Expert Systems with Applications 39 (2012) 4233–4239

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Expert Systems with Applications

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A template-based Tabu Search algorithm for the Consistent Vehicle Routing Problem

C.D. Tarantilis ⇑, F. Stavropoulou, P.P. RepoussisCenter for Operations Research & Decision Systems, Management Science Laboratory, Department of Management Science & Technology,Athens University of Economics & Business, Athens, Greece

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

Keywords:Vehicle routingTabu SearchDistribution logistics

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⇑ Corresponding author. Address: Patision 74 st., G+30 2108203805; fax: +30 2108828078.

E-mail addresses: tarantil@aueb.gr (C.D.(F. Stavropoulou), prepousi@aueb.gr (P.P. Repoussis).

This paper presents a generic template-based solution framework and its application to the so-calledConsistent Vehicle Routing Problem (ConVRP). The ConVRP is an NP-hard combinatorial optimizationproblem and involves the design of a set of minimum cost vehicle routes to service a set of customerswith known demands over multiple days. Customers may receive service either once or with a predefinedfrequency; however frequent customers must receive consistent service, i.e., must be visited by the samedriver over approximately the same time throughout the planning period. The proposed solution frame-work adopts a two-level master–slave decomposition scheme. Initially, a master template route scheduleis constructed in an effort to determine the service sequence and assignment of frequent customers tovehicles. On return, the master template is used as the basis to design the actual vehicle routes and ser-vice schedules for both frequent and non-frequent customers over multiple days. To this end, a TabuSearch improvement method is employed that operates on a dual mode basis and modifies both the tem-plate routes and the actual daily schedules in a sequential fashion. Computational experiments on bench-mark data sets illustrate the competitiveness of the proposed approach compared to existing results.

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1. Introduction Vehicle Routing Problem (CVRP) (Tarantilis, 2005) and involves the

The design and implementation of periodic delivery systemshas become a crucial concern for modern companies seeking toprovide high quality and low cost services to their customers. Evi-dently, optimizing repetitive delivery operations over multipledays can add up to significant cost savings, and thus improve pro-ductivity and competitiveness. Periodic deliveries occur in a widerange of real life applications, including among others refuse andmunicipal waste collection, mail collection and delivery, scheduledretail and wholesale delivery and distribution, vending machinereplenishment, and elevator repair and maintenance (Francis,Smilowitz, & Tzur, 2008). Therefore, in practical terms studyingsuch operational planning problems definitely seems worthwhile,apart from the theoretical and computational research challengesarising due to their combinatorial nature.

In broad terms, multi-period vehicle routing and schedulingproblems deal with the optimum assignment and service sequenceof a set of customer orders to a fleet of vehicles over multiple days(the term ‘‘day’’ is used as a general unit of time throughout thispaper). In the literature, these problems are typically modeled asPeriodic Vehicle Routing Problems (PVRP) (Beltrami & Bodin,1974). The PVRP is a generalization of the well-known Capacitated

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design of a set of vehicle routes, over a predefined planning hori-zon, in order to service a set of customers with known demandand frequencies of service (i.e. customers can be visited accordingto different day combinations). Typically, the objective is to mini-mize the total traveling cost, expressed in terms of one-time (e.g.fleet size) and recurring costs (e.g. distance traveled), while satisfy-ing operational constraints (e.g. vehicle capacity and visit require-ments). Note that besides the daily routing decisions (i.e.assignment and service sequence of each customer on vehicleroutes), a schedule from a candidate set of schedules for each cus-tomer must be also selected.

Contrary to the above described periodic delivery operationalsetting, in recent years more and more companies tend to focusand invest on brand loyalty and customer relationship manage-ment, and thereafter, they are interested in implementing cus-tomer-oriented rather than demand-oriented approaches. Forexample, there are numerous real life applications in which cus-tomers need to be visited by the same service provider (i.e. vehiclecrew and driver). Furthermore, in many cases customers need to beserviced according to a predefined visiting sequence or a certainservice time consistency (e.g. a minimum variance of service timesover multiple days). Typical real life paradigms that depict thesetype of consistent service considerations can be found in parceldeliveries and courier services, home care and nursing servicesfor the elderly and cleaning services. In these cases, the main effortis to gain competitive advantage by forming bonds with thecustomers.

