Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 2

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2 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 2 Outline Vehicle routing problem; How interior point methods can help; Interior point branch-price-and-cut: central primal-dual solutions; Results for vehicle routing variants; Conclusion and future developments.

3 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 3 Vehicle routing problem One of the most studied combinatorial optimization problems;

4 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 3 Vehicle routing problem One of the most studied combinatorial optimization problems; Theoretical reason: very difficult to solve by standard methods; tough testbed for new algorithms; the literature growth is almost perfectly exponential with a 6.09% annual growth rate (Eksioglu et al. 2009; Braekers et al., 2016)

5 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 3 Vehicle routing problem One of the most studied combinatorial optimization problems; Theoretical reason: very difficult to solve by standard methods; tough testbed for new algorithms; the literature growth is almost perfectly exponential with a 6.09% annual growth rate (Eksioglu et al. 2009; Braekers et al., 2016) Practical reason: models important real-life situations, faced by many companies around the world; impact our day-to-day lives; Airline companies; train and bus schedules; freight transportation... We want it at the best price and on time!

6 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 4 VRP with time windows (VRPTW)

7 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 4 VRP with time windows (VRPTW)

8 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 4 VRP with time windows (VRPTW)

9 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 5 Vehicle routing problem To effectively solve vehicle routing problems, we need to rely on a variety of formulations and state-of-the-art solution methods; Heuristic, exact and hybrid methods; Vehicle flow formulation (x k ij) and set partitioning formulation (λ p); (completely different, but the same actually! See: Munari, A generalized formulation for vehicle routing problems, ArXiv, 2016) Branch-and-cut; column generation and branch-and-price; Auxiliary techniques: dynamic programming; implicit enumeration; and many others to speed up the generation of routes and/or valid inequalities; Typical things that you need to solve large-scale problems. 5

10 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 6 Large-scale optimization problems A formulation that challenges state-of-the-art implementations; Special structure in the coefficient matrix, which allows a reformulation (e.g. Dantzig-Wolfe decomposition, Lagrangian relaxation, Benders decomposition, etc); Geoffrion (1970); Vanderbeck and Wolsey (2010);

11 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 7 Large-scale optimization problems

12 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 7 Large-scale optimization problems Master problem Decomposition Subproblems

13 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 7 Large-scale optimization problems Master problem Column generation Decomposition Subproblems

14 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 7 Large-scale optimization problems Master problem Column generation TOO MANY VARIABLES Decomposition Subproblems

15 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 8 Large-scale discrete optimization problems Z n 8

16 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 8 Large-scale discrete optimization problems Z n Master Rn Subproblems 8

17 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 8 Large-scale discrete optimization problems Column generation Z n Master Rn Subproblems 8

18 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 8 Large-scale discrete optimization problems Column generation Z n Master Rn Subproblems 8

19 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 8 Large-scale discrete optimization problems Column generation Z n Master Rn Subproblems Branch-and-price 8

20 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 9 Large-scale optimization problems In column generation and branch-and-price, we typically have to solve hundreds of thousands of linear programming (LP) problems in sequence; This way, it is important to use a fast LP method;

21 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 9 Large-scale optimization problems In column generation and branch-and-price, we typically have to solve hundreds of thousands of linear programming (LP) problems in sequence; This way, it is important to use a fast LP method; Are we really interested in an optimal solution of these problems? What informations would be relevant in this context?

22 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 10 Column generation

23 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 10 Column generation We are interested in solving a linear programming problem with a huge number of columns, called the Master Problem (MP): z := min s.t. c jλ j, j N a jλ j = b, j N λ j 0, j N. N is too big; The columns (c j, a j) A are not known explicitly; We know how to generate them!

