Kenneth E. Rosing

Heuristic concentration and Tabu search: A head to head comparison

Earlier this year two papers applying the metaheuristics Tabu search (TS) and Heuristic concentration (HC) to the p-median problem were published in consecutive volumes of this journal. Here we apply the method of HC to some of the data sets which were used for computational experience in the paper on TS. For these examples, which we regard as being of a particularly challenging character, HC discovers the superior solution (superior to TS) in about 95% of the cases, and, where the optimal solution is known, the optimal solution in about 80% of the cases. No general conclusion on the relative times for the two approaches can be drawn.

A network location-allocation model trading off flow capturing and p -median objectives

The flow capturing and thep-median location—allocation models deal quite differently with demand for service in a network. Thep-median model assumes that demand is expressed at nodes and locates facilities to minimize the total distance between such demand nodes and the nearest facility. The flow-capturing model assumes that demand is expressed on links and locates facilities to maximize the one-time exposure of such traffic to facilities. Demand in a network is often of both types: it is expressed by passing flows and by consumers centred in residential areas, aggregated as nodes. We here present a hybrid model with the dual objective of serving both types of demand. We use this model to examine the tradeoff between serving the two types of demand in a small test network using synthetic demand data. A major result is the counter-intuitive finding that thep-median model is more susceptible to impairment by the flow capturing objective than is the flow capturing model to thep-median objective. The results encourage us to apply the model to a real-world network using actual traffic data.

A gamma heuristic for the p-median problem

Heuristic concentration (HC) is a two-stage metaheuristic that can be applied to a wide variety of combinatorial problems. It is particularly suited to location problems in which the number of facilities is given in advance. In such settings, the first stage of HC repeatedly applies some random-start interchange (or other) heuristic to produce a number of alternative facility configurations. A subset of the best of these alternatives is collected and the union of the facility sites in them is called a concentration set (CS). Among the component elements of the CS are likely to be included those sites which are members of the optimal solution. In earlier studies the second stage of HC has consisted of an exact procedure to extract the best possible solution from the CS. In this paper we demonstrate, for the p-median problem, the use of two sequentially active heuristics in the second stage of HC. That is, we offer two additional layers of heuristics to improve solutions which are found in the first stage of HC. Thus we are describing a variant of the HC metaheuristic consisting of three layers of heuristics which are utilized in sequence. We propose for this procedure the name of Gamma Heuristic.

Applying the flow-capturing location-allocation model to an authentic network: Edmonton, Canada

Traditional location-allocation models aim to locate network facilities to optimally serve demand expressed as weights at nodes. For some types of facilities demand is not expressed at nodes, but as passing network traffic. The flow-capturing location-allocation model responds to this type of demand and seeks to maximize one-time exposure of such traffic to facilities. This new model has previously been investigated only with small and contrived problems. In this paper, we apply the flow-capturing location-allocation model to morning-peak traffic in Edmonton, Canada. We explore the effectiveness of exact, vertex substitution, and greedy solution procedures; the first two are computationally demanding, the greedy is very efficient and extremely robust. We hypothesize that the greedy algorithms robustness is enhanced by the structured flow present in an authentic urban road network. The flow-capturing model was derived to overcome flow cannibalization, wasteful redundant flow-capturing; we demonstrate that this is an important consideration in an authentic network. We conclude that real-world testing is an important aspect of location model development.

Heuristic concentration for the p-median: an example demonstrating how and why it works

We map certain combinatorial aspects of the p-median problem and explore their effects on the efficacy of a common (1-opt) interchange heuristic and of heuristic concentration (HC) for the problems solution. Although the problems combinatorial characteristics exist in abstract space, its data exist in two-dimensional space and are therefore mappable. By simultaneously analysing the problems patterns in geographic space and its combinatorial characteristics in abstract space, we provide new insight into what demand node configurations cause problems for the interchange heuristic and how HC overcomes these problems.

An empirical investigation of the effectiveness of a vertex substitution heuristic

Many problems require the identification of a subset of points to minimize (or maximize) some function. A vertex substitution heuristic (VSH) employs a strategy of one-by-one replacement to approximate, or perhaps find, the optimal set. The Teitz and Bart heuristic is the archetype of this procedure and is the heuristic most frequently used for the solution of the p-median problem. One study of the performance of this heuristic with increasing numbers of facilities (p) in problems with a very small number of demand nodes (n) has been published. However, no study satisfactorily indicates the relative effectiveness of this heuristic method with increasing values of n or p. In this paper we compare optimal and heuristic solutions for ninety problems varying the values of n and p systematically. The results indicate that there is a definite reduction in the effectiveness of the heuristic with increasing values of n or p.

Network distance characteristics that affect computational effort in p-median location problems

Abstract In solving location models, the effort expended and the quality of the solutions obtained often varies significantly from one problem instance to another. Our conjecture is that this is not a random occurrence but, instead can be correlated with characteristics of the network upon which the model is constructed. In this paper, we examine a variety of approaches for generating networks for the p-median model. The types of networks studied include those based on Euclidean inter-point distances, shortest paths developed from a predefined network, and randomly generated distance matrices. In addition to the process by which the network is constructed, we consider the distribution of distances, symmetry and the satisfaction of the triangle inequality. All of these characteristics influence the effort required to solve the p-median model. We have found, however, that the triangle inequality has the most significant impact.

The Robustness of Two Common Heuristics for the p-Median Problem

Optimal p-median solutions were computed for six test problems on a network of forty-nine demand nodes and compared with solutions from two heuristic algorithms. Comparison of the optimal solutions with those from the Teitz and Bart heuristic indicates that this heuristic is very robust. Tests of the Maranzana heuristic, however, indicate that it is efficient only for small values of p (numbers of facilities) and that its robustness decreases rapidly as problem size increases.

Heuristic Concentration: A Study of Stage One

Heuristic concentration (HC) is a metaheuristic for the solution of certain combinatorial problems. In stage one, a concentration set (CS), consisting of nodes likely to be in the optimal solution, is developed by multiple runs of an interchange heuristic. In stage two, a good, usually optimal, solution is constructed by selecting the best nodes from the CS. The CS is effective when it is small but comprehensive. Both of these characteristics depend upon: (1) the robustness of the heuristic; (2) the number of times it is run, q; and (3) the number of “best” solutions used to create the CS, m. Stage two is thus totally dependent upon the efficiency of stage one for the improved, and at least potentially optimal, solution. Proper values for the parameters m and q increase the probability of selecting correct elements to construct the optimal solution in stage two and to decrease the work in its identification. After a consideration of the robustness of two alternative interchange heuristics I will concentrate on the appropriate values for the parameters m and q. This is an empirical examination and the p-median problem is used throughout.

Testing a Bicriterion Location-Allocation Model with Real-World Network Traffic: The Case of Edmonton, Canada

The p-median model in a network treats demand for service as weights expressed at nodes. The flow-capturing location-allocation model treats demands expressed by traffic (OD) flows. We consider a bicriterion model, a hybrid of these two, which trades off node- and flow-based demand, and apply it to realworld data. These are journey to work data in a system of 177 traffic zones in Edmonton, Canada. The system is relatively complex, comprising 703 nodes, 2198 links and 23,350 OD pairs.