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This paper deals with knapsack problem in fuzzy nature, where both the objective function and constraints are considered to be fuzzy. Three different models for fuzzy knapsack problem are proposed including, expected value model, chance-constrained model, and dependent-chance model. Credibility ranking method is applied to convert the fuzzy models into a crisp equivalent linear one considering triangular and trapezoidal fuzzy numbers. The solution of the fuzzy problem is obtained with respect to different satisfaction degrees in the objective function and constraints. Several numerical examples are given to demonstrate different models and concepts. The proposed approaches are applied to model and to solve a fuzzy pre-disaster investment decision problem.

One of the main topics discussed in a supply chain is the production-distribution problem. Producing and distributing the products plays a key role in reducing the costs of the chain. To design a supply chain, a network of efficient management and production-distribution decisions is essential. Accordingly, providing an appropriate mathematical model for such problems can be helpful in designing and managing supply chain networks. Mathematical formulations must be drawn close to the real world due to the importance of supply chain networks. This makes those formulations more complicated. In this study, a novel multi-objective formulation is devised for the production-distribution problem of a supply chain that consists of several suppliers, manufacturers, distributors, and different customers. Also, a Mixed Integer Linear Programming (MILP) mathematical model is proposed for designing a multi-objective and multi-period supply chain network. In addition, grey flexible linear programming (GFLP) is done for a multi-objective production-distribution problem in a supply chain network. The network is designed for the first time to cope with the uncertain nature of costs, demands, and capacity parameters. In this regard, due to the NP-hardness and complexity of problems and the necessity of using meta-heuristic algorithms, NSGA-II and Fast PGA algorithm are applied and compared in terms of several criteria that emphasize the quality and diversity of the solutions.

The first author would like to appreciate from the research grant of University of Mazandaran. The research of Jose Luis Verdegay is supported in part by the project TIN2017-86647-P (Spanish Ministry of Economy and Competitiveness) which includes FEDER funds from the European Union., Recently fuzzy interval flexible linear programs have attracted many interests. These models are an extension of the classical linear programming which deal with crisp parameters. However, in most of the real-world applications, the nature of the parameters of the decisionmaking problems is generally imprecise. Such uncertainties can lead to increased complexities in the related optimisation efforts. Simply ignoring these uncertainties is considered undesired as it may result in inferior or wrong decisions. Therefore, inexact linear programming methods are desired under uncertainty. In this paper,weconcentrate a fuzzy flexible linear programming model with flexible constraints and the interval objective function and then propose a new solving approach based on solving an associated multi-objective model. Finally, numerical example is included to illustrate the mentioned solving process., University of Mazandaran, Spanish Ministry of Economy and Competitiveness TIN2017-86647-P, European Commission

One of the main topics discussed in a supply chain is the production-distribution problem. Producing and distributing the products plays a key role in reducing the costs of the chain. To design a supply chain, a network of efficient management and production-distribution decisions is essential. Accordingly, providing an appropriate mathematical model for such problems can be helpful in designing and managing supply chain networks. Mathematical formulations must be drawn close to the real world due to the importance of supply chain networks. This makes those formulations more complicated. In this study, a novel multi-objective formulation is devised for the production-distribution problem of a supply chain that consists of several suppliers, manufacturers, distributors, and different customers. Also, a Mixed Integer Linear Programming (MILP) mathematical model is proposed for designing a multi-objective and multi-period supply chain network. In addition, grey flexible linear programming (GFLP) is done for a multi-objective production-distribution problem in a supply chain network. The network is designed for the first time to cope with the uncertain nature of costs, demands, and capacity parameters. In this regard, due to the NP-hardness and complexity of problems and the necessity of using meta-heuristic algorithms, NSGA-II and Fast PGA algorithm are applied and compared in terms of several criteria that emphasize the quality and diversity of the solutions.

Neutrosophic set theory is a generalization of the intuitionistic fuzzy set which can be considered as a powerful tool to express the indeterminacy and inconsistent information that exist commonly in engineering applications and real meaningful science activities. In this paper an interval neutrosophic linear programming (INLP) model will be presented, where its parameters are represented by triangular interval neutrosophic numbers (TINNs) and call it INLP problem. Afterward, by using a ranking function we present a technique to convert the INLP problem into a crisp model and then solve it by standard methods.

The aim of this paper is to introduce a formulation of linear programming problems involving intuitionistic fuzzy variables. Here, we will focus on duality and a simplex-based algorithm for these p...