24 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 11 Column generation Restricted Master Problem (RMP): z RMP := min s.t. with N N. c jλ j, j N a jλ j = b, j N (u) λ j 0, j N. Let ū be a dual optimal solution of the RMP;

25 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 11 Column generation Restricted Master Problem (RMP): z RMP := min s.t. with N N. c jλ j, j N a jλ j = b, j N (u) λ j 0, j N. Let ū be a dual optimal solution of the RMP; Pricing subproblem (oracle): z SP := min{0, c j u T a j (c j, a j) A}. (cj, a j) are the variables in the subproblem; If zsp < 0, then new columns are generated;

26 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 12 Standard column generation Optimal solutions, typically obtained by the simplex method: Extreme points of the RMP; Bang-bang: they oscillate too much between consecutive iterations; u j+1 is typically far from u j ; Heading-in and tailing-off; Degeneracy; see Vanderbeck (2005); Lubbecke and Desrosiers (2005); Extreme points result in slow convergence of the method. 12

27 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 13 Oscillation in a VRP instance u j u j+1 2, for each iteration j: Munari, P.; Gondzio, J. Column generation and branch-and-price with interior point methods. Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, v. 3 (1), 2015.

28 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 13 Column generation variants Stabilization techniques: avoid extreme solutions! use a point in the interior of the feasible set;

29 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 14 Column generation variants Stability center and/or safety region in the dual space; (a)

30 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 14 Column generation variants Stability center and/or safety region in the dual space; (a)

31 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 14 Column generation variants Stability center and/or safety region in the dual space; (a)

32 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 14 Column generation variants Stability center and/or safety region in the dual space; (a)

33 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 14 Column generation variants Stability center and/or safety region in the dual space; (a)

34 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 15 Column generation variants Stabilization techniques: avoid extreme solutions! use a point in the interior of the feasible set; Different strategies: dynamic boxes and penalties (Marsten et al., 1975, du Merle 1999; Ben Amor et al. 2009); smoothing (Wentges, 1997; Neame, 1999; Pessoa, 2013); bundle and nonlinear penalties (Frangioni, 2002; Briant et al., 2004) interior points (Goffin and Vial, 2002; Rosseau et al., 2003); and many others; 15

35 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 16 Column generation variants Most of them: modify the master problem or require additional control on the dual solutions; Add variables, bounds, constraints, penalties,... The master problem may become more difficult to solve; Some of them may be difficult to implement; Several parameters to tune.

36 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 16 Column generation variants Most of them: modify the master problem or require additional control on the dual solutions; Add variables, bounds, constraints, penalties,... The master problem may become more difficult to solve; Some of them may be difficult to implement; Several parameters to tune. They all agree on one thing: Column generation is more efficient when based on well-centered interior points of the feasible set;

37 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 16 Column generation variants Most of them: modify the master problem or require additional control on the dual solutions; Add variables, bounds, constraints, penalties,... The master problem may become more difficult to solve; Some of them may be difficult to implement; Several parameters to tune. They all agree on one thing: Column generation is more efficient when based on well-centered interior points of the feasible set; So, why not using a primal-dual interior point method? This is straightforward: does not require any changes in the RMP nor additional control (naturally stable solutions).

38 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 17 Interior point method

39 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 17 Interior point method

40 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 17 Interior point method

41 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 18 Primal-dual column generation method (PDCGM)

42 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 18 Primal-dual column generation method (PDCGM) Primal-dual interior point method to get primal-dual solutions (Gondzio and Sarkissian, 1996; Gondzio et al., 2013; Munari and Gondzio, 2013; Gondzio et al., 2016);

43 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 18 Primal-dual column generation method (PDCGM) Primal-dual interior point method to get primal-dual solutions (Gondzio and Sarkissian, 1996; Gondzio et al., 2013; Munari and Gondzio, 2013; Gondzio et al., 2016); Suboptimal solution ( λ, ũ) (ε-optimal solution): we stop the interior point method with optimality tolerance ε.

44 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 18 Primal-dual column generation method (PDCGM) Primal-dual interior point method to get primal-dual solutions (Gondzio and Sarkissian, 1996; Gondzio et al., 2013; Munari and Gondzio, 2013; Gondzio et al., 2016); Suboptimal solution ( λ, ũ) (ε-optimal solution): we stop the interior point method with optimality tolerance ε. The distance to optimality ε is dynamically adjusted according to the relative gap; ε = min{ε max, gap/d} gap = (UB LB)/(1 + UB ); D: degree of optimality (fixed, D > 1);

