Improved Linear Programming Models for Discriminant Analysis*

Discriminant analysis is an important tool for practical problem solving. Classical statistical applications have been joined recently by applications in the fields of management science and artificial intelligence. In a departure from the methodology of statistics, a series of proposals have appeared for capturing the goals of discriminant analysis in a collection of linear programming formulations. The evolution of these formulations has brought advances that have removed a number of initial shortcomings and deepened our understanding of how these models differ in essential ways from other familiar classes of LP formulations. We will demonstrate, however, that the full power of the LP discriminant analysis models has not been achieved, due to a previously undetected distortion that inhibits the quality of solutions generated. The purpose of this paper is to show how to eliminate this distortion and thereby increase the scope and flexibility of these models. We additionally show how these outcomes open the door to special model manipulations and simplifications, including the use of a successive goal method for establishing a series of conditional objectives to achieve improved discrimination.     

Notes and Communications RESOLVING CERTAIN DIFFICULTIES AND IMPROVING THE CLASSIFICATION POWER OF LP DISCRIMINANT ANALYSIS FORMULATIONS

In certain settings, difficulties arise that limit the effectiveness of LP formulations for the discriminant problem. Explanations and possible remedies have been offered, but these have had only limited success. We provide a simple way to overcome these problems based on an appropriate use and interpretation of normalizations. In addition, we demonstrate a normalization that is invariant under all translations of the problem data, providing a stability property not shared by previous approaches. Finally, we discuss the possibility of using more general models to improve discrimination.     

A Variable Selection Criterion in the Linear Programming Approaches to Discriminant Analysis*

There are numerous variable selection rules in classical discriminant analysis. These rules enable a researcher to distinguish significant variables from nonsignificant ones and thus provide a parsimonious classification model based solely on significant variables. Prominent among such rules are the forward and backward stepwise variable selection criteria employed in statistical software packages such as Statistical Package for the Social Sciences and BMDP Statistical Software. No such criterion currently exists for linear programming (LP) approaches to discriminant analysis. In this paper, a criterion is developed to distinguish significant from nonsignificant variables for use in LP models. This criterion is based on the “jackknife” methodology. Examples are presented to illustrate implementation of the proposed criterion.     

Unacceptable Solutions and the Hybrid Discriminant Model

A new family of linear programming (LP) discriminant models, the hybrid discriminant model, was recently presented with claims that it avoids all the problems of its predecessor LP models. However, this note shows that it does not avoid unacceptable solutions.     

EVALUATING ALTERNATIVE LINEAR PROGRAMMING MODELS TO SOLVE THE TWO-GROUP DISCRIMINANT PROBLEM

The two-group discriminant problem has applications in many areas, for example, differentiating between good credit risks and poor ones, between promising new firms and those likely to fail, or between patients with strong prospects for recovery and those highly at risk. To expand our tools for dealing with such problems, we propose a class of nonpara-metric discriminant procedures based on linear programming (LP). Although these procedures have attracted considerable attention recently, only a limited number of computational studies have examined the relative merits of alternative formulations. In this paper we provide a detailed study of three contrasting formulations for the two-group problem. The experimental design provides a variety of test conditions involving both normal and nonnormal populations. Our results establish the LP model which seeks to minimize the sum of deviations beyond the two-group boundary as a promising alternative to more conventional linear discriminant techniques.     

Mathematical Programming Formulations for the Discriminant Problem: An Old Dog Does New Tricks*

In recent years, much research has been done on the application of mathematical programming (MP) techniques to the discriminant problem. While promising results have been obtained, many of these techniques are plagued by a number of problems associated with the model formulation including unbounded, improper, and unacceptable solutions as well as solution instability under linear transformation of the data. In attempting to solve these problems, numerous formulations have been proposec involving additional variables and/or normalization constraints. While effective, these models can also become quite complex. In this paper we demonstrate that a simple, well-known special case of Hand's [13] original formulation provides an implicit normalization which avoids the problems for which various complicated remedies have been devised. While other researchers have made use of this formulation, its properties have not previously been fully recognized.     

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.     

Necessary and Sufficient Conditions of Unacceptable Solutions in LP Discriminant Analysis*

Characterization of unacceptable solutions for linear programming (LP) discriminant models have been discussed in the literature and the results presented so far are not satisfactory. This paper establishes necessary and sufficient conditions of unacceptable solutions for a number of LP models, addresses the practical implications of these conditions, and discusses the relationship between unacceptable solutions and multiple solutions.     

