Integer Linear Programming Refining Procedures for Vehicle Routing Problems Paolo Toth DEIS, University of Bologna, Italy. IASI - CNR, Roma, March 9, 2010

Outline - The Distance-Constrained Capacitated Vehicle Routing Problem (DCVRP). - The Open Vehicle Routing Problem (OVRP). - An ILP improvement procedure for VRPs. - Computational results for OVRP. - Computational results for DCVRP. - Conclusions.

Based on the papers: T., Tramontani, “An ILP Local Search for Capacitated Vehicle Routing Problems”, from The Vehicle Routing Problem: Latest Advances and New Challenges (Golden, Raghavan, Wasil, Eds.), Springer, Salari, T., Tramontani, “An ILP Improvement Procedure for the Open Vehicle Routing Problem”, Computers & Operations Research (to appear).

Given a “depot”, a set of n “customers” (each having a positive demand), and a fleet of m identical vehicles (each having a “maximum distance” D and a “capacity” Q): Constraints: -Each customer must be visited by exactly one “route”. -Each route must start from the depot, visit a subset of customers and return to the depot. -Each vehicle can perform at most one route. -Each route must have a “global demand” not exceeding Q. Capacitated Vehicle Routing Problem (CVRP)

Given a “depot”, a set of n “customers” (each having a positive demand), and a fleet of m identical vehicles (each having a “maximum distance” D and a “capacity” Q): Constraints: -Each customer must be visited by exactly one “route”. -Each route must start from the depot, visit a subset of customers and return to the depot. -Each vehicle can perform at most one route. -Each route must have a “global demand” not exceeding Q. -Each route must have a “global cost” (distance traveled, duration) not exceeding D. Distance-Constrained Capacitated Vehicle Routing Problem (DCVRP)

Objectives: -Minimize the number of used vehicles as first objective, and then the global traveling cost. -Minimize the global traveling cost. DCVRP is strongly NP-Hard: generalization of the Bin Packing Problem (traveling costs equal to 0) and of the Traveling Salesman Problem (m = 1). Distance-Constrained Capacitated Vehicle Routing Problem (2)

-n = 13, m = 3 Depot Customers + Distance-Constrained Capacitated Vehicle Routing Problem (3)

Open Vehicle Routing Problem (OVRP) -A variant of the “classical” Distance-Constrained Capacitated Vehicle Routing Problem in which the vehicles are not required to return to the depot after completing their service. Depot Customers + Final Customers

Given a “depot”, a set of n “customers” (each having a positive demand), and a fleet of m identical vehicles (each having a “maximum distance” D and a “capacity” Q): Constraints: -Each customer must be visited by exactly one “open route” (path). -Each route must start from the depot and visit a subset of customers. -Each vehicle can perform at most one route. -Each route must have a “global demand” not exceeding Q. -Each route must have a “global cost” (distance traveled, duration) not exceeding D. Open Vehicle Routing Problem (1)

Objectives: -Minimize the number of used vehicles as first objective and then the global traveling cost. -Minimize the global traveling cost. OVRP is strongly NP-Hard: generalization of the Bin Packing Problem (traveling costs equal to 0) and of the Shortest Hamiltonian Path Problem (m = 1). Open Vehicle Routing Problem (2)

If a directed graph is considered: OVRP is a special case of the classical DCVRP (by setting to zero the cost of each arc entering the depot). If an undirected graph is considered: DCVRP is a special case of OVRP: any DCVRP instance on n customers in a complete undirected graph can be transformed into an OVRP instance on n customers, but no transformation exists in the reverse direction (Letchford – Lysgaard - Eglese, JORS, 2007). Open Vehicle Routing Problem (3)

-Companies not owning a vehicle fleet: customers are served by hired vehicles which are not required to come back to the depot (Tarantilis et al., JORS 2005). -Pick up and delivery applications where each vehicle starts from the depot, delivers to a set of customers and then it is required to visit the same customers in reverse order, picking up items to be back-hauled to the depot (Schrage, Networks, 1981). -Planning of train services and school bus routes (Fu- Eglese-Li, JORS, 2005). Open Vehicle Routing Problem Applications

