Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 2
|
|
- David Burke
- 5 years ago
- Views:
Transcription
1
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
More informationInteger Programming Reformulations: Dantzig-Wolfe & Benders Decomposition the Coluna Software Platform
Integer Programming Reformulations: Dantzig-Wolfe & Benders Decomposition the Coluna Software Platform François Vanderbeck B. Detienne, F. Clautiaux, R. Griset, T. Leite, G. Marques, V. Nesello, A. Pessoa,
More informationStabilization in Column Generation: numerical study
1 / 26 Stabilization in Column Generation: numerical study Artur Pessoa 3 Ruslan Sadykov 1,2 Eduardo Uchoa 3 François Vanderbeck 2,1 1 INRIA Bordeaux, France 2 Univ. Bordeaux I, France 3 Universidade Federal
More informationAcceleration and Stabilization Techniques for Column Generation
Acceleration and Stabilization Techniques for Column Generation Zhouchun Huang Qipeng Phil Zheng Department of Industrial Engineering & Management Systems University of Central Florida Sep 26, 2014 Outline
More informationBranch-and-Price-and-Cut for the Split Delivery Vehicle Routing Problem with Time Windows
Branch-and-Price-and-Cut for the Split Delivery Vehicle Routing Problem with Time Windows Guy Desaulniers École Polytechnique de Montréal and GERAD Column Generation 2008 Aussois, France Outline Introduction
More informationInterior-Point versus Simplex methods for Integer Programming Branch-and-Bound
Interior-Point versus Simplex methods for Integer Programming Branch-and-Bound Samir Elhedhli elhedhli@uwaterloo.ca Department of Management Sciences, University of Waterloo, Canada Page of 4 McMaster
More informationColumn Generation. ORLAB - Operations Research Laboratory. Stefano Gualandi. June 14, Politecnico di Milano, Italy
ORLAB - Operations Research Laboratory Politecnico di Milano, Italy June 14, 2011 Cutting Stock Problem (from wikipedia) Imagine that you work in a paper mill and you have a number of rolls of paper of
More informationBranch and Price for the Vehicle Routing Problem with Discrete Split Deliveries and Time Windows
Branch and Price for the Vehicle Routing Problem with Discrete Split Deliveries and Time Windows Matteo Salani Ilaria Vacca December 24, 2009 Report TRANSP-OR 091224 Transport and Mobility Laboratory Ecole
More informationLecture 8: Column Generation
Lecture 8: Column Generation (3 units) Outline Cutting stock problem Classical IP formulation Set covering formulation Column generation A dual perspective Vehicle routing problem 1 / 33 Cutting stock
More informationNotes on Dantzig-Wolfe decomposition and column generation
Notes on Dantzig-Wolfe decomposition and column generation Mette Gamst November 11, 2010 1 Introduction This note introduces an exact solution method for mathematical programming problems. The method is
More informationPartial Path Column Generation for the Vehicle Routing Problem with Time Windows
Partial Path Column Generation for the Vehicle Routing Problem with Time Windows Bjørn Petersen & Mads Kehlet Jepsen } DIKU Department of Computer Science, University of Copenhagen Universitetsparken 1,
More informationLarge-scale optimization and decomposition methods: outline. Column Generation and Cutting Plane methods: a unified view
Large-scale optimization and decomposition methods: outline I Solution approaches for large-scaled problems: I Delayed column generation I Cutting plane methods (delayed constraint generation) 7 I Problems
More informationAn Integer Cutting-Plane Procedure for the Dantzig-Wolfe Decomposition: Theory
An Integer Cutting-Plane Procedure for the Dantzig-Wolfe Decomposition: Theory by Troels Martin Range Discussion Papers on Business and Economics No. 10/2006 FURTHER INFORMATION Department of Business
More informationInteger program reformulation for robust branch-and-cut-and-price
Integer program reformulation for robust branch-and-cut-and-price Marcus Poggi de Aragão Informática PUC-Rio Eduardo Uchoa Engenharia de Produção Universidade Federal Fluminense Outline of the talk Robust
More informationA Column Generation Based Heuristic for the Dial-A-Ride Problem
