Integer Linear Programming Refining Procedures for

Vehicle Routing Problems

Paolo TothDEIS, 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, 2008.

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

DepotDepot

++

++

++

++ ++

++

++

++

++

++

++

++

++

CustomersCustomers++

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.

DepotDepot

++

++

++

++ ++

++

++

++

++

++

++

++

++

CustomersCustomers++

Final CustomersFinal 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:1) Selection phase: Randomly select a set F of customers.

2) Extraction phase:2) Extraction phase: Extract the customers in F and build a restricted solution z(F)

by short-cutting the extracted customers. Each edge in Each edge in z(F)z(F) is viewed as an is viewed as an insertion point insertion point ii which can which can

allocate one or more customers in allocate one or more customers in FF. Denote with . Denote with I I the set of the set of all the insertion points.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 phase3) Recombination phase:

For each insertion point insertion point ii inin I, g I, generate a pool Si 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:4) Reallocation phase: Reallocate all the extracted customers to the restricted

solution (through the possible insertion of a sequence of Si 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)1) RanRandomdom-Alternate scheme-Alternate scheme: for any route, select in a random : for any route, select in a random

way all the customers in odd position or all the customers in way all the customers in odd position or all the customers in even position.even position.

2)2) Scattered schemeScattered scheme: each customer is selected with a probability : each customer is selected with a probability pp; this scheme allows for the removal of consecutive ; this scheme allows for the removal of consecutive customers (route subsequences)customers (route subsequences)

3)3) Neighborhood schemeNeighborhood scheme: given a “: given a “seedseed” customer ” customer rr, then , then rr is is selected and other customers selected and other customers vv are selected with a are selected with a probability inversely proportional to the distance of probability inversely proportional to the distance of v v from from rr (so that ( (so that (p n)p n) customers are selected on average). customers are selected on average).

The The seedseed customer is iteratively randomly chosen. customer is iteratively randomly chosen.

Computational experiments: schemes Computational experiments: schemes 1)1) and and 2)2) lead to lead to strong improvements of strong improvements of “bad initial solutions”“bad initial solutions”, scheme , scheme 3)3) better for better for “good initial solutions”“good initial solutions”..

Reallocation Model NNotations and definitionsotations 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) ).

- Si : subset of the sequences s which can be allocated to

insertion point i (for each insertion point i in I ); Si (v): subset of Si 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.

si

rq rc

Reallocation Model (3)

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si Fvxi

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Subject to:

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ri Ss

sisi

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rrcDx 6,

7,,1,0 isi Ssix

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otherwise

ipoinsertiontoallocatedisSssequenceifx i

si

Recombination phase

1)1) InitializationInitialization

For each insertion point i I, initialize subset Si 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 RCmax .

- For each insertion point i I, solve the column generation problem associated with i, adding to Si all the feasible sequences corresponding to elementary paths in F whose associated variables have a reduced cost less than RCmax .

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 vF, 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 vP, 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 RCmax .

Pw

Pw

Overall Improvement Procedure1. (Initialization) Set kt := 0 and kp := 0. Take the starting solution Z0 as the

incumbent solution, and initialize the current solution Zc as Zc := Z0 .2. (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 Zc and build the corresponding restricted OVRP solution Zc(F), obtained by shortcutting the extracted customers (I = corresponding insertion point set).

4. (Reallocation) Define the sequence sets Si (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 Zc and Z0 .

.5. (Termination) Set kt := kt + 1. If kt = Ktmax then Stop.

6. (Perturbation) If Zc has been improved in the last iteration, set kp := 0;otherwise set kp := kp + 1. If kp = Kpmax , “perturb” the current solution Zc

and set kp := 0. In any case, repeat from Step 2.

Perturbation Procedure- If the current solution is not improved after a given number

Kpmax of consecutive iterations, a random perturbation is performed.

- Randomly extract np customers from the current solution

Zc (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: 50 - 199);

- Fisher (Operations Res., 1994; instances F11-F12, n: 71 - 134); - Li – Golden – Wasil (Computers & O.R., 2007; large instances

O1-O8, n: 200 - 480).

- 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:

- RCmax = 1 (threshold for the reduced costs) ,- p = 0.5 (probability for a customer to be extracted), - Ktmax = 5000 (max. number of main iterations), - Kpmax = 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).

