- Email: [email protected]
A Hybrid Branch-and-Cut Approach for the Capacitated Vehicle Routing Problem
21st European Symposium on Computer Aided Process Engineering – ESCAPE 21 E.N. Pistikopoulos, M.C. Georgiadis and A. Kokossis (Editors) © 2011 Elsevier B.V. All rights reserved.
A Hybrid Branch-and-Cut Approach for the Capacitated Vehicle Routing Problem Chrysanthos E. Gounarisa, Panagiotis P. Repoussisb, Christos D. Tarantilisb, and Christodoulos A. Floudasa a
Computer-Aided Systems Laboratory, Department of Chemical and Biological Engineering, Princeton University, NJ 08544, USA b Center for Operations Research & Decision Systems, Department of Management Science & Technology, Athens University of Economics & Business, Athens 11362, GR
Abstract This paper presents a hybrid optimization approach that combines deterministic and metaheuristic algorithms for the Capacitated Vehicle Routing Problem (CVRP). The approach combines a new branch-and-cut framework, that utilizes a two-commodity flow representation and novel heuristic-based procedures to separate various classes of cuts, with a subordinate Adaptive Memory Programming metaheuristic algorithm for the identification of high quality solutions. New local-scope cuts are suggested to exclude infeasible or suboptimal solutions, break problem symmetries, and tighten constraints. Computational experiments illustrate the potential of the new approach. Keywords: Vehicle Routing, Distribution Logistics, Branch-and-Cut
1. Introduction The Vehicle Routing Problem (VRP) deals with the optimal assignment and service sequence of a set of customers to a fleet of vehicles and is one of the most studied combinatorial optimization problems in the operations research literature (Laporte, 2009). However, unlike the Traveling Salesman Problem, where 1000-customer instances can be solved to optimality on a routinely basis, instances of VRP with more than one hundred customers can be hard to solve (Baldacci et al., 2010). In this paper, we address the Capacitated Vehicle Routing Problem (CVRP). Given a homogeneous fleet of capacitated vehicles, the objective is to design a set of least cost round-trip routes to serve a set of customers with known demand. Previous methods for solving the CVRP include branch-and-cut (Lysgaard et al., 2004), branch-and-cut-andprice (Fukasawa et al., 2006), and set partitioning approaches (Baldacci et al., 2008). Heuristic methods, such as iterative improvement, evolutionary algorithms, and hybrid metaheuristic schemes have also made significant contributions; however, most of them fail to provide a good compromise between solution quality and computational speed. Our goal is to develop a novel hybrid optimization method that combines –in a cooperative fashion– algorithms that provide theoretical guarantee of reaching optimal solutions with metaheuristic algorithms, which typically exhibit superior performance in regards to the speed of obtaining good quality solutions. In particular, we aim at exploiting synergies between an Adaptive Memory Programming (AMP) metaheuristic algorithm and a Branch-and-Cut (BC) solution framework. The former generates and continuously updates (via information from the relaxation solutions at each node of the BC tree) a reference set of high quality diversified solutions. This pool of elite solutions is then used for updating the incumbent and for guiding the BC tree search.
C.E. Gounaris et al.
508
2. Two-Commodity Network Flow Formulation Let V0 ^0,1, , N , N 1` be a node set and A ^(i, j ) : 0 d i j d N 1` be the resulting undirected arc set. The set V V0 \ ^0, N 1` represents the N customers, while nodes 0 and N 1 represent duplicate instances of the single depot (for departure and arrival of vehicles, respectively). A cost cij t 0 is associated with each arc (i, j ) A .
Furthermore, there exists a homogeneous fleet of K vehicles with maximum carrying capacity Q . Each customer i V requires qi units of product (0 qi d Q) . The solution of the CVRP calls for the determination of a set of vehicle routes with a minimum total cost, such that each customer is visited only once by exactly one vehicle, all available vehicles are used, each vehicle route starts and ends at the depot, and the cumulative customer demand satisfied by each route does not exceed the capacity of the vehicle. Baldacci et al. (2004) were the first to describe the CVRP with a twocommodity network flow formulation. We use their formulation in a slightly sparser form. For each of the undirected arcs (i, j ) A , a binary variable [ ij indicates if the arc is traversed or not (in either direction), while two flow variables, xij and x ji , represent the vehicle’s load and residual capacity (empty space). Eqs.(1-7) express the CVRP: min ¦¦ cij[ ij (1) [ ,x
s.t.
i
j !i
¦[
ji
j i
¦ [ij
s.t . xij x ji s.t .
