An Ant Colony Algorithm hybridized with insertion heuristics for the Time Dependent Vehicle Routing Problem with Time Windows

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1 An Ant Colony Algorithm hybridized with insertion heuristics for the Time Dependent Vehicle Routing Problem with Time Windows S.R.Balseiro,a,1, I.Loiseau b, J.Ramonet c a Graduate School of Business, Columbia University b Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires c Facultad de Ingenieria, Universidad de Buenos Aires Abstract This paper presents an Ant Colony System algorithm hybridized with insertion heuristics for the Time Dependent Vehicle Routing Problem with Time Windows (TDVRPTW). In the TDVRPTW a fleet of vehicles must deliver goods to a set of customers, time window constraints of the customers must be respected and the fact that the travel time between to points depends on the time of departure has to be taken into account. The latter assumption is particularly important in an urban context where the traffic plays a significant role. A shortcoming of Ant Colony algorithms for capacitated routing problems is that, at the final stages of the algorithm, ants tend to create infeasible solutions with unrouted clients. Hence, we propose enhancing the algorithm with an aggressive insertion heuristic relying on the minimum delay metric. Computational results confirm the benefits of more involved insertion heuristics. Moreover, the resulting algorithm turns out to be competitive, matching or improving the best known results in several benchmark problems. Key words: Ant Colony System, Insertion Heuristics, Minimum Delay, Time-dependent, Vehicle routing problem with time windows 1. Introduction In the last decades, due to the increasing demand for better service and tighter requirements from the customers, physical distribution became more complex, and a key process within the chain value. Scheduling and routing problems were first studied at the end of the 50 s, and since then hundreds of models and algorithms have been proposed for the diferent versions of these problems. The Vehicle Routing Problem (VRP) requires the determination of an optimal set of routes for a fleet vehicles that needs to serve a set of customers. The problem was first introduced by Dantzig and Ramser in 1959 [10]. Since then, attention has been devoted to more complex variants of the VRP appearing in real life. Additional constraints were considered, such as limits on the capacity of the vehicles, time windows for the customers to be served, and limits on the time a driver can work or on the lengths of the routes, just to name a few. In this paper we deal with the case where traveling times vary during the planning horizon, vehicle capacity is limited and clients impose time windows constraints. In the Time Dependent Vehicle Routing Problem with Time Windows (TDVRPTW) a set of customers must be served by a fleet of vehicles located in a single depot. A quantity of goods d i should be delivered to each client i = 1,..., n, whose service takes an amount of time s i. The customers impose a time window [a i, b i ] during which they can be served, i.e., the service can start at a i at the earliest and no later than b i. Corresponding author. Tel.: ; fax: addresses: srb2155@columbia.edu (S.R.Balseiro), irene@dc.uba.ar (I.Loiseau), juan ramonet@yahoo.com.ar (J.Ramonet) 1 This work was developed while the author was affiliated with Universidad de Buenos Aires. Preprint submitted to Computers & Operations Research September 12, 2010

2 A vehicle is permitted to arrive at a client i before a i, but it must wait until that time to deliver the goods. Since deliveries cannot be split, a client is always served by a single vehicle. Finally, the vehicles have a maximum capacity C and the routes start and finish in a unique depot. Any resident of a large metropolitan area has experienced that, in an urban environment, the time taken to travel between two points does not depend solely on the distance: traffic fluctuations seriously affect travel speeds, resulting in great variations in traveling times. There are many causes for this phenomenon. Some are completely random such as road accidents or weather conditions. Others causes such as rush hours occur daily and are related to the traveling direction and time of day: generally during the morning the traffic flows to centric areas of the city while during the evening it drains to the suburbs. Finally, the variations may be periodical or even seasonal. The novelty of the TDVRPTW, introduced by Malandraki [24], with respect to the well-studied VRPTW is the inclusion of time dependence, i.e., the time taken to travel an arc is a function of the departure time. If this time dependence is ignored sub-optimal solutions could be obtained or even solutions could turn out to be infeasible, since vehicles might end up arriving later than expected leading to the violation of time windows. In Ant Colony Optimization (ACO), a group of simple artificial ants, which cooperate via low-level communications, are coordinated to solve complex optimization problems [14]. Many ACO algorithms have been proposed to solve different types of combinatorial optimization problems, including several VRP variants. A weakness of Ant Colony algorithms for capacitated routing problems is that ants tend to create infeasible solutions with unrouted clients. Most implementations return feasibility to the solutions by applying a simple ad hoc post insertion procedure, and when the post insertion fails to route all the clients, the partial solution is discarded. Usually this aggravates as solutions improve and constraints tighten. We propose hybridizing the metaheuristic with an aggressive insertion procedure based on the minimum delay technique to increase the number of feasible solutions at the final stages of the algorithm. In this paper we present a Multiple Ant Colony System algorithm hybridized with Insertion Heuristics (MACS-IH), and study the impact on this algorithm of the more involved insertion heuristics. The organization of this paper is as follows. In Section 2 we review the literature and existing algorithms for the TDVRPTW. In Section 3 we formulate the problem as Mixed Integer Linear Program. Next, in Section 4 we describe Ant Colony Optimization, and in particular the MACS-IH algorithm. In Section 5 we explain the local search procedure, and in Section 6 the post insertion heuristic. Section 7 reports the test performed to tune the new algorithm and computational results on instances for the VRPTW with and without time dependence. We conclude with some final remarks and future work in Section Literature Review Since the TDVRPTW is a generalization of the VRPTW, it belongs to the class of NP-hard problems [26]. Most real world sized problems cannot be solved to optimality within a reasonable time by existing exact approachs. Heuristic algorithms instead can find solutions of very high quality in much shorter time. Several state-of-the-art metaheuristics have been developed for the VRPTW, such as tabu search [29, 8], simulated annealing [7, 9], genetic algorithms [30, 4], evolution strategies [19] and ant colony systems [16]. Toth and Vigo [31] and the more recent book by Golden et al. [18] offer comprehensive review about the Vehicle Routing Problem. Considerably less research has been devoted to the TDVRPTW. In the seminal work of Malandraki [24] she presented a Mixed Integer Linear Programming (MILP) formulation, some simple constructive heuristics algorithms and a cutting plane method for TDVRP, i.e., a special case without time windows. Later, Ichoua et al. [20] proposed a parallel tabu search for TDVRP with soft time windows, which minimizes a weighted sum of the total time and lateness. In their paper, they study time dependence models and point out that previous models did not verify the first in first out (FIFO) property. This property states that when two identical vehicles traverse the same arc, the one leaving afterwards should always arrive later. Furthermore, they tested their model in a dynamic environment. 2

