Computers & Industrial Engineering 61 (2011) 1318–1323

Contents lists available at SciVerse ScienceDirect

Computers & Industrial Engineering

journal homepage: www.elsevier .com/ locate/caie

Modeling and analysis of single item multi-period procurement lot-sizingproblem considering rejections and late deliveries

Devendra Choudhary a,⇑, Ravi Shankar b

a Department of Mechanical Engineering, Govt. Engineering College Ajmer, Badliya Circle, NH-08, Ajmer, Rajasthan 305 002, Indiab Department of Management Studies, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 March 2011Received in revised form 4 August 2011Accepted 5 August 2011Available online 17 August 2011

Keywords:Integer programmingLot-sizingInventory managementQuantity discountsSupply chain

0360-8352/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.cie.2011.08.005

⇑ Corresponding author. Mobile: +91 09413689065E-mail addresses: dceca@rediffmail.com (D. Choud

(R. Shankar).

Integer linear programming approach has been used to solve a multi-period procurement lot-sizing prob-lem for a single product that is procured from a single supplier considering rejections and late deliveriesunder all-unit quantity discount environment. The intent of proposed model is two fold. First, we aim toestablish tradeoffs among cost objectives and determine appropriate lot-size and its timing to minimizetotal cost over the decision horizon considering quantity discount, economies of scale in transactions andinventory management. Second, the optimization model has been used to analyze the effect of variationsin problem parameters such as rejection rate, demand, storage capacity and inventory holding cost for amulti-period procurement lot-sizing problem. This analysis helps the decision maker to figure out oppor-tunities to significantly reduce cost. An illustration is included to demonstrate the effectiveness of theproposed model. The proposed approach provides flexibility to decision maker in multi-period procure-ment lot-sizing decisions through tradeoff curves and sensitivity analysis.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Multi-period procurement lot-sizing decision seeks best trade-offs among multiple cost objectives to determine appropriate lot-size and its timing to minimize total cost over the decision horizon.The multiple cost objectives are purchasing cost, transaction(ordering and transportation) cost, inventory holding cost and/orshortage cost. Supplier offers discounts, which tend to encouragebuyer to procure larger quantities to obtain operating advantagessuch as economies of scale and reducing the cost of ordering andtransportation. In such a scenario, product could be carried for-ward to a future period, incurring inventory holding cost. Thismeans that in each period either procurement takes place or buyerhas inventory carried forward from the preceding period. Smallerlot-size procurement strategy reduces inventory holding cost butincreases purchasing cost and transaction cost. Procurement of lar-ger lot-size reduces purchasing cost and transactions cost but leadsto higher inventory cost. Supply chain risks such as rejections andlate deliveries also affect the procurement lot-sizing decisions.Therefore, decision maker considers tradeoffs among purchasingcost, transaction cost, inventory holding cost and/or shortage costin multi-period procurement lot-sizing decisions to minimize totalcost over decision horizon.

ll rights reserved.

.hary), ravi1@dms.iitd.ernet.in

Material Requirement Planning (MRP) involves procurementlot-sizing decisions to be made when demand is both stable as wellas lumpy and the approach is spread over a finite time horizon. Inthe restricted case, when demand is stable and known over thedecision horizon, the simple static EOQ model can find the opti-mum solution. Both methods fail to consider realistic constraintsregarding supplier capacity, rejections, late deliveries and timedependent variations in problem parameters. The exact solutionin more general situations has been obtainable by Dynamic Pro-gramming (DP). Wagner and Whitin (1958) presented a dynamicprogramming solution algorithm for single product, multi-periodinventory lot-sizing problem. Even though DP algorithms(Aggarwal & Park, 1993; Federgruen & Tzur, 1991; Heady & Zhu,1994; Silver & Meal, 1973) provide an optimal solution, these areconsidered difficult to understand and require high computationalresources.

To our knowledge, there is no multi-period linear program-ming model available in the literature for procurement lot-sizingproblem which can substitute EOQ model and DP model toovercome their limitations, and also considers price breaks andrealistic constraints as well as supports Material RequirementPlanning.