2, i V
j !i
¦[
0j
j
¦[
i ( N 1)
i
Q[ij , (i, j ) A
¦x
Q qi , i V
¦x
¦q
ij
&
K
(2 & 3) (4) (5)
j
s.t .
j
0j
i
i
&
¦x
i ( N 1)
0
i
(6 & 7)
3. Strengthening Inequalities & Separation Algorithms 3.1. Commodity Flow Inequalities Constraints that strengthen the bounds of the flow variable bound strengthening constraints are appended to the formulation from the onset and are similar to the flow inequalities suggested by Baldacci et al. (2004). They can be expressed as follows: ª º xoj t «¦ qi ( K 1)Q »[oj , j V (8) ¬ i ¼ (9) xij t q j[ij & x ji t qi[ij , (i, j ) V 3.2. Local Scope Inequalities After examining the structure of the relaxation solution at each node, it is possible to infer that certain solution segments (i.e., collections of arcs P ) lead to infeasible, suboptimal, or otherwise undesirable solutions. Such segments can be disallowed through the addition locally at the node level of an appropriate cut of the type: (10) ¦[ij d| P | 1 i , j P
We focus on solution segments that correspond to fully formed paths, that is, collections of consecutively joint arcs P ^ i, j : [ij 1`, augmented by adjacent fractional arcs,
Hybrid Branch-and-Cut for the CVRP
509
0 [ij 1 . We search for undesirable paths through a structured approach that grows
such paths iteratively, and we disallow formation of augmented paths due to 5 reasons: (a) subtour elimination – cyclical routes that do not include the depot; (b) capacity restrictions – routes that exceed vehicle capacity; (c) path dominance – suboptimal routes for which there exists a lower cost ordering of the customers; (d) symmetry breaking – non-nominal routes; that is, routes that begin by visiting a customer that is lexicographically higher than the last customer to be visited by this route; and, (e) demand restrictions – routes that are about to terminate before they have satisfied a minimum amount of demand. A separate class of local scope cuts results by inferring that a coefficient of a variable in a constraint can be suitably increased or decreased so as to tighten this constraint. In particular, we attempt to lift Eqs.(9) by replacing the coefficient of the binary variable (right hand side) with the cumulative load of the fully formed path connected to node i through –and including– node j , under the condition that this path (denoted Pj ) remains intact, and vice-versa: xij t qPj ([ij | Pj |
¦[
n ,m Pj
nm
) & x ji t qPi ([ij | Pi |
¦[
n ,m Pi
nm
)
(11 & 12)
3.3. Global Scope Inequalities There also exist 5 classes of cutting planes that are globally valid for the CVRP. These are the so-called Rounded Capacity (RC), Homogeneous Multistar (HM), Framed Capacity (FC), Strengthened Comb (SC) and Hypotour (HI) inequalities (see Naddef and Rinaldi, 2002, for detailed explanation). Due to their vast number, only those that are violated at a given node relaxation solution are taken into consideration. To this end, we have developed metaheuristic-based algorithms for their efficient separation (i.e., identifying which instances are in fact violated). The emphasis is given on RC and HM inequalities, whose separation is done concurrently through a new Tabu Search (TS) algorithm that improves upon the search framework presented in Augerat et al. (1999). For FC inequalities, a novel multi-restart TS algorithm, combined with a partition generation mechanism and edge-exchange neighborhood search, is used. The SC separation procedure proposed by Lysgaard et al. (2004) is used, enhanced with a TS procedure for expanding the teeth. Finally, the HI separation procedure proposed by Lysgaard et al. (2004) is adopted. The details of the above separation algorithms are omitted for conciseness of this paper.