3 Fleischmann et al. [15] developed a savings, sequential insertion and 2-opt local search algorithms for the TDVRPTW. The highlight of their work is the extension of the concept of time window of a route [27] to a time dependent problem. The latter allows the heuristics to check feasibility of insertions and other moves in constant time. The paper concludes with computational results using travel time data from a traffic information system in Berlin. Recently, Donati et al. [11] presented an Ant Colony metaheuristics for the TDVRPTW employing the Multiple Ant Colony System framework. They apply the algorithm to a real road network in Padua, Italy. Since its shares some resemblance with our algorithm, it will be described in detail in the following sections. The new insertion procedures we incorporate were motivated on the solution techniques for the m- VRPTW, a variant of the VRPTW in which the objective is to maximize the number of served clients with m or fewer vehicles. [21] Lau et al. proposed a Tabu Search algorithm with a holding list containing the unrouted clients for the m-vrptw. They define an extended neighborhood space that allows relocations and exchanges with the holding list. In [22], Lim and Zhang present another competitive Tabu Search algorithm for the WRPTW by considering local search operators that seek to minimize the minimum delay (MDL) of the unrouted clients. This metric measures the minimal delay caused by inserting an unrouted customer back into the solution [19, 3], and will be further explained in Section Problem Formulation The TDVRPTW involves the minimization of the following objectives in order of priority: (1) number of routes and (2) total time (alternatively distance or travel time). Thus, a solution that employs fewer vehicles is always better than other using more in spite of the total time and only when two solutions have the same number of routes their total times are compared. This objective is very common in practice and extensively used in the literature [31]. It arises, for instance, when fixed cost associated with vehicles and drivers are considered. Because these costs usually have a greater impact than per-mile transportation cost, practitioners prefer an operation with fewer routes. To take into account time dependence, Malandraki and Daskin [25] propose dividing the time horizon in M slices and assigning a constant travel time t m ij for each arc (i, j) and slice m. Therefore, the travel time of each arc is a piecewise step function. However, this representation does not satisfy the FIFO property [20]. The fulfillment of the FIFO property can be accomplished by assigning a speed distribution to each arc and then obtaining the time distribution by integration. The resulting time distribution is continuous, and with slop greater than 1, thus satisfying the FIFO property [15]. Ichoua et al. [20] suggests adopting a step function for the speed during the day mainly because of its simplicity and its ease of determination in real situations. In this model the resulting time distribution is a piecewise linear function that satisfies the FIFO property. An example of a speed and travel time distributions are presented in Figure 1. (a) Speed Distribution (b) Travel Time Distribution Figure 1: An example of a speed distribution and the induced travel time distribution for an arc with a distance of 0.5 units. The scheduling horizon is divided in three periods, the first and last are rush hours with slower traveling speeds. 3

4 We present here a MILP formulation based on the vehicle flow model introduced by Malandraki and Daskin [25]. While their formulation minimizes only the total traveled time, ours considers a hierarchical objective function that first minimizes the number of routes and then the total time. This function reflects real problems were diminisshing the number of vechicles is the first goal. In this new formulation clients impose hard time windows constraints. Also, in [25] the arc travel times are a step function of the departure time, while we consider piecewise linear travel time distributions satisfying the FIFO property. The problem with n clients is defined over the directed graph G = (V, X). The graph contains 2κ copies of the depot, where κ is an upper bound on the number of vehicles. Each route starts on one of the first κ dummy depots (numbered from 1 κ to 0) and ends on any of the last κ copies (numbered from n + 1 to n+κ). Throughout the formulation N = {1,..., n} is the set of all clients while the sets D O = {1 κ,..., 0} and D E = {n + 1,..., n + κ} represent the dummy origin and end copies of the depot respectively. The forward star + (i) is the set of vertices j such that (i, j) X and the backward star (i) is the set of vertices i such that (i, j) X. The time horizon is divided in M time slices, each one delimited by its end time T m. Travel times are a piecewise linear function of the departure time. Within slice m the travel time of arc (i, j) when departing at time t is αij m + βm ij t. In order to comply with the FIFO property coefficients αm ij, and βm ij are chosen so that the function is continuous over all slices and βij m 1. A binary variable x m ij is assigned to each arc and time slice, so that xm ij is 1 if a vehicle travels through arc (i, j) during time slice m in the solution. Each node is assigned a positive variable t i that holds the departure time from node i. The variable w i registers the load delivered by the visiting vehicle when arrived at node i. min subject to B i D O i (j) m M i (j) m M j + (i) m M j + (i) m M j + (i) m M x m ij + i D E t i (1) x m ij = 1 j N (2) x m ij 1 j D O (3) x m ij = 1 i N (4) x m ij 1 i D E (5) t i + αij m + βij m t i + s j t j B ( 1 x m ) ij (i, j) X, m M (6) t i T m B ( 1 x m ) ij (i, j) X, m M (7) t i T m 1 x m ij (i, j) X, m M (8) a i + s i t i b i + s i i V (9) ( ) w i + d i w j B 1 m M x m ij (i, j) X (10) w i C i D E (11) x m ij {0, 1} (i, j) X, m M (12) t i 0, w i 0 i V (13) In the objective function of the problem (1) the first term is the number vehicles used and it is multiplied by a large constant B to establish the objective s hierarchical nature. The second term of the objective is the total time of all routes, which is the sum of the travel times, service times and waiting times. Constraints 4

