A Hybrid Algorithm for the Capacitated Vehicle Routing Problem with Three Dimensional Loading Constraints

Transcription

1 A Hybrid Algorithm for the Capacitated Vehicle Routing Problem with Three Dimensional Loading Constraints Andreas Bortfeldt Diskussionsbeitrag Nr. 460 Dezember 2010 Diskussionsbeiträge der Fakultät für Wirtschaftswissenschaft der FernUniversität in Hagen Herausgegeben vom Dekan der Fakultät Alle Rechte liegen bei den Autoren

2 A Hybrid Algorithm for the Capacitated Vehicle Routing Problem with Three Dimensional Loading Constraints Andreas Bortfeldt Abstract: The capacitated vehicle routing problem with three dimensional loading constraints combines capacitated vehicle routing and three dimensional loading with additional packing constraints which concern, for example, unloading operations. An efficient hybrid algorithm including a tabu search algorithm for routing and a tree search algorithm for loading is introduced. Computational results are presented for all publicly available test instances. Most of the best solutions previously reported in literature have been improved while the computational effort is drastically reduced compared to other methods. Key words: Capacitated vehicle routing, loading, three dimensional packing, tabu search, tree search. Fakultät für Wirtschaftswissenschaft, FernUniversität in Hagen Profilstr. 8, D Hagen, BRD Tel.: 02331/ Fax: 02331/ E Mail: andreas.bortfeldt@fernuni hagen.de 2

3 1 Introduction A Hybrid Algorithm for the Capacitated Vehicle Routing Problem with Three Dimensional Loading Constraints Andreas Bortfeldt University of Hagen, Department of Information Systems, Hagen, Germany This paper deals with the capacitated vehicle routing problem with three dimensional (3D) loading constraints (3L CVRP) that represents a combination of vehicle routing and 3D container loading, which was introduced by Gendreau et al. (2006). The 3L CVRP results from the capacitated vehicle routing problem if the goods to be shipped to the customers are rectangular 3D boxes of a given size that are to be packed into identical rectangular loading spaces of the vehicles used. Hence, a problem solution comprises a set of routes and a packing plan for each route. Such a packing plan must include placements of all boxes of the customers that are visited on the given route. Moreover, each packing plan has to meet some packing constraints, e.g., a weight limit for the boxes in a vehicle has to be observed. Compared to the original capacitated vehicle routing problem the 3L CVRP allows for a more detailed modeling of mixed cargo transportation by vehicles. First, with shipments of packaged goods the practicability of a planned route can only be guaranteed unerringly if the route is accompanied by a 3D packing plan. In other words, modelling a practical use case as 3L CVRP has the advantage that calculated routing plans can definitely be implemented. Moreover, several frequently occurring packing constraints, e.g., the loading of fragile goods, can only be considered as part of 3L CVRP (cf. Iori and Martello 2010). Thus the 3L CVRP is of high practical relevance. On the other hand, the 3L CVRP is an extremely hard and challenging optimization problem since it results as a combination of two problems routing of vehicles and loading boxes into them that are already NP hard and difficult to solve in practice. In this paper a hybrid algorithm for solving the 3L CVRP is proposed, consisting of a tabu search algorithm (TSA) for vehicle routing and a tree search algorithm (TRSA) for packing boxes. In comparison with all available 3L CVRP solution procedures the hybrid algorithm will show its strength in terms of solution quality and efficiency. The remainder of the paper is organized as follows: In Section 2 the 3L CVRP is formulated, while in Section 3 the relevant literature is reviewed. In Section 4 the hybrid algorithm is described and in Section 5 the method is compared to other algorithms from literature by numerical experiments. Finally, some conclusions are drawn in Section 6. 2 Problem formulation We are given v max identical vehicles with a rectangular loading space with length L, width W and height H. Let V = {0,1,...,n} be a set of n+1 nodes that correspond to a depot (node 0) and n customers (nodes 1,...,n). Let E be a set of undirected edges (i,j) that connect all node pairs and let G = (V,E) be the resulting graph. Let a distance c ij (c ij > 0) be assigned to each edge (i,j) (0 i < j n). Each customer i (i = 1,...,n) is to be supplied with a set of m i rectangular packing pieces (boxes) I ik (k = 1,...,m i ) which are initially located at the depot. Let the box I ik have the length l ik, the width w ik, and the height h ik (i = 1,...,n, k = 1,...,m i ). Let the loading space of each vehicle be embedded in the first octant of a Cartesian coordinates system in such a way that the length, width and height of the loading space lie parallel to the x, y and z axes. The placement of a box I ik in a loading space is given by the coordinates x ik, y ik and z ik of the corner of the box that is closest to the origin of the coordinates system; in addition, an orientation index o ik indicates which of the six possible spatial orientations is selected (i = 1,...,n, k = 1,...,m i ). A spatial orientation of a box is given by a one to one mapping of the three box dimensions and the 3

