Weak efficiency in multiobjective variational problems under generalized convexity

In this paper, we provide new pseudoinvexity conditions on the involved functionals of a multiobjective variational problem, such that all vector Kuhn-Tucker or Fritz John points are weakly efficient solutions if and only if these conditions are fulfilled. We relate weakly efficient solutions to optimal solutions of weighting problems. We improve recent papers, and we generalize pseudoinvexity conditions used in multiobjective mathematical programming, so as some of their characterization results. The new conditions and results are illustrated with an example.     

Fuzzy goal programming for multiobjective transportation problems

Several fuzzy approaches can be considered for solving multiobjective transportation problem. This paper presents a fuzzy goal programming approach to determine an optimal compromise solution for the multiobjective transportation problem. We assume that each objective function has a fuzzy goal. Also we assign a special type of nonlinear (hyperbolic) membership function to each objective function to describe each fuzzy goal. The approach focuses on minimizing the negative deviation variables from 1 to obtain a compromise solution of the multiobjective transportation problem. We show that the proposed method and the fuzzy programming method are equivalent. In addition, the proposed approach can be applied to solve other multiobjective mathematical programming problems. A numerical example is given to illustrate the efficiency of the proposed approach.     

An Algorithm for Solving the Linear Goal Programming Problem by Solving its Dual

Goal programming, and in particular lexicographic goal programming (i.e. goal programming within a so-called ‘pre-emptive priority’ structure or having non-Archimedean weights), has become one of the most widely used of the approaches for multi-objective mathematical programming. While also applicable to non-linear or integer models, most of the literature has considered the lexicographic linear goal-programming model and its solution via primal simplex-based methods. However, in many cases, enhanced efficiency (and significant additional flexibility) may be gained via an investigation of the dual of this problem. In this paper we consider an algorithm for solving such a dual and also indicate how it may be implemented on conventional (i.e. single objective) simplex software.     

On computational methods for solutions of multiobjective linear production programming games

In this paper we consider a production model in which multiple decision makers pool resources to produce finished goods. Such a production model, which is assumed to be linear, can be formulated as a multiobjective linear programming problem. It is shown that a multi-commodity game arises from the multiobjective linear production programming problem with multiple decision makers and such a game is referred to as a multiobjective linear production programming game. The characteristic sets in the game can be obtained by finding the set of all the Pareto extreme points of the multiobjective programming problem. It is proven that the core of the game is not empty, and points in the core are computed by using the duality theory of multiobjective linear programming problems. Moreover, the least core and the nucleolus of the game are examined. Finally, we consider a situation that decision makers first optimize their multiobjective linear production programming problem and then they examine allocation of profits and/or costs. Computational methods are developed and illustrative numerical examples are given.     

Invex Functions and Generalized Convexity in Multiobjective Programming

Martin (Ref. 1) studied the optimality conditions of invex functions for scalar programming problems. In this work, we generalize his results making them applicable to vectorial optimization problems. We prove that the equivalence between minima and stationary points or Kuhn–Tucker points (depending on the case) remains true if we optimize several objective functions instead of one objective function. To this end, we define accurately stationary points and Kuhn–Tucker optimality conditions for multiobjective programming problems. We see that the Martin results cannot be improved in mathematical programming, because the new types of generalized convexity that have appeared over the last few years do not yield any new optimality conditions for mathematical programming problems.     

Measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming

One way of solving multiple objective mathematical programming problems is finding discrete representations of the efficient set. A modified goal of finding good discrete representations of the efficient set would contribute to the practicality of vector maximization algorithms. We define coverage, uniformity and cardinality as the three attributes of quality of discrete representations and introduce a framework that includes these attributes in which discrete representations can be evaluated, compared to each other, and judged satisfactory or unsatisfactory by a Decision Maker. We provide simple mathematical programming formulations that can be used to compute the coverage error of a given discrete representation. Our formulations are practically implementable when the problem under study is a multiobjective linear programming problem. We believe that the interactive algorithms along with the vector maximization methods can make use of our framework and its tools. Received April 7, 1998 / Revised version received March 1999?Published online November 9, 1999     

Nondominated equilibrium solutions of a multiobjective two-person nonzero-sum game in extensive form and corresponding mathematical programming problem

In most of studies on multiobjective noncooperative games, games are represented in normal form and a solution concept of Pareto equilibrium solutions which is an extension of Nash equilibrium solutions has been focused on. However, for analyzing economic situations and modeling real world applications, we often see cases where the extensive form representation of games is more appropriate than the normal form representation. In this paper, in a multiobjective two-person nonzero-sum game in extensive form, we employ the sequence form of strategy representation to define a nondominated equilibrium solution which is an extension of a Pareto equilibrium solution, and provide a necessary and sufficient condition that a pair of realization plans, which are strategies of players in sequence form, is a nondominated equilibrium solution. Using the necessary and sufficient condition, we formulate a mathematical programming problem yielding nondominated equilibrium solutions. Finally, giving a numerical example, we demonstrate that nondominated equilibrium solutions can be obtained by solving the formulated mathematical programming problem.     

