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A template-based Tabu Search algorithm for the Consistent Vehicle Routing Problem
Expert Systems with Applications 39 (2012) 4233–4239
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
A template-based Tabu Search algorithm for the Consistent Vehicle Routing Problem C.D. Tarantilis ⇑, F. Stavropoulou, P.P. Repoussis Center 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 Keywords: Vehicle routing Tabu Search Distribution logistics
i n f o
a b s t r a c t This paper presents a generic template-based solution framework and its application to the so-called Consistent Vehicle Routing Problem (ConVRP). The ConVRP is an NP-hard combinatorial optimization problem and involves the design of a set of minimum cost vehicle routes to service a set of customers with known demands over multiple days. Customers may receive service either once or with a predefined frequency; however frequent customers must receive consistent service, i.e., must be visited by the same driver over approximately the same time throughout the planning period. The proposed solution framework adopts a two-level master–slave decomposition scheme. Initially, a master template route schedule is constructed in an effort to determine the service sequence and assignment of frequent customers to vehicles. On return, the master template is used as the basis to design the actual vehicle routes and service schedules for both frequent and non-frequent customers over multiple days. To this end, a Tabu Search improvement method is employed that operates on a dual mode basis and modifies both the template routes and the actual daily schedules in a sequential fashion. Computational experiments on benchmark data sets illustrate the competitiveness of the proposed approach compared to existing results. 2011 Elsevier Ltd. All rights reserved.
1. Introduction The design and implementation of periodic delivery systems has become a crucial concern for modern companies seeking to provide high quality and low cost services to their customers. Evidently, optimizing repetitive delivery operations over multiple days can add up to significant cost savings, and thus improve productivity and competitiveness. Periodic deliveries occur in a wide range of real life applications, including among others refuse and municipal waste collection, mail collection and delivery, scheduled retail and wholesale delivery and distribution, vending machine replenishment, and elevator repair and maintenance (Francis, Smilowitz, & Tzur, 2008). Therefore, in practical terms studying such operational planning problems definitely seems worthwhile, apart from the theoretical and computational research challenges arising due to their combinatorial nature. In broad terms, multi-period vehicle routing and scheduling problems deal with the optimum assignment and service sequence of 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 this paper). In the literature, these problems are typically modeled as Periodic Vehicle Routing Problems (PVRP) (Beltrami & Bodin, 1974). The PVRP is a generalization of the well-known Capacitated ⇑ Corresponding author. Address: Patision 74 st., GR11362 Athens, Greece. Tel.: +30 2108203805; fax: +30 2108828078. E-mail addresses: [email protected] (C.D. Tarantilis), [email protected] (F. Stavropoulou), [email protected] (P.P. Repoussis). 0957-4174/$ - see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.09.111
Vehicle Routing Problem (CVRP) (Tarantilis, 2005) and involves the design of a set of vehicle routes, over a predefined planning horizon, in order to service a set of customers with known demand and frequencies of service (i.e. customers can be visited according to different day combinations). Typically, the objective is to minimize the total traveling cost, expressed in terms of one-time (e.g. fleet size) and recurring costs (e.g. distance traveled), while satisfying operational constraints (e.g. vehicle capacity and visit requirements). Note that besides the daily routing decisions (i.e. assignment and service sequence of each customer on vehicle routes), a schedule from a candidate set of schedules for each customer must be also selected. Contrary to the above described periodic delivery operational setting, in recent years more and more companies tend to focus and invest on brand loyalty and customer relationship management, and thereafter, they are interested in implementing customer-oriented rather than demand-oriented approaches. For example, there are numerous real life applications in which customers need to be visited by the same service provider (i.e. vehicle crew and driver). Furthermore, in many cases customers need to be serviced according to a predefined visiting sequence or a certain service time consistency (e.g. a minimum variance of service times over multiple days). Typical real life paradigms that depict these type of consistent service considerations can be found in parcel deliveries and courier services, home care and nursing services for the elderly and cleaning services. In these cases, the main effort is to gain competitive advantage by forming bonds with the customers.
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A first attempt in the literature towards modeling and solving multi-period vehicle routing and scheduling problems with consistent customer service constraints has been put forward by Groër, Golden, and Wasil (2009). In specific, they introduced the so-called Consistent Vehicle Routing Problem (ConVRP), motivated by a reallife parcel delivery application. To this end, a multi-start solution construction framework is proposed, combined with a Record-toRecord travel local search metaheuristic algorithm (Li, Golden, & Wasil, 2005) and savings-based construction heuristics. The ConVRP is an NP-hard combinatorial optimization problem in the strong sense and involves the design of minimum cost routes in order to service a set of customers with known demand over multiple days via a homogeneous fleet of depot returning capacitated vehicles. Customers may receive service either once (non frequent customers) or with a predefined frequency (frequent customers); however frequent customers must receive consistent service throughout the planning period, such that (a) their visiting sequence remains the same (or similarly the maximum service time difference between the earliest and latest vehicle arrival times over multiple days does not exceed a maximum time limit) and (b) the service is performed by the same vehicle. On the other hand, during each day each customer must be visited only once by exactly one vehicle, while each vehicle has a maximum carrying capacity and operates for no more than a maximum time limit. The goal is to minimize the total distance traveled by the vehicles such that all customer requirements are fulfilled without violating capacity, route duration and consistent service constraints. Towards this new and emerging line of research, the main contribution and aim of this paper is to design and develop a generic and flexible template-based solution framework for the ConVRP. Following the concept of template route schedules, the proposed approach adopts a two-level decomposition scheme and solves a master and a slave sub-problem in a sequential fashion. In particular, the master sub-problem is concerned with the design of a template route schedule in order to determine the assignment of frequent customers to vehicles and their visiting sequences, while the corresponding slave sub-problem seeks to determine the actual daily vehicle routes for both frequent and non-frequent customers on the basis of the template route schedule. From the implementation viewpoint, effective savings and insertion based construction heuristics are utilized, coupled with a Tabu Search (TS) algorithm that operates on a dual mode basis. Given an initial template route schedule, TS is applied in an effort to minimize the total distance traveled, considering only frequent customers (master mode). On return, the corresponding daily schedules for both frequent and non-frequent customers are constructed and further improved by TS (slave mode). In this case, TS is applied to the actual daily schedules and takes into account, apart from vehicle capacities and route duration restrictions, the precedence constraints (i.e. assignment and visiting sequence requirements) dictated by the corresponding template route schedule. For the evaluation of the proposed template-based solution approach, computational experiments on benchmark data sets of the literature are reported. Compared to existing results, it proved to be highly competitive and improved the best reported cumulative and mean results over all problem instances with very reasonable computational requirements for practical applications. To this end, competitive advantage of proposed solution framework is its fairly simple algorithmic structure, its flexibility to accommodate various types of consistent service constraints and the small number of parameters introduced. The remainder of this paper is organized as follows. Section 3 presents the proposed solution framework and provides detailed descriptions of all algorithmic components and mechanisms. Computational experiments assessing the quality of the proposed approach along with a comparative performance analysis are
