An Effective Multi-restart Deterministic Annealing Metaheuristic for the Fleet Size and Mix Vehicle Routing Problem with Time Windows

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1 An Effective Multi-restart Deterministic Annealing Metaheuristic for the Fleet Size and Mix Vehicle Routing Problem with Time Windows Olli Bräysy Agora Innoroad Laboratory, Agora Center, P.O. Box 35, FI University of Jyväsylä, Finland, Wout Dullaert Institute of Transport and Maritime Management Antwerp, University of Antwerp, Keizerstraat 64, B-2000 Antwerp, Belgium Geir Hasle SINTEF ICT, Department of Applied Mathematics, P.O. Box 124 Blindern, NO-0314 Oslo Norway David Mester Institute of Evolution, Mathematical and Population Genetics Laboratory, University of Haifa, Haifa, Israel Michel Gendreau Département d informatique et de recherche opérationnelle and Centre de recherche sur les transports, Université de Montréal, C.P. 6128, Succursale Centre-ville, H3C 3J7 Montréal, Canada Abstract The purpose of this paper is to present a new deterministic annealing metaheuristic for the fleet size and mix vehicle routing problem with time windows. The objective is to service, at minimal total cost, a set of customers within their time windows by a heterogeneous capacitated vehicle fleet. First, we motivate and define the problem. We then proceed to give a mathematical formulation of the most studied variant in the literature, in the form of a mixed-integer linear program. We also suggest an industrially relevant, alternative definition that also leads to a linear mixed-integer formulation. The suggested metaheuristic solution method solves both problem variants and comprises three phases. In the Phase 1, high quality initial solutions are generated by means of a savings-based heuristic combining diversification strategies with learning mechanisms. In Phase 2, an attempt is made to reduce the number of routes in the initial solution with a new local search procedure, and in Phase 3 the solution from Phase 2 is further improved by a set of four local search operators that are embedded in a deterministic annealing framewor to guide the improvement process. Some new implementation strategies are also suggested for efficient time window feasibility checs. Extensive computational experiments on the 168 benchmar instances of Liu and Shen show that the suggested method outperforms the previously published results and found 167 best-nown solutions. Experimental results are also given for the new problem variant. Keywords: vehicle routing, time windows, heterogeneous fleet, fleet dimensioning. 1. Introduction The Vehicle Routing Problem (VRP) is a ey to efficient transportation management and supply-chain coordination. In broad terms, it deals with the optimal assignment of a set of transportation orders to a fleet of vehicles, and the sequencing of stops for each vehicle. The VRP has a large number of real-life applications and comes in many guises, depending on the type of operation, the time frame for decision maing, the objective, and the types of constraint that must

2 be adhered to. The objective usually reflects the minimization of total transportation costs. Often, and particularly in tactical route design and fleet dimensioning, the objective models a combination of driving costs for the routing plan (routing costs) and fleet acquisition / depreciation costs (vehicle costs). For a general treatment of the VRP, we refer to the textboo by Toth and Vigo (2001). For a literature survey of VRP extensions, we refer to Bräysy et al. (2007a and 2007b). Although often assumed in VRP theory, a vehicle fleet is rarely homogeneous in real life. A fleet manager typically controls vehicles that differ in their equipment, carrying capacity, speed, and cost structure. There are many types of motivation for eeping a diversified fleet. Some customers may require vehicles with costly equipment. Large vehicles may be more cost effective, but the road networ and the nature of customers premises often impose physical restrictions on vehicle size and weight. Some types of vehicle may not be allowed in urban areas due to environmental concerns. Vehicles of different carrying capacity give flexibility to allocate capacity according to varying demand in a more cost effective way. The Fleet Size and Mix Vehicle Routing Problem (FSMVRP) is a VRP where the homogeneous fleet assumption of the traditional VRP has been lifted. A number of vehicle types with different capacities and acquisition costs are given. The objective is to find a fleet composition and a corresponding routing plan that minimizes the sum of routing and vehicle costs. Practical applications of FSMVRP are abundant and include deliveries to grocery stores, the pet food and flour delivery problem, and the mail collection problem. In the literature one can separate several variants of the problem according to how the variable costs and fleet size are issued, and whether there are limits on the number of vehicles of each type. For reviews on the FSMVRP, we refer the reader to Salhi and Rand (1993), Osman and Salhi (1996), and Lee et al. (2006). In this paper, we study the FSMVRP with time windows (FSMVRPTW), where each customer has a time window within which service must start. It is a natural extension of the recently much studied VRPTW (see Bräysy and Gendreau, 2005a,b for a recent survey). Time windows are critical in many applications, as more and more importance is given to customer service and timeliness. We focus on the FSMVRPTW variant that was first defined by Liu and Shen (1999). Here, the fleet is heterogeneous because each vehicle might have its own fixed cost and its own capacity. Driving speed and customer service times are identical for all vehicles. In contrast with the VRPTW, the routing cost for a solution is defined as the sum of en route time over all vehicles used. By en route time we mean the time that elapses from the instant vehicle leaves the depot to the time it returns. One must note that there is no assumption on depot start times. If we were to assume a fixed depot start time for vehicles, the problem would be different. For Liu and Shen FSMVRPTW instances, an optimal solution will have zero waiting time before the customer visited first on any one tour. Moreover, since all customers must be served, the sum of service times in any feasible solution is constant. In Liu and Shen (1999), and in most of the subsequent literature, the total service time is subtracted from the total en route time in reports of experimental results. The main contribution of this paper is to present a new efficient multi-start deterministic annealing (MSDA) heuristic for the FSMVRPTW. As we shall see, the Liu and Shen FSMVRPTW definition leads to a mixed-integer linear program (MILP) formulation. We propose a new and industrially relevant variant of the problem that is more similar to the VRPTW and focuses on the sum of arc travel costs rather than on en route times. This alternative definition leads to a mixedinteger linear program (MILP) formulation. Our method wors for both FSMVRPTW variants. The remainder of the paper is organized as follows. In Subsection 1.1, a mathematical formulation of the Liu and Shen variant of the FSMVRPTW is given. We also give our alternative definition and present a corresponding mathematical formulation. Subsection 1.2 presents a literature review. The algorithmic approach is described in Section 2, and some important details on the implementation are given in Section 3. Section 4 describes our experimental study and results. Conclusions are given in Section 5. 2

