research article designing a fresh food supply chain...
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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 506531 8 pageshttpdxdoiorg1011552013506531
Research ArticleDesigning a Fresh Food Supply Chain Network An Applicationof Nonlinear Programming
Yu-Chung Tsao
Department of Industrial Management National Taiwan University of Science and Technology No 43 Section 4Keelung Road Darsquoan District Taipei 106 Taiwan
Correspondence should be addressed to Yu-Chung Tsao yctsaomailntustedutw
Received 30 July 2013 Accepted 19 November 2013
Academic Editor Chong Lin
Copyright copy 2013 Yu-Chung Tsao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In todayrsquos business environment many fresh food companies have complex supply networks to distribute their products Forexample agricultural products are distributed through a multiechelon supply chain which includes agricultural associationagricultural produce marketing corporations (APMCs) markets and so forth In this paper a fresh produce supply network modelis designed to determine the optimal service area for APMCs the replenishment cycle time of APMCs and the freshness-keepingeffort while maximizing the total profit The objective is to address the integrated facility location inventory allocation andfreshness-keeping effort problems This paper develops an algorithm to solve the nonlinear problem provides numerical analysisto illustrate the proposed solution procedure and discusses the effects of various system parameters on the decisions and totalprofits A real case of an agricultural product supply chain in Taiwan is used to verify the model Results of this study can serve asa reference for business managers and administrators
1 Introduction
The demand and distribution volume of fresh food hasincreased year by year as consumption levels have increasedIn 2009 the total production volume of vegetables reachedsix hundred million tons in China Compared with theproduction volume in 2008 it had increased by 2684 tonsHowever the distribution of fresh food products such asagricultural produce and live seafood are challenging andrisky operations due to the perishable nature of such prod-ucts Having a robust distribution system to distribute freshfood products not only reduces the food deterioration butalso increases the customer service level
For food supply chain management Georgiadis et al[1] presented a system dynamics-based methodologicalapproach for mapping and analyzing multiechelon food sup-ply chains Apaiah and Hendrix [2] determined the locationof primary production ingredient processing product pro-duction areas andmodes of transportation for Novel ProteinFoods in The Netherlands Osvald and Stirn [3] developedan algorithm to solve the vehicle routing problem with time
windows and time-dependent travel-times for fresh vegeta-bles Cai et al [4] determined the retailerrsquos optimal orderquantity level of freshness-keeping effort selling price andwholesale prices for fresh produce Hong et al [5] discusseda financially viable business model for a Radio FrequencyIdentification (RFID) application versus a food traceabilitysystemThey conducted a case study of RFID implementationin a chain of convenience stores in Taiwan Wang and Li [6]proposed dynamic product quality evaluation based upon apricing model for perishable food supply chains Cai et al[7] considered a supply chain in which a producer suppliesa fresh product through a third-party logistics provider to adistant market where a distributor purchases and sells it toend customers
So far few studies have considered the food supply net-work design problem that is simultaneously determining theintegrated location allocation and inventory decisions Forsupply network design Shen [8] conducted a complete reviewof the supply design literature and discussed future researchtopics Recently Park et al [9] considered a single-sourcingnetwork design problem for a three-level supply chain
2 Journal of Applied Mathematics
Nagurney [10] proposed a framework for supply networkdesign and redesign that allows for the determination of theoptimal levels of capacity and operational production flowsThere are a few research papers on the retaile-tail businessmodels For instance Bretthauer et al [11] determined theappropriate locations of online sales to minimize the totalcosts and satisfy in-store and online demand Tsao and Lu [12]developed a supply network design which considers distancediscount and quantity discount for transportation costsKlibi and Martel [13] studied various modeling approachesto design resilient supply networks for the location-trans-portation problem under uncertainty Nasr considered trans-shipment and safety stock under supply interruption in aproduction system The issue of supply network design ispopular in this field of research
Designing an appropriate distribution network for freshfood is important in reducing the logistics cost and satis-fying the customer service level However none consideredthe perishable nature of fresh food in the supply networkdesign problem that is determining the facility location andinventory allocation decisions simultaneously Distributionof fresh products falls in the category of cold chains in whichthe products are kept at low temperature [4] Preventing thedeterioration of perishable products is therefore an importantmission especially for fresh food products Investing moreresources in preservative efforts by using better packagingmore powerful cooling facilities or faster modes of trans-portation helps to keep spoilage to aminimum [14] Howeverdetermining the level of the freshness-keeping effort is animportant decision Determining the best trade-off betweencosts and benefits is a critical problem for the fresh foodsupply chain This paper incorporates the freshness-keepingeffort decision into the supply network design model Themodel simultaneously determines the optimal location allo-cation inventory and freshness-keeping effort decisions tomaximize the total profit for a fresh food product
The contributions of this paper to the literature are asfollows First this is the first supply network design paper toconsider fresh food products In particular this study consid-ers the freshness-keeping effort decision Second this paperuses a continuous approximation method to model the totalcost The proposed solution defines the input data in termsof continuous functions and is capable of formulating thesefunctions for a data set of any size This is very importantfor dealing with practical problemsThird this study presentsa solution approach for solving the supply network designproblem described above We also conducted a numericalanalysis to discuss the impacts of the changing parametersand provide the management implications Fourth in thispaper we applied our procedure to an agricultural productdistribution system in TaiwanWe show that the model couldbe applied in actual practice
2 Model Formulation
In this paper we considered the supply network design foragricultural productsThemodel can also be applied to otherfresh food products The network studied in this paper is
a multiechelon supply chain with farmers selling goods tofarmersrsquo associations Then farmersrsquo associations sell andtransport agricultural products to agricultural produce-mar-keting corporations (APMCs)The APMCs are located at thethird level help consolidate shipments arriving from farm-ersrsquo associations and deliver them to retailers (traditionalmarkets supermarkets and hyper markets) The retailersdownstreammeet the demands of end customers Goods flowfrom upper-stream facilities to the downstream facilities (seeFigure 1)
The mathematical model in this study is based on thefollowing assumptions
(1) Demand per unit time for retail store in cluster 119894 isan independent and identically distributed Poissonprocess with rate 120582
119894
(2) Each APMCrsquos service area is close to circular Serviceregions have somewhat irregular shapes as opposedto circles hexagons or squares in the economicsliterature This irregular service area is shown to havelittle effect on the optimal solution [15] MoreovereachAPMC is located in the center of the service area
(3) Each retailer is assigned to a particular APMC andserved only by that APMC when the retailer is in theAPMCrsquos service area
(4) The perishable rate of the fresh food is decreasing asthe freshness-keeping effort increases
(5) The freshness-keeping effort cost is not only convexbut also increasing in the freshness-keeping effort
This study uses the following notations
119879 joint replenishment cycle time (decision variable)119860119894 service area for each APMC in cluster 119894 where 119894 =
1 2 119873 (decision variable)120591 freshness-keeping effort (decision variable)119876 ordering quantity120579 perishable rate (or deteriorating rate) of the freshfood 120579 = 120572 minus 120573120591 where 120572 and 120573 are constants119865 facility cost of opening each APMC120575119894 store density in cluster 119894
120585 length of the planning horizon120582119894 demand rate for retail store in cluster 119894
119862119891 fixed cost per inbound shipment
119862V variable cost per item for each inbound shipment119862119879 outbound transportation cost per unit distance
per item119891119903 constant that depends on the distance metric and
shape of the APMC service region119896(120591) freshness-keeping effort cost per unit item119896(120591) = 119886 + 119887120591
2 where 119886 and 119887 are constants119877 ordering cost for APMCℎ inventory holding cost for APMC
Journal of Applied Mathematics 3
middot middot middot middot middot middot
middot middot middotmiddot middot middot
middot middot middot middot middot middot middot middot middot
Farmer Farmer Farmer
Farmersrsquo association
Agricultural products marketing corporation
Hypermarket Hypermarket Market Market Supermarket Supermarket
Consumers
Figure 1 Agricultural product supply chain network
119862119894 service area in cluster 119894
119901 unit selling price
119888 unit purchasing cost
This study uses an approximation technique [16] to dividethe network into smaller regions over which the discretevariable can be modeled using the slow varying functionsUsing the method the given service region is covered withclusters 119894 119894 = 1 2 119873 Clusters i 119894 = 1 2 119873 existwithin the given service region such that the store density isnearly constant over each cluster This model calculates thecomponents of the total network cost as follows
(1) The total revenue is sum119873119894=1
119901120585120582119894120575119894119862119894
(2) The total purchasing cost is sum119873119894=1
119888120585120582119894120575119894119862119894
(3) The total facility cost is given by multiplying thefacility cost of opening each APMC with the numberof APMC s namely sum119873
119894=1119865(119862119894119860119894)
(4) Let 119862119891
be the fixed cost per inbound shipmentand 119862V the variable cost per item for each inboundshipmentThen the total inbound transportation costis sum119873119894=1
(119862119891+ 119862V119876)119879
(5) Assuming ldquoclose to circularrdquo service regions with thefacility at the center the average distance traveled byeach item is 119891
119903radic119860119894[15] The total outbound trans-
portation cost is sum119873119894=1
119888119879119891119903radic119860119894120585120582119894120575119894119862119894
(6) The total cost for freshness-keeping effort is sum119873119894=1
(119886 +
1198871205912)120585120582119894120575119894119862119894
(7) The total ordering cost is sum119873119894=1
(119877119879)(119862119894119860119894)
(8) Depletion of inventory can occur both as a result ofdemand and due to deterioration during the replen-ishment cycle Variations in the inventory level 119868(119905)with respect to time 119905 can be expressed as
119889119868 (119905)
119889119905= minus120579119868 (119905) minus 120585120582
119894120575119894119862119894 0 le 119905 le 119879 (1)
with the boundary condition 119868(119879) = 0 The solutionto (1) can now be expressed as
119868 (119905) =120585120582119894120575119894119862119894
120579[119890120579(119879minus119905)
minus 1] 0 le 119905 le 119879 (2)
Note that the quantity ordered during each replenish-ment cycle is
119876 =120585120582119894120575119894119862119894
120579(119890120579119879
minus 1) =120585120582119894120575119894119862119894
120572 minus 120573120591[119890(120572minus120573120591)119879
minus 1] (3)
The total inventory holding cost is sum119873119894=1
(ℎ119879) int119879
0119868(119905)119889119905 =
sum119873
119894=1(ℎ120585120582119894120575119894119862119894120579119879) int
119879
0[119890120579(119879minus119905)
minus 1]119889119905 = sum119873
119894=1(ℎ120585120582119894120575119894119862119894(120572 minus
120573120591)2119879)[119890(120572minus120573120591)119879
minus (120572 minus 120573120591)119879 minus 1]
4 Journal of Applied Mathematics
Therefore the total network profit is
Π(1198601 1198602 119860
119873 119879 120591)
=
119873
sum
119894=1
(119901 minus 119888) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119865119862119894
119860119894
+
119873
sum
119894=1
119888119879119891119903radic119860119894120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119862119891+ 119862V120585120582119894120575119894119862119894 [119890
(120572minus120573120591)119879minus 1] (120572 minus 120573120591)
119879
minus
119873
sum
119894=1
(119886 + 1198871205912) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119877
119879
119862119894
119860119894
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894
(120572 minus 120573120591)2119879
[119890(120572minus120573120591)119879
minus (120572 minus 120573120591) 119879 minus 1]
(4)
3 Decision-Making Approach
By using a truncated Taylor series expansion (119890120582119879 asymp 1 + 120582119879 +
(12)12058221198792) we can rewrite (4) as follows
Π(1198601 1198602 119860
119873 119879 120591)
asymp
119873
sum
119894=1
(119901 minus 119888) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119865119862119894
119860119894
+
119873
sum
119894=1
119888119879119891119903radic119860119894120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119862119891
119879+ 119862V120585120582119894120575119894119862119894 [1 +
(120572 minus 120573120591) 119879
2]
minus
119873
sum
119894=1
(119886 + 1198871205912) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119877
119879
119862119894
119860119894
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894119879
2
(5)
The problem is to determine the optimal service areareplenishment cycle time and freshness-keeping effort tomaximize the profit function in (5)
1205972Π
1205971198602
119894
= minus (2119865119862119894
1198603
119894
) minus (2119877119862119894
1198791198603
119894
)
minus (minus119888119879119891119903120585120582119894120575119894119862119894
411986032
119894
) 119894 = 1 2 119873
1205972Π
1205971205912= minus
119873
sum
119894=1
2119887120585120582119894120575119894119862119894lt 0
(6)
Since the facility opening cost 119865 is large 1205972Π1205971198602
119894lt 0
is satisfied in general case Then from 1205972Π120597119860
119894120597120591 = 0 and
1205972Π120597119860
119894120597119860119896
= 0 119894 = 1 2 119873 119896 = 1 2 119873 119894 = 119896 theHessian matrix is
119867119873+1
=
[[[[[[[[[[[[[[[[[[[[
[
1205972Π
1205971198602
1
0 0 0
01205972Π
1205971198602
2
0
0
0 01205972Π
1205971198602
119873
0
0 0 01205972Π
1205971205912
]]]]]]]]]]]]]]]]]]]]
]
(7)
Since 1205972Π120597119860
2
119894lt 0 and 120597
2Π1205971205912lt 0 for 119894 = 1 2 119873
we know that (minus1)119895sdot |119863119895| gt 0 is satisfied for all 119895 119895 =
1 2 119873 + 1 From the maximum theorem Winston [17]we know that Π(119860
1 1198602 119860
119873 120591 | 119879) is a concave function
of (1198601 1198602 119860
119873 120591)Thismeans that the optimal119860
119894(119879) and
120591(119879) can be obtained by solving 120597Π120597119860119894= 0 and 120597Π120597120591 = 0
119894 = 1 2 119873 To solve this problem we first determine theclosed form of the retail price 119860
119894(119879) and 120591(119879) that maximize
Π(1198601 1198602 119860
119873 120591 | 119879) Solving 120597Π120597119860
119894= 0 and 120597Π120597120591 =
0 simultaneously leads to
119860lowast
119894(119879) = [
2(119877 + 119865119879)
1198621198791198911199031205851205821198941205751198941198792
]
23
(8)
120591lowast(119879) =
119862V120573119879
4119887 (9)
Equations (8) and (9) leads to Properties 1 and 2
Property 1 (a) The service area for each APMC 119860119894increases
as the joint replenishment cycle time 119879 or the outboundtransportation cost 119862
119879decreases
(b) The service area for each APMC 119860119894increases as the
facility cost 119865 or the ordering cost 119877 increases
Property 2 (a) The freshness-keeping effort 120591 increases asthe joint replenishment cycle time 119879 the perishable rateparameter 120573 or the variable inbound shipment cost 119862Vincreases
(b) The freshness-keeping effort 120591 increases as thefreshness-keeping effort cost parameter 119887 decreases
Substituting 120591lowast(119879) and 119860
lowast
119894(119879) 119894 = 1 2 119873 into the
corresponding Π(1198601 1198602 119860
119873 119879 120591) reduces the model to
a single variable function Π(119879)
Π (119879) =
119873
sum
119894=1
(119901 minus 119888) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119865119862119894[1198621198791198911199031205851205821198941205751198941198792
