A new Method for Solving Sensitivity Analysis for Fuzzy Linear Programming Problems

In this paper, we discusses the problem that involving the conflict fuzzy multi-objective in transportation planning which is one of special case in multi-objective linear programming. To find the simultaneously optimal solution of the problem, we use the Interactive Fuzzy Multi-Objective Linear Programming (IFMOLP) method. This method can reduce the fuzzy multi-objective linear programming problem into deterministic single objective linear programming which can be solved using simplex method. Beside that , with IFMOLP method, decision maker (DM) can establish interactively the goal from the objective function to produce the pareto optimal solution so that.

Using different dominance criteria in stochastic fuzzy linear multiobjective programming: A case of fuzzy weighted objective function

In this paper, we propose a method of solving the fully fuzzy linear fractional programming problems, Express all the parameters and variables are triangular fuzzy numbers. Convert all the triangular fuzzy numbers in their parametric form, we convert the fractional programming problem in to a single objective linear programming problem in parametric form. We put new fuzzy arithmetic and fuzzy ranking, we obtain the optimal solution the given fully fuzzy linear fractional programming problem without converting to its equivalent crisp linear programming problem. A numerical example is provided to illustrate the efficiency of the proposed method.

Many planning problems involve choosing a set of optimal decisions for a system in the face of uncertainty of elements that may play a central role in the way the system is analyzed and operated. During the past decade, there has been a renewed interest in the modelling, analysis, and solution of such problems due to a remarkable development of both new theoretical results and novel computational techniques in stochastic optimization. A prominent approach is to develop upper and lower bounding approximations to the problem along with procedures to sharpen bounds until an acceptable tolerance is satisfied. The contributions of this dissertation are concerned with the latter approach. The thesis first studies the stochastic linear programming problem with randomness in both the objective coefficients and the constraints. A convex concave saddle property of the value function is utilized to derive new bounding techniques which generalize previously known results. These approximations require discretizing bounded domains of the random variables in such a way that tight upper and lower bounds result. Such techniques will prove attractive with the recent advances in large scale linear programming. The above results are also extended to obtain new upper and lower bounds when the domains of random variables are unbounded. While these bounds are tight, the approximating models are large-scale deterministic linear programs. In particular, with a proposed order-cone decomposition for the domains, these linear programs are well-structured, thus enabling one to use efficient techniques for solution, such as parallel computation. The thesis next considers convex stochastic programs. Using aggregation concepts from the deterministic literature, new bounds are developed for the problem which are computable using standard convex programming algorithms. Finally, the discussion is focused on a stochastic convex program arising in a certain resource allocation problem. Exploiting the problem structure, bounds are developed via the Karush-Kuhn-Tucker conditions. Rather than discretizing domains, these approximations advocate replacing difficult multidimensional integrals by a series of simple univariate integrals. Such practice allows one to preserve differentiability properties so that smooth convex programming methods can be applied for solution.

In this paper, an efficient approach is presented to solve fully fuzzy linear fractional programming problems. In this approach, first by Alpha-cuts on the objective function and the constraints, the problem is transforming into a bi-objective linear programming problem, and then the resulting problem is converted into two crisp linear programming problems. A numerical example is presented to illustrate the proposed approach. The obtained results demonstrate the acceptable performance of the proposed solution approach.

A class of optimal shape design problems is studied in this paper. The boundary shape of a domain is determined such that the solution of the underlying partial differential equation matches, as well as possible, a given desired state. In the original optimal shape design problem, the variable domain is parameterized by a class of functions in such a way that the optimal design problem is changed to an optimal control problem on a fixed domain. Then, the resulting distributed control problem is embedded in a measure theoretical form, in fact, an infinite-dimensional linear programming problem. The optimal measure representing the optimal shape is approximated by a solution of a finite-dimensional linear programming problem. The method is evaluated via a numerical example.

We present an O â nL -iteration homogeneous and self-dual linear programming LP algorithm. The algorithm possesses the following features: â ¢ It solves the linear programming problem without any r...

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

Vendor Selection (VS) problem is considered as a critical decision that effects on the financial position for any firms. Hence the purchasing managers consider the selection process of the right vendors is a major function of the outsourcing process. Cost, quality, and services are three common criteria for VS problem. In the real cases the parameters of any problems are usually vague or not completely known, so that the Fuzzy Set Theory (FST) is one of the most powerful existing tools capable of addressing ambiguity and blurring information in parameters. A fuzzy parameters for VS problem are dealt with Linear Membership Function (LMF) through using cut that convert a fuzzy numbers to Interval Multiple Objective Linear Programming (IMOLP). The formulation of IMOLP is determined based on the value of , then IMOLP model is divided to upper and lower sub-models. A compromise interval solution will obtain from solving the sub-models as conventionally linear programming. The proposed approach is clarified with an illustrative example.

