and M J Rosenblatt ldquoTheories and realitiesof quantity discounts an exploratory studyrdquo Production andOperations Management vol 7 no 4 pp 352ndash369 1998

[14] C L Munson and J Hu ldquoIncorporating quantity discountsand their inventory impacts into the centralized purchasingdecisionrdquoEuropean Journal ofOperational Research vol 201 no2 pp 581ndash592 2010

[15] H Y Kang and A H I Lee ldquoInventory replenishment modelusing fuzzy multiple objective programming a case study

Mathematical Problems in Engineering 11

of a high-tech company in Taiwanrdquo Applied Soft ComputingJournal vol 10 no 4 pp 1108ndash1118 2010

[16] A Mendoza and J A Ventura ldquoIncorporating quantity dis-counts to the EOQ model with transportation costsrdquo Interna-tional Journal of Production Economics vol 113 no 2 pp 754ndash765 2008

[17] J Moussourakis and C Haksever ldquoA practical model forordering in multi-product multi-constraint inventory systemswith all-units quantity discountsrdquo International Journal ofInformation and Management Sciences vol 19 no 2 pp 263ndash283 2008

[18] P A Rubin and W C Benton ldquoA generalized framework forquantity discount pricing schedulesrdquo Decision Sciences vol 34no 1 pp 173ndash188 2003

[19] G Zhang and L Ma ldquoOptimal acquisition policy with quantitydiscounts and uncertain demandsrdquo International Journal ofProduction Research vol 47 no 9 pp 2409ndash2425 2009

[20] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[21] A H I Lee and H Y Kang ldquoA mixed 0-1 integer programmingfor inventory model a case study of TFT-LCD manufacturingcompany in Taiwanrdquo Kybernetes vol 37 no 1 pp 66ndash82 2008

[22] J Shi and G Zhang ldquoMulti-product budget-constrained acqui-sition and pricing with uncertain demand and supplier quantitydiscountsrdquo International Journal of Production Economics vol128 no 1 pp 322ndash331 2010

[23] A A Taleizadeh S T A Niaki and R Nikousokhan ldquoCon-straint multiproduct joint-replenishment inventory controlproblem using uncertain programmingrdquo Applied Soft Comput-ing vol 11 pp 5143ndash5154 2011

[24] J Rezaei and M Davoodi ldquoA deterministic multi-item inven-tory model with supplier selection and imperfect qualityrdquoApplied Mathematical Modelling vol 32 no 10 pp 2106ndash21162008

[25] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[26] S P Chen and Y H Ho ldquoAnalysis of the newsboy problemwith fuzzy demands and incremental discountsrdquo InternationalJournal of Production Economics vol 129 no 1 pp 169ndash177 2011

[27] Y J Lin and C H Ho ldquoIntegrated inventory model withquantity discount and price-sensitive demandrdquo Top vol 19 no1 pp 177ndash188 2011

[28] S Krichen A Laabidi and F B Abdelaziz ldquoSingle suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in paymentsrdquo Computers andIndustrial Engineering vol 60 no 1 pp 164ndash172 2011

[29] J Shi G Zhang and J Sha ldquoJointly pricing and ordering fora multi-product multi-constraint newsvendor problem withsupplier quantity discountsrdquo Applied Mathematical Modellingvol 35 no 6 pp 3001ndash3011 2011

[30] H Shin and W C Benton ldquoA quantity discount approach tosupply chain coordinationrdquo European Journal of OperationalResearch vol 180 no 2 pp 601ndash616 2007

[31] H Gurnani and Y Gerchak ldquoCoordination in decentralizedassembly systems with uncertain component yieldsrdquo EuropeanJournal of Operational Research vol 176 no 3 pp 1559ndash15762007

[32] Y He and J Zhang ldquoRandom yield risk sharing in a two-levelsupply chainrdquo International Journal of Production Economicsvol 112 no 2 pp 769ndash781 2008

[33] Y He and J Zhang ldquoRandom yield supply chain with a yielddependent secondary marketrdquo European Journal of OperationalResearch vol 206 no 1 pp 221ndash230 2010

[34] C C Hsieh and C H Wu ldquoCapacity allocation ordering andpricing decisions in a supply chain with demand and supplyuncertaintiesrdquo European Journal of Operational Research vol184 no 2 pp 667ndash684 2008

[35] C C Hsieh and C H Wu ldquoCoordinated decisions for substi-tutable products in a common retailer supply chainrdquo EuropeanJournal of Operational Research vol 196 no 1 pp 273ndash2882009

[36] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[37] P Kelle S Transchel and S Minner ldquoBuyer-supplier coopera-tion and negotiation support with random yield considerationrdquoInternational Journal of Production Economics vol 118 no 1 pp152ndash159 2009

[38] B Keren ldquoThe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmegavol 37 no 4 pp 801ndash810 2009

[39] P Kouvelis and J M Milner ldquoSupply chain capacity andoutsourcing decisions the dynamic interplay of demand andsupply uncertaintyrdquo IIE Transactions vol 34 no 8 pp 717ndash7282002

[40] X Yan M Zhang and K Liu ldquoA note on coordination indecentralized assembly systems with uncertain componentyieldsrdquo European Journal of Operational Research vol 205 no2 pp 469ndash478 2010

[41] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[42] K Zimmer ldquoSupply chain coordination with uncertain just-in-time deliveryrdquo International Journal of Production Economicsvol 77 no 1 pp 1ndash15 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 2 Numerical results for all models (119898 = 015 119899 = 01 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4256 4197 1014 74585 146830 221415

C centralized decision mdash mdash 4351 4351 1 mdash mdash 222105

QD quantity discount withthe supplier as a dominantparty

01 3853 4415 4387 1006 75275 146830 22210502 4221 4307 4251 1013 75275 146830 22210503 4231 4307 4251 1013 75275 146830 22210504 4236 4307 4251 1013 75275 146830 222105

QD quantity discount withthe manufacturer as adominant party

01 4004 4310 4267 101 74585 147520 22210502 4033 4397 4381 1004 74585 147520 22210503 4173 4310 4270 1009 74585 147520 22210504 4197 4392 4379 1003 74585 147520 222105

6 The Design of Quantity Discount Contracts

The actual producing activities can be boiled down to theissue of the Stackelberg game problem Quantity discountcontracts under both the supplier as the dominant side andthe manufacturer as the dominant side are to be discussedseparately The principle of designing the pact is as follows