2(119877 + 119865119879)]
23
+
119873
sum
119894=1
119888119879119891119903[
2(119877 + 119865119879)
1198621198791198911199031205851205821198941205751198941198792
]
13
120585120582119894120575119894119862119894
Journal of Applied Mathematics 5
Step 1 Find an initial solution 119879119894=1
by inspectionStep 2 Calculate Π
1015840(119879) and Π
10158401015840(119879)
Step 3 Let 119879119894+1
= 119879119894minus
Π1015840(119879)
Π10158401015840(119879)
Step 4 If 1003816100381610038161003816119879119894+1 minus 119879119894
1003816100381610038161003816 le 120576 where 120576 is the tolerance error then go to Step 5 otherwise let 119894 = 119894 + 1 and go to Step 3Step 5 Let 119879lowast = 119879
119894+1and Π
lowast(119879lowast) = Π(119879
119894+1)
Step 6 Determine 119860lowast
119894(119879lowast) and 120591
lowast(119879lowast) by (8) and (9) respectively
Algorithm 1
minus
119873
sum
119894=1
119862119891
119879+ 119862V120585120582119894120575119894119862119894
times[1 + (120572 minus 120573
119873
sum
119894=1
119862V120585120582119894120575119894119862119894120573119879
4119887120585120582119894120575119894119862119894
)119879
2]
minus
119873
sum
119894=1
[
[
119886 + 119887(
119873
sum
119894=1
119862V120585120582119894120575119894119862119894120573119879
4119887120585120582119894120575119894119862119894
)
2
]
]
120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119877119862119894
119879[1198621198791198911199031205851205821198941205751198941198792
2 (119877 + 119865119879)]
23
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894119879
2
(10)
Based on the discussion above Algorithm 1 based onNewtonrsquos method determines the optimal values for 119879
lowast 119860lowast
119894
and 120591lowast
4 Numerical Study
This section presents a numerical study to illustrate the pro-posed solution approach and provide quantitative insightsThe goals of the numerical study in this study are as follows
(1) to illustrate the procedures of the solution approach(2) to discuss the effects of the related parameters on deci-
sions and cost
41 Numerical Example To illustrate the algorithmdescribedabove we applied our procedure to an agricultural productdistribution system in Taiwan Since Taiwan joined theWorldTrade Organization (WTO) local agriculture has facedgreater competition due to the reduction of customs dutiesand international trade Therefore having an efficient agri-cultural food supply network is important for Taiwan Asshown in Figure 1 farmers sell their agricultural products toagricultural associations Then the agricultural product isdistributed from agricultural associations to agriculturalproduct-marketing corporations (APMCs)TheAPMCs con-solidate shipments arriving from the agricultural associationsand deliver them to the retailers (traditional markets super-markets and hypermarkets) We consider three cities TaipeiCity New Taipei City and Keelung City in northern Taiwanthey are Cluster 1 Cluster 2 and Cluster 3 respectively
in our example In this case we wanted to determine theoptimal APMC service areas the optimal number of APMCsin each cluster the optimal freshness-keeping effort andthe optimal joint replenishment cycle time In practice therelated parameters can be determined by regression analysisusing historical transaction data Please note that the dataused here have been scaled to protect confidentiality Pleasealso note that the freshness-keeping effort cost just needsto satisfy the requirement The freshness-keeping effort costis not only convex but also increasing in the freshness-keeping effort One may get other freshness-keeping effortcost form from the regression analysis However our modeland solution approach can also be used if the freshness-keeping effort cost satisfies the above requirement
The parameters of the product are 119865 = 100000 119901 = 100119888 = 50 119862
1= 10000 119862
2= 8000 119862
3= 6000 ℎ = 1 119877 = 30
119862119879
= 10 1205821= 11 120582
2= 10 120582
3= 9 120585 = 12 120575
1= 006 120575
2=
005 1205753= 004 119891
119903= 001 119862
119891= 1000 119862V = 5 120572 = 005 120573 =
001 119886 = 3 119887 = 01 After applying the algorithm in Section 3the optimal joint replenishment cycle time is119879lowast = 02346 theoptimal APMC service area in cluster119862
1is119860lowast1= 399859 the
optimal APMC service area in cluster1198622is119860lowast2= 481160 the
optimal APMC service area in cluster1198623is119860lowast3= 598964 the
optimal freshness-keeping effort is 120591lowast = 00293 and the totalnetwork profit is Πlowast = 4853230 Figure 2 shows the graphicillustrations of Π versus different 119879 The algorithm can findthe optimal solution in a very short time
42 Numerical Analysis Several numerical analyses are con-ducted to gain management insights into the structures ofthe proposed policies The following numerical analyses areused to show the effects of 119865 ℎ 119877 119862V 119862119879 119886 119887 120572 and 120573 on theoptimal decisions and the total network profit The results ofTables 1 2 and 3 are as follows
(1) When the facility cost 119865 increases the optimal jointreplenishment cycle time 119879
lowast the optimal freshness-keeping effort 120591
lowast and the total network profit Πlowast
decrease but the optimal APMC service area 119860lowast
119894
increases This verifies Property 1 (b) When thefacility cost increases it is reasonable to increase theAPMC service area to reduce the number of APMCsopening
(2) When the inventory holding cost ℎ increases the opti-mal joint replenishment cycle time 119879
lowast the optimalfreshness-keeping effort 120591
lowast and the total network
6 Journal of Applied Mathematics
Table 1 Effects of different system parameters
Parameter 119879lowast
119860lowast
1119860lowast
2119860lowast
3120591lowast
Πlowast
119865 = 80000 02356 344660 414737 516279 00294 4964300119865 = 100000 02346 399859 481160 598964 00293 4853230119865 = 120000 02340 451475 543270 676282 00292 4756130ℎ = 025 02876 399796 481084 598870 00360 4853460ℎ = 05 02346 399859 481160 598964 00293 4853230ℎ = 075 02031 399911 481223 599043 00254 484907119877 = 20 02327 399747 481025 598797 00291 4853460119877 = 30 02346 399859 481160 598964 00293 4853230119877 = 40 02365 399968 481292 599129 00296 4853020119888119907= 3 02519 399835 481131 598929 00189 5161330
119888119907= 5 02346 399859 481160 598964 00293 4853230
119888119907= 7 02205 399880 481186 598997 00386 4545270
119888119879= 5 02325 634741 763799 950804 00291 5427370
119888119879= 10 02346 399859 481160 598964 00293 4853230
119888119879= 15 02364 305147 367191 457092 00296 4371680
Table 2 Effects of a and b
a b005 0 1 015
2
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
1198603
lowast= 598964 119860
3
lowast= 598964 119860
3
lowast= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 5006370 Π
lowast= 5006350 Π
lowast= 5006350
3
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 4853250 Π
lowast= 4853230 Π
lowast= 4853230
4
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 4700130 Π
lowast= 4700110 Π
lowast= 4700110
profit Πlowast decrease but the optimal APMC service
area 119860lowast
119894increases It is reasonable that when the
inventory holding cost increases the company willdecrease the replenishment cycle time in an effortto lower inventory costs Also the freshness-keepingeffort can be decreased with a shorter replenish-ment cycle time They will also reduce the numberof APMCs to hold less inventory as the inventoryholding cost increases
Table 3 Effects of 120572 and 120573
120572120573
0005 001 0015
003
119879lowast= 02519 119879
lowast= 02521 119879
lowast= 02525
119860lowast
1= 399835 119860
lowast
1= 399835 119860
lowast
1= 399835
119860lowast
2= 481132 119860
lowast
2= 481131 119860
lowast
2= 481131
119860lowast
3= 598929 119860
lowast
3= 598929 119860
lowast
3= 598928
120591lowast= 00157 120591
lowast= 00315 120591
lowast= 00473
Πlowast= 4855080 Π
lowast= 4855100 Π
lowast= 4855110
005
119879lowast= 02345 119879
lowast= 02346 119879
lowast= 02349
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399858
119860lowast
2= 481160 119860
lowast
2= 481160 119860
lowast
2= 481159
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598963
120591lowast= 00147 120591
lowast= 00293 120591
lowast= 00440
Πlowast= 4853220 Π
lowast= 4853230 Π
lowast= 4853250
007
119879lowast= 02202 119879
lowast= 02204 119879
lowast= 02206
119860lowast
1= 399881 119860
lowast
1= 399881 119860
lowast
1= 399880
119860lowast
2= 481186 119860
lowast
2= 481186 119860
lowast
2= 481186
119860lowast
3= 598997 119860
lowast
3= 598997 119860
lowast
3= 598997
120591lowast= 00138 120591
lowast= 00275 120591
lowast= 00414
Πlowast= 4851490 Π
lowast= 4851490 Π
lowast= 4851510
(3) When the ordering cost 119877 increases the optimaljoint replenishment cycle time119879lowast the optimal APMCservice area 119860
lowast
119894 and the optimal freshness-keeping
effort 120591lowast all increase but the total network profit
Πlowast decreases This also verifies Property 1 (b) If
the ordering cost increases it is reasonable that thecompany will increase the replenishment cycle timeto reduce replenishment frequency The companywill also likely decrease the number of APMCs to
Journal of Applied Mathematics 7
018 022 024 026 028 03
Π
T
48532 times 106
48528 times 106
48526 times 106
48524 times 106
Figure 2 Graphic illustrations of Π versus 119879
reduce the replenishment times as the ordering costincreases
(4) When the variable inbound shipment cost 119862Vincreases the optimal joint replenishment cycle time119879lowast and the total network profit Π
lowast decrease butthe optimal APMC service area 119860
lowast
119894and the optimal
freshness-keeping effort 120591lowast increase This verifies
Property 2 (a) The number of APMCs will decreaseto reduce the inbound transportation times as thevariable inbound shipment cost increases
(5) When the outbound transportation cost119862119879increases
the optimal DC service area119860lowast
119894and the total network
profit Πlowast decrease but the optimal joint replenish-
ment cycle time119879lowast and the optimal freshness-keepingeffort 120591
lowast increase This also verifies Property 1 (a)When the outbound transportation cost increases itis reasonable to increase the joint replenishment cycletime to reduce the outbound transportation times
(6) When the freshness-keeping effort cost increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases It is reasonable to decrease
the freshness-keeping effort and the replenishmentcycle time when the freshness-keeping effort costincreases
(7) When the perishable rate of the fresh food increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases When the perishable rate
of the fresh food increases the company will decreasethe cycle time to reduce the deterioration of thefresh food When the freshness-keeping effort hasmore service on reducing the deterioration of thefresh food companies are willing to increase theirfreshness-keeping effort
5 Conclusions
This study considers a supply network design problem foragricultural productsThemodel can also be applied to other
fresh food products The agricultural produce marketingcorporations (APMCs) help consolidate shipments arrivingfrom farmersrsquo associations and deliver them to traditionalmarkets supermarkets and hypermarkets The proposedmethod integrates sales revenue purchasing costs facilitycosts inventory costs transportation costs ordering costsand freshness-keeping effort costs The key decisions includewhere to locate the APMCs how to assign retailers toAPMCs the joint replenishment cycle time at APMCs andthe freshness-keeping effort such that the total networkprofit is maximized An easy-to-use algorithm is providedfor solving the nonlinear programming problem Numericalstudies illustrate the solution procedures and the effectsof the facility cost inventory holding cost transportationcost ordering cost and perishable rate of the fresh food ondecisions and profits The results of this study are a usefulreference for managerial decision making and administra-tion Further research on this topic may consider when thefreshness-keeping effort has an influence on demand Thiswould include the market demand for fresh food dependenton the freshness of the product
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his gratitude to the editor and theanonymous reviewers for their detailed comments and valu-able suggestions to improve the exposition of this paper Thispaper is supported in part by the National Science Councilunder Grant NSC 102-2410-H-011-029-MY3 and 102-2221-E-011-159-MY3
References
[1] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005
[2] R K Apaiah and E M T Hendrix ldquoDesign of a supply chainnetwork for pea-based novel protein foodsrdquo Journal of FoodEngineering vol 70 no 3 pp 383ndash391 2005
[3] A Osvald and L Z Stirn ldquoA vehicle routing algorithm for thedistribution of fresh vegetables and similar perishable foodrdquoJournal of Food Engineering vol 85 no 2 pp 285ndash295 2008
[4] X Cai J Chen Y Xiao and X Xu ldquoOptimization andcoordination of fresh product supply chains with freshness-keeping effortrdquo Production and OperationsManagement vol 19no 3 pp 261ndash278 2010
[5] I-H Hong J-F Dang Y-H Tsai et al ldquoAn RFID applicationin the food supply chain a case study of convenience stores inTaiwanrdquo Journal of Food Engineering vol 106 no 2 pp 119ndash1262011
[6] XWang andD Li ldquoAdynamic product quality evaluation basedpricing model for perishable food supply chainsrdquo Omega vol40 no 6 pp 906ndash917 2012
8 Journal of Applied Mathematics
[7] X Cai J Chen Y Xiao X Xu andG Yu ldquoFresh-product supplychain management with logistics outsourcingrdquo Omega vol 41no 4 pp 752ndash765 2013
[8] Z-J M Shen ldquoIntegrated supply chain design models asurvey and future research directionsrdquo Journal of Industrial andManagement Optimization vol 3 no 1 pp 1ndash27 2007
[9] S Park T-E Lee and C S Sung ldquoA three-level supplychain network design model with risk-pooling and lead timesrdquoTransportation Research E vol 46 no 5 pp 563ndash581 2010
[10] A Nagurney ldquoOptimal supply chain network design andredesign at minimal total cost and with demand satisfactionrdquoInternational Journal of Production Economics vol 128 no 1 pp200ndash208 2010
[11] K M Bretthauer S Mahar and M A VenakataramananldquoInventory and distribution strategies for retaile-tail organiza-tionsrdquo Computers and Industrial Engineering vol 58 no 1 pp119ndash132 2010
[12] Y-C Tsao and J-C Lu ldquoA supply chain network design consid-ering transportation cost discountsrdquo Transportation Research Evol 48 no 2 pp 401ndash414 2012
[13] W Klibi and A Martel ldquoModeling approaches for the designof resilient supply networks under disruptionsrdquo InternationalJournal of Production Economics vol 135 no 2 pp 882ndash8982012
[14] S Doris ldquoKeeping freshness in fresh-cut productrdquo AgriculturalResearch vol 46 no 2 pp 12ndash14 1998
[15] A Dasci and V Verter ldquoA continuous model for production-distribution system designrdquo European Journal of OperationalResearch vol 129 no 2 pp 287ndash298 2001
[16] Y C Tsao D Mangotra J C Lu and M Dong ldquoA continuousapproximation approach for the integrated facility-inventoryallocation problemrdquo European Journal of Operational Researchvol 222 pp 216ndash228 2012
[17] W L Winston Operations Research Applications and Algo-rithms ThomsonBrooksCole Belmont Mass USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
Nagurney [10] proposed a framework for supply networkdesign and redesign that allows for the determination of theoptimal levels of capacity and operational production flowsThere are a few research papers on the retaile-tail businessmodels For instance Bretthauer et al [11] determined theappropriate locations of online sales to minimize the totalcosts and satisfy in-store and online demand Tsao and Lu [12]developed a supply network design which considers distancediscount and quantity discount for transportation costsKlibi and Martel [13] studied various modeling approachesto design resilient supply networks for the location-trans-portation problem under uncertainty Nasr considered trans-shipment and safety stock under supply interruption in aproduction system The issue of supply network design ispopular in this field of research
Designing an appropriate distribution network for freshfood is important in reducing the logistics cost and satis-fying the customer service level However none consideredthe perishable nature of fresh food in the supply networkdesign problem that is determining the facility location andinventory allocation decisions simultaneously Distributionof fresh products falls in the category of cold chains in whichthe products are kept at low temperature [4] Preventing thedeterioration of perishable products is therefore an importantmission especially for fresh food products Investing moreresources in preservative efforts by using better packagingmore powerful cooling facilities or faster modes of trans-portation helps to keep spoilage to aminimum [14] Howeverdetermining the level of the freshness-keeping effort is animportant decision Determining the best trade-off betweencosts and benefits is a critical problem for the fresh foodsupply chain This paper incorporates the freshness-keepingeffort decision into the supply network design model Themodel simultaneously determines the optimal location allo-cation inventory and freshness-keeping effort decisions tomaximize the total profit for a fresh food product
