DEGENERACY AND THE (MIS)INTERPRETATION OF SENSITIVITY ANALYSIS IN LINEAR PROGRAMMING

Under conditions of degeneracy, sensitivity analysis information such as cost and right-hand-side ranges, which are produced by standard linear programming procedures used in commercial codes, can be misleading. From a managerial perspective, the interpretation of such information can be erroneous. In this paper we present these problems, explain their occurence with elementary examples, and discuss procedures for their resolution.     

ON USING LINEAR PROGRAMMING IN DISCRIMINANT PROBLEMS

This commentary evaluates the usefulness of the Freed and Glover [6] linear programming approach to the discriminant problem, relates linear programming to other parametric and nonparametric approaches, and evaluates the linear programming approach.     

THE EFFECTS OF DEGENERACY ON COST-COEFFICIENT RANGES AND AN ALGORITHM TO RESOLVE INTERPRETATION PROBLEMS

The problems of determining valid shadow prices and of performing a valid sensitivity analysis for managerial purposes are considered. For primal degenerate optima, earlier results on finding shadow prices are used to propose a clear and efficient method for finding valid cost-coefficient ranges. An example is used to demonstrate how managerial information can be obtained at degenerate optima.     

A Note on “Discrete Sequential Search with Group Activities”

This note comments on a paper published by Wagner and Davis [Decision Sciences (2001), 32(4), 557–573]. These authors present an integer‐programming model for the single‐item discrete sequential search problem with group activities. Based on their experiments, they conjecture that the problem can be solved as a linear program. In this note, we provide a counterexample for which the optimal value of the linear program they propose is different from the optimal value of the integer‐programming model, hence contradicting their conjecture for the specific linear program that they specify. To the best of our knowledge, the conjecture settled in this note was still an open question.     

ESTIMATION OF MISCLASSIFICATION PROBABILITIES IN THE LINEAR PROGRAMMING APPROACHES TO THE TWO-GROUP DISCRIMINANT PROBLEM

Expressions for misclassification probabilities are derived under a contaminated multivariate normal model for the linear-programming approaches to the two-group discriminant problem.     

SOME DIFFICULTIES AND IMPROVEMENTS IN APPLYING LINEAR PROGRAMMING FORMULATIONS TO THE DISCRIMINANT PROBLEM

In this paper, we discuss some disturbing features of two linear programming (LP) approaches to the discriminant problem. Specifically, we show that both approaches are sensitive to the choice of origin for the data although, intuitively, placement of origin should have no effect on the method of assigning cases to groups. In addition, we show that these LP approaches may lead to discriminant functions which assign all cases to the same group. We show that the usual statistical approach to this problem does not share these difficulties, and we make recommendations for implementing these LP approaches which help to alleviate the difficulties.     

THE TIMING OF LABOR TRANSFERS IN DUAL RESOURCE-CONSTRAINED SYSTEMS: “PUSH” VS. “PULL” RULES*

Altough the dual resource-constrained (DRC) system has been studied, the decision rule used to determine when workers are eligible for transfer largely has been ignored. Some earlier studies examined the impact of this rule [5] [12] [15] but did not include labor-transfer times in their models. Gunther [6] incorporated labour-transfer times into his model, but the model involved only one worker and two machines. No previous study has examined decision rules that initiate labor transfers based on labor needs (“pull” rules). Labor transfers always have been initiated based on lack of need (“push” rules). This study examines three “pull” variations of the “When” labor-assignment decision rule. It compares their performances to the performances of two “push” rules and a comparable machine-limited system. A nonparametric statistical test, Jonckheere's S statistic, is used to test for significance of the rankings of the rules: a robust parametric multiple-comparison statistical test, Tukey's B statistic, is used to test the differences. One “pull” and one “push” decision rule provide similar performances and top the rankings consistently. Decision rules for determining when labor should be transferred from one work area to another are valuable aids for managers. This especially is true for the ever-increasing number of managers operating in organizations that recognize the benefits of a cross-trained work force. Recently there has been much interest in cross-training workers, perhaps because one of the mechanisms used in just-in-time systems to handle unbalanced work loads is to have cross-trained workers who can be shifted as demand dictates [8]. If management is to take full advantage of a cross-trained work force, it need to know when to transfer workers.     

Notes and Communications A NON-DUAL APPROACH TO SENSITIVITY ANALYSIS—THE RIGHT-HAND-SIDE CASE

Linear programming and sensitivity analysis in linear programming have long been integral parts of business school curricula, both at the undergraduate and graduate levels. Most business school students are required to take an introductory course in management science emphasizing linear programming and sensitivity analysis. This note is educational and presents a non-dual approach to restoring the optimal solution to a linear program when changes in the right-hand-side vector render the current solution infeasible and thus nonoptimal.     

