Lagrangian-informed mixed integer programming reformulations

La programmation linéaire en nombres entiers est une approche robuste qui permet de résoudre rapidement de grandes instances de problèmes d'optimisation discrète. Toutefois, les problèmes gagnent constamment en complexité et imposent parfois de fortes limites sur le temps de calcul. Il devient alors nécessaire de développer des méthodes spécialisées afin de résoudre approximativement ces problèmes, tout en calculant des bornes sur leurs valeurs optimales afin de prouver la qualité des solutions obtenues. Nous proposons d'explorer une approche de reformulation en nombres entiers guidée par la relaxation lagrangienne. Après l'identification d'une forte relaxation lagrangienne, un processus systématique permet d'obtenir une seconde formulation en nombres entiers. Cette reformulation, plus compacte que celle de Dantzig et Wolfe, comporte exactement les mêmes solutions entières que la formulation initiale, mais en améliore la borne linéaire: elle devient égale à la borne lagrangienne. L'approche de reformulation permet d'unifier et de généraliser des formulations et des méthodes de borne connues. De plus, elle offre une manière simple d'obtenir des reformulations de moins grandes tailles en contrepartie de bornes plus faibles. Ces reformulations demeurent de grandes tailles. C'est pourquoi nous décrivons aussi des méthodes spécialisées pour en résoudre les relaxations linéaires. Finalement, nous appliquons l'approche de reformulation à deux problèmes de localisation. Cela nous mène à de nouvelles formulations pour ces problèmes; certaines sont de très grandes tailles, mais nos méthodes de résolution spécialisées les rendent pratiques.Integer linear programming is a robust and efficient approach to solve large-scale instances of combinatorial problems. However, problems constantly gain in complexity and sometimes impose strong constraints on computation times. We must then develop specialised methods to compute heuristic primal solutions to the problem and derive lower bounds on the optimal value, and thus prove the quality of our primal solutions. We propose to guide a reformulation approach for mixed integer programs with Lagrangian relaxations. After the identification of a strong relaxation, a mechanical process leads to a second integer formulation. This reformulation is equivalent to the initial one, but its linear relaxation is equivalent to the strong Lagrangian dual. We will show that the reformulation approach unifies and generalises prior formulations and lower bounding approaches, and that it exposes a simple mechanism to reduce the size of reformulations in return for weaker bounds. Nevertheless, our reformulations are large. We address this issue by solving their linear relaxations with specialised methods. Finally, we apply the reformulation approach to two location problems. This yields novel formulations for both problems; some are very large but, thanks to the aforementioned specialised methods, still practical

Comparison of Formulations for the Two-Level Uncapacitated Facility Location Problem with Single Assignment Constraints

International audienceWe consider the two-level uncapacitated facility location problem with single assignment constraints (TUFLP-S), an extension of the uncapacitated facility location problem. We present six mixed-integer programming models for the TUFLP-S based on reformulation techniques and on the relaxation of the integrality of some of the variables associated with location decisions. We compare the models by carrying out extensive computational experiments on large, hard, artificial instances, as well as on instances derived from an industrial application in freight transportation

Formulations, Bounds and Heuristic Methods for a Two-Echelon Adaptive Location-Distribution Problem

We consider a two-echelon location-distribution problem arising from an actual application in fast delivery service. This problem belongs to the class of adaptive logistics problems, as the locations of the facilities (typically, parking spaces) are revised on a daily basis according to demand variations. We present and compare two formulations for this problem: an arc-based model and a path-based model. Since these formulations cannot be solved in reasonable time for large-scale instances, we introduce a heuristic method based on a variable neighborhood search approach

Models and Relaxations for Two-Level Uncapacitated Facility Location with Single-Assignment

National audienceIn this talk, we study and compare several formulations for the two-level uncapacitatedfacility location problem with single assignment constraints (TUFLP-S), an extension of theuncapacitated facility location problem (UFLP) [4]. The UFLP consists in selecting a set ofdepots from potential locations in order to minimize an objective function that includes fixedcosts associated with each depot and transportation costs from any depot to each customer.In the two-level uncapacitated facility location problem (TUFLP), the single set of locations issubstituted with two tiers of locations (depots and satellites), and the path to each customermust begin at a depot and transit by a satellite. The objective function includes fixed costsassociated with the depots and the satellites, fixed costs for establishing connections betweendepots and satellites, and transportation costs from any depot to each customer, i.e., each pathof the form depot-satellite-customer has a corresponding transportation cost. The TUFLP-Simposes the additional restriction that each satellite can be connected to at most one depot.These single assignment constraints appear in a number of applications, most notably in trans-portation [5] and telecommunications [2]. Note also that, for a large class of TUFLP instancesfor which the single assignment constraints are not explicitly enforced, there is an optimalsolution that satisfies these constraints, due to the structure of the objective function [2].First, we present a general formulation for the TUFLP [1], and we adapt this model toderive an initial MIP formulation for the TUFLP-S. We then propose five additional MIPformulations based on reformulation techniques and on the relaxation of the integrality of someof the variables associated with location decisions. One of these formulations was previouslyconsidered in [3] to derive a Lagrangian relaxation for the TUFLP-S. We theoretically comparethe LP relaxations of these models. Then, we compare the models by solving a large numberof various instances with a state-of-the-art MIP solver. The results show that, whenever fixedcosts at the depots (at the satellites) are significant, it is beneficial to keep the integrality of thecorresponding binary variables, but to relax the integrality of the binary variables associatedwith the satellites (with the depots). In our experiments, poor results are obtained by thereformulation that minimizes the number of binary variables by relaxing the integrality of thetwo types of location variables