Decision Power and Solutions of LP Discriminant Models: Rejoinder

A major problem of LP discriminant analysis is the validity of the solution. This note shows that to comprehend the effectiveness of the solution, conditions for unacceptable solutions need to be tightly characterized.     

A Note on Discriminant Analysis Using LAD

Using a regression approach to discriminant analysis is often incorrect because it forces the use of a binary dependent variable which violates virtually any distributional assumption for a linear model. However, assuming a Laplace distribution in an LP framework leads to a theoretical foundation for MSD discriminant analysis.     

AN APPROACH FOR IMPROVED INTERPRETATION OF MULTIVARIATE ANALYSIS *

This paper proposes a method, canonical rotation analysis, which facilitates the substantive interpretation of results in multivariate analysis. Canonical rotation analysis is developed as a model which integrates multivariate least squares approaches and the varimax rotation criterion. The generalized applicability of the model to canonical correlation, multiple discriminant analysis, and multivariate analysis of variance is developed. The advantages and limitations of canonical rotation analysis are discussed and illustrated in the context of an industrial marketing research problem.     

Allocation of Returned Products among Different Recovery Options through an Opportunity Cost–Based Dynamic Approach*

In a make‐to‐order product recovery environment, we consider the allocation decision for returned products decision under stochastic demand of a firm with three options: refurbishing to resell, parts harvesting, and recycling. We formulate the problem as a multiperiod Markov decision process (MDP) and present a linear programming (LP) approximation that provides an upper bound on the optimal objective function value of the MDP model. We then present two solution approaches to the MDP using the LP solution: a static approach that uses the LP solution directly and a dynamic approach that adopts a revenue management perspective and employs bid‐price controls technique where the LP is resolved after each demand arrival. We calculate the bid prices based on the shadow price interpretation of the dual variables for the inventory constraints and accept a demand if the marginal value is higher than the bid price. Since the need for solving the LP at each demand arrival requires a very efficient solution procedure, we present a transportation problem formulation of the LP via variable redefinitions and develop a one‐pass optimal solution procedure for it. We carry out an extensive numerical analysis to compare the two approaches and find that the dynamic approach provides better performance in all of the tested scenarios. Furthermore, the solutions obtained are within 2% of the upper bound on the optimal objective function value of the MDP model.     

Minimizing the maximum travel time in a combined model of facility location and network design

This paper presents a combined Facility Location/Network Design Problem which simultaneously considers the location of facilities and the design of its underlying network so as to minimize the maximum customer-facility travel time. The model generalizes the classical p-center problem and has various applications in regional planning, distribution, telecommunications, emergency systems, and other areas. Two mixed integer programming formulations are presented and compared. Several valid inequalities are derived for the most promising of these formulations to strengthen its LP relaxation bound and to reduce the enumeration tree. Numerical results of a series of computational experiments for instances with up to 100 nodes and 500 candidate links are reported.     

Evaluating productive performance: A new approach based on the product-mix problem consistent with Data Envelopment Analysis

We propose a new approach to estimate technical coefficients from a set of Decision Making Units (DMUs) under the assumption that their production plans are set by process engineers through Linear Programming (LP) techniques. The idea behind this approach is that most manufacturing and agricultural firms routinely resort to LP-based modeling in their decision making processes in order to plan output production and, therefore, this particularity should be taken into account when estimating their technical efficiency. A usual model of LP for these sectors is the so-called product-mix problem, which we relate to a standard Data Envelopment Analysis (DEA) model in terms of the Directional Distance Function. In this paper, we finally show how to estimate the technical coefficients of a sample of Andalusian farms in Spain and how this information can be seen as a complement to the usual by-products associated with estimating technical efficiency by DEA.     

Combining Neural Networks and Statistical Predictions to Solve the Classification Problem in Discriminant Analysis*

A number of recent studies have compared the performance of neural networks (NNs) to a variety of statistical techniques for the classification problem in discriminant analysis. The empirical results of these comparative studies indicate that while NNs often outperform the more traditional statistical approaches to classification, this is not always the case. Thus, decision makers interested in solving classification problems are left in a quandary as to what tool to use on a particular data set. We present a new approach to solving classification problems by combining the predictions of a well-known statistical tool with those of an NN to create composite predictions that are more accurate than either of the individual techniques used in isolation.     

