Hanan Luss

Operations Research and Capacity Expansion Problems: A Survey

Planning for the expansion of production capacity is of vital importance in many applications within the private and public sectors. Examples can be found in heavy process industries, communication networks, electrical power services, and water resource systems. In all of these applications, the expansion of production capacity requires the commitment of substantial capital resources over long periods of time. Capacity expansion planning consists primarily of determining future expansion times, sizes, and locations, as well as the types of production facilities. Since the late 1950s, operations research methodology has been used to develop various models and solution approaches suitable for different applications. In this paper, we attempt to unify the existing literature on capacity expansion problems, emphasizing modeling approaches, algorithmic solutions, and relevant applications. The paper includes an extensive list of references covering a broad spectrum of capacity expansion problems.

On Equitable Resource Allocation Problems: a Lexicographic Minimax Approach

In this expository paper, we review a variety of resource allocation problems in which it is desirable to allocate limited resources equitably among competing activities. Applications for such problems are found in diverse areas, including distribution planning, production planning and scheduling, and emergency services location. Each activity is associated with a performance function, representing, for example, the weighted shortfall of the selected activity level from a specified target. A resource allocation solution is called equitable if no performance function value can be improved without either violating a constraint or degrading an already equal or worse-off (i.e., larger) performance function value that is associated with a different activity. A lexicographic minimax solution determines this equitable solution; that is, it determines the lexicographically smallest vector whose elements, the performance function values, are sorted in nonincreasing order. The problems reviewed include large-scale allocation problems with multiple knapsack resource constraints, multiperiod allocation problems for storable resources, and problems with substitutable resources. The solution of large-scale problems necessitates the design of efficient algorithms that take advantage of special mathematical structures. Indeed, efficient algorithms for many models will be described. We expect that this paper will help practitioners to formulate and solve diverse resource allocation problems, and motivate researchers to explore new models and algorithmic approaches.

Technical Note—Allocation of Effort Resources among Competing Activities

This paper presents problems of resource allocation among many activities, such as allocating a given marketing budget among sales territories, where the return function for each territory attains different parameters. Applying the Kuhn-Tucker conditions, we derive “single pass” algorithms for different concave payoff functions, maximizing total returns under a given amount of available effort. Various extensions are also mentioned.

Maintenance Policies When Deterioration Can be Observed by Inspections

In this paper we examine maintenance policies for systems in which the degree of deterioration can be observed through inspections. We develop a Markovian model in which the holding times in the various states are exponentially distributed. The costs incurred include costs of inspections, state occupancy costs, costs of preventive repairs, and the costs for repairing a failed system either at an inspection event or immediately after the occurrence of a malfunction. The policies examined include the scheduling of inspections and preventive repairs so that the total expected cost per time unit is minimized.

Resource allocation among competing activities: a lexicographic minimax approach

We examine an allocation problem in which the objective is to allocate resources among competing activities so as to balance weighted deviations from given demands. A lexicographic minimax algorithm that solves successive problems by A minimax optimizer is developed. The algorithm is extremely fast and can readily solve large-scale problems that may be encountered in applications, e.g., in production planning.

Algorithms for Separable Nonlinear Resource Allocation Problems

We consider a simple resource allocation problem with a single resource constraint. The objective function is composed of separable, convex performance functions, one for each activity. Likewise, the constraint has separable, convex resource-usage functions, one for each activity. The objective is to minimize the sum of the performance functions, subject to satisfying the resource constraint and nonnegativity constraints. This problem extends the well-studied problem in which the resource constraint is linear. We present several algorithms to solve the problem. These algorithms extend approaches developed for the linearly constrained problem. They can readily solve large problems and find the optimal solution in a number of iterations that does not exceed the number of variables. We provide several examples for illustration purposes, present computational results, and highlight the similarities and differences among the algorithms.

A lexicographic minimax algorithm for multiperiod resource allocation

Resource allocation problems are typically formulated as mathematical programs with some special structure that facilitates the development of efficient algorithms. We consider a multiperiod problem in which excess resources in one period can be used in subsequent periods. The objective minimizes lexicographically the nonincreasingly sorted vector of weighted deviations of cumulative activity levels from cumulative demands. To this end, we first develop a new minimax algorithm that minimizes the largest weighted deviation among all cumulative activity levels. The minimax algorithm handles resource constraints, ordering constraints, and lower and upper bounds. At each iteration, it fixes certain variables at their lower bounds, and sets groups of other variables equal to each other as long as no lower bounds are violated. The algorithm takes advantage of the problems special structure; e.g., each term in the objective is a linear decreasing function of only one variable. This algorithm solves large problems very fast, orders of magnitude faster than well known linear programming packages. (The latter are, of course, not designed to solve such minimax problems efficiently.) The lexicographic procedure repeatedly employs the minimax algorithm described above to solve problems, each of the same format but with smaller dimension.

A Dual-Based Algorithm for Multiproduct Uncapacitated Facility Location

The multiproduct uncapacitated facility location problem (MUFLP) is a generalization of the classic uncapacitated facility location problem (UFLP). In MUFLP, different products are required by the customers. In addition to the fixed cost for opening a facility, there is an added fixed cost for handling a particular product. Assignment costs are incurred for satisfying a customers requirement for each of the separate products. The objective is to minimize the total fixed costs plus assignment costs subject to satisfying all customer requirements. We propose a new dual-based algorithm for MUFLP that extends previous work for UFLP that has proved so successful. Dual ascent and dual adjustment procedures generate a good feasible solution to the dual of the linear programming relaxation of MUFLP. A feasible primal solution to MUFLP can then be constructed based on this dual solution. Like the analogous procedures for UFLP, these procedures can be used either as a stand-alone heuristic or, else, they can be incorporated into a branch-and-bound algorithm. Computational experience is given for both randomly generated and nonrandom problems with quite favorable results.

Multifacility-type capacity expansion planning: Algorithms and complexities

We examine several variations of a capacity expansion model with multiple facility types, and with the flexibility for converting capacity from one facility type to another. Applications can be found in communication networks and production facilities. The models include costs of expansions, conversions, excess capacities and capacity shortages, and assume cost functions that are either linear or concave. We find the optimal solution by exploiting properties of extreme flows in networks and employing a dynamic programming algorithm. Specializing our results permits us to obtain different model variations by imposing certain conditions on the cost functions, and to derive the computational complexities for these variations. Typically, applications require extensive sensitivity studies that may use up substantial computer resources. Our analysis unifies previous work, extends the results, and provides intuition concerning the difficulties involved in applying the models.

Design of Stacked Self-Healing Rings Using a Genetic Algorithm

Ring structures in telecommunications are taking on increasing importance because of their “self-healing” properties. We consider a ring design problem in which several stacked self-healing rings (SHRs) follow the same route, and, thus, pass through the same set of nodes. Traffic can be exchanged among these stacked rings at a designated hub node. Each non-hub node may be connected to multiple rings. It is necessary to determine to which rings each node should be connected, and how traffic should be routed on the rings. The objective is to optimize the tradeoff between the costs for connecting nodes to rings and the costs for routing demand on multiple rings. We describe a genetic algorithm that finds heuristic solutions for this problem. The initial generation of solutions includes randomly-generated solutions, complemented by “seed” solutions obtained by applying a greedy randomized adaptive search procedure (GRASP) to two related problems. Subsequent generations are created by recombining pairs of “parent” solutions. Computational experiments compare the genetic algorithm with a commercial integer programming package.