Michael M. Kostreva

Equitable aggregations and multiple criteria analysis

Abstract In the past decade, increasing interest in equity issues resulted in new methodologies in the area of operations research. This paper deals with the concept of equitably efficient solutions to multiple criteria optimization problems. Multiple criteria optimization usually starts with an assumption that the criteria are incomparable. However, many applications arise from situations which present equitable criteria. Moreover, some aggregations of criteria are often applied to select efficient solutions in multiple criteria analysis. The latter enforces comparability of criteria (possibly rescaled). This paper presents aggregations which can be used to derive equitably efficient solutions to both linear and nonlinear multiple optimization problems. An example with equitable solutions to a capital budgeting problem is analyzed in detail. An equitable form of the reference point method is introduced and analyzed.

The generalized Leontief input-output model and its application to the choice of new technology

The Leontief input-output model is generalized and formulated as a generalized linear complementarity problem. Conditions for existence of solutions are given, and solution techniques are reviewed. An application of the model to choosing new technologies is suggested.

Nurse scheduling on a microcomputer

Abstract A computer package for solving the problem of shift scheduling for nurses in a hospital unit has been developed. There are two main modules in the package: 1. (1) the survey module, which is designed to query the nurses and the supervisor to obtain information about nurse preferences and unit requirements and to produce the input file for the scheduler 2. (2) the scheduler module, which creates and solves (approximately) an optimization problem and hence generates the desired shift schedules for the individual nurses. A description of the package and its implementation on an Apple Macintosh computer is given here.

A generalization of the norm-relaxed method of feasible directions

This paper generalizes the norm-relaxed method of feasible directions which has roots in the classical method of feasible directions (MFD). This generalized algorithm introduces some parameters which can be adjusted to speed up the convergence. An investigation of its global convergence is included and the numerical performance is shown on 15 test problems. Its global convergence is guaranteed under a rather mild and standard assumption. The numerical tests show that this new algorithm converges faster than the previous norm-relaxed MFD for most test problems by selecting a set of suitable parameters even though it only has the same guaranteed convergence rate as the previous norm-relaxed MFD.

Relocation problems arising in conservation biology

Abstract An early motivation for this study was the problem of relocation of scarce or endangered species of animals for breeding and/or reintroduction to establish new populations in the wild. In this paper, we introduce single and multiple objective optimization models which are designed to comprehend a wide variety of objectives which are of interest to conservation and wildlife managers. We present the models in a general way and point out special features relative to ecology as they arise. Thus, the models may be used for relocation decisions analysis in diverse fields, not only in conservation and ecology. After presentation of the models in such a general way, we reformulate the models to make use of the special structure present. Such reformulation reduces the number of decision variables and constraints and, in general, makes solutions easy to obtain. By easy to obtain, we mean that tools from linear and mixed-integer programming together with elementary sorting procedures provide the basis for solving the models. In order to illustrate the capabilities of the models and solution techniques developed, we present the results of their application to the real-life relocation problem arising while analyzing restoration of the globally endangered Przewalskis horse population.

Some existence theorems of Nash and Berge Equilibria

Abstract Existence theorems of Nash and Berge equilibria are introduced and proved. Radjefs results have been extended. The concepts of S-system and (S,R,M)-system as defined earlier are employed to prove the existence of Berge equilibria.

Approximate mean value analysis algorithms for queuing networks: existence, uniqueness, and convergence results

This paper is concerned with the properties of nonlinear equations associated with the Scheweitzer-Bard (S-B) approximate mean value analysis (MVA) heuristic for closed product-form queuing networks. Three forms of nonlinear S-B approximate MVA equations in multiclass networks are distinguished: Schweitzer, minimal, and the nearly decoupled forms. The approximate MVA equations have enabled us to: (a) derive bounds on the approximate throughput; (b) prove the existence and uniqueness of the S-B throughput solution, and the convergence of the S-B approximation algorithm for a wide class of monotonic, single-class networks; (c) establish the existence of the S-B solution for multiclass, monotonic networks; and (d) prove the asymptotic (i.e., as the number of customers of each class tends to ∞) uniqueness of the S-B throughput solution, and (e) the convergence of the gradient projection and the primal-dual algorithms to solve the asymptotic versions of the minimal, the Schweitzer, and the nearly decoupled forms of MVA equations for multiclass networks with single server and infinite server nodes. The convergence is established by showing that the approximate MVA equations are the gradient vector of a convex function, and by using results from convex programming and the convex duality theory.

Issues in Evaluation of Complex Fire Models

Several methods of evaluation of the predictive capability have been applied to fire models, but with limited utility. These range from explicit evaluation of the equations used in simple models such as ASET to pointwise evaluation of complex models from numerous computer runs of a model. This paper presents a discussion of the issues involved in conducting an analysis of a complex room fire model. Examples using currently available room fire models are presented. For the models and test cases examined, heat release rate and heat transfer effects dominate the behavior of the models. For simple models like the ASET model, analytical techniques can be readily applied. For more complex fire models, obtaining an overall assessment of the model increases in complexity with the complexity of the model, requiring evaluation of numerous model inputs and outputs. Thus, more directed local investigations are currently the only tractable means of evaluation. Areas for additional research are identified.

Linear complementarity problems and multiple objective programming

An equivalence is demonstrated between solving a linear complementarity problem with general data and finding a certain subset of the efficient points of a multiple objective programming problem. A new multiple objective programming based approach to solving linear complementarity problems is presented. Results on existence, uniqueness and computational complexity are included.

Using a circadian rhythms model to evaluate shift schedules

As long as the need for shiftwork exists, so will the demand for an optimal scheduling strategy that balances the needs of both industry and the shiftworker. One aim is for schedules that require workers to be on duty during the times that they are most naturally alert and awake. Czeisler et al. (1982) developed a set of circadian rhythm based guidelines intended to aid in designing such shiftwork schedules. This paper takes research one step further by testing such empirical criteria in a mathematical setting. The two-oscillator model of free-run human circadian rhythms developed by Kronauer et al. (1982) was modified to represent the circadian rhythms of a shiftworker on a pre-selected shiftwork schedule. Numerical simulations were used to compare the circadian rhythms produced from a variety of shiftwork schedules to the free-run rhythms. Shift schedules that resulted in circadian rhythms closest to the free-run rhythms were identified as preferred schedules. The numerical results supported Czeislers findings (1982), indicating the best shift schedules adopt a slow, forward-shifting rotation pattern, rotate shifts after 2-week periods and allow an average of 2 days off per week.