PurposeFor extending the common definitions and concepts of grey system theory to the optimization subject, a dual problem is proposed for the primal grey linear programming problem.Design/methodology/approachThe authors discuss the solution concepts of primal and dual of grey linear programming problems without converting them to classical linear programming problems. A numerical example is provided to illustrate the theory developed.FindingsBy using arithmetic operations between interval grey numbers, the authors prove the complementary slackness theorem for grey linear programming problem and the associated dual problem.Originality/valueComplementary slackness theorem for grey linear programming is first presented and proven. After that, a dual simplex method in grey environment is introduced and then some useful concepts are presented.

The current paper focuses on a multiobjective transportation problem which is a special type of vector minimum problem. In order to deal with such a problem, a new model based on fuzzy goal program...

Because of the intricate nature of real-world scenarios, experts could encounter many ambiguities throughout the decision-making (DM) process. Adopting a DM strategy in conditions of indeterminacy so that the decision makers are limited to a small number of experts is always helpful in real life. Neutrosophic conception is a convenient technique for handling inconsistent, ambiguous, and uncertain values. This research presents an autocratic DM strategy based on Neutrosophic Sets (NSs) to address these ambiguities. The essential component of the suggested technique is the conversion of diverse management decision and weight matrices into a unified evaluation matrix. Supplier Selection (SS) is a multi-criteria decision-making problem where a limited number of alternative suppliers are evaluated using a limited set of criteria. The suggested methodology based on different score functions is applied to SS issues involving construction materials. The numerical illustrations indicate the success of the introduced method in selecting the best supplier with the least computational complexity. The important point obtained in this research is that adopting a suitable score function appropriate to the characteristics of the data plays an important role in the decision-making process.

In this paper, due to increasing competition in the business world, which makes decision makers dealing with multiple options/information for optimal decisions on a single task, we will look at mul...

Data Envelopment Analysis (DEA) is a widely used technique for measuring the relative efficiencies of Decision Making Units (DMUs) with multiple deterministic 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. Hence, we formulate a new DEA model to deal with fuzzy stochastic DEA models. The contributions of the present study are fivefold: (1) We formulate a deterministic linear model according to the probability–possibility approach for solving input-oriented fuzzy stochastic DEA model, (2) In contrast to the existing approach, which is infeasible for some threshold values; the proposed approach is feasible for all threshold values, (3) We apply the cross-efficiency technique to increase the discrimination power of the proposed fuzzy stochastic DEA model and to rank the efficient DMUs, (4) We solve two numerical examples to illustrate the proposed approach and to describe the effects of threshold values on the efficiency results, and (5) We present a pilot study for the NATO enlargement problem to demonstrate the applicability of the proposed model.

In this paper, a multi-objective model of transportation problem with multi-choice parameters is presented. The alternative choices of the multi-choice parameter are considered as random variables. Due to the conflicting objective functions and the multi-choice of parameters, the problem cannot be solved directly. So, we first consider the interpolating polynomials for the multi -choice parameters. Next, we use the expected value model and the chance constraint approach to convert the original problem into a crisp form which is equivalent to it. Finally, a fuzzy approach is applied to find a compromise solution.

Data envelopment analysis (DEA) is a widely used mathematical programming technique for measuring the relative efficiency of decision-making units which consume multiple inputs to produce multiple outputs. Although precise input and output data are fundamentally used in classical DEA models, real-life problems often involve uncertainties characterized by fuzzy and/or random input and output data. We present a new input-oriented dual DEA model with fuzzy and random input and output data and propose a deterministic equivalent model with linear constraints to solve the model. The main contributions of this paper are fourfold: (1) we extend the concept of a normal distribution for fuzzy stochastic variables and propose a DEA model for problems characterized by fuzzy stochastic variables; (2) we transform the proposed DEA model with fuzzy stochastic variables into a deterministic equivalent linear form; (3) the proposed model which is linear and always feasible can overcome the nonlinearity and infeasibility in the existing fuzzy stochastic DEA models; (4) we present a case study in the banking industry to exhibit the applicability of the proposed method and feasibility of the obtained solutions.

In this paper, we consider a Fuzzy Stochastic Linear Fractional Programming problem (FSLFP). In this problem, the coefficients and scalars in the objective function are the triangular fuzzy number ...