(1) In the Stackelberg game the dominant side fulfills thequantity discount coordination of the supply chainthrough the determination of the quantity discountqualification point and the price discount rate

(2) Considering the rationality of the follower the dom-inant side for the acceptance of the strategy by thefollower should ensure that the interest of the followershould be no less than the interest gained under thenoncoordinated pattern

(3) Based on the fulfillment of the above conditionsthe expected profit of the dominant side should bemaximized

61 The Quantity Discount Contract with the Supplier as theDominant Side Under the uncertainty of the two-echelonyields and demand provided that the supplier holds thedominant position with themanufacturer being the followerthe supplier is in the position to determine the parameterpackage of the quantity discount including the quantitydiscount qualification point 119902lowast and the price discount rate120572 so as to realize the coordination in the supply chain Themodel is as follows

max 119864 (120587QD1 )

st Equation (19)119864 (120587QD2 ) ge 119864 (1205871198802 )

(20)

For the first item as the supplier holds the dominantposition the goal is to achieve the maximized profit of thesupplier in the decision making The second item represents

the ordering strategy of the manufacturer the third itemguarantees the profits of the manufacturer should not be lessthan that under independent and noncoordinated decisionmaking pattern

62 The Quantity Discount Contract with the Manufactureras the Dominant Party Under the uncertainty of the two-echelon yields and demand in supply chains given thatthe manufacturer is in the dominant position in quantitydiscount and coordination with the supplier in a secondaryposition the manufacturer determines the parameter pack-age of quantity discount including the quantity discountqualification point and the price discount rate so as to realizecoordination in supply chains The model is shown as below

max 119864 (120587QD2 )

st Equation (15)119864 (120587QD1 ) ge 119864 (1205871198801 )

(21)

The first item shows that the goal to be achieved is tomaximize the profit of the manufacturer in the decisionmaking the second item shows that the ordering strategyis provided by the supplier the third item shows that thesupplierrsquos profit level should be kept no less than the levelunder the independent andnoncoordinated decision-makingpattern

7 Numerical Examples and Discussions

The parameters of a certain supply chain are set as below1198880 = 20 ℎ0 = 5 1199030 = 15 1198881 = 10 ℎ1 = 7 1199031 = 301199010 = 40 1199011 = 95 119863 = 400 The output of the supplier andthe output of the manufacturer random variables for marketdemand of the finished products are uniformly distributedover [1minus119898 1+119898] [1minus119899 1+119899] and [1minus 119897 1+ 119897] respectively

Tables 2ndash4 present the decision making for productionand ordering and the expected profits under an independent

Mathematical Problems in Engineering 9

Table 3 Numerical results for all models (119898 = 02 119899 = 02 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4345 4293 1012 71085 141007 212093

C centralized decision mdash mdash 4537 4537 1 mdash mdash 216542

QD quantity discount withthe supplier as a dominantparty

01 4125 4592 4558 1007 75535 141007 21654202 4330 4590 4558 1007 75535 141007 21654203 4433 4515 4460 1012 75535 141007 21654204 4440 4515 4460 1012 75535 141007 216542

QD quantity discount withthe manufacturer as adominant party

01 3517 4534 4534 1 71869 144673 21654202 4058 4533 4532 1 72759 143783 21654203 4222 4532 4531 1 73007 143535 21654204 4302 4532 4531 1 73123 143419 216542

Table 4 Numerical results for all models (119898 = 015 119899 = 01 119897 = 03)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4291 4232 1014 75196 125231 200427

C centralized decision mdash mdash 4387 4387 1 mdash mdash 201123

QD quantity discount withthe supplier as a dominantparty

01 4227 4342 4286 1013 75892 125231 20112302 4256 4342 4286 1013 75892 125231 20112303 4266 4342 4286 1013 75892 125231 20112304 4271 4342 4286 1013 75892 125231 201123

QD quantity discount withthe manufacturer as adominant party

01 4036 4346 4305 101 75196 125927 20112302 4162 4346 4305 101 75196 125927 20112303 4207 4346 4305 101 75196 125927 20112304 4231 4346 4305 101 75196 125927 201123

and noncoordinated decision-making pattern and the cen-tralized decision making pattern under the the uncertaintyof the two-echelon yields and demand they also respectivelypresent quantity discount coordination decisionmaking withthe supplier as the dominant role solving by model (20) andthe quantity discount coordination decision making with themanufacturer as the dominant role solving by model (21)

With Tables 2ndash4 the following conclusions can be drawn

(1) Under an independent decision-making patternthere exists the Nash equilibrium between the sup-plier and the manufacturer and the profit under theNash equilibrium is less than that gained under thecentralized decision-making pattern under the quan-tity discount pact themanufacturer has increased theordering quantity of the raw material the ratios ofthe planned output of the supplier to the orderingquantity of the manufacturer have decreased

(2) Under the quantity discount pact with the supplierin the dominant position the supplier sets different

combinations of the quantity discount qualificationpoint and the price discount rate under the situationwhere the profit of the manufacturer under the quan-tity discount coordination pact exceeds that underindependent and noncoordinated decision-makingpattern In this sense the profit of the supply chainbased on quantity discount reaches the profit level ofthe chain under a centralized pattern The supplierand the manufacturer can select a certain combina-tion and set forth a quantity discount coordinationpact

(3) Under the quantity discount coordination with themanufacturer in the dominant position the manu-facturer sets different combinations as well With theprofit of the supply chain under the quantity discountcoordination pact exceeding the profit under theindependent and noncoordinated decision-makingpattern the chain profit based on quantity discountreaches the profit level gained in a centralized patternThe supplier and themanufacturer can select a certain

10 Mathematical Problems in Engineering

combination and set forth a quantity discount coordi-nation pact

(4) With a comparison of Tables 2 3 and 4 it can befound that the larger the uncertainty of two-echelonyields and demand the larger the contribution of thequantity discount coordination pact to the profit ofthe supply chains With 119898 = 015 119899 = 01 and 119897 =02 the quantity discount coordination pact helpsincrease the total profit by 03 With 119898 = 015119899 = 01 and 119897 = 03 the quantity discountcoordination pact helps increase the total profit by04 With 119898 = 02 119899 = 02 and 119897 = 02 thequantity discount coordination pact helps increasethe total profit by 21 This shows that the quantitydiscount coordination contract can effectively reducethe negative effect by the uncertainty of two-echelonyields and demand upon the efficiency and benefit ofthe supply chains

