Interactive Multiple Objective Linear Programming Research Articles

An interactive MOLP method for solving output-oriented DEA problems with undesirable factors

Data Envelopment Analysis (DEA) and Multiple Objective Linear Programming (MOLP) are widely used for performance assessment in organizations. Although DEA and MOLP are similar in structure, DEA is used to assess and analyze past performance and MOLP is used to predict future performance. Several equivalence models between output-oriented DEA models and MOLP models have been proposed in the literature. However these models are not applicable to performance evaluation problems with undesirable outputs. We propose an interactive method for solving output-oriented DEA models with undesirable outputs. We show that the output-oriented BCC model of Seiford and Zhu [47] can be equivalently stated as the maximization of the minimum of several objectives over the production possibility set, which in turn is a scalarization of a multi-objective linear program. We then employ the well-known Zionts-Wallenius procedure to solve the multi-objective optimization problem. We present an example to demonstrate the applicability of the proposed method and exhibit the efficacy of the procedures and algorithms.

Integrating DEA-oriented performance assessment and target setting using interactive MOLP methods

Data envelopment analysis (DEA) and multiple objective linear programming (MOLP) are tools that can be used in management control and planning. Whilst these two types of model are similar in structure, DEA is directed to assessing past performances as part of management control function and MOLP to planning future performance targets. This paper is devoted to investigating equivalence models and interactive tradeoff analysis procedures in MOLP, such that DEA-oriented performance assessment and target setting can be integrated in a way that the decision makers' preferences can be taken into account in an interactive fashion. Three equivalence models are investigated between the output-oriented dual DEA model and the minimax reference point formulations, namely the super-ideal point model, the ideal point model and the shortest distance model. These models can be used to support efficiency analysis in the same way as the conventional DEA model does and also support tradeoff analysis for setting target values by individuals or groups. A case study is conducted to illustrate how DEA-oriented efficiency analysis can be conducted using the MOLP methods and how such performance assessment can be integrated into an interactive procedure for setting realistic target values.

Using interactive multiobjective methods to solve DEA problems with value judgements

Data envelopment analysis (DEA) is a performance measurement tool that was initially developed without consideration of the decision maker (DM)'s preference structures. Ever since, there has been a wide literature incorporating DEA with value judgements such as the goal and target setting models. However, most of these models require prior judgements on target or weight setting. This paper will establish an equivalence model between DEA and multiple objective linear programming (MOLP) and show how a DEA problem can be solved interactively without any prior judgements by transforming it into an MOLP formulation. Various interactive multiobjective models would be used to solve DEA problems with the aid of PROMOIN, an interactive multiobjective programming software tool. The DM can then search along the efficient frontier to locate the most preferred solution where resource allocation and target levels based on the DM's value judgements can be set. An application on the efficiency analysis of retail banks in the UK is examined. Comparisons of the results among the interactive MOLP methods are investigated and recommendations on which method may best fit the data set and the DM's preferences will be made.

Global Optimization in Practice: An Application to Interactive MultipleObjective Linear Programming

A multiple objective linear programming problem (P) involves the simultaneous maximization of two or more conflicting linear objective functions over a nonempty polyhedron X. Many of the most popular methods for solving this type of problem, including many well-known interactive methods, involve searching the efficient set X_E of the problem. Generally, however, X_E is a complicated, nonconvex set. As a result, concepts and methods from global optimization may be useful in searching X_E. In this paper, we will explain in theory, and show via an actual application to citrus rootstock selection in Florida, how the potential usefulness of the well-known interactive method STEM for solving problem (P) in this way, can depend crucially upon how accurately certain global optimization problems involving minimizations over X_E are solved. In particular, we will show both in theory and in practice that the choice of whether to use the popular but unreliable ’’payoff table‘‘ approach or to use one of the lesser known, more accurate global optimization methods to solve these problems can determine whether STEM succeeds or fails as a decision aid. Several lessons and conclusions of transferable value derived from this research are also given.

