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The time–cost trade-off has been recognized as a very significant aspect of construction management. Generally, time–cost trade-off can be modeled as a fuzzy linear programming problem with symmetric or non-symmetric fuzzy numbers. However, it was successfully solved when the fuzzy membership functions are only symmetric. In the present work, a novel approach is introduced to solve fuzzy linear programming problem with non-symmetric fuzzy membership functions by transforming it to its corresponding nearest symmetric one. The transformed problem is then converted to its crisp linear programming problem and then solved by the standard primal simplex method. Two examples are presented to show the effectiveness of the proposed approach, and the results are discussed.

Minimum-cost capacitated fuzzy network is formulated as a fuzzy linear programming problem. A novel fuzzy linear programming formulation for minimum-cost capacitated fuzzy network where the total resource constraints are fuzzy is proposed. The proposed model is then implemented to minimize the operations cost of American Airlines. The research is helpful to identify the most profitable destinations for the American Airlines. Twelve origin/destination pairs are taken into considerations namely Atlantic (A), Latin American (L), Pacific (P) and Domestic (D). Flight operations capacity, Available Seat Miles ASM, is taken as a measure of capacity. The goal is to minimize the flight operations cost while ensuring maximum flight operations capacities to all destinations. This is followed by perspective data analytics for “What American Airline should do to be more profitable?” Perspective analytics suggest the airline to extend flight operations capacity in certain origin/destination pairs, whereas to maintain the previous approximate average capacity for those having high operations costs. The solution of the proposed fuzzy model suggests that flight operations capacity ASM can be significantly increased by 22345148 (000) with relatively small increase 1539356 (000) USD in operations cost. The fuzzy model is superior for it emphasizes to increase flight operations capacity ASM for the origin/destination pairs with minimum flight operations costs.

In this study, we first show that the existing arithmetic operations of trapezoidal fuzzy numbers do not satisfy the basic properties. Then, for trapezoidal fuzzy numbers, we define new arithmetic operations. Furthermore, we demonstrate that the existing Simplex method for addressing fully fuzzy linear programming problems has some drawbacks. Finally, we provide a new strategy for solving fully fuzzy linear programming problems and compare our results with the existing methods. © 2022, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional. 2-s2.0-85123434717

We investigate various types of fuzzy linear programming problems based on models and solution methods. First, we review fuzzy linear programming problems with fuzzy decision variables and fuzzy linear programming problems with fuzzy parameters (fuzzy numbers in the definition of the objective function or constraints) along with the associated duality results. Then, we review the fully fuzzy linear programming problems with all variables and parameters being allowed to be fuzzy. Most methods used for solving such problems are based on ranking functions, $$\alpha $$-cuts, using duality results or penalty functions. In these methods, authors deal with crisp formulations of the fuzzy problems. Recently, some heuristic algorithms have also been proposed. In these methods, some authors solve the fuzzy problem directly, while others solve the crisp problems approximately.

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.

In this article, we consider some well-known approaches for solving fuzzy linear programming (FLP) problems. We present some of the difficulties of these approaches. Then, crisp linear programming problems are suggested for solving FLP problems. A new algorithm is also given. The proposed approach has advantages over the other methods. For example, we can achieve higher membership degrees for objective function and constraints. Moreover, we show that the fuzzy optimal solutions obtained by the proposed approach are efficient enough. Also, we see that unlike the previous methods, our method finds efficient solutions by solving only one crisp linear problem instead of solving two or three crisp problems. Finally some numerical examples are presented to show the efficiency of the given approach over the other approaches.

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.

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.

Several authors have proposed different methods for solving fuzzy minimum cost flow (MCF) problems. In this paper, some single and multi-objective fuzzy MCF problems are chosen which cannot be solved by using any of the existing methods and a new method is proposed for solving such type of problems. The main advantage of the proposed method over existing methods is that the fuzzy MCF problems which can be solved by using the existing methods can also be solved by the proposed method. But, there exist several fuzzy MCF problems which can be solved only by using the proposed method i.e., it is not possible to solve these problems by using the existing methods. To illustrate the proposed method and also to show the advantages of the proposed method over existing methods some single and multi-objective fuzzy MCF problems which cannot be solved by using the existing methods are solved by using the proposed method and the obtained results are discussed.

