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Considering the double uncertainty caused by the ambiguity of statistical data and the ambiguity produced by the subjective assessment of decision makers, the crisp values of criteria may be insufficient to model the multi-criteria decision-making (MCDM) problem in the real world. This paper proposes a fuzzy best-worst method (FBWM) based on a dual-interval solution algorithm to extend the best-worst method (the most recent MCDM method) to fuzzy environments. The reference comparisons for the best criteria and for the worst criteria are represented by fuzzy numbers. Then, according to the BWM method, a nonlinear constrained optimization problem with fuzzy parameters is formulated. We decompose the membership function in fuzzy numbers into several interval numbers of special form and solve the aforementioned fuzzy BWM problem by compound interval algorithm to obtain fuzzy weights of different criteria. Meanwhile, an integral type-reduced method is proposed for determining the fuzzy consistency ratio in order to assess the reliability of the FBWM results. The viability of the new algorithm to expand the BWM method into fuzzy environments has been validated through three numerical examples. In contrast to the existing FBWM method, the proposed method avoids the arithmetic operation between fuzzy numbers during the solution process, directly transfers the uncertain information in the membership function corresponding to the fuzzy comparison vector to the result, and generates fuzzy weight value, which indicates that the proposed algorithm is able to obtain accurate BWM results in fuzzy environments. The results of the study provide new solution ideas for multi-criteria optimization problems under uncertainty. [ABSTRACT FROM AUTHOR]

One of the special cases of type-2 fuzzy sets are the interval type-2 fuzzy sets, which are less complicated and easier to understand than T2FSs. In this study, we explore the interval type-2 fuzzy linear programming problem with the resources vector that have imprecision of the vagueness type. These types of vagueness are expressed via membership functions. First, we review the three available methods, including the Figueroa and Sarani methods. Then, using the three ideas of Verdegay, Werners, and Guu and Wu for solving fuzzy linear programming problems with vagueness in the resources vector, we propose three new methods for solving interval type-2 fuzzy linear programming problems with vagueness in the resources vector. Finally, we demonstrate the effectiveness of our proposed methods by solving an example and comparing the results obtained with each other and with those of previous methods.

In this study, Interval number and Fuzzy number Linear Programming problems are addressed. We deal with some convenient methods to solve the addressed problems. We focus on preparing an improving approach based on Tang Shaocheng method to solve linear programming problems with interval or fuzzy numbers. Also, some applications of our proposed approach in fractional programming are presented. Particularly, we propose an algorithm using our improved approach to solve a fuzzy multi-objective linear fractional programming problem. Finally, we illustrate the proposed approach using numerical and some real life examples. [ABSTRACT FROM AUTHOR]

This research addresses the Inventory-Routing Problem (IRP) with priority customers under conditions of uncertainty. The problem involves devising a plan to supply a set of customers from a facility, establishing precedence rules, and making transportation and inventory management decisions while considering uncertainty in customer demand. Priority in deliveries is assigned to customers in the health sector in accordance with government regulations. Companies must uphold product quality for health sector customers while simultaneously aiming to minimize total costs. We propose a fuzzy mixed-integer linear programming model to solve the IRP with priority customers and uncertainty in demand rates, modeled using triangular fuzzy numbers. To convert the fuzzy linear programming model into a deterministic one, an ordering method for fuzzy numbers is applied. The model is implemented using CPLEX, and a set of benchmark instances is solved. The primary results indicate that prioritizing customers leads to an increase in total costs due to visiting these customers first on each route created. Additionally, routes without priority customers exhibit higher average vehicle capacity utilization. In most cases, the solutions obtained maintain quality within the specified