Research ArticleA Two-Phase Heuristic Algorithm for the Common FrequencyRouting Problem with Vehicle Type Choice in the Milk Run

Yu Lin Tianyi Xu and Zheyong Bian

College of Management amp Economics Tianjin University 92 Weijin Road Nankai District Tianjin 300072 China

Correspondence should be addressed to Tianyi Xu 598775393qqcom

Received 12 August 2015 Revised 23 September 2015 Accepted 27 September 2015

Academic Editor Yuri Vladimirovich Mikhlin

Copyright copy 2015 Yu Lin 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

High frequency and small lot size are characteristics of milk runs and are often used to implement the just-in-time (JIT) strategyin logistical systems The common frequency problem which simultaneously involves planning of the route and frequency hasbeen extensively researched in milk run systems In addition vehicle type choice in the milk run system also has a significantinfluence on the operating costTherefore in this paper we simultaneously consider vehicle routing planning frequency planningand vehicle type choice in order to optimize the sum of the cost of transportation inventory and dispatch To this end we develop amathematicalmodel to describe the common frequency problemwith vehicle type choice Since the problem isNP hard we developa two-phase heuristic algorithm to solve the model More specifically an initial satisfactory solution is first generated through agreedy heuristic algorithm to maximize the ratio of the superior arc frequency to the inferior arc frequency Following this a tabusearch (TS) with limited search scope is used to improve the initial satisfactory solution Numerical examples with different sizesestablish the efficacy of our model and our proposed algorithm

1 Introduction

A just-in-time (JIT) supply systemmanaged parts transporta-tion between suppliers and a manufacturer operating underthe JIT discipline [1] By simulating the JIT process from dif-ferent perspectives researchers showed that the JIT strategycould significantly improve efficiency and reduce cost [2ndash4] With progress in research the study of JIT has becomemore specialized For example in the implementation of JITproduction for the manufacturer a minimum inventory ofraw materials is required to meet production needs whichin turn requires that the manufacturer supply parts in smalland multifrequency batches according to operational partsconsumption (speed) For inbound logistics the popularmilkrun is well suited to manufacturersrsquo need for JIT productionbecause of its characteristics of high frequency and small lotsize which enable it to help reduce the cost of inventory andtransportation Therefore many manufacturers use the milkrun as themainmode of transportation for inbound logistics

The milk run originated from the traditional systemof milk distribution and sales in Western culture In thissystem a milkman simultaneously supplied customers with

full bottles ofmilk and picked up the empty ones according toa predefined route Over time the high frequency and smalllot sizes involved in this procedure made it attractive for usein manufacturing worldwide since it was conducive to JITproduction The method has since developed into a popularone for collecting and delivering goods for multiple suppliersand manufactures using freight cars [5] With respect tothe milk run mode researchers [6ndash9] currently focused onthe vehicle routing problem (VRP) Dantzig and Ramser[10] first introduced the idea of the VRP Since then addi-tional scholars have conducted research in this field Withsubsequent research on the problem addressing practicalapplications according to varying constraints the VRP nowhas several formulations For instance the vehicle routingproblem with time windows (VRPTW) adds the constraintof a hard or soft time window based on the VRP whichhas encouraged various solutions [11ndash14] In addition to thetime window measuring the cost of inventory is a crucialfactor for decision makers Chien et al [15] first used the costof inventory as a factor in the vehicle routing problem andclaimed that inventory allocation and the VRP were signifi-cant logistical decisions Based on this premise Chuah [16]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 404868 13 pageshttpdxdoiorg1011552015404868

2 Mathematical Problems in Engineering

discovered that frequency was affected by the inventoryin the VRP and proposed a common frequency routing(CFR) problem Based on the traditional VRP the CFRproblem simultaneously considers the relationship betweenfrequency and inventory Moreover Chuah and Yingling [17]considered the amount in the inventory required to balancethe relationship between inventory and frequency becauselow inventories increase the frequency of milk runs whereashigh inventories have the opposite effect Chuah subsequently[18] undertook a comprehensive study of the CFR problemwhere he discussed the effects of various factors on theproblem and proposed a gradual change in kanban levels toattain optimal cost Further research led to the discovery thatfixed frequency decisions could change inventory cost in theCFR problem and that frequency became a decision variablein JIT systems [1] The multiple vehicle routing problem(MVRP) has been a focus of VRP research in addition to theCFR problem Chan et al [19] formulated a multiple depotmultiple vehicle location routing problem with a robustlocation routing strategy to solve the MVRP Gintner et al[20] considered theMVRPwithmultiple depots an issue thatarose in public transport bus routes and proposed a two-phase method to assign buses to cover a given set of trips tosolve an optimal scheduling problem

Although the traditional CFR problem considers plan-ning of the route and frequency it does not consider theproblemofmultiple types of vehicles in theMVRPand simplyuses vehicle load as a constraint However different vehicletypes have different vehicle load capacities and vehicle loadcan influence pickup frequencywhich in turnhas a significantimpact on inventory cost and transportation cost Thereforethe decision regarding the choice of vehicle is significantSome scholars have addressed this problem Blanton andWainwright [21] used a genetic algorithm to research theproblem of scheduling vehicles of multiple types Ahn andRakha [22] investigated the effects of the choice of route ondifferent types of vehicles usingmicroscopic andmacroscopicemission estimation tools The results showed that froma perspective that considers the environments as well asenergy consumption the shortest route is not always optimalCavalcante and Roorda [23] considered the choice of vehicleas a discrete variable to solve the discrete model problemHowever their work only considered transportation costinfluenced by choice of vehicle without an analysis of theeffect of the frequency plan on the inventory cost andtransportation cost

The abovementioned method shows that the VRP inmilk runs is now being considered in the context of JITsupply systems with ever-increasing constraints and practi-cal orientation However no comprehensive study has yetbeen conducted to simultaneously consider route decisionfrequency plan and vehicle type choice Therefore thispaper proposes the common frequency routing problemwith vehicle type choice (CFR-VTC) We consider the dis-patched vehicles the pickup frequency and routing as theobjective of minimizing the cost of transportation inventoryand dispatch We propose a two-phase heuristic algorithmcalled two-phase tabu search (TS) with limited search scope(TP-TSLSS) The effectiveness of this algorithm is verified

Time

Inventory

Safetyinventory

0

Inventory 1

Inventory 1Inventory 2

average value

Inventory 2average value

Figure 1 Relationship between parts inventory and time

via numerical examples In comparison with the simulatedannealing algorithm (SA) and TS our TP-TSLSS can signifi-cantly improve the efficiency of the search process and obtainmore stable and accurate solutions in a relatively short timeby generating an initial satisfactory solution and limiting thesearch scope Moreover we confirm that the multiple vehicletype mode incurs lower total cost than the same vehicle typemode

The remainder of this paper is organized as follows inSection 2 we describe the CFR-VTC problem and establishthe mathematical model for it The two-phase tabu searchalgorithm with limited search scope (TP-TSLSS) to solve thismodel is proposed in Section 3 Fifty-five numerical examplesare employed in the experiment to demonstrate the effec-tiveness of the proposed algorithm and four transportationmodes are compared to demonstrate efficacy of the CFR-VTCmodel in Section 4 Finally we draw conclusions and suggestdirections for future work in Section 5

2 Formulation

21 Problem Analysis A logistics network system is com-posed of a manufacturer and multiple suppliers To ensureJIT production the manufacturer uses multiple vehicle typesfor high-frequency pickups and small lot sizes in the milkrun With a production line that consumes parts linearlyvehicle type arrangement route frequency and correspond-ing vehicle type planning are required to minimize totaltransportation inventory and dispatch costs

Figure 1 shows the relationship between the parts inven-tory and time In the same period the frequency of inventory2 in the figure is higher than that of inventory 1 However theaverage value of inventory 2 is less than that of inventory 1These results show that higher frequencies incur lower inven-tories but an increase in frequency increases transportationcost Additionally different vehicle type arrangements willproduce different dispatch costs for a stable supply frequencyis determined according to the carrying capacity of eachvehicle type Therefore a trade-off point must exist for

Mathematical Problems in Engineering 3

the arrangement of vehicle type pickup frequency and trans-portation route that can minimize the cost of transportationinventory and dispatch Based on the above analysis vehicletype pickup frequency and transportation route are selectedas decision variables in our mathematic model

We make the following assumptions

Assumption 1 Each vehicle type possesses a different carry-ing capacity Each route requires only one vehicle type toperform the relevant transportation task in the milk run oneach day

Assumption 2 There is a sufficient number of vehicles tocomplete the transportation task The demand of the man-ufacturer by supplier 119894 can only be transported by one truckin the completion of one task

22 Notation

(1) Parameters

119881 = 119894 | 119894 = 0 1 2 119899 is set of suppliers and themanufacturer

0 represents the manufacturer 1 119899 represent thesuppliers119880 = 119906 | 119906 = 0 1 119886 is set of vehicle types120578 is cost of unit transportation distance120573119894is cost of unit parts inventory of supplier 119894

120574119906is the dispatch cost of vehicle type 119906

119888119894119895is the distance between supplier 119894 and 119895

119902119906is carrying capacity of vehicle type 119906

119889119894is the pickup quantity for supplier 119894

119905119894is unloading time at supplier 119894

119905119894119895is travel time from supplier 119894 to supplier 119895

119879 is the maximum allowable cycle time of each routefor a given frequency

(2) Decision Variables

119898 is the number of routes119870 = 119896 | 119896 = 0 1 2 119898 is set of routes

119909119894119895119896119906

=

1 if type 119906 vehicle passes from supplier 119894 to supplier 119895 in route 119896

0 otherwise119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

119910119894119896119906

=

1 if type 119906 vehicle picks up at supplier 119894 in route 119896

0 otherwise119894 isin 119881 119896 isin 119870 119906 isin 119880

119887119896119906=

1 if type 119906 vehicle is dispatched for route 119896

0 otherwise119896 isin 119870 119906 isin 119880

(1)

(3) Variables That Are Dependent on the Decision Variables

119891119896 the frequency of route 119896 depends on 119910

119894119896119906 The

definition is provided in (3)119901119894119896 the pickup quantity of supplier 119894 for a given

frequency depends on 119910119894119896119906The definition is provided

in (4)

23 The Proposed Mathematical Model for CFR-VTC Con-sider

min 119911 = sum

119894isin119881

sum

119895isin119881

sum

119896isin119870

sum

119906isin119880

120578 sdot 119891119896sdot 119888119894119895sdot 119909119894119895119896119906

+ sum

119894isin119881

sum

119896isin119870

120573119894sdot 119901119894119896

+ sum

119896isin119870

sum

119906isin119880

120574119906sdot 119887119896119906

(2)

where

119891119896= lceilsum

119906

sum119894119910119894119896119906

sdot 119889119894

119902119906

rceil 119894 isin 119881 119896 isin 119870 119906 isin 119880 (3)

ldquolceilrceilrdquo means rounding up

119901119894119896= [

sum119906119910119894119896119906

sdot 119889119894

119891119896

] 119894 isin 119881 119896 isin 119870 119906 isin 119880 (4)

subject to

sum

119906

sum

119894

119909119894119895119896119906

= sum

119906

119910119895119896119906

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(5)

sum

119906

sum

119895

119909119894119895119896119906

= sum

119906

119910119894119896119906

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(6)

sum

119906

sum

119896

119910119894119896119906

= 1

119894 isin 119881 119896 isin 119870 119906 isin 119880

(7)

4 Mathematical Problems in Engineering

sum

119906

sum

119894isin119882

sum

119895isin119882

119909119894119895119896119906

le |119882| minus 1

forall119882 sube 119881 0 notin 119882 |119882| ge 2 119896 isin 119870 119906 isin 119880

(8)

sum

119906

sum

119894

119905119894sdot 119910119894119896119906

+sum

119906

sum

119894

sum

119895

119905119894119895sdot 119909119894119895119896119906

le 119879

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(9)

119909119894119895119896119906

= 0 or 1

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(10)

119910119894119896119906

= 0 or 1

119894 isin 119881 119896 isin 119870 119906 isin 119880

(11)

119887119896119906= 0 or 1

119896 isin 119870 119906 isin 119880

(12)

sgn(sum119894

119910119894119896119906) = 119887

119896119906

119894 isin 119881 119896 isin 119870 119906 isin 119880

(13)

The first part of the objective function above addressestransportation cost the second part addresses inventory costand the third computes the cost of dispatched vehiclesThe transportation cost depends on the length of a givenroute as well as the frequency of the route The inventorycost depends on the pickup quantity for the suppliers ineach route and the route frequency Each vehicle type has acorresponding dispatch cost therefore the third part of thefunction depends on the vehicle type choice

The following is a detailed description of constraints (5) to(13) Equations (5) to (7) ensure that each supplier distributesin only one route and the path of each vehicle dispatchedforms a loop Constraint (8) ensures that the subloop isavoided Equation (9) ensures that the cycle time of eachroute for one frequency is less than the maximum allowablecycle time 119879 The value of 119879 depends on the real situationEquations (10) to (12) indicate the scope of variables Equation(13) ensures that each route has only one vehicle dispatched

3 The Design of the Two-Phase HeuristicAlgorithm

CFR-VTC as an extension of VRP is NP hard Therefore wepropose a two-phase heuristic algorithm called two-phase TSwith limited search scope (TP-TSLSS) to solve the problemIn the first phase an initial satisfactory solution is generatedthrough a greedy heuristic algorithm to maximize the ratio1199112of the superior arc frequency to the inferior arc frequency

The following steps are used to generate the initial satisfactorysolution (1) the arcs are divided into the superior arc andthe inferior arc according to the 80-20 rule (the superiorarc and the inferior are described in Section 33) (2) thegreedy heuristic algorithm is then employed to maximizethe ratio 119911

2of the frequency of the superior arc to that of

the inferior arc through the four neighborhood selectionapproaches described in Section 32 (3) the final solutionof the first phase with the optimal 119911max

2in the number of

iterations is exported In the second phase the solution fromthe first phase is improved using the TS To improve searchefficiency we limit the scope of the search in the algorithmto render the 119911

2of the candidate solution greater than the

product of the 119911max2

of the initial satisfactory solution anda coefficient 120572 Calculating the 119911

2value is simpler than

calculating the objective function value 119911 Consequently theTP-TSLSS greatly enhances the search efficiency by limitingthe scope of 119911

2rather than directly using the objective

function value 119911 for the search

31 Representation and Evaluation of Solution Weuse119909 and Vto represent the route and vehicle type respectively and theseform an intact solution We set the manufacturer code to 0and represent the supplier numbers with 119894 (119894 = 1 2 119899)We randomly select the suppliers and the manufacturer asa feasible route We first randomly generate 119898 routes (119898 isin

[2 119899 minus 1] where 119898 is an integer) and randomly distribute 119899suppliers into these routes For example solution 119909 includesroutes 119901

10 minus 2 minus 3 minus 5 minus 0 119901

20 minus 7 minus 1 minus 9 minus 8 minus 0 until

all suppliers have finished selecting These routes constitutesolution 119909 Second we calculate the sum of 119905

119894and 119905119894119895of each

route 1199011 1199012 of solution 119909 and we examine whether the

total time of each route meets the time constraint of (9) Ifnot we randomly split the route whose total time exceeds119879 For example if the total time of route 119901

2exceeds 119879 the

route 11990120 minus 7 minus 1 minus 9 minus 8 minus 0 is randomly split as 119901

210 minus

7 minus 1 minus 0 and 119901220 minus 9 minus 8 minus 0 Then recalculate the total

time of each new route respectively and repeat the samestep until all the routes meet the time constraint of (9) Inthat way the new solution adjusted constitutes the feasiblesolution Finally we randomly select the vehicle type 119906

119895(119895 =

1 2 119898) corresponding to each route of feasible solutionand set [119906

1 1199062 119906

119898] as solution V for each vehicle type

We use the objective function value to evaluate the qualityof solutions

32 Neighborhood Solution Selection Method There are fourmethods to select a neighborhood solution

(1) Randomly select a route and two suppliers on theroute Reverse the traversal order of the two suppliersFor instance randomly select route 119901

10 minus 7 minus 1 minus 9 minus

8 minus 0 Randomly reverse the order of 7 minus 1 minus 9 minus 8 1199011

becomes 0 minus 8 minus 9 minus 1 minus 7 minus 0

(2) Randomly select two routes and a supplier fromone of these two routes Insert the chosen supplierrandomly into the other route If there is no supplierremaining in the route from where the supplier wastaken set the route to null For instance randomlyselect routes119901

10minus2minus3minus5minus0 and119901

20minus7minus1minus9minus8minus0

Randomly select supplier 2 from 1199011to be inserted

into 1199012 1199011becomes 0 minus 3 minus 5 minus 0 and 119901

2becomes

0 minus 7 minus 1 minus 2 minus 9 minus 8 minus 0

Mathematical Problems in Engineering 5

Randomly generate an initial solution of (1199090 V0) which meets the time constraint in (9) Set the current solution 119909now = 1199090 Vnow = V

0

then calculate 1199112(119909now Vnow) Set it = 1 (it records the current number of iterations)

Do while it lt 119872If rand lt 119901

1

Generate a neighborhood solution (1199091 V1) through the first neighborhood method

Do while the time of the new solution (1199091 V1) is not less than 119879

Regenerate a neighborhood solution (1199091 V1) through the first neighborhood method

End doIf 1199112(1199091 V1) gt 1199112(119909now Vnow)

119909now = 1199091Vnow = V

1

End ifEnd ifExecute the other three neighborhood methods using a similar processit = it + 1

End doOutput the initial satisfactory solution (1199091015840

0 V10158400) = (119909now Vnow) and define 119911max

2= 1199112(1199091015840

0 V10158400)

Algorithm 1

(3) Randomly select a route and a supplier from it Addthe supplier to another route For instance say thereare five routes 119901

1 1199012 1199013 1199014 and 119901

5 Randomly select

route 11990110minus2minus3minus5minus0 and supplier 2 and move the

latter into 1199016 1199016becomes 0 minus 2 minus 0

(4) Randomly select a vehicle type of solution V allowingthe vehicle type to changewithin the allowable vehicletype range For instance V = [2 2 3 4 5] becomes[2 2 3 5 5]

33 The First-Phase Algorithm 80-20 rule is originally pro-posed by the Italian economist Pareto so it is also calledParetorsquos law Commonly it is stated that 20 of all causesbring about 80 of all effects [24ndash26] In this paper we firstdefine an arc set 119860 that records the distances between anytwo nodes which contain the suppliers and themanufacturerThen according to the 80-20 rule we define 20 of arcs in set119860 with the shortest length as the superior arc set 119860

1and the

other arcs are defined as the inferior arc set 1198602 The lengths

of all arcs in 1198601are smaller than those of the arcs in 119860

2

that is |1198861| lt |119886

2| 1198861isin 1198601 1198862isin 1198602 The number of arcs

in 1198602is four times that of the number in 119860

1 For a specific

solution (119909 V) we define 119891119904 as the sum of frequencies of allsuperior arcs and 119891119894 as the sum of frequencies of all inferiorarcs 119911

2(119909 V) is defined as 119891119904119891119894 The objective of the first-

phase algorithm is to maximize 1199112 We set the number of

iterations to119872The greedy heuristic algorithm is designed as shown in

Algorithm 1

34 The Second-Phase Algorithm Osman [27] Taillard et al[28] and Alonso et al [29] applied TS to solve the VRP andverified its effectiveness In this paper we limit the scope ofsearch to ensure that the 119911

2value of every candidate solution

is greater than 120572sdot119911max2

(where 120572 is a coefficient slightly smallerin value than 1 We find that 120572 isin [07 09] is an appropriate

range) We call the second-phase algorithm TS with limitedsearch scope (TSLSS) We set the number of iterations as NI

The TSLSS is shown in Algorithm 2

35 The Process of Two-Phase TS with Limited Search ScopeWe combine the first-phase algorithm in Section 33 and thesecond-phase algorithm in Section 34 as the two-phase TSwith limited search scope (TP-TSLSS)

Figure 2 shows the specific process

4 Computational Experiments

In this section our proposed two-phase heuristic algorithmis programmed in MATLAB R2011b to solve 55 smallmedium and large numerical problemsThese computationalexperiments are run on an Acer 4820TG computer with anIntel i5 CPU (24GHz) and 4GB (347 available) of memory

41 Experiments with Varying Supplier Size Weuse a numberof examples involving varying numbers of suppliers to vali-date ourmodel and algorithmWhen the number of suppliersis below 100 both the horizontal and vertical coordinatesof suppliers are drawn from the uniform distribution inthe range [0 100] using the randi() function in MATLABMoreover the coordinates of the suppliers are drawn from theuniform distribution in the range [0 200] when the numberof suppliers is between 100 and 200 Finally when the numberof suppliers is between 200 and 300 the coordinates aredrawn from the uniform distribution in the range [0 300]The horizontal and vertical coordinates are represented by119883 and 119884 respectively The pickup quantity is randomly gen-erated from the uniform distribution in the range [80 150]whereas the inventory cost of each supplierrsquos parts is subjectto uniform distribution in the range [0 1] The dispatch costfor each vehicle type is uniformly distributed from 100 to400 We set the coordinates of the manufacture to (50 50)

6 Mathematical Problems in Engineering

Set the total number of iterations performed by the TS (NI) Set tabulist = Set the initial solution to (11990910158400 V10158400) which is generated in

the first phase Let the optimal solution be 119909best = 1199091015840

0 Vbest = V1015840

0and the current solution 119909now = 119909

1015840

0 Vnow = V1015840

0

Let 119899 = 0 (record the number of current iterations) Set the number of candidate solutions (ℎ) in the TSDo while 119899 lt NI

Employ the four neighborhood methods to generate h candidate solutions (1199091 V1) (1199092 V2) (119909

ℎ Vℎ) satisfying the time

constraint in (9) with 1199112values greater than 120572 sdot 119911max

2 Calculate the objective function values of these solutions

119911(1199091 V1) 119911(119909

2 V2) 119911(119909

ℎ Vℎ)

Obtain (119909opt Vopt) whose objective function value is optimalIf min119911(119909

1 V1) 119911(119909

2 V2) 119911(119909

ℎ Vℎ) lt 119911(119909best Vbest)

(119909now Vnow) = (119909opt Vopt)Update the tabulist

ElseDo while (119909opt Vopt) is in the tabulistSelect the suboptimal solution from the candidate solutions and record it as (119909opt Vopt)

End do(119909now Vnow) = (119909opt Vopt)Update the tabulist

End if119899 = 119899 + 1

End do

Algorithm 2

The algorithms described in Section 3 are implemented to testthese numerical instances

42 Algorithm Comparison All of the 55 examples are gen-erated using the method described in Section 41 Similarproblems have been solved in some of the literature [27ndash29] using TS algorithms whereas other studies [30ndash32] haveemployed simulated annealing algorithms (SA) to solve themThus to demonstrate the effectiveness of our model and theTP-TSLSS algorithm each example in this study is run fivetimes using TP-TSLSS TS and the SA We then analyzeand compare the experiment results obtained by the threealgorithms Stop criteria of SA and TS are defined as followsSA stops when the temperature reaches a specific valueaccording to the scale of each instance and TS stops aftera specific number of iterations are performed by TS Thenumber of iterations performed by TS is twice as TP-TSLSS

Table 1 shows the results of the first phase of the two-phase algorithm SN represents the number of suppliersColumn ldquo119911(119909

0 V0)rdquo represents the average objective function

value of the initial solution ldquo119911(1199091015840 V1015840)rdquo represents the averageobjective function value of solutions obtained by the first-phase algorithm and ldquo1199051015840rdquo represents the time taken by thefirst-phase algorithm (the unit of time used is second)Following the execution of the first-phase algorithm wefind that the 119911

2value increases significantly within a short

time Consequently the objective function value 119911 can beimproved significantly through the optimization of the first-phase algorithm within a short time

Table 2 shows the comparison results of the optimalobjective function values of solutions obtained by the threealgorithms We choose examples 23 31 36 and 41 to showthe convergence of TS and TP-TSLSS in Figures 3 4 5 and 6

The horizontal axis represents the number of iterations andthe vertical axis represents the average objective functionvalue The column ldquoaveragerdquo represents the average objectivefunction value of the optimal solution in each algorithmldquo119905rdquo represents the CPU time for each algorithm and ldquo120590rdquorepresents the standard deviation

We draw the following conclusions from a comparison ofSA TS and TP-TSLSS shown in Table 2 and the convergencegraphs (Figures 3 4 5 and 6)

(1) When the scales of the instances are small theobjective function values of the solutions obtainedby SA are greater than those obtained by the othertwo algorithms Therefore the performance of SAis inferior to that of TS and TP-TSLSS in solvingsmall-scale problems By contrast SA outperformsTS but is worse than TP-TSLSS when the number ofsuppliers exceeds 30 A comparison of TP-TSLSS andTS shows that the former yields solutionswith smallerobjective function values than the latter except fora few small-scale instances Moreover Figures 3 45 and 6 show that the number of iterations of TSis twice that of TP-TSLSS but TS is inferior to TP-TSLSS The performance of TP-TSLSS is better thanthat of the other two algorithms for most instancesThe advantage of TP-TSLSS becomesmore obvious asthe scale of problems increases

(2) The computational time of TP-TSLSS is the shortestfollowed by SA and TS

(3) Solutions obtained by TP-TSLSS have the smalleststandard deviationThe comparison of standard devi-ation values proves that TP-TSLSS is the most robustfollowed byTSwithmoderate robustness and SAwith

Mathematical Problems in Engineering 7

The first phase algorithm start

Randomly generate the initial solution

Randomly generate the new

If it satisfies the time constraint (9)

Yes

No

If it satisfies the time constraint (9)

No

Yes

Yes No

Yes

No

The second phase algorithm start

solution of the first phase

solutions that meet the timeconstraint (9) in the

function value is the smallest among the candidate solutions

If the selected solution

If the selectedsolution

Select the suboptimal

Accept the new solution and replace the current solution

Accept the new solution and replace the current solution

Update the tabulist

Stop

Yes

No

YesNo

No

Yes

Yes

No

(x0 0)

neighborhood of N(xnow now)

Calculate z2(x1 1)

If z2(x1 1) gt z2(xnow now)

Accept (x1 1) set Reject (x1 1)

it = it + 1

it le M

(xnow now)

set zmax2 = z2(xnow now)

neighborhood of N(xnow now)

If the z2 value of eachcandidate solution gt 120572 middot z

max2

Select the solution (xoptopt) whose objective

(xopt opt) is better than (xbest best)

(xopt opt) is

candidate solution as (xoptopt)

n = n + 1

If n lt NI

in tabulist

Randomly generate h candidate

Set xnow = x0 now = 0 Calculatez2(xnow now) it = 1

solution (x1 1) in the

(xnow now) = (x1 1)

Output (x9984000 9984000) =

Tabulist H = Φ set the

9984000 xnow = x

9984000 and

algorithm xbest = x9984000 best =

now = 9984000 n = 0

Figure 2 The TP-TSLSS process

the least Therefore the two-phase algorithm (TP-TSLSS) can obtain better and more stable solutions ina relatively short time

To verify the effectiveness of the two-phase heuristicalgorithm we select an optimal solution for TP-TSLSS froman example involving 30 suppliers Figure 7 shows the pathdiagram

In addition we take the 119905-test of statistical analysis tocompare difference in objective function values of the three

algorithms 119879 test employs 119905 distribution theory to infer theoccurring probability of differences and to examine whetherthe differences are significant in two averages [33 34] Thispaper takes 119905-test to compare the differences in averageobjective function values of the three algorithms First werun 10 times for all 55 examples We employ the SPSS 18 totake normality test for the objective function values of the 55examples We find that except the objective function valuesof the five numerical examples with the scale of 10 suppliersthe rest of the numerical examplesrsquo objective function values

8 Mathematical Problems in Engineering

Table 1 Results obtained for the first-phase algorithm

Instance SN 119911(1199091015840 V1015840) 119911(1199090 V0) 1199112(1199091015840 V1015840) 119911

2(1199090 V0) 1199051015840 (s)

1 10 18391 30698 2 01667 0322 10 21128 2582 24444 02308 0313 10 23407 38168 23333 025 0344 10 21757 26998 30435 05455 0295 10 2490 29351 07059 02444 036 20 3158 57114 48182 02535 0347 20 50645 65045 132381 03488 0338 20 3600 58948 68667 01227 0319 20 43496 57095 103529 03793 0310 20 32278 52732 88571 01333 03211 30 42922 73623 149 02202 03612 30 68309 76401 71695 02818 03513 30 6574 72963 55 00896 03614 30 64024 65789 380147 023 03715 30 48815 92522 245 04107 03616 50 12523 14449 397647 0244 03617 50 11190 14649 185238 03557 03518 50 11945 14672 1297 03254 03719 50 11919 12149 91 025 03820 50 10930 17825 211429 02551 03621 70 11737 17540 479091 02878 03422 70 15508 18462 545625 01973 03923 70 12484 20227 1003333 02653 03824 70 15890 20555 29 03419 03725 70 14998 19819 172857 02903 03526 90 18475 20596 450556 03529 03827 90 19643 20328 472356 03427 0428 90 18792 19873 46729 02957 03929 90 16268 20921 8746 03268 03830 90 20846 28943 3667 03832 04131 110 25081 46116 2068 04035 08332 110 30848 45420 1857 04689 07833 110 27515 42895 1939 05073 07934 110 28371 37652 185 04971 07335 110 23627 39712 3046 04837 07836 150 38053 59964 180213 02545 09237 150 36892 56279 3263 02384 08938 150 39604 62874 20481 01953 09339 150 40285 58164 16306 02491 09240 150 37681 63692 2617 01902 0941 200 86334 111050 55 02611 210242 200 88274 113130 35806 02194 208743 200 85145 113670 41758 02874 207444 200 89732 112823 528 02391 212345 200 87692 135713 4825 0261 208446 250 111760 145270 39955 02601 402947 250 111830 146220 4316 02206 418148 250 118713 149254 38421 02059 401749 250 116915 153873 3617 01973 398250 250 120317 148148 3291 02391 405951 300 131980 169670 44194 02328 597252 300 129468 163200 4841 0304 629253 300 139642 161726 4163 02845 581754 300 130613 159212 40157 02174 618355 300 129862 163944 49153 0316 6084

0 50 100 150 200 250 300 350 40006

08

1

12

14

16

18

2

TSTP-TSLSS

times104

Figure 3 Convergence graph of 70 suppliers

0 100 200 300 400 500 60015

2

25

3

35

4

45

TSTP-TSLSS

times104

Figure 4 Convergence graph of 110 suppliers

obtained by all the three algorithms have a good normalityThen we take two hypothesis tests at 005 confidence levelconditions The first hypothesis test is designed for SA andTP-TSLSS with the null hypothesis 119867 120583

1le 1205830 The second

hypothesis test is designed for TS and TP-TSLSS with thenull hypothesis 119867 120583

2le 1205830 Finally we take independent

sample 119905-test for the two hypotheses respectively The 119901

values obtained are listed in Table 3For the comparison of SA and TP-TSLSS all 119901 values of

the examples with 20 or more suppliers are less than 005therefore the null hypotheses of these examples are rejecteddemonstrating that TP-TSLSS is better than SA

For the comparison of TS and TP-TSLSS apart from 119901

values of examples of 20 suppliers scale (TS and TP-TSLSS)being greater than 005 119901 values of the remaining examplesare less than 005 therefore the null hypotheses of these

Mathematical Problems in Engineering 9

Table 2 Results obtained for the three algorithms (SA TS and TP-TSLSS)

Instance SN SA TS TP-TSLSSAverage 119905SA (s) 120590SA Average 119905TS (s) 120590TS Average 119905TP-TSLSS (s) 120590TP-TSLSS

1 10 16191 1654 8731 13862 3247 0 13862 1463 02 10 15743 1526 7631 1388 3258 0 1388 1419 03 10 1491 1594 4020 14219 3282 0 14219 1346 04 10 15825 1631 7218 14778 3354 0 14778 1322 05 10 16049 1682 7194 14875 3196 4031 14382 1407 06 20 25689 1727 8256 23333 4664 4107 23549 1686 32187 20 26021 1714 841 26698 4561 562 25649 1811 35918 20 24928 1793 8646 25106 4628 479 24217 1765 3459 20 26183 1737 8729 27656 4619 601 25714 1879 331410 20 23105 1749 8439 22758 4594 504 22423 1814 36211 30 34219 2015 1183 35278 5492 7660 33173 1825 427112 30 36798 2125 11038 34214 5135 7135 33082 1819 41613 30 34192 2062 10949 35073 5284 7058 33019 1746 408414 30 35193 2048 11164 36465 5175 7563 3396 1822 402315 30 38276 2116 12645 42145 5428 7937 37361 1707 42516 50 6425 2485 41072 59709 6514 12047 54051 2116 555717 50 60158 2472 38264 61101 6628 11062 5998 2392 583518 50 57028 2493 34183 57413 6604 8459 56276 2216 519419 50 56026 2419 33963 58708 6592 9246 52185 2145 574920 50 60924 2562 32974 63686 6585 8794 5686 2281 538521 70 85106 4057 37016 90797 7849 18584 76153 2136 1170322 70 82179 3947 38294 8820 7632 17294 78465 2073 1075223 70 82462 3929 41174 84594 7940 16848 73865 2491 1047324 70 88171 4019 38265 89905 7835 17182 81702 2615 1014925 70 81096 4095 34793 82468 7529 1749 80516 2491 1135226 90 10492 6418 40327 10737 10673 27232 92756 4215 1772827 90 11600 6429 41046 12446 10428 23472 90794 3881 1713628 90 108433 6316 40684 136475 10582 25819 89615 3974 1738529 90 116923 6285 40958 114367 10418 24174 92359 4039 1748230 90 102395 6193 40715 108691 10837 22964 95725 4139 1698331 110 20487 8972 77967 211168 13248 54139 180161 5797 3382932 110 187321 8017 78924 204292 13082 52764 117423 5826 3103433 110 190355 8526 83592 198163 13569 51074 185354 5494 3409334 110 190241 8438 93164 192525 13194 50986 179394 5618 358535 110 193482 8525 76492 216241 13074 51196 173106 5769 3792936 150 30813 18038 13484 27982 35922 97346 24094 15006 8353537 150 29813 18157 12467 30583 35542 10048 24695 14678 8214638 150 30847 18262 13295 31764 35686 98683 28174 14903 7824839 150 29139 18037 14829 30269 35742 90474 27108 14286 7958340 150 28742 17984 12793 29264 35291 98134 26495 14982 8497341 200 50954 24044 32316 56340 41249 28743 48246 19151 1576342 200 58192 23859 35935 58861 41465 27631 52407 20018 1639443 200 56102 24168 38193 57928 41672 24926 52887 19427 1582644 200 57149 24027 37194 59218 41839 27428 50961 18737 1782445 200 56291 23939 36913 58316 41962 29726 51620 19472 1629746 250 67194 38054 46911 75935 63028 36877 62171 32006 2428747 250 62719 37937 48512 77501 63542 35825 59954 31078 2592648 250 61461 37846 49236 78519 63887 36014 58917 31403 2174549 250 70192 37735 43926 76110 64041 35826 63625 31786 2852850 250 64192 37342 47295 71926 63993 35714 62128 32682 2394851 300 82155 52069 73979 92167 91249 58212 78166 46351 4232652 300 84124 51053 74924 91135 89465 56149 77640 44718 3872653 300 88307 51749 71486 90114 90672 59376 75783 48227 3741954 300 83491 51283 73951 87910 91839 52864 78629 41937 4173655 300 76529 50959 75056 88412 90962 58251 73210 42672 43945