The contributions of this paper to the literature are asfollows First this is the first supply network design paper toconsider fresh food products In particular this study consid-ers the freshness-keeping effort decision Second this paperuses a continuous approximation method to model the totalcost The proposed solution defines the input data in termsof continuous functions and is capable of formulating thesefunctions for a data set of any size This is very importantfor dealing with practical problemsThird this study presentsa solution approach for solving the supply network designproblem described above We also conducted a numericalanalysis to discuss the impacts of the changing parametersand provide the management implications Fourth in thispaper we applied our procedure to an agricultural productdistribution system in TaiwanWe show that the model couldbe applied in actual practice
2 Model Formulation
In this paper we considered the supply network design foragricultural productsThemodel can also be applied to otherfresh food products The network studied in this paper is
a multiechelon supply chain with farmers selling goods tofarmersrsquo associations Then farmersrsquo associations sell andtransport agricultural products to agricultural produce-mar-keting corporations (APMCs)The APMCs are located at thethird level help consolidate shipments arriving from farm-ersrsquo associations and deliver them to retailers (traditionalmarkets supermarkets and hyper markets) The retailersdownstreammeet the demands of end customers Goods flowfrom upper-stream facilities to the downstream facilities (seeFigure 1)
The mathematical model in this study is based on thefollowing assumptions
(1) Demand per unit time for retail store in cluster 119894 isan independent and identically distributed Poissonprocess with rate 120582
119894
(2) Each APMCrsquos service area is close to circular Serviceregions have somewhat irregular shapes as opposedto circles hexagons or squares in the economicsliterature This irregular service area is shown to havelittle effect on the optimal solution [15] MoreovereachAPMC is located in the center of the service area
(3) Each retailer is assigned to a particular APMC andserved only by that APMC when the retailer is in theAPMCrsquos service area
(4) The perishable rate of the fresh food is decreasing asthe freshness-keeping effort increases
(5) The freshness-keeping effort cost is not only convexbut also increasing in the freshness-keeping effort
This study uses the following notations
119879 joint replenishment cycle time (decision variable)119860119894 service area for each APMC in cluster 119894 where 119894 =
1 2 119873 (decision variable)120591 freshness-keeping effort (decision variable)119876 ordering quantity120579 perishable rate (or deteriorating rate) of the freshfood 120579 = 120572 minus 120573120591 where 120572 and 120573 are constants119865 facility cost of opening each APMC120575119894 store density in cluster 119894
120585 length of the planning horizon120582119894 demand rate for retail store in cluster 119894
119862119891 fixed cost per inbound shipment
119862V variable cost per item for each inbound shipment119862119879 outbound transportation cost per unit distance
per item119891119903 constant that depends on the distance metric and
shape of the APMC service region119896(120591) freshness-keeping effort cost per unit item119896(120591) = 119886 + 119887120591
2 where 119886 and 119887 are constants119877 ordering cost for APMCℎ inventory holding cost for APMC
Journal of Applied Mathematics 3
middot middot middot middot middot middot
middot middot middotmiddot middot middot
middot middot middot middot middot middot middot middot middot
Farmer Farmer Farmer
Farmersrsquo association
Agricultural products marketing corporation
Hypermarket Hypermarket Market Market Supermarket Supermarket
Consumers
Figure 1 Agricultural product supply chain network
119862119894 service area in cluster 119894
119901 unit selling price
119888 unit purchasing cost
This study uses an approximation technique [16] to dividethe network into smaller regions over which the discretevariable can be modeled using the slow varying functionsUsing the method the given service region is covered withclusters 119894 119894 = 1 2 119873 Clusters i 119894 = 1 2 119873 existwithin the given service region such that the store density isnearly constant over each cluster This model calculates thecomponents of the total network cost as follows
(1) The total revenue is sum119873119894=1
119901120585120582119894120575119894119862119894
(2) The total purchasing cost is sum119873119894=1
119888120585120582119894120575119894119862119894
(3) The total facility cost is given by multiplying thefacility cost of opening each APMC with the numberof APMC s namely sum119873
119894=1119865(119862119894119860119894)
(4) Let 119862119891
be the fixed cost per inbound shipmentand 119862V the variable cost per item for each inboundshipmentThen the total inbound transportation costis sum119873119894=1
(119862119891+ 119862V119876)119879
(5) Assuming ldquoclose to circularrdquo service regions with thefacility at the center the average distance traveled byeach item is 119891
119903radic119860119894[15] The total outbound trans-
portation cost is sum119873119894=1
119888119879119891119903radic119860119894120585120582119894120575119894119862119894
(6) The total cost for freshness-keeping effort is sum119873119894=1
(119886 +
1198871205912)120585120582119894120575119894119862119894
(7) The total ordering cost is sum119873119894=1
(119877119879)(119862119894119860119894)
(8) Depletion of inventory can occur both as a result ofdemand and due to deterioration during the replen-ishment cycle Variations in the inventory level 119868(119905)with respect to time 119905 can be expressed as
119889119868 (119905)
119889119905= minus120579119868 (119905) minus 120585120582
119894120575119894119862119894 0 le 119905 le 119879 (1)
with the boundary condition 119868(119879) = 0 The solutionto (1) can now be expressed as
119868 (119905) =120585120582119894120575119894119862119894
120579[119890120579(119879minus119905)
minus 1] 0 le 119905 le 119879 (2)
Note that the quantity ordered during each replenish-ment cycle is
119876 =120585120582119894120575119894119862119894
120579(119890120579119879
minus 1) =120585120582119894120575119894119862119894
120572 minus 120573120591[119890(120572minus120573120591)119879
minus 1] (3)
The total inventory holding cost is sum119873119894=1
(ℎ119879) int119879
0119868(119905)119889119905 =
sum119873
119894=1(ℎ120585120582119894120575119894119862119894120579119879) int
119879
0[119890120579(119879minus119905)
minus 1]119889119905 = sum119873
119894=1(ℎ120585120582119894120575119894119862119894(120572 minus
120573120591)2119879)[119890(120572minus120573120591)119879
minus (120572 minus 120573120591)119879 minus 1]
4 Journal of Applied Mathematics
Therefore the total network profit is
Π(1198601 1198602 119860
119873 119879 120591)
=
119873
sum
119894=1
(119901 minus 119888) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119865119862119894
119860119894
+
119873
sum
119894=1
119888119879119891119903radic119860119894120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119862119891+ 119862V120585120582119894120575119894119862119894 [119890
(120572minus120573120591)119879minus 1] (120572 minus 120573120591)
119879
minus
119873
sum
119894=1
(119886 + 1198871205912) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119877
119879
119862119894
119860119894
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894
(120572 minus 120573120591)2119879
[119890(120572minus120573120591)119879
minus (120572 minus 120573120591) 119879 minus 1]
(4)
3 Decision-Making Approach
By using a truncated Taylor series expansion (119890120582119879 asymp 1 + 120582119879 +
(12)12058221198792) we can rewrite (4) as follows
Π(1198601 1198602 119860
119873 119879 120591)
asymp
119873
sum
119894=1
(119901 minus 119888) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119865119862119894
119860119894
+
119873
sum
119894=1
119888119879119891119903radic119860119894120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119862119891
119879+ 119862V120585120582119894120575119894119862119894 [1 +
(120572 minus 120573120591) 119879
2]
minus
119873
sum
119894=1
(119886 + 1198871205912) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119877
119879
119862119894
119860119894
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894119879
2
(5)
The problem is to determine the optimal service areareplenishment cycle time and freshness-keeping effort tomaximize the profit function in (5)
1205972Π
1205971198602
119894
= minus (2119865119862119894
1198603
119894
) minus (2119877119862119894
1198791198603
119894
)
minus (minus119888119879119891119903120585120582119894120575119894119862119894
411986032
119894
) 119894 = 1 2 119873
1205972Π
1205971205912= minus
119873
sum
119894=1
2119887120585120582119894120575119894119862119894lt 0
(6)
Since the facility opening cost 119865 is large 1205972Π1205971198602
119894lt 0
is satisfied in general case Then from 1205972Π120597119860
119894120597120591 = 0 and
1205972Π120597119860
119894120597119860119896
= 0 119894 = 1 2 119873 119896 = 1 2 119873 119894 = 119896 theHessian matrix is
119867119873+1
=
[[[[[[[[[[[[[[[[[[[[
[
1205972Π
1205971198602
1
0 0 0
01205972Π
1205971198602
2
0
0
0 01205972Π
1205971198602
119873
0
0 0 01205972Π
1205971205912
]]]]]]]]]]]]]]]]]]]]
]
(7)
Since 1205972Π120597119860
2
119894lt 0 and 120597
2Π1205971205912lt 0 for 119894 = 1 2 119873
we know that (minus1)119895sdot |119863119895| gt 0 is satisfied for all 119895 119895 =
1 2 119873 + 1 From the maximum theorem Winston [17]we know that Π(119860
1 1198602 119860
119873 120591 | 119879) is a concave function
of (1198601 1198602 119860
119873 120591)Thismeans that the optimal119860
119894(119879) and
120591(119879) can be obtained by solving 120597Π120597119860119894= 0 and 120597Π120597120591 = 0
119894 = 1 2 119873 To solve this problem we first determine theclosed form of the retail price 119860
119894(119879) and 120591(119879) that maximize
Π(1198601 1198602 119860
119873 120591 | 119879) Solving 120597Π120597119860
119894= 0 and 120597Π120597120591 =
0 simultaneously leads to
119860lowast
119894(119879) = [
2(119877 + 119865119879)
1198621198791198911199031205851205821198941205751198941198792
]
23
(8)
120591lowast(119879) =
119862V120573119879
4119887 (9)
Equations (8) and (9) leads to Properties 1 and 2
Property 1 (a) The service area for each APMC 119860119894increases
as the joint replenishment cycle time 119879 or the outboundtransportation cost 119862
119879decreases
(b) The service area for each APMC 119860119894increases as the
facility cost 119865 or the ordering cost 119877 increases
Property 2 (a) The freshness-keeping effort 120591 increases asthe joint replenishment cycle time 119879 the perishable rateparameter 120573 or the variable inbound shipment cost 119862Vincreases
(b) The freshness-keeping effort 120591 increases as thefreshness-keeping effort cost parameter 119887 decreases
Substituting 120591lowast(119879) and 119860
lowast
119894(119879) 119894 = 1 2 119873 into the
corresponding Π(1198601 1198602 119860
119873 119879 120591) reduces the model to
a single variable function Π(119879)
Π (119879) =
119873
sum
119894=1
(119901 minus 119888) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119865119862119894[1198621198791198911199031205851205821198941205751198941198792
2(119877 + 119865119879)]
23
+
119873
sum
119894=1
119888119879119891119903[
2(119877 + 119865119879)
1198621198791198911199031205851205821198941205751198941198792
]
13
120585120582119894120575119894119862119894
Journal of Applied Mathematics 5
Step 1 Find an initial solution 119879119894=1
by inspectionStep 2 Calculate Π
1015840(119879) and Π
10158401015840(119879)
Step 3 Let 119879119894+1
= 119879119894minus
Π1015840(119879)
Π10158401015840(119879)
Step 4 If 1003816100381610038161003816119879119894+1 minus 119879119894
1003816100381610038161003816 le 120576 where 120576 is the tolerance error then go to Step 5 otherwise let 119894 = 119894 + 1 and go to Step 3Step 5 Let 119879lowast = 119879
119894+1and Π
lowast(119879lowast) = Π(119879
119894+1)
Step 6 Determine 119860lowast
119894(119879lowast) and 120591
lowast(119879lowast) by (8) and (9) respectively
Algorithm 1
minus
119873
sum
119894=1
119862119891
119879+ 119862V120585120582119894120575119894119862119894
times[1 + (120572 minus 120573
119873
sum
119894=1
119862V120585120582119894120575119894119862119894120573119879
4119887120585120582119894120575119894119862119894
)119879
2]
minus
119873
sum
119894=1
[
[
119886 + 119887(
119873
sum
119894=1
119862V120585120582119894120575119894119862119894120573119879
4119887120585120582119894120575119894119862119894
)
2
]
]
120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119877119862119894
119879[1198621198791198911199031205851205821198941205751198941198792
2 (119877 + 119865119879)]
23
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894119879
2
(10)
Based on the discussion above Algorithm 1 based onNewtonrsquos method determines the optimal values for 119879
lowast 119860lowast
119894
and 120591lowast
4 Numerical Study
This section presents a numerical study to illustrate the pro-posed solution approach and provide quantitative insightsThe goals of the numerical study in this study are as follows
(1) to illustrate the procedures of the solution approach(2) to discuss the effects of the related parameters on deci-
sions and cost
41 Numerical Example To illustrate the algorithmdescribedabove we applied our procedure to an agricultural productdistribution system in Taiwan Since Taiwan joined theWorldTrade Organization (WTO) local agriculture has facedgreater competition due to the reduction of customs dutiesand international trade Therefore having an efficient agri-cultural food supply network is important for Taiwan Asshown in Figure 1 farmers sell their agricultural products toagricultural associations Then the agricultural product isdistributed from agricultural associations to agriculturalproduct-marketing corporations (APMCs)TheAPMCs con-solidate shipments arriving from the agricultural associationsand deliver them to the retailers (traditional markets super-markets and hypermarkets) We consider three cities TaipeiCity New Taipei City and Keelung City in northern Taiwanthey are Cluster 1 Cluster 2 and Cluster 3 respectively
in our example In this case we wanted to determine theoptimal APMC service areas the optimal number of APMCsin each cluster the optimal freshness-keeping effort andthe optimal joint replenishment cycle time In practice therelated parameters can be determined by regression analysisusing historical transaction data Please note that the dataused here have been scaled to protect confidentiality Pleasealso note that the freshness-keeping effort cost just needsto satisfy the requirement The freshness-keeping effort costis not only convex but also increasing in the freshness-keeping effort One may get other freshness-keeping effortcost form from the regression analysis However our modeland solution approach can also be used if the freshness-keeping effort cost satisfies the above requirement
The parameters of the product are 119865 = 100000 119901 = 100119888 = 50 119862
1= 10000 119862
2= 8000 119862
3= 6000 ℎ = 1 119877 = 30
119862119879
= 10 1205821= 11 120582
2= 10 120582
3= 9 120585 = 12 120575
1= 006 120575
2=
005 1205753= 004 119891
119903= 001 119862
119891= 1000 119862V = 5 120572 = 005 120573 =
001 119886 = 3 119887 = 01 After applying the algorithm in Section 3the optimal joint replenishment cycle time is119879lowast = 02346 theoptimal APMC service area in cluster119862
1is119860lowast1= 399859 the
optimal APMC service area in cluster1198622is119860lowast2= 481160 the
optimal APMC service area in cluster1198623is119860lowast3= 598964 the
optimal freshness-keeping effort is 120591lowast = 00293 and the totalnetwork profit is Πlowast = 4853230 Figure 2 shows the graphicillustrations of Π versus different 119879 The algorithm can findthe optimal solution in a very short time
42 Numerical Analysis Several numerical analyses are con-ducted to gain management insights into the structures ofthe proposed policies The following numerical analyses areused to show the effects of 119865 ℎ 119877 119862V 119862119879 119886 119887 120572 and 120573 on theoptimal decisions and the total network profit The results ofTables 1 2 and 3 are as follows
(1) When the facility cost 119865 increases the optimal jointreplenishment cycle time 119879
lowast the optimal freshness-keeping effort 120591
lowast and the total network profit Πlowast
decrease but the optimal APMC service area 119860lowast
119894
increases This verifies Property 1 (b) When thefacility cost increases it is reasonable to increase theAPMC service area to reduce the number of APMCsopening
(2) When the inventory holding cost ℎ increases the opti-mal joint replenishment cycle time 119879
lowast the optimalfreshness-keeping effort 120591
lowast and the total network
6 Journal of Applied Mathematics
Table 1 Effects of different system parameters
Parameter 119879lowast
119860lowast
1119860lowast
2119860lowast
3120591lowast
Πlowast