45 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 18 Primal-dual column generation method (PDCGM) Primal-dual interior point method to get primal-dual solutions (Gondzio and Sarkissian, 1996; Gondzio et al., 2013; Munari and Gondzio, 2013; Gondzio et al., 2016); Suboptimal solution ( λ, ũ) (ε-optimal solution): we stop the interior point method with optimality tolerance ε. The distance to optimality ε is dynamically adjusted according to the relative gap; ε = min{ε max, gap/d} gap = (UB LB)/(1 + UB ); D: degree of optimality (fixed, D > 1); We save time and stop with a well-centered dual solution! 17

46 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 19 Non-optimal solutions from interior point method (b)

47 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 19 Non-optimal solutions from interior point method

48 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 19 Non-optimal solutions from interior point method (c)

49 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 20 PDCGM: Algorithm

50 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 20 PDCGM: Algorithm 1. Input: Initial RMP; parameters κ, ε max, D > 1, δ > 0,. 2. set LB =, UB =, gap =, ε = 0.5; 3. while (gap > δ) do 4. find a well-centered ε-optimal solution ( λ, ũ) of the RMP; 5. UB = min{ub, z RMP }; 6. call the oracle with the query point ũ; 7. LB = max{lb, κ z SP + b T ũ}; 8. gap = (UB LB)/(1 + UB ); 9. ε = min{ε max, gap/d}; 10. if ( z SP < 0) then add the new columns into the RMP; 11. end(while)

51 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 20 PDCGM: Algorithm 1. Input: Initial RMP; parameters κ, ε max, D > 1, δ > 0,. 2. set LB =, UB =, gap =, ε = 0.5; 3. while (gap > δ) do 4. find a well-centered ε-optimal solution ( λ, ũ) of the RMP; 5. UB = min{ub, z RMP }; 6. call the oracle with the query point ũ; 7. LB = max{lb, κ z SP + b T ũ}; 8. gap = (UB LB)/(1 + UB ); 9. ε = min{ε max, gap/d}; 10. if ( z SP < 0) then add the new columns into the RMP; 11. end(while)

52 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 21 PDCGM: well-centered solutions Primal-dual interior point method with symmetric neighborhood (Gondzio, 2015): ( λ, ũ) is well-centered in the feasible set: γ µ (c j ũ T a j) λ j (1/γ) µ, j N, for some γ (0.1, 1], where µ = (1/ N )(c T ũ T A) λ; Natural way of stabilizing dual solutions. 20

53 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 22 PDCGM: Convergence Theorem Let z be the optimal value of the MP. Given the optimality tolerance δ > 0, the primal-dual column generation method converges in a finite number of steps to a primal feasible solution ˆλ of the MP with objective value z that satisfies: ( z z ) < δ(1 + z ). Gondzio, J.; González-Brevis, P. and Munari, P. New developments in the Primal-Dual Column Generation Technique, European Journal of Operational Research 224, pp , 2013;

54 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 23 PDCGM: Algorithm Implementation in C, using interior point solver HOPDM; Publicly available code Source-code examples are provided for different applications: Cutting stock problem; Vehicle routing problem; Capacitated lot sizing problem with setup times; Multiple kernel learning; Two-stage stochastic programming; Multicommodity network flow. Gondzio, J.; González-Brevis, P.; Munari, P. Large-Scale Optimization with the Primal-Dual Column Generation Method. Mathematical Programming Computation, v. 8 (1), p , 2016.

55 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 24 Interior point branch-price-and-cut (IPBPC)

56 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 24 Interior point branch-price-and-cut (IPBPC)

57 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 24 Interior point branch-price-and-cut (IPBPC)

58 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 24 Interior point branch-price-and-cut (IPBPC)

59 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 24 Interior point branch-price-and-cut (IPBPC)

60 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 24 Interior point branch-price-and-cut (IPBPC)

61 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 24 Interior point branch-price-and-cut (IPBPC)

62 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 24 Interior point branch-price-and-cut (IPBPC)

63 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 24 Interior point branch-price-and-cut (IPBPC)

64 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 24 Interior point branch-price-and-cut (IPBPC)

65 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 25 Interior point branch-price-and-cut (IPBPC) The primal-dual interior point algorithm will be used to provide well-centered, suboptimal solutions:

66 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 25 Interior point branch-price-and-cut (IPBPC) The primal-dual interior point algorithm will be used to provide well-centered, suboptimal solutions: Column generation;

67 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 25 Interior point branch-price-and-cut (IPBPC) The primal-dual interior point algorithm will be used to provide well-centered, suboptimal solutions: Column generation; Valid inequalities;

68 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 25 Interior point branch-price-and-cut (IPBPC) The primal-dual interior point algorithm will be used to provide well-centered, suboptimal solutions: Column generation; Valid inequalities; Branching (early termination; central branching).

