Possibilistic Linear Programming Problems Research Articles

Measuring Cost Efficiency with New Fuzzy DEA Models

In conventional cost efficiency models, input and output data and their corresponding input prices are fundamentally known for each decision-making unit. However, the observed values of the input and output data in real problems are sometimes imprecise. This study shows that the cost efficiency evaluation method can be improved to account for situations where input–output data and their corresponding input prices are fuzzy numbers. A new fuzzy cost-minimizing model known as a possibilistic linear programming problem is proposed to evaluate the cost efficiency. In this model data information is considered as triangular fuzzy numbers. Using the \(\alpha\)-level-based approach, the model is transformed into an interval programming problem which is solved as a crisp parametric linear programming model. In addition, new definitions of the fuzzy cost efficiency and cost efficient unit are suggested. Finally, a numerical example is presented to illustrate the proposed method.

Possibilistic Linear Programming Problems involving Normal Random Variables

A new solution procedure of possibilistic linear programming problem is developed involving the right hand side parameters of the constraints as normal random variables with known means and variances and the objective function coefficients are considered as triangular possibility distribution. In order to solve the proposed problem, convert the problem into a crisp equivalent deterministic multi-objective mathematical programming problem and then solved by using fuzzy programming method. A numerical example is presented to illustrate the solution procedure and developed methodology.

An Approach to Solve a Possibilistic Linear Programming Problem

The objective of the paper is to deal with a kind of possibilistic linear programming (PLP) problem involving multiple objectives of conflicting nature. In particular, we have considered a multi objective linear programming (MOLP) problem whose objective is to simultaneously minimize cost and maximize profit in a supply chain where cost and profit coefficients, and related parameters such as available supply, forecast demand and budget are fuzzy with trapezoidal fuzzy numbers. An example is given to illustrate the strategy used to solve the aforesaid PLP problem.

An improvement in possibilistic linear programming approach for imprecise production/transportation planning decisions

In many existing applications of possibilistic linear programming (PLP) model, the decision makers (DM) are asked to specify the positive ideal solution (PIS) and negative ideal solution (NIS) for defining the linear membership function proposed by Zimmermann. In practice, this is a real difficulty for an ordinary DM without possessing acute sense of arithmetic operation and adequate comprehension of PLP model. Furthermore, this task is obviously subjective and will influence the search route of the solution process via the linear membership functions. This study provides an objective method to specify appropriate PIS and NIS for defining efficient linear membership functions so as to rapidly approach a better compromise solution, and also presents a simple way for further improvement of the compromise solution to reach a satisfactory solution. The proposed approaches aim to facilitate the decision making process and promote computational efficiency as well as improve the compromise solution for the PLP method. Additionally, an integrated production/transportation planning decision (PTPD) problem is used to demonstrate the feasibility of applying the proposed methods to real industrial case. Not only the simplicity and objectivity in specifying PIS and NIS but also the computational efficiency, the results of this study show definite advantage of the proposed methods in solving PLP problem.

Management of surgical waiting lists through a Possibilistic Linear Multiobjective Programming problem

This study attempts to apply a management science technique to improve the efficiency of Hospital Administration. We aim to design the performance of the surgical services at a Public Hospital that allows the Decision-Maker to plan surgical scheduling over one year in order to reduce waiting lists. Real decision problems usually involve 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 apply a Possibilistic Linear Multiobjective Programming method, developed by the authors, for solving a hospital management problem using a Fuzzy Compromise Programming approach.

Oblique fuzzy vectors and their use in possibilistic linear programming

In this paper, we propose oblique fuzzy vectors to treat the interactivity among fuzzy numbers. Oblique fuzzy vectors are extensions of fuzzy numbers and vectors of non-interactive fuzzy numbers. The interactivity among fuzzy numbers can be treated by a non-singular matrix in an oblique fuzzy vector. We discuss characterization of an oblique fuzzy vector and the tractability of manipulation of oblique fuzzy vectors in fuzzy linear functions. Moreover, we discuss possibilistic linear programming problems with oblique fuzzy vectors. It is shown that the possibilistic linear programming problems are reduced to linear programming problems with a special structure.

Possibilistic Linear Programming with Fuzzy If-Then Rule Coefficients

In this paper, we propose a scenario decomposition approach for the treatment of interactive fuzzy numbers. Scenario decomposed fuzzy numbers (SDFNs) reflect a fact that we may have different estimations of possible ranges of uncertain variables depending on scenarios, which are expressed by fuzzy if-then rules. The properties of SDFNs are investigated. Possibilistic linear programming problems with SDFNs are formulated by two different approaches, fractile and modality optimization approaches. It is shown that the problems are reduced to linear programming problems in fractile optimization models with the necessity measures and that the problems can be solved by a linear programming technique and a bisection method in modality optimization models with necessity measures. A simple numerical example is given.

Formulation of general possibilistic linear programming problems for complex industrial systems

The current models and methods for PLP are usually restricted on some special types and usually the same type of possibilistic distribution. This paper focuses on linear programming problems with general possibilistic resources (GRPLP) and linear programming problems with general possibilistic objective coefficients (GOPLP). By introducing some new concepts of the largest most possible point, the smallest most possible point, the most optimistic point and the most possible decision, a new approach for formulating possibilistic constraints through general possibilistic resources and a satisfactory solution method will be discussed in this paper. A most possible decision method for GRPLP and a dual approach for GOPLP will be proposed for solving complex industrial decision problems.