This paper applied an integer linear programming approachto solve multi-period procurement lot-sizing problem forsingle product and single supplier considering rejections andlate delivery performance under all-unit quantity discountenvironment.

D. Choudhary, R. Shankar / Computers & Industrial Engineering 61 (2011) 1318–1323 1319

The purpose of this paper is to:

� Develop a mathematical model to establish tradeoffs amongcost objectives and determine appropriate lot-size to pro-cure and its period to minimize total cost over the decisionhorizon.

� Investigate the effect of variation in problem parameterssuch as rejection rate, demand, storage capacity and inven-tory holding cost on total cost.

The paper is further organized as follows. Section 2 presents abrief literature review of the existing quantitative approaches re-lated to procurement lot-sizing problem. In Section 3, an integerlinear programming formulation is developed for multi period pro-curement lot-sizing problem considering all-unit quantity dis-counts. Section 4 presents an illustration with solution todemonstrate the effectiveness of the proposed approach. Finally,conclusions are provided in Section 5.

2. Literature review

Brahimi, Dauzere-Peres, Najid, and Nordli (2006) presented asurvey of the single item lot-sizing problem for its uncapacitatedand capacitated versions. Karimi, Fatemi Ghomi, and Wilson(2003) discussed a number of important characteristics of lot-siz-ing models, including the planning horizon, number of levels, num-ber of products, capacity or resource constraints, deterioration ofitems, demand, setup structure and shortage. Ben-Daya, Darwish,and Ertogral (2008) and Robinson, Narayanan, and Sahin (2009)proposed different models and classifications of the lot-sizingproblem.

Smith, Robles, and Cárdenas-Barrón (2009) formulated andsolved a single item joint pricing and production decision problemover a multi-period time horizon. The objective is to maximizeprofits considering capacity and inventory constraints. They con-sider decision variables, such as sales price, production quantity,and sales amount for a single item. Buffa and Jackson (1983) pro-posed a goal programming model considering price, quality anddelivery goals to schedule purchase for single product over a de-fined planning horizon.

Pratsini (2000) proposed the lot-sizing model with setup learn-ing for the single level, multi-item, capacity constrained case. Hedeveloped a heuristic to analyze the effects of setup learning ona production schedule. The study revealed that setup learningcan have unexpected results on a product depending on the rela-tive value of its setup to holding cost ratio compared with the ra-tios of the other products. Benton (1991) developed a non-linearmodel and a heuristic solution approach for supplier selectionand lot sizing under conditions of multiple items, multiple suppli-ers, resource limitations and all-unit quantity discounts. The objec-tive is to minimize the total cost (purchasing, inventory andordering costs) subject to an inventory investment constraint andshortage related constraints. In their article, Raza and Akgunduz(2008) presented a comparative study of heuristic algorithms oneconomic lot scheduling problem (ELSP). They showed that Simu-lated Annealing algorithm finds the best solution to these ELSPproblems, and outperforms other meta-heuristic technique suchas Dobson’s heuristic, hybrid GA, Neighborhood Search heuristicsand Tabu Search.

Polatoglu and Sahin (2000) suggested a multi-period purchas-ing policy where demand in each period is considered as a randomvariable, the probability distribution of which may depend on priceand period. Chaudhry, Forst, and Zydiak (1993) proposed a mathe-matical formulation to minimize the purchasing cost for individual

item over a single period considering capacity constraints, deliveryperformance and quality with quantity discounts. Bender, Brown,Isaac, and Shapiro (1985) described a procurement problem facedby IBM to minimize the sum of purchasing, transportation andinventory cost over the planning horizon for multiple products,multiple time periods and quantity discounts.

