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

In this study, we present a technique to solve LR -type fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints. We define LR -type bipolar fuzzy numbers and their arithmetic operations. We discuss multiplication of LR -type bipolar fuzzy numbers. Furthermore, we develop a method to solve LR -type FBFLPPs with equality constraints involving LR -type bipolar fuzzy numbers as parameters and variables. Moreover, we define ranking for LR -type bipolar fuzzy numbers which transform the LR -type FBFLPP into a crisp linear programming problem. Finally, we consider numerical examples to illustrate the proposed method.

In large and complex systems it is often very difficult to find exact solutions to the Linear Programming problems with the standard techniques, since the necessary data cannot be easily determined precisely and therefore estimates of them are used in practice. Consequently, fuzzy techniques have been developed for achieving better results. In this work we introduce a simple method for solving Fuzzy Linear Programming problems with the help of Triangular Fuzzy Numbers and we apply it for solving business problems.

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

There are several daily life problems where we have to deal with the uncertainties and we are forced to solve the uncertain linear programming models. Certain methods have been presented for dealing with linear programming problems based on fuzzy sets and intuitionistic fuzzy sets which are characterized by membership degree, membership and non-membership degrees, respectively. In this study, we first extend the concept of crisp linear programming problem in Pythagorean fuzzy environment based on triangular Pythagorean fuzzy numbers. The profit/cost coefficients in objective function, input/output coefficients and right-hand side coefficients and decision variables of a linear programming problem are considered as triangular Pythagorean fuzzy numbers. Further, we present methods for solving fully Pythagorean fuzzy linear programming problems for non-negative and unrestricted triangular Pythagorean fuzzy numbers with equality constraints. We also apply the proposed technique to solve practical models.

The Yin-Yang bipolar fuzzy set is a powerful mathematical tool for depicting fuzziness and vagueness. We first extend the concept of crisp linear programming problem in a bipolar fuzzy environment based on bipolar fuzzy numbers. We first define arithmetic operations of unrestricted bipolar fuzzy numbers and multiplication of an unrestricted trapezoidal bipolar fuzzy number (TrBFN) with non-negative TrBFN. We then propose a method for solving fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints in which the coefficients are unrestricted triangular bipolar fuzzy numbers and decision variables are nonnegative triangular bipolar fuzzy numbers. Furthermore, we present a method for solving FBFLPPs with equality constraints in which the coefficients and decision variables are unrestricted TrBFNs. The FBFLPP is transformed into a crisp linear programming problem, and then, it is solved to achieve the exact bipolar fuzzy optimal solution. We illustrate the proposed methodologies with several numerical examples.

Our purpose in this paper is to present a proper diet model for diabetics in the fuzzy environment. In previous studies many efforts were made regarding preparation of a diet model appropriate for different age groups .but in most studies the daily nutrient intake decisions were made based on crisp data. By prescribing a diet based on crisp data, some of realities are neglected. For the same reason, we dealt with human diet problem through fuzzy approach. Also due to the importance of nutrition for individuals, especially patients with diabetic diseases, we have provided nutrition model as multi-objective fuzzy linear programming problem in which minimization of saturated fats, Sugar, and maximization of carbohydrate and total fiber are considered as our objectives. Results indicated uncertainty about amount of nutrients and their intake affects diet quality making it more realistic.

Dubey et al. [40] have shown that solving an Atanassov's I-fuzzy Linear Programming Problem represented by Atanassov's I-fuzzy sets with linear membership and non-membership functions is equivalent...

Предложены детерминированные эквиваленты задач нечеткого линейного программирования c двухсторонними ограничениями на детерминированные значения переменных, которые могут принимать как положительные, так и отрицательные значения, а коэффициенты функции цели и линейные функции ограничений, а также левые и правые граничные их значения — нечеткие множества, преставленные fuzzy-интервалами с LR-представлением функций принадлежности общего вида. На основе рассмотренных в работе правил ранжирования и оценки доминирования нечетких множеств установлены условия безусловного (строгого) и условного (нестрогого) выполнения ограничений. Различного вида детерминированные эквиваленты задачи, в которых требуется выполнение fuzzy-ограничений с различной степенью строгости, предусматривают решения не одной, а некоторого множества задач линейного программирования. Выбирается наилучшее из допустимых решений.