Solving the Classification Problem in Discriminant Analysis Via Linear and Nonlinear Programming Methods*

This paper demonstrates the feasibility of applying nonlinear programming methods to solve the classification problem in discriminant analysis. The application represents a useful extension of previously proposed linear programming-based solutions for discriminant analysis. The analysis of data obtained by conducting a Monte Carlo simulation experiment shows that these new procedures are promising. Future research that should promote application of the proposed methods for solving classification problems in a business decision-making environment is discussed.     

The Role of Society in Engineering Risk Analysis: A Capabilities-Based Approach

This article proposes a new conceptual framework in engineering risk analysis to account for the net impact of hazards on individuals in a society. It analyzes four limitations of prevailing approaches to risk analysis and suggests a way to overcome them. These limitations are a result of how societal impacts are characteristically accounted for and valued. Prevailing approaches typically focus too narrowly on the consequences of natural or man-made hazards, not accounting for the broader societal impacts of such hazards. Such approaches lack a uniform and consistent metric for accounting for the impact of the nonquantifiable consequences (like psychological trauma or societal impacts) and rely upon implicit and potentially inaccurate value judgments when evaluating risks. To overcome these limitations, we propose an alternative, Capabilities-Based Approach to the treatment of society in risk analysis. A similar approach is currently used by the United Nations to quantitatively measure the degree of development in countries around the world. In a Capabilities-Based Approach, the potential benefits and losses due to a hazard are measured and compared in a uniform way by using individual capabilities (functionings individuals are able, still able, or unable to achieve) as a metric. This Capabilities-Based Approach provides a foundation for identifying and quantifying the broader, complex societal consequences of hazards and is based on explicit, value judgments. The Capabilities-Based Approach can accommodate different methods or techniques for risk determination and for risk evaluation and can be used in assessing risk in diverse types of hazards (natural or man-made) and different magnitudes that range from minor to catastrophic. In addition, implementing a Capabilities-Based Approach contributes to the development of a single standard for public policy decision making, since a Capabilities-Based Approach is already in use in development economics and policy.     

A robust approach to the share-of-choice product design problem

A critical issue when solving the share-of-choice product design problem is the reliability of the optimal solution in the presence of partworth uncertainty. Existing approaches use point estimates of an individual's partworth utilities as input to the product optimization stage, ignoring within-person variability in estimates. Post-optimality sensitivity analysis is occasionally performed to assess the degree to which a solution is negatively impacted by partworth uncertainty. We propose a robust optimization model that explicitly captures variation in partworth estimates during the optimization process. Using a large, commercial dataset, we benchmark our model's performance against its deterministic counterpart. We also present inferential theory to guide the selection of model parameters controlled by the analyst. Results reveal that the new approach produces robust solutions in the face of measurement error. Out-of-sample coverage for individuals drawn from the target population is significantly higher than corresponding solutions from published methods.     

DNMA: A double normalization-based multiple aggregation method for multi-expert multi-criteria decision making

This paper develops a comprehensive algorithm for multi-expert multi-criteria decision making problems considering quantitative and qualitative criteria in forms of benefit, cost or target types. We focus on using probabilistic linguistic term sets to express the qualitative evaluations due to their excellence in expressing complex individual and collective linguistic assessments. Firstly, we develop a target-based linear normalization technique and a target-based vector normalization technique. A weight adjustment method is proposed to achieve the tradeoff between criteria after normalization. Given that the two target-based normalization techniques have different advantages, we then propose a ranking method, which consists three subordinate models, based on these two target-based normalization approaches and three aggregation techniques. Reliable results of a multi-expert multi-criteria decision making problem are determined by integrating the subordinate utility values and the ranks of alternatives. The proposed method is implemented to solve the green enterprise ranking problems and the excavation scheme selection problem for shallow buried tunnels, respectively. The advantages of the proposed method are emphasized through comparative analyses with other ranking methods.     

Multiple criteria decision making for maritime investment policy in the developing countries

This model for multiple criteria decision making enables one to select an optimal investment policy for developing the marine industry in developing countries. The model is composed of: the mathematical model of national economy (LP model) for optimizing an investment scheme under economic and technological constraints; the model of I-O analysis for estimating the change of industrial structures induced by the investment; and the decision making model to which the results of the preceding models are fed back so as to enable the policy planner to select a feasible policy based on his own multiple criteria such as the value of GDP growth rate, international payments, unemployment rate, etc. Due consideration is paid in this study to the role of the marine industries, e.g. shipping and ports, which can be said to be the pivot of economic development in a developing country in connection with its foreign trade policy. A case study is carried out to verify this hypothesis.