4 three coordinate directions. A packing plan P for a loading space comprises one or more placements and is regarded as feasible if the following three conditions hold: (FP1) each placed box lies completely within the loading space; (FP2) any two boxes that are placed in the same truck loading space do not overlap; (FP3) each placed box lies parallel to the surface areas of the loading space. Figure 1 shows a loading space with placed boxes. front z back I21 door I11 I12 y I22 x Figure 1. Loading space of a vehicle with placed boxes. A feasible route R is a sequence of three or more nodes that starts and ends at the depot and a customer must not occur more than once in the sequence. A solution of the 3L CVRP is a set of v ordered pairs (R l, P l ), where R l is a route and P l is a packing plan (l = 1,...,v). In order to be feasible, a solution must fulfil the following three conditions: (F1) all routes R l and packing plans P l are feasible (l = 1,...,v); (F2) each customer i belongs to exactly one route R l (i = 1,...,n); (F3) the packing plan P l for a route R l contains exactly one placement for each customer i of R l and each of its boxes I ik (k = 1,...,m i ). Additionally, the following packing constraints are involved: (C1) Unloading sequence constraint. When customer i (i = 1,...,n) is visited, it must be possible to unload all his boxes I ik (k = 1,...,m i ) exclusively by using movements parallel to the longitudinal axis of the loading space. Consequently, no box demanded by another customer that is visited later than customer i must be placed above a box I ik (of customer i) or between I ik and the rear of the vehicle (cf. Figure 1). This constraint is also referred to as the LIFO (last in, first out) constraint. (C2) Weight constraint. Each box I ik has a positive weight d ik (i = 1,..,n, k = 1,...,m i ) and the total weight of all the boxes placed in a vehicle must not exceed a maximum load weight D. (C3) Orientation constraint. The orientation of all boxes is fixed with respect to height while horizontal 90 rotations are permitted. (C4) Support constraint. If a box is not placed on the floor, a certain percentage a of its base area is to be supported by other boxes. If b and c are the horizontal dimensions of a box above the floor, a portion of the base area of size a b c is to be placed on other boxes. (C5) Stacking constraint. A fragility flag f ik is assigned to each box I ik (i = 1,..,n, k = 1,...,m i ). If a box is fragile (f ik = 1) only other fragile boxes may be placed on its top surface, whereas both fragile and non fragile boxes may be stacked on a non fragile box (f ik = 0). The 3L CVRP consists of determining a feasible solution with at most v max routes that minimises the total travel distance (of all routes) and meets the constraints (C1) to (C5). Figure 2 illustrates a simple 3L CVRP instance and the routes of a possible solution (while edges that do not belong to these routes are not shown). 4

5 I 1 2 I12 I I I 31 I I33 I51 I52 Figure 2. I 41 legend: 4 0 I : depot VRLP instance with routes of a solution. : customer : selected, directed edge 3 Literature review The CVRP and related vehicle routing problems, e.g. the VRP with time windows (VRPTW), belong to the most intensively studied problems in combinatorial optimisation. The reader is referred to the books by Toth and Vigo (2002) and Golden et al. (2008) for a comprehensive survey on vehicle routing. Successful exact methods were proposed by Fukasawa et al. (2006), Baldacci et al. (2008) and Baldacci and Mingozzi (2009). Effective meta heuristic methods were recently developed by Prince (2004), Mester and Bräysy (2007) and Pisinger and Ropke (2007). More recent surveys of meta heuristic methods for the VRP were contributed by Cordeau and Laporte (2004) and Cordeau et al. (2005) while Baldacci et al. (2007, 2010) reviewed exact approaches for vehicle routing problems. An up to date typology of cutting and packing (C&P) problems was developed by Wäscher et al. (2007); this paper also includes an extensive bibliography of more recent C&P research work. Several 3D packing problems seem to be relevant in the context of the development of solution methods for the 3L CVRP. The 3D bin packing problem (3D BPP) requires packing a set of boxes in a minimum number of identical containers, while in the 3D strip packing problem (3D SPP) all given boxes have to be packed completely into a container with a fixed cross section so that the variable container length is minimised. Finally, in the 3D knapsack problem (3D KP) with one container to be loaded a subset of the given boxes of maximum total volume (or value) is to be loaded into the container. All these problems are NP hard and only relatively small instances (in terms of the number of items) have been solved to optimality so far. Exact methods for the 3D BPP were proposed by Martello et al. (2000, 2007) while Fekete et al. (2007) devised a more general approach for exactly solving multidimensional packing problems. Heuristic methods for the 3D BPP were developed by Faroe et al. (2003) and Crainic et al. (2008) while Bortfeldt and Mack (2007) and Allen et al. (2010) proposed heuristics for the 3D SPP. Recent heuristic methods for the 3D KP were devised by Parreño et al. (2007, 2008) and Fanslau and Bortfeldt (2010). Effective heuristics for C&P problems are mostly based on meta heuristic paradigms or tree search approaches. Iori and Martello (2010) survey the state of the art in the field of routing problems with loading constraints. Together with the 3L CVRP, the respective 2D problem variant (2L CVRP) and other 5