4234 C.D. Tarantilis et al. / Expert Systems with Applications 39 (2012) 4233–4239

A first attempt in the literature towards modeling and solvingmulti-period vehicle routing and scheduling problems with consis-tent customer service constraints has been put forward by Groër,Golden, and Wasil (2009). In specific, they introduced the so-calledConsistent Vehicle Routing Problem (ConVRP), motivated by a real-life parcel delivery application. To this end, a multi-start solutionconstruction framework is proposed, combined with a Record-to-Record travel local search metaheuristic algorithm (Li, Golden, &Wasil, 2005) and savings-based construction heuristics.

The ConVRP is an NP-hard combinatorial optimization problemin the strong sense and involves the design of minimum cost routesin order to service a set of customers with known demand overmultiple days via a homogeneous fleet of depot returning capaci-tated vehicles. Customers may receive service either once (nonfrequent customers) or with a predefined frequency (frequentcustomers); however frequent customers must receive consistentservice throughout the planning period, such that (a) their visitingsequence remains the same (or similarly the maximum servicetime difference between the earliest and latest vehicle arrivaltimes over multiple days does not exceed a maximum time limit)and (b) the service is performed by the same vehicle. On the otherhand, during each day each customer must be visited only once byexactly one vehicle, while each vehicle has a maximum carryingcapacity and operates for no more than a maximum time limit.The goal is to minimize the total distance traveled by the vehiclessuch that all customer requirements are fulfilled without violatingcapacity, route duration and consistent service constraints.

Towards this new and emerging line of research, the main con-tribution and aim of this paper is to design and develop a genericand flexible template-based solution framework for the ConVRP.Following the concept of template route schedules, the proposedapproach adopts a two-level decomposition scheme and solves amaster and a slave sub-problem in a sequential fashion. In partic-ular, the master sub-problem is concerned with the design of atemplate route schedule in order to determine the assignment offrequent customers to vehicles and their visiting sequences, whilethe corresponding slave sub-problem seeks to determine the actualdaily vehicle routes for both frequent and non-frequent customerson the basis of the template route schedule. From the implementa-tion viewpoint, effective savings and insertion based constructionheuristics are utilized, coupled with a Tabu Search (TS) algorithmthat operates on a dual mode basis. Given an initial template routeschedule, TS is applied in an effort to minimize the total distancetraveled, considering only frequent customers (master mode). Onreturn, the corresponding daily schedules for both frequent andnon-frequent customers are constructed and further improved byTS (slave mode). In this case, TS is applied to the actual daily sched-ules and takes into account, apart from vehicle capacities and routeduration restrictions, the precedence constraints (i.e. assignmentand visiting sequence requirements) dictated by the correspondingtemplate route schedule.

For the evaluation of the proposed template-based solution ap-proach, computational experiments on benchmark data sets of theliterature are reported. Compared to existing results, it proved tobe highly competitive and improved the best reported cumulativeand mean results over all problem instances with very reasonablecomputational requirements for practical applications. To this end,competitive advantage of proposed solution framework is its fairlysimple algorithmic structure, its flexibility to accommodate vari-ous types of consistent service constraints and the small numberof parameters introduced.

The remainder of this paper is organized as follows. Section 3presents the proposed solution framework and provides detaileddescriptions of all algorithmic components and mechanisms. Com-putational experiments assessing the quality of the proposed ap-proach along with a comparative performance analysis are

presented in Section 4. Finally, in Section 5 conclusions are drawnand pointers for future research are provided.

2. Problem description & notation

The ConVRP can be defined on a complete directed graph G =(N,A), where N = {0,1, . . . ,n} is the node set and A = {(i, j) :i, j 2 N, i – j} is the arc set. Node 0 represents the origin and desti-nation depot and each node of Nc = Nn{0} corresponds to a cus-tomer (n denotes the total number of customers). Each arc(i, j) 2 A is linked to a travel cost cij, while the travel time (or equiv-alently travel distance) matrix M = (cij) is symmetric, i.e., cij = cji.Additionally, let K be the set of vehicles. Each vehicle k 2 K has aknown capacity of Q units and operates for at most B time unitsduring each day d 2 D, where D denotes the days of servicerequirements.

The service days and the corresponding demands of each cus-tomer are known in advance. In particular, each customer i 2 Nc

is associated with a non-negative demand qid and a non-negativeservice duration sid for each day d 2 D. Furthermore, binary indica-tors wid show if a customer i requires service on day d, while pre-defined service frequency fi is linked to every customer. As such,the customer set Nc can be divided into two disjoint subsets, i.e.,the set of frequent Nf and the set of non-frequent customers Nnf

that require service only once during the planning horizon.The solution DS of a ConVRP instance can be seen as a collection

of jDj daily schedules dsd (one for every d 2 D). Each dsd consists ofa set of vehicle routes. Given the actual service requirements andspecifications for each customer as described above, the goal isto find a set of least cost daily schedules such that:

– The total traveling time (or alternatively the total traveling dis-tance) incurred by the vehicles over multiple days, includingservice times at customers’ locations, is minimized.

