Undoubtedly, humans have been analyzing the effectiveness of locational decisions since they inhabited their first cave. We use the term “facility” here in its broadest sense. That is, it is meant to include entities such as air and maritime ports, factories, warehouses, retail outlets, schools, hospitals, day-care centers, bus stops, subway stations, electronic switching centers, computer concentrators and terminals, rain gages, emergency warning sirens, and satellites, to name but a few that have been analyzed in the research literature. The ubiquity of locational decision-making has led to a strong interest in location analysis and modeling within the operations research and management science community. The long and voluminous history of location research results from several factors. First, location decisions are frequently made at all levels of human organization from individuals and households to firms, government agencies and even international agencies. Second, such decisions are often strategic in nature. That is, they involve large sums of capital resources and their economic effects are long term. In the private sector they have a major influence on the ability of a firm to compete in the market place. In the public sector they influence the efficiency by which jurisdictions provide public services and the ability of these jurisdictions to attract households and other economic activity. Third, they frequently impose economic externalities. Such externalities include pollution, congestion, and economic development, among others. Fourth, location models are often extremely difficult to solve, at least optimally. Even the most basic models are computationally intractable for large problem instances. In fact, the computational complexity of location models is a major reason that the widespread interest in formulating and implementing such models did not occur until the advent of high-speed digital computers. Finally, location models are application specific. That is, their structural form (the objectives, constraints and variables) is determined by the particular location problem under study. Consequently, there does not exist a general location model that is appropriate for all potential or existing applications.

3 Discrete Network Location Models

Interactive multiple objective optimization: survey l—continuous case

The Weber Problem

The single-period inventory models have wide applications in the real world in assisting the decision maker to determine the optimal quantity to order. Due to lack of historical data, the demand has to be subjectively determined in many cases. In this paper, a single-period inventory model for cases of fuzzy demand is constructed. The costs considered include the procurement cost, shortage cost, and holding cost. For different fuzzy total cost resulted from different order quantity, a method for ranking fuzzy numbers is adopted to find the optimal order quantity in terms of the cost. When the profit gained from selling one item is less (greater) than the loss incurred due to one unsold item, the optimal order quantity lies in the range defined for the left-shape (right-shape) function of the fuzzy demand. If the unit profit is equal to the unit loss, then all quantities with a membership grade 1 are optimal to be ordered. The methodology of this paper can be applied to construct other inventory models with fuzzy demand.

A single-period inventory model with fuzzy demand

This paper considers a general form of the single facility minisum location problem also referred to as the Fermat-Weber problem, where distances are measured by an lp norm. An iterative solution algorithm is given which generalizes the well-known Weiszfeld procedure for Euclidean distances. Global convergence of the algorithm is proven for any value of the parameter p in the closed interval [1, 2], provided an iterate does not coincide with a singular point of the iteration functions. However, for p > 2, the descent property of the algorithm and as a result, global convergence, are no longer guaranteed. These results generalize the work of Kuhn for Euclidean p = 2 distances.

Global Convergence of a Generalized Iterative Procedure for the Minisum Location Problem with lp Distances

In this paper, a man-machine interactive method is presented as an aid in solving the bicriterion mathematical programming problem. It is assumed that the two objective functions are real-valued functions of the decision variables which are themselves constrained to some compact and nonempty set. The overall utility function is assumed to be unknown explicitly to the decision-maker but is assumed to be a real-valued function defined on the pairs of feasible values of the objective functions and monotone non-decreasing in each argument. The decision-maker is required only to provide yes or no answers to questions regarding the desirability of increase or decrease in objective function values of solutions that he will not accept as optimal. Convergence of the method is indicated and a numerical example is presented in order to demonstrate its applicability.

An Interactive Method as an Aid in Solving Bicriterion Mathematical Programming Problems

The effect of axis rotation on distance estimation

A simple method is presented for determining ‘closed-form’ solutions for an optimum (s, S) ordering policy for the single-period inventory problem with a set-up cost of ordering and the uncertain total demand over the period represented by a triangular probability density function. The distribution reflecting the decision maker's degree of belief that all values of total demand outside of two (possibly ‘soft’) limits are barely credible and all values within the limits have uniform increasing probability density towards a (possibly ‘soft’) modal value. The importance of the closed form solutions obtained is that they remove the need for enumeration over alternative values of s in determining the optimum (s, S) ordering policy, also they can be encoded in an algorithm simple to implement and they allow easy sensitivity analysis of the results to perturbations in the estimates of the problem parameters. A numerical example is presented to illustrate the essential features of the method.

The Single-Period Inventory Problem with Triangular Demand Distribution

This paper applies a single-facility location model to the distribution network covered by the Brantford, Ontario branch of Thibodeau-Finch Transport Ltd. The model is used to determine whether or not the present terminal location should be maintained for the current demand structure and for a projected demand structure. Optimal locations are obtained for both demand structures using a distance function that is tailored to the actual road network over which the firm's vehicles travel. It is found that the transportation costs associated with the optimal locations are sufficiently close to those with the present location to conclude that Thibodeau-Finch should not consider relocation among its strategic options. The paper includes relevant data for verifying the conclusions drawn in support of the status quo.

https://link.springer.com/content/pdf/10.1057%2Fjors.1985.26.pdf

John Walker

Papers

An Interactive Method as an Aid in Solving Bicriterion Mathematical Programming Problems

The effect of axis rotation on distance estimation

The Single-Period Inventory Problem with Triangular Demand Distribution

Terminal Location Problem: A Case Study Supporting the Status Quo

An interactive method as an aid in solving multi-objective mathematical programming problems