4. Hybrid Branch-and-Cut Approach 4.1. Adaptive Memory Programming (AMP) Metaheuristic Initially, a reference set (pool) of high quality solutions is generated via an AMP algorithm. This is achieved via the repeated construction of provisional solutions out of promising building blocks identified during the search, while updating these adaptive memory components based on the progress and experience gained (Tarantilis, 2005; Repoussis et al., 2009). For this purpose, a knowledge extraction mechanism is utilized, coupled with a probabilistic construction heuristic and a TS algorithm. Overall, the proposed learning mechanism considers two properties: solution quality and appearance frequency for each pair of customers visited consecutively during a route. During branch-and-cut, at the end of each node’s processing (right before branching), the algorithm utilizes information from the LP relaxation in an effort to
510
C.E. Gounaris et al.
provide a new incumbent. Initially, a Path-Relinking algorithm generates an integer feasible solution [ int that is as “close” as possible to the node’s fractional solution [ f , in terms of Hamming distance d ¦int(1 [ijf ) ¦int[ijf . Next, a new provisional H ( i , j ):^[ij 1` ( i , j ):^[ij 0` solution is generated via reconstructing part of [ int using frequently observed components from the AMP pool. This solution is further improved via TS and, if particular criteria are met, the reference set and memory structures are updated. 4.2. Branch & Cut Framework Given that high quality initial upper bounds are provided through our metaheuristic framework, the priority of the branch-and-cut implementation is on improving the lower bound and on minimizing the number of subproblems (nodes) to be considered until the gap is closed. To this end, we adopt a best-bound-first node selection strategy. After obtaining the standard LP relaxation at each node, the cutting plane phase proceeds as follows: we first search for any augmented paths that need to be disallowed due to suboptimality, infeasibility, or non-nominality of the solution. Next, we check for potential to lift any flow variables and, lastly, we check for global cut violations (with emphasis on RC/HM). If at least one cut is identified at any of these three stages, we reoptimize the LP and repeat the process without continuing with the next stage(s). If no cuts are identified whatsoever, we proceed with branching the node. Let set S V and let G (S ) be the sum of [ ij of all arcs in the corresponding cut-set. As branching rule, we use the disjunction ^G (S ) 2` ^G (S ) t 4` . Among candidate sets for which G ( S ) | 3 , we select the one with the largest total demand.
5. Computational Studies We applied our framework to the standard benchmark data sets that were also used by Lysgaard et al. (2004), where a BC framework based on a different representation of the CVRP –called the vehicle flow formulation– is presented. Table 1 exhibits the root node gap for the 10 hardest problems we attempted, including two 100-customer instances. On average, our commodity flow-based method performs equally well with the vehicle flow-based method. The average root note gap can be improved if we enable CPLEX options for adding generic MIP cuts (e.g., Gomory cuts), however we have observed that doing so deteriorates the overall performance of the algorithm at later nodes. Therefore, for runs to full optimality, we disable cuts identified by CPLEX. Table 2 presents the time necessary to fully close the gap and the number of tree nodes that had to be explored for a set of medium-difficulty problems. The largest instance solved to guaranteed optimality was P-76-6, which involves 75 customers. Two instances were solved at root node by both methods. Our framework required fewer BC nodes for 9 of the remaining 13 instances. The improvement was very substantial for problems A-45-7 and B-50-8, two tight instances that are known to be hard to solve.
6. Conclusions This paper presents a hybrid BC framework for the exact solution of the CVRP that is based on the two-commodity flow formulation, systematic use of local scope cuts, new metaheuristic-based separation techniques for known classes of cuts, as well as an AMP metaheuristic algorithm for identification of high quality integer feasible solutions and acceleration of the search. Computational experiments on benchmark data sets illustrate the potential of the proposed approach.