10 Mathematical Problems in Engineering

0 100 200 300 400 500 600 700 800 900 10002

25

3

35

4

45

5

55

6times10

4

TSTP-TSLSS

Figure 5 Convergence graph of 150 suppliers

0 100 200 300 400 500 600 700 800 900 1000 11004

5

6

7

8

9

10

11times10

4

TSTP-TSLSS

Figure 6 Convergence graph of 200 suppliers

examples are rejected demonstrating TP-TSLSS is better thanTS when the suppliers are more than 30

43 Results Analysis of the Different Modes General studiesof the routing problems did not consider the effect of vehicletype choice on transportation frequency and inventory costwhile we take the vehicle type choice into consideration Toexamine this effect we make a comparison of the generaltransportation mode (Mode 1) with the CFR-VTC mode(Mode 2) for different scales of instances

Mode 1 Only one vehicle type is allowed for the transporta-tion task

Mode 2 All vehicle types are available

Figure 8 shows a comparison of the results for the twomodes (Mode 1 includes three cases only vehicle type 1 onlyvehicle type 2 and only vehicle type 3 in which the vehicle

17

23

306

1127

2021

2719

1018

9

16

14

11

29

4

528

2 3

826 24

22

1513

250

00 50 100

50

100

Vehicle 3 frequency 5Vehicle 2 fre

quency 4

Vehicle 2 frequency 8

X

Y

Figure 7 The path diagram of TP-TSLSS

types are not required for decision Mode 2 signifies that allthe three vehicle types are involved in the decision)The loadcapacity of vehicle type 1 is the smallest whose value is 100and that of vehicle type 2 is moderate whose value is 300whereas the capacity of vehicle type 3 is the largest whosevalue is 600 The horizontal axis in Figure 8 represents thenumber of suppliers and the vertical axis shows the four typesof costmdashtransportation cost inventory cost dispatch costand total cost The four cost trends are shown in Figure 8

The transportation cost graph in Figure 8 shows thatthe case involving only vehicle type 3 records the lowesttransportation cost because a vehicle with a large loadcapacity requires less frequency to complete the pickup taskHowever the inventory cost and dispatch cost in this caseare the highest when the scales of instances are large becauseof low frequency and high load capacity By contrast thecase involving only vehicle type 1 has the advantage of lowinventory and dispatch costs while the transportation costin this case is the highest However the total cost of Mode2 is lower than all the three cases in Mode 1 Therefore wefind that we can balance the cost of transportation inventoryand dispatch by a reasonable choice of vehicle to acquire theoptimal total cost Mode 2 has greater cost advantages thanMode 1

5 Conclusions

In this paper we considered the common frequency routingproblem with vehicle type choice in milk runs in logisticalsystems We developed a mathematical model to describe thevehicle type choice frequency planning and vehicle routeplanning in the milk run system To solve this model wedeveloped a two-phase TS algorithm with limited searchscope (TP-TSLSS) that increased the search efficiency Theproposed TP-TSLSS algorithm was tested on 55 numerical

Mathematical Problems in Engineering 11

0

20000

40000

60000

80000

100000

120000

10 20 30 50 70 90 110 150 200 250 300

Total cost

Vehicle type 1

Vehicle type 3Vehicle type 2

Vehicle type 1

Vehicle type 3Vehicle type 2

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 20 30 50 70 90 110 150 200 250 300

Transportation cost

Multiple vehicle types

Multiple vehicle types

Multiple vehicle types

Vehicle type 1

Vehicle type 3Vehicle type 2

Multiple vehicle typesVehicle type 1

Vehicle type 3Vehicle type 2

0

2000

4000

6000

8000

10000

12000

14000

10 20 30 50 70 90 110 150 200 250 300

Inventory cost

0

5000

10000

15000

20000

25000

30000

10 20 30 50 70 90 110 150 200 250 300

Dispatch cost

Figure 8 A comparison of the mode results

examples with varying scales for verification and was com-pared with the TS and the SA methods The results showedthat our TP-TSLSS obtained better solutions in a shortertime and a more stable manner By comparing differenttransportationmodes we concluded that consideration of thevehicle type choice could help save on cost of transportationinventory and dispatch

This was the first study to comprehensively considerfrequency planning vehicle type choice and routing plan-ning Although the consideration of a variety of vehicle

types significantly increased the complexity of the originalCFR problem the TP-TSLSS algorithm could reduce searchtime and improve accuracy through limiting search scopeIn the future we intended to extend this model to casesinvolving multiple manufacturers The problem will be morecomplex and the decision might depend on additionalfactors Moreover multiple depots could be considered froma practical perspective Finally we planned to develop evenmore efficient and accurate algorithms to solve these prob-lems

12 Mathematical Problems in Engineering

Table 3 119901 value of 119905-test

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

1 10 mdash mdash2 10 mdash mdash3 10 mdash mdash4 10 mdash mdash5 10 mdash mdash6 20 000031 0167 20 00000028 0188 20 0000026 0219 20 00043 01210 20 0000030 02811 30 00051 0008512 30 00032 002713 30 00017 001014 30 00072 0003615 30 0015 0002916 50 00000022 0000002517 50 00000063 00004118 50 0012 0006119 50 00000027 0000003520 50 00000012 0000001821 70 000023 000001022 70 000014 0000002923 70 0000092 0000001224 70 0000011 00000006325 70 000047 000002126 90 0000016 000004727 90 00000024 00000001828 90 00000051 00000001029 90 0000030 000002830 90 0000029 000001131 110 0000089 000006232 110 00000033 00000001033 110 000032 00002934 110 0000092 000005935 110 0000075 0000003336 150 00000013 0000001437 150 00000014 0000002038 150 0000067 000005139 150 000018 000005340 150 000051 000004741 200 0000069 00000001542 200 000000098 00000008343 200 0000074 0000001944 200 00000023 00000001245 200 0000082 0000003846 250 000000080 000000003747 250 000000015 0000000013

Table 3 Continued

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

48 250 000000011 0000000009849 250 00000064 00000001350 250 0000094 00000002751 300 00000011 00000006052 300 000000081 00000005653 300 000000042 00000001754 300 00000098 00000008255 300 0000020 0000000041

Conflict of Interests

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

References

[1] K H Chuah and J C Yingling ldquoRouting for a just-in-timesupply pickup and delivery systemrdquo Transportation Science vol39 no 3 pp 328ndash339 2005

[2] L Zhuang ldquoTowards a more economic production and just-in-time delivery systemrdquo International Journal of ProductionEconomics vol 36 no 3 pp 307ndash313 1994

[3] M Savsar and A Al-Jawini ldquoSimulation analysis of just-in-time production systemsrdquo International Journal of ProductionEconomics vol 42 no 1 pp 67ndash78 1995

[4] R R Lummus ldquoA simulation analysis of sequencing alternativesfor JIT lines using kanbansrdquo Journal of OperationsManagementvol 13 no 3 pp 183ndash191 1995

[5] S J Sadjadi M Jafari and T Amini ldquoA new mathematicalmodeling and a genetic algorithm search for milk run problem(an auto industry supply chain case study)rdquo The InternationalJournal of Advanced Manufacturing Technology vol 44 no 1-2pp 194ndash200 2009

[6] M Jafari-Eskandari S J Sadjadi M S Jabalameli and ABozorgi-Amiri ldquoA robust optimization approach for the MilkRun problem (An auto industry supply chain case study)rdquo inProceedings of the IEEE International Conference on Computersamp Industrial Engineering (CIE rsquo09) pp 1076ndash1081 TroyesFrance July 2009

[7] M J Eskandari A R Aliahmadi and G H H Khaleghi ldquoArobust optimisation approach for the milk run problem withtime windows with inventory uncertainty an auto industrysupply chain case studyrdquo International Journal of Rapid Man-ufacturing vol 1 no 3 pp 334ndash347 2010

[8] T Du F KWang and P-Y Lu ldquoA real-time vehicle-dispatchingsystem for consolidating milk runsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 43 no 5 pp565ndash577 2007

[9] J Ma and G Sun ldquoMutation Ant colony algorithm of milk-run vehicle routing problemwith fastest completion time basedon dynamic optimizationrdquo Discrete Dynamics in Nature andSociety vol 2013 Article ID 418436 6 pages 2013

[10] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

discovered that frequency was affected by the inventoryin the VRP and proposed a common frequency routing(CFR) problem Based on the traditional VRP the CFRproblem simultaneously considers the relationship betweenfrequency and inventory Moreover Chuah and Yingling [17]considered the amount in the inventory required to balancethe relationship between inventory and frequency becauselow inventories increase the frequency of milk runs whereashigh inventories have the opposite effect Chuah subsequently[18] undertook a comprehensive study of the CFR problemwhere he discussed the effects of various factors on theproblem and proposed a gradual change in kanban levels toattain optimal cost Further research led to the discovery thatfixed frequency decisions could change inventory cost in theCFR problem and that frequency became a decision variablein JIT systems [1] The multiple vehicle routing problem(MVRP) has been a focus of VRP research in addition to theCFR problem Chan et al [19] formulated a multiple depotmultiple vehicle location routing problem with a robustlocation routing strategy to solve the MVRP Gintner et al[20] considered theMVRPwithmultiple depots an issue thatarose in public transport bus routes and proposed a two-phase method to assign buses to cover a given set of trips tosolve an optimal scheduling problem

Although the traditional CFR problem considers plan-ning of the route and frequency it does not consider theproblemofmultiple types of vehicles in theMVRPand simplyuses vehicle load as a constraint However different vehicletypes have different vehicle load capacities and vehicle loadcan influence pickup frequencywhich in turnhas a significantimpact on inventory cost and transportation cost Thereforethe decision regarding the choice of vehicle is significantSome scholars have addressed this problem Blanton andWainwright [21] used a genetic algorithm to research theproblem of scheduling vehicles of multiple types Ahn andRakha [22] investigated the effects of the choice of route ondifferent types of vehicles usingmicroscopic andmacroscopicemission estimation tools The results showed that froma perspective that considers the environments as well asenergy consumption the shortest route is not always optimalCavalcante and Roorda [23] considered the choice of vehicleas a discrete variable to solve the discrete model problemHowever their work only considered transportation costinfluenced by choice of vehicle without an analysis of theeffect of the frequency plan on the inventory cost andtransportation cost

The abovementioned method shows that the VRP inmilk runs is now being considered in the context of JITsupply systems with ever-increasing constraints and practi-cal orientation However no comprehensive study has yetbeen conducted to simultaneously consider route decisionfrequency plan and vehicle type choice Therefore thispaper proposes the common frequency routing problemwith vehicle type choice (CFR-VTC) We consider the dis-patched vehicles the pickup frequency and routing as theobjective of minimizing the cost of transportation inventoryand dispatch We propose a two-phase heuristic algorithmcalled two-phase tabu search (TS) with limited search scope(TP-TSLSS) The effectiveness of this algorithm is verified

Time

Inventory

Safetyinventory

0

Inventory 1

Inventory 1Inventory 2

average value

Inventory 2average value

Figure 1 Relationship between parts inventory and time

via numerical examples In comparison with the simulatedannealing algorithm (SA) and TS our TP-TSLSS can signifi-cantly improve the efficiency of the search process and obtainmore stable and accurate solutions in a relatively short timeby generating an initial satisfactory solution and limiting thesearch scope Moreover we confirm that the multiple vehicletype mode incurs lower total cost than the same vehicle typemode

The remainder of this paper is organized as follows inSection 2 we describe the CFR-VTC problem and establishthe mathematical model for it The two-phase tabu searchalgorithm with limited search scope (TP-TSLSS) to solve thismodel is proposed in Section 3 Fifty-five numerical examplesare employed in the experiment to demonstrate the effec-tiveness of the proposed algorithm and four transportationmodes are compared to demonstrate efficacy of the CFR-VTCmodel in Section 4 Finally we draw conclusions and suggestdirections for future work in Section 5

2 Formulation

21 Problem Analysis A logistics network system is com-posed of a manufacturer and multiple suppliers To ensureJIT production the manufacturer uses multiple vehicle typesfor high-frequency pickups and small lot sizes in the milkrun With a production line that consumes parts linearlyvehicle type arrangement route frequency and correspond-ing vehicle type planning are required to minimize totaltransportation inventory and dispatch costs

Figure 1 shows the relationship between the parts inven-tory and time In the same period the frequency of inventory2 in the figure is higher than that of inventory 1 However theaverage value of inventory 2 is less than that of inventory 1These results show that higher frequencies incur lower inven-tories but an increase in frequency increases transportationcost Additionally different vehicle type arrangements willproduce different dispatch costs for a stable supply frequencyis determined according to the carrying capacity of eachvehicle type Therefore a trade-off point must exist for

Mathematical Problems in Engineering 3

the arrangement of vehicle type pickup frequency and trans-portation route that can minimize the cost of transportationinventory and dispatch Based on the above analysis vehicletype pickup frequency and transportation route are selectedas decision variables in our mathematic model

We make the following assumptions

Assumption 1 Each vehicle type possesses a different carry-ing capacity Each route requires only one vehicle type toperform the relevant transportation task in the milk run oneach day

Assumption 2 There is a sufficient number of vehicles tocomplete the transportation task The demand of the man-ufacturer by supplier 119894 can only be transported by one truckin the completion of one task

22 Notation

(1) Parameters

119881 = 119894 | 119894 = 0 1 2 119899 is set of suppliers and themanufacturer

0 represents the manufacturer 1 119899 represent thesuppliers119880 = 119906 | 119906 = 0 1 119886 is set of vehicle types120578 is cost of unit transportation distance120573119894is cost of unit parts inventory of supplier 119894

120574119906is the dispatch cost of vehicle type 119906

119888119894119895is the distance between supplier 119894 and 119895

119902119906is carrying capacity of vehicle type 119906

119889119894is the pickup quantity for supplier 119894

119905119894is unloading time at supplier 119894

119905119894119895is travel time from supplier 119894 to supplier 119895

119879 is the maximum allowable cycle time of each routefor a given frequency

(2) Decision Variables

119898 is the number of routes119870 = 119896 | 119896 = 0 1 2 119898 is set of routes

119909119894119895119896119906

=

1 if type 119906 vehicle passes from supplier 119894 to supplier 119895 in route 119896

0 otherwise119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

119910119894119896119906

=

1 if type 119906 vehicle picks up at supplier 119894 in route 119896

0 otherwise119894 isin 119881 119896 isin 119870 119906 isin 119880

119887119896119906=

1 if type 119906 vehicle is dispatched for route 119896

0 otherwise119896 isin 119870 119906 isin 119880

(1)

(3) Variables That Are Dependent on the Decision Variables

119891119896 the frequency of route 119896 depends on 119910

119894119896119906 The

definition is provided in (3)119901119894119896 the pickup quantity of supplier 119894 for a given

frequency depends on 119910119894119896119906The definition is provided

in (4)

23 The Proposed Mathematical Model for CFR-VTC Con-sider

min 119911 = sum

119894isin119881

sum

119895isin119881

sum

119896isin119870

sum

119906isin119880

120578 sdot 119891119896sdot 119888119894119895sdot 119909119894119895119896119906

+ sum

119894isin119881

sum

119896isin119870

120573119894sdot 119901119894119896

+ sum

119896isin119870

sum

119906isin119880

120574119906sdot 119887119896119906

(2)

where

119891119896= lceilsum

119906

sum119894119910119894119896119906

sdot 119889119894

119902119906

rceil 119894 isin 119881 119896 isin 119870 119906 isin 119880 (3)

ldquolceilrceilrdquo means rounding up

119901119894119896= [

sum119906119910119894119896119906

sdot 119889119894

119891119896

] 119894 isin 119881 119896 isin 119870 119906 isin 119880 (4)

subject to

sum

119906

sum

119894

119909119894119895119896119906

= sum

119906

119910119895119896119906

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(5)

sum

119906

sum

119895

119909119894119895119896119906

= sum

119906

119910119894119896119906

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(6)

sum

119906

sum

119896

119910119894119896119906

= 1

119894 isin 119881 119896 isin 119870 119906 isin 119880

(7)

4 Mathematical Problems in Engineering

sum

119906

sum

119894isin119882

sum

119895isin119882

119909119894119895119896119906

le |119882| minus 1

forall119882 sube 119881 0 notin 119882 |119882| ge 2 119896 isin 119870 119906 isin 119880

(8)

sum

119906

sum

119894

119905119894sdot 119910119894119896119906

+sum

119906

sum

119894

sum

119895

119905119894119895sdot 119909119894119895119896119906

le 119879

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(9)

119909119894119895119896119906

= 0 or 1

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(10)

119910119894119896119906

= 0 or 1

119894 isin 119881 119896 isin 119870 119906 isin 119880

(11)

119887119896119906= 0 or 1

119896 isin 119870 119906 isin 119880

(12)

sgn(sum119894

119910119894119896119906) = 119887

119896119906

119894 isin 119881 119896 isin 119870 119906 isin 119880

(13)

The first part of the objective function above addressestransportation cost the second part addresses inventory costand the third computes the cost of dispatched vehiclesThe transportation cost depends on the length of a givenroute as well as the frequency of the route The inventorycost depends on the pickup quantity for the suppliers ineach route and the route frequency Each vehicle type has acorresponding dispatch cost therefore the third part of thefunction depends on the vehicle type choice

The following is a detailed description of constraints (5) to(13) Equations (5) to (7) ensure that each supplier distributesin only one route and the path of each vehicle dispatchedforms a loop Constraint (8) ensures that the subloop isavoided Equation (9) ensures that the cycle time of eachroute for one frequency is less than the maximum allowablecycle time 119879 The value of 119879 depends on the real situationEquations (10) to (12) indicate the scope of variables Equation(13) ensures that each route has only one vehicle dispatched

3 The Design of the Two-Phase HeuristicAlgorithm

CFR-VTC as an extension of VRP is NP hard Therefore wepropose a two-phase heuristic algorithm called two-phase TSwith limited search scope (TP-TSLSS) to solve the problemIn the first phase an initial satisfactory solution is generatedthrough a greedy heuristic algorithm to maximize the ratio1199112of the superior arc frequency to the inferior arc frequency

The following steps are used to generate the initial satisfactorysolution (1) the arcs are divided into the superior arc andthe inferior arc according to the 80-20 rule (the superiorarc and the inferior are described in Section 33) (2) thegreedy heuristic algorithm is then employed to maximizethe ratio 119911

2of the frequency of the superior arc to that of

the inferior arc through the four neighborhood selectionapproaches described in Section 32 (3) the final solutionof the first phase with the optimal 119911max

2in the number of

iterations is exported In the second phase the solution fromthe first phase is improved using the TS To improve searchefficiency we limit the scope of the search in the algorithmto render the 119911

2of the candidate solution greater than the

product of the 119911max2

of the initial satisfactory solution anda coefficient 120572 Calculating the 119911

2value is simpler than

calculating the objective function value 119911 Consequently theTP-TSLSS greatly enhances the search efficiency by limitingthe scope of 119911

2rather than directly using the objective

function value 119911 for the search

31 Representation and Evaluation of Solution Weuse119909 and Vto represent the route and vehicle type respectively and theseform an intact solution We set the manufacturer code to 0and represent the supplier numbers with 119894 (119894 = 1 2 119899)We randomly select the suppliers and the manufacturer asa feasible route We first randomly generate 119898 routes (119898 isin

[2 119899 minus 1] where 119898 is an integer) and randomly distribute 119899suppliers into these routes For example solution 119909 includesroutes 119901

10 minus 2 minus 3 minus 5 minus 0 119901

20 minus 7 minus 1 minus 9 minus 8 minus 0 until

all suppliers have finished selecting These routes constitutesolution 119909 Second we calculate the sum of 119905

119894and 119905119894119895of each

route 1199011 1199012 of solution 119909 and we examine whether the

total time of each route meets the time constraint of (9) Ifnot we randomly split the route whose total time exceeds119879 For example if the total time of route 119901

2exceeds 119879 the

route 11990120 minus 7 minus 1 minus 9 minus 8 minus 0 is randomly split as 119901

210 minus

7 minus 1 minus 0 and 119901220 minus 9 minus 8 minus 0 Then recalculate the total

time of each new route respectively and repeat the samestep until all the routes meet the time constraint of (9) Inthat way the new solution adjusted constitutes the feasiblesolution Finally we randomly select the vehicle type 119906

119895(119895 =

1 2 119898) corresponding to each route of feasible solutionand set [119906

1 1199062 119906

119898] as solution V for each vehicle type

We use the objective function value to evaluate the qualityof solutions

32 Neighborhood Solution Selection Method There are fourmethods to select a neighborhood solution

(1) Randomly select a route and two suppliers on theroute Reverse the traversal order of the two suppliersFor instance randomly select route 119901

10 minus 7 minus 1 minus 9 minus

8 minus 0 Randomly reverse the order of 7 minus 1 minus 9 minus 8 1199011

becomes 0 minus 8 minus 9 minus 1 minus 7 minus 0

(2) Randomly select two routes and a supplier fromone of these two routes Insert the chosen supplierrandomly into the other route If there is no supplierremaining in the route from where the supplier wastaken set the route to null For instance randomlyselect routes119901

10minus2minus3minus5minus0 and119901

20minus7minus1minus9minus8minus0

Randomly select supplier 2 from 1199011to be inserted

into 1199012 1199011becomes 0 minus 3 minus 5 minus 0 and 119901

2becomes

0 minus 7 minus 1 minus 2 minus 9 minus 8 minus 0

Mathematical Problems in Engineering 5

Randomly generate an initial solution of (1199090 V0) which meets the time constraint in (9) Set the current solution 119909now = 1199090 Vnow = V

0

then calculate 1199112(119909now Vnow) Set it = 1 (it records the current number of iterations)

Do while it lt 119872If rand lt 119901

1

Generate a neighborhood solution (1199091 V1) through the first neighborhood method

Do while the time of the new solution (1199091 V1) is not less than 119879

Regenerate a neighborhood solution (1199091 V1) through the first neighborhood method

End doIf 1199112(1199091 V1) gt 1199112(119909now Vnow)

119909now = 1199091Vnow = V

1

End ifEnd ifExecute the other three neighborhood methods using a similar processit = it + 1

End doOutput the initial satisfactory solution (1199091015840

0 V10158400) = (119909now Vnow) and define 119911max

2= 1199112(1199091015840

0 V10158400)

Algorithm 1

(3) Randomly select a route and a supplier from it Addthe supplier to another route For instance say thereare five routes 119901

1 1199012 1199013 1199014 and 119901

5 Randomly select

route 11990110minus2minus3minus5minus0 and supplier 2 and move the

latter into 1199016 1199016becomes 0 minus 2 minus 0

(4) Randomly select a vehicle type of solution V allowingthe vehicle type to changewithin the allowable vehicletype range For instance V = [2 2 3 4 5] becomes[2 2 3 5 5]

33 The First-Phase Algorithm 80-20 rule is originally pro-posed by the Italian economist Pareto so it is also calledParetorsquos law Commonly it is stated that 20 of all causesbring about 80 of all effects [24ndash26] In this paper we firstdefine an arc set 119860 that records the distances between anytwo nodes which contain the suppliers and themanufacturerThen according to the 80-20 rule we define 20 of arcs in set119860 with the shortest length as the superior arc set 119860

1and the

other arcs are defined as the inferior arc set 1198602 The lengths

of all arcs in 1198601are smaller than those of the arcs in 119860

2

that is |1198861| lt |119886

2| 1198861isin 1198601 1198862isin 1198602 The number of arcs

in 1198602is four times that of the number in 119860

1 For a specific

solution (119909 V) we define 119891119904 as the sum of frequencies of allsuperior arcs and 119891119894 as the sum of frequencies of all inferiorarcs 119911

2(119909 V) is defined as 119891119904119891119894 The objective of the first-

phase algorithm is to maximize 1199112 We set the number of

iterations to119872The greedy heuristic algorithm is designed as shown in

Algorithm 1

34 The Second-Phase Algorithm Osman [27] Taillard et al[28] and Alonso et al [29] applied TS to solve the VRP andverified its effectiveness In this paper we limit the scope ofsearch to ensure that the 119911

2value of every candidate solution

is greater than 120572sdot119911max2

(where 120572 is a coefficient slightly smallerin value than 1 We find that 120572 isin [07 09] is an appropriate

range) We call the second-phase algorithm TS with limitedsearch scope (TSLSS) We set the number of iterations as NI

The TSLSS is shown in Algorithm 2

35 The Process of Two-Phase TS with Limited Search ScopeWe combine the first-phase algorithm in Section 33 and thesecond-phase algorithm in Section 34 as the two-phase TSwith limited search scope (TP-TSLSS)

Figure 2 shows the specific process

4 Computational Experiments

In this section our proposed two-phase heuristic algorithmis programmed in MATLAB R2011b to solve 55 smallmedium and large numerical problemsThese computationalexperiments are run on an Acer 4820TG computer with anIntel i5 CPU (24GHz) and 4GB (347 available) of memory

41 Experiments with Varying Supplier Size Weuse a numberof examples involving varying numbers of suppliers to vali-date ourmodel and algorithmWhen the number of suppliersis below 100 both the horizontal and vertical coordinatesof suppliers are drawn from the uniform distribution inthe range [0 100] using the randi() function in MATLABMoreover the coordinates of the suppliers are drawn from theuniform distribution in the range [0 200] when the numberof suppliers is between 100 and 200 Finally when the numberof suppliers is between 200 and 300 the coordinates aredrawn from the uniform distribution in the range [0 300]The horizontal and vertical coordinates are represented by119883 and 119884 respectively The pickup quantity is randomly gen-erated from the uniform distribution in the range [80 150]whereas the inventory cost of each supplierrsquos parts is subjectto uniform distribution in the range [0 1] The dispatch costfor each vehicle type is uniformly distributed from 100 to400 We set the coordinates of the manufacture to (50 50)

6 Mathematical Problems in Engineering

Set the total number of iterations performed by the TS (NI) Set tabulist = Set the initial solution to (11990910158400 V10158400) which is generated in

the first phase Let the optimal solution be 119909best = 1199091015840

0 Vbest = V1015840

0and the current solution 119909now = 119909

1015840

0 Vnow = V1015840

0

Let 119899 = 0 (record the number of current iterations) Set the number of candidate solutions (ℎ) in the TSDo while 119899 lt NI

Employ the four neighborhood methods to generate h candidate solutions (1199091 V1) (1199092 V2) (119909

ℎ Vℎ) satisfying the time

constraint in (9) with 1199112values greater than 120572 sdot 119911max

2 Calculate the objective function values of these solutions

119911(1199091 V1) 119911(119909

2 V2) 119911(119909

ℎ Vℎ)

Obtain (119909opt Vopt) whose objective function value is optimalIf min119911(119909

1 V1) 119911(119909

2 V2) 119911(119909

ℎ Vℎ) lt 119911(119909best Vbest)

(119909now Vnow) = (119909opt Vopt)Update the tabulist

ElseDo while (119909opt Vopt) is in the tabulistSelect the suboptimal solution from the candidate solutions and record it as (119909opt Vopt)

End do(119909now Vnow) = (119909opt Vopt)Update the tabulist

End if119899 = 119899 + 1

End do

Algorithm 2

The algorithms described in Section 3 are implemented to testthese numerical instances

42 Algorithm Comparison All of the 55 examples are gen-erated using the method described in Section 41 Similarproblems have been solved in some of the literature [27ndash29] using TS algorithms whereas other studies [30ndash32] haveemployed simulated annealing algorithms (SA) to solve themThus to demonstrate the effectiveness of our model and theTP-TSLSS algorithm each example in this study is run fivetimes using TP-TSLSS TS and the SA We then analyzeand compare the experiment results obtained by the threealgorithms Stop criteria of SA and TS are defined as followsSA stops when the temperature reaches a specific valueaccording to the scale of each instance and TS stops aftera specific number of iterations are performed by TS Thenumber of iterations performed by TS is twice as TP-TSLSS

Table 1 shows the results of the first phase of the two-phase algorithm SN represents the number of suppliersColumn ldquo119911(119909

0 V0)rdquo represents the average objective function

value of the initial solution ldquo119911(1199091015840 V1015840)rdquo represents the averageobjective function value of solutions obtained by the first-phase algorithm and ldquo1199051015840rdquo represents the time taken by thefirst-phase algorithm (the unit of time used is second)Following the execution of the first-phase algorithm wefind that the 119911

2value increases significantly within a short

time Consequently the objective function value 119911 can beimproved significantly through the optimization of the first-phase algorithm within a short time

Table 2 shows the comparison results of the optimalobjective function values of solutions obtained by the threealgorithms We choose examples 23 31 36 and 41 to showthe convergence of TS and TP-TSLSS in Figures 3 4 5 and 6

The horizontal axis represents the number of iterations andthe vertical axis represents the average objective functionvalue The column ldquoaveragerdquo represents the average objectivefunction value of the optimal solution in each algorithmldquo119905rdquo represents the CPU time for each algorithm and ldquo120590rdquorepresents the standard deviation

We draw the following conclusions from a comparison ofSA TS and TP-TSLSS shown in Table 2 and the convergencegraphs (Figures 3 4 5 and 6)

(1) When the scales of the instances are small theobjective function values of the solutions obtainedby SA are greater than those obtained by the othertwo algorithms Therefore the performance of SAis inferior to that of TS and TP-TSLSS in solvingsmall-scale problems By contrast SA outperformsTS but is worse than TP-TSLSS when the number ofsuppliers exceeds 30 A comparison of TP-TSLSS andTS shows that the former yields solutionswith smallerobjective function values than the latter except fora few small-scale instances Moreover Figures 3 45 and 6 show that the number of iterations of TSis twice that of TP-TSLSS but TS is inferior to TP-TSLSS The performance of TP-TSLSS is better thanthat of the other two algorithms for most instancesThe advantage of TP-TSLSS becomesmore obvious asthe scale of problems increases

(2) The computational time of TP-TSLSS is the shortestfollowed by SA and TS

(3) Solutions obtained by TP-TSLSS have the smalleststandard deviationThe comparison of standard devi-ation values proves that TP-TSLSS is the most robustfollowed byTSwithmoderate robustness and SAwith

Mathematical Problems in Engineering 7

The first phase algorithm start

Randomly generate the initial solution

Randomly generate the new

If it satisfies the time constraint (9)

Yes

No

If it satisfies the time constraint (9)

No

Yes

Yes No

Yes

No

The second phase algorithm start

solution of the first phase

solutions that meet the timeconstraint (9) in the

function value is the smallest among the candidate solutions

If the selected solution

If the selectedsolution

Select the suboptimal

Accept the new solution and replace the current solution

Accept the new solution and replace the current solution

Update the tabulist

Stop

Yes

No

YesNo

No

Yes

Yes

No

(x0 0)

neighborhood of N(xnow now)

Calculate z2(x1 1)

If z2(x1 1) gt z2(xnow now)

Accept (x1 1) set Reject (x1 1)

it = it + 1

it le M

(xnow now)

set zmax2 = z2(xnow now)

neighborhood of N(xnow now)

If the z2 value of eachcandidate solution gt 120572 middot z

max2

Select the solution (xoptopt) whose objective

(xopt opt) is better than (xbest best)

(xopt opt) is

candidate solution as (xoptopt)

n = n + 1

If n lt NI

in tabulist

Randomly generate h candidate

Set xnow = x0 now = 0 Calculatez2(xnow now) it = 1

solution (x1 1) in the

(xnow now) = (x1 1)

Output (x9984000 9984000) =

Tabulist H = Φ set the

9984000 xnow = x

9984000 and

algorithm xbest = x9984000 best =

now = 9984000 n = 0

Figure 2 The TP-TSLSS process

the least Therefore the two-phase algorithm (TP-TSLSS) can obtain better and more stable solutions ina relatively short time

