Branch and Cut and Price for the Time Dependent Vehicle Routing Problem with Time Windows

Transcription

1 Branch and Cut and Price or the Time Dependent Vehicle Routing Problem with Time Windows Said Dabia, Stean Röpke, Tom van Woensel, Ton de Kok Beta Working Paper series 361 BETA publicatie WP 361 (working paper) ISBN ISSN NUR 804 Eindhoven November 2011

2 TRANSPORTATION SCIENCE Vol. 00, No. 0, Xxxxx 0000, pp issn eissn INFORMS doi /xxxx c 0000 INFORMS Branch and Cut and Price or the Time Dependent Vehicle Routing Problem with Time Windows Said Dabia Eindhoven University o Technology, School o Industrial Engineering, Eindhoven, The Netherlands, s.dabia@tue.nl Stean Røpke Denmark University o Technology, Department o Transport, Copenhagen, Denmark, sr@transport.dtu.dk Tom Van Woensel Eindhoven University o Technology, School o Industrial Engineering, Eindhoven, The Netherlands, t.v.woensel@tue.nl, Ton De Kok Eindhoven University o Technology, School o Industrial Engineering, Eindhoven, The Netherlands, a.g.d.kok@tue.nl In this paper, we consider the Time-Dependent Vehicle Routing Problem with Time Windows (TDVRPTW). Travel times are time-dependent, meaning that depending on the departure time rom a customer a dierent travel time is incurred. Because o time-dependency, vehicles dispatch times rom the depot are crucial as road congestion might be avoided. Due to its complexity, all existing solutions to the TDVRPTW are based on (meta-) heuristics and no exact methods are known or this problem. In this paper, we propose the irst exact method to solve the TDVRPTW. The MIP ormulation is decomposed into a master problem that is solved by means o column generation, and a pricing problem. To insure integrality, the resulting algorithm is embedded in a Branch and Cut ramework. We aim to determine the set o routes with the least total travel time. Furthermore, or each vehicle, the best dispatch time rom the depot is calculated. Key words : vehicle routing problem; column generation; time-dependent travel times; branch and cut History : 1. Introduction The vehicle routing problem with time windows (VRPTW) concerns the determination o a set o routes starting and ending at a depot, in which the demand o a set o geographically scattered customers is ulilled. Each route is traversed by a vehicle with a ixed and inite capacity, and each customer must be visited exactly once. The total demand delivered in each route should not exceed the vehicle s capacity. At customers time windows are imposed, meaning that service at a customer is only allowed to start within its time window. The solution to the VRPTW consists o the set o routes with the least traveled distance. Due to its practical relevance, the VRPTW has been extensively studied in the literature (Toth and Vigo 2002). Consequently, many (meta-) heuristics and exact methods have been successully developed to solve it. However, most o the existing models are time-independent, meaning that a vehicle is assumed to travel with constant speed throughout its operating period. Because o road congestion, vehicles hardly travel with constant speed. Obviously, solutions derived rom timeindependent models to the VRPTW could be ineasible when implemented in real-lie. In act, in real-lie road congestion results in tremendous delays. Consequently, it is unlikely that a vehicle respects customers time windows. Thereore, it is highly important to consider time-dependent travel times when dealing with the VRPTW. In this paper, we consider the time-dependent vehicle routing problem with time windows (TDVRPTW). We take road congestion into account by assuming time-dependent travel times: 1

3 Author: Branch and Cut and Price or the TDVRPTW 2 Transportation Science 00(0), pp , c 0000 INFORMS depending on the departure time at a customer a dierent travel time is incurred. We divide the planning horizon into time zones (e.g. morning, aternoon, etc.) where a dierent speed is associated with each o these zones. The resulting stepwise speed unction is translated into travel time unctions that satisy the First-In First-Out (FIFO) principle (see also Ichoua et al. (2003)). Because o the time-dependency, the vehicles dispatch times rom the depot are crucial. In act, a later dispatch time rom the depot might result in a reduced travel time as congestion might be avoided. In this paper, we aim to determine the set o routes with the least total travel time. Furthermore, or each vehicle, the best dispatch time rom the depot is calculated. Despite numerous publications dealing with the vehicle routing problem, very ew addressed the inherent time-dependent nature o this problem. Additionally, to our knowledge, all existing algorithms are based on (meta-) heuristics, and no exact approach has been provided or the TDVRPTW. In this paper, we solve the TDVRPTW exactly. We use the low arc ormulation o the VRPTW which is decomposed into a master problem (set partitioning problem) and a pricing problem. While the master problem remains unchanged, compared to that o the VRPTW (as time-dependency is implicitly included in the set o easible solutions) the pricing problem is translated into a time-dependent elementary shortest path problem with resource constraints (TDESPPRC), where time windows and capacity are the constrained resources. The relaxation o the master problem is solved by means o column generation. To guarantee integrality, the resulting column generation algorithm is embedded in a branch-and-bound ramework. Furthermore, in each node, we use cutting planes in the pricing problem to obtain better lower bounds and hence reduce the size o branching trees. This results in a branch-and-cut-and-price (BCP) algorithm. Timedependency in travel times increases the complexity o the pricing problem. In act, the set o easible solutions increases as the cost o a generated column (i.e. route) does not depend only on the visited customers, but also on the vehicles dispatch time rom the depot. The pricing problem in case o the VRPTW is usually solved by means o a labeling algorithm (Desrochers 1986). However, the labeling algorithm designed or the VRPTW is incapable to deal with timedependency in travel times and needs to be adapted. In this paper, we develop a time-dependent labeling (TDL) algorithm such that in each label the arrival time unction (i.e. unction o the departure time rom the depot) o the corresponding partial path is stored. the TDL generates columns that have negative reduced cost together with their best dispatch time rom the depot. To accelerate the BCP algorithm, two heuristics based on the TDL algorithm are designed to quickly ind columns with negative reduced cost. Moreover, new dominance criteria are introduced to discard labels that do not lead to routes in the inal optimal solution. Furthermore, we relax the pricing problem by allowing non-elementary paths. The resulting pricing problem is a timedependent shortest path problem with resource constraints (TDSPPRC). Although the TDSPPRC results in worse lower bounds, it is easier to solve and integrality is still guaranteed by branch-andbound. Moreover, TDSPPRC should work well or instances with tight time windows. The pricing problem is explained in more details in section 5. Over the last decades, BCP proved to be the most successul exact method or the VRPTW. Hence, our choice or a BCP ramework to solve the TDVRPTW is well motivated. The main contributions o this paper are summarized as ollows. First, we present an exact method or the TDVRPTW. We propose a branch-and-cut-and price algorithm to determine the set o routes with the least total travel time. Contrary to the VRPTW, the pricing problem is translated into a TDESPPRC and solved by a time-dependent labeling algorithm. Second, we capture road congestion by incorporating time-dependent travel times. Because o time dependency, vehicles dispatch times rom the depot are crucial. In this paper, dispatch times rom the depot are also optimized. In the literature as well as in practice, dispatch time optimization is approached as a post-processing step, i.e. given the routes, the optimal dispatch times are determined (Kok et al. 2007). In this paper, the scheduling (dispatch time optimization) and routing are simultaneously perormed. Third,...

