Research Article Integrated Inventory Routing Problem with ...downloads.hindawi.com/journals/mpe/2014/537409.pdfResearch Article Integrated Inventory Routing Problem with Quality Time

Research ArticleIntegrated Inventory Routing Problem with Quality TimeWindows and Loading Cost for Deteriorating Items underDiscrete Time

Tao Jia12 Xiaofan Li12 Nengmin Wang12 and Ran Li12

1 School of Management Xirsquoan JiaoTong University No 28 Xianning Road Xirsquoan Shaanxi 710049 China2The Key Lab of the Ministry of Education for Process Control and Efficiency Engineering No 28 Xianning RoadXirsquoan Shaanxi 710049 China

Correspondence should be addressed to Tao Jia jiataomailxjtueducn

Received 21 September 2013 Revised 26 November 2013 Accepted 26 November 2013 Published 29 January 2014

Academic Editor Yunqiang Yin

Copyright copy 2014 Tao Jia et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate an integrated inventory routing problem (IRP) in which one supplier with limited production capacity distributesa single item to a set of retailers using homogeneous vehicles In the objective function we consider a loading cost which is oftenneglected in previous research Considering the deterioration in the products we set a soft time window during the transportationstage and a hard timewindowduring the sales stage and to prevent jams andwaiting cost the time interval of two successive vehiclesreturning to the supplierrsquos facilities is required not to be overly short Combining all of these factors a two-echelon supply chainmixed integer programming model under discrete time is proposed and a two-phase algorithm is developed The first phase usestabu search to obtain the retailersrsquo orderingmatrixThe second phase is to generate production scheduling and distribution routingadopting a saving algorithm and a neighbourhood search respectively Computational experiments are conducted to illustrate theeffectiveness of the proposed model and algorithm

1 Introduction

Perishable items exist widely in our daily life such as freshproducts fruits vegetables and seafood which decreasein quantity (or utility) during their delivery and storagestages Deteriorating items can be classified by differentcriteria According to the study of Raafat [1] two classesof deteriorating items are mainly of concern (1) items withcontinuous decay for example inventory models that aresubject to exponential decay and (2) items with a fixed shelflife Corresponding to theoretical research two methods arealways used in operations management practice to calculatethe deteriorating costs of these two classes of deterioratingitems that is the ldquoweight discountrdquo and ldquoshelf life windowrdquoThe ldquoWeight discountrdquo is to calculate the deterioration (quan-tityweight) loss in the storage and transportation stages it isa weight discount rate that is generally negotiated by sellersand buyers based on the delivered amount of goods thepreservation conditions or the storage and transportation

time In the real world this accounting method is alwaysused in Chinese seafood cold chains For most deterioratingitems that are sold in the supermarkets retailers can considerthe quality corruption and set a fixed shelf-life windowOnce they exceed the expiration date the goods will berejected and their values will drop to zero instantaneouslyLukasse and Polderdijk [2] studied the quality degradation ofmushrooms and established a piecewise nonlinear functionof mushroom quality decline Considering the randomnessof the perishable food delivery speed and the time-varyingtemperatures Hsu et al [3] divided the corruption intotransportation corruption and customer-site corruption thatresults from handling under normal temperatures Perishablefood had different spoilage rates in the above processesand the customer site corruption rate was larger Rong etal [4] studied the distribution plan of a single productwith a known demand rate The objective of the productionand distribution problem was to determine the varyingdeterioration that was caused by the time and temperature in

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 537409 14 pageshttpdxdoiorg1011552014537409

2 Mathematical Problems in Engineering

different transportation and storage processes If the qualityrequirements were not satisfied then the goods would beabandoned and a penalty cost would be incurred

Transportation is an important logistics activity in thedeteriorating items distribution channel and it is also a keydecision that connects the whole supply network as an effec-tive integrated system Moreover for perishable productsthe transportation process is more difficult to manipulatebecause of the changing environmental factors thus moreattention should be paid to the transportation stage In a jointdecision-making supply chain the order quantity from thenetwork nodes is also an important factor that influences thetransportation decision Thus to achieve the minimal totalrelevant cost of the channel themanagers should consider thetransportation cost (the space value of the logistics) and theinventory cost (the time value of the logistics) simultaneouslyThe inventory routing problem (IRP) just seeks to determinesimultaneously an optimal inventory and distribution strat-egy that minimises the total cost thus it is one of the hotspots in recent academic research Federgruen and Zipkin [5]are amongst the first to study a single-day inventory routingproblem Customersrsquo demands were assumed to be a randomvariable A limited inventory at the distribution centre canlead to shortages at the customer sites as a result the holdingand shortage costs were considered in the decision alongwiththe transportation costs Some of the available methods forthe deterministic vehicle routing problem had been extendedto this caseThis paper clearly identified the research categoryof the inventory routing problem and is widely regarded asthe first real IRP model in academia Anily and Federgruen[6] proposed a class of strategies in which a collection ofregions (sets of retailers) was specified that cover all outlets ifan outlet belonged to several regions then a specific fractionof its salesoperations was assigned to each of these regionsEach time one of the retailers in a given region received adelivery this delivery was made by a vehicle that visited allof the other outlets in the region as well (in an efficient route)to achieve economies of scale This strategy was widely usedin follow-up studies of IRP Kleywegt et al [7] established anIRP model in which customersrsquo demands on different dayswere independent random vectors with a joint probabilitydistribution119865 that did not changewith timeThey formulatedthe inventory routing problem as a Markov decision processand proposed approximation methods to find good solutionsthat had reasonable computational effort Aghezzaf et al[8] used the concept of a vehicle multitour which impliesthat a vehiclersquos travel plan might possibly contain morethan one tour and vehicles could be reloaded Sales pointswithin a partition were divided into regions that were similarto the regions in the study by Anily and Federgruen [6]To solve the nonlinear mixed integer formulation of thisproblem a column generation-based approximation methodwas suggested

In some of the papers on the IRP the planning timehorizon is divided into integer periods according to somenatural properties of time that is the study of the inventoryrouting problem under discrete time A discrete time modelis conducive to solving a problem that has varying retailerdemand (changing with time) and the delivery time interval

must be integer multiples of a unit period thus it is easy tomanipulate in operations management practice The unit forthe discrete time is called one period and the definition ofa period depending on the context of the problem mightbe an hour a day or a week Bard and Nananukul [9]established an IRP model with discrete time By assessing aseries of heuristics algorithms they first developed a two-step procedure that estimated daily delivery quantities andthen solved the vehicle routing problem for each day overthe planning horizon Abdelmaguid et al [10] studied aninventory routing problem with backlogging under discretetime Based on a two-phase approach transitions that usethe idea of exchanging customersrsquo delivery amounts betweenperiods were conducted and the transitions could furtherreduce the total cost by balancing the inventory costsbackordering costs and transportation costs In most studiesof inventory routing models with discrete time the deliverieswere assumed to be finished within one single period butSavelsbergh and Song [11] investigated an inventory routingproblem with designed delivery tours that spanned severaldays covering enormous geographical areas and involvingproduct pickups at different facilities They developed aninteger programming-based optimisation algorithm that wasembedded in a local search procedure to improve solutionsthat were produced by a randomised greedy heuristic

With the increasing fierce global competition and con-tinuous advancement of information technology the idea ofhaving a decision that is integrated between the upstreamand downstream enterprises within a supply chain is widelyaccepted Based on this idea research about the jointeconomic lot-sizing problem (JELP) for supply chains hasbecome a hotspot in the area of inventory managementtheory referring to the review by Ben-Daya et al [12] fordetails Lacking consideration about distribution routinginformation makes the JELP model more suitable for themedium- and long-term planning process for enterprisesFurthermore focusing on retailersrsquo lot-sizing and distribu-tion routing the integrated inventory routing problems takesuppliersrsquo production costs and inventory costs into accountthus the coordinated production-distribution schedulingcan further reduce the total relevant cost of the supply chainLei et al [13] were the first to formulate the production-inventory-distribution-routing problem as a mixed-integerprogram and illustrated the benefits that were brought by thecoordination Bard and Nananukul [14] adopted a branch-and-price framework to solve the integrated inventory rout-ing problem A hybrid methodology that combined thecolumn generation heuristic bounded heuristic algorithmand tabu search algorithm was developedThe result demon-strated that the performance of hybrid methodology wasmuch better than that of eitherCPLEXor the standard branchand price algorithm

In this paper we consider an integrated production-distributionmodel It is assumed that no deterioration occursat the supplierrsquos factory (or facilities) because of good pro-tection and production conditions However the perishablegoods have different quality degradation processes duringthe transportation and sales stages Besides the retailers andcustomers have their own acceptable quality standards For

Mathematical Problems in Engineering 3

example the shelf life of bread is 3 days (or 3 periods) afterits production If the storage time is more than 3 days thenthe customers will absolutely not accept it however for theretailers the appointed time for delivery might be within oneday after the breadrsquos production because they need time to sellit Thus different time windows are set in the transportationand sales stages The purpose of setting the transportationtime window and the sales time window (shelf-life limit) isto make it convenient to describe deteriorating itemsrsquo qualitycontrol standards for enterprisesThe time windows here canalso be generalised to other decision contexts for exampleMcDonaldrsquos asks the Xia Hui Logistics Company (a Chinesefirm) to make delivery to each of its stores between 11 pmand 1 am which can also be considered in this model

In general IRP and its variants the transportation costusually includes the fixed cost and variable cost The fixedcost occurs once a vehicle is used and the variable cost is inproportion to the sumof the distance However an importantcost is often neglected in the variable cost that is the loadingcost The loading cost is an important part of the charge forthe consumption of gas and it changes due to the amount ofload in the vehicle [15] In reality the vehicle load varies fromone customer to another within a given route and the loadcost will decrease because of unloading at each customer siteWhen deciding the vehicle distribution paths neglecting theloading cost will lead to a larger gap between reality and theanalytical model In this paper we consider an integrated IRPmodel with a loading cost

Moreover when planning the distribution routing andassigning the vehicle loading tasks we also consider thetime interval between two successive vehiclesrsquo returning tothe supplierrsquos factory or a distribution centre Because themodel is based on a discrete time vehiclesrsquo dispatching andreturning in each period will not overlap If the time intervalis too short a vehicle jam and insufficiency of handlingequipment and personnel could occur and cause muchchaos at the shipment facilities for example the deliveryplatform which is not a preferred scenario for the managerAdditionally a suitable time interval can enable multipledeliveries by one vehicle and the number of vehicles usedwilldecline

In this paper we propose a new formulation of aproduction-distribution-inventory IRP model with qualitytime windows and loading cost under discrete time In thetransportation and storage stages different time windowsare set and if the time interval between two successivevehiclesrsquo returning to the supplierrsquos factory is too short thenan extra cost would be incurred The objective of the modelis to determine the supplierrsquos production plan the deliveryschedule and the vehicle routing in such a way that the totalrelevant cost is minimised while satisfying both the qualityand delivery requirements

The remainder of this paper is organised as followsIn Section 2 a mathematical model is formulated for theproposed problem along with the problem assumptionsCombining the features of the model a two-phase algorithmis developed in Section 3 In Section 4 the computationalresults of the algorithm and sensitivity analysis are presentedFinally Section 5 provides several conclusions

2 Problem Description and Notation

21 Problem Description We study a supply chain systemthat is composed of one supplier and a set of retailers Eachretailer faces a deterministic and independent demand for asingle item per time period The planning horizon considers120591 periods and the length of each period is 119879 Deliveries toretailers aremade by a homogeneous fleet of vehicles from thesupplier The production capacity of the supplier is limitedThe item has different shelf lives in the transportation stageand sales stage The demand at each retailer is relativelysmall compared with the vehicle capacity and the retailersare located closely which enables one vehicle to cover manyretailers within one trip The delivery in each period is anindependent vehicle routing problem Our study is underdiscrete time In one period the supply chain has threeseries stages production distribution and sales They arecalled production in period 119901 distribution in period 119901and sales in period 119901 respectively Because the productionand distribution have a lead time the starting time of theproduction and distribution can be extrapolated backwardbased on the sales time

To be more practical the vehiclersquos loading cost in thetransportation is consideredThe lower bound of the intervalof two successive vehiclesrsquo returning time is set to avoid jamsin the platform If the actual interval is less than the thresholdvalue then an extra penalty cost would be incurred Hencewe generate the production schedule and distribution routingover the planning horizon to minimise the total costs whilemeeting the time windows and other constraints such as thetime interval

The problem that we consider can be formulated asa mixed integer programming model and the modellingmethod and process can be referred from previous researchBell et al [16] consider the problem of gas distributionand they determine a set of routes that have at most fourcustomers they use amixed integer problem (MIP) approachto define the quantities to deliver the set of vehicles to useand the dates of visit Savelsbergh and Song [17] considerthe IRP that considers the delivery of one product to twotypes of customers To solve this problem they present threeheuristics that define the routes and the quantities to deliverThey also solve an MIP to redefine the quantities to deliverOther similar research adopts a mixed integer programmingmodel to study relative problems that can be found inresearches by Resende and Ribeiro [18] Lei et al [13] Boudiaet al [19 20] Bard and Nananukul [14] and Moin et al [21]Based on their research we develop our model as follows

22 Assumptions

(1) The production capacity of the supplier is limited butsufficient to satisfy the retailersrsquo demands At the endof each period inventory is measured to calculate theinventory costs

(2) The number of homogeneous vehicles is unlimitedEach of these vehicles couldmake atmost one trip perperiod


(3) The inventory capacities of the retailers are not lim-ited and shortages are not permitted The demandsof the retailers could not be divided

(4) The loading andunloading periods are not consideredhere alone

(5) All of the initial inventories are zero as well as the endof the planning horizon

23 Notation

(1) Parameters

119891119904 setup cost of the supplier119862 maximal production capacity of the supplier in oneperiodℎ119904 unit holding cost of the supplier119898119901 production quantity in period 119901

119891V fixed cost when a vehicle is used119888V driving cost per unit distance119905max 1 the longest driving time limit for a vehicletravelling from the supplierrsquos facilities to any of theretailers within which there is no penalty119905max 2 the longest driving time limit for a vehicletravelling from the supplierrsquos facilities to any of theretailers119887 deteriorating costunit productunit time when thearriving time is between 119905max 1 and 119905max 2119879 the time length of one period119904119901 the speed of the vehicles in period 119901119889119894119895 the distance from site 119894 to 119895

120595 the capacity of the vehicles119894119905V the lower bound time interval of two successivevehiclesrsquo returning to the supplierrsquos facilities119888119901 when the actual interval is less than the thresholdvalue 119894119905V the cost per unit time119888119892 the cost of delivering product per unit weight andper unit distance119881119901 the number of delivery paths in period 119901ℎ119888 the unit holding cost of the retailers119889119901

119894 the demand of retailer 119894 in period 119901

120593 the unloading cost of the retailers120576 the upper bound of the periods that goods can besaved at retailersrsquo stores119878 the set of retailers119878+ the set of retailers and the supplier119878V119901 the set of retailers that have been served byvehicle V in period 119901119881 the set of vehicles119867 the set of periods in the planning horizon