– The accumulated service of a vehicle route (total quantity car-ried) is less than or equal to the maximum carrying capacity Q.

– The total duration time of each individual vehicle route does notexceed a predefined upper limit B (e.g. depot operating hours).

– Each customer i 2 Nc is visited only once by exactly one vehicleon each day d that service can take place.

– Frequent customers i 2 Nf need to be visited by the same vehiclek 2 K throughout the planning horizon.

– The visiting sequence of frequent customers i 2 Nf must remainthe same for each d 2 D and/or the maximum service time dif-ference between the earliest and latest vehicle arrival timesover multiple days does not exceed a maximum time limit L.

Following the ConVRP definition provided by Groër et al. (2009),we set all vehicle departure times from the depot to zero. Clearly, ifthis restriction is relaxed, time consistency violations can beavoided to some extent by delaying the vehicles’ departure times.

3. Template-based solution framework

3.1. Motivation and basic concept

Periodic delivery problems can be seen as series of vehicle routingproblems, in which vehicle routes must be constructed over a planningperiod of D days. Considering the ConVRP, the service days and the cor-responding demands of each customer are known in advance. How-ever, consistent customer service constraints impose that frequentcustomers must be visited by the same vehicle at approximatelythe same time during the days they require service throughout theplanning horizon. Therefore, contrary to traditional PVRP instances,the ConVRP cannot be decomposed into jDj separate CVRP problems

C.D. Tarantilis et al. / Expert Systems with Applications 39 (2012) 4233–4239 4235

(assuming that a service schedule is selected for each customer),since the daily vehicle routing plans are interrelated.

In an effort to deal and achieve the desirable consistency overmultiple days, the rational of template route schedules, introducedby Groër et al. (2009), is utilized. A template route contains onlyfrequent customers and it can be used as a master plan to deter-mine their visiting sequence in the actual daily vehicle routingschedules. To work around this concept, the proposed solution ap-proach adopts a two-level master–slave decomposition scheme.The master sub-problem is concerned with the design of a tem-plate route schedule in order to determine the service sequenceand assignment of frequent customers to vehicle routes. On theother hand, the slave sub-problem uses the template as the basisto design the actual daily vehicle routes and service schedules forboth frequent and non-frequent customers over multiple days.

On the basis of the above, the ConVRP can be decomposed into amaster and a slave sub-problem that needs to be solved in asequential fashion. For this purpose, effective construction heuris-tics are utilized, while an innovative TS improvement method isalso employed that operates on a dual mode basis and modifiesboth the template routes and the daily schedules. At first, an initialfeasible template route schedule T is generated via a generalizedinsertion-based merging construction heuristic, considering all fre-quent customers simultaneously. To this end, the master TS mode(mTS) is applied to minimize the total traveling time. Although alllocal moves considered at the master mode modify directly thecurrent T, the neighborhood evaluation and the associated feasibil-ity checks are based on the resulting daily schedules. Upon termi-nation of the template improvement level, the correspondingpartially constructed daily vehicle routing plans are populatedwith the remaining set of non-frequent customers. Given the final-ized set of daily schedules DS, the slave TS mode (sTS) is triggeredfor further improvement (post optimization). In an effort to pre-serve the feasibility, precedence constraints are also considered,in terms of customer assignments to vehicles and visiting sequencerestrictions, as dictated by the corresponding T.

Below, the algorithmic structure of the proposed template-based solution framework is provided.

Template-based solution framework

//Parameters: b, zm, zs, vm, vs//DS 0;// Generate an empty set of daily schedulesNf Build set ();//Frequent customersNnf Build set ();//Non frequent customers//Master LevelT 0;// Generate an empty template route scheduleT Merging Heuristic (Nf,b);// Template InitializationT mTS (T,zm,vm);// Template Improvement//Slave LevelWhile day < D do {

dsday 0;// Generate an empty daily scheduledsday Initialization (T,day,Nf);// Adaptation of Tdsday Insertion Heuristic (day,Nnf);// Completion of dsday

dsday sTS (dsday,zs,vs);// Post OptimizationDS Add (dsday);

}Return DS;

3.2. Template initialization

The goal during the template initialization phase is to generateinitial template schedules considering the whole set of frequentcustomers, regardless their required service frequency. For this

purpose, a generalized insertion-based merging construction heu-ristic is utilized. At first, a solution is generated considering thatone vehicle is assigned to each customer. Next, at each iterationtwo routes are selected and merged, according to a greedy function,until no further improvement (i.e. positive saving) can be obtained.