To verify the effectiveness of the two-phase heuristicalgorithm we select an optimal solution for TP-TSLSS froman example involving 30 suppliers Figure 7 shows the pathdiagram

In addition we take the 119905-test of statistical analysis tocompare difference in objective function values of the three

algorithms 119879 test employs 119905 distribution theory to infer theoccurring probability of differences and to examine whetherthe differences are significant in two averages [33 34] Thispaper takes 119905-test to compare the differences in averageobjective function values of the three algorithms First werun 10 times for all 55 examples We employ the SPSS 18 totake normality test for the objective function values of the 55examples We find that except the objective function valuesof the five numerical examples with the scale of 10 suppliersthe rest of the numerical examplesrsquo objective function values

8 Mathematical Problems in Engineering

Table 1 Results obtained for the first-phase algorithm

Instance SN 119911(1199091015840 V1015840) 119911(1199090 V0) 1199112(1199091015840 V1015840) 119911

2(1199090 V0) 1199051015840 (s)

1 10 18391 30698 2 01667 0322 10 21128 2582 24444 02308 0313 10 23407 38168 23333 025 0344 10 21757 26998 30435 05455 0295 10 2490 29351 07059 02444 036 20 3158 57114 48182 02535 0347 20 50645 65045 132381 03488 0338 20 3600 58948 68667 01227 0319 20 43496 57095 103529 03793 0310 20 32278 52732 88571 01333 03211 30 42922 73623 149 02202 03612 30 68309 76401 71695 02818 03513 30 6574 72963 55 00896 03614 30 64024 65789 380147 023 03715 30 48815 92522 245 04107 03616 50 12523 14449 397647 0244 03617 50 11190 14649 185238 03557 03518 50 11945 14672 1297 03254 03719 50 11919 12149 91 025 03820 50 10930 17825 211429 02551 03621 70 11737 17540 479091 02878 03422 70 15508 18462 545625 01973 03923 70 12484 20227 1003333 02653 03824 70 15890 20555 29 03419 03725 70 14998 19819 172857 02903 03526 90 18475 20596 450556 03529 03827 90 19643 20328 472356 03427 0428 90 18792 19873 46729 02957 03929 90 16268 20921 8746 03268 03830 90 20846 28943 3667 03832 04131 110 25081 46116 2068 04035 08332 110 30848 45420 1857 04689 07833 110 27515 42895 1939 05073 07934 110 28371 37652 185 04971 07335 110 23627 39712 3046 04837 07836 150 38053 59964 180213 02545 09237 150 36892 56279 3263 02384 08938 150 39604 62874 20481 01953 09339 150 40285 58164 16306 02491 09240 150 37681 63692 2617 01902 0941 200 86334 111050 55 02611 210242 200 88274 113130 35806 02194 208743 200 85145 113670 41758 02874 207444 200 89732 112823 528 02391 212345 200 87692 135713 4825 0261 208446 250 111760 145270 39955 02601 402947 250 111830 146220 4316 02206 418148 250 118713 149254 38421 02059 401749 250 116915 153873 3617 01973 398250 250 120317 148148 3291 02391 405951 300 131980 169670 44194 02328 597252 300 129468 163200 4841 0304 629253 300 139642 161726 4163 02845 581754 300 130613 159212 40157 02174 618355 300 129862 163944 49153 0316 6084

0 50 100 150 200 250 300 350 40006

08

1

12

14

16

18

2

TSTP-TSLSS

times104

Figure 3 Convergence graph of 70 suppliers

0 100 200 300 400 500 60015

2

25

3

35

4

45

TSTP-TSLSS

times104

Figure 4 Convergence graph of 110 suppliers

obtained by all the three algorithms have a good normalityThen we take two hypothesis tests at 005 confidence levelconditions The first hypothesis test is designed for SA andTP-TSLSS with the null hypothesis 119867 120583

1le 1205830 The second

hypothesis test is designed for TS and TP-TSLSS with thenull hypothesis 119867 120583

2le 1205830 Finally we take independent

sample 119905-test for the two hypotheses respectively The 119901

values obtained are listed in Table 3For the comparison of SA and TP-TSLSS all 119901 values of

the examples with 20 or more suppliers are less than 005therefore the null hypotheses of these examples are rejecteddemonstrating that TP-TSLSS is better than SA

For the comparison of TS and TP-TSLSS apart from 119901

values of examples of 20 suppliers scale (TS and TP-TSLSS)being greater than 005 119901 values of the remaining examplesare less than 005 therefore the null hypotheses of these

Mathematical Problems in Engineering 9

Table 2 Results obtained for the three algorithms (SA TS and TP-TSLSS)

Instance SN SA TS TP-TSLSSAverage 119905SA (s) 120590SA Average 119905TS (s) 120590TS Average 119905TP-TSLSS (s) 120590TP-TSLSS

1 10 16191 1654 8731 13862 3247 0 13862 1463 02 10 15743 1526 7631 1388 3258 0 1388 1419 03 10 1491 1594 4020 14219 3282 0 14219 1346 04 10 15825 1631 7218 14778 3354 0 14778 1322 05 10 16049 1682 7194 14875 3196 4031 14382 1407 06 20 25689 1727 8256 23333 4664 4107 23549 1686 32187 20 26021 1714 841 26698 4561 562 25649 1811 35918 20 24928 1793 8646 25106 4628 479 24217 1765 3459 20 26183 1737 8729 27656 4619 601 25714 1879 331410 20 23105 1749 8439 22758 4594 504 22423 1814 36211 30 34219 2015 1183 35278 5492 7660 33173 1825 427112 30 36798 2125 11038 34214 5135 7135 33082 1819 41613 30 34192 2062 10949 35073 5284 7058 33019 1746 408414 30 35193 2048 11164 36465 5175 7563 3396 1822 402315 30 38276 2116 12645 42145 5428 7937 37361 1707 42516 50 6425 2485 41072 59709 6514 12047 54051 2116 555717 50 60158 2472 38264 61101 6628 11062 5998 2392 583518 50 57028 2493 34183 57413 6604 8459 56276 2216 519419 50 56026 2419 33963 58708 6592 9246 52185 2145 574920 50 60924 2562 32974 63686 6585 8794 5686 2281 538521 70 85106 4057 37016 90797 7849 18584 76153 2136 1170322 70 82179 3947 38294 8820 7632 17294 78465 2073 1075223 70 82462 3929 41174 84594 7940 16848 73865 2491 1047324 70 88171 4019 38265 89905 7835 17182 81702 2615 1014925 70 81096 4095 34793 82468 7529 1749 80516 2491 1135226 90 10492 6418 40327 10737 10673 27232 92756 4215 1772827 90 11600 6429 41046 12446 10428 23472 90794 3881 1713628 90 108433 6316 40684 136475 10582 25819 89615 3974 1738529 90 116923 6285 40958 114367 10418 24174 92359 4039 1748230 90 102395 6193 40715 108691 10837 22964 95725 4139 1698331 110 20487 8972 77967 211168 13248 54139 180161 5797 3382932 110 187321 8017 78924 204292 13082 52764 117423 5826 3103433 110 190355 8526 83592 198163 13569 51074 185354 5494 3409334 110 190241 8438 93164 192525 13194 50986 179394 5618 358535 110 193482 8525 76492 216241 13074 51196 173106 5769 3792936 150 30813 18038 13484 27982 35922 97346 24094 15006 8353537 150 29813 18157 12467 30583 35542 10048 24695 14678 8214638 150 30847 18262 13295 31764 35686 98683 28174 14903 7824839 150 29139 18037 14829 30269 35742 90474 27108 14286 7958340 150 28742 17984 12793 29264 35291 98134 26495 14982 8497341 200 50954 24044 32316 56340 41249 28743 48246 19151 1576342 200 58192 23859 35935 58861 41465 27631 52407 20018 1639443 200 56102 24168 38193 57928 41672 24926 52887 19427 1582644 200 57149 24027 37194 59218 41839 27428 50961 18737 1782445 200 56291 23939 36913 58316 41962 29726 51620 19472 1629746 250 67194 38054 46911 75935 63028 36877 62171 32006 2428747 250 62719 37937 48512 77501 63542 35825 59954 31078 2592648 250 61461 37846 49236 78519 63887 36014 58917 31403 2174549 250 70192 37735 43926 76110 64041 35826 63625 31786 2852850 250 64192 37342 47295 71926 63993 35714 62128 32682 2394851 300 82155 52069 73979 92167 91249 58212 78166 46351 4232652 300 84124 51053 74924 91135 89465 56149 77640 44718 3872653 300 88307 51749 71486 90114 90672 59376 75783 48227 3741954 300 83491 51283 73951 87910 91839 52864 78629 41937 4173655 300 76529 50959 75056 88412 90962 58251 73210 42672 43945

10 Mathematical Problems in Engineering

0 100 200 300 400 500 600 700 800 900 10002

25

3

35

4

45

5

55

6times10

4

TSTP-TSLSS

Figure 5 Convergence graph of 150 suppliers

0 100 200 300 400 500 600 700 800 900 1000 11004

5

6

7

8

9

10

11times10

4

TSTP-TSLSS

Figure 6 Convergence graph of 200 suppliers

examples are rejected demonstrating TP-TSLSS is better thanTS when the suppliers are more than 30

43 Results Analysis of the Different Modes General studiesof the routing problems did not consider the effect of vehicletype choice on transportation frequency and inventory costwhile we take the vehicle type choice into consideration Toexamine this effect we make a comparison of the generaltransportation mode (Mode 1) with the CFR-VTC mode(Mode 2) for different scales of instances

Mode 1 Only one vehicle type is allowed for the transporta-tion task

Mode 2 All vehicle types are available

Figure 8 shows a comparison of the results for the twomodes (Mode 1 includes three cases only vehicle type 1 onlyvehicle type 2 and only vehicle type 3 in which the vehicle

17

23

306

1127

2021

2719

1018

9

16

14

11

29

4

528

2 3

826 24

22

1513

250

00 50 100

50

100

Vehicle 3 frequency 5Vehicle 2 fre

quency 4

Vehicle 2 frequency 8

X

Y

Figure 7 The path diagram of TP-TSLSS

types are not required for decision Mode 2 signifies that allthe three vehicle types are involved in the decision)The loadcapacity of vehicle type 1 is the smallest whose value is 100and that of vehicle type 2 is moderate whose value is 300whereas the capacity of vehicle type 3 is the largest whosevalue is 600 The horizontal axis in Figure 8 represents thenumber of suppliers and the vertical axis shows the four typesof costmdashtransportation cost inventory cost dispatch costand total cost The four cost trends are shown in Figure 8

The transportation cost graph in Figure 8 shows thatthe case involving only vehicle type 3 records the lowesttransportation cost because a vehicle with a large loadcapacity requires less frequency to complete the pickup taskHowever the inventory cost and dispatch cost in this caseare the highest when the scales of instances are large becauseof low frequency and high load capacity By contrast thecase involving only vehicle type 1 has the advantage of lowinventory and dispatch costs while the transportation costin this case is the highest However the total cost of Mode2 is lower than all the three cases in Mode 1 Therefore wefind that we can balance the cost of transportation inventoryand dispatch by a reasonable choice of vehicle to acquire theoptimal total cost Mode 2 has greater cost advantages thanMode 1

5 Conclusions

In this paper we considered the common frequency routingproblem with vehicle type choice in milk runs in logisticalsystems We developed a mathematical model to describe thevehicle type choice frequency planning and vehicle routeplanning in the milk run system To solve this model wedeveloped a two-phase TS algorithm with limited searchscope (TP-TSLSS) that increased the search efficiency Theproposed TP-TSLSS algorithm was tested on 55 numerical

Mathematical Problems in Engineering 11

0

20000

40000

60000

80000

100000

120000

10 20 30 50 70 90 110 150 200 250 300

Total cost

Vehicle type 1

Vehicle type 3Vehicle type 2

Vehicle type 1

Vehicle type 3Vehicle type 2

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 20 30 50 70 90 110 150 200 250 300

Transportation cost

Multiple vehicle types

Multiple vehicle types

Multiple vehicle types

Vehicle type 1

Vehicle type 3Vehicle type 2

Multiple vehicle typesVehicle type 1

Vehicle type 3Vehicle type 2

0

2000

4000

6000

8000

10000

12000

14000

10 20 30 50 70 90 110 150 200 250 300

Inventory cost

0

5000

10000

15000

20000

25000

30000

10 20 30 50 70 90 110 150 200 250 300

Dispatch cost

Figure 8 A comparison of the mode results

examples with varying scales for verification and was com-pared with the TS and the SA methods The results showedthat our TP-TSLSS obtained better solutions in a shortertime and a more stable manner By comparing differenttransportationmodes we concluded that consideration of thevehicle type choice could help save on cost of transportationinventory and dispatch

This was the first study to comprehensively considerfrequency planning vehicle type choice and routing plan-ning Although the consideration of a variety of vehicle

types significantly increased the complexity of the originalCFR problem the TP-TSLSS algorithm could reduce searchtime and improve accuracy through limiting search scopeIn the future we intended to extend this model to casesinvolving multiple manufacturers The problem will be morecomplex and the decision might depend on additionalfactors Moreover multiple depots could be considered froma practical perspective Finally we planned to develop evenmore efficient and accurate algorithms to solve these prob-lems

12 Mathematical Problems in Engineering

Table 3 119901 value of 119905-test

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

1 10 mdash mdash2 10 mdash mdash3 10 mdash mdash4 10 mdash mdash5 10 mdash mdash6 20 000031 0167 20 00000028 0188 20 0000026 0219 20 00043 01210 20 0000030 02811 30 00051 0008512 30 00032 002713 30 00017 001014 30 00072 0003615 30 0015 0002916 50 00000022 0000002517 50 00000063 00004118 50 0012 0006119 50 00000027 0000003520 50 00000012 0000001821 70 000023 000001022 70 000014 0000002923 70 0000092 0000001224 70 0000011 00000006325 70 000047 000002126 90 0000016 000004727 90 00000024 00000001828 90 00000051 00000001029 90 0000030 000002830 90 0000029 000001131 110 0000089 000006232 110 00000033 00000001033 110 000032 00002934 110 0000092 000005935 110 0000075 0000003336 150 00000013 0000001437 150 00000014 0000002038 150 0000067 000005139 150 000018 000005340 150 000051 000004741 200 0000069 00000001542 200 000000098 00000008343 200 0000074 0000001944 200 00000023 00000001245 200 0000082 0000003846 250 000000080 000000003747 250 000000015 0000000013

Table 3 Continued

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

48 250 000000011 0000000009849 250 00000064 00000001350 250 0000094 00000002751 300 00000011 00000006052 300 000000081 00000005653 300 000000042 00000001754 300 00000098 00000008255 300 0000020 0000000041

Conflict of Interests

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

References

[1] K H Chuah and J C Yingling ldquoRouting for a just-in-timesupply pickup and delivery systemrdquo Transportation Science vol39 no 3 pp 328ndash339 2005

[2] L Zhuang ldquoTowards a more economic production and just-in-time delivery systemrdquo International Journal of ProductionEconomics vol 36 no 3 pp 307ndash313 1994

[3] M Savsar and A Al-Jawini ldquoSimulation analysis of just-in-time production systemsrdquo International Journal of ProductionEconomics vol 42 no 1 pp 67ndash78 1995

[4] R R Lummus ldquoA simulation analysis of sequencing alternativesfor JIT lines using kanbansrdquo Journal of OperationsManagementvol 13 no 3 pp 183ndash191 1995

[5] S J Sadjadi M Jafari and T Amini ldquoA new mathematicalmodeling and a genetic algorithm search for milk run problem(an auto industry supply chain case study)rdquo The InternationalJournal of Advanced Manufacturing Technology vol 44 no 1-2pp 194ndash200 2009

[6] M Jafari-Eskandari S J Sadjadi M S Jabalameli and ABozorgi-Amiri ldquoA robust optimization approach for the MilkRun problem (An auto industry supply chain case study)rdquo inProceedings of the IEEE International Conference on Computersamp Industrial Engineering (CIE rsquo09) pp 1076ndash1081 TroyesFrance July 2009

[7] M J Eskandari A R Aliahmadi and G H H Khaleghi ldquoArobust optimisation approach for the milk run problem withtime windows with inventory uncertainty an auto industrysupply chain case studyrdquo International Journal of Rapid Man-ufacturing vol 1 no 3 pp 334ndash347 2010

[8] T Du F KWang and P-Y Lu ldquoA real-time vehicle-dispatchingsystem for consolidating milk runsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 43 no 5 pp565ndash577 2007

[9] J Ma and G Sun ldquoMutation Ant colony algorithm of milk-run vehicle routing problemwith fastest completion time basedon dynamic optimizationrdquo Discrete Dynamics in Nature andSociety vol 2013 Article ID 418436 6 pages 2013

[10] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

the arrangement of vehicle type pickup frequency and trans-portation route that can minimize the cost of transportationinventory and dispatch Based on the above analysis vehicletype pickup frequency and transportation route are selectedas decision variables in our mathematic model

We make the following assumptions

Assumption 1 Each vehicle type possesses a different carry-ing capacity Each route requires only one vehicle type toperform the relevant transportation task in the milk run oneach day

Assumption 2 There is a sufficient number of vehicles tocomplete the transportation task The demand of the man-ufacturer by supplier 119894 can only be transported by one truckin the completion of one task

22 Notation

(1) Parameters

119881 = 119894 | 119894 = 0 1 2 119899 is set of suppliers and themanufacturer

0 represents the manufacturer 1 119899 represent thesuppliers119880 = 119906 | 119906 = 0 1 119886 is set of vehicle types120578 is cost of unit transportation distance120573119894is cost of unit parts inventory of supplier 119894

120574119906is the dispatch cost of vehicle type 119906

119888119894119895is the distance between supplier 119894 and 119895

119902119906is carrying capacity of vehicle type 119906

119889119894is the pickup quantity for supplier 119894

119905119894is unloading time at supplier 119894

119905119894119895is travel time from supplier 119894 to supplier 119895

119879 is the maximum allowable cycle time of each routefor a given frequency

(2) Decision Variables

119898 is the number of routes119870 = 119896 | 119896 = 0 1 2 119898 is set of routes

119909119894119895119896119906

=

1 if type 119906 vehicle passes from supplier 119894 to supplier 119895 in route 119896

0 otherwise119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

119910119894119896119906

=

1 if type 119906 vehicle picks up at supplier 119894 in route 119896

0 otherwise119894 isin 119881 119896 isin 119870 119906 isin 119880

119887119896119906=

1 if type 119906 vehicle is dispatched for route 119896

0 otherwise119896 isin 119870 119906 isin 119880

(1)

(3) Variables That Are Dependent on the Decision Variables

119891119896 the frequency of route 119896 depends on 119910

119894119896119906 The

definition is provided in (3)119901119894119896 the pickup quantity of supplier 119894 for a given

frequency depends on 119910119894119896119906The definition is provided

in (4)

23 The Proposed Mathematical Model for CFR-VTC Con-sider

min 119911 = sum

119894isin119881

sum

119895isin119881

sum

119896isin119870

sum

119906isin119880

120578 sdot 119891119896sdot 119888119894119895sdot 119909119894119895119896119906

+ sum

119894isin119881

sum

119896isin119870

120573119894sdot 119901119894119896

+ sum

119896isin119870

sum

119906isin119880

120574119906sdot 119887119896119906

(2)

where

119891119896= lceilsum

119906

sum119894119910119894119896119906

sdot 119889119894

119902119906

rceil 119894 isin 119881 119896 isin 119870 119906 isin 119880 (3)

ldquolceilrceilrdquo means rounding up

119901119894119896= [

sum119906119910119894119896119906

sdot 119889119894

119891119896

] 119894 isin 119881 119896 isin 119870 119906 isin 119880 (4)

subject to

sum

119906

sum

119894

119909119894119895119896119906

= sum

119906

119910119895119896119906

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(5)

sum

119906

sum

119895

119909119894119895119896119906

= sum

119906

119910119894119896119906

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(6)

sum

119906

sum

119896

119910119894119896119906

= 1

119894 isin 119881 119896 isin 119870 119906 isin 119880

(7)

4 Mathematical Problems in Engineering

sum

119906

sum

119894isin119882

sum

119895isin119882

119909119894119895119896119906

le |119882| minus 1

forall119882 sube 119881 0 notin 119882 |119882| ge 2 119896 isin 119870 119906 isin 119880

(8)

sum

119906

sum

119894

119905119894sdot 119910119894119896119906

+sum

119906

sum

119894

sum

119895

119905119894119895sdot 119909119894119895119896119906

le 119879

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(9)

119909119894119895119896119906

= 0 or 1

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(10)

119910119894119896119906

= 0 or 1

119894 isin 119881 119896 isin 119870 119906 isin 119880

(11)

119887119896119906= 0 or 1

119896 isin 119870 119906 isin 119880

(12)

sgn(sum119894

119910119894119896119906) = 119887

119896119906

119894 isin 119881 119896 isin 119870 119906 isin 119880

(13)

The first part of the objective function above addressestransportation cost the second part addresses inventory costand the third computes the cost of dispatched vehiclesThe transportation cost depends on the length of a givenroute as well as the frequency of the route The inventorycost depends on the pickup quantity for the suppliers ineach route and the route frequency Each vehicle type has acorresponding dispatch cost therefore the third part of thefunction depends on the vehicle type choice

The following is a detailed description of constraints (5) to(13) Equations (5) to (7) ensure that each supplier distributesin only one route and the path of each vehicle dispatchedforms a loop Constraint (8) ensures that the subloop isavoided Equation (9) ensures that the cycle time of eachroute for one frequency is less than the maximum allowablecycle time 119879 The value of 119879 depends on the real situationEquations (10) to (12) indicate the scope of variables Equation(13) ensures that each route has only one vehicle dispatched

3 The Design of the Two-Phase HeuristicAlgorithm

CFR-VTC as an extension of VRP is NP hard Therefore wepropose a two-phase heuristic algorithm called two-phase TSwith limited search scope (TP-TSLSS) to solve the problemIn the first phase an initial satisfactory solution is generatedthrough a greedy heuristic algorithm to maximize the ratio1199112of the superior arc frequency to the inferior arc frequency

The following steps are used to generate the initial satisfactorysolution (1) the arcs are divided into the superior arc andthe inferior arc according to the 80-20 rule (the superiorarc and the inferior are described in Section 33) (2) thegreedy heuristic algorithm is then employed to maximizethe ratio 119911

2of the frequency of the superior arc to that of

the inferior arc through the four neighborhood selectionapproaches described in Section 32 (3) the final solutionof the first phase with the optimal 119911max

2in the number of

iterations is exported In the second phase the solution fromthe first phase is improved using the TS To improve searchefficiency we limit the scope of the search in the algorithmto render the 119911

2of the candidate solution greater than the

product of the 119911max2

of the initial satisfactory solution anda coefficient 120572 Calculating the 119911

2value is simpler than

calculating the objective function value 119911 Consequently theTP-TSLSS greatly enhances the search efficiency by limitingthe scope of 119911

2rather than directly using the objective

function value 119911 for the search

31 Representation and Evaluation of Solution Weuse119909 and Vto represent the route and vehicle type respectively and theseform an intact solution We set the manufacturer code to 0and represent the supplier numbers with 119894 (119894 = 1 2 119899)We randomly select the suppliers and the manufacturer asa feasible route We first randomly generate 119898 routes (119898 isin

[2 119899 minus 1] where 119898 is an integer) and randomly distribute 119899suppliers into these routes For example solution 119909 includesroutes 119901

10 minus 2 minus 3 minus 5 minus 0 119901

20 minus 7 minus 1 minus 9 minus 8 minus 0 until

all suppliers have finished selecting These routes constitutesolution 119909 Second we calculate the sum of 119905

119894and 119905119894119895of each

route 1199011 1199012 of solution 119909 and we examine whether the

total time of each route meets the time constraint of (9) Ifnot we randomly split the route whose total time exceeds119879 For example if the total time of route 119901

2exceeds 119879 the

route 11990120 minus 7 minus 1 minus 9 minus 8 minus 0 is randomly split as 119901

210 minus

7 minus 1 minus 0 and 119901220 minus 9 minus 8 minus 0 Then recalculate the total

time of each new route respectively and repeat the samestep until all the routes meet the time constraint of (9) Inthat way the new solution adjusted constitutes the feasiblesolution Finally we randomly select the vehicle type 119906

119895(119895 =

1 2 119898) corresponding to each route of feasible solutionand set [119906

1 1199062 119906

119898] as solution V for each vehicle type

We use the objective function value to evaluate the qualityof solutions

32 Neighborhood Solution Selection Method There are fourmethods to select a neighborhood solution

(1) Randomly select a route and two suppliers on theroute Reverse the traversal order of the two suppliersFor instance randomly select route 119901

10 minus 7 minus 1 minus 9 minus

8 minus 0 Randomly reverse the order of 7 minus 1 minus 9 minus 8 1199011

becomes 0 minus 8 minus 9 minus 1 minus 7 minus 0

(2) Randomly select two routes and a supplier fromone of these two routes Insert the chosen supplierrandomly into the other route If there is no supplierremaining in the route from where the supplier wastaken set the route to null For instance randomlyselect routes119901

10minus2minus3minus5minus0 and119901

20minus7minus1minus9minus8minus0

Randomly select supplier 2 from 1199011to be inserted

into 1199012 1199011becomes 0 minus 3 minus 5 minus 0 and 119901

2becomes

0 minus 7 minus 1 minus 2 minus 9 minus 8 minus 0

Mathematical Problems in Engineering 5

Randomly generate an initial solution of (1199090 V0) which meets the time constraint in (9) Set the current solution 119909now = 1199090 Vnow = V

0

then calculate 1199112(119909now Vnow) Set it = 1 (it records the current number of iterations)

Do while it lt 119872If rand lt 119901

1

Generate a neighborhood solution (1199091 V1) through the first neighborhood method

Do while the time of the new solution (1199091 V1) is not less than 119879

Regenerate a neighborhood solution (1199091 V1) through the first neighborhood method

End doIf 1199112(1199091 V1) gt 1199112(119909now Vnow)

119909now = 1199091Vnow = V

1

End ifEnd ifExecute the other three neighborhood methods using a similar processit = it + 1

End doOutput the initial satisfactory solution (1199091015840

0 V10158400) = (119909now Vnow) and define 119911max

2= 1199112(1199091015840

0 V10158400)

Algorithm 1

(3) Randomly select a route and a supplier from it Addthe supplier to another route For instance say thereare five routes 119901

1 1199012 1199013 1199014 and 119901

5 Randomly select

route 11990110minus2minus3minus5minus0 and supplier 2 and move the

latter into 1199016 1199016becomes 0 minus 2 minus 0

(4) Randomly select a vehicle type of solution V allowingthe vehicle type to changewithin the allowable vehicletype range For instance V = [2 2 3 4 5] becomes[2 2 3 5 5]

33 The First-Phase Algorithm 80-20 rule is originally pro-posed by the Italian economist Pareto so it is also calledParetorsquos law Commonly it is stated that 20 of all causesbring about 80 of all effects [24ndash26] In this paper we firstdefine an arc set 119860 that records the distances between anytwo nodes which contain the suppliers and themanufacturerThen according to the 80-20 rule we define 20 of arcs in set119860 with the shortest length as the superior arc set 119860

1and the

other arcs are defined as the inferior arc set 1198602 The lengths

of all arcs in 1198601are smaller than those of the arcs in 119860

2

that is |1198861| lt |119886

2| 1198861isin 1198601 1198862isin 1198602 The number of arcs

in 1198602is four times that of the number in 119860

1 For a specific

solution (119909 V) we define 119891119904 as the sum of frequencies of allsuperior arcs and 119891119894 as the sum of frequencies of all inferiorarcs 119911

2(119909 V) is defined as 119891119904119891119894 The objective of the first-

phase algorithm is to maximize 1199112 We set the number of

iterations to119872The greedy heuristic algorithm is designed as shown in

Algorithm 1

34 The Second-Phase Algorithm Osman [27] Taillard et al[28] and Alonso et al [29] applied TS to solve the VRP andverified its effectiveness In this paper we limit the scope ofsearch to ensure that the 119911

2value of every candidate solution

is greater than 120572sdot119911max2

(where 120572 is a coefficient slightly smallerin value than 1 We find that 120572 isin [07 09] is an appropriate

range) We call the second-phase algorithm TS with limitedsearch scope (TSLSS) We set the number of iterations as NI

The TSLSS is shown in Algorithm 2

35 The Process of Two-Phase TS with Limited Search ScopeWe combine the first-phase algorithm in Section 33 and thesecond-phase algorithm in Section 34 as the two-phase TSwith limited search scope (TP-TSLSS)

Figure 2 shows the specific process

4 Computational Experiments

In this section our proposed two-phase heuristic algorithmis programmed in MATLAB R2011b to solve 55 smallmedium and large numerical problemsThese computationalexperiments are run on an Acer 4820TG computer with anIntel i5 CPU (24GHz) and 4GB (347 available) of memory

41 Experiments with Varying Supplier Size Weuse a numberof examples involving varying numbers of suppliers to vali-date ourmodel and algorithmWhen the number of suppliersis below 100 both the horizontal and vertical coordinatesof suppliers are drawn from the uniform distribution inthe range [0 100] using the randi() function in MATLABMoreover the coordinates of the suppliers are drawn from theuniform distribution in the range [0 200] when the numberof suppliers is between 100 and 200 Finally when the numberof suppliers is between 200 and 300 the coordinates aredrawn from the uniform distribution in the range [0 300]The horizontal and vertical coordinates are represented by119883 and 119884 respectively The pickup quantity is randomly gen-erated from the uniform distribution in the range [80 150]whereas the inventory cost of each supplierrsquos parts is subjectto uniform distribution in the range [0 1] The dispatch costfor each vehicle type is uniformly distributed from 100 to400 We set the coordinates of the manufacture to (50 50)

6 Mathematical Problems in Engineering

Set the total number of iterations performed by the TS (NI) Set tabulist = Set the initial solution to (11990910158400 V10158400) which is generated in

the first phase Let the optimal solution be 119909best = 1199091015840

0 Vbest = V1015840

0and the current solution 119909now = 119909

1015840

0 Vnow = V1015840

0

Let 119899 = 0 (record the number of current iterations) Set the number of candidate solutions (ℎ) in the TSDo while 119899 lt NI

Employ the four neighborhood methods to generate h candidate solutions (1199091 V1) (1199092 V2) (119909

ℎ Vℎ) satisfying the time

constraint in (9) with 1199112values greater than 120572 sdot 119911max

2 Calculate the objective function values of these solutions

119911(1199091 V1) 119911(119909

2 V2) 119911(119909

ℎ Vℎ)

Obtain (119909opt Vopt) whose objective function value is optimalIf min119911(119909

1 V1) 119911(119909

2 V2) 119911(119909

ℎ Vℎ) lt 119911(119909best Vbest)

(119909now Vnow) = (119909opt Vopt)Update the tabulist

ElseDo while (119909opt Vopt) is in the tabulistSelect the suboptimal solution from the candidate solutions and record it as (119909opt Vopt)

End do(119909now Vnow) = (119909opt Vopt)Update the tabulist

End if119899 = 119899 + 1

End do

Algorithm 2

The algorithms described in Section 3 are implemented to testthese numerical instances

42 Algorithm Comparison All of the 55 examples are gen-erated using the method described in Section 41 Similarproblems have been solved in some of the literature [27ndash29] using TS algorithms whereas other studies [30ndash32] haveemployed simulated annealing algorithms (SA) to solve themThus to demonstrate the effectiveness of our model and theTP-TSLSS algorithm each example in this study is run fivetimes using TP-TSLSS TS and the SA We then analyzeand compare the experiment results obtained by the threealgorithms Stop criteria of SA and TS are defined as followsSA stops when the temperature reaches a specific valueaccording to the scale of each instance and TS stops aftera specific number of iterations are performed by TS Thenumber of iterations performed by TS is twice as TP-TSLSS

Table 1 shows the results of the first phase of the two-phase algorithm SN represents the number of suppliersColumn ldquo119911(119909

0 V0)rdquo represents the average objective function

value of the initial solution ldquo119911(1199091015840 V1015840)rdquo represents the averageobjective function value of solutions obtained by the first-phase algorithm and ldquo1199051015840rdquo represents the time taken by thefirst-phase algorithm (the unit of time used is second)Following the execution of the first-phase algorithm wefind that the 119911

2value increases significantly within a short

time Consequently the objective function value 119911 can beimproved significantly through the optimization of the first-phase algorithm within a short time

Table 2 shows the comparison results of the optimalobjective function values of solutions obtained by the threealgorithms We choose examples 23 31 36 and 41 to showthe convergence of TS and TP-TSLSS in Figures 3 4 5 and 6

The horizontal axis represents the number of iterations andthe vertical axis represents the average objective functionvalue The column ldquoaveragerdquo represents the average objectivefunction value of the optimal solution in each algorithmldquo119905rdquo represents the CPU time for each algorithm and ldquo120590rdquorepresents the standard deviation

We draw the following conclusions from a comparison ofSA TS and TP-TSLSS shown in Table 2 and the convergencegraphs (Figures 3 4 5 and 6)

(1) When the scales of the instances are small theobjective function values of the solutions obtainedby SA are greater than those obtained by the othertwo algorithms Therefore the performance of SAis inferior to that of TS and TP-TSLSS in solvingsmall-scale problems By contrast SA outperformsTS but is worse than TP-TSLSS when the number ofsuppliers exceeds 30 A comparison of TP-TSLSS andTS shows that the former yields solutionswith smallerobjective function values than the latter except fora few small-scale instances Moreover Figures 3 45 and 6 show that the number of iterations of TSis twice that of TP-TSLSS but TS is inferior to TP-TSLSS The performance of TP-TSLSS is better thanthat of the other two algorithms for most instancesThe advantage of TP-TSLSS becomesmore obvious asthe scale of problems increases

