Fuzzy Linear Programming Research Articles

Empirical (α, β)-acceptable optimal values to full fuzzy linear fractional programming problems

In this paper we aim to provide empirical solutions to a special class of full fuzzy linear fractional programming problems. We use trapezoidal fuzzy numbers to describe the parameters and derive empirical shape of the membership of the goal function optimal values of the problem. Our approach essentially follows the extension principle, and is based on solving crisp quadratic optimization problems. The model we propose treats in different ways, through two independent parameters, the objective function coefficients and coefficients in the constraints. To illustrate our theory, we solve a relevant instance and compare our numerical results with the numerical results recalled from the recent literature.

On multi-objective extensions of the classical assignment model with fuzzy parameters and fuzzy goals

This study advances the limitations of current fuzzy multi-objective assignment models by exploring some formulations with fuzzy parameters and fuzzy goals. The first formulation expresses the coefficients of the objective functions and constraints as fuzzy sets. Two crucial fuzzy transformations were adopted, along with the computational process of the epsilon-constrained multi-objective optimisation. On the other hand, the second formulation assumes the fuzzy coefficients of objective functions and constraints and extends such fuzziness by introducing fuzzy constraints. Lastly, the coefficients of the objective functions and the constraints are expressed as crisp sets while allowing permissible constraint violations. The symmetric fuzzy linear programming solution concepts in the domain literature were adopted as part of the computational process in arriving at a model solution. Actual case examples aided these formulations to gain insights into their computational complexity, efficiency, scalability, and flexibility.

Finding compromise solutions for fully fuzzy multi-objective linear programming problems by using game theory approach

Solving multi-objective linear programming (MOLP) problems and fully fuzzy multi-objective linear programming (FFMOLP) problems involves the trade-off process among several objectives. A new algorithm extended where FFMOLP problems are solved using a 2-player zero-sum game approach to deal with this case. Firstly, The FFMOLP problem is separated into a certain number of fully fuzzy linear programming (FFLP) problems and each is solved by applying any method. After forming a ratio matrix, a game theory approach is applied for finding the weights of objective functions and a weighted LP problem is constructed by these weights. Solving the weighted LP problem, a fuzzy compromise solution of the FFMOLP problem is found. Constructing different ratio matrices, it is also possible to obtain more than one compromise solution to be offered to the decision-maker(s). Some examples are given to show the applicability of the algorithm.

NEW IMPROVED METHOD FOR SOLVING THE FUZZY LINEAR PROGRAMMING PROBLEMS WITH VARIABLES GIVEN AS FUZZY NUMBERS

The present paper aims to propose an alternative solution approach in obtaining the fuzzy optimal solution to a fuzzy linear programming problem with variables given as fuzzy numbers with minimum uncertainty. In this paper, the fuzzy linear programming problems with variables given as fuzzy numbers is transformed into equivalent interval linear programming problems with variables given as interval numbers. The solutions to these interval linear programming problems with variables given as interval numbers are then obtained with the help of linear programming technique. A set of six random numerical examples has been solved using the proposed approach.

A new approach to solve fully intuitionistic fuzzy linear programming problem with unrestricted decision variables

In many real-world problems, one may encounter uncertainty in the input data. The fuzzy set theory fits well to handle such situations. However, it is not always possible to determine with full satisfaction the membership and non-membership degrees associated with an element of the fuzzy set. The intuitionistic fuzzy sets play a key role in dealing with the hesitation factor along-with the uncertainty involved in the problem and hence, provides more flexibility in the decision-making process. In this article, we introduce a new ordering on the set of intuitionistic fuzzy numbers and propose a simple approach for solving the fully intuitionistic fuzzy linear programming problems with mixed constraints and unrestricted variables where the parameters and decision variables of the problem are represented by intuitionistic fuzzy numbers. The proposed method converts the problem into a crisp non-linear programming problem and further finds the intuitionistic fuzzy optimal solution to the problem. Some of the key significance of the proposed study are also pointed out along-with the limitations of the existing studies. The approach is illustrated step-by-step with the help of a numerical example and further, a production planning problem is also demonstrated to show the applicability of the study in practical situations. Finally, the efficiency of the proposed algorithm is analyzed with the existing studies based on various computational parameters.

Interval-Type Fuzzy Linear Fractional Programming Problem in Neutrosophic Environment: A Fuzzy Mathematical Programming Approach

Interval-Type Fuzzy Linear Fractional Programming Problem in Neutrosophic Environment: A Fuzzy Mathematical Programming Approach

Fuzzy solution of fully fuzzy multi-objective linear fractional programming problems

In this study, we present a novel method for solving fully fuzzy multi-objective linear fractional programming problems without transforming to equivalent crisp problems. First, we calculate the fuzzy optimal value for each fractional objective function and then we convert the fully fuzzy multi-objective linear fractional programming problem to a single objective fuzzy linear fractional programming problem and find its fuzzy optimal solution which inturn yields a fuzzy Pareto optimal solution for the given fully fuzzy multi-objective linear fractional programming problem. To demonstrate the proposed strategy, a numerical example is provided.

