Fuzzy Linear Programming Research Articles

Solving Fuzzy Linear Programming as Multi Objective Linear Programming Problem

The constraints and the objective function of the fuzzy linear programming problem are converted into the multi-objective optimization problem (i.e.,) into an equivalent crisp linear problem.Finally, the multi-objective linear programming problem is converted into the weighted linear programming problem to show that they are independent of weights and obtained the complete optimal solution.

A weighted fuzzy linear programming model in economic input–output analysis: an application to risk management of energy system disruptions

Climate change exposes economic systems to numerous risks, including reduced agricultural production and electric power supply shortages. The interdependent nature of economic systems causes disruptions in any sector to cascade to other sectors via forward and backward linkages. This work develops an optimization model with which allocation of scarce goods or resources can be optimized; the model uses an overall index of satisfaction of fuzzy economic output goals under conditions of scarcity caused by climatic disruptions. The proposed model includes a vulnerability measure that integrates information elicited from expert judgment. A case study based on a scenario of drought-induced electricity shortage in the Philippine economy is examined. Results show that trade, transportation and service-oriented industries suffer losses in gross domestic product in the Philippine case.

A new data envelopment analysis in fully fuzzy environment on the base of the degree of certainty of information

Data Envelopment Analysis (DEA) is a linear programming (LP) based technique for evaluating the comparative efficiency of decision-making units (DMUs) based on multiple inputs and outputs. Conventional DEA forms require precise data. However in real problems, it is not easy to measure inputs and ou tputs in an exact way to find precise data. Although, fuzzy set theory has been introduced as a powerful approach to quantify vague data and several authors have suggested various fuzzy DEA models, there is a key flaw in previous approaches. When coping with real information, fuzziness is not sufficient to consider and a reliability of the information is very vital too. A Z-number has extra ability to depict the uncertain information. This concept relates to the topic of reliability of information. Z-number can portray fuzziness and reliability of information concurrently. In this paper, we consider a different fuzzy DEA form, where all the inputs and outputs and also their weights are Z-numbers. This Z-numbers DEA model turned into fully fuzzy LP on the basis of fuzzy expectation. Finally, we transform the fully fuzzified DEA model to the classical LP model. This method has very straightforward calculations and the key benefit of the proposed method is its low computational intricacy.

A fuzzy technique for supply chain network design with quantity discounts

This paper proposes a hierarchical technique for Supply Chain Network (SCN) efficiency maximisation under uncertainty composed of three steps. The first step extends a previous fuzzy cross-efficiency Data Envelopment Analysis approach, originally intended for suppliers’ selection, in order to evaluate and rank all the actors in each SCN stage under conflicting nondeterministic criteria. Afterwards, a fuzzy linear integer programming model is stated and solved for each pair of subsequent SCN stages to determine the quantities required from each stakeholder to maximise the overall SCN efficiency while satisfying the estimated demand and respecting the nodes capacity. Finally, a heuristics is applied to limit the exchange of small quantities in the SCN, in which the trade is not economically convenient according to quantity discounts. An illustrative example from the literature shows the technique effectiveness.

A new methodological development for solving linear bilevel integer programming problems in hybrid fuzzy environment

This paper deals with fuzzy goal programming approach to solve fuzzy linear bilevel integer programming problems with fuzzy probabilistic constraints following Pareto distrib ution and Frechet distribution. In the proposed approach a new chance constrained programming methodology is developed from the view point of managing those probabilistic constraints in a hybrid fuzzy environment. A method of defuzzification of fuzzy numbers using α cut has been adopted to reduce the problem into a linear bilevel integer programming problem. The individual optimal value of the objective of each DM is found in isolation to construct the fuzzy membership goals. Finally, fuzzy goal programming approach is used to achieve maximum degree of each of the membership goals by minimizing under deviational variables in the decision making environment. To demonstrate the efficiency of the proposed approach , a numerical example is provided.