Variable neighborhood approaches for a multi-echelon capacitated location-distribution problem

International audienc

Models and Relaxations for Two-Level Uncapacitated Facility Location with Single-Assignment

National audienceIn this talk, we study and compare several formulations for the two-level uncapacitatedfacility location problem with single assignment constraints (TUFLP-S), an extension of theuncapacitated facility location problem (UFLP) [4]. The UFLP consists in selecting a set ofdepots from potential locations in order to minimize an objective function that includes fixedcosts associated with each depot and transportation costs from any depot to each customer.In the two-level uncapacitated facility location problem (TUFLP), the single set of locations issubstituted with two tiers of locations (depots and satellites), and the path to each customermust begin at a depot and transit by a satellite. The objective function includes fixed costsassociated with the depots and the satellites, fixed costs for establishing connections betweendepots and satellites, and transportation costs from any depot to each customer, i.e., each pathof the form depot-satellite-customer has a corresponding transportation cost. The TUFLP-Simposes the additional restriction that each satellite can be connected to at most one depot.These single assignment constraints appear in a number of applications, most notably in trans-portation [5] and telecommunications [2]. Note also that, for a large class of TUFLP instancesfor which the single assignment constraints are not explicitly enforced, there is an optimalsolution that satisfies these constraints, due to the structure of the objective function [2].First, we present a general formulation for the TUFLP [1], and we adapt this model toderive an initial MIP formulation for the TUFLP-S. We then propose five additional MIPformulations based on reformulation techniques and on the relaxation of the integrality of someof the variables associated with location decisions. One of these formulations was previouslyconsidered in [3] to derive a Lagrangian relaxation for the TUFLP-S. We theoretically comparethe LP relaxations of these models. Then, we compare the models by solving a large numberof various instances with a state-of-the-art MIP solver. The results show that, whenever fixedcosts at the depots (at the satellites) are significant, it is beneficial to keep the integrality of thecorresponding binary variables, but to relax the integrality of the binary variables associatedwith the satellites (with the depots). In our experiments, poor results are obtained by thereformulation that minimizes the number of binary variables by relaxing the integrality of thetwo types of location variables

Models and Relaxations for Two-Level Uncapacitated Facility Location with Single-Assignment

National audienceIn this talk, we study and compare several formulations for the two-level uncapacitatedfacility location problem with single assignment constraints (TUFLP-S), an extension of theuncapacitated facility location problem (UFLP) [4]. The UFLP consists in selecting a set ofdepots from potential locations in order to minimize an objective function that includes fixedcosts associated with each depot and transportation costs from any depot to each customer.In the two-level uncapacitated facility location problem (TUFLP), the single set of locations issubstituted with two tiers of locations (depots and satellites), and the path to each customermust begin at a depot and transit by a satellite. The objective function includes fixed costsassociated with the depots and the satellites, fixed costs for establishing connections betweendepots and satellites, and transportation costs from any depot to each customer, i.e., each pathof the form depot-satellite-customer has a corresponding transportation cost. The TUFLP-Simposes the additional restriction that each satellite can be connected to at most one depot.These single assignment constraints appear in a number of applications, most notably in trans-portation [5] and telecommunications [2]. Note also that, for a large class of TUFLP instancesfor which the single assignment constraints are not explicitly enforced, there is an optimalsolution that satisfies these constraints, due to the structure of the objective function [2].First, we present a general formulation for the TUFLP [1], and we adapt this model toderive an initial MIP formulation for the TUFLP-S. We then propose five additional MIPformulations based on reformulation techniques and on the relaxation of the integrality of someof the variables associated with location decisions. One of these formulations was previouslyconsidered in [3] to derive a Lagrangian relaxation for the TUFLP-S. We theoretically comparethe LP relaxations of these models. Then, we compare the models by solving a large numberof various instances with a state-of-the-art MIP solver. The results show that, whenever fixedcosts at the depots (at the satellites) are significant, it is beneficial to keep the integrality of thecorresponding binary variables, but to relax the integrality of the binary variables associatedwith the satellites (with the depots). In our experiments, poor results are obtained by thereformulation that minimizes the number of binary variables by relaxing the integrality of thetwo types of location variables

Models, Relaxations and Heuristics for two-level uncapacitated facility location with single-assignment

International audienc

Models, Relaxations and Heuristics for two-level uncapacitated facility location with single-assignment

International audienc