A Novel Approach to Solve Fully Fuzzy Linear Programming Problems with Modified Triangular Fuzzy Numbers

Recently, new methods have been recommended to solve fully fuzzy linear programming (FFLP) issues. Likewise, the present study examines a new approach to solve FFLP issues through fuzzy decision parameters and variables using triangular fuzzy numbers. The strategy, which is based on alpha-cut theory and modified triangular fuzzy numbers, is suggested to obtain the optimal fully fuzzy solution for real-world problems. In this method, the problem is considered as a fully fuzzy problem and then is solved by applying the new definition presented for the triangular fuzzy number to optimize decision variables and the objective function. Several numerical examples are solved to illustrate the above method.

Inner product of fuzzy vectors

The inner product of vectors of non-normal fuzzy intervals will be studied in this paper by using extension principle and expression in decomposition theorem. The membership functions of inner product will be different with respect to these two different methodologies. Since the non-normal fuzzy interval is more general than the normal fuzzy interval, the corresponding membership functions will become more complicated. Therefore, we shall establish their relationship including the equivalence and fuzziness based on the \(\alpha \)-level sets. The potential application of inner product of fuzzy vectors is to study the fuzzy linear programming problems.

An approach to optimize the cost of transportation problem based on triangular fuzzy programming problem

In this article, we address a fully fuzzy triangular linear fractional programming (FFLFP) problem under the condition that all the parameters and decision variables are characterized by triangular fuzzy numbers. Utilizing the computation of triangular fuzzy numbers and Lexicographic order (LO), the FFLFP problem is changed over to a multi-objective function. Consequently, the problem is changed into a multi-objective crisp problem. This paper outfits another idea for diminishing the computational complexity, in any case without losing its viability crisp LFP issues. Lead from real-life problems, a couple of mathematical models are considered to survey the legitimacy, usefulness and applicability of our method. Finally, some mathematical analysis along with one case study is given to show the novel strategies are superior to the current techniques.

Type 2 Fully Fuzzy Linear Programming

Since its inception, fuzzy linear programming (FLP) has proved to be a more powerful tool than classical linear programming to optimize real-life problems dealing with uncertainty. However, the proposed models are partially fuzzy; in other words, they suppose that only some aspects can be uncertain, while others have to be crisp. Furthermore, the few methods that deal with fully fuzzy problems use Type 1 fuzzy membership function, while Type 2 fuzzy logic captures the uncertainty in a more suitable way. This work presents a fully fuzzy linear programming approach in which all parameters are represented by unrestricted Interval Type 2 fuzzy numbers (IT2FN) and variables by positive IT2FN. The treated comparative results show that the proposed achieves a better optimized function while permitting consideration of both equality and inequality constraints.

Exploring the Optimal Allocation Decision-Making of Expenditure Budget in Hospitals Under Multi-Objective Constraints: Evidence from Urban Public Hospitals, China

Hospitals in many countries face the need for balancing different categories of expenditures to achieve multiple goals within a limited budget. This study established a two-stage fuzzy linear programming (FLP) estimation model to explore the optimal allocation decision-making of expenditure budget under the multi-objective constraints. Taking all urban public hospitals in Henan province of China as a sample, the optimal allocation decision-making of total expenditure budget was tested with the human resources expenditures (HE) as the dependent variable. And the outcome was compared with the actual expenditure data of these hospitals between 2010 and 2016. The study found that when the HE achieves the maximum and minimum feasible scale, the expenditure scales of the budget allocation categories including pharmaceutical expenditures, medical supplies expenditures, and other expenditures were all within a reasonable range. Among them, the observed promoting space for HE was 3.78 billion yuan. The results show that the FLP method can help urban public hospitals to make better total expenditure budget allocation decisions, which can maintain their reasonable expenditure structure under the hospitals’ development goals and the government’s regulatory requirements.

Solving fully fuzzy linear programming problems by controlling the variation range of variables

This paper deals with a fully fuzzy linear programming problem (FFLP) in which the coefficients of decision variables, the right-hand coefficients and variables are characterized by fuzzy numbers. A method of obtaining optimal fuzzy solutions is proposed by controlling the left and right sides of the fuzzy variables according to the fuzzy parameters. By using fuzzy controlled solutions, we avoid unexpected answers. Finally, two numerical examples are solved to demonstrate how the proposed model can provide a better optimal solution than that of other methods using several ranking functions.

Analysis of Probability Distribution due to Aspiration Level in Fuzzy Matrix Game

In this paper we studied zero-sum matrix game. These games have been represented in form of fuzzy linear programming problems which consists the aspiration level and tolerance level. The aspiration levels are determined on the basis of payoff matrix instead of using probability distribution over the strategies. For a particular tolerance level, the aspiration level is studied and the result are illustrated graphically.

