Simple Plant Location Problem Research Articles

Massively parallel analog tabu search using neural networks applied to simple plant location problems

Neural networks and tabu search are two very significant techniques which have emerged recently for the solution of discrete optimization problems. Neural networks possess the desirable quality of implementability in massively parallel hardware while the tabu search metaheuristic shows great promise as a powerful global search method. Tabu Neural Network (TANN) integrates an analog version of the short term memory component of tabu search with neural networks to generate a massively parallel, analog global search strategy that is hardware implementable. In TANN, both the choice of the element to enter the tabu list as well as the maintenance of the decision elements in tabu status is accomplished via neuronal activities. In this paper we apply TANN to the simple plant location problem. Comparisons with the Hopfield-Tank network show an average improvement of about 85% in the quality of solutions obtained.

An algorithm for optimal shipments with given frequencies

This article deals with the problem of minimizing the transportation and inventory cost associated with the shipment of several products from a source to a destination, when a finite set of shipping frequencies is available. A mixed-integer programming model—shown to be NP-hard—is formulated for that problem. The computational complexity of some similar models applied to different problems is also investigated. In particular, whereas the capacitated plant location problem with operational cost in product form is NP-hard, the simple plant location problem with the same characteristics can be solved in polynomial time. A branch-and-bound algorithm is finally worked out, and some computational results are presented. © 1996 John Wiley & Sons, Inc.

Dual search procedures for the exact formulation of the simple plant location problem with spatial interaction

Abstract We present an exact formulation of the simple plant location problem with spatial interaction. The formulation is based on the derivation of the spatially interactive travel behavior, often described through a spatial “gravity” model. By using techniques from separable programming, we can derive an exact formulation of the problem, i.e. without using any approximation of the function that describes the spatially interactive travel behavior. The resulting model is a pure zero-one location model. We also present a dual ascent and adjustment procedure for the linear programming relaxation of the exact formulation of the simple plant location problem with spatial interaction. Furthermore, we present a solution method based on Lagrangean relaxation and subgradient optimization. The two dual search procedures are computationally tested on problems with a maximal size of 50 × 100.

A dual ascent procedure for the exact formulation of the simple plant location problem with spatial interaction

We present a dual ascent and adjustment procedure for the linear programming relaxation of the exact formulation of the simple plant location problem with spatial interaction, which is a pure zero-one location model (an extension of the classical simple plant location problem). The procedure is based on the same principles as the dual ascent method of Erlenkotter for the uncapacitated facility location problem, with some important technical differences. Some computational results are given

FEASIBLE REGION REDUCTION CUTS FOR THE SIMPLE PLANT LOCATION PROBLEM

We introduce a conceptually new method of generating strong cuts for the simple plant location problem. Based on the simple structure of the problem, we generate inequalities which even cut off part of the integer feasible region but still provide valid and sharp lower bounds of the problem when added to its linear programming relaxation. Also shown is the relation between such inequalities and the conventional valid inequalities of the problem.

On some optimization problems on k-trees and partial k-trees

A k-tree is a graph that can be reduced to the k-complete graph by sequentially removing k-degree vertices with completely connected neighbors. Partial k-trees are graphs embeddable in a k-tree with the same vertex set. In this paper we develop efficient algorithms for several path distance optimization problems on partial k-trees, and k-cable distance optimization problems on k-trees. Specifically, we develop a linear time algorithm to find shortest simple paths from a given vertex to all other vertices in a partial k-tree, we compute the diameter of a partial k-tree with equal edge lengths in linear time, and we construct an O( n k + 2 ) algorithm to solve the simple plant location problem in an n-vertex partial k-tree. Then, we analyze some cable distance optimization problems in k-trees. We derive some properties of cable distance in k-trees and we present a new characterization of a k-path in k-trees. Finally, we develop algorithms to solve a certain k-cable decomposition problem in k-trees in O( n 2) time and to compute the k-cable diameter of a k-tree with equal edge lengths in linear time.

Technical Note—On the Equivalence of Dual Methods for Two Location Problems

Separate algorithms have been reported in the literature to find dual solutions to the linear programming relaxations of the simple plant location problem, and the location covering problem. We explore the connection between these two algorithms.

Solving makespan minimization problems with Lagrangean decomposition

The problem of minimizing makespan in a single stage manufacturing process with parallel machines and multiple job types can be decomposed into as many simple plant location problems as there are job types and as many constrained 0–1 knapsack problems as there are machines. None of these subproblems has the integrality property, therefore the Lagrangean dual bound obtained is stronger than the LP relaxation bound as well as the associated Lagrangean relaxation bounds. We show on an example that the bound can indeed be stronger.

Hub facility location with fixed costs

In thep-hub location model, it is usual to ignore the fixed costs of opening facilities. In contrast, the simple plant location problem includes fixed facility costs and thereby makes thenumber of facilities one of the decision variables. This paper introduces fixed facility costs into a hub location model, thereby making the number of hubs a decision variable. As an initial effort toward solving the hub model with fixed costs, a two-step procedure is devised.Step 1 involves an estimate of a good upper bound on the solution, using heuristics.Step 2 involves the computation of a tight lower bound on the solution; as a by-product, an improvement of the best solution from Step 1 may be found. Computational results are discussed for sample problems.