A Column Generation Based Heuristic for the Dial-A-Ride Problem Nastaran Rahmani 1, Boris Detienne 2,3, Ruslan Sadykov 3,2, François Vanderbeck 2,3 1 Kedge Business School, 680 Cours de la Libération,
More information3.10 Column generation method
3.10 Column generation method Many relevant decision-making (discrete optimization) problems can be formulated as ILP problems with a very large (exponential) number of variables. Examples: cutting stock,
More informationPart 4. Decomposition Algorithms
In the name of God Part 4. 4.4. Column Generation for the Constrained Shortest Path Problem Spring 2010 Instructor: Dr. Masoud Yaghini Constrained Shortest Path Problem Constrained Shortest Path Problem
More informationPartial Path Column Generation for the Elementary Shortest Path Problem with Resource Constraints
Partial Path Column Generation for the Elementary Shortest Path Problem with Resource Constraints Mads Kehlet Jepsen & Bjørn Petersen Department of Computer Science, University of Copenhagen Universitetsparken
More informationColumn Generation for Extended Formulations
1 / 28 Column Generation for Extended Formulations Ruslan Sadykov 1 François Vanderbeck 2,1 1 INRIA Bordeaux Sud-Ouest, France 2 University Bordeaux I, France ISMP 2012 Berlin, August 23 2 / 28 Contents
More informationA Capacity Scaling Procedure for the Multi-Commodity Capacitated Network Design Problem. Ryutsu Keizai University Naoto KATAYAMA
A Capacity Scaling Procedure for the Multi-Commodity Capacitated Network Design Problem Ryutsu Keizai University Naoto KATAYAMA Problems 2006 1 Multi-Commodity Network Design Problem The basic model for
More information3.10 Column generation method
3.10 Column generation method Many relevant decision-making problems can be formulated as ILP problems with a very large (exponential) number of variables. Examples: cutting stock, crew scheduling, vehicle
More informationModels and Cuts for the Two-Echelon Vehicle Routing Problem
Models and Cuts for the Two-Echelon Vehicle Routing Problem Guido Perboli Roberto Tadei Francesco Masoero Department of Control and Computer Engineering, Politecnico di Torino Corso Duca degli Abruzzi,
More informationDecomposition-based Methods for Large-scale Discrete Optimization p.1
Decomposition-based Methods for Large-scale Discrete Optimization Matthew V Galati Ted K Ralphs Department of Industrial and Systems Engineering Lehigh University, Bethlehem, PA, USA Départment de Mathématiques
More informationLecture 9: Dantzig-Wolfe Decomposition
Lecture 9: Dantzig-Wolfe Decomposition (3 units) Outline Dantzig-Wolfe decomposition Column generation algorithm Relation to Lagrangian dual Branch-and-price method Generated assignment problem and multi-commodity
More informationRobust Optimization for the Vehicle Routing Problem with Multiple Deliverymen
Robust Optimization for the Vehicle Routing Problem with Multiple Deliverymen Jonathan De La Vega, Pedro Munari and Reinaldo Morabito Production Engineering Department, Federal University of São Carlos,
More informationStabilized Branch-and-cut-and-price for the Generalized Assignment Problem
Stabilized Branch-and-cut-and-price for the Generalized Assignment Problem Alexandre Pigatti, Marcus Poggi de Aragão Departamento de Informática, PUC do Rio de Janeiro {apigatti, poggi}@inf.puc-rio.br
More informationColumn Generation. MTech Seminar Report. Soumitra Pal Roll No: under the guidance of
Column Generation MTech Seminar Report by Soumitra Pal Roll No: 05305015 under the guidance of Prof. A. G. Ranade Computer Science and Engineering IIT-Bombay a Department of Computer Science and Engineering
More informationExtended Formulations, Lagrangian Relaxation, & Column Generation: tackling large scale applications
Extended Formulations, Lagrangian Relaxation, & Column Generation: tackling large scale applications François Vanderbeck University of Bordeaux INRIA Bordeaux-Sud-Ouest part : Defining Extended Formulations
More informationNetwork Flows. 6. Lagrangian Relaxation. Programming. Fall 2010 Instructor: Dr. Masoud Yaghini
In the name of God Network Flows 6. Lagrangian Relaxation 6.3 Lagrangian Relaxation and Integer Programming Fall 2010 Instructor: Dr. Masoud Yaghini Integer Programming Outline Branch-and-Bound Technique