69 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 25 Interior point branch-price-and-cut (IPBPC) The primal-dual interior point algorithm will be used to provide well-centered, suboptimal solutions: Column generation; Valid inequalities; Branching (early termination; central branching). More stable primal and dual solutions; Deeper columns and cuts; Speed up solution times. Very few attempts in the literature (du Merle et al., 1999; Elhedhli and Goffin, 2004, Munari and Gondzio, 2013);

70 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 26 Interior point branch-price-and-cut (IPBPC) Oracle: two types of subproblems; ORACLE Pricing subproblem new column (s) primal and dual solutions Separation subproblem new cut (s) We start calling the separation subproblem as soon as the gap falls below a tolerance ε c (= 0.1), at every K c iterations;

71 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 26 Interior point branch-price-and-cut (IPBPC) Oracle: two types of subproblems; ORACLE Pricing subproblem new column (s) primal and dual solutions Separation subproblem new cut (s) We start calling the separation subproblem as soon as the gap falls below a tolerance ε c (= 0.1), at every K c iterations; Early branching: stop CG with a loose tolerance ε b (= 10 3 ) and branch!

72 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 27 Interior point branch-price-and-cut (IPBPC) Two-steps: Preprocessing step Quickly obtain a suboptimal solution of the MP Branch? yes Branch node Step 1 To quickly solve the master problem: looser optimality tolerance; heuristics in the pricing subproblem. 25

73 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 27 Interior point branch-price-and-cut (IPBPC) Two-steps: Preprocessing step Quickly obtain a suboptimal solution of the MP Branch? no Find an optimal solution of the MP yes Branch node Step 1 Step 2 To quickly solve the master problem: looser optimality tolerance; heuristics in the pricing subproblem. 25

74 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 27 Interior point branch-price-and-cut (IPBPC) Two-steps: Preprocessing step Quickly obtain a suboptimal solution of the MP Branch? no Find an optimal solution of the MP yes Branch node yes Branch? no Prune node Step 1 Step 2 To quickly solve the master problem: looser optimality tolerance; heuristics in the pricing subproblem. 25

75 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 28 VRP with time windows (VRPTW)

76 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 29 VRP with time windows (VRPTW) Set partitioning formulation: min s.t. c rλ r r R a riλ r = 1, i = 1,..., n, r R λ r {0, 1}, r R. R: set of feasible routes; Routes are generated by solving a Resource Constrained Elementary Shortest Path Problem (subproblem).

77 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 30 VRP with time windows (VRPTW) IPBPC implementation for the VRPTW; RMP: primal-dual interior point method (HOPDM); Subproblem: label-setting algorithm with improvements (Feillet et al., 2004; Righini and Salani, 2008; Desaulniers et al. 2008); Valid inequalities: Subset row cuts (Jepsen et al., 2008);

78 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 30 VRP with time windows (VRPTW) IPBPC implementation for the VRPTW; RMP: primal-dual interior point method (HOPDM); Subproblem: label-setting algorithm with improvements (Feillet et al., 2004; Righini and Salani, 2008; Desaulniers et al. 2008); Valid inequalities: Subset row cuts (Jepsen et al., 2008); Solomon s instances (standard benchmark for VRPTW);

79 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 30 VRP with time windows (VRPTW) IPBPC implementation for the VRPTW; RMP: primal-dual interior point method (HOPDM); Subproblem: label-setting algorithm with improvements (Feillet et al., 2004; Righini and Salani, 2008; Desaulniers et al. 2008); Valid inequalities: Subset row cuts (Jepsen et al., 2008); Solomon s instances (standard benchmark for VRPTW); Comparison to a state-of-the-art simplex-based BPC by Desaulniers, Lessard and Hadjar (2008).