On the basis of the above, six different merging combinationscan be obtained. Let two routes r0 = {0,w, . . . ,i, j, . . . ,z,0} andr00 = {0,u, . . . , l,0}, where indexes i, j, w, z, u, l denote frequent cus-tomers. The first combination is to insert r00 after r0, i.e.,r̂ ¼ f0;w; . . . ; i; j; . . . ; z;u; . . . ; l;0g, with a corresponding savingc0u + cz0 � czu. The second is to insert r00 in front of r0, i.e.,r̂ ¼ f0;u; . . . ; l;w; . . . ; i; j; . . . ; z;0g, with a saving c0w + cl0 � clw. Thethird is to insert r00 within all positions between consecutive cus-tomers i, j, i.e., r̂ ¼ f0;w; . . . ; i;u; . . . ; l; j; . . . ; z;0g. The resulting sav-ing formula can be defined as cij + c0u + cl0 � ciu � clj. In a mannersimilar, three additional merging combinations emerge by revers-ing the visiting sequence of route r0.

Due to route duration constraints, one may benefit if a sequen-tial manner of construction, similar to traditional insertion-basedconstruction methods, is followed. In fact, if during early stagesof construction merging is performed mostly between routes withmany customers, the final output tends to be poor, especially interms of the number of vehicles deployed. To this end, an effortis made to favor merging combinations between single customerroutes and routes containing more than two customers, by multi-plying the corresponding saving with a parameter b that will reg-ulate the degree of sequential construction.

During the construction of the template route schedule, onlyfeasible merging combinations are considered with respect to theresulting daily schedules. Clearly, template routes contain a mixof frequent customers with different combinations of daily servicerequirements. However, it is straightforward to check the feasibil-ity of the resulting daily schedules. Let aid denote the arrival time ofthe vehicle at the location of customer i on day d. The template T isfeasible, in terms of vehicle capacity and route duration con-straints, if the following inequalities hold throughout the planninghorizon:

Pi2rqidwid 6 Q8d 2 D; r 2 T (vehicle capacity); aid + wid(-

sid + ci0) 6 Bwid "d 2 D, i 2 Nf (route duration); andjaida � aidb

j 6 L� Bðwida þwidb� 2Þ 8i 2 Nf ; da; db 2 D; a – b (ser-

vice time consistency).

3.3. Generation of daily schedules

Given the master template route schedule, the correspondingdaily vehicle routing schedules can be determined in a sequentialfashion for each day of the planning horizon as follows. Initially,the template route schedule is adopted and the frequent customersthat do not require service on that particular day are removed. Bydoing so, the assignment to vehicles and the visiting sequence offrequent customers are preserved. Subsequently, the non-frequentcustomers that may require service are identified and designatedas unrouted customers. To this end, a parallel construction heuris-tic is employed to insert the remaining unrouted customers and tofinalize the partially built daily schedules.

The proposed parallel construction heuristic builds iteratively asolution by selecting and adding one by one unrouted customers tothe partially built vehicle routes, until a feasible daily schedule isgenerated for both frequent and non-frequent customers. At eachiteration, the selection of customers and insertion positions be-tween adjacent routed customers is made according to a greedycriterion that measures the insertion cost. From the implementa-tion viewpoint, only feasible insertion positions are considered.However, if no feasible insertion positions can be found for theremaining unrouted customers, new vehicle routes are initialized.

An illustrative example of the dependence between the masterand the slave subproblems is the following. Let a planning horizon

4236 C.D. Tarantilis et al. / Expert Systems with Applications 39 (2012) 4233–4239

of three days d1, d2 and d3, and two disjoint sets of frequent cus-tomers Nf = {w,h, i, j, l,s,u} and non frequent customers Nnf = {y,e,x}.Assume that the master template route schedule consists of tworoutes, e.g., tr1 = {0,w, l, i,s,0} and tr2 = {0,h, j,u,0}. During the firstday, the following set of customers requires service, e.g., {h, i, j, -l,u,y,e}. Therefore, the resulting partial daily schedule for d1 con-sists of the following partially constructed vehicle routes, i.e.,dr1 = {0,h, j,u,0} and dr2 = {0, l, i,0}. On this basis, the final step isto find the least cost insertion positions for non-frequent custom-ers y and e. To this end, the resulting complete daily schedule of d1

may take the following form, e.g., dr1 = {0,h, j,u,0} anddr2 = {0, l,y,e, i,0}.