6 related problems are also examined. Generally the literature is still very limited and there are only few solution procedures for the 3L CVRP: Gendreau et al. (2006) suggested a two stage tabu search algorithm (TSA). The outer TSA serves for planning the routes, while the inner TSA solves a 3D strip packing problem for loading a vehicle in accordance with a given customer sequence. A move of the outer TSA is defined through shifting a customer to another route. For each move tested by the outer TSA the inner TSA is to be invoked again for all changed routes. Tarantilis et al. (2009) designed a hybrid procedure combining the strategies tabu search and guided local search and they use a collection of plain packing heuristics. Fuellerer et al. (2010) developed an ant colony algorithm (ACO), using fast but effective packing heuristics and different heuristic information measures for routing and packing. Procedures for the 3L CVRP often result as enhancements of 2L CVRP solution procedures. Metaheuristic procedures for the 2D case were proposed by Gendreau et al. (2008), Fuellerer et al. (2009) and Zachariades et al. (2009). Lori et al. (2003) presented an exact branch and cut algorithm for the 2L CVRP. Moura and Oliveira (2008) considered a different 3D routing problem with loading constraints that is derived from the VRPTW, i.e. time windows are included. The most usual vector valued objective function for the VRPTW is also adopted: the number of vehicles is to be minimized with higher priority, whereas the total travel distance is to be minimized with lower priority. Finally, the weight constraint (C2) and the stacking constraint (C5) are not taken into account (while the other packing constraints (C1), (C3) and (C4) are observed, cf. Section 2). The authors proposed two heuristics that are in part based on earlier published methods for the VRPTW and 3D KP. The paper by Moura and Oliveira reveals that there are several relevant variants of the routing problem with 3D loading constraints for which side constraints regarding routing and loading may vary. 4 The hybrid algorithm The new hybrid algorithm for solving the 3L CVRP consists of a tabu search algorithm (TSA) for routing vehicles and a subordinate tree search algorithm (TRSA) for loading the boxes of all customers of a route into the loading space of a vehicle. 4.1 The tabu search algorithm for routing vehicles The overall procedure of the TSA is shown in Figure 3. First, an initial solution is generated at random (to be described below) and the best solution sbest is initialized accordingly. As mentioned above, a solution comprises a set of routes and associated packing plans. The tabu list is initialized as empty, has a constant length tllength and contains the inverse moves of the last tllength implemented moves. The following loop is cycled until a time bound timelimit is exceeded, or a maximum number of iterations maxiter is reached. Per iteration the best non tabu move is determined with regard to the current solution s and then applied to s. Subsequently the tabu list and, if necessary, the best solution are updated. Finally, the best found solution is output. route_vehicles (in: instance data, parameters, out: best solution sbest) generate initial solution s at random; set best solution sbest := s; set tabulist := ; repeat determine best non tabu move mbest; apply mbest to s: s := s mbest; update tabulist and sbest where necessary; until (timelimit exceeded or maximum number of iterations maxiter reached); output(sbest). Figure 3. Overall procedure of TSA for routing vehicles. Generally a feasible move m with regard to a current solution s has the following components: a set of one or more old routes of s that are eliminated by the move, a set of one or more new routes 6