Instances Prev.best sol. m Initial solution

Cost Time

C2 567.14 10 567.14 7.8

C3 639.74* 8 641.88 23.2

C4 733.13 12 738.94 6.8

C5 869.25 17 878.95 61.9

C6 412.96 6 412.96 0.6

C7 568.49 11 568.49 6.0

C8 644.63 9 646.31 81.9

C9 756.14 14 761.28 46.6

C10 875.07 17 903.10 51.9

C11 682.12 7 717.15 23.1

C12 534.24* 10 534.71 4.2

C13 896.50 12 917.90 82.1

C14 591.87 11 600.66 2.5

F12 769.66 7 777.07 28.4

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.

Instances Prev.best sol. m Initial solution Best solution

Average solution

Average best time

Average final time

Cost Time

C2 567.14 10 567.14 7.8 567.14 567.14 84.5 84.5

C3 639.74* 8 641.88 23.2 639.74* 640.28 91.6 121.5

C4 733.13 12 738.94 6.8 733.13 733.13 4.7 166.3

C5 869.25 17 878.95 61.9 868.81 868.81 16.0 234.4

C6 412.96 6 412.96 0.6 412.96 412.96 44.8 44.8

C7 568.49 11 568.49 6.0 568.49 568.49 82.7 82.7

C8 644.63 9 646.31 81.9 644.63 644.63 1.2 147.9

C9 756.14 14 761.28 46.6 756.14 756.24 124.8 268.0

C10 875.07 17 903.10 51.9 877.47 879.01 330.8 490.6

C11 682.12 7 717.15 23.1 682.83 683.94 167.2 231.9

C12 534.24* 10 534.71 4.2 534.24* 534.24 54.6 101.1

C13 896.50 12 917.90 82.1 894.19 896.35 663.5 1178.7

C14 591.87 11 600.66 2.5 591.87 591.87 237.8 367.1

F12 769.66 7 777.07 28.4 769.55 770.05 64.4 159.7

Average deviation (time) 1.18 (30.5) 0.00 0.06 (140.6) (262.8)

Pentium IV 3 GHz Pentium 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.

Instances Prev.best sol. m Initial solution Best solution

Average solution

Average best time

Average final time

Cost Source

C2 567.14 10 567.14 Fu..., Pisinger..., Fleszar... 567.14 567.14 84.5 84.5

C4 733.13 12 733.13 Fu..., Pisinger..., 733.13 733.13 146.0 146.0

C5 869.24 17 869.24 Derigs-Reuter 868.93 868.94 143.8 263.4

C6 412.96 6 412.96 Fu..., Pisinger..., Fleszar... 412.96 412.96 44.8 44.8

C7 583.19

568.49

10

11

583.19

568.49

Pisinger-Ropke

Fu-Eglese_Li

583.19

568.49

583.19

568.49

81.2

82.7

81.2

82.7

C8 644.63 9 644.63 Fleszar-Osman-Hindi 644.63 644.63 138.0 138.0

C9 757.84

756.14

13

14

757.84

756.14

Pisinger-Ropke

Derigs-Reuter

757.69

756.14

757.72

756.14

76.5

234.8

391.7

234.8

C10 875.07 17 875.07 Derigs-Reuter 874.71 874.71 2.3 394.4

C11 682.12 7 682.12 Pisinger-Ropke, Fleszar... 682.12 682.12 178.3 178.3

C13 904.04

896.50

11

12

904.04

917.90

Fleszar-Osman-Hindi

Fu-Eglese_Li

899.16

894.19

899.16

896.35

45.9

663.5

1003.9

1178.7

C14 591.87

581.81

11

12

591.87

581.81

Pisinger-Ropke, Fleszar...

Derigs-Reuter591.87

581.81

591.87

581.81

285.8

324.6

285.8

324.6

F12 769.66 7 769.66 Fleszar-Osman-Hindi 769.55 769.59 99.6 142.0

Average deviation (time) 0.25 -0.06 0.05 (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

Instances Prev.best sol. m Initial solution Best solution

Average solution

Average best time

Average final time

Cost Time

O1 6018.52 5 6018.52 467 6018.52 6018.52 178.4 178.4

O2 4584.55 9 4584.69 549 4573.53 4573.53 156.1 318.8

O3 7731.46 7 7731.46 4047 7731.46 7731.46 295.4 295.4

O4 7260.59 10 7260.59 927 7251.74 7254.51 220.0 422.0

O5 9167.19 9 9167.19 1186 9156.74 9159.60 238.6 504.5

O6 9803.80 9 9805.45 1231 9804.25 9805.21 558.1 604.0

O7 10348.57 10 10348.57 3190 10344.37 10344.37 229.4 661.9

O8 12420.16 10 12420.16 1969 12420.16 12420.16 645.9 645.9

Average deviation (time) 0.00 (1685.5) -0.06 -0.05 (315.2) (453.9)