Exact Algorithms (no distance constraints, no empty route): - Letchford – Lysgaard – Eglese (JORS, 2007): branch- and-cut, - Pessoa – Poggi de Aragao – Uchoa (“The VRP: Latest Advances and New Challenges”, Golden, Raghavan, Wasil, eds, Springer, 2008): branch-and-cut-and-price. OVRP Literature

Heuristic Algorithms (distance constraints, minimize the number of routes and then the global cost): - Brandao (EJOR, 2004): tabu search heuristics, - Fu – Eglese – Li (JORS, 2005 and 2006): tabu search heuristics, - Li – Golden - Wasil (Computers & O.R., 2007): record to record travel heuristic, - Pisinger – Ropke (Computers & O.R., 2007): adaptive large neighborhood search heuristic following a destruct-and-repair paradigm, - Derigs – Reuter (JORS, 2008): tabu search heuristics, - Fleszar - Osman - Hindi (EJOR, 2009): variable neighborhood search heuristic, - Li – Tian – Leung (JORS, 2009): ant colony optimization heuristic. OVRP Literature (2)

Heuristic Algorithms ( no distance constraints, minimize the global cost): - Sariklis - Powell (JORS, 2000): two phase heuristic, - Tarantilis – Diakoulaki – Kiranoudis (EJOR, 2004): population based heuristic, - Tarantilis – Iannou – Kiranoudis - Prastacos (JORS, 2005): threshold accepting metaheuristic. OVRP Literature (3)

An ILP improvement procedure for OVRP (General description of the algorithm) Given a feasible initial solution z for OVRP: 1) Selection phase: Randomly select a set F of customers. 2) Extraction phase: Extract the customers in F and build a restricted solution z(F) by short-cutting the extracted customers. Each edge in z(F) is viewed as an insertion point i which can allocate one or more customers in F. Denote with I the set of all the insertion points. ( For each restricted route, add to z(F) an insertion point (artificial arc with cost 0) connecting the last customer of the route with the depot.

General description of the algorithm (2) 3) Recombination phase 3) Recombination phase: insertion point i in I, g For each insertion point i in I, generate a pool S i of sequences through the recombination of the customers in F (pool of elementary paths connecting subsets of customers in F ), by using a Column Generation Procedure. 4) Reallocation phase: Reallocate all the extracted customers to the restricted solution (through the possible insertion of a sequence of S i into insertion point i in I ) in an optimal way (i.e., by minimizing the global re-insertion cost), by solving an ILP model (Reallocation Model). The previous 4 phases are iteratively executed.

Similar framework proposed for the CVRP in: - De Franceschi - Fischetti - T. (Mathematical Programming, 2006). Presented at IASI-CNR, May 17, 2005 (Mini-Workshop in Discrete Optimization, in honor of Egon Balas). Other heuristic algorithms based on the optimal solution of ILP models:... Fischetti – Lodi (Mathematical Programming, 2004): general MIPs (“Local Branching”), Danna – Rothberg – Le Pape (Mathematical Programming, 2005): general MIPs, Archetti – Speranza- Savelsbergh ( Transportation Science, 2008): Split Delivery VRP, Hewitt – Nemhauser – Savesbergh (INFORMS Journal oj Computing, 2009): Capacitated Fixed Charge Network Flow Problem....

Initial solution

Addition of the final arcs

Selection phase

Extraction phase Restricted Solution

Recombination phase

Allocation phase 1

Allocation phase 2

Elimination of the final arcs

Selection Criteria (choice of F) 1)Random-Alternate scheme: for any route, select in a random way all the customers in odd position or all the customers in even position. 2)Scattered scheme: each customer is selected with a probability p; this scheme allows for the removal of consecutive customers (route subsequences) 3)Neighborhood scheme: given a “seed” customer r, then r is selected and other customers v are selected with a probability inversely proportional to the distance of v from r (so that (p n) customers are selected on average). The seed customer is iteratively randomly chosen. The seed customer is iteratively randomly chosen. Computational experiments: schemes 1) and 2) lead to strong improvements of “bad initial solutions”, scheme 3) better for “good initial solutions”. Computational experiments: schemes 1) and 2) lead to strong improvements of “bad initial solutions”, scheme 3) better for “good initial solutions”.