3.4. Tabu Search improvement method

The template and the corresponding daily vehicle routingschedules are improved, in terms of distance minimization, bymeans of a dual-mode TS algorithm. The master TS mode (mTS)is applied to the template route schedule and considers explicitlythe feasibility of the master vehicle routing plan with respect tothe resulting daily schedules, while the slave TS mode (sTS) is ap-plied to the actual daily vehicle routing schedules and takes intoaccount the assignment and visiting sequence requirements dic-tated by the associated template route schedule. In both cases,the solution space is explored on the basis of simple edge-ex-change neighborhood structures. However the degrees of freedomof mTS are much higher compared to sTS, since the feasible regionof allowable move operators of the latter is greatly reduced.

In broad terms, TS seeks to explore the solution space by mov-ing at each iteration from a solution s to the best admissible solu-tion s0 in a subset Uy(s) of a neighborhood structure y. The shortterm memory records the most recently visited solutions and pre-vents revisiting them for a predefined number of iterations v (tabutenure) in order to avoid cycling. The tabu status of a neighboringsolution can be overridden, only if predefined aspiration criteriaare met (i.e. new local optimum solutions). The overall procedureiterates until some termination conditions are met (maximumnumber of iterations z without observing any further improvementin our case) and the best encountered local optimum solution s⁄ isreturned.

The neighborhood structures considered in the proposed imple-mentation are based on traditional O(n2) edge-exchange localmoves, namely intra- and inter-route 2-Opt, 2-Opt⁄, 1–0 Relocateand 1–1 Exchange (swap) (Tarantilis, Kiranoudis, & Vassiliadis,2002). The oscillation among these multiple neighborhood struc-tures is purely stochastic with equal selection probability. At eachiteration, given the allowed set of neighbors Uy(s), the best admis-sible neighbor s0 replaces the current solution s, while both the for-ward and reversal attributes (i.e. edges being added or deleted) ofthe corresponding local moves are stored within the tabu list. Forthe evaluation of the neighborhood structures, a lexicographicordering search is followed and acceleration mechanisms sug-gested by Repoussis, Tarantilis, and Ioannou (2009) that reducethe evaluation complexity are also incorporated. To this end,emphasis is given on direct feasibility gains based on the charac-teristics of the proposed decomposition scheme.

Regarding the slave TS mode, one may benefit from the inherentprecedence constraints, since the possible removal and re-inser-tion positions of frequent customers are very few and only non-fre-quent customers can freely change positions. Let two templateroutes, e.g., tr1 = {0,w, l, i,s,0} and tr2 = {0,h, j,u,0}, and the corre-sponding daily vehicle routing schedule, e.g., dr1 = {0,h, j,u,0} anddr2 = {0, l,y,e, i,0} (see also Section 3.3). Non-frequent customers yand e do not follow any precedence constraints and all feasible in-tra and inter-route local moves are allowed. On the other hand, fre-quent customers follow the assignment and precedence ordering

restrictions dictated by the template route schedule. For example,customer i can be relocated at positions strictly after frequent cus-tomer l on the same route, while non feasible intra-route localmove exist for frequent customers h, j and u.

Clearly, the feasibility checks of a local move on the daily sched-ules, in terms of route duration, vehicle capacity and service timeconsistency, can be made in almost constant time. On the otherhand, considering the mTS mode the computational effort of feasi-bility checks slightly increases, because a local move on the tem-plate route schedule affects the feasibility of the resulting dailyschedules. Thus, one has to measure explicitly the effect of localmoves for each day of the planning horizon (see also Section 3.2).

4. Computational experiments

4.1. Benchmark data set

For the evaluation of the proposed solution approach, severalcomputational experiments are performed using the benchmarkdata set generated for the ConVRP by Groër et al. (2009). This dataset consists of 12 medium-scale problem instances, divided intotwo groups. The first group contains 7 problem instances, whichconsider only vehicle capacity constraints. The second group ofproblem instances includes both vehicle capacity and route dura-tion constraints. The number of customers ranges from 50 to199, while a 5-day planning horizon with a non fixed fleet size isassumed for all problem instances. Finally, it is worth to mentionthat the objective function values refer to the sum of all the timesspent along each vehicle route (total en route time), including dis-tance traveled and service times at customers’ locations.