4 Author: Branch and Cut and Price or the TDVRPTW Transportation Science 00(0), pp , c 0000 INFORMS 3 The paper is organized as ollows Literature Review An abundant number o publications is devoted to the vehicle routing problem (see Laporte (1992), Toth and Vigo (2002), and Laporte (2007) or good reviews). Speciically, the VRPTW has been extensively studied. For good reviews on the VRPTW, the reader is reerred to Bräysy and Gendreau (2005a), and Bräysy and Gendreau (2005b). The majority o these publications assume a time-independent environment where vehicles travel with a constant speed throughout their operating period. Perceiving that vehicles operate in a stochastic and dynamic environment, more researchers moved their eort towards the optimization o the time-dependent vehicle routing problems. Nevertheless, literature on this subject remains scarce. In the context o dynamic vehicle routing, we mention the work o Bertsimas and Simchi-Levi (1996), Bertsimas and Ryzin (1991) and Bertsimas and Ryzin (1993a) where a probabilistic analysis o the vehicle routing problem with stochastic demand and service time is provided. Malandraki and Dial (1996), Hill and Benton (1992) and Ichoua et al. (2003) tackle the vehicle routing problem where vehicles travel time depends on the time o the day, and Malandraki and Daskin (1992) considers a time-dependent traveling salesman problem. Time-dependent travel times has been modeled by dividing the planning horizon into a number o zones, where a dierent speed is associated with each o these time zones (see Ichoua et al. (2003) and Jabali et al. (2009)). In Van Woensel et al. (2008), traic congestion is captured using a queuing approach. Malandraki and Dial (1996) and Malandraki and Daskin (1992) models travel time using stepwise unction, such that dierent time zones are assigned dierent travel times. Fleischmann et al. (2004) emphasized that modeling travel times as such leads to the undesired eect o passing. That is, a later start time might lead to an earlier arrival time. As in Ichoua et al. (2003), we consider travel time unctions that adhere to the FIFO principle. Such travel time unctions does not allow passing. While several successul (meta-) heuristics and exact algorithms have been developed to solve the VRPTW, algorithms designed to deal with the TDVRPTW are somewhat limited to (meta-) heuristics. In act, most o the existing algorithms are based on tabu search (Ichoua et al. (2003), Van Woensel et al. (2008), Jabali et al. (2009) and Maden et al. (2010)). In Malandraki and Dial (1996) mixed integer linear ormulations the time-dependent vehicle routing problem are presented and several heuristics based on nearest neighbor and cutting planes are provided. Donati et al. (2008) proposes an algorithm based on a multi ant colony system and Haghani and Jung (2005) presents a genetic algorithm. In Hashimoto et al. (2008) a local search algorithm or the TDVRPTW is developed and a dynamic programming is embedded in the local search to determine the optimal starting or each route. Androutsopoulos and Zograos (2009) considers a multi-criteria routing problem, they propose an approach based on the decomposition o the problem into a sequence o elementary itinerary subproblems that are solved by means o dynamic programming. Malandraki and Daskin (1992) presents a restricted dynamic programming or the time-dependent traveling salesman problem. In each iteration o the dynamic programming, only a subset with a predeined size and consisting o the best solutions is kept and used to compute solutions in the next iteration. Tang (2008) emphasizes the diiculty o implementing route improvement procedures in case o time-dependent travel times and proposes eicient ways to deal with that issue. In this paper, we attempt to solve the TDVRPTW to optimality using column generation. To the best o our knowledge, this is the irst time an exact method or the TDVRPTW is presented. Column generation has been successully implemented or the VRPTW. For a overview o column generation algorithms, the reader is reerred to Lübbecke and Desrosiers (2005). in the context o the VRPTW, Kohl et al. (1999) designed an eicient column generation algorithm where they applied subtour elimination constraints and 2-path cuts. This has been improved by Cook and Rich (1999) by applying k-path cuts. Jespen et al. (2008) proposes a column generation algorithm