5 (2) and (4) force each client to be visited exactly once while (3) and (5) allow the vehicles to stay in the deposit if not needed. Constraint (6) computes the departure time for each node depending on the time slice the vehicle left and the source node. Constraints (7) and (8) link the departure time t i with the time slice so that the proper piece of the traveling time function is employed. Then (9) verifies that the time windows (defined on arrival time) are respected. (10) computes the load of the vehicles as they advance through the routes, while (11) imposes the maximum vehicle capacity. Finally, in (12) the decision variables x m ij are restricted to be binary, and in (13) the time and load variables are set to be nonnegative. This formulation employs O(Mn 2 ) non-negative integer variables and O(Mn 2 ) constraints. 4. Ant Colony Optimization Similarly to other insect societies, ant colonies are distributed systems that despite of the simplicity of their individuals present a highly organized social structure. Ants coordinate their activities through stimergy: an indirect communication accomplished by modifying the environment in which they move [14]. When ants move they leave a trail of a chemical substance called pheromone. While an isolated ant moves randomly, if it detects a trail of pheromone left by another ant, it may choose to follow that trail and strengthen it with its own pheromone. As more ants follow the trail, more pheromone is deposited, and more attractive it becomes for future ants. This positive feedback is an example of the organizational behavior of the colony. The motivation behind Ant Colony Optimization (ACO) lies in the fact that the same organizational principia that successfully govern ants can be exploited to coordinate a group of artificial agents cooperating to solve complex optimization problems [13]. Our MACS-IH algorithm, is based on the Multiple Ant Colony System (MACS) framework proposed by Dorigo and Gamardella [12]. In MACS computational ants are independent agents that, like real ants, communicate exclusively through pheromone trails. In order to tackle the optimization of multiple objectives, two ant colonies are maintained in parallel, each dealing with a single objective. MACS has been successfully applied to many VRP variants [13, 11, 16], and results presented by authors show that it is comparable to state-of-the-art algorithms, both in calculation time and quality of solutions. MACS-IH coordinates the activities of the two colonies so as to optimize both objectives simultaneously. The first colony, denoted by ACS-VEI, attempts to reduce the number of routes. The second colony, denoted by ACS-TIME, optimizes the feasible solutions found by ACS-VEI by minimizing the total time. Each colony has a fixed number of ants, denoted by k. These colonies work independently; each has its own best solution and pheromone matrix, but they communicate through the exchange of pheromone updates. In the remainder of this section we describe the algorithm from the highest to the lowest level of detail. We begin by explaining the main procedure and how the colonies are coordinated. Then, we dive into the colonies, and describe how they operate. We conclude by showing how individual ants behave and solutions are constructed The main procedure Figure 2 presents a flowchart of the whole algorithm while a pseudocode for the main procedure is shown in Algorithm 1. The algorithm starts with an initial solution, denoted by Ψ 0. Three different constructive heuristics were implemented to generate the initial solution, namely: Solomon I1, Sequential Nearest Neighbor and Parallel Nearest Neighbor [28]. Ψ 0 is taken as the best of the solutions generated by those heuristics. At any point in time the algorithm maintains an upper bound on the minimum number of vehicles needed, denoted by v, and a feasible solution with exactly v vehicles, and the best-so-far total time, denoted by Ψ TIME. Initially, Ψ TIME is set to Ψ 0, and v to the number of vehicles in that solution. The colony ACS-TIME works on solutions with the v vehicles, and attempts to improve the total time of Ψ TIME. Simultaneously, the colony ACS-VEI tries to reduce the number of vehicles by minimizing the number of unrouted clients in solutions with v 1 vehicles. The colony maintains an infeasible solution with v 1 vehicles, and the least-so-far number of unserved clients, denoted by Ψ VEI. If ACS-VEI obtains a solution with no clients unserved, that is unrouted(ψ VEI ) = 0, it has succeeded in finding a feasible solution with 5

6 v 1 vehicles. At this point v is decreased by one, and both colonies are restarted with the reduced number of vehicles. The initial solution for ACS-VEI is created by discarding the shortest route from the initial solution of ACS-TIME. The algorithm runs until some stopping criteria is met. For instance, when solutions show no more improvement, a maximum number of iterations is reached, or the total execution time elapsed exceeds a given limit. The algorithm returns the best-so-far feasible solution Ψ TIME. ACS-TIME ACS-VEI Constructive Phase with v vehicles Constructive Phase with v -1 vehicles for each solution k new solutions for each solution k new solutions INSERTIONHEURISTIC All served LOCALSEARCH v v - 1 Restart colonies All served INSERTIONHEURISTIC Fewer unserved clients or less MDL else Less total time e else Replace Ψ VEI Replace Ψ TIME GLOBALPHERMONEUPDATE Ψ TIME GLOBALPHERMONEUPDATE Ψ TIME and Ψ VEI Repeat Repeat Figure 2: Flowchart of the Multiple Ant Colony System Algorithm hybridized with Insertion Heuristics. Two MACS-IH ant colonies are maintained in parallel. Algorithm 1 Pseudocode for the MACS-IH main procedure 1: function MACS-IH(Ψ 0 ) 2: v vehicles(ψ 0 ) Initial number of vehicles 3: Ψ TIME Ψ 0 Initial solutions 4: Ψ VEI new solution(v 1) 5: repeat 6: ACS-TIME.Iterate(v) Iterate ACS-TIME colony with v vehicles 7: ACS-VEI.Iterate(v 1) Iterate ACS-VEI colony with v 1 vehicles 8: if unrouted(ψ VEI ) = 0 then ACS-VEI found a feasible solution with v 1 vehicles? 9: v v 1 Decrease the number of vehicles 10: Ψ TIME Ψ VEI Update the solutions 11: Ψ VEI new solution(v 1) 12: end if 13: until termination criterion is met 14: return Ψ TIME 15: end function 6