A goal programming model for bank assets and liabilities management

Assets and liabilities management is one of the most important issues in bank strategic planning. In the past, this problem has often been addressed through conventional mathematical programming, i.e. linear programming.However, bank management typically involves several conflicting goals, such as the maximisation of returns, minimisation of risk, expansion of deposits and loans, etc. The complexity of this problem can be captured more adequately by multiobjective mathematical programming. This paper discusses the construction and application at the Commercial Bank of Greece of a goal programming model that takes into account the essential institutional, financial, legal and bank policy considerations.     

Solving a multiobjective possibilistic problem through compromise programming

Real decision problems usually consider several objectives that have parameters which are often given by the decision maker in an imprecise way. It is possible to handle these kinds of problems through multiple criteria models in terms of possibility theory.Here we propose a method for solving these kinds of models through a fuzzy compromise programming approach.To formulate a fuzzy compromise programming problem from a possibilistic multiobjective linear programming problem the fuzzy ideal solution concept is introduced. This concept is based on soft preference and indifference relationships and on canonical representation of fuzzy numbers by means of their α-cuts. The accuracy between the ideal solution and the objective values is evaluated handling the fuzzy parameters through their expected intervals and a definition of discrepancy between intervals is introduced in our analysis.     

Multiobjective data envelopment analysis

Data envelopment analysis (DEA) is popularly used to evaluate relative efficiency among public or private firms. Most DEA models are established by individually maximizing each firm's efficiency according to its advantageous expectation by a ratio. Some scholars have pointed out the interesting relationship between the multiobjective linear programming (MOLP) problem and the DEA problem. They also introduced the common weight approach to DEA based on MOLP. This paper proposes a new linear programming problem for computing the efficiency of a decision-making unit (DMU). The proposed model differs from traditional and existing multiobjective DEA models in that its objective function is the difference between inputs and outputs instead of the outputs/inputs ratio. Then an MOLP problem, based on the introduced linear programming problem, is formulated for the computation of common weights for all DMUs. To be precise, the modified Chebychev distance and the ideal point of MOLP are used to generate common weights. The dual problem of this model is also investigated. Finally, this study presents an actual case study analysing R&D efficiency of 10 TFT-LCD companies in Taiwan to illustrate this new approach. Our model demonstrates better performance than the traditional DEA model as well as some of the most important existing multiobjective DEA models.     

COTS selection using fuzzy interactive approach

In this paper, we introduce fuzzy mathematical programming (FMP) for decision-making related to software creation by selecting optimal commercial-off-the-shelf (COTS) products in a modular software system. Each module in such software systems have different alternatives with variations in their properties, for example, quality, reliability, execution time, size and cost. Due to these variations, component-based software developers generally deals with the problem of selecting appropriate COTS products. The development of COTS-based systems largely depends on the success of the selection process. Various crisp optimization models of COTS products selection have been proposed in literature. However, in real COTS products selection problem, it is difficult to estimate precisely the values of various model parameters due to lack of sufficient data and also because of measurement errors. Hence, instead of crisp optimization model, if we use flexible optimization model then we might obtain results which are more preferred by the decision maker. In this study, we use multiple methodologies such as quality model, analytical hierarchy process and FMP to develop fuzzy multiobjective optimization model of the COTS products selection. To determine a preferred compromise solution for the multiobjective optimization problem, an interactive fuzzy approach is used.     

A class of multiobjective linear programming models with random rough coefficients

In the present paper, we concentrate on dealing with a class of multiobjective programming problems with random rough coefficients. We first discuss how to turn a constrained model with random rough variables into crisp equivalent models. Then an interactive algorithm which is similar to the interactive fuzzy satisfying method is introduced to obtain the decision maker’s satisfying solution. In addition, the technique of random rough simulation is applied to deal with general random rough objective functions and random rough constraints which are usually hard to convert into their crisp equivalents. Furthermore, combined with the techniques of random rough simulation, a genetic algorithm using the compromise approach is designed for solving a random rough multiobjective programming problem. Finally, illustrative examples are given in order to show the application of the proposed models and algorithms.     

Solving the multiobjective possibilistic linear programming problem

In conventional multiobjective decision making problems, the estimation of the parameters of the model is often a problematic task. Normally they are either given by the decision maker (DM), who has imprecise information and/or expresses his considerations subjectively, or by statistical inference from past data and their stability is doubtful. Therefore, it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets for which a lot of fuzzy approaches to multiobjective programming have been developed. In this paper we propose a method to solve a multiobjective linear programming problem involving fuzzy parameters (FP-MOLP), whose possibility distributions are given by fuzzy numbers, estimated from the information provided by the DM. As the parameters, intervening in the model, are fuzzy the solutions will be also fuzzy. We propose a new Pareto Optimal Solution concept for fuzzy multiobjective programming problems. It is based on the extension principle and the joint possibility distribution of the fuzzy parameters of the problem. The method relies on α-cuts of the fuzzy solution to generate its possibility distributions. These ideas are illustrated with a numerical example.     