(2) Variables

119868119901 inventory of the supplier at the beginning ofperiod 119901119905119901

119894 the time that the vehicles arrive at site i (including

the supplier)119905119901

Vlast the time that vehicle V arrives at its last retailerin period 119901119905119901

119894119895 the driving time from 119894 to 119895 in period 119901 (119905119901

119894119895=

119889119894119895119904119901)

end119905119901119894 the returning time of vehicle 119894 in period 119901 to

the supplierrsquos facilities119897V119901119894119895 the load of vehicle V on arc (119894 119895) in period 119901

119868119901

119894 the inventory of retailer 119894 at the beginning of

period 119901119908119901

119894 the amount of products delivered to retailer 119894 at

the beginning of period 119901119860119901

119894 the set of successive periods from period p

(including period 119901) to the period of the next arrivalto retailer 119894 decided by the 0-1 matrix 119911119901

119894

119909V119901119894119895 1 if vehicle V drives from site 119894 to 119895 in period 119901 0

otherwise (decision variable)119910V119901 1 if vehicle V is used in period 119901 0 otherwise

(decision variable)119911119901

119894 1 if there is an arrival at retailer 119894 in period 119901 0

otherwise (decision variable)119906119901 1 if there is production in period 119901 0 otherwise

(decision variable)

3 Model Formulation

31 Cost Analysis The total costs include the supplierrsquos cost119885119878 the transportation cost 119885119879 the retailersrsquo cost 119885119862 and the

cost of returning the time interval penalty 119885119868The supplierrsquos cost includes the production setup cost

sum119901isin119867

119891119904

119906119901 and the inventory holding cost sum

119901isin119867ℎ119904

119868119901+1

where sum119901isin119867

ℎ119904

119868119901+1

= sum119901isin119867

ℎ119904

(119868119901

+ 119898119901

minus sum119894isin119878119908119901

119894) The

inventory of the supplier at the beginning of period 119901 is119868119901

= 119868119901minus1

minus sum119894isin119878119908119901minus1

119894+ 119898119901minus1 119898119901 le 119862119906119901 which means that

the inventory of the supplier at the beginning of period 119901 isdetermined by the inventory at the beginning of period 119901minus1the production quantity and the delivery amount in period119901

The transportation cost includes the vehiclefixed usage cost sum

119901isin119867sumVisin119881 119891

V119910V119901 the driving cost

sum119901isin119867

sumVisin119881sum119895isin119878+ sum119894isin119878+ 119888V119889119894119895119909V119901119894119895 the loading cost

sum119901isin119867

sum119894isin119878+ sum119895isin119878+ 119888119892

119889119894119895119909V119901119894119895119897V119901119894119895 and the cost of arriving

overtime at the retailers sum119901isin119867

sumVisin119881sum119894isin119878 119887119908119901

119894max119905119901

119894119911119901

119894minus

119905max 1 0 Deteriorating item quality is a fuzzy conceptThe actual quality of the deteriorating items is changingor decreasing due to their deteriorating characteristics forexample the amount of bacterial flora is changing Howeverin reality these items are still regarded as qualified products


because they are within the time window If the delivery timeto the retailers exceeds the appointed time (still within thetime window) the retailers might not completely reject theproducts but they might allow the suppliers to provide somecompensation Therefore we set two thresholds 119905max 1 and119905max 2 (119905max 2 ge 119905max 1) and the vehicle driving time should notexceed 119905max 2 When the driving time is smaller than 119905max 1no extra penalty cost will be calculated but if the travellingtime 119905119901

119894is between 119905max 1 and 119905max 2 then extra compensation

cost will be incurredThe cost of each retailer includes the unloading cost120593 and

the inventory holding costsum119901isin119867

ℎ119888

119879(119868119901

119894minus 119889119901

119894) The inventory

of retailer 119894 at the beginning of period 119901 is equal to theinventory at the beginning of period 119901 minus 1 plus the productsthat arrived at the beginning of period 119901 and minus theinventory consumption in period119901minus1 that is 119868119901

119894= 119868119901minus1

119894+119908119901

119894minus

119889119901minus1

119894 The delivery of products to retailer 119894 at the beginning of

period119901 is the sumof the demandswithin the vehiclesrsquo arrivaltime intervals and the interval here is expressed as integermultiples of a unit period that is 119908119901

119894= 119911119901

119894sum119901isin119860119901

119894

119889119901

119894 In the

problem retailersrsquo demands of many periods can be met byone delivery which results in overstock and inventory cost atthe retailersTherefore the retailersrsquo cost is an important partof the objective function Moreover the production decisionand distribution decision will change the retailersrsquo inventorystatus thus consideration of the retailersrsquo cost is a morecomprehensive and practical method

The cost of the two successive vehiclesrsquo returning timeinterval penalty issum

119901isin119867sum119894isin119881119901 sum119895isin119881119901 119894 = 119895

119888119901max119894119905Vminus|end119905119901

119894minus

end119905119901119895| 0 Assume that the lower bound of the time interval

of the vehiclesrsquo returning to the supplierrsquos facilities is 119894119905V Ifthe actual interval is less than the threshold value that is|end119905119901119894minus end119905119901

119895| lt 119894119905V then the cost of the returning time

interval penalty will be calculated

32 Mathematical Model The objective function and con-straints are listed as follows over the planning horizon

min119885 = sum119901isin119867

(119891s119906119901

+ ℎs119868119901

) +sum

119894isin119878

sum

119901isin119867

[ℎc119879 (119868119901

119894minus 119889119901

119894) + 120593119911

119901

119894]

+ sum

119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

+ sum

119901isin119867

sum

119894isin119881119901

sum

119895isin119881119901 119894 = 119895

119888119901max 119894119905Vminus10038161003816100381610038161003816end119905

119901

119894minus end119905119901

119895

10038161003816100381610038161003816 0

(1)

subject to

119898+120576119894

sum

119901=119898

119911119901

119894ge 1 forall119898 (119898 + 120576

119894) isin 119867 forall119894 isin 119878 (2)

119911119901

119894= sum

Visin119881sum

119895isin119878+

119909V119901119895119894 forall119894 isin 119878 119901 isin 119867 (3)

sum

Visin119881sum

119894isin119878+

119909V119901119894119895le 1 forall119895 isin 119878 119901 isin 119867 (4)

sum

119894isin119878+

119909V119901119894119895minus sum

119896isin119878+

119909V119901119895119896= 0 forallV isin 119881 119895 isin 119878 119901 isin 119867 (5)

119910V119901le sum

119894isin119878

sum

119895isin119878+

119909V119901119894119895 forallV isin 119881 119901 isin 119867 (6)

119909V119901119894119895minus 119910

V119901le 0 forallV isin 119881 119894 isin 119878+ 119895 isin 119878+ 119901 isin 119867 (7)

119908119901

119894= 119911119901

119894sum

119901isin119860119901

119894

119889119901

119894 forall119894 isin 119878 119901 isin 119867 (8)

sum

119894isin119878V119901

119908119901

119894le 120595

V forall119901 isin 119867 V isin 119881 (9)

119905119901

119894119895=

119889119894119895

119904119901 forall119894 isin 119878

+

119895 isin 119878+

(10)

119905119901

119895119909V119901119894119895minus 119905119901

119894119909V119901119894119895= 119905119901

119894119895119909V119901119894119895 forall119894 isin 119878

+

119895 isin 119878+

V isin 119881 119901 isin 119867(11)

119905119901

119894119911119901

119894le 119905max 2 forall119894 isin 119878 (12)

sum

119895isin119878

119894 = 119895

sum

Visin119881(119897

V119901119895119894119909V119901119895119894minus 119897

V119901119894119895119909V119901119894119895) = 119889119901

119894 forall119894 isin 119878 119901 isin 119867

(13)

1198681

119894= 0 forall119894 isin 119878 (14)

119868119901

119894= 119868119901minus1

119894+ 119908119901

119894minus 119889119901minus1

119894 forall119894 isin 119878 119901 isin 119867 1 (15)

1198681

= 0 (16)

119868119901

= 119868119901minus1

minussum

119894isin119878

119908119901minus1

119894+ 119898119901minus1

forall119901 isin 119867 1 (17)

119898119901

le 119862119906119901

forall119901 isin 119867 (18)

sum

119894isin119878

1199081

119894le 1198981

(19)

sum

119894isin119878

119908119901

119894le 119868119901

+ 119898119901

forall119901 isin 119867 1 (20)

The model is a mixed integer programming formulation(because there are 0-1 decision variables and continuousdecision variables) Constraint (2) represents that the timeinterval between the vehicles that service a retailer cannotexceed a certain value to avoid shortages it means that afterthe arrival of deteriorating items to retailer 119894 in period mthere must be at least one other arrival to retailer 119894 beforethe period 119898 + 120576

119894 Constraint (3) indicates that when there

is a product arrival at retailer 119894 a vehicle must be arrangedto serve the retailer in this period Constraint (4) shows thatthe demand of a retailer cannot be divided Route continuityis enforced by constraint (5) Constraints (6) and (7) indicatethe constraint relationship between 119910V119901 and 119909V119901

119894119895 Constraint

(8) guarantees that there are no shortages at retailers Heresum119901isin119860119901

119894

119889119901

119894represents the sum of the demands of retailer 119894


between its two successive arrivals119860119901119894denotes the set of peri-

ods between two successive deliveries to retailer 119894 fromperiod119901 For example retailer 1 is served in periods 1 and 4 and nowperiod 119901 is period 1 and1198601

1represents the period set 1 2 3

The demands in 11986011will be delivered to retailer 1 in period 1

because the next delivery will occur in period 4 and shortagesare not permitted Constraint (9) is the vehicle capacityconstraint Constraint (10) defines the time that it takes for thevehicles to drive from retailer 119894 to 119895 in one period Constraint(11) states the relationship between the absolute time 119905119901

119894and

the relative time 119905119901119894119895 Constraint (12) is the upper bound of

the delivery time Constraint (13) shows the logic relationshipbetween the demand of retailer 119894 and the vehicle loads on thetwo arcs that link retailer 119894 Constraints (14) and (16) representthat the initial inventories of the retailers and supplier are zeroin period 0 Constraints (15) and (17) are the inventory flowbalance equations Constraint (18) bounds the production ineach period to the capacity of the supplier Constraints (19)and (20) indicate the maximal delivery quantity in period 119901

4 Solution Algorithm

41 Solution Procedure Outline The IRP is NP-hard becauseit includes the vehicle routing problem (VRP) as a sub-problem It might not be possible to calculate its exact valueespecially for large problem sizes An efficient heuristic how-ever can be used to approximate it this approach has beenimplemented in previous research see for example Popovicet al [22] and Boudia et al [19 20]Therefore in this sectionwe propose a heuristic algorithm for this NP-hard problem

A key decision in solving the IRP is whether to deliverto retailer 119894 in period 119901 because a binary variable which wedefine as 119911119901

119894 effectively separates the production distribution

and inventory problems Thus in this paper we define theproblem of solving the arrival matrix 119911119901

119894as themain problem

and the problems of supplier decisions and vehicle routingdecisions are two sub-problems SUB1 and SUB2

After giving an arrival matrix for all of the retailersand all of the periods the amount delivered to retailer 119894 inperiod119901 can be calculated according to the retailersrsquo demandsin each period because the shortages are not permittedObtaining the delivery period and amounts the suppliercan meet the production scheduling (production periodproduction quantity and relative cost) At the same timethe best routing solution for these delivery amounts can beobtained by solving 120591 separate capacitated vehicle routingproblems Therefore identifying the arrival matrix is a keyissue because what is left are production scheduling andvehicle routing problems

42 Algorithm Implementation

(1)Master ProblemThemodel of the master problem is givenas follows

min119885 = sum119894isin119878

sum

119901isin119867

[ℎ119888

119879 (119868119901

119894minus 119889119901

119894) + 120593119911

119901

119894] + 119885119878

+ 119885119879

+ 119885119868

(21)

subject to constraints (2) (8) (14) (15) and

119908119901

119894le 120595


119901

sum

119895=1

sum

119894isin119878

119908119895

119894le 119901119862 (23)

Constraint (22) indicates that the demand of the retailersmust not be divided which means that the delivery quantityto a retailer cannot exceed the vehicle capacity Constraint(23) enforces that the supplierrsquos production capacity con-straint must be met (as is stated in assumption (1))

To solve the main problem we proposed a tabu searchalgorithm (TS) to decide the binary variable 119911119901

119894 Many appli-

cations such as job shop scheduling graph coloring TSPand other vehicle routing problems have been successfullytackled by TS see for example Glover and Laguna [23]for an overview and Fallahi et al [24] In the followingthe implemented TS is completely described by specifyingthe other components namely the initial solution solutionevaluation neighbourhood definition and stop condition

Initial Solution The solution of the TS is an arrival matrix119911119901

119894 which represents whether to deliver to retailer 119894 in period

119901 When the master problem is not optimised we initialisethe solution as all 119911119901

119894= 1 in other words all of the retailers

receive a delivery in each period

Solution Evaluation After obtaining an arrival matrix and ifthe generated matrix 119911119901

119894meets the relative constraints SUB1

and SUB2 are solved to obtain the total cost This cost is usedto evaluate the solutionrsquos performance

Neighbourhood Definition A neighbourhood is a set ofpoints that can be reached from the current solution byperforming one or more moves For this problem we definethe neighbourhood of the current solution as all feasiblearrival matrixes 119911119901

119894 in other words the values are changed

for some of the 119911119901119894in the current solutionWhen choosing the

changed 119911119901119894 a variable neighbourhood search is presented At

the early stage of the search a number of retailers (not in thetabu list) are randomly selected from the chosen period ascandidates (the number is referred to as count) then num thenumber of retailers is randomly chosen from the candidatesto make a conversion simultaneously (apparently num is notgreater than count) In the later stages of the search num isset to a small value to make a more accurate search

Tabu listThe tabu list stores forbiddenmoves to avoid havingthe local search procedure trapped in a local optimum Thelength of the tabu list in our algorithm is not constant Thetabu list can store a certain number of retailers and thenumber is related to the number of retailers and periods Ifthe current solution can produce a better result then these119911119901

119894that are in the solution will be added to the tabu list At

the same time the tabu status of the same amount of 119911119901119894

will be overridden according to the first-in-first-out principleEach entry on the tabu list is a combination of a customer


and period which is represented by the 2-dimensional vector(customer 119894 and period 119901)

Stop Condition If the search does not find a better solutionwithin the predetermined steps such as 500 steps then it willstop The steps can be changed according to the size of theproblem

Algorithm DescriptionThe solution procedure for the masterproblem is listed as follows

Step 1 Initialisation For every retailer 119894 and period119901 let 119911119901119894=

1 Solve SUB1 and SUB2 and save the results as 119878best 119878rem and119878this Calculate the total costs and save them as119862best and119862reminitialise the tabu list 119879119860119861119880 = and countnobetter=0

Step 2 Tabu search

Substep 1 Let 119904119886V119890119887119890119904119905 = minusinfin 119870 = and1198701=

Substep 2 Select one period 119901 randomly from the set119867 1in which not all of the retailers are set as tabu Choose thecount size of the different retailers from those that are not intabu