Ustun and Demirtas (2008) proposed an integration of ANP andachievement scalarizing function to choose the best suppliers andto find the optimal order quantities and inventory levels. Liao andKuhn (2004) presented a multi-objective optimization model forsingle item assuming that all suppliers’ lots simultaneously arriveat the beginning of each replenishment period. The objectives arethe minimization of total cost, the total quality rejections and totallate deliveries subject to capacity and demand constraints. Demir-tas and Ustun (2009) developed an integrated ANP and GP ap-proach to solve multi-period inventory lot-sizing scenario, forsingle product and multiple suppliers. A multi objective mixedinteger linear programming model is proposed to achieve fourgoals: budget, aggregate quality, total value of purchasing and de-mand over the planning horizon. Rezaei and Davoodi (2011) pro-posed two multi objective mixed integer non-linear models formulti period lot-sizing problems involving multiple products andmultiple suppliers. Each model is constructed on three objectivefunctions: cost, quality and delivery. In first model shortages arenot allowed while second model considered that demand duringthe stock-out period is back ordered.

3. Model formulation

The proposed model deals with procurement lot-sizing problemin which there is a time-varying demand for a single product overmulti-periods.

3.1. Model parameters and decision variables

Following parameters and decision variables are to be adoptedfor mathematical formulation.

Parameters:

dt buyer’s demand of the product in period t pmt cost of procuring one unit of product between price

break level m and m + 1 in period t

bm quantity at which all-unit price breaks occur ot cost of ordering in period t tmt cost of transportation of lot-size at price break level m in

period t

qmt percentage of rejected products delivered by supplier at

price break level m in period t

lmt percentage of products late delivered by supplier at price

break level m in period t

Ct supplier capacity in period t ht unit inventory holding cost of the product in period t w buyer’s storage capacity It intermediate variable indicates inventory of the product,

carried over from period t to period t + 1

Decision variables: xmt lot-size (number of units of product) that buyer procures

from supplier at price break level m in period t

ymt binary variable used in separating price levels m for

product in a transaction between buyer and supplier.This also separates transportation cost per procured lot-size between price break level m and m + 1 in period t

zt

binary variable indicating whether supplier is ordered ornot in period t

1320 D. Choudhary, R. Shankar / Computers & Industrial Engineering 61 (2011) 1318–1323

3.2. Model assumptions

The proposed model is constrained by the followingassumptions:

a. Demand of the item is constant and known with certaintyfor each period over planning horizon.

b. Shortage or backordering is not allowed.c. Capacity of supplier is finite.d. Ordering cost applies for each period in which an order is

placed to the supplier.e. Carrier-size dependent transportation cost applies for each

period in which lot is procured from supplier.f. Inventory holding cost applies when product is carried

across a period in the planning horizon.g. Late delivery is assumed to receive in following period.h. Rejected products are disposed in the same period at scrap

value and are excluded from inventory which is carriedacross a period in the planning horizon.

i. Late deliveries are assumed of perfect quality.

3.3. Mathematical formulation

With above parameters and decision variables, a mathematicalformulation may be stated as follows:

Min Z ¼ ½Z1; Z2; Z3� ð1Þ

Z1 ¼X

m

Xt

pmtxmt ð1aÞ

Z2 ¼X

t

otzt þX

m

Xt

tmtymt ð1bÞ

Z3 ¼X

t

htIt ð1cÞ

Table 1Buyer’s product related data for illustration.

Period (t) Holding cost

1 2 3 4 5 6 ht

Demand 400 400 400 400 400 400 5

Subject to

It�1 þX

m

xmt þX

m

lmkxmk � dt PX

m

lmtxmt þX

m

qmtxmt 8t

and k ¼ t � 1 ð2Þ

It ¼ It�1 þX

m

xmt þX

m

lmkxmk �X

m

lmtxmt �X

m

qmtxmt

� dt 8t and k ¼ t � 1 ð3Þ

xmt 6XT

k¼t

dk

!zt 8m; 8t ð4Þ

bm�1;tymt 6 xmt 6 bmtymt 8m; 8t ð5Þ

xMt 6 CtyMt 8t ð6ÞX

m

ymt ¼ zt 8t ð7Þ

It 6 w 8t ð8Þ

It P 0 8t ð9Þ

xmt P 0 and integer 8m; 8t ð10Þ

ymt 2 f0; 1g 8m; 8t ð11Þ

zt 2 f0; 1g 8t ð12Þ

The objective of the model is to minimize the buyer’s total cost.The objective function (1) consists of three parts: (1a) the purchas-ing cost, (1b) the transaction cost, and (1c) the inventory holdingcost for the remaining inventory in each period. The first term inequation (1b) represents the ordering cost while second termmeans that if a lot-size is procured between price break level mand m + 1 in period t then there is fixed transportation cost associ-ated with chosen carrier depending on carrier size. It may be notedthat ordering cost depends on whether procurement takes place ornot; therefore binary variable zt is used in expression of orderingcost. Similarly, selection of appropriate carrier depends on lot-sizeprocured between price break level m and m + 1 in period t; there-fore binary variable ymt is used in the expression of transportationcost.