7 4.2. The ACS-VEI and ACS-TIME colonies An iteration of a colony is composed of a constructive and an update phase. The constructive phase involves the production of new solutions, while in the update phase these new solutions are improved and the pheromone is deposited. In the constructive phase first all the ants are located in the depot, and then ants move, in parallel and one step at a time, appending new clients to their partial solutions. By the end of the constructive phase each ant has built a complete solution to the problem with at most v vehicles (or v 1 in the case of ACS-VEI). The details of the constructive phase are given in Section 4.3. As we shall see both colonies differ mainly in their update phase. We proceed by explaining first colony ACS-TIME, and then ACS-VEI ACS-TIME Algorithm 2 provides pseudocode for a complete ACS-TIME s iteration. In the update phase of ACS- TIME infeasible solutions first undergo an InsertionHeuristic procedure to insert unserved clients. If the procedure fails to accommodate all clients, the infeasible solution is discarded. Afterwards, a LocalSearch procedure is applied to improve the solution s total time. Both heuristics play a very important role in the performance of the metaheuristic. We defer the description of these heuristics until sections 6 and 5 respectively. Finally, the total time of the ant s solution is compared to that of the incumbent Ψ TIME, and if better the new solution is set as the incumbent. Computational ants do not deposit pheromone as they move, instead pheromone is deposited once the solution is constructed and its quality known. This way, strayed ants are not followed and poor solutions discouraged. In this model each arc (i, j) of the graph has associated a non-negative pheromone trail, denoted by τ ij, which records how efficient the arc was in the past. In ACS-TIME pheromone is deposited by the GlobalPhermoneUpdate procedure at the end of the iteration, in the arcs belonging to Ψ TIME according to the following formula τ ij (1 ρ) τ ij + ρ time(ψ TIME ) (i, j) Ψ TIME, where time(ψ TIME ) is the total time of the solution, and ρ is parameter between 0 and 1. Algorithm 2 Pseudo code for an iteration of the ACS-TIME colony 1: procedure ACS-TIME.iterate(v) Constructive phase 2: for all a ants : a.initialize(v) Initialize all ants with v vehicles 3: repeat 4: for all a ants : a.iterate Move all ants one step at a time in parallel 5: until all ants are done Update phase 6: for all Ψ {a.ψ : a ants} do 7: if unrouted(ψ) > 0 then 8: InsertionHeuristic(Ψ) Try to insert unserved clients 9: ignore ant is some clients remain unserved 10: end if 11: LocalSearch(Ψ) Perform a local search to reduce total time 12: if time(ψ) < time(ψ TIME ) then 13: Ψ TIME Ψ A better solution was found 14: end if 15: end for 16: GlobalPheromoneUpdate(Ψ TIME ) Global pheromone update according to Ψ TIME 17: end procedure 7

8 ACS-VEI Algorithm 3 provides pseudocode for a complete ACS-VEI s iteration. ACS-VEI differs from the previous colony in what follows. First, the colony maintains counters {in i } i N, which register the number of times that a client i has not been included in a solution [16]. These counters are used to make clients that are frequently not served more attractive to the ants, and hence increase their likelihood of being served. Whenever a new better solution is found in is reset. Second, because ACS-VEI is not concerned about the solutions time, the local search is not applied to the ants solutions. Third, a solution is considered better when the number of unrouted clients is lower. When two solutions have the same number of unrouted clients, the solution with lower MDL is considered better. The MDL metric quantifies how much the constraints would be violated if the unserved clients were to be inserted. Hence, the lower the MDL, the closer the clients are to be served. This metric is described in Section 6. ACS-VEI s pheromone matrix is updated with two solutions: Ψ VEI and Ψ TIME. The rationale for this double update is that if pheromone is only deposited according to Ψ VEI, then arcs incident to its unrouted nodes would receive no new pheromone, and thus reducing the probability of including these nodes in successive iterations. Algorithm 3 Pseudo code for an iteration of the ACS-VEI colony 1: procedure ACS-VEI.iterate(v) Constructive phase 2: for all a ants : a.initialize(v) Initialize all ants with v vehicles 3: repeat 4: for all a ants : a.iterate Move all ants one step at a time in parallel 5: until all ants are done Update phase 6: for all Ψ {a.ψ : a ants} do 7: for all i unrouted(ψ) do 8: in i in i + 1 Keep track of unserved customers 9: end for 10: InsertionHeuristic(Ψ) Try to insert unserved clients 11: if unrouted(ψ) < unrouted(ψ VEI ) or 12: (unrouted(ψ) = unrouted(ψ VEI ) and mdl(ψ) < mdl(ψ VEI )) then 13: Ψ VEI Ψ A better solution was found 14: in 0 15: end if 16: end for 17: GlobalPheromoneUpdate(Ψ TIME ) Double global pheromone update 18: GlobalPheromoneUpdate(Ψ VEI ) 19: end procedure 4.3. The constructive phase In the constructive phase each ant creates a new solution by moving in parallel through the graph. Algorithm 4 describes how ants create the solutions. The procedure Ant.Initilize initializes an ant by adding v empty routes to its solution and locating it in the depot. Ant.Iterate performs an iteration at the ant level. Broadly, in one iteration ant moves to an adjacent node in the graph, and appends this node to its partial solution. As it advances through the graph adding clients, it registers which clients have been visited, and updates the load and travel time of the partial route. At some point, due to the capacity or time windows constraints, the ant can no longer accommodate new clients to the route. When this happens, it returns to depot, picks a new vehicle and starts a new route. These steps continue until no more vehicles are available, thus finalizing the ants constructive phase. DecisionRule is the procedure employed by ants to choose the next customer to visit. Like real ants, their computational counterparts move randomly but biased towards the strongest pheromone trails. The 8