Possibility distributions of fuzzy decision variables obtained from possibilistic linear programming problems

In this paper, several kinds of possibility distributions of fuzzy variables are studied in possibilistic linear programming problems to reflect the inherent fuzziness in fuzzy decision problems. Interval and triangular possibility distributions are used to express the non-interactive cases between the fuzzy decision variables, and exponential possibility distributions are used to represent the interrelated cases. Possibilistic linear programming problems based on exponential possibility distributions become non-linear optimization problems. In order to solve optimization problems easily, algorithms for obtaining center vectors and distribution matrices in sequence are proposed. By the proposed algorithms, the possibility distribution of fuzzy decision variables can be obtained.

可能性多目的線形計画問題に対する可能的有効解と感度分析

本論文では, 従来の多目的計画問題における目的関数の係数をファジィ(可能性)係数へと拡張した可能性多目的線形計画問題を取り扱う.この問題の解集合のひとつである可能的有効解集合はファジィ集合となるため, ある実行可能解に関し, その可能的有効性の度合い(メンバシップグレード)を計算する事が必要となる.一般にこの値は非線形計画問題を解くことにより得られるが, 本論文では, 可能性係数を実用上頻繁に利用される台形型ファジィ数に限定し, その双対問題を考えることにより可能的有効性の度合いを計算する方法を提案する.また, その計算の際に得られる情報は感度分析的な利用が可能なため, 可能的有効解集合を決定した後に, 意思決定者によるファジィ数の再設定の検討を行うことが可能となる.

A possibilistic linear program is equivalent to a stochastic linear program in a special case

In this paper, we discuss an equivalent condition between a possibilistic linear programming problem with a quadratic membership function and a stochastic linear programming problem with a multivariate normal distribution. First, the possibilistic linear programming problem is formulated as a usual mathematical programming problem based on maximizing the necessity measure. Next, the stochastic linear programming problem is formulated as a usual mathematical programming problem by introducing the flexible programming approach. Since these two formulated programming problems are similar in their forms, an equivalent condition between these problems is given. Lastly, the equivalence is examined from a geometric viewpoint.

Possible and necessary optimality tests in possibilistic linear programming problems

In this paper, two kinds of optimality concepts are introduced into fuzzy linear programming problems in order to discuss the rationality of an obtained solution. One is possible optimality evaluating the possibility degree to which the solution is optimal. The other, the dual concept, is necessary optimality evaluating the necessity degree to which the solution is optimal. When possibility distributions are defined by closed intervals, degrees of possible and necessary optimalities become binary. Necessary and sufficient conditions for possible and necessary optimalities are induced in this case. Extending the results, methods of calculating the degrees of possible and necessary optimalities are proposed. Some numerical examples are given to demonstrate the proposed methods.

An extension to possibilistic linear programming

This paper proposes an extension to the solution of possibilistic linear programming problems with fuzzy number parameters (introduced by Buckley [ Fuzzy Sets and Systems 31 (1989) 329–341]). The method relies on α-cuts of the fuzzy number parameters to generate a succession of pairs of classical linear programs. The solutions of these linear programs produce a convex possibility distribution of the optimal objective values with the corresponding levels of the decision variables. The approach is illustrated with an imprecise water quality management problem.

A stochastic possibilistic programming model for bank hedging decision problems

Abstract For a practical bank hedging decision optimization problem, interest rates and price of futures contract may involve both fuzziness and randomness. For subjective nature of satisfaction, maximum desired values of loan demand, deposit supply and ratio of desired loan to deposit are often fuzzy. In this study, we consider and solve a stochastic possibilistic programming model of bank hedging decision problems with the above characters. We first use the expected value to obtain an auxiliary possibilistic linear programming problem which is further resolved by use of β-level cut. An (crisp) auxiliary bi-objective linear programming model is then proposed and solved by our augmented maximum approach. For illustration purpose, a numerical bank hedging decision problem is solved.

A new approach to some possibilistic linear programming problems

In practice, the unit costs/profits of new products or new projects, the lending and borrowing interest rates, and cash flows are always imprecise. We provide an auxiliary multiple objective linear programming model to solve a linear programming problem with imprecise objective and/or constraint coefficients. Our strategy is to maximize the most possible value of the imprecise profit. At the same time, we would like to minimize the risk of obtaining lower profit and maximize the possibility of obtaining higher profit. This strategy is equivalent to the practical considerations of financial problems. In this paper, a numeric investment problem is solved for illustrating the new approach.

Stability in possibilistic linear programming with continuous fuzzy number parameters

We prove that possibilistic linear programming problems (introduced by Buckley in [2]) with continuous fuzzy number parameters are well-posed, i.e. small changes of the membership function of the parameters may cause only a small deviation in the possibilistic distribution of the objective function.

Solving possibilistic linear programming problems

We present a solution to certain possibilistic linear programming problems which have been previously introduced in [1].