Method for Optimizing the Dual of Linear Fuzzy Programming Problems

This article’s goal is to support the existence of the dual in a Linear Fuzzy Real environment and focus on its application to Linear fuzzy program problems. This concept will apply to linear fuzzy programming problems that contain fuzzy constraints with a crisp objective function, crisp constraints with a fuzzy objective function, or fuzzy constraints with a fuzzy objective function. It is also proposed here that optimizing fuzzy constraints and objectives of the dual linear program that consist of a triplet and are much like triangular fuzzy numbers, but differ in that they are a hybrid fuzzy number that contains characteristics that are both fuzzy and crisp.

Selecting new product development team members with knowledge sharing approach

Purpose – The purpose of this paper is to provide a method for selection of the new product development (NPD) project team members, in such a way to maximize the expertise level of team members and at the same time, optimize knowledge sharing in the organization. Design/methodology/approach – According to the motivation-opportunity-ability framework, knowledge sharing antecedents were determined. Then, the problem of selecting appropriate members of the project team was formulated as a bi-objective integer non-linear programming model. Due to the uncertainty conditions in the evaluation of candidates, the fuzzy sets approach was used for modeling. To solve the problem, first, the non-linear programming model was converted to a linear model. Subsequently, the fuzzy bi-objective linear programming problem was solved by using an approximate algorithm. Findings – Results of applying the proposed method to an Iranian ship-building company showed its effectiveness in selecting appropriate members of the project team. Practical implications – With the aid of the proposed approach, project managers will be able to form effective project teams that while increasing the success probability of the project, facilitate the maintenance of knowledge acquired during the project lifecycle. Originality/value – This paper, for the first time, has tried to provide a method for selecting the NPD project team members, in a way that while selecting candidates with highest expertise, maximizes the sharing of knowledge among them.

A note on “Solving intuitionistic fuzzy linear programming problems by ranking function”

Suresh et al. (Solving intuitionistic fuzzy linear programming problems by ranking function, Journal of Intelligent and Fuzzy Systems, 27, 3081-3087, 2014) proposed the ranking function for comparing triangular intuitionistic fuzzy numbers (TIFNs) and applied this ranking function to solve different types of intuitionistic fuzzy linear programming problems (IFLPPs). In this note, it is pointed out that the ranking function proposed by authors is not valid. Hence, the results of intuitionistic fuzzy linear programming problems, obtained by using this ranking function, are also not valid. Further, the exact ranking function is obtained by modifying existing ranking function and using the exact ranking function, the exact results of intuitionistic fuzzy linear programming problems considered by Suresh et al., are obtained.

Some deterministic models of fuzzy linear programming problems

We consider deterministic equivalents of various formulations of linear programming prob-lems, in which the coefficients of the objective function, constraints and the boundary values of the variables of the problem and the right-hand side are represented by fuzzy sets. The methods for comparing the fuzzy sets and selecting the best ones are proposed. The problem of finding the vec-tor of variables as a vector of real numbers is reduced to solving the one-criterion or multicriteria problem with the significantly large number of constraints. In solving the problem as a vector of Fuzzy-sets, the equivalent problem was determined – a sequence of linear programming problems. The formulated problems can be solved by the simplex method.

A mathematical model for solving fully fuzzy linear programming problem with trapezoidal fuzzy numbers

In this paper, an efficient method is introduced to solve fully fuzzy linear programming problems. The proposed method is derived from the multi-objective linear programming problem and lexicographic ordering method. Theoretical analysis for the proposed method has been provided. Moreover, some numerical experiments are given to show the preference of the proposed methods and are compared with some available methods.

Fuzzy linear programming approach for solving transportation problems with interval-valued trapezoidal fuzzy numbers

Transportation problem (TP) is an important network structured linear programming problem that arises in several contexts and has deservedly received a great deal of attention in the literature. The central concept in this problem is to find the least total transportation cost of a commodity in order to satisfy demands at destinations using available supplies at origins in a crisp environment. In real life situations, the decision maker may not be sure about the precise values of the coefficients belonging to the transportation problem. The aim of this paper is to introduce a formulation of TP involving interval-valued trapezoidal fuzzy numbers for the transportation costs and values of supplies and demands. We propose a fuzzy linear programming approach for solving interval-valued trapezoidal fuzzy numbers transportation problem based on comparison of interval-valued fuzzy numbers by the help of signed distance ranking. To illustrate the proposed approach an application example is solved. It is demonstrated that study of interval-valued trapezoidal fuzzy numbers transportation problem gives rise to the same expected results as those obtained for TP with trapezoidal fuzzy numbers.