Aspiration-Based Search Algorithm (ABSALG) for Multiple Objective Linear Programming Problems: Theory and Comparative Tests

We develop an interactive method for multiple objective linear programming based on aspiration levels of a decision maker. The method assumes an unknown pseudoconcave preference structure of a decision maker throughout the decision process, and the decision maker's ability to select a preferred solution from p + 1 alternatives, where p is the number of objectives. In addition to presenting the supporting theory and algorithm, we perform a comparative study using a fictitious decision maker, comparing our approach to those of Steuer and Choo (Steuer, R. E., E. Choo. 1983. An interactive weighted Tchebycheff procedure for multiple objective programming. Math. Prog. 26 326–344.) and Reeves and Franz (Reeves, G. R., L. Franz. 1985. A simplified interactive multiple objective linear programming procedure. Comp. & Oper. Res. 12 589–601.). All three methods are interactive. During an iteration, each method presents several solution alternatives to the decision maker simultaneously. Our approach utilizes a Tchebycheff function that facilitates attainment of an optimum at a nonextreme point solution. The statistics collected in the comparative study provide insights into the nature of the algorithms and the behavior of the solution techniques with different categories of problem structure and different underlying utility functions.

Tchebycheff norms in multi-objective linear programming

Interactive multiple objective linear programming techniques relying upon weighted combined objective functions have been criticized because they are limited to generating model extreme points. Tchebycheff norms provide the ability to reach other points on the nondominated surface, by minimizing the maximum deviation from targets. Rather than being an L 1 metric as used in conventional linear programming, or an L 2 metric used in least-squares regression, Tchebycheff norms minimize an L ∞ metric. Ideal points can be estimated from payoff tables, and Tchebycheff norms used to minimize the distance from the current estimate of the ideal point. This paper compares two Tchebycheff methods for multiple objective linear programming.

Using efficient feasible directions in interactive multiple objective linear programming

In this paper we show how to incorporate efficient feasible directions into interactive line search algorithms for multiple objective linear programming problems. The resulting new line search is guaranteed to generate points that lie in the efficient set. In addition, the new line search automatically corrects possible errors in judgment that the decision maker may inadvertently commit in giving some of his responses. These two improvements are achieved without scarificing the key property of typical interactive line search algorithms, namely that they yield a new point at each iteration more preferred by the decision maker than the current point. The only additional computational requirement of the new line search is the solution of a single linear programming problem at each iteration. For these reasons, we advocate the incorporation of efficient feasible directions into interactive line search algorithms for multiple objective linear programs in the manner shown in this paper.

Computational efficiency and interactive molp algorithms: An implementation of the simolp procedure

This paper presents an investigation of computational considerations that are relevant to implementing interactive procedures for solving large scale multiobjective linear programming (MOLP) problems. Four approaches to implementing the simplified interactive multiple objective linear programming (SIMOLP) procedure of Reeves and Franz [4. Computers Opns Res. 12, 589–601 (1985)] are compared. Statistics on the number of simplex iterations and CPU time required for each approach arc presented for a set of 75 randomly generated problems. The results suggest several techniques that may be applicable to interactive algorithms in general. Significant, systematic reductions in the computational requirements are obtained for a wide variety of test problems.

Preference Structure Representation Using Convex Cones in Multicriteria Integer Programming

A new efficient system of representing the decision-maker's preference structure in solving multicriteria integer programming problems is developed. The problem is solved by an interactive branch-and-bound method that employs the procedure of Zionts and Wallenius (Zionts, S., J. Wallenius. 1983. An interactive multiple objective linear programming method for a class of underlying nonlinear utility functions. Management Sci. 29(5).) for multicriteria linear programming. The decision-maker's underlying utility function is assumed to be pseudoconcave, and his pairwise comparisons of decision alternatives are used to determine his preference structure in terms of certain convex cones in the objective function space and constraints on the weights on the objectives in the weight space. The two forms of preference structure representation are interrelated, and their underlying theory is developed. The primary objective of a representation scheme is exactness, and, in this respect, it is shown that the constraints on the weights are not adequate for representing nonlinear utility functions. On the other hand, the convex cones exactly represent any quasiconcave utility function and clearly avoid the approximations and inaccuracies in other utility assessment systems. Accordingly, an efficient ordered representation using convex cones is developed. An algorithmic framework for multicriteria integer programming that integrates the representation using convex cones with the branch-and-bound solution procedure is developed. Computational experience with bicriteria problems having up to 80 variables and 40 constraints is presented.