In this note, it is shown that there is error in the existing technique (Khan, I.U., Ahmad, T., Maan, N: A simplified novel technique for solving fully fuzzy linear programming problems. J. Optim. Theory Appl. 159:536---546, 2013).

In this paper, shortcomings and limitations of the existing methods for solving fuzzy linear programming (FLP) problems are pointed out. To overcome the limitations as well as to resolve the shortcomings, a new method is proposed for solving FLP problems. To show the advantage of the proposed method over existing methods, a FLP problem is solved by the existing methods as well as the proposed method, and the obtained results are compared.

This paper presents the Multi Objective Fuzzy Linear Programming (MOFLP) model, which deals with the fuzziness in resources and decision variables, closer to the real world situation. The MOFLP model is developed and applied to the Jayakwadi Project Stage-I built across the river Godavari in the State of Maharashtra, India. It focuses on the four objectives namely maximization of Net Benefits, maximization of Crop Production, maximization of Employment Generation and maximization of Manure Utilization. The level of satisfaction λ = 0.625 is worked out for compromised solution for four conflicting objectives under fuzzy environment. The MOFLP model compromised solution for irrigation planning provides Net Benefits 1522.75 (Million Rupees), Crop Production 322504.40 (Tons), Employment Generation 29.27 (Million Man Days) and Manure Utilization 147229.40 (Tons) respectively and irrigation intensity 68.50 %. The results obtained are promising for sustainable development in irrigation sector and closer to the true picture of the real world problem as it incorporates the fuzziness in both resources and decision variables simultaneously.

In traditional approach, failure mode and effects analysis determines the risk priories of failure modes through the risk priority number which is determined by multiplication of three risk factors namely, failure occurrence (O), failure severity (S) and failure detection ability (D). In this approach, different weights of risk factors were not taken into consideration so that the three risk factors were assumed to have the same weight. This may not be realistic in real applications. In this paper we treat the risk factors as fuzzy variables and evaluate them using fuzzy linguistic terms and fuzzy ratings. As a result, fuzzy risk priority numbers (FRPNs) are proposed for prioritization of failure modes. The FRPNs are defined as fuzzy geometric means of the fuzzy ratings for O, S and D and can be computed using alpha-level sets and linear programming models. A numerical example is provided to examine the results of this model.

We propose a bi-objective cell formation problem with demand of products expressed in a number of probabilistic scenarios. To deal with the uncertain demand of products, a framework of two-stage stochastic programming model is presented. The proposed model considers minimizing the sum of the miscellaneous costs (machine constant cost, expected machine variable cost, cell fixed-charge cost, and expected intercell movement cost) and expected total cell loading variation. Because of conflicting objectives, we develop a two-phase fuzzy linear programming approach for solving bi-objective cell formation problem. To show the effectiveness of the proposed approach, numerical examples are solved and the results are compared with the two existing approaches in the literature. The computational results show that the proposed fuzzy method achieves lower objective functions as well as higher satisfaction degrees.

The problem of irrigation planning becomes more complex by considering an uncertainty. The uncertainties can be tackled by formulating the problem of irrigation planning as Fuzzy Linear Programming (FLP). FLP models can incorporate the scenario of real world problem. In the present study, Multi Objective Fuzzy Linear Programming (MOFLP) irrigation planning model is formulated for deriving the optimal cropping pattern plan for the case study of Jayakwadi project in the Godavari river sub basin in Maharashtra State, India. Four conflicting objectives are considered such as Net Benefits (NB), Crop/Yield Production (CP), Employment Generation/Labour Requirement (EG) and Manure Utilization (MU). Four different cases are considered to incorporate the uncertainty in MOFLP model. To include the uncertainty in irrigation planning problem only objectives are taken as fuzzy and constraints are crisp in nature in Case-I. To consider the uncertainty involved in availability of resources, in Case-II the stipulations are fuzzy. The technological coefficients are fuzzy in Case-III. The Case-IV includes both technological coefficients and stipulations fuzzy. The level of satisfaction (λ) works out to be 0.58, 0.50, 0.50 and 0.28 respectively for Case-I to IV. The results obtained in Case-IV are more realistic and promising as it involves the uncertainty in technological coefficients and stipulations simultaneously.

