The multi-facility min-max Weber problem

An algorithm is presented to solve the problem of the locating a given number of facilities on the plane amongst given customers so that the maximum weighted distance from any facility to the customers it services is minimised. The algorithm successfully overcomes the allocation aspects of this problem by generating partitions of customers using a method originally designed for graph colouring embedded within a modified bisection search. Problems of 50 customers and three facilities can be solved in entirely acceptable computer times.     

Solving the multi-commodity capacitated multi-facility Weber problem using Lagrangean relaxation and a subgradient-like algorithm

The Multi-commodity Capacitated Multi-facility Weber Problem (MCMWP) is concerned with locating I-capacitated facilities in the plane in order to satisfy the demands of J customers for K commodities so that the total transportation cost is minimized. We propose a Lagrangean relaxation scheme and a subgradient-like algorithm based on the relaxation of the capacity and commodity bundle constraints. The resulting subproblem is a variant of the well-known Multi-facility Weber Problem and it can be solved by using column generation and branch-and-price on an equivalent set covering formulation, which is accurate but extremely inefficient. Therefore, we devise different strategies to increase the efficiency. They mainly benefit from the effective usage of the lower and upper bounds on the optimal value of the Lagrangean subproblem. On the basis of extensive computational tests, we can say that they increase the efficiency considerably and result in accurate Lagrangean heuristics.     

Efficient heuristics for the rectilinear distance capacitated multi-facility Weber problem

In this paper, we consider the capacitated multi-facility Weber problem with rectilinear distance. This problem is concerned with locating m capacitated facilities in the Euclidean plane to satisfy the demand of n customers with the minimum total transportation cost. The demand and location of each customer are known a priori and the transportation cost between customers and facilities is proportional to the rectilinear distance separating them. We first give a new mixed integer linear programming formulation of the problem by making use of a well-known necessary condition for the optimal facility locations. We then propose new heuristic solution methods based on this formulation. Computational results on benchmark instances indicate that the new methods can provide very good solutions within a reasonable amount of computation time.     

Solving covering problems and the uncapacitated plant location problem on trees

Given a tree network on n vertices, a neighborhood subtree is defined as the set of all points on the tree within a certain radius of a given point, called the center. It is shown that for any two neighborhood subtrees containing the same endpoint of a longest path in the tree one is contained in the other. This result is then used to obtain O(n²) algorithms for the minimum cost covering problem and the minimum cost operating problem as well as an O(n³) algorithm for the uncapacitated plant location problem on the tree.     

Solving the uncapacitated multiple allocation hub location problem by means of a dual-ascent technique

This paper deals with the uncapacitated multiple allocation hub location problem. The dual problem of a four-indexed formulation is considered and a heuristic method, based on a dual-ascent technique, is designed. This heuristic, which is reinforced with several specifical subroutines and does not require any external linear problem solver, is the core tool embedded in an exact branch-and-bound framework. Besides, the heuristic provides the branch-and-bound algorithm with good lower bounds for the nodes of the branching tree. The results of the computational experience (with the classical CAB and AP data sets) are included, showing the great effectiveness of this approach: instances with up to 120 nodes are solved.     

Solving the deterministic and stochastic uncapacitated facility location problem: from a heuristic to a simheuristic

The uncapacitated facility location problem (UFLP) is a popular combinatorial optimization problem with practical applications in different areas, from logistics to telecommunication networks. While most of the existing work in the literature focuses on minimizing total cost for the deterministic version of the problem, some degree of uncertainty (e.g., in the customers’ demands or in the service costs) should be expected in real-life applications. Accordingly, this paper proposes a simheuristic algorithm for solving the stochastic UFLP (SUFLP), where optimization goals other than the minimum expected cost can be considered. The development of this simheuristic is structured in three stages: (i) first, an extremely fast savings-based heuristic is introduced; (ii) next, the heuristic is integrated into a metaheuristic framework, and the resulting algorithm is tested against the optimal values for the UFLP; and (iii) finally, the algorithm is extended by integrating it with simulation techniques, and the resulting simheuristic is employed to solve the SUFLP. Some numerical experiments contribute to illustrate the potential uses of each of these solving methods, depending on the version of the problem (deterministic or stochastic) as well as on whether or not a real-time solution is required.     

Solving an inverse parametric control problem for one-dimensional maps

The article analytically investigates one-dimensional maps and their properties under periodic perturbations. A method is proposed to find parametric perturbations that stabilize given cycles of one-dimensional maps. By solving the inverse problem we find the existence regions of stable cycles of parametrically perturbed quadratic maps and circle maps of standard form.     

On formulations of the stochastic uncapacitated lot-sizing problem

We consider two formulations of a stochastic uncapacitated lot-sizing problem. We show that by adding (?,S) inequalities to the one with the smaller number of variables, both formulations give the same LP bound. Then we show that for two-period problems, adding another class of inequalities gives the convex hull of integral solutions.     