Substep 3 If countnobetter lt 20 then produce a randomnumber num as the number of retailers in a group else let119899119906119898 = 1 Put the selected count groups of retailers from set119870 into set 119870

1

Substep 4 Choose one group from set 1198701and change the

selected retailersrsquo 119911119901119894value in solution 119878rem in the current

period If the newly generated matrix 119911119901119894meets the relative

constraints then solve SUB1 and SUB2 and save the resultsas 119878this and the total cost as 119862this Let 119904119886V119890 = 119862rem minus 119862this If119904119886V119890 gt 119904119886V119890119887119890119904119905 then let 119904119886V119890 = 119904119886V119890119887119890119904119905 If 119870

1= then go

to Substep 4 otherwise go to Substep 5

Substep 5 If 119904119886V119890119887119890119904119905 = minus infin then put the transferred 119911119901119894of

solution 119878now into the tabu list according to the first-in-first-out principle and let 119878rem = 119878now and 119862rem = 119862now

Substep 6 If 119862best minus 119862rem gt 0 then let countnobetter= 0 119878best = 119878rem and 119862best = 119862rem otherwise letcountnobetter++

Substep 7 If countnobetter lt 500 then go to Substep 1otherwise break out of the cycle and go to Step 3

Step 3 Output the results Output the final results 119878best andthe total cost 119862best

(2) Supplierrsquos Production Decision Problem (SUB1)Themodelof SUB1 is listed as follows

min119885119878 = sum119901isin119867

(119891119904

119906119901

+ ℎ119904

119868119901

) (24)

subject to constraints (8) (16) (17) and (18)

SUB1 decides the 0-1 variable 119906119901 and the continuousvariable119898119901 which is to determinewhen and howmany itemsthe supplier prodeces Because the structure of the solutionsis simple we adopt the saving algorithm and save a value thatis equal to the setup cost minus the inventory cost The latestproduced principle is used in the production arrangementwhichmeans that whenwe consider the production in period119901 only when the production capacity is insufficient willwe arrange the excess production quantity to the nearestsatisfactory period

The flowchart of the solution procedure for SUB1 isdepicted in Figure 1

(3) Vehicle Routing Problem (SUB2) The model of SUB2 islisted as follows

min119885119879 = sum119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

(25)

subject to constraints (3)sim(6) and (9)sim(13)SUB2 decides 0-1 for the variable 119909V119901

119894119895and adopts a

neighbourhood search The neighbourhood structure is theimproved exchange of a 2-opt path When exchanging thepaths some retailers in one path are inserted into the otherpath according to a minimal insert cost criterion where thecost of this inserting method is

sum

119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

+ sum

119901isin119867

sum

119894isin119881119901

sum

119895isin119881119901 119894 = 119895

119888119901max 119894119905V minus 10038161003816100381610038161003816end119905

119901

119894minus end119905119901

119895

10038161003816100381610038161003816 0

(26)

To speed up obtaining the optimal solution the algorithmdeveloped by Taillard et al [25] is referred to when choosingthe exchanged paths thus paths that have a nearer distancewill become easier to be selected The CLOSE matrix isdefined in this algorithm as the closeness of two paths andthe exchange probability of near paths will be larger than thatof far paths

Definition 1The CLOSE matrix is a two-dimensional matrixin which the number of both rows and columns is equal tothe number of paths (or the number of vehicles used) Thismatrix represents the degree of closeness between two givenpaths CLOSE[119896][119897] refers to the value of closeness betweenpath 119896 (which contains 119898 retailers that belong to 119878

119896) and

path 119897 (which contains 119899 retailers that belong to 119878119897) in other


Start

InitializationThe supplier will produce in each period

p = 2

up = 1

Yes

Yes

Yes

Yes

Yes

No

No

No

NoNo

Let up = 0 and arrange the production in period pto previous periods according to the latest production principle

The production capacity inprevious periods is sufficient

Calculate the save value = production setup cost-added inventory cost

Is save value the lowest so far

Save the value and the solution

p = p + 1

p isin H

The save value = 0

Stop and output the results

This solution is notavailable

Figure 1 Flowchart of the solution procedure for SUB1

words CLOSE[119896][119897] = (sum119894isinS119896

sum119895isinS119897

119889119894119895)(119898 times 119899) When 119896 = 119897

CLOSE[119896][119897] = +infinWhen choosing the exchanged paths based on theCLOSE

matrix the probability of selecting the pair path 119896 and path 119897is directly proportional to 1CLOSE[119896][119897] (repeated selectionis permitted)


5 Computational Experiments and Analysis

51 Numerical Example An example is adopted to test theeffectiveness of the proposed model and the impact of someof the parameters on the decisions The relative parametersare as follows The supply chain is made of one supplierand 12 retailers The retailersrsquo demands are given in Table 1The distance between any two nodes is within 25 and theplanning time horizon in this example consists of sevenperiods each representing one day Table 2 provides the

distance matrix for the nodes The vehicle velocities in eachperiod are 35 40 40 40 35 30 and 30 respectively Thesupplierrsquos production setup cost is 1000 per order and theinventory cost is 008 per day per unit The daily productioncapacity is 1500 The fixed usage cost of the vehicles is 10and the driving cost is 1 per kilometre The longest drivingtime threshold value without the penalty 119905max 1 is 2 hoursand 005 would be penalised per hour per unit after 2 hoursbut within 4 hours that is 119905max 2 = 4 The vehicle capacity is150 units The cost of delivering a product of per unit weightand per unit distance is 005 The lower bound of the timeinterval of the vehiclesrsquo returning to the supplierrsquos facilities119894119905V is 05 hour if the actual interval is less than the thresholdvalue then the cost of a unit time is 5The holding cost of theretailers is 01 per day per unit and the unloading cost is 50The deteriorating itemrsquos shelf life at the retailer sites is 3 days

The model is coded and run on the platform of VisualC++ 60 All of the tests are performed on the Intel(R) Core(TM)2 Duo at 200GHz with 2GBRAMunder theMicrosoft


Start

p = 1

Initialization Assign a vehicle to each of the retailers who havezp

i= 1 Save the cost as countnobetter = 0

Choose the exchanged paths according to the CLOSE matrix

Select one pair and the exchange paths according to the minimal insert cost criterion

Do new paths meet constraints (9) and (12)

Save the solution and calculate the save value

The save value is the lowest so far

Save the value

All pairs of paths been exchanged

The lowest value lt

Countnobetter ++

Countnobetter lt 500

Yes

Yes

YesYes

Yes

Yes

No

No

No

No

No

No

p = p + 1

p isin H


Countnobetter = 0= the lowest value

Cbest

Cbest Cbest


Table 1 Retailersrsquo demand in each period

Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 223 303 379 311 295 212 421 132 121 204 153 2512 248 379 512 423 344 270 557 256 129 223 170 2823 352 513 688 587 458 415 833 548 182 279 223 3134 363 525 713 603 458 427 840 567 193 281 229 3275 308 452 578 525 401 357 679 433 155 252 201 3056 251 377 499 438 346 277 530 251 134 225 177 2767 398 589 732 659 480 479 757 677 213 303 252 344


Table 2 Distance matrix of nodes

Nodes Nodes0 1 2 3 4 5 6 7 8 9 10 11 12

0 0 55 17 38 34 67 67 66 61 85 108 142 1511 0 43 9 85 123 88 67 111 138 163 197 2062 0 55 42 81 79 72 68 95 121 156 1633 0 39 39 59 73 55 68 81 11 1234 0 45 91 97 27 53 8 117 1225 0 94 111 38 34 42 75 856 0 27 113 126 134 155 1737 0 122 14 153 177 1938 0 29 6 99 109 0 33 72 7110 0 39 4311 0 2512 0

Table 3 Arrival matrix of retailers

Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 1 0 0 1 1 03 1 1 1 0 0 0 0 0 1 0 0 04 0 1 1 1 1 1 1 1 0 0 0 15 0 0 0 0 1 1 1 0 0 1 1 16 1 0 1 1 0 0 0 1 1 0 0 07 0 1 0 0 1 0 1 0 1 0 0 0

Table 4 Supplierrsquos production vectors

Vectors ValuePeriod 1 2 3 4 5 6 7119906119901 1 0 0 1 0 1 0119898119901 13718 0 0 12229 0 6078 0

Table 5 Distribution paths of vehicles

Period Paths Transportationcost

1 0-2-1-0 0-10-12-0 0-3-0 0-4-00-6-7-0 0-8-9-11-0 0-5-0 458879

2 0-7-0 0-10-11-0 1959693 0-1-0 0-2-0 0-3-9-0 1371554 0-8-5-0 0-7-6-0 0-2-0 0-4-12-0 0-3-0 3057665 0-6-0 0-10-11-0 0-12-0 0-7-0 0-5-0 4095696 0-1-0 0-8-9-0 0-4-0 0-3-0 1771197 0-2-7-0 0-5-9-0 119629

Windows 7 operation system The results are presented inTables 3 4 5 and 6

The original solution is the following in each period thesupplier produces and each retailer receives a delivery Eachvehicle will serve only one retailer per trip The total cost

Table 6 Cost distribution

Value

Supplierrsquos cost Production setup cost 3000inventory cost 125968 Total 312597

Transportationcost See Table 4 Total 180409

Retailersrsquo cost Unloading cost 2100inventory cost 21117 Total 231117

Cost of timeinterval penalty Total 220

Total cost 746123

of the initial solution is 162872 (which is referred to as alower bound of 1) Second when only the master problemis not optimised which means that all of the retailers areserved in each period the supplier makes his productiondecision with the saving algorithm and vehicle routes aremade with the neighbourhood search The total cost of thissolution is 981152 which is another low-value limitationof our numerical example (referred to as lower bound 2)This strategy is common in practice for example a supplierdelivers different types of goods to stores once a day and onlythe production scheduling and vehicle routes are optimisedFinally when the master problem and two subproblems areall optimised the result is our final solution and the total


Table 7 Results of different shelf-life 120576 in the retail stage

120576Largest delivery

intervalNumber ofdeliveries Retailersrsquo cost Supplierrsquos cost

(production periods)Transportation

costCost of time

interval penalty Total cost

1 1 84 4200 319893 (1 3 5) 222759 185 9811522 2 52 272522 313474 (1 3 6) 190471 205 7969683 3 42 231117 312597 (1 4 6) 180409 220 7461234 4 36 21096 309867 (1 4 5) 174071 250 7198985 5 37 210083 310735 (1 4 6) 174471 230 7182896 5 37 211246 30961 (1 4 6) 17373 270 7215267 6 37 213764 311514 (1 3 6) 176761 200 722039

Table 8 Results of different upper bounds for the vehicle driving time 119905max 1 and 119905max 2

119905max1 119905max2 Retailersrsquo cost Supplierrsquos cost(production periods)

Transportationcost

Cost of timeinterval penalty

Drivingdistance Total cost

05 25 236385 309729 (1 4 5) 183889 230 4509 75300308 28 229261 308132 (1 3 5) 182608 275 4482 74750111 31 23415 308947 (1 3 5) 181876 230 4465 74797313 33 230983 311138 (1 3 5) 182952 195 4532 74457317 37 231117 312597 (1 4 6) 180409 220 4419 7461232 4 231117 312597 (1 4 6) 180409 220 4419 746123

cost is 746123 The optimised total cost is 5419 lower thanthe lower bound of 1 and 2395 lower than the lower boundof 2 The final results demonstrate that a mixed strategythat considers the supplier retailers and distribution routessimultaneously can save a substantial amount of cost in thedistribution channel

From Tables 3-4 we can see that in the three productionperiods the retailersrsquo goods arrival rates are all larger thanthose in the nonproduction periods The average arrivalnumber in the production periods is 833 compared to425 in the nonproduction periods It can be concludedthat production decisions and arrival decisions are closelyconnected

If the shelf life of a considered deteriorating item isunlimited then the total cost will be 630671 which is 24084lower than that in the case (323) Thus accounting forthe corruption characteristics of the perishable productswill change the decisions in the supply chain additionallyconsidering the supplier costs and shelf life of the perishablegoods in the model will have an obvious impact on thesolutions and the total costs

52 Sensitivity Analysis The solution algorithm that is pro-posed above is derived from several existing mature algo-rithms that is a neighbourhood search tabu search and sav-ing algorithm as a result in this section sensitivity analysiswill be conducted mainly to illustrate the effectiveness of theproposed model The parameters that are considered hereinclude the storage life at the retailer sites the upper boundsof the vehicle driving time the unit loading cost the vehiclesrsquoreturn time interval and the driving cost per unit distance

Table 7 shows that with an increase in the shelf life at theretailer sites it is more likely that a lower total relevant cost

will be attained However due to the constraints of the othercosts in the model even if the shelf life at the retailer siteskeeps rising the decrease in the total cost is limited All of theabove shows that the improvement in the perishable goodsshelf management at the retailersrsquo sites could have a positiveinfluence on the total cost while the sales stage is only partof the deterioration in the itemsrsquo life cycles the impact on thetotal integrated cost is limited to a certain range

Table 8 indicates that when the upper bounds of thevehicle driving time 119905max 1 and 119905max 2 are small their increasecould reduce the transportation cost thus reducing thetotal cost When the upper bounds are larger a furtherimprovementwould have little impactThen we can illustratethat improving the preservation condition of the vehiclescould reduce the total cost of the supply chain and improvethe goodsrsquo quality which is of great benefit in the food supplychain It is clear that the impacts of the vehiclersquos largest drivingtime limit 119905max 2 on the decision variables could achievesimilar results by changing the vehicle velocity as a resultthe sensitivity analysis of the vehicle velocity is omitted here

Table 9 shows that with an increase in the unit loadingcost the ratio of the unit loading cost to the vehiclersquos fixedusage cost also increases as well as the total number ofvehicles and the transportation cost it has little influence onthe supplierrsquos and retailersrsquo costs The variation extent of thetransportation cost is not as obvious as for the unit loadingcost This finding arises because more vehicles are used toavoid having a distance that is too long and a load that is toolarge on a vehicle which significantly restrains the increasingtrends in the transportation cost

From Table 10 we can see that as the value of 119894119905Vincreases that is the supplierrsquos facilities have even toughercriteria on the vehiclesrsquo returning time the cost of the time


Table 9 Results of different unit loading costs of vehicles 119888119892

119888119892 Vehicle numbers Retailersrsquo cost Supplierrsquos cost

(production periods) Transportation cost Total cost

0 26 227073 310153 (1 3 5) 7055 624276003 26 235268 310703 (1 4 6) 138368 700839005 28 231117 312597 (1 4 6) 180409 746123008 28 230055 312561 (1 4 6) 244352 81196801 28 230745 310843 (1 4 5) 284931 848519015 29 226472 30948 (1 3 5) 394428 9533802 31 229913 312012 (1 3 5) 497812 106624

Table 10 Results of different intervals of vehiclesrsquo returning time 119894119905119907

