Discrete Optimization
Selecting hierarchical facilities in a service-operations environment

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Abstract

This paper presents and solves a hierarchical model for the location of service facilities and further looks at the operational issues associated with managing such facilities. By the very nature of the demand they serve, service systems require that timely service be readily available to those who need it. We argue that the location–allocation of such facilities often involves several layers of service. When all of a facility’s resources are needed to meet each demand for service, and demand cannot be queued, the need for a backup unit may be required. Effective siting decisions must address both the need for a backup response facility for each demand point and a reasonable limit on each facility’s workload. The paper develops an integer linear programming model for locating facilities offering several layers of service. A Lagrangian relaxation methodology coupled with a heuristic is employed. Results of extensive computational experiments are presented to demonstrate the viability of the approach.

Introduction

The usefulness of location modeling to decision-makers allocating scarce resources has been increasing the past two decades as analytical models with a variety of flexible objective functions and related solution techniques have been developed (Brandeau and Chiu, 1989; Schilling et al., 1993). In many location contexts, the quality of service to customers (from the facilities that are being located) depends on the distance of the customer and the facility to which the customer is assigned. Customers are generally, though not always, assigned to the nearest facility. Often, service is deemed adequate if the customer is within a given distance of the facility and is deemed inadequate if the distance exceeds some critical value.

Associated with each demand node (we consider customer zones as demand sites) is a subset of candidate facility nodes that can serve or cover the demand node. Often, demand nodes are said to be covered if the shortest path distance between the demand node and the facility is less than or equal to a specific coverage distance. The concept of coverage has proved to be a useful and appealing measure of performance for facility siting decisions where a minimum threshold of service is desired (Church and ReVelle, 1974).

Most formulations of location covering models make the following assumption: “A demand is considered covered if and only if there exists at least one available unit within a specified distance or travel time of its location.” (Current and Storbeck, 1988). In spatial analysis, proximity in terms of distance or time is a fundamental metric, and many siting models seek to optimize it. However, decision-makers often base decisions on the “satisfactory” rather than the best possible. Among public services, a number of context-free models have been developed for the location of facilities and the allocation of different levels of services to them (Bianchi and Church, 1988; Batta and Mannur, 1990; Church et al., 1991; Daskin, 1982; Eaton et al., 1985).

Often, facilities are hierarchical in terms of the types (or levels) of services being offered. In this paper, we use the words “types” and “levels” of service interchangeably. For example, in designing a location plan for banking facilities, one would typically have to locate three types of facilities. At the first level are automatic teller machines or drive-in banks that allow clients to deposit their funds as well as receive cash, while providing a statement of current account balance. In addition to such basic services, branch offices at the next hierarchy allow patrons to obtain all of these services as well as a variety of other services including maintaining a safety deposit box, applying for residential loans, and purchasing government bonds. Finally, at the third level are main bank offices that typically provide all services available at branch offices as well as handle applications for large corporate loans at the main bank office. As such, the referral of clients from lower-level to higher-level may occur depending on the type and extent of services needed by the clients.

Postal services offer at least two levels of hierarchical facilities. At the first level (lowest) are post boxes at which customers may simply deposit mail. At the next level are post offices where postal patrons may deposit mail, obtain money orders, process certified mail, and obtain passport applications, in addition to buying stamps. Further, a number of “behind the scenes” operations such as sorting mail for an entire city may occur only at a main post office.

As indicated by the examples above, in hierarchical facility location problems, facilities at different levels of the hierarchy are distinguished by the services they provide. In addition, there must be some sort of link between the facilities being located. For example, in postal services, there is a need to delineate which branch offices or main post offices are responsible for the collection of mail from each of the post boxes that are located in the region under consideration.

In this research, we propose an integer-programming model that would solve a comprehensive hierarchical location–allocation problem to site service facilities. The objective of the model is to maximize demand coverage while it locates a given number of facilities and allocates different levels of service to the open facilities. We incorporate this model as part of a decision support system (DSS) that supports an efficient heuristic solution procedure to locate hierarchical service facilities and allocate different types of services to them. In Section 2, we provide a background for work in this area and then propose the model and solution procedure to tackle problems that portray a hierarchical structure. Summary and conclusions are provided in Section 8.

Section snippets

Background

In the literature, facility location models have been well studied. Schilling et al. (1993) provide a review that introduces the notion of coverage as applied to facility location problems. The problem setting in our paper incorporates this notion of coverage as applied to real world problems that portray hierarchical facilities.

In the context of hierarchical facilities, systems could be classified by the way in which services are offered and the region to which services are provided by the

Motivation for the model

The hierarchical service location–allocation model provided in this study is motivated by three important considerations:

1. By the very nature of demand they serve, service systems require timely service to be readily available to those who need it. As such the service facility location–allocation problem is hierarchical in nature. Nevertheless, a vast majority of the existing literature has neglected to consider the functional dependence of different hierarchies of services (please refer to

Solution methodology

The procedure that we propose to solve Model P is based on the solution to the Lagrangian dual (Fisher, 1985). A Lagrangian relaxation is obtained by dualizing (moving) certain complicating constraints into the objective function and penalizing their absence from the feasible region through multipliers. For given values of multipliers, the solution to the Lagrangian relaxation provides an upper bound on the optimal objective value of Model P. As the multipliers are updated, the bounds

A heuristic solution procedure

An automatic solution to problem P is not available. We propose a heuristic solution that uses the Lagrangian problem solution as a starting point in generating feasible solutions for the original problem. This procedure is developed as an integral part of the subgradient optimization procedure.

Step 1: From the solution to the two subproblems, we have opened at most Q facilities and allocated basic service to each of those facilities and opened at most T facilities and allocated specialized

Computation study and results

This section presents our extensive experimentation with the Lagrangian relaxation algorithm that is embedded in the DSS as well as a comparison with the solution from CPLEX. The algorithm is coded in Pascal and its performance is compared with that of CPLEX. Both CPLEX and the heuristic are run on a SPARC-10 workstation.

This section contains a comparison of the CPLEX and heuristic procedure using two measures of performance: solution quality (amount of deviation from optimality) and computing

Managerial implications and limitations

The DSS we propose provides a means by which location and allocation strategies can be evaluated by a company in the service industry. From the quality of the location–allocation solutions, we can conduct extensive managerial analysis. We can abstract the following situations that would be of interest to managers:

(a) The % of demand that is within the basic distance standard: In Table 2, we report the % of demand covered within the basic distance as well as the % of demand nodes that are

Summary and conclusions

In this paper we have presented a hierarchical model embedded in a decision support system for the service location–allocation problem that is hierarchical in nature. The service quality function incorporating the travel distance to uncovered points is similar to a quality loss function employed in a traditional manufacturing setting. An effective solution procedure was developed in conjunction with a Lagrangian relaxation of the model. Results of computational experiments for problems ranging

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