As a consequence of more extensive collaboration between countries, the need for better humanitarian relief assistance has become a significant challenge to the international community. In case of a disaster exceeding the national response capacity of the affected country, donor countries provide the relief items, which are then consolidated at collection points to be shipped to points of delivery in the stricken country. After the items are transported to the point of delivery, the responsible authority in distributing aids to vulnerable populations then becomes the national relief agencies. In this context, we propose an international relief planning model that can handle the uncertain information while maximizing the credibility of the international agencies in the most cost efficient way.

A two-step infinite α-cuts fuzzy linear programming (TSIFP) method is developed in this study. The introduction of infinite α-cuts to conventional fuzzy linear programming frameworks makes it possible to generate more reliable optimal results than conventional fuzzy linear programming, where finite α-cuts were assumed to be sufficient in representing all fuzzy information of the membership functions. In contrast to the previous studies, the proposed TSIFP can be noted as the first attempt in solving FLP without any unreasonable simplification and assumption. An agricultural irrigation system is then provided for demonstrating its applicability. The results show that reasonable solutions and allocation strategies are obtained. As a typical finite α-cuts fuzzy linear programming method, fuzzy robust linear programming (FRLP) is further considered to solve the same problem; results from this method are then compared with those from TSIFP. It is indicated that, due to the constraints being relaxed in FRLP, more water beyond the system’s capacity would be over-allocated for pursuing higher system benefits, implying the unreliability of FRLP in being extended to real-world practices. Two scenario analyses under different α-level cutting means for FRLP are also investigated.

This paper deals with a methodology for solving a Chance Constrained Fuzzy linear programming problem. The methodology is applied to the Chance Constrained model where the constraints have two different types of fuzzy inequalities and the method is justified through numerical examples.

A class of fuzzy linear programming (FLP) problems based on fuzzy relations is introduced, the concepts of feasible and α -efficient solutions are defined. The class of crisp (classical) LP problems and interval LP problems can be embedded into the class of FLP ones. Moreover, for FLP problems a new concept of duality is introduced and the weak and strong duality theorems are derived. The previous results are applied to the special case of interval LP and compared to the existing literature.

Multiobjective fuzzy linear programming (MOFLP) irrigation planning model is formulated for the evaluation of management strategy for the case study of Jayakwadi irrigation project, Maharashtra, India. Three conflicting objectives net benefits, agricultural production and labour employment are considered in the irrigation planning scenario. All three criteria are to be maximised and the last two are sustainability related. All three objective functions are quantified by linear membership functions in a fuzzy multi objective framework. It is observed from MOFLP solution that net benefits, agricultural production and labour employment are 2.031×109 Rupees, 2.1186×106 tons, 3.5858×107 man-days respectively with degree of truth (λ) 0.5715. Analysis of results indicated that net benefits, agricultural production, labour employment have decreased by 4.13, 5.39 and 3.4% as compared to ideal values in the crisp linear programming (LP) model.

A robust game theoretic approach for constructing effective international concords for conflict solving is discussed. Ann-person cooperative game in characteristic function form is used for international conflict solving via formation of coalitions. The nucleolus and the augmented nucleolus as solution concepts of the game are derived on its alternative forms, and robustness of the solutions when the evaluation of the coalition values is varied is examined. For solving this problem, parametric linear programming is used. This approach provides in a resemblant form an-alternative device to solving the fuzzy linear programming based on interval analysis.

Because of its complexity from both a legal and economic standpoint, the problem of optimal waste load allocation is multiobjective by nature and should be treated accordingly. To perform this task, an optimization technique known as fuzzy linear programming is utilized in solving a multiple-discharge, two-objective waste load allocation problem. The two objectives considered are: (1) the maximization of waste discharge and (2) the minimization of the largest difference in equity measure between the various dischargers. Results from this study reveal that fuzzy linear programming is a valuable tool for solving the multiple-objective water quality management problems. Moreover, it is shown that the selection of a linear or logistic membership function in providing preference criteria between the two objects, has no effect on the ‘best compromising solution’.