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Linear programming (LP) has long proved its merit as the most flexible and most widely used technique for resource allocation problems in various fields. To solve an LP problem, we have traditionally considered crisp values for the parameters, which are unrealistic in real-world decision-making under uncertainty. The fuzzy set theory has been used to model the imprecise parameter values in LP problems to overcome this shortcoming, resulting in a fuzzy LP (FLP) problem. This paper proposes a new method for solving fuzzy variable linear programming (FVLP) problems in which the decision variables and resource vectors are fuzzy numbers. We show how to use the standard simplex algorithm to solve this problem by converting the fuzzy problem into a crisp one once a linear ranking function is chosen. The novelty of the proposed model resides in that it requires less effort on fuzzy computations as opposed to the existing fuzzy methods. Furthermore, to solve the FVLP problem using the existing methods, fuzzy arithmetic operations and the solution to fuzzy systems of equations are required. By contrast, only arithmetic operations of real numbers and the solution to crisp systems of equations are required to solve the same problem with the method proposed in this study. Finally, a transportation case study in the coal industry is presented to demonstrate the applicability of the proposed algorithm.

One of the important issues under discussion connected with traffic on the roads is improving transportation. In this regard, spatial information, including the shortest path, is of particular importance due to the reduction of economic and environmental costs. Here, the constrained shortest path (CSP) problem which has an important application in location-based online services is considered. The aim of this problem is to find a path with the lowest cost where the traversal time of the path does not exceed from a predetermined time bound. Since precise prediction of cost and time of the paths is not possible due to traffic and weather conditions, this paper discusses the CSP problems with fuzzy cost and fuzzy time. After formulating the CSP problem an efficient algorithm for finding the constrained optimal path is designed. The application of the proposed model is presented on a location-based online service called Snap.

In recent years, numerous researchers examined and analyzed several different types of uncertainty in shortest path (SP) problems. However, those SP problems in which the costs of arcs are expressed in terms of mixed interval-valued fuzzy numbers are less addressed. Here, for solving such uncertain SP problems, first a new procedure is extended to approximate the summation of mixed interval-valued fuzzy numbers using alpha cuts. Then, an extended distance function is introduced for comparing the path weights. Finally, we intend to use a modified artificial bee colony (MABC) algorithm to find the interval-valued membership function of SP in such mixed interval-valued fuzzy network. The proposed algorithm is illustrated via two applications of SP problems in wireless sensor networks and then the results are compared with those derived from genetic and particle swarm optimization (PSO) algorithms, based on three indexes convergence iteration, convergence time and run time. The obtained results confirm that the MABC algorithm has less convergence iteration, convergence time and implementation time compared to GA and PSO algorithm.

Data envelopment analysis (DEA) is a prominent technique for evaluating relative efficiency of a set of entities called decision making units (DMUs) with homogeneous structures. In order to implement a comprehensive assessment, undesirable factors should be included in the efficiency analysis. The present study endeavors to propose a novel approach for solving DEA model in the presence of undesirable outputs in which all input/output data are represented by triangular fuzzy numbers. To this end, two virtual fuzzy DMUs called fuzzy ideal DMU (FIDMU) and fuzzy anti-ideal DMU (FADMU) are introduced into proposed fuzzy DEA framework. Then, a lexicographic approach is used to find the best and the worst fuzzy efficiencies of FIDMU and FADMU, respectively. Moreover, the resulting fuzzy efficiencies are used to measure the best and worst fuzzy relative efficiencies of DMUs to construct a fuzzy relative closeness index. To address the overall assessment, a new approach is proposed for ranking fuzzy relative closeness indexes based on which the DMUs are ranked. The developed framework greatly reduces the complexity of computation compared with commonly used existing methods in the literature. To validate the proposed methodology and proposed ranking method, a numerical example is illustrated and compared the results with an existing approach.

This paper is concerned with the solution procedure of a fully fuzzy linear programming (FFLP) problem in which all parameters are represented in terms of triangular fuzzy numbers. According to the...