Computationally Efficient Methods for Calculating the Ideal Solution for MOLP Problems

This paper presents the results of an investigation into computational considerations that are relevant to large-scale multiobjective linear programming (MOLP) problems. Four approaches to obtaining a representation of the ideal solution are compared. Statistics on the number of simplex iterations and CPU time required are analysed for a set of randomly generated multiobjective linear programming problems. Recommendations are made based on the analysis of these results which are applicable to many MOLP solution algorithms.     

Tackling an MCDM-problem with the help of some results from fuzzy set theory

The problem to be addressed and tackled in this paper arose as a byproduct from some efforts at solving problems involving multiple goals by linking linear and goal programming models. The critical issue was that some forms for interdependence among the goals could not be handled in the programming models. Here we will deal with a set of goals — with realistic counterparts in a Finnish plywood industry — in which a subset of the goals are (i) conflicting, another subset (ii) unilaterally supporting and a third subset (iii) mutually supporting. It is furthermore observed that the elements of a studied set of goals may be partly independent and partly interdependent, which makes the context a fullfledged MCDM-problem. It is tackled with a technique which is based on the theory of fuzzy sets, the conceptual framework for fuzzy decisions and the algorithms developed for fuzzy mathematical programming. The resulting fuzzy multiobjective programming model is simplified and tested with the help of a fairly complex numerical example.     

A review of operations research models in invasive species management: state of the art,challenges, and future directions

Invasive species are a major threat to the economy, the environment, health, and thus human well-being. The international community, including the United Nations’ Global Invasive Species Program (GISP), National Invasive Species Council (NISC), and Center for Invasive Species Management (CISM), has called for a rapid control of invaders in order to minimize their adverse impacts. The effective management of invasive species is a highly complex problem requiring the development of decision tools that help managers prioritize actions most efficiently by considering corresponding bio-economic costs, impacts on ecosystems, and benefits of control. Operations research methods, such as mathematical programming models, are powerful tools for evaluating different management strategies and providing optimal decisions for allocating limited resources to control invaders. In this paper, we summarize the mathematical models applied to optimize invasive species prevention, surveillance, and control. We first define key concepts in invasive species management (ISM) in a framework that characterizes biological invasions, associated economic and environmental costs, and their management. We then present a spatio-temporal optimization model that illustrates various biological and economic aspects of an ISM problem. Next, we classify the relevant literature with respect to modeling methods: optimal control, stochastic dynamic programming, linear programming, mixed-integer programming, simulation models, and others. We further classify the ISM models with respect to the solution method used, their focus and objectives, and the specific application considered. We discuss limitations of the existing research and provide several directions for further research in optimizing ISM planning. Our review highlights the fact that operations research could play a key role in ISM and environmental decision-making, in particular closing the gap between the decision-support needs of managers and the decision-making tools currently available to management.     

The linearization method: Principal concepts and perspective directions

The linearization method, for solving the general problem of nonlinear programming and its various modifications, is considered. On the basic ideas of the linearization method, the algorithms for solving the various problems of mathematical programming are constructed for (a) solving systems of equalities and inequalities, (b) multiobjective programming and (c) complementary problem.     

Some results on symmetric duality of multiobjective programmes with generalized (F,ρ) invexity

In this present article we have given some multiobjective programming problems with their symmetric duals and have derived weak and strong duality results with respect to such programs. Moreover, we have also used most general type of invexity assumptions involved with the functions which are related to the programming problems. It is to be pointed out that the objective functions in such programs contain terms like support functions which in turn are able to give results on particular classes of programs involving quadratic terms. Our results in particular give as special cases some earlier results on symmetric duals given in the current literature.     

Multiobjective programming with new invexities

(Φ, ρ)-invexity and (Φ, ρ) ^w -invexity generalize known invexity type properties and have been introduced with the intent of extending most of theoretical results in mathematical programming. Here, we push this approach further, to obtain authentic extensions of previously known optimality and duality results in multiobjective programming.     

Multiobjective mathematical programming models for acid rain control

Although acid rain control is inherently multiobjective, previous optimization approaches have generally been single-objective, often acting to minimize aggregate abatement cost or emission reductions. Using an updated, least-cost deposition-constrained deterministic model as a basic framework, three multiobjective models are developed that consist of formulations which permit deviations about target deposition levels, the addition of constraints to effect measures of equity and models to enforce restrictions on aggregate emission reduction tonnage. The deposition deviation model shows that large abatement cost savings can be realized if the hard upper bound on maximum allowable deposition limit is preferentially relaxed. The socalled equity model develops strategies that attempt to balance within each state and province, the disparity between fractional emission and fractional deposition reductions. The aggregate emission reduction model shows some of the effects associated with the imposition of a common type of acid rain proposal. Our intent is to demonstrate that the incorporation of multiobjectivity into mathematical programming models for optimizing acid rain control constitutes an important step toward the identification of more representative, more useful and hopefully, scientifically and politically acceptable abatement strategies.