A solution algorithm for non-convex mixed integer optimization problems with only few continuous variables

Geometric branch-and-bound techniques are well-known solution algorithms for non-convex continuous global optimization problems with box constraints. Several approaches can be found in the literature differing mainly in the bounds used.     

A decomposition approach for facility location and relocation problem with uncertain number of future facilities

In this paper, we discuss two challenges of long term facility location problem that occur simultaneously; future demand change and uncertain number of future facilities. We introduce a mathematical model that minimizes the initial and expected future weighted travel distance of customers. Our model allows relocation for the future instances by closing some of the facilities that were located initially and opening new ones, without exceeding a given budget. We present an integer programming formulation of the problem and develop a decomposition algorithm that can produce near optimal solutions in a fast manner. We compare the performance of our mathematical model against another method adapted from the literature and perform sensitivity analysis. We present numerical results that compare the performance of the proposed decomposition algorithm against the exact algorithm for the problem.     

The directed profitable location Rural Postman Problem

In this paper we introduce an extension of the well known Rural Postman Problem, which combines arc routing with profits and facility location. Profitable arcs must be selected, facilities located at both end-points of the selected arcs, and a tour identified so as to maximize the difference between the profit collected along the arcs and the cost of traversing the arcs and installing the facilities. We analyze properties of the problem, present a mathematical programming formulation and a branch-and-cut algorithm. In an extensive computational experience the algorithm could solve instances with up to 140 vertices and 190 arcs and up to 50 vertices and 203 arcs.     

The Aircraft Maintenance Base Location Problem

Aviation authorities such as the Federal Aviation Administration (FAA) provide stringent guidelines for aircraft maintenance, with violations leading to significant penalties for airlines. Moreover, poorly maintained aircraft can lead to mass cancellation of flights, causing tremendous inconvenience to passengers and resulting in a significant erosion in brand image for the airline in question. Aircraft maintenance operations of a complex and extended nature can only be performed at designated maintenance bases. Aircraft maintenance planning literature has focused on developing good tail-number routing plans, while assuming that the locations of the maintenance bases themselves are fixed. This paper considers an inverse optimization problem, viz., locating a minimal number of maintenance bases on an Euler tour, while ensuring that all required aircraft maintenance activities can be performed with a stipulated periodicity. The Aircraft Maintenance Base Location Problem (AMBLP) is shown to be NP-complete and a new lower bound is developed for the problem. The performance of four simple “quick and dirty” heuristics for obtaining feasible solutions to AMBLP is analyzed.     

A heuristic for BILP problems: The Single Source Capacitated Facility Location Problem

In the Single Source Capacitated Facility Location Problem (SSCFLP) each customer has to be assigned to one facility that supplies its whole demand. The total demand of customers assigned to each facility cannot exceed its capacity. An opening cost is associated with each facility, and is paid if at least one customer is assigned to it. The objective is to minimize the total cost of opening the facilities and supply all the customers. In this paper we extend the Kernel Search heuristic framework to general Binary Integer Linear Programming (BILP) problems, and apply it to the SSCFLP. The heuristic is based on the solution to optimality of a sequence of subproblems, where each subproblem is restricted to a subset of the decision variables. The subsets of decision variables are constructed starting from the optimal values of the linear relaxation. Variants based on variable fixing are proposed to improve the efficiency of the Kernel Search framework. The algorithms are tested on benchmark instances and new very large-scale test problems. Computational results demonstrate the effectiveness of the approach. The Kernel Search algorithm outperforms the best heuristics for the SSCFLP available in the literature. It found the optimal solution for 165 out of the 170 instances with a proven optimum. The error achieved in the remaining instances is negligible. Moreover, it achieved, on 100 new very large-scale instances, an average gap equal to 0.64% computed with respect to a lower bound or the optimum, when available. The variants based on variable fixing improved the efficiency of the algorithm with minor deteriorations of the solution quality.     

A continuous approximation approach for the integrated facility-inventory allocation problem

In today’s retail business many companies have a complex distribution network with several national and regional distribution centers. This article studies an integrated facility location and inventory allocation problem for designing a distribution network with multiple distribution centers and retailers. The key decisions are where to locate the regional distribution centers (RDCs), how to assign retail stores to RDCs and what should be the inventory policy at the different locations such that the total network cost is minimized. Due to the complexity of the problem, a continuous approximation (CA) model is used to represent the network. Nonlinear programming techniques are developed to solve the optimization problems. The main contribution of this work lies in developing a new CA modeling technique when the discrete data cannot be modeled by a continuous function and applying this technique to solve an integrated facility location-allocation and inventory-management problem. Our methodology is illustrated with the network from a leading US retailer. Numerical analysis suggests that the total cost is significantly lower in the case of the integrated model as compared with the non-integrated model, where the location-allocation and inventory-management problems are considered separately. This paper also studies the effects of changing parameter values on the optimal solutions and to point out some management implications.     

A combined facility location and network design problem with multi-type of capacitated links

This article presents a mixed-integer model to optimize the location of facilities and the underlying transportation network at the same time to minimize the total transportation and operating costs. In this problem, it is assumed that for connecting two nodes, there are several types of links in which their capacity, transportation and construction costs are different. The developed model has various applications in telecommunication, emergency, regional planning, pipeline network, energy management, distribution, to just name a few. To solve the model effectively, this paper also proposes a fix-and-optimize heuristic based on the evolutionary fire-fly algorithm. Finally, to validate the model and evaluate the algorithm’s performance, a series of test instances with up to 100 nodes and 600 candidate links with three different levels of quality are reported.     

Competitive facility location and design problem

We develop a spatial interaction model that seeks to simultaneously optimize location and design decisions for a set of new facilities. The facilities compete for customer demand with pre-existing competitive facilities and with each other. The customer demand is assumed to be elastic, expanding as the utility of the service offered by the facilities increases. Increases in the utility can be achieved by increasing the number of facilities, design improvements, or locating facilities closer to the customer.     

From stars to comets: Improved local search for universal facility location

We improve the approximation ratio for the Universal Facility Location Problem to 6.702 by a local search algorithm with an extended pivot operation.