5 Author: Branch and Cut and Price or the TDVRPTW 4 Transportation Science 00(0), pp , c 0000 INFORMS by applying subset-row inequalities to the master problem (set partitioning). Although, adding subset-row inequalities to the master problem increases the complexity o the pricing problem, Jespen et al. (2008) shows that better lower bounds can be obtained rom the linear relaxation o the master problem. To accelerate the pricing problem solution, Desaulniers et al. (2008) proposes a tabu search heuristic or the ESPPRC. Furthermore, elmentarity is relaxed or a subset o nodes and generalized k-inequalities are introduced. Recently, Baldacci et al. (2010) introduce a new route relaxation, called ng-route, used to solve the pricing problem. Their ramework proves to be very eective in solving diicult instances o the VRPTW with wide time windows. Fleischmann et al. (2004) argued that existing algorithms or the VRPTW ail to solve the TDVRPTW. One major drawback o the existing algorithms is the incapability to deal with the dynamic nature o travel times. Thereore, existing algorithms or the VRPTW can not be applied to the TDVRPTW without a radical modiication o their structure. In this paper, a branch-and-cut-and-price ramework is modiied such that time-dependent travel times can be incorporated. 3. Problem Description We consider a graph G(V, A) on which the problem is deined. V = {0, 1,..., n, n+1} is the set o all nodes such that V c = V/{0, n+1} represents the set o customers that need to be served. Moreover, 0 is the start deport and n + 1 is the end depot. A = {(i, j) : i j and i, j V } is the set o all arcs between the nodes. Let K be the set o homogeneous vehicles such that each vehicle has a inite capacity Q and q i demand o customer i V c. We assume q 0 = q n+1 = 0 and K is unbounded. Let a i and b i be respectively the opening and closing time o node s i time window. At node i, a service time s i is needed. We denote t i departure time rom node i V and τ ij (t i ) travel time rom node i to node j which depend on the departure time at node i. Table 1 summarizes the notation used in this paper. Table 1 Variable V V c K Q t i t l i(l) q i s i x ijk γ + (S) γ (S) τ ij(t i) δ v(l) (t j ) Ω s p e p c p a ip π i Notation used in this paper. Description : Set o nodes : Set o customers : Set o vehicles : Capacity o a vehicle : Departure time at node i : Latest possible departure time at a node i visited on the partial path represented by L : Demand at nodei : Service time at node i : Binary variable. Is one i and only i arc (i, j) is traversed by vehicle k : Arcs originating rom the set S V. We write γ + (i) instead o γ + ({i}) : Arcs ending in the set S V. We write γ (i) instead o γ ({i}) : Travel time rom node i to node j when departure time at i is t i : Piecewise linear unction measuring the arrival at the current node v(l) o the partial path represented by L when departure at the start node j is t j : Set o all easible routes : Start time o route p Ω : End time o route p Ω : cost o route p Ω. It is deined as as e p s p : Is one i node i is visited by path p and zero otherwise : Dual variable associated with row i o the master problem : Reduced cost o route p Ω c p [a i, b i] : Time window at node i X : Size o the set X

6 Author: Branch and Cut and Price or the TDVRPTW Transportation Science 00(0), pp , c 0000 INFORMS Travel Time and Arrival Time Functions We divide the planning horizon into time zones where a dierent speed is associated with each o these zones. The resulting stepwise speed unction is translated into travel time unctions that satisy the First-In First-Out (FIFO) principle. Usually traic networks have a morning and an aternoon congestion period. Thereore, we consider speed proiles that have two periods with relatively low speeds. In the rest o the planning horizon, speeds are relatively high. This complies with data collected or a Belgian highway (Van Woensel and Vandaele (2006)). Figure 1 depicts the speed proile or each start time or an arbitrary link. Moreover, it shows how the speed proile is translated into a travel time unction. We call the points a, b, c, d and e where speeds change speed breakpoints. Speed breakpoints are also breakpoints in the travel time unction. The other travel time breakpoints are determined as the start time to arrive exactly at a speed breakpoint (e.g, a is the start time to exactly arrive at time a) using the procedure as described in Ichoua et al. (2003). While the slopes in the travel time unction mean that the traveled distance is traversed using Figure 1 Speed and travel time unctions. several speeds, the horizontal segments mean that it is traversed using only one speed. Clearly, or large distances we might have travel time unctions without any horizontal segments. Travel time unctions are stepwise linear unctions in which every two consecutive travel time breakpoints deine a zone. Given any start time within a zone, travel time can easily be computed using the breakpoints deining that zone. Thereore, travel time unctions can be completely represented by their breakpoints. Given a partial path P i starting at the depot 0 and ending at some node i, the arrival time at i depends on the dispatch time t 0 at the depot. Due to the FIFO property o the travel time unctions, a later dispatch at the depot will result in a later arrival at node i. Thereore, i route P i is uneasible or some dispatch time t 0 at the depot (i.e. time windows are violated), P i will be uneasible or any dispatch time at the depot that is later than t 0. Moreover, I we deine δ i (t 0 ) as the arrival time unction at node i given a dispatch time t 0 at the depot, δ i (t 0 ) will be nondecreasing in t 0. We call the parent node j o node i, the node that is visited directly beore node i on route P i. δ j (t 0 ) is the arrival time at j given a dispatch time t 0 at the depot, and τ ji (δ j (t 0 )) is the incurred travel time rom j to i. Consequently, or every i V, δ i (t 0 ) is recursively calculated as ollows: δ 0 (t 0 ) = t 0 and δ i (t 0 ) = δ j (t 0 ) + τ ji (δ j (t 0 )) (1) Where δ 0 (t 0 ) is a sort o dummy unction representing the arrival time at the depot given a dispatch time t 0 at the same depot. Formula (1) shows that an arrival time unction is the sum o two linear stepwise unctions (travel time unction and arrival time unction o the parent node), hence it is also a linear stepwise unction. Figure 2 depicts the recursive calculation o the arrival time unctions using equation (1). Again, we can completely represent an arrival time unction using the arrival time unction breakpoints resulting rom either breakpoints o travel time unctions,

