Showing papers in "Annals of Operations Research in 1986"

NLPQL: A fortran subroutine solving constrained nonlinear programming problems

Abstract: NLPQL is a FORTRAN implementation of a sequential quadratic programming method for solving nonlinearly constrained optimization problems with differentiable objective and constraint functions. At each iteration, the search direction is the solution of a quadratic programming subproblem. This paper discusses the organization of NLPQL, including the formulation of the subproblem and the information that must be provided by a user. A summary is given of the performance of different algorithmic options of NLPQL on a collection of test problems (115 hand-selected or application problems, 320 randomly generated problems). The performance of NLPQL is compared with that of some other available codes.

An exact algorithm for solving a capacitated location-routing problem

Abstract: In location-routing problems, the objective is to locate one or many depots within a set of sites (representing customer locations or cities) and to construct delivery routes from the selected depot or depots to the remaining sites at least system cost. The objective function is the sum of depot operating costs, vehicle acquisition costs and routing costs. This paper considers one such problem in which a weight is assigned to each site and where sites are to be visited by vehicles having a given capacity. The solution must be such that the sum of the weights of sites visited on any given route does not exceed the capacity of the visiting vehicle. The formulation of an integer linear program for this problem involves degree constraints, generalized subtour elimination constraints, and chain barring constraints. An exact algorithm, using initial relaxation of most of the problem constraints, is presented which is capable of solving problems with up to twenty sites within a reasonable number of iterations.

Discrete space location-allocation solutions from genetic algorithms

Abstract: Genetic algorithms are adaptive sampling strategies based on information processing models from population genetics. Because they are able to sample a population broadly, they have the potential to out-perform existing heuristics for certain difficult classes of location problems. This paper describes reproductive plans in the context of adaptive optimization methods, and details the three genetic operators which are the core of the reproductive design. An algorithm is presented to illustrate applications to discrete-space location problems, particularly thep-median. The algorithm is unlikely to compete in terms of efficiency with existingp-median heuristics. However, it is highly general and can be fine-tuned to maximize computational efficiency for any specific problem class.

Discrete stochastic location models

Abstract: 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.

p-Median theorems for competitive locations

Abstract: In competitive location theory, one wishes to optimally choose the locations ofr facilities to compete againstp existing facilities for providing service (or goods) to the customers who are at given discrete points (or nodes). One normally assumes that: (a) the level of demand of each customer is fixed (i.e. this demand is not a function of how far a customer is from a facility), and (b) the customer always uses the closest available facility. In this paper we study competitive locations when one or both of the above assumptions have been relaxed. In particular, we show that for each case and under certain assumptions, there exists a set of optimal locations which consists entirely of nodes.

Spatial competition facility location models: Definition, formulation and solution approach

Abstract: Models are presented for locating a firm's production facilities and determining production levels at these facilities so as to maximize the firm's profit. These models take into account the changes in price at each of the spatially separated markets that would result from the increase in supply provided by the new facilities and also from the response of competing firms. Two different models of spatial competition are presented to represent the competitive market situation in which the firm's production facilities are being located. These models are formulated as variational inequalities; recent sensitivity analysis results for variational inequalities are used to develop derivatives of the prices at each of the spatially separated markets with respect to the production levels at each of the new facilities. These derivatives are used to develop a linear approximation of the implicit function relating prices to productions. A heuristic solution procedure making use of this approximation is proposed.

Theoretical links between median and coverage location problems

Abstract: The relationship between the maximal covering problem and the P-median problem is reviewed. It is shown that two multiple coverage models, themaximumexpectedcoverageproblem (MECP) and thebackupcoverageproblem (BACOP), are special cases of thevectorassignmentP-medianproblem (VAPMP). This relationship is utilized to solve both MECP and BACOP on test sets from the literature. Computational experience is reported.

The variance equity measure in locational decision theory

Abstract: Classical location theories and models were initially developed for the private sector, so that the related operational research literature has emphasized performance measures of efficiency and effectiveness. For public sector applications, measures of equity become important, yet such measures have received little formal treatment. This paper suggests a locational equity measure, the variance measure, and investigates its properties for tree networks. A fast algorithm (O(M)) to locate the minimum variance point on a tree network is developed, and some numerical results illustrate the variance optimal location.