3 1.1. Problem Formulation We base our formulation of the Liu and Shen FSMVRPTW variant described above on the well-nown vehicle flow arc-based formulation for the VRPTW, see for instance Cordeau et al. N= 0 1, K, n n+ 1. Nodes 0 and n + 1 (2001). Let G= ( N,A ) be a digraph, where { } { } { } represent the depot for tour start and tour finish, respectively. = { 1,, n} C K is the set of customers. A N N represents the travel possibilities between nodes. Normally, N is complete, except that no arcs are incident to node 0 and no arcs are incident from node ( n + 1). V = { 1, K, K} is the set of available vehicles. There is a travel cost cij, ( i, j) A, V that may or may not be dependent on the vehicle. There are vehicle specific acquisition / depreciation costs e and capacities q. t ij is the vehicle independent travel time between nodes. For each customer i, there is a time window ai, bi within which service must start, and s i is the vehicle independent customer service time.,, i, j A, V, There are two types of variables in the formulation. For each triplet ( i j ) with ( ) x ij is a binary decision variable that expresses whether vehicle travels directly from customer i to customer j ; there are K( n+ 2) 2 such variables. For each pair (, ) V, i with i N, y i is a real variable that determines the exact start time of service at this node if it is served by that vehicle; there are K( n+ 2) such variables. The formulation is shown below. 3

4 ( + ) minimize ex + y y x (1) 0 j n j V j C V j C subject to: V j N i C j N i N i N 0 j ij in, + 1 = 1, i C (2) d x q, V (3) x i j N ij = 1, V (4) x x = 0, h C, V (5) x ih x j N hj = 1, V (6) x ( y + s + t y ) 0, i, j A, V (7) ij i i ij j i ij i j N i0 ii n+ 1, i ( ) a x y, i N, V (8) y b x, i N, V (9) i i ij j N x = 0 i N, V (10) x = 0 i N, V (11) x = 0 i N, V (12) x {0,1}, ( i, j) A, V (13) ij In the objective (1), the sum of x 0 j over customers indicates whether vehicle is used. In case its value is 1, the constant vehicle cost plus the en route cost for the tour accrue. The first, linear term represents the sum of all vehicle costs, whereas the second, non-linear term is the sum of en route costs. Constraints (2) state that all customers must be visited by a vehicle. (3) are the vehicle capacity constraints. Constraints (4) enforce that each vehicle must leave the depot exactly once, and (6) that each vehicle must arrive at the depot exactly once. (5) ensure that if a vehicle arrives at a customer, it also departs from that customer. Together, (4), (5), and (6) are in-degree and outdegree (or flow balance) constraints on the graph that ensure a connected round trip (possibly empty) for each vehicle. There are K( n+ 2) such constraints. For a given vehicle, one of the ( n + 2 ) flow balance constraints is linearly dependent on the remaining ones and may be removed. (7) guarantee that the arrival times at two consecutive customers allow for service and travel time. (8) and (9) are the time window constraints. (10), (11), and (12) mae sure that tours start at node 0 and finish at node ( n + 1) and there are no arcs that loop bac on the same node. The second, en route time term of the objective (1) may be rewritten as follows: ( 1 0) 0 ( 1 0) 0 ( 1 0)( 1 yn+ y x j = yn+ y x j = yn+ y x0, n+ 1) V j C V j C V 4