Pentium IV 2.8 GHz Pentium 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)

Instances Prev.best sol. m Initial solution Best solution

Average solution

Average best time

Average final time

Cost Time

C1 416.06* 5 467.80 0.0 416.06* 416.84 21.8 66.3

C2 564.06 11 657.07 0.1 564.06 564.06 46.7 83.8

C3 639.74* 8 768.93 0.1 642.14 642.63 61.9 117.8

C4 733.13 12 1069.38 0.2 738.05 742.94 101.5 178.0

C5 869.24 17 1449.20 0.4 879.89 884.95 211.8 272.4

C6 412.96 6 444.98 0.0 412.96 415.29 7.8 48.0

C7 568.49 11 654.27 0.1 568.49 568.69 32.7 78.6

C8 644.63 9 752.98 0.1 645.16 645.64 80.4 157.7

C9 756.14 14 896.61 0.2 756.38 757.48 221.7 299.4

C10 875.07 17 983.97 0.4 886.75 894.35 628.8 713.7

C11 682.12 7 835.93 0.1 689.24 691.03 149.8 219.8

C12 534.24* 10 545.25 0.1 534.24* 534.24* 39.8 96.9

C13 896.50 12 1025.11 0.1 902.87 906.10 723.4 1133.4

C14 581.81 12 641.66 0.1 581.92 581.92 248.8 386.9

F11 177.00* 4 201.27 0.1 177.00* 177.00* 48.8 89.2

F12 769.66 7 919.22 0.2 782.66 783.93 68.9 176.1

Average deviation (time) 19.67 (0.1) 0.45 0.70 (168.4) (257.4)

Pentium IV 3.4 GHz 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 DCVRP28 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:

- RCmax = 1 (threshold for the reduced costs) ,

- p = 0.5 (probability for a customer to be extracted),

- Ktmax = 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

Instances Prev.best sol. m Initial solution Final solution New best sol.

Cost Source Cost CPU time

C1 521* 5 521* Gendreau … _ 5 _

C2 830* 10 832 Gendreau … 831 25 _

C3 815* 8 815* Gendreau … _ 51 _

C4 820* 10 824 Gendreau … 820 38 _

C5 1034* 7 1035 Gendreau … 1034 63 _

C6 548 6 548 Gendreau … _ 5 _

C7 905 11 907 Gendreau … 905 30 _

C8 856 9 856 Gendreau … _ 48 _

C9 865 11 866 Gendreau … 865 69 _

C10 1526 11 1529 Gendreau … 1524 337 1524

C11 1015 12 1016 Taillard 1015 283 _

C12 1289 17 1316 Gendreau … 1288 5940 1285 (16)

C13 1147 14 1180 Gendreau … 1146 734 1145

C14 1392 18 1404 Gendreau … 1385 1384 1378

14 “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.

Instances Prev.best sol. m Initial solution Final solution New best sol.

Cost Source Cost CPU time

G1 707.79 22 707.79 Mester-Braysy _ 5225 _

G2 859.11 26 859.11 Mester-Braysy _ 4259 _

G3 583.39 14 583.39 Mester-Braysy 582.64 19903 582.64

G4 997.52 27 998.73 Mester-Braysy 998.69 9440 _

G5 1081.31 30 1081.31 Mester-Braysy _ 7194 _

G6 741.56 16 742.04 Mester-Braysy 739.53 45516 739.53

G7 1366.86 33 1366.86 Mester-Braysy 1366.54 12717 1366.54

G8 1345.23 33 1345.23 Mester-Braysy 1343.47 31629 1343.47

G9 918.42 18 918.45 Mester-Braysy 916.62 623207 916.62

G10 1820.09 38 1821.15 Mester-Braysy 1820.94 15721 _

G11 1622.69 37 1622.69 Mester-Braysy 1622.39 36867 1622.39

G12 1107.19 19 1107.19 Mester-Braysy _ 23792 _

V1 1067* 14 1076 T,-Vigo 1067 139 _

T1 24422.50 25 24435.50 Taillard 24421.11 8271 24421.11

14 “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, …