Hybrid Branch-and-Cut for the CVRP
511
Table 1. Root node performance (% gap) Benchmark LP This This Problems relaxation paper paper+ A-69-9 9.04 3.87 3.30 A-80-10 8.42 3.02 2.39 B-68-9 10.64 1.14 1.14 E-76-8 6.06 2.34 2.17 E-76-10 7.02 3.62 3.33 E-101-8 5.65 1.57 1.55 E-101-14 6.82 3.75 3.20 P-60-15 6.56 3.91 3.48 P-65-10 6.37 3.12 2.48 P-76-5 4.03 1.56 1.55 Average 7.06 2.79 2.46 + CPLEX v11.0 generic MIP cuts enabled
Lysgaard et al., 2004 3.85 3.03 1.10 2.33 3.63 1.52 3.75 3.95 3.14 1.51 2.78
Table 2. Runs to full optimality Benchmark This paper Problems t (sec) # nodes A-44-6 86 125 A-45-7 2,835 2,084 A-48-7 5,147 203 A-55-9 145 156 B-43-6 39 100 B-44-7 3 1 B-45-6 152 276 B-50-7 2 1 B-50-8 4,523 1,503 B-52-7 3 3 B-57-7 99 49 B-64-9 21 7 E-51-5 13 8 P-50-7 78 131 P-76-4 143 105 Note: Optimum solution not provided as input
Lysgaard et al., 2004 t (sec) # nodes 620 211 19,414 4,170 372 113 468 152 125 63 8 1 299 159 11 1 31,026 5,694 25 15 441 168 42 13 59 17 805 263 535 141
References P. Augerat, J.M. Belenguer, E. Benavent, A. Corberán, D. Naddef, 1999, Separating capacity inequalities in the CVRP using tabu search. European Journal of Operational Research, 106, 546–557. R. Baldacci, E. Hadjiconstantinou, A. Mingozzi, 2004, An exact algorithm for the capacitated vehicle routing problem based on a two-commodity network flow formulation, Operations Research, 52, 5, 723-738. R. Baldacci, N. Christofides, A. Mingozzi., 2008, An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts, Mathematical Programming Ser.A, 115,:351-385. R. Baldacci, P. Toth, D. Vigo, 2010, Exact algorithms for routing problems under vehicle capacity constraints, Annals of Operations Research, 175, 1, 213-245. R. Fukasawa, H. Longo, J. Lysgaard, M.P. de Aragao, M. Reis, E. Uchoa, R.F. Werneck, 2006, Robust branch-and-cut-and-price for the capacitated vehicle routing problem, Mathematical Programming Ser.A, 106, 491-511. G. Laporte, 2009, Fifty years of vehicle routing, Transportation Science, 43, 4, 408-416. J. Lysgaard, A.N. Letchford, R.W. Eglese, 2004, A new branch-and-cut algorithm for the capacitated vehicle routing problem, Mathematical Programming Ser.A, 100, 423-445. D. Naddef, G. Rinaldi, 2002, Branch-and-cut algorithms for the capacitated VRP. In: P. Toth and P. Vigo (Eds.), The Vehicle Routing Problem, SIAM Monographs on Discrete Mathematics and Applications, SIAM, Philadelphia, 53-81. P.P. Repoussis, C.D. Tarantilis, G. Ioannou, 2009, Arc-guided evolutionary algorithm for the vehicle routing problem with time windows, IEEE Transactions on Evolutionary Computation, 13, 3, 624-647. C.D. Tarantilis, 2005, Solving the vehicle routing problem with adaptive memory programming methodology, Computers and Operations Research, 32, 9, 2309-2327.
A Hybrid Branch-and-Cut Approach for the Capacitated Vehicle Routing Problem Chrysanthos E. Gounarisa, Panagiotis P. Repoussisb, Christos D. Tarantilisb, and Christodoulos A. Floudasa a
Computer-Aided Systems Laboratory, Department of Chemical and Biological Engineering, Princeton University, NJ 08544, USA b Center for Operations Research & Decision Systems, Department of Management Science & Technology, Athens University of Economics & Business, Athens 11362, GR
Abstract This paper presents a hybrid optimization approach that combines deterministic and metaheuristic algorithms for the Capacitated Vehicle Routing Problem (CVRP). The approach combines a new branch-and-cut framework, that utilizes a two-commodity flow representation and novel heuristic-based procedures to separate various classes of cuts, with a subordinate Adaptive Memory Programming metaheuristic algorithm for the identification of high quality solutions. New local-scope cuts are suggested to exclude infeasible or suboptimal solutions, break problem symmetries, and tighten constraints. Computational experiments illustrate the potential of the new approach. Keywords: Vehicle Routing, Distribution Logistics, Branch-and-Cut
1. Introduction The Vehicle Routing Problem (VRP) deals with the optimal assignment and service sequence of a set of customers to a fleet of vehicles and is one of the most studied combinatorial optimization problems in the operations research literature (Laporte, 2009). However, unlike the Traveling Salesman Problem, where 1000-customer instances can be solved to optimality on a routinely basis, instances of VRP with more than one hundred customers can be hard to solve (Baldacci et al., 2010). In this paper, we address the Capacitated Vehicle Routing Problem (CVRP). Given a homogeneous fleet of capacitated vehicles, the objective is to design a set of least cost round-trip routes to serve a set of customers with known demand. Previous methods for solving the CVRP include branch-and-cut (Lysgaard et al., 2004), branch-and-cut-andprice (Fukasawa et al., 2006), and set partitioning approaches (Baldacci et al., 2008). Heuristic methods, such as iterative improvement, evolutionary algorithms, and hybrid metaheuristic schemes have also made significant contributions; however, most of them fail to provide a good compromise between solution quality and computational speed. Our goal is to develop a novel hybrid optimization method that combines –in a cooperative fashion– algorithms that provide theoretical guarantee of reaching optimal solutions with metaheuristic algorithms, which typically exhibit superior performance in regards to the speed of obtaining good quality solutions. In particular, we aim at exploiting synergies between an Adaptive Memory Programming (AMP) metaheuristic algorithm and a Branch-and-Cut (BC) solution framework. The former generates and continuously updates (via information from the relaxation solutions at each node of the BC tree) a reference set of high quality diversified solutions. This pool of elite solutions is then used for updating the incumbent and for guiding the BC tree search.