5 If vehicle is not used, x 0, n + 1 is 1 and we get a zero contribution for these vehicles, as intended. On the other hand, if 0, n 1 yn+ 1 y0. Since we are minimizing, the en route time difference will be made as small as possible given the temporal constraints (7), (8), and (9). Hence, we may replace the non-linear en route time term with y y. The objective becomes the mixed-integer linear expression: ( n+ 1 0) V x + is 0, vehicle is used, and we get a contribution ( ) min ( n+ ) ex + y y 0 j 1 0 V j C V (1') The quadratic travel time constraint (7) can be linearized by use of the big M tric. For a general introduction, see for instance Bazaraa et al. (2005). For a similar use in a VRPTW formulation, see Cordeau et al. (2001). In our case, the linearization of (7) becomes: where ( ) ( y + s + t y ) 1 x M ( i, j) A, V (7') M ij may be replaced by: i i ij j ij ij max ( b + s + t a,0) ( i, j) A, V i i ij j Hence, the objective and all constraints are mixed-integer linear forms, and we may conclude that the Liu and Shen variant of the FSMVRPTW has a MILP formulation. Commercial solvers can therefore be used to solve small problem instances. We propose a new variant of the FSMVRPTW which is a simple generalization of the VRPTW and still leads to a MILP formulation. Rather than focusing on total en route time, this variant focuses on the sum of arc travel costs over all arcs in the solution. The above formulation is also valid for the new problem, except that the objective then becomes: min ex + cx (1 b) 0 j ij ij V j C V (, i j) A The alternative objective (1b), together with (2)-(6), (7 ), and (8)-(13), constitutes a MILP. As a special case, all arc costs may be vehicle independent. Note that the new variant may also be time oriented (arc cost may be set to travel time), but the objective does not include waiting time. Still, the problem has industrial relevance. A richer FSMVRPTW variant that considers both waiting time and arc costs is easy to define just by combining objectives (1 ) and (1b): 0 + ( + 1 0) + min ex y y cx (1 c) j n ij ij V j C V V (, i j) A 1.2. Literature review The FSMVRPTW has been much less studied than the FSMVRP. As discussed above, the first wor was reported by Liu and Shen (1999). For their variant, they developed a savings-based construction heuristic and an improvement heuristic inspired by the wor of Golden et al. (1984) for the FSMVRP. The approach was tested on a new benchmar developed by the authors from the well-nown Solomon s benchmar set for the VRPTW (Solomon, 1987). Three sets of instances were created, each representing a different cost structure for vehicle types. 5

6 Dullaert et al. (2002) proposed a sequential insertion heuristic based on Solomon s I1 heuristic for the VRPTW and tested their approach on the Liu and Shen benchmar, with competitive results. Privé et al. (2006) present a case study on the soft drin distribution and collection of recyclable containers. The problem is modeled as FSMVRPTW with additional volume constraint and with a modified objective function that considers in addition revenues from selling recyclable material. Three construction heuristics and an improvement procedure are suggested for the problem. Dell Amico et al. (2006) recently proposed a solution approach based on a regret-based parallel insertion procedure, and subsequent improvement by ruin and recreate (Schrimpf et al., 2000, and Pisinger and Røpe, 2006). Their approach outperforms earlier results on the Liu and Shen benchmar. Calvete et al. (2006) apply a two-phase approach for FSMVRP with hard and soft time windows and multiple objectives. In the first phase feasible routes are enumerated and the total penalty incurred by each route due to deviations from targets is computed. With these data, in the second phase a set-partitioning problem is solved in order to get the best set of feasible routes. Tavaoli-Moghaddam et al. (2006) consider a model where the cost of the routes is independent of the route lengths and only the depot has a time window. The depot time window is used to restrict the route lengths. The model is solved by a hybrid simulated annealing based on nearest neighborhood heuristic. Dondo and Cerdá (2006) present a three-phase approach for FSMVRPTW with multiple depots. Phase I aims to identify a set of cost-effective feasible clusters while Phase II assigns clusters to vehicles and sequences them on each tour by using a cluster-based MILP formulation. Ordering nodes within clusters and scheduling vehicle arrival times at customer locations for each tour through solving a small MILP model is performed in Phase III. 2. Solution approach The proposed solution approach consists of three phases. In Phase 1 high quality initial solutions are generated by means of a savings-based heuristic combining diversification strategies with learning mechanisms (subsection 2.1.). In Phase 2 the focus is on reducing the number of vehicles with a new local search procedure (subsection 2.2.) and in Phase 3, a set of four local search operators are embedded in a deterministic annealing framewor to guide the improvement process of solution from Phase 2 (subsection 2.3.). All phases are repeated n init times, i.e., several initial solutions are created and improved separately Phase 1: Constructing initial solutions At the beginning of the search, initial solutions are created by a probabilistic modification of the savings heuristic (Clare and Wright, 1964). As in the original savings heuristic, the search is started by serving each customer by way of a separate route. However, after the first construction and improvement phase, some of the arcs in the final solution of the previous improvement phase, S current, are ept instead of starting with single customer routes in Phase 1. For more details, see subsection 2.3. The initial route construction heuristic features three main modifications compared to the original savings heuristic (See Algorithm 1). First, the heuristic is implemented from an insertion point of view to combine the strengths of savings and insertion-based heuristics. As in Liu and Shen (1999), when merging two routes R 1 and R 2, the search is not limited to inserting R 1 either in front of or after R 2, but all positions between consecutive customers within route R 2 are considered. Second, the algorithm accepts not only the best savings Δ 1, but also the second Δ 2 and third best savings Δ 3 with given probabilities, p 1, p2, p3 to allow for some diversification during the search. Here one must note that Δ 2 and Δ 3 are accepted only if they also improve the objective value, i.e., are positive. Third, each route is initialized with the smallest possible vehicle type. Savings are calculated by maing the difference between the sum of fixed vehicle costs F and the total en route 6