9 probability p ij of moving to client j from the current node i is computed based on two values: a heuristic value η ij that yields closer clients more attractive, and the pheromone level τ ij. The formula for the transition probability is τ ij ηβ α ij if j N p ij = l N τ α i, i il ηβ il (14) 0 otherwise, where the positive exponents α and β are parameters determining the relative importance of η versus τ, and N i denotes the set of feasible nearest neighbors of node i, that is, clients j + (i) such that (i) they are not visited, (ii) their demand does not exceed the remaining capacity of the route, and (iii) they can be reached before the end of their time window. The heuristic value η ij is computed as follows [11] { ) } 1 η ij = max 1, (max{a ij (t i ), a j } t i )(b j t i in j, (15) where A ij (t) denotes the arrival time to j when departing from i at time t, and t i denotes the departure time from i, which is equivalent to time(ψ r ) at that point. Within the maximum of (15) the first factor takes into account the distance to the customer, the second the urgency of delivery, and in ACS-VEI the discounting term in j renders more attractive clients that are frequently not served. For ACS-TIME in is set to zero. DecisionRule employs a pseudo-random proportional mechanism to choose the next customer: the neighbor with the highest probability is chosen with a probability q 0 while with a probability 1 q 0 the next node is chosen randomly according to (14). As the parameter q 0 closes to one the search is narrowed towards the best solution, whereas lower values promote the exploration of new solutions. Thus, the parameter controls the behavior of the search shifting from exploitation to exploration. Another important mechanism of the algorithm is evaporation: like in nature, pheromone s intensity decreases as time goes by. The purpose of evaporation is reducing the influence of pheromone deposited in the first iterations whose solutions quality is usually poor. Hence pristine solutions are forgotten in favor of more promising new ones and the convergence towards a premature sub-optimum is avoided. In ACS pheromone evaporation occurs locally during the constructive phase whenever an ant crosses an arc using the formula τ ij (1 ξ) τ ij + ξ τ 0, where ξ is a parameter between 0 and 1, and τ 0 is the initial level of pheromone, which is set to τ 0 = (N time(ψ 0 )) 1. 9

10 Algorithm 4 Ant s algorithm 1: procedure Ant.Initialize(v) Initialize an ant with v vehicles 2: add v empty routes to solution Ψ 3: r 0 Set current route to the first one 4: i depot Locate ant in depot 5: end procedure 6: procedure Ant.Iterate Move ant one step 7: j DecisionRule(i) Choose the next customer j 8: Ψ r Ψ r {j} Add the new customer j to the route Ψ r 9: time(ψ r ) max{a ij (time(ψ r )), a j } + s j Update the route s total time 10: load(ψ r ) load(ψ r ) + d j Update the route s load 11: visited j true Register client j as visited 12: τ ij ξτ ij + (1 ξ)τ 0 Pheromone evaporation 13: i j Move the ant 14: if j = depot then 15: r r + 1 Current route is finished 16: end if 17: if r = v then 18: done true Finish when no more vehicles available 19: end if 20: end procedure 5. Local Search The LocalSearch procedure attempts to improve the total time of a candidate solution by iteratively moving to a neighbor solution. The neighboring solutions are those obtained from applying some local search (LS) operator. Most LS operators can be defined in terms of Lin s λ-opt [23], which involves the removal of λ arcs from a solution and the subsequent reconnection of the resulting segments in every possible way. In this section we describe the operators implemented, and the improvements made to increase the efficiency of the feasibility check and the evaluation of new solutions. A survey by Bräysy and Gendreau [5] examines several local search algorithms Operators Since LS plays such an important role in the algorithm s performance, several neighbor operators where implemented. These are classified as single-route or multi-route depending on whether they operate on one or two routes respectively. Next, we define the operators implemented followed by the resulting routes as in Lim and Zhang [22]. Let r = v 0,..., v i 1, v i, v i+1,..., v j 1, v j, v j+1,..., v m and r = v 0,..., v p 1, v p, v p+1,..., v q 1, v q, v q+1,..., v n be two different routes from the candidate solution. For the sake of clarity, the changes introduced in the resulting routes are underlined. (a) Single-route (i) Relocate. Moves a customer v i before customer v j within the route (v j may be before v i ). r = v 0,..., v i 1, v i+1,..., v j 1, v i, v j, v j+1,..., v m (ii) Exchange. Exchanges two clients v i and v j within a route. r = v 0,..., v i 1, v j, v i+1,..., v j 1, v i, v j+1,..., v m (iii) 2-opt. Inverts the order of a sequence of clients of length w (with w 4). r = v 0,..., v i 1, v i+w 1,..., v i, v i+w,..., v m 10