In this paper, based on existing results, decision making about portfolio investment schemes is discussed, ordering method of fuzzy numbers of interval value is shown, corresponding auxiliary models are established and solutions are provided with theories of fuzzy mathematics, optimization theory and numerical calculation, etc. Then it applies software programming to solve the portfolio investment situation between investors in savings and four securities according to the established models. The result shows that investors can choose the risk coefficient that they can bear to reach the maximum value of expected returns. The greater the risk coefficient, the greater the income, the smaller the risk coefficient and the smaller the income. Investors can determine their own portfolio strategy according to their own conditions in order to meet their own interests.

As a result of uncertainty and complexity for environments of decision-making, it is more suitable for decision makers to use hesitant fuzzy linguistic information. In this paper, a novel group decision making (GDM) model based on fuzzy linear programming is proposed for incomplete comparative expressions with hesitant fuzzy linguistic term set (HFLTSs). We establish an equivalence theorem of additive consistency between 2-tuple fuzzy linguistic preference relation (FLPR) and corresponding fuzzy preference relation. Based on this framework, a fuzzy linear programming is established to address incomplete comparative expressions with HFLTSs. It is more important that the proposed fuzzy linear programming has a double action, finding the highest consistent incomplete 2-tuple FLPR and increasing inconsistent 2-tuple FLPR to the additive consistent 2-tuple FLPR based on given incomplete comparative expressions with HFLTSs. By this means, a novel GDM model is constructed based on importance induced ordered weighted averaging operator. Finally, an investment decision-making in real-world is solved by the proposed model, which shows the result of GDM is effectiveness.

This note tries to answer issues raised in Bhardwaj and Kumar (J Optim Theory Appl 163(2): 685---696, 2014). The research summarizes that the results obtained in Khan et al. (J Optim Theory Appl 159: 536---546, 2013) are sound and correct and it fulfills all the necessary requirements of its scope and objectives.

This paper studies two person zero sum matrix games with Atanassov’s I-fuzzy goals. The indeterminacy factor in the goals of each player is resolved using Yager’s indeterminacy resolution principle [38] and Hurwicz optimism-pessimism criterion [20]. This game is shown to be equivalent to two fuzzy linear programming problems involving piecewise linear membership functions. The crisp equivalent of theses problems are formulated using Yang et al. [39] and Inuiguchi et al. [21] approaches.

In recent years, the decision-making models with hesitant fuzzy preference relations (HFPRs) have received a lot of attention by some researchers. Meanwhile, the previous studies normally adopt normalization technical means to ensure the same number for all elements, which biases original information of decision-makers. In order to overcome this problem, in this paper, the multiplicative consistency of HFPRs is defined and the highest consistent reduced HFPRs are obtained by means of fuzzy linear programming method from given HFPRs. The proposed regression method eliminates the unreasonable information and retains the reasonable information from a given HFPR. In addition, the proposed method overcomes drawbacks of Zhu and Xu’s regression method and is more simple and effective. On account of the obtained reduced HFPRs by the proposed regression method, a GDM model is established. Finally, a supplier selection problem was researched to present the effectiveness and pragmatism of the proposed approach, which proved that the method could offer beneficial insights into the GDM procedure.

A study was undertaken in Bhagwanpur distributary of Vaishali Branch Canal in Gandak Canal Command Area, Bihar to optimally allocate land area under different crops (rice and maize in kharif, wheat, lentil, potato in rabi and green gram in summer) in such a manner that maximizes net return, maximizes crop production and minimizes labour requirement employing simplex linear programming method and Multi-Objective Fuzzy Linear Programming (MOFLP) method. Maximum net return, maximum agricultural production, and minimum labour required under defined constraints (including 10% affinity level of farmers to rice and wheat crops) as obtained employing Simplex method were ` 3.7 × 108, 5.06 × 107 Kg and 66,092 man-days, respectively, whereas Multi-Objective Fuzzy Linear Programming (MOFLP) method yielded compromised solution with net return, crop production and labour required as ` 2.4 × 108, 3.3 × 107Kg and 1,79,313 man-days, respectively. As the affinity level of farmers to rice and wheat crops increased from 10% to 40%, maximum net return and maximum production as obtained from simplex linear programming method and MOFLP followed a decreasing trend and minimum labour required followed an increasing trend. MOFLP may be considered as one of the best capable ways of providing a compromised solution, which can fulfill all the objectives at a time.