Reallocation Model Notations and definitions : - z(F): Restricted Solution obtained by extracting the customers in F from the initial solution. - : set of routes in the restricted solution. -I = I (z, F): set of edges in the restricted solution (set of insertion points in z(F) ). -S i : subset of the sequences s which can be allocated to insertion point i (for each insertion point i in I ); S i (v): subset of S i containing customer v (for each v in F). -q(s): global demand of sequence s.

Reallocation Model (2) - : extra cost for assigning sequence s to insertion point i. -I(r): set of insertion points associated with restricted route r. - and : global demand and cost, respectively, of restricted route r.

Reallocation Model (3) Subject to:

Recombination phase 1)Initialization For each insertion point i  I, initialize subset S i with: - the “basic” sequence extracted from i ; - the feasible singleton sequence (single customer v in F) with the minimum insertion cost; Initialize the LP Relaxation of the Reallocation Model (LRM) with the initial pool of variables corresponding to the current sequences, and solve LRM.

-For each insertion point i  I, add to the pool of variables all the feasible sequences {v1, v2} (v1, v2  F ) having reduced cost less than a given threshold RC max. -For each insertion point i  I, solve the column generation problem associated with i, adding to S i all the feasible sequences corresponding to elementary paths in F whose associated variables have a reduced cost less than RC max. Recombination phase: Column Generation Procedure

- For any insertion point i  I, the column generation problem associated with i in LRM is a “Resource Constrained Elementary Shortest Path Problem” (RCESPP), which usually arises in the Set Partitioning formulation of CVRP. - For each insertion point i  I, we solve the corresponding RCESPP through a greedy heuristic, with the aim of finding as many variables with small reduced cost as possible. Recombination phase: Column Generation Procedure (2)

Recombination phase: Column generation (Heuristic Algorithm) Given an insertion point i = (a,b) and a starting feasible path P = {a,v,b}, with v  F, s.t. the insertion of v between a and b has the minimum reduced cost. 1)Evaluate all the 1-1 feasible exchanges between each extracted customer and each customer v  P, and select the best one (minimum reduced cost); if such an exchange leads to an improvement, perform it and repeat 1. 2)Evaluate the feasible insertion of each extracted customer in each edge (v1,v2)  P, and select the best one. Force such an insertion even if it leads to a worsening of the current path, and repeat from Step 1). If no feasible insertion exists then stop. At any time a new path (sequence) is generated, the corresponding variable is added to the pool of variables if its reduced cost is smaller than RC max.

Overall Improvement Procedure 1. (Initialization) Set kt := 0 and kp := 0. Take the starting solution Z 0 as the incumbent solution, and initialize the current solution Z c as Z c := Z (Customer selection) Build the set F by selecting each customer with a probability p. 3. (Customer extraction) Extract the customers of F from the current solution Z c and build the corresponding restricted OVRP solution Z c (F), obtained by shortcutting the extracted customers (I = corresponding insertion point set). 4. (Reallocation) Define the sequence sets S i (i  I ) as previously described (column generation on LRM). Build the corresponding Reallocation Model and solve it by using a general-purpose ILP solver. Once an optimal (or near-optimal) ILP solution has been found, build the corresponding new OVRP solution and possibly update Z c and Z (Termination) Set kt := kt + 1. If kt = Kt max then Stop. 6. (Perturbation) If Z c has been improved in the last iteration, set kp := 0; otherwise set kp := kp + 1. If kp = Kp max, “perturb” the current solution Z c and set kp := 0. In any case, repeat from Step 2.

Perturbation Procedure -If the current solution is not improved after a given number Kp max of consecutive iterations, a random perturbation is performed. -Randomly extract np customers from the current solution Z c (with np randomly generated in a given interval). -Reinsert each extracted customer, in turn, in its best feasible position different from the original one. -If a customer cannot be inserted in any currently non-empty route (due to the capacity and/or distance constraints), a new route is created to allocate the customer.

