A Multi-Exchange Heuristic for the Single-Source Capacitated Facility Location Problem

The fleet-assignment model (FAM) for an airline assigns fleet types to the set of flight legs that satisfies a variety of constraints and minimizes the cost of the assignment. A through connection at a station is a connection between an arrival flight and a departure flight at the station, both of which have the same fleet type assigned to them, which ensures that the same plane flies both legs. Typically, passengers are willing to pay a premium for through connections. The through-assignment model (TAM) identifies a set of profitable throughs between arrival and departure flights flown by the same fleet type at each station to maximize the through benefits. TAM is usually solved after obtaining the solution from FAM. In this sequential approach, TAM cannot change the fleeting to get a better through assignment, and FAM does not take into account the through benefits. The goal of the combined through-fleet-assignment model (ctFAM) is to come up with a fleeting and through assignment that achieves the maximum combined benefit of the integrated model. We give a mixed integer-programming (MIP) formulation of ctFAM that is too large to be solved to even near optimality within allowable time for the data obtained by a major U.S. airline. We thus focus on neighborhood search algorithms for solving ctFAM, in which we start with the solution obtained by the previous sequential approach (that is, solving FAM first, followed by TAM) and improve it successively. Our approach is based on generalizing the swap-based neighborhood search approach of Talluri (1996) for FAM, which proceeds by swapping the fleet assignment of two flight paths flown by two different plane types that originate and terminate at the same stations and the same times. An important feature of our approach is that the size of our neighborhood is very large; hence the suggested algorithm is in the category of very large-scale neighborhood (VLSN) search algorithms. Another important feature of our approach is that we use integer programming to identify improved neighbors. We provide computational results that indicate that the neighborhood search approach for ctFAM provides substantial savings over the sequential approach of solving FAM and TAM.

A Very Large-Scale Neighborhood Search Algorithm for the Combined Through-Fleet-Assignment Model

Optimization problems solved using very large scale neighborhood (VLSN) search algorithms include scheduling problems, the capacitated minimum spanning tree problem, the traveling salesman problem, and weapon-target assignment (WTA). This correspondence paper presents an enhanced VLSN search algorithm for obtaining feasible solutions and constructing improvement graphs. This enhanced VLSN search algorithm solves the constrained WTA (CWTA) problem, in which the number of interceptors available to each weapon and the number of interceptors allowed to fire at each target have upper bounds. The proposed enhanced VLSN search algorithm can solve a CWTA problem with 100 targets and 100 weapons (where the upper bound on the number of interceptors for each weapon is one and both the lower and upper bounds on the number of interceptors for each target are equal to one) within an average of 3 s. This study demonstrates that the proposed Enhanced VLSN is superior to existing approaches.

The past few decades have witnessed numerous applications of operations research in logistics, and these applications have resulted in substantial cost savings. However, the U.S. railroad industry has not benefited from the advances, and most of the planning and scheduling processes do not use modeling and optimization. Indeed, most of the planning and scheduling problems arising in railroads, which involve billions of dollars of resources annually, are currently being solved manually. The main reason for not using OR models and methodologies is the mathematical difficulty of these problems, which prevented the development of decision tools that railroads can use to obtain implementable solutions. However, now this situation is gradually changing. We are developing cutting-edge operations research algorithms, by using state-of-the-art ideas from linear and integer programming, network flows, discrete optimization, heuristics, and very large-scale neighborhood (VLSN) search, that railroads have already started using and from which they have started deriving immense benefits. This chapter gives an overview of the railroad planning and scheduling problems, including the railroad blocking problem, train scheduling problem, yard location problem, train dispatching problem, locomotive scheduling problem, and crew scheduling problem. Some of these problems are very large-scale integer programming problems containing billions or even trillions of integer variables. We will describe algorithms that can solve these problems to near-optimality within one to two hours of computational time. We present computational results of these algorithms on the data provided by several U.S. railroads, demonstrating potential benefits from tens to hundreds of millions annually.

Tariff concessions in production sourcing

An iterated tabu search heuristic for the Single Source Capacitated Facility Location Problem

A branch-and-price algorithm for the robust single-source capacitated facility location problem under demand uncertainty

We address the problem of accelerating column generation (CG) for set-covering formulations via dual optimal inequalities (DOIs). We study two novel classes of DOIs, which are referred to as Flexible DOIs (F-DOIs) and Smooth-DOIs (S-DOIs), respectively (and jointly as SF-DOIs). F-DOIs provide rebates for covering items more than necessary. S-DOIs describe the payment of a penalty to permit the undercoverage of items in exchange for the overinclusion of other items. Unlike other classes of DOIs from the literature, the S-DOIs and F-DOIs rely on very little problem-specific knowledge and, as such, have the potential to be applied to a vast number of problem domains. In particular, we discuss the application of the new DOIs to three relevant problems: the single-source capacitated facility location problem, the capacitated p-median problem, and the capacitated vehicle-routing problem. We provide computational evidence of the strength of the new inequalities by embedding them within a column-generation solver for these problems. Substantial speedups can be observed as when compared with a nonstabilized variant of the same CG procedure to achieve the linear-relaxation lower bound on problems with dense columns and structured assignment costs.