(5) We also find out that undermost scenarios the supplychain benefits from the yields and demand risksreduction and generates a higher profit ContrastingTables 2 and 3 we find out that sometimes in thequantity discount contract the supplierrsquos profit canincrease as yieldsrsquo randomness increases And con-trasting Tables 2 and 4 we find out that in the unco-ordinated case and quantity discount contract thesupplierrsquos profit can increase as demand randomnessincreases

8 Conclusions

This paper explores the quantity discount coordinationmod-els in the fashion supply chain with uncertain yields andrandom demand The planned production outputs of thesupplier and the ordering quantity of the manufacturer havebeen set as the decision-making parameters The indepen-dent and noncoordinated and centralized decision-makingpatterns have been established under the uncertainty of two-echelon yields and demandThrough the analysis it has beenfound that the profit of supply chains under independentand noncoordinated decision-making pattern is less than thatunder the centralized pattern which proves the necessityof the coordination of supply chains Based on this twoparametersmdashquantity discount qualification point and pricediscount ratiomdashhave been introduced and quantity discountcoordination models have been established The quantitydiscount coordination pacts with both the supplier and themanufacturer in the dominant position have been set forth

The result reveals that the quantity discount coordinationpact proposed by this paper can promote the manufacturerrsquosordering quantity of the raw material which contributesto the result that the profit of the supply chain based onquantity discount reaches the profit level of the supply chainunder the centralized decision-making pattern The largerthe uncertainty of two-echelon yields and demand the largercontribution of the pact to the benefit of the supply chainwill be The quantity discount coordination pact under theuncertainty of two-echelon yields and demand can reduce the

negative influence brought by the uncertainty and thereforecan contribute to the purpose of the coordination of thesupply chain

Acknowledgments

The authors would like to thank the referees for theirconstructive comments and suggestions that have greatlyimproved the quality of the paper The research was sup-ported by the National Natural Science Foundation of China(Grant no 71201164)

References

[1] D J Thomas and P M Griffin ldquoCoordinated supply chainmanagementrdquo European Journal of Operational Research vol94 no 1 pp 1ndash15 1996

[2] N Agrawal S A Smith andA A Tsay ldquoMulti-vendor sourcingin a retail supply chainrdquo Production and Operations Manage-ment vol 11 no 2 pp 157ndash181 2002

[3] C Chiang and W C Benton ldquoSole sourcing versus dualsourcing under stochastic demands and lead timesrdquo NavalResearch Logistics vol 41 no 5 pp 609ndash624 1994

[4] J D Hong and J C Hayya ldquoJust-in-time purchasing singleor multiple sourcingrdquo International Journal of ProductionEconomics vol 27 no 2 pp 175ndash181 1992

[5] M Treleven and S B Schweikhart ldquoA riskbenefit analysisof sourcing strategies single vs multiple sourcingrdquo Journal ofOperations Management vol 7 no 3-4 pp 93ndash114 1988

[6] G P Cachon and P H Zipkin ldquoCompetitive and cooperativeinventory policies in a two-stage supply chainrdquo ManagementScience vol 45 no 7 pp 936ndash953 1999

[7] M Leng and M Parlar ldquoGame theoretic applications in supplychainmanagement a reviewrdquo INFOR vol 43 no 3 pp 187ndash2202005

[8] MMoses and S Seshadri ldquoPolicy mechanisms for supply chaincoordinationrdquo IIETransactions vol 32 no 3 pp 245ndash262 2000

[9] H J Peng and M H Zhou ldquoProduction and ordering decisionin supply chain with uncertainty in two-echelon yields anddemandrdquo Journal of Systems Engineering vol 25 no 5 pp 622ndash628 2010 (Chinese)

[10] C T Chang ldquoAn acquisition policy for a single item multi-supplier system with real-world constraintsrdquo Applied Mathe-matical Modelling vol 30 no 1 pp 1ndash9 2006

[11] C T Chang C L Chin and M F Lin ldquoOn the single itemmulti-supplier system with variable lead-time price-quantitydiscount and resource constraintsrdquo Applied Mathematics andComputation vol 182 no 1 pp 89ndash97 2006

[12] J Li and L Liu ldquoSupply chain coordination with quantitydiscount policyrdquo International Journal of Production Economicsvol 101 no 1 pp 89ndash98 2006

[13] C L Munson and M J Rosenblatt ldquoTheories and realitiesof quantity discounts an exploratory studyrdquo Production andOperations Management vol 7 no 4 pp 352ndash369 1998

[14] C L Munson and J Hu ldquoIncorporating quantity discountsand their inventory impacts into the centralized purchasingdecisionrdquoEuropean Journal ofOperational Research vol 201 no2 pp 581ndash592 2010

[15] H Y Kang and A H I Lee ldquoInventory replenishment modelusing fuzzy multiple objective programming a case study

Mathematical Problems in Engineering 11

of a high-tech company in Taiwanrdquo Applied Soft ComputingJournal vol 10 no 4 pp 1108ndash1118 2010

[16] A Mendoza and J A Ventura ldquoIncorporating quantity dis-counts to the EOQ model with transportation costsrdquo Interna-tional Journal of Production Economics vol 113 no 2 pp 754ndash765 2008

[17] J Moussourakis and C Haksever ldquoA practical model forordering in multi-product multi-constraint inventory systemswith all-units quantity discountsrdquo International Journal ofInformation and Management Sciences vol 19 no 2 pp 263ndash283 2008

[18] P A Rubin and W C Benton ldquoA generalized framework forquantity discount pricing schedulesrdquo Decision Sciences vol 34no 1 pp 173ndash188 2003

[19] G Zhang and L Ma ldquoOptimal acquisition policy with quantitydiscounts and uncertain demandsrdquo International Journal ofProduction Research vol 47 no 9 pp 2409ndash2425 2009

[20] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[21] A H I Lee and H Y Kang ldquoA mixed 0-1 integer programmingfor inventory model a case study of TFT-LCD manufacturingcompany in Taiwanrdquo Kybernetes vol 37 no 1 pp 66ndash82 2008

[22] J Shi and G Zhang ldquoMulti-product budget-constrained acqui-sition and pricing with uncertain demand and supplier quantitydiscountsrdquo International Journal of Production Economics vol128 no 1 pp 322ndash331 2010

[23] A A Taleizadeh S T A Niaki and R Nikousokhan ldquoCon-straint multiproduct joint-replenishment inventory controlproblem using uncertain programmingrdquo Applied Soft Comput-ing vol 11 pp 5143ndash5154 2011