More informationColumn Generation for Bi-Objective Vehicle Routing Problems with a Min-Max Objective
Author manuscript, published in "ATMOS - 13th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems - 2013 33 (2013) 137--149" DOI : 10.4230/OASIcs.ATMOS.2013.137 Column
More informationOn the exact solution of a large class of parallel machine scheduling problems
1 / 23 On the exact solution of a large class of parallel machine scheduling problems Teobaldo Bulhões 2 Ruslan Sadykov 1 Eduardo Uchoa 2 Anand Subramanian 3 1 Inria Bordeaux and Univ. Bordeaux, France
More informationCut-First Branch-and-Price-Second for the CARP Workshop on Large Scale Optimization 2012 Vevey, Switzerland
Cut-First Branch-and-Price-Second for the CARP Workshop on Large Scale Optimization 2012 Vevey, Switzerland Claudia Bode and Stefan Irnich {claudia.bode,irnich}@uni-mainz.de Chair for Logistics Management
More informationOutline. Relaxation. Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING. 1. Lagrangian Relaxation. Lecture 12 Single Machine Models, Column Generation
Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING 1. Lagrangian Relaxation Lecture 12 Single Machine Models, Column Generation 2. Dantzig-Wolfe Decomposition Dantzig-Wolfe Decomposition Delayed Column
More informationA Hub Location Problem with Fully Interconnected Backbone and Access Networks
A Hub Location Problem with Fully Interconnected Backbone and Access Networks Tommy Thomadsen Informatics and Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby Denmark tt@imm.dtu.dk
More informationAn Integrated Column Generation and Lagrangian Relaxation for Flowshop Scheduling Problems
Proceedings of the 2009 IEEE International Conference on Systems, Man, and Cybernetics San Antonio, TX, USA - October 2009 An Integrated Column Generation and Lagrangian Relaxation for Flowshop Scheduling
More informationFeasibility Pump Heuristics for Column Generation Approaches
1 / 29 Feasibility Pump Heuristics for Column Generation Approaches Ruslan Sadykov 2 Pierre Pesneau 1,2 Francois Vanderbeck 1,2 1 University Bordeaux I 2 INRIA Bordeaux Sud-Ouest SEA 2012 Bordeaux, France,
More informationCut and Column Generation
F A C U L T Y O F S C I E N C E U N I V E R S I T Y O F C O P E N H A G E N PhD thesis Simon Spoorendonk Cut and Column Generation Academic advisor: David Pisinger Submitted: 31/10/08 Preface This Ph.D.
More informationA Node-Flow Model for 1D Stock Cutting: Robust Branch-Cut-and-Price
A Node-Flow Model for 1D Stock Cutting: Robust Branch-Cut-and-Price Gleb Belov University of Dresden Adam N. Letchford Lancaster University Eduardo Uchoa Universidade Federal Fluminense August 4, 2011
More informationIntroduction to Bin Packing Problems
Introduction to Bin Packing Problems Fabio Furini March 13, 2015 Outline Origins and applications Applications: Definition: Bin Packing Problem (BPP) Solution techniques for the BPP Heuristic Algorithms
More informationBenders Decomposition Methods for Structured Optimization, including Stochastic Optimization
Benders Decomposition Methods for Structured Optimization, including Stochastic Optimization Robert M. Freund April 29, 2004 c 2004 Massachusetts Institute of echnology. 1 1 Block Ladder Structure We consider
More informationSolving Elementary Shortest-Path Problems as Mixed-Integer Programs
Gutenberg School of Management and Economics Discussion Paper Series Solving Elementary Shortest-Path Problems as Mixed-Integer Programs Michael Drexl and Stefan Irnich Januar 2012 Discussion paper number
More informationLagrangian Relaxation in MIP
Lagrangian Relaxation in MIP Bernard Gendron May 28, 2016 Master Class on Decomposition, CPAIOR2016, Banff, Canada CIRRELT and Département d informatique et de recherche opérationnelle, Université de Montréal,
More informationA Mixed-Integer Linear Program for the Traveling Salesman Problem with Structured Time Windows
A Mixed-Integer Linear Program for the Traveling Salesman Problem with Structured Time Windows Philipp Hungerländer Christian Truden 5th January 2017 Abstract In this extended abstract we introduce the