presented in Section 4. Finally, in Section 5 conclusions are drawn and 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 destination depot and each node of Nc = Nn{0} corresponds to a customer (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 equivalently 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 a known capacity of Q units and operates for at most B time units during each day d 2 D, where D denotes the days of service requirements. The service days and the corresponding demands of each customer are known in advance. In particular, each customer i 2 Nc is associated with a non-negative demand qid and a non-negative service duration sid for each day d 2 D. Furthermore, binary indicators wid show if a customer i requires service on day d, while predefined 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 of a set of vehicle routes. Given the actual service requirements and specifications for each customer as described above, the goal is to find a set of least cost daily schedules such that: – The total traveling time (or alternatively the total traveling distance) incurred by the vehicles over multiple days, including service times at customers’ locations, is minimized. – The accumulated service of a vehicle route (total quantity carried) is less than or equal to the maximum carrying capacity Q. – The total duration time of each individual vehicle route does not exceed a predefined upper limit B (e.g. depot operating hours). – Each customer i 2 Nc is visited only once by exactly one vehicle on each day d that service can take place. – Frequent customers i 2 Nf need to be visited by the same vehicle k 2 K throughout the planning horizon. – The visiting sequence of frequent customers i 2 Nf must remain the same for each d 2 D and/or the maximum service time difference between the earliest and latest vehicle arrival times over 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, if this restriction is relaxed, time consistency violations can be avoided 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 routing problems, in which vehicle routes must be constructed over a planning period of D days. Considering the ConVRP, the service days and the corresponding demands of each customer are known in advance. However, consistent customer service constraints impose that frequent customers must be visited by the same vehicle at approximately the same time during the days they require service throughout the planning horizon. Therefore, contrary to traditional PVRP instances, the ConVRP cannot be decomposed into jDj separate CVRP problems
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(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 over multiple days, the rational of template route schedules, introduced by Groër et al. (2009), is utilized. A template route contains only frequent customers and it can be used as a master plan to determine their visiting sequence in the actual daily vehicle routing schedules. To work around this concept, the proposed solution approach adopts a two-level master–slave decomposition scheme. The master sub-problem is concerned with the design of a template route schedule in order to determine the service sequence and assignment of frequent customers to vehicle routes. On the other hand, the slave sub-problem uses the template as the basis to design the actual daily vehicle routes and service schedules for both frequent and non-frequent customers over multiple days. On the basis of the above, the ConVRP can be decomposed into a master and a slave sub-problem that needs to be solved in a sequential fashion. For this purpose, effective construction heuristics are utilized, while an innovative TS improvement method is also employed that operates on a dual mode basis and modifies both the template routes and the daily schedules. At first, an initial feasible template route schedule T is generated via a generalized insertion-based merging construction heuristic, considering all frequent customers simultaneously. To this end, the master TS mode (mTS) is applied to minimize the total traveling time. Although all local moves considered at the master mode modify directly the current T, the neighborhood evaluation and the associated feasibility checks are based on the resulting daily schedules. Upon termination of the template improvement level, the corresponding partially constructed daily vehicle routing plans are populated with the remaining set of non-frequent customers. Given the finalized set of daily schedules DS, the slave TS mode (sTS) is triggered for further improvement (post optimization). In an effort to preserve the feasibility, precedence constraints are also considered, in terms of customer assignments to vehicles and visiting sequence restrictions, as dictated by the corresponding T. Below, the algorithmic structure of the proposed templatebased solution framework is provided. Template-based solution framework //Parameters: b, zm, zs, vm, vs// DS 0;// Generate an empty set of daily schedules Nf Build set ();//Frequent customers Nnf Build set ();//Non frequent customers //Master Level T 0;// Generate an empty template route schedule T Merging Heuristic (Nf,b);// Template Initialization T mTS (T, zm, vm);// Template Improvement //Slave Level While day < D do { dsday 0;// Generate an empty daily schedule dsday Initialization (T, day, Nf);// Adaptation of T dsday Insertion Heuristic (day,Nnf);// Completion of dsday dsday sTS (dsday, zs, vs);// Post Optimization DS Add (dsday); } Return DS;
3.2. Template initialization The goal during the template initialization phase is to generate initial template schedules considering the whole set of frequent customers, regardless their required service frequency. For this
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purpose, a generalized insertion-based merging construction heuristic is utilized. At first, a solution is generated considering that one vehicle is assigned to each customer. Next, at each iteration two 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 combinations can be obtained. Let two routes r0 = {0, w, . . . ,i, j, . . . ,z, 0} and r00 = {0, u, . . . , l, 0}, where indexes i, j, w, z, u, l denote frequent customers. The first combination is to insert r00 after r0 , i.e., ^r ¼ f0; w; . . . ; i; j; . . . ; z; u; . . . ; l; 0g, with a corresponding saving c0u + cz0 czu. The second is to insert r00 in front of r0 , i.e., ^r ¼ f0; u; . . . ; l; w; . . . ; i; j; . . . ; z; 0g, with a saving c0w + cl0 clw. The third is to insert r00 within all positions between consecutive customers i, j, i.e., ^r ¼ f0; w; . . . ; i; u; . . . ; l; j; . . . ; z; 0g. The resulting saving formula can be defined as cij + c0u + cl0 ciu clj. In a manner similar, three additional merging combinations emerge by reversing the visiting sequence of route r0 . Due to route duration constraints, one may benefit if a sequential manner of construction, similar to traditional insertion-based construction methods, is followed. In fact, if during early stages of construction merging is performed mostly between routes with many customers, the final output tends to be poor, especially in terms of the number of vehicles deployed. To this end, an effort is made to favor merging combinations between single customer routes and routes containing more than two customers, by multiplying the corresponding saving with a parameter b that will regulate the degree of sequential construction. During the construction of the template route schedule, only feasible merging combinations are considered with respect to the resulting daily schedules. Clearly, template routes contain a mix of frequent customers with different combinations of daily service requirements. However, it is straightforward to check the feasibility of the resulting daily schedules. Let aid denote the arrival time of the vehicle at the location of customer i on day d. The template T is feasible, in terms of vehicle capacity and route duration constraints, if the following inequalities hold throughout the planning P horizon: i2r qid wid 6 Q 8d 2 D; r 2 T (vehicle capacity); aid + wid(sid + ci0) 6 Bwid "d 2 D, i 2 Nf (route duration); and jaida aidb j 6 L Bðwida þ widb 2Þ 8i 2 N f ; da ; db 2 D; a – b (service time consistency). 3.3. Generation of daily schedules Given the master template route schedule, the corresponding daily vehicle routing schedules can be determined in a sequential fashion for each day of the planning horizon as follows. Initially, the template route schedule is adopted and the frequent customers that do not require service on that particular day are removed. By doing so, the assignment to vehicles and the visiting sequence of frequent customers are preserved. Subsequently, the non-frequent customers that may require service are identified and designated as unrouted customers. To this end, a parallel construction heuristic is employed to insert the remaining unrouted customers and to finalize the partially built daily schedules. The proposed parallel construction heuristic builds iteratively a solution by selecting and adding one by one unrouted customers to the partially built vehicle routes, until a feasible daily schedule is generated for both frequent and non-frequent customers. At each iteration, the selection of customers and insertion positions between adjacent routed customers is made according to a greedy criterion that measures the insertion cost. From the implementation viewpoint, only feasible insertion positions are considered. However, if no feasible insertion positions can be found for the remaining unrouted customers, new vehicle routes are initialized. An illustrative example of the dependence between the master and the slave subproblems is the following. Let a planning horizon