Pythagorean fuzzy set as an extension of fuzzy set has been presented to handle the uncertainty in real-world decision-making problems. In this work, we formulate a shortest path (SP) problem in an interval-valued Pythagorean fuzzy environment. Here, the costs related to arcs are taken in the form of interval-valued Pythagorean fuzzy numbers (IVPFNs). The main contributions of this paper are fourfold: (1) the interval-valued Pythagorean fuzzy optimality conditions in directed networks are described to design of solution algorithm. (2) To do this, an improved score function is used to compare the costs between different paths with their arc costs represented by IVPFNs. (3) Based on these optimality conditions and the improved score function, the traditional Dijkstra algorithm is extended to find the cost of interval-valued Pythagorean fuzzy SP (IVPFSP) and corresponding IVPFSP. (4) Finally, a small sized telecommunication network is provided to illustrate the potential application of the proposed method.

An efficient method to handle the uncertain parameters of a linear programming (LP) problem is to express the uncertain parameters by fuzzy numbers which are more realistic, and create a conceptual and theoretical framework for dealing with imprecision and vagueness. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand side, and/or the elements of the coefficient matrix. The aim of this article is to introduce a formulation of FLP problems involving interval-valued trapezoidal fuzzy numbers for the decision variables and the right-hand-side of the constraints. We propose a new method for solving this kind of FLP problems based on comparison of interval-valued fuzzy numbers by the help of signed distance ranking. To do this, we first define an auxiliary problem, having only interval-valued trapezoidal fuzzy cost coefficients, and then study the relationships between these problems leading to a solution for the primary problem. It is demonstrated that study of LP problems with interval-valued trapezoidal fuzzy variables gives rise to the same expected results as those obtained for LP with trapezoidal fuzzy variables.

In the article, Veeramani and Sumathi [10] presented an interesting algorithm to solve a fully fuzzy linear fractional programming (FFLFP) problem with all parameters as well as decision variables as triangular fuzzy numbers. They transformed the FFLFP problem under consideration into a bi-objective linear programming (LP) problem, which is then converted into two crisp LP problems. In this paper, we show that they have used an inappropriate property for obtaining non-negative fuzzy optimal solution of the same problem which may lead to the erroneous results. Using a numerical example, we show that the optimal fuzzy solution derived from the existing model may not be non-negative. To overcome this shortcoming, a new constraint is added to the existing fuzzy model that ensures the fuzzy optimal solution of the same problem is a non-negative fuzzy number. Finally, the modified solution approach is extended for solving FFLFP problems with trapezoidal fuzzy parameters and illustrated with the help of a numerical example.

In this paper, a well-known network-structured problem called the transportation problem (TP) is considered in an uncertain environment. The transportation costs, supply and demand are represented by trapezoidal intuitionistic fuzzy numbers (TrIFNs) which are the more generalized form of trapezoidal fuzzy numbers involving a degree of acceptance and a degree of rejection. We formulate the intuitionistic fuzzy TP (IFTP) and propose a solution approach to solve the problem. The IFTP is converted into a deterministic linear programming (LP) problem, which is solved using standard LP algorithms. The main contributions of this paper are fivefold: (1) we convert the formulated IFTP into a deterministic classical LP problem based on ordering of TrIFNs using accuracy function; (2) in contrast to most existing approaches, which provide a crisp solution, we propose a new approach that provides an intuitionistic fuzzy optimal solution; (3) in contrast to existing methods that include negative parts in the obtained intuitionistic fuzzy optimal solution and intuitionistic fuzzy optimal cost, we propose a new method that provides non-negative intuitionistic fuzzy optimal solution and optimal cost; (4) we discuss about the advantages of the proposed method over the existing methods for solving IFTPs; (5) we demonstrate the feasibility and richness of the obtained solutions in the context of two application examples.

In this study, we show that ranking fuzzy numbers with the area method using circumcenter of centroids presented by Rao and Shankar failed to rank effectively the generalized fuzzy numbers. By proving a theorem and using some numerical examples, we demonstrate that their proposed method cannot rank consistently some fuzzy numbers or is not consistent with human intuition.