Our purpose in this paper is to present a proper diet model for cardiovascular patients in the fuzzy environment. In previous studies many efforts were made regarding preparation of a diet model appropriate for different age groups. But in most studies the daily nutrient intake decisions were made based on crisp data. By prescribing a diet based on crisp data, some of realities are neglected. For the same reason, we dealt with human diet problem through fuzzy approach. Also due to the importance of nutrition for individuals, especially patients with cardiovascular diseases, we have provided nutrition model as multi-objective fuzzy linear programming problem in which minimization of saturated fats, salt, monosaccharide and amount of received calery are considered as our objectives. First we review literature of fuzzy mathematics and then we will discuss nutrition model of cardiovascular patients in the fuzzy environment. Results indicated uncertainty about amount of nutrients and their intake affects diet quality making it more realistic.

The paper presents a new method for solving fully fuzzy linear programming problems with inequality constraints and parameterized fuzzy numbers, by means of solving multiobjective linear programming problems. The equivalence is proven between the set of nondominated solutions of the fully fuzzy linear programming problem and the set of weakly efficient solutions of the considered and related multiobjective linear problem. The whole set of nondominated solutions for a fully fuzzy linear programming problem is explicitly obtained by means of a finite generator set., The first author was partially supported by the research project MTM2017-89577-P (MINECO, Spain), and the second author was partially supported by Spanish Ministry of Economy and Competitiveness through grants AYA2016-75931-C2-1-P, AYA2015-68012-C2-1, AYA2014-57490-P, AYA2013-40611-P, and from the Consejería de Educación y Ciencia (Junta de Andalucía) through TIC-101, TIC-4075 and TIC-114.

According to several researches, it must be used a healthy diet included of calories, carbohydrates, low fat and enough protein for an athlete. This study mainly focused on the function of diet-model in quantitative pattern by using the optimized linear programming. Though, it is the fact that the quantity of diet nutrition is not exactly determinable for reaching to mere level. Such uncertainty in medical references causes relative diet level which contradicted decisions upon exact data. So, supplying the optimized diet for athletes in fuzzy set is the final goal of this article. to maximize the use of calorie and minimize the fat are the general goals in order to reach to linear programming by considering the least and utmost allowed level of nutrient as the limitations of the model to find solution for this problem which results in the reaching of more reasonable and real-like levels as documentary references.

Classical linear regression has been used to measure the relationship between rainfall data and altitude in different meteorological stations, in order to evaluate a linear relation. The values of rainfall are supposed as dependent variables and the values of elevation of each station as independent variables. It has long been known that a classical statistical relationship exists between annual rainfall and the station elevation which in many cases is linear as the one examined in this article. However classical linear regression makes rigid assumptions about the statistical properties of the model, accepting the error terms as random variables, and the violation of this assumption could affect the validity of the classical linear regression. Fuzzy regression assumes ambiguous and imprecise parameters and data. For this reason it may be more effective than classical regression. In this paper we evaluate the relationship between annual rainfall data and the elevation of each station in Thessaly’s meteorological stations, using fuzzy linear regression with trapezoidal membership functions. In this possibilistic model the dependent measured elevations are crisp, and the independent observed rainfall values as well as the parameters of the model are fuzzy.

So far, many methods have been proposed to solve interval type-2 fuzzy linear programming problems with uncertainty of kind Vagueness. However, there are no models of interval type-2 fuzzy linear programming problems with uncertainty of kind Ambiguity in which all or part of the parameters are ambiguous. In this paper, while expressing an interval type-2 fuzzy linear programming problem with uncertainty of kind Ambiguity, we will introduce a method for solving it based on the nearest interval approximation.

In this paper, we define the additive consistency of hesitant fuzzy preference relation (HFPR). The highest consistent FPR is obtained from a given HFPR using fuzzy linear programming method. More specifically, the proposed method is more simple and reasonable than previous method.

In view of the traditional evaluation index of sintering basic characteristics ignoring process information, several high temperature characteristic numbers, which contain process information, were used to evaluate the sintering basic characteristics more comprehensively in this paper. The evaluation model of iron ore properties was established based on the high temperature characteristic numbers with employing fuzzy mathematics. Then, with the constraint of sinter properties and the raw materials, the ore blending model was built, whose target was to obtain the lowest cost and the best sinter properties. A multi-objective fuzzy linear programming was used to solve the model in order to get the best ore matching schemes. The ore blending model was then applied to the practical production, which was proved to be dependable.