An Efficacious Multi-Objective Fuzzy Linear Programming Approach for Optimal Power Flow Considering Distributed Generation

This paper proposes a new formulation for the multi-objective optimal power flow (MOOPF) problem for meshed power networks considering distributed generation. An efficacious multi-objective fuzzy linear programming optimization (MFLP) algorithm is proposed to solve the aforementioned problem with and without considering the distributed generation (DG) effect. A variant combination of objectives is considered for simultaneous optimization, including power loss, voltage stability, and shunt capacitors MVAR reserve. Fuzzy membership functions for these objectives are designed with extreme targets, whereas the inequality constraints are treated as hard constraints. The multi-objective fuzzy optimal power flow (OPF) formulation was converted into a crisp OPF in a successive linear programming (SLP) framework and solved using an efficient interior point method (IPM). To test the efficacy of the proposed approach, simulations are performed on the IEEE 30-busand IEEE 118-bus test systems. The MFLP optimization is solved for several optimization cases. The obtained results are compared with those presented in the literature. A unique solution with a high satisfaction for the assigned targets is gained. Results demonstrate the effectiveness of the proposed MFLP technique in terms of solution optimality and rapid convergence. Moreover, the results indicate that using the optimal DG location with the MFLP algorithm provides the solution with the highest quality.

MODELING AND SOLVING TIME-COST TRADE-OFF PROBLEM IN THE FUZZY ENVIRONMENT (CASE STUDY)

Time and cost are considered among the main criteria for projects and managers are always looking for project completion in the shortest possible time and at lowest cost in order to achieve success in projects. One of the fundamental challenges in this respect is the selection of an appropriate approach to reach the mentioned goals. In this research, after examining the studies conducted regarding trade-off and assessing the shortcomings of modeling, a trade-off model has been initially provided by adding the factor of cost to the revised programming model by Pritsker. In the next step, in order to consider implementation-dependent conditions between activities in reality, a relationship has been added to the original model. Further, in order for the results of modeling to get closer to reality, the original model was evaluated in a fuzzy environment and to this end, fuzzy linear programming method has been used in a non-symmetric state. To solve the model, GAMS software has been applied. At the end, the proposed model has been implemented in a real project to ensure the accuracy of the results of modeling. In order to allow a better comparison, the results associated with both states of certainty and uncertainty conditions were compared to the results obtained from the implementation of the project in reality. The results indicate the high efficiency of the proposed model under uncertainty conditions. Additionally, given the executive modes intended for activities, it was determined that increasing and decreasing the project completion time with regard to the changes in the project budget amounts are only possible to some extent.

Intuitionistic fuzzy linear programming and duality: a level sets approach

The paper is concerned with linear programming problems whose input data may be intuitionistic fuzzy (IF) while the values of variables are always real numbers. We propose rather general approach to this type of problems based on level sets, and present recent results for problems in which the notions of feasibility and optimality are based on the IF relations. Special attention is devoted to the weak and strong duality.

Solving the linear fractional programming problem in a fuzzy environment: Numerical approach

The fuzzy linear fractional programming problem is an important planning tool in different areas such as engineering, business, finance, and economics. In this study, we propose the use of the (α, r) acceptable optimal value for a linear fractional programming problem with fuzzy coefficients and fuzzy decision variables, as well as developing a method for computing them. To obtain acceptable (α, r) optimal values, we take an α-cut on the objective function and r-cut on the constraints. We then formulate an equivalent bi-objective linear fractional programming problem to calculate the upper and lower bounds of the fully fuzzy LFP problem. Using the upper and lower bounds obtained, we construct the membership functions of the optimal values numerically. We illustrate the proposed procedure using numerical and real life examples.