7 time RT (travel time, waiting time and service time) for the current routes and the merged route. Vehicle sizes are updated whenever needed, and are always set to the smallest vehicle available capable of serving the customers on the route. To save time the calculated savings are stored in a matrix, which is updated during the search based on the merges performed. Merges are attempted until no further improvement can be found. Algorithm 1 if S current = null Initialize routes by servicing each customer in a separate route by a vehicle of the smallest vehicle type else // Restart procedure 2 // Create a new starting point based on the arcs present in S current and on the arc frequency information. while // permanent loop until no more improvement is found for all routes r 1 for all routes r 2 for all insertion positions in route r 2 Calculate savings in fixed vehicle costs F and en route time RT and update best savings 1 Δ,, Δ information 2 3 end for all end for end for Δ if best savings Δ > 1 0 Generate a random number r between 0 and 1 to determine using probabilities p1, p2, p3 whether routes are to be merged according to the routes and insertion positions associated with the positive savings of Δ, Δ Δ. end if else brea 1 2, 3 end while 2.2. Phase 2: Route elimination The route elimination phase is based on a depletion procedure called ELIM, which again uses simple insertions, as explained below. In ELIM, all routes of the current solution are considered for depletion, in random order. For a given route, ELIM removes all customers and tries to insert them in the remaining routes. The criticality measure ς i defined in (14) determines the sequence in which customer i is reinserted. The most critical customer is reinserted first. Here, we use the notation of subsection 1.1. The η function performs normalization over all customers to the [ 0,1 ] interval. η ( di ) ( b a ) ςi = + η η i i ( c ) According to (14), the larger the demand, the narrower the time window and the longer the distance from the depot, the higher is the criticality of a customer. For a given customer, all positions in the remaining routes are tried. The best feasible insertion according to the total cost objective is selected. The algorithm starts by calculating the 0i (14) 7

8 total savings, Sav, obtained by eliminating the currently selected route. It then eeps trac of the total increase in total costs resulting from reinsertions, ϑ. If the total increase exceeds the total saving or if any of the customers cannot be inserted in another route, the algorithm reverses the insertions performed from the current route and proceeds to the next route in sequence. If all the removed customers have been inserted in other routes at a lower cost, the route is eliminated. In the Depletion Phase, the ELIM procedure described above is run until quiescence. Algorithm 2 for all routes r 1 Calculate Sav r 1 and set ϑ = 0 for all customers c in r 1 (in the order of their criticality) Set Savings = for all routes r 2 Calculate savings in fixed vehicle costs: Δ F = F1 + F2 Fnew for all insertion positions p in route r 2 Calculate savings in en route time Δ ST : if insertion position p is feasible: Δ ST = ST1 + ST2 STnew end if else set ΔST = end else Δ = ΔF + ΔST > Savings if ( ) Savings = ΔF + ΔST Set r best = r 2 and end if end for all end for all > if Savings ϑ = ϑ + (new cost of rbest if ϑ < Sav r1 p best = p - old cost of r best ) insert customer c in route rbest and insertion position p best end if else reinsert customers c from r 1 in original positions (bactrac) brea end else endif else brea end else end for all reinsert customers c from r 1 in original positions (bactrac) Update bactrac information end for all 8

9 2.3. Phase 3: Local search improvement The solution from Phase 2 is further improved by local search and metaheuristics. Four local search operators are utilized: the ELIM route elimination operator described above, a route splitting operator called SPLIT, along with the ICROSS and IOPT operators suggested in Bräysy (2003). The SPLIT neighborhood consists of all solutions that result from splitting a single route in the current solution into two parts at any point. We employ it in a greedy, first-accept fashion, simply by looping through all routes and all customers in them, splitting the selected route into two parts at the position of the current customer. The move is made if the split reduces total cost. Cross-exchanges (Taillard et al., 1997) wor by relocating or exchanging segments of consecutive customers from different routes while preserving orientation. When a segment is removed from Route 1, it is inserted in the most promising position in Route 2. The segment removed from Route 2 is inserted at the location where the segment from Route 1 was removed. The maximum segment length is limited to l s customers. In addition to the cross-exchange moves, the ICROSS neighborhood also consists of moves where the current segment is reversed before insertion. Relative to Bräysy (2003), ICROSS has been extended with the adjustment of vehicle types. The sum of fixed vehicle costs and en route time is used for move evaluation. For details on implementation techniques to reduce computation time for the traditional ICROSS operator, the reader is referred to Bräysy et al. (2004). The IOPT intra-tour operator is a generalization of Or-opt (Or, 1976). It considers segments of any length and also includes moves where the segment is reversed before it is relocated (Bräysy, 2003). As IOPT is an intra-tour operator, move assessment is limited to en route time. The four local search operators are employed as follows (Algorithm 3): ICROSS, IOPT, ELIM, and SPLIT (in that order) are repeated for a given number of iterations n improve. In the beginning of each iteration, the routes are randomly ordered to diversify the search. Because a first-accept strategy is used, the order in which routes are considered affects output. To increase the efficiency of the improvement phase, ICROSS and IOPT do not consider a route pair or a single route if no improvement was found last time and the routes have not changed since. There is little gain to be made from invoing both the route elimination and route splitting operator in each iteration. Therefore, route elimination is executed only every second iteration and route splitting every third iteration. Because route splitting is not applied in the same iteration when route elimination was already invoed, route splitting is in fact executed less than every third iteration. Algorithm 3 Set i last= 0, threshold T = 0 and cost of S current, C ( S current ) = for i = 1to nimprove Shuffle routes in random order for all route pair combinations ( r, s) Apply ICROSS end for all for all routes r Apply IOPT if i mod( 2) = 0 and i 2 Apply ELIM until no improvement end if else if i mod( 3) = 0 Apply SPLIT until no improvement 9