(2) The computational time of TP-TSLSS is the shortestfollowed by SA and TS

(3) Solutions obtained by TP-TSLSS have the smalleststandard deviationThe comparison of standard devi-ation values proves that TP-TSLSS is the most robustfollowed byTSwithmoderate robustness and SAwith

Mathematical Problems in Engineering 7

The first phase algorithm start

Randomly generate the initial solution

Randomly generate the new

If it satisfies the time constraint (9)

Yes

No

If it satisfies the time constraint (9)

No

Yes

Yes No

Yes

No

The second phase algorithm start

solution of the first phase

solutions that meet the timeconstraint (9) in the

function value is the smallest among the candidate solutions

If the selected solution

If the selectedsolution

Select the suboptimal

Accept the new solution and replace the current solution

Accept the new solution and replace the current solution

Update the tabulist

Stop

Yes

No

YesNo

No

Yes

Yes

No

(x0 0)

neighborhood of N(xnow now)

Calculate z2(x1 1)

If z2(x1 1) gt z2(xnow now)

Accept (x1 1) set Reject (x1 1)

it = it + 1

it le M

(xnow now)

set zmax2 = z2(xnow now)

neighborhood of N(xnow now)

If the z2 value of eachcandidate solution gt 120572 middot z

max2

Select the solution (xoptopt) whose objective

(xopt opt) is better than (xbest best)

(xopt opt) is

candidate solution as (xoptopt)

n = n + 1

If n lt NI

in tabulist

Randomly generate h candidate

Set xnow = x0 now = 0 Calculatez2(xnow now) it = 1

solution (x1 1) in the

(xnow now) = (x1 1)

Output (x9984000 9984000) =

Tabulist H = Φ set the

9984000 xnow = x

9984000 and

algorithm xbest = x9984000 best =

now = 9984000 n = 0

Figure 2 The TP-TSLSS process

the least Therefore the two-phase algorithm (TP-TSLSS) can obtain better and more stable solutions ina relatively short time

To verify the effectiveness of the two-phase heuristicalgorithm we select an optimal solution for TP-TSLSS froman example involving 30 suppliers Figure 7 shows the pathdiagram

In addition we take the 119905-test of statistical analysis tocompare difference in objective function values of the three

algorithms 119879 test employs 119905 distribution theory to infer theoccurring probability of differences and to examine whetherthe differences are significant in two averages [33 34] Thispaper takes 119905-test to compare the differences in averageobjective function values of the three algorithms First werun 10 times for all 55 examples We employ the SPSS 18 totake normality test for the objective function values of the 55examples We find that except the objective function valuesof the five numerical examples with the scale of 10 suppliersthe rest of the numerical examplesrsquo objective function values

8 Mathematical Problems in Engineering

Table 1 Results obtained for the first-phase algorithm

Instance SN 119911(1199091015840 V1015840) 119911(1199090 V0) 1199112(1199091015840 V1015840) 119911

2(1199090 V0) 1199051015840 (s)

1 10 18391 30698 2 01667 0322 10 21128 2582 24444 02308 0313 10 23407 38168 23333 025 0344 10 21757 26998 30435 05455 0295 10 2490 29351 07059 02444 036 20 3158 57114 48182 02535 0347 20 50645 65045 132381 03488 0338 20 3600 58948 68667 01227 0319 20 43496 57095 103529 03793 0310 20 32278 52732 88571 01333 03211 30 42922 73623 149 02202 03612 30 68309 76401 71695 02818 03513 30 6574 72963 55 00896 03614 30 64024 65789 380147 023 03715 30 48815 92522 245 04107 03616 50 12523 14449 397647 0244 03617 50 11190 14649 185238 03557 03518 50 11945 14672 1297 03254 03719 50 11919 12149 91 025 03820 50 10930 17825 211429 02551 03621 70 11737 17540 479091 02878 03422 70 15508 18462 545625 01973 03923 70 12484 20227 1003333 02653 03824 70 15890 20555 29 03419 03725 70 14998 19819 172857 02903 03526 90 18475 20596 450556 03529 03827 90 19643 20328 472356 03427 0428 90 18792 19873 46729 02957 03929 90 16268 20921 8746 03268 03830 90 20846 28943 3667 03832 04131 110 25081 46116 2068 04035 08332 110 30848 45420 1857 04689 07833 110 27515 42895 1939 05073 07934 110 28371 37652 185 04971 07335 110 23627 39712 3046 04837 07836 150 38053 59964 180213 02545 09237 150 36892 56279 3263 02384 08938 150 39604 62874 20481 01953 09339 150 40285 58164 16306 02491 09240 150 37681 63692 2617 01902 0941 200 86334 111050 55 02611 210242 200 88274 113130 35806 02194 208743 200 85145 113670 41758 02874 207444 200 89732 112823 528 02391 212345 200 87692 135713 4825 0261 208446 250 111760 145270 39955 02601 402947 250 111830 146220 4316 02206 418148 250 118713 149254 38421 02059 401749 250 116915 153873 3617 01973 398250 250 120317 148148 3291 02391 405951 300 131980 169670 44194 02328 597252 300 129468 163200 4841 0304 629253 300 139642 161726 4163 02845 581754 300 130613 159212 40157 02174 618355 300 129862 163944 49153 0316 6084

0 50 100 150 200 250 300 350 40006

08

1

12

14

16

18

2

TSTP-TSLSS

times104

Figure 3 Convergence graph of 70 suppliers

0 100 200 300 400 500 60015

2

25

3

35

4

45

TSTP-TSLSS

times104

Figure 4 Convergence graph of 110 suppliers

obtained by all the three algorithms have a good normalityThen we take two hypothesis tests at 005 confidence levelconditions The first hypothesis test is designed for SA andTP-TSLSS with the null hypothesis 119867 120583

1le 1205830 The second

hypothesis test is designed for TS and TP-TSLSS with thenull hypothesis 119867 120583

2le 1205830 Finally we take independent

sample 119905-test for the two hypotheses respectively The 119901

values obtained are listed in Table 3For the comparison of SA and TP-TSLSS all 119901 values of

the examples with 20 or more suppliers are less than 005therefore the null hypotheses of these examples are rejecteddemonstrating that TP-TSLSS is better than SA

For the comparison of TS and TP-TSLSS apart from 119901

values of examples of 20 suppliers scale (TS and TP-TSLSS)being greater than 005 119901 values of the remaining examplesare less than 005 therefore the null hypotheses of these

Mathematical Problems in Engineering 9

Table 2 Results obtained for the three algorithms (SA TS and TP-TSLSS)

Instance SN SA TS TP-TSLSSAverage 119905SA (s) 120590SA Average 119905TS (s) 120590TS Average 119905TP-TSLSS (s) 120590TP-TSLSS

1 10 16191 1654 8731 13862 3247 0 13862 1463 02 10 15743 1526 7631 1388 3258 0 1388 1419 03 10 1491 1594 4020 14219 3282 0 14219 1346 04 10 15825 1631 7218 14778 3354 0 14778 1322 05 10 16049 1682 7194 14875 3196 4031 14382 1407 06 20 25689 1727 8256 23333 4664 4107 23549 1686 32187 20 26021 1714 841 26698 4561 562 25649 1811 35918 20 24928 1793 8646 25106 4628 479 24217 1765 3459 20 26183 1737 8729 27656 4619 601 25714 1879 331410 20 23105 1749 8439 22758 4594 504 22423 1814 36211 30 34219 2015 1183 35278 5492 7660 33173 1825 427112 30 36798 2125 11038 34214 5135 7135 33082 1819 41613 30 34192 2062 10949 35073 5284 7058 33019 1746 408414 30 35193 2048 11164 36465 5175 7563 3396 1822 402315 30 38276 2116 12645 42145 5428 7937 37361 1707 42516 50 6425 2485 41072 59709 6514 12047 54051 2116 555717 50 60158 2472 38264 61101 6628 11062 5998 2392 583518 50 57028 2493 34183 57413 6604 8459 56276 2216 519419 50 56026 2419 33963 58708 6592 9246 52185 2145 574920 50 60924 2562 32974 63686 6585 8794 5686 2281 538521 70 85106 4057 37016 90797 7849 18584 76153 2136 1170322 70 82179 3947 38294 8820 7632 17294 78465 2073 1075223 70 82462 3929 41174 84594 7940 16848 73865 2491 1047324 70 88171 4019 38265 89905 7835 17182 81702 2615 1014925 70 81096 4095 34793 82468 7529 1749 80516 2491 1135226 90 10492 6418 40327 10737 10673 27232 92756 4215 1772827 90 11600 6429 41046 12446 10428 23472 90794 3881 1713628 90 108433 6316 40684 136475 10582 25819 89615 3974 1738529 90 116923 6285 40958 114367 10418 24174 92359 4039 1748230 90 102395 6193 40715 108691 10837 22964 95725 4139 1698331 110 20487 8972 77967 211168 13248 54139 180161 5797 3382932 110 187321 8017 78924 204292 13082 52764 117423 5826 3103433 110 190355 8526 83592 198163 13569 51074 185354 5494 3409334 110 190241 8438 93164 192525 13194 50986 179394 5618 358535 110 193482 8525 76492 216241 13074 51196 173106 5769 3792936 150 30813 18038 13484 27982 35922 97346 24094 15006 8353537 150 29813 18157 12467 30583 35542 10048 24695 14678 8214638 150 30847 18262 13295 31764 35686 98683 28174 14903 7824839 150 29139 18037 14829 30269 35742 90474 27108 14286 7958340 150 28742 17984 12793 29264 35291 98134 26495 14982 8497341 200 50954 24044 32316 56340 41249 28743 48246 19151 1576342 200 58192 23859 35935 58861 41465 27631 52407 20018 1639443 200 56102 24168 38193 57928 41672 24926 52887 19427 1582644 200 57149 24027 37194 59218 41839 27428 50961 18737 1782445 200 56291 23939 36913 58316 41962 29726 51620 19472 1629746 250 67194 38054 46911 75935 63028 36877 62171 32006 2428747 250 62719 37937 48512 77501 63542 35825 59954 31078 2592648 250 61461 37846 49236 78519 63887 36014 58917 31403 2174549 250 70192 37735 43926 76110 64041 35826 63625 31786 2852850 250 64192 37342 47295 71926 63993 35714 62128 32682 2394851 300 82155 52069 73979 92167 91249 58212 78166 46351 4232652 300 84124 51053 74924 91135 89465 56149 77640 44718 3872653 300 88307 51749 71486 90114 90672 59376 75783 48227 3741954 300 83491 51283 73951 87910 91839 52864 78629 41937 4173655 300 76529 50959 75056 88412 90962 58251 73210 42672 43945

10 Mathematical Problems in Engineering

0 100 200 300 400 500 600 700 800 900 10002

25

3

35

4

45

5

55

6times10

4

TSTP-TSLSS

Figure 5 Convergence graph of 150 suppliers

0 100 200 300 400 500 600 700 800 900 1000 11004

5

6

7

8

9

10

11times10

4

TSTP-TSLSS

Figure 6 Convergence graph of 200 suppliers

examples are rejected demonstrating TP-TSLSS is better thanTS when the suppliers are more than 30

43 Results Analysis of the Different Modes General studiesof the routing problems did not consider the effect of vehicletype choice on transportation frequency and inventory costwhile we take the vehicle type choice into consideration Toexamine this effect we make a comparison of the generaltransportation mode (Mode 1) with the CFR-VTC mode(Mode 2) for different scales of instances

Mode 1 Only one vehicle type is allowed for the transporta-tion task

Mode 2 All vehicle types are available

Figure 8 shows a comparison of the results for the twomodes (Mode 1 includes three cases only vehicle type 1 onlyvehicle type 2 and only vehicle type 3 in which the vehicle

17

23

306

1127

2021

2719

1018

9

16

14

11

29

4

528

2 3

826 24

22

1513

250

00 50 100

50

100

Vehicle 3 frequency 5Vehicle 2 fre

quency 4

Vehicle 2 frequency 8

X

Y

Figure 7 The path diagram of TP-TSLSS

types are not required for decision Mode 2 signifies that allthe three vehicle types are involved in the decision)The loadcapacity of vehicle type 1 is the smallest whose value is 100and that of vehicle type 2 is moderate whose value is 300whereas the capacity of vehicle type 3 is the largest whosevalue is 600 The horizontal axis in Figure 8 represents thenumber of suppliers and the vertical axis shows the four typesof costmdashtransportation cost inventory cost dispatch costand total cost The four cost trends are shown in Figure 8

The transportation cost graph in Figure 8 shows thatthe case involving only vehicle type 3 records the lowesttransportation cost because a vehicle with a large loadcapacity requires less frequency to complete the pickup taskHowever the inventory cost and dispatch cost in this caseare the highest when the scales of instances are large becauseof low frequency and high load capacity By contrast thecase involving only vehicle type 1 has the advantage of lowinventory and dispatch costs while the transportation costin this case is the highest However the total cost of Mode2 is lower than all the three cases in Mode 1 Therefore wefind that we can balance the cost of transportation inventoryand dispatch by a reasonable choice of vehicle to acquire theoptimal total cost Mode 2 has greater cost advantages thanMode 1

5 Conclusions

In this paper we considered the common frequency routingproblem with vehicle type choice in milk runs in logisticalsystems We developed a mathematical model to describe thevehicle type choice frequency planning and vehicle routeplanning in the milk run system To solve this model wedeveloped a two-phase TS algorithm with limited searchscope (TP-TSLSS) that increased the search efficiency Theproposed TP-TSLSS algorithm was tested on 55 numerical

Mathematical Problems in Engineering 11

0

20000

40000

60000

80000

100000

120000

10 20 30 50 70 90 110 150 200 250 300

Total cost

Vehicle type 1

Vehicle type 3Vehicle type 2

Vehicle type 1

Vehicle type 3Vehicle type 2

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 20 30 50 70 90 110 150 200 250 300

Transportation cost

Multiple vehicle types

Multiple vehicle types

Multiple vehicle types

Vehicle type 1

Vehicle type 3Vehicle type 2

Multiple vehicle typesVehicle type 1

Vehicle type 3Vehicle type 2

0

2000

4000

6000

8000

10000

12000

14000

10 20 30 50 70 90 110 150 200 250 300

Inventory cost

0

5000

10000

15000

20000

25000

30000

10 20 30 50 70 90 110 150 200 250 300

Dispatch cost

Figure 8 A comparison of the mode results

examples with varying scales for verification and was com-pared with the TS and the SA methods The results showedthat our TP-TSLSS obtained better solutions in a shortertime and a more stable manner By comparing differenttransportationmodes we concluded that consideration of thevehicle type choice could help save on cost of transportationinventory and dispatch

This was the first study to comprehensively considerfrequency planning vehicle type choice and routing plan-ning Although the consideration of a variety of vehicle

types significantly increased the complexity of the originalCFR problem the TP-TSLSS algorithm could reduce searchtime and improve accuracy through limiting search scopeIn the future we intended to extend this model to casesinvolving multiple manufacturers The problem will be morecomplex and the decision might depend on additionalfactors Moreover multiple depots could be considered froma practical perspective Finally we planned to develop evenmore efficient and accurate algorithms to solve these prob-lems

12 Mathematical Problems in Engineering

Table 3 119901 value of 119905-test

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

1 10 mdash mdash2 10 mdash mdash3 10 mdash mdash4 10 mdash mdash5 10 mdash mdash6 20 000031 0167 20 00000028 0188 20 0000026 0219 20 00043 01210 20 0000030 02811 30 00051 0008512 30 00032 002713 30 00017 001014 30 00072 0003615 30 0015 0002916 50 00000022 0000002517 50 00000063 00004118 50 0012 0006119 50 00000027 0000003520 50 00000012 0000001821 70 000023 000001022 70 000014 0000002923 70 0000092 0000001224 70 0000011 00000006325 70 000047 000002126 90 0000016 000004727 90 00000024 00000001828 90 00000051 00000001029 90 0000030 000002830 90 0000029 000001131 110 0000089 000006232 110 00000033 00000001033 110 000032 00002934 110 0000092 000005935 110 0000075 0000003336 150 00000013 0000001437 150 00000014 0000002038 150 0000067 000005139 150 000018 000005340 150 000051 000004741 200 0000069 00000001542 200 000000098 00000008343 200 0000074 0000001944 200 00000023 00000001245 200 0000082 0000003846 250 000000080 000000003747 250 000000015 0000000013

Table 3 Continued

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

48 250 000000011 0000000009849 250 00000064 00000001350 250 0000094 00000002751 300 00000011 00000006052 300 000000081 00000005653 300 000000042 00000001754 300 00000098 00000008255 300 0000020 0000000041

Conflict of Interests

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

References

[1] K H Chuah and J C Yingling ldquoRouting for a just-in-timesupply pickup and delivery systemrdquo Transportation Science vol39 no 3 pp 328ndash339 2005

[2] L Zhuang ldquoTowards a more economic production and just-in-time delivery systemrdquo International Journal of ProductionEconomics vol 36 no 3 pp 307ndash313 1994

[3] M Savsar and A Al-Jawini ldquoSimulation analysis of just-in-time production systemsrdquo International Journal of ProductionEconomics vol 42 no 1 pp 67ndash78 1995

[4] R R Lummus ldquoA simulation analysis of sequencing alternativesfor JIT lines using kanbansrdquo Journal of OperationsManagementvol 13 no 3 pp 183ndash191 1995

[5] S J Sadjadi M Jafari and T Amini ldquoA new mathematicalmodeling and a genetic algorithm search for milk run problem(an auto industry supply chain case study)rdquo The InternationalJournal of Advanced Manufacturing Technology vol 44 no 1-2pp 194ndash200 2009

[6] M Jafari-Eskandari S J Sadjadi M S Jabalameli and ABozorgi-Amiri ldquoA robust optimization approach for the MilkRun problem (An auto industry supply chain case study)rdquo inProceedings of the IEEE International Conference on Computersamp Industrial Engineering (CIE rsquo09) pp 1076ndash1081 TroyesFrance July 2009

[7] M J Eskandari A R Aliahmadi and G H H Khaleghi ldquoArobust optimisation approach for the milk run problem withtime windows with inventory uncertainty an auto industrysupply chain case studyrdquo International Journal of Rapid Man-ufacturing vol 1 no 3 pp 334ndash347 2010

[8] T Du F KWang and P-Y Lu ldquoA real-time vehicle-dispatchingsystem for consolidating milk runsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 43 no 5 pp565ndash577 2007

[9] J Ma and G Sun ldquoMutation Ant colony algorithm of milk-run vehicle routing problemwith fastest completion time basedon dynamic optimizationrdquo Discrete Dynamics in Nature andSociety vol 2013 Article ID 418436 6 pages 2013

[10] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

sum

119906

sum

119894isin119882

sum

119895isin119882

119909119894119895119896119906

le |119882| minus 1

forall119882 sube 119881 0 notin 119882 |119882| ge 2 119896 isin 119870 119906 isin 119880

(8)

sum

119906

sum

119894

119905119894sdot 119910119894119896119906

+sum

119906

sum

119894

sum

119895

119905119894119895sdot 119909119894119895119896119906

le 119879

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(9)

119909119894119895119896119906

= 0 or 1

119894 119895 isin 119881 119896 isin 119870 119906 isin 119880

(10)

119910119894119896119906

= 0 or 1

119894 isin 119881 119896 isin 119870 119906 isin 119880

(11)

119887119896119906= 0 or 1

119896 isin 119870 119906 isin 119880

(12)

sgn(sum119894

119910119894119896119906) = 119887

119896119906

119894 isin 119881 119896 isin 119870 119906 isin 119880

(13)

The first part of the objective function above addressestransportation cost the second part addresses inventory costand the third computes the cost of dispatched vehiclesThe transportation cost depends on the length of a givenroute as well as the frequency of the route The inventorycost depends on the pickup quantity for the suppliers ineach route and the route frequency Each vehicle type has acorresponding dispatch cost therefore the third part of thefunction depends on the vehicle type choice

The following is a detailed description of constraints (5) to(13) Equations (5) to (7) ensure that each supplier distributesin only one route and the path of each vehicle dispatchedforms a loop Constraint (8) ensures that the subloop isavoided Equation (9) ensures that the cycle time of eachroute for one frequency is less than the maximum allowablecycle time 119879 The value of 119879 depends on the real situationEquations (10) to (12) indicate the scope of variables Equation(13) ensures that each route has only one vehicle dispatched

3 The Design of the Two-Phase HeuristicAlgorithm

CFR-VTC as an extension of VRP is NP hard Therefore wepropose a two-phase heuristic algorithm called two-phase TSwith limited search scope (TP-TSLSS) to solve the problemIn the first phase an initial satisfactory solution is generatedthrough a greedy heuristic algorithm to maximize the ratio1199112of the superior arc frequency to the inferior arc frequency

The following steps are used to generate the initial satisfactorysolution (1) the arcs are divided into the superior arc andthe inferior arc according to the 80-20 rule (the superiorarc and the inferior are described in Section 33) (2) thegreedy heuristic algorithm is then employed to maximizethe ratio 119911

2of the frequency of the superior arc to that of

the inferior arc through the four neighborhood selectionapproaches described in Section 32 (3) the final solutionof the first phase with the optimal 119911max

2in the number of

iterations is exported In the second phase the solution fromthe first phase is improved using the TS To improve searchefficiency we limit the scope of the search in the algorithmto render the 119911

2of the candidate solution greater than the

product of the 119911max2

of the initial satisfactory solution anda coefficient 120572 Calculating the 119911

2value is simpler than

calculating the objective function value 119911 Consequently theTP-TSLSS greatly enhances the search efficiency by limitingthe scope of 119911

2rather than directly using the objective

function value 119911 for the search

31 Representation and Evaluation of Solution Weuse119909 and Vto represent the route and vehicle type respectively and theseform an intact solution We set the manufacturer code to 0and represent the supplier numbers with 119894 (119894 = 1 2 119899)We randomly select the suppliers and the manufacturer asa feasible route We first randomly generate 119898 routes (119898 isin

[2 119899 minus 1] where 119898 is an integer) and randomly distribute 119899suppliers into these routes For example solution 119909 includesroutes 119901

10 minus 2 minus 3 minus 5 minus 0 119901

20 minus 7 minus 1 minus 9 minus 8 minus 0 until

all suppliers have finished selecting These routes constitutesolution 119909 Second we calculate the sum of 119905

119894and 119905119894119895of each

route 1199011 1199012 of solution 119909 and we examine whether the

total time of each route meets the time constraint of (9) Ifnot we randomly split the route whose total time exceeds119879 For example if the total time of route 119901

2exceeds 119879 the

route 11990120 minus 7 minus 1 minus 9 minus 8 minus 0 is randomly split as 119901

210 minus

7 minus 1 minus 0 and 119901220 minus 9 minus 8 minus 0 Then recalculate the total

time of each new route respectively and repeat the samestep until all the routes meet the time constraint of (9) Inthat way the new solution adjusted constitutes the feasiblesolution Finally we randomly select the vehicle type 119906

119895(119895 =

1 2 119898) corresponding to each route of feasible solutionand set [119906

1 1199062 119906

119898] as solution V for each vehicle type

We use the objective function value to evaluate the qualityof solutions

32 Neighborhood Solution Selection Method There are fourmethods to select a neighborhood solution

(1) Randomly select a route and two suppliers on theroute Reverse the traversal order of the two suppliersFor instance randomly select route 119901

10 minus 7 minus 1 minus 9 minus

8 minus 0 Randomly reverse the order of 7 minus 1 minus 9 minus 8 1199011

becomes 0 minus 8 minus 9 minus 1 minus 7 minus 0

(2) Randomly select two routes and a supplier fromone of these two routes Insert the chosen supplierrandomly into the other route If there is no supplierremaining in the route from where the supplier wastaken set the route to null For instance randomlyselect routes119901

10minus2minus3minus5minus0 and119901

20minus7minus1minus9minus8minus0

Randomly select supplier 2 from 1199011to be inserted

into 1199012 1199011becomes 0 minus 3 minus 5 minus 0 and 119901

2becomes

0 minus 7 minus 1 minus 2 minus 9 minus 8 minus 0

Mathematical Problems in Engineering 5

Randomly generate an initial solution of (1199090 V0) which meets the time constraint in (9) Set the current solution 119909now = 1199090 Vnow = V

0

then calculate 1199112(119909now Vnow) Set it = 1 (it records the current number of iterations)

Do while it lt 119872If rand lt 119901

1

Generate a neighborhood solution (1199091 V1) through the first neighborhood method

Do while the time of the new solution (1199091 V1) is not less than 119879

Regenerate a neighborhood solution (1199091 V1) through the first neighborhood method

End doIf 1199112(1199091 V1) gt 1199112(119909now Vnow)

119909now = 1199091Vnow = V

1

End ifEnd ifExecute the other three neighborhood methods using a similar processit = it + 1

End doOutput the initial satisfactory solution (1199091015840

0 V10158400) = (119909now Vnow) and define 119911max

2= 1199112(1199091015840

0 V10158400)

Algorithm 1

(3) Randomly select a route and a supplier from it Addthe supplier to another route For instance say thereare five routes 119901

1 1199012 1199013 1199014 and 119901

5 Randomly select

route 11990110minus2minus3minus5minus0 and supplier 2 and move the

latter into 1199016 1199016becomes 0 minus 2 minus 0

(4) Randomly select a vehicle type of solution V allowingthe vehicle type to changewithin the allowable vehicletype range For instance V = [2 2 3 4 5] becomes[2 2 3 5 5]

33 The First-Phase Algorithm 80-20 rule is originally pro-posed by the Italian economist Pareto so it is also calledParetorsquos law Commonly it is stated that 20 of all causesbring about 80 of all effects [24ndash26] In this paper we firstdefine an arc set 119860 that records the distances between anytwo nodes which contain the suppliers and themanufacturerThen according to the 80-20 rule we define 20 of arcs in set119860 with the shortest length as the superior arc set 119860

1and the

other arcs are defined as the inferior arc set 1198602 The lengths

of all arcs in 1198601are smaller than those of the arcs in 119860

2

that is |1198861| lt |119886

2| 1198861isin 1198601 1198862isin 1198602 The number of arcs

in 1198602is four times that of the number in 119860

1 For a specific

solution (119909 V) we define 119891119904 as the sum of frequencies of allsuperior arcs and 119891119894 as the sum of frequencies of all inferiorarcs 119911

2(119909 V) is defined as 119891119904119891119894 The objective of the first-

phase algorithm is to maximize 1199112 We set the number of

iterations to119872The greedy heuristic algorithm is designed as shown in

Algorithm 1

34 The Second-Phase Algorithm Osman [27] Taillard et al[28] and Alonso et al [29] applied TS to solve the VRP andverified its effectiveness In this paper we limit the scope ofsearch to ensure that the 119911

2value of every candidate solution

is greater than 120572sdot119911max2

(where 120572 is a coefficient slightly smallerin value than 1 We find that 120572 isin [07 09] is an appropriate

range) We call the second-phase algorithm TS with limitedsearch scope (TSLSS) We set the number of iterations as NI

The TSLSS is shown in Algorithm 2

35 The Process of Two-Phase TS with Limited Search ScopeWe combine the first-phase algorithm in Section 33 and thesecond-phase algorithm in Section 34 as the two-phase TSwith limited search scope (TP-TSLSS)

Figure 2 shows the specific process

4 Computational Experiments

In this section our proposed two-phase heuristic algorithmis programmed in MATLAB R2011b to solve 55 smallmedium and large numerical problemsThese computationalexperiments are run on an Acer 4820TG computer with anIntel i5 CPU (24GHz) and 4GB (347 available) of memory

41 Experiments with Varying Supplier Size Weuse a numberof examples involving varying numbers of suppliers to vali-date ourmodel and algorithmWhen the number of suppliersis below 100 both the horizontal and vertical coordinatesof suppliers are drawn from the uniform distribution inthe range [0 100] using the randi() function in MATLABMoreover the coordinates of the suppliers are drawn from theuniform distribution in the range [0 200] when the numberof suppliers is between 100 and 200 Finally when the numberof suppliers is between 200 and 300 the coordinates aredrawn from the uniform distribution in the range [0 300]The horizontal and vertical coordinates are represented by119883 and 119884 respectively The pickup quantity is randomly gen-erated from the uniform distribution in the range [80 150]whereas the inventory cost of each supplierrsquos parts is subjectto uniform distribution in the range [0 1] The dispatch costfor each vehicle type is uniformly distributed from 100 to400 We set the coordinates of the manufacture to (50 50)

6 Mathematical Problems in Engineering

Set the total number of iterations performed by the TS (NI) Set tabulist = Set the initial solution to (11990910158400 V10158400) which is generated in

the first phase Let the optimal solution be 119909best = 1199091015840

0 Vbest = V1015840

0and the current solution 119909now = 119909

1015840

0 Vnow = V1015840

0

Let 119899 = 0 (record the number of current iterations) Set the number of candidate solutions (ℎ) in the TSDo while 119899 lt NI

Employ the four neighborhood methods to generate h candidate solutions (1199091 V1) (1199092 V2) (119909

ℎ Vℎ) satisfying the time

constraint in (9) with 1199112values greater than 120572 sdot 119911max

2 Calculate the objective function values of these solutions

119911(1199091 V1) 119911(119909

2 V2) 119911(119909

ℎ Vℎ)

Obtain (119909opt Vopt) whose objective function value is optimalIf min119911(119909

1 V1) 119911(119909

2 V2) 119911(119909

ℎ Vℎ) lt 119911(119909best Vbest)

(119909now Vnow) = (119909opt Vopt)Update the tabulist

ElseDo while (119909opt Vopt) is in the tabulistSelect the suboptimal solution from the candidate solutions and record it as (119909opt Vopt)

End do(119909now Vnow) = (119909opt Vopt)Update the tabulist

End if119899 = 119899 + 1

End do

Algorithm 2

The algorithms described in Section 3 are implemented to testthese numerical instances

42 Algorithm Comparison All of the 55 examples are gen-erated using the method described in Section 41 Similarproblems have been solved in some of the literature [27ndash29] using TS algorithms whereas other studies [30ndash32] haveemployed simulated annealing algorithms (SA) to solve themThus to demonstrate the effectiveness of our model and theTP-TSLSS algorithm each example in this study is run fivetimes using TP-TSLSS TS and the SA We then analyzeand compare the experiment results obtained by the threealgorithms Stop criteria of SA and TS are defined as followsSA stops when the temperature reaches a specific valueaccording to the scale of each instance and TS stops aftera specific number of iterations are performed by TS Thenumber of iterations performed by TS is twice as TP-TSLSS

Table 1 shows the results of the first phase of the two-phase algorithm SN represents the number of suppliersColumn ldquo119911(119909

0 V0)rdquo represents the average objective function

value of the initial solution ldquo119911(1199091015840 V1015840)rdquo represents the averageobjective function value of solutions obtained by the first-phase algorithm and ldquo1199051015840rdquo represents the time taken by thefirst-phase algorithm (the unit of time used is second)Following the execution of the first-phase algorithm wefind that the 119911

2value increases significantly within a short

time Consequently the objective function value 119911 can beimproved significantly through the optimization of the first-phase algorithm within a short time

Table 2 shows the comparison results of the optimalobjective function values of solutions obtained by the threealgorithms We choose examples 23 31 36 and 41 to showthe convergence of TS and TP-TSLSS in Figures 3 4 5 and 6

The horizontal axis represents the number of iterations andthe vertical axis represents the average objective functionvalue The column ldquoaveragerdquo represents the average objectivefunction value of the optimal solution in each algorithmldquo119905rdquo represents the CPU time for each algorithm and ldquo120590rdquorepresents the standard deviation

We draw the following conclusions from a comparison ofSA TS and TP-TSLSS shown in Table 2 and the convergencegraphs (Figures 3 4 5 and 6)

(1) When the scales of the instances are small theobjective function values of the solutions obtainedby SA are greater than those obtained by the othertwo algorithms Therefore the performance of SAis inferior to that of TS and TP-TSLSS in solvingsmall-scale problems By contrast SA outperformsTS but is worse than TP-TSLSS when the number ofsuppliers exceeds 30 A comparison of TP-TSLSS andTS shows that the former yields solutionswith smallerobjective function values than the latter except fora few small-scale instances Moreover Figures 3 45 and 6 show that the number of iterations of TSis twice that of TP-TSLSS but TS is inferior to TP-TSLSS The performance of TP-TSLSS is better thanthat of the other two algorithms for most instancesThe advantage of TP-TSLSS becomesmore obvious asthe scale of problems increases

(2) The computational time of TP-TSLSS is the shortestfollowed by SA and TS

(3) Solutions obtained by TP-TSLSS have the smalleststandard deviationThe comparison of standard devi-ation values proves that TP-TSLSS is the most robustfollowed byTSwithmoderate robustness and SAwith

Mathematical Problems in Engineering 7

The first phase algorithm start

Randomly generate the initial solution

Randomly generate the new

If it satisfies the time constraint (9)

Yes

No

If it satisfies the time constraint (9)

No

Yes

Yes No

Yes

No

The second phase algorithm start

solution of the first phase

solutions that meet the timeconstraint (9) in the

function value is the smallest among the candidate solutions

If the selected solution

If the selectedsolution

Select the suboptimal

Accept the new solution and replace the current solution

Accept the new solution and replace the current solution

Update the tabulist

Stop

Yes

No

YesNo

No

Yes

Yes

No

(x0 0)

neighborhood of N(xnow now)

Calculate z2(x1 1)

If z2(x1 1) gt z2(xnow now)

Accept (x1 1) set Reject (x1 1)

it = it + 1

it le M

(xnow now)

set zmax2 = z2(xnow now)

neighborhood of N(xnow now)