7 that replace the old ones and feasible packing plans for all new routes. Clearly, a move is only feasible if there is a feasible packing plan for each new route, placing all boxes of all customers of that route and observing the constraints (C1) to (C5). Thus producing a feasible move requires generating a feasible packing plan for each new route. Consider a solution with the routes (0,1,2,3,0) and (0,4,5,6,0) and suppose a move m swaps the customers 3 and 6. Then new feasible packing plans have to be generated for the routes (0,1,2,6,0) and (0,4,5,3,0) to ensure that the boxes of the customers 1,2 and 6 and of the customers 4,5 and 3, respectively, can be stored together in one loading space in a feasible way. If this turns out to be impossible the move m must be discarded. On the other hand, to apply a feasible move m to a solution s means replacing the old routes by the new ones, including the associated packing plans. Altogether four move types are applied in the TSA: (1) An inter swap move swaps two customers of two different routes. (2) An inter shift move shifts one customer from one route to any position in a second route. (3) An intra swap move swaps to customers of the same route. (4) An intra shift move shifts one customer to any different position within the same route. Note that moves of these types cannot lead to a greater number of vehicles (number of routes) if applied to a given solution. The four move types are not used simultaneously in the tabu search. Instead a two phase approach is adopted that was successfully implemented in VRPTW algorithms (cf., e.g., Homberger and Gehring, 2005). The two phases can be characterized as follows: The first phase starts after the initial solution has been generated and takes just as long as the number of vehicles v in the best solution sbest exceeds the given threshold v max (see Section 2). Hence in the first phase the goal is to reduce the number of vehicles until the threshold v max is reached(by the best solution). In this phase only the inter swap and inter shift move types are applied since the other move types cannot contribute to the phase goal. Per iteration all feasible moves of both these types are tried. In both the move types two old routes are replaced generally by two new routes. A route r 1 is said to be lighter than a route r 2 if the box volume stowed in the loading space of r 1 is smaller (not greater) than the box volume stowed in the loading space of r 2.The move quality is then measured by the stowed box volume of the lighter route of the move that should be minimum. Thus a move m 1 is better than a move m 2 if m 1 includes the lightest of the four new routes (of both moves). By means of this quality criterion the stepwise reduction of the total box volume and consequently of the number of customers is promoted in one or more routes and one can hope that entire routes can be saved in the end. Of course, a route can be finally eliminated only by means of an inter shift move that relocates the last customer of a route to another one. In the first phase the incumbent best solution sbest is replaced by the solution of an iteration s if s reduces the number of vehicles of sbest or if s reduces the total travel distance (retaining the number of vehicles). The second phase begins after the threshold v max has been reached for the first time. Now the goal is to reduce the total travel distance and all feasible moves of all four move types (introduced above) are tried per iteration. The quality of a move is now measured by the total travel distance of the resulting set of routes that should be as small as possible. Note that a move or even the best move with regard to the current solution s can also lead to an increased total travel distance. In the second phase of the search the incumbent best solution sbest is replaced by the solution of an iteration s if the total travel distance is reduced or if the number of vehicles is reduced and the total travel distance is not increased. Hence a reduction of the number of vehicles might also occur in the second search phase. Determining the best move of an iteration could be done in the following manner: first all feasible moves (including packing plans) are produced and then the moves are evaluated and the best move is selected (following the slogan Packing first Evaluating second ). However, this procedure would result in huge effort for generating packing plans for new routes. Consider a 3L CVRP instance with 100 customers and suppose that there is a current solution s with ten routes of ten customers each. It can easily be seen that independent of the search phase (i.e. the permitted move types) more than vehicles have to be packed experimentally only in one iteration. Therefore, another 7