119894119905119907 Driving distance Retailersrsquo cost Supplierrsquos cost(production periods) Transportation cost Cost of time interval

penalty Total cost

0 437 230527 308956 (1 3 5) 179719 0 71920203 4494 23003 309192 (1 4 6) 181748 195 7404705 4419 231117 312597 (1 4 6) 180409 220 74612308 4555 235768 310155 (1 4 6) 184253 265 7566761 4394 234704 312683 (1 4 6) 181926 255 75481312 4579 23448 314667 (1 4 6) 182735 225 75438215 4383 236642 308349 (1 3 5) 180522 280 7535132 4508 234047 313886 (1 4 6) 181584 260 755517

Table 11 Results of different driving costs per unit distance 119888V

119888V Driving distance Retailersrsquo cost Supplierrsquos cost Transportation cost Cost of time interval

penalty Total cost

025 4386 234611 310482 146507 230 714605 4527 226242 31041 158853 205 7160051 4419 231117 312597 180409 220 74612315 461 231057 311007 205359 240 7714232 4531 230277 312784 22819 215 7927515 4268 236455 31163 349259 175 91484510 4336 23922 311318 573219 215 11452620 4071 248607 31081 94919 190 152761

interval penalty increases as well as the total cost of thesystem Therefore if the facilities have sufficient capacity toaccommodate more vehicles simultaneously then we willhave a higher chance to decrease the total relevant cost of thesupply chain

From Table 11 the results reveal that when the drivingcost per unit distance 119888V increases the transportation costincreases and in addition the total relevant cost increasesBecause of other constraints of the proposed model thevariation extent of decreasing the travelling distance of thevehicles is not as obvious as that of the driving cost per unitdistance

When 119888119892 and 119894119905V are equal to 0 the proposed model istransformed into the classical IRP with time windows andthe total cost is 606011

6 Conclusions

Accounting for the loading cost and the vehiclesrsquo return-ing time interval a new formulation of a production-distribution-inventory IRPmodel under discrete time is pro-posed When making integrated decisions for deterioratingitems in a physical distribution channel it is reasonable toconsider time windows that are in a transit stage as well as asales stage to control the product quality With the objectiveof minimising the total relevant cost this model determinesthe supplierrsquos production plan the retailersrsquo delivery time andthe vehicle routing in each period over the planning horizonBased on the features of the problem a two-phase algorithmis developed A numerical example and sensitivity analysishave also been conducted to demonstrate the effectiveness


of the model and algorithm The impact of the relevantparameter changes is also illustrated on the decision makingand the total relevant costs

The results show that the vehiclesrsquo loading cost andreturn time interval do have obvious impacts on the decisionvariables and considering the joint total relevant cost of thesupply chain will make the proposed model a useful tool inthe operation of the production and distribution systemswithdeteriorating items

The primary contributions of this paper lie in the con-sideration of the vehiclersquos loading cost and returning timeinterval as well as in the deteriorating itemsrsquo quality controlin an integrated supply chain However several factors suchas multiple products split pickup and other factors are notconsidered These factors can be explored in future studiesand their consideration would help to make the inventoryrouting problem more realistic additionally testing theeffectiveness of the proposed algorithm will be the primaryundertaking in our future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors greatly thank the editor and the anonymousreferees for their insightful comments and suggestions whichhave significantly improved this paper The research pre-sented in this paper was supported by the National NaturalScience Foundation Project of China (71271168) the NationalSocial Science Foundation Project of China (1282D070) theFundamental Research Funds for the Central Universitiesthe Program for New Century Excellent Talents in University(NCET-13-0460) and the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1173)

References

[1] F Raafat ldquoSurvey of literature on continuously deterioratinginventory modelsrdquo The Journal of the Operational ResearchSociety vol 42 no 1 pp 27ndash37 1991

[2] L J S Lukasse and J J Polderdijk ldquoPredictive modellingof post-harvest quality evolution in perishables applied tomushroomsrdquo Journal of Food Engineering vol 59 no 2-3 pp191ndash198 2003

[3] C-I Hsu S-F Hung and H-C Li ldquoVehicle routing problemwith time-windows for perishable food deliveryrdquo Journal ofFood Engineering vol 80 no 2 pp 465ndash475 2007

[4] A Rong R Akkerman and M Grunow ldquoAn optimizationapproach for managing fresh food quality throughout thesupply chainrdquo International Journal of Production Economicsvol 131 no 1 pp 421ndash429 2011

[5] A Federgruen and P Zipkin ldquoA combined vehicle routing andinventory allocation problemrdquo Operations Research vol 32 no5 pp 1019ndash1037 1984

[6] S Anily and A Federgruen ldquoOne warehouse multiple retailersystems with vehicle routing costsrdquo Management Science vol36 no 1 pp 92ndash114 1990

[7] A J Kleywegt V S Nori and M W P Savelsbergh ldquoThestochastic inventory routing problem with direct deliveriesrdquoTransportation Science vol 36 no 1 pp 94ndash118 2002

[8] E-H Aghezzaf B Raa and H Van Landeghem ldquoModelinginventory routing problems in supply chains of high consump-tion productsrdquo European Journal of Operational Research vol169 no 3 pp 1048ndash1063 2006

[9] J F Bard and N Nananukul ldquoHeuristics for a multiperiodinventory routing problem with production decisionsrdquo Com-puters amp Industrial Engineering vol 57 no 3 pp 713ndash723 2009

[10] T F Abdelmaguid MM Dessouky and F Ordonez ldquoHeuristicapproaches for the inventory-routing problem with backlog-gingrdquo Computers amp Industrial Engineering vol 56 no 4 pp1519ndash1534 2009

[11] M Savelsbergh and J-H Song ldquoAn optimization algorithmfor the inventory routing problem with continuous movesrdquoComputers amp Operations Research vol 35 no 7 pp 2266ndash22822008

[12] M Ben-Daya M Darwish and K Ertogral ldquoThe joint eco-nomic lot sizing problem review and extensionsrdquo EuropeanJournal of Operational Research vol 185 no 2 pp 726ndash7422008

[13] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[14] J F Bard and N Nananukul ldquoA branch-and-price algorithmfor an integrated production and inventory routing problemrdquoComputers amp Operations Research vol 37 no 12 pp 2202ndash22172010

[15] J Tang J Zhang and Z Pan ldquoA scatter search algorithmfor solving vehicle routing problem with loading costrdquo ExpertSystems with Applications vol 37 no 6 pp 4073ndash4083 2010

[16] W J Bell L M Dalberto M L Fisher et al ldquoImproving thedistribution of industrial gases with an on-line computerizedrouting and scheduling optimizerrdquo Interfaces vol 13 no 6 pp4ndash23 1983

[17] M Savelsbergh and J-H Song ldquoInventory routing with contin-uous movesrdquo Computers amp Operations Research vol 34 no 6pp 1744ndash1763 2007

[18] M G C Resende and C C Ribeiro ldquoGRASP with path-relinking recent advances and applicationsrdquo in Metaheuris-tics Progress as Real Problem Solvers vol 32 of Opera-tions ResearchComputer Science Interfaces Series pp 29ndash63Springer New York NY USA 2005

[19] M Boudia M A O Louly and C A Prins ldquoA memeticalgorithm with population management for a production-distribution problemrdquo Information Control Problems in Manu-facturing vol 12 no 1 pp 541ndash546 2006

[20] M Boudia M A O Louly and C Prins ldquoA reactive GRASPand path relinking for a combined production-distributionproblemrdquo Computers amp Operations Research vol 34 no 11 pp3402ndash3419 2007

[21] N H Moin S Salhi and N A B Aziz ldquoAn efficient hybridgenetic algorithm for themulti-productmulti-period inventoryrouting problemrdquo International Journal of Production Eco-nomics vol 133 no 1 pp 334ndash343 2011

[22] D Popovic M Vidovic and G Radivojevic ldquoVariable neigh-borhood search heuristic for the Inventory Routing Problem infuel deliveryrdquo Expert Systems with Applications vol 39 no 18pp 13390ndash13398 2012

[23] F Glover and M Laguna Tabu Search Kluwer AcademicPublishers Boston Mass USA 1997


[24] A E Fallahi C Prins and R W Calvo ldquoA memetic algorithmand a tabu search for the multi-compartment vehicle routingproblemrdquo Computers amp Operations Research vol 35 no 5 pp1725ndash1741 2008

[25] E Taillard P BadeauM Gendreau F Guertin and J-Y PotvinldquoA tabu search heuristic for the vehicle routing problem withsoft time windowsrdquo Transportation Science vol 31 no 2 pp170ndash186 1997

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Stochastic AnalysisInternational Journal of


different transportation and storage processes If the qualityrequirements were not satisfied then the goods would beabandoned and a penalty cost would be incurred

Transportation is an important logistics activity in thedeteriorating items distribution channel and it is also a keydecision that connects the whole supply network as an effec-tive integrated system Moreover for perishable productsthe transportation process is more difficult to manipulatebecause of the changing environmental factors thus moreattention should be paid to the transportation stage In a jointdecision-making supply chain the order quantity from thenetwork nodes is also an important factor that influences thetransportation decision Thus to achieve the minimal totalrelevant cost of the channel themanagers should consider thetransportation cost (the space value of the logistics) and theinventory cost (the time value of the logistics) simultaneouslyThe inventory routing problem (IRP) just seeks to determinesimultaneously an optimal inventory and distribution strat-egy that minimises the total cost thus it is one of the hotspots in recent academic research Federgruen and Zipkin [5]are amongst the first to study a single-day inventory routingproblem Customersrsquo demands were assumed to be a randomvariable A limited inventory at the distribution centre canlead to shortages at the customer sites as a result the holdingand shortage costs were considered in the decision alongwiththe transportation costs Some of the available methods forthe deterministic vehicle routing problem had been extendedto this caseThis paper clearly identified the research categoryof the inventory routing problem and is widely regarded asthe first real IRP model in academia Anily and Federgruen[6] proposed a class of strategies in which a collection ofregions (sets of retailers) was specified that cover all outlets ifan outlet belonged to several regions then a specific fractionof its salesoperations was assigned to each of these regionsEach time one of the retailers in a given region received adelivery this delivery was made by a vehicle that visited allof the other outlets in the region as well (in an efficient route)to achieve economies of scale This strategy was widely usedin follow-up studies of IRP Kleywegt et al [7] established anIRP model in which customersrsquo demands on different dayswere independent random vectors with a joint probabilitydistribution119865 that did not changewith timeThey formulatedthe inventory routing problem as a Markov decision processand proposed approximation methods to find good solutionsthat had reasonable computational effort Aghezzaf et al[8] used the concept of a vehicle multitour which impliesthat a vehiclersquos travel plan might possibly contain morethan one tour and vehicles could be reloaded Sales pointswithin a partition were divided into regions that were similarto the regions in the study by Anily and Federgruen [6]To solve the nonlinear mixed integer formulation of thisproblem a column generation-based approximation methodwas suggested

In some of the papers on the IRP the planning timehorizon is divided into integer periods according to somenatural properties of time that is the study of the inventoryrouting problem under discrete time A discrete time modelis conducive to solving a problem that has varying retailerdemand (changing with time) and the delivery time interval

must be integer multiples of a unit period thus it is easy tomanipulate in operations management practice The unit forthe discrete time is called one period and the definition ofa period depending on the context of the problem mightbe an hour a day or a week Bard and Nananukul [9]established an IRP model with discrete time By assessing aseries of heuristics algorithms they first developed a two-step procedure that estimated daily delivery quantities andthen solved the vehicle routing problem for each day overthe planning horizon Abdelmaguid et al [10] studied aninventory routing problem with backlogging under discretetime Based on a two-phase approach transitions that usethe idea of exchanging customersrsquo delivery amounts betweenperiods were conducted and the transitions could furtherreduce the total cost by balancing the inventory costsbackordering costs and transportation costs In most studiesof inventory routing models with discrete time the deliverieswere assumed to be finished within one single period butSavelsbergh and Song [11] investigated an inventory routingproblem with designed delivery tours that spanned severaldays covering enormous geographical areas and involvingproduct pickups at different facilities They developed aninteger programming-based optimisation algorithm that wasembedded in a local search procedure to improve solutionsthat were produced by a randomised greedy heuristic

With the increasing fierce global competition and con-tinuous advancement of information technology the idea ofhaving a decision that is integrated between the upstreamand downstream enterprises within a supply chain is widelyaccepted Based on this idea research about the jointeconomic lot-sizing problem (JELP) for supply chains hasbecome a hotspot in the area of inventory managementtheory referring to the review by Ben-Daya et al [12] fordetails Lacking consideration about distribution routinginformation makes the JELP model more suitable for themedium- and long-term planning process for enterprisesFurthermore focusing on retailersrsquo lot-sizing and distribu-tion routing the integrated inventory routing problems takesuppliersrsquo production costs and inventory costs into accountthus the coordinated production-distribution schedulingcan further reduce the total relevant cost of the supply chainLei et al [13] were the first to formulate the production-inventory-distribution-routing problem as a mixed-integerprogram and illustrated the benefits that were brought by thecoordination Bard and Nananukul [14] adopted a branch-and-price framework to solve the integrated inventory rout-ing problem A hybrid methodology that combined thecolumn generation heuristic bounded heuristic algorithmand tabu search algorithm was developedThe result demon-strated that the performance of hybrid methodology wasmuch better than that of eitherCPLEXor the standard branchand price algorithm

In this paper we consider an integrated production-distributionmodel It is assumed that no deterioration occursat the supplierrsquos factory (or facilities) because of good pro-tection and production conditions However the perishablegoods have different quality degradation processes duringthe transportation and sales stages Besides the retailers andcustomers have their own acceptable quality standards For











22 Assumptions







23 Notation

(1) Parameters






(2) Variables






119894119895=

119889119894119895119904119901)



119868119901


period 119901119908119901





119894






(decision variable)

3 Model Formulation



sum119901isin119867

119891119904


119901isin119867ℎ119904

119868119901+1


ℎ119904

119868119901+1

= sum119901isin119867

ℎ119904

(119868119901

+ 119898119901

minus sum119894isin119878119908119901

119894) The


= 119868119901minus1





119901isin119867sumVisin119881 119891


sum119901isin119867


sum119901isin119867

sum119894isin119878+ sum119895isin119878+ 119888119892




119894max119905119901

119894119911119901

119894minus







ℎ119888

119879(119868119901

119894minus 119889119901



119894= 119868119901minus1

119894+119908119901

119894minus

119889119901minus1



119894= 119911119901

119894sum119901isin119860119901

119894

119889119901

119894 In the




119888119901max119894119905Vminus|end119905119901

119894minus






min119885 = sum119901isin119867

(119891s119906119901

+ ℎs119868119901

) +sum

119894isin119878

sum

119901isin119867

[ℎc119879 (119868119901

119894minus 119889119901

119894) + 120593119911

119901

119894]

+ sum

119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

+ sum

119901isin119867

sum

119894isin119881119901

sum

119895isin119881119901 119894 = 119895

119888119901max 119894119905Vminus10038161003816100381610038161003816end119905

119901

119894minus end119905119901

119895

10038161003816100381610038161003816 0

(1)

subject to

119898+120576119894

sum

119901=119898

119911119901

119894ge 1 forall119898 (119898 + 120576

119894) isin 119867 forall119894 isin 119878 (2)