119865 = 80000 02356 344660 414737 516279 00294 4964300119865 = 100000 02346 399859 481160 598964 00293 4853230119865 = 120000 02340 451475 543270 676282 00292 4756130ℎ = 025 02876 399796 481084 598870 00360 4853460ℎ = 05 02346 399859 481160 598964 00293 4853230ℎ = 075 02031 399911 481223 599043 00254 484907119877 = 20 02327 399747 481025 598797 00291 4853460119877 = 30 02346 399859 481160 598964 00293 4853230119877 = 40 02365 399968 481292 599129 00296 4853020119888119907= 3 02519 399835 481131 598929 00189 5161330
119888119907= 5 02346 399859 481160 598964 00293 4853230
119888119907= 7 02205 399880 481186 598997 00386 4545270
119888119879= 5 02325 634741 763799 950804 00291 5427370
119888119879= 10 02346 399859 481160 598964 00293 4853230
119888119879= 15 02364 305147 367191 457092 00296 4371680
Table 2 Effects of a and b
a b005 0 1 015
2
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
1198603
lowast= 598964 119860
3
lowast= 598964 119860
3
lowast= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 5006370 Π
lowast= 5006350 Π
lowast= 5006350
3
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 4853250 Π
lowast= 4853230 Π
lowast= 4853230
4
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 4700130 Π
lowast= 4700110 Π
lowast= 4700110
profit Πlowast decrease but the optimal APMC service
area 119860lowast
119894increases It is reasonable that when the
inventory holding cost increases the company willdecrease the replenishment cycle time in an effortto lower inventory costs Also the freshness-keepingeffort can be decreased with a shorter replenish-ment cycle time They will also reduce the numberof APMCs to hold less inventory as the inventoryholding cost increases
Table 3 Effects of 120572 and 120573
120572120573
0005 001 0015
003
119879lowast= 02519 119879
lowast= 02521 119879
lowast= 02525
119860lowast
1= 399835 119860
lowast
1= 399835 119860
lowast
1= 399835
119860lowast
2= 481132 119860
lowast
2= 481131 119860
lowast
2= 481131
119860lowast
3= 598929 119860
lowast
3= 598929 119860
lowast
3= 598928
120591lowast= 00157 120591
lowast= 00315 120591
lowast= 00473
Πlowast= 4855080 Π
lowast= 4855100 Π
lowast= 4855110
005
119879lowast= 02345 119879
lowast= 02346 119879
lowast= 02349
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399858
119860lowast
2= 481160 119860
lowast
2= 481160 119860
lowast
2= 481159
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598963
120591lowast= 00147 120591
lowast= 00293 120591
lowast= 00440
Πlowast= 4853220 Π
lowast= 4853230 Π
lowast= 4853250
007
119879lowast= 02202 119879
lowast= 02204 119879
lowast= 02206
119860lowast
1= 399881 119860
lowast
1= 399881 119860
lowast
1= 399880
119860lowast
2= 481186 119860
lowast
2= 481186 119860
lowast
2= 481186
119860lowast
3= 598997 119860
lowast
3= 598997 119860
lowast
3= 598997
120591lowast= 00138 120591
lowast= 00275 120591
lowast= 00414
Πlowast= 4851490 Π
lowast= 4851490 Π
lowast= 4851510
(3) When the ordering cost 119877 increases the optimaljoint replenishment cycle time119879lowast the optimal APMCservice area 119860
lowast
119894 and the optimal freshness-keeping
effort 120591lowast all increase but the total network profit
Πlowast decreases This also verifies Property 1 (b) If
the ordering cost increases it is reasonable that thecompany will increase the replenishment cycle timeto reduce replenishment frequency The companywill also likely decrease the number of APMCs to
Journal of Applied Mathematics 7
018 022 024 026 028 03
Π
T
48532 times 106
48528 times 106
48526 times 106
48524 times 106
Figure 2 Graphic illustrations of Π versus 119879
reduce the replenishment times as the ordering costincreases
(4) When the variable inbound shipment cost 119862Vincreases the optimal joint replenishment cycle time119879lowast and the total network profit Π
lowast decrease butthe optimal APMC service area 119860
lowast
119894and the optimal
freshness-keeping effort 120591lowast increase This verifies
Property 2 (a) The number of APMCs will decreaseto reduce the inbound transportation times as thevariable inbound shipment cost increases
(5) When the outbound transportation cost119862119879increases
the optimal DC service area119860lowast
119894and the total network
profit Πlowast decrease but the optimal joint replenish-
ment cycle time119879lowast and the optimal freshness-keepingeffort 120591
lowast increase This also verifies Property 1 (a)When the outbound transportation cost increases itis reasonable to increase the joint replenishment cycletime to reduce the outbound transportation times
(6) When the freshness-keeping effort cost increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases It is reasonable to decrease
the freshness-keeping effort and the replenishmentcycle time when the freshness-keeping effort costincreases
(7) When the perishable rate of the fresh food increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases When the perishable rate
of the fresh food increases the company will decreasethe cycle time to reduce the deterioration of thefresh food When the freshness-keeping effort hasmore service on reducing the deterioration of thefresh food companies are willing to increase theirfreshness-keeping effort
5 Conclusions
This study considers a supply network design problem foragricultural productsThemodel can also be applied to other
fresh food products The agricultural produce marketingcorporations (APMCs) help consolidate shipments arrivingfrom farmersrsquo associations and deliver them to traditionalmarkets supermarkets and hypermarkets The proposedmethod integrates sales revenue purchasing costs facilitycosts inventory costs transportation costs ordering costsand freshness-keeping effort costs The key decisions includewhere to locate the APMCs how to assign retailers toAPMCs the joint replenishment cycle time at APMCs andthe freshness-keeping effort such that the total networkprofit is maximized An easy-to-use algorithm is providedfor solving the nonlinear programming problem Numericalstudies illustrate the solution procedures and the effectsof the facility cost inventory holding cost transportationcost ordering cost and perishable rate of the fresh food ondecisions and profits The results of this study are a usefulreference for managerial decision making and administra-tion Further research on this topic may consider when thefreshness-keeping effort has an influence on demand Thiswould include the market demand for fresh food dependenton the freshness of the product
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his gratitude to the editor and theanonymous reviewers for their detailed comments and valu-able suggestions to improve the exposition of this paper Thispaper is supported in part by the National Science Councilunder Grant NSC 102-2410-H-011-029-MY3 and 102-2221-E-011-159-MY3
References
[1] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005
[2] R K Apaiah and E M T Hendrix ldquoDesign of a supply chainnetwork for pea-based novel protein foodsrdquo Journal of FoodEngineering vol 70 no 3 pp 383ndash391 2005
[3] A Osvald and L Z Stirn ldquoA vehicle routing algorithm for thedistribution of fresh vegetables and similar perishable foodrdquoJournal of Food Engineering vol 85 no 2 pp 285ndash295 2008
[4] X Cai J Chen Y Xiao and X Xu ldquoOptimization andcoordination of fresh product supply chains with freshness-keeping effortrdquo Production and OperationsManagement vol 19no 3 pp 261ndash278 2010
[5] I-H Hong J-F Dang Y-H Tsai et al ldquoAn RFID applicationin the food supply chain a case study of convenience stores inTaiwanrdquo Journal of Food Engineering vol 106 no 2 pp 119ndash1262011
[6] XWang andD Li ldquoAdynamic product quality evaluation basedpricing model for perishable food supply chainsrdquo Omega vol40 no 6 pp 906ndash917 2012
8 Journal of Applied Mathematics
[7] X Cai J Chen Y Xiao X Xu andG Yu ldquoFresh-product supplychain management with logistics outsourcingrdquo Omega vol 41no 4 pp 752ndash765 2013
[8] Z-J M Shen ldquoIntegrated supply chain design models asurvey and future research directionsrdquo Journal of Industrial andManagement Optimization vol 3 no 1 pp 1ndash27 2007
[9] S Park T-E Lee and C S Sung ldquoA three-level supplychain network design model with risk-pooling and lead timesrdquoTransportation Research E vol 46 no 5 pp 563ndash581 2010
[10] A Nagurney ldquoOptimal supply chain network design andredesign at minimal total cost and with demand satisfactionrdquoInternational Journal of Production Economics vol 128 no 1 pp200ndash208 2010
[11] K M Bretthauer S Mahar and M A VenakataramananldquoInventory and distribution strategies for retaile-tail organiza-tionsrdquo Computers and Industrial Engineering vol 58 no 1 pp119ndash132 2010
[12] Y-C Tsao and J-C Lu ldquoA supply chain network design consid-ering transportation cost discountsrdquo Transportation Research Evol 48 no 2 pp 401ndash414 2012
[13] W Klibi and A Martel ldquoModeling approaches for the designof resilient supply networks under disruptionsrdquo InternationalJournal of Production Economics vol 135 no 2 pp 882ndash8982012
[14] S Doris ldquoKeeping freshness in fresh-cut productrdquo AgriculturalResearch vol 46 no 2 pp 12ndash14 1998
[15] A Dasci and V Verter ldquoA continuous model for production-distribution system designrdquo European Journal of OperationalResearch vol 129 no 2 pp 287ndash298 2001
[16] Y C Tsao D Mangotra J C Lu and M Dong ldquoA continuousapproximation approach for the integrated facility-inventoryallocation problemrdquo European Journal of Operational Researchvol 222 pp 216ndash228 2012
[17] W L Winston Operations Research Applications and Algo-rithms ThomsonBrooksCole Belmont Mass USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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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
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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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
Journal of Applied Mathematics 3
middot middot middot middot middot middot
middot middot middotmiddot middot middot
middot middot middot middot middot middot middot middot middot
Farmer Farmer Farmer
Farmersrsquo association
Agricultural products marketing corporation
Hypermarket Hypermarket Market Market Supermarket Supermarket
Consumers
Figure 1 Agricultural product supply chain network
119862119894 service area in cluster 119894
119901 unit selling price
119888 unit purchasing cost
This study uses an approximation technique [16] to dividethe network into smaller regions over which the discretevariable can be modeled using the slow varying functionsUsing the method the given service region is covered withclusters 119894 119894 = 1 2 119873 Clusters i 119894 = 1 2 119873 existwithin the given service region such that the store density isnearly constant over each cluster This model calculates thecomponents of the total network cost as follows
(1) The total revenue is sum119873119894=1
119901120585120582119894120575119894119862119894
(2) The total purchasing cost is sum119873119894=1
119888120585120582119894120575119894119862119894
(3) The total facility cost is given by multiplying thefacility cost of opening each APMC with the numberof APMC s namely sum119873
119894=1119865(119862119894119860119894)
(4) Let 119862119891
be the fixed cost per inbound shipmentand 119862V the variable cost per item for each inboundshipmentThen the total inbound transportation costis sum119873119894=1
(119862119891+ 119862V119876)119879
(5) Assuming ldquoclose to circularrdquo service regions with thefacility at the center the average distance traveled byeach item is 119891
119903radic119860119894[15] The total outbound trans-
portation cost is sum119873119894=1
119888119879119891119903radic119860119894120585120582119894120575119894119862119894
(6) The total cost for freshness-keeping effort is sum119873119894=1
(119886 +
1198871205912)120585120582119894120575119894119862119894
(7) The total ordering cost is sum119873119894=1
(119877119879)(119862119894119860119894)
(8) Depletion of inventory can occur both as a result ofdemand and due to deterioration during the replen-ishment cycle Variations in the inventory level 119868(119905)with respect to time 119905 can be expressed as
119889119868 (119905)
119889119905= minus120579119868 (119905) minus 120585120582
119894120575119894119862119894 0 le 119905 le 119879 (1)
with the boundary condition 119868(119879) = 0 The solutionto (1) can now be expressed as
119868 (119905) =120585120582119894120575119894119862119894
120579[119890120579(119879minus119905)
minus 1] 0 le 119905 le 119879 (2)
Note that the quantity ordered during each replenish-ment cycle is
119876 =120585120582119894120575119894119862119894
120579(119890120579119879
minus 1) =120585120582119894120575119894119862119894
120572 minus 120573120591[119890(120572minus120573120591)119879
minus 1] (3)
The total inventory holding cost is sum119873119894=1
(ℎ119879) int119879
0119868(119905)119889119905 =
sum119873
119894=1(ℎ120585120582119894120575119894119862119894120579119879) int
119879
0[119890120579(119879minus119905)
minus 1]119889119905 = sum119873
119894=1(ℎ120585120582119894120575119894119862119894(120572 minus
120573120591)2119879)[119890(120572minus120573120591)119879
minus (120572 minus 120573120591)119879 minus 1]
4 Journal of Applied Mathematics
Therefore the total network profit is
Π(1198601 1198602 119860
119873 119879 120591)
=
119873
sum
119894=1
(119901 minus 119888) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119865119862119894
119860119894
+
119873
sum
119894=1
119888119879119891119903radic119860119894120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119862119891+ 119862V120585120582119894120575119894119862119894 [119890
(120572minus120573120591)119879minus 1] (120572 minus 120573120591)
119879
minus
119873
sum
119894=1
(119886 + 1198871205912) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119877
119879
119862119894
119860119894
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894
(120572 minus 120573120591)2119879
[119890(120572minus120573120591)119879
minus (120572 minus 120573120591) 119879 minus 1]
(4)
3 Decision-Making Approach
By using a truncated Taylor series expansion (119890120582119879 asymp 1 + 120582119879 +
(12)12058221198792) we can rewrite (4) as follows
Π(1198601 1198602 119860
119873 119879 120591)
asymp
119873
sum
119894=1
(119901 minus 119888) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119865119862119894
119860119894
+
119873
sum
119894=1
119888119879119891119903radic119860119894120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119862119891
119879+ 119862V120585120582119894120575119894119862119894 [1 +
(120572 minus 120573120591) 119879
2]
minus
119873
sum
119894=1
(119886 + 1198871205912) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119877
119879
119862119894
119860119894
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894119879
2
(5)
The problem is to determine the optimal service areareplenishment cycle time and freshness-keeping effort tomaximize the profit function in (5)
1205972Π
1205971198602
119894
= minus (2119865119862119894
1198603
119894
) minus (2119877119862119894
1198791198603
119894
)
minus (minus119888119879119891119903120585120582119894120575119894119862119894
411986032
119894
) 119894 = 1 2 119873
1205972Π
1205971205912= minus
119873
sum
119894=1
2119887120585120582119894120575119894119862119894lt 0
(6)
Since the facility opening cost 119865 is large 1205972Π1205971198602
119894lt 0
is satisfied in general case Then from 1205972Π120597119860
119894120597120591 = 0 and
1205972Π120597119860
119894120597119860119896
= 0 119894 = 1 2 119873 119896 = 1 2 119873 119894 = 119896 theHessian matrix is
119867119873+1
=
[[[[[[[[[[[[[[[[[[[[
[
1205972Π
1205971198602
1
0 0 0
01205972Π
1205971198602
2
0
0
0 01205972Π
1205971198602
119873
0
0 0 01205972Π
1205971205912
]]]]]]]]]]]]]]]]]]]]
]
(7)
Since 1205972Π120597119860
2
119894lt 0 and 120597
2Π1205971205912lt 0 for 119894 = 1 2 119873
we know that (minus1)119895sdot |119863119895| gt 0 is satisfied for all 119895 119895 =
1 2 119873 + 1 From the maximum theorem Winston [17]we know that Π(119860
1 1198602 119860
119873 120591 | 119879) is a concave function
of (1198601 1198602 119860
119873 120591)Thismeans that the optimal119860
119894(119879) and
120591(119879) can be obtained by solving 120597Π120597119860119894= 0 and 120597Π120597120591 = 0
119894 = 1 2 119873 To solve this problem we first determine theclosed form of the retail price 119860
119894(119879) and 120591(119879) that maximize
Π(1198601 1198602 119860
119873 120591 | 119879) Solving 120597Π120597119860
119894= 0 and 120597Π120597120591 =
0 simultaneously leads to
119860lowast
119894(119879) = [
2(119877 + 119865119879)
1198621198791198911199031205851205821198941205751198941198792
]
23
(8)
120591lowast(119879) =
119862V120573119879
4119887 (9)
Equations (8) and (9) leads to Properties 1 and 2
Property 1 (a) The service area for each APMC 119860119894increases
as the joint replenishment cycle time 119879 or the outboundtransportation cost 119862
119879decreases
(b) The service area for each APMC 119860119894increases as the
facility cost 119865 or the ordering cost 119877 increases
Property 2 (a) The freshness-keeping effort 120591 increases asthe joint replenishment cycle time 119879 the perishable rateparameter 120573 or the variable inbound shipment cost 119862Vincreases
(b) The freshness-keeping effort 120591 increases as thefreshness-keeping effort cost parameter 119887 decreases
Substituting 120591lowast(119879) and 119860
lowast
119894(119879) 119894 = 1 2 119873 into the
corresponding Π(1198601 1198602 119860
119873 119879 120591) reduces the model to