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of three days d1, d2 and d3, and two disjoint sets of frequent customers 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 two routes, e.g., tr1 = {0, w, l, i, s, 0} and tr2 = {0, h, j, u, 0}. During the first day, the following set of customers requires service, e.g., {h, i, j, l, u, y, e}. Therefore, the resulting partial daily schedule for d1 consists 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 is to find the least cost insertion positions for non-frequent customers 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} and dr2 = {0, l, y, e, i, 0}. 3.4. Tabu Search improvement method The template and the corresponding daily vehicle routing schedules are improved, in terms of distance minimization, by means of a dual-mode TS algorithm. The master TS mode (mTS) is applied to the template route schedule and considers explicitly the feasibility of the master vehicle routing plan with respect to the resulting daily schedules, while the slave TS mode (sTS) is applied to the actual daily vehicle routing schedules and takes into account the assignment and visiting sequence requirements dictated by the associated template route schedule. In both cases, the solution space is explored on the basis of simple edge-exchange neighborhood structures. However the degrees of freedom of mTS are much higher compared to sTS, since the feasible region of allowable move operators of the latter is greatly reduced. In broad terms, TS seeks to explore the solution space by moving at each iteration from a solution s to the best admissible solution s0 in a subset Uy(s) of a neighborhood structure y. The short term memory records the most recently visited solutions and prevents revisiting them for a predefined number of iterations v (tabu tenure) in order to avoid cycling. The tabu status of a neighboring solution can be overridden, only if predefined aspiration criteria are met (i.e. new local optimum solutions). The overall procedure iterates until some termination conditions are met (maximum number of iterations z without observing any further improvement in our case) and the best encountered local optimum solution s⁄ is returned. The neighborhood structures considered in the proposed implementation are based on traditional O(n2) edge-exchange local moves, namely intra- and inter-route 2-Opt, 2-Opt⁄, 1–0 Relocate and 1–1 Exchange (swap) (Tarantilis, Kiranoudis, & Vassiliadis, 2002). The oscillation among these multiple neighborhood structures is purely stochastic with equal selection probability. At each iteration, given the allowed set of neighbors Uy(s), the best admissible neighbor s0 replaces the current solution s, while both the forward and reversal attributes (i.e. edges being added or deleted) of the corresponding local moves are stored within the tabu list. For the evaluation of the neighborhood structures, a lexicographic ordering search is followed and acceleration mechanisms suggested by Repoussis, Tarantilis, and Ioannou (2009) that reduce the evaluation complexity are also incorporated. To this end, emphasis is given on direct feasibility gains based on the characteristics of the proposed decomposition scheme. Regarding the slave TS mode, one may benefit from the inherent precedence constraints, since the possible removal and re-insertion positions of frequent customers are very few and only non-frequent customers can freely change positions. Let two template routes, e.g., tr1 = {0, w, l, i, s, 0} and tr2 = {0, h, j, u, 0}, and the corresponding daily vehicle routing schedule, e.g., dr1 = {0, h, j, u, 0} and dr2 = {0, l, y, e, i, 0} (see also Section 3.3). Non-frequent customers y and e do not follow any precedence constraints and all feasible intra and inter-route local moves are allowed. On the other hand, frequent 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 customer l on the same route, while non feasible intra-route local move exist for frequent customers h, j and u. Clearly, the feasibility checks of a local move on the daily schedules, in terms of route duration, vehicle capacity and service time consistency, can be made in almost constant time. On the other hand, considering the mTS mode the computational effort of feasibility checks slightly increases, because a local move on the template route schedule affects the feasibility of the resulting daily schedules. Thus, one has to measure explicitly the effect of local moves 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, several computational experiments are performed using the benchmark data set generated for the ConVRP by Groër et al. (2009). This data set consists of 12 medium-scale problem instances, divided into two groups. The first group contains 7 problem instances, which consider only vehicle capacity constraints. The second group of problem instances includes both vehicle capacity and route duration constraints. The number of customers ranges from 50 to 199, while a 5-day planning horizon with a non fixed fleet size is assumed for all problem instances. Finally, it is worth to mention that the objective function values refer to the sum of all the times spent along each vehicle route (total en route time), including distance traveled and service times at customers’ locations. 4.2. Parameter settings and termination conditions Towards the design of approaches with the least possible number of user-defined parameters, the proposed solution framework incorporates three parameters, namely the degree of sequential construction b, the tabu list size v, and the number of maximum number of TS iterations without observing any further improvement z (termination condition). Based on computational experience, these parameters are relatively insensitive to the characteristics of the problem instances considered, while using simple adjustments one can determine very well performing parameter settings with modest effort within reasonable value ranges. In this paper, several intuitively selected combinations were experimentally tested, choosing the one that yielded the best average output. Below, suitable value ranges for each parameter are provided. Regarding the generalized insertion-based merging construction heuristic, parameter b needs to be determined. Clearly, b must be set greater than or equal to 1 in order to boost the degree of sequential construction. However, large values of b may affect the impact of the other types of savings. To this end, a value range between 1 and 1.5 was found to perform well for most ConVRP problem instances, especially in terms of total number of vehicles employed. In what follows, b was fixed to 1.3 at all simulation runs. Two parameters must be defined considering the TS improvement method for each mode; the tabu list size v and the number of maximum number of TS iterations z. Between the master and the slave TS modes, the most labor intensive and critical to the overall performance is the former. As such, regarding the tabu list size vm for mTS, a range between 30 and 60 is typically used by standard TS implementations of the literature and seems to fit well for ConVRP problem instances. Contrary, a rather smaller range between 10 and 20 suits for vs, due to the very limited search capacity of sTS. On the other hand, as z increases the efficiency of local
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search also increases. However, a balance between efficiency and effectiveness is needed, since large values of z may result in excessive computational time consumption. To this end, values of zm and zs up to 20,000 and 500 iterations, respectively, seem to provide a good compromise (see also Section 4.4 for an effectiveness analysis w.r.t. zm). The proposed template-based Tabu Search algorithm was implemented in C++ (Visual Studio 2008). All computational experiments reported in subsequent sections were performed on a 2.8 GHz Intel Xeon personal computer, assuming 5 simulation runs for each problem instance with fixed parameter settings as follows: 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 according to the sum of total traveling times along each vehicle route, including service times at customers’ locations. Additionally, a possible secondary measure of solution quality is the total number of vehicles. However, these two objectives can be either conflicting or complementary. For this reason, all comparisons with existing results are made only towards the total traveling times. Given the definition of consistent service constraints regarding frequent customers, there are two alternatives; either to ensure the same visiting sequence throughout the planning horizon or to set a maximum time limit L w.r.t. the maximum service time difference between the earliest and latest vehicle arrival times lmax (i.e. lmax 6 L). Thereafter, in order for the competition between different solution approaches for ConVRP instances to be fair and objective, both types of consistent service constraints are imposed either separately or simultaneously. In the former case, only visiting sequence restrictions are imposed, while in the latter case, besides visiting sequence restrictions for frequent customers, the maximum arrival time differential lmax reported by Groër et al. (2009) is set as the maximum time limit L and it is enforced as an additional constraint. Table 1 summarizes the results obtained on the benchmark data sets of Groër et al. (2009). The first line lists the authors using the following abbreviations: ConRTR stands for (Groër et al., 2009) and TTS stands for the proposed template-based Tabu Search algorithm. Initially, the first set of columns illustrate the total traveling time (TT), the total number of vehicles (NV) and the maximum arrival time differential lmax reported by (Groër et al., 2009) for each problem instance. The second set of columns (TTS-L) reports the best results obtained considering that L equals lmax of Groër et al. (2009). To this end, %Gap denotes the percentage gap w.r.t. ConRTR, while CT stands for the total computational time in seconds. The third set of columns (TTS-F) reports the best results obtained