Computational Results for OVRP 24 Benchmark instances from the literature, taken from: - Christofides – Mingozzi – T. (“Combinatorial Optimization”, Christofides – Mingozzi – T. - Sandi, eds, Wiley, 1979; instances C1-C14, n: ); - Fisher (Operations Res., 1994; instances F11-F12, n: ); - Li – Golden – Wasil (Computers & O.R., 2007; large instances O1-O8, n: ). -C1-C5, C11-C12, F11-F12 and O1-O8 instances have only capacity constraints; -C6-C10 and C13-C14 are the same instances as C1-C5 and C11-C12, respectively, but with both capacity and distance constraints (modified for OVRP: D = 0.9 (original duration)), and a larger number of vehicles.

Computational Results for OVRP (2) -Algorithm coded in C. -Test on a Pentium IV, 3.4 GHz with 1 GByte RAM. -Times expressed in seconds. -ILOG Cplex 10.0 as LP and ILP solver. -5 runs executed for each instance (with 5 different random number generator seeds). -Parameters : -RC max = 1 (threshold for the reduced costs), - p = 0.5 (probability for a customer to be extracted), -Kt max = 5000 (max. number of main iterations), -Kp max = 50 (max. number of iterations without improvement), -np randomly generated between 15 and 25 (number of customers extracted for the perturbation).

Computational Results for OVRP (3) Very good solutions (sometimes the best known solution!) considered as “Initial Solutions”. -Provided by: Fu – Eglese – Li (JORS, 2005 and 2006), Pisinger – Ropke (Computers & O.R., 2007), Derigs – Reuter (JORS, 2008), Fleszar – Osman - Hindi (EJOR, 2009).

InstancesPrev.best sol.mInitial solution CostTime C C * C C C C C C C C C * C C F Average deviation (time)1.18(30.5) Pentium IV 3 GHz Computational results on the 16 “classical” instances, starting from the solutions by Fu-Eglese-Li Provably optimal solutions. Initial solutions optimal for C1 and F11.

InstancesPrev.best sol.mInitial solutionBest solution Average solution Average best time Average final time CostTime C C * * C C C C C C C C C * * C C F Average deviation (time)1.18(30.5) (140.6)(262.8) Pentium IV 3 GHzPentium IV 3.4 GHz Computational results on the 16 “classical” instances, starting from the solutions by Fu-Eglese-Li Provably optimal solutions. Final solution cost equal to the previous best known one. 3 new best solutions. Initial solutions optimal for C1 and F11.

InstancesPrev.best sol.mInitial solutionBest solution Average solution Average best time Average final time CostSource C Fu..., Pisinger..., Fleszar C Fu..., Pisinger..., C Derigs-Reuter C Fu..., Pisinger..., Fleszar C Pisinger-Ropke Fu-Eglese_Li C Fleszar-Osman-Hindi C Pisinger-Ropke Derigs-Reuter C Derigs-Reuter C Pisinger-Ropke, Fleszar C Fleszar-Osman-Hindi Fu-Eglese_Li C Pisinger-Ropke, Fleszar... Derigs-Reuter F Fleszar-Osman-Hindi Average deviation (time) (169.3)(320.2) Computational results on the 16 “classical” instances, starting from the best available solutions 6 new best solutions (over 12). Initial solutions optimal for C1, C3, C12, F11

InstancesPrev.best sol.mInitial solutionBest solution Average solution Average best time Average final time CostTime O O O O O O O O Average deviation (time)0.00(1685.5) (315.2)(453.9) Pentium IV 2.8 GHzPentium IV 3.4 GHz Computational results on the 8 “large-size” instances, starting from the solutions by Derigs-Reuter (JORS, 2008) 4 new best solutions (over 8)

InstancesPrev.best sol.mInitial solutionBest solution Average solution Average best time Average final time CostTime C * * C C * C C C C C C C C C * * C C F * * F Average deviation (time)19.67(0.1) (168.4)(257.4) Pentium IV 3.4 GHz Computational results on the 16 “classical” instances, starting from “bad quality” initial solutions Provably optimal solutions. Final solution cost equal to the previous best known one (6 over 16).