C.E. Gounaris et al.
508
2. Two-Commodity Network Flow Formulation Let V0 ^0,1, , N , N 1` be a node set and A ^(i, j ) : 0 d i j d N 1` be the resulting undirected arc set. The set V V0 \ ^0, N 1` represents the N customers, while nodes 0 and N 1 represent duplicate instances of the single depot (for departure and arrival of vehicles, respectively). A cost cij t 0 is associated with each arc (i, j ) A .
Furthermore, there exists a homogeneous fleet of K vehicles with maximum carrying capacity Q . Each customer i V requires qi units of product (0 qi d Q) . The solution of the CVRP calls for the determination of a set of vehicle routes with a minimum total cost, such that each customer is visited only once by exactly one vehicle, all available vehicles are used, each vehicle route starts and ends at the depot, and the cumulative customer demand satisfied by each route does not exceed the capacity of the vehicle. Baldacci et al. (2004) were the first to describe the CVRP with a twocommodity network flow formulation. We use their formulation in a slightly sparser form. For each of the undirected arcs (i, j ) A , a binary variable [ ij indicates if the arc is traversed or not (in either direction), while two flow variables, xij and x ji , represent the vehicle’s load and residual capacity (empty space). Eqs.(1-7) express the CVRP: min ¦¦ cij[ ij (1) [ ,x
s.t.
i
j !i
¦[
ji
j i
¦ [ij
s.t . xij x ji s.t .
2, i V
j !i
¦[
0j
j
¦[
i ( N 1)
i
Q[ij , (i, j ) A
¦x
Q qi , i V
¦x
¦q
ij
&
K
(2 & 3) (4) (5)
j
s.t .
j
0j
i
i
&
¦x
i ( N 1)
0
i
(6 & 7)
3. Strengthening Inequalities & Separation Algorithms 3.1. Commodity Flow Inequalities Constraints that strengthen the bounds of the flow variable bound strengthening constraints are appended to the formulation from the onset and are similar to the flow inequalities suggested by Baldacci et al. (2004). They can be expressed as follows: ª º xoj t «¦ qi ( K 1)Q »[oj , j V (8) ¬ i ¼ (9) xij t q j[ij & x ji t qi[ij , (i, j ) V 3.2. Local Scope Inequalities After examining the structure of the relaxation solution at each node, it is possible to infer that certain solution segments (i.e., collections of arcs P ) lead to infeasible, suboptimal, or otherwise undesirable solutions. Such segments can be disallowed through the addition locally at the node level of an appropriate cut of the type: (10) ¦[ij d| P | 1 i , j P
We focus on solution segments that correspond to fully formed paths, that is, collections of consecutively joint arcs P ^ i, j : [ij 1`, augmented by adjacent fractional arcs,
Hybrid Branch-and-Cut for the CVRP
509
0 [ij 1 . We search for undesirable paths through a structured approach that grows
such paths iteratively, and we disallow formation of augmented paths due to 5 reasons: (a) subtour elimination – cyclical routes that do not include the depot; (b) capacity restrictions – routes that exceed vehicle capacity; (c) path dominance – suboptimal routes for which there exists a lower cost ordering of the customers; (d) symmetry breaking – non-nominal routes; that is, routes that begin by visiting a customer that is lexicographically higher than the last customer to be visited by this route; and, (e) demand restrictions – routes that are about to terminate before they have satisfied a minimum amount of demand. A separate class of local scope cuts results by inferring that a coefficient of a variable in a constraint can be suitably increased or decreased so as to tighten this constraint. In particular, we attempt to lift Eqs.(9) by replacing the coefficient of the binary variable (right hand side) with the cumulative load of the fully formed path connected to node i through –and including– node j , under the condition that this path (denoted Pj ) remains intact, and vice-versa: xij t qPj ([ij | Pj |
¦[
n ,m Pj
nm
) & x ji t qPi ([ij | Pi |
¦[
n ,m Pi
nm
)
(11 & 12)
3.3. Global Scope Inequalities There also exist 5 classes of cutting planes that are globally valid for the CVRP. These are the so-called Rounded Capacity (RC), Homogeneous Multistar (HM), Framed Capacity (FC), Strengthened Comb (SC) and Hypotour (HI) inequalities (see Naddef and Rinaldi, 2002, for detailed explanation). Due to their vast number, only those that are violated at a given node relaxation solution are taken into consideration. To this end, we have developed metaheuristic-based algorithms for their efficient separation (i.e., identifying which instances are in fact violated). The emphasis is given on RC and HM inequalities, whose separation is done concurrently through a new Tabu Search (TS) algorithm that improves upon the search framework presented in Augerat et al. (1999). For FC inequalities, a novel multi-restart TS algorithm, combined with a partition generation mechanism and edge-exchange neighborhood search, is used. The SC separation procedure proposed by Lysgaard et al. (2004) is used, enhanced with a TS procedure for expanding the teeth. Finally, the HI separation procedure proposed by Lysgaard et al. (2004) is adopted. The details of the above separation algorithms are omitted for conciseness of this paper.