10 end if end for all Calculate the cost of the new solution C( S n ) If C ( S n ) < C( Scurrent ) Set S = S and last improvement i last = i current n ( S n ) C( Sbest if C < ) Set S = S > best if i nlim it Update arc frequencies matrix end if end if end if else if (i i last n and = 0 Set i last = i, T r T end if else if T 0 Set threshold T end else if else T T T = end else end else Δ n T ) // Restart Procedure 1 = max, S n = Scurrent and update maximum distance limit = r T max end for By embedding the local searches in a Deterministic Annealing (DA) framewor, moves in ICROSS, IOPT and route elimination are sometimes accepted even if they worsen the objective value, as long as the worsening is within the current value of the threshold limit. For SPLIT, worsening is not allowed as we found that it resulted in too large diversification. Simulated Annealing (SA) (Aarts et al., 1997), although being successfully applied to a wide range of combinatorial problems, often becomes inherently slow because of its randomized local search strategies. Deterministic annealing methods (such as Threshold Accepting by Duec and Scheurer, 1990, and Duec, 1993) try to overcome this deficiency while preserving the advantages of SA. Instead of performing a stochastic search over a combinatorial search space, DA combines advantages of simulated annealing with the efficiency of a deterministic procedure and has been applied successfully to a variety of combinatorial optimization problems (see e.g. Duec and Scheurer 1990, Yagiura and Ibarai 2001, Tsuchiya et al. 2001, and Schellewald et al. 2006). Our Deterministic Annealing procedure starts with threshold T = 0 (no worsening allowed) and is repeated with that value until a local minimum is reached. If no more improvements have been found for a given number of iterations n, Restart Procedure 1 is commenced. The threshold is set to a maximum value and the search is restarted from the current best solution, S current. The maximum value is r Tmax where r is a random number in the range [0, 1] and Tmax is a user-defined parameter. At each non-improving iteration, the threshold is reduced by Δ T units until zero is reached again. Then the search is repeated with zero threshold until no more improvements can be found. To escape the new local optimum, the threshold is set again to a new maximum threshold value r T. max 10

11 It should be noted that two types of restarts and two types of best solutions are used. In addition to Restart Procedure 1, there is also Restart Procedure 2 which involves creating a new initial solution with the modified savings algorithm (see Algorithm 1). S current refers to the best solution obtained so far from the current initial solution and S best to the overall best solution found during the entire search. To avoid too much intensification, we found that it is better to restart from S current than S best with both restart types. To limit the search efforts and to speed up the algorithm (for ICROSS and IOPT), numerous different strategies were tested. It appeared that limiting the search to eliminate the longest arcs or waiting time, or to the closest neighboring customers within a fixed distance limit (see, e.g., Berger et al., 2003, and Bräysy et al., 2004) often reduced solution quality too much. We therefore developed a new learning approach that proved to wor well. The idea is to focus only on moves that involve customers that are close to each other. The difficulty lies however in identifying a good limit of closeness. Therefore a distance limit is initialized in the first solution and updated during the search. In the initial solution the distance limit is set to the distance of the farthest customer from the depot to represent some ind of potential maximum arc length in a solution, multiplied by a uniform number r (between 0 and 1). The distance limit is redefined at each restart involving the creation of a new initial solution (restart 2). The search on the first initial solution is used to gather information about the maximum distances that have still made improvements possible. When woring on the subsequent initial solutions, the maximum distance that has still enabled improvement is used as the new reference and multiplied by r to get the new distance limit. When executing ICROSS and IOPT, moves involving nodes within the current distance limit are the only ones taen into consideration. When creating subsequent initial solutions, the algorithm uses information gathered during the search in a frequency-based long-term memory of arcs that appear in high quality solutions This learning mechanism is based on identifying the common arcs in S best. Given that in the beginning it taes several iterations to obtain the first good solution (a local optimum has been found), good solutions are considered as the new S best solutions obtained after iteration n limit. Every time we find a new incumbent S best, we update the long-term memory. When the iteration limit n improve for the current initial solution is reached, a starting point is created for the savings heuristic to create a new initial solution based on the above arc frequency information in the long-term memory. Here the following strategy is used. The frequencies of the arcs of the solution of the last iteration S current are evaluated. If an arc is not present in b% (say, e.g., 70%) of the best solutions, it is removed. From the remainder of the arcs in the solution (common in enough best solutions), arcs are randomly deleted or ept according to a given user-defined probability, p d. The resulting set of arcs forms the starting point for the savings algorithm. The creation of initial solutions and improvement of them is repeated for a given number of times, n init, specified by the user. 3. Implementation techniques To enable quic solution updates, solutions are maintained in a double-lined list in an array data structure. In practice this means having two arrays of size n (the number of customers in the problem), one for successors and one for predecessors. For example, the value in position i of the successors next table refers to the customer serviced after i. In addition to the tables, information about the first and last customers of each route is maintained. An opportunistic strategy is used to speed up checing constraint feasibility and objective function values. This means that features that are easiest or quicest to calculate or that are most liely to violate a feasibility condition are checed first. Following this, the feasibility of the capacity constraint is checed first. This can be done in constant time by maintaining information 11