Sensitivity of Linear Programs to Related Changes in Multiple Inputs*

In a 1991 paper in Decision Sciences, Camm and Burwell presented smart modeling techniques for obtaining the sensitivity of linear programs to simultaneous common changes in common inputs. This note addresses sensitivity analysis of simultaneous related changes in multiple inputs, and shows how related multi-parametric analysis can be performed with software that has no or only limited parametric programming capability.     

REPLY TO: “ON THE SENSITIVITY OF BRAND-CHOICE SIMULATIONS TO ATTRIBUTE IMPORTANCE WEIGHTS”

Our reply to Curry, Louviere, and Augustine's critique of our earlier paper focuses on differences in motivation between our research and theirs. Our interest in the problem relates to the possible incorporation of self-explicated evaluations in conjoint data collection methods; subsequent to the appearance of our original paper, we have developed hybrid models that combine elements of self-explicated (compositional) and conjoint (decompositional) data collection procedures. As far as we can surmise from their critique, Curry, Louviere, and Augustine are concerned with much broader strategic issues relating share of choices in the consumer population to changes in the shape of attribute weight distributions, shape of the Pareto tradeoff boundary, and so on.     

Notes and Communications “A DECISION THEORY EXAMPLE IN FOOTBALL”: A COMMENT*

Janssen and Daniel analyzed the choice between a one- or a two-point conversion for a particular game situation in college football. Their decision criteria was maximum expected utility based on a von Neumann-Morgenstern utility function defined over the games outcomes. An alternative approach based on a stochastic dominance criterion is presented that does not rely on knowledge of the relative importance of tying vs. winning; rather, it relies on a notion of consistency in the sequential problem.     

A Note on “An Efficient Zero-One Formulation of the Multilevel Lot-Sizing Problem”

In a recent paper, McKnew, Saydam, and Coleman [3] presented a novel zero-one integer programming formulation of the multilevel dynamic, deterministic lot-sizing problem in assembly systems. They stated that “the relaxed linear programming solution to this formulation will always be integer’ [3, p. 280] since the constraint matrix is totally unimodular. In this note, we point out that the constraint matrix is not totally unimodular and therefore the authors’claim that a linear relaxation of the zero-one integer formulation always yields an integer solution is not true.     

SOME COMMENTS ON “A SYSTEMS PERSPECTIVE ON MATERIALS REQUIREMENTS PLANNING”*

We believe there are some serious deficiencies in the Morecroft article on MRP problems which appeared in a recent issue of this journal. The article, at the most, shows that MRP, inappropriately imposed on an existing production system, can result in undesirable system performance. This does not represent a new or startling conclusion. Four problem areas in the article are discussed, and suggested alternative approaches are offered.     

COMMENTS ON “COST INCREASES DUE TO DEMAND UNCERTAINTY IN MRP LOT SIZING”: A VERIFICATION OF ORDERING PROBABILITIES

In a recent article in this journal, De Bodt and Van Wassenhove [1] presented analytic derivations related to lot-sizing behavior under uncertainty. Although their models appear to have been verified in the aggregate by simulation experiments, detailed justifications for several of the derivations are missing. The present paper looks at De Bodt and Van Wassenhove's analysis and provides verifications of (and corrections to) the ordering probabilities and order cycles used by the authors to estimate the cost effects of forecast errors in the particular operating environment studied. The probabilities simulated in this study also generate additional insight into the “system nervousness” caused by lot-sizing and forecast errors.     

Cognitive Limitations and Investment “Myopia”*

Optimization of investment decisions in an uncertain and dynamically evolving environment is difficult due to the limitations of the decision-maker's cognitive capacity. Thus, actual investment decisions may deviate from the dynamically optimal decision rule. This paper investigates how a potential investment rule bias affects the expected payoff from a project that has an uncertain development time and an uncertain completion cost. The result shows that the presence of a potential bias in the adopted decision rule dissipates project value and that the dissipating effect is greater for a longer term project if the completion cost is an increasing function of the time to completion.     

A Response to Xiao's “Necessary and Sufficient Conditions of Unacceptable Solutions in LP Discriminant Analysis”: Something is Amiss

Characterization of unacceptable solutions of linear programming methods for linear discriminant problems have been studied by many researchers. This note shows that a recent correction was not a correction but a tightening of prior results.