Current best known solutions for OVRP 10 new best solutions (over 30 instances for which the current best known solution is not proved to be optimal).

Computational Results for DCVRP 28 Benchmark instances from the literature proposed by: - Christofides – Mingozzi – T. (“Combinatorial Optimization”, Christofides – Mingozzi – T. - Sandi, eds, Wiley, 1979; instances C1-C14, n: 50 – 199, rounded integer costs); - Golden – Wasil – Kelly - Chao (“Fleet Management and Logistics”, Crainic - Laporte, eds, Kluwer, 1998; instances G1-G12, n: 241 – 484, real costs); - Vigo (Vigo web page; instance V1, n: 100, integer costs); - Taillard (Taillard web page; instance T1, n: 385, real costs). -C1-C5, C11-C12, G1-G12, V1, T1 instances have only capacity constraints; -C6-C10 and C13-C14 are the same instances as C1-C5 and C11-C12, respectively, but with both capacity and distance constraints (and a larger number of vehicles).

-Algorithm coded in C. -Test on a Pentium M, 1.86 GHz with 1 GByte RAM. -Times expressed in seconds. -ILOG Cplex 10.0 as LP and ILP solver. -Parameters : -RC max = 1 (threshold for the reduced costs), - p = 0.5 (probability for a customer to be extracted), -Kt max = 5000 (max. number of main iterations). Computational Results for DCVRP

Very good solutions considered as “Initial Solutions”, provided by: Taillard (Networks, 1993),, Gendreau - Hertz - Laporte (Man. Science, 1999), T. - Vigo (INFORMS Journal on Computing, 2003). Mester - Braysy (Computers & O.R., 2007). Other Best Solutions: Rochat – Taillard (Journal of Heuristics, 1995), Xu – Kelly (Transportation Science, 1996). Prins (Computers & Operations Research, 2004), Wassan (Journal of the Operational Research Society, 2006), De Franceschi - Fischetti - T. (Mathematical Programming, 2006). Pisinger –Ropke (Computers & Operations Research, 2007), Computational Results for DCVRP

InstancesPrev.best sol.mInitial solutionFinal solutionNew best sol. CostSourceCostCPU time C1521*5 Gendreau …_5_ C2830*10832Gendreau …83125_ C3815*8 Gendreau …_51_ C4820*10824Gendreau …82038_ C51034*71035Gendreau …103463_ C65486 Gendreau …_5_ C Gendreau …90530_ C88569 Gendreau …_48_ C Gendreau …86569_ C Gendreau … C Taillard _ C Gendreau … (16) C Gendreau … C Gendreau … “classical” instances, starting from the best available solutions, Rounded integer costs Provably optimal solutions. Final solution cost equal to the previous best known one. 4 new best solutions.

InstancesPrev.best sol.mInitial solutionFinal solutionNew best sol. CostSourceCostCPU time G Mester-Braysy_5225_ G Mester-Braysy_4259_ G Mester-Braysy G Mester-Braysy _ G Mester-Braysy_7194_ G Mester-Braysy G Mester-Braysy G Mester-Braysy G Mester-Braysy G Mester-Braysy _ G Mester-Braysy G Mester-Braysy_23792_ V11067*141076T,-Vigo _ T Taillard “large” instances, starting from the best available solutions, Real costs Provably optimal solutions. Final solution cost equal to the previous best known one. 7 new best solutions.

Conclusions - For both OVRP and DCVRP the proposed method is very effective in improving the starting solution, even if it is of very- good quality. Future directions: - More sophisticated criteria for extracting the customers from the current solution (Selection Phase). - The overall procedure can be considered as a general framework and it could be extended to cover other variants of Vehicle Routing Problems such as: Vehicle Routing Problems with Heterogenous Fleet, Multi-Depot Vehicle Routing Problems, Multi-Trip Vehicle Routing Problems, …