4. Hybrid Branch-and-Cut Approach 4.1. Adaptive Memory Programming (AMP) Metaheuristic Initially, a reference set (pool) of high quality solutions is generated via an AMP algorithm. This is achieved via the repeated construction of provisional solutions out of promising building blocks identified during the search, while updating these adaptive memory components based on the progress and experience gained (Tarantilis, 2005; Repoussis et al., 2009). For this purpose, a knowledge extraction mechanism is utilized, coupled with a probabilistic construction heuristic and a TS algorithm. Overall, the proposed learning mechanism considers two properties: solution quality and appearance frequency for each pair of customers visited consecutively during a route. During branch-and-cut, at the end of each node’s processing (right before branching), the algorithm utilizes information from the LP relaxation in an effort to
510
C.E. Gounaris et al.
provide a new incumbent. Initially, a Path-Relinking algorithm generates an integer feasible solution [ int that is as “close” as possible to the node’s fractional solution [ f , in terms of Hamming distance d ¦int(1 [ijf ) ¦int[ijf . Next, a new provisional H ( i , j ):^[ij 1` ( i , j ):^[ij 0` solution is generated via reconstructing part of [ int using frequently observed components from the AMP pool. This solution is further improved via TS and, if particular criteria are met, the reference set and memory structures are updated. 4.2. Branch & Cut Framework Given that high quality initial upper bounds are provided through our metaheuristic framework, the priority of the branch-and-cut implementation is on improving the lower bound and on minimizing the number of subproblems (nodes) to be considered until the gap is closed. To this end, we adopt a best-bound-first node selection strategy. After obtaining the standard LP relaxation at each node, the cutting plane phase proceeds as follows: we first search for any augmented paths that need to be disallowed due to suboptimality, infeasibility, or non-nominality of the solution. Next, we check for potential to lift any flow variables and, lastly, we check for global cut violations (with emphasis on RC/HM). If at least one cut is identified at any of these three stages, we reoptimize the LP and repeat the process without continuing with the next stage(s). If no cuts are identified whatsoever, we proceed with branching the node. Let set S V and let G (S ) be the sum of [ ij of all arcs in the corresponding cut-set. As branching rule, we use the disjunction ^G (S ) 2` ^G (S ) t 4` . Among candidate sets for which G ( S ) | 3 , we select the one with the largest total demand.
5. Computational Studies We applied our framework to the standard benchmark data sets that were also used by Lysgaard et al. (2004), where a BC framework based on a different representation of the CVRP –called the vehicle flow formulation– is presented. Table 1 exhibits the root node gap for the 10 hardest problems we attempted, including two 100-customer instances. On average, our commodity flow-based method performs equally well with the vehicle flow-based method. The average root note gap can be improved if we enable CPLEX options for adding generic MIP cuts (e.g., Gomory cuts), however we have observed that doing so deteriorates the overall performance of the algorithm at later nodes. Therefore, for runs to full optimality, we disable cuts identified by CPLEX. Table 2 presents the time necessary to fully close the gap and the number of tree nodes that had to be explored for a set of medium-difficulty problems. The largest instance solved to guaranteed optimality was P-76-6, which involves 75 customers. Two instances were solved at root node by both methods. Our framework required fewer BC nodes for 9 of the remaining 13 instances. The improvement was very substantial for problems A-45-7 and B-50-8, two tight instances that are known to be hard to solve.