Constraints (2) and (3) are the inventory balance equations.They express that the initial inventory (It�1) added to the currentperiod’s procurement (xmt) plus previous period’s late deliveredproducts are used to satisfy the demand (dt) without allowingshortages due to late deliveries (lmtxmt) and imperfect quality(qmtxmt) items associated with current period’s procurement. Whatremains after deducting current period’s late deliveries and imper-fect quality products is kept in stock at the end of period (It). Con-strains (4) ensure that buyer cannot place a procurement orderwithout charging an appropriate ordering cost. Constraints (5)–(7) describe the supplier’s all-unit price break schemes. Constraints(5) ensure that a procured lot from the supplier at a specific pricebreak is in the discount interval offered. Constraints (6) representthe restriction on procurement lot size due to capacity of the sup-plier. Capacity of supplier is assumed to be greater than quantity atwhich price break level M is offered i.e. Ct > bMt. Constraints (7) en-sure only one price break level is used for the lot-size if it is pro-cured in period t. It may be noted that binary variable zt is usedto activate both the optimal procurement lot size in Constraints(4) and optimal price break level in Constraints (7). Constrains(8) represent the restriction on inventory at the end of perioddue to buyer’s space limitation. Finally, Constrains (9)–(12) areused to force non-negative integer values and binary restrictionsin the model.

Computational difficulty increases with the number of vari-ables. Moreover, adding more price breaks will make the modelharder to solve because of additional variables required in formu-lation. However, problems of a reasonable size can be solved usingany commercial software within a few seconds of computer time.

4. An illustration

The effectiveness of proposed integer linear programming mod-el, presented in this paper is demonstrated through an illustration.Demand data of product for six periods are given in Table 1. Prod-uct demand is considered stable over planning horizon. Buyer’sspace availability is assumed to be unlimited. Table 2 provides sup-plier related data for numerical illustration.

The linear programming software LINGO is used to solve thisproblem. While the model can be used in a number of ways, in thisillustration it is used to analyze tradeoffs among cost objectivesand to evaluate the effect of variation in problem parameters suchas holding cost, demand, quality performance etc. on lot-sizingdecision.

Table 2Supplier data related to product for illustration.

Quantity level pmt tmt lmt qmt ot Ct

Q 6 450 20 3550 0.10 0.05 500 1400450 < Q 6 750 19 5000750 < Q 18 8000

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

Sol. 3 Sol. 5 Sol. 8 Sol. 6 Sol. 1 Sol. 9 Sol. 7 Sol. 2

Cos

t

Purchasing cost

Transaction cost

Inventory cost

Total cost

& 4

Fig. 1. Graphical display of solutions for multiple cost objectives.

D. Choudhary, R. Shankar / Computers & Industrial Engineering 61 (2011) 1318–1323 1321

4.1. Analysis of tradeoffs among cost objectives

It is evident from Table 3 that optimal procurement lot-size isobtained by striking best tradeoffs among multiple cost objectives.Variation in priority of multiple cost objectives will cause variationin procured lot-size in each period as shown in Table 3. The pro-curement lot-size in each period is sensitive with the priority ofmultiple objectives.

The first three solutions in Table 3 are obtained consideringonly one of the objectives and the resulting Solutions 1, 2 and 3are the solutions which minimize Z1, Z2 and Z3, respectively. InSolutions 1 and 2, a few large size lots are procured to take advan-tage of price discounts and to minimize transaction cost, respec-tively. Product is to be procured in small lots to minimizeinventory holding cost as shown in Solution 3. Two-objective solu-tions are also generated with only two of the objectives to analyzethe tradeoffs among the objectives and results are obtained inSolution 4 to Solution 6 in Table 3. The last three solutions in Table3 are obtained for three objectives. Solution 9 was obtained by set-ting equal weights to each objective. Solution 4 reflects similar re-sults as those obtained in Solution 2.