4.2. Parameter settings and termination conditions

Towards the design of approaches with the least possible num-ber of user-defined parameters, the proposed solution frameworkincorporates three parameters, namely the degree of sequentialconstruction b, the tabu list size v, and the number of maximumnumber of TS iterations without observing any further improve-ment z (termination condition). Based on computational experi-ence, these parameters are relatively insensitive to thecharacteristics of the problem instances considered, while usingsimple adjustments one can determine very well performingparameter settings with modest effort within reasonable valueranges. In this paper, several intuitively selected combinationswere experimentally tested, choosing the one that yielded the bestaverage output. Below, suitable value ranges for each parameterare provided.

Regarding the generalized insertion-based merging construc-tion heuristic, parameter b needs to be determined. Clearly, b mustbe set greater than or equal to 1 in order to boost the degree ofsequential construction. However, large values of b may affectthe impact of the other types of savings. To this end, a value rangebetween 1 and 1.5 was found to perform well for most ConVRPproblem instances, especially in terms of total number of vehiclesemployed. In what follows, b was fixed to 1.3 at all simulation runs.

Two parameters must be defined considering the TS improve-ment method for each mode; the tabu list size v and the numberof maximum number of TS iterations z. Between the master andthe slave TS modes, the most labor intensive and critical to theoverall performance is the former. As such, regarding the tabu listsize vm for mTS, a range between 30 and 60 is typically used bystandard TS implementations of the literature and seems to fit wellfor ConVRP problem instances. Contrary, a rather smaller range be-tween 10 and 20 suits for vs, due to the very limited search capacityof sTS. On the other hand, as z increases the efficiency of local

C.D. Tarantilis et al. / Expert Systems with Applications 39 (2012) 4233–4239 4237

search also increases. However, a balance between efficiency andeffectiveness is needed, since large values of z may result in exces-sive computational time consumption. To this end, values of zm andzs up to 20,000 and 500 iterations, respectively, seem to provide agood compromise (see also Section 4.4 for an effectiveness analysisw.r.t. zm).

The proposed template-based Tabu Search algorithm wasimplemented in C++ (Visual Studio 2008). All computational exper-iments reported in subsequent sections were performed on a2.8 GHz Intel Xeon personal computer, assuming 5 simulation runsfor each problem instance with fixed parameter settings as fol-lows: b = 1.3, zm = 20,000, vm = 56, zs = 500 and vs = 12.

4.3. Comparative analysis with existing results

Computational results for ConVRP instances are ranked accord-ing to the sum of total traveling times along each vehicle route,including service times at customers’ locations. Additionally, a pos-sible secondary measure of solution quality is the total number ofvehicles. However, these two objectives can be either conflicting orcomplementary. For this reason, all comparisons with existing re-sults are made only towards the total traveling times.

Given the definition of consistent service constraints regardingfrequent customers, there are two alternatives; either to ensure thesame visiting sequence throughout the planning horizon or to set amaximum time limit L w.r.t. the maximum service time differencebetween the earliest and latest vehicle arrival times lmax (i.e.lmax 6 L). Thereafter, in order for the competition between differentsolution approaches for ConVRP instances to be fair and objective,both types of consistent service constraints are imposed either sep-arately or simultaneously. In the former case, only visiting se-quence restrictions are imposed, while in the latter case, besidesvisiting sequence restrictions for frequent customers, the maxi-mum arrival time differential lmax reported by Groër et al. (2009)is set as the maximum time limit L and it is enforced as an addi-tional constraint.

Table 1 summarizes the results obtained on the benchmark datasets of Groër et al. (2009). The first line lists the authors using thefollowing abbreviations: ConRTR stands for (Groër et al., 2009) andTTS stands for the proposed template-based Tabu Search algo-rithm. Initially, the first set of columns illustrate the total travelingtime (TT), the total number of vehicles (NV) and the maximum ar-rival time differential lmax reported by (Groër et al., 2009) for eachproblem instance. The second set of columns (TTS-L) reports thebest results obtained considering that L equals lmax of Groër et al.(2009). To this end, %Gap denotes the percentage gap w.r.t. Con-RTR, while CT stands for the total computational time in seconds.The third set of columns (TTS-F) reports the best results obtained

Table 1Comparative analysis on Groër et al. (2009) benchmark data sets.