12 on the current capacity use and the maximum available vehicle capacity (for details, see Campbell and Savelsbergh, 2004). Then (only) if the capacity constraint is feasible, the algorithm checs whether arrival times are met for the newly inserted customer or the first customer of the reinserted segment. By maintaining information of the current arrival times at each customer this can also be done in constant time (see Campbell and Savelsbergh, 2004). In the next step, we chec whether arrival times are met at the customer served after the newly inserted customer or segment. By retaining information on the latest possible arrival times at each customer and by inserting only one customer, this feasibility chec can also be done in constant time in the same way as for the first customer. However, should a segment be inserted, one should loop through the whole segment. Because this looping can become computationally expensive in the case of large problems and long segments, it is avoided by the following strategy. Consider inserting segment ( r, s) between i and j in route ( 0, i, r, s, j,...,0) starting and ending at the depot 0. Let u j denote the arrival time at customer j and let v j denote the start of service at a,. If the vehicle arrives before the earliest possible time at customer j, within its time window [ ] which service may start j b j a j, the vehicle has to wait for w j units of time. It is assumed that the segment is extracted from an existing solution, i.e., the customers in the segment were originally in a given order and the original arrival times at each customer are nown. The new arrival time to the first customer of a given segment can easily be calculated in constant time. Given the above, the tas is to find out the arrival time at the last customer of the segment. It is also necessary to calculate the following four values in advance (as they are calculated once, outside the main search loop, the computational effort is limited): the total service time within the segment ( σ rs ), the total travel time within the segment ( τ rs ), total waiting time of the segment ( wrs ) and the so-called overswing. Overswing ( ο rs ) measures slac in the segment by examining how much earlier it is possible to arrive at the first customer of the segment, before there will be waiting time at one or more customers within the segment. Overswing only needs to be considered if there is no waiting time at the segment. If there is waiting time, overswing is set to 0. In practice, overswing is defined as the difference between the current arrival time at customer and the early time window of a customer. We need to loop through the segment and find the minimum overswing value for the segment. Note that the service time, travel time and waiting time need to be calculated only when relocating segments in ICROSS and IOPT. For the savings operations the corresponding values for each route are maintained all the time (as each route is considered to be a route segment with a fixed order of customers), so there is no need to calculate them. Furthermore, one is often interested in checing the total duration or in this case the en route time of the route, which requires calculating the new arrival time at depot. By maintaining the information on the total waiting, service and travel times of the route, and calculating cumulative values en route, one can calculate the corresponding values in the end part of the route in constant time (obvious for service time and travel time, for waiting time we refer to Bräysy et al., 2004) and using the idea below to determine the new arrival time at depot. First it is necessary to chec whether arrival at the first customer of the segment is earlier or later than before. new If arrival time is earlier, u < u, new and if w > 0, u = u, rs s s r new new new and if w rs = 0 and ur ur ors, us = us ( ur ur ), r 12