[24] J Rezaei and M Davoodi ldquoA deterministic multi-item inven-tory model with supplier selection and imperfect qualityrdquoApplied Mathematical Modelling vol 32 no 10 pp 2106ndash21162008

[25] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[26] S P Chen and Y H Ho ldquoAnalysis of the newsboy problemwith fuzzy demands and incremental discountsrdquo InternationalJournal of Production Economics vol 129 no 1 pp 169ndash177 2011

[27] Y J Lin and C H Ho ldquoIntegrated inventory model withquantity discount and price-sensitive demandrdquo Top vol 19 no1 pp 177ndash188 2011

[28] S Krichen A Laabidi and F B Abdelaziz ldquoSingle suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in paymentsrdquo Computers andIndustrial Engineering vol 60 no 1 pp 164ndash172 2011

[29] J Shi G Zhang and J Sha ldquoJointly pricing and ordering fora multi-product multi-constraint newsvendor problem withsupplier quantity discountsrdquo Applied Mathematical Modellingvol 35 no 6 pp 3001ndash3011 2011

[30] H Shin and W C Benton ldquoA quantity discount approach tosupply chain coordinationrdquo European Journal of OperationalResearch vol 180 no 2 pp 601ndash616 2007

[31] H Gurnani and Y Gerchak ldquoCoordination in decentralizedassembly systems with uncertain component yieldsrdquo EuropeanJournal of Operational Research vol 176 no 3 pp 1559ndash15762007

[32] Y He and J Zhang ldquoRandom yield risk sharing in a two-levelsupply chainrdquo International Journal of Production Economicsvol 112 no 2 pp 769ndash781 2008

[33] Y He and J Zhang ldquoRandom yield supply chain with a yielddependent secondary marketrdquo European Journal of OperationalResearch vol 206 no 1 pp 221ndash230 2010

[34] C C Hsieh and C H Wu ldquoCapacity allocation ordering andpricing decisions in a supply chain with demand and supplyuncertaintiesrdquo European Journal of Operational Research vol184 no 2 pp 667ndash684 2008

[35] C C Hsieh and C H Wu ldquoCoordinated decisions for substi-tutable products in a common retailer supply chainrdquo EuropeanJournal of Operational Research vol 196 no 1 pp 273ndash2882009

[36] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[37] P Kelle S Transchel and S Minner ldquoBuyer-supplier coopera-tion and negotiation support with random yield considerationrdquoInternational Journal of Production Economics vol 118 no 1 pp152ndash159 2009

[38] B Keren ldquoThe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmegavol 37 no 4 pp 801ndash810 2009

[39] P Kouvelis and J M Milner ldquoSupply chain capacity andoutsourcing decisions the dynamic interplay of demand andsupply uncertaintyrdquo IIE Transactions vol 34 no 8 pp 717ndash7282002

[40] X Yan M Zhang and K Liu ldquoA note on coordination indecentralized assembly systems with uncertain componentyieldsrdquo European Journal of Operational Research vol 205 no2 pp 469ndash478 2010

[41] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[42] K Zimmer ldquoSupply chain coordination with uncertain just-in-time deliveryrdquo International Journal of Production Economicsvol 77 no 1 pp 1ndash15 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 3 Numerical results for all models (119898 = 02 119899 = 02 119897 = 02)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4345 4293 1012 71085 141007 212093

C centralized decision mdash mdash 4537 4537 1 mdash mdash 216542

QD quantity discount withthe supplier as a dominantparty

01 4125 4592 4558 1007 75535 141007 21654202 4330 4590 4558 1007 75535 141007 21654203 4433 4515 4460 1012 75535 141007 21654204 4440 4515 4460 1012 75535 141007 216542

QD quantity discount withthe manufacturer as adominant party

01 3517 4534 4534 1 71869 144673 21654202 4058 4533 4532 1 72759 143783 21654203 4222 4532 4531 1 73007 143535 21654204 4302 4532 4531 1 73123 143419 216542

Table 4 Numerical results for all models (119898 = 015 119899 = 01 119897 = 03)Model

Quantity discountparameters

Decision making forproduction and order Expected profit

120572 119902lowast 1199020 1199021 11990201199021 119864(1205871) 119864(1205872) 119864(1205870)U independent andnoncoordinated decision mdash mdash 4291 4232 1014 75196 125231 200427

C centralized decision mdash mdash 4387 4387 1 mdash mdash 201123

QD quantity discount withthe supplier as a dominantparty

01 4227 4342 4286 1013 75892 125231 20112302 4256 4342 4286 1013 75892 125231 20112303 4266 4342 4286 1013 75892 125231 20112304 4271 4342 4286 1013 75892 125231 201123

QD quantity discount withthe manufacturer as adominant party

01 4036 4346 4305 101 75196 125927 20112302 4162 4346 4305 101 75196 125927 20112303 4207 4346 4305 101 75196 125927 20112304 4231 4346 4305 101 75196 125927 201123

and noncoordinated decision-making pattern and the cen-tralized decision making pattern under the the uncertaintyof the two-echelon yields and demand they also respectivelypresent quantity discount coordination decisionmaking withthe supplier as the dominant role solving by model (20) andthe quantity discount coordination decision making with themanufacturer as the dominant role solving by model (21)

With Tables 2ndash4 the following conclusions can be drawn

(1) Under an independent decision-making patternthere exists the Nash equilibrium between the sup-plier and the manufacturer and the profit under theNash equilibrium is less than that gained under thecentralized decision-making pattern under the quan-tity discount pact themanufacturer has increased theordering quantity of the raw material the ratios ofthe planned output of the supplier to the orderingquantity of the manufacturer have decreased

(2) Under the quantity discount pact with the supplierin the dominant position the supplier sets different

combinations of the quantity discount qualificationpoint and the price discount rate under the situationwhere the profit of the manufacturer under the quan-tity discount coordination pact exceeds that underindependent and noncoordinated decision-makingpattern In this sense the profit of the supply chainbased on quantity discount reaches the profit level ofthe chain under a centralized pattern The supplierand the manufacturer can select a certain combina-tion and set forth a quantity discount coordinationpact

(3) Under the quantity discount coordination with themanufacturer in the dominant position the manu-facturer sets different combinations as well With theprofit of the supply chain under the quantity discountcoordination pact exceeding the profit under theindependent and noncoordinated decision-makingpattern the chain profit based on quantity discountreaches the profit level gained in a centralized patternThe supplier and themanufacturer can select a certain