Formulation of a balanced diet, which provides all nutritional requirements of livestock in accordance with its special physiological conditions, is not possible totally. According to the frequency and salient increasing of the breeding center in the country and the shortage of forage and food materials considering the available resources, diet optimization is considered as an essential program. Linear programming used to in diet formulation. Fuzzy linear programming presents a new perspective for solving of different problems such as livestock diet formulation. The main feature of fuzzy forms is their mathematical design, which allows the decision makers to work with intervals. In the formulation of a balanced diet by using fuzzy linear programming, the economic power of stockbreeder and the satisfaction percentage of animal nutrient requirements will be considered and determined. According to the decision maker’s idea and with special accuracy, the diet problem can formulate in accordance with t...

Purpose The purpose of this paper is to discuss the animal diet problem in grey environment which is adapted to the real situations. In particular, a new approach to solve these problems is proposed. Design/methodology/approach With the objective to produce the least-cost diet, in the traditional model for optimizing the diet problem, the price of foods, the nutrients requirements and the necessity of foods requirement have been considered as grey interval numbers. Grey linear programming approach has been employed to solve the grey diet problem. Grey linear programming with flexibility in selection of the coefficients can be more effective for solving the diet problems. In this research, only the positioned method has been used. The grey diet model is solved by using GAMS software based on the positioned method. Findings The main contribution of this work is to introduce a new model in the practical case that is concerned with diet problem under a kind of uncertainty environment and furthermore, proposing a novel method to solve the formulated problem. In this way, using a grey model and applying all restrictions, the least cost for one kilogram of total mixed ration was 6,893-10,163 Rials, and at this level, cow’s nutrient requirement was met. Based on the numerical examination, which was done on the real case, the achieved results have showed that the uncertainty of foods requirement and nutrients requirements had slight effect on the animal budget diet. Originality/value This problem must be viewed from another perspective because of the uncertainty regarding the amount of nutrients per unit of foods and the diversity of animals’ daily needs to receive them. In particular, a new method to optimize the fully mixed diet of lactating cows in early lactation that are readily available in the northeast of Iran in uncertainty environment has been proposed.

‎cross-efficiency evaluation as an extension of the data envelopment analysis (DEA) has found an appropriate function in ranking decision making units (DMU). However, DEA suffers from a potential flaw, that is, the existence of multiple optimal solutions. Different methods have been proposed to obtain a unique solution (based on a specific criterion). One of such methods in the allocation of weights for the ranking of the DMUs is Wang et al.’s method [Wang, Y. M., Chin, K. S., Jiang, P. (2011). Weight determination in the cross-efficiency evaluation. Computers $\&‎$‎ Industrial Engineering, 61, 497-502. [32]]. This method aims to choose a solution among the multiple solutions by increasing the contribution of inputs in their use of resources as well as by increasing the contribution of the outputs in production and to the extent possible to covertly prevent the selection of zero solutions in inputs and outputs. In the present article, we argue that such selection of weights is not appropriate because in the cross-efficiency evaluation of the DMUs, we always search for the weights that use the minimum resources to increase the production. Therefore, we suggest that the selection of weights among the multiple weights should be determined by decreasing the contribution of inputs in the use of resources and increasing the contribution of outputs in the production yet it should overtly prevent the selection of zero solutions to the extent possible. ‎To this end‎, ‎some examples are illustrated which show how it differs with other methods‎.

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. The classical DEA models were initially formulated only for desirable inputs and outputs. However, undesirable outputs may be present in the production process which needs to be minimized. In addition, in real-world problems, the observed values of the input and output data are often vague or random. Indeed, decision makers may encounter a hybrid uncertain environment where fuzziness and randomness coexist in a problem. In order to deal with the above problems, this paper proposes fuzzy stochastic DEA model with undesirable outputs. Three fuzzy DEA models with respect to probability–possibility, probability–necessity and probability–credibility constraints are applied. The contribution of this paper is fourfold: (1) the proposed approach considers the impact of undesirable outputs on the performance of DMUs; (2) unlike the existing methods, the proposed solution approach provides efficiency scores in the range of zero and one for all DMUs; (3) the proposed approach analyzes the influence of the presence of both fuzzily imprecision and probabilistic uncertainty in the data over the efficiency results; and (4) a case study in the banking industry is presented to exhibit the efficacy of the procedures and demonstrate the applicability of the proposed model.

Good distribution of commodities from depots to predefined customers is an important factor on total costs, so this problem is expanded by many scholars to be more realistic, and then good management leads to achieve lower costs and more satisfaction for customers and companies. Several kinds of vehicle routing problems (VRPs) have been proposed in this area. In this paper, we combine three types of VRPs: multi-depot VRP, close and open VRP and heterogeneous VRP (HVRP) considering that each vehicle does not need to come back to the original depot. The goal of the problem is to minimize total costs and exceed from an average traveled distance for each type of vehicle internal drivers to make a fair between internal drivers. Thus, we use an augmented $\in -$ constraint method to achieve several optimal solutions. Finally, we use aggregate AHP-TOPSIS to obtain the best solution for the presented model.

