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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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

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OptimizationJournal of

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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

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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

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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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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 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

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

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

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

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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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