More informationto work with) can be solved by solving their LP relaxations with the Simplex method I Cutting plane algorithms, e.g., Gomory s fractional cutting
Summary so far z =max{c T x : Ax apple b, x 2 Z n +} I Modeling with IP (and MIP, and BIP) problems I Formulation for a discrete set that is a feasible region of an IP I Alternative formulations for the
More informationBenders Decomposition for the Uncapacitated Multicommodity Network Design Problem
Benders Decomposition for the Uncapacitated Multicommodity Network Design Problem 1 Carlos Armando Zetina, 1 Ivan Contreras, 2 Jean-François Cordeau 1 Concordia University and CIRRELT, Montréal, Canada
More informationDecomposition methods in optimization
Decomposition methods in optimization I Approach I: I Partition problem constraints into two groups: explicit and implicit I Approach II: I Partition decision variables into two groups: primary and secondary
More informationAn Adaptive Partition-based Approach for Solving Two-stage Stochastic Programs with Fixed Recourse
An Adaptive Partition-based Approach for Solving Two-stage Stochastic Programs with Fixed Recourse Yongjia Song, James Luedtke Virginia Commonwealth University, Richmond, VA, ysong3@vcu.edu University
More informationInteger Programming ISE 418. Lecture 8. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 8 Dr. Ted Ralphs ISE 418 Lecture 8 1 Reading for This Lecture Wolsey Chapter 2 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.4.1, II.4.2, II.5.4 Duality for Mixed-Integer
More informationClassification of Dantzig-Wolfe Reformulations for MIP s
Classification of Dantzig-Wolfe Reformulations for MIP s Raf Jans Rotterdam School of Management HEC Montreal Workshop on Column Generation Aussois, June 2008 Outline and Motivation Dantzig-Wolfe reformulation
More informationA Horizon Decomposition approach for the Capacitated Lot-Sizing Problem with Setup Times
Submitted to INFORMS Journal on Computing manuscript (Please, provide the mansucript number!) A Horizon Decomposition approach for the Capacitated Lot-Sizing Problem with Setup Times Fragkos Ioannis Rotterdam
More informationApplications. Stephen J. Stoyan, Maged M. Dessouky*, and Xiaoqing Wang
Introduction to Large-Scale Linear Programming and Applications Stephen J. Stoyan, Maged M. Dessouky*, and Xiaoqing Wang Daniel J. Epstein Department of Industrial and Systems Engineering, University of
More informationAN INTEGRATED COLUMN GENERATION AND LAGRANGIAN RELAXATION FOR SOLVING FLOWSHOP PROBLEMS TO MINIMIZE THE TOTAL WEIGHTED TARDINESS
International Journal of Innovative Computing, Information and Control ICIC International c 2011 ISSN 1349-4198 Volume 7, Number 11, November 2011 pp. 6453 6471 AN INTEGRATED COLUMN GENERATION AND LAGRANGIAN
More informationBenders Decomposition
Benders Decomposition Yuping Huang, Dr. Qipeng Phil Zheng Department of Industrial and Management Systems Engineering West Virginia University IENG 593G Nonlinear Programg, Spring 2012 Yuping Huang (IMSE@WVU)
More informationMulticommodity Flows and Column Generation
Lecture Notes Multicommodity Flows and Column Generation Marc Pfetsch Zuse Institute Berlin pfetsch@zib.de last change: 2/8/2006 Technische Universität Berlin Fakultät II, Institut für Mathematik WS 2006/07
More informationVehicle Routing and MIP
CORE, Université Catholique de Louvain 5th Porto Meeting on Mathematics for Industry, 11th April 2014 Contents: The Capacitated Vehicle Routing Problem Subproblems: Trees and the TSP CVRP Cutting Planes
More informationis called an integer programming (IP) problem. model is called a mixed integer programming (MIP)
INTEGER PROGRAMMING Integer Programming g In many problems the decision variables must have integer values. Example: assign people, machines, and vehicles to activities in integer quantities. If this is
More informationOutline. Outline. Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING. 1. Scheduling CPM/PERT Resource Constrained Project Scheduling Model
Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING Lecture 3 and Mixed Integer Programg Marco Chiarandini 1. Resource Constrained Project Model 2. Mathematical Programg 2 Outline Outline 1. Resource Constrained