If the z2 value of eachcandidate solution gt 120572 middot z

max2

Select the solution (xoptopt) whose objective

(xopt opt) is better than (xbest best)

(xopt opt) is

candidate solution as (xoptopt)

n = n + 1

If n lt NI

in tabulist

Randomly generate h candidate

Set xnow = x0 now = 0 Calculatez2(xnow now) it = 1

solution (x1 1) in the

(xnow now) = (x1 1)

Output (x9984000 9984000) =

Tabulist H = Φ set the

9984000 xnow = x

9984000 and

algorithm xbest = x9984000 best =

now = 9984000 n = 0

Figure 2 The TP-TSLSS process

the least Therefore the two-phase algorithm (TP-TSLSS) can obtain better and more stable solutions ina relatively short time

To verify the effectiveness of the two-phase heuristicalgorithm we select an optimal solution for TP-TSLSS froman example involving 30 suppliers Figure 7 shows the pathdiagram

In addition we take the 119905-test of statistical analysis tocompare difference in objective function values of the three

algorithms 119879 test employs 119905 distribution theory to infer theoccurring probability of differences and to examine whetherthe differences are significant in two averages [33 34] Thispaper takes 119905-test to compare the differences in averageobjective function values of the three algorithms First werun 10 times for all 55 examples We employ the SPSS 18 totake normality test for the objective function values of the 55examples We find that except the objective function valuesof the five numerical examples with the scale of 10 suppliersthe rest of the numerical examplesrsquo objective function values

8 Mathematical Problems in Engineering

Table 1 Results obtained for the first-phase algorithm

Instance SN 119911(1199091015840 V1015840) 119911(1199090 V0) 1199112(1199091015840 V1015840) 119911

2(1199090 V0) 1199051015840 (s)

1 10 18391 30698 2 01667 0322 10 21128 2582 24444 02308 0313 10 23407 38168 23333 025 0344 10 21757 26998 30435 05455 0295 10 2490 29351 07059 02444 036 20 3158 57114 48182 02535 0347 20 50645 65045 132381 03488 0338 20 3600 58948 68667 01227 0319 20 43496 57095 103529 03793 0310 20 32278 52732 88571 01333 03211 30 42922 73623 149 02202 03612 30 68309 76401 71695 02818 03513 30 6574 72963 55 00896 03614 30 64024 65789 380147 023 03715 30 48815 92522 245 04107 03616 50 12523 14449 397647 0244 03617 50 11190 14649 185238 03557 03518 50 11945 14672 1297 03254 03719 50 11919 12149 91 025 03820 50 10930 17825 211429 02551 03621 70 11737 17540 479091 02878 03422 70 15508 18462 545625 01973 03923 70 12484 20227 1003333 02653 03824 70 15890 20555 29 03419 03725 70 14998 19819 172857 02903 03526 90 18475 20596 450556 03529 03827 90 19643 20328 472356 03427 0428 90 18792 19873 46729 02957 03929 90 16268 20921 8746 03268 03830 90 20846 28943 3667 03832 04131 110 25081 46116 2068 04035 08332 110 30848 45420 1857 04689 07833 110 27515 42895 1939 05073 07934 110 28371 37652 185 04971 07335 110 23627 39712 3046 04837 07836 150 38053 59964 180213 02545 09237 150 36892 56279 3263 02384 08938 150 39604 62874 20481 01953 09339 150 40285 58164 16306 02491 09240 150 37681 63692 2617 01902 0941 200 86334 111050 55 02611 210242 200 88274 113130 35806 02194 208743 200 85145 113670 41758 02874 207444 200 89732 112823 528 02391 212345 200 87692 135713 4825 0261 208446 250 111760 145270 39955 02601 402947 250 111830 146220 4316 02206 418148 250 118713 149254 38421 02059 401749 250 116915 153873 3617 01973 398250 250 120317 148148 3291 02391 405951 300 131980 169670 44194 02328 597252 300 129468 163200 4841 0304 629253 300 139642 161726 4163 02845 581754 300 130613 159212 40157 02174 618355 300 129862 163944 49153 0316 6084

0 50 100 150 200 250 300 350 40006

08

1

12

14

16

18

2

TSTP-TSLSS

times104

Figure 3 Convergence graph of 70 suppliers

0 100 200 300 400 500 60015

2

25

3

35

4

45

TSTP-TSLSS

times104

Figure 4 Convergence graph of 110 suppliers

obtained by all the three algorithms have a good normalityThen we take two hypothesis tests at 005 confidence levelconditions The first hypothesis test is designed for SA andTP-TSLSS with the null hypothesis 119867 120583

1le 1205830 The second

hypothesis test is designed for TS and TP-TSLSS with thenull hypothesis 119867 120583

2le 1205830 Finally we take independent

sample 119905-test for the two hypotheses respectively The 119901

values obtained are listed in Table 3For the comparison of SA and TP-TSLSS all 119901 values of

the examples with 20 or more suppliers are less than 005therefore the null hypotheses of these examples are rejecteddemonstrating that TP-TSLSS is better than SA

For the comparison of TS and TP-TSLSS apart from 119901

values of examples of 20 suppliers scale (TS and TP-TSLSS)being greater than 005 119901 values of the remaining examplesare less than 005 therefore the null hypotheses of these

Mathematical Problems in Engineering 9

Table 2 Results obtained for the three algorithms (SA TS and TP-TSLSS)

Instance SN SA TS TP-TSLSSAverage 119905SA (s) 120590SA Average 119905TS (s) 120590TS Average 119905TP-TSLSS (s) 120590TP-TSLSS

1 10 16191 1654 8731 13862 3247 0 13862 1463 02 10 15743 1526 7631 1388 3258 0 1388 1419 03 10 1491 1594 4020 14219 3282 0 14219 1346 04 10 15825 1631 7218 14778 3354 0 14778 1322 05 10 16049 1682 7194 14875 3196 4031 14382 1407 06 20 25689 1727 8256 23333 4664 4107 23549 1686 32187 20 26021 1714 841 26698 4561 562 25649 1811 35918 20 24928 1793 8646 25106 4628 479 24217 1765 3459 20 26183 1737 8729 27656 4619 601 25714 1879 331410 20 23105 1749 8439 22758 4594 504 22423 1814 36211 30 34219 2015 1183 35278 5492 7660 33173 1825 427112 30 36798 2125 11038 34214 5135 7135 33082 1819 41613 30 34192 2062 10949 35073 5284 7058 33019 1746 408414 30 35193 2048 11164 36465 5175 7563 3396 1822 402315 30 38276 2116 12645 42145 5428 7937 37361 1707 42516 50 6425 2485 41072 59709 6514 12047 54051 2116 555717 50 60158 2472 38264 61101 6628 11062 5998 2392 583518 50 57028 2493 34183 57413 6604 8459 56276 2216 519419 50 56026 2419 33963 58708 6592 9246 52185 2145 574920 50 60924 2562 32974 63686 6585 8794 5686 2281 538521 70 85106 4057 37016 90797 7849 18584 76153 2136 1170322 70 82179 3947 38294 8820 7632 17294 78465 2073 1075223 70 82462 3929 41174 84594 7940 16848 73865 2491 1047324 70 88171 4019 38265 89905 7835 17182 81702 2615 1014925 70 81096 4095 34793 82468 7529 1749 80516 2491 1135226 90 10492 6418 40327 10737 10673 27232 92756 4215 1772827 90 11600 6429 41046 12446 10428 23472 90794 3881 1713628 90 108433 6316 40684 136475 10582 25819 89615 3974 1738529 90 116923 6285 40958 114367 10418 24174 92359 4039 1748230 90 102395 6193 40715 108691 10837 22964 95725 4139 1698331 110 20487 8972 77967 211168 13248 54139 180161 5797 3382932 110 187321 8017 78924 204292 13082 52764 117423 5826 3103433 110 190355 8526 83592 198163 13569 51074 185354 5494 3409334 110 190241 8438 93164 192525 13194 50986 179394 5618 358535 110 193482 8525 76492 216241 13074 51196 173106 5769 3792936 150 30813 18038 13484 27982 35922 97346 24094 15006 8353537 150 29813 18157 12467 30583 35542 10048 24695 14678 8214638 150 30847 18262 13295 31764 35686 98683 28174 14903 7824839 150 29139 18037 14829 30269 35742 90474 27108 14286 7958340 150 28742 17984 12793 29264 35291 98134 26495 14982 8497341 200 50954 24044 32316 56340 41249 28743 48246 19151 1576342 200 58192 23859 35935 58861 41465 27631 52407 20018 1639443 200 56102 24168 38193 57928 41672 24926 52887 19427 1582644 200 57149 24027 37194 59218 41839 27428 50961 18737 1782445 200 56291 23939 36913 58316 41962 29726 51620 19472 1629746 250 67194 38054 46911 75935 63028 36877 62171 32006 2428747 250 62719 37937 48512 77501 63542 35825 59954 31078 2592648 250 61461 37846 49236 78519 63887 36014 58917 31403 2174549 250 70192 37735 43926 76110 64041 35826 63625 31786 2852850 250 64192 37342 47295 71926 63993 35714 62128 32682 2394851 300 82155 52069 73979 92167 91249 58212 78166 46351 4232652 300 84124 51053 74924 91135 89465 56149 77640 44718 3872653 300 88307 51749 71486 90114 90672 59376 75783 48227 3741954 300 83491 51283 73951 87910 91839 52864 78629 41937 4173655 300 76529 50959 75056 88412 90962 58251 73210 42672 43945

10 Mathematical Problems in Engineering

0 100 200 300 400 500 600 700 800 900 10002

25

3

35

4

45

5

55

6times10

4

TSTP-TSLSS

Figure 5 Convergence graph of 150 suppliers

0 100 200 300 400 500 600 700 800 900 1000 11004

5

6

7

8

9

10

11times10

4

TSTP-TSLSS

Figure 6 Convergence graph of 200 suppliers

examples are rejected demonstrating TP-TSLSS is better thanTS when the suppliers are more than 30

43 Results Analysis of the Different Modes General studiesof the routing problems did not consider the effect of vehicletype choice on transportation frequency and inventory costwhile we take the vehicle type choice into consideration Toexamine this effect we make a comparison of the generaltransportation mode (Mode 1) with the CFR-VTC mode(Mode 2) for different scales of instances

Mode 1 Only one vehicle type is allowed for the transporta-tion task

Mode 2 All vehicle types are available

Figure 8 shows a comparison of the results for the twomodes (Mode 1 includes three cases only vehicle type 1 onlyvehicle type 2 and only vehicle type 3 in which the vehicle

17

23

306

1127

2021

2719

1018

9

16

14

11

29

4

528

2 3

826 24

22

1513

250

00 50 100

50

100

Vehicle 3 frequency 5Vehicle 2 fre

quency 4

Vehicle 2 frequency 8

X

Y

Figure 7 The path diagram of TP-TSLSS

types are not required for decision Mode 2 signifies that allthe three vehicle types are involved in the decision)The loadcapacity of vehicle type 1 is the smallest whose value is 100and that of vehicle type 2 is moderate whose value is 300whereas the capacity of vehicle type 3 is the largest whosevalue is 600 The horizontal axis in Figure 8 represents thenumber of suppliers and the vertical axis shows the four typesof costmdashtransportation cost inventory cost dispatch costand total cost The four cost trends are shown in Figure 8

The transportation cost graph in Figure 8 shows thatthe case involving only vehicle type 3 records the lowesttransportation cost because a vehicle with a large loadcapacity requires less frequency to complete the pickup taskHowever the inventory cost and dispatch cost in this caseare the highest when the scales of instances are large becauseof low frequency and high load capacity By contrast thecase involving only vehicle type 1 has the advantage of lowinventory and dispatch costs while the transportation costin this case is the highest However the total cost of Mode2 is lower than all the three cases in Mode 1 Therefore wefind that we can balance the cost of transportation inventoryand dispatch by a reasonable choice of vehicle to acquire theoptimal total cost Mode 2 has greater cost advantages thanMode 1

5 Conclusions

In this paper we considered the common frequency routingproblem with vehicle type choice in milk runs in logisticalsystems We developed a mathematical model to describe thevehicle type choice frequency planning and vehicle routeplanning in the milk run system To solve this model wedeveloped a two-phase TS algorithm with limited searchscope (TP-TSLSS) that increased the search efficiency Theproposed TP-TSLSS algorithm was tested on 55 numerical

Mathematical Problems in Engineering 11

0

20000

40000

60000

80000

100000

120000

10 20 30 50 70 90 110 150 200 250 300

Total cost

Vehicle type 1

Vehicle type 3Vehicle type 2

Vehicle type 1

Vehicle type 3Vehicle type 2

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 20 30 50 70 90 110 150 200 250 300

Transportation cost

Multiple vehicle types

Multiple vehicle types

Multiple vehicle types

Vehicle type 1

Vehicle type 3Vehicle type 2

Multiple vehicle typesVehicle type 1

Vehicle type 3Vehicle type 2

0

2000

4000

6000

8000

10000

12000

14000

10 20 30 50 70 90 110 150 200 250 300

Inventory cost

0

5000

10000

15000

20000

25000

30000

10 20 30 50 70 90 110 150 200 250 300

Dispatch cost

Figure 8 A comparison of the mode results

examples with varying scales for verification and was com-pared with the TS and the SA methods The results showedthat our TP-TSLSS obtained better solutions in a shortertime and a more stable manner By comparing differenttransportationmodes we concluded that consideration of thevehicle type choice could help save on cost of transportationinventory and dispatch

This was the first study to comprehensively considerfrequency planning vehicle type choice and routing plan-ning Although the consideration of a variety of vehicle

types significantly increased the complexity of the originalCFR problem the TP-TSLSS algorithm could reduce searchtime and improve accuracy through limiting search scopeIn the future we intended to extend this model to casesinvolving multiple manufacturers The problem will be morecomplex and the decision might depend on additionalfactors Moreover multiple depots could be considered froma practical perspective Finally we planned to develop evenmore efficient and accurate algorithms to solve these prob-lems

12 Mathematical Problems in Engineering

Table 3 119901 value of 119905-test

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

1 10 mdash mdash2 10 mdash mdash3 10 mdash mdash4 10 mdash mdash5 10 mdash mdash6 20 000031 0167 20 00000028 0188 20 0000026 0219 20 00043 01210 20 0000030 02811 30 00051 0008512 30 00032 002713 30 00017 001014 30 00072 0003615 30 0015 0002916 50 00000022 0000002517 50 00000063 00004118 50 0012 0006119 50 00000027 0000003520 50 00000012 0000001821 70 000023 000001022 70 000014 0000002923 70 0000092 0000001224 70 0000011 00000006325 70 000047 000002126 90 0000016 000004727 90 00000024 00000001828 90 00000051 00000001029 90 0000030 000002830 90 0000029 000001131 110 0000089 000006232 110 00000033 00000001033 110 000032 00002934 110 0000092 000005935 110 0000075 0000003336 150 00000013 0000001437 150 00000014 0000002038 150 0000067 000005139 150 000018 000005340 150 000051 000004741 200 0000069 00000001542 200 000000098 00000008343 200 0000074 0000001944 200 00000023 00000001245 200 0000082 0000003846 250 000000080 000000003747 250 000000015 0000000013

Table 3 Continued

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

48 250 000000011 0000000009849 250 00000064 00000001350 250 0000094 00000002751 300 00000011 00000006052 300 000000081 00000005653 300 000000042 00000001754 300 00000098 00000008255 300 0000020 0000000041

Conflict of Interests

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

References

[1] K H Chuah and J C Yingling ldquoRouting for a just-in-timesupply pickup and delivery systemrdquo Transportation Science vol39 no 3 pp 328ndash339 2005

[2] L Zhuang ldquoTowards a more economic production and just-in-time delivery systemrdquo International Journal of ProductionEconomics vol 36 no 3 pp 307ndash313 1994

[3] M Savsar and A Al-Jawini ldquoSimulation analysis of just-in-time production systemsrdquo International Journal of ProductionEconomics vol 42 no 1 pp 67ndash78 1995

[4] R R Lummus ldquoA simulation analysis of sequencing alternativesfor JIT lines using kanbansrdquo Journal of OperationsManagementvol 13 no 3 pp 183ndash191 1995

[5] S J Sadjadi M Jafari and T Amini ldquoA new mathematicalmodeling and a genetic algorithm search for milk run problem(an auto industry supply chain case study)rdquo The InternationalJournal of Advanced Manufacturing Technology vol 44 no 1-2pp 194ndash200 2009

[6] M Jafari-Eskandari S J Sadjadi M S Jabalameli and ABozorgi-Amiri ldquoA robust optimization approach for the MilkRun problem (An auto industry supply chain case study)rdquo inProceedings of the IEEE International Conference on Computersamp Industrial Engineering (CIE rsquo09) pp 1076ndash1081 TroyesFrance July 2009

[7] M J Eskandari A R Aliahmadi and G H H Khaleghi ldquoArobust optimisation approach for the milk run problem withtime windows with inventory uncertainty an auto industrysupply chain case studyrdquo International Journal of Rapid Man-ufacturing vol 1 no 3 pp 334ndash347 2010

[8] T Du F KWang and P-Y Lu ldquoA real-time vehicle-dispatchingsystem for consolidating milk runsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 43 no 5 pp565ndash577 2007

[9] J Ma and G Sun ldquoMutation Ant colony algorithm of milk-run vehicle routing problemwith fastest completion time basedon dynamic optimizationrdquo Discrete Dynamics in Nature andSociety vol 2013 Article ID 418436 6 pages 2013

[10] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Randomly generate an initial solution of (1199090 V0) which meets the time constraint in (9) Set the current solution 119909now = 1199090 Vnow = V

0

then calculate 1199112(119909now Vnow) Set it = 1 (it records the current number of iterations)

Do while it lt 119872If rand lt 119901

1

Generate a neighborhood solution (1199091 V1) through the first neighborhood method

Do while the time of the new solution (1199091 V1) is not less than 119879

Regenerate a neighborhood solution (1199091 V1) through the first neighborhood method

End doIf 1199112(1199091 V1) gt 1199112(119909now Vnow)

119909now = 1199091Vnow = V

1

End ifEnd ifExecute the other three neighborhood methods using a similar processit = it + 1

End doOutput the initial satisfactory solution (1199091015840

0 V10158400) = (119909now Vnow) and define 119911max

2= 1199112(1199091015840

0 V10158400)

Algorithm 1

(3) Randomly select a route and a supplier from it Addthe supplier to another route For instance say thereare five routes 119901

1 1199012 1199013 1199014 and 119901

5 Randomly select

route 11990110minus2minus3minus5minus0 and supplier 2 and move the

latter into 1199016 1199016becomes 0 minus 2 minus 0

(4) Randomly select a vehicle type of solution V allowingthe vehicle type to changewithin the allowable vehicletype range For instance V = [2 2 3 4 5] becomes[2 2 3 5 5]

33 The First-Phase Algorithm 80-20 rule is originally pro-posed by the Italian economist Pareto so it is also calledParetorsquos law Commonly it is stated that 20 of all causesbring about 80 of all effects [24ndash26] In this paper we firstdefine an arc set 119860 that records the distances between anytwo nodes which contain the suppliers and themanufacturerThen according to the 80-20 rule we define 20 of arcs in set119860 with the shortest length as the superior arc set 119860

1and the

other arcs are defined as the inferior arc set 1198602 The lengths

of all arcs in 1198601are smaller than those of the arcs in 119860

2

that is |1198861| lt |119886

2| 1198861isin 1198601 1198862isin 1198602 The number of arcs

in 1198602is four times that of the number in 119860

1 For a specific

solution (119909 V) we define 119891119904 as the sum of frequencies of allsuperior arcs and 119891119894 as the sum of frequencies of all inferiorarcs 119911

2(119909 V) is defined as 119891119904119891119894 The objective of the first-

phase algorithm is to maximize 1199112 We set the number of

iterations to119872The greedy heuristic algorithm is designed as shown in

Algorithm 1

34 The Second-Phase Algorithm Osman [27] Taillard et al[28] and Alonso et al [29] applied TS to solve the VRP andverified its effectiveness In this paper we limit the scope ofsearch to ensure that the 119911

2value of every candidate solution

is greater than 120572sdot119911max2

(where 120572 is a coefficient slightly smallerin value than 1 We find that 120572 isin [07 09] is an appropriate

range) We call the second-phase algorithm TS with limitedsearch scope (TSLSS) We set the number of iterations as NI

The TSLSS is shown in Algorithm 2

35 The Process of Two-Phase TS with Limited Search ScopeWe combine the first-phase algorithm in Section 33 and thesecond-phase algorithm in Section 34 as the two-phase TSwith limited search scope (TP-TSLSS)

Figure 2 shows the specific process

4 Computational Experiments

In this section our proposed two-phase heuristic algorithmis programmed in MATLAB R2011b to solve 55 smallmedium and large numerical problemsThese computationalexperiments are run on an Acer 4820TG computer with anIntel i5 CPU (24GHz) and 4GB (347 available) of memory

41 Experiments with Varying Supplier Size Weuse a numberof examples involving varying numbers of suppliers to vali-date ourmodel and algorithmWhen the number of suppliersis below 100 both the horizontal and vertical coordinatesof suppliers are drawn from the uniform distribution inthe range [0 100] using the randi() function in MATLABMoreover the coordinates of the suppliers are drawn from theuniform distribution in the range [0 200] when the numberof suppliers is between 100 and 200 Finally when the numberof suppliers is between 200 and 300 the coordinates aredrawn from the uniform distribution in the range [0 300]The horizontal and vertical coordinates are represented by119883 and 119884 respectively The pickup quantity is randomly gen-erated from the uniform distribution in the range [80 150]whereas the inventory cost of each supplierrsquos parts is subjectto uniform distribution in the range [0 1] The dispatch costfor each vehicle type is uniformly distributed from 100 to400 We set the coordinates of the manufacture to (50 50)

6 Mathematical Problems in Engineering

Set the total number of iterations performed by the TS (NI) Set tabulist = Set the initial solution to (11990910158400 V10158400) which is generated in

the first phase Let the optimal solution be 119909best = 1199091015840

0 Vbest = V1015840

0and the current solution 119909now = 119909

1015840

0 Vnow = V1015840

0

Let 119899 = 0 (record the number of current iterations) Set the number of candidate solutions (ℎ) in the TSDo while 119899 lt NI

Employ the four neighborhood methods to generate h candidate solutions (1199091 V1) (1199092 V2) (119909

ℎ Vℎ) satisfying the time

constraint in (9) with 1199112values greater than 120572 sdot 119911max

2 Calculate the objective function values of these solutions

119911(1199091 V1) 119911(119909

2 V2) 119911(119909

ℎ Vℎ)

Obtain (119909opt Vopt) whose objective function value is optimalIf min119911(119909

1 V1) 119911(119909

2 V2) 119911(119909

ℎ Vℎ) lt 119911(119909best Vbest)

(119909now Vnow) = (119909opt Vopt)Update the tabulist

ElseDo while (119909opt Vopt) is in the tabulistSelect the suboptimal solution from the candidate solutions and record it as (119909opt Vopt)

End do(119909now Vnow) = (119909opt Vopt)Update the tabulist

End if119899 = 119899 + 1

End do

Algorithm 2

The algorithms described in Section 3 are implemented to testthese numerical instances

42 Algorithm Comparison All of the 55 examples are gen-erated using the method described in Section 41 Similarproblems have been solved in some of the literature [27ndash29] using TS algorithms whereas other studies [30ndash32] haveemployed simulated annealing algorithms (SA) to solve themThus to demonstrate the effectiveness of our model and theTP-TSLSS algorithm each example in this study is run fivetimes using TP-TSLSS TS and the SA We then analyzeand compare the experiment results obtained by the threealgorithms Stop criteria of SA and TS are defined as followsSA stops when the temperature reaches a specific valueaccording to the scale of each instance and TS stops aftera specific number of iterations are performed by TS Thenumber of iterations performed by TS is twice as TP-TSLSS

Table 1 shows the results of the first phase of the two-phase algorithm SN represents the number of suppliersColumn ldquo119911(119909

0 V0)rdquo represents the average objective function

value of the initial solution ldquo119911(1199091015840 V1015840)rdquo represents the averageobjective function value of solutions obtained by the first-phase algorithm and ldquo1199051015840rdquo represents the time taken by thefirst-phase algorithm (the unit of time used is second)Following the execution of the first-phase algorithm wefind that the 119911

2value increases significantly within a short

time Consequently the objective function value 119911 can beimproved significantly through the optimization of the first-phase algorithm within a short time

Table 2 shows the comparison results of the optimalobjective function values of solutions obtained by the threealgorithms We choose examples 23 31 36 and 41 to showthe convergence of TS and TP-TSLSS in Figures 3 4 5 and 6

The horizontal axis represents the number of iterations andthe vertical axis represents the average objective functionvalue The column ldquoaveragerdquo represents the average objectivefunction value of the optimal solution in each algorithmldquo119905rdquo represents the CPU time for each algorithm and ldquo120590rdquorepresents the standard deviation

We draw the following conclusions from a comparison ofSA TS and TP-TSLSS shown in Table 2 and the convergencegraphs (Figures 3 4 5 and 6)

(1) When the scales of the instances are small theobjective function values of the solutions obtainedby SA are greater than those obtained by the othertwo algorithms Therefore the performance of SAis inferior to that of TS and TP-TSLSS in solvingsmall-scale problems By contrast SA outperformsTS but is worse than TP-TSLSS when the number ofsuppliers exceeds 30 A comparison of TP-TSLSS andTS shows that the former yields solutionswith smallerobjective function values than the latter except fora few small-scale instances Moreover Figures 3 45 and 6 show that the number of iterations of TSis twice that of TP-TSLSS but TS is inferior to TP-TSLSS The performance of TP-TSLSS is better thanthat of the other two algorithms for most instancesThe advantage of TP-TSLSS becomesmore obvious asthe scale of problems increases

(2) The computational time of TP-TSLSS is the shortestfollowed by SA and TS

(3) Solutions obtained by TP-TSLSS have the smalleststandard deviationThe comparison of standard devi-ation values proves that TP-TSLSS is the most robustfollowed byTSwithmoderate robustness and SAwith

Mathematical Problems in Engineering 7

The first phase algorithm start

Randomly generate the initial solution

Randomly generate the new

If it satisfies the time constraint (9)

Yes

No

If it satisfies the time constraint (9)

No

Yes

Yes No

Yes

No

The second phase algorithm start

solution of the first phase

solutions that meet the timeconstraint (9) in the

function value is the smallest among the candidate solutions

If the selected solution

If the selectedsolution

Select the suboptimal

Accept the new solution and replace the current solution

Accept the new solution and replace the current solution

Update the tabulist

Stop

Yes

No

YesNo

No

Yes

Yes

No

(x0 0)

neighborhood of N(xnow now)

Calculate z2(x1 1)

If z2(x1 1) gt z2(xnow now)

Accept (x1 1) set Reject (x1 1)

it = it + 1

it le M

(xnow now)

set zmax2 = z2(xnow now)

neighborhood of N(xnow now)

If the z2 value of eachcandidate solution gt 120572 middot z

max2

Select the solution (xoptopt) whose objective

(xopt opt) is better than (xbest best)

(xopt opt) is

candidate solution as (xoptopt)

n = n + 1

If n lt NI

in tabulist

Randomly generate h candidate

Set xnow = x0 now = 0 Calculatez2(xnow now) it = 1

solution (x1 1) in the

(xnow now) = (x1 1)

Output (x9984000 9984000) =

Tabulist H = Φ set the

9984000 xnow = x

9984000 and

algorithm xbest = x9984000 best =

now = 9984000 n = 0

Figure 2 The TP-TSLSS process

the least Therefore the two-phase algorithm (TP-TSLSS) can obtain better and more stable solutions ina relatively short time

To verify the effectiveness of the two-phase heuristicalgorithm we select an optimal solution for TP-TSLSS froman example involving 30 suppliers Figure 7 shows the pathdiagram

In addition we take the 119905-test of statistical analysis tocompare difference in objective function values of the three

algorithms 119879 test employs 119905 distribution theory to infer theoccurring probability of differences and to examine whetherthe differences are significant in two averages [33 34] Thispaper takes 119905-test to compare the differences in averageobjective function values of the three algorithms First werun 10 times for all 55 examples We employ the SPSS 18 totake normality test for the objective function values of the 55examples We find that except the objective function valuesof the five numerical examples with the scale of 10 suppliersthe rest of the numerical examplesrsquo objective function values

8 Mathematical Problems in Engineering

Table 1 Results obtained for the first-phase algorithm

Instance SN 119911(1199091015840 V1015840) 119911(1199090 V0) 1199112(1199091015840 V1015840) 119911

2(1199090 V0) 1199051015840 (s)

1 10 18391 30698 2 01667 0322 10 21128 2582 24444 02308 0313 10 23407 38168 23333 025 0344 10 21757 26998 30435 05455 0295 10 2490 29351 07059 02444 036 20 3158 57114 48182 02535 0347 20 50645 65045 132381 03488 0338 20 3600 58948 68667 01227 0319 20 43496 57095 103529 03793 0310 20 32278 52732 88571 01333 03211 30 42922 73623 149 02202 03612 30 68309 76401 71695 02818 03513 30 6574 72963 55 00896 03614 30 64024 65789 380147 023 03715 30 48815 92522 245 04107 03616 50 12523 14449 397647 0244 03617 50 11190 14649 185238 03557 03518 50 11945 14672 1297 03254 03719 50 11919 12149 91 025 03820 50 10930 17825 211429 02551 03621 70 11737 17540 479091 02878 03422 70 15508 18462 545625 01973 03923 70 12484 20227 1003333 02653 03824 70 15890 20555 29 03419 03725 70 14998 19819 172857 02903 03526 90 18475 20596 450556 03529 03827 90 19643 20328 472356 03427 0428 90 18792 19873 46729 02957 03929 90 16268 20921 8746 03268 03830 90 20846 28943 3667 03832 04131 110 25081 46116 2068 04035 08332 110 30848 45420 1857 04689 07833 110 27515 42895 1939 05073 07934 110 28371 37652 185 04971 07335 110 23627 39712 3046 04837 07836 150 38053 59964 180213 02545 09237 150 36892 56279 3263 02384 08938 150 39604 62874 20481 01953 09339 150 40285 58164 16306 02491 09240 150 37681 63692 2617 01902 0941 200 86334 111050 55 02611 210242 200 88274 113130 35806 02194 208743 200 85145 113670 41758 02874 207444 200 89732 112823 528 02391 212345 200 87692 135713 4825 0261 208446 250 111760 145270 39955 02601 402947 250 111830 146220 4316 02206 418148 250 118713 149254 38421 02059 401749 250 116915 153873 3617 01973 398250 250 120317 148148 3291 02391 405951 300 131980 169670 44194 02328 597252 300 129468 163200 4841 0304 629253 300 139642 161726 4163 02845 581754 300 130613 159212 40157 02174 618355 300 129862 163944 49153 0316 6084

0 50 100 150 200 250 300 350 40006

08

1

12

14

16

18

2

TSTP-TSLSS

times104

Figure 3 Convergence graph of 70 suppliers

0 100 200 300 400 500 60015

2

25

3

35

4

45

TSTP-TSLSS

times104

Figure 4 Convergence graph of 110 suppliers

obtained by all the three algorithms have a good normalityThen we take two hypothesis tests at 005 confidence levelconditions The first hypothesis test is designed for SA andTP-TSLSS with the null hypothesis 119867 120583

1le 1205830 The second

hypothesis test is designed for TS and TP-TSLSS with thenull hypothesis 119867 120583

2le 1205830 Finally we take independent

sample 119905-test for the two hypotheses respectively The 119901

values obtained are listed in Table 3For the comparison of SA and TP-TSLSS all 119901 values of

the examples with 20 or more suppliers are less than 005therefore the null hypotheses of these examples are rejecteddemonstrating that TP-TSLSS is better than SA

For the comparison of TS and TP-TSLSS apart from 119901

values of examples of 20 suppliers scale (TS and TP-TSLSS)being greater than 005 119901 values of the remaining examplesare less than 005 therefore the null hypotheses of these

Mathematical Problems in Engineering 9

Table 2 Results obtained for the three algorithms (SA TS and TP-TSLSS)

Instance SN SA TS TP-TSLSSAverage 119905SA (s) 120590SA Average 119905TS (s) 120590TS Average 119905TP-TSLSS (s) 120590TP-TSLSS