11 (iv) Or-opt. Moves a sequence of clients v i,..., v i+w 1 of length w to another position (with w 3). r = v 0,..., v i 1, v i+w,..., v j 1, v i,..., v i+w 1, v j, v m (b) Multi-route (i) Relocate. Moves a customer v i from r before a customer v p in route r. r = v 0,..., v i 1, v i+1,..., v m r = v 0,..., v p 1, v i, v p,..., v n (ii) Exchange. Exchanges customers v i in r with v p in route r. (iii) 2-opt*. Exchanges the tails of two routes. r = v 0,..., v i 1, v p, v i+1,..., v m r = v 0,..., v p 1, v i, v p+1,..., v n r = v 0,..., v i 1, v p,..., v n r = v 0,..., v p 1, v i,..., v m (iv) Cross. Exchanges two sequences of length w 1, and w 2 (with w 1 4, w 2 4). r = v 0,..., v i 1, v p,..., v p+w 2 1, v i+1,..., v m r = v 0,..., v p 1, v i,..., v i+w1 1, v p+w 2,..., v n To produce solutions of good quality in the least amount of time, the simpler operators Relocate and Exchange (single- and multi-route) are applied first. When no more progress is obtained, the local search switches to the more complex operators 2-opt, Or-opt, 2-opt*, and Cross. The motivation behind this strategy lies in the fact that usually in the first iterations there are promising moves that can be attained by reduced neighborhood operators, which are more efficient Feasibility Check In order to check the feasibility of solutions generated by most LS operators in O(1) time, the route variables earliest departure and latest arrival are maintained for every customer in the solution [6]. For a route r = v 0, v 1, v 2,..., v m the earliest departure ed vi records the earliest time the vehicle can depart from v i after servicing all the previous clients in the route including v i. If A ij (t) is the arrival time to client j when departing from i at time t, these values can be computed as follows ed v0 = a v0, ed vi = max ( A i 1,i ( edvi 1 ), avi ) + svi, for i = 1,..., m. (16) Analogously, the latest arrival la vi is the latest time the vehicle can arrive to v i so that it can service all the next clients in the route including v i, and it is computed as la vm = b vm, la vi = min ( A 1 i,i+1 ( lavi+1 ) svi, b vi ), for i = 0,..., m 1, where A 1 ij (t) gives the time of departure from i so as to reach j by time t. So, to check the feasibility of inserting a client v between v p and v p+1 in r we only need to check that the following conditions are verified load (r) + d v C, (17a) ( ) edvp bv, (17b) A vp,v A v,vp+1 (ed v ) la vp, 11 (17c)

12 where ed v = max ( ( ) ) A vp,v edvp, av + sv is the earliest departure from the candidate client v. Equation (17a) verifies that the route has enough capacity, (17b) checks that v s time window is not violated, and (17c) that the time windows of all subsequent are respected. The route variables need to be updated whenever a move is performed. Also, in order to check the feasibility of insertions of sequence of clients, such as Or-opt or Cross, we employ the concept of Time Window of a Route as described in Fleischmann et al. [15]. Detailed formulas for the time window of a route with time dependence can be found in Balseiro [1] Evaluation Due to time dependence, travel times need to be propagated to the end of the route in order to evaluate a move. They are calculated using equation (16). Since the evaluation of each move is computationally expensive, some optimizations were implemented to reduce the numbers of evaluations needed. To improve scalability, the neighborhood space is restricted to solutions in which the newly connected nodes verify a Nearest Neighbor (NN) relationship. Solomon [28] proposes three metrics to measure the closeness of customers. Given two clients i and j, t ij measures the mean travel time between the two customers averaged over all slices, T ij measures the travel time to j when departing from i the earliest possible, and v ij measures the urgency of delivery to customer j as expressed by the the time remaining until the vehicle s last possible service start. A new neighborhood criterion M ij was designed as the sum of the average travel time between the clients and the minimum waiting time to start service. It is computed as M ij = t ij + max (0, a j t ij s i b i ). This new criteria was introduced because previous metrics failed to penalize clients that were close in distance, but distant in time. For example, consider two close clients with their time windows on opposite ends of the time horizon. After serving the earliest client, a vehicle should wait a long time before serving the other. Thus, they should not be considered neighbors. The final score c ij is taken as the convex combination of the metrics: c ij = δ 1 t ij + δ 2 T ij + δ 3 v ij + δ 4 M ij, (18) where the weights δ i are parameters to be determined. Clients are sorted in ascending order according to c ij and the first k = 30 are selected as NN of i. Finally, instead of applying the first or best improvement in the neighborhood of a solution, the local search examines all moves and keeps the best ones, thus allowing multiple improvements to be applied in a single sweep. However, not all the improvements can be usually performed since some of them could be incompatible: for instance, trying to relocate the same client to different positions. Besides, moves are not necessarily independent because applying an improvement could render another compatible move useless, i.e. yielding no improvement. A greedy criteria was employed to determine the order in which the moves are applied. The best move of the list is applied, the remaining moves are reevaluated, and those incompatible and useless are removed. The procedure is repeated until no more improvements are available. 6. Insertion Heuristics Insertion heuristics attempt to incorporate the largest number of unrouted clients into an infeasible solution. Since ants inevitably produce solutions with unrouted clients, the success of the whole algorithm depends to a great extent on the performance of the post-insertion. Previous implementations of ACO algorithms for the VRPTW [16] and TDVRPTW [11] rely only on a simple Direct Insertion (DI) to reduce the number of unserved clients. The DI procedure attempts to insert clients one at a time until no more clients can be incorporated due to capacity or time windows constraints. When DI terminates, some clients might remain unrouted. However, this does not imply that those clients are unroutable: probably they could be inserted after performing some 12