80 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 31 Nodes_100 Number of nodes Instance C101 C102 C103 C104 C105 C106 C107 C108 C109 RC101 RC102 RC103 RC104 RC105 RC106 RC107 RC108 R101 R102 R103 R104 R105 R106 R107 R108 R109 R110 R111 R112 DLH08 IPBPC Nodes

81 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 32 Comparing to a simplex-based BPC Number of nodes DLH08 IPBPC Ratio C RC R

82 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 33 Cuts_100 Number of valid inequalities Instance C101 C102 C103 C104 C105 C106 C107 C108 C109 RC101 RC102 RC103 RC104 RC105 RC106 RC107 RC108 R101 R102 R103 R104 R105 R106 R107 R108 R109 R110 R111 R112 DLH08 IPBPC Valid inequalities

83 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 34 Comparing to a simplex-based BPC Number of valid inequalities DLH08 IPBPC Ratio C RC R

84 Pedro Munari - COA 2017, February 10th, University of Edinburgh, CPUtime_100 Scotland, UK 35 CPU time Instance C101 C102 C103 C104 C105 C106 C107 C108 C109 RC101 RC102 RC103 RC104 RC105 RC106 RC107 RC108 R101 R102 R103 R104 R105 R106 R107 R108 R109 R110 R111 R112 DLH08 IPBPC Seconds

85 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 36 Comparing to a simplex-based BPC CPU time (sec) DLH08 IPBPC Ratio C RC R

86 Solving challenging vehicle routing problems: you better follow the central path Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 37 Author's personal copy IPBPC: impact of suboptimal solutions 30

87 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 38 Interior point branch-price-and-cut (IPBPC) Oracle: two types of subproblems; ORACLE Pricing subproblem new column (s) primal and dual solutions Separation subproblem new cut (s) We start calling the separation subproblem as soon as the gap falls below a tolerance ε c (= 0.1), at every K c iterations; Early branching: stop CG with a loose tolerance ε b (= 10 3 ) and branch!

88 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 39 VRPTW with multiple deliverymen (VRPTWMD)

89 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 39 VRPTW with multiple deliverymen (VRPTWMD)

90 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 39 VRPTW with multiple deliverymen (VRPTWMD)

91 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 39 VRPTW with multiple deliverymen (VRPTWMD)

92 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 40 VRPTW with multiple deliverymen (VRPTWMD) Pureza, Morabito and Reimann (2012):

93 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 40 VRPTW with multiple deliverymen (VRPTWMD) Pureza, Morabito and Reimann (2012): Vehicle flow (compact) formulation and metaheuristics;

94 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 40 VRPTW with multiple deliverymen (VRPTWMD) Pureza, Morabito and Reimann (2012): Vehicle flow (compact) formulation and metaheuristics; Objective function: L ω 1 l=1 j C x l 0j + ω 2 L lx l 0j + ω 3 L l=1 j C l=1 i N j N d ijx l ij (nb of vehicles) (nb of deliverymen) (distance)

95 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 40 VRPTW with multiple deliverymen (VRPTWMD) Pureza, Morabito and Reimann (2012): Vehicle flow (compact) formulation and metaheuristics; Objective function: L ω 1 l=1 j C x l 0j + ω 2 L lx l 0j + ω 3 L l=1 j C l=1 i N j N d ijx l ij (nb of vehicles) (nb of deliverymen) (distance) We propose a set partitioning formulation and an interior point branch-price-and-cut method;

96 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 41 VRPTWMD: Set partitioning formulation min s.t. L l=1 L l=1 p P l c l pλ l p a l piλ l p = 1, i = 1,..., n, p P l L lλ l p D, p P l l=1 λ l p {0, 1}, l = 1,..., L, p P l. P l : set of all feasible routes in mode l, l = 1,..., L; λ l p: 1, if the p-th route in mode l is chosen; 0, o.w. 35

97 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 42 VRPTWMD: Set partitioning formulation Columns: visited customers and mode (number of deliverymen) a l p = l customer 1 is not visited customer 2 is visited route mode Cost of a route p in mode l: c l p = ω 1 L x l p0j + ω 2 L lx l p0j + ω 3 l=1 j C l=1 j C l=1 i N j N L c ijx l pij, where x l pij = 1 if and only if route p P l visits node i and goes directly to node j.