More informationStrengthened Benders Cuts for Stochastic Integer Programs with Continuous Recourse
Strengthened Benders Cuts for Stochastic Integer Programs with Continuous Recourse Merve Bodur 1, Sanjeeb Dash 2, Otay Günlü 2, and James Luedte 3 1 Department of Mechanical and Industrial Engineering,
More informationIntroduction to Mathematical Programming IE406. Lecture 21. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 21 Dr. Ted Ralphs IE406 Lecture 21 1 Reading for This Lecture Bertsimas Sections 10.2, 10.3, 11.1, 11.2 IE406 Lecture 21 2 Branch and Bound Branch
More informationResource Constrained Project Scheduling Linear and Integer Programming (1)
DM204, 2010 SCHEDULING, TIMETABLING AND ROUTING Lecture 3 Resource Constrained Project Linear and Integer Programming (1) Marco Chiarandini Department of Mathematics & Computer Science University of Southern
More informationImproved BCP for Capacitated Vehicle Routing
Improved Branch-Cut-and-Price for Capacitated Vehicle Routing Diego Pecin Artur Pessoa Marcus Poggi Eduardo Uchoa PUC - Rio de Janeiro Universidade Federal Fluminense January, 0 Aussois-0 Pecin, Pessoa,
More informationPartial Path Column Generation for the Vehicle Routing Problem
Partial Path Column Generation for the Vehicle Routing Problem Report 12.2009 DTU Management Engineering Mads Kehlet Jepsen Bjørn Petersen November 2009 Partial Path Column Generation for the Vehicle Routing
More informationInteger linear programming models for a cement delivery problem
Integer linear programming models for a cement delivery problem Alain Hertz Département de mathématiques et de génie industriel École Polytechnique de Montréal alain.hertz@gerad.ca Marc Uldry and Marino
More informationParallel PIPS-SBB Multi-level parallelism for 2-stage SMIPS. Lluís-Miquel Munguia, Geoffrey M. Oxberry, Deepak Rajan, Yuji Shinano
Parallel PIPS-SBB Multi-level parallelism for 2-stage SMIPS Lluís-Miquel Munguia, Geoffrey M. Oxberry, Deepak Rajan, Yuji Shinano ... Our contribution PIPS-PSBB*: Multi-level parallelism for Stochastic
More informationLogic, Optimization and Data Analytics
Logic, Optimization and Data Analytics John Hooker Carnegie Mellon University United Technologies Research Center, Cork, Ireland August 2015 Thesis Logic and optimization have an underlying unity. Ideas
More informationDevelopment of the new MINLP Solver Decogo using SCIP - Status Report
Development of the new MINLP Solver Decogo using SCIP - Status Report Pavlo Muts with Norman Breitfeld, Vitali Gintner, Ivo Nowak SCIP Workshop 2018, Aachen Table of contents 1. Introduction 2. Automatic
More informationSection Notes 9. Midterm 2 Review. Applied Math / Engineering Sciences 121. Week of December 3, 2018
Section Notes 9 Midterm 2 Review Applied Math / Engineering Sciences 121 Week of December 3, 2018 The following list of topics is an overview of the material that was covered in the lectures and sections
More informationColumn Generation in Integer Programming with Applications in Multicriteria Optimization
Column Generation in Integer Programming with Applications in Multicriteria Optimization Matthias Ehrgott Department of Engineering Science The University of Auckland, New Zealand email: m.ehrgott@auckland.ac.nz
More informationIntroduction to integer programming II
Introduction to integer programming II Martin Branda Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Computational Aspects of Optimization
More informationColumn Generation. i = 1,, 255;
Column Generation The idea of the column generation can be motivated by the trim-loss problem: We receive an order to cut 50 pieces of.5-meter (pipe) segments, 250 pieces of 2-meter segments, and 200 pieces
More informationAn Optimization-Based Heuristic for the Split Delivery Vehicle Routing Problem
An Optimization-Based Heuristic for the Split Delivery Vehicle Routing Problem Claudia Archetti (1) Martin W.P. Savelsbergh (2) M. Grazia Speranza (1) (1) University of Brescia, Department of Quantitative
More informationBranch-and-Price algorithm for Vehicle Routing Problem: tutorial