Solving nonlinear integer programs with large-scale optimization software

Abstract: This paper describes recent experience in tackling large nonlinear integer programming problems using the MINOS large-scale optimization software. A technique is presented for extending the constrained search approach used in MINOS to exploring integer-feasible solutions once a continuous optimal solution is obtained. Computational experience with this approach is described for two classes of problems: quadratic assignment problems and pipeline network design problems.

Minisum hierarchical location-allocation problems on a network: A survey

Abstract: In practice, the systems that provide services/products generally consist of two or more levels of facilities. At present, only a few of these systems have been modelled and solved. However, the limited research published in this area is scattered in various journals and is not easily accessible. Our objective in this article is to provide a brief survey of the minisumk-hierarchical location-allocation problems on a network.

A hierarchical location-allocation model with allocations based on facility size

Abstract: In this paper, I address the location of successively inclusive hierarchical facility systems. Location analysts have traditionally generated such systems under two unrealistic assumptions — first, that facilities can be located independently at each level, and secondly, that patrons will invariably attend the closest facility offering a particular level of service. In this paper, I employ a heuristic method which allows all levels to be located simultaneously. Further, I introduce an objective function based on a negative exponential adaption of Reilly's “retail gravitation law” which accounts for the differential attractivess of facilities at different levels.

Relaxation techniques for strictly convex network problems

Abstract: Gauss—Seidel type relaxation techniques are applied in the context of strictly convex pure networks with separable cost functions. The algorithm is an extension of the Bertsekas—Tseng approach for solving the linear network problem and its dual as a pair of monotropic programming problems. The method is extended to cover the class of generalized network problems. Alternative internal tactics for the dual problem are examined. Computational experiments —aimed at the improved efficiency of the algorithm — are presented.

Locating an obnoxious facility within a polygonal region

Abstract: The purpose of this paper is to review some of the many instances in which a location analyst must make a decision as to where an obnoxious facility should be placed. An obnoxious facility is one that can be defined as a facility which has undesirable interactions with existing facilities. Examples include equipment which emit pollutants such as particulates, noise and radiation or warehouses that contain flammable materials. Other obnoxious facilities include machines that are potential sources of hazards to nearby machines and workers. The interaction between the obnoxious facility and each existing facility is reflected through a weighting factor. The feasible region is considered to be continuous in the form of a convex or nonconvex simple polygon. Since the new facility is to be located away from the existing facilities, an appropriate criterion for optimization is the MAXIMIN or the MAXISUM criterion. Algorithms are reviewed for two common metrics under both criteria, i.e. Euclidean and rectilinear.

“Spatial nodes” in discrete location problems

Abstract: Discrete location models often assume an underlying network where demands originate at point nodes. To apply these models to planar regions with continuously distributed demand, the region is usually partitioned into “zones” and the demand from each zone is assumed to originate at a point, usually the zone centroid. Thus, the point node in the underlying network represents a spatial zone with a finite area. This paper examines the effect of approximating these “spatial nodes” by point nodes. In some problem scenarios, the approximation does not affect the solution. However, especially when the locational criterion includes the consideration of intra-zonal travel cost variances (e.g. travel time variance) and demands may originate anywhere within zones of nonzero area, point nodes do not give an accurate evaluation of the performance of a locational design. To illustrate the application of the concept of spatial nodes, a model is formulated for locatingp (fire-fighting) units in a region having continuously distributed demand with the objective of minimizing a nonlinear function of arrival times of the first and second closest units to any (fire) incident. A heuristic “site-substitution” procedure is presented that solves the formulated model.