A note on “Fully fuzzy fixed charge multi-item solid transportation problem”

Giri et al. [P.K. Giri, M.K. Maiti, M. Maiti, Fully fuzzy fixed charge multi-item solid transportation problem, Applied Soft Computing, 27 (2015) 77–91] proposed an approach for solving mathematical programming of fully fuzzy fixed charge multi-item solid transportation problems (FFFCMISTP) and claimed that it is better to use their proposed approach as compared to the existing method [A. Kumar, J. Kaur, P. Singh, A new method for solving fully fuzzy linear programming problems, Applied Mathematical Modelling, 35 (2011) 817–823]. The aim of this note is to point out that Giri et al. have used some mathematical incorrect assumptions in their proposed approach and hence the claim of Giri et al. is not valid.

A Note on “A New Method for Solving Fuzzy Linear Programming Problems Based on the Fuzzy Linear Complementary Problem (FLCP)”

Mottaghi et al. (Int J Fuzzy Syst 17:236–245, 2015) proposed some results by considering a pair of fuzzy linear programming problems as a primal—dual pair. In this paper, it is pointed out that the pair of fuzzy linear programming problems, considered by Mottaghi et al. as a primal—dual pair, is not a primal—dual pair. Hence, all the results, proposed by Mottaghi et al. by considering the pair of fuzzy linear programming problems as a primal—dual pair, are not valid.

Fuzzy multi-objective linear programming for a stochastic hub maximal covering problem with uncertain shipments

When there are many origins and destinations in a network, the use of hubs helps to benefit from the economies of scale in transportation and considerable reduction in the number of required links. We propose a bi-objective mathematical model for a stochastic hub maximal covering problem formulated under two real-world assumptions. In many networks, there are some risks for the shipments among the nodes. In order to consider such risks, a second objective, in addition to maximising the utility of covered nodes, is aimed at maximising the safety of weakest path in the network. Also, similar to the real world situation, it is assumed that the transport times are normal random variables. An efficient nonlinear mixed-integer model is developed and after linearising, a fuzzy multi-objective linear programming method is applied to solve it. Notably, the single objective version of our model has smaller number of variables and equations than the previous ones. The computational results by establishing 24 test problems extracted from the Turkish dataset confirm: 1) the validity of proposed formulations for single objective hub maximal covering problem; 2) the satisfying performance of proposed bi-objective model in improving the safety of designed network.

Uncertain and multi-objective programming models for crop planting structure optimization

Crop planting structure optimization is a significant way to increase agricultural economic benefits and improve agricultural water management. The complexities of fluctuating stream conditions, varying economic profits, and uncertainties and errors in estimated modeling parameters, as well as the complexities among economic, social, natural resources and environmental aspects, have led to the necessity of developing optimization models for crop planting structure which consider uncertainty and multi-objectives elements. In this study, three single-objective programming models under uncertainty for crop planting structure optimization were developed, including an interval linear programming model, an inexact fuzzy chance-constrained programming (IFCCP) model and an inexact fuzzy linear programming (IFLP) model. Each of the three models takes grayness into account. Moreover, the IFCCP model considers fuzzy uncertainty of parameters/variables and stochastic characteristics of constraints, while the IFLP model takes into account the fuzzy uncertainty of both constraints and objective functions. To satisfy the sustainable development of crop planting structure planning, a fuzzy-optimizationtheory-based fuzzy linear multi-objective programming model was developed, which is capable of reflecting both uncertainties and multi-objective. In addition, a multiobjective fractional programming model for crop structure optimization was also developed to quantitatively express the multi-objective in one optimization model with the numerator representing maximum economic benefits and the denominator representing minimum crop planting area allocation. These models better reflect actual situations, considering the uncertainties and multi-objectives of crop planting structure optimization systems. The five models developed were then applied to a real case study in Minqin County, north-west China. The advantages, the applicable conditions and the solution methods of each model are expounded. Detailed analysis of results of each model and their comparisons demonstrate the feasibility and applicability of the models developed, therefore decision makers can choose the appropriate model when making decisions.

A Mathematical Model for Fuzzyp-Median Problem with Fuzzy Weights and Variables

We investigate thep-median problem with fuzzy variables and weights of vertices. The fuzzy equalities and inequalities transform to crisp cases by using some technique used in fuzzy linear programming. We show that the fuzzy objective function also can be replaced by crisp functions. Therefore an auxiliary linear programming model is obtained for the fuzzyp-median problem. The results are compared with two previously proposed methods.