10 Mathematical Problems in Engineering

combination and set forth a quantity discount coordi-nation pact

(4) With a comparison of Tables 2 3 and 4 it can befound that the larger the uncertainty of two-echelonyields and demand the larger the contribution of thequantity discount coordination pact to the profit ofthe supply chains With 119898 = 015 119899 = 01 and 119897 =02 the quantity discount coordination pact helpsincrease the total profit by 03 With 119898 = 015119899 = 01 and 119897 = 03 the quantity discountcoordination pact helps increase the total profit by04 With 119898 = 02 119899 = 02 and 119897 = 02 thequantity discount coordination pact helps increasethe total profit by 21 This shows that the quantitydiscount coordination contract can effectively reducethe negative effect by the uncertainty of two-echelonyields and demand upon the efficiency and benefit ofthe supply chains

(5) We also find out that undermost scenarios the supplychain benefits from the yields and demand risksreduction and generates a higher profit ContrastingTables 2 and 3 we find out that sometimes in thequantity discount contract the supplierrsquos profit canincrease as yieldsrsquo randomness increases And con-trasting Tables 2 and 4 we find out that in the unco-ordinated case and quantity discount contract thesupplierrsquos profit can increase as demand randomnessincreases

8 Conclusions

This paper explores the quantity discount coordinationmod-els in the fashion supply chain with uncertain yields andrandom demand The planned production outputs of thesupplier and the ordering quantity of the manufacturer havebeen set as the decision-making parameters The indepen-dent and noncoordinated and centralized decision-makingpatterns have been established under the uncertainty of two-echelon yields and demandThrough the analysis it has beenfound that the profit of supply chains under independentand noncoordinated decision-making pattern is less than thatunder the centralized pattern which proves the necessityof the coordination of supply chains Based on this twoparametersmdashquantity discount qualification point and pricediscount ratiomdashhave been introduced and quantity discountcoordination models have been established The quantitydiscount coordination pacts with both the supplier and themanufacturer in the dominant position have been set forth

The result reveals that the quantity discount coordinationpact proposed by this paper can promote the manufacturerrsquosordering quantity of the raw material which contributesto the result that the profit of the supply chain based onquantity discount reaches the profit level of the supply chainunder the centralized decision-making pattern The largerthe uncertainty of two-echelon yields and demand the largercontribution of the pact to the benefit of the supply chainwill be The quantity discount coordination pact under theuncertainty of two-echelon yields and demand can reduce the

negative influence brought by the uncertainty and thereforecan contribute to the purpose of the coordination of thesupply chain

Acknowledgments

The authors would like to thank the referees for theirconstructive comments and suggestions that have greatlyimproved the quality of the paper The research was sup-ported by the National Natural Science Foundation of China(Grant no 71201164)

References

[1] D J Thomas and P M Griffin ldquoCoordinated supply chainmanagementrdquo European Journal of Operational Research vol94 no 1 pp 1ndash15 1996

[2] N Agrawal S A Smith andA A Tsay ldquoMulti-vendor sourcingin a retail supply chainrdquo Production and Operations Manage-ment vol 11 no 2 pp 157ndash181 2002

[3] C Chiang and W C Benton ldquoSole sourcing versus dualsourcing under stochastic demands and lead timesrdquo NavalResearch Logistics vol 41 no 5 pp 609ndash624 1994

[4] J D Hong and J C Hayya ldquoJust-in-time purchasing singleor multiple sourcingrdquo International Journal of ProductionEconomics vol 27 no 2 pp 175ndash181 1992

[5] M Treleven and S B Schweikhart ldquoA riskbenefit analysisof sourcing strategies single vs multiple sourcingrdquo Journal ofOperations Management vol 7 no 3-4 pp 93ndash114 1988

[6] G P Cachon and P H Zipkin ldquoCompetitive and cooperativeinventory policies in a two-stage supply chainrdquo ManagementScience vol 45 no 7 pp 936ndash953 1999

[7] M Leng and M Parlar ldquoGame theoretic applications in supplychainmanagement a reviewrdquo INFOR vol 43 no 3 pp 187ndash2202005

[8] MMoses and S Seshadri ldquoPolicy mechanisms for supply chaincoordinationrdquo IIETransactions vol 32 no 3 pp 245ndash262 2000

[9] H J Peng and M H Zhou ldquoProduction and ordering decisionin supply chain with uncertainty in two-echelon yields anddemandrdquo Journal of Systems Engineering vol 25 no 5 pp 622ndash628 2010 (Chinese)

[10] C T Chang ldquoAn acquisition policy for a single item multi-supplier system with real-world constraintsrdquo Applied Mathe-matical Modelling vol 30 no 1 pp 1ndash9 2006

[11] C T Chang C L Chin and M F Lin ldquoOn the single itemmulti-supplier system with variable lead-time price-quantitydiscount and resource constraintsrdquo Applied Mathematics andComputation vol 182 no 1 pp 89ndash97 2006

[12] J Li and L Liu ldquoSupply chain coordination with quantitydiscount policyrdquo International Journal of Production Economicsvol 101 no 1 pp 89ndash98 2006

[13] C L Munson and M J Rosenblatt ldquoTheories and realitiesof quantity discounts an exploratory studyrdquo Production andOperations Management vol 7 no 4 pp 352ndash369 1998

[14] C L Munson and J Hu ldquoIncorporating quantity discountsand their inventory impacts into the centralized purchasingdecisionrdquoEuropean Journal ofOperational Research vol 201 no2 pp 581ndash592 2010

[15] H Y Kang and A H I Lee ldquoInventory replenishment modelusing fuzzy multiple objective programming a case study

Mathematical Problems in Engineering 11

of a high-tech company in Taiwanrdquo Applied Soft ComputingJournal vol 10 no 4 pp 1108ndash1118 2010

[16] A Mendoza and J A Ventura ldquoIncorporating quantity dis-counts to the EOQ model with transportation costsrdquo Interna-tional Journal of Production Economics vol 113 no 2 pp 754ndash765 2008

[17] J Moussourakis and C Haksever ldquoA practical model forordering in multi-product multi-constraint inventory systemswith all-units quantity discountsrdquo International Journal ofInformation and Management Sciences vol 19 no 2 pp 263ndash283 2008

[18] P A Rubin and W C Benton ldquoA generalized framework forquantity discount pricing schedulesrdquo Decision Sciences vol 34no 1 pp 173ndash188 2003

[19] G Zhang and L Ma ldquoOptimal acquisition policy with quantitydiscounts and uncertain demandsrdquo International Journal ofProduction Research vol 47 no 9 pp 2409ndash2425 2009