The conventional DEA models such as CCR and BBC models require precise input and output data, which may not always be available in real world applications. However, in real life problems, inputs and outputs are often imprecise. To deal with imprecise data, the notion of fuzziness has been introduced in DEA and so the DEA has been extended to fuzzy DEA (FDEA). In this chapter, the main approaches for solving FDEA models are classified into five groups and the mathematical approaches of each category are described briefly. Then, R codes for each FDEA model are provided. Finally, numerical examples are provided to illustrate the main advantages of R in FDEA models.

The aim of this chapter is to study linear programming problem with fuzzy coefficients as one of the convenient model of fuzzy linear programs. In this way, we first introduce a general form of Fuzzy Number Linear programming (FNLP) problems and then give the fundamental concepts which are useful in throughout of the chapter.

Since the fuzziness may appear in many ways for the parameters of linear programming models, hence the definition of fuzzy linear programming is not unique. One of these models is Semi-Fully Fuzzy Linear Programming (SFFLP) problem where the coefficients in the objective function, the right hand side vector and the decision variables are a kind of fuzzy numbers, simultaneity. This chapter is assigned to these type of problems.

The shortest path (SP) problem is a network optimization problem with a wide range of applications in business and engineering. Conventional network problems assume precise values for the weights of the edges. However, these weights are often vague and ambiguous in practical applications. Several heuristics have been proposed to find the shortest path (SP) weight and the corresponding SP on a network with fuzzy arc weights. These heuristics largely use α -cuts and the least squares method. We propose an artificial bee colony (ABC) algorithm to solve the fuzzy SP (FSP) problems with fuzzy arc weights. The performance of the proposed ABC algorithm is compared with the performance of other competing algorithms with two SP problems taken from the literature. We present a wireless sensor network (WSN) problem and demonstrate the applicability of the proposed method and exhibit the efficiency of the procedures and algorithms.

Transportation problems have various applications in logistics and supply chains for reducing costs. In real-life situations, the parameters of transportation problems may not be known precisely because of uncontrollable factors. Herein, we propose a new method for solving fuzzy transportation problems (FTPs) in which the transportation costs and supply and demand are represented by non-negative LR flat fuzzy numbers. The FTP is converted into four transportation problems, which are solved using standard transportation simplex algorithms. The advantages of the proposed method over existing methods are discussed in the context of two application examples. The results show that the proposed method is simpler and computationally more efficient than existing methods in the literature.

The Shortest Path (SP) problem is one of the most widely used problems in network optimization which has a wide range of applications in various fields of science and engineering such as communication, transportation, routing and scheduling. The aim of this problem is to find a minimum cost path between two specified nodes. In the present communication, we consider a modified version of the SP known as constraint SP (CSP) problem with an additional constraint that establishes an upper limit on the travel time for the path. The objective of the CSP problem is to determine a minimum cost path between two specified nodes that the traversal time of the path does not exceed from a specified time. Traditional CSP problems assume the arc weights represented by time and cost are specified precisely. However, these weights can fluctuate with traffic conditions, weather, or payload. For this reason, being able to deal with vague and imprecise data may greatly contribute to the application of CSP problems. Here, we first formulate a CSP problem in a directed network where the arc weights represented by cost and time are intuitionistic trapezoidal fuzzy numbers. We then develop an approach for solving the intuitionistic fuzzy CSP problem under consideration. Finally, we present a small numerical example to illustrate the proposed approach.

In this chapter, the non-simplex based approaches for solving several kinds of LPPs with fuzzy parameters are explored. In such approaches, the fuzzy constraints are first converted to crisp ones based on arithmetic operations on fuzzy numbers and then the standard methods are used for solving the crisp problems. Such approaches increase the number of functional constraints and thus directly affect the computational time of the simplex method.

The aim of this chapter is to study the simplex based approaches for solving several kinds of LP problems with fuzzy parameters. In the LP problems with fuzzy parameters that can be solved by the use of simplex approaches, some or all parameters of the problems under consideration may be fuzzy numbers and the inequalities may be interpreted in terms of fuzzy rankings. Here, we first classify such fuzzy LP problems into five general groups and then discuss the solution methodologies for each one.