a single variable function Π(119879)
Π (119879) =
119873
sum
119894=1
(119901 minus 119888) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119865119862119894[1198621198791198911199031205851205821198941205751198941198792
2(119877 + 119865119879)]
23
+
119873
sum
119894=1
119888119879119891119903[
2(119877 + 119865119879)
1198621198791198911199031205851205821198941205751198941198792
]
13
120585120582119894120575119894119862119894
Journal of Applied Mathematics 5
Step 1 Find an initial solution 119879119894=1
by inspectionStep 2 Calculate Π
1015840(119879) and Π
10158401015840(119879)
Step 3 Let 119879119894+1
= 119879119894minus
Π1015840(119879)
Π10158401015840(119879)
Step 4 If 1003816100381610038161003816119879119894+1 minus 119879119894
1003816100381610038161003816 le 120576 where 120576 is the tolerance error then go to Step 5 otherwise let 119894 = 119894 + 1 and go to Step 3Step 5 Let 119879lowast = 119879
119894+1and Π
lowast(119879lowast) = Π(119879
119894+1)
Step 6 Determine 119860lowast
119894(119879lowast) and 120591
lowast(119879lowast) by (8) and (9) respectively
Algorithm 1
minus
119873
sum
119894=1
119862119891
119879+ 119862V120585120582119894120575119894119862119894
times[1 + (120572 minus 120573
119873
sum
119894=1
119862V120585120582119894120575119894119862119894120573119879
4119887120585120582119894120575119894119862119894
)119879
2]
minus
119873
sum
119894=1
[
[
119886 + 119887(
119873
sum
119894=1
119862V120585120582119894120575119894119862119894120573119879
4119887120585120582119894120575119894119862119894
)
2
]
]
120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119877119862119894
119879[1198621198791198911199031205851205821198941205751198941198792
2 (119877 + 119865119879)]
23
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894119879
2
(10)
Based on the discussion above Algorithm 1 based onNewtonrsquos method determines the optimal values for 119879
lowast 119860lowast
119894
and 120591lowast
4 Numerical Study
This section presents a numerical study to illustrate the pro-posed solution approach and provide quantitative insightsThe goals of the numerical study in this study are as follows
(1) to illustrate the procedures of the solution approach(2) to discuss the effects of the related parameters on deci-
sions and cost
41 Numerical Example To illustrate the algorithmdescribedabove we applied our procedure to an agricultural productdistribution system in Taiwan Since Taiwan joined theWorldTrade Organization (WTO) local agriculture has facedgreater competition due to the reduction of customs dutiesand international trade Therefore having an efficient agri-cultural food supply network is important for Taiwan Asshown in Figure 1 farmers sell their agricultural products toagricultural associations Then the agricultural product isdistributed from agricultural associations to agriculturalproduct-marketing corporations (APMCs)TheAPMCs con-solidate shipments arriving from the agricultural associationsand deliver them to the retailers (traditional markets super-markets and hypermarkets) We consider three cities TaipeiCity New Taipei City and Keelung City in northern Taiwanthey are Cluster 1 Cluster 2 and Cluster 3 respectively
in our example In this case we wanted to determine theoptimal APMC service areas the optimal number of APMCsin each cluster the optimal freshness-keeping effort andthe optimal joint replenishment cycle time In practice therelated parameters can be determined by regression analysisusing historical transaction data Please note that the dataused here have been scaled to protect confidentiality Pleasealso note that the freshness-keeping effort cost just needsto satisfy the requirement The freshness-keeping effort costis not only convex but also increasing in the freshness-keeping effort One may get other freshness-keeping effortcost form from the regression analysis However our modeland solution approach can also be used if the freshness-keeping effort cost satisfies the above requirement
The parameters of the product are 119865 = 100000 119901 = 100119888 = 50 119862
1= 10000 119862
2= 8000 119862
3= 6000 ℎ = 1 119877 = 30
119862119879
= 10 1205821= 11 120582
2= 10 120582
3= 9 120585 = 12 120575
1= 006 120575
2=
005 1205753= 004 119891
119903= 001 119862
119891= 1000 119862V = 5 120572 = 005 120573 =
001 119886 = 3 119887 = 01 After applying the algorithm in Section 3the optimal joint replenishment cycle time is119879lowast = 02346 theoptimal APMC service area in cluster119862
1is119860lowast1= 399859 the
optimal APMC service area in cluster1198622is119860lowast2= 481160 the
optimal APMC service area in cluster1198623is119860lowast3= 598964 the
optimal freshness-keeping effort is 120591lowast = 00293 and the totalnetwork profit is Πlowast = 4853230 Figure 2 shows the graphicillustrations of Π versus different 119879 The algorithm can findthe optimal solution in a very short time
42 Numerical Analysis Several numerical analyses are con-ducted to gain management insights into the structures ofthe proposed policies The following numerical analyses areused to show the effects of 119865 ℎ 119877 119862V 119862119879 119886 119887 120572 and 120573 on theoptimal decisions and the total network profit The results ofTables 1 2 and 3 are as follows
(1) When the facility cost 119865 increases the optimal jointreplenishment cycle time 119879
lowast the optimal freshness-keeping effort 120591
lowast and the total network profit Πlowast
decrease but the optimal APMC service area 119860lowast
119894
increases This verifies Property 1 (b) When thefacility cost increases it is reasonable to increase theAPMC service area to reduce the number of APMCsopening
(2) When the inventory holding cost ℎ increases the opti-mal joint replenishment cycle time 119879
lowast the optimalfreshness-keeping effort 120591
lowast and the total network
6 Journal of Applied Mathematics
Table 1 Effects of different system parameters
Parameter 119879lowast
119860lowast
1119860lowast
2119860lowast
3120591lowast
Πlowast
119865 = 80000 02356 344660 414737 516279 00294 4964300119865 = 100000 02346 399859 481160 598964 00293 4853230119865 = 120000 02340 451475 543270 676282 00292 4756130ℎ = 025 02876 399796 481084 598870 00360 4853460ℎ = 05 02346 399859 481160 598964 00293 4853230ℎ = 075 02031 399911 481223 599043 00254 484907119877 = 20 02327 399747 481025 598797 00291 4853460119877 = 30 02346 399859 481160 598964 00293 4853230119877 = 40 02365 399968 481292 599129 00296 4853020119888119907= 3 02519 399835 481131 598929 00189 5161330
119888119907= 5 02346 399859 481160 598964 00293 4853230
119888119907= 7 02205 399880 481186 598997 00386 4545270
119888119879= 5 02325 634741 763799 950804 00291 5427370
119888119879= 10 02346 399859 481160 598964 00293 4853230
119888119879= 15 02364 305147 367191 457092 00296 4371680
Table 2 Effects of a and b
a b005 0 1 015
2
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
1198603
lowast= 598964 119860
3
lowast= 598964 119860
3
lowast= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 5006370 Π
lowast= 5006350 Π
lowast= 5006350
3
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 4853250 Π
lowast= 4853230 Π
lowast= 4853230
4
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 4700130 Π
lowast= 4700110 Π
lowast= 4700110
profit Πlowast decrease but the optimal APMC service
area 119860lowast
119894increases It is reasonable that when the
inventory holding cost increases the company willdecrease the replenishment cycle time in an effortto lower inventory costs Also the freshness-keepingeffort can be decreased with a shorter replenish-ment cycle time They will also reduce the numberof APMCs to hold less inventory as the inventoryholding cost increases
Table 3 Effects of 120572 and 120573
120572120573
0005 001 0015
003
119879lowast= 02519 119879
lowast= 02521 119879
lowast= 02525
119860lowast
1= 399835 119860
lowast
1= 399835 119860
lowast
1= 399835
119860lowast
2= 481132 119860
lowast
2= 481131 119860
lowast
2= 481131
119860lowast
3= 598929 119860
lowast
3= 598929 119860
lowast
3= 598928
120591lowast= 00157 120591
lowast= 00315 120591
lowast= 00473
Πlowast= 4855080 Π
lowast= 4855100 Π
lowast= 4855110
005
119879lowast= 02345 119879
lowast= 02346 119879
lowast= 02349
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399858
119860lowast
2= 481160 119860
lowast
2= 481160 119860
lowast
2= 481159
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598963
120591lowast= 00147 120591
lowast= 00293 120591
lowast= 00440
Πlowast= 4853220 Π
lowast= 4853230 Π
lowast= 4853250
007
119879lowast= 02202 119879
lowast= 02204 119879
lowast= 02206
119860lowast
1= 399881 119860
lowast
1= 399881 119860
lowast
1= 399880
119860lowast
2= 481186 119860
lowast
2= 481186 119860
lowast
2= 481186
119860lowast
3= 598997 119860
lowast
3= 598997 119860
lowast
3= 598997
120591lowast= 00138 120591
lowast= 00275 120591
lowast= 00414
Πlowast= 4851490 Π
lowast= 4851490 Π
lowast= 4851510
(3) When the ordering cost 119877 increases the optimaljoint replenishment cycle time119879lowast the optimal APMCservice area 119860
lowast
119894 and the optimal freshness-keeping
effort 120591lowast all increase but the total network profit
Πlowast decreases This also verifies Property 1 (b) If
the ordering cost increases it is reasonable that thecompany will increase the replenishment cycle timeto reduce replenishment frequency The companywill also likely decrease the number of APMCs to
Journal of Applied Mathematics 7
018 022 024 026 028 03
Π
T
48532 times 106
48528 times 106
48526 times 106
48524 times 106
Figure 2 Graphic illustrations of Π versus 119879
reduce the replenishment times as the ordering costincreases
(4) When the variable inbound shipment cost 119862Vincreases the optimal joint replenishment cycle time119879lowast and the total network profit Π
lowast decrease butthe optimal APMC service area 119860
lowast
119894and the optimal
freshness-keeping effort 120591lowast increase This verifies
Property 2 (a) The number of APMCs will decreaseto reduce the inbound transportation times as thevariable inbound shipment cost increases
(5) When the outbound transportation cost119862119879increases
the optimal DC service area119860lowast
119894and the total network
profit Πlowast decrease but the optimal joint replenish-
ment cycle time119879lowast and the optimal freshness-keepingeffort 120591
lowast increase This also verifies Property 1 (a)When the outbound transportation cost increases itis reasonable to increase the joint replenishment cycletime to reduce the outbound transportation times
(6) When the freshness-keeping effort cost increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases It is reasonable to decrease
the freshness-keeping effort and the replenishmentcycle time when the freshness-keeping effort costincreases
(7) When the perishable rate of the fresh food increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases When the perishable rate
of the fresh food increases the company will decreasethe cycle time to reduce the deterioration of thefresh food When the freshness-keeping effort hasmore service on reducing the deterioration of thefresh food companies are willing to increase theirfreshness-keeping effort
5 Conclusions
This study considers a supply network design problem foragricultural productsThemodel can also be applied to other
fresh food products The agricultural produce marketingcorporations (APMCs) help consolidate shipments arrivingfrom farmersrsquo associations and deliver them to traditionalmarkets supermarkets and hypermarkets The proposedmethod integrates sales revenue purchasing costs facilitycosts inventory costs transportation costs ordering costsand freshness-keeping effort costs The key decisions includewhere to locate the APMCs how to assign retailers toAPMCs the joint replenishment cycle time at APMCs andthe freshness-keeping effort such that the total networkprofit is maximized An easy-to-use algorithm is providedfor solving the nonlinear programming problem Numericalstudies illustrate the solution procedures and the effectsof the facility cost inventory holding cost transportationcost ordering cost and perishable rate of the fresh food ondecisions and profits The results of this study are a usefulreference for managerial decision making and administra-tion Further research on this topic may consider when thefreshness-keeping effort has an influence on demand Thiswould include the market demand for fresh food dependenton the freshness of the product
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his gratitude to the editor and theanonymous reviewers for their detailed comments and valu-able suggestions to improve the exposition of this paper Thispaper is supported in part by the National Science Councilunder Grant NSC 102-2410-H-011-029-MY3 and 102-2221-E-011-159-MY3
References
[1] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005
[2] R K Apaiah and E M T Hendrix ldquoDesign of a supply chainnetwork for pea-based novel protein foodsrdquo Journal of FoodEngineering vol 70 no 3 pp 383ndash391 2005
[3] A Osvald and L Z Stirn ldquoA vehicle routing algorithm for thedistribution of fresh vegetables and similar perishable foodrdquoJournal of Food Engineering vol 85 no 2 pp 285ndash295 2008
[4] X Cai J Chen Y Xiao and X Xu ldquoOptimization andcoordination of fresh product supply chains with freshness-keeping effortrdquo Production and OperationsManagement vol 19no 3 pp 261ndash278 2010
[5] I-H Hong J-F Dang Y-H Tsai et al ldquoAn RFID applicationin the food supply chain a case study of convenience stores inTaiwanrdquo Journal of Food Engineering vol 106 no 2 pp 119ndash1262011
[6] XWang andD Li ldquoAdynamic product quality evaluation basedpricing model for perishable food supply chainsrdquo Omega vol40 no 6 pp 906ndash917 2012
8 Journal of Applied Mathematics
[7] X Cai J Chen Y Xiao X Xu andG Yu ldquoFresh-product supplychain management with logistics outsourcingrdquo Omega vol 41no 4 pp 752ndash765 2013
[8] Z-J M Shen ldquoIntegrated supply chain design models asurvey and future research directionsrdquo Journal of Industrial andManagement Optimization vol 3 no 1 pp 1ndash27 2007
[9] S Park T-E Lee and C S Sung ldquoA three-level supplychain network design model with risk-pooling and lead timesrdquoTransportation Research E vol 46 no 5 pp 563ndash581 2010
[10] A Nagurney ldquoOptimal supply chain network design andredesign at minimal total cost and with demand satisfactionrdquoInternational Journal of Production Economics vol 128 no 1 pp200ndash208 2010
[11] K M Bretthauer S Mahar and M A VenakataramananldquoInventory and distribution strategies for retaile-tail organiza-tionsrdquo Computers and Industrial Engineering vol 58 no 1 pp119ndash132 2010
[12] Y-C Tsao and J-C Lu ldquoA supply chain network design consid-ering transportation cost discountsrdquo Transportation Research Evol 48 no 2 pp 401ndash414 2012
[13] W Klibi and A Martel ldquoModeling approaches for the designof resilient supply networks under disruptionsrdquo InternationalJournal of Production Economics vol 135 no 2 pp 882ndash8982012
[14] S Doris ldquoKeeping freshness in fresh-cut productrdquo AgriculturalResearch vol 46 no 2 pp 12ndash14 1998
[15] A Dasci and V Verter ldquoA continuous model for production-distribution system designrdquo European Journal of OperationalResearch vol 129 no 2 pp 287ndash298 2001
[16] Y C Tsao D Mangotra J C Lu and M Dong ldquoA continuousapproximation approach for the integrated facility-inventoryallocation problemrdquo European Journal of Operational Researchvol 222 pp 216ndash228 2012
[17] W L Winston Operations Research Applications and Algo-rithms ThomsonBrooksCole Belmont Mass USA 2004
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
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
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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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
4 Journal of Applied Mathematics
Therefore the total network profit is
Π(1198601 1198602 119860
119873 119879 120591)
=
119873
sum
119894=1
(119901 minus 119888) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119865119862119894
119860119894
+
119873
sum
119894=1
119888119879119891119903radic119860119894120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119862119891+ 119862V120585120582119894120575119894119862119894 [119890
(120572minus120573120591)119879minus 1] (120572 minus 120573120591)
119879
minus
119873
sum
119894=1
(119886 + 1198871205912) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119877
119879
119862119894
119860119894
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894
(120572 minus 120573120591)2119879
[119890(120572minus120573120591)119879
minus (120572 minus 120573120591) 119879 minus 1]
(4)
3 Decision-Making Approach
By using a truncated Taylor series expansion (119890120582119879 asymp 1 + 120582119879 +
(12)12058221198792) we can rewrite (4) as follows
Π(1198601 1198602 119860
119873 119879 120591)
asymp
119873
sum
119894=1
(119901 minus 119888) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119865119862119894
119860119894
+
119873
sum
119894=1
119888119879119891119903radic119860119894120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119862119891
119879+ 119862V120585120582119894120575119894119862119894 [1 +
(120572 minus 120573120591) 119879
2]
minus
119873
sum
119894=1
(119886 + 1198871205912) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119877
119879
119862119894
119860119894
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894119879