13 changes to the solution. For instance, inserting a new client might require the previous relocation of another client out of the target route. In the present algorithm, the InsertionHeuristic procedure combines three new insertion heuristics: (i) Local Search + Insertion (LSI), (ii) Local Search + MDL (LSMDL), (iii) Local Search + MFT (LSMFT). The LSI procedure combines the local search s algorithmic framework with an insertion procedure. Given a particular LS operator, it generates all neighboring solutions, and attempts to insert the unrouted clients in them. If a candidate solution where a client can be inserted is found, the search moves to the new solution, and the client is inserted. These steps are repeated until no more insertions can be performed. Because LSI explores more solutions, it succeeds in incorporating more clients to an infeasible solution than DI. LSI improvements are accepted only if a node can be inserted in a neighboring solution. Hence, it fails to include tougher clients that require multiple successive changes in the solution before they can be served. For example, before a node could be inserted it might be necessary to relocate another client to a second route, invert a sequence of clients, and exchange two tails. The minimum delay metric (MDL), which measures how hard is to insert a client, was introduced to address this problem. MDL was first described by Homberger and Gehring [19], later employed successfully by Bent and Van Hentenryck [3], and then extended in Lim and Zhang [22]. Our procedure LSMDL incorporates the MDL metric to the local search s framework. We first describe how the metric is computed, and then its implementation in the local search. MDL measures how hard is to insert a client by quantifying the incidence of the three causes preventing a client from insertion as described in equations (17). These are (i) the violation of the capacity of the vehicle, (ii) the violation of the unrouted client s time window, and (iii) the violation of a subsequent client s time window. Given a solution Ψ and an unrouted client v, the calculation of mdl(v, Ψ) involves finding the insertion position for v that minimally violates the capacity and time windows constraints. First, the { MDL of} a customer is taken as the minimum MDL among all routes r Ψ, that is, mdl(v, Ψ) = min r Ψ mdl (v, r), where we denote by mdl(v, r) the MDL of a customer v with route r. In turn, given a route r, the MDL is taken as the maximum between the penalty for exceeding the route s capacity, denoted by mdl load, and the total penalty for violating time windows, denoted by mdl time. Hence, we have that mdl(v, r) = max (mdl load (v, r), mdl time (v, r)). Ideally, the load penalty mdl load should be equal to the extra capacity required to accommodate the client in the route, i.e., the total violation of condition (17a). Unfortunately, capacity and time have different units. So it is approximated as the additional time necessary to insert it, assuming a constant demand to service ratio among clients. The resulting formula is { 0, if load (r) + dv C r, mdl load (v, r) = load(r)+d v C r load(r)+d v (b 0 a 0 ), otherwise. The penalty for time windows mdl time quantifies the violation of the conditions (17b) and (17c). Its computation involves finding the position p in route r such that the insertion of v between clients v p and v p+1 minimally violates the sum of the three penalties p b, p v and p a. Penalty p b measures the impact of arriving later than permitted to the unrouted client itself, that is, the violation of b v as in condition (17b). Penalty p v accounts for arriving later than permitted to any subsequent client v p+1, v p+2,..., v m in the route, that is, the violation of condition of la vp+1 as in (17c). Notice that penalties p v and p b suffice to determine whether the client can be inserted. Nevertheless, Lim and Zhang [22] propose a third penalty p a that accounts for the violation of the v s early time window 13

14 a v caused by inserting v before v p+1, v p+2,..., v m. The resulting formulas are mdl time (v, r) = min {p b (v, v p, r) + p a (v, v p, r) + p v (v, v p, r)}, v p r p b (v, v p, r) = max { ( ) A vp,v edvp bv, 0 }, p v (v, v p, r) = max { A v,vp+1 (ed v ) la vp, 0 }, { ( p a (v, v p, r) = max a v A 1 ( ) ) } v,v lavp+1 p+1 sv, 0, ( ) where p a measures the violation of the constraint A 1 v,v lavp+1 p+1 sv a v. It is not hard to see that (17c) implies the latter constraint, but the reverse does not hold. The rationale of including p a is symmetry. Removing a client before the optimal insertion point could potentially decrease both p b and p v, while relocating one after would only decrease p b. Hence, without p a the LS would give priority to the relocation of those clients before the insertion point. Figure 3 illustrates the computation of the three penalties when inserting client v in the two-client route 0, v p, v p+1, 0. Time is represented in the horizontal axis, while the different clients are in the vertical axis. Forward arrows show the earliest departures from each customer when departing from the depot at earliest time a 0. Similarly, backward arrows show the latest arrivals to each customer when arriving to the depot at the latest time b 0. Even when the vehicle departs from v p at the earliest time ed vp, it arrives to v after the end of its time window b v, and incurs a penalty p b. After being serviced, it departs from v at time ed v and it arrives to v p+1 after the latest admissible arrival la vp+1 incurring in a penalty p v. Notice that la vp+1 is the latest it can arrive to make it on time to the depot. Now consider the backward traveling. In order to arrive to v p+1 by the latest time la vp+1, it would have to begin service at client v before the early time window a v. Thus, it incurs a penalty p a. v p v v p+1 a 0 b 0 0[ ] a vp ed vp b vp [ ] la vp a v b v [ ] ed v a vp+1 b vp+1 [ ] la vp+1 Service p b p v p a [ ] Time window Fwd. travel Bwd. travel Figure 3: Computation of the three penalties when inserting client v in the two-client route 0, v p, v p+1, 0. Time is represented in the horizontal axis, while the different clients are in the vertical axis. Figure based on Homberger and Gehring [19] The LSMDL procedure applies the local search framework to reduce the total MDL of the solution. We take the total MDL of the solution mdl(ψ) as the sum of the individual MDL of all unrouted clients, that is, mdl (Ψ) = v unrouted(ψ) mdl (v, Ψ). LSMDL computes the MDL of all neighboring solutions, and searches for a candidate solution with lower MDL. When such a solution is found, the incumbent is replaced, and these steps are repeated until no more improvements are achieved. The algorithm employs a data structure called Ejection Pool [22] that keeps all unrouted clients sorted by minimum delay in increasing order. If at any point some unrouted client v verifies that mdl(v, Ψ) = 0, then v is safely inserted in Ψ. To improve the performance of the algorithm, the optimal insertion point is stored along with the minimum delay of each unrouted clients. Finally, if the LSMDL fails to reduce the minimum delay of the solution, as a secondary objective the heuristic tries to maximize the solution s maximal free time (MFT) [22]. The MFT measures the maximum contiguous waiting time within a route. The maximization of this secondary objective helps to create more 14