Boolean and Graph Theoretic Formulations of the Simple Plant Location Problem

A formulation of the simple plant location problem as the minimization of a pseudo-Boolean function is transformed into a set covering problem and into a weighted vertex packing problem on a graph, called the “plant location graph.” For the special case of a simple plant location problem derived from a tree network, the plant location graph is shown to be weakly triangulated, so that a maximum weighted vertex packing can be found in polynomial-time by existing algorithms.

On the exact solution of large-scale simple plant location problems

The simple plant location problem deals with the selection of facility sites from a set of possible locations with the intention of serving a given set of customers from the chosen locations. The objective is to minimize the sum of the fixed charges for establishing the facilities and the transportation costs corresponding to the supply of the demands. We show how to modify a primal-dual version of Erlenkotter's exact algorithm to get an improved procedure. Computational experience with large-scale problems indicates that, compared with Erlenkotter's code, an implementation of the modified algorithm is faster by more than one order of magnitude. For real-life problems with 1500 potential facility sites the increase of computational speed may even exceed the factor of 100.

A Lagrangean dual ascent algorithm for simple plant location problems

We propose to strengthen the separable Lagrangean relaxation of the Simple Plant Location Problem (SPLP) by using Benders inequalities generated during a Lagrangean dual ascent procedure. These inequalities are expressed in terms of the 0–1 variables only, and they can be used as knapsack constraints in the pure integer part of the Lagrangean relaxation. We show how coupling this technique with a good primal heuristic can substantially reduce integrality gaps.

The design of bus route systems — An interactive location-allocation approach

An interactive modelling approach is developed to solve the practical problem of bus route network design. Possible bus routes are identified with facilities which can be “located”. Zones or pairs of zones in the urban area are identified with customers who will be “allocated” to the established facilities. It is shown that the classical Set Covering Problem is useful under the assumption of fixed demand; the Simple Plant Location Problem is effective under the assumption of demand which is sensitive to the level of bus service provided.

A facility location problem with clients' preference orderings

The Simple Plant Location Problem (SPLP) consists in identifying a pattern of facilities among a set of potential sites so that the sum of the costs for opening the facilities and for supplying a group of geographically dispersed clients therefrom is minimized. It is posited in this framework that the locator rules on the allocation of the clients to the facilities. When the clients are free to decide upon which facility they will attend, their choices do not necessarily match the most economical supply pattern from the point of view of the locator. We suppose here that each client has a preference ordering on the list of potential sites whereby he determines his allocation out of any set of open facilities. This preference ordering depends on characteristics of the client, of the sites, and of the trip between the client's location and the sites. Mathematically, we translate the orderings into a set of constraints to be appended to the SPLP. This new problem, termed SPLPO, has a wide range of applications as shown by a series of examples. We present an algorithm for solving the SPLPO and extensive computational experiments are reported.

Discrete stochastic location models

In this paper, we study how the two classical location models, the simple plant location problem and thep-median problem, are transformed in a two-stage stochastic program with recourse when uncertainty on demands, variable production and transportation costs, and selling prices is introduced. We also discuss the relation between the stochastic version of the SPLP and the stochastic version of thep-median.

A Cross Decomposition Algorithm for Capacitated Facility Location

We present an implementation of the Cross Decomposition method to solve the capacitated facility location problem. The method unifies Benders Decomposition and Lagrangean relaxation into a single framework that involves successive solutions to a transportation problem and a simple plant location problem. Computational tests on problems with as many as 100 candidate facility locations and 200 customers provide evidence that the approach is superior to several other methods. The specialized Cross Decomposition program is, on a common set of test problems, about ten times faster than several recently developed programs. We also present a simple algorithm to strengthen Benders cuts.

A tree search algorithm for the multi-commodity location problem

The multi-commodity location problem is an extension of the simple plant location problem. The problem is to decide on locations of facilities to meet customer demands for several commodities in such a way that total fixed plus variable costs are minimized. Only one commodity may be supplied from any location. In this paper a primal and a dual heuristic for producing good bounds are presented. A method of improving these bounds by using a new Lagrangean relaxation for the problem is also presented. Computational results with problems taken from the literature are provided.

The simple plant location problem: Survey and synthesis

With emphasis on the simple plant location problem, (SPLP), we consider an important family of discrete, deterministic, single-criterion, NP-hard, and widely applicable optimization problems. The introductory discussion on problem formulation aspects is followed by the establishment of relationships between SPLP and set packing, set covering and set partitioning problems which all are among those structures in integer programming having the most wide-spread applications. An extensive discourse on solution properties and computational techniques, spanning from early heuristics to the presumably most novel exact methods is then provided. Other subjects of concern include a subfamily of SPLP's solvable in polynomial time, analyses of approximate algorithms, transformability of p-CENTER and p-MEDIAN to SPLP, and structural properties of the SPLP polytope. Along the way we attempt to synthesize these findings and relate them to other areas of integer programming.

A Primal Approach to the Simple Plant Location Problem

The most successful algorithms for solving simple plant location problems are presently dual-based procedures. However, primal procedures have distinct practical advantages (e.g., in sensitivity analysis). We propose a primal subgradient algorithm to solve the well-known strong linear programming relaxation of the problem. Typically this algorithm converges very fast to a point whose objective value is close to the integer optimum and where most of the decision variables have been fixed either to 0 or to 1. To fix the values of the remaining variables we use a greedy-interchange algorithm. Thus we propose thiss approach as a heuristic. Computational experience shows that an optimal solution is discovered with high frequency.

Some facets of the simple plant location polytope

We relate the simple plant location problem to the vertex packing problem and derive several classes of facets of their associated integer polytopes.