without considering any additional constraints towards lmax (i.e. L is set to infinity). Finally, the bottom section of the table provides the corresponding average results over all problem instances. Based on the computational results provided in Table 1, TTS seems to be highly competitive compared to the current state-of-the-art for the ConVRP. In particular, all existing results were improved both in terms of total traveling times and total number of vehicles. Considering the TTS-L configuration, with additional constraints towards the maximum service time differential of frequent customers, cost reductions up to 4.76% are observed w.r.t. ConRTR. Furthermore, the average number of vehicles deployed is reduced from 10.50 to 9.00 (a total of 19 vehicles), while the customer service, expressed in terms of lmax, is improved by 17% from 46.39 to 39.33 (average values for all problem instances). Similar are the figures considering the TTS-F configuration. In particular, costs reductions up to 5.34% are obtained, in terms of total traveling time, while the total number of vehicle is further reduced to an average of 8.92 vehicles per problem instance. Finally, it is worth to highlight that in both configurations new best solutions are obtained for 11 out of 12 problem instances, while the average results over multiple simulation runs revealed very small variations (close to 0.6% worst case). Another point of interest is to examine the effect of service time restrictions. As expected, when this type of consistent service constraint is relaxed better results are obtained. In particular, cost reductions close to 0.8% on average are observed between TTS-L and TTS-F configurations, while the resulting average lmax is increased from 39.33 to 50.66, respectively. To this end, it seems that considerable improvements can be achieved with a rather small compromise towards the maximum allowable service time limits. Therefore, depending on the nature of services provided to customers, alternative cost effective operational settings emerge for different types and mixes of consistent service constraints. 4.4. Efficiency analysis Competitive advantage of the proposed template-based Tabu Search algorithm is its fairly simple algorithmic structure and the small number of parameters introduced. In particular, the only user-depend parameter that is critical to the performance and the search capability of the proposed solution approach, at the cost of the expense in computational time, is the total number of mTS iterations without observing any improvement. Therefore, it is important to examine the computational time consumption w.r.t. the number of iterations and the corresponding improvements observed. Table 2 summarizes the mean and cumulative results obtained for different values of z regarding mTS, ranging from 2500 to 20,000. In particular, MNV, MTT and MCT stand for mean number
Table 1 Comparative analysis on Groër et al. (2009) benchmark data sets. Problem instance
Pr.01 Pr.02 Pr.03 Pr.04 Pr.05 Pr.06 Pr.07 Pr.08 Pr.09 Pr.10 Pr.11 Pr.12 Average
ConRTR
TTS-L
TTS-F
TT
NV
lmax
TT
NV
lmax
%Gap
CT
TT
NV
lmax
%Gap
CT
2282.14 3872.86 3628.22 4952.91 6416.77 4084.24 7126.07 7456.19 11033.54 13916.80 4753.89 3861.35
5 11 7 12 16 5 12 9 14 18 7 10
24.38 34.26 22.87 27.53 26.93 63.47 83.96 73.04 106.43 60.17 16.10 17.58
2210.56 3622.71 3451.10 4572.00 5732.62 4096.87 6752.36 7279.39 10585.10 13120.40 4721.09 3607.88
4 9 6 10 13 5 10 8 13 16 6 8
21.99 27.75 21.92 25.15 19.99 55.38 63.28 62.01 84.76 57.17 15.68 16.91
3.14 6.46 4.88 7.69 10.66 0.31 5.24 2.37 4.06 5.72 0.69 6.56
80 93 369 388 550 70 161 539 947 1052 480 172
2180.39 3594.56 3419.74 4572.06 5720.86 4096.87 6752.36 7279.39 10560.90 12989.80 4543.27 3607.88
4 9 6 10 13 5 10 8 13 15 6 8
27.22 42.23 28.49 27.51 51.78 55.38 63.28 62.01 106.43 74.72 51.98 16.91
4.46 7.19 5.75 7.69 10.85 0.31 5.24 2.37 4.28 6.66 4.43 6.56
38 72 170 745 554 45 104 194 909 911 175 125
46.39
5812.67
39.33
4.76
408
5776.51
50.66
5.43
337
6115.42
10.50
9.00
8.92
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Table 2 Efficiency 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. MTT MNV MCT CTT CNV
3.87 5847.99 9.17 84 70175.84 110
4.09 5840.70 9.08 118 70088.38 109
4.40 5822.63 9.00 158 69871.57 108
4.71 5815.57 9.00 215 69786.84 108
4.71 5815.57 9.00 251 69786.84 108
4.72 5815.42 9.00 319 69785.01 108
4.72 5815.42 9.00 373 69785.01 108
4.76 5812.67 9.00 408 69752.08 108
Fig. 1. % Improvement ratio vs z (mTS termination condition).
of vehicles, mean traveling time and mean computational time, while CNV and CTD stand for cumulative number of vehicles and cumulative traveling time over all problem instances. Additionally, the second row (% Impr.) shows the percentage improvement of TTS-L configuration w.r.t. the results reported by ConRTR (see Table 1). Based on the result provided in Table 2, it is evident that even for very small values of z and computational times less than 1.5 min, high quality solutions are obtained (improvement close to 3.87%). Similar are the figures concerning the total number of vehicles. On the other hand, as moving towards larger values of z, the performance of the proposed solution approach also increases. However, from a point and after this is made at the expense of computational time. Note that only a 0.04% improvement is achieved from 10,000 to 20,000 iterations, while the CT consumption of the latter is twice as much the consumption of the former. Fig. 1 illustrates the % improvement over the corresponding computational time consumption w.r.t. the number of iterations. To this end, values below 15,000 iterations seem to provide good compromise between efficiency and effectiveness for practical applications. Note that experimentally the effect of z values regarding sTS is minor, because of the structure of the benchmark data sets and the small number of non-frequent customers.
5. Conclusions Modern companies tend to design and follow customer-oriented rather than demand-oriented approaches, especially for those active in the service sectors, in order to achieve the goals of sustainable development. Towards this emerging field of research, this paper deals with multi-period vehicle routing problems considering several different types of consistent service constraints. More specifically, a generic template-based Tabu Search algorithm is proposed for the so-called Consistent Vehicle Routing Problem. The latter involves the design of a set of minimum cost vehicle routes to service a set of customers with known demands over a planning horizon of multiple days. Customers may
receive service either once or with a predefined frequency; however frequent customers must receive consistent service, such that the customers’ visiting sequence remains the same and/or the maximum service time difference between the earliest and latest vehicle arrival times does not exceed a maximum time limit, and the service is performed by the same vehicle throughout the planning horizon. The proposed solution approach adopts a two-level master– slave decomposition scheme based on the fact that the vehicle routing plans over multiple days are highly interrelated due to consistent service constraints and cannot be treated separately. In specific, the master sub-problem is concerned with the design of a template route schedule in order to determine the service sequence and assignment of frequent customers to vehicle routes. On return, the slave sub-problem uses the template as the basis to design the actual daily vehicle routes and service schedules for both frequent and non-frequent customers over multiple days. To this end, a Tabu Search algorithm is employed to minimize the total traveling time. The latter operates on a dual mode basis and modifies both the template routes and the actual daily schedules in a sequential fashion. Experimental results on benchmark data sets of the literature demonstrated the competitiveness of the proposed solution approach. Compared to existing results, it proved to be highly competitive improving the best reported cumulative and mean results over all problem instances with very reasonable computational requirements and fixed parameter settings. Furthermore, apart from cost improvements in terms of total traveling times, it managed to reduce significantly the corresponding fleet size. To this end, the small number of user-depended parameters and the fairly simple and generic algorithmic structure indicate its applicability towards real life applications. A worth pursing research direction is towards the design of adaptive memory programming approaches based on the concept of template route schedules. On the other hand, as research moves towards more realistic and rich problems, the development of medium and large scale benchmark data sets with varying mixes of frequent and non-frequent customers and different operational settings is of great interest.
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Acknowledgments Support from the Senate Committee of the Athens University of Economic and Business for the ‘‘Basic Research Funding Program (BRFP)’’ is gratefully acknowledged. References Beltrami, E. J., & Bodin, L. D. (1974). Networks and vehicle routing for municipal waste collection. Networks, 4(1), 65–94. Francis, P. M., Smilowitz, K. R., & Tzur, M. (2008). The period vehicle routing problem and its extensions. In B. Golden et al. (Eds.), The vehicle routing problem (pp. 73–102). New York: Springer Science+Business Media LLC.