Use of the displaced worst compromise in interactive multiobjective programming

The development and application of an interactive multiple-objective linear programming algorithm are considered. The displaced worst compromise is used as a reference point, to select the most preferred decision alternative. This point represents the undesired target values, which a decision-maker would prefer to avoid. To reflect the dynamics of a decisionmaker's preferences, use is made of the concept of a displaced point which captures changes of reference during the decision process. An illustrative example is presented to show the process of reaching a final compromise decision.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An interactive multiple objective linear programming method based on piecewise-linear additive value functions

S. Zionts and J. Wallenius (1976) have discussed how their method for interactive multiple-objective linear programming can be extended to handle pseudoconcave utility functions. They do not, however, specify how an implied interactive search over faces of the simplex is to be done. E. Jacquet-Lagreze and J. Siskos (1982) have introduced the idea of approximating a utility function by a piecewise-linear additive form, i.e. a separable programming approach. It is shown how these two concepts can be brought together in an interactive method for multiple-objective linear programming that avoids the unspecified search of Zionts and Wallenius but is nevertheless able to approximate the true preferred solution when the utility function is concave. This method is applied to two hypothetical problems, and its performance is compared with that of the Zionts-Wallenius procedure. The major advantage of the proposed method is that it uses a single procedure, requiring only pairwise preference statements from the decision-maker.

Interactive multiple objective linear programming system implemented on a microcomputer

Linear programming is one of the most widely used Operations Research/Management Science techniques. Recently, multiple objective decision making has been well established as a practical approach to seek a satisfactory solution to a decision making problem. Much attention has been focused on a microcomputer as an economical management tool. In this paper we propose an interactive goal attainment method using the eigenvector algorithm for solving a multiple objective linear programming problem interactively on microcomputers. In the software package Micro-LPS based on the method proposed, we design a conversational and user-friendly system in which the user commands are involved.

A simplified interactive multiple objective linear programming procedure

Differences in decision making styles and decision settings have motivated the development of a variety of interactive approaches to assist decision makers (DMs) in determining preferred solutions to multiple criteria decision making problems. This paper presents a Simplified Interactive Multiple Objective Linear Programming (SIMOLP) procedure which is designed to appeal to a broad range of DMs because of its simplicity and flexibility. Specifically, at any stage in the interactive process, the procedure attempts to 1. (1) minimize the inputs required from the DMs; 2. (2) provide DMs with a representative set of decision alternatives while simultaneously limiting the number of alternatives considered; and 3. (3) allow DMs to change their minds about previously eliminated alternatives. The procedure is illustrated with both graphical and numerical examples and computational experience is discussed.

An Interactive Multiple Objective Linear Programming Method for a Class of Underlying Nonlinear Utility Functions

This paper develops a method for interactive multiple objective linear programming assuming an unknown pseudo concave utility function satisfying certain general properties. The method is an extension of our earlier method published in this journal (Zionts, S., Wallenius, J. 1976. An interactive programming method for solving the multiple criteria problem. Management Sci. 22 (6) 652–663.). Various technical problems present in predecessor versions have been resolved. In addition to presenting the supporting theory and algorithm, we discuss certain options in implementation and summarize our practical experience with several versions of the method.

An interactive multiple-objective linear programming approach to a problem in forest management

In many situations it is under legislative mandate to manage publicly owned forest resources for multiple uses (e.g., timber production, hunting, grazing). The major obstacle that has been encountered in applying previously developed mathematical programming procedures to multiple-use forest management has been the difficulty in assessing the appropriate criterion weights required. To avoid the criterion weight estimation problem, an interactive multiple-objective linear programming approach, which does not require criterion weights of any kind, was developed in response to the needs of the multiple-use forest management problem. The procedure uses a combination of linear programming and vector-maximum techniques. At each iteration the cone generated by the gradients of the multiple objectives is contracted. On the last two iterations the most acceptable efficient extreme point is identified with the aid of a filtering device. As illustrated, the method has been applied to prepare preliminary management plans for a 10 000-acre subunit of a national forest.

An Interactive Multiple-Objective Linear Programming Approach to a Problem in Forest Management

In many situations it is under legislative mandate to manage publicly owned forest resources for multiple uses (e.g., timber production, hunting, grazing). The major obstacle that has been encountered in applying previously developed mathematical programming procedures to multiple-use forest management has been the difficulty in assessing the appropriate criterion weights required. To avoid the criterion weight estimation problem, an interactive multiple-objective linear programming approach, which does not require criterion weights of any kind, was developed in response to the needs of the multiple-use forest management problem. The procedure uses a combination of linear programming and vector-maximum techniques. At each iteration the cone generated by the gradients of the multiple objectives is contracted. On the last two iterations the most acceptable efficient extreme point is identified with the aid of a filtering device. As illustrated, the method has been applied to prepare preliminary management plans for a 10,000-acre sub-unit of a national forest.