2
(5)
The problem is to determine the optimal service areareplenishment cycle time and freshness-keeping effort tomaximize the profit function in (5)
1205972Π
1205971198602
119894
= minus (2119865119862119894
1198603
119894
) minus (2119877119862119894
1198791198603
119894
)
minus (minus119888119879119891119903120585120582119894120575119894119862119894
411986032
119894
) 119894 = 1 2 119873
1205972Π
1205971205912= minus
119873
sum
119894=1
2119887120585120582119894120575119894119862119894lt 0
(6)
Since the facility opening cost 119865 is large 1205972Π1205971198602
119894lt 0
is satisfied in general case Then from 1205972Π120597119860
119894120597120591 = 0 and
1205972Π120597119860
119894120597119860119896
= 0 119894 = 1 2 119873 119896 = 1 2 119873 119894 = 119896 theHessian matrix is
119867119873+1
=
[[[[[[[[[[[[[[[[[[[[
[
1205972Π
1205971198602
1
0 0 0
01205972Π
1205971198602
2
0
0
0 01205972Π
1205971198602
119873
0
0 0 01205972Π
1205971205912
]]]]]]]]]]]]]]]]]]]]
]
(7)
Since 1205972Π120597119860
2
119894lt 0 and 120597
2Π1205971205912lt 0 for 119894 = 1 2 119873
we know that (minus1)119895sdot |119863119895| gt 0 is satisfied for all 119895 119895 =
1 2 119873 + 1 From the maximum theorem Winston [17]we know that Π(119860
1 1198602 119860
119873 120591 | 119879) is a concave function
of (1198601 1198602 119860
119873 120591)Thismeans that the optimal119860
119894(119879) and
120591(119879) can be obtained by solving 120597Π120597119860119894= 0 and 120597Π120597120591 = 0
119894 = 1 2 119873 To solve this problem we first determine theclosed form of the retail price 119860
119894(119879) and 120591(119879) that maximize
Π(1198601 1198602 119860
119873 120591 | 119879) Solving 120597Π120597119860
119894= 0 and 120597Π120597120591 =
0 simultaneously leads to
119860lowast
119894(119879) = [
2(119877 + 119865119879)
1198621198791198911199031205851205821198941205751198941198792
]
23
(8)
120591lowast(119879) =
119862V120573119879
4119887 (9)
Equations (8) and (9) leads to Properties 1 and 2
Property 1 (a) The service area for each APMC 119860119894increases
as the joint replenishment cycle time 119879 or the outboundtransportation cost 119862
119879decreases
(b) The service area for each APMC 119860119894increases as the
facility cost 119865 or the ordering cost 119877 increases
Property 2 (a) The freshness-keeping effort 120591 increases asthe joint replenishment cycle time 119879 the perishable rateparameter 120573 or the variable inbound shipment cost 119862Vincreases
(b) The freshness-keeping effort 120591 increases as thefreshness-keeping effort cost parameter 119887 decreases
Substituting 120591lowast(119879) and 119860
lowast
119894(119879) 119894 = 1 2 119873 into the
corresponding Π(1198601 1198602 119860
119873 119879 120591) reduces the model to
a single variable function Π(119879)
Π (119879) =
119873
sum
119894=1
(119901 minus 119888) 120585120582119894120575119894119862119894minus
119873
sum
119894=1
119865119862119894[1198621198791198911199031205851205821198941205751198941198792
2(119877 + 119865119879)]
23
+
119873
sum
119894=1
119888119879119891119903[
2(119877 + 119865119879)
1198621198791198911199031205851205821198941205751198941198792
]
13
120585120582119894120575119894119862119894
Journal of Applied Mathematics 5
Step 1 Find an initial solution 119879119894=1
by inspectionStep 2 Calculate Π
1015840(119879) and Π
10158401015840(119879)
Step 3 Let 119879119894+1
= 119879119894minus
Π1015840(119879)
Π10158401015840(119879)
Step 4 If 1003816100381610038161003816119879119894+1 minus 119879119894
1003816100381610038161003816 le 120576 where 120576 is the tolerance error then go to Step 5 otherwise let 119894 = 119894 + 1 and go to Step 3Step 5 Let 119879lowast = 119879
119894+1and Π
lowast(119879lowast) = Π(119879
119894+1)
Step 6 Determine 119860lowast
119894(119879lowast) and 120591
lowast(119879lowast) by (8) and (9) respectively
Algorithm 1
minus
119873
sum
119894=1
119862119891
119879+ 119862V120585120582119894120575119894119862119894
times[1 + (120572 minus 120573
119873
sum
119894=1
119862V120585120582119894120575119894119862119894120573119879
4119887120585120582119894120575119894119862119894
)119879
2]
minus
119873
sum
119894=1
[
[
119886 + 119887(
119873
sum
119894=1
119862V120585120582119894120575119894119862119894120573119879
4119887120585120582119894120575119894119862119894
)
2
]
]
120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119877119862119894
119879[1198621198791198911199031205851205821198941205751198941198792
2 (119877 + 119865119879)]
23
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894119879
2
(10)
Based on the discussion above Algorithm 1 based onNewtonrsquos method determines the optimal values for 119879
lowast 119860lowast
119894
and 120591lowast
4 Numerical Study
This section presents a numerical study to illustrate the pro-posed solution approach and provide quantitative insightsThe goals of the numerical study in this study are as follows
(1) to illustrate the procedures of the solution approach(2) to discuss the effects of the related parameters on deci-
sions and cost
41 Numerical Example To illustrate the algorithmdescribedabove we applied our procedure to an agricultural productdistribution system in Taiwan Since Taiwan joined theWorldTrade Organization (WTO) local agriculture has facedgreater competition due to the reduction of customs dutiesand international trade Therefore having an efficient agri-cultural food supply network is important for Taiwan Asshown in Figure 1 farmers sell their agricultural products toagricultural associations Then the agricultural product isdistributed from agricultural associations to agriculturalproduct-marketing corporations (APMCs)TheAPMCs con-solidate shipments arriving from the agricultural associationsand deliver them to the retailers (traditional markets super-markets and hypermarkets) We consider three cities TaipeiCity New Taipei City and Keelung City in northern Taiwanthey are Cluster 1 Cluster 2 and Cluster 3 respectively
in our example In this case we wanted to determine theoptimal APMC service areas the optimal number of APMCsin each cluster the optimal freshness-keeping effort andthe optimal joint replenishment cycle time In practice therelated parameters can be determined by regression analysisusing historical transaction data Please note that the dataused here have been scaled to protect confidentiality Pleasealso note that the freshness-keeping effort cost just needsto satisfy the requirement The freshness-keeping effort costis not only convex but also increasing in the freshness-keeping effort One may get other freshness-keeping effortcost form from the regression analysis However our modeland solution approach can also be used if the freshness-keeping effort cost satisfies the above requirement
The parameters of the product are 119865 = 100000 119901 = 100119888 = 50 119862
1= 10000 119862
2= 8000 119862
3= 6000 ℎ = 1 119877 = 30
119862119879
= 10 1205821= 11 120582
2= 10 120582
3= 9 120585 = 12 120575
1= 006 120575
2=
005 1205753= 004 119891
119903= 001 119862
119891= 1000 119862V = 5 120572 = 005 120573 =
001 119886 = 3 119887 = 01 After applying the algorithm in Section 3the optimal joint replenishment cycle time is119879lowast = 02346 theoptimal APMC service area in cluster119862
1is119860lowast1= 399859 the
optimal APMC service area in cluster1198622is119860lowast2= 481160 the
optimal APMC service area in cluster1198623is119860lowast3= 598964 the
optimal freshness-keeping effort is 120591lowast = 00293 and the totalnetwork profit is Πlowast = 4853230 Figure 2 shows the graphicillustrations of Π versus different 119879 The algorithm can findthe optimal solution in a very short time
42 Numerical Analysis Several numerical analyses are con-ducted to gain management insights into the structures ofthe proposed policies The following numerical analyses areused to show the effects of 119865 ℎ 119877 119862V 119862119879 119886 119887 120572 and 120573 on theoptimal decisions and the total network profit The results ofTables 1 2 and 3 are as follows
(1) When the facility cost 119865 increases the optimal jointreplenishment cycle time 119879
lowast the optimal freshness-keeping effort 120591
lowast and the total network profit Πlowast
decrease but the optimal APMC service area 119860lowast
119894
increases This verifies Property 1 (b) When thefacility cost increases it is reasonable to increase theAPMC service area to reduce the number of APMCsopening
(2) When the inventory holding cost ℎ increases the opti-mal joint replenishment cycle time 119879
lowast the optimalfreshness-keeping effort 120591
lowast and the total network
6 Journal of Applied Mathematics
Table 1 Effects of different system parameters
Parameter 119879lowast
119860lowast
1119860lowast
2119860lowast
3120591lowast
Πlowast
119865 = 80000 02356 344660 414737 516279 00294 4964300119865 = 100000 02346 399859 481160 598964 00293 4853230119865 = 120000 02340 451475 543270 676282 00292 4756130ℎ = 025 02876 399796 481084 598870 00360 4853460ℎ = 05 02346 399859 481160 598964 00293 4853230ℎ = 075 02031 399911 481223 599043 00254 484907119877 = 20 02327 399747 481025 598797 00291 4853460119877 = 30 02346 399859 481160 598964 00293 4853230119877 = 40 02365 399968 481292 599129 00296 4853020119888119907= 3 02519 399835 481131 598929 00189 5161330
119888119907= 5 02346 399859 481160 598964 00293 4853230
119888119907= 7 02205 399880 481186 598997 00386 4545270
119888119879= 5 02325 634741 763799 950804 00291 5427370
119888119879= 10 02346 399859 481160 598964 00293 4853230
119888119879= 15 02364 305147 367191 457092 00296 4371680
Table 2 Effects of a and b
a b005 0 1 015
2
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
1198603
lowast= 598964 119860
3
lowast= 598964 119860
3
lowast= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 5006370 Π
lowast= 5006350 Π
lowast= 5006350
3
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 4853250 Π
lowast= 4853230 Π
lowast= 4853230
4
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 4700130 Π
lowast= 4700110 Π
lowast= 4700110
profit Πlowast decrease but the optimal APMC service
area 119860lowast
119894increases It is reasonable that when the
inventory holding cost increases the company willdecrease the replenishment cycle time in an effortto lower inventory costs Also the freshness-keepingeffort can be decreased with a shorter replenish-ment cycle time They will also reduce the numberof APMCs to hold less inventory as the inventoryholding cost increases
Table 3 Effects of 120572 and 120573
120572120573
0005 001 0015
003
119879lowast= 02519 119879
lowast= 02521 119879
lowast= 02525
119860lowast
1= 399835 119860
lowast
1= 399835 119860
lowast
1= 399835
119860lowast
2= 481132 119860
lowast
2= 481131 119860
lowast
2= 481131
119860lowast
3= 598929 119860
lowast
3= 598929 119860
lowast
3= 598928
120591lowast= 00157 120591
lowast= 00315 120591
lowast= 00473
Πlowast= 4855080 Π
lowast= 4855100 Π
lowast= 4855110
005
119879lowast= 02345 119879
lowast= 02346 119879
lowast= 02349
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399858
119860lowast
2= 481160 119860
lowast
2= 481160 119860
lowast
2= 481159
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598963
120591lowast= 00147 120591
lowast= 00293 120591
lowast= 00440
Πlowast= 4853220 Π
lowast= 4853230 Π
lowast= 4853250
007
119879lowast= 02202 119879
lowast= 02204 119879
lowast= 02206
119860lowast
1= 399881 119860
lowast
1= 399881 119860
lowast
1= 399880
119860lowast
2= 481186 119860
lowast
2= 481186 119860
lowast
2= 481186
119860lowast
3= 598997 119860
lowast
3= 598997 119860
lowast
3= 598997
120591lowast= 00138 120591
lowast= 00275 120591
lowast= 00414
Πlowast= 4851490 Π
lowast= 4851490 Π
lowast= 4851510
(3) When the ordering cost 119877 increases the optimaljoint replenishment cycle time119879lowast the optimal APMCservice area 119860
lowast
119894 and the optimal freshness-keeping
effort 120591lowast all increase but the total network profit
Πlowast decreases This also verifies Property 1 (b) If
the ordering cost increases it is reasonable that thecompany will increase the replenishment cycle timeto reduce replenishment frequency The companywill also likely decrease the number of APMCs to
Journal of Applied Mathematics 7
018 022 024 026 028 03
Π
T
48532 times 106
48528 times 106
48526 times 106
48524 times 106
Figure 2 Graphic illustrations of Π versus 119879
reduce the replenishment times as the ordering costincreases
(4) When the variable inbound shipment cost 119862Vincreases the optimal joint replenishment cycle time119879lowast and the total network profit Π
lowast decrease butthe optimal APMC service area 119860
lowast
119894and the optimal
freshness-keeping effort 120591lowast increase This verifies
Property 2 (a) The number of APMCs will decreaseto reduce the inbound transportation times as thevariable inbound shipment cost increases
(5) When the outbound transportation cost119862119879increases
the optimal DC service area119860lowast
119894and the total network
profit Πlowast decrease but the optimal joint replenish-
ment cycle time119879lowast and the optimal freshness-keepingeffort 120591
lowast increase This also verifies Property 1 (a)When the outbound transportation cost increases itis reasonable to increase the joint replenishment cycletime to reduce the outbound transportation times
(6) When the freshness-keeping effort cost increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases It is reasonable to decrease
the freshness-keeping effort and the replenishmentcycle time when the freshness-keeping effort costincreases
(7) When the perishable rate of the fresh food increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases When the perishable rate
of the fresh food increases the company will decreasethe cycle time to reduce the deterioration of thefresh food When the freshness-keeping effort hasmore service on reducing the deterioration of thefresh food companies are willing to increase theirfreshness-keeping effort
5 Conclusions
This study considers a supply network design problem foragricultural productsThemodel can also be applied to other
fresh food products The agricultural produce marketingcorporations (APMCs) help consolidate shipments arrivingfrom farmersrsquo associations and deliver them to traditionalmarkets supermarkets and hypermarkets The proposedmethod integrates sales revenue purchasing costs facilitycosts inventory costs transportation costs ordering costsand freshness-keeping effort costs The key decisions includewhere to locate the APMCs how to assign retailers toAPMCs the joint replenishment cycle time at APMCs andthe freshness-keeping effort such that the total networkprofit is maximized An easy-to-use algorithm is providedfor solving the nonlinear programming problem Numericalstudies illustrate the solution procedures and the effectsof the facility cost inventory holding cost transportationcost ordering cost and perishable rate of the fresh food ondecisions and profits The results of this study are a usefulreference for managerial decision making and administra-tion Further research on this topic may consider when thefreshness-keeping effort has an influence on demand Thiswould include the market demand for fresh food dependenton the freshness of the product
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his gratitude to the editor and theanonymous reviewers for their detailed comments and valu-able suggestions to improve the exposition of this paper Thispaper is supported in part by the National Science Councilunder Grant NSC 102-2410-H-011-029-MY3 and 102-2221-E-011-159-MY3
References
[1] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005
[2] R K Apaiah and E M T Hendrix ldquoDesign of a supply chainnetwork for pea-based novel protein foodsrdquo Journal of FoodEngineering vol 70 no 3 pp 383ndash391 2005
[3] A Osvald and L Z Stirn ldquoA vehicle routing algorithm for thedistribution of fresh vegetables and similar perishable foodrdquoJournal of Food Engineering vol 85 no 2 pp 285ndash295 2008
[4] X Cai J Chen Y Xiao and X Xu ldquoOptimization andcoordination of fresh product supply chains with freshness-keeping effortrdquo Production and OperationsManagement vol 19no 3 pp 261ndash278 2010
[5] I-H Hong J-F Dang Y-H Tsai et al ldquoAn RFID applicationin the food supply chain a case study of convenience stores inTaiwanrdquo Journal of Food Engineering vol 106 no 2 pp 119ndash1262011
[6] XWang andD Li ldquoAdynamic product quality evaluation basedpricing model for perishable food supply chainsrdquo Omega vol40 no 6 pp 906ndash917 2012
8 Journal of Applied Mathematics
[7] X Cai J Chen Y Xiao X Xu andG Yu ldquoFresh-product supplychain management with logistics outsourcingrdquo Omega vol 41no 4 pp 752ndash765 2013
[8] Z-J M Shen ldquoIntegrated supply chain design models asurvey and future research directionsrdquo Journal of Industrial andManagement Optimization vol 3 no 1 pp 1ndash27 2007
[9] S Park T-E Lee and C S Sung ldquoA three-level supplychain network design model with risk-pooling and lead timesrdquoTransportation Research E vol 46 no 5 pp 563ndash581 2010
[10] A Nagurney ldquoOptimal supply chain network design andredesign at minimal total cost and with demand satisfactionrdquoInternational Journal of Production Economics vol 128 no 1 pp200ndash208 2010
[11] K M Bretthauer S Mahar and M A VenakataramananldquoInventory and distribution strategies for retaile-tail organiza-tionsrdquo Computers and Industrial Engineering vol 58 no 1 pp119ndash132 2010