Branch-and-Price algorithm for Vehicle Routing Problem: tutorial Kyuree AHN Department of Industrial and Systems Engineering KAIST, Republic of Korea Friday, May 17, 2017 Presentation Overview Problem
More informationThe two-machine flowshop total completion time problem: A branch-and-bound based on network-flow formulation
The two-machine flowshop total completion time problem: A branch-and-bound based on network-flow formulation Boris Detienne 1, Ruslan Sadykov 1, Shunji Tanaka 2 1 : Team Inria RealOpt, University of Bordeaux,
More informationA generic view of Dantzig Wolfe decomposition in mixed integer programming
Operations Research Letters 34 (2006) 296 306 Operations Research Letters www.elsevier.com/locate/orl A generic view of Dantzig Wolfe decomposition in mixed integer programming François Vanderbeck a,,
More informationKNAPSACK PROBLEMS WITH SETUPS
7 e Conférence Francophone de MOdélisation et SIMulation - MOSIM 08 - du 31 mars au 2 avril 2008 - Paris - France Modélisation, Optimisation et Simulation des Systèmes : Communication, Coopération et Coordination
More informationLinear Programming Duality
Summer 2011 Optimization I Lecture 8 1 Duality recap Linear Programming Duality We motivated the dual of a linear program by thinking about the best possible lower bound on the optimal value we can achieve
More informationReformulation and Decomposition of Integer Programs
Reformulation and Decomposition of Integer Programs François Vanderbeck 1 and Laurence A. Wolsey 2 (Reference: CORE DP 2009/16) (1) Université Bordeaux 1 & INRIA-Bordeaux (2) Université de Louvain, CORE.
More informationA Priori Route Evaluation for the Lateral Transhipment Problem (ARELTP) with Piecewise Linear Profits
1 / 47 A Priori Route Evaluation for the Lateral Transhipment Problem (ARELTP) with Piecewise Linear Profits Martin Romauch 1 Richard Hartl 1 Thibaut Vidal 2 1 University of Vienna 2 PUC-Rio, Rio de Janeiro,
More informationDecomposition Methods for Integer Programming
Decomposition Methods for Integer Programming J.M. Valério de Carvalho vc@dps.uminho.pt Departamento de Produção e Sistemas Escola de Engenharia, Universidade do Minho Portugal PhD Course Programa Doutoral
More informationA computational study of enhancements to Benders Decomposition in uncapacitated multicommodity network design
A computational study of enhancements to Benders Decomposition in uncapacitated multicommodity network design 1 Carlos Armando Zetina, 1 Ivan Contreras, 2 Jean-François Cordeau 1 Concordia University and
More informationTotally unimodular matrices. Introduction to integer programming III: Network Flow, Interval Scheduling, and Vehicle Routing Problems
Totally unimodular matrices Introduction to integer programming III: Network Flow, Interval Scheduling, and Vehicle Routing Problems Martin Branda Charles University in Prague Faculty of Mathematics and
More informationAn Exact Algorithm for the Steiner Tree Problem with Delays
Electronic Notes in Discrete Mathematics 36 (2010) 223 230 www.elsevier.com/locate/endm An Exact Algorithm for the Steiner Tree Problem with Delays Valeria Leggieri 1 Dipartimento di Matematica, Università
More informationSolution of Large-scale LP Problems Using MIP Solvers: Repeated Assignment Problem
The Eighth International Symposium on Operations Research and Its Applications (ISORA 09) Zhangjiajie, China, September 20 22, 2009 Copyright 2009 ORSC & APORC, pp. 190 197 Solution of Large-scale LP Problems
More informationAn improved Benders decomposition algorithm for the logistics facility location problem with capacity expansions
DOI 10.1007/s10479-011-1050-9 An improved Benders decomposition algorithm for the logistics facility location problem with capacity expansions Lixin Tang Wei Jiang Georgios K.D. Saharidis Springer Science+Business
More informationOn the Approximate Linear Programming Approach for Network Revenue Management Problems
On the Approximate Linear Programming Approach for Network Revenue Management Problems Chaoxu Tong School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14853,
More informationLecture 8: Column Generation