A review of the results reveals a few interesting observations.Smaller lot-size reduces inventory holding cost but increases pur-chasing cost and transaction cost. Larger lot-size reduces purchas-ing cost and transaction cost, and leads to higher inventory cost.Therefore, decision maker considers tradeoffs among purchasingcost, transaction cost and inventory holding cost in procurementlot-sizing decisions to minimize total cost.

Fig. 1 shows the graphical display of solutions for the illustra-tion. Decision makers may choose the best compromising solution

Table 3Computational results to indicate tradeoffs among multiple cost objectives.

Solutions 1 2

Weights W1 1 0W2 0 1W3 0 0

Objectives Purchasing cost 45,486 45,486Transaction cost 25,500 17,000Inventory cost 5621 22,106Total cost 76,607 84,592

Multi-period procurement lot size x11 0 0x12 0 0x13 0 0x14 0 0x15 0 0x16 0 0x21 0 0x22 0 0x23 0 0x24 0 0x25 0 0x26 0 0x31 843 1127x32 0 1400x33 933 0x34 0 0x35 751 0x36 0 0

by considering realistic constraints such as supplier’s capacity,buyer’s storage capacity and transportation availability. Solutions6–8 are such compromising solutions.

Fig. 2 shows the graphical results for Solution 1–6 and Solution8 (Table 3) indicating tradeoffs among purchasing cost, transactioncost and inventory holding cost.

4.2. Analysis to evaluate the effect of variation in problem parameters

In this section, we investigate the effect of variation in problemparameters on multi-period lot-sizing decision. Table 4 shows thatthe five parameters are varied from current state. Problem set 1represents variation in holding cost. In Problem set 2 and 3, de-mand is considered lumpy. It may be noted that total demand dur-ing decision horizon is same. Capacity of supplier is taken less thanfrom the current capacity in Problem set 4. In Problem set 5,buyer’s storage capacity is restricted to 250 units. As shown inProblem set 6, rejection rate of supplier is assumed to be 50% betterthan current rejection rate while quality rejections increase by 50%in Problem set 7.

3 4 5 6 7 8 9

0 0.5 0.5 0 0.5 0.25 0.330 0.5 0 0.5 0.25 0.25 0.331 0 0.5 0.5 0.25 0.5 0.33

50,909 45,486 47,668 48,463 45,486 49,319 47,13625,750 17,000 33,100 20,550 17,000 23,150 18,05093 22,106 1828 4531 11,656 1933 809476,752 84,592 82,596 73,544 74,142 74,402 73,280

0 0 0 0 0 0 0415 0 0 0 0 415 0422 0 378 0 0 441 0421 0 0 450 0 450 450440 0 0 0 0 0 0400 0 16 0 0 0 0471 0 471 577 0 471 00 0 454 750 0 0 00 0 0 0 0 0 00 0 451 0 0 0 00 0 0 750 0 750 7500 0 0 0 0 0 00 1127 0 0 1327 0 13270 1400 0 0 0 0 00 0 0 0 0 0 00 0 0 0 1200 0 00 0 758 0 0 0 00 0 0 0 0 0 0

0

10000

20000

30000

40000

50000

60000

Purchasing cost Transaction cost Inventory cost

Cos

t

Sol. 3

Sol. 5

Sol. 6

Sol. 1

Sol. 2 & 4

Sol. 9

Fig. 2. Solutions indicating tradeoffs among cost objectives.

Table 4Description of variation in problem parameters.

Problem sets Description Current state (CS) New value

1 Holding cost 5 82 Demand Stable Lumpy1a

3 Demand Stable Lumpy2b

4 Supplier capacity 1400 10005 Buyer storage capacity Unlimited 2506 Quality rate 0.05 0.0257 Quality rate 0.05 0.075

a Lumpy1 means (300, 500, 300, 500, 300, 500).b Lumpy2 means (200, 200, 800, 200, 200, 800).