[12] Y-C Tsao and J-C Lu ldquoA supply chain network design consid-ering transportation cost discountsrdquo Transportation Research Evol 48 no 2 pp 401ndash414 2012
[13] W Klibi and A Martel ldquoModeling approaches for the designof resilient supply networks under disruptionsrdquo InternationalJournal of Production Economics vol 135 no 2 pp 882ndash8982012
[14] S Doris ldquoKeeping freshness in fresh-cut productrdquo AgriculturalResearch vol 46 no 2 pp 12ndash14 1998
[15] A Dasci and V Verter ldquoA continuous model for production-distribution system designrdquo European Journal of OperationalResearch vol 129 no 2 pp 287ndash298 2001
[16] Y C Tsao D Mangotra J C Lu and M Dong ldquoA continuousapproximation approach for the integrated facility-inventoryallocation problemrdquo European Journal of Operational Researchvol 222 pp 216ndash228 2012
[17] W L Winston Operations Research Applications and Algo-rithms ThomsonBrooksCole Belmont Mass USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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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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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
Step 1 Find an initial solution 119879119894=1
by inspectionStep 2 Calculate Π
1015840(119879) and Π
10158401015840(119879)
Step 3 Let 119879119894+1
= 119879119894minus
Π1015840(119879)
Π10158401015840(119879)
Step 4 If 1003816100381610038161003816119879119894+1 minus 119879119894
1003816100381610038161003816 le 120576 where 120576 is the tolerance error then go to Step 5 otherwise let 119894 = 119894 + 1 and go to Step 3Step 5 Let 119879lowast = 119879
119894+1and Π
lowast(119879lowast) = Π(119879
119894+1)
Step 6 Determine 119860lowast
119894(119879lowast) and 120591
lowast(119879lowast) by (8) and (9) respectively
Algorithm 1
minus
119873
sum
119894=1
119862119891
119879+ 119862V120585120582119894120575119894119862119894
times[1 + (120572 minus 120573
119873
sum
119894=1
119862V120585120582119894120575119894119862119894120573119879
4119887120585120582119894120575119894119862119894
)119879
2]
minus
119873
sum
119894=1
[
[
119886 + 119887(
119873
sum
119894=1
119862V120585120582119894120575119894119862119894120573119879
4119887120585120582119894120575119894119862119894
)
2
]
]
120585120582119894120575119894119862119894
minus
119873
sum
119894=1
119877119862119894
119879[1198621198791198911199031205851205821198941205751198941198792
2 (119877 + 119865119879)]
23
minus
119873
sum
119894=1
ℎ120585120582119894120575119894119862119894119879
2
(10)
Based on the discussion above Algorithm 1 based onNewtonrsquos method determines the optimal values for 119879
lowast 119860lowast
119894
and 120591lowast
4 Numerical Study
This section presents a numerical study to illustrate the pro-posed solution approach and provide quantitative insightsThe goals of the numerical study in this study are as follows
(1) to illustrate the procedures of the solution approach(2) to discuss the effects of the related parameters on deci-
sions and cost
41 Numerical Example To illustrate the algorithmdescribedabove we applied our procedure to an agricultural productdistribution system in Taiwan Since Taiwan joined theWorldTrade Organization (WTO) local agriculture has facedgreater competition due to the reduction of customs dutiesand international trade Therefore having an efficient agri-cultural food supply network is important for Taiwan Asshown in Figure 1 farmers sell their agricultural products toagricultural associations Then the agricultural product isdistributed from agricultural associations to agriculturalproduct-marketing corporations (APMCs)TheAPMCs con-solidate shipments arriving from the agricultural associationsand deliver them to the retailers (traditional markets super-markets and hypermarkets) We consider three cities TaipeiCity New Taipei City and Keelung City in northern Taiwanthey are Cluster 1 Cluster 2 and Cluster 3 respectively
in our example In this case we wanted to determine theoptimal APMC service areas the optimal number of APMCsin each cluster the optimal freshness-keeping effort andthe optimal joint replenishment cycle time In practice therelated parameters can be determined by regression analysisusing historical transaction data Please note that the dataused here have been scaled to protect confidentiality Pleasealso note that the freshness-keeping effort cost just needsto satisfy the requirement The freshness-keeping effort costis not only convex but also increasing in the freshness-keeping effort One may get other freshness-keeping effortcost form from the regression analysis However our modeland solution approach can also be used if the freshness-keeping effort cost satisfies the above requirement
The parameters of the product are 119865 = 100000 119901 = 100119888 = 50 119862
1= 10000 119862
2= 8000 119862
3= 6000 ℎ = 1 119877 = 30
119862119879
= 10 1205821= 11 120582
2= 10 120582
3= 9 120585 = 12 120575
1= 006 120575
2=
005 1205753= 004 119891
119903= 001 119862
119891= 1000 119862V = 5 120572 = 005 120573 =
001 119886 = 3 119887 = 01 After applying the algorithm in Section 3the optimal joint replenishment cycle time is119879lowast = 02346 theoptimal APMC service area in cluster119862
1is119860lowast1= 399859 the
optimal APMC service area in cluster1198622is119860lowast2= 481160 the
optimal APMC service area in cluster1198623is119860lowast3= 598964 the
optimal freshness-keeping effort is 120591lowast = 00293 and the totalnetwork profit is Πlowast = 4853230 Figure 2 shows the graphicillustrations of Π versus different 119879 The algorithm can findthe optimal solution in a very short time
42 Numerical Analysis Several numerical analyses are con-ducted to gain management insights into the structures ofthe proposed policies The following numerical analyses areused to show the effects of 119865 ℎ 119877 119862V 119862119879 119886 119887 120572 and 120573 on theoptimal decisions and the total network profit The results ofTables 1 2 and 3 are as follows
(1) When the facility cost 119865 increases the optimal jointreplenishment cycle time 119879
lowast the optimal freshness-keeping effort 120591
lowast and the total network profit Πlowast
decrease but the optimal APMC service area 119860lowast
119894
increases This verifies Property 1 (b) When thefacility cost increases it is reasonable to increase theAPMC service area to reduce the number of APMCsopening
(2) When the inventory holding cost ℎ increases the opti-mal joint replenishment cycle time 119879
lowast the optimalfreshness-keeping effort 120591
lowast and the total network
6 Journal of Applied Mathematics
Table 1 Effects of different system parameters
Parameter 119879lowast
119860lowast
1119860lowast
2119860lowast
3120591lowast
Πlowast
119865 = 80000 02356 344660 414737 516279 00294 4964300119865 = 100000 02346 399859 481160 598964 00293 4853230119865 = 120000 02340 451475 543270 676282 00292 4756130ℎ = 025 02876 399796 481084 598870 00360 4853460ℎ = 05 02346 399859 481160 598964 00293 4853230ℎ = 075 02031 399911 481223 599043 00254 484907119877 = 20 02327 399747 481025 598797 00291 4853460119877 = 30 02346 399859 481160 598964 00293 4853230119877 = 40 02365 399968 481292 599129 00296 4853020119888119907= 3 02519 399835 481131 598929 00189 5161330
119888119907= 5 02346 399859 481160 598964 00293 4853230
119888119907= 7 02205 399880 481186 598997 00386 4545270
119888119879= 5 02325 634741 763799 950804 00291 5427370
119888119879= 10 02346 399859 481160 598964 00293 4853230
119888119879= 15 02364 305147 367191 457092 00296 4371680
Table 2 Effects of a and b
a b005 0 1 015
2
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
1198603
lowast= 598964 119860
3
lowast= 598964 119860
3
lowast= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 5006370 Π
lowast= 5006350 Π
lowast= 5006350
3
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 4853250 Π
lowast= 4853230 Π
lowast= 4853230
4
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 4700130 Π
lowast= 4700110 Π
lowast= 4700110
profit Πlowast decrease but the optimal APMC service
area 119860lowast
119894increases It is reasonable that when the
inventory holding cost increases the company willdecrease the replenishment cycle time in an effortto lower inventory costs Also the freshness-keepingeffort can be decreased with a shorter replenish-ment cycle time They will also reduce the numberof APMCs to hold less inventory as the inventoryholding cost increases
Table 3 Effects of 120572 and 120573
120572120573
0005 001 0015
003
119879lowast= 02519 119879
lowast= 02521 119879
lowast= 02525
119860lowast
1= 399835 119860
lowast
1= 399835 119860
lowast
1= 399835
119860lowast
2= 481132 119860
lowast
2= 481131 119860
lowast
2= 481131
119860lowast
3= 598929 119860
lowast
3= 598929 119860
lowast
3= 598928
120591lowast= 00157 120591
lowast= 00315 120591
lowast= 00473
Πlowast= 4855080 Π
lowast= 4855100 Π
lowast= 4855110
005
119879lowast= 02345 119879
lowast= 02346 119879
lowast= 02349
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399858
119860lowast
2= 481160 119860
lowast
2= 481160 119860
lowast
2= 481159
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598963
120591lowast= 00147 120591
lowast= 00293 120591
lowast= 00440
Πlowast= 4853220 Π
lowast= 4853230 Π
lowast= 4853250
007
119879lowast= 02202 119879
lowast= 02204 119879
lowast= 02206
119860lowast
1= 399881 119860
lowast
1= 399881 119860
lowast
1= 399880
119860lowast
2= 481186 119860
lowast
2= 481186 119860
lowast
2= 481186
119860lowast
3= 598997 119860
lowast
3= 598997 119860
lowast
3= 598997
120591lowast= 00138 120591
lowast= 00275 120591
lowast= 00414
Πlowast= 4851490 Π
lowast= 4851490 Π
lowast= 4851510
(3) When the ordering cost 119877 increases the optimaljoint replenishment cycle time119879lowast the optimal APMCservice area 119860
lowast
119894 and the optimal freshness-keeping
effort 120591lowast all increase but the total network profit
Πlowast decreases This also verifies Property 1 (b) If
the ordering cost increases it is reasonable that thecompany will increase the replenishment cycle timeto reduce replenishment frequency The companywill also likely decrease the number of APMCs to
Journal of Applied Mathematics 7
018 022 024 026 028 03
Π
T
48532 times 106
48528 times 106
48526 times 106
48524 times 106
Figure 2 Graphic illustrations of Π versus 119879
reduce the replenishment times as the ordering costincreases
(4) When the variable inbound shipment cost 119862Vincreases the optimal joint replenishment cycle time119879lowast and the total network profit Π
lowast decrease butthe optimal APMC service area 119860
lowast
119894and the optimal
freshness-keeping effort 120591lowast increase This verifies
Property 2 (a) The number of APMCs will decreaseto reduce the inbound transportation times as thevariable inbound shipment cost increases
(5) When the outbound transportation cost119862119879increases
the optimal DC service area119860lowast
119894and the total network
profit Πlowast decrease but the optimal joint replenish-
ment cycle time119879lowast and the optimal freshness-keepingeffort 120591
lowast increase This also verifies Property 1 (a)When the outbound transportation cost increases itis reasonable to increase the joint replenishment cycletime to reduce the outbound transportation times
(6) When the freshness-keeping effort cost increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases It is reasonable to decrease
the freshness-keeping effort and the replenishmentcycle time when the freshness-keeping effort costincreases
(7) When the perishable rate of the fresh food increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases When the perishable rate
of the fresh food increases the company will decreasethe cycle time to reduce the deterioration of thefresh food When the freshness-keeping effort hasmore service on reducing the deterioration of thefresh food companies are willing to increase theirfreshness-keeping effort
5 Conclusions
This study considers a supply network design problem foragricultural productsThemodel can also be applied to other
fresh food products The agricultural produce marketingcorporations (APMCs) help consolidate shipments arrivingfrom farmersrsquo associations and deliver them to traditionalmarkets supermarkets and hypermarkets The proposedmethod integrates sales revenue purchasing costs facilitycosts inventory costs transportation costs ordering costsand freshness-keeping effort costs The key decisions includewhere to locate the APMCs how to assign retailers toAPMCs the joint replenishment cycle time at APMCs andthe freshness-keeping effort such that the total networkprofit is maximized An easy-to-use algorithm is providedfor solving the nonlinear programming problem Numericalstudies illustrate the solution procedures and the effectsof the facility cost inventory holding cost transportationcost ordering cost and perishable rate of the fresh food ondecisions and profits The results of this study are a usefulreference for managerial decision making and administra-tion Further research on this topic may consider when thefreshness-keeping effort has an influence on demand Thiswould include the market demand for fresh food dependenton the freshness of the product
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his gratitude to the editor and theanonymous reviewers for their detailed comments and valu-able suggestions to improve the exposition of this paper Thispaper is supported in part by the National Science Councilunder Grant NSC 102-2410-H-011-029-MY3 and 102-2221-E-011-159-MY3
References
[1] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005
[2] R K Apaiah and E M T Hendrix ldquoDesign of a supply chainnetwork for pea-based novel protein foodsrdquo Journal of FoodEngineering vol 70 no 3 pp 383ndash391 2005
[3] A Osvald and L Z Stirn ldquoA vehicle routing algorithm for thedistribution of fresh vegetables and similar perishable foodrdquoJournal of Food Engineering vol 85 no 2 pp 285ndash295 2008
[4] X Cai J Chen Y Xiao and X Xu ldquoOptimization andcoordination of fresh product supply chains with freshness-keeping effortrdquo Production and OperationsManagement vol 19no 3 pp 261ndash278 2010
[5] I-H Hong J-F Dang Y-H Tsai et al ldquoAn RFID applicationin the food supply chain a case study of convenience stores inTaiwanrdquo Journal of Food Engineering vol 106 no 2 pp 119ndash1262011
[6] XWang andD Li ldquoAdynamic product quality evaluation basedpricing model for perishable food supply chainsrdquo Omega vol40 no 6 pp 906ndash917 2012
8 Journal of Applied Mathematics
[7] X Cai J Chen Y Xiao X Xu andG Yu ldquoFresh-product supplychain management with logistics outsourcingrdquo Omega vol 41no 4 pp 752ndash765 2013
[8] Z-J M Shen ldquoIntegrated supply chain design models asurvey and future research directionsrdquo Journal of Industrial andManagement Optimization vol 3 no 1 pp 1ndash27 2007
[9] S Park T-E Lee and C S Sung ldquoA three-level supplychain network design model with risk-pooling and lead timesrdquoTransportation Research E vol 46 no 5 pp 563ndash581 2010
[10] A Nagurney ldquoOptimal supply chain network design andredesign at minimal total cost and with demand satisfactionrdquoInternational Journal of Production Economics vol 128 no 1 pp200ndash208 2010
[11] K M Bretthauer S Mahar and M A VenakataramananldquoInventory and distribution strategies for retaile-tail organiza-tionsrdquo Computers and Industrial Engineering vol 58 no 1 pp119ndash132 2010
[12] Y-C Tsao and J-C Lu ldquoA supply chain network design consid-ering transportation cost discountsrdquo Transportation Research Evol 48 no 2 pp 401ndash414 2012
[13] W Klibi and A Martel ldquoModeling approaches for the designof resilient supply networks under disruptionsrdquo InternationalJournal of Production Economics vol 135 no 2 pp 882ndash8982012
[14] S Doris ldquoKeeping freshness in fresh-cut productrdquo AgriculturalResearch vol 46 no 2 pp 12ndash14 1998
[15] A Dasci and V Verter ldquoA continuous model for production-distribution system designrdquo European Journal of OperationalResearch vol 129 no 2 pp 287ndash298 2001
[16] Y C Tsao D Mangotra J C Lu and M Dong ldquoA continuousapproximation approach for the integrated facility-inventoryallocation problemrdquo European Journal of Operational Researchvol 222 pp 216ndash228 2012
[17] W L Winston Operations Research Applications and Algo-rithms ThomsonBrooksCole Belmont Mass USA 2004
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
6 Journal of Applied Mathematics
Table 1 Effects of different system parameters
Parameter 119879lowast
119860lowast
1119860lowast
2119860lowast
3120591lowast
Πlowast
119865 = 80000 02356 344660 414737 516279 00294 4964300119865 = 100000 02346 399859 481160 598964 00293 4853230119865 = 120000 02340 451475 543270 676282 00292 4756130ℎ = 025 02876 399796 481084 598870 00360 4853460ℎ = 05 02346 399859 481160 598964 00293 4853230ℎ = 075 02031 399911 481223 599043 00254 484907119877 = 20 02327 399747 481025 598797 00291 4853460119877 = 30 02346 399859 481160 598964 00293 4853230119877 = 40 02365 399968 481292 599129 00296 4853020119888119907= 3 02519 399835 481131 598929 00189 5161330
119888119907= 5 02346 399859 481160 598964 00293 4853230
119888119907= 7 02205 399880 481186 598997 00386 4545270
119888119879= 5 02325 634741 763799 950804 00291 5427370
119888119879= 10 02346 399859 481160 598964 00293 4853230
119888119879= 15 02364 305147 367191 457092 00296 4371680
Table 2 Effects of a and b
a b005 0 1 015
2