Fuzzy sets as defined by Zadeh (Inf. Con. 8(3):338–353, 1965 [1]) provide a useful, effective and operative tool for handling imprecise data. It not only provides a powerful representation of measurement uncertainty, but also gives a meaningful representation of vague concepts expressed in natural language. The purpose of this chapter is to review the basic definitions and results on fuzzy sets, fuzzy numbers, fuzzy arithmetic and fuzzy ranking methods.

Many researchers have focused on a Transportation Problem (TP) in uncertain environment because of its importance to various applications. This paper is concerned with the solution procedure of a TP in which transportation costs are represented in terms of intuitionistic triangular fuzzy numbers and supplies and demands are real numbers. We first formulate the intuitionistic fuzzy TP (IFTP) and then propose a new solution technique to solve the problem. Based on the proposed approach, the IFTP is converted into a Multi Objective Linear Programming (MOLP) problem with five objective functions. Then, a lexicographic approach is used to obtain the efficient solution of the resulting MOLP problem. The optimization process confirms that the optimum intuitionistic fuzzy transportation cost preserves the form of an intuitionistic triangular fuzzy number. A simple numerical example is included to illustrate of the proposed technique. The obtained results confirm the reliability and applicability of the proposed approach.

Network flows over time (also called Dynamic Network Flows) are considered as generalized form of standard (static) networks by adding an element of time. The factor of transit time on the arcs which specify the amount of time it takes for one unit of the flows to pass through a particular arc, make these networks different to traditional form of networks. While in most of network flow problems, transit times and capacities are given constant, stochastic or Fuzzy numbers, in this paper we suppose that transit times, capacities and consequently flows on arcs fall within specific ranges expressed as compact intervals. This vagueness in transit times, capacities and flows could arise in a number of ways: 1) when determining the exact capacity or transit time quantities is hard and there may happen errors in determining them which intervals reflect the measurement errors; 2) when it is impossible or even it is not needed to produce neither a distribution nor fuzzy functions but transit times, capacities and flows lie within specific ranges, or vary in time within these ranges. While determining the exact time for traveling in specific path in network transportation system is extremely difficult or almost impossible, determination of interval which can describe the range of required time for travelling in the network would be easy and fully operational. Our contribution in this paper is producing maximum flow and quickest s − t flow algorithms, first to network flows over time with interval-valued capacities and then to T-length bounded network flows over time with interval-valued transit times and capacities.

Transportation problems have wide applications in logistics and supply chain for reducing the cost. Effective algorithms have been proposed to solve the transportation problem in the case when all of the parameters, namely the supply, demand values and the unit transportation costs, are given in a precise way. However, in real applications, there are many diverse situations due to uncertainty. So, it is significant to investigate the transportation problem under uncertain environment. In this paper, a two-step method is proposed for solving fuzzy transportation problem (FTP) where all of the parameters are represented by non-negative triangular fuzzy numbers. The first is to employ fuzzy arithmetic to convert FTP into a linear programming with fuzzy costs and crisp constraints. The second is to use a new decomposition technique to transform the resulting problem into three crisp bounded transportation problems. The advantages of the proposed method over the existing methods are discussed by an application example. The obtained results show that the method proposed in this study is simpler and computationally more efficient than some existing methods commonly used in the literature.

In the article (Sudhagar and Ganesan in Optim Eng 10.1007/s11081-012-9202-6) Sudhagar and Ganesan presented an algorithm to solve a fuzzy transportation problem without converting it into a crisp transportation problem based upon score method of ranking fuzzy numbers. Using an example we show that their method will not always lead to a fuzzy optimal solution. Specially, we give a non negative fuzzy solution for the numerical example solved by them which has less total fuzzy transportation cost.