[20] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[21] A H I Lee and H Y Kang ldquoA mixed 0-1 integer programmingfor inventory model a case study of TFT-LCD manufacturingcompany in Taiwanrdquo Kybernetes vol 37 no 1 pp 66ndash82 2008

[22] J Shi and G Zhang ldquoMulti-product budget-constrained acqui-sition and pricing with uncertain demand and supplier quantitydiscountsrdquo International Journal of Production Economics vol128 no 1 pp 322ndash331 2010

[23] A A Taleizadeh S T A Niaki and R Nikousokhan ldquoCon-straint multiproduct joint-replenishment inventory controlproblem using uncertain programmingrdquo Applied Soft Comput-ing vol 11 pp 5143ndash5154 2011

[24] J Rezaei and M Davoodi ldquoA deterministic multi-item inven-tory model with supplier selection and imperfect qualityrdquoApplied Mathematical Modelling vol 32 no 10 pp 2106ndash21162008

[25] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[26] S P Chen and Y H Ho ldquoAnalysis of the newsboy problemwith fuzzy demands and incremental discountsrdquo InternationalJournal of Production Economics vol 129 no 1 pp 169ndash177 2011

[27] Y J Lin and C H Ho ldquoIntegrated inventory model withquantity discount and price-sensitive demandrdquo Top vol 19 no1 pp 177ndash188 2011

[28] S Krichen A Laabidi and F B Abdelaziz ldquoSingle suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in paymentsrdquo Computers andIndustrial Engineering vol 60 no 1 pp 164ndash172 2011

[29] J Shi G Zhang and J Sha ldquoJointly pricing and ordering fora multi-product multi-constraint newsvendor problem withsupplier quantity discountsrdquo Applied Mathematical Modellingvol 35 no 6 pp 3001ndash3011 2011

[30] H Shin and W C Benton ldquoA quantity discount approach tosupply chain coordinationrdquo European Journal of OperationalResearch vol 180 no 2 pp 601ndash616 2007

[31] H Gurnani and Y Gerchak ldquoCoordination in decentralizedassembly systems with uncertain component yieldsrdquo EuropeanJournal of Operational Research vol 176 no 3 pp 1559ndash15762007

[32] Y He and J Zhang ldquoRandom yield risk sharing in a two-levelsupply chainrdquo International Journal of Production Economicsvol 112 no 2 pp 769ndash781 2008

[33] Y He and J Zhang ldquoRandom yield supply chain with a yielddependent secondary marketrdquo European Journal of OperationalResearch vol 206 no 1 pp 221ndash230 2010

[34] C C Hsieh and C H Wu ldquoCapacity allocation ordering andpricing decisions in a supply chain with demand and supplyuncertaintiesrdquo European Journal of Operational Research vol184 no 2 pp 667ndash684 2008

[35] C C Hsieh and C H Wu ldquoCoordinated decisions for substi-tutable products in a common retailer supply chainrdquo EuropeanJournal of Operational Research vol 196 no 1 pp 273ndash2882009

[36] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[37] P Kelle S Transchel and S Minner ldquoBuyer-supplier coopera-tion and negotiation support with random yield considerationrdquoInternational Journal of Production Economics vol 118 no 1 pp152ndash159 2009

[38] B Keren ldquoThe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmegavol 37 no 4 pp 801ndash810 2009

[39] P Kouvelis and J M Milner ldquoSupply chain capacity andoutsourcing decisions the dynamic interplay of demand andsupply uncertaintyrdquo IIE Transactions vol 34 no 8 pp 717ndash7282002

[40] X Yan M Zhang and K Liu ldquoA note on coordination indecentralized assembly systems with uncertain componentyieldsrdquo European Journal of Operational Research vol 205 no2 pp 469ndash478 2010

[41] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[42] K Zimmer ldquoSupply chain coordination with uncertain just-in-time deliveryrdquo International Journal of Production Economicsvol 77 no 1 pp 1ndash15 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

combination and set forth a quantity discount coordi-nation pact

(4) With a comparison of Tables 2 3 and 4 it can befound that the larger the uncertainty of two-echelonyields and demand the larger the contribution of thequantity discount coordination pact to the profit ofthe supply chains With 119898 = 015 119899 = 01 and 119897 =02 the quantity discount coordination pact helpsincrease the total profit by 03 With 119898 = 015119899 = 01 and 119897 = 03 the quantity discountcoordination pact helps increase the total profit by04 With 119898 = 02 119899 = 02 and 119897 = 02 thequantity discount coordination pact helps increasethe total profit by 21 This shows that the quantitydiscount coordination contract can effectively reducethe negative effect by the uncertainty of two-echelonyields and demand upon the efficiency and benefit ofthe supply chains

(5) We also find out that undermost scenarios the supplychain benefits from the yields and demand risksreduction and generates a higher profit ContrastingTables 2 and 3 we find out that sometimes in thequantity discount contract the supplierrsquos profit canincrease as yieldsrsquo randomness increases And con-trasting Tables 2 and 4 we find out that in the unco-ordinated case and quantity discount contract thesupplierrsquos profit can increase as demand randomnessincreases

8 Conclusions

This paper explores the quantity discount coordinationmod-els in the fashion supply chain with uncertain yields andrandom demand The planned production outputs of thesupplier and the ordering quantity of the manufacturer havebeen set as the decision-making parameters The indepen-dent and noncoordinated and centralized decision-makingpatterns have been established under the uncertainty of two-echelon yields and demandThrough the analysis it has beenfound that the profit of supply chains under independentand noncoordinated decision-making pattern is less than thatunder the centralized pattern which proves the necessityof the coordination of supply chains Based on this twoparametersmdashquantity discount qualification point and pricediscount ratiomdashhave been introduced and quantity discountcoordination models have been established The quantitydiscount coordination pacts with both the supplier and themanufacturer in the dominant position have been set forth

The result reveals that the quantity discount coordinationpact proposed by this paper can promote the manufacturerrsquosordering quantity of the raw material which contributesto the result that the profit of the supply chain based onquantity discount reaches the profit level of the supply chainunder the centralized decision-making pattern The largerthe uncertainty of two-echelon yields and demand the largercontribution of the pact to the benefit of the supply chainwill be The quantity discount coordination pact under theuncertainty of two-echelon yields and demand can reduce the

negative influence brought by the uncertainty and thereforecan contribute to the purpose of the coordination of thesupply chain