Truck dispatching in surface mines -Application of fuzzy linear programming

Material handling in surface mines accounts for around 50% of the operational cost. Optimum truck dispatching plays a critical role in the reduction of this operational cost in truck and shovel surface mines. Researchers in this field have presented several mathematical models to solve the truck dispatching problem optimally. However, a critical survey of the literature has shown that three significant drawbacks exist in the available truck dispatching models. The published models underestimate the importance of the interaction between truck fleet, shovel fleet, and the processing plants. They also disregard goals set by strategic-level plans. Moreover, none of the available models account for the uncertainty associated with the input parameters. In this paper we present a new truck dispatching model that covers all of these drawbacks, using a fuzzy linear programming method. The performance of the developed model was evaluated through implementatin in an active surface mining operation. The results show a significant improvement in production and fleet utilization.

Wetland Restoration Planning Approach Based on Interval Fuzzy Linear Programming under Uncertainty.

When planning wetland restoration projects, the planting area allocation and the costs of the restoration measures are two major issues faced by decision makers. In this study, a framework based on the interval fuzzy linear programming (IFLP) method is introduced for the first time to plan wetland restoration projects. The proposed framework can not only effectively deal with interval and fuzzy uncertainties that exist in the planning process of wetland restorations but also handle trade-offs between ecological environment benefits and economic cost. This framework was applied to a real-world wetland restoration planning problem in the northeast of China to verify its validity and examine the credibility of the constraints. The optimized results obtained from the framework that we have developed indicate that higher ecological and social benefits can be obtained with optimal restoration costs after using the wetland restoration decision-making framework. The optimal restoration measure allocation schemes obtained by IFLP under different credibility levels can help decision makers generate a range of alternatives, which can also provide decision suggestions to local managers to generate a satisfactory decision-making plan. Furthermore, a comparison was made between the IFLP model and ILP model in this study. The comparison results indicate that the IFLP model provides more information regarding ecological environment and economic trade-offs between the system objective, certainty, and reliability. This framework provides managers with an effective way to plan wetland restoration projects, while transference of the model may help solve similar problems.

A New Interval-Valued Hesitant Fuzzy-Based Optimization Method

An interval-valued hesitant fuzzy set (IVHFS) is a best tool to address uncertainty and hesitation of a production planning problem (PPP) which appears in engineering, agriculture, and industrial sectors. Often, a PPP is formulated as a multiobjective linear programming problem (MOLPP) and therefore, it is very necessary to develop a suitable and realistic method to deal MOLPP with uncertainty and hesitation. In this paper, we define a set of possible interval-valued hesitant fuzzy degrees for all objectives, and using this, a MOLPP is converted into a interval-valued hesitant fuzzy linear programming (IVHFLPP). Further, we introduce a new optimization technique based on a new operation of IVHFS, and later it is implemented in a computational method to search a Pareto optimal solution of the considered problem. Further, a PPP is solved by using the proposed method and the result shows the superiority of the proposed computational method over the existing methods.

Solving semi-fully fuzzy linear programming problems

Solving semi-fully fuzzy linear programming problems

Optimal Excess Commuting Evaluation Based on Local Minimal Costs

Excess commuting refers to the value of unnecessary commuting or distance costs. Traditional commuting distance models adapt the most efficient scenario with people working in the nearest workplace geographically. Even though there have been some attempts to include constraints with commuter attributes and neighborhood features, problems arise with traditional geographical space and the subjectivity of these predefined characteristics. In this paper, we propose a method to calculate theoretical local minimal costs, which considers preferences that are inherently behavioral based on current work–home trips in the process of reassigning the work–home configuration. Our method is based on a feature space with a higher dimension and with the enlargement of attributes and relations of and between commuters and neighborhoods. Additionally, our solution is arrived at innovatively by improved Fuzzy C-Means clustering and linear programming. Unlike traditional clustering algorithms, our improved method adapts entropy information and selects the initial parameters based on the actual data rather than on prior knowledge. Using the real origin–destination matrix, theoretical minimal costs are calculated within each cluster, referred to as local minimal costs, and the average sum of local minimal costs is our theoretical minimal cost. The difference between the expected minimal cost and the actual cost is the excess commuting. Using our method, experimental results show that only 13% of the daily commuting distance in Wuhan could be avoided, and the theoretical distance is approximately 1.06 km shorter than the actual commuting distance.

LR-type fully Pythagorean fuzzy linear programming problems with equality constraints

A Pythagorean fuzzy set is a powerful model for depicting fuzziness and uncertainty. This model is more flexible and practical as compared to an intuitionistic fuzzy model. This research article presents a new model called LR-type fully Pythagorean fuzzy linear programming problem. We consider the notions of LR-type Pythagorean fuzzy number, ranking for LR-type Pythagorean fuzzy numbers and arithmetic operations for unrestricted LR-type Pythagorean fuzzy numbers. We propose a method to solve LR-type fully Pythagorean fuzzy linear programming problems with equality constraints. We describe our proposed method with numerical examples including diet problem.