All the variables are considered fuzzy, including input and output data and efficiency scores.A lexicographic multi-objective linear programming optimization approach is proposed.Fuzzy input and output targets are computed.A super-efficiency fuzzy DEA model is also formulated to rank fuzzy efficient DMUs. There is an extensive literature in data envelopment analysis (DEA) aimed at evaluating the relative efficiency of a set of decision-making units (DMUs). Conventional DEA models use definite and precise data while real-life problems often consist of some ambiguous and vague information, such as linguistic terms. Fuzzy sets theory can be effectively used to handle data ambiguity and vagueness in DEA problems. This paper proposes a novel fully fuzzified DEA (FFDEA) approach where, in addition to input and output data, all the variables are considered fuzzy, including the resulting efficiency scores. A lexicographic multi-objective linear programming (MOLP) approach is suggested to solve the fuzzy models proposed in this study. The contribution of this paper is fivefold: (1) both fuzzy Constant and Variable Returns to Scale models are considered to measure fuzzy efficiencies; (2) a classification scheme for DMUs, based on their fuzzy efficiencies, is defined with three categories; (3) fuzzy input and output targets are computed for improving the inefficient DMUs; (4) a super-efficiency FFDEA model is also formulated to rank the fuzzy efficient DMUs; and (5) the proposed approach is illustrated, and compared with existing methods, using a dataset from the literature.

Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature.

In a recent paper, Kaur and Kumar (2012) proposed a new method based on ranking function for solving fuzzy transportation problem (FTP) by assuming that the values of transportation costs are represented by generalized trapezoidal fuzzy numbers. Here it is shown that once the ranking function is chosen, the FTP is converted into crisp one, which is easily solved by the standard transportation algorithms. The main contribution here is the reduction of the computational complexity of the existing method. By solving two application examples, it is shown that it is possible to find a same optimal solution without solving any FTP. Since the proposed approach is based on classical approach it is very easy to understand and to apply on real life transportation problems for the decision makers.

There are several methods in the literature to deal with the sensitivity analysis of fuzzy linear programming (FLP) problems. In this paper, the shortcomings of some existing methods are pointed out and to overcome these shortcomings, a new method is proposed to handle the sensitivity analysis of that kind of FLP problems in which the elements of the coefficient matrix of the constraints are represented by real numbers, and the remaining parameters are represented by symmetric trapezoidal fuzzy numbers. To the best of our knowledge, till now there is no method in the literature to deal with the sensitivity analysis of such FLP problems without converting them to the crisp linear programming problems. The advantages of the proposed method over existing methods are discussed. To illustrate the approach a numerical example is examined by using the proposed method and the obtained results are discussed. The method is easy to understand and to apply for dealing with the sensitivity analysis of the same problems occurring in real life situations.

In recent years various attempts have been made to study the solution of fuzzy variable linear programming problems by introducing and solving certain auxiliary problems or directly by the use of dual simplex method based on a certain linear ranking function. But these methods are not applicable for situations in which some or all variables are restricted to lie within fuzzy lower and fuzzy upper bounds. In this paper, as a natural extension of the results in crisp linear programming and based on a certain linear ranking function we obtain some new results leading to a new method to overcome this shortcoming. To show the application of our method a real life problem is solved by using the proposed method. Also, one application of this approach in solving maximum flow problem with fuzzy capacity is dealt with.

There are several methods, in the literature, for solving fuzzy variable linear programming problems fuzzy linear programming in which the right-hand-side vectors and decision variables are represented by trapezoidal fuzzy numbers. In this paper, the shortcomings of some existing methods are pointed out and to overcome these shortcomings a new method based on the bounded dual simplex method is proposed to determine the fuzzy optimal solution of that kind of fuzzy variable linear programming problems in which some or all variables are restricted to lie within lower and upper bounds. To illustrate the proposed method, an application example is solved and the obtained results are given. The advantages of the proposed method over existing methods are discussed. Also, one application of this algorithm in solving bounded transportation problems with fuzzy supplies and demands is dealt with. The proposed method is easy to understand and to apply for determining the fuzzy optimal solution of bounded fuzzy variable linear programming problems occurring in real-life situations.