1 10 16191 1654 8731 13862 3247 0 13862 1463 02 10 15743 1526 7631 1388 3258 0 1388 1419 03 10 1491 1594 4020 14219 3282 0 14219 1346 04 10 15825 1631 7218 14778 3354 0 14778 1322 05 10 16049 1682 7194 14875 3196 4031 14382 1407 06 20 25689 1727 8256 23333 4664 4107 23549 1686 32187 20 26021 1714 841 26698 4561 562 25649 1811 35918 20 24928 1793 8646 25106 4628 479 24217 1765 3459 20 26183 1737 8729 27656 4619 601 25714 1879 331410 20 23105 1749 8439 22758 4594 504 22423 1814 36211 30 34219 2015 1183 35278 5492 7660 33173 1825 427112 30 36798 2125 11038 34214 5135 7135 33082 1819 41613 30 34192 2062 10949 35073 5284 7058 33019 1746 408414 30 35193 2048 11164 36465 5175 7563 3396 1822 402315 30 38276 2116 12645 42145 5428 7937 37361 1707 42516 50 6425 2485 41072 59709 6514 12047 54051 2116 555717 50 60158 2472 38264 61101 6628 11062 5998 2392 583518 50 57028 2493 34183 57413 6604 8459 56276 2216 519419 50 56026 2419 33963 58708 6592 9246 52185 2145 574920 50 60924 2562 32974 63686 6585 8794 5686 2281 538521 70 85106 4057 37016 90797 7849 18584 76153 2136 1170322 70 82179 3947 38294 8820 7632 17294 78465 2073 1075223 70 82462 3929 41174 84594 7940 16848 73865 2491 1047324 70 88171 4019 38265 89905 7835 17182 81702 2615 1014925 70 81096 4095 34793 82468 7529 1749 80516 2491 1135226 90 10492 6418 40327 10737 10673 27232 92756 4215 1772827 90 11600 6429 41046 12446 10428 23472 90794 3881 1713628 90 108433 6316 40684 136475 10582 25819 89615 3974 1738529 90 116923 6285 40958 114367 10418 24174 92359 4039 1748230 90 102395 6193 40715 108691 10837 22964 95725 4139 1698331 110 20487 8972 77967 211168 13248 54139 180161 5797 3382932 110 187321 8017 78924 204292 13082 52764 117423 5826 3103433 110 190355 8526 83592 198163 13569 51074 185354 5494 3409334 110 190241 8438 93164 192525 13194 50986 179394 5618 358535 110 193482 8525 76492 216241 13074 51196 173106 5769 3792936 150 30813 18038 13484 27982 35922 97346 24094 15006 8353537 150 29813 18157 12467 30583 35542 10048 24695 14678 8214638 150 30847 18262 13295 31764 35686 98683 28174 14903 7824839 150 29139 18037 14829 30269 35742 90474 27108 14286 7958340 150 28742 17984 12793 29264 35291 98134 26495 14982 8497341 200 50954 24044 32316 56340 41249 28743 48246 19151 1576342 200 58192 23859 35935 58861 41465 27631 52407 20018 1639443 200 56102 24168 38193 57928 41672 24926 52887 19427 1582644 200 57149 24027 37194 59218 41839 27428 50961 18737 1782445 200 56291 23939 36913 58316 41962 29726 51620 19472 1629746 250 67194 38054 46911 75935 63028 36877 62171 32006 2428747 250 62719 37937 48512 77501 63542 35825 59954 31078 2592648 250 61461 37846 49236 78519 63887 36014 58917 31403 2174549 250 70192 37735 43926 76110 64041 35826 63625 31786 2852850 250 64192 37342 47295 71926 63993 35714 62128 32682 2394851 300 82155 52069 73979 92167 91249 58212 78166 46351 4232652 300 84124 51053 74924 91135 89465 56149 77640 44718 3872653 300 88307 51749 71486 90114 90672 59376 75783 48227 3741954 300 83491 51283 73951 87910 91839 52864 78629 41937 4173655 300 76529 50959 75056 88412 90962 58251 73210 42672 43945

10 Mathematical Problems in Engineering

0 100 200 300 400 500 600 700 800 900 10002

25

3

35

4

45

5

55

6times10

4

TSTP-TSLSS

Figure 5 Convergence graph of 150 suppliers

0 100 200 300 400 500 600 700 800 900 1000 11004

5

6

7

8

9

10

11times10

4

TSTP-TSLSS

Figure 6 Convergence graph of 200 suppliers

examples are rejected demonstrating TP-TSLSS is better thanTS when the suppliers are more than 30

43 Results Analysis of the Different Modes General studiesof the routing problems did not consider the effect of vehicletype choice on transportation frequency and inventory costwhile we take the vehicle type choice into consideration Toexamine this effect we make a comparison of the generaltransportation mode (Mode 1) with the CFR-VTC mode(Mode 2) for different scales of instances

Mode 1 Only one vehicle type is allowed for the transporta-tion task

Mode 2 All vehicle types are available

Figure 8 shows a comparison of the results for the twomodes (Mode 1 includes three cases only vehicle type 1 onlyvehicle type 2 and only vehicle type 3 in which the vehicle

17

23

306

1127

2021

2719

1018

9

16

14

11

29

4

528

2 3

826 24

22

1513

250

00 50 100

50

100

Vehicle 3 frequency 5Vehicle 2 fre

quency 4

Vehicle 2 frequency 8

X

Y

Figure 7 The path diagram of TP-TSLSS

types are not required for decision Mode 2 signifies that allthe three vehicle types are involved in the decision)The loadcapacity of vehicle type 1 is the smallest whose value is 100and that of vehicle type 2 is moderate whose value is 300whereas the capacity of vehicle type 3 is the largest whosevalue is 600 The horizontal axis in Figure 8 represents thenumber of suppliers and the vertical axis shows the four typesof costmdashtransportation cost inventory cost dispatch costand total cost The four cost trends are shown in Figure 8

The transportation cost graph in Figure 8 shows thatthe case involving only vehicle type 3 records the lowesttransportation cost because a vehicle with a large loadcapacity requires less frequency to complete the pickup taskHowever the inventory cost and dispatch cost in this caseare the highest when the scales of instances are large becauseof low frequency and high load capacity By contrast thecase involving only vehicle type 1 has the advantage of lowinventory and dispatch costs while the transportation costin this case is the highest However the total cost of Mode2 is lower than all the three cases in Mode 1 Therefore wefind that we can balance the cost of transportation inventoryand dispatch by a reasonable choice of vehicle to acquire theoptimal total cost Mode 2 has greater cost advantages thanMode 1

5 Conclusions

In this paper we considered the common frequency routingproblem with vehicle type choice in milk runs in logisticalsystems We developed a mathematical model to describe thevehicle type choice frequency planning and vehicle routeplanning in the milk run system To solve this model wedeveloped a two-phase TS algorithm with limited searchscope (TP-TSLSS) that increased the search efficiency Theproposed TP-TSLSS algorithm was tested on 55 numerical

Mathematical Problems in Engineering 11

0

20000

40000

60000

80000

100000

120000

10 20 30 50 70 90 110 150 200 250 300

Total cost

Vehicle type 1

Vehicle type 3Vehicle type 2

Vehicle type 1

Vehicle type 3Vehicle type 2

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 20 30 50 70 90 110 150 200 250 300

Transportation cost

Multiple vehicle types

Multiple vehicle types

Multiple vehicle types

Vehicle type 1

Vehicle type 3Vehicle type 2

Multiple vehicle typesVehicle type 1

Vehicle type 3Vehicle type 2

0

2000

4000

6000

8000

10000

12000

14000

10 20 30 50 70 90 110 150 200 250 300

Inventory cost

0

5000

10000

15000

20000

25000

30000

10 20 30 50 70 90 110 150 200 250 300

Dispatch cost

Figure 8 A comparison of the mode results

examples with varying scales for verification and was com-pared with the TS and the SA methods The results showedthat our TP-TSLSS obtained better solutions in a shortertime and a more stable manner By comparing differenttransportationmodes we concluded that consideration of thevehicle type choice could help save on cost of transportationinventory and dispatch

This was the first study to comprehensively considerfrequency planning vehicle type choice and routing plan-ning Although the consideration of a variety of vehicle

types significantly increased the complexity of the originalCFR problem the TP-TSLSS algorithm could reduce searchtime and improve accuracy through limiting search scopeIn the future we intended to extend this model to casesinvolving multiple manufacturers The problem will be morecomplex and the decision might depend on additionalfactors Moreover multiple depots could be considered froma practical perspective Finally we planned to develop evenmore efficient and accurate algorithms to solve these prob-lems

12 Mathematical Problems in Engineering

Table 3 119901 value of 119905-test

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

1 10 mdash mdash2 10 mdash mdash3 10 mdash mdash4 10 mdash mdash5 10 mdash mdash6 20 000031 0167 20 00000028 0188 20 0000026 0219 20 00043 01210 20 0000030 02811 30 00051 0008512 30 00032 002713 30 00017 001014 30 00072 0003615 30 0015 0002916 50 00000022 0000002517 50 00000063 00004118 50 0012 0006119 50 00000027 0000003520 50 00000012 0000001821 70 000023 000001022 70 000014 0000002923 70 0000092 0000001224 70 0000011 00000006325 70 000047 000002126 90 0000016 000004727 90 00000024 00000001828 90 00000051 00000001029 90 0000030 000002830 90 0000029 000001131 110 0000089 000006232 110 00000033 00000001033 110 000032 00002934 110 0000092 000005935 110 0000075 0000003336 150 00000013 0000001437 150 00000014 0000002038 150 0000067 000005139 150 000018 000005340 150 000051 000004741 200 0000069 00000001542 200 000000098 00000008343 200 0000074 0000001944 200 00000023 00000001245 200 0000082 0000003846 250 000000080 000000003747 250 000000015 0000000013

Table 3 Continued

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

48 250 000000011 0000000009849 250 00000064 00000001350 250 0000094 00000002751 300 00000011 00000006052 300 000000081 00000005653 300 000000042 00000001754 300 00000098 00000008255 300 0000020 0000000041

Conflict of Interests

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

References

[1] K H Chuah and J C Yingling ldquoRouting for a just-in-timesupply pickup and delivery systemrdquo Transportation Science vol39 no 3 pp 328ndash339 2005

[2] L Zhuang ldquoTowards a more economic production and just-in-time delivery systemrdquo International Journal of ProductionEconomics vol 36 no 3 pp 307ndash313 1994

[3] M Savsar and A Al-Jawini ldquoSimulation analysis of just-in-time production systemsrdquo International Journal of ProductionEconomics vol 42 no 1 pp 67ndash78 1995

[4] R R Lummus ldquoA simulation analysis of sequencing alternativesfor JIT lines using kanbansrdquo Journal of OperationsManagementvol 13 no 3 pp 183ndash191 1995

[5] S J Sadjadi M Jafari and T Amini ldquoA new mathematicalmodeling and a genetic algorithm search for milk run problem(an auto industry supply chain case study)rdquo The InternationalJournal of Advanced Manufacturing Technology vol 44 no 1-2pp 194ndash200 2009

[6] M Jafari-Eskandari S J Sadjadi M S Jabalameli and ABozorgi-Amiri ldquoA robust optimization approach for the MilkRun problem (An auto industry supply chain case study)rdquo inProceedings of the IEEE International Conference on Computersamp Industrial Engineering (CIE rsquo09) pp 1076ndash1081 TroyesFrance July 2009

[7] M J Eskandari A R Aliahmadi and G H H Khaleghi ldquoArobust optimisation approach for the milk run problem withtime windows with inventory uncertainty an auto industrysupply chain case studyrdquo International Journal of Rapid Man-ufacturing vol 1 no 3 pp 334ndash347 2010

[8] T Du F KWang and P-Y Lu ldquoA real-time vehicle-dispatchingsystem for consolidating milk runsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 43 no 5 pp565ndash577 2007

[9] J Ma and G Sun ldquoMutation Ant colony algorithm of milk-run vehicle routing problemwith fastest completion time basedon dynamic optimizationrdquo Discrete Dynamics in Nature andSociety vol 2013 Article ID 418436 6 pages 2013

[10] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Set the total number of iterations performed by the TS (NI) Set tabulist = Set the initial solution to (11990910158400 V10158400) which is generated in

the first phase Let the optimal solution be 119909best = 1199091015840

0 Vbest = V1015840

0and the current solution 119909now = 119909

1015840

0 Vnow = V1015840

0

Let 119899 = 0 (record the number of current iterations) Set the number of candidate solutions (ℎ) in the TSDo while 119899 lt NI

Employ the four neighborhood methods to generate h candidate solutions (1199091 V1) (1199092 V2) (119909

ℎ Vℎ) satisfying the time

constraint in (9) with 1199112values greater than 120572 sdot 119911max

2 Calculate the objective function values of these solutions

119911(1199091 V1) 119911(119909

2 V2) 119911(119909

ℎ Vℎ)

Obtain (119909opt Vopt) whose objective function value is optimalIf min119911(119909

1 V1) 119911(119909

2 V2) 119911(119909

ℎ Vℎ) lt 119911(119909best Vbest)

(119909now Vnow) = (119909opt Vopt)Update the tabulist

ElseDo while (119909opt Vopt) is in the tabulistSelect the suboptimal solution from the candidate solutions and record it as (119909opt Vopt)

End do(119909now Vnow) = (119909opt Vopt)Update the tabulist

End if119899 = 119899 + 1

End do

Algorithm 2

The algorithms described in Section 3 are implemented to testthese numerical instances

42 Algorithm Comparison All of the 55 examples are gen-erated using the method described in Section 41 Similarproblems have been solved in some of the literature [27ndash29] using TS algorithms whereas other studies [30ndash32] haveemployed simulated annealing algorithms (SA) to solve themThus to demonstrate the effectiveness of our model and theTP-TSLSS algorithm each example in this study is run fivetimes using TP-TSLSS TS and the SA We then analyzeand compare the experiment results obtained by the threealgorithms Stop criteria of SA and TS are defined as followsSA stops when the temperature reaches a specific valueaccording to the scale of each instance and TS stops aftera specific number of iterations are performed by TS Thenumber of iterations performed by TS is twice as TP-TSLSS

Table 1 shows the results of the first phase of the two-phase algorithm SN represents the number of suppliersColumn ldquo119911(119909

0 V0)rdquo represents the average objective function

value of the initial solution ldquo119911(1199091015840 V1015840)rdquo represents the averageobjective function value of solutions obtained by the first-phase algorithm and ldquo1199051015840rdquo represents the time taken by thefirst-phase algorithm (the unit of time used is second)Following the execution of the first-phase algorithm wefind that the 119911

2value increases significantly within a short

time Consequently the objective function value 119911 can beimproved significantly through the optimization of the first-phase algorithm within a short time

Table 2 shows the comparison results of the optimalobjective function values of solutions obtained by the threealgorithms We choose examples 23 31 36 and 41 to showthe convergence of TS and TP-TSLSS in Figures 3 4 5 and 6

The horizontal axis represents the number of iterations andthe vertical axis represents the average objective functionvalue The column ldquoaveragerdquo represents the average objectivefunction value of the optimal solution in each algorithmldquo119905rdquo represents the CPU time for each algorithm and ldquo120590rdquorepresents the standard deviation

We draw the following conclusions from a comparison ofSA TS and TP-TSLSS shown in Table 2 and the convergencegraphs (Figures 3 4 5 and 6)

(1) When the scales of the instances are small theobjective function values of the solutions obtainedby SA are greater than those obtained by the othertwo algorithms Therefore the performance of SAis inferior to that of TS and TP-TSLSS in solvingsmall-scale problems By contrast SA outperformsTS but is worse than TP-TSLSS when the number ofsuppliers exceeds 30 A comparison of TP-TSLSS andTS shows that the former yields solutionswith smallerobjective function values than the latter except fora few small-scale instances Moreover Figures 3 45 and 6 show that the number of iterations of TSis twice that of TP-TSLSS but TS is inferior to TP-TSLSS The performance of TP-TSLSS is better thanthat of the other two algorithms for most instancesThe advantage of TP-TSLSS becomesmore obvious asthe scale of problems increases

(2) The computational time of TP-TSLSS is the shortestfollowed by SA and TS

(3) Solutions obtained by TP-TSLSS have the smalleststandard deviationThe comparison of standard devi-ation values proves that TP-TSLSS is the most robustfollowed byTSwithmoderate robustness and SAwith

Mathematical Problems in Engineering 7

The first phase algorithm start

Randomly generate the initial solution

Randomly generate the new

If it satisfies the time constraint (9)

Yes

No

If it satisfies the time constraint (9)

No

Yes

Yes No

Yes

No

The second phase algorithm start

solution of the first phase

solutions that meet the timeconstraint (9) in the

function value is the smallest among the candidate solutions

If the selected solution

If the selectedsolution

Select the suboptimal

Accept the new solution and replace the current solution

Accept the new solution and replace the current solution

Update the tabulist

Stop

Yes

No

YesNo

No

Yes

Yes

No

(x0 0)

neighborhood of N(xnow now)

Calculate z2(x1 1)

If z2(x1 1) gt z2(xnow now)

Accept (x1 1) set Reject (x1 1)

it = it + 1

it le M

(xnow now)

set zmax2 = z2(xnow now)

neighborhood of N(xnow now)

If the z2 value of eachcandidate solution gt 120572 middot z

max2

Select the solution (xoptopt) whose objective

(xopt opt) is better than (xbest best)

(xopt opt) is

candidate solution as (xoptopt)

n = n + 1

If n lt NI

in tabulist

Randomly generate h candidate

Set xnow = x0 now = 0 Calculatez2(xnow now) it = 1

solution (x1 1) in the

(xnow now) = (x1 1)

Output (x9984000 9984000) =

Tabulist H = Φ set the

9984000 xnow = x

9984000 and

algorithm xbest = x9984000 best =

now = 9984000 n = 0

Figure 2 The TP-TSLSS process

the least Therefore the two-phase algorithm (TP-TSLSS) can obtain better and more stable solutions ina relatively short time

To verify the effectiveness of the two-phase heuristicalgorithm we select an optimal solution for TP-TSLSS froman example involving 30 suppliers Figure 7 shows the pathdiagram

In addition we take the 119905-test of statistical analysis tocompare difference in objective function values of the three

algorithms 119879 test employs 119905 distribution theory to infer theoccurring probability of differences and to examine whetherthe differences are significant in two averages [33 34] Thispaper takes 119905-test to compare the differences in averageobjective function values of the three algorithms First werun 10 times for all 55 examples We employ the SPSS 18 totake normality test for the objective function values of the 55examples We find that except the objective function valuesof the five numerical examples with the scale of 10 suppliersthe rest of the numerical examplesrsquo objective function values

8 Mathematical Problems in Engineering

Table 1 Results obtained for the first-phase algorithm

Instance SN 119911(1199091015840 V1015840) 119911(1199090 V0) 1199112(1199091015840 V1015840) 119911

2(1199090 V0) 1199051015840 (s)

1 10 18391 30698 2 01667 0322 10 21128 2582 24444 02308 0313 10 23407 38168 23333 025 0344 10 21757 26998 30435 05455 0295 10 2490 29351 07059 02444 036 20 3158 57114 48182 02535 0347 20 50645 65045 132381 03488 0338 20 3600 58948 68667 01227 0319 20 43496 57095 103529 03793 0310 20 32278 52732 88571 01333 03211 30 42922 73623 149 02202 03612 30 68309 76401 71695 02818 03513 30 6574 72963 55 00896 03614 30 64024 65789 380147 023 03715 30 48815 92522 245 04107 03616 50 12523 14449 397647 0244 03617 50 11190 14649 185238 03557 03518 50 11945 14672 1297 03254 03719 50 11919 12149 91 025 03820 50 10930 17825 211429 02551 03621 70 11737 17540 479091 02878 03422 70 15508 18462 545625 01973 03923 70 12484 20227 1003333 02653 03824 70 15890 20555 29 03419 03725 70 14998 19819 172857 02903 03526 90 18475 20596 450556 03529 03827 90 19643 20328 472356 03427 0428 90 18792 19873 46729 02957 03929 90 16268 20921 8746 03268 03830 90 20846 28943 3667 03832 04131 110 25081 46116 2068 04035 08332 110 30848 45420 1857 04689 07833 110 27515 42895 1939 05073 07934 110 28371 37652 185 04971 07335 110 23627 39712 3046 04837 07836 150 38053 59964 180213 02545 09237 150 36892 56279 3263 02384 08938 150 39604 62874 20481 01953 09339 150 40285 58164 16306 02491 09240 150 37681 63692 2617 01902 0941 200 86334 111050 55 02611 210242 200 88274 113130 35806 02194 208743 200 85145 113670 41758 02874 207444 200 89732 112823 528 02391 212345 200 87692 135713 4825 0261 208446 250 111760 145270 39955 02601 402947 250 111830 146220 4316 02206 418148 250 118713 149254 38421 02059 401749 250 116915 153873 3617 01973 398250 250 120317 148148 3291 02391 405951 300 131980 169670 44194 02328 597252 300 129468 163200 4841 0304 629253 300 139642 161726 4163 02845 581754 300 130613 159212 40157 02174 618355 300 129862 163944 49153 0316 6084

0 50 100 150 200 250 300 350 40006

08

1

12

14

16

18

2

TSTP-TSLSS

times104

Figure 3 Convergence graph of 70 suppliers

0 100 200 300 400 500 60015

2

25

3

35

4

45

TSTP-TSLSS

times104

Figure 4 Convergence graph of 110 suppliers

obtained by all the three algorithms have a good normalityThen we take two hypothesis tests at 005 confidence levelconditions The first hypothesis test is designed for SA andTP-TSLSS with the null hypothesis 119867 120583

1le 1205830 The second

hypothesis test is designed for TS and TP-TSLSS with thenull hypothesis 119867 120583

2le 1205830 Finally we take independent

sample 119905-test for the two hypotheses respectively The 119901

values obtained are listed in Table 3For the comparison of SA and TP-TSLSS all 119901 values of

the examples with 20 or more suppliers are less than 005therefore the null hypotheses of these examples are rejecteddemonstrating that TP-TSLSS is better than SA

For the comparison of TS and TP-TSLSS apart from 119901

values of examples of 20 suppliers scale (TS and TP-TSLSS)being greater than 005 119901 values of the remaining examplesare less than 005 therefore the null hypotheses of these

Mathematical Problems in Engineering 9

Table 2 Results obtained for the three algorithms (SA TS and TP-TSLSS)

Instance SN SA TS TP-TSLSSAverage 119905SA (s) 120590SA Average 119905TS (s) 120590TS Average 119905TP-TSLSS (s) 120590TP-TSLSS

1 10 16191 1654 8731 13862 3247 0 13862 1463 02 10 15743 1526 7631 1388 3258 0 1388 1419 03 10 1491 1594 4020 14219 3282 0 14219 1346 04 10 15825 1631 7218 14778 3354 0 14778 1322 05 10 16049 1682 7194 14875 3196 4031 14382 1407 06 20 25689 1727 8256 23333 4664 4107 23549 1686 32187 20 26021 1714 841 26698 4561 562 25649 1811 35918 20 24928 1793 8646 25106 4628 479 24217 1765 3459 20 26183 1737 8729 27656 4619 601 25714 1879 331410 20 23105 1749 8439 22758 4594 504 22423 1814 36211 30 34219 2015 1183 35278 5492 7660 33173 1825 427112 30 36798 2125 11038 34214 5135 7135 33082 1819 41613 30 34192 2062 10949 35073 5284 7058 33019 1746 408414 30 35193 2048 11164 36465 5175 7563 3396 1822 402315 30 38276 2116 12645 42145 5428 7937 37361 1707 42516 50 6425 2485 41072 59709 6514 12047 54051 2116 555717 50 60158 2472 38264 61101 6628 11062 5998 2392 583518 50 57028 2493 34183 57413 6604 8459 56276 2216 519419 50 56026 2419 33963 58708 6592 9246 52185 2145 574920 50 60924 2562 32974 63686 6585 8794 5686 2281 538521 70 85106 4057 37016 90797 7849 18584 76153 2136 1170322 70 82179 3947 38294 8820 7632 17294 78465 2073 1075223 70 82462 3929 41174 84594 7940 16848 73865 2491 1047324 70 88171 4019 38265 89905 7835 17182 81702 2615 1014925 70 81096 4095 34793 82468 7529 1749 80516 2491 1135226 90 10492 6418 40327 10737 10673 27232 92756 4215 1772827 90 11600 6429 41046 12446 10428 23472 90794 3881 1713628 90 108433 6316 40684 136475 10582 25819 89615 3974 1738529 90 116923 6285 40958 114367 10418 24174 92359 4039 1748230 90 102395 6193 40715 108691 10837 22964 95725 4139 1698331 110 20487 8972 77967 211168 13248 54139 180161 5797 3382932 110 187321 8017 78924 204292 13082 52764 117423 5826 3103433 110 190355 8526 83592 198163 13569 51074 185354 5494 3409334 110 190241 8438 93164 192525 13194 50986 179394 5618 358535 110 193482 8525 76492 216241 13074 51196 173106 5769 3792936 150 30813 18038 13484 27982 35922 97346 24094 15006 8353537 150 29813 18157 12467 30583 35542 10048 24695 14678 8214638 150 30847 18262 13295 31764 35686 98683 28174 14903 7824839 150 29139 18037 14829 30269 35742 90474 27108 14286 7958340 150 28742 17984 12793 29264 35291 98134 26495 14982 8497341 200 50954 24044 32316 56340 41249 28743 48246 19151 1576342 200 58192 23859 35935 58861 41465 27631 52407 20018 1639443 200 56102 24168 38193 57928 41672 24926 52887 19427 1582644 200 57149 24027 37194 59218 41839 27428 50961 18737 1782445 200 56291 23939 36913 58316 41962 29726 51620 19472 1629746 250 67194 38054 46911 75935 63028 36877 62171 32006 2428747 250 62719 37937 48512 77501 63542 35825 59954 31078 2592648 250 61461 37846 49236 78519 63887 36014 58917 31403 2174549 250 70192 37735 43926 76110 64041 35826 63625 31786 2852850 250 64192 37342 47295 71926 63993 35714 62128 32682 2394851 300 82155 52069 73979 92167 91249 58212 78166 46351 4232652 300 84124 51053 74924 91135 89465 56149 77640 44718 3872653 300 88307 51749 71486 90114 90672 59376 75783 48227 3741954 300 83491 51283 73951 87910 91839 52864 78629 41937 4173655 300 76529 50959 75056 88412 90962 58251 73210 42672 43945

10 Mathematical Problems in Engineering

0 100 200 300 400 500 600 700 800 900 10002

25

3

35

4

45

5

55

6times10

4

TSTP-TSLSS

Figure 5 Convergence graph of 150 suppliers

0 100 200 300 400 500 600 700 800 900 1000 11004

5

6

7

8

9

10

11times10

4

TSTP-TSLSS

Figure 6 Convergence graph of 200 suppliers

examples are rejected demonstrating TP-TSLSS is better thanTS when the suppliers are more than 30

43 Results Analysis of the Different Modes General studiesof the routing problems did not consider the effect of vehicletype choice on transportation frequency and inventory costwhile we take the vehicle type choice into consideration Toexamine this effect we make a comparison of the generaltransportation mode (Mode 1) with the CFR-VTC mode(Mode 2) for different scales of instances

Mode 1 Only one vehicle type is allowed for the transporta-tion task

Mode 2 All vehicle types are available

Figure 8 shows a comparison of the results for the twomodes (Mode 1 includes three cases only vehicle type 1 onlyvehicle type 2 and only vehicle type 3 in which the vehicle

17

23

306

1127

2021

2719

1018

9

16

14

11

29

4

528

2 3

826 24

22

1513

250

00 50 100

50

100

Vehicle 3 frequency 5Vehicle 2 fre

quency 4

Vehicle 2 frequency 8

X

Y

Figure 7 The path diagram of TP-TSLSS

types are not required for decision Mode 2 signifies that allthe three vehicle types are involved in the decision)The loadcapacity of vehicle type 1 is the smallest whose value is 100and that of vehicle type 2 is moderate whose value is 300whereas the capacity of vehicle type 3 is the largest whosevalue is 600 The horizontal axis in Figure 8 represents thenumber of suppliers and the vertical axis shows the four typesof costmdashtransportation cost inventory cost dispatch costand total cost The four cost trends are shown in Figure 8

The transportation cost graph in Figure 8 shows thatthe case involving only vehicle type 3 records the lowesttransportation cost because a vehicle with a large loadcapacity requires less frequency to complete the pickup taskHowever the inventory cost and dispatch cost in this caseare the highest when the scales of instances are large becauseof low frequency and high load capacity By contrast thecase involving only vehicle type 1 has the advantage of lowinventory and dispatch costs while the transportation costin this case is the highest However the total cost of Mode2 is lower than all the three cases in Mode 1 Therefore wefind that we can balance the cost of transportation inventoryand dispatch by a reasonable choice of vehicle to acquire theoptimal total cost Mode 2 has greater cost advantages thanMode 1

5 Conclusions

In this paper we considered the common frequency routingproblem with vehicle type choice in milk runs in logisticalsystems We developed a mathematical model to describe thevehicle type choice frequency planning and vehicle routeplanning in the milk run system To solve this model wedeveloped a two-phase TS algorithm with limited searchscope (TP-TSLSS) that increased the search efficiency Theproposed TP-TSLSS algorithm was tested on 55 numerical

Mathematical Problems in Engineering 11

0

20000

40000

60000

80000

100000

120000

10 20 30 50 70 90 110 150 200 250 300

Total cost

Vehicle type 1

Vehicle type 3Vehicle type 2

Vehicle type 1

Vehicle type 3Vehicle type 2

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 20 30 50 70 90 110 150 200 250 300

Transportation cost

Multiple vehicle types

Multiple vehicle types

Multiple vehicle types

Vehicle type 1

Vehicle type 3Vehicle type 2

Multiple vehicle typesVehicle type 1

Vehicle type 3Vehicle type 2

0

2000

4000

6000

8000

10000

12000

14000

10 20 30 50 70 90 110 150 200 250 300

Inventory cost

0

5000

10000

15000

20000

25000

30000

10 20 30 50 70 90 110 150 200 250 300

Dispatch cost

Figure 8 A comparison of the mode results

examples with varying scales for verification and was com-pared with the TS and the SA methods The results showedthat our TP-TSLSS obtained better solutions in a shortertime and a more stable manner By comparing differenttransportationmodes we concluded that consideration of thevehicle type choice could help save on cost of transportationinventory and dispatch

This was the first study to comprehensively considerfrequency planning vehicle type choice and routing plan-ning Although the consideration of a variety of vehicle

types significantly increased the complexity of the originalCFR problem the TP-TSLSS algorithm could reduce searchtime and improve accuracy through limiting search scopeIn the future we intended to extend this model to casesinvolving multiple manufacturers The problem will be morecomplex and the decision might depend on additionalfactors Moreover multiple depots could be considered froma practical perspective Finally we planned to develop evenmore efficient and accurate algorithms to solve these prob-lems

12 Mathematical Problems in Engineering

Table 3 119901 value of 119905-test

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

1 10 mdash mdash2 10 mdash mdash3 10 mdash mdash4 10 mdash mdash5 10 mdash mdash6 20 000031 0167 20 00000028 0188 20 0000026 0219 20 00043 01210 20 0000030 02811 30 00051 0008512 30 00032 002713 30 00017 001014 30 00072 0003615 30 0015 0002916 50 00000022 0000002517 50 00000063 00004118 50 0012 0006119 50 00000027 0000003520 50 00000012 0000001821 70 000023 000001022 70 000014 0000002923 70 0000092 0000001224 70 0000011 00000006325 70 000047 000002126 90 0000016 000004727 90 00000024 00000001828 90 00000051 00000001029 90 0000030 000002830 90 0000029 000001131 110 0000089 000006232 110 00000033 00000001033 110 000032 00002934 110 0000092 000005935 110 0000075 0000003336 150 00000013 0000001437 150 00000014 0000002038 150 0000067 000005139 150 000018 000005340 150 000051 000004741 200 0000069 00000001542 200 000000098 00000008343 200 0000074 0000001944 200 00000023 00000001245 200 0000082 0000003846 250 000000080 000000003747 250 000000015 0000000013

Table 3 Continued

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

48 250 000000011 0000000009849 250 00000064 00000001350 250 0000094 00000002751 300 00000011 00000006052 300 000000081 00000005653 300 000000042 00000001754 300 00000098 00000008255 300 0000020 0000000041

Conflict of Interests

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

References

[1] K H Chuah and J C Yingling ldquoRouting for a just-in-timesupply pickup and delivery systemrdquo Transportation Science vol39 no 3 pp 328ndash339 2005

[2] L Zhuang ldquoTowards a more economic production and just-in-time delivery systemrdquo International Journal of ProductionEconomics vol 36 no 3 pp 307ndash313 1994

[3] M Savsar and A Al-Jawini ldquoSimulation analysis of just-in-time production systemsrdquo International Journal of ProductionEconomics vol 42 no 1 pp 67ndash78 1995

[4] R R Lummus ldquoA simulation analysis of sequencing alternativesfor JIT lines using kanbansrdquo Journal of OperationsManagementvol 13 no 3 pp 183ndash191 1995

[5] S J Sadjadi M Jafari and T Amini ldquoA new mathematicalmodeling and a genetic algorithm search for milk run problem(an auto industry supply chain case study)rdquo The InternationalJournal of Advanced Manufacturing Technology vol 44 no 1-2pp 194ndash200 2009

[6] M Jafari-Eskandari S J Sadjadi M S Jabalameli and ABozorgi-Amiri ldquoA robust optimization approach for the MilkRun problem (An auto industry supply chain case study)rdquo inProceedings of the IEEE International Conference on Computersamp Industrial Engineering (CIE rsquo09) pp 1076ndash1081 TroyesFrance July 2009

[7] M J Eskandari A R Aliahmadi and G H H Khaleghi ldquoArobust optimisation approach for the milk run problem withtime windows with inventory uncertainty an auto industrysupply chain case studyrdquo International Journal of Rapid Man-ufacturing vol 1 no 3 pp 334ndash347 2010

[8] T Du F KWang and P-Y Lu ldquoA real-time vehicle-dispatchingsystem for consolidating milk runsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 43 no 5 pp565ndash577 2007

[9] J Ma and G Sun ldquoMutation Ant colony algorithm of milk-run vehicle routing problemwith fastest completion time basedon dynamic optimizationrdquo Discrete Dynamics in Nature andSociety vol 2013 Article ID 418436 6 pages 2013

[10] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

The first phase algorithm start

Randomly generate the initial solution

Randomly generate the new

If it satisfies the time constraint (9)

Yes

No

If it satisfies the time constraint (9)