7 Author: Branch and Cut and Price or the TDVRPTW 6 Transportation Science 00(0), pp , c 0000 INFORMS breakpoints o the arrival time unction o the parent node, or rom time windows. The cost o a path is equal to its duration δ i (t 0 ) t 0. Clearly, the departure time t 0 rom the depot that results in the shortest path duration belongs to a breakpoint. That is: t 0 = min t 0 B i {δ i (t 0 ) t 0 } (2) Figure 2 Arrival time unctions. B i is the set o breakpoints o the arrival time unction δ i (t 0 ) The Mathematical Formulation I ω ik is the departure time o vehicle k at customer i and x ijk is a binary variable that takes the value 1 i and only i arc (i, j) is traversed by vehicle k, the objective unction or the TDVRPTW is as ollows: τ ij (ω ik )x ijk (3) k K (i,j) A For every arc (i, j), we denote Z ij as the set o zones o the corresponding travel unction τ ij (t i ). A zone Z m Z ij, is deined by two consecutive travel time breakpoints, Z m = [r m, r m+1 [. A slope θ m and an intersection η m with the y-axis can be calculated using r m, r m+1, τ ij (r m ) and τ ij (r m+1 ). Thereore, or some Z m Z ij, the travel time τ ij (ω ik ) rom i to j given departure time ω ik at i is: The objective unction can be re-written as ollows: τ ij (ω ik ) = θ m ω ik + η m (4) Z ij k K (i,j) A m=1 (θ m ω ik + η m )x m ijk (5) Where, x m ijk is a binary variable that takes the value 1 i and only arc (i, j) is traversed by vehicle k and departure time rom customer i is within zone Z m. Obviously, the non-linear term ω ik x m ijk will appear in the objective unction. However, i we deine the variable: ω m ik = { ωik i x m ijk = 1 0 otherwise (6) ω ik x m ijk can be replaced by ω m ik. Furthermore, we denote Z + ij and Z ij respectively as the set o zones with positive slope and the set o zones with absolutely negative slope. The MIP ormulation or TDVRPTW can be written as ollows: min z = k K Z ij (i,j) A m=1 (θ m ω m ik + η m x m ijk) (7)

8 Author: Branch and Cut and Price or the TDVRPTW Transportation Science 00(0), pp , c 0000 INFORMS 7 subject to: x k (γ + (i)) = 1 i V/{n + 1} (8) k K x k (γ + (0)) = 1 k K (9) x k (γ + (j)) = x k (γ (j)) k K, j V/{n + 1} (10) x k (γ (n + 1)) = 1 k K (11) (1 + θ m )ω m ik s i + η m ω m jk s j + (1 x m ijk)m k K, (i, j) A, m Z ij (12) ω m ik ω ik (1 x m ijk)m k K, (i, j) A, m Z ij + (13) ω m ik min(ω ik, Mx m ijk) k K, (i, j) A, m Z ij (14) a i + s i ω m ik b i + s i k K, i V (15) q i x k (γ + (i)) Q k K (16) i N x m ijk {0, 1} k K, (i, j) A, m Z ij (17) r m ω m ik < r m+1 k K, i V, m Z ij (18) When departure time is within a zone with positive slope, w m ik will appear with a positive sign in the objective unction (7), and the optimization will attempt to set it as low as possible to reduce travel time. This is taken care o by means o constraint (13). However, when departure time is within a zone with a negative slope, w m ik will appear with negative sign in the objective unction, and the optimization will attempt to set it as large as possible through constraint (14). Obviously, the number o decision variable has increased. However, we don t have to decide on all o them. In act, due to the FIFO assumption, waiting at customers will not result in better solutions. Thereore, we only have to decide on departure time at the depot. Departure times at customers take place immediately ater inishing service which is computable given the sequence o visited customers. 4. Column Generation To derive the set partitioning ormulation or the TDVRPTW, we deine Ω as the set o easible paths satisying constraints (9)-(18) (the index k is dropped since we are considering a homogeneous leet). A easible path is deined by the sequence o customers visited along it and the dispatch time at the depot. To each path p Ω, we associate the cost c p which is simply its duration. Hence: c p = e p s p (19) Where e p and s p are respectively the end time and the start time o path p. Furthermore, i y p is a binary variable that takes the value 1 i and only i the path p is included in the solution, the TDVRPTW is ormulated as the ollowing set partitioning problem: min z M = p Ω c p y p (20) subject to: a ip y p = 1 i V (21) p Ω y p {0, 1} p Ω. (22) The objective unction (20) minimize the duration o the chosen routes. Constraint (21) guarantees that each node is visited only once. Solving the LP-relaxation, resulting rom relaxing the integrality constraints o the variables y p, o the master problem provides a lower bound on its optimal

9 Author: Branch and Cut and Price or the TDVRPTW 8 Transportation Science 00(0), pp , c 0000 INFORMS value. The set o easible paths Ω is usually very large making it hard to solve the LP-relaxation o the master problem. Thereore, we have recourse to column generation. In column generation, a restricted master problem is solved by considering only a subset Ω Ω o easible paths. Additional paths with negative reduced cost are generated ater solving a pricing subproblem. The pricing problem or the TDVRPTW is (the index k is dropped): min z P = τ ij (ω i )x ij (23) (i,j) A subject to constraints (9)-(18). Furthermore, τ ij (ω i ) = τ ij (ω i ) π i is the arc reduced cost, where π i is the dual variable associated with servicing node i. In the master problem, π i results rom the constraint corresponding to node i in the set o constraints (21). The objective unction o the pricing problem can be expressed as: z P = e p s p i V c a ip π i (24) or in the variables x ij as: z P = e p s p i V c π i j γ + (i) x ij (25) The problem with the objective unction (24) and constraints (9)-(18) is called the time-dependent elementary shortest path problem with resource constraints (TDESPPRC). In this paper the only resources we consider are time windows. Capacity is relaxed in the pricing problem and handled using valid inequalities. Thereore, a easible solution to the pricing problem must only respect time windows.in the next section the pricing problem is addressed in more details and it is shown how it is solved by means o a time-dependent labeling algorithm Capacity Cuts 5. The Pricing Problem Solving the pricing problem involves inding columns (i.e. routes) with negative reduced cost that improve the objective unction o master problem. In case o the TDVRPTW, this corresponds to solving the TDESPPRC and inding paths with negative cost. The TDESPPRC is a generalization o the ESPPRC in which costs are time-dependent. In this paper, we solve the pricing problem by means o a time-dependent labeling (TDL) algorithm which is a modiication o the labeling algorithm applied to the ESPPRC. To speed up the TDL algorithm, a bi-directional search is perormed in which labels are extended both orward rom the depot (i.e. node 0) to its successors, and backward rom the depot (i.e. node n+1) to its predecessors. While orward labels are extended to some ixed time t m (e.g. the middle o the planning horizon) but not urther, backward labels are extended to, but are allowed to directly cross, t m. Forward and backward labels are inally merged to construct complete tours. The running time o a labeling algorithm depends on the length o partial paths associated with its labels. A bi-directional search avoids generating long paths and thereore limits running times The Forward TDL Algorithm In the orward TDL algorithm, labels are extended rom the depot (i.e. node 0) to its successors. The extension to a node is allowed i it is easible and i the earliest arrival time (including waiting and service time) at that node is no urther than t m. We associate the ollowing components to a Label L : The set o easible extensions E(L ) o L is the set o partial paths that when departing at node v(l ) at time δ v(l )(0), they reach the depot (i.e. node n+1) without violating time windows.