6. Conclusions This paper presents a hybrid BC framework for the exact solution of the CVRP that is based on the two-commodity flow formulation, systematic use of local scope cuts, new metaheuristic-based separation techniques for known classes of cuts, as well as an AMP metaheuristic algorithm for identification of high quality integer feasible solutions and acceleration of the search. Computational experiments on benchmark data sets illustrate the potential of the proposed approach.
Hybrid Branch-and-Cut for the CVRP
511
Table 1. Root node performance (% gap) Benchmark LP This This Problems relaxation paper paper+ A-69-9 9.04 3.87 3.30 A-80-10 8.42 3.02 2.39 B-68-9 10.64 1.14 1.14 E-76-8 6.06 2.34 2.17 E-76-10 7.02 3.62 3.33 E-101-8 5.65 1.57 1.55 E-101-14 6.82 3.75 3.20 P-60-15 6.56 3.91 3.48 P-65-10 6.37 3.12 2.48 P-76-5 4.03 1.56 1.55 Average 7.06 2.79 2.46 + CPLEX v11.0 generic MIP cuts enabled
Lysgaard et al., 2004 3.85 3.03 1.10 2.33 3.63 1.52 3.75 3.95 3.14 1.51 2.78
Table 2. Runs to full optimality Benchmark This paper Problems t (sec) # nodes A-44-6 86 125 A-45-7 2,835 2,084 A-48-7 5,147 203 A-55-9 145 156 B-43-6 39 100 B-44-7 3 1 B-45-6 152 276 B-50-7 2 1 B-50-8 4,523 1,503 B-52-7 3 3 B-57-7 99 49 B-64-9 21 7 E-51-5 13 8 P-50-7 78 131 P-76-4 143 105 Note: Optimum solution not provided as input
Lysgaard et al., 2004 t (sec) # nodes 620 211 19,414 4,170 372 113 468 152 125 63 8 1 299 159 11 1 31,026 5,694 25 15 441 168 42 13 59 17 805 263 535 141
References P. Augerat, J.M. Belenguer, E. Benavent, A. Corberán, D. Naddef, 1999, Separating capacity inequalities in the CVRP using tabu search. European Journal of Operational Research, 106, 546–557. R. Baldacci, E. Hadjiconstantinou, A. Mingozzi, 2004, An exact algorithm for the capacitated vehicle routing problem based on a two-commodity network flow formulation, Operations Research, 52, 5, 723-738. R. Baldacci, N. Christofides, A. Mingozzi., 2008, An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts, Mathematical Programming Ser.A, 115,:351-385. R. Baldacci, P. Toth, D. Vigo, 2010, Exact algorithms for routing problems under vehicle capacity constraints, Annals of Operations Research, 175, 1, 213-245. R. Fukasawa, H. Longo, J. Lysgaard, M.P. de Aragao, M. Reis, E. Uchoa, R.F. Werneck, 2006, Robust branch-and-cut-and-price for the capacitated vehicle routing problem, Mathematical Programming Ser.A, 106, 491-511. G. Laporte, 2009, Fifty years of vehicle routing, Transportation Science, 43, 4, 408-416. J. Lysgaard, A.N. Letchford, R.W. Eglese, 2004, A new branch-and-cut algorithm for the capacitated vehicle routing problem, Mathematical Programming Ser.A, 100, 423-445. D. Naddef, G. Rinaldi, 2002, Branch-and-cut algorithms for the capacitated VRP. In: P. Toth and P. Vigo (Eds.), The Vehicle Routing Problem, SIAM Monographs on Discrete Mathematics and Applications, SIAM, Philadelphia, 53-81. P.P. Repoussis, C.D. Tarantilis, G. Ioannou, 2009, Arc-guided evolutionary algorithm for the vehicle routing problem with time windows, IEEE Transactions on Evolutionary Computation, 13, 3, 624-647. C.D. Tarantilis, 2005, Solving the vehicle routing problem with adaptive memory programming methodology, Computers and Operations Research, 32, 9, 2309-2327.