In this paper, we generalize the bounded dual simplex algorithm for solving minimum cost flow problem with fuzzy cost, which its aim is to find the least fuzzy cost of a commodity through a capacitated network in order to satisfy demands at certain nodes using available supplies at other nodes. This algorithm begins with dual feasibility and iterates between dual and primal problems until optimality is achieved. Here, we use the linear ranking functions to compare fuzzy numbers. By using the proposed method the optimal solution of minimum cost flow problems with fuzzy costs can be easily obtained. To illustrate the proposed method a numerical example is solved and the obtained results are discussed.

Fuzzy Linear Programming (FLP), as one of the main branches of operation research, is concerned with the optimal allocation of limited resources to several competing activities on the basis of given criteria of optimality in fuzzy environment. Numerous researchers have studied various properties of FLP problems and proposed different approaches for solving them. This work presents a survey on models and methods for solving FLP problems. The solution approaches are divided into four areas: (1) Linear Programming (LP) problems with fuzzy inequalities and crisp objective function, (2) LP problems with crisp inequalities and fuzzy objective function, (3) LP problems with fuzzy inequalities and fuzzy objective function and (4) LP problems with fuzzy parameters. In the first area, the imprecise right-hand-side of the constraints is specified with a fuzzy set and a monotone membership function expressing the individual satisfaction of the decision maker. In the second area, a fuzzy goal and corresponding tolerance is defined for each coefficient of decision variables in the objective function. In the third area, both the coefficients of decision variables in the objective function and the right-hand-side of the constraints are specified by fuzzy goals and corresponding tolerances. In the fourth area, some or all parameters of the LP problem are represented in terms of fuzzy numbers. In this contribution, some of the most common models and procedures for solving FLP problems are analyzed. The solution approaches are illustrated with numerical examples.

Cost efficiency (CE) evaluates the ability to produce current outputs at minimal cost, given its input prices. In ordinary CE model, the input prices are assumed to be definite. In recent years, various attempts have been made to measuring CE when the input prices are as trapezoidal fuzzy numbers. The main contribution of this paper is to provide a new approach for generalizing the CE of decision making units in data envelopment analysis when the input prices are trapezoidal fuzzy numbers, where concepts of fuzzy linear programming problems and CE, are directly used. Here, the author used the linear ranking functions to compare fuzzy numbers. The proposed method is illustrated with two application examples and proves to be persuasive and acceptable in real world systems.

In this paper, we generalize the concept of sensitivity analysis in fuzzy number linear programming (FLNP) problems by applying fuzzy simplex algorithms and using the general linear ranking functions on fuzzy numbers. The purpose of sensitivity analysis is to determine changes in the optimal solution of FNLP problem resulting from changes in the data. If the change affects the optimality of the basis, we perform primal pivots to achieve optimality by use of the fuzzy primal simplex method. Whenever the change destroys the feasibility of the optimal basis, we perform dual pivots to achieve feasibility by use of the fuzzy dual simplex method.

In the real word, there are many problems which have linear programming models and sometimes it is necessary to formulate these models with parameters of uncertainty. Many numbers from these problems are linear programming problems with fuzzy variables. Some authors considered these problems and have developed various methods for solving these problems. Recently, Mahdavi-Amiri and Nasseri (2007) considered linear programming problems with trapezoidal fuzzy data and/or variables and stated a fuzzy simplex algorithm to solve these problems. Moreover, they developed the duality results in fuzzy environment and presented a dual simplex algorithm for solving linear programming problems with trapezoidal fuzzy variables. Here, the authors show that this presented dual simplex algorithm directly using the primal simplex tableau algorithm tenders the capability for sensitivity (or post optimality) analysis using primal simplex tableaus.

Two existing methods for solving a class of fuzzy linear programming (FLP) problems involving symmetric trapezoidal fuzzy numbers without converting them to crisp linear programming problems are the fuzzy primal simplex method proposed by Ganesan and Veeramani [1] and the fuzzy dual simplex method proposed by Ebrahimnejad and Nasseri [2]. The former method is not applicable when a primal basic feasible solution is not easily at hand and the later method needs to an initial dual basic feasible solution. In this paper, we develop a novel approach namely the primal-dual simplex algorithm to overcome mentioned shortcomings. A numerical example is given to illustrate the proposed approach.