13 new new and if w = 0 and u u > o, u = u o. rs new If arrival time is later, u > u, new and if w = 0, u = u + σ + τ, rs s new new and if w u u ), u = u, rs r ( r r r r r s new new new and if w < u u ), u = u + u u ) w. rs ( r r s rs rs r s s rs s s rs ( r r rs SPLIT moves are always feasible. With IOPT and ICROSS, moves are checed in given order, starting from the beginning of the routes. At the same time, we accumulate the waiting time (the waiting times at each customer are recorded to speed up this calculation) and travel time and service time until the current point. Loops are broen immediately once it is nown that no feasible solutions can be obtained at the later positions in the route, based on time window feasibility. The main idea is that if arrive time is too late at a customer in the segment, insertion positions for that segment later in the other route can be ignored (either the segment length or its starting point is modified depending on whether we arrive too late at the first or other customer in the segment). Finally, we calculate the change in the objective value and chec whether the arrival time deadline is met at all customers within a reinserted/relocated segment (if there is more than one customer). Note that the feasibility of the segment can be checed faster than the objective value, but it is evaluated last because the violations are rare once all the previous conditions are met. 4. Computational experiments Computational experiments were performed to assess the performance of the proposed algorithm. We first describe the instances considered in these experiments, as well as the parameter values used. This is followed by a presentation and a discussion of the results obtained with our deterministic annealing metaheuristic Problem data and experimental setting The proposed heuristic algorithm was implemented in Java (JDK 5.0) and tested on an AMD Athlon (512 MB RAM) computer. The computational experiments were performed on the benchmar instances proposed by Liu and Shen (1999), derived from the well-nown VRPTW instances of Solomon (1987). Solomon s problem sets for the VRPTW consist of 56 instances of 100 customers with randomly generated coordinates (set R), clustered coordinates (set C) or both (semi-clustered RC set). Subsets R1, C1 and RC1 have a short scheduling horizon and small vehicle capacities. By contrast, subsets R2, C2 and RC2 may have routes with a larger number of customers. For each six subsets Liu and Shen introduced several vehicle types with different capacities and costs. In addition, three different vehicle cost structures A, B and C were suggested, resulting in total to 168 problem instances. Here cost structure A refers to the largest vehicle costs and C to the smallest. In the following, unless explicitly stated otherwise, we use the same objective and reporting procedure as Liu and Shen (1999) and Dell Amico et al. (2006). The total cost to be minimized is given by the total fixed cost of the vehicles used and the total en route time. Here the total en route time refers to the difference between the arrival time at depot and the departure time from depot, excluding the service times at customers. It is important to note, however, that service times are considered when checing temporal feasibility of the solutions. They are only deducted in the 13

14 presented result tables from the total en route time obtained in the solutions. To minimize the waiting time of the solutions, adjustment of the departure time from the depot is allowed during execution of the algorithm. Three separate test runs were carried out for each of the 168 problems. The following parameter values were used: number of iterations, n = 1000 ; number of initial solutions, n init = 2 ; maximum threshold, t max = ; maximum segment length in ICROSS and IOPT, l s = 3; The threshold change step, Δt = ; restart limit n = 40 iterations; and arc frequencies are recorded from n lim it = 5 iteration. Probabilities in savings heuristic are p = , p = 0. 25, 2 p 3 = 0.25 and in the type 2 restart, b = 0. 7 and p d = These parameter values were selected following a sensitivity analysis of a few intuitively selected values. The sensitivity analysis was based on six test problems under cost structure A. The six problems were selected from the set to be as different as possible, as well as more difficult to solve than average. The selected problems were: R104, R208, C103, C204, RC101, and RC202. Figure 1 illustrates the percentage difference with respect to the minimum value obtained with different maximum thresholds, for different maximum threshold values. It should be noted here that the actual threshold value is a random proportion of the maximum threshold, as described in Section 2. The figure is based on the average total cost over all six problems. 2,5 improve Figure 1. Sensitivity analysis of Maximum threshold 2 1,5 % 1 0,5 0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1 0,2 0,3 Maximum threshold According to the figure, 0.04 appears to be the best value and therefore it was selected for the remainder of the experiments. The differences between smaller values appear small, whereas larger values seem to reduce solution quality. Figure 2 presents the percentage difference with respect to the minimum value for different number of initial solutions, using problems RC101 and C

15 % 2 1,8 1,6 1,4 1,2 1 0,8 0,6 0,4 0,2 0 Figure 2. Sensitivity analysis of Initial solutions Initial solutions RC101 C103 According to Figure 2, it would appear to be important to create several initial solutions, but the gain from additional solutions becomes small after five to six solutions. Figure 3 illustrates the convergence of the algorithm over 5000 iterations. The vertical axis gives the percentage difference to the minimum total cost, based on iterations. This illustration is provided for the total cost of each of the six main problem groups in the Liu and Shen (1999) benchmar. 15

16 Figure 3. Sensitivity analysis of Number of iterations C1 C2 R1 R2 RC1 RC Figure 3 indicates that most of the improvement is obtained during the first iterations. On the other hand, the algorithm seems to be able to find small improvements up to several thousands of iterations. The difference to the minimum cost solution is also significantly larger for the R2 and C2 sets, even after iterations. The latter observation motivated a further investigation of MSDA performance up to iterations for all instances, as discussed in 4.2 below Computational results Table 1 reports the algorithm s performance on the Liu and Shen (1999) benchmar for three test runs. The total cost for the different problem instances hardly differs over the three runs. In fact, the largest deviation is only 1.62% ((total cost lowest total cost)/lowest total cost), which indicates a good robustness of the algorithm. The average capacity utilization over all problems is high: 96.0% for cost structure A, 93.6% for cost structure B, and 93.0% for cost structure C. Table 1. Comparison between three independent runs Cost structure A Cost structure B Cost structure C Problem Run 1 Run 2 Run 3 Run 1 Run 2 Run 3 Run 1 Run 2 Run 3 R C RC R C RC CPU/ s To compare the algorithm s performance to previous results reported in the literature, MSDA Medium is defined as the best result over three runs using the parameter setting described in 4.1 above. MSDA Best refers to results for the same parameter setting as MSDA Medium, but with iterations. MSDA Quic refers to a single run using one initial solution and 200 iterations, other parameter values remaining the same. 16