Problem instance ConRTR TTS-L

TT NV lmax TT NV lm

Pr.01 2282.14 5 24.38 2210.56 4 21Pr.02 3872.86 11 34.26 3622.71 9 27Pr.03 3628.22 7 22.87 3451.10 6 21Pr.04 4952.91 12 27.53 4572.00 10 25Pr.05 6416.77 16 26.93 5732.62 13 19Pr.06 4084.24 5 63.47 4096.87 5 55Pr.07 7126.07 12 83.96 6752.36 10 63Pr.08 7456.19 9 73.04 7279.39 8 62Pr.09 11033.54 14 106.43 10585.10 13 84Pr.10 13916.80 18 60.17 13120.40 16 57Pr.11 4753.89 7 16.10 4721.09 6 15Pr.12 3861.35 10 17.58 3607.88 8 16

Average 6115.42 10.50 46.39 5812.67 9.00 39

without considering any additional constraints towards lmax (i.e. Lis set to infinity). Finally, the bottom section of the table providesthe corresponding average results over all problem instances.

Based on the computational results provided in Table 1, TTS seemsto be highly competitive compared to the current state-of-the-art forthe ConVRP. In particular, all existing results were improved both interms of total traveling times and total number of vehicles. Consideringthe TTS-L configuration, with additional constraints towards the max-imum service time differential of frequent customers, cost reductionsup to 4.76% are observed w.r.t. ConRTR. Furthermore, the average num-ber of vehicles deployed is reduced from 10.50 to 9.00 (a total of 19vehicles), while the customer service, expressed in terms of lmax, is im-proved by 17% from 46.39 to 39.33 (average values for all problem in-stances). Similar are the figures considering the TTS-F configuration. Inparticular, costs reductions up to 5.34% are obtained, in terms of totaltraveling time, while the total number of vehicle is further reducedto an average of 8.92 vehicles per problem instance. Finally, it is worthto highlight that in both configurations new best solutions are ob-tained for 11 out of 12 problem instances, while the average resultsover multiple simulation runs revealed very small variations (closeto 0.6% worst case).

Another point of interest is to examine the effect of service timerestrictions. As expected, when this type of consistent service con-straint is relaxed better results are obtained. In particular, costreductions close to 0.8% on average are observed between TTS-Land TTS-F configurations, while the resulting average lmax is in-creased from 39.33 to 50.66, respectively. To this end, it seems thatconsiderable improvements can be achieved with a rather smallcompromise towards the maximum allowable service time limits.Therefore, depending on the nature of services provided to custom-ers, alternative cost effective operational settings emerge for differ-ent types and mixes of consistent service constraints.

4.4. Efficiency analysis

Competitive advantage of the proposed template-based TabuSearch algorithm is its fairly simple algorithmic structure and thesmall number of parameters introduced. In particular, the onlyuser-depend parameter that is critical to the performance and thesearch capability of the proposed solution approach, at the cost ofthe expense in computational time, is the total number of mTS iter-ations without observing any improvement. Therefore, it is impor-tant to examine the computational time consumption w.r.t. thenumber of iterations and the corresponding improvementsobserved.

Table 2 summarizes the mean and cumulative results obtainedfor different values of z regarding mTS, ranging from 2500 to20,000. In particular, MNV, MTT and MCT stand for mean number

TTS-F

ax %Gap CT TT NV lmax %Gap CT

.99 �3.14 80 2180.39 4 27.22 �4.46 38

.75 �6.46 93 3594.56 9 42.23 �7.19 72

.92 �4.88 369 3419.74 6 28.49 �5.75 170

.15 �7.69 388 4572.06 10 27.51 �7.69 745

.99 �10.66 550 5720.86 13 51.78 �10.85 554

.38 0.31 70 4096.87 5 55.38 0.31 45

.28 �5.24 161 6752.36 10 63.28 �5.24 104

.01 �2.37 539 7279.39 8 62.01 �2.37 194

.76 �4.06 947 10560.90 13 106.43 �4.28 909

.17 �5.72 1052 12989.80 15 74.72 �6.66 911

.68 �0.69 480 4543.27 6 51.98 �4.43 175

.91 �6.56 172 3607.88 8 16.91 �6.56 125

.33 �4.76 408 5776.51 8.92 50.66 �5.43 337

Table 2Efficiency analysis on Groër et al. (2009) benchmark data sets.

z (mTS) 2500 5000 7500 10,000 12,500 15,000 17,500 20,000

% Impr. 3.87 4.09 4.40 4.71 4.71 4.72 4.72 4.76MTT 5847.99 5840.70 5822.63 5815.57 5815.57 5815.42 5815.42 5812.67MNV 9.17 9.08 9.00 9.00 9.00 9.00 9.00 9.00MCT 84 118 158 215 251 319 373 408CTT 70175.84 70088.38 69871.57 69786.84 69786.84 69785.01 69785.01 69752.08CNV 110 109 108 108 108 108 108 108

Fig. 1. % Improvement ratio vs z (mTS termination condition).