Evolving hyper-heuristics for the uncapacitated examination timetabling problem

This paper reports on the use of an evolutionary algorithm (EA) to search a space of heuristic combinations for the uncapacitated examination timetabling problem. The representation used by an EA has an effect on the difficulty of the search and hence the overall success of the system. The paper examines three different representations of heuristic combinations for this problem and compares their performance on a set of benchmark problems for the uncapacitated examination timetabling problem. The study has revealed that certain representations do result in a better performance and generalization of the hyper-heuristic. An EA-based hyper-heuristic combining the use of all three representations (CEA) was implemented and found to generalize better than the EA using each of the representations separately.     

Constructive heuristics for the uncapacitated continuous location-allocation problem

The multisource location-allocation problem in continuous space is investigated. Two constructive heuristic techniques are proposed to solve this problem. Both methods are based on designing suitable schemes for the generation of the initial solutions. The first considers the furthest distance rule and is enhanced by schemes borrowed from tabu search such as constructing the forbidden regions and freeing strategy. The second considers the discrete solutions found when solving the p-median problem. Some results on existing test problems are presented.     

A generalized Weiszfeld method for the multi-facility location problem

An iterative method is proposed for the K facilities location problem. The problem is relaxed using probabilistic assignments, depending on the distances to the facilities. The probabilities, that decompose the problem into K single-facility location problems, are updated at each iteration together with the facility locations. The proposed method is a natural generalization of the Weiszfeld method to several facilities.     

A projection method for the uncapacitated facility location problem

Several algorithms already exist for solving the uncapacitated facility location problem. The most efficient are based upon the solution of the strong linear programming relaxation. The dual of this relaxation has a condensed form which consists of minimizing a certain piecewise linear convex function. This paper presents a new method for solving the uncapacitated facility location problem based upon the exact solution of the condensed dual via orthogonal projections. The amount of work per iteration is of the same order as that of a simplex iteration for a linear program inm variables and constraints, wherem is the number of clients. For comparison, the underlying linear programming dual hasmn + m + n variables andmn +n constraints, wheren is the number of potential locations for the facilities. The method is flexible as it can handle side constraints. In particular, when there is a duality gap, the linear programming formulation can be strengthened by adding cuts. Numerical results for some classical test problems are included.     

Lower bounds for the two-stage uncapacitated facility location problem

In the two-stage uncapacitated facility location problem, a set of customers is served from a set of depots which receives the product from a set of plants. If a plant or depot serves a product, a fixed cost must be paid, and there are different transportation costs between plants and depots, and depots and customers. The objective is to locate plants and depots, given both sets of potential locations, such that each customer is served and the total cost is as minimal as possible. In this paper, we present a mixed integer formulation based on twice-indexed transportation variables, and perform an analysis of several Lagrangian relaxations which are obtained from it, trying to determine good lower bounds on its optimal value. Computational results are also presented which support the theoretical potential of one of the relaxations.     

A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem

This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical (ℓ,S) inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the (ℓ,S) inequalities to a general class of valid inequalities, called the inequalities, and we establish necessary and sufficient conditions which guarantee that the inequalities are facet-defining. A separation heuristic for inequalities is developed and incorporated into a branch-and-cut algorithm. A computational study verifies the usefulness of the inequalities as cuts. This research has been supported in part by the National Science Foundation under Award number DMII-0121495.     

Degeneracy in the multi-source Weber problem

Received June 10, 1996 / Revised version received May 20, 1997 Published online October 21, 1998     

Algorithms for the single-source uncapacitated minimum concave-cost network flow problem

We investigate algorithms, applications, and complexity issues for the single-source uncapacitated (SSU) version of the minimum concave-cost network flow problem (MCNFP). We present applications arising from production planning, and prove complexity results for both global and local search. We formally state the local search algorithm of Gallo and Sodini [5], and present alternative local search algorithms. Computational results are provided to compare the various local search algorithms proposed and the effects of initial solution techniques.     

Solving a large nonlinear programming problem on a vector processing computer

In an earlier paper, the authors formulated an acoustic antenna array design problem as a nonlinear program. Computation times for large problems (about 400 sensors) were in the range of 8 to 10 hours on a Vax 11/780. Most of this time was spent in computing the objective function and its gradient. This paper describes how these computations (and these only) were recoded to exploit the vector processing capabilities of a Cray 1-M computer. Run times are reduced to less than one minute. The results have implications for many nonlinear programs whose function evaluations are very time consuming.     

An exact cooperative method for the uncapacitated facility location problem

In this paper, we present a cooperative primal-dual method to solve the uncapacitated facility location problem exactly. It consists of a primal process, which performs a variation of a known and effective tabu search procedure, and a dual process, which performs a lagrangian branch-and-bound search. Both processes cooperate by exchanging information which helps them find the optimal solution. Further contributions include new techniques for improving the evaluation of the branch-and-bound nodes: decision-variable bound tightening rules applied at each node, and a subgradient caching strategy to improve the bundle method applied at each node.     

A note on the Weber location problem

In this note, we collect some interesting and useful results about the Weber problem. We investigate an accelerated Weiszfeld procedure which increases the step size and find a formula for the step size that empirically produces the fastest convergence rate. We also derive an estimate for the optimal cost of the system.