69000

70000

71000

72000

73000

74000

75000

76000

77000

78000

CS 1 2 3 4 5 6 7

Tot

al c

ost

Problem sets

Fig. 3. Effect of variation in problem parameters on total cost.

1322 D. Choudhary, R. Shankar / Computers & Industrial Engineering 61 (2011) 1318–1323

The optimal solutions for all problem sets of numerical exampleare summarized in Table 5. For Problem set 1, due to high inven-tory holding cost, product is frequently procured in small lots incomparison with current state. In Problem set 2 and 3, due to lum-py demand, the total cost is less in comparison with current state.Consideration of multi-period planning horizon allows the buyer tominimize total cost in lumpy demand scenario by adjusting lot-size and number of orders accordingly. Problem set 4 and 5 show

Table 5Computational results for different problem sets.

Problem sets CS 1 2 3

Purchasing cost 47,136 49,319 48,421 4Transaction cost 18,050 23,150 20,550 2Inventory cost 8094 3092 3623 5Total cost 73,280 75,561 72,594 7x11 0 0 408 4x12 0 415 0 0x13 0 441 0 0x14 450 450 0 0x15 0 0 0 0x16 0 0 0 0x21 0 471 0 0x22 0 0 750 0x23 0 0 0 0x24 0 0 619 0x25 750 750 750 0x26 0 0 0 0x31 1327 0 0 0x32 0 0 0 0x33 0 0 0 1x34 0 0 0 0x35 0 0 0 1x36 0 0 0 0

the effect of variation in supplier’s capacity and buyer’s spaceavailability on multi-period procurement lot-sizing decisions,respectively. Similarly, from Problem set 6, 50% improvement inquality rate of supplier from current state leads to decrease in totalcost of buyer by an amount 1688. Problem set 7 shows that totalcost increases due to increase in rejections. These findings helpthe buyer to decide on how much and where to invest in buyer–supplier coordination process.

Fig. 3 shows that procurement lot-sizing model proposed in thispaper is sensitive to the variation in problem parameters. The com-putational results suggest that the proposed model captures allrealistic constraints in multi-period procurement lot-sizing deci-sion making process and analyzes tradeoffs in cost objectives.

6. Conclusions

Multi-period procurement lot-sizing decisions simultaneouslydetermine what quantity is to be procured and in which period itshould be procured so as to minimize total cost by striking trade-offs among purchasing cost, inventory holding cost and transactioncost. Lot-sizing decision is also influenced by quantity discounts,quality and delivery performance. This paper presents an integerlinear programming approach for procurement lot-sizing problemin the real world situation. By formulating the multi-period lot-siz-ing problem as an integer linear programming, we have captured

4 5 6 7

6,330 48,463 50,889 45,966 48,0071,050 20,550 25,750 18,050 19,500259 4531 103 7576 72492,639 73,544 76,742 71,592 74,75622 0 0 0 0

0 415 0 00 422 0 0450 421 450 00 443 0 00 396 0 0577 471 0 0750 0 0 00 0 0 00 0 0 547750 0 750 7500 0 0 00 0 1262 12980 0 0 0

105 0 0 0 00 0 0 0

000 0 0 0 00 0 0 0

D. Choudhary, R. Shankar / Computers & Industrial Engineering 61 (2011) 1318–1323 1323

the realistic constraint at all levels over a finite planning horizon.The proposed model can support MRP system in realisticsituations.

The computation analysis shows that problem of reasonablesize can be solved using any commercial software in a few secondsof computer time via the proposed formulation. However, as thenumber of quantity discount levels and/or periods increases, themodel could become very large with hundreds of binary variablesand may become computationally intractable. Hence, the future re-search work may explore exact solution approaches such as branchand bound or cutting plane methods, or heuristics and approxi-mate algorithms.

Future research might also include several products. If therewere two products, both managed by a single supplier, consoli-dated shipments of a mixed load could be dispatched to buyer.Two or more products may allow additional economies in inven-tory or transportation decisions. Further, the proposed model fo-cuses on the buyer’s benefits. The issue of coordination betweenbuyer and supplier can also be studied to optimize the whole sup-ply chain benefits in a multi-period procurement lot-sizing deci-sion making process.