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Groër, C., Golden, B. L., & Wasil, E. (2009). The consistent vehicle routing problem. INFORMS Manufacturing & Service Operations Management, 11(4), 630–643. Li, F., Golden, B. L., & Wasil, E. (2005). Very large-scale vehicle routing: New test problems, algorithms and results. Computers & Operations Research, 32, 1197–1212. Repoussis, P. P., Tarantilis, C. D., & Ioannou, G. (2009). An arc-guided evolutionary algorithm for the vehicle routing problem with time windows. IEEE Transactions on Evolutionary Computation, 13(3), 624–647. Tarantilis, C. D. (2005). Solving the vehicle routing problem with adaptive memory programming methodology. Computers and Operations Research, 32(9), 2309–2327. Tarantilis, C. D., Kiranoudis, C. T., & Vassiliadis, V. S. (2002). A list based threshold accepting algorithm for the capacitated vehicle routing problem. International Journal of Computer Mathematics, 79(5), 537–553.
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
A template-based Tabu Search algorithm for the Consistent Vehicle Routing Problem C.D. Tarantilis ⇑, F. Stavropoulou, P.P. Repoussis Center 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 Keywords: Vehicle routing Tabu Search Distribution logistics
i n f o
a b s t r a c t This paper presents a generic template-based solution framework and its application to the so-called Consistent Vehicle Routing Problem (ConVRP). The ConVRP is an NP-hard combinatorial optimization problem and involves the design of a set of minimum cost vehicle routes to service a set of customers with known demands over multiple days. Customers may receive service either once or with a predefined frequency; however frequent customers must receive consistent service, i.e., must be visited by the same driver over approximately the same time throughout the planning period. The proposed solution framework adopts a two-level master–slave decomposition scheme. Initially, a master template route schedule is constructed in an effort to determine the service sequence and assignment of frequent customers to vehicles. On return, the master template is used as the basis to design the actual vehicle routes and service schedules for both frequent and non-frequent customers over multiple days. To this end, a Tabu Search improvement method is employed that operates on a dual mode basis and modifies both the template routes and the actual daily schedules in a sequential fashion. Computational experiments on benchmark data sets illustrate the competitiveness of the proposed approach compared to existing results. 2011 Elsevier Ltd. All rights reserved.
1. Introduction The design and implementation of periodic delivery systems has become a crucial concern for modern companies seeking to provide high quality and low cost services to their customers. Evidently, optimizing repetitive delivery operations over multiple days can add up to significant cost savings, and thus improve productivity and competitiveness. Periodic deliveries occur in a wide range of real life applications, including among others refuse and municipal waste collection, mail collection and delivery, scheduled retail and wholesale delivery and distribution, vending machine replenishment, and elevator repair and maintenance (Francis, Smilowitz, & Tzur, 2008). Therefore, in practical terms studying such operational planning problems definitely seems worthwhile, apart from the theoretical and computational research challenges arising due to their combinatorial nature. In broad terms, multi-period vehicle routing and scheduling problems deal with the optimum assignment and service sequence of 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 this paper). In the literature, these problems are typically modeled as Periodic Vehicle Routing Problems (PVRP) (Beltrami & Bodin, 1974). The PVRP is a generalization of the well-known Capacitated ⇑ Corresponding author. Address: Patision 74 st., GR11362 Athens, Greece. Tel.: +30 2108203805; fax: +30 2108828078. E-mail addresses: [email protected] (C.D. Tarantilis), [email protected] (F. Stavropoulou), [email protected] (P.P. Repoussis). 0957-4174/$ - see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.09.111
Vehicle Routing Problem (CVRP) (Tarantilis, 2005) and involves the design of a set of vehicle routes, over a predefined planning horizon, in order to service a set of customers with known demand and frequencies of service (i.e. customers can be visited according to different day combinations). Typically, the objective is to minimize the total traveling cost, expressed in terms of one-time (e.g. fleet size) and recurring costs (e.g. distance traveled), while satisfying operational constraints (e.g. vehicle capacity and visit requirements). Note that besides the daily routing decisions (i.e. assignment and service sequence of each customer on vehicle routes), a schedule from a candidate set of schedules for each customer must be also selected. Contrary to the above described periodic delivery operational setting, in recent years more and more companies tend to focus and invest on brand loyalty and customer relationship management, and thereafter, they are interested in implementing customer-oriented rather than demand-oriented approaches. For example, there are numerous real life applications in which customers need to be visited by the same service provider (i.e. vehicle crew and driver). Furthermore, in many cases customers need to be serviced according to a predefined visiting sequence or a certain service time consistency (e.g. a minimum variance of service times over multiple days). Typical real life paradigms that depict these type of consistent service considerations can be found in parcel deliveries and courier services, home care and nursing services for the elderly and cleaning services. In these cases, the main effort is to gain competitive advantage by forming bonds with the customers.
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A first attempt in the literature towards modeling and solving multi-period vehicle routing and scheduling problems with consistent customer service constraints has been put forward by Groër, Golden, and Wasil (2009). In specific, they introduced the so-called Consistent Vehicle Routing Problem (ConVRP), motivated by a reallife parcel delivery application. To this end, a multi-start solution construction framework is proposed, combined with a Record-toRecord travel local search metaheuristic algorithm (Li, Golden, & Wasil, 2005) and savings-based construction heuristics. The ConVRP is an NP-hard combinatorial optimization problem in the strong sense and involves the design of minimum cost routes in order to service a set of customers with known demand over multiple days via a homogeneous fleet of depot returning capacitated vehicles. Customers may receive service either once (non frequent customers) or with a predefined frequency (frequent customers); however frequent customers must receive consistent service throughout the planning period, such that (a) their visiting sequence remains the same (or similarly the maximum service time difference between the earliest and latest vehicle arrival times over multiple days does not exceed a maximum time limit) and (b) the service is performed by the same vehicle. On the other hand, during each day each customer must be visited only once by exactly one vehicle, while each vehicle has a maximum carrying capacity and operates for no more than a maximum time limit. The goal is to minimize the total distance traveled by the vehicles such that all customer requirements are fulfilled without violating capacity, route duration and consistent service constraints. Towards this new and emerging line of research, the main contribution and aim of this paper is to design and develop a generic and flexible template-based solution framework for the ConVRP. Following the concept of template route schedules, the proposed approach adopts a two-level decomposition scheme and solves a master and a slave sub-problem in a sequential fashion. In particular, the master sub-problem is concerned with the design of a template route schedule in order to determine the assignment of frequent customers to vehicles and their visiting sequences, while the corresponding slave sub-problem seeks to determine the actual daily vehicle routes for both frequent and non-frequent customers on the basis of the template route schedule. From the implementation viewpoint, effective savings and insertion based construction heuristics are utilized, coupled with a Tabu Search (TS) algorithm that operates on a dual mode basis. Given an initial template route schedule, TS is applied in an effort to minimize the total distance traveled, considering only frequent customers (master mode). On return, the corresponding daily schedules for both frequent and non-frequent customers are constructed and further improved by TS (slave mode). In this case, TS is applied to the actual daily schedules and takes into account, apart from vehicle capacities and route duration restrictions, the precedence constraints (i.e. assignment and visiting sequence requirements) dictated by the corresponding template route schedule. For the evaluation of the proposed template-based solution approach, computational experiments on benchmark data sets of the literature are reported. Compared to existing results, it proved to be highly competitive and improved the best reported cumulative and mean results over all problem instances with very reasonable computational requirements for practical applications. To this end, competitive advantage of proposed solution framework is its fairly simple algorithmic structure, its flexibility to accommodate various types of consistent service constraints and the small number of parameters introduced. The remainder of this paper is organized as follows. Section 3 presents the proposed solution framework and provides detailed descriptions of all algorithmic components and mechanisms. Computational experiments assessing the quality of the proposed approach along with a comparative performance analysis are