8 procedure to determine the best move per iteration is chosen that is shown in Figure 4 and commented below. select_best_move (in: solution s, phase, out: best move mbest) generate all non tabu moves m = m(s, phase) without new packing plans; evaluate quality of all moves regarding to phase; sort moves by quality descending; cross move list and generate packing plans for new routes per move until nmsel feasible moves are found; select best move mbest at random among the nmsel moves; return mbest. Figure 4. Determining the best move of an iteration. The procedure is based on the fact that evaluating moves is possible before packing plans for the new routes are available. While packing a loading space with boxes is computationally expensive evaluating a move using a phase specific criterion (see above) is a cheap operation. Therefore much effort can be saved if the search for the best move is organized following the motto Evaluating first Packing second : First all potential moves are generated as pure VRP moves without generating packing plans, evaluated by the move quality criteria introduced above and sorted by move quality in descending order. Packing plans have then to be generated only for the best potential moves until the first (and consequently best) feasible move is found. However, to diversify the search (and being taught by experiments) the packing checks are continued until a small set of nmsel (e.g. nmsel = 3) feasible and high quality moves is found and the best move (to be returned) is then determined at random among the nmsel found feasible moves. Of course, it may occur that dozens or even hundreds of potential moves have to be checked in terms of packing before the first nmsel feasible moves are found. Nevertheless, compared to the first Packing first Evaluating second approach relatively few loading spaces are packed per iteration. On the other hand, the move concept seems to be flexible since multiple move types are applied simultaneously and no potential (VRP) moves are excluded arbitrarily from the search. To further accelerate the search all routes that have ever been tried are collected in an unsorted vector called route cache. A single entry of that cache includes a route (series of customers) and a packing flag stating whether the boxes of all customers can be stowed together in a loading space or not (by means of the packing algorithm used). The route cache is completed by customer indices, one for each customer. The index of a customer i comprises all positions of the route cache at which a route can be found that visits customer i. Whenever a route is tested in terms of packing at first the route is searched for in the route cache, using the index of the first customer in the given route, and if the route is found a call of the packing algorithm is unnecessary and can be saved. Otherwise, i.e. if the route is tested for the first time, the packing algorithm is called and the route is inserted in the cache (and some customer indices) together with the outcome of the packing test. The initial solution of the tabu search is generated by means of a multi start randomized savings procedure (cf., e.g., Toth and Vigo 2002). First a link list is provided, including all potential links of any two customers i and j (1 i < j n). The link list is sorted in descendent order by the saving calculated as saving(i,j) := c i0 + c j0 c ij. Afterwards ntrials solutions are generated at random using the link list and the best solution is taken as the initial solution of the tabu search. If the number of vehicles (or routes) in the best solution sbest exceeds the prescribed maximum number v max then sbest and a newly generated solution s are compared according the comparison rule specified for the first phase of the tabu search (see above); otherwise the rule for the second phase is applied. Each of the ntrials solutions s is initialized by solution s 0 consisting of n shuffle routes 0 i 0 (1 i n). Then the link list is passed through and each of the potential links is chosen at random, i.e. with probability 0.5. A chosen link between customers i and j is examined under two aspects. First a routing check is carried out to guarantee that neither customer i nor j is connected with more than two customers and no cycle results by the link. Second a packing check is performed to ensure that the boxes of all customers of the resulting route can be stored in one loading space observing all 8

9 specified constraints. Only if the outcome of both checks is positive the link is implemented, i.e. customers i and j and the respective partial routes are connected. In each trial a complete solution results from solution s 0 by all chosen and implemented links. 4.2 The tree search algorithm for packing boxes For a given route and the related set of boxes of all visited customers the tree search algorithm (TRSA) tries to generate a so called complete solution, i.e. a packing plan by which all boxes are stowed. The TRSA performs a depth first search and each node of the search tree has three components: (1) a (partial) solution currentsolution of zero or more implemented placements, (2) a set freeboxes of free (not yet packed) boxes and (3) a list potentialplacements of potential placements that could be carried out in addition. An implemented or potential placement is characterized by a box and the related customer, the coordinates of the lower left back corner (called reference point) of the placed box (see Fig. 1) and its spatial orientation. The depth first search is carried out by the recursive procedure add_placement shown in Figure 5. Before the procedure is called for the first time currentsolution is set empty, freeboxes is filled by all boxes and the list potentialplacements is filled by all feasible box placements in the lower left back corner of the loading space LxWxH. The procedure checks first whether the search can be aborted. The search is finished if currentsolution is a complete solution that is returned. It is also finished if the number of calls of add_placement exceeds a given limit maxapcalls. Moreover, the current instance of the procedure is aborted if there is at least one free box without a potential placement since in most cases a complete solution, comprising all given boxes, can no longer be achieved. add_placement (inout: currentsolution, in: freeboxes, potentialplacements) // abort search if appropriate if all boxes stowed in currentsolution or number of procedure calls > maxapcalls then stop endif; // abort search on given path if appropriate if there is at least one free box without placement in potentialplacements then return; endif; provide list currentplacements with potential placements that are currently to be tried; for i := 1 to currentplacements do currentsolution := currentsolution { currentplacements(i) }; // add placement to solution freeboxes := freeboxes \ { currentplacements(i).box }; // update free boxes potentialplacements := update(potentialplacements); // update potential placements add_placement (currentsolution, freeboxes, potentialplacements ); // recursive call endfor. Figure 5. Packing procedure add_placement. Candidates for the next placement are chosen from the list potentialplacements and provided in the list currentplacements. The selection and sorting of the candidate placements is governed by two filtering rules concerning the boxes and reference points of placements. First a ranking for boxes is specified: given two boxes b 1 and b 2 box b 1 has a higher rank if the related customer is to be visited later than the customer related to box b 2 or if the boxes belong to the same customer and box b 1 has greater volume than box b 2. By means of the first filtering rule it is ensured that only potential placements are selected that contain boxes of relatively high rank compared to all free boxes. Hereby the chance is increased that the LIFO constraint (C1) can be observed for all given boxes and a complete solution can be achieved. Note that the LIFO constraint requires that (roughly spoken) boxes of customers to be visited later have to be packed earlier (see Figure 1). Moreover, the filtering rule favors packing of bulky boxes at an early stage and if the LIFO constraint is ignored this is the only content of that rule since the box rank is then defined exclusively by the box volume. A linear order < p is specified for all points within a loading space. Given two points p 1 = (x 1, y 1, z 1 ) and p 2 = (x 2, y 2, z 2 ) the condition p 1 < p p 2 holds if x 1 < x 2 or x 1 = x 2 and z 1 < z 2 or x 1 = x 2 and z 1 = z 2 and y 1 < y 2. The second filtering rule guarantees that only potential placements are chosen for allocating the next box that have small reference points in the sense of order < p. Hereby the loading space is filled primarily from left to right, then from bottom to top and finally from front (driver s cabin) to back 9