Acknowledgments

The authors are grateful to the editor and two anonymous ref-erees for their valuable suggestions, which have been of great helpfor the improvement of the paper.

References

Wagner, H. M., & Whitin, T. M. (1958). Dynamic version of the economic lot-sizemodel. Management Science, 5, 89–96.

Silver, E. A., & Meal, H. C. (1973). A heuristic for selecting lot size quantities for thecase of a deterministic time varying rate and discrete opportunities forreplenishment. Production and Inventory Management, 14, 64–74.

Federgruen, A., & Tzur, M. (1991). A simple forward algorithm to solve generaldynamic lot sizing models with n periods in O(n log n) or O(n) time.Management Science, 37, 909–925.

Aggarwal, A., & Park, J. K. (1993). Improved algorithms for economic lot-sizeproblems. Operations Research, 45, 49–71.

Heady, R. B., & Zhu, Z. (1994). An improved implementation of the Wagner–Whitinalgorithm. Production and Operations Management, 3, 55–63.

Brahimi, N., Dauzere-Peres, S., Najid, N. M., & Nordli, A. (2006). Single item lot sizingproblems. European Journal of Operational Research, 168, 1–16.

Karimi, B., Fatemi Ghomi, S. M. T., & Wilson, J. M. (2003). The capacitated lot sizingproblem: A review of models and algorithms. Omega, 31(5), 365–378.

Ben-Daya, M., Darwish, M., & Ertogral, K. (2008). The joint economic lot sizingproblem: Review and extensions. European Journal of Operational Research,185(2), 726–742.

Robinson, P., Narayanan, A., & Sahin, F. (2009). Coordinated deterministic dynamicdemand lot-sizing problem: A review of models and algorithms. Omega, 37(1),3–15.

Smith, N. R., Robles, J. L., & Cárdenas-Barrón, L. E. (2009). Optimal pricing andproduction master planning in a multiperiod horizon considering capacity andinventory constraints. Mathematical Problems in Engineering, 2009, 1–15.

Buffa, F. P., & Jackson, W. M. (1983). A goal programming model for purchaseplanning. Journal of Purchasing and Materials Management, 19, 27–34.

Pratsini, E. (2000). The capacitated dynamic lot size problem with variabletechnology. Computers & Industrial Engineering, 38(4), 493–504.

Benton, W. C. (1991). Quantity discount decisions under conditions of multipleitems, multiple suppliers and resource limitation. International Journal ofProduction Research, 29, 1953–1961.

Raza, A. S., & Akgunduz, A. (2008). A comparative study of heuristic algorithms oneconomic lot scheduling problem. Computers & Industrial Engineering, 55(1),94–109.

Polatoglu, H., & Sahin, I. (2000). Optimal procurement policies under price-dependent demand. International Journal of Production Economics, 65, 141–171.

Chaudhry, S. S., Forst, F. G., & Zydiak, J. L. (1993). Vendor selection with price breaks.European Journal of Operational Research, 70, 52–66.

Bender, P. S., Brown, R. W., Isaac, M. H., & Shapiro, J. F. (1985). Improving purchasingproductivity at IBM with a normative decision support system. Interfaces, 15,106–115.

Ustun, O., & Demirtas, E. A. (2008). Multi-period lot-sizing with supplier selectionusing achievement scalarizing functions. Computers & Industrial Engineering,54(4), 918–931.

Liao, Z., & Kuhn, A. (2004). Operational integration of supplier selection andprocurement lot sizing in supply chain. In Proceedings of global project andmanufacturing management symposium 2004, Siegen Germany.

Demirtas, E. A., & Ustun, O. (2009). Analytic network process and multi-period goalprogramming integration in purchasing decisions. Computers & IndustrialEngineering, 56(2), 677–690.

Rezaei, J., & Davoodi, M. (2011). Multi objective models for lot-sizing with supplierselection. International Journal of Production Economics, 130(1), 77–86.