10 Author: Branch and Cut and Price or the TDVRPTW Transportation Science 00(0), pp , c 0000 INFORMS 9 v(l ) the current node visited on the partial path represented by L c(l ) the sum o the dual variables associated with nodes visited along the partial path represented by L δ v(l )(t 0 ) arrival time at v(l ) through the partial path represented by L when the departure time at the depot is t 0. It includes both waiting time and service time at v(l ) S(L ) set o nodes visited along the partial path represented by L I L E(L ), we denote L L as the label resulting rom extending L by L. I label L is the parent label o label L, the arrival time unction associated with label L is extended as ollows: Furthermore, we have: δ v(l )(t 0 ) = δ v(l )(t 0 ) + τ v(l )v(l )(δ v(l )(t 0 )) (26) S(L ) = S(L ) {v(l )} and c(l ) = c(l ) π v(l ) (27) Where π v(l ) is the dual variable corresponding to visiting node v(l ). Given the FIFO assumption, the earliest arrival time at v(l ) corresponds to the earliest possible dispatch time at the depot, t 0 = 0: δ v(l )(0) = δ v(l )(0) + τ v(l )v(l )(δ v(l )(0)) (28) The extension o label L to label L is easible i: δ v(l )(0) min(t m, b v(l ) + s v(l )) (29) In case o the ESPPRC, only the arrival time corresponding to a departure time t 0 = 0 rom the depot is stored. Obviously, in case o the TDESPPRC, computing and storing arrival time unctions is more complicated. The TDL algorithm is a complete enumeration in which, or every label, all possible extensions are derived and stored. It ends when all labels are processed. However, the number o labels that can be processed might be very large. Consequently, the labeling algorithm might be computationally very expensive. To reduce the number o labels, dominance criteria are introduced. In case o the orward TDL algorithm, dominance is deined as ollows: Deinition 1. Label L 2 is dominated by label L 1 i: 1. E(L 2 ) E(L 1 ) 2. c(l 1 L) c(l 2 L), L E(L 2 ) Deinition 1 states that any easible extension o label L 2 is also easible or label L 1. Furthermore, extending L 1 should always result in a better route. However, it is not straightorward to veriy the conditions o Deinition 1 as it requires the computation and the evaluation o all easible extensions o both labels L 1 and L 2. Thereore, suicient dominance criteria that that are computationally less expensive are desirable. In Proposition 1, the suicient conditions (3.), (4.) and (5.) are introduced. Condition (3.) is needed because o the elementarity o paths. Condition (4.), in addition to the FIFO assumption, guarantees that time windows o nodes visited along any easible extension o L 2 are respected when reached through L 1. Conditions (5.) ensures that no cheaper route can be obtained by extending L 2 regardless o departure time at the depot. I we denote t l 0(L ) as the latest easible start time at the depot o the partial path represented by label L, Proposition 1 is ormally stated as ollows:

11 Author: Branch and Cut and Price or the TDVRPTW 10 Transportation Science 00(0), pp , c 0000 INFORMS Proposition 1. Label L 2 is dominated by label L 1 i: 1. v(l 1 ) = v(l 2 ) 2. c(l 1 ) c(l 2 ) 3. S(L 1 ) S(L 2 ) 4. δ v(l 1 )(t 0 ) δ v(l 2 )(t 0 ), t 0 [0, t l 0(L 2 )] 5. t l 0(L 2 ) t l 0(L 1 ) Proo o Proposition1: First we prove that E(L 2 ) E(L 1 ). Let L E(L 2 ), then S(L) S(L 2 ) =. As S(L 1 ) S(L 2 ), we should also have S(L) S(L 1 ) =. Now we will show that customers time windows along the partial path represented by L are respected when reached trough L 1. Let i be a node visited on the parrtial path represented by L, and L i L be the partial path with i as the current node and the same start node as L. Furthermore, let t 0 t l 0(L 2 ) be some start time at the depot. Now we will show that c(l 1 L) c(l 2 L) δ v(l 1 L i )(t 0 ) = δ v(l 1 )(t 0 ) + δ v(li )(δ v(l 1 )(t 0 )) δ v(l 2 )(t 0 ) + δ v(li )(δ v(l 2 )(t 0 )) = δ v(l 2 L i )(t 0 ) b i δ v(l 1 L)(t 0 ) = δ v(l 1 )(t 0 ) + δ v(l) (δ v(l 1 )(t 0 )) δ v(l 2 )(t 0 ) + δ v(l) (δ v(l 2 )(t 0 )) = δ v(l 2 L)(t 0 ) Furthermore, we know that: c(l 1 ) c(l 2 ). Hence, We conclude that or all t 0 t l 0(L 2 ): Hence, and since t l 0(L 2 ) t l 0(L 1 ), c(l 1 L) = c(l 1 ) + c(l) c(l 2 ) + c(l) = c(l 2 L) δ v(l 1 L)(t 0 ) t 0 + c(l 1 L) δ v(l 2 L)(t 0 ) t 0 + c(l 2 L) { } { } min δ v(l t 0 t l 1 0 (L1 ) L)(t 0 ) t 0 + c(l 1 L) min δ v(l t 0 t l 2 0 (L2 ) L)(t 0 ) t 0 + c(l 2 L) Dominance as introduced in Proposition 1 is weak and will probably not suiciently reduce the number o labels processed by the TDL algorithm. In act, S(L 1 ) S(L 2 ) implies c(l 1 ) c(l 2 ) which contradicts the second condition. Hence, conditions (2.) and (3.) are only both true in case o equality. Furthermore, very cheap labels representing partial paths with a very long duration, that does not lead to a route in the optimal solution will probably not be dominated. In Figure 3, the numbers associated with the arcs represent travel times and the numbers associated with the nodes represents dual variables. Because o Condition (2.), the label representing partial path P 2 will not be dominated by the one representing partial path P 1. However, a path s reduced cost is equal to its duration reduced by the sum o the dual variables corresponding to the nodes visited