4238 C.D. Tarantilis et al. / Expert Systems with Applications 39 (2012) 4233–4239

of vehicles, mean traveling time and mean computational time,while CNV and CTD stand for cumulative number of vehicles andcumulative traveling time over all problem instances. Additionally,the second row (% Impr.) shows the percentage improvement ofTTS-L configuration w.r.t. the results reported by ConRTR (seeTable 1).

Based on the result provided in Table 2, it is evident that evenfor very small values of z and computational times less than1.5 min, high quality solutions are obtained (improvement closeto 3.87%). Similar are the figures concerning the total number ofvehicles. On the other hand, as moving towards larger values ofz, the performance of the proposed solution approach also in-creases. However, from a point and after this is made at the ex-pense of computational time. Note that only a 0.04%improvement is achieved from 10,000 to 20,000 iterations, whilethe CT consumption of the latter is twice as much the consumptionof the former.

Fig. 1 illustrates the % improvement over the correspondingcomputational time consumption w.r.t. the number of iterations.To this end, values below 15,000 iterations seem to provide goodcompromise between efficiency and effectiveness for practicalapplications. Note that experimentally the effect of z values regard-ing sTS is minor, because of the structure of the benchmark datasets and the small number of non-frequent customers.

5. Conclusions

Modern companies tend to design and follow customer-ori-ented rather than demand-oriented approaches, especially forthose active in the service sectors, in order to achieve the goalsof sustainable development. Towards this emerging field of re-search, this paper deals with multi-period vehicle routing prob-lems considering several different types of consistent serviceconstraints. More specifically, a generic template-based TabuSearch algorithm is proposed for the so-called Consistent VehicleRouting Problem. The latter involves the design of a set of mini-mum cost vehicle routes to service a set of customers with knowndemands over a planning horizon of multiple days. Customers may

receive service either once or with a predefined frequency; how-ever frequent customers must receive consistent service, such thatthe customers’ visiting sequence remains the same and/or themaximum service time difference between the earliest and latestvehicle arrival times does not exceed a maximum time limit, andthe service is performed by the same vehicle throughout the plan-ning horizon.

The proposed solution approach adopts a two-level master–slave decomposition scheme based on the fact that the vehiclerouting plans over multiple days are highly interrelated due to con-sistent service constraints and cannot be treated separately. In spe-cific, the master sub-problem is concerned with the design of atemplate route schedule in order to determine the service se-quence and assignment of frequent customers to vehicle routes.On return, the slave sub-problem uses the template as the basisto design the actual daily vehicle routes and service schedules forboth frequent and non-frequent customers over multiple days. Tothis end, a Tabu Search algorithm is employed to minimize the to-tal traveling time. The latter operates on a dual mode basis andmodifies both the template routes and the actual daily schedulesin a sequential fashion.

Experimental results on benchmark data sets of the literaturedemonstrated the competitiveness of the proposed solution ap-proach. Compared to existing results, it proved to be highly com-petitive improving the best reported cumulative and meanresults over all problem instances with very reasonable computa-tional requirements and fixed parameter settings. Furthermore,apart from cost improvements in terms of total traveling times, itmanaged to reduce significantly the corresponding fleet size. Tothis end, the small number of user-depended parameters and thefairly simple and generic algorithmic structure indicate its applica-bility towards real life applications.

A worth pursing research direction is towards the design ofadaptive memory programming approaches based on the conceptof template route schedules. On the other hand, as research movestowards more realistic and rich problems, the development ofmedium and large scale benchmark data sets with varying mixesof frequent and non-frequent customers and different operationalsettings is of great interest.

C.D. Tarantilis et al. / Expert Systems with Applications 39 (2012) 4233–4239 4239

Acknowledgments

Support from the Senate Committee of the Athens University ofEconomic and Business for the ‘‘Basic Research Funding Program(BRFP)’’ is gratefully acknowledged.

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