presented in Section 4. Finally, in Section 5 conclusions are drawn and 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 destination depot and each node of Nc = Nn{0} corresponds to a customer (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 equivalently 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 a known capacity of Q units and operates for at most B time units during each day d 2 D, where D denotes the days of service requirements. The service days and the corresponding demands of each customer are known in advance. In particular, each customer i 2 Nc is associated with a non-negative demand qid and a non-negative service duration sid for each day d 2 D. Furthermore, binary indicators wid show if a customer i requires service on day d, while predefined 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 of a set of vehicle routes. Given the actual service requirements and specifications for each customer as described above, the goal is to find a set of least cost daily schedules such that: – The total traveling time (or alternatively the total traveling distance) incurred by the vehicles over multiple days, including service times at customers’ locations, is minimized. – The accumulated service of a vehicle route (total quantity carried) is less than or equal to the maximum carrying capacity Q. – The total duration time of each individual vehicle route does not exceed a predefined upper limit B (e.g. depot operating hours). – Each customer i 2 Nc is visited only once by exactly one vehicle on each day d that service can take place. – Frequent customers i 2 Nf need to be visited by the same vehicle k 2 K throughout the planning horizon. – The visiting sequence of frequent customers i 2 Nf must remain the same for each d 2 D and/or the maximum service time difference between the earliest and latest vehicle arrival times over 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, if this restriction is relaxed, time consistency violations can be avoided 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 routing problems, in which vehicle routes must be constructed over a planning period of D days. Considering the ConVRP, the service days and the corresponding demands of each customer are known in advance. However, consistent customer service constraints impose that frequent customers must be visited by the same vehicle at approximately the same time during the days they require service throughout the planning horizon. Therefore, contrary to traditional PVRP instances, the ConVRP cannot be decomposed into jDj separate CVRP problems
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(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 over multiple days, the rational of template route schedules, introduced by Groër et al. (2009), is utilized. A template route contains only frequent customers and it can be used as a master plan to determine their visiting sequence in the actual daily vehicle routing schedules. To work around this concept, the proposed solution approach adopts a two-level master–slave decomposition scheme. The master sub-problem is concerned with the design of a template route schedule in order to determine the service sequence and assignment of frequent customers to vehicle routes. On the other hand, the slave sub-problem uses the template as the basis to design the actual daily vehicle routes and service schedules for both frequent and non-frequent customers over multiple days. On the basis of the above, the ConVRP can be decomposed into a master and a slave sub-problem that needs to be solved in a sequential fashion. For this purpose, effective construction heuristics are utilized, while an innovative TS improvement method is also employed that operates on a dual mode basis and modifies both the template routes and the daily schedules. At first, an initial feasible template route schedule T is generated via a generalized insertion-based merging construction heuristic, considering all frequent customers simultaneously. To this end, the master TS mode (mTS) is applied to minimize the total traveling time. Although all local moves considered at the master mode modify directly the current T, the neighborhood evaluation and the associated feasibility checks are based on the resulting daily schedules. Upon termination of the template improvement level, the corresponding partially constructed daily vehicle routing plans are populated with the remaining set of non-frequent customers. Given the finalized set of daily schedules DS, the slave TS mode (sTS) is triggered for further improvement (post optimization). In an effort to preserve the feasibility, precedence constraints are also considered, in terms of customer assignments to vehicles and visiting sequence restrictions, as dictated by the corresponding T. Below, the algorithmic structure of the proposed templatebased solution framework is provided. Template-based solution framework //Parameters: b, zm, zs, vm, vs// DS 0;// Generate an empty set of daily schedules Nf Build set ();//Frequent customers Nnf Build set ();//Non frequent customers //Master Level T 0;// Generate an empty template route schedule T Merging Heuristic (Nf,b);// Template Initialization T mTS (T, zm, vm);// Template Improvement //Slave Level While day < D do { dsday 0;// Generate an empty daily schedule dsday Initialization (T, day, Nf);// Adaptation of T dsday Insertion Heuristic (day,Nnf);// Completion of dsday dsday sTS (dsday, zs, vs);// Post Optimization DS Add (dsday); } Return DS;
3.2. Template initialization The goal during the template initialization phase is to generate initial template schedules considering the whole set of frequent customers, regardless their required service frequency. For this
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purpose, a generalized insertion-based merging construction heuristic is utilized. At first, a solution is generated considering that one vehicle is assigned to each customer. Next, at each iteration two 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 combinations can be obtained. Let two routes r0 = {0, w, . . . ,i, j, . . . ,z, 0} and r00 = {0, u, . . . , l, 0}, where indexes i, j, w, z, u, l denote frequent customers. The first combination is to insert r00 after r0 , i.e., ^r ¼ f0; w; . . . ; i; j; . . . ; z; u; . . . ; l; 0g, with a corresponding saving c0u + cz0 czu. The second is to insert r00 in front of r0 , i.e., ^r ¼ f0; u; . . . ; l; w; . . . ; i; j; . . . ; z; 0g, with a saving c0w + cl0 clw. The third is to insert r00 within all positions between consecutive customers i, j, i.e., ^r ¼ f0; w; . . . ; i; u; . . . ; l; j; . . . ; z; 0g. The resulting saving formula can be defined as cij + c0u + cl0 ciu clj. In a manner similar, three additional merging combinations emerge by reversing the visiting sequence of route r0 . Due to route duration constraints, one may benefit if a sequential manner of construction, similar to traditional insertion-based construction methods, is followed. In fact, if during early stages of construction merging is performed mostly between routes with many customers, the final output tends to be poor, especially in terms of the number of vehicles deployed. To this end, an effort is made to favor merging combinations between single customer routes and routes containing more than two customers, by multiplying the corresponding saving with a parameter b that will regulate the degree of sequential construction. During the construction of the template route schedule, only feasible merging combinations are considered with respect to the resulting daily schedules. Clearly, template routes contain a mix of frequent customers with different combinations of daily service requirements. However, it is straightforward to check the feasibility of the resulting daily schedules. Let aid denote the arrival time of the vehicle at the location of customer i on day d. The template T is feasible, in terms of vehicle capacity and route duration constraints, if the following inequalities hold throughout the planning P horizon: i2r qid wid 6 Q 8d 2 D; r 2 T (vehicle capacity); aid + wid(sid + ci0) 6 Bwid "d 2 D, i 2 Nf (route duration); and jaida aidb j 6 L Bðwida þ widb 2Þ 8i 2 N f ; da ; db 2 D; a – b (service time consistency). 3.3. Generation of daily schedules Given the master template route schedule, the corresponding daily vehicle routing schedules can be determined in a sequential fashion for each day of the planning horizon as follows. Initially, the template route schedule is adopted and the frequent customers that do not require service on that particular day are removed. By doing so, the assignment to vehicles and the visiting sequence of frequent customers are preserved. Subsequently, the non-frequent customers that may require service are identified and designated as unrouted customers. To this end, a parallel construction heuristic is employed to insert the remaining unrouted customers and to finalize the partially built daily schedules. The proposed parallel construction heuristic builds iteratively a solution by selecting and adding one by one unrouted customers to the partially built vehicle routes, until a feasible daily schedule is generated for both frequent and non-frequent customers. At each iteration, the selection of customers and insertion positions between adjacent routed customers is made according to a greedy criterion that measures the insertion cost. From the implementation viewpoint, only feasible insertion positions are considered. However, if no feasible insertion positions can be found for the remaining unrouted customers, new vehicle routes are initialized. An illustrative example of the dependence between the master and the slave subproblems is the following. Let a planning horizon