Two key issues in local search algorithms are the definition of a neighborhood and the way to examine it. In this chapter we consider techniques for examining very large neighborhoods, in particular, ways for exactly searching them. We first illustrate such techniques using three paradigmatic examples. In the largest part of the chapter, we focus on the development and experimental study of very largescale neighborhood search algorithms for two coloring problems. The first example concerns the well-known (vertex) graph coloring problem. Despite initial promising results on the use of very large-scale neighborhoods, our final conclusion was negative: the usage of the proposed very large-scale neighborhoods did not help to improve the performance of effective stochastic local search algorithms. The second example, the graph set T-coloring problem, yielded more positive results. In this case, a very large-scale neighborhood that was specially tailored for this problem and that can be efficiently searched, resulted to be an essential component of a new state-of-the-art algorithm for various instance classes.

Very large-scale neighborhood search

The capacitated constraint is a common variant in the field of combinatorial optimization. In the capcitated facility location problem, there is a limitation of service for each facility, that is, capacity. From the view of the approximation algorithm, this variant makes the algorithm difficult to design. Actually, the integral gap of the standard linear program for the capacitated facility location problem is infinite. That is, there is no LP-based constant approximation algorithm. In this work, we consider a special case of the uniform open cost for the capacitated facility location problem. Moreover, all clients are needed to be served. There is another variant of facility location problem called as “robust”. In this paper, we also consider one of the robust forms, named cpacitated facility location problem with penalties. We obtain an LP-based 5.732-approximation algorithm for a uniform capacitated facility location problem with penalties.

The Capacitated Facility Location Problem (CFLP) is to locate a set of facilities with capacity constraints, to satisfy at the minimum cost the order-demands of a set of clients. A multi-source version of the problem is considered in which each client can be served by more than one facility. In this paper we present a reformulation of the CFLP based on Mixed Dicut Inequalities, a family of minimum knapsack inequalities of a mixed type, containing both binary and continuous (flow) variables. By aggregating flow variables, any Mixed Dicut Inequality turns into a binary minimum knapsack inequality with a single continuous variable. We will refer to the convex hull of the feasible solutions of this minimum knapsack problem as the Mixed Dicut polytope. We observe that the Mixed Dicut polytope is a rich source of valid inequalities for the CFLP: basic families of valid CFLP inequalities, like Variable Upper Bounds, Cover, Flow Cover and Effective Capacity Inequalities, are valid for the Mixed Dicut polytope. Furthermore we observe that new families of valid inequalities for the CFLP can be derived by the lifting procedures studied for the minimum knapsack problem with a single continuous variable. To deal with large-scale instances, we have developed a Branch-and-Cut-and-Price algorithm, where the separation algorithm consists of the complete enumeration of the facets of the Mixed Dicut polytope for a set of candidate Mixed Dicut Inequalities. We observe that our procedure returns inequalities that dominate most of the known classes of inequalities presented in the literature. We report on computational experience with instances up to 1000 facilities and 1000 clients to validate the approach.

A novel matheuristic approach is presented and tested on a well-known optimisation problem, namely, capacitated facility location problem (CFLP). The algorithm combines local search and mathematical programming. While the local search algorithm is used to select a subset of promising facilities, mathematical programming strategies are used to solve the subproblem to optimality. Proposed local search is influenced by instance-specific information such as installation cost and the distance between customers and facilities. The algorithm is tested on large instances of the CFLP, where neither local search nor mathematical programming is able to find good quality solutions within acceptable computational times. Our approach is shown to be a very competitive alternative to solve large-scale instances for the CFLP.

This article considers a variant of the capacitated facility location problem, where each customer has a non-negative demand and each facility has a non-negative capacity limit and a service distance beyond which it cannot serve customers. A fixed number of facilities are opened from potential locations. The open facilities can only provide service to the customers incurring service costs. Due to the capacity limits and service distance of facilities, all the customers may not be serviced by the open facilities, and these customers can be outsourced incurring outsourcing costs. Each customer has its own non-negative outsourcing cost. The goal is to minimize the sum of service and outsourcing costs. This problem is termed the Capacitated Facility Location Problem with Service Distance and Outsourcing (CFLPSDO). A 0-1 integer linear programming formulation for CFLPSDO is presented in this article. For experimentation, 45 instances are generated from real-world maximal covering location problem instances. These are solved using the exact solver CPLEX to create benchmark solutions. Then, a customized Artificial Bee Colony (ABC) algorithm-based metaheuristic, with two operators for enhanced search space exploration and exploitation, is employed and found to produce optimal solutions efficiently.

In the universal facility location problem, we are given a set of clients and facilities. Our goal is to find an assignment such that the total connection and facility cost is minimized. The connection cost is proportional to the distance between each client and its assigned facility, whereas the facility cost is a nondecreasing function with respect to the total number of clients assigned to the facility. The universal facility location problem generalizes several classical facility location problems, including the uncapacitated facility location problem and the capacitated facility location problem (both hard and soft capacities). This work considers the universal facility location problem with linear penalties, where each client can be rejected for service with a penalty. The objective is to minimize the total connection, facility and penalty cost. We present a $$(5.83+\epsilon )$$ -approximation local search algorithm for this problem.