Quadratic Programming (QP) is a mathematical modeling technique designed to optimize the usage of limited resources and has been widely applied to solve real world problems. In conventional quadratic programming model the parameters are known constants. However in many practical situations, it is not reasonable to require that the constraints or the objective function in quadratic programming problems be specified in precise, crisp terms. In such situations, it is desirable to use some type of Fuzzy Quadratic Programming (FQP) problem. In this paper a new approach is proposed to derive the fuzzy objective value of fuzzy quadratic programming problem, where the constraints coefficients and the right-hand sides are all triangular fuzzy numbers. The proposed method is solved using MATLABTM toolbox and the numerical results are presented.

Designing a multi-echelon multi-product multi-period supply chain model.Considering new product development effects in supply chain configuration.Developing a priority based genetic algorithm to find the suitable solution at reasonable time. New product development has become increasingly important recently due to highly competitive market place and economic reasons. Development and production of new products in the planning horizon require an efficient and responsiveness supply chain network. As new products appear in the market, the old products could become obsolete, and then phased out. A generously persuasive parameter for new product and developed product problems in a supply chain is the time which the developed products are introduced and the old products are phased out and also the time new products are introduced in the planning horizon in order to maximum the total profit.With consideration of the factors noted above, this study proposes to design a multi echelon multi product multi period supply chain model which incorporates product development and new product production and their effects on supply chain configuration.In terms of the solution technique, to overcome NP-hardness of the proposed model, priority based genetic algorithm is applied to find the suitable time for introducing developed and new product in the planning horizon, production schedule and design of supply chain network in order to maximum the total profit in a reasonable computational time. The accuracy of the proposed genetic algorithm is validated on small, medium and large instances that have been solved using the software LINGO, in order to evaluate the performance of the algorithm. Then, the implementation of the fuzzy crossover and mutation controllers is described. It is able to regulate the rates of crossover and mutation operators during the search process. Finally, a comparison is done on conventional GA and the controlled GA.

This study aims to design a multi-echelon, multi-objective supply chain model that incorporates new product development and its effects on supply chain configuration. To survive in a highly competitive industry, strategies to either collaborate or compete with rival firms within a network should be considered in the new product development process, and it is crucial to pay great attention to customers’ needs and interests. Considering the imprecise nature of some critical parameters plays an important role in making suitable strategic decisions. This fact requires considering uncertainties of the environment such as customer demands and supplier capacities. In this study, a supply chain involving multiple suppliers, manufacturers, distributors and customers and addressing a multi-objective, multi-period and multi-product aggregate procurement and production planning problem is considered. The first objective function aims to maximize the profit of the supply chain, including that associated with new product development. The second objective function considers customer satisfaction, and the third one maximizes the production of the developed and new products. To address real-world planning problems involving noisy, incomplete or erroneous data, the supplier capacity parameters of the supply chain and demand fluctuations are subject to uncertainty, which is modeled by fuzzy stochastic programming. Finally, the proposed multi-objective model is solved as a single-objective mixed integer programming model by applying the revised multi-choice goal programming method. In addition, a numerical example is provided to demonstrate the applicability of the proposed model.

In the real world, 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.

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.

One of the most interesting models of the linear programming in uncertainty environment is the flexible linear programming problem. It is shown that by using a suitable membership function for their constraints we can obtain an equivalent linear programming problem with fuzzy variables (FVLP).

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.

Goal Programming(GP) is one of the most important techniques to solve multiple objective programming problem, which for each target an aspirational level is considered by the decision maker. If several aspiration level are considered for each target, then there will be the issue of multi- choice goal programming problem which was first proposed by chang.T.C., in 2007. The first research on fuzzy multi-choice GP was presented by Bankian-Tabrizi, B. and etc. al., In the year (2012), they used triangular fuzzy numbers as the aspirational level of target. In this paper, a new model of flexible fuzzy multi-choice GP is considered which goals and constraints are of fuzzy flexible type, and in this paper a new method is presented to solve problem using linear multi-parametric Programming while the minimum degree membership for constraints and goals are considered by decision maker. For further understanding, we present a numerical example and solve it using the above method.