119911119901

119894= sum

Visin119881sum

119895isin119878+


sum

Visin119881sum

119894isin119878+


sum

119894isin119878+

119909V119901119894119895minus sum

119896isin119878+


119910V119901le sum

119894isin119878

sum

119895isin119878+


119909V119901119894119895minus 119910


119908119901

119894= 119911119901

119894sum

119901isin119860119901

119894

119889119901

119894 forall119894 isin 119878 119901 isin 119867 (8)

sum

119894isin119878V119901

119908119901

119894le 120595


119905119901

119894119895=

119889119894119895

119904119901 forall119894 isin 119878

+

119895 isin 119878+

(10)

119905119901

119895119909V119901119894119895minus 119905119901

119894119909V119901119894119895= 119905119901

119894119895119909V119901119894119895 forall119894 isin 119878

+

119895 isin 119878+

V isin 119881 119901 isin 119867(11)

119905119901

119894119911119901


sum

119895isin119878

119894 = 119895

sum

Visin119881(119897

V119901119895119894119909V119901119895119894minus 119897

V119901119894119895119909V119901119894119895) = 119889119901

119894 forall119894 isin 119878 119901 isin 119867

(13)

1198681

119894= 0 forall119894 isin 119878 (14)

119868119901

119894= 119868119901minus1

119894+ 119908119901

119894minus 119889119901minus1

119894 forall119894 isin 119878 119901 isin 119867 1 (15)

1198681

= 0 (16)

119868119901

= 119868119901minus1

minussum

119894isin119878

119908119901minus1

119894+ 119898119901minus1

forall119901 isin 119867 1 (17)

119898119901

le 119862119906119901

forall119901 isin 119867 (18)

sum

119894isin119878

1199081

119894le 1198981

(19)

sum

119894isin119878

119908119901

119894le 119868119901

+ 119898119901

forall119901 isin 119867 1 (20)






119894

119889119901








119894and













min119885 = sum119894isin119878

sum

119901isin119867

[ℎ119888

119879 (119868119901

119894minus 119889119901

119894) + 120593119911

119901

119894] + 119885119878

+ 119885119879

+ 119885119868

(21)


119908119901

119894le 120595


119901

sum

119895=1

sum

119894isin119878

119908119895

119894le 119901119862 (23)



119894 Many appli-

























Step 2 Tabu search




1





1= then go








min119885119878 = sum119901isin119867

(119891119904

119906119901

+ ℎ119904

119868119901

) (24)





min119885119879 = sum119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

(25)




sum

119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

+ sum

119901isin119867

sum

119894isin119881119901

sum

119895isin119881119901 119894 = 119895

119888119901max 119894119905V minus 10038161003816100381610038161003816end119905

119901

119894minus end119905119901

119895

10038161003816100381610038161003816 0

(26)



119896) and



Start


p = 2

up = 1

Yes

Yes

Yes

Yes

Yes

No

No

No

NoNo






p = p + 1

p isin H

The save value = 0






119889119894119895)(119898 times 119899) When 119896 = 119897









Start

p = 1








Save the value


The lowest value lt

Countnobetter ++


Yes

Yes

YesYes

Yes

Yes

No

No

No

No

No

No

p = p + 1

p isin H



Cbest

Cbest Cbest



Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 223 303 379 311 295 212 421 132 121 204 153 2512 248 379 512 423 344 270 557 256 129 223 170 2823 352 513 688 587 458 415 833 548 182 279 223 3134 363 525 713 603 458 427 840 567 193 281 229 3275 308 452 578 525 401 357 679 433 155 252 201 3056 251 377 499 438 346 277 530 251 134 225 177 2767 398 589 732 659 480 479 757 677 213 303 252 344



Nodes Nodes0 1 2 3 4 5 6 7 8 9 10 11 12

0 0 55 17 38 34 67 67 66 61 85 108 142 1511 0 43 9 85 123 88 67 111 138 163 197 2062 0 55 42 81 79 72 68 95 121 156 1633 0 39 39 59 73 55 68 81 11 1234 0 45 91 97 27 53 8 117 1225 0 94 111 38 34 42 75 856 0 27 113 126 134 155 1737 0 122 14 153 177 1938 0 29 6 99 109 0 33 72 7110 0 39 4311 0 2512 0


Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 1 0 0 1 1 03 1 1 1 0 0 0 0 0 1 0 0 04 0 1 1 1 1 1 1 1 0 0 0 15 0 0 0 0 1 1 1 0 0 1 1 16 1 0 1 1 0 0 0 1 1 0 0 07 0 1 0 0 1 0 1 0 1 0 0 0





1 0-2-1-0 0-10-12-0 0-3-0 0-4-00-6-7-0 0-8-9-11-0 0-5-0 458879

2 0-7-0 0-10-11-0 1959693 0-1-0 0-2-0 0-3-9-0 1371554 0-8-5-0 0-7-6-0 0-2-0 0-4-12-0 0-3-0 3057665 0-6-0 0-10-11-0 0-12-0 0-7-0 0-5-0 4095696 0-1-0 0-8-9-0 0-4-0 0-3-0 1771197 0-2-7-0 0-5-9-0 119629




Value





Total cost 746123







costCost of time


1 1 84 4200 319893 (1 3 5) 222759 185 9811522 2 52 272522 313474 (1 3 6) 190471 205 7969683 3 42 231117 312597 (1 4 6) 180409 220 7461234 4 36 21096 309867 (1 4 5) 174071 250 7198985 5 37 210083 310735 (1 4 6) 174471 230 7182896 5 37 211246 30961 (1 4 6) 17373 270 7215267 6 37 213764 311514 (1 3 6) 176761 200 722039



Transportationcost



05 25 236385 309729 (1 4 5) 183889 230 4509 75300308 28 229261 308132 (1 3 5) 182608 275 4482 74750111 31 23415 308947 (1 3 5) 181876 230 4465 74797313 33 230983 311138 (1 3 5) 182952 195 4532 74457317 37 231117 312597 (1 4 6) 180409 220 4419 7461232 4 231117 312597 (1 4 6) 180409 220 4419 746123














0 26 227073 310153 (1 3 5) 7055 624276003 26 235268 310703 (1 4 6) 138368 700839005 28 231117 312597 (1 4 6) 180409 746123008 28 230055 312561 (1 4 6) 244352 81196801 28 230745 310843 (1 4 5) 284931 848519015 29 226472 30948 (1 3 5) 394428 9533802 31 229913 312012 (1 3 5) 497812 106624



penalty Total cost

0 437 230527 308956 (1 3 5) 179719 0 71920203 4494 23003 309192 (1 4 6) 181748 195 7404705 4419 231117 312597 (1 4 6) 180409 220 74612308 4555 235768 310155 (1 4 6) 184253 265 7566761 4394 234704 312683 (1 4 6) 181926 255 75481312 4579 23448 314667 (1 4 6) 182735 225 75438215 4383 236642 308349 (1 3 5) 180522 280 7535132 4508 234047 313886 (1 4 6) 181584 260 755517



penalty Total cost

025 4386 234611 310482 146507 230 714605 4527 226242 31041 158853 205 7160051 4419 231117 312597 180409 220 74612315 461 231057 311007 205359 240 7714232 4531 230277 312784 22819 215 7927515 4268 236455 31163 349259 175 91484510 4336 23922 311318 573219 215 11452620 4071 248607 31081 94919 190 152761




6 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















22 Assumptions







23 Notation

(1) Parameters






(2) Variables






119894119895=

119889119894119895119904119901)



119868119901


period 119901119908119901





119894






(decision variable)

3 Model Formulation



sum119901isin119867

119891119904


119901isin119867ℎ119904

119868119901+1


ℎ119904

119868119901+1

= sum119901isin119867

ℎ119904

(119868119901

+ 119898119901

minus sum119894isin119878119908119901

119894) The


= 119868119901minus1





119901isin119867sumVisin119881 119891


sum119901isin119867


sum119901isin119867

sum119894isin119878+ sum119895isin119878+ 119888119892




119894max119905119901

119894119911119901

119894minus







ℎ119888

119879(119868119901

119894minus 119889119901



119894= 119868119901minus1

119894+119908119901

119894minus

119889119901minus1



119894= 119911119901

119894sum119901isin119860119901

119894

119889119901

119894 In the




119888119901max119894119905Vminus|end119905119901

119894minus






min119885 = sum119901isin119867

(119891s119906119901

+ ℎs119868119901

) +sum

119894isin119878

sum

119901isin119867

[ℎc119879 (119868119901

119894minus 119889119901

119894) + 120593119911

119901

119894]

+ sum

119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

+ sum

119901isin119867

sum

119894isin119881119901

sum

119895isin119881119901 119894 = 119895

119888119901max 119894119905Vminus10038161003816100381610038161003816end119905

119901

119894minus end119905119901

119895

10038161003816100381610038161003816 0

(1)

subject to

119898+120576119894

sum

119901=119898

119911119901

119894ge 1 forall119898 (119898 + 120576

119894) isin 119867 forall119894 isin 119878 (2)

119911119901

119894= sum

Visin119881sum

119895isin119878+


sum

Visin119881sum

119894isin119878+


sum

119894isin119878+

119909V119901119894119895minus sum

119896isin119878+


119910V119901le sum

119894isin119878

sum

119895isin119878+


119909V119901119894119895minus 119910


119908119901

119894= 119911119901

119894sum

119901isin119860119901

119894

119889119901

119894 forall119894 isin 119878 119901 isin 119867 (8)

sum

119894isin119878V119901

119908119901

119894le 120595


119905119901

119894119895=

119889119894119895

119904119901 forall119894 isin 119878

+

119895 isin 119878+

(10)

119905119901

119895119909V119901119894119895minus 119905119901

119894119909V119901119894119895= 119905119901

119894119895119909V119901119894119895 forall119894 isin 119878

+

119895 isin 119878+

V isin 119881 119901 isin 119867(11)

119905119901

119894119911119901


sum

119895isin119878

119894 = 119895

sum

Visin119881(119897

V119901119895119894119909V119901119895119894minus 119897

V119901119894119895119909V119901119894119895) = 119889119901

119894 forall119894 isin 119878 119901 isin 119867

(13)

1198681

119894= 0 forall119894 isin 119878 (14)

119868119901

119894= 119868119901minus1

119894+ 119908119901

119894minus 119889119901minus1

119894 forall119894 isin 119878 119901 isin 119867 1 (15)

1198681

= 0 (16)

119868119901

= 119868119901minus1

minussum

119894isin119878

119908119901minus1

119894+ 119898119901minus1

forall119901 isin 119867 1 (17)

119898119901

le 119862119906119901

forall119901 isin 119867 (18)

sum

119894isin119878

1199081

119894le 1198981

(19)

sum

119894isin119878

119908119901

119894le 119868119901

+ 119898119901

forall119901 isin 119867 1 (20)






119894

119889119901








119894and













min119885 = sum119894isin119878

sum

119901isin119867

[ℎ119888

119879 (119868119901

119894minus 119889119901

119894) + 120593119911

119901

119894] + 119885119878

+ 119885119879

+ 119885119868

(21)


119908119901

119894le 120595


119901

sum

119895=1

sum

119894isin119878

119908119895

119894le 119901119862 (23)



119894 Many appli-

























Step 2 Tabu search




1





1= then go








min119885119878 = sum119901isin119867

(119891119904

119906119901

+ ℎ119904

119868119901

) (24)





min119885119879 = sum119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

(25)




sum

119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

+ sum

119901isin119867

sum

119894isin119881119901

sum

119895isin119881119901 119894 = 119895

119888119901max 119894119905V minus 10038161003816100381610038161003816end119905

119901

119894minus end119905119901

119895

10038161003816100381610038161003816 0

(26)



119896) and



Start


p = 2

up = 1

Yes

Yes

Yes

Yes

Yes

No

No

No

NoNo






p = p + 1

p isin H

The save value = 0






119889119894119895)(119898 times 119899) When 119896 = 119897









Start

p = 1








Save the value


The lowest value lt

Countnobetter ++


Yes

Yes

YesYes

Yes

Yes

No

No

No

No

No

No

p = p + 1

p isin H



Cbest

Cbest Cbest



Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 223 303 379 311 295 212 421 132 121 204 153 2512 248 379 512 423 344 270 557 256 129 223 170 2823 352 513 688 587 458 415 833 548 182 279 223 3134 363 525 713 603 458 427 840 567 193 281 229 3275 308 452 578 525 401 357 679 433 155 252 201 3056 251 377 499 438 346 277 530 251 134 225 177 2767 398 589 732 659 480 479 757 677 213 303 252 344



Nodes Nodes0 1 2 3 4 5 6 7 8 9 10 11 12

0 0 55 17 38 34 67 67 66 61 85 108 142 1511 0 43 9 85 123 88 67 111 138 163 197 2062 0 55 42 81 79 72 68 95 121 156 1633 0 39 39 59 73 55 68 81 11 1234 0 45 91 97 27 53 8 117 1225 0 94 111 38 34 42 75 856 0 27 113 126 134 155 1737 0 122 14 153 177 1938 0 29 6 99 109 0 33 72 7110 0 39 4311 0 2512 0


Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 1 0 0 1 1 03 1 1 1 0 0 0 0 0 1 0 0 04 0 1 1 1 1 1 1 1 0 0 0 15 0 0 0 0 1 1 1 0 0 1 1 16 1 0 1 1 0 0 0 1 1 0 0 07 0 1 0 0 1 0 1 0 1 0 0 0





1 0-2-1-0 0-10-12-0 0-3-0 0-4-00-6-7-0 0-8-9-11-0 0-5-0 458879

2 0-7-0 0-10-11-0 1959693 0-1-0 0-2-0 0-3-9-0 1371554 0-8-5-0 0-7-6-0 0-2-0 0-4-12-0 0-3-0 3057665 0-6-0 0-10-11-0 0-12-0 0-7-0 0-5-0 4095696 0-1-0 0-8-9-0 0-4-0 0-3-0 1771197 0-2-7-0 0-5-9-0 119629




Value





Total cost 746123







costCost of time


1 1 84 4200 319893 (1 3 5) 222759 185 9811522 2 52 272522 313474 (1 3 6) 190471 205 7969683 3 42 231117 312597 (1 4 6) 180409 220 7461234 4 36 21096 309867 (1 4 5) 174071 250 7198985 5 37 210083 310735 (1 4 6) 174471 230 7182896 5 37 211246 30961 (1 4 6) 17373 270 7215267 6 37 213764 311514 (1 3 6) 176761 200 722039



Transportationcost



05 25 236385 309729 (1 4 5) 183889 230 4509 75300308 28 229261 308132 (1 3 5) 182608 275 4482 74750111 31 23415 308947 (1 3 5) 181876 230 4465 74797313 33 230983 311138 (1 3 5) 182952 195 4532 74457317 37 231117 312597 (1 4 6) 180409 220 4419 7461232 4 231117 312597 (1 4 6) 180409 220 4419 746123