No

Yes

Yes No

Yes

No

The second phase algorithm start

solution of the first phase

solutions that meet the timeconstraint (9) in the

function value is the smallest among the candidate solutions

If the selected solution

If the selectedsolution

Select the suboptimal

Accept the new solution and replace the current solution

Accept the new solution and replace the current solution

Update the tabulist

Stop

Yes

No

YesNo

No

Yes

Yes

No

(x0 0)

neighborhood of N(xnow now)

Calculate z2(x1 1)

If z2(x1 1) gt z2(xnow now)

Accept (x1 1) set Reject (x1 1)

it = it + 1

it le M

(xnow now)

set zmax2 = z2(xnow now)

neighborhood of N(xnow now)

If the z2 value of eachcandidate solution gt 120572 middot z

max2

Select the solution (xoptopt) whose objective

(xopt opt) is better than (xbest best)

(xopt opt) is

candidate solution as (xoptopt)

n = n + 1

If n lt NI

in tabulist

Randomly generate h candidate

Set xnow = x0 now = 0 Calculatez2(xnow now) it = 1

solution (x1 1) in the

(xnow now) = (x1 1)

Output (x9984000 9984000) =

Tabulist H = Φ set the

9984000 xnow = x

9984000 and

algorithm xbest = x9984000 best =

now = 9984000 n = 0

Figure 2 The TP-TSLSS process

the least Therefore the two-phase algorithm (TP-TSLSS) can obtain better and more stable solutions ina relatively short time

To verify the effectiveness of the two-phase heuristicalgorithm we select an optimal solution for TP-TSLSS froman example involving 30 suppliers Figure 7 shows the pathdiagram

In addition we take the 119905-test of statistical analysis tocompare difference in objective function values of the three

algorithms 119879 test employs 119905 distribution theory to infer theoccurring probability of differences and to examine whetherthe differences are significant in two averages [33 34] Thispaper takes 119905-test to compare the differences in averageobjective function values of the three algorithms First werun 10 times for all 55 examples We employ the SPSS 18 totake normality test for the objective function values of the 55examples We find that except the objective function valuesof the five numerical examples with the scale of 10 suppliersthe rest of the numerical examplesrsquo objective function values

8 Mathematical Problems in Engineering

Table 1 Results obtained for the first-phase algorithm

Instance SN 119911(1199091015840 V1015840) 119911(1199090 V0) 1199112(1199091015840 V1015840) 119911

2(1199090 V0) 1199051015840 (s)

1 10 18391 30698 2 01667 0322 10 21128 2582 24444 02308 0313 10 23407 38168 23333 025 0344 10 21757 26998 30435 05455 0295 10 2490 29351 07059 02444 036 20 3158 57114 48182 02535 0347 20 50645 65045 132381 03488 0338 20 3600 58948 68667 01227 0319 20 43496 57095 103529 03793 0310 20 32278 52732 88571 01333 03211 30 42922 73623 149 02202 03612 30 68309 76401 71695 02818 03513 30 6574 72963 55 00896 03614 30 64024 65789 380147 023 03715 30 48815 92522 245 04107 03616 50 12523 14449 397647 0244 03617 50 11190 14649 185238 03557 03518 50 11945 14672 1297 03254 03719 50 11919 12149 91 025 03820 50 10930 17825 211429 02551 03621 70 11737 17540 479091 02878 03422 70 15508 18462 545625 01973 03923 70 12484 20227 1003333 02653 03824 70 15890 20555 29 03419 03725 70 14998 19819 172857 02903 03526 90 18475 20596 450556 03529 03827 90 19643 20328 472356 03427 0428 90 18792 19873 46729 02957 03929 90 16268 20921 8746 03268 03830 90 20846 28943 3667 03832 04131 110 25081 46116 2068 04035 08332 110 30848 45420 1857 04689 07833 110 27515 42895 1939 05073 07934 110 28371 37652 185 04971 07335 110 23627 39712 3046 04837 07836 150 38053 59964 180213 02545 09237 150 36892 56279 3263 02384 08938 150 39604 62874 20481 01953 09339 150 40285 58164 16306 02491 09240 150 37681 63692 2617 01902 0941 200 86334 111050 55 02611 210242 200 88274 113130 35806 02194 208743 200 85145 113670 41758 02874 207444 200 89732 112823 528 02391 212345 200 87692 135713 4825 0261 208446 250 111760 145270 39955 02601 402947 250 111830 146220 4316 02206 418148 250 118713 149254 38421 02059 401749 250 116915 153873 3617 01973 398250 250 120317 148148 3291 02391 405951 300 131980 169670 44194 02328 597252 300 129468 163200 4841 0304 629253 300 139642 161726 4163 02845 581754 300 130613 159212 40157 02174 618355 300 129862 163944 49153 0316 6084

0 50 100 150 200 250 300 350 40006

08

1

12

14

16

18

2

TSTP-TSLSS

times104

Figure 3 Convergence graph of 70 suppliers

0 100 200 300 400 500 60015

2

25

3

35

4

45

TSTP-TSLSS

times104

Figure 4 Convergence graph of 110 suppliers

obtained by all the three algorithms have a good normalityThen we take two hypothesis tests at 005 confidence levelconditions The first hypothesis test is designed for SA andTP-TSLSS with the null hypothesis 119867 120583

1le 1205830 The second

hypothesis test is designed for TS and TP-TSLSS with thenull hypothesis 119867 120583

2le 1205830 Finally we take independent

sample 119905-test for the two hypotheses respectively The 119901

values obtained are listed in Table 3For the comparison of SA and TP-TSLSS all 119901 values of

the examples with 20 or more suppliers are less than 005therefore the null hypotheses of these examples are rejecteddemonstrating that TP-TSLSS is better than SA

For the comparison of TS and TP-TSLSS apart from 119901

values of examples of 20 suppliers scale (TS and TP-TSLSS)being greater than 005 119901 values of the remaining examplesare less than 005 therefore the null hypotheses of these

Mathematical Problems in Engineering 9

Table 2 Results obtained for the three algorithms (SA TS and TP-TSLSS)

Instance SN SA TS TP-TSLSSAverage 119905SA (s) 120590SA Average 119905TS (s) 120590TS Average 119905TP-TSLSS (s) 120590TP-TSLSS

1 10 16191 1654 8731 13862 3247 0 13862 1463 02 10 15743 1526 7631 1388 3258 0 1388 1419 03 10 1491 1594 4020 14219 3282 0 14219 1346 04 10 15825 1631 7218 14778 3354 0 14778 1322 05 10 16049 1682 7194 14875 3196 4031 14382 1407 06 20 25689 1727 8256 23333 4664 4107 23549 1686 32187 20 26021 1714 841 26698 4561 562 25649 1811 35918 20 24928 1793 8646 25106 4628 479 24217 1765 3459 20 26183 1737 8729 27656 4619 601 25714 1879 331410 20 23105 1749 8439 22758 4594 504 22423 1814 36211 30 34219 2015 1183 35278 5492 7660 33173 1825 427112 30 36798 2125 11038 34214 5135 7135 33082 1819 41613 30 34192 2062 10949 35073 5284 7058 33019 1746 408414 30 35193 2048 11164 36465 5175 7563 3396 1822 402315 30 38276 2116 12645 42145 5428 7937 37361 1707 42516 50 6425 2485 41072 59709 6514 12047 54051 2116 555717 50 60158 2472 38264 61101 6628 11062 5998 2392 583518 50 57028 2493 34183 57413 6604 8459 56276 2216 519419 50 56026 2419 33963 58708 6592 9246 52185 2145 574920 50 60924 2562 32974 63686 6585 8794 5686 2281 538521 70 85106 4057 37016 90797 7849 18584 76153 2136 1170322 70 82179 3947 38294 8820 7632 17294 78465 2073 1075223 70 82462 3929 41174 84594 7940 16848 73865 2491 1047324 70 88171 4019 38265 89905 7835 17182 81702 2615 1014925 70 81096 4095 34793 82468 7529 1749 80516 2491 1135226 90 10492 6418 40327 10737 10673 27232 92756 4215 1772827 90 11600 6429 41046 12446 10428 23472 90794 3881 1713628 90 108433 6316 40684 136475 10582 25819 89615 3974 1738529 90 116923 6285 40958 114367 10418 24174 92359 4039 1748230 90 102395 6193 40715 108691 10837 22964 95725 4139 1698331 110 20487 8972 77967 211168 13248 54139 180161 5797 3382932 110 187321 8017 78924 204292 13082 52764 117423 5826 3103433 110 190355 8526 83592 198163 13569 51074 185354 5494 3409334 110 190241 8438 93164 192525 13194 50986 179394 5618 358535 110 193482 8525 76492 216241 13074 51196 173106 5769 3792936 150 30813 18038 13484 27982 35922 97346 24094 15006 8353537 150 29813 18157 12467 30583 35542 10048 24695 14678 8214638 150 30847 18262 13295 31764 35686 98683 28174 14903 7824839 150 29139 18037 14829 30269 35742 90474 27108 14286 7958340 150 28742 17984 12793 29264 35291 98134 26495 14982 8497341 200 50954 24044 32316 56340 41249 28743 48246 19151 1576342 200 58192 23859 35935 58861 41465 27631 52407 20018 1639443 200 56102 24168 38193 57928 41672 24926 52887 19427 1582644 200 57149 24027 37194 59218 41839 27428 50961 18737 1782445 200 56291 23939 36913 58316 41962 29726 51620 19472 1629746 250 67194 38054 46911 75935 63028 36877 62171 32006 2428747 250 62719 37937 48512 77501 63542 35825 59954 31078 2592648 250 61461 37846 49236 78519 63887 36014 58917 31403 2174549 250 70192 37735 43926 76110 64041 35826 63625 31786 2852850 250 64192 37342 47295 71926 63993 35714 62128 32682 2394851 300 82155 52069 73979 92167 91249 58212 78166 46351 4232652 300 84124 51053 74924 91135 89465 56149 77640 44718 3872653 300 88307 51749 71486 90114 90672 59376 75783 48227 3741954 300 83491 51283 73951 87910 91839 52864 78629 41937 4173655 300 76529 50959 75056 88412 90962 58251 73210 42672 43945

10 Mathematical Problems in Engineering

0 100 200 300 400 500 600 700 800 900 10002

25

3

35

4

45

5

55

6times10

4

TSTP-TSLSS

Figure 5 Convergence graph of 150 suppliers

0 100 200 300 400 500 600 700 800 900 1000 11004

5

6

7

8

9

10

11times10

4

TSTP-TSLSS

Figure 6 Convergence graph of 200 suppliers

examples are rejected demonstrating TP-TSLSS is better thanTS when the suppliers are more than 30

43 Results Analysis of the Different Modes General studiesof the routing problems did not consider the effect of vehicletype choice on transportation frequency and inventory costwhile we take the vehicle type choice into consideration Toexamine this effect we make a comparison of the generaltransportation mode (Mode 1) with the CFR-VTC mode(Mode 2) for different scales of instances

Mode 1 Only one vehicle type is allowed for the transporta-tion task

Mode 2 All vehicle types are available

Figure 8 shows a comparison of the results for the twomodes (Mode 1 includes three cases only vehicle type 1 onlyvehicle type 2 and only vehicle type 3 in which the vehicle

17

23

306

1127

2021

2719

1018

9

16

14

11

29

4

528

2 3

826 24

22

1513

250

00 50 100

50

100

Vehicle 3 frequency 5Vehicle 2 fre

quency 4

Vehicle 2 frequency 8

X

Y

Figure 7 The path diagram of TP-TSLSS

types are not required for decision Mode 2 signifies that allthe three vehicle types are involved in the decision)The loadcapacity of vehicle type 1 is the smallest whose value is 100and that of vehicle type 2 is moderate whose value is 300whereas the capacity of vehicle type 3 is the largest whosevalue is 600 The horizontal axis in Figure 8 represents thenumber of suppliers and the vertical axis shows the four typesof costmdashtransportation cost inventory cost dispatch costand total cost The four cost trends are shown in Figure 8

The transportation cost graph in Figure 8 shows thatthe case involving only vehicle type 3 records the lowesttransportation cost because a vehicle with a large loadcapacity requires less frequency to complete the pickup taskHowever the inventory cost and dispatch cost in this caseare the highest when the scales of instances are large becauseof low frequency and high load capacity By contrast thecase involving only vehicle type 1 has the advantage of lowinventory and dispatch costs while the transportation costin this case is the highest However the total cost of Mode2 is lower than all the three cases in Mode 1 Therefore wefind that we can balance the cost of transportation inventoryand dispatch by a reasonable choice of vehicle to acquire theoptimal total cost Mode 2 has greater cost advantages thanMode 1

5 Conclusions

In this paper we considered the common frequency routingproblem with vehicle type choice in milk runs in logisticalsystems We developed a mathematical model to describe thevehicle type choice frequency planning and vehicle routeplanning in the milk run system To solve this model wedeveloped a two-phase TS algorithm with limited searchscope (TP-TSLSS) that increased the search efficiency Theproposed TP-TSLSS algorithm was tested on 55 numerical

Mathematical Problems in Engineering 11

0

20000

40000

60000

80000

100000

120000

10 20 30 50 70 90 110 150 200 250 300

Total cost

Vehicle type 1

Vehicle type 3Vehicle type 2

Vehicle type 1

Vehicle type 3Vehicle type 2

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 20 30 50 70 90 110 150 200 250 300

Transportation cost

Multiple vehicle types

Multiple vehicle types

Multiple vehicle types

Vehicle type 1

Vehicle type 3Vehicle type 2

Multiple vehicle typesVehicle type 1

Vehicle type 3Vehicle type 2

0

2000

4000

6000

8000

10000

12000

14000

10 20 30 50 70 90 110 150 200 250 300

Inventory cost

0

5000

10000

15000

20000

25000

30000

10 20 30 50 70 90 110 150 200 250 300

Dispatch cost

Figure 8 A comparison of the mode results

examples with varying scales for verification and was com-pared with the TS and the SA methods The results showedthat our TP-TSLSS obtained better solutions in a shortertime and a more stable manner By comparing differenttransportationmodes we concluded that consideration of thevehicle type choice could help save on cost of transportationinventory and dispatch

This was the first study to comprehensively considerfrequency planning vehicle type choice and routing plan-ning Although the consideration of a variety of vehicle

types significantly increased the complexity of the originalCFR problem the TP-TSLSS algorithm could reduce searchtime and improve accuracy through limiting search scopeIn the future we intended to extend this model to casesinvolving multiple manufacturers The problem will be morecomplex and the decision might depend on additionalfactors Moreover multiple depots could be considered froma practical perspective Finally we planned to develop evenmore efficient and accurate algorithms to solve these prob-lems

12 Mathematical Problems in Engineering

Table 3 119901 value of 119905-test

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

1 10 mdash mdash2 10 mdash mdash3 10 mdash mdash4 10 mdash mdash5 10 mdash mdash6 20 000031 0167 20 00000028 0188 20 0000026 0219 20 00043 01210 20 0000030 02811 30 00051 0008512 30 00032 002713 30 00017 001014 30 00072 0003615 30 0015 0002916 50 00000022 0000002517 50 00000063 00004118 50 0012 0006119 50 00000027 0000003520 50 00000012 0000001821 70 000023 000001022 70 000014 0000002923 70 0000092 0000001224 70 0000011 00000006325 70 000047 000002126 90 0000016 000004727 90 00000024 00000001828 90 00000051 00000001029 90 0000030 000002830 90 0000029 000001131 110 0000089 000006232 110 00000033 00000001033 110 000032 00002934 110 0000092 000005935 110 0000075 0000003336 150 00000013 0000001437 150 00000014 0000002038 150 0000067 000005139 150 000018 000005340 150 000051 000004741 200 0000069 00000001542 200 000000098 00000008343 200 0000074 0000001944 200 00000023 00000001245 200 0000082 0000003846 250 000000080 000000003747 250 000000015 0000000013

Table 3 Continued

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

48 250 000000011 0000000009849 250 00000064 00000001350 250 0000094 00000002751 300 00000011 00000006052 300 000000081 00000005653 300 000000042 00000001754 300 00000098 00000008255 300 0000020 0000000041

Conflict of Interests

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

References

[1] K H Chuah and J C Yingling ldquoRouting for a just-in-timesupply pickup and delivery systemrdquo Transportation Science vol39 no 3 pp 328ndash339 2005

[2] L Zhuang ldquoTowards a more economic production and just-in-time delivery systemrdquo International Journal of ProductionEconomics vol 36 no 3 pp 307ndash313 1994

[3] M Savsar and A Al-Jawini ldquoSimulation analysis of just-in-time production systemsrdquo International Journal of ProductionEconomics vol 42 no 1 pp 67ndash78 1995

[4] R R Lummus ldquoA simulation analysis of sequencing alternativesfor JIT lines using kanbansrdquo Journal of OperationsManagementvol 13 no 3 pp 183ndash191 1995

[5] S J Sadjadi M Jafari and T Amini ldquoA new mathematicalmodeling and a genetic algorithm search for milk run problem(an auto industry supply chain case study)rdquo The InternationalJournal of Advanced Manufacturing Technology vol 44 no 1-2pp 194ndash200 2009

[6] M Jafari-Eskandari S J Sadjadi M S Jabalameli and ABozorgi-Amiri ldquoA robust optimization approach for the MilkRun problem (An auto industry supply chain case study)rdquo inProceedings of the IEEE International Conference on Computersamp Industrial Engineering (CIE rsquo09) pp 1076ndash1081 TroyesFrance July 2009

[7] M J Eskandari A R Aliahmadi and G H H Khaleghi ldquoArobust optimisation approach for the milk run problem withtime windows with inventory uncertainty an auto industrysupply chain case studyrdquo International Journal of Rapid Man-ufacturing vol 1 no 3 pp 334ndash347 2010

[8] T Du F KWang and P-Y Lu ldquoA real-time vehicle-dispatchingsystem for consolidating milk runsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 43 no 5 pp565ndash577 2007

[9] J Ma and G Sun ldquoMutation Ant colony algorithm of milk-run vehicle routing problemwith fastest completion time basedon dynamic optimizationrdquo Discrete Dynamics in Nature andSociety vol 2013 Article ID 418436 6 pages 2013

[10] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 1 Results obtained for the first-phase algorithm

Instance SN 119911(1199091015840 V1015840) 119911(1199090 V0) 1199112(1199091015840 V1015840) 119911

2(1199090 V0) 1199051015840 (s)

1 10 18391 30698 2 01667 0322 10 21128 2582 24444 02308 0313 10 23407 38168 23333 025 0344 10 21757 26998 30435 05455 0295 10 2490 29351 07059 02444 036 20 3158 57114 48182 02535 0347 20 50645 65045 132381 03488 0338 20 3600 58948 68667 01227 0319 20 43496 57095 103529 03793 0310 20 32278 52732 88571 01333 03211 30 42922 73623 149 02202 03612 30 68309 76401 71695 02818 03513 30 6574 72963 55 00896 03614 30 64024 65789 380147 023 03715 30 48815 92522 245 04107 03616 50 12523 14449 397647 0244 03617 50 11190 14649 185238 03557 03518 50 11945 14672 1297 03254 03719 50 11919 12149 91 025 03820 50 10930 17825 211429 02551 03621 70 11737 17540 479091 02878 03422 70 15508 18462 545625 01973 03923 70 12484 20227 1003333 02653 03824 70 15890 20555 29 03419 03725 70 14998 19819 172857 02903 03526 90 18475 20596 450556 03529 03827 90 19643 20328 472356 03427 0428 90 18792 19873 46729 02957 03929 90 16268 20921 8746 03268 03830 90 20846 28943 3667 03832 04131 110 25081 46116 2068 04035 08332 110 30848 45420 1857 04689 07833 110 27515 42895 1939 05073 07934 110 28371 37652 185 04971 07335 110 23627 39712 3046 04837 07836 150 38053 59964 180213 02545 09237 150 36892 56279 3263 02384 08938 150 39604 62874 20481 01953 09339 150 40285 58164 16306 02491 09240 150 37681 63692 2617 01902 0941 200 86334 111050 55 02611 210242 200 88274 113130 35806 02194 208743 200 85145 113670 41758 02874 207444 200 89732 112823 528 02391 212345 200 87692 135713 4825 0261 208446 250 111760 145270 39955 02601 402947 250 111830 146220 4316 02206 418148 250 118713 149254 38421 02059 401749 250 116915 153873 3617 01973 398250 250 120317 148148 3291 02391 405951 300 131980 169670 44194 02328 597252 300 129468 163200 4841 0304 629253 300 139642 161726 4163 02845 581754 300 130613 159212 40157 02174 618355 300 129862 163944 49153 0316 6084

0 50 100 150 200 250 300 350 40006

08

1

12

14

16

18

2

TSTP-TSLSS

times104

Figure 3 Convergence graph of 70 suppliers

0 100 200 300 400 500 60015

2

25

3

35

4

45

TSTP-TSLSS

times104

Figure 4 Convergence graph of 110 suppliers

obtained by all the three algorithms have a good normalityThen we take two hypothesis tests at 005 confidence levelconditions The first hypothesis test is designed for SA andTP-TSLSS with the null hypothesis 119867 120583

1le 1205830 The second

hypothesis test is designed for TS and TP-TSLSS with thenull hypothesis 119867 120583

2le 1205830 Finally we take independent

sample 119905-test for the two hypotheses respectively The 119901

values obtained are listed in Table 3For the comparison of SA and TP-TSLSS all 119901 values of

the examples with 20 or more suppliers are less than 005therefore the null hypotheses of these examples are rejecteddemonstrating that TP-TSLSS is better than SA

For the comparison of TS and TP-TSLSS apart from 119901

values of examples of 20 suppliers scale (TS and TP-TSLSS)being greater than 005 119901 values of the remaining examplesare less than 005 therefore the null hypotheses of these

Mathematical Problems in Engineering 9

Table 2 Results obtained for the three algorithms (SA TS and TP-TSLSS)

Instance SN SA TS TP-TSLSSAverage 119905SA (s) 120590SA Average 119905TS (s) 120590TS Average 119905TP-TSLSS (s) 120590TP-TSLSS

1 10 16191 1654 8731 13862 3247 0 13862 1463 02 10 15743 1526 7631 1388 3258 0 1388 1419 03 10 1491 1594 4020 14219 3282 0 14219 1346 04 10 15825 1631 7218 14778 3354 0 14778 1322 05 10 16049 1682 7194 14875 3196 4031 14382 1407 06 20 25689 1727 8256 23333 4664 4107 23549 1686 32187 20 26021 1714 841 26698 4561 562 25649 1811 35918 20 24928 1793 8646 25106 4628 479 24217 1765 3459 20 26183 1737 8729 27656 4619 601 25714 1879 331410 20 23105 1749 8439 22758 4594 504 22423 1814 36211 30 34219 2015 1183 35278 5492 7660 33173 1825 427112 30 36798 2125 11038 34214 5135 7135 33082 1819 41613 30 34192 2062 10949 35073 5284 7058 33019 1746 408414 30 35193 2048 11164 36465 5175 7563 3396 1822 402315 30 38276 2116 12645 42145 5428 7937 37361 1707 42516 50 6425 2485 41072 59709 6514 12047 54051 2116 555717 50 60158 2472 38264 61101 6628 11062 5998 2392 583518 50 57028 2493 34183 57413 6604 8459 56276 2216 519419 50 56026 2419 33963 58708 6592 9246 52185 2145 574920 50 60924 2562 32974 63686 6585 8794 5686 2281 538521 70 85106 4057 37016 90797 7849 18584 76153 2136 1170322 70 82179 3947 38294 8820 7632 17294 78465 2073 1075223 70 82462 3929 41174 84594 7940 16848 73865 2491 1047324 70 88171 4019 38265 89905 7835 17182 81702 2615 1014925 70 81096 4095 34793 82468 7529 1749 80516 2491 1135226 90 10492 6418 40327 10737 10673 27232 92756 4215 1772827 90 11600 6429 41046 12446 10428 23472 90794 3881 1713628 90 108433 6316 40684 136475 10582 25819 89615 3974 1738529 90 116923 6285 40958 114367 10418 24174 92359 4039 1748230 90 102395 6193 40715 108691 10837 22964 95725 4139 1698331 110 20487 8972 77967 211168 13248 54139 180161 5797 3382932 110 187321 8017 78924 204292 13082 52764 117423 5826 3103433 110 190355 8526 83592 198163 13569 51074 185354 5494 3409334 110 190241 8438 93164 192525 13194 50986 179394 5618 358535 110 193482 8525 76492 216241 13074 51196 173106 5769 3792936 150 30813 18038 13484 27982 35922 97346 24094 15006 8353537 150 29813 18157 12467 30583 35542 10048 24695 14678 8214638 150 30847 18262 13295 31764 35686 98683 28174 14903 7824839 150 29139 18037 14829 30269 35742 90474 27108 14286 7958340 150 28742 17984 12793 29264 35291 98134 26495 14982 8497341 200 50954 24044 32316 56340 41249 28743 48246 19151 1576342 200 58192 23859 35935 58861 41465 27631 52407 20018 1639443 200 56102 24168 38193 57928 41672 24926 52887 19427 1582644 200 57149 24027 37194 59218 41839 27428 50961 18737 1782445 200 56291 23939 36913 58316 41962 29726 51620 19472 1629746 250 67194 38054 46911 75935 63028 36877 62171 32006 2428747 250 62719 37937 48512 77501 63542 35825 59954 31078 2592648 250 61461 37846 49236 78519 63887 36014 58917 31403 2174549 250 70192 37735 43926 76110 64041 35826 63625 31786 2852850 250 64192 37342 47295 71926 63993 35714 62128 32682 2394851 300 82155 52069 73979 92167 91249 58212 78166 46351 4232652 300 84124 51053 74924 91135 89465 56149 77640 44718 3872653 300 88307 51749 71486 90114 90672 59376 75783 48227 3741954 300 83491 51283 73951 87910 91839 52864 78629 41937 4173655 300 76529 50959 75056 88412 90962 58251 73210 42672 43945

10 Mathematical Problems in Engineering

0 100 200 300 400 500 600 700 800 900 10002

25

3

35

4

45

5

55

6times10

4

TSTP-TSLSS

Figure 5 Convergence graph of 150 suppliers

0 100 200 300 400 500 600 700 800 900 1000 11004

5

6

7

8

9

10

11times10

4

TSTP-TSLSS

Figure 6 Convergence graph of 200 suppliers

examples are rejected demonstrating TP-TSLSS is better thanTS when the suppliers are more than 30

43 Results Analysis of the Different Modes General studiesof the routing problems did not consider the effect of vehicletype choice on transportation frequency and inventory costwhile we take the vehicle type choice into consideration Toexamine this effect we make a comparison of the generaltransportation mode (Mode 1) with the CFR-VTC mode(Mode 2) for different scales of instances

Mode 1 Only one vehicle type is allowed for the transporta-tion task

Mode 2 All vehicle types are available

Figure 8 shows a comparison of the results for the twomodes (Mode 1 includes three cases only vehicle type 1 onlyvehicle type 2 and only vehicle type 3 in which the vehicle

17

23

306

1127

2021

2719

1018

9

16

14

11

29

4

528

2 3

826 24

22

1513

250

00 50 100

50

100

Vehicle 3 frequency 5Vehicle 2 fre

quency 4

Vehicle 2 frequency 8

X

Y

Figure 7 The path diagram of TP-TSLSS

types are not required for decision Mode 2 signifies that allthe three vehicle types are involved in the decision)The loadcapacity of vehicle type 1 is the smallest whose value is 100and that of vehicle type 2 is moderate whose value is 300whereas the capacity of vehicle type 3 is the largest whosevalue is 600 The horizontal axis in Figure 8 represents thenumber of suppliers and the vertical axis shows the four typesof costmdashtransportation cost inventory cost dispatch costand total cost The four cost trends are shown in Figure 8

The transportation cost graph in Figure 8 shows thatthe case involving only vehicle type 3 records the lowesttransportation cost because a vehicle with a large loadcapacity requires less frequency to complete the pickup taskHowever the inventory cost and dispatch cost in this caseare the highest when the scales of instances are large becauseof low frequency and high load capacity By contrast thecase involving only vehicle type 1 has the advantage of lowinventory and dispatch costs while the transportation costin this case is the highest However the total cost of Mode2 is lower than all the three cases in Mode 1 Therefore wefind that we can balance the cost of transportation inventoryand dispatch by a reasonable choice of vehicle to acquire theoptimal total cost Mode 2 has greater cost advantages thanMode 1

5 Conclusions

In this paper we considered the common frequency routingproblem with vehicle type choice in milk runs in logisticalsystems We developed a mathematical model to describe thevehicle type choice frequency planning and vehicle routeplanning in the milk run system To solve this model wedeveloped a two-phase TS algorithm with limited searchscope (TP-TSLSS) that increased the search efficiency Theproposed TP-TSLSS algorithm was tested on 55 numerical

Mathematical Problems in Engineering 11

0

20000

40000

60000

80000

100000

120000

10 20 30 50 70 90 110 150 200 250 300

Total cost

Vehicle type 1

Vehicle type 3Vehicle type 2

Vehicle type 1

Vehicle type 3Vehicle type 2

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 20 30 50 70 90 110 150 200 250 300

Transportation cost

Multiple vehicle types

Multiple vehicle types

Multiple vehicle types

Vehicle type 1

Vehicle type 3Vehicle type 2

Multiple vehicle typesVehicle type 1

Vehicle type 3Vehicle type 2

0

2000

4000

6000

8000

10000

12000

14000

10 20 30 50 70 90 110 150 200 250 300

Inventory cost

0

5000

10000

15000

20000

25000

30000

10 20 30 50 70 90 110 150 200 250 300

Dispatch cost

Figure 8 A comparison of the mode results

examples with varying scales for verification and was com-pared with the TS and the SA methods The results showedthat our TP-TSLSS obtained better solutions in a shortertime and a more stable manner By comparing differenttransportationmodes we concluded that consideration of thevehicle type choice could help save on cost of transportationinventory and dispatch

This was the first study to comprehensively considerfrequency planning vehicle type choice and routing plan-ning Although the consideration of a variety of vehicle

types significantly increased the complexity of the originalCFR problem the TP-TSLSS algorithm could reduce searchtime and improve accuracy through limiting search scopeIn the future we intended to extend this model to casesinvolving multiple manufacturers The problem will be morecomplex and the decision might depend on additionalfactors Moreover multiple depots could be considered froma practical perspective Finally we planned to develop evenmore efficient and accurate algorithms to solve these prob-lems

12 Mathematical Problems in Engineering

Table 3 119901 value of 119905-test

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

1 10 mdash mdash2 10 mdash mdash3 10 mdash mdash4 10 mdash mdash5 10 mdash mdash6 20 000031 0167 20 00000028 0188 20 0000026 0219 20 00043 01210 20 0000030 02811 30 00051 0008512 30 00032 002713 30 00017 001014 30 00072 0003615 30 0015 0002916 50 00000022 0000002517 50 00000063 00004118 50 0012 0006119 50 00000027 0000003520 50 00000012 0000001821 70 000023 000001022 70 000014 0000002923 70 0000092 0000001224 70 0000011 00000006325 70 000047 000002126 90 0000016 000004727 90 00000024 00000001828 90 00000051 00000001029 90 0000030 000002830 90 0000029 000001131 110 0000089 000006232 110 00000033 00000001033 110 000032 00002934 110 0000092 000005935 110 0000075 0000003336 150 00000013 0000001437 150 00000014 0000002038 150 0000067 000005139 150 000018 000005340 150 000051 000004741 200 0000069 00000001542 200 000000098 00000008343 200 0000074 0000001944 200 00000023 00000001245 200 0000082 0000003846 250 000000080 000000003747 250 000000015 0000000013

Table 3 Continued

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

48 250 000000011 0000000009849 250 00000064 00000001350 250 0000094 00000002751 300 00000011 00000006052 300 000000081 00000005653 300 000000042 00000001754 300 00000098 00000008255 300 0000020 0000000041

Conflict of Interests

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

References

[1] K H Chuah and J C Yingling ldquoRouting for a just-in-timesupply pickup and delivery systemrdquo Transportation Science vol39 no 3 pp 328ndash339 2005

[2] L Zhuang ldquoTowards a more economic production and just-in-time delivery systemrdquo International Journal of ProductionEconomics vol 36 no 3 pp 307ndash313 1994

[3] M Savsar and A Al-Jawini ldquoSimulation analysis of just-in-time production systemsrdquo International Journal of ProductionEconomics vol 42 no 1 pp 67ndash78 1995

[4] R R Lummus ldquoA simulation analysis of sequencing alternativesfor JIT lines using kanbansrdquo Journal of OperationsManagementvol 13 no 3 pp 183ndash191 1995

[5] S J Sadjadi M Jafari and T Amini ldquoA new mathematicalmodeling and a genetic algorithm search for milk run problem(an auto industry supply chain case study)rdquo The InternationalJournal of Advanced Manufacturing Technology vol 44 no 1-2pp 194ndash200 2009

[6] M Jafari-Eskandari S J Sadjadi M S Jabalameli and ABozorgi-Amiri ldquoA robust optimization approach for the MilkRun problem (An auto industry supply chain case study)rdquo inProceedings of the IEEE International Conference on Computersamp Industrial Engineering (CIE rsquo09) pp 1076ndash1081 TroyesFrance July 2009

[7] M J Eskandari A R Aliahmadi and G H H Khaleghi ldquoArobust optimisation approach for the milk run problem withtime windows with inventory uncertainty an auto industrysupply chain case studyrdquo International Journal of Rapid Man-ufacturing vol 1 no 3 pp 334ndash347 2010

[8] T Du F KWang and P-Y Lu ldquoA real-time vehicle-dispatchingsystem for consolidating milk runsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 43 no 5 pp565ndash577 2007

