An Interactive Framework for Multi-Person,Multiobjective Decisions

Group decision making in the presence of multiple conflicting objectives is complex and difficult. This paper describes and evaluates an iterative technique to facilitate multiple objective decision making by multiple decision makers. The proposed method augments an interactive multiobjective optimization procedure with a preference ranking tool and a consensus ranking heuristic. Two multiple objective linear programming (MOLP) solution approaches, the SIMOLP method of Reeves and Franz [39] and the interactive weighted Tchebycheff procedure of Steuer and Choo [49], are recommended optimization strategies to be used independently or in concert. Computational experience suggests that the proposed framework is an effective decision-making tool. The procedure quickly located excellent compromise solutions in a series of test problems with hypothetical decision makers. In addition, human decision makers gave positive evaluations of the procedure and the production plans the procedure provided for a resource allocation case problem.     

ON FUZZY GOAL PROGRAMMING*

This paper illustrates how the goal programming problem with fuzzy goals having linear membership functions may be formulated as a single goal programming problem. Also, a previously defined method for dealing with fuzzy weights for each of the goals is re-examined.     

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.     

A NOTE ON DECISION-MAKING CRITERIA FOR ALGORITHM SELECTION: REDUCING GOAL PROGRAMMING COMPUTATIONAL EFFORT

Most linear goal programming (LPG) algorithms are based on a simplex-type solution method that begins with an initial simplex tableau with solution-set variables (basic variables) consisting of all-negative deviational variables or all-positive deviational variables. This note (1) demonstrates how computational solution effort can be reduced by selecting the appropriate all-negative or all-positive deviational variable algorithm and (2) describes a procedure that can be used to aid decision makers in selecting the appropriate algorithm for all types of applied goal programming (GP) models. Results of this study reveal the procedure as accurate and providing computational time savings when applied to a large sample of GP problems.     

Use and Abuse of Weights in Multiple Objective Decision Support Models*

This paper examines difficulties with the use of weights to solve multiple objective decision support models: misunderstanding of the meaning of weights, issues of commensurability and, most important, the likely inability of weights alone to isolate the decision maker's most-preferred point. The constraint method is shown to be an attractive alternative.     

A NOTE ON USER SOLUTION STRATEGY FOR MIXED-INTEGER LINEAR PROGRAMMING MODELS

Large-scale mixed-integer linear programming (MILP) models may easily prove extraordinarily difficult to solve, even with efficient commercially implemented MILP solution codes. Drawing on experience gained in solving and analyzing three intertemporal investment planning MILP models for electric power supply, this note offers several practical suggestions for reducing computer solution times for general production-allocation MILP models. Solution time reduction stems from judicious use of the powerful computational capabilities of existing commercial linear programming codes in conjunction with information known or to be learned by the practitioner about the model's structure.     

Optimal Design of a Two-Level Transportation Network with a Different Unit Cost for Each Secondary Link*

The objective of this paper is to further develop Singhal's (1990) framework for designing a two-level hierarchical transportation network consisting of a trunk or primary link and several feeder or secondary links. Secondary links are perpendicular lines from each of the given points and the primary link is a straight line or curve connecting the feet of two extreme perpendicular lines. The problem and the associated strategic and operational considerations such as cost, time, feasibility, and preferred regions for the primary link in the context of rural highway planning are discussed. Two-level networks are also common in electricity transmission, pipelines, and telecommunication design. The core of the framework is a model for finding the path of a primary link such that a weighted sum of the lengths of the perpendicular lines from each point to a linear primary link and the distance between the feet of the two extreme perpendicular lines is minimized. The analysis shows that for almost every problem there exists a wide range of solutions for which the total cost is only slightly higher than that of the optimal solution. This offers considerable flexibility to the decision maker. These solutions can be evaluated in view of the broader objectives and constraints that are not included in the model. The use of computer graphics and the option of a nonlinear or piecewise linear primary link are also discussed.     

MODIFIED STEPPING-STONE METHOD AS A TEACHING AID FOR CAPACITATED TRANSPORTATION PROBLEMS

This paper extends the standard stepping-stone method to the case of capacitated transportation problems. The purpose is to offer an alternative to network-oriented or simplex-based methods (such as the out-of-kilter and upper-bounding techniques) often used in handling the capacitated transportation problem but generally beyond the scope of introductory decision sciences courses. The proposed procedure is based on the simple stepping-stone method and requires only a few modifications in the cost/assignment matrix and some alterations in the pivoting rules. As such, it is much easier to understand and apply than the above-mentioned solution algorithms. It therefore can serve as an effective tool for teaching capacitated transportation problems. Two numerical examples illustrate applications of the proposed method, procedures for sensitivity analysis, and comparisons with linear programming.     

Review of Empirical Studies in Multiobjective Mathematical Programming: Subject Reflection of Nonlinear Utility and Learning

Multiple objective programming provides a means of aiding decision makers facing complex decisions where trade-offs among conflicting objectives must be reconciled. Interactive multiobjective programming provides a means for decision makers to learn what these trade-offs involve, while the mathematical program generates solutions that seek improvement of the implied utility of the decision maker. A variety of multiobjective programming techniques have been presented in the multicriteria decision-making literature. This study reviews published studies with human subjects where some of these techniques were applied. While all of the techniques have the ability to support decision makers under conditions of multiple objectives, a number of features in applying these systems have been tested by these studies. A general evolution of techniques is traced, starting with methods relying upon linear combinations of value, to more recent methods capable of reflecting nonlinear trade-offs of value. Support of nonlinear utility and enhancing decision-maker learning are considered.     