12 Author: Branch and Cut and Price or the TDVRPTW Transportation Science 00(0), pp , c 0000 INFORMS 11 along that path. Thereore, extending P 1 clearly results in a better inal route. Another pitall o Proposition 1 is that cheap labels are not able to dominate more expensive labels with, or some departure time at the depot, a shorter duration. In Figure 4, because o Condition (4.), the label representing partial path P 2, with cost -100, will not be dominated by the one representing partial path P 1 with cost The range o dispatch times at the depot, in which partial path P 2 has a shorter duration, has a width o 500 time units. Clearly, or any starting time at the depot in this range, it is possible to ind an earlier (but no more than 500 time units earlier) starting time at the depot that results in the same arrival time at the end node or both P 1 and P 2. Leaving the depot earlier might increase P 1 s duration. However, given P 1 s new start time, its duration will be no more than 500 time units longer than P 2 s duration. Thereore, the extension o P 1 will result in a better inal route. Figure 3 Figure 4

13 Author: Branch and Cut and Price or the TDVRPTW 12 Transportation Science 00(0), pp , c 0000 INFORMS In Proposition 2, we improve dominance in two directions. First, or every label L, we extend S(L ) to the set S(L ) by adding nodes that are unreachable rom v(l ). The triangle inequality is not satisied or time varying travel times as traveling directly to a node is not necessarily the shortest path. Consequently, a node that can not be directly reached rom the end node might be indirectly reached via a diverted route. However, i we calculate the earliest arrival time to all nodes as in ormula (28) and take the minimum, all nodes with a close time smaller than that minimum will not be reachable rom v(l ). This can be done quickly, although we might ail to ind all unreachable nodes. Second, we relax Condition (2.) by adding the quantity φ to the cost c(l 2 ) o label L 2. φ is a real number related to how much the start time o the partial path represented by label L 1, can be postponed (in case φ is positive) or expedited (in case φ is negative) and still arrive at the end node at the same time as when reaching the end node through the partial path represented by label L 2. φ is illustrated in Figure 4. For every label L, let δ 1 (t v(l ) a) = max{t t l 0(L ) : δ v(l )(t) = t a }. The unction δ 1 (t v(l) a) is deined on the domain = {t a R : t t l 0(L ) : δ v(l) (t) = t a }. Proposition 2 is stated as ollows: A δ 1 v(l ) Proposition 2. Label L 2 is dominated by label L 1 i: 1. v(l 1 ) = v(l 2 ) 2. c(l 1 ) c(l 2 ) + φ 3. S(L 1 ) S(L 2 ) 4. δ v(l 1 )(0) δ v(l 2 )(0) φ = min { t l 0(L 1 ) t l 0(L 2 ), min t A { }} δ 1 v(l 1 )(t) δ 1 v(l 2 )(t) and A = A δ 1 v(l 1 ) Aδ 1 v(l 2 ) Proo o Proposition 2: We will prove Proposition 2 or the case φ 0. Similarly to Proposition 1, and by using the act that δ v(l 1 )(0) δ v(l 2 )(0) and S(L 1 ) S(L 2 ), we can prove that any easible extension to L 2 is also easible or L 1. Let L E(L 2 ), and t 0 t l 0(L 2 ) be some start time at the depot. Now, let t be such that: { δ 1 t v(l = 1 )(t 0) t 0 i δ v(l 2 )(t 0 ) A δ 1 v(l 1 ) t l 0(L 1 ) t l 0(L 2 ) otherwise t is illustrated in Figure 5, and can also be written as: { δ 1 t v(l = 1 )(t 0) δ 1 v(l 1 )(δ v(l 1 ) (t 0 )) i δ v(l 2 )(t 0 ) A δ 1 v(l 1 ) t l 0(L 1 ) t l 0(L 2 ) otherwise Postponing the start time o L 1 at the depot by t (i.e. the start time at the depot is t 0 + t instead o t 0 ) results in a arrival time at the current node that is smaller than arrival time at the same current node reached through L 2, and when the start time at the depot is t 0. Furthermore, t 0 + t t l 0(L 1 ). Thereore: δ v(l 2 )(t 0 ) δ v(l 1 )(t 0 + t ) Consequently: δ v(l 2 L)(t 0 ) = δ v(l 2 )(t 0 ) + δ v(l) (δ v(l 2 )(t 0 )) δ v(l 1 )(t 0 + t ) + δ v(l) (δ v(l 1 )(t 0 + t )) = δ v(l 1 L)(t 0 + t )