10 (door), cf. Figure 1. Limiting the set of admitted reference points helps to reduce the redundancy of the search while the fact that the next packed box may be allocated at some different points in the loading space strengthens the flexibility of the search, as some experiments show. The list potentialplacements is always kept sorted in ascending order by the reference point (according to order < p ). Additionally, placements with the same reference point are sorted in descending order by the box rank. The list currentplacements is then derived from potentialplacements in two steps: (1) Let maxrank be the highest rank of a free box. According to the first filtering rule all potential placements are collected in a temporary list for which the inequality maxrank rank(box) maxboxrankdifference holds; the box belongs to a placement and maxboxrankdifference is a parameter. (2) The temporary list is then passed through. According to the second filtering rule all potential placements of that list are inserted in turn into the list currentplacements until all placements were transferred that belong to the first (smallest) maxrefpoints reference points; maxrefpoints is a further parameter. All placements of list currentplacements are then implemented alternatively. For each placement the current solution, the set of free boxes and the list of potential placements are updated accordingly before the procedure add_placement is called recursively to extend the solution once again. To update the list potentialplacements first all potential placements are removed that are not compatible with the placement just implemented. To generate new potential placements additional points in the loading space are determined that could act as reference points. These potential reference points are generated as extreme points following the correspondent approach by Crainic et al. (2008). Very briefly: if a box was packed then six new extreme points are found as each of three defined box corners is projected into the directions of two defined coordinate axes. For each new extreme point all feasible placements of free boxes are constructed (if any) and inserted into potentialplacements. Clearly, to be feasible a new potential placement must be compatible with all placements in currentsolution. 5 Computational experiments The hybrid algorithm, called hereafter VRLH1, was implemented in C and tested on an Intel 3.17 GHz PC (Core 2 Duo E8500) with 2.0 GB RAM. Two sets of 3L VRP instances were used for the experiments. The first set was introduced by Gendreau et al. (2006) and consists of 27 instances with customer numbers (n) between 15 and 100. The second instance set was proposed by Tarantilis et al. (2009) and includes 12 instances with customer numbers between 50 and 125. Throughout the test the parameter setting shown in Table 1 was used for method VRLH1; time values are always specified in CPU seconds (s). Table 1. Parameter setting for method VRLH1. Parameter Description Value Parameter Description Value maxiter timelimit tllength nmsel Max. no. of TS iterations Max. time for TS without generation of initial solution Max. length of tabu list Required no. of feasible solutions per iteration 2500, if n < 50, 10000, if n s, if n < 30, 300 s, if 30 n < s, if n 50 maxapcalls maxboxrankdiff Max. no. of calls of procedure add_placement Max. tolerated rank difference of boxes n/5 maxrefpoints Max. number of admitted reference points 3 ntrials No. of trials for generating the initial solution