Acknowledgments

The authors would like to thank the referees for theirconstructive comments and suggestions that have greatlyimproved the quality of the paper The research was sup-ported by the National Natural Science Foundation of China(Grant no 71201164)

References

[1] D J Thomas and P M Griffin ldquoCoordinated supply chainmanagementrdquo European Journal of Operational Research vol94 no 1 pp 1ndash15 1996

[2] N Agrawal S A Smith andA A Tsay ldquoMulti-vendor sourcingin a retail supply chainrdquo Production and Operations Manage-ment vol 11 no 2 pp 157ndash181 2002

[3] C Chiang and W C Benton ldquoSole sourcing versus dualsourcing under stochastic demands and lead timesrdquo NavalResearch Logistics vol 41 no 5 pp 609ndash624 1994

[4] J D Hong and J C Hayya ldquoJust-in-time purchasing singleor multiple sourcingrdquo International Journal of ProductionEconomics vol 27 no 2 pp 175ndash181 1992

[5] M Treleven and S B Schweikhart ldquoA riskbenefit analysisof sourcing strategies single vs multiple sourcingrdquo Journal ofOperations Management vol 7 no 3-4 pp 93ndash114 1988

[6] G P Cachon and P H Zipkin ldquoCompetitive and cooperativeinventory policies in a two-stage supply chainrdquo ManagementScience vol 45 no 7 pp 936ndash953 1999

[7] M Leng and M Parlar ldquoGame theoretic applications in supplychainmanagement a reviewrdquo INFOR vol 43 no 3 pp 187ndash2202005

[8] MMoses and S Seshadri ldquoPolicy mechanisms for supply chaincoordinationrdquo IIETransactions vol 32 no 3 pp 245ndash262 2000

[9] H J Peng and M H Zhou ldquoProduction and ordering decisionin supply chain with uncertainty in two-echelon yields anddemandrdquo Journal of Systems Engineering vol 25 no 5 pp 622ndash628 2010 (Chinese)

[10] C T Chang ldquoAn acquisition policy for a single item multi-supplier system with real-world constraintsrdquo Applied Mathe-matical Modelling vol 30 no 1 pp 1ndash9 2006

[11] C T Chang C L Chin and M F Lin ldquoOn the single itemmulti-supplier system with variable lead-time price-quantitydiscount and resource constraintsrdquo Applied Mathematics andComputation vol 182 no 1 pp 89ndash97 2006

[12] J Li and L Liu ldquoSupply chain coordination with quantitydiscount policyrdquo International Journal of Production Economicsvol 101 no 1 pp 89ndash98 2006

[13] C L Munson and M J Rosenblatt ldquoTheories and realitiesof quantity discounts an exploratory studyrdquo Production andOperations Management vol 7 no 4 pp 352ndash369 1998

[14] C L Munson and J Hu ldquoIncorporating quantity discountsand their inventory impacts into the centralized purchasingdecisionrdquoEuropean Journal ofOperational Research vol 201 no2 pp 581ndash592 2010

[15] H Y Kang and A H I Lee ldquoInventory replenishment modelusing fuzzy multiple objective programming a case study

Mathematical Problems in Engineering 11

of a high-tech company in Taiwanrdquo Applied Soft ComputingJournal vol 10 no 4 pp 1108ndash1118 2010

[16] A Mendoza and J A Ventura ldquoIncorporating quantity dis-counts to the EOQ model with transportation costsrdquo Interna-tional Journal of Production Economics vol 113 no 2 pp 754ndash765 2008

[17] J Moussourakis and C Haksever ldquoA practical model forordering in multi-product multi-constraint inventory systemswith all-units quantity discountsrdquo International Journal ofInformation and Management Sciences vol 19 no 2 pp 263ndash283 2008

[18] P A Rubin and W C Benton ldquoA generalized framework forquantity discount pricing schedulesrdquo Decision Sciences vol 34no 1 pp 173ndash188 2003

[19] G Zhang and L Ma ldquoOptimal acquisition policy with quantitydiscounts and uncertain demandsrdquo International Journal ofProduction Research vol 47 no 9 pp 2409ndash2425 2009

[20] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[21] A H I Lee and H Y Kang ldquoA mixed 0-1 integer programmingfor inventory model a case study of TFT-LCD manufacturingcompany in Taiwanrdquo Kybernetes vol 37 no 1 pp 66ndash82 2008

[22] J Shi and G Zhang ldquoMulti-product budget-constrained acqui-sition and pricing with uncertain demand and supplier quantitydiscountsrdquo International Journal of Production Economics vol128 no 1 pp 322ndash331 2010

[23] A A Taleizadeh S T A Niaki and R Nikousokhan ldquoCon-straint multiproduct joint-replenishment inventory controlproblem using uncertain programmingrdquo Applied Soft Comput-ing vol 11 pp 5143ndash5154 2011

[24] J Rezaei and M Davoodi ldquoA deterministic multi-item inven-tory model with supplier selection and imperfect qualityrdquoApplied Mathematical Modelling vol 32 no 10 pp 2106ndash21162008

[25] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[26] S P Chen and Y H Ho ldquoAnalysis of the newsboy problemwith fuzzy demands and incremental discountsrdquo InternationalJournal of Production Economics vol 129 no 1 pp 169ndash177 2011

[27] Y J Lin and C H Ho ldquoIntegrated inventory model withquantity discount and price-sensitive demandrdquo Top vol 19 no1 pp 177ndash188 2011

[28] S Krichen A Laabidi and F B Abdelaziz ldquoSingle suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in paymentsrdquo Computers andIndustrial Engineering vol 60 no 1 pp 164ndash172 2011

[29] J Shi G Zhang and J Sha ldquoJointly pricing and ordering fora multi-product multi-constraint newsvendor problem withsupplier quantity discountsrdquo Applied Mathematical Modellingvol 35 no 6 pp 3001ndash3011 2011

[30] H Shin and W C Benton ldquoA quantity discount approach tosupply chain coordinationrdquo European Journal of OperationalResearch vol 180 no 2 pp 601ndash616 2007

[31] H Gurnani and Y Gerchak ldquoCoordination in decentralizedassembly systems with uncertain component yieldsrdquo EuropeanJournal of Operational Research vol 176 no 3 pp 1559ndash15762007

[32] Y He and J Zhang ldquoRandom yield risk sharing in a two-levelsupply chainrdquo International Journal of Production Economicsvol 112 no 2 pp 769ndash781 2008