[9] J Ma and G Sun ldquoMutation Ant colony algorithm of milk-run vehicle routing problemwith fastest completion time basedon dynamic optimizationrdquo Discrete Dynamics in Nature andSociety vol 2013 Article ID 418436 6 pages 2013

[10] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 2 Results obtained for the three algorithms (SA TS and TP-TSLSS)

Instance SN SA TS TP-TSLSSAverage 119905SA (s) 120590SA Average 119905TS (s) 120590TS Average 119905TP-TSLSS (s) 120590TP-TSLSS

1 10 16191 1654 8731 13862 3247 0 13862 1463 02 10 15743 1526 7631 1388 3258 0 1388 1419 03 10 1491 1594 4020 14219 3282 0 14219 1346 04 10 15825 1631 7218 14778 3354 0 14778 1322 05 10 16049 1682 7194 14875 3196 4031 14382 1407 06 20 25689 1727 8256 23333 4664 4107 23549 1686 32187 20 26021 1714 841 26698 4561 562 25649 1811 35918 20 24928 1793 8646 25106 4628 479 24217 1765 3459 20 26183 1737 8729 27656 4619 601 25714 1879 331410 20 23105 1749 8439 22758 4594 504 22423 1814 36211 30 34219 2015 1183 35278 5492 7660 33173 1825 427112 30 36798 2125 11038 34214 5135 7135 33082 1819 41613 30 34192 2062 10949 35073 5284 7058 33019 1746 408414 30 35193 2048 11164 36465 5175 7563 3396 1822 402315 30 38276 2116 12645 42145 5428 7937 37361 1707 42516 50 6425 2485 41072 59709 6514 12047 54051 2116 555717 50 60158 2472 38264 61101 6628 11062 5998 2392 583518 50 57028 2493 34183 57413 6604 8459 56276 2216 519419 50 56026 2419 33963 58708 6592 9246 52185 2145 574920 50 60924 2562 32974 63686 6585 8794 5686 2281 538521 70 85106 4057 37016 90797 7849 18584 76153 2136 1170322 70 82179 3947 38294 8820 7632 17294 78465 2073 1075223 70 82462 3929 41174 84594 7940 16848 73865 2491 1047324 70 88171 4019 38265 89905 7835 17182 81702 2615 1014925 70 81096 4095 34793 82468 7529 1749 80516 2491 1135226 90 10492 6418 40327 10737 10673 27232 92756 4215 1772827 90 11600 6429 41046 12446 10428 23472 90794 3881 1713628 90 108433 6316 40684 136475 10582 25819 89615 3974 1738529 90 116923 6285 40958 114367 10418 24174 92359 4039 1748230 90 102395 6193 40715 108691 10837 22964 95725 4139 1698331 110 20487 8972 77967 211168 13248 54139 180161 5797 3382932 110 187321 8017 78924 204292 13082 52764 117423 5826 3103433 110 190355 8526 83592 198163 13569 51074 185354 5494 3409334 110 190241 8438 93164 192525 13194 50986 179394 5618 358535 110 193482 8525 76492 216241 13074 51196 173106 5769 3792936 150 30813 18038 13484 27982 35922 97346 24094 15006 8353537 150 29813 18157 12467 30583 35542 10048 24695 14678 8214638 150 30847 18262 13295 31764 35686 98683 28174 14903 7824839 150 29139 18037 14829 30269 35742 90474 27108 14286 7958340 150 28742 17984 12793 29264 35291 98134 26495 14982 8497341 200 50954 24044 32316 56340 41249 28743 48246 19151 1576342 200 58192 23859 35935 58861 41465 27631 52407 20018 1639443 200 56102 24168 38193 57928 41672 24926 52887 19427 1582644 200 57149 24027 37194 59218 41839 27428 50961 18737 1782445 200 56291 23939 36913 58316 41962 29726 51620 19472 1629746 250 67194 38054 46911 75935 63028 36877 62171 32006 2428747 250 62719 37937 48512 77501 63542 35825 59954 31078 2592648 250 61461 37846 49236 78519 63887 36014 58917 31403 2174549 250 70192 37735 43926 76110 64041 35826 63625 31786 2852850 250 64192 37342 47295 71926 63993 35714 62128 32682 2394851 300 82155 52069 73979 92167 91249 58212 78166 46351 4232652 300 84124 51053 74924 91135 89465 56149 77640 44718 3872653 300 88307 51749 71486 90114 90672 59376 75783 48227 3741954 300 83491 51283 73951 87910 91839 52864 78629 41937 4173655 300 76529 50959 75056 88412 90962 58251 73210 42672 43945

10 Mathematical Problems in Engineering

0 100 200 300 400 500 600 700 800 900 10002

25

3

35

4

45

5

55

6times10

4

TSTP-TSLSS

Figure 5 Convergence graph of 150 suppliers

0 100 200 300 400 500 600 700 800 900 1000 11004

5

6

7

8

9

10

11times10

4

TSTP-TSLSS

Figure 6 Convergence graph of 200 suppliers

examples are rejected demonstrating TP-TSLSS is better thanTS when the suppliers are more than 30

43 Results Analysis of the Different Modes General studiesof the routing problems did not consider the effect of vehicletype choice on transportation frequency and inventory costwhile we take the vehicle type choice into consideration Toexamine this effect we make a comparison of the generaltransportation mode (Mode 1) with the CFR-VTC mode(Mode 2) for different scales of instances

Mode 1 Only one vehicle type is allowed for the transporta-tion task

Mode 2 All vehicle types are available

Figure 8 shows a comparison of the results for the twomodes (Mode 1 includes three cases only vehicle type 1 onlyvehicle type 2 and only vehicle type 3 in which the vehicle

17

23

306

1127

2021

2719

1018

9

16

14

11

29

4

528

2 3

826 24

22

1513

250

00 50 100

50

100

Vehicle 3 frequency 5Vehicle 2 fre

quency 4

Vehicle 2 frequency 8

X

Y

Figure 7 The path diagram of TP-TSLSS

types are not required for decision Mode 2 signifies that allthe three vehicle types are involved in the decision)The loadcapacity of vehicle type 1 is the smallest whose value is 100and that of vehicle type 2 is moderate whose value is 300whereas the capacity of vehicle type 3 is the largest whosevalue is 600 The horizontal axis in Figure 8 represents thenumber of suppliers and the vertical axis shows the four typesof costmdashtransportation cost inventory cost dispatch costand total cost The four cost trends are shown in Figure 8

The transportation cost graph in Figure 8 shows thatthe case involving only vehicle type 3 records the lowesttransportation cost because a vehicle with a large loadcapacity requires less frequency to complete the pickup taskHowever the inventory cost and dispatch cost in this caseare the highest when the scales of instances are large becauseof low frequency and high load capacity By contrast thecase involving only vehicle type 1 has the advantage of lowinventory and dispatch costs while the transportation costin this case is the highest However the total cost of Mode2 is lower than all the three cases in Mode 1 Therefore wefind that we can balance the cost of transportation inventoryand dispatch by a reasonable choice of vehicle to acquire theoptimal total cost Mode 2 has greater cost advantages thanMode 1

5 Conclusions

In this paper we considered the common frequency routingproblem with vehicle type choice in milk runs in logisticalsystems We developed a mathematical model to describe thevehicle type choice frequency planning and vehicle routeplanning in the milk run system To solve this model wedeveloped a two-phase TS algorithm with limited searchscope (TP-TSLSS) that increased the search efficiency Theproposed TP-TSLSS algorithm was tested on 55 numerical

Mathematical Problems in Engineering 11

0

20000

40000

60000

80000

100000

120000

10 20 30 50 70 90 110 150 200 250 300

Total cost

Vehicle type 1

Vehicle type 3Vehicle type 2

Vehicle type 1

Vehicle type 3Vehicle type 2

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 20 30 50 70 90 110 150 200 250 300

Transportation cost

Multiple vehicle types

Multiple vehicle types

Multiple vehicle types

Vehicle type 1

Vehicle type 3Vehicle type 2

Multiple vehicle typesVehicle type 1

Vehicle type 3Vehicle type 2

0

2000

4000

6000

8000

10000

12000

14000

10 20 30 50 70 90 110 150 200 250 300

Inventory cost

0

5000

10000

15000

20000

25000

30000

10 20 30 50 70 90 110 150 200 250 300

Dispatch cost

Figure 8 A comparison of the mode results

examples with varying scales for verification and was com-pared with the TS and the SA methods The results showedthat our TP-TSLSS obtained better solutions in a shortertime and a more stable manner By comparing differenttransportationmodes we concluded that consideration of thevehicle type choice could help save on cost of transportationinventory and dispatch

This was the first study to comprehensively considerfrequency planning vehicle type choice and routing plan-ning Although the consideration of a variety of vehicle

types significantly increased the complexity of the originalCFR problem the TP-TSLSS algorithm could reduce searchtime and improve accuracy through limiting search scopeIn the future we intended to extend this model to casesinvolving multiple manufacturers The problem will be morecomplex and the decision might depend on additionalfactors Moreover multiple depots could be considered froma practical perspective Finally we planned to develop evenmore efficient and accurate algorithms to solve these prob-lems

12 Mathematical Problems in Engineering

Table 3 119901 value of 119905-test

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

1 10 mdash mdash2 10 mdash mdash3 10 mdash mdash4 10 mdash mdash5 10 mdash mdash6 20 000031 0167 20 00000028 0188 20 0000026 0219 20 00043 01210 20 0000030 02811 30 00051 0008512 30 00032 002713 30 00017 001014 30 00072 0003615 30 0015 0002916 50 00000022 0000002517 50 00000063 00004118 50 0012 0006119 50 00000027 0000003520 50 00000012 0000001821 70 000023 000001022 70 000014 0000002923 70 0000092 0000001224 70 0000011 00000006325 70 000047 000002126 90 0000016 000004727 90 00000024 00000001828 90 00000051 00000001029 90 0000030 000002830 90 0000029 000001131 110 0000089 000006232 110 00000033 00000001033 110 000032 00002934 110 0000092 000005935 110 0000075 0000003336 150 00000013 0000001437 150 00000014 0000002038 150 0000067 000005139 150 000018 000005340 150 000051 000004741 200 0000069 00000001542 200 000000098 00000008343 200 0000074 0000001944 200 00000023 00000001245 200 0000082 0000003846 250 000000080 000000003747 250 000000015 0000000013

Table 3 Continued

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

48 250 000000011 0000000009849 250 00000064 00000001350 250 0000094 00000002751 300 00000011 00000006052 300 000000081 00000005653 300 000000042 00000001754 300 00000098 00000008255 300 0000020 0000000041

Conflict of Interests

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

References

[1] K H Chuah and J C Yingling ldquoRouting for a just-in-timesupply pickup and delivery systemrdquo Transportation Science vol39 no 3 pp 328ndash339 2005

[2] L Zhuang ldquoTowards a more economic production and just-in-time delivery systemrdquo International Journal of ProductionEconomics vol 36 no 3 pp 307ndash313 1994

[3] M Savsar and A Al-Jawini ldquoSimulation analysis of just-in-time production systemsrdquo International Journal of ProductionEconomics vol 42 no 1 pp 67ndash78 1995

[4] R R Lummus ldquoA simulation analysis of sequencing alternativesfor JIT lines using kanbansrdquo Journal of OperationsManagementvol 13 no 3 pp 183ndash191 1995

[5] S J Sadjadi M Jafari and T Amini ldquoA new mathematicalmodeling and a genetic algorithm search for milk run problem(an auto industry supply chain case study)rdquo The InternationalJournal of Advanced Manufacturing Technology vol 44 no 1-2pp 194ndash200 2009

[6] M Jafari-Eskandari S J Sadjadi M S Jabalameli and ABozorgi-Amiri ldquoA robust optimization approach for the MilkRun problem (An auto industry supply chain case study)rdquo inProceedings of the IEEE International Conference on Computersamp Industrial Engineering (CIE rsquo09) pp 1076ndash1081 TroyesFrance July 2009

[7] M J Eskandari A R Aliahmadi and G H H Khaleghi ldquoArobust optimisation approach for the milk run problem withtime windows with inventory uncertainty an auto industrysupply chain case studyrdquo International Journal of Rapid Man-ufacturing vol 1 no 3 pp 334ndash347 2010

[8] T Du F KWang and P-Y Lu ldquoA real-time vehicle-dispatchingsystem for consolidating milk runsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 43 no 5 pp565ndash577 2007

[9] J Ma and G Sun ldquoMutation Ant colony algorithm of milk-run vehicle routing problemwith fastest completion time basedon dynamic optimizationrdquo Discrete Dynamics in Nature andSociety vol 2013 Article ID 418436 6 pages 2013

[10] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 100 200 300 400 500 600 700 800 900 10002

25

3

35

4

45

5

55

6times10

4

TSTP-TSLSS

Figure 5 Convergence graph of 150 suppliers

0 100 200 300 400 500 600 700 800 900 1000 11004

5

6

7

8

9

10

11times10

4

TSTP-TSLSS

Figure 6 Convergence graph of 200 suppliers

examples are rejected demonstrating TP-TSLSS is better thanTS when the suppliers are more than 30

43 Results Analysis of the Different Modes General studiesof the routing problems did not consider the effect of vehicletype choice on transportation frequency and inventory costwhile we take the vehicle type choice into consideration Toexamine this effect we make a comparison of the generaltransportation mode (Mode 1) with the CFR-VTC mode(Mode 2) for different scales of instances

Mode 1 Only one vehicle type is allowed for the transporta-tion task

Mode 2 All vehicle types are available

Figure 8 shows a comparison of the results for the twomodes (Mode 1 includes three cases only vehicle type 1 onlyvehicle type 2 and only vehicle type 3 in which the vehicle

17

23

306

1127

2021

2719

1018

9

16

14

11

29

4

528

2 3

826 24

22

1513

250

00 50 100

50

100

Vehicle 3 frequency 5Vehicle 2 fre

quency 4

Vehicle 2 frequency 8

X

Y

Figure 7 The path diagram of TP-TSLSS

types are not required for decision Mode 2 signifies that allthe three vehicle types are involved in the decision)The loadcapacity of vehicle type 1 is the smallest whose value is 100and that of vehicle type 2 is moderate whose value is 300whereas the capacity of vehicle type 3 is the largest whosevalue is 600 The horizontal axis in Figure 8 represents thenumber of suppliers and the vertical axis shows the four typesof costmdashtransportation cost inventory cost dispatch costand total cost The four cost trends are shown in Figure 8

The transportation cost graph in Figure 8 shows thatthe case involving only vehicle type 3 records the lowesttransportation cost because a vehicle with a large loadcapacity requires less frequency to complete the pickup taskHowever the inventory cost and dispatch cost in this caseare the highest when the scales of instances are large becauseof low frequency and high load capacity By contrast thecase involving only vehicle type 1 has the advantage of lowinventory and dispatch costs while the transportation costin this case is the highest However the total cost of Mode2 is lower than all the three cases in Mode 1 Therefore wefind that we can balance the cost of transportation inventoryand dispatch by a reasonable choice of vehicle to acquire theoptimal total cost Mode 2 has greater cost advantages thanMode 1

5 Conclusions

In this paper we considered the common frequency routingproblem with vehicle type choice in milk runs in logisticalsystems We developed a mathematical model to describe thevehicle type choice frequency planning and vehicle routeplanning in the milk run system To solve this model wedeveloped a two-phase TS algorithm with limited searchscope (TP-TSLSS) that increased the search efficiency Theproposed TP-TSLSS algorithm was tested on 55 numerical

Mathematical Problems in Engineering 11

0

20000

40000

60000

80000

100000

120000

10 20 30 50 70 90 110 150 200 250 300

Total cost

Vehicle type 1

Vehicle type 3Vehicle type 2

Vehicle type 1

Vehicle type 3Vehicle type 2

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 20 30 50 70 90 110 150 200 250 300

Transportation cost

Multiple vehicle types

Multiple vehicle types

Multiple vehicle types

Vehicle type 1

Vehicle type 3Vehicle type 2

Multiple vehicle typesVehicle type 1

Vehicle type 3Vehicle type 2

0

2000

4000

6000

8000

10000

12000

14000

10 20 30 50 70 90 110 150 200 250 300

Inventory cost

0

5000

10000

15000

20000

25000

30000

10 20 30 50 70 90 110 150 200 250 300

Dispatch cost

Figure 8 A comparison of the mode results

examples with varying scales for verification and was com-pared with the TS and the SA methods The results showedthat our TP-TSLSS obtained better solutions in a shortertime and a more stable manner By comparing differenttransportationmodes we concluded that consideration of thevehicle type choice could help save on cost of transportationinventory and dispatch

This was the first study to comprehensively considerfrequency planning vehicle type choice and routing plan-ning Although the consideration of a variety of vehicle

types significantly increased the complexity of the originalCFR problem the TP-TSLSS algorithm could reduce searchtime and improve accuracy through limiting search scopeIn the future we intended to extend this model to casesinvolving multiple manufacturers The problem will be morecomplex and the decision might depend on additionalfactors Moreover multiple depots could be considered froma practical perspective Finally we planned to develop evenmore efficient and accurate algorithms to solve these prob-lems

12 Mathematical Problems in Engineering

Table 3 119901 value of 119905-test

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

1 10 mdash mdash2 10 mdash mdash3 10 mdash mdash4 10 mdash mdash5 10 mdash mdash6 20 000031 0167 20 00000028 0188 20 0000026 0219 20 00043 01210 20 0000030 02811 30 00051 0008512 30 00032 002713 30 00017 001014 30 00072 0003615 30 0015 0002916 50 00000022 0000002517 50 00000063 00004118 50 0012 0006119 50 00000027 0000003520 50 00000012 0000001821 70 000023 000001022 70 000014 0000002923 70 0000092 0000001224 70 0000011 00000006325 70 000047 000002126 90 0000016 000004727 90 00000024 00000001828 90 00000051 00000001029 90 0000030 000002830 90 0000029 000001131 110 0000089 000006232 110 00000033 00000001033 110 000032 00002934 110 0000092 000005935 110 0000075 0000003336 150 00000013 0000001437 150 00000014 0000002038 150 0000067 000005139 150 000018 000005340 150 000051 000004741 200 0000069 00000001542 200 000000098 00000008343 200 0000074 0000001944 200 00000023 00000001245 200 0000082 0000003846 250 000000080 000000003747 250 000000015 0000000013

Table 3 Continued

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

48 250 000000011 0000000009849 250 00000064 00000001350 250 0000094 00000002751 300 00000011 00000006052 300 000000081 00000005653 300 000000042 00000001754 300 00000098 00000008255 300 0000020 0000000041

Conflict of Interests

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

References

[1] K H Chuah and J C Yingling ldquoRouting for a just-in-timesupply pickup and delivery systemrdquo Transportation Science vol39 no 3 pp 328ndash339 2005

[2] L Zhuang ldquoTowards a more economic production and just-in-time delivery systemrdquo International Journal of ProductionEconomics vol 36 no 3 pp 307ndash313 1994

[3] M Savsar and A Al-Jawini ldquoSimulation analysis of just-in-time production systemsrdquo International Journal of ProductionEconomics vol 42 no 1 pp 67ndash78 1995

[4] R R Lummus ldquoA simulation analysis of sequencing alternativesfor JIT lines using kanbansrdquo Journal of OperationsManagementvol 13 no 3 pp 183ndash191 1995

[5] S J Sadjadi M Jafari and T Amini ldquoA new mathematicalmodeling and a genetic algorithm search for milk run problem(an auto industry supply chain case study)rdquo The InternationalJournal of Advanced Manufacturing Technology vol 44 no 1-2pp 194ndash200 2009

[6] M Jafari-Eskandari S J Sadjadi M S Jabalameli and ABozorgi-Amiri ldquoA robust optimization approach for the MilkRun problem (An auto industry supply chain case study)rdquo inProceedings of the IEEE International Conference on Computersamp Industrial Engineering (CIE rsquo09) pp 1076ndash1081 TroyesFrance July 2009

[7] M J Eskandari A R Aliahmadi and G H H Khaleghi ldquoArobust optimisation approach for the milk run problem withtime windows with inventory uncertainty an auto industrysupply chain case studyrdquo International Journal of Rapid Man-ufacturing vol 1 no 3 pp 334ndash347 2010

[8] T Du F KWang and P-Y Lu ldquoA real-time vehicle-dispatchingsystem for consolidating milk runsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 43 no 5 pp565ndash577 2007

[9] J Ma and G Sun ldquoMutation Ant colony algorithm of milk-run vehicle routing problemwith fastest completion time basedon dynamic optimizationrdquo Discrete Dynamics in Nature andSociety vol 2013 Article ID 418436 6 pages 2013

[10] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0

20000

40000

60000

80000

100000

120000

10 20 30 50 70 90 110 150 200 250 300

Total cost

Vehicle type 1

Vehicle type 3Vehicle type 2

Vehicle type 1

Vehicle type 3Vehicle type 2

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 20 30 50 70 90 110 150 200 250 300

Transportation cost

Multiple vehicle types

Multiple vehicle types

Multiple vehicle types

Vehicle type 1

Vehicle type 3Vehicle type 2

Multiple vehicle typesVehicle type 1

Vehicle type 3Vehicle type 2

0

2000

4000

6000

8000

10000

12000

14000

10 20 30 50 70 90 110 150 200 250 300

Inventory cost

0

5000

10000

15000

20000

25000

30000

10 20 30 50 70 90 110 150 200 250 300

Dispatch cost

Figure 8 A comparison of the mode results

examples with varying scales for verification and was com-pared with the TS and the SA methods The results showedthat our TP-TSLSS obtained better solutions in a shortertime and a more stable manner By comparing differenttransportationmodes we concluded that consideration of thevehicle type choice could help save on cost of transportationinventory and dispatch

This was the first study to comprehensively considerfrequency planning vehicle type choice and routing plan-ning Although the consideration of a variety of vehicle

types significantly increased the complexity of the originalCFR problem the TP-TSLSS algorithm could reduce searchtime and improve accuracy through limiting search scopeIn the future we intended to extend this model to casesinvolving multiple manufacturers The problem will be morecomplex and the decision might depend on additionalfactors Moreover multiple depots could be considered froma practical perspective Finally we planned to develop evenmore efficient and accurate algorithms to solve these prob-lems

12 Mathematical Problems in Engineering

Table 3 119901 value of 119905-test

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

1 10 mdash mdash2 10 mdash mdash3 10 mdash mdash4 10 mdash mdash5 10 mdash mdash6 20 000031 0167 20 00000028 0188 20 0000026 0219 20 00043 01210 20 0000030 02811 30 00051 0008512 30 00032 002713 30 00017 001014 30 00072 0003615 30 0015 0002916 50 00000022 0000002517 50 00000063 00004118 50 0012 0006119 50 00000027 0000003520 50 00000012 0000001821 70 000023 000001022 70 000014 0000002923 70 0000092 0000001224 70 0000011 00000006325 70 000047 000002126 90 0000016 000004727 90 00000024 00000001828 90 00000051 00000001029 90 0000030 000002830 90 0000029 000001131 110 0000089 000006232 110 00000033 00000001033 110 000032 00002934 110 0000092 000005935 110 0000075 0000003336 150 00000013 0000001437 150 00000014 0000002038 150 0000067 000005139 150 000018 000005340 150 000051 000004741 200 0000069 00000001542 200 000000098 00000008343 200 0000074 0000001944 200 00000023 00000001245 200 0000082 0000003846 250 000000080 000000003747 250 000000015 0000000013

Table 3 Continued

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

48 250 000000011 0000000009849 250 00000064 00000001350 250 0000094 00000002751 300 00000011 00000006052 300 000000081 00000005653 300 000000042 00000001754 300 00000098 00000008255 300 0000020 0000000041

Conflict of Interests

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

References

[1] K H Chuah and J C Yingling ldquoRouting for a just-in-timesupply pickup and delivery systemrdquo Transportation Science vol39 no 3 pp 328ndash339 2005

[2] L Zhuang ldquoTowards a more economic production and just-in-time delivery systemrdquo International Journal of ProductionEconomics vol 36 no 3 pp 307ndash313 1994

[3] M Savsar and A Al-Jawini ldquoSimulation analysis of just-in-time production systemsrdquo International Journal of ProductionEconomics vol 42 no 1 pp 67ndash78 1995

[4] R R Lummus ldquoA simulation analysis of sequencing alternativesfor JIT lines using kanbansrdquo Journal of OperationsManagementvol 13 no 3 pp 183ndash191 1995

[5] S J Sadjadi M Jafari and T Amini ldquoA new mathematicalmodeling and a genetic algorithm search for milk run problem(an auto industry supply chain case study)rdquo The InternationalJournal of Advanced Manufacturing Technology vol 44 no 1-2pp 194ndash200 2009

[6] M Jafari-Eskandari S J Sadjadi M S Jabalameli and ABozorgi-Amiri ldquoA robust optimization approach for the MilkRun problem (An auto industry supply chain case study)rdquo inProceedings of the IEEE International Conference on Computersamp Industrial Engineering (CIE rsquo09) pp 1076ndash1081 TroyesFrance July 2009

[7] M J Eskandari A R Aliahmadi and G H H Khaleghi ldquoArobust optimisation approach for the milk run problem withtime windows with inventory uncertainty an auto industrysupply chain case studyrdquo International Journal of Rapid Man-ufacturing vol 1 no 3 pp 334ndash347 2010

[8] T Du F KWang and P-Y Lu ldquoA real-time vehicle-dispatchingsystem for consolidating milk runsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 43 no 5 pp565ndash577 2007

[9] J Ma and G Sun ldquoMutation Ant colony algorithm of milk-run vehicle routing problemwith fastest completion time basedon dynamic optimizationrdquo Discrete Dynamics in Nature andSociety vol 2013 Article ID 418436 6 pages 2013

[10] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Table 3 119901 value of 119905-test

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

1 10 mdash mdash2 10 mdash mdash3 10 mdash mdash4 10 mdash mdash5 10 mdash mdash6 20 000031 0167 20 00000028 0188 20 0000026 0219 20 00043 01210 20 0000030 02811 30 00051 0008512 30 00032 002713 30 00017 001014 30 00072 0003615 30 0015 0002916 50 00000022 0000002517 50 00000063 00004118 50 0012 0006119 50 00000027 0000003520 50 00000012 0000001821 70 000023 000001022 70 000014 0000002923 70 0000092 0000001224 70 0000011 00000006325 70 000047 000002126 90 0000016 000004727 90 00000024 00000001828 90 00000051 00000001029 90 0000030 000002830 90 0000029 000001131 110 0000089 000006232 110 00000033 00000001033 110 000032 00002934 110 0000092 000005935 110 0000075 0000003336 150 00000013 0000001437 150 00000014 0000002038 150 0000067 000005139 150 000018 000005340 150 000051 000004741 200 0000069 00000001542 200 000000098 00000008343 200 0000074 0000001944 200 00000023 00000001245 200 0000082 0000003846 250 000000080 000000003747 250 000000015 0000000013

Table 3 Continued

Instance SN 119901 value(SA and TP-TSLSS)

119901 value(TS and TP-TSLSS)

48 250 000000011 0000000009849 250 00000064 00000001350 250 0000094 00000002751 300 00000011 00000006052 300 000000081 00000005653 300 000000042 00000001754 300 00000098 00000008255 300 0000020 0000000041

Conflict of Interests

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

References

[1] K H Chuah and J C Yingling ldquoRouting for a just-in-timesupply pickup and delivery systemrdquo Transportation Science vol39 no 3 pp 328ndash339 2005

[2] L Zhuang ldquoTowards a more economic production and just-in-time delivery systemrdquo International Journal of ProductionEconomics vol 36 no 3 pp 307ndash313 1994

[3] M Savsar and A Al-Jawini ldquoSimulation analysis of just-in-time production systemsrdquo International Journal of ProductionEconomics vol 42 no 1 pp 67ndash78 1995

[4] R R Lummus ldquoA simulation analysis of sequencing alternativesfor JIT lines using kanbansrdquo Journal of OperationsManagementvol 13 no 3 pp 183ndash191 1995

[5] S J Sadjadi M Jafari and T Amini ldquoA new mathematicalmodeling and a genetic algorithm search for milk run problem(an auto industry supply chain case study)rdquo The InternationalJournal of Advanced Manufacturing Technology vol 44 no 1-2pp 194ndash200 2009

[6] M Jafari-Eskandari S J Sadjadi M S Jabalameli and ABozorgi-Amiri ldquoA robust optimization approach for the MilkRun problem (An auto industry supply chain case study)rdquo inProceedings of the IEEE International Conference on Computersamp Industrial Engineering (CIE rsquo09) pp 1076ndash1081 TroyesFrance July 2009

[7] M J Eskandari A R Aliahmadi and G H H Khaleghi ldquoArobust optimisation approach for the milk run problem withtime windows with inventory uncertainty an auto industrysupply chain case studyrdquo International Journal of Rapid Man-ufacturing vol 1 no 3 pp 334ndash347 2010

[8] T Du F KWang and P-Y Lu ldquoA real-time vehicle-dispatchingsystem for consolidating milk runsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 43 no 5 pp565ndash577 2007

[9] J Ma and G Sun ldquoMutation Ant colony algorithm of milk-run vehicle routing problemwith fastest completion time basedon dynamic optimizationrdquo Discrete Dynamics in Nature andSociety vol 2013 Article ID 418436 6 pages 2013

[10] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[11] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics (AI rsquo03) pp 97ndash102 Innsbruck Austria February2003

[12] B Kallehauge ldquoFormulations and exact algorithms for thevehicle routing problem with time windowsrdquo Computers andOperations Research vol 35 no 7 pp 2307ndash2330 2008

[13] A LeBouthillier andTGCrainic ldquoA cooperative parallelmeta-heuristic for the vehicle routing problem with time windowsrdquoComputers andOperations Research vol 32 no 7 pp 1685ndash17082005

[14] G B Alvarenga G R Mateus and G de Tomi ldquoA geneticand set partitioning two-phase approach for the vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 34 no 6 pp 1561ndash1584 2007

[15] T W Chien A Balakrishnan and R T Wong ldquoAn integratedinventory allocation and vehicle routing problemrdquo Transporta-tion Science vol 23 no 2 pp 67ndash76 1989

[16] KHChuahOptimal common frequency routing for just-in-timesupply delivery systems [MS thesis] University of KentuckyLexington Ky USA 2000

[17] K H Chuah and J C Yingling ldquoAnalyzing inventorytrans-portation cost tradeoffs for milkrun parts delivery systems tolarge JIT assembly plantsrdquo SAE Technical Paper 2001-01-2600SAE International 2001

[18] K H ChuahOptimization and simulation of just-in-time supplypickup and delivery systems [A Dissertation for the Degree ofDoctor of Philosophy] University of Kentucky Lexington KyUSA 2004

[19] Y Chan W B Carter and M D Burnes ldquoA multiple-depotmultiple-vehicle location-routing problem with stochasticallyprocessed demandsrdquo Computers and Operations Research vol28 no 8 pp 803ndash826 2001

[20] V Gintner N Kliewer and L Suhl ldquoSolving large multiple-depot multiple-vehicle-type bus scheduling problems in prac-ticerdquo OR Spectrum vol 27 no 4 pp 507ndash523 2005

[21] J L Jr Blanton and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquo inProceedings of the 5th International Conference on Genetic Algo-rithms pp 452ndash459 Morgan Kaufmann Urbana-ChampaignIll USA June 1993

[22] K Ahn and H Rakha ldquoThe effects of route choice decisionson vehicle energy consumption and emissionsrdquo TransportationResearch Part D Transport and Environment vol 13 no 3 pp151ndash167 2008

[23] R Cavalcante andM J Roorda ldquoAdisaggregate urban shipmentsizevehicle-type choice modelrdquo in Proceedings of the 89thAnnual Meeting of the Transportation Research Board 10-3878Washington DC USA January 2010

[24] Q L Burrell ldquoThe 8020 rule library lore or statistical lawrdquoJournal of Documentation vol 41 no 1 pp 24ndash39 1985

[25] A Ultsch Proof of Paretorsquos 8020 Law and Precise Limitsfor ABC-Analysis Data Bionics Research Group University ofMarburg Lahn Germany 2002

[26] M Hardy ldquoParetorsquos lawrdquoTheMathematical Intelligencer vol 32no 3 pp 38ndash43 2010

[27] I H Osman ldquoMetastrategy simulated annealing and tabusearch algorithms for the vehicle routing problemrdquo Annals ofOperations Research vol 41 no 4 pp 421ndash451 1993

[28] 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

[29] F Alonso M J Alvarez and J E Beasley ldquoA tabu searchalgorithm for the periodic vehicle routing problem with mul-tiple vehicle trips and accessibility restrictionsrdquo Journal of theOperational Research Society vol 59 no 7 pp 963ndash976 2008

[30] G Laporte ldquoThe vehicle routing problem an overview of exactand approximate algorithmsrdquo European Journal of OperationalResearch vol 59 no 3 pp 345ndash358 1992

[31] R Tavakkoli-Moghaddam N Safaei and Y Gholipour ldquoAhybrid simulated annealing for capacitated vehicle routingproblems with the independent route lengthrdquo Applied Mathe-matics and Computation vol 176 no 2 pp 445ndash454 2006

[32] S D Hosseini M Akbarpour Shirazi and B Karimi ldquoCross-docking and milk run logistics in a consolidation network ahybrid of harmony search and simulated annealing approachrdquoJournal of Manufacturing Systems vol 33 no 4 pp 567ndash5772014

[33] H Hotelling ldquoA generalized T test and measure of multivariatedispersionrdquo in Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability Held at the StatisticalLaboratory Department of Mathematics University of Califor-nia July 31ndashAugust 12 1950 pp 23ndash41 University of CaliforniaPress Berkeley Calif USA 1951

[34] D C Montgomery Statistical Quality Control John Wiley ampSons 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of