In this paper we present a Fuzzy linear Mathematical programming approach for optimal allocation of land under cultivation. Fuzzy Mathematical programming approach is more realistic and flexible optimal solution for the agricultural land cultivation problem. In this study we have discussed how to deal with decision making problems that are described by Fuzzy linear programming (Flp) models and formulated with the elements of uncertainty. This form of approximation can be convenient and sufficient for making good decisions. BIBECHANA 13 (2016) 72-76

This paper is to deal with a kind of fuzzy multi-objective linear programming (FMOLP) problem with symmetric trapezoidal fuzzy numbers. Several methods has been proposed in order to obtain fuzzy efficient solution to FMOLP problem. In this paper, we change FMOLP problem into the complete stratified fuzzy linear programming problem, then use the stratified simplex method to obtain the fuzzy optimal solution directly without converting them to crisp linear programming problem. We then prove that this fuzzy optimal solution is the fuzzy efficient solution of the original FMOLP problem. A numerical example is given to illustrate the proposed method.

Angelov’s approach is the most common approach to solve Atanassov’s I-fuzzy linear programming problems. Here certain other variants of this basic model are studied. These variants are motivated by a recent study (Chandra and Aggarwal 2014) on fuzzy linear programming problems and follow very naturally from an associated bi-objective programming problem. This bi-objective programming problem aims to achieve a trade-off between the twin objectives of ‘meeting the aspiration level’ and ‘satisfaction of the given constraints’ which are defined in terms of their respective score functions. As an outcome of this study a new two phase approach to solve Atanassov’s I-fuzzy linear programming problems is obtained. Further, two additional formulations are also proposed on similar lines which also provide an efficient solution of the associated bi-objective optimization problem. These efficient solutions have a natural interpretation for the given I-fuzzy linear programming problem. A small numerical example is included as an illustration.

This note provides a counterexample to illustrate the incorrectness of the proof of Proposition 3.3 that was presented by Wu (Fuzzy Optim Decis Mak 2:61---73, 2003). The original proof of Proposition 3.3 by Wu can only be correct when the extra assumption $$\mu _{\widetilde{y}_i}(0)= 1$$μy~i(0)=1 is added. The correct proof of Proposition 3.3 is also presented in this note.

Ezzati et al. (2013) proposed a method for comparing triangular fuzzy numbers and using it, propose a new algorithm to find the fuzzy optimal solution of fully fuzzy linear programming problems with equality constraints. Ezzati et al. also claimed that the fuzzy optimal solution of fully fuzzy linear programming problems with inequality constraints can also be obtained by the same algorithm by transforming it into fully fuzzy linear programming problems with equality constraints. In this note, it is proved that the fully fuzzy linear programming problems with inequality constraints cannot be transformed into fully fuzzy linear programming problems with equality constraints and hence, the algorithm, proposed by Ezzati et al. to find the fuzzy optimal solution of fully fuzzy linear programming problems with equality constraints, cannot be used for finding the fuzzy optimal solution of fully fuzzy linear programming problems with inequality constraints.

Recently, two new algorithms have been proposed to solve a fully fuzzy linear programming (FFLP) problem by Lotfi et al. [F.H. Lotfi, T. Allahviranloo, M.A. Jondabeha, L. Alizadeh, Solving a fully fuzzy linear programming using lexicography method and fuzzy approximate solution, Appl. Math. Model. 33 (2009) 3151–3156] and Kumar et al. [A. Kumar, J. Kaur, P. Singh, A new method for solving fully fuzzy linear programming problems, Appl. Math. Model. 33 (2011) 817–823]. In this paper, based on a new lexicographic ordering on triangular fuzzy numbers, a novel algorithm is proposed to solve the FFLP problem by converting it to its equivalent a multi-objective linear programming (MOLP) problem and then it is solved by the lexicographic method. By a theorem, it is shown that the lexicographic optimal solution of MOLP problem can be considered as an optimal solution of the FFLP problem. Then, a simple example and two real problems, as two case studies, will be used to illustrate our algorithm and compare it with the existing methods.