Duopoly competition in networks

Abstract: This paper presents a new approach to modeling competition between firms in network-based industries, i.e. industries where the firms' technology decisions correspond to choices of networks. Industries having this structure include transportation, telecommunications, and some service industries. Competition is studied between two firms who make both network design decisions and price decisions for services. This situation is modeled as a game, an equilibrium solution corresponding to a Nash equilibrium is defined, and properties of the solution are characterized. Necessary and sufficient conditions are shown for equilibrium solutions and existence of equilibrium solutions is demonstrated. Among the results is that each firm will maximize its own profit by minimizing total industry cost of providing services. An example demonstrating results is presented.

Locating the two-median of a tree network with continuous link demands

Abstract: Typical formulations of thep-median problem on a network assume discrete nodal demands. However, for many problems, demands are better represented by continuous functions along the links, in addition to nodal demands. For such problems, optimal server locations need not occur at nodes, so that algorithms of the kind developed for the discrete demand case can not be used. In this paper we show how the 2-median of a tree network with continuous link demands can be found using an algorithm based on sequential location and allocation. We show that the algorithm will converge to a local minimum and then present a procedure for finding the global minimum solution.

The design of branch and bound algorithms for a class of nonlinear integer programs

Abstract: This paper is concerned with computational experimentation leading to the design of effective branch and bound algorithms for an important class of nonlinear integer programming problems, namely linearly constrained problems, which are used to model several real-world situations. The main contribution here is a study of the effect of node and branching variable selection and storage reduction strategies on overall computational effort for this class of problems, as well as the generation of a set of adequate test problems. Several node and branching variable strategies are compared in the context of a pure breadth-first enumeration, as well as in a special breadth and depth enumeration combination approach presented herein. Also, the effect of using updated pseudocosts is briefly addressed. Computational experience is presented on a set of eighteen suitably-sized nonlinear test problems, as well as on some random linear integer programs. Some of the new rules proposed are demonstrated to be significantly superior to previously suggested strategies; interestingly, even for linear integer programming problems.

Dynamic location problems

Abstract: A class of dynamic location problems is introduced. The relationship between a static problem and its corresponding dynamic one is studied. We concentrate on two types of dynamic problems. The first is the global optimization problem, in which one looks for the all-times optimum. The second is the steady-state problem in which one seeks to determine the steady-state behavior of the system if one exists. General approaches to these problems are discussed.

Evaluating dredging and offshore loading locations for u.s. coal exports using the local logistics system

Abstract: The formulation and use of a mixed integer mathematical programming location-allocation model, the Coal Logistics System (COLS), is presented in this study. COLS is used to evaluate the potential for reducing water-borne coal transportation costs, and concomitantly the costs of delivering coal to European markets. This cost reduction is accomplished through the use of supercolliers which would require the dredging of channels at selected ports or the use of offshore loading sites at East and Gulf Coast ports or both. The model developed and the analysis presented in this paper are intended to aid in the determination of the location and extent of these activities, and to indicate the size of the potential reduction in the costs of U.S. export coal. In order to be able to accommodate these deeper draft vessels at East and Gulf Coast ports, expensive improvements would need to be undertaken which may include the deepening of harbor channels to the depths required for 120 000 dead weight ton (dwt) or larger supercolliers. Since dredging requires large initial investments and has significant long-term maintenance costs, excess capacity could represent an inefficient use of both U.S. revenues and the ports' own funds. The use of offshore loading moorings to permit the “topping off” of supercolliers by self-unloading colliers at the deepwater sections of harbor channels has been proposed as a way of reducing trans-ocean transportation costs and avoiding the large investments and time delays associated with dredging activities. The Coal Logistics System was modified and extended to enable the evaluation of these two port improvement options to be undertaken in a consistent and realistic manner.

A complementary pivoting algorithm for linear network problems

Abstract: In this paper, an algorithm is presented for the transshipment problem that is an adaption of the method used by Jones, Saigal, and Schneider for solving singlecommodity, spatial-equilibrium problems. The approach uses a variable-dimension strategy in which a sequence of subproblems is formed by solving the problem ‘one-node-at-a-time’. The algorithm is tested on uncapacitated transportation problems. Although the computational results are not directly comparable to other methods (since the algorithm is implemented in C under UNIX), the results show that the method is very effective and may be competitive with the best available algorithms for linear network problems.