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of three days d1, d2 and d3, and two disjoint sets of frequent customers 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 two routes, e.g., tr1 = {0, w, l, i, s, 0} and tr2 = {0, h, j, u, 0}. During the first day, the following set of customers requires service, e.g., {h, i, j, l, u, y, e}. Therefore, the resulting partial daily schedule for d1 consists 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 is to find the least cost insertion positions for non-frequent customers 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} and dr2 = {0, l, y, e, i, 0}. 3.4. Tabu Search improvement method The template and the corresponding daily vehicle routing schedules are improved, in terms of distance minimization, by means of a dual-mode TS algorithm. The master TS mode (mTS) is applied to the template route schedule and considers explicitly the feasibility of the master vehicle routing plan with respect to the resulting daily schedules, while the slave TS mode (sTS) is applied to the actual daily vehicle routing schedules and takes into account the assignment and visiting sequence requirements dictated by the associated template route schedule. In both cases, the solution space is explored on the basis of simple edge-exchange neighborhood structures. However the degrees of freedom of mTS are much higher compared to sTS, since the feasible region of allowable move operators of the latter is greatly reduced. In broad terms, TS seeks to explore the solution space by moving at each iteration from a solution s to the best admissible solution s0 in a subset Uy(s) of a neighborhood structure y. The short term memory records the most recently visited solutions and prevents revisiting them for a predefined number of iterations v (tabu tenure) in order to avoid cycling. The tabu status of a neighboring solution can be overridden, only if predefined aspiration criteria are met (i.e. new local optimum solutions). The overall procedure iterates until some termination conditions are met (maximum number of iterations z without observing any further improvement in our case) and the best encountered local optimum solution s⁄ is returned. The neighborhood structures considered in the proposed implementation are based on traditional O(n2) edge-exchange local moves, namely intra- and inter-route 2-Opt, 2-Opt⁄, 1–0 Relocate and 1–1 Exchange (swap) (Tarantilis, Kiranoudis, & Vassiliadis, 2002). The oscillation among these multiple neighborhood structures is purely stochastic with equal selection probability. At each iteration, given the allowed set of neighbors Uy(s), the best admissible neighbor s0 replaces the current solution s, while both the forward and reversal attributes (i.e. edges being added or deleted) of the corresponding local moves are stored within the tabu list. For the evaluation of the neighborhood structures, a lexicographic ordering search is followed and acceleration mechanisms suggested by Repoussis, Tarantilis, and Ioannou (2009) that reduce the evaluation complexity are also incorporated. To this end, emphasis is given on direct feasibility gains based on the characteristics of the proposed decomposition scheme. Regarding the slave TS mode, one may benefit from the inherent precedence constraints, since the possible removal and re-insertion positions of frequent customers are very few and only non-frequent customers can freely change positions. Let two template routes, e.g., tr1 = {0, w, l, i, s, 0} and tr2 = {0, h, j, u, 0}, and the corresponding daily vehicle routing schedule, e.g., dr1 = {0, h, j, u, 0} and dr2 = {0, l, y, e, i, 0} (see also Section 3.3). Non-frequent customers y and e do not follow any precedence constraints and all feasible intra and inter-route local moves are allowed. On the other hand, frequent 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 customer l on the same route, while non feasible intra-route local move exist for frequent customers h, j and u. Clearly, the feasibility checks of a local move on the daily schedules, in terms of route duration, vehicle capacity and service time consistency, can be made in almost constant time. On the other hand, considering the mTS mode the computational effort of feasibility checks slightly increases, because a local move on the template route schedule affects the feasibility of the resulting daily schedules. Thus, one has to measure explicitly the effect of local moves 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, several computational experiments are performed using the benchmark data set generated for the ConVRP by Groër et al. (2009). This data set consists of 12 medium-scale problem instances, divided into two groups. The first group contains 7 problem instances, which consider only vehicle capacity constraints. The second group of problem instances includes both vehicle capacity and route duration constraints. The number of customers ranges from 50 to 199, while a 5-day planning horizon with a non fixed fleet size is assumed for all problem instances. Finally, it is worth to mention that the objective function values refer to the sum of all the times spent along each vehicle route (total en route time), including distance traveled and service times at customers’ locations. 4.2. Parameter settings and termination conditions Towards the design of approaches with the least possible number of user-defined parameters, the proposed solution framework incorporates three parameters, namely the degree of sequential construction b, the tabu list size v, and the number of maximum number of TS iterations without observing any further improvement z (termination condition). Based on computational experience, these parameters are relatively insensitive to the characteristics of the problem instances considered, while using simple adjustments one can determine very well performing parameter settings with modest effort within reasonable value ranges. In this paper, several intuitively selected combinations were experimentally tested, choosing the one that yielded the best average output. Below, suitable value ranges for each parameter are provided. Regarding the generalized insertion-based merging construction heuristic, parameter b needs to be determined. Clearly, b must be set greater than or equal to 1 in order to boost the degree of sequential construction. However, large values of b may affect the impact of the other types of savings. To this end, a value range between 1 and 1.5 was found to perform well for most ConVRP problem instances, especially in terms of total number of vehicles employed. In what follows, b was fixed to 1.3 at all simulation runs. Two parameters must be defined considering the TS improvement method for each mode; the tabu list size v and the number of maximum number of TS iterations z. Between the master and the slave TS modes, the most labor intensive and critical to the overall performance is the former. As such, regarding the tabu list size vm for mTS, a range between 30 and 60 is typically used by standard TS implementations of the literature and seems to fit well for ConVRP problem instances. Contrary, a rather smaller range between 10 and 20 suits for vs, due to the very limited search capacity of sTS. On the other hand, as z increases the efficiency of local
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search also increases. However, a balance between efficiency and effectiveness is needed, since large values of z may result in excessive computational time consumption. To this end, values of zm and zs up to 20,000 and 500 iterations, respectively, seem to provide a good compromise (see also Section 4.4 for an effectiveness analysis w.r.t. zm). The proposed template-based Tabu Search algorithm was implemented in C++ (Visual Studio 2008). All computational experiments reported in subsequent sections were performed on a 2.8 GHz Intel Xeon personal computer, assuming 5 simulation runs for each problem instance with fixed parameter settings as follows: 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 according to the sum of total traveling times along each vehicle route, including service times at customers’ locations. Additionally, a possible secondary measure of solution quality is the total number of vehicles. However, these two objectives can be either conflicting or complementary. For this reason, all comparisons with existing results are made only towards the total traveling times. Given the definition of consistent service constraints regarding frequent customers, there are two alternatives; either to ensure the same visiting sequence throughout the planning horizon or to set a maximum time limit L w.r.t. the maximum service time difference between the earliest and latest vehicle arrival times lmax (i.e. lmax 6 L). Thereafter, in order for the competition between different solution approaches for ConVRP instances to be fair and objective, both types of consistent service constraints are imposed either separately or simultaneously. In the former case, only visiting sequence restrictions are imposed, while in the latter case, besides visiting sequence restrictions for frequent customers, the maximum arrival time differential lmax reported by Groër et al. (2009) is set as the maximum time limit L and it is enforced as an additional constraint. Table 1 summarizes the results obtained on the benchmark data sets of Groër et al. (2009). The first line lists the authors using the following abbreviations: ConRTR stands for (Groër et al., 2009) and TTS stands for the proposed template-based Tabu Search algorithm. Initially, the first set of columns illustrate the total traveling time (TT), the total number of vehicles (NV) and the maximum arrival time differential lmax reported by (Groër et al., 2009) for each problem instance. The second set of columns (TTS-L) reports the best results obtained considering that L equals lmax of Groër et al. (2009). To this end, %Gap denotes the percentage gap w.r.t. ConRTR, while CT stands for the total computational time in seconds. The third set of columns (TTS-F) reports the best results obtained