0 26 227073 310153 (1 3 5) 7055 624276003 26 235268 310703 (1 4 6) 138368 700839005 28 231117 312597 (1 4 6) 180409 746123008 28 230055 312561 (1 4 6) 244352 81196801 28 230745 310843 (1 4 5) 284931 848519015 29 226472 30948 (1 3 5) 394428 9533802 31 229913 312012 (1 3 5) 497812 106624



penalty Total cost

0 437 230527 308956 (1 3 5) 179719 0 71920203 4494 23003 309192 (1 4 6) 181748 195 7404705 4419 231117 312597 (1 4 6) 180409 220 74612308 4555 235768 310155 (1 4 6) 184253 265 7566761 4394 234704 312683 (1 4 6) 181926 255 75481312 4579 23448 314667 (1 4 6) 182735 225 75438215 4383 236642 308349 (1 3 5) 180522 280 7535132 4508 234047 313886 (1 4 6) 181584 260 755517



penalty Total cost

025 4386 234611 310482 146507 230 714605 4527 226242 31041 158853 205 7160051 4419 231117 312597 180409 220 74612315 461 231057 311007 205359 240 7714232 4531 230277 312784 22819 215 7927515 4268 236455 31163 349259 175 91484510 4336 23922 311318 573219 215 11452620 4071 248607 31081 94919 190 152761




6 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













23 Notation

(1) Parameters






(2) Variables






119894119895=

119889119894119895119904119901)



119868119901


period 119901119908119901





119894






(decision variable)

3 Model Formulation



sum119901isin119867

119891119904


119901isin119867ℎ119904

119868119901+1


ℎ119904

119868119901+1

= sum119901isin119867

ℎ119904

(119868119901

+ 119898119901

minus sum119894isin119878119908119901

119894) The


= 119868119901minus1





119901isin119867sumVisin119881 119891


sum119901isin119867


sum119901isin119867

sum119894isin119878+ sum119895isin119878+ 119888119892




119894max119905119901

119894119911119901

119894minus







ℎ119888

119879(119868119901

119894minus 119889119901



119894= 119868119901minus1

119894+119908119901

119894minus

119889119901minus1



119894= 119911119901

119894sum119901isin119860119901

119894

119889119901

119894 In the




119888119901max119894119905Vminus|end119905119901

119894minus






min119885 = sum119901isin119867

(119891s119906119901

+ ℎs119868119901

) +sum

119894isin119878

sum

119901isin119867

[ℎc119879 (119868119901

119894minus 119889119901

119894) + 120593119911

119901

119894]

+ sum

119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

+ sum

119901isin119867

sum

119894isin119881119901

sum

119895isin119881119901 119894 = 119895

119888119901max 119894119905Vminus10038161003816100381610038161003816end119905

119901

119894minus end119905119901

119895

10038161003816100381610038161003816 0

(1)

subject to

119898+120576119894

sum

119901=119898

119911119901

119894ge 1 forall119898 (119898 + 120576

119894) isin 119867 forall119894 isin 119878 (2)

119911119901

119894= sum

Visin119881sum

119895isin119878+


sum

Visin119881sum

119894isin119878+


sum

119894isin119878+

119909V119901119894119895minus sum

119896isin119878+


119910V119901le sum

119894isin119878

sum

119895isin119878+


119909V119901119894119895minus 119910


119908119901

119894= 119911119901

119894sum

119901isin119860119901

119894

119889119901

119894 forall119894 isin 119878 119901 isin 119867 (8)

sum

119894isin119878V119901

119908119901

119894le 120595


119905119901

119894119895=

119889119894119895

119904119901 forall119894 isin 119878

+

119895 isin 119878+

(10)

119905119901

119895119909V119901119894119895minus 119905119901

119894119909V119901119894119895= 119905119901

119894119895119909V119901119894119895 forall119894 isin 119878

+

119895 isin 119878+

V isin 119881 119901 isin 119867(11)

119905119901

119894119911119901


sum

119895isin119878

119894 = 119895

sum

Visin119881(119897

V119901119895119894119909V119901119895119894minus 119897

V119901119894119895119909V119901119894119895) = 119889119901

119894 forall119894 isin 119878 119901 isin 119867

(13)

1198681

119894= 0 forall119894 isin 119878 (14)

119868119901

119894= 119868119901minus1

119894+ 119908119901

119894minus 119889119901minus1

119894 forall119894 isin 119878 119901 isin 119867 1 (15)

1198681

= 0 (16)

119868119901

= 119868119901minus1

minussum

119894isin119878

119908119901minus1

119894+ 119898119901minus1

forall119901 isin 119867 1 (17)

119898119901

le 119862119906119901

forall119901 isin 119867 (18)

sum

119894isin119878

1199081

119894le 1198981

(19)

sum

119894isin119878

119908119901

119894le 119868119901

+ 119898119901

forall119901 isin 119867 1 (20)






119894

119889119901








119894and













min119885 = sum119894isin119878

sum

119901isin119867

[ℎ119888

119879 (119868119901

119894minus 119889119901

119894) + 120593119911

119901

119894] + 119885119878

+ 119885119879

+ 119885119868

(21)


119908119901

119894le 120595


119901

sum

119895=1

sum

119894isin119878

119908119895

119894le 119901119862 (23)



119894 Many appli-

























Step 2 Tabu search




1





1= then go








min119885119878 = sum119901isin119867

(119891119904

119906119901

+ ℎ119904

119868119901

) (24)





min119885119879 = sum119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

(25)




sum

119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

+ sum

119901isin119867

sum

119894isin119881119901

sum

119895isin119881119901 119894 = 119895

119888119901max 119894119905V minus 10038161003816100381610038161003816end119905

119901

119894minus end119905119901

119895

10038161003816100381610038161003816 0

(26)



119896) and



Start


p = 2

up = 1

Yes

Yes

Yes

Yes

Yes

No

No

No

NoNo






p = p + 1

p isin H

The save value = 0






119889119894119895)(119898 times 119899) When 119896 = 119897









Start

p = 1








Save the value


The lowest value lt

Countnobetter ++


Yes

Yes

YesYes

Yes

Yes

No

No

No

No

No

No

p = p + 1

p isin H



Cbest

Cbest Cbest



Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 223 303 379 311 295 212 421 132 121 204 153 2512 248 379 512 423 344 270 557 256 129 223 170 2823 352 513 688 587 458 415 833 548 182 279 223 3134 363 525 713 603 458 427 840 567 193 281 229 3275 308 452 578 525 401 357 679 433 155 252 201 3056 251 377 499 438 346 277 530 251 134 225 177 2767 398 589 732 659 480 479 757 677 213 303 252 344



Nodes Nodes0 1 2 3 4 5 6 7 8 9 10 11 12

0 0 55 17 38 34 67 67 66 61 85 108 142 1511 0 43 9 85 123 88 67 111 138 163 197 2062 0 55 42 81 79 72 68 95 121 156 1633 0 39 39 59 73 55 68 81 11 1234 0 45 91 97 27 53 8 117 1225 0 94 111 38 34 42 75 856 0 27 113 126 134 155 1737 0 122 14 153 177 1938 0 29 6 99 109 0 33 72 7110 0 39 4311 0 2512 0


Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 1 0 0 1 1 03 1 1 1 0 0 0 0 0 1 0 0 04 0 1 1 1 1 1 1 1 0 0 0 15 0 0 0 0 1 1 1 0 0 1 1 16 1 0 1 1 0 0 0 1 1 0 0 07 0 1 0 0 1 0 1 0 1 0 0 0





1 0-2-1-0 0-10-12-0 0-3-0 0-4-00-6-7-0 0-8-9-11-0 0-5-0 458879

2 0-7-0 0-10-11-0 1959693 0-1-0 0-2-0 0-3-9-0 1371554 0-8-5-0 0-7-6-0 0-2-0 0-4-12-0 0-3-0 3057665 0-6-0 0-10-11-0 0-12-0 0-7-0 0-5-0 4095696 0-1-0 0-8-9-0 0-4-0 0-3-0 1771197 0-2-7-0 0-5-9-0 119629




Value





Total cost 746123







costCost of time


1 1 84 4200 319893 (1 3 5) 222759 185 9811522 2 52 272522 313474 (1 3 6) 190471 205 7969683 3 42 231117 312597 (1 4 6) 180409 220 7461234 4 36 21096 309867 (1 4 5) 174071 250 7198985 5 37 210083 310735 (1 4 6) 174471 230 7182896 5 37 211246 30961 (1 4 6) 17373 270 7215267 6 37 213764 311514 (1 3 6) 176761 200 722039



Transportationcost



05 25 236385 309729 (1 4 5) 183889 230 4509 75300308 28 229261 308132 (1 3 5) 182608 275 4482 74750111 31 23415 308947 (1 3 5) 181876 230 4465 74797313 33 230983 311138 (1 3 5) 182952 195 4532 74457317 37 231117 312597 (1 4 6) 180409 220 4419 7461232 4 231117 312597 (1 4 6) 180409 220 4419 746123














0 26 227073 310153 (1 3 5) 7055 624276003 26 235268 310703 (1 4 6) 138368 700839005 28 231117 312597 (1 4 6) 180409 746123008 28 230055 312561 (1 4 6) 244352 81196801 28 230745 310843 (1 4 5) 284931 848519015 29 226472 30948 (1 3 5) 394428 9533802 31 229913 312012 (1 3 5) 497812 106624



penalty Total cost

0 437 230527 308956 (1 3 5) 179719 0 71920203 4494 23003 309192 (1 4 6) 181748 195 7404705 4419 231117 312597 (1 4 6) 180409 220 74612308 4555 235768 310155 (1 4 6) 184253 265 7566761 4394 234704 312683 (1 4 6) 181926 255 75481312 4579 23448 314667 (1 4 6) 182735 225 75438215 4383 236642 308349 (1 3 5) 180522 280 7535132 4508 234047 313886 (1 4 6) 181584 260 755517



penalty Total cost

025 4386 234611 310482 146507 230 714605 4527 226242 31041 158853 205 7160051 4419 231117 312597 180409 220 74612315 461 231057 311007 205359 240 7714232 4531 230277 312784 22819 215 7927515 4268 236455 31163 349259 175 91484510 4336 23922 311318 573219 215 11452620 4071 248607 31081 94919 190 152761




6 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














ℎ119888

119879(119868119901

119894minus 119889119901



119894= 119868119901minus1

119894+119908119901

119894minus

119889119901minus1



119894= 119911119901

119894sum119901isin119860119901

119894

119889119901

119894 In the




119888119901max119894119905Vminus|end119905119901

119894minus






min119885 = sum119901isin119867

(119891s119906119901

+ ℎs119868119901

) +sum

119894isin119878

sum

119901isin119867

[ℎc119879 (119868119901

119894minus 119889119901

119894) + 120593119911

119901

119894]

+ sum

119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

+ sum

119901isin119867

sum

119894isin119881119901

sum

119895isin119881119901 119894 = 119895

119888119901max 119894119905Vminus10038161003816100381610038161003816end119905

119901

119894minus end119905119901

119895

10038161003816100381610038161003816 0

(1)

subject to

119898+120576119894

sum

119901=119898

119911119901

119894ge 1 forall119898 (119898 + 120576

119894) isin 119867 forall119894 isin 119878 (2)

119911119901

119894= sum

Visin119881sum

119895isin119878+


sum

Visin119881sum

119894isin119878+


sum

119894isin119878+

119909V119901119894119895minus sum

119896isin119878+


119910V119901le sum

119894isin119878

sum

119895isin119878+


119909V119901119894119895minus 119910


119908119901

119894= 119911119901

119894sum

119901isin119860119901

119894

119889119901

119894 forall119894 isin 119878 119901 isin 119867 (8)

sum

119894isin119878V119901

119908119901

119894le 120595


119905119901

119894119895=

119889119894119895

119904119901 forall119894 isin 119878

+

119895 isin 119878+

(10)

119905119901

119895119909V119901119894119895minus 119905119901

119894119909V119901119894119895= 119905119901

119894119895119909V119901119894119895 forall119894 isin 119878

+

119895 isin 119878+

V isin 119881 119901 isin 119867(11)

119905119901

119894119911119901


sum

119895isin119878

119894 = 119895

sum

Visin119881(119897

V119901119895119894119909V119901119895119894minus 119897

V119901119894119895119909V119901119894119895) = 119889119901

119894 forall119894 isin 119878 119901 isin 119867

(13)

1198681

119894= 0 forall119894 isin 119878 (14)

119868119901

119894= 119868119901minus1

119894+ 119908119901

119894minus 119889119901minus1

119894 forall119894 isin 119878 119901 isin 119867 1 (15)

1198681

= 0 (16)

119868119901

= 119868119901minus1

minussum

119894isin119878

119908119901minus1

119894+ 119898119901minus1

forall119901 isin 119867 1 (17)

119898119901

le 119862119906119901

forall119901 isin 119867 (18)

sum

119894isin119878

1199081

119894le 1198981

(19)

sum

119894isin119878

119908119901

119894le 119868119901

+ 119898119901

forall119901 isin 119867 1 (20)






119894

119889119901








119894and













min119885 = sum119894isin119878

sum

119901isin119867

[ℎ119888

119879 (119868119901

119894minus 119889119901

119894) + 120593119911

119901

119894] + 119885119878

+ 119885119879

+ 119885119868

(21)


119908119901

119894le 120595


119901

sum

119895=1

sum

119894isin119878

119908119895

119894le 119901119862 (23)



119894 Many appli-

























Step 2 Tabu search




1





1= then go








min119885119878 = sum119901isin119867

(119891119904

119906119901

+ ℎ119904

119868119901

) (24)





min119885119879 = sum119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

(25)




sum

119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

+ sum

119901isin119867

sum

119894isin119881119901

sum

119895isin119881119901 119894 = 119895

119888119901max 119894119905V minus 10038161003816100381610038161003816end119905

119901

119894minus end119905119901

119895

10038161003816100381610038161003816 0

(26)



119896) and



Start


p = 2

up = 1

Yes

Yes

Yes

Yes

Yes

No

No

No

NoNo






p = p + 1

p isin H

The save value = 0






119889119894119895)(119898 times 119899) When 119896 = 119897









Start

p = 1








Save the value


The lowest value lt

Countnobetter ++


Yes

Yes

YesYes

Yes

Yes

No

No

No

No

No

No

p = p + 1

p isin H



Cbest

Cbest Cbest



Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 223 303 379 311 295 212 421 132 121 204 153 2512 248 379 512 423 344 270 557 256 129 223 170 2823 352 513 688 587 458 415 833 548 182 279 223 3134 363 525 713 603 458 427 840 567 193 281 229 3275 308 452 578 525 401 357 679 433 155 252 201 3056 251 377 499 438 346 277 530 251 134 225 177 2767 398 589 732 659 480 479 757 677 213 303 252 344



Nodes Nodes0 1 2 3 4 5 6 7 8 9 10 11 12

0 0 55 17 38 34 67 67 66 61 85 108 142 1511 0 43 9 85 123 88 67 111 138 163 197 2062 0 55 42 81 79 72 68 95 121 156 1633 0 39 39 59 73 55 68 81 11 1234 0 45 91 97 27 53 8 117 1225 0 94 111 38 34 42 75 856 0 27 113 126 134 155 1737 0 122 14 153 177 1938 0 29 6 99 109 0 33 72 7110 0 39 4311 0 2512 0


Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 1 0 0 1 1 03 1 1 1 0 0 0 0 0 1 0 0 04 0 1 1 1 1 1 1 1 0 0 0 15 0 0 0 0 1 1 1 0 0 1 1 16 1 0 1 1 0 0 0 1 1 0 0 07 0 1 0 0 1 0 1 0 1 0 0 0





1 0-2-1-0 0-10-12-0 0-3-0 0-4-00-6-7-0 0-8-9-11-0 0-5-0 458879

2 0-7-0 0-10-11-0 1959693 0-1-0 0-2-0 0-3-9-0 1371554 0-8-5-0 0-7-6-0 0-2-0 0-4-12-0 0-3-0 3057665 0-6-0 0-10-11-0 0-12-0 0-7-0 0-5-0 4095696 0-1-0 0-8-9-0 0-4-0 0-3-0 1771197 0-2-7-0 0-5-9-0 119629




Value





Total cost 746123







costCost of time


1 1 84 4200 319893 (1 3 5) 222759 185 9811522 2 52 272522 313474 (1 3 6) 190471 205 7969683 3 42 231117 312597 (1 4 6) 180409 220 7461234 4 36 21096 309867 (1 4 5) 174071 250 7198985 5 37 210083 310735 (1 4 6) 174471 230 7182896 5 37 211246 30961 (1 4 6) 17373 270 7215267 6 37 213764 311514 (1 3 6) 176761 200 722039



Transportationcost



05 25 236385 309729 (1 4 5) 183889 230 4509 75300308 28 229261 308132 (1 3 5) 182608 275 4482 74750111 31 23415 308947 (1 3 5) 181876 230 4465 74797313 33 230983 311138 (1 3 5) 182952 195 4532 74457317 37 231117 312597 (1 4 6) 180409 220 4419 7461232 4 231117 312597 (1 4 6) 180409 220 4419 746123














0 26 227073 310153 (1 3 5) 7055 624276003 26 235268 310703 (1 4 6) 138368 700839005 28 231117 312597 (1 4 6) 180409 746123008 28 230055 312561 (1 4 6) 244352 81196801 28 230745 310843 (1 4 5) 284931 848519015 29 226472 30948 (1 3 5) 394428 9533802 31 229913 312012 (1 3 5) 497812 106624



penalty Total cost

0 437 230527 308956 (1 3 5) 179719 0 71920203 4494 23003 309192 (1 4 6) 181748 195 7404705 4419 231117 312597 (1 4 6) 180409 220 74612308 4555 235768 310155 (1 4 6) 184253 265 7566761 4394 234704 312683 (1 4 6) 181926 255 75481312 4579 23448 314667 (1 4 6) 182735 225 75438215 4383 236642 308349 (1 3 5) 180522 280 7535132 4508 234047 313886 (1 4 6) 181584 260 755517



penalty Total cost

025 4386 234611 310482 146507 230 714605 4527 226242 31041 158853 205 7160051 4419 231117 312597 180409 220 74612315 461 231057 311007 205359 240 7714232 4531 230277 312784 22819 215 7927515 4268 236455 31163 349259 175 91484510 4336 23922 311318 573219 215 11452620 4071 248607 31081 94919 190 152761




6 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119894and













min119885 = sum119894isin119878

sum

119901isin119867

[ℎ119888

119879 (119868119901

119894minus 119889119901

119894) + 120593119911

119901

119894] + 119885119878

+ 119885119879

+ 119885119868

(21)


119908119901

119894le 120595


119901

sum

119895=1

sum

119894isin119878

119908119895

119894le 119901119862 (23)



119894 Many appli-

























Step 2 Tabu search




1





1= then go








min119885119878 = sum119901isin119867

(119891119904

119906119901

+ ℎ119904

119868119901

) (24)





min119885119879 = sum119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

(25)




sum

119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

+ sum

119901isin119867

sum

119894isin119881119901

sum

119895isin119881119901 119894 = 119895

119888119901max 119894119905V minus 10038161003816100381610038161003816end119905

119901

119894minus end119905119901

119895

10038161003816100381610038161003816 0

(26)



119896) and



Start


p = 2

up = 1

Yes

Yes

Yes

Yes

Yes

No

No

No

NoNo






p = p + 1

p isin H

The save value = 0






119889119894119895)(119898 times 119899) When 119896 = 119897









Start

p = 1








Save the value


The lowest value lt

Countnobetter ++


Yes

Yes

YesYes

Yes

Yes

No

No

No

No

No

No

p = p + 1

p isin H



Cbest

Cbest Cbest



Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 223 303 379 311 295 212 421 132 121 204 153 2512 248 379 512 423 344 270 557 256 129 223 170 2823 352 513 688 587 458 415 833 548 182 279 223 3134 363 525 713 603 458 427 840 567 193 281 229 3275 308 452 578 525 401 357 679 433 155 252 201 3056 251 377 499 438 346 277 530 251 134 225 177 2767 398 589 732 659 480 479 757 677 213 303 252 344



Nodes Nodes0 1 2 3 4 5 6 7 8 9 10 11 12

0 0 55 17 38 34 67 67 66 61 85 108 142 1511 0 43 9 85 123 88 67 111 138 163 197 2062 0 55 42 81 79 72 68 95 121 156 1633 0 39 39 59 73 55 68 81 11 1234 0 45 91 97 27 53 8 117 1225 0 94 111 38 34 42 75 856 0 27 113 126 134 155 1737 0 122 14 153 177 1938 0 29 6 99 109 0 33 72 7110 0 39 4311 0 2512 0


Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 1 0 0 1 1 03 1 1 1 0 0 0 0 0 1 0 0 04 0 1 1 1 1 1 1 1 0 0 0 15 0 0 0 0 1 1 1 0 0 1 1 16 1 0 1 1 0 0 0 1 1 0 0 07 0 1 0 0 1 0 1 0 1 0 0 0





1 0-2-1-0 0-10-12-0 0-3-0 0-4-00-6-7-0 0-8-9-11-0 0-5-0 458879

2 0-7-0 0-10-11-0 1959693 0-1-0 0-2-0 0-3-9-0 1371554 0-8-5-0 0-7-6-0 0-2-0 0-4-12-0 0-3-0 3057665 0-6-0 0-10-11-0 0-12-0 0-7-0 0-5-0 4095696 0-1-0 0-8-9-0 0-4-0 0-3-0 1771197 0-2-7-0 0-5-9-0 119629




Value





Total cost 746123







costCost of time


1 1 84 4200 319893 (1 3 5) 222759 185 9811522 2 52 272522 313474 (1 3 6) 190471 205 7969683 3 42 231117 312597 (1 4 6) 180409 220 7461234 4 36 21096 309867 (1 4 5) 174071 250 7198985 5 37 210083 310735 (1 4 6) 174471 230 7182896 5 37 211246 30961 (1 4 6) 17373 270 7215267 6 37 213764 311514 (1 3 6) 176761 200 722039



Transportationcost



05 25 236385 309729 (1 4 5) 183889 230 4509 75300308 28 229261 308132 (1 3 5) 182608 275 4482 74750111 31 23415 308947 (1 3 5) 181876 230 4465 74797313 33 230983 311138 (1 3 5) 182952 195 4532 74457317 37 231117 312597 (1 4 6) 180409 220 4419 7461232 4 231117 312597 (1 4 6) 180409 220 4419 746123














0 26 227073 310153 (1 3 5) 7055 624276003 26 235268 310703 (1 4 6) 138368 700839005 28 231117 312597 (1 4 6) 180409 746123008 28 230055 312561 (1 4 6) 244352 81196801 28 230745 310843 (1 4 5) 284931 848519015 29 226472 30948 (1 3 5) 394428 9533802 31 229913 312012 (1 3 5) 497812 106624



penalty Total cost

0 437 230527 308956 (1 3 5) 179719 0 71920203 4494 23003 309192 (1 4 6) 181748 195 7404705 4419 231117 312597 (1 4 6) 180409 220 74612308 4555 235768 310155 (1 4 6) 184253 265 7566761 4394 234704 312683 (1 4 6) 181926 255 75481312 4579 23448 314667 (1 4 6) 182735 225 75438215 4383 236642 308349 (1 3 5) 180522 280 7535132 4508 234047 313886 (1 4 6) 181584 260 755517



penalty Total cost

025 4386 234611 310482 146507 230 714605 4527 226242 31041 158853 205 7160051 4419 231117 312597 180409 220 74612315 461 231057 311007 205359 240 7714232 4531 230277 312784 22819 215 7927515 4268 236455 31163 349259 175 91484510 4336 23922 311318 573219 215 11452620 4071 248607 31081 94919 190 152761




6 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Step 2 Tabu search




1





1= then go








min119885119878 = sum119901isin119867

(119891119904

119906119901

+ ℎ119904

119868119901

) (24)





min119885119879 = sum119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

(25)




sum

119901isin119867

sum

Visin119881119891V119910V119901+ sum

119901isin119867

sum

Visin119881sum

119895isin119878+

sum

119894isin119878+

119888V119889119894119895119909V119901119894119895

+ sum

119901isin119867

sum

Visin119881sum

119894isin119878

119887119908119901

119894max 119905119901

119894119911119901

119894minus 119905max 1 0

+ sum

119901isin119867

sum

119894isin119878+

sum

119895isin119878+

119888119892

119889119894119895119909V119901119894119895119897V119901119894119895

+ sum

119901isin119867

sum

119894isin119881119901

sum

119895isin119881119901 119894 = 119895

119888119901max 119894119905V minus 10038161003816100381610038161003816end119905

119901

119894minus end119905119901

119895

10038161003816100381610038161003816 0

(26)



119896) and



Start


p = 2

up = 1

Yes

Yes

Yes

Yes

Yes

No

No

No

NoNo






p = p + 1

p isin H

The save value = 0






119889119894119895)(119898 times 119899) When 119896 = 119897









Start

p = 1








Save the value


The lowest value lt

Countnobetter ++


Yes

Yes

YesYes

Yes

Yes

No

No

No

No

No

No

p = p + 1

p isin H



Cbest

Cbest Cbest



Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 223 303 379 311 295 212 421 132 121 204 153 2512 248 379 512 423 344 270 557 256 129 223 170 2823 352 513 688 587 458 415 833 548 182 279 223 3134 363 525 713 603 458 427 840 567 193 281 229 3275 308 452 578 525 401 357 679 433 155 252 201 3056 251 377 499 438 346 277 530 251 134 225 177 2767 398 589 732 659 480 479 757 677 213 303 252 344



Nodes Nodes0 1 2 3 4 5 6 7 8 9 10 11 12

0 0 55 17 38 34 67 67 66 61 85 108 142 1511 0 43 9 85 123 88 67 111 138 163 197 2062 0 55 42 81 79 72 68 95 121 156 1633 0 39 39 59 73 55 68 81 11 1234 0 45 91 97 27 53 8 117 1225 0 94 111 38 34 42 75 856 0 27 113 126 134 155 1737 0 122 14 153 177 1938 0 29 6 99 109 0 33 72 7110 0 39 4311 0 2512 0


Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 1 0 0 1 1 03 1 1 1 0 0 0 0 0 1 0 0 04 0 1 1 1 1 1 1 1 0 0 0 15 0 0 0 0 1 1 1 0 0 1 1 16 1 0 1 1 0 0 0 1 1 0 0 07 0 1 0 0 1 0 1 0 1 0 0 0





1 0-2-1-0 0-10-12-0 0-3-0 0-4-00-6-7-0 0-8-9-11-0 0-5-0 458879

2 0-7-0 0-10-11-0 1959693 0-1-0 0-2-0 0-3-9-0 1371554 0-8-5-0 0-7-6-0 0-2-0 0-4-12-0 0-3-0 3057665 0-6-0 0-10-11-0 0-12-0 0-7-0 0-5-0 4095696 0-1-0 0-8-9-0 0-4-0 0-3-0 1771197 0-2-7-0 0-5-9-0 119629




Value





Total cost 746123







costCost of time


1 1 84 4200 319893 (1 3 5) 222759 185 9811522 2 52 272522 313474 (1 3 6) 190471 205 7969683 3 42 231117 312597 (1 4 6) 180409 220 7461234 4 36 21096 309867 (1 4 5) 174071 250 7198985 5 37 210083 310735 (1 4 6) 174471 230 7182896 5 37 211246 30961 (1 4 6) 17373 270 7215267 6 37 213764 311514 (1 3 6) 176761 200 722039



Transportationcost



05 25 236385 309729 (1 4 5) 183889 230 4509 75300308 28 229261 308132 (1 3 5) 182608 275 4482 74750111 31 23415 308947 (1 3 5) 181876 230 4465 74797313 33 230983 311138 (1 3 5) 182952 195 4532 74457317 37 231117 312597 (1 4 6) 180409 220 4419 7461232 4 231117 312597 (1 4 6) 180409 220 4419 746123














0 26 227073 310153 (1 3 5) 7055 624276003 26 235268 310703 (1 4 6) 138368 700839005 28 231117 312597 (1 4 6) 180409 746123008 28 230055 312561 (1 4 6) 244352 81196801 28 230745 310843 (1 4 5) 284931 848519015 29 226472 30948 (1 3 5) 394428 9533802 31 229913 312012 (1 3 5) 497812 106624



penalty Total cost

0 437 230527 308956 (1 3 5) 179719 0 71920203 4494 23003 309192 (1 4 6) 181748 195 7404705 4419 231117 312597 (1 4 6) 180409 220 74612308 4555 235768 310155 (1 4 6) 184253 265 7566761 4394 234704 312683 (1 4 6) 181926 255 75481312 4579 23448 314667 (1 4 6) 182735 225 75438215 4383 236642 308349 (1 3 5) 180522 280 7535132 4508 234047 313886 (1 4 6) 181584 260 755517



penalty Total cost

025 4386 234611 310482 146507 230 714605 4527 226242 31041 158853 205 7160051 4419 231117 312597 180409 220 74612315 461 231057 311007 205359 240 7714232 4531 230277 312784 22819 215 7927515 4268 236455 31163 349259 175 91484510 4336 23922 311318 573219 215 11452620 4071 248607 31081 94919 190 152761




6 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start


p = 2

up = 1

Yes

Yes

Yes

Yes

Yes

No

No

No

NoNo






p = p + 1

p isin H

The save value = 0






119889119894119895)(119898 times 119899) When 119896 = 119897









Start

p = 1








Save the value


The lowest value lt

Countnobetter ++


Yes

Yes

YesYes

Yes

Yes

No

No

No

No

No

No

p = p + 1

p isin H



Cbest

Cbest Cbest



Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 223 303 379 311 295 212 421 132 121 204 153 2512 248 379 512 423 344 270 557 256 129 223 170 2823 352 513 688 587 458 415 833 548 182 279 223 3134 363 525 713 603 458 427 840 567 193 281 229 3275 308 452 578 525 401 357 679 433 155 252 201 3056 251 377 499 438 346 277 530 251 134 225 177 2767 398 589 732 659 480 479 757 677 213 303 252 344