98 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 43 VRPTWMD: Exact hybrid method

99 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 43 VRPTWMD: Exact hybrid method Hybrid: math programming + heuristics (matheuristic); They have been proposed for many different types of problems, in special for VRP variants (Archetti and Speranza, 2014); Combine the best of two worlds! (Metaheuristic helps BPC with UB) We propose combining the IPBPC (exact method) with two metaheuristics: ILS: Iterated Local Search; LNS: Large Neighbourhood Search; These metaheuristics have been successfully used to find feasible solutions of the VRPTWMD and many other variants. Álvarez, A. and Munari, P. Metaheuristic approaches for the vehicle routing problem with time windows and multiple deliverymen. Journal of Management & Production, v. 23, p , 2016.

100 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 44 VRPTW with multiple deliverymen (VRPTWMD) Exact hybrid method! 40

101 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 45 VRPTW with multiple deliverymen (VRPTWMD) Computational experiments Solomon s instances (VRPTW): 100 customers; R1 (12), C1 (9), RC1 (8); R2 (11), C2 (8), RC2 (8); larger capacities and time windows Service times (Pureza et al., 2012): s l i = min{2 di, T max{wa i, t 0i} t i,n+1} l ω 1 = 1, ω 2 = 0.1, ω 3 = (vehicles, deliverymen, travel costs); Linux PC with Intel Core i7 3.1 GHz CPU, 16 GB of RAM; Time limit: 1 hour.

102 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 46 VRPTW with multiple deliverymen (VRPTWMD) Average results for each class Hybrid IPBPC IPBPC/Hybrid Class Objective Objective Ratio (%) C R RC C R RC Álvarez, A. and Munari, P. An exact hybrid method for the vehicle routing problem with time windows and multiple deliverymen. Computers & Operations Research, (Accepted)

103 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 47 VRPTW with multiple deliverymen (VRPTWMD) Best results from the literature (from metaheuristics) Best results IPBPC/Best Hybrid/Best Class Objective nv nd Dist Ratio (%) Ratio (%) C R RC C R RC Álvarez, A. and Munari, P. An exact hybrid method for the vehicle routing problem with time windows and multiple deliverymen. Computers & Operations Research, (Accepted)

104 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 48 VRPTW with multiple deliverymen (VRPTWMD) Hybrid method: source of best solutions Source 600 sec 3600 sec Best % Best % MH initial MIP initial Integer RMP MIP heur MH polish MH initial : Initial solution provided by metaheuristics, before starting the BPC; MIP initial : MIP heuristic, using initial columns only (before starting the BPC); Integer RMP: Integer solution found from the linear relaxation of the RMP; MIP heur: MIP heuristic at the end of the node; MH polish: Metaheuristics after finding a new incumbent of the BPC. 43

105 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 49 Conclusions and future developments Similar to most combinatorial optimization problems, VRP requires sophisticated solution methods to work well in practice; Interior point methods offer advantageous features when integrated to these methods; Natural way of stabilizing column generation and improving cut generation and branching; Reductions in iterations, nodes and CPU time; In addition, hybrid methods that combine exact and heuristic approaches seem to be a good option in practice, to solve real-life problems; Wide range of applications may benefit from these tools.

106 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 50 VRP under uncertainty Uncertainty on demand, travel times, service times; Robust optimization and Stochastic programming; Very active area in the last few years! (Oyola et al., 2016; Gendreau et al., 2016) How to effectively incorporate uncertainty to set partitioning formulations, to be able to solve large-scale problems? Ongoing project in collaboration with Jacek Gondzio and Douglas Alem;

107 Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 51 Obrigado :) Acknowledgments munari@dep.ufscar.br

A generalized formulation for vehicle routing problems

Working paper A generalized formulation for vehicle routing problems Pedro Munari a, Twan Dollevoet b, Remy Spliet b a Production Engineering Department, Federal University of São Carlos, Brazil b Econometric