[33] Y He and J Zhang ldquoRandom yield supply chain with a yielddependent secondary marketrdquo European Journal of OperationalResearch vol 206 no 1 pp 221ndash230 2010

[34] C C Hsieh and C H Wu ldquoCapacity allocation ordering andpricing decisions in a supply chain with demand and supplyuncertaintiesrdquo European Journal of Operational Research vol184 no 2 pp 667ndash684 2008

[35] C C Hsieh and C H Wu ldquoCoordinated decisions for substi-tutable products in a common retailer supply chainrdquo EuropeanJournal of Operational Research vol 196 no 1 pp 273ndash2882009

[36] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[37] P Kelle S Transchel and S Minner ldquoBuyer-supplier coopera-tion and negotiation support with random yield considerationrdquoInternational Journal of Production Economics vol 118 no 1 pp152ndash159 2009

[38] B Keren ldquoThe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmegavol 37 no 4 pp 801ndash810 2009

[39] P Kouvelis and J M Milner ldquoSupply chain capacity andoutsourcing decisions the dynamic interplay of demand andsupply uncertaintyrdquo IIE Transactions vol 34 no 8 pp 717ndash7282002

[40] X Yan M Zhang and K Liu ldquoA note on coordination indecentralized assembly systems with uncertain componentyieldsrdquo European Journal of Operational Research vol 205 no2 pp 469ndash478 2010

[41] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[42] K Zimmer ldquoSupply chain coordination with uncertain just-in-time deliveryrdquo International Journal of Production Economicsvol 77 no 1 pp 1ndash15 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

of a high-tech company in Taiwanrdquo Applied Soft ComputingJournal vol 10 no 4 pp 1108ndash1118 2010

[16] A Mendoza and J A Ventura ldquoIncorporating quantity dis-counts to the EOQ model with transportation costsrdquo Interna-tional Journal of Production Economics vol 113 no 2 pp 754ndash765 2008

[17] J Moussourakis and C Haksever ldquoA practical model forordering in multi-product multi-constraint inventory systemswith all-units quantity discountsrdquo International Journal ofInformation and Management Sciences vol 19 no 2 pp 263ndash283 2008

[18] P A Rubin and W C Benton ldquoA generalized framework forquantity discount pricing schedulesrdquo Decision Sciences vol 34no 1 pp 173ndash188 2003

[19] G Zhang and L Ma ldquoOptimal acquisition policy with quantitydiscounts and uncertain demandsrdquo International Journal ofProduction Research vol 47 no 9 pp 2409ndash2425 2009

[20] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[21] A H I Lee and H Y Kang ldquoA mixed 0-1 integer programmingfor inventory model a case study of TFT-LCD manufacturingcompany in Taiwanrdquo Kybernetes vol 37 no 1 pp 66ndash82 2008

[22] J Shi and G Zhang ldquoMulti-product budget-constrained acqui-sition and pricing with uncertain demand and supplier quantitydiscountsrdquo International Journal of Production Economics vol128 no 1 pp 322ndash331 2010

[23] A A Taleizadeh S T A Niaki and R Nikousokhan ldquoCon-straint multiproduct joint-replenishment inventory controlproblem using uncertain programmingrdquo Applied Soft Comput-ing vol 11 pp 5143ndash5154 2011

[24] J Rezaei and M Davoodi ldquoA deterministic multi-item inven-tory model with supplier selection and imperfect qualityrdquoApplied Mathematical Modelling vol 32 no 10 pp 2106ndash21162008

[25] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[26] S P Chen and Y H Ho ldquoAnalysis of the newsboy problemwith fuzzy demands and incremental discountsrdquo InternationalJournal of Production Economics vol 129 no 1 pp 169ndash177 2011

[27] Y J Lin and C H Ho ldquoIntegrated inventory model withquantity discount and price-sensitive demandrdquo Top vol 19 no1 pp 177ndash188 2011

[28] S Krichen A Laabidi and F B Abdelaziz ldquoSingle suppliermultiple cooperative retailers inventory model with quantitydiscount and permissible delay in paymentsrdquo Computers andIndustrial Engineering vol 60 no 1 pp 164ndash172 2011

[29] J Shi G Zhang and J Sha ldquoJointly pricing and ordering fora multi-product multi-constraint newsvendor problem withsupplier quantity discountsrdquo Applied Mathematical Modellingvol 35 no 6 pp 3001ndash3011 2011

[30] H Shin and W C Benton ldquoA quantity discount approach tosupply chain coordinationrdquo European Journal of OperationalResearch vol 180 no 2 pp 601ndash616 2007

[31] H Gurnani and Y Gerchak ldquoCoordination in decentralizedassembly systems with uncertain component yieldsrdquo EuropeanJournal of Operational Research vol 176 no 3 pp 1559ndash15762007

[32] Y He and J Zhang ldquoRandom yield risk sharing in a two-levelsupply chainrdquo International Journal of Production Economicsvol 112 no 2 pp 769ndash781 2008

[33] Y He and J Zhang ldquoRandom yield supply chain with a yielddependent secondary marketrdquo European Journal of OperationalResearch vol 206 no 1 pp 221ndash230 2010

[34] C C Hsieh and C H Wu ldquoCapacity allocation ordering andpricing decisions in a supply chain with demand and supplyuncertaintiesrdquo European Journal of Operational Research vol184 no 2 pp 667ndash684 2008

[35] C C Hsieh and C H Wu ldquoCoordinated decisions for substi-tutable products in a common retailer supply chainrdquo EuropeanJournal of Operational Research vol 196 no 1 pp 273ndash2882009

[36] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[37] P Kelle S Transchel and S Minner ldquoBuyer-supplier coopera-tion and negotiation support with random yield considerationrdquoInternational Journal of Production Economics vol 118 no 1 pp152ndash159 2009

[38] B Keren ldquoThe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmegavol 37 no 4 pp 801ndash810 2009

[39] P Kouvelis and J M Milner ldquoSupply chain capacity andoutsourcing decisions the dynamic interplay of demand andsupply uncertaintyrdquo IIE Transactions vol 34 no 8 pp 717ndash7282002

[40] X Yan M Zhang and K Liu ldquoA note on coordination indecentralized assembly systems with uncertain componentyieldsrdquo European Journal of Operational Research vol 205 no2 pp 469ndash478 2010

[41] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[42] K Zimmer ldquoSupply chain coordination with uncertain just-in-time deliveryrdquo International Journal of Production Economicsvol 77 no 1 pp 1ndash15 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of