Recently Ganesan and Veeramani introduced a new approach for solving a kind of linear programming problems involving symmetric trapezoidal fuzzy numbers without converting them to the crisp linear programming problems. But their approach is not efficient for situations in which some or all variables are restricted to lie within fuzzy lower and fuzzy upper bounds. In this paper, by a natural extension of their approach we obtain some new results leading to a new method to overcome this shortcoming.

The most popular approach to handle the challenge of solving fuzzy linear programming problems is to convert the fuzzy linear programming into the corresponding deterministic linear programming. Mahdavi-Amiri and Nasseri [15,16] developed the fuzzy dual simplex algorithm to fuzzy linear programming with fuzzy parameters. In this paper, we use the complementary slackness to solve it without the need of a simplex tableau.

There are several methods in the literature for solving fuzzy transportation problem (FTP) in which the transportation costs and supply and demand are fuzzy numbers. In this paper, the shortcomings of existing methods are pointed out and to overcome these shortcomings a novel algorithm, based on a new lexicographic ordering on triangular fuzzy numbers, is proposed to solve FTP by converting it into a multi-objective linear programming (MOLP) problem with three-objective functions. The advantages of the proposed method over the existing methods are discussed. The main contribution of this paper is providing a new technique to reduce the computational complexity of recently introduced fuzzy transportation models. Finally, to show the application of proposed method an application example and an existing real world FTP are solved by using the proposed method.

In conventional transportation problems it is assume that the decision maker has exact information about the coefficients belonging to the problem. In real world applications, values of transportation cost, supply and demand of the product may not be known precisely due to uncontrollable factors. To deal with such situations, fuzzy set theory is applied in literature to solve the transportation problems. In this paper, some existing methods for solving fuzzy transportation problem (FTP) are reviewed. Moreover, the shortcomings of some existing methods are pointed out and to overcome these shortcomings, a new method is proposed for finding that kind of FTP in which the transportation costs and values of supplies and demands are represented by non-negative triangular fuzzy numbers. The FTP is converted into three crisp transportation problems and these crisp problems are solved with the standard transportation simplex algorithms. The advantages of the proposed method over the existing methods are discussed by an application example. The obtained results show that the method proposed in this study is simpler and computationally more efficient than some existing methods commonly used in the literature.

Data envelopment analysis (DEA) is a widely used technique for measuring the relative efficiencies of decision making units (DMUs) with multiple inputs and multiple outputs. However, in real-world problems, the observed values of the input and output data are often vague or random. Indeed, decision makers (DMs) may encounter a hybrid uncertain environment where fuzziness and randomness coexist in a problem. This paper proposes a normal distribution with fuzzy components to deal with fuzziness and randomness of input and output data. The proposed normal distribution introduces a class of fuzzy random variables. We propose a DEA model problems characterised by fuzzy stochastic variables. Unlike the existing approaches, the proposed fuzzy stochastic DEA model is transformed into a deterministic model with linear constraints. A case study in the insurance industry is presented to exhibit the efficacy and the applicability of the proposed model.

In a recently quoted paper [N. Mahdavi-Amiri and S.H. Nasseri, Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables, Fuzzy Sets and Systems 158 (2007)1961-1978], has been established the dual of linear programming problem with trapezoidal fuzzy variables as a fuzzy number linear programming, using certain general linear ranking functions. In this paper, we define a new dual problem for the linear programming problem with trapezoidal fuzzy variables as a linear programming problem with trapezoidal fuzzy variables and hence deduce the duality results such as weak duality, strong duality and complementary slackness theorems. In throughout of the paper, we apply the trapezoidal fuzzy numbers as a convenient fuzzy numbers which is more practical in comparison of other numbers (and also as a general form of triangular fuzzy numbers) and moreover use the certain linear ranking functions for the sake of illustrating the performance of our approach, but it is no restrict at all to use these linear ranking functions. In particular, we emphasize that the new duality definition and the associated duality results lead us to provide the primal and dual simplex algorithms for solving the linear programming problems with trapezoidal fuzzy numbers.