Lecture 8: Column Generation (3 units) Outline Cutting stock problem Classical IP formulation Set covering formulation Column generation A dual perspective 1 / 24 Cutting stock problem 2 / 24 Problem description
More informationImprovements to Benders' decomposition: systematic classification and performance comparison in a Transmission Expansion Planning problem
Improvements to Benders' decomposition: systematic classification and performance comparison in a Transmission Expansion Planning problem Sara Lumbreras & Andrés Ramos July 2013 Agenda Motivation improvement
More informationThe Stochastic Vehicle Routing Problem
The Stochastic Vehicle Routing Problem Research Paper Vrije Universiteit Amsterdam Faculty of Sciences Study programme Business Analytics De Boelelaan 08a 08 HV Amsterdam Dimitry Erkin, email:dimitry.erkin@gmail.com
More informationInteger Linear Programming Modeling
DM554/DM545 Linear and Lecture 9 Integer Linear Programming Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. Assignment Problem Knapsack Problem
More informationWeighted Acyclic Di-Graph Partitioning by Balanced Disjoint Paths
Weighted Acyclic Di-Graph Partitioning by Balanced Disjoint Paths H. Murat AFSAR Olivier BRIANT Murat.Afsar@g-scop.inpg.fr Olivier.Briant@g-scop.inpg.fr G-SCOP Laboratory Grenoble Institute of Technology
More informationDecomposition and Reformulation in Integer Programming
and Reformulation in Integer Programming Laurence A. WOLSEY 7/1/2008 / Aussois and Reformulation in Integer Programming Outline 1 Resource 2 and Reformulation in Integer Programming Outline Resource 1
More informationwhere X is the feasible region, i.e., the set of the feasible solutions.
3.5 Branch and Bound Consider a generic Discrete Optimization problem (P) z = max{c(x) : x X }, where X is the feasible region, i.e., the set of the feasible solutions. Branch and Bound is a general semi-enumerative
More informationIntroduction into Vehicle Routing Problems and other basic mixed-integer problems
Introduction into Vehicle Routing Problems and other basic mixed-integer problems Martin Branda Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical
More informationPricing Routines for Vehicle Routing with Time Windows on Road Networks
Pricing Routines for Vehicle Routing with Time Windows on Road Networks Adam N. Letchford Saeideh D. Nasiri Amar Oukil Published in Computers & Operations Research, July 2014 Abstract Several very effective
More informationInteger Programming ISE 418. Lecture 16. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 16 Dr. Ted Ralphs ISE 418 Lecture 16 1 Reading for This Lecture Wolsey, Chapters 10 and 11 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.3.7, II.5.4 CCZ Chapter 8
More informationA Branch-and-Cut Algorithm for the Stochastic Uncapacitated Lot-Sizing Problem
Yongpei Guan 1 Shabbir Ahmed 1 George L. Nemhauser 1 Andrew J. Miller 2 A Branch-and-Cut Algorithm for the Stochastic Uncapacitated Lot-Sizing Problem December 12, 2004 Abstract. This paper addresses a
More information1 Solution of a Large-Scale Traveling-Salesman Problem... 7 George B. Dantzig, Delbert R. Fulkerson, and Selmer M. Johnson
Part I The Early Years 1 Solution of a Large-Scale Traveling-Salesman Problem............ 7 George B. Dantzig, Delbert R. Fulkerson, and Selmer M. Johnson 2 The Hungarian Method for the Assignment Problem..............
More informationBenders Decomposition Methods for Structured Optimization, including Stochastic Optimization
Benders Decomposition Methods for Structured Optimization, including Stochastic Optimization Robert M. Freund May 2, 2001 Block Ladder Structure Basic Model minimize x;y c T x + f T y s:t: Ax = b Bx +
More informationChapter 3: Discrete Optimization Integer Programming
Chapter 3: Discrete Optimization Integer Programming Edoardo Amaldi DEIB Politecnico di Milano edoardo.amaldi@polimi.it Website: http://home.deib.polimi.it/amaldi/opt-16-17.shtml Academic year 2016-17
More informationColumn generation for extended formulations
EURO J Comput Optim (2013) 1:81 115 DOI 10.1007/s13675-013-0009-9 ORIGINAL PAPER Column generation for extended formulations Ruslan Sadykov François Vanderbeck Received: 5 May 2012 / Accepted: 21 December
More information