Fuzzy sets theory has been applied to many disciplines such as control theory and management sciences, mathematical modeling, industrial applications and etc. We usually face some difficulties when a such real-world problems are formulated into a mathematical programming problem. One of the difficulties is caused by the uncertainty in knowledge, information and/or decision makers preferences.

The use of weight restrictions in data envelopment analysis (DEA) is considered an appropriate method to avoid zero inputs and outputs weights. Due to a number of problems associated with the use of weight restrictions, the production trade-off method in DEA is often preferred. An important issue is that in most common DEA models, the internal structure of the production units is ignored, and the units are often considered as black-boxes. The current study aims to estimate the production trade-offs in two-stage network data envelopment analysis (NDEA) to observe its likely impact on efficiency evaluation and discrimination of units and subunits. Also, the probable effect of such the trade-offs are shown on the overall efficiency decomposition to divisional efficiencies and production possibility set (PPS) in two-stage NDEA. Finally, a numerical example is used to explain the results and compare the possible effects of different production trade-offs scenarios in two-stage NDEA with standard DEA models.

An information system network (ISN) can be modeled as a stochastic-flow network (SFN). There are several algorithms to evaluate reliability of an SFN in terms of Minimal Cuts (MCs). The existing algorithms commonly first find all the upper boundary points (called d-MCs) in an SFN, and then determine the reliability of the network using some approaches such as inclusion-exclusion method, sum of disjoint products, etc. However, most of the algorithms have been compared via complexity results or through one or two benchmark networks. Thus, comparing those algorithms through random test problems can be desired. Here, the authors first state a simple improved algorithm. Then, by generating a number of random test problems and implementing the algorithms in MATLAB, the proposed algorithm is demonstrated to be more efficient than some existing ones in medium-sized networks. The performance profile introduced by Dolan and More is used for analyzing the output of programs.

Some new concepts in regards to α -feasibility and α -efficiency of solutions in fuzzy mathematical programming problems are introduced in this paper, where α is a vector of distinct satisfaction degrees. Based on the defined concepts, a new method is suggested to solve fuzzy mathematical programming problems. In this sense, the proposed approach enables decision makers to take into account more flexible solutions by allowing desired distinct satisfactions in constraints. In the case of linear problems with fuzzy constraints, multi-parametric programming is employed to obtain the optimal solution as an affine function of distinct satisfaction degrees. In particular, it proves that the obtained solution is convex and continuous. Therefore, the different optimal solutions can be obtained by a simple substituting the new values of satisfaction parameters into the parametric profiles without any further optimization calculations, which is desirable for online optimization and sensitivity analysis of the profit to satisfaction parameters.

Ordering fuzzy quantities and their comparison play a key tool in many applied models in the world and in particular decision-making procedures. However a huge number of researches is attracted to this filed but until now there is any unique accepted method to rank the fuzzy quantities. In fact, each proposed method may has some shortcoming. So we are going to present a novel method based on the angle of the reference functions to cover a wide range of fuzzy quantities by over coming the draw backs of some existing methods. In the mentioned firstly, the angle between the left and right membership functions (the reference functions) of every fuzzy set is called Angle of Fuzzy Set (AFS), and then in order to extend ranking of two fuzzy sets the angle of fuzzy sets and α-cuts is used. The method is illustrated by some numerical examples and in particular the results of ranking by the proposed method and some common and existing methods for ranking fuzzy sets is compared to verify the advantage of the new approach. In particular, based on the results of comparison of our method with well known methods which are exist in the literature, we will see that against of most existing ranking approaches, our proposed approach can rank fuzzy numbers that have the same mode and symmetric spreads. In fact, the proposed method in this paper can effectively rank symmetric fuzzy numbers as well as the effective methods which are appeared in the literature. Moreover, unlike of most existing ranking approaches, our proposed approach can rank non-normal fuzzy sets. Finally, we emphasize that the concept of fuzzy ordering is one of key role in establishing the numerical algorithms in operations research such as fuzzy primal simplex algorithms, fuzzy dual simplex algorithms and as well as discussed in the works of Ebrahimnejad and Nasseri and coworkers [1] , [2] , [3] , [4] , [5] , [6] , [7] .

The objective of this paper is to deal with a kind of fuzzy linear programming problem involving triangular fuzzy numbers. Then some interesting and fundamental results are achieved which in turn lead to a solution of fuzzy linear programming models without converting the problems to the crisp linear programming models. Finally, the theoretical results are also supported by a real case study in a banking system. The same idea is emphasized to be also useful when a general LR fuzzy numbers is given.

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.