17 Table 2 gives a general overview of the performance of the MSDA algorithm, compared to the approaches of Liu and Shen (1999), Dullaert et al. (2000) and Dell Amico et al. (2006). To allow a better comparison, we report total cost values and en route time values. The average CPU time refers to an average time per instance over all 168 problems and the specified number of runs of the heuristic. Table 2. Comparison between different approaches for the Liu and Shen (1999) variant Data set Liu & Shen 99 Dullaert et al. 02 Dell Amico et al. 06 MSDA Quic MSDA Quic MSDA Medium MSDA Medium MSDA Best MSDA Best total cost en route total cost en route total en route total en route total cost time time cost time cost time R1A R1B R1C C1A C1B C1C RC1A RC1B RC1C R2A R2B R2C C2A C2B C2C RC2A RC2B RC2C Average Computer P 233 N/A P 600 AMD 2600 AMD 2600 AMD 2600 AMD 2600 AMD 2600 AMD 2600 Runs Average CPU seconds per instance N/A According to Table 2, MSDA Medium is consistently better than the competing methods with respect to total cost objective in all data sets. MSDA Medium also found 157 best-nown solutions to the 168 test problems. Six of the best-nown solutions are matches with previous best-nown solutions and 151 are new best-nown solutions. MSDA Medium, however, requires slightly more computational effort (considering the speed of the computers) compared to other methods. Based on Dongarra (2006) one may conclude that the computer used for the MSDA experiments is approximately 10 times faster than the one used by Dell Amico et al. (2006) and about 30 times faster than the one used by Liu and Shen (1999). Therefore one run of MSDA Medium (see Table 1) can be considered to be comparable to Dell Amico et al. (2006) and MSDA Quic to be about twice as fast as Liu and Shen (1999). MSDA Quic gives also better solution quality on the average over all data sets with respect to previous methods but it is outperformed in several individual data sets. In terms of en route time, both MSDA Quic and MSDA Medium are better than Dullaert et al. (2002) on average, although the Dullaert et al. method can be assumed to be much faster since it utilizes only a simple cheapest insertion heuristic. With roughly ten times more CPU time, MSDA Best yields an average performance improvement of some 1.1 % over MSDA Medium and 167 best-nown solutions (two of them are matches with previous best-nown solutions). 17

18 Table 3 reports on the savings of the new best solutions found by MSDA compared to the best-solutions previously reported in the literature (total cost of previous best nown best solutions MSDA)*100/previous best nown). Cost savings tend to differ over the cost structures, without exposing a clear pattern related to the level of vehicle costs. MSDA does offer significantly higher cost savings for the R2, C2 and RC2 subsets compared to the R1, C1 and RC1 subsets. The smallest improvement was obtained for the C1 set for cost structure C, the largest for RC2 for the same low vehicle cost structure. These findings may well indicate a varying power of the MSDA heuristic over the different subsets or important differences in the quality of the previously bestnown results. For detailed results on the new best-nown solutions, see the Appendix. Table 3. Cost savings in percentages compared to previous best nown solutions A B C R C RC R C RC A separate test run was carried out where the en route time objective (1 ) was replaced with the total travel cost objective (1b). As arc travel cost we used the vehicle independent arc distance. These results are given in Table 4. Apart from the objective function, no adjustments were made to the algorithm or parameter values. Again, detailed results are found in the Appendix. Table 4. MSDA Medium results for the total arc distance variant 18

19 Data set MSDA Medium distance MSDA Medium total cost (with distance) R1A R1B R1C C1A C1B C1C RC1A RC1B RC1C R2A R2B R2C C2A C2B C2C RC2A RC2B RC2C Average Computer Average CPU sec AMD AMD Summary and Conclusions The Fleet Size and Mix Vehicle Routing Problem with Time Windows (FSMVRPTW) deals with the minimization of a combination of fixed vehicle costs and routing costs for a heterogeneous fleet. It is an industrially important problem that has not been studied much in the literature. The problem was introduced by Liu and Shen (1999). The authors created a FSMVRPTW benchmar based on Solomon s VRPTW instances (Solomon, 1987). Their model has the objective of minimizing the sum of fixed fleet costs and total en route time and leads to a mixed-integer linear program formulation. With slight variations, the problem has also been studied by Dullaert et al. (2002) and Dell Amico et al. (2006). For the Liu and Shen FSMVRPTW, there has been some confusion in the literature on how to report objective values. In the results from our experimental investigation reported here, we have tried to remove ambiguities. Also, we have defined a new variant of the FSMVRPTW that is a straightforward extension of the VRPTW. Instead of the en route time objective component, there is a total travel cost component. This also leads to a mixedinteger linear program formulation. In our view, this alternative problem definition is industrially relevant. We have developed a three phase multi-restart deterministic annealing metaheuristic called MSDA for solving both variants of the FSMVRPTW. The first phase is designed to generate a number of diverse initial solutions in a short time, utilizing a probabilistic extension of the savings heuristic by Clare and Wright (1964). Depletion of tours is the overall goal of phase two. It is built upon a greedy procedure that tries to eliminate routes successively by removing all orders and

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