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
1198603
lowast= 598964 119860
3
lowast= 598964 119860
3
lowast= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 5006370 Π
lowast= 5006350 Π
lowast= 5006350
3
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 4853250 Π
lowast= 4853230 Π
lowast= 4853230
4
119879lowast= 02349 119879
lowast= 02346 119879
lowast= 02346
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399859
119860lowast
2= 481159 119860
lowast
2= 481160 119860
lowast
2= 481160
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598964
120591lowast= 00587 120591
lowast= 00293 120591
lowast= 00195
Πlowast= 4700130 Π
lowast= 4700110 Π
lowast= 4700110
profit Πlowast decrease but the optimal APMC service
area 119860lowast
119894increases It is reasonable that when the
inventory holding cost increases the company willdecrease the replenishment cycle time in an effortto lower inventory costs Also the freshness-keepingeffort can be decreased with a shorter replenish-ment cycle time They will also reduce the numberof APMCs to hold less inventory as the inventoryholding cost increases
Table 3 Effects of 120572 and 120573
120572120573
0005 001 0015
003
119879lowast= 02519 119879
lowast= 02521 119879
lowast= 02525
119860lowast
1= 399835 119860
lowast
1= 399835 119860
lowast
1= 399835
119860lowast
2= 481132 119860
lowast
2= 481131 119860
lowast
2= 481131
119860lowast
3= 598929 119860
lowast
3= 598929 119860
lowast
3= 598928
120591lowast= 00157 120591
lowast= 00315 120591
lowast= 00473
Πlowast= 4855080 Π
lowast= 4855100 Π
lowast= 4855110
005
119879lowast= 02345 119879
lowast= 02346 119879
lowast= 02349
119860lowast
1= 399859 119860
lowast
1= 399859 119860
lowast
1= 399858
119860lowast
2= 481160 119860
lowast
2= 481160 119860
lowast
2= 481159
119860lowast
3= 598964 119860
lowast
3= 598964 119860
lowast
3= 598963
120591lowast= 00147 120591
lowast= 00293 120591
lowast= 00440
Πlowast= 4853220 Π
lowast= 4853230 Π
lowast= 4853250
007
119879lowast= 02202 119879
lowast= 02204 119879
lowast= 02206
119860lowast
1= 399881 119860
lowast
1= 399881 119860
lowast
1= 399880
119860lowast
2= 481186 119860
lowast
2= 481186 119860
lowast
2= 481186
119860lowast
3= 598997 119860
lowast
3= 598997 119860
lowast
3= 598997
120591lowast= 00138 120591
lowast= 00275 120591
lowast= 00414
Πlowast= 4851490 Π
lowast= 4851490 Π
lowast= 4851510
(3) When the ordering cost 119877 increases the optimaljoint replenishment cycle time119879lowast the optimal APMCservice area 119860
lowast
119894 and the optimal freshness-keeping
effort 120591lowast all increase but the total network profit
Πlowast decreases This also verifies Property 1 (b) If
the ordering cost increases it is reasonable that thecompany will increase the replenishment cycle timeto reduce replenishment frequency The companywill also likely decrease the number of APMCs to
Journal of Applied Mathematics 7
018 022 024 026 028 03
Π
T
48532 times 106
48528 times 106
48526 times 106
48524 times 106
Figure 2 Graphic illustrations of Π versus 119879
reduce the replenishment times as the ordering costincreases
(4) When the variable inbound shipment cost 119862Vincreases the optimal joint replenishment cycle time119879lowast and the total network profit Π
lowast decrease butthe optimal APMC service area 119860
lowast
119894and the optimal
freshness-keeping effort 120591lowast increase This verifies
Property 2 (a) The number of APMCs will decreaseto reduce the inbound transportation times as thevariable inbound shipment cost increases
(5) When the outbound transportation cost119862119879increases
the optimal DC service area119860lowast
119894and the total network
profit Πlowast decrease but the optimal joint replenish-
ment cycle time119879lowast and the optimal freshness-keepingeffort 120591
lowast increase This also verifies Property 1 (a)When the outbound transportation cost increases itis reasonable to increase the joint replenishment cycletime to reduce the outbound transportation times
(6) When the freshness-keeping effort cost increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases It is reasonable to decrease
the freshness-keeping effort and the replenishmentcycle time when the freshness-keeping effort costincreases
(7) When the perishable rate of the fresh food increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases When the perishable rate
of the fresh food increases the company will decreasethe cycle time to reduce the deterioration of thefresh food When the freshness-keeping effort hasmore service on reducing the deterioration of thefresh food companies are willing to increase theirfreshness-keeping effort
5 Conclusions
This study considers a supply network design problem foragricultural productsThemodel can also be applied to other
fresh food products The agricultural produce marketingcorporations (APMCs) help consolidate shipments arrivingfrom farmersrsquo associations and deliver them to traditionalmarkets supermarkets and hypermarkets The proposedmethod integrates sales revenue purchasing costs facilitycosts inventory costs transportation costs ordering costsand freshness-keeping effort costs The key decisions includewhere to locate the APMCs how to assign retailers toAPMCs the joint replenishment cycle time at APMCs andthe freshness-keeping effort such that the total networkprofit is maximized An easy-to-use algorithm is providedfor solving the nonlinear programming problem Numericalstudies illustrate the solution procedures and the effectsof the facility cost inventory holding cost transportationcost ordering cost and perishable rate of the fresh food ondecisions and profits The results of this study are a usefulreference for managerial decision making and administra-tion Further research on this topic may consider when thefreshness-keeping effort has an influence on demand Thiswould include the market demand for fresh food dependenton the freshness of the product
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his gratitude to the editor and theanonymous reviewers for their detailed comments and valu-able suggestions to improve the exposition of this paper Thispaper is supported in part by the National Science Councilunder Grant NSC 102-2410-H-011-029-MY3 and 102-2221-E-011-159-MY3
References
[1] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005
[2] R K Apaiah and E M T Hendrix ldquoDesign of a supply chainnetwork for pea-based novel protein foodsrdquo Journal of FoodEngineering vol 70 no 3 pp 383ndash391 2005
[3] A Osvald and L Z Stirn ldquoA vehicle routing algorithm for thedistribution of fresh vegetables and similar perishable foodrdquoJournal of Food Engineering vol 85 no 2 pp 285ndash295 2008
[4] X Cai J Chen Y Xiao and X Xu ldquoOptimization andcoordination of fresh product supply chains with freshness-keeping effortrdquo Production and OperationsManagement vol 19no 3 pp 261ndash278 2010
[5] I-H Hong J-F Dang Y-H Tsai et al ldquoAn RFID applicationin the food supply chain a case study of convenience stores inTaiwanrdquo Journal of Food Engineering vol 106 no 2 pp 119ndash1262011
[6] XWang andD Li ldquoAdynamic product quality evaluation basedpricing model for perishable food supply chainsrdquo Omega vol40 no 6 pp 906ndash917 2012
8 Journal of Applied Mathematics
[7] X Cai J Chen Y Xiao X Xu andG Yu ldquoFresh-product supplychain management with logistics outsourcingrdquo Omega vol 41no 4 pp 752ndash765 2013
[8] Z-J M Shen ldquoIntegrated supply chain design models asurvey and future research directionsrdquo Journal of Industrial andManagement Optimization vol 3 no 1 pp 1ndash27 2007
[9] S Park T-E Lee and C S Sung ldquoA three-level supplychain network design model with risk-pooling and lead timesrdquoTransportation Research E vol 46 no 5 pp 563ndash581 2010
[10] A Nagurney ldquoOptimal supply chain network design andredesign at minimal total cost and with demand satisfactionrdquoInternational Journal of Production Economics vol 128 no 1 pp200ndash208 2010
[11] K M Bretthauer S Mahar and M A VenakataramananldquoInventory and distribution strategies for retaile-tail organiza-tionsrdquo Computers and Industrial Engineering vol 58 no 1 pp119ndash132 2010
[12] Y-C Tsao and J-C Lu ldquoA supply chain network design consid-ering transportation cost discountsrdquo Transportation Research Evol 48 no 2 pp 401ndash414 2012
[13] W Klibi and A Martel ldquoModeling approaches for the designof resilient supply networks under disruptionsrdquo InternationalJournal of Production Economics vol 135 no 2 pp 882ndash8982012
[14] S Doris ldquoKeeping freshness in fresh-cut productrdquo AgriculturalResearch vol 46 no 2 pp 12ndash14 1998
[15] A Dasci and V Verter ldquoA continuous model for production-distribution system designrdquo European Journal of OperationalResearch vol 129 no 2 pp 287ndash298 2001
[16] Y C Tsao D Mangotra J C Lu and M Dong ldquoA continuousapproximation approach for the integrated facility-inventoryallocation problemrdquo European Journal of Operational Researchvol 222 pp 216ndash228 2012
[17] W L Winston Operations Research Applications and Algo-rithms ThomsonBrooksCole Belmont Mass USA 2004
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
Journal of Applied Mathematics 7
018 022 024 026 028 03
Π
T
48532 times 106
48528 times 106
48526 times 106
48524 times 106
Figure 2 Graphic illustrations of Π versus 119879
reduce the replenishment times as the ordering costincreases
(4) When the variable inbound shipment cost 119862Vincreases the optimal joint replenishment cycle time119879lowast and the total network profit Π
lowast decrease butthe optimal APMC service area 119860
lowast
119894and the optimal
freshness-keeping effort 120591lowast increase This verifies
Property 2 (a) The number of APMCs will decreaseto reduce the inbound transportation times as thevariable inbound shipment cost increases
(5) When the outbound transportation cost119862119879increases
the optimal DC service area119860lowast
119894and the total network
profit Πlowast decrease but the optimal joint replenish-
ment cycle time119879lowast and the optimal freshness-keepingeffort 120591
lowast increase This also verifies Property 1 (a)When the outbound transportation cost increases itis reasonable to increase the joint replenishment cycletime to reduce the outbound transportation times
(6) When the freshness-keeping effort cost increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases It is reasonable to decrease
the freshness-keeping effort and the replenishmentcycle time when the freshness-keeping effort costincreases
(7) When the perishable rate of the fresh food increasesthe optimal joint replenishment cycle time 119879
lowast theoptimal freshness-keeping effort 120591
lowast and the totalnetwork profit Πlowast decrease but the optimal APMCservice area 119860
lowast
119894increases When the perishable rate
of the fresh food increases the company will decreasethe cycle time to reduce the deterioration of thefresh food When the freshness-keeping effort hasmore service on reducing the deterioration of thefresh food companies are willing to increase theirfreshness-keeping effort
5 Conclusions
This study considers a supply network design problem foragricultural productsThemodel can also be applied to other
fresh food products The agricultural produce marketingcorporations (APMCs) help consolidate shipments arrivingfrom farmersrsquo associations and deliver them to traditionalmarkets supermarkets and hypermarkets The proposedmethod integrates sales revenue purchasing costs facilitycosts inventory costs transportation costs ordering costsand freshness-keeping effort costs The key decisions includewhere to locate the APMCs how to assign retailers toAPMCs the joint replenishment cycle time at APMCs andthe freshness-keeping effort such that the total networkprofit is maximized An easy-to-use algorithm is providedfor solving the nonlinear programming problem Numericalstudies illustrate the solution procedures and the effectsof the facility cost inventory holding cost transportationcost ordering cost and perishable rate of the fresh food ondecisions and profits The results of this study are a usefulreference for managerial decision making and administra-tion Further research on this topic may consider when thefreshness-keeping effort has an influence on demand Thiswould include the market demand for fresh food dependenton the freshness of the product
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his gratitude to the editor and theanonymous reviewers for their detailed comments and valu-able suggestions to improve the exposition of this paper Thispaper is supported in part by the National Science Councilunder Grant NSC 102-2410-H-011-029-MY3 and 102-2221-E-011-159-MY3
References
[1] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005
[2] R K Apaiah and E M T Hendrix ldquoDesign of a supply chainnetwork for pea-based novel protein foodsrdquo Journal of FoodEngineering vol 70 no 3 pp 383ndash391 2005
[3] A Osvald and L Z Stirn ldquoA vehicle routing algorithm for thedistribution of fresh vegetables and similar perishable foodrdquoJournal of Food Engineering vol 85 no 2 pp 285ndash295 2008
[4] X Cai J Chen Y Xiao and X Xu ldquoOptimization andcoordination of fresh product supply chains with freshness-keeping effortrdquo Production and OperationsManagement vol 19no 3 pp 261ndash278 2010
[5] I-H Hong J-F Dang Y-H Tsai et al ldquoAn RFID applicationin the food supply chain a case study of convenience stores inTaiwanrdquo Journal of Food Engineering vol 106 no 2 pp 119ndash1262011
[6] XWang andD Li ldquoAdynamic product quality evaluation basedpricing model for perishable food supply chainsrdquo Omega vol40 no 6 pp 906ndash917 2012
8 Journal of Applied Mathematics
[7] X Cai J Chen Y Xiao X Xu andG Yu ldquoFresh-product supplychain management with logistics outsourcingrdquo Omega vol 41no 4 pp 752ndash765 2013
[8] Z-J M Shen ldquoIntegrated supply chain design models asurvey and future research directionsrdquo Journal of Industrial andManagement Optimization vol 3 no 1 pp 1ndash27 2007
[9] S Park T-E Lee and C S Sung ldquoA three-level supplychain network design model with risk-pooling and lead timesrdquoTransportation Research E vol 46 no 5 pp 563ndash581 2010
[10] A Nagurney ldquoOptimal supply chain network design andredesign at minimal total cost and with demand satisfactionrdquoInternational Journal of Production Economics vol 128 no 1 pp200ndash208 2010
[11] K M Bretthauer S Mahar and M A VenakataramananldquoInventory and distribution strategies for retaile-tail organiza-tionsrdquo Computers and Industrial Engineering vol 58 no 1 pp119ndash132 2010
[12] Y-C Tsao and J-C Lu ldquoA supply chain network design consid-ering transportation cost discountsrdquo Transportation Research Evol 48 no 2 pp 401ndash414 2012
[13] W Klibi and A Martel ldquoModeling approaches for the designof resilient supply networks under disruptionsrdquo InternationalJournal of Production Economics vol 135 no 2 pp 882ndash8982012
[14] S Doris ldquoKeeping freshness in fresh-cut productrdquo AgriculturalResearch vol 46 no 2 pp 12ndash14 1998
[15] A Dasci and V Verter ldquoA continuous model for production-distribution system designrdquo European Journal of OperationalResearch vol 129 no 2 pp 287ndash298 2001
[16] Y C Tsao D Mangotra J C Lu and M Dong ldquoA continuousapproximation approach for the integrated facility-inventoryallocation problemrdquo European Journal of Operational Researchvol 222 pp 216ndash228 2012
[17] W L Winston Operations Research Applications and Algo-rithms ThomsonBrooksCole Belmont Mass USA 2004
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 Journal of Applied Mathematics
[7] X Cai J Chen Y Xiao X Xu andG Yu ldquoFresh-product supplychain management with logistics outsourcingrdquo Omega vol 41no 4 pp 752ndash765 2013
[8] Z-J M Shen ldquoIntegrated supply chain design models asurvey and future research directionsrdquo Journal of Industrial andManagement Optimization vol 3 no 1 pp 1ndash27 2007
[9] S Park T-E Lee and C S Sung ldquoA three-level supplychain network design model with risk-pooling and lead timesrdquoTransportation Research E vol 46 no 5 pp 563ndash581 2010
[10] A Nagurney ldquoOptimal supply chain network design andredesign at minimal total cost and with demand satisfactionrdquoInternational Journal of Production Economics vol 128 no 1 pp200ndash208 2010
[11] K M Bretthauer S Mahar and M A VenakataramananldquoInventory and distribution strategies for retaile-tail organiza-tionsrdquo Computers and Industrial Engineering vol 58 no 1 pp119ndash132 2010
[12] Y-C Tsao and J-C Lu ldquoA supply chain network design consid-ering transportation cost discountsrdquo Transportation Research Evol 48 no 2 pp 401ndash414 2012
[13] W Klibi and A Martel ldquoModeling approaches for the designof resilient supply networks under disruptionsrdquo InternationalJournal of Production Economics vol 135 no 2 pp 882ndash8982012
[14] S Doris ldquoKeeping freshness in fresh-cut productrdquo AgriculturalResearch vol 46 no 2 pp 12ndash14 1998
[15] A Dasci and V Verter ldquoA continuous model for production-distribution system designrdquo European Journal of OperationalResearch vol 129 no 2 pp 287ndash298 2001
[16] Y C Tsao D Mangotra J C Lu and M Dong ldquoA continuousapproximation approach for the integrated facility-inventoryallocation problemrdquo European Journal of Operational Researchvol 222 pp 216ndash228 2012
[17] W L Winston Operations Research Applications and Algo-rithms ThomsonBrooksCole Belmont Mass USA 2004
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