15 space for future insertions, improving in turn the performance of the LSMDL. As in [22], the MFT of a route r is computed as { mft (r) = max lavp+1 A ( )} v p,v edvp p+1. 0 p<m 7. Computational Results Three groups of computational experiments were performed: the first to determine the best set of parameters for the MACS-IH algorithm and the lasts two to benchmark its performance without and with time dependence respectively Test problems Tests were performed on Solomon s instances for the VRPTW [28]. This classic set of problems contains 56 euclidean instances of 100 clients with time windows but no time dependence. Instances are grouped into six sets according to their attributes. In sets R1 and R2 customer coordinates were randomly generated by a uniform distribution, in sets C1 and C2 they are clustered in groups, and mixed in problems of type RC1 and RC2. Problem sets R1, C1 and RC1 have a short scheduling horizon and lower vehicle capacity, allowing only a few customers to be served by the same vehicle. Conversely, the sets R2, C2 and RC2 have a long scheduling horizon and higher vehicle capacity, thus permitting many clients to be serviced by the same route. Ichoua et al. [20] provide a time-dependent extension of Solomon s instances. They divide the scheduling horizon evenly in three periods T {1, 2, 3}: morning rush hour, middle of the day and evening rush hour. Additionally, arcs are randomly grouped in three different categories c {1, 2, 3} using a symmetric category matrix. Each category is assigned a distinct step speed function, so that category 1 has the lowest speeds while category 3 has the highest speeds. All instances use the same category matrix. Furthermore, they adjust travel speeds to create 3 different types of scenarios, called 1, 2, and 3. Scenario 1 is the one with the lowest time-dependence while scenario 3 has the highest degree of time-dependence. However, they maintain the average speed in approximately 1, so that the average difficulty is the same as in Solomon s problems. These three scenarios are applied to every instance. Ichoua et al. [20] present an algorithm for a variant of the TDVRPTW with soft time windows. When hard time windows are imposed some instances become infeasible, because it becomes impossible to service some clients and return to the depot in time. Unfortunately, we could not find any time-dependent instances designed for the problem with hard time windows. In order to test the algorithm in all instances of these authors and to introduce the least amount of change, we propose a new category matrix. To return feasibility to all instances, the categories of a few arcs departing from the depot were increased to a higher level. Hence, after increasing the travel speed of these arcs, all clients can be served in time again. The algorithm was implemented in C++ and all the experiments reported in this section were carried out on a Intel Centrino 1.7 GHz PC with 512 Mb RAM Parameters Settings In this subsection we describe the experiments conducted to find a good set of parameters for the nearest neighbor metric proposed in Section 5.3 and the test the different insertion heuristics described in Section Nearest Neighbor Metric Different values of parameters were tested to find a robust set of weights δ 1, δ 2, δ 3 and δ 4 for the nearest neighbor metric c ij in Equation (18). Recall that weights are normalized so that they sum up to one. The candidate parameters were chosen so as to the test all possible combinations of the metrics with uniform weights. First, the incidence of each individual metric was tested, that is, for every i = 1,..., 4 we set δ i = 1 and zero elsewhere. Second, we test every pair of metric with uniform weights, i.e. δ i = δ j = 1 2 for i < j. 15

16 Third, we test every triple, i.e. δ i = δ j = δ k = 1 3 for i < j < k. Finally, we test for the case when all weights are uniformly distributed, i.e. δ i = 1 4 for all i. The total number of vectors to test was 24 1 = 15. Given a set of parameters, four different solutions were constructed using a (i) sequential NN heuristic, (ii) sequential NN heuristic plus local search, (iii) parallel NN heuristic and (iv) parallel NN heuristic plus local search. The algorithm was tested with the 56 Time Dependent Solomon instances (with the new category matrix) and the 3 speed scenarios. Thus, 672 solutions were constructed for each parameter set. Results are presented in Table 1. Results show that, on overall, the best metrics are the second one (T ij ) and the fourth (M ij ). Moreover, the introduction of the new metric M ij improves the quality of the solutions produced by the constructive heuristics and the local search. The third metric (v ij ), which measures the urgency of delivery, is the one that has the least impact. Weights of δ i = 1 4 were adopted. Parameters NN Sequential + LS NN Parallel + LS Best δ 1 δ 2 δ 3 δ 4 Routes T.Time T.Time Routes T.Time T.Time Routes T.Time 1/4 1/4 1/4 1/ /3 1/3 1/ /3 1/3 0 1/ /2 0 1/ /2 1/ /2 1/ / / /3 0 1/3 1/ /2 1/ /3 1/3 1/ /2 0 1/ Table 1: Results for the Nearest Neighbor Metric tests. For each set of parameters, the results shown are the averages over 56 time dependent Solomon instances, new category matrix, and all speed scenarios. Since the local search does not modify the number of routes in the solution, only the average total time is reported for those solutions. The last column reports the average over the best of the four solutions constructed for each instance. Results are sorted according the routes in the Best column Insertion Heuristics Two experiments were conducted to assess the impact of the new insertion heuristics in the algorithm. The first experiment compares the effectiveness of proposed insertion heuristics in solutions with unserved customers, while the second measures the impact of the heuristics in the Ant Colony algorithm. Four different variants of InsertionHeuristic were tested: (i) only DI, (ii) LSI, (iii) combination of LSI and LSDMDL, (iv) combination of LSI, LSMDL, and LSMFT. In the last three variants two local search cycles were executed, the first cycle with Relocate and Exchange (single route and multi-route) and the other with 2-opt*, 2-opt and Cross. In the first experiment the insertion heuristics were tested on solutions with limited number of vehicles. Starting from the number of vehicles known to be sufficient to serve all customers, we gradually reduced the fleet size until the number of vehicles was halved. Given a fixed fleet of vehicles, a solution was constructed with Solomon I1 heuristic, and then InsertionHeuristic was applied to minimize the number of unserved customers. Table 2 reports the number of served customers in each intermediate solution for the RC1 time dependent Solomon instances (with the new category matrix and second speed scenario). Experiments on other Solomon instances yield similar behavior. Average results for all problems are shown in Table 3. The results demonstrate the strength of the insertion heuristics to maximize the number of customers served. According to Table 3, LSI reduces the number of unserved customers by 2.5 on average when compared to DI. The incorporation of the MDL metric further reduces the number of unserved customers 16

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