11 In Table 2 the new method VRLH1 is compared to the TSA by Gendreau et al. (2006), the hybrid algorithm by Tarantilis et al. (2009) and the ACO algorithm by Fuellerer et al. (2010) using the first instance set. Table 2. Results for the 27 instances by Gendreau et al. (2006). I (n) Gendreau et al. (2006) Tarantilis et al. (2009) Fuellerer et al. (2010) VRLH1 ttd ttot ttd ttot ttd ttd avg ttot ttd ttd avg ttot 1 (15) (15) (20) (20) (21) (21) (22) (22) (25) (29) (29) (30) (32) (32) (32) (35) (40) (44) (50) (71) (75) (75) (75) (75) (100) (100) (100) avg g (%) nbest nv CPU Pentium iv (3.0 GHz) Pentium iv (2.8 GHz) Pentium iv (3.2 GHz) Intel (3.17 GHz) nruns 1? tr (%) Each instance is characterized by its index (I) and the number of customers (n). For each method the best total travel distance ttd per instance is shown. For the ACO algorithm and VRLH1 the average total travel distance ttd avg over the computed runs is also presented (see line nruns for the number of runs). Note that the TSA is a deterministic method while no mean values are available for the hybrid method by Tarantilis et al. Furthermore, the total computation times ttot are given in seconds of the respective CPU (see line CPU). For the ACO algorithm and VRLH1 the total computation times are averaged over the computed runs. At the end the total travel distances (ttd and ttd avg ) and the total computation times ttot are averaged over the 27 instances. The resulting averaged distance values are denoted by ttd and ttd avg. The relative distance improvements by VRLH1 are given by gap values g (see line g). The gap g between VRLH1 and the ACO algorithm is calculated according to g = (ttd avg (VRLH1) ttd avg (ACO))/ ttd avg (ACO) (in %) and the gap g between VRLH1 and the TSA is calculated as g = (ttd avg (VRLH1) ttd (TSA))/ ttd (TSA) (in %). Finally, the gap between VRLH1 and the hybrid algorithm (HA) by Tarantilis et al. is computed as g = (ttd (VRLH1) ttd (HA))/ ttd (HA) (in %). For each method the number of achieved best ttd values nbest and the total number of used vehicles nv in the best solutions is shown (best ttd values are set in Italic). Finally, the mean total computation time of VRLH1 over all instances is given as a time rate (percentage) tr of the mean total computation time of the compared methods; e.g., taking the TSA the time rate tr is calculated as tr(tsa) = (mean total computation time (VRLH1)) / (mean total computation time (TSA)) (in %). 11

12 In Table 3 the new method VRLH1 is compared to the hybrid algorithm by Tarantilis et al. (2009) by means of the second instance set. Table 3 is designed as Table 2 and all explanations given before remain valid. Table 3. Results for the 12 instances by Tarantilis et al. (2009). I (n) Tarantilis et al. (2009) 12 VRLH1 ttd ttot ttd ttd avg ttot 1 (50) (50) (50) (75) (75) (75) (100) (100) (100) (125) (125) (125) avg g (%) -3.8 nbest 1 11 nv CPU Pentium iv (2.8 GHz) Intel (3.17 GHz) nruns? 10 tr (%) 5.4 Both tables present a more or less uniform picture. The new hybrid method VRLH1 is clearly superior to the three competing methods in terms of solution quality. Taking all 39 benchmark instances VRLH1 achieved 35 new best solutions and only for two instances (instance 17 of set 1 and instance 12 of set 2) one of the compared methods did yield a better result. The mean improvements (gaps) regarding the total travel distance depend on the instance set and the compared method and range between 0.8% (set 1, ACO algorithm) and 8% (set 1, TSA). Moreover, the hybrid method VRLH1 was able to save many vehicles (or routes) in particular for the first instance set. While performing better with regard to all aspects of solution quality the computational effort needed by VRLH1 is much lower than for the compared methods. This becomes evident if the mean total computation times (ttd) and the time rates (tr) are looked at. Of course, the time rates are only significant if the used processors are of comparable strength. This is certainly the case for the PC used by Fuellerer et al. while the PCs used by Tarantilis et al. and by Gendreau et al. are a bit slower than the one used for VRLH1. Nevertheless, it can be stated that on average the total computation times consumed by VRLH1 amount to only 10 20% of the total computation times needed by the competing methods. Hence, the new method is first of all a significantly more efficient method compared to the other algorithms. In further experiments the impact of the constraints (C1), (C4) and (C5) on the solution quality was investigated. The results for the first instance set (introduced above) are shown in the Table 4 and the results for the second set are given in Table 5. In both tables results are presented for five different loading configurations. In the first loading configuration, called standard configuration, all constraints have to be respected (as in tables 2 and 3) while in the following three configurations the fragility constraint (C5), LIFO constraint (C1) and support constraint (C4), respectively, are ignored. In the last configuration all these constraints are dropped. For each configuration and instance the algorithm VRLH1 was performed ten times and the average total travel distance (ttd avg ) and the mean total computation time (ttot) over all runs are shown. Besides average values of ttd avg (denoted by ttd avg ) and ttot over all instances averaged gap values g are calculated for the last four configurations. The gap value for a configuration i (i = 2,3,4,5) represents the relative improvement