We present an approach for ordering fuzzy numbers, then we utilize it to suggest two models for solving the fuzzy linear programming (FLP) problem. In this research we present a new approach for ordering two fuzzy numbers and show some practical advantages of it over other methods. We also utilize the suggested approach to introduce two techniques to solve the FLP problem. In the first technique, we convert the fuzzy linear program into a crisp bicriteria linear program. In this new model the fuzzy objective function is replaced by two crisp objectives to be optimized, and each fuzzy constraint is replaced by two or three crisp constraints depending on the level of certainty /spl alpha/-whether /spl alpha/ has or has not been assigned. In the second technique we find the possibility distribution of the objective function over all possible feasible decisions directly.

<p>There exist several methods for solving fuzzy linear or nonlinear programming problems under positivity fuzzy variables and restricted fuzzy coefficients. Due to the limitation of these methods, they can’t be applied for solving fully fuzzy linear or non- linear programming problems with unrestricted fuzzy coefficients and fuzzy variables. In this paper an efficient method to find the fuzzy optimal solution for fully fuzzy quadratic programming (FFQP) problem with unrestricted variables and parameters has been proposed. All the coefficients and decision variables of both objective functions and the constraints are triangular fuzzy numbers. The proposed method is based on converted FFQP problem into crisp quadratic programming (CQP) problem. Finally an illustrative numerical example has been given to clarify the proposed solution method.</p>

The difficulty in solving either multi-objective linear programming (MOLP) problems or fuzzy multi-objective linear programming (FMOLP) problems is the trade-off among objectives. To deal with this difficulty, we proposed a new algorithm for solving FMOLP problems by using zero-sum game. First, FMOLP problem given is converted to a crisp MOLP problem by using ranking function, and then a payoff matrix is constructed to find the weights of each objective function of the MOLP problem. After that, each weight is multiplied with the corresponding objective function, a single-objective LP problem is obtained and thus, FMOLP problem is solved. The proposed algorithm is illustrated by numerical examples for the FMLOP problems having fuzzy constraints or not.

In this paper by using the lexicography method, we proposed a new model to solve fully fuzzy linear programming problem with L-R fuzzy number and find the fuzzy optimal solution of it. Our method has graceful structure and is easy to implement compared with some existing methods. To illustrate our method, a numerical example is solved.

<abstract><p>The primary goal of optimization theory is to formulate solutions for real-life challenges that play a fundamental role in our daily lives. One of the most significant issues within this framework is the Linear Fractional Programming Problem (LFrPP). In practical situations, such as production planning and financial decision-making, it is often feasible to express objectives as a ratio of two distinct objectives. To enhance the efficacy of these problems in representing real-world scenarios, it is reasonable to utilize fuzzy sets for expressing variables and parameters. In this research, we have worked on the Fully Fuzzy Linear Fractional Linear Programming Problem (FFLFrLPP). In our approach to problem-solving, we simplified the intricate structure of the FFLFrLPP into a crisp Linear Programming Problem (LPP) while accommodating the inherent fuzziness. Notably, unlike literature, our proposed technique avoided variable transformation, which is highly competitive in addressing fuzzy-based problems. Our methodology also distinguishes itself from the literature in preserving fuzziness throughout the process, from problem formulation to solution. In this study, we conducted a rigorous evaluation of our proposed methodology by applying it to a selection of numerical examples and production problems sourced from the existing literature. Our findings revealed significant improvements in performance when compared to established solution approaches. Additionally, we presented comprehensive statistical analyses showcasing the robustness and effectiveness of our algorithms when addressing large-scale problem instances. This research underscores the innovative contributions of our methods to the field, further advancing the state-of-the-art in problem-solving techniques.</p></abstract>

A suggested algorithm to solve triangular fuzzy rough integer linear programming (TFRILP) problems with α-level is introduced in this paper in order to find rough value optimal solutions and decision rough integer variables, where all parameters and decision variables in the constraints and the objective function are triangular fuzzy rough numbers. In real-life situations, the parameters of a linear programming problem model may not be defined precisely, because of the current market globalization and some other uncontrollable factors. In order to solve this problem, a proper methodology is adopted to solve the TFRILP problems by the slice-sum method with the branch-and-bound technique, through which two fuzzy integers linear programming (FILP) problems with triangular fuzzy interval coefficients and variables were constructed. One of these problems is an FILP problem, where all of its coefficients are the upper approximation interval and represent rather satisfactory solutions; the other is an FILP problem, where all of its coefficients are the lower approximation interval and represent completely satisfactory solutions. Moreover, α-level at α = 0.5 is adopted to find some other rough value optimal solutions and decision rough integer variables. Integer programming is used, since a lot of the linear programming problems require that the decision variables be integers. In addition, the motivation behind this study is to enable the decision makers to make the right decision considering the proposed solutions, while dealing with the uncertain and imprecise data. A flowchart is also provided to illustrate the problem-solving steps. Finally, two numerical examples are given to clarify the obtained results.