Nodes Nodes0 1 2 3 4 5 6 7 8 9 10 11 12

0 0 55 17 38 34 67 67 66 61 85 108 142 1511 0 43 9 85 123 88 67 111 138 163 197 2062 0 55 42 81 79 72 68 95 121 156 1633 0 39 39 59 73 55 68 81 11 1234 0 45 91 97 27 53 8 117 1225 0 94 111 38 34 42 75 856 0 27 113 126 134 155 1737 0 122 14 153 177 1938 0 29 6 99 109 0 33 72 7110 0 39 4311 0 2512 0


Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 1 0 0 1 1 03 1 1 1 0 0 0 0 0 1 0 0 04 0 1 1 1 1 1 1 1 0 0 0 15 0 0 0 0 1 1 1 0 0 1 1 16 1 0 1 1 0 0 0 1 1 0 0 07 0 1 0 0 1 0 1 0 1 0 0 0





1 0-2-1-0 0-10-12-0 0-3-0 0-4-00-6-7-0 0-8-9-11-0 0-5-0 458879

2 0-7-0 0-10-11-0 1959693 0-1-0 0-2-0 0-3-9-0 1371554 0-8-5-0 0-7-6-0 0-2-0 0-4-12-0 0-3-0 3057665 0-6-0 0-10-11-0 0-12-0 0-7-0 0-5-0 4095696 0-1-0 0-8-9-0 0-4-0 0-3-0 1771197 0-2-7-0 0-5-9-0 119629




Value





Total cost 746123







costCost of time


1 1 84 4200 319893 (1 3 5) 222759 185 9811522 2 52 272522 313474 (1 3 6) 190471 205 7969683 3 42 231117 312597 (1 4 6) 180409 220 7461234 4 36 21096 309867 (1 4 5) 174071 250 7198985 5 37 210083 310735 (1 4 6) 174471 230 7182896 5 37 211246 30961 (1 4 6) 17373 270 7215267 6 37 213764 311514 (1 3 6) 176761 200 722039



Transportationcost



05 25 236385 309729 (1 4 5) 183889 230 4509 75300308 28 229261 308132 (1 3 5) 182608 275 4482 74750111 31 23415 308947 (1 3 5) 181876 230 4465 74797313 33 230983 311138 (1 3 5) 182952 195 4532 74457317 37 231117 312597 (1 4 6) 180409 220 4419 7461232 4 231117 312597 (1 4 6) 180409 220 4419 746123














0 26 227073 310153 (1 3 5) 7055 624276003 26 235268 310703 (1 4 6) 138368 700839005 28 231117 312597 (1 4 6) 180409 746123008 28 230055 312561 (1 4 6) 244352 81196801 28 230745 310843 (1 4 5) 284931 848519015 29 226472 30948 (1 3 5) 394428 9533802 31 229913 312012 (1 3 5) 497812 106624



penalty Total cost

0 437 230527 308956 (1 3 5) 179719 0 71920203 4494 23003 309192 (1 4 6) 181748 195 7404705 4419 231117 312597 (1 4 6) 180409 220 74612308 4555 235768 310155 (1 4 6) 184253 265 7566761 4394 234704 312683 (1 4 6) 181926 255 75481312 4579 23448 314667 (1 4 6) 182735 225 75438215 4383 236642 308349 (1 3 5) 180522 280 7535132 4508 234047 313886 (1 4 6) 181584 260 755517



penalty Total cost

025 4386 234611 310482 146507 230 714605 4527 226242 31041 158853 205 7160051 4419 231117 312597 180409 220 74612315 461 231057 311007 205359 240 7714232 4531 230277 312784 22819 215 7927515 4268 236455 31163 349259 175 91484510 4336 23922 311318 573219 215 11452620 4071 248607 31081 94919 190 152761




6 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

p = 1








Save the value


The lowest value lt

Countnobetter ++


Yes

Yes

YesYes

Yes

Yes

No

No

No

No

No

No

p = p + 1

p isin H



Cbest

Cbest Cbest



Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 223 303 379 311 295 212 421 132 121 204 153 2512 248 379 512 423 344 270 557 256 129 223 170 2823 352 513 688 587 458 415 833 548 182 279 223 3134 363 525 713 603 458 427 840 567 193 281 229 3275 308 452 578 525 401 357 679 433 155 252 201 3056 251 377 499 438 346 277 530 251 134 225 177 2767 398 589 732 659 480 479 757 677 213 303 252 344



Nodes Nodes0 1 2 3 4 5 6 7 8 9 10 11 12

0 0 55 17 38 34 67 67 66 61 85 108 142 1511 0 43 9 85 123 88 67 111 138 163 197 2062 0 55 42 81 79 72 68 95 121 156 1633 0 39 39 59 73 55 68 81 11 1234 0 45 91 97 27 53 8 117 1225 0 94 111 38 34 42 75 856 0 27 113 126 134 155 1737 0 122 14 153 177 1938 0 29 6 99 109 0 33 72 7110 0 39 4311 0 2512 0


Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 1 0 0 1 1 03 1 1 1 0 0 0 0 0 1 0 0 04 0 1 1 1 1 1 1 1 0 0 0 15 0 0 0 0 1 1 1 0 0 1 1 16 1 0 1 1 0 0 0 1 1 0 0 07 0 1 0 0 1 0 1 0 1 0 0 0





1 0-2-1-0 0-10-12-0 0-3-0 0-4-00-6-7-0 0-8-9-11-0 0-5-0 458879

2 0-7-0 0-10-11-0 1959693 0-1-0 0-2-0 0-3-9-0 1371554 0-8-5-0 0-7-6-0 0-2-0 0-4-12-0 0-3-0 3057665 0-6-0 0-10-11-0 0-12-0 0-7-0 0-5-0 4095696 0-1-0 0-8-9-0 0-4-0 0-3-0 1771197 0-2-7-0 0-5-9-0 119629




Value





Total cost 746123







costCost of time


1 1 84 4200 319893 (1 3 5) 222759 185 9811522 2 52 272522 313474 (1 3 6) 190471 205 7969683 3 42 231117 312597 (1 4 6) 180409 220 7461234 4 36 21096 309867 (1 4 5) 174071 250 7198985 5 37 210083 310735 (1 4 6) 174471 230 7182896 5 37 211246 30961 (1 4 6) 17373 270 7215267 6 37 213764 311514 (1 3 6) 176761 200 722039



Transportationcost



05 25 236385 309729 (1 4 5) 183889 230 4509 75300308 28 229261 308132 (1 3 5) 182608 275 4482 74750111 31 23415 308947 (1 3 5) 181876 230 4465 74797313 33 230983 311138 (1 3 5) 182952 195 4532 74457317 37 231117 312597 (1 4 6) 180409 220 4419 7461232 4 231117 312597 (1 4 6) 180409 220 4419 746123














0 26 227073 310153 (1 3 5) 7055 624276003 26 235268 310703 (1 4 6) 138368 700839005 28 231117 312597 (1 4 6) 180409 746123008 28 230055 312561 (1 4 6) 244352 81196801 28 230745 310843 (1 4 5) 284931 848519015 29 226472 30948 (1 3 5) 394428 9533802 31 229913 312012 (1 3 5) 497812 106624



penalty Total cost

0 437 230527 308956 (1 3 5) 179719 0 71920203 4494 23003 309192 (1 4 6) 181748 195 7404705 4419 231117 312597 (1 4 6) 180409 220 74612308 4555 235768 310155 (1 4 6) 184253 265 7566761 4394 234704 312683 (1 4 6) 181926 255 75481312 4579 23448 314667 (1 4 6) 182735 225 75438215 4383 236642 308349 (1 3 5) 180522 280 7535132 4508 234047 313886 (1 4 6) 181584 260 755517



penalty Total cost

025 4386 234611 310482 146507 230 714605 4527 226242 31041 158853 205 7160051 4419 231117 312597 180409 220 74612315 461 231057 311007 205359 240 7714232 4531 230277 312784 22819 215 7927515 4268 236455 31163 349259 175 91484510 4336 23922 311318 573219 215 11452620 4071 248607 31081 94919 190 152761




6 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Nodes Nodes0 1 2 3 4 5 6 7 8 9 10 11 12

0 0 55 17 38 34 67 67 66 61 85 108 142 1511 0 43 9 85 123 88 67 111 138 163 197 2062 0 55 42 81 79 72 68 95 121 156 1633 0 39 39 59 73 55 68 81 11 1234 0 45 91 97 27 53 8 117 1225 0 94 111 38 34 42 75 856 0 27 113 126 134 155 1737 0 122 14 153 177 1938 0 29 6 99 109 0 33 72 7110 0 39 4311 0 2512 0


Period Retailer1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 1 0 0 1 1 03 1 1 1 0 0 0 0 0 1 0 0 04 0 1 1 1 1 1 1 1 0 0 0 15 0 0 0 0 1 1 1 0 0 1 1 16 1 0 1 1 0 0 0 1 1 0 0 07 0 1 0 0 1 0 1 0 1 0 0 0





1 0-2-1-0 0-10-12-0 0-3-0 0-4-00-6-7-0 0-8-9-11-0 0-5-0 458879

2 0-7-0 0-10-11-0 1959693 0-1-0 0-2-0 0-3-9-0 1371554 0-8-5-0 0-7-6-0 0-2-0 0-4-12-0 0-3-0 3057665 0-6-0 0-10-11-0 0-12-0 0-7-0 0-5-0 4095696 0-1-0 0-8-9-0 0-4-0 0-3-0 1771197 0-2-7-0 0-5-9-0 119629




Value





Total cost 746123







costCost of time


1 1 84 4200 319893 (1 3 5) 222759 185 9811522 2 52 272522 313474 (1 3 6) 190471 205 7969683 3 42 231117 312597 (1 4 6) 180409 220 7461234 4 36 21096 309867 (1 4 5) 174071 250 7198985 5 37 210083 310735 (1 4 6) 174471 230 7182896 5 37 211246 30961 (1 4 6) 17373 270 7215267 6 37 213764 311514 (1 3 6) 176761 200 722039



Transportationcost



05 25 236385 309729 (1 4 5) 183889 230 4509 75300308 28 229261 308132 (1 3 5) 182608 275 4482 74750111 31 23415 308947 (1 3 5) 181876 230 4465 74797313 33 230983 311138 (1 3 5) 182952 195 4532 74457317 37 231117 312597 (1 4 6) 180409 220 4419 7461232 4 231117 312597 (1 4 6) 180409 220 4419 746123














0 26 227073 310153 (1 3 5) 7055 624276003 26 235268 310703 (1 4 6) 138368 700839005 28 231117 312597 (1 4 6) 180409 746123008 28 230055 312561 (1 4 6) 244352 81196801 28 230745 310843 (1 4 5) 284931 848519015 29 226472 30948 (1 3 5) 394428 9533802 31 229913 312012 (1 3 5) 497812 106624



penalty Total cost

0 437 230527 308956 (1 3 5) 179719 0 71920203 4494 23003 309192 (1 4 6) 181748 195 7404705 4419 231117 312597 (1 4 6) 180409 220 74612308 4555 235768 310155 (1 4 6) 184253 265 7566761 4394 234704 312683 (1 4 6) 181926 255 75481312 4579 23448 314667 (1 4 6) 182735 225 75438215 4383 236642 308349 (1 3 5) 180522 280 7535132 4508 234047 313886 (1 4 6) 181584 260 755517



penalty Total cost

025 4386 234611 310482 146507 230 714605 4527 226242 31041 158853 205 7160051 4419 231117 312597 180409 220 74612315 461 231057 311007 205359 240 7714232 4531 230277 312784 22819 215 7927515 4268 236455 31163 349259 175 91484510 4336 23922 311318 573219 215 11452620 4071 248607 31081 94919 190 152761




6 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














costCost of time


1 1 84 4200 319893 (1 3 5) 222759 185 9811522 2 52 272522 313474 (1 3 6) 190471 205 7969683 3 42 231117 312597 (1 4 6) 180409 220 7461234 4 36 21096 309867 (1 4 5) 174071 250 7198985 5 37 210083 310735 (1 4 6) 174471 230 7182896 5 37 211246 30961 (1 4 6) 17373 270 7215267 6 37 213764 311514 (1 3 6) 176761 200 722039



Transportationcost



05 25 236385 309729 (1 4 5) 183889 230 4509 75300308 28 229261 308132 (1 3 5) 182608 275 4482 74750111 31 23415 308947 (1 3 5) 181876 230 4465 74797313 33 230983 311138 (1 3 5) 182952 195 4532 74457317 37 231117 312597 (1 4 6) 180409 220 4419 7461232 4 231117 312597 (1 4 6) 180409 220 4419 746123














0 26 227073 310153 (1 3 5) 7055 624276003 26 235268 310703 (1 4 6) 138368 700839005 28 231117 312597 (1 4 6) 180409 746123008 28 230055 312561 (1 4 6) 244352 81196801 28 230745 310843 (1 4 5) 284931 848519015 29 226472 30948 (1 3 5) 394428 9533802 31 229913 312012 (1 3 5) 497812 106624



penalty Total cost

0 437 230527 308956 (1 3 5) 179719 0 71920203 4494 23003 309192 (1 4 6) 181748 195 7404705 4419 231117 312597 (1 4 6) 180409 220 74612308 4555 235768 310155 (1 4 6) 184253 265 7566761 4394 234704 312683 (1 4 6) 181926 255 75481312 4579 23448 314667 (1 4 6) 182735 225 75438215 4383 236642 308349 (1 3 5) 180522 280 7535132 4508 234047 313886 (1 4 6) 181584 260 755517



penalty Total cost

025 4386 234611 310482 146507 230 714605 4527 226242 31041 158853 205 7160051 4419 231117 312597 180409 220 74612315 461 231057 311007 205359 240 7714232 4531 230277 312784 22819 215 7927515 4268 236455 31163 349259 175 91484510 4336 23922 311318 573219 215 11452620 4071 248607 31081 94919 190 152761




6 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0 26 227073 310153 (1 3 5) 7055 624276003 26 235268 310703 (1 4 6) 138368 700839005 28 231117 312597 (1 4 6) 180409 746123008 28 230055 312561 (1 4 6) 244352 81196801 28 230745 310843 (1 4 5) 284931 848519015 29 226472 30948 (1 3 5) 394428 9533802 31 229913 312012 (1 3 5) 497812 106624



penalty Total cost

0 437 230527 308956 (1 3 5) 179719 0 71920203 4494 23003 309192 (1 4 6) 181748 195 7404705 4419 231117 312597 (1 4 6) 180409 220 74612308 4555 235768 310155 (1 4 6) 184253 265 7566761 4394 234704 312683 (1 4 6) 181926 255 75481312 4579 23448 314667 (1 4 6) 182735 225 75438215 4383 236642 308349 (1 3 5) 180522 280 7535132 4508 234047 313886 (1 4 6) 181584 260 755517



penalty Total cost

025 4386 234611 310482 146507 230 714605 4527 226242 31041 158853 205 7160051 4419 231117 312597 180409 220 74612315 461 231057 311007 205359 240 7714232 4531 230277 312784 22819 215 7927515 4268 236455 31163 349259 175 91484510 4336 23922 311318 573219 215 11452620 4071 248607 31081 94919 190 152761




6 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Integrated Inventory Routing Problem with ...downloads.hindawi.com/journals/mpe/2014/537409.pdfResearch Article Integrated Inventory Routing Problem with Quality Time