13 of the total travel distances compared to the standard configuration (i = 1)and is calculated according to g(i) = (ttd avg (i)) ttd avg (1))/ ttd avg (1) (in %). For each loading configuration the total number of used vehicles over all best solutions is indicated in line nv. Table 4. Results of method VRLH1 for the 27 instances by Gendreau et al. (2006) and five loading configurations. I (n) All constraints No fragility No LIFO No support 3D loading only ttd avg ttot ttd avg ttot ttd avg ttot ttd avg ttot ttd avg ttot 1 (15) (15) (20) (20) (21) (21) (22) (22) (25) (29) (29) (30) (32) (32) (32) (35) (40) (44) (50) (71) (75) (75) (75) (75) (100) (100) (100) avg g (%) nv Table 5. Results of method VRLH1 for the 12 instances by Tarantilis et al. (2009) and five loading configurations. I (n) All constraints No fragility No LIFO No support 3D loading only ttd avg ttot ttd avg ttot ttd avg ttot ttd avg ttot ttd avg ttot 1 (50) (50) (50) (75) (75) (75) (100) (100) (100) (125) (125) (125) avg g (%) nv The results on the impact of the considered constraints (fragility, LIFO and support) are similar to the findings of other authors (cf., e.g., Gendreau et al., 2006, and Fuellerer et al., 2010). If these constraints are dropped individually or collectively, considerably better total travel distances are achieved. The improvements, measured by the gaps g, depend on the instance set and on the selected constraint(s). Significantly higher improvements were found for the second test set provided by Tarantilis et al. than for the first set by Gendreau et al. Regarding the constraints, the smallest effect results if the fragility restriction is ignored while the greatest improvements nearly 10% for 13

14 the first set and 14% for the second set are found if all three constraints are abandoned. In Tables 4 and 5 it is also shown that dropping the constraints leads to smaller numbers of vehicles being used and to lower computation times as well. In Table 6 the results for the five loading configurations (defined above) that were achieved by VRLH1 and the competing methods are roughly compared. The first two columns specify the test set and loading configuration. Then, the best total travel distance (ttd) and the total computation time are shown if available (since computation times for the algorithm of Tarantilis et al. are only known for the first configuration no times are indicated for this algorithm here, cf. Tables 2 and 3). In addition, the average total travel distance over the performed runs (ttd avg ) is presented for the ACO algorithm and VRLH1 for the first set. All values were averaged over the instances of the respective test set and best distance values are set in Italic. Table 6. Average results of competing methods for two instance sets and five loading configurations. Set Loading configuration Gendreau et al. (2006) Tarantilis et al. (2009) Fuellerer et al. (2010) VRLH1 ttd ttot ttd ttd ttd avg ttot ttd ttd avg ttot 1 All constraints No fragility No LIFO No support D loading only All constraints No fragility No LIFO No support D loading only The results in Table 6 can be summarized as follows. For all ten combinations of a test set and a loading configuration the hybrid method VRLH1 achieves the smallest (averaged) best total travel distance ttd and for eight combinations VRLH1 shows also the smallest (averaged) mean total travel distance over all runs ttd avg. However, the ACO algorithm performs better in terms of the mean total travel distance ttd avg if the LIFO constraint is dropped. Clearly, the LIFO constraint reduces the feasible placements to a great extent; hence, controlling the tree search for packing boxes (within VRLH1) is much easier if this constraint is in force. In general, the method VRLH1 proves to be more effective and efficient than the competing methods even if the constraints are relaxed. Again, the computing times needed by VRLH1 amount mostly to only 10 20% of the times needed by the competing algorithms (even if different strengths of the used processors are taken into account). 6 Conclusions In this article a new algorithm for the CVRP with three dimensional loading constraints is presented. The hybrid algorithm includes a tabu search algorithm for routing and a tree search algorithm for packing boxes into the loading space of a vehicle. Computational results were presented for all publicly available test instances and the hybrid algorithm has been compared to all available solution procedures for the 3L CVRP. The new method turns out to be superior to the competing methods in terms of solution quality. For 35 of 39 benchmark instances new best solutions were calculated. Moreover, the hybrid algorithm proved to be very efficient since the total computation times are reduced by up to one order of magnitude compared to the other methods. There are two key success factors. On the one hand, the effort for packing boxes is considerably reduced as moves in the routing procedure are evaluated before checking whether they are feasible in terms of packing. On 14

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