14 Author: Branch and Cut and Price or the TDVRPTW Transportation Science 00(0), pp , c 0000 INFORMS 13 Figure 5 Now we will show that c(l 1 L) c(l 2 L) Obviously φ t, Hence, Furthermore, we know that: φ c(l 1 ) c(l 2 ). Hence, δ v(l 1 L)(t 0 + t ) (t 0 + t ) δ v(l 2 L)(t 0 ) t 0 t δ v(l 2 L)(t 0 ) t 0 φ δ v(l 1 L)(t 0 + t ) (t 0 + t ) + c(l 1 L) δ v(l 2 L)(t 0 ) t 0 + c(l 2 L) We conclude that or all t 0 t l 0(L 2 ), there exists t 0 = t 0 + t t l 0(L 1 ) such that: δ v(l 1 L)( t 0 ) ( t 0 ) + c(l 1 L) δ v(l 2 L)(t 0 ) t 0 + c(l 2 L) Hence, { } { } min δ v(l t 0 t l 1 0 (L1 ) L)(t 0 ) t 0 + c(l 1 L) min δ v(l t 0 t l 2 0 (L2 ) L)(t 0 ) t 0 + c(l 2 L) 5.2. The Backward TDL Algorithm In the backward TDL algorithm, labels are extended rom the depot (i.e. node n + 1) to its predecessors. The extension o a label is allowed i it is easible and i the latest possible departure time at the end node is no urther than t m. To a Label L b, we associate the ollowing components: v(l b ) the irst node visited on the partial path represented by L b c(l b ) the sum o the dual variables associated with nodes visited along the partial path represented by L b δ n+1 (t v(lb )) arrival time at the depot through the partial path represented by L b and when leaving node v(l b ) at time t v(lb ) S(L b ) set o nodes visited along the partial path represented by L b The set o easible extensions E(L b ) o L b is the set o partial paths departing at the depot (i.e. node 0) at some time t 0 0 and reaching node v(l b ) at some time t v(lb ) > t 0 (t v(lb ) includes

15 Author: Branch and Cut and Price or the TDVRPTW 14 Transportation Science 00(0), pp , c 0000 INFORMS waiting and service at v(l b )) without violating time windows. Going back to the depot through the partial path represented by label L b should be easible given that the departure time at v(l b ) is t v(lb ). I label L b is the parent label o label L b, the arrival time unction corresponding to label L b is computed as ollows: Furthermore, we have: δ n+1 (t v(lb )) = δ n+1 (t v(l b ) = t v(lb ) + τ v(lb )v(l b ) (t v(lb ))) (30) S(L b ) = S(L b) {v(l b )} and c(l b ) = c(l b) π v(lb ) (31) The latest departure time t l v(l b ) at node v(l b), such that the arrival at node v(l b) is exactly its latest possible departure time, can be calculated using the procedure as described in Ichoua et al. (2003). The extension o L b with node v(l b ) is easible i: t l v(l b ) a v(lb ) + s v(lb ) and t l v(l b ) t m (32) Again, as illustrated in Figure 6, arrival time unctions are non-decreasing linear stepwise unctions. Moreover, arrival time unctions are completely deined by their breakpoints. Arrival time unction breakpoints result rom travel time unctions breakpoints, breakpoints calculated as departure time at the start node to hit a breakpoint on the arrival time unction o the destination node, or rom time windows. Furthermore, dominance can be deined in the same way as in the case o the orward TDL algorithm. To avoid redundancy, we only present the improved dominance criteria as it is slightly dierent. Figure 6 The arrival time unction. In Proposition 3, S(L b ) denotes the set o visited nodes along the partial path represented by label L b extended by nodes that are unreachable rom v(l b ). In act, the latest departure rom all nodes, such that arrival time at v(l b ) is its latest possible start time, is calculated using the procedure as described in Ichoua et al. (2003), and the maximum is taken. All nodes with an opening time (service time included) larger than that maximum will not be reachable rom v(l b ). Furthermore, we relax Condition (2.) by adding the quantity φ b to the cost c(l 2 b). φ b is a real number related to, given a departure time at node v(l 1 b), how early (in case φ b is negative) or late

16 Author: Branch and Cut and Price or the TDVRPTW Transportation Science 00(0), pp , c 0000 INFORMS 15 (in case φ b is positive) arrival at the depot takes place when traversing the partial path represented by label L 1 b instead o the partial path represented by label L 2 b. Note that φ b is conceptually dierent rom φ as it is related to arrival time at the end node (i.e. the depot) instead o departure time at the depot. In the orward search, we can not relate φ to the arrival time at the end node as this might be dierent rom the depot. Thereore, any gains in terms o arrival time does not guarantee a gain in the inal complete tour. In act, gains can easily be lost by possible waiting time due to time windows. I denote D δlb as the deinition domain o the arrival time unction δ v(l )(t 0 ), we state Proposition 3 as ollows: Proposition 3. Label L 2 b is dominated by label L 1 b i: 1. v(l 1 b) = v(l 2 b) 2. c(l 1 b) v(l 2 b) + φ b 3. S(L 1 b) S(L 2 b) 4. δ 1 φ b = min n+1(t l v(l 1 b )) δ 1 n+1(t l v(l 2)) b { δ n+1 (t l v(l 1)) δ n+1(t l v(l b 2 )), min b t D Proo: see appendix { δ n+1 (t v(l 1 b ) = t) δ n+1 (t v(l 2 b ) = t)} } and D = D δ L 1 b Dδ L 2 b 5.3. Merging Forward and backward Labels Ater all orward and backward labels are processed, they are joined to construct easible tours with negative reduced cost. A orward label L and a backward label L b are joined i v(l ) = v(l b ), S(L ) S(L b )/{i} =, and there exists at least one possible dispatch time t 0 at the depot or which δ n+1 (t v(lb ) = δ v(l )(t 0 )) is deined. The attributes o label L resulting rom merging a orward L and a backward label L b are calculated as ollows: v(l) = n + 1 c(l) = c(l ) + c(l b ) S(L) = S(L ) S(L b ) B L = B L BL 1 b B L is the set o breakpoints deining the arrival time unction δ v(l) (t 0 ) associated with label L. It is the union o the set B L corresponding the breakpoints o the arrival time unction δ v(lt )(t 0 ) associated with label L, and B L 1 = {δ 1 (t v(l b ) v(l b )) : t v(lb ) B Lb } where B Lb is the set o breakpoints deining the arrival time unction δ n+1 (t v(lb )) associated with label L b. Proposition 4. For every route R in the optimal solution, there exist a orward path P backward path P b such that the route R is obtained by merging P and P b. and 5.4. The Pricing Problem Heuristics Branch-and-price algorithms can be accelerated using heuristics to solve the pricing problem. In act, the heuristic will search or paths with negative reduced cost and add them to the master problem. When the heuristics ails to ind any more paths with negative reduced cost, the exact algorithm is called. Ideally, or every node in the branching tree, the exact algorithm is called only once to check that no more paths with negative reduced cost exist. In our BCP ramework, we use two heuristics. First, a greedy heuristic that extend each label to the node with the smallest travel