without considering any additional constraints towards lmax (i.e. L is set to infinity). Finally, the bottom section of the table provides the corresponding average results over all problem instances. Based on the computational results provided in Table 1, TTS seems to be highly competitive compared to the current state-of-the-art for the ConVRP. In particular, all existing results were improved both in terms of total traveling times and total number of vehicles. Considering the TTS-L configuration, with additional constraints towards the maximum service time differential of frequent customers, cost reductions up to 4.76% are observed w.r.t. ConRTR. Furthermore, the average number of vehicles deployed is reduced from 10.50 to 9.00 (a total of 19 vehicles), while the customer service, expressed in terms of lmax, is improved by 17% from 46.39 to 39.33 (average values for all problem instances). Similar are the figures considering the TTS-F configuration. In particular, costs reductions up to 5.34% are obtained, in terms of total traveling time, while the total number of vehicle is further reduced to an average of 8.92 vehicles per problem instance. Finally, it is worth to highlight that in both configurations new best solutions are obtained for 11 out of 12 problem instances, while the average results over multiple simulation runs revealed very small variations (close to 0.6% worst case). Another point of interest is to examine the effect of service time restrictions. As expected, when this type of consistent service constraint is relaxed better results are obtained. In particular, cost reductions close to 0.8% on average are observed between TTS-L and TTS-F configurations, while the resulting average lmax is increased from 39.33 to 50.66, respectively. To this end, it seems that considerable improvements can be achieved with a rather small compromise towards the maximum allowable service time limits. Therefore, depending on the nature of services provided to customers, alternative cost effective operational settings emerge for different types and mixes of consistent service constraints. 4.4. Efficiency analysis Competitive advantage of the proposed template-based Tabu Search algorithm is its fairly simple algorithmic structure and the small number of parameters introduced. In particular, the only user-depend parameter that is critical to the performance and the search capability of the proposed solution approach, at the cost of the expense in computational time, is the total number of mTS iterations without observing any improvement. Therefore, it is important to examine the computational time consumption w.r.t. the number of iterations and the corresponding improvements observed. Table 2 summarizes the mean and cumulative results obtained for different values of z regarding mTS, ranging from 2500 to 20,000. In particular, MNV, MTT and MCT stand for mean number
Table 1 Comparative analysis on Groër et al. (2009) benchmark data sets. Problem instance
Pr.01 Pr.02 Pr.03 Pr.04 Pr.05 Pr.06 Pr.07 Pr.08 Pr.09 Pr.10 Pr.11 Pr.12 Average
ConRTR
TTS-L
TTS-F
TT
NV
lmax
TT
NV
lmax
%Gap
CT
TT
NV
lmax
%Gap
CT
2282.14 3872.86 3628.22 4952.91 6416.77 4084.24 7126.07 7456.19 11033.54 13916.80 4753.89 3861.35
5 11 7 12 16 5 12 9 14 18 7 10
24.38 34.26 22.87 27.53 26.93 63.47 83.96 73.04 106.43 60.17 16.10 17.58
2210.56 3622.71 3451.10 4572.00 5732.62 4096.87 6752.36 7279.39 10585.10 13120.40 4721.09 3607.88
4 9 6 10 13 5 10 8 13 16 6 8
21.99 27.75 21.92 25.15 19.99 55.38 63.28 62.01 84.76 57.17 15.68 16.91
3.14 6.46 4.88 7.69 10.66 0.31 5.24 2.37 4.06 5.72 0.69 6.56
80 93 369 388 550 70 161 539 947 1052 480 172
2180.39 3594.56 3419.74 4572.06 5720.86 4096.87 6752.36 7279.39 10560.90 12989.80 4543.27 3607.88
4 9 6 10 13 5 10 8 13 15 6 8
27.22 42.23 28.49 27.51 51.78 55.38 63.28 62.01 106.43 74.72 51.98 16.91
4.46 7.19 5.75 7.69 10.85 0.31 5.24 2.37 4.28 6.66 4.43 6.56
38 72 170 745 554 45 104 194 909 911 175 125
46.39
5812.67
39.33
4.76
408
5776.51
50.66
5.43
337
6115.42
10.50
9.00
8.92
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Table 2 Efficiency 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. MTT MNV MCT CTT CNV
3.87 5847.99 9.17 84 70175.84 110
4.09 5840.70 9.08 118 70088.38 109
4.40 5822.63 9.00 158 69871.57 108
4.71 5815.57 9.00 215 69786.84 108
4.71 5815.57 9.00 251 69786.84 108
4.72 5815.42 9.00 319 69785.01 108
4.72 5815.42 9.00 373 69785.01 108
4.76 5812.67 9.00 408 69752.08 108
Fig. 1. % Improvement ratio vs z (mTS termination condition).
of vehicles, mean traveling time and mean computational time, while CNV and CTD stand for cumulative number of vehicles and cumulative traveling time over all problem instances. Additionally, the second row (% Impr.) shows the percentage improvement of TTS-L configuration w.r.t. the results reported by ConRTR (see Table 1). Based on the result provided in Table 2, it is evident that even for very small values of z and computational times less than 1.5 min, high quality solutions are obtained (improvement close to 3.87%). Similar are the figures concerning the total number of vehicles. On the other hand, as moving towards larger values of z, the performance of the proposed solution approach also increases. However, from a point and after this is made at the expense of computational time. Note that only a 0.04% improvement is achieved from 10,000 to 20,000 iterations, while the CT consumption of the latter is twice as much the consumption of the former. Fig. 1 illustrates the % improvement over the corresponding computational time consumption w.r.t. the number of iterations. To this end, values below 15,000 iterations seem to provide good compromise between efficiency and effectiveness for practical applications. Note that experimentally the effect of z values regarding sTS is minor, because of the structure of the benchmark data sets and the small number of non-frequent customers.
5. Conclusions Modern companies tend to design and follow customer-oriented rather than demand-oriented approaches, especially for those active in the service sectors, in order to achieve the goals of sustainable development. Towards this emerging field of research, this paper deals with multi-period vehicle routing problems considering several different types of consistent service constraints. More specifically, a generic template-based Tabu Search algorithm is proposed for the so-called Consistent Vehicle Routing Problem. The latter involves the design of a set of minimum cost vehicle routes to service a set of customers with known demands over a planning horizon of multiple days. Customers may
receive service either once or with a predefined frequency; however frequent customers must receive consistent service, such that the customers’ visiting sequence remains the same and/or the maximum service time difference between the earliest and latest vehicle arrival times does not exceed a maximum time limit, and the service is performed by the same vehicle throughout the planning horizon. The proposed solution approach adopts a two-level master– slave decomposition scheme based on the fact that the vehicle routing plans over multiple days are highly interrelated due to consistent service constraints and cannot be treated separately. In specific, the master sub-problem is concerned with the design of a template route schedule in order to determine the service sequence and assignment of frequent customers to vehicle routes. On return, the slave sub-problem uses the template as the basis to design the actual daily vehicle routes and service schedules for both frequent and non-frequent customers over multiple days. To this end, a Tabu Search algorithm is employed to minimize the total traveling time. The latter operates on a dual mode basis and modifies both the template routes and the actual daily schedules in a sequential fashion. Experimental results on benchmark data sets of the literature demonstrated the competitiveness of the proposed solution approach. Compared to existing results, it proved to be highly competitive improving the best reported cumulative and mean results over all problem instances with very reasonable computational requirements and fixed parameter settings. Furthermore, apart from cost improvements in terms of total traveling times, it managed to reduce significantly the corresponding fleet size. To this end, the small number of user-depended parameters and the fairly simple and generic algorithmic structure indicate its applicability towards real life applications. A worth pursing research direction is towards the design of adaptive memory programming approaches based on the concept of template route schedules. On the other hand, as research moves towards more realistic and rich problems, the development of medium and large scale benchmark data sets with varying mixes of frequent and non-frequent customers and different operational settings is of great interest.
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Acknowledgments Support from the Senate Committee of the Athens University of Economic and Business for the ‘‘Basic Research Funding Program (BRFP)’’ is gratefully acknowledged. References Beltrami, E. J., & Bodin, L. D. (1974). Networks and vehicle routing for municipal waste collection. Networks, 4(1), 65–94. Francis, P. M., Smilowitz, K. R., & Tzur, M. (2008). The period vehicle routing problem and its extensions. In B. Golden et al. (Eds.), The vehicle routing problem (pp. 73–102). New York: Springer Science+Business Media LLC.
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