In this paper we propose an approach which makes it possible to search non-dominated and only non-dominated solutions in multiple-objective linear programming. The approach is based on the use of a reference direction...
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Various difficulties arise in using decision set-based vector maximization methods to solve a multiple-objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years hav...
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Various difficulties arise in using decision set-based vector maximization methods to solve a multiple-objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years have begun to develop tools for analyzing and solving problem (MOLP) in outcome space, rather than in decision space. In this article, we present and validate a new hybrid vector maximization approach for solving problem (MOLP) in outcome space. The approach systematically integrates a simplicial partitioning technique into an outer approximation procedure to yield an algorithm that generates the set of all efficient extreme points in the outcome set of problem (MOLP) in a finite number of iterations. Some key potential practical and computational advantages of the approach are indicated.
We introduce in this paper a new multiple-objective linear programming (MOLP) algorithm. The algorithm is based on the single-objective path-following primal-dual linearprogramming algorithm and combines it with aspi...
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The problem (P) of optimizing a linear function d(T)x over the efficient set for a multiple-objectivelinear program (M) is difficult because the efficient set is typically nonconvex. Given the objective function dire...
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The problem (P) of optimizing a linear function d(T)x over the efficient set for a multiple-objectivelinear program (M) is difficult because the efficient set is typically nonconvex. Given the objective function direction d and the set of domination directions D, if d(T) pi greater than or equal to 0 for all nonzero pi epsilon D), then a technique for finding an optimal solution of (P) is presented in Section 2. Otherwise, given a current efficient point ($) over cap chi, if there is no adjacent efficient edge yielding an increase in d(T)x, then a cutting plane d(T)x = dT chi is used to obtain a multiple-objectivelinear program (($) over bar M) with a reduced feasible set and an efficient set ($) over bar E To find a better efficient point, we solve the problem (I-i) of maximizing c(i)(T) chi, over the reduced feasible set in (($) over bar M) sequentially for i. If there is a x(i) epsilon ($) over bar E that is an optimal solution of (I-i) for some i and d(T) chi(i) > dT ($) over cap chi, then we can choose chi(i) as a current efficient point. Pivoting on the reduced feasible set allows us to find a better efficient point or to show that the current efficient point ($) over cap chi is optimal for (P). Two algorithms for solving (P) in a finite sequence of pivots are presented along with a numerical example.
This paper presents a new, interactive multi-objectivelinear-programming procedure to aid decision-makers in setting up goals for desired outputs. The procedure relies on empirical production functions generated by t...
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This paper presents a new, interactive multi-objectivelinear-programming procedure to aid decision-makers in setting up goals for desired outputs. The procedure relies on empirical production functions generated by the use of data envelopment analysis. It presents the decision-maker with a set of alternative efficient points in order either to compare different sets of inputs in terms of their effectiveness for goal achievement, or to set goals against which future management performance may be measured. With each iteration the new information provided by the decision-maker is used to adjust the procedure, leading to points which have greater effectiveness utility for the decision-maker. A numerical example is provided along with guidelines for future applications.
Often, the coefficients of a linearprogramming problem represent estimates of true values of data or are subject to systematic variations. In such cases, it is useful to perturb the original data and to either comput...
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Often, the coefficients of a linearprogramming problem represent estimates of true values of data or are subject to systematic variations. In such cases, it is useful to perturb the original data and to either compute, estimate, or otherwise describe the values of the functionf which gives the optimal value of the linear program for each perturbation. If the right-hand derivative off at a chosen point exists and is calculated, then the values off in a neighborhood of that point can be estimated. However, if the optimal solution set of either the primal problem or the dual problem is unbounded, then this derivative may not exist. In this note, we show that, frequently, even if the primal problem or the dual problem has an unbounded optimal solution set, the nature of the values off at points near a given point can be investigated. To illustrate the potential utility of our results, their application to two types of problems is also explained.
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