Multi-level Multi-objective Quadratic Fractional Programming Problem with Fuzzy Parameters: A FGP Approach

The motivation behind this paper is to present multi-level multi-objective quadratic fractional programming (ML-MOQFP) problem with fuzzy parameters in the constraints. ML-MOQFP problem is an important class of non-linear fractional programming problem. These type of problems arise in many fields such as production planning, financial and corporative planning, health care and hospital planning. Firstly, the concept of the -cut and fuzzy partial order relation are applied to transform the set of fuzzy constraints into a common crisp set. Then, the quadratic fractional objective functions in each level are transformed into non-linear objective functions based on a proposed transformation. Secondly, in the proposed model, separate non-linear membership functions for each objective function of the ML-MOQFP problem are defined. Then, the fuzzy goal programming (FGP) approach is utilized to obtain a compromise solution for the ML-MOQFP problem by minimizing the sum of the negative deviational variables. Finally, an illustrative numerical example is given to demonstrate the applicability and performance of the proposed approach

Consistency test and weight generation for additive interval fuzzy preference relations

Some simple yet pragmatic methods of consistency test are developed to check whether an interval fuzzy preference relation is consistent. Based on the definition of additive consistent fuzzy preference relations proposed by Tanino (Fuzzy Sets Syst 12:117–131, 1984), a study is carried out to examine the correspondence between the element and weight vector of a fuzzy preference relation. Then, a revised approach is proposed to obtain priority weights from a fuzzy preference relation. A revised definition is put forward for additive consistent interval fuzzy preference relations. Subsequently, linear programming models are established to generate interval priority weights for additive interval fuzzy preference relations. A practical procedure is proposed to solve group decision problems with additive interval fuzzy preference relations. Theoretic analysis and numerical examples demonstrate that the proposed methods are more accurate than those in Xu and Chen (Eur J Oper Res 184:266–280, 2008b)

Optimal Solution of Nonlinear Transportation Costs and Applied of the Fussy Transportation Model

Transportation costs are central issue for the resource allocation problems, which deal with principally cost minimization of transportation from supply to demand areas. Many researchers used linear programming methods to study minimization of transportation cost, because linear models also help to analyze and develop planning methods. Lately, fuzzy linear programming methods have been studied and introduced as a complementary. procedure. However, with this method, each unit cost along the transportation cost surface is regularly fixed. Generally, each transport cost unit is a function of traffic condition and distance under study, Based on these facts, this paper examines the practical issues of one aspect of the transportation problem in relation to the nonlinear cost function associated with traffic condition and distance. First, this paper shows that an optimal solution to a linear programming of transportation problem applies the Lagrange multiplier method. The results indicate that the fuzzy optimal solution ( for a case that introduces the Lagrange optimal solution to the fuzzy linear programming ) is a nonlinear transportation cost function. Second, a fuzzy optimal solution using an ordinary simplex method was applied. The results confirmed that the convergence iterating value for the MODI method finally astringed. This indicates that the fuzzy solution model can be applied for an optimal solution using the repetition of the MODI method

Linear, integer separable and fuzzy programming problems: a united approach towards automatic reformulation

For mathematical programming (MP) to have greater impact as a decision tool, MP software systems must offer suitable support in terms of model communication and modelling techniques. In this paper modelling techniques that allow logical restrictions to be modelled in integer programming terms are described and their implications discussed. In addition it is demonstrated that many classes of non-linearities which are not variable separable may be after suitable algebraic manipulation put in a variable separable form. The methods of reformulating the fuzzy linear programming problem as a Max-Min problem is also introduced. It is shown that analysis of bounds plays a key role in the following four important contexts: model reduction, reformulation of logical restrictions as 0-1 mixed integer programs, reformulation of nonlinear programs as variable separable programs and reformulation of fuzzy linear programs. It is observed that as well as incorporating an interface between the modeller and the optimiser there is a need to make available to the modeller software facilities which support the model reformulation techniques described here

Fuzzy linear programming: review and implementation

Most of the time we encounter problems where quantities cannot be expressed by crisp numbers; instead, the use of vague qualifications is preferred. The Fuzzy Set Theory (FST), developed by Lotfi A. Zadeh, is an effective framework which can be used in the solution of such problems. In FST, the objects belong to a set to a degree between [0,1]. The degree of belongingness is referred to as the membership degree. FTS states that utilizing this membership degree is more profitable than just partitioning the objects dichotomously as in classical bivalent (two-valued) set theory. In most of the problems, where linear programming can be applied , the decision maker choses to state the inequalities and the coefficients used in objective function and constraints by vague expressions. Fuzzy Linear Programming, based on the FST, is developed to model and solve such problems. In this thesis, the proposed aproach, in the most general case, first compares the fuzzy left-hand side with the fuzzy right-hand side and the fuzzy objective function with a fuzzy goal by means of a membership function based on the fuzzy relation using "min" function. After determination of these membership functions associated with the constraints and the objective function, a new auxiliary problem is formed. The obtained auxiliary problem is a non-linear fractional programming problem with its nominator are defined by linear functions. The optimal solution of such a problem can be found by solving a sequence of linear programs. In this study, the solution approaches present in the literature for fuzzy linear programming are categorized, some points which are unclear is identified and tried to be improved, and finally the proposed solution methodology is applied to so-called fuzzy analytical hierarchy process

Multiple fuzzy reasoning approach to fuzzy mathematical programming problems

We suggest solving fuzzy mathematical programming problems via the use of multiple fuzzy reasoning techniques. We show that our approach gives Buckley’s solution [1] to possibilistic mathematical programs when the inequality relations are understood in possibilistic sense

Strict Solution Method for Linear Programming Problem with Ellipsoidal Distributions under Fuzziness

This paper considers a linear programming problem with ellipsoidal distributions including fuzziness. Since this problem is not well-defined due to randomness and fuzziness, it is hard to solve it directly. Therefore, introducing chance constraints, fuzzy goals and possibility measures, the proposed model is transformed into the deterministic equivalent problems. Furthermore, since it is difficult to solve the main problem analytically and efficiently due to nonlinear programming, the solution method is constructed introducing an appropriate parameter and performing the equivalent transformations