17 Author: Branch and Cut and Price or the TDVRPTW 16 Transportation Science 00(0), pp , c 0000 INFORMS time. Second, a truncated labeling heuristic in which only a limited number o labels is stored. Moreover, or the truncated heuristic, relaxed dominance criteria are used. In act, we relax the condition on the sets o visited customers. Furthermore, we dominate label L 2 by label L 1 i: { } { } min δv(l1 )(t 0 ) t 0 min δv(l2 )(t 0 ) t 0 t 0 B L1 t 0 B L2 (33) and { } { } min δv(l1 )(t 0 ) t 0 + c(l1 ) min δv(l2 )(t 0 ) t 0 + c(l2 ) (34) t 0 B L1 t 0 B L2 { } Where B Li is the set o breakpoints deining δ v(li )(t 0 ), i = 1, 2. min t0 B δv(li Li )(t 0 ) t 0 is the minimum duration o the partial path represented by label L i, i = 1, 2. The number o stored labels can be increased each time the heuristic ails to ind paths with negative reduced cost (e.g. we start with 250, then we increase the number o labels to 500 labels and inally to 1000 labels). 6. Computational Results The BCP algorithm is implemented on a (mention properties o the machine). The open source ramework COIN is used to solve the linear programming relaxation o the master problem. For our numerical study, we use the well known Solomon s data sets (Solomon (1987)) that ollow a naming convention o DT m.n. D is the geographic distribution o the customers which can be R (Random), C (Clustered) or RC (Randomly Clustered). T is the instance type which can be either 1 (instances with tight time windows) or 2 (instances with wide time windows). m denotes the number o the instance and n the number o customers that need to be served. Road congestion is taken into account by assuming that vehicles travel through the network using dierent speed proiles. We consider speed proiles with two congested periods. Speeds in the rest o the planning horizon (i.e. the depot s time window) are relatively high. We consider speed proiles that comply with data rom real lie. Furthermore, we assume three types o links: ast, normal and slow. Slow links might represent links within the city center, ast links might represent highways and normal links might represent the transition rom highways to city centers. Moreover, without loss o generality, we assume that breakpoints are the same or all speed proiles as congestion tends to happen around the same time regardless o the link s type (e.g. rush hours).the choice or a link type is done randomly and remains the same or all instances. The ollowing speed proiles are considered: Table 2 Speed Proiles. Zone1 Zone2 Zone3 Zone4 Zone5 Fast Normal Slow Speed breakpoints are such that: a = 0.2b n+1, b = 0.3b n+1, c = 0.7b n+1, d = 0.8b n+1 and e = b n+1. a, b, c, d and e are depicted in Figure 1, and b n+1 is the upper bound o the depot s time window. Travel time breakpoints are calculated using the procedure as described in Ichoua et al. (2003). Figures 7 and 8 illustrate respectively two travel time unctions or a link rom an R instance and a link rom an RC instance.

18 Author: Branch and Cut and Price or the TDVRPTW Transportation Science 00(0), pp , c 0000 INFORMS 17 Figure 7 Travel time unction or an R instance. Figure 8 Travel time unction or an RC instance TDESPPRC vs. TDSPPRC 6.2. Bi-directional TDL vs. Monodirectional TDL 7. Conclusions and Future Research Acknowledgments The research o Said Dabia has been unded by TRANSUMO, project number Appendix. In progress Reerences Androutsopoulos, K. N., K. G. Zograos Solving the multi-criteria time-dependent routing and scheduling in a multimodal ixed scheduled network. European Journal o Operational Research Baldacci, R., A. Mingozzi, R. Roberti New route relaxation and pricing strategies or the vehicle routing problem. Working paper, the university o Bologna. Bertsimas, D. J., G. Van Ryzin A stochastic and dynamic vehicle routing problem in the euclidian plane. Operations Research Bertsimas, D. J., G. Van Ryzin. 1993a. Stochastic and dynamic vehicle routing problems in the euclidean plane with multiple capcitated vehicles. Operations Research Bertsimas, D. J., D. Simchi-Levi A new generation o vehicle routing research: robust algorithms, addressing uncertainty. Operations Research 44(2) Bräysy, O., M. Gendreau. 2005a. Vehicle routing problem with time windows, part i: Route construction and local search algorithms. Transportation Science 39(1) Bräysy, O., M. Gendreau. 2005b. Vehicle routing problem with time windows, part ii: Metaheuristics. Transportation Science 39(1)

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