A parameterized hessian quadratic programming problem

Abstract: We present a general active set algorithm for the solution of a convex quadratic programming problem having a parametrized Hessian matrix. The parametric Hessian matrix is a positive semidefinite Hessian matrix plus a real parameter multiplying a symmetric matrix of rank one or two. The algorithm solves the problem for all parameter values in the open interval upon which the parametric Hessian is positive semidefinite. The algorithm is general in that any of several existing quadratic programming algorithms can be extended in a straightforward manner for the solution of the parametric Hessian problem.

Spatial price equilibrium, distribution center location and successive over-relaxation

Abstract: A spatial price equilibrium problem is modeled which allows piecewise linear convex flow costs, and a capacity limit on the trade flow between each supply/demand pair of regions. Alternatively, the model determines the locations of intermediate distribution centers in a market economy composed of separate regions, each with approximately linear supply and demand functions. Equilibrium prices, regional supply and demand quantities, and commodity flows are determined endogenously. The model has a quadratic programming formulation which is then reduced by exploiting the structure. The reduced model is particularly well suited to solution using successive over-relaxation.

Advanced basis construction in linear programming

Abstract: This paper considers basis construction in a linear program when the number of activities with basic status is not equal to the number of rows in the particular scenario. This arises when starting with an ‘advanced basis’, obtained from a solution to another scenario. The goal here is to provide a triangular-seeking algorithm that takes advantage of structural properties during the construction of a basis agenda. For completeness, some facts, which are known but have not been published, are given about choosing an advanced basis and about spikes.

Residential location and school planning in a tightening urban economy

Abstract: Facing worse fiscal plight, many municipalities in Sweden must today carefully reexamine their activities In urban planning, this has resulted in a growing interest in how the urban development could be designed to support and facilitate the efficient use of existing public investments This paper focuses on the school sector as being one of the most costly A location-allocation model of the capacitated facility location type is formulated A set of potential schools consisting of existing and new ones are considered The school-age children are assigned to a subset of these schools so as to minimize the sum of the capital costs of this subset and the transportation costs of the children The model is applied to the municipality of Uppsala in Sweden Different future settlement structures proposed by the planners as well as different housing allocations generated by a separate optimization model are evaluated

Integrating modeling, algorithm design, and computational implementation to solve a large-scale nonlinear mixed integer programming problem

Abstract: This paper describes the formulation of a nonlinear mixed integer programming model for a large-scale product development and distribution problem and the design and computational implementation of a special purpose algorithm to solve the model. The results described demonstrate that integrating the art of modeling with the sciences of solution methodology and computer implementation provides a powerful approach for attacking difficult problems. The efforts described here were successful because they capitalized on the wealth of existing modeling technology and algorithm technology, the availability of efficient and reliable optimization, matrix generation and graphics software, and the speed of large-scale computer hardware. The model permitted the combined use of decomposition, general linear programming and network optimization within a branch and bound algorithm to overcome mathematical complexity. The computer system reliably found solutions with considerably better objective function values 30 to 50 times faster than had been achieved using general purpose optimization software alone. Throughout twenty months of daily use, the system was credited with providing insights and suggesting strategies that led to very large dollar savings.

Review of recent results about theκ-centrum of a tree

Abstract: This paper is concerned with theκ-centrum of a graph This concept, related to a particular location problem, generalizes that of the center and that of the median of a graph: theκ-centrum is the set of points for which the sum of the (weighted) distances from theκ farthest vertices is minimized The paper will review some recent results about this problem In particular, some properties of cardinality, connectivity and, more generally, of the structure of theκ-centrum of a weighted tree will be presented

Modifying the forrest-Tomlin and saunders updates for linear programming problems with variable upper bounds

Abstract: The author previously described a modification of the simplex method to handle variable upper bounds implicitly. This method can easily be used when the representation of the basis inverse (e.g. a triangular decomposition of the basis) is maintained as a dense matrix in core, but appears to cause difficulties for large problems where secondary storage and packed matrices may be employed. Here we show how the Fonest—Tomlin and Saunders updating schemes, which are designed for such large problems, can be adapted.