Recently, Ebrahimnejad (2011) generalised the primal-dual simplex algorithm for solving a class of fuzzy linear programming (FLP) problems involving symmetric trapezoidal fuzzy numbers without converting them to crisp linear programming problems. In this paper, we describe the fuzzy primal-dual algorithm in tableau form and illustrate our approach by a numerical example. If there is no uncertainty among parameters then the proposed approach gives the same result as in crisp FLP problems. Since the proposed method is a direct extension of classical method so it is very easy to understand and apply the proposed method to find the fuzzy optimal solution of FLP problems occurring in the real life situations.

Solving fuzzy linear programming problems have received a great deal of attention. Recently, Ganesan and Veeramani (2006) developed a new method for solving a kind of these problems involving symmetric trapezoidal fuzzy numbers without converting them to the crisp linear programming problems based on primal simplex method. But their method has no efficient when a primal basic feasible solution is not at hand. In this paper, we develop a new dual simplex algorithm to overcome this shortcoming by using the duality results which has been proposed by Nasseri and Mahdavi-Amiri (2009) and Nasseri et al. (2010). This algorithm starts with a dual basic feasible solution, but primal basic infeasible solution and walks to an optimal solution by moving among adjacent dual basic feasible solution.

There are two important approaches based on linear ranking functions for solving linear programming problems with cost coefficients as an auxiliary problem to obtain a fuzzy solution of fuzzy variable linear programming problem. The first approach uses the primal simplex method that assumes an initial primal feasible basic solution is at hand. The second approach is based on dual simplex method that begins with a basic dual feasible basic solution and proceeds by pivoting through a series of dual basic solutions until the associated complementary primal basic fuzzy solution is feasible. In this paper, we propose a new method called the primal-dual algorithm, which is similar to the dual simplex method and begins with dual feasibility and proceeds to obtain primal feasibility while maintaining complementary slackness. An important difference between the dual simplex method and the primal-dual method is that the primal-dual algorithm does not require a dual feasible solution to be basic. This algorithm is useful specially for solving minimum fuzzy cost flow problem in which finding an initial dual feasible solution turns out to be a trivial task.

Linear programming models play an important role in management, economic, data envelopment analysis, operations research and many industrial applications. In many practical situations there is a kind of ambiguity in the parameters of these models which can be expressed by means of fuzzy numbers. In the literature of fuzzy mathematical programming there are many types of the fuzzy linear programming problems. But in this paper, we deal with a kind of linear programming which includes the triangular fuzzy numbers in its parameters. For finding the solution of these problems, we propose a revised simplex algorithm for an extended linear programming problem which is equivalent to the original fuzzy linear programming problem. An illustrative example is presented to clarify the proposed approach. A fuzzy DEA model has been also considered as a practical application to illustrate the effectiveness of the proposed approach. (Received 20 December 2009; Revised 29 September 2010; Accepted 11 November 2010)

Fuzzy number linear programming problems have recently attracted some interests. Some authors used the concept of comparison of fuzzy numbers for solving fuzzy linear programming problems. In effect, most convenient methods are based on the concept of comparison of fuzzy numbers by use of ranking functions. But the existing methods are not useful for situations in which some or all variables are restricted to lie within fuzzy lower and fuzzy upper bounds. In this paper, we introduce a new method called the bounded dual simplex method for bounded fuzzy number linear programming problems which is useful for these situations. This algorithm constructs a dual feasible basic solution after obtaining a working basic and from there moves towards attaining primal feasibility while maintaining dual feasibility throughout.

Linear programming problems with fuzzy variables have been investigated by many researchers in the recent literature. Some methods to solve these problems, such as the primal simplex method and the dual simplex method, are based on the concept of comparison of fuzzy numbers by using ranking functions. In this paper, we give a new primal-dual algorithm for solving linear programming problems with fuzzy variables by using duality results, which was proposed by Mahdavi-Amiri and Nasseri (2007). This algorithm will be useful for sensitivity analysis when the activity vectors change for basic columns.