An Efficient Zero-One Formulation of the Multilevel Lot-Sizing Problem

This paper presents a zero-one linear formulation of the multilevel lot-sizing problem for materials requirement planning systems without capacity constraints. The model is an efficient statement of the problem and has a structure that is particularly convenient for research work. In addition, it is demonstrated that the relaxed linear programming solution to this formulation will always be integer. The results of a rather large computational history are reported along with a variable reduction methodology that allows for the solution of reasonably sized research problems.     

Modelling the Telecommunication Pricing Decision

The local telecommunication pricing decision on residential, business, and private lines is modeled. This study demonstrates that when regulators use a weighted sum linear goal programming model to determine prices, adjustments can be made to Ramsey prices that reflect the regulators’ concerns about the trade-off of fairness with efficiency. An actual example shows that when regulators’ preferences were taken into account, residential prices were 14 percent lower than efficient optimal Ramsey prices.     

A Multiple Objective Embedded Network Model for Human Resource Planning and an Implementation of the Tchebycheff Method

A multiple objective embedded network model is proposed to model a variety of human resource planning problems including executive succession planning, compensation planning, training program design, diversity management and human systems design. The Tchebycheff Method, an interactive multiple objective programming solution procedure developed by Steuer and Choo [32], is implemented using NETSIDE, a computer routine for solving network problems with side constraints developed by Kennington and Whisman [17]. This paper demonstrates how the network structure common to many types of human resource planning problems can be exploited to improve solution efficiency, and how our approach extends the use of network models in human resource planning by including multiple objectives and extranetwork constraints. An illustrative example demonstrating the modeling and solution approach is presented, and the potential applications of these approaches in two specific areas of human resource planning are discussed.     

Interactive Multiple Objective Decision Support for Sales Force Sizing and Deployment*

An interactive decision aid is introduced for the deployment of two sales resources: salespeople and sales support staff. The aid consists of a normative sales resource allocation model with five objectives and an interactive multiple objective programming solution procedure. The specific decision problem addressed involves the assignment of salespeople and sales support people to customer accounts and the allocation of the time they spend on these accounts. The authors contribute to the existing sales resource modeling literature by dealing with the deployment of two sales resources and interactively solving this problem with respect to five short-run and long-run objectives of the firm. This approach differs from existing sales force modeling efforts in which the solution is found noninteractively by optimizing a single sales resource model with respect to a single objective, often short-run sales. An application of the decision aid to the deployment problem of an industrial sales force manager is presented. Furthermore, useful extensions of the basic sales resource allocation model are discussed.     

ON FUZZY GOAL PROGRAMMING—SOME COMMENTS

This paper pertains to goal programming with fuzzy goals and fuzzy priorities. Hannan [1], in his paper on fuzzy goal programming, alludes to the difficulty of handling fuzzy priorities and further notes that a method that this author proposed [2] may lead to incorrect results. In this note, the general problem of goal programming with fuzzy priorities is reexamined, along with the solution to the specific example presented in my original paper [2]. It is shown that the method for handling fuzzy priorities originally proposed by this author does indeed capture the relative importance of goals.     

PORTFOLIO MODELING IN MULTIPLE-CRITERIA SITUATIONS UNDER UNCERTAINTY

There have been many models for portfolio selection, but most do not explicitly include uncertainty and multiple objectives. This paper presents an approach that includes these aspects using a form of stochastic integer programming with recourse. The method involves the use of a time-based decision tree structure called a “project tree.” Using this basic format, an illustrative six-project example is presented and analyzed. Various forms of objectives are discussed, ranging from the maximization of expected portfolio value to the maximization of the minimum weighted portfolio deviation from two goals. In each case, formulated numerical problems are given, and the solutions derived are presented. The approach is shown to be very flexible and capable of handling a variety of situations and objectives.     

Branch-and-Bound Strategies for the Log Bucking Problem

Tree bucking is the initial production process in converting felled trees into useable wood products. This process has been previously modelled as a dynamic programming problem. Unlike other production problems that have been modelled as dynamic programming problems, there have been no serious attempts to formulate this problem as a branch-and-bound model and then examine the model's performance. This research develops the tree bucking problem as a branch-and-bound model to be tested by varying several parameters. The testing is performed in two phases: (1) a sensitivity analysis is performed to test two key parameters used by the model, and (2) branching strategies are tested on various problem scenarios. The size of the solution sets searched by the technique vary from as low as 40 to as many as 41,000 possible combinations.     

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.     

A PREEMPTIVE VALUE-FUNCTION METHOD APPROACH FOR MULTIOBJECTIVE LINEAR PROGRAMMING PROBLEMS

Many common decision-making situations require integration of a number of objectives or goals into a single objective function to be optimized. A number of techniques for performing this analysis are presented in the literature. These procedures generally require a tradeoff between realism and tractability. The present approach uses the idea of preemptive priority levels with an approximate value function at each level. An efficient experimental design is used to assess directly a quadratic approximation of the multiattribute value functions. An existing algorithm is then used to solve the resulting problem. This procedure is shown to give good results when compared to both the value-function method and goal programming.     

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.     

Optimality conditions for a bilevel matroid problem

In bilevel programming there are two decision makers, the leader and the follower, who act in a hierarchy. In this paper we deal with a weighted matroid problem where each of the decision makers has a different set of weights. The independent set of the matroid that is chosen by the follower determines the payoff to both the leader and the follower according to their different weights. The leader can increase his payoff by changing the weights of the follower, thus influencing the follower’s decision, but he has to pay a penalty for this. We want to find an optimum strategy for the leader. This is a bilevel programming problem with continuous variables in the upper level and a parametric weighted matroid problem in the lower level. We analyze the structure of the lower level problem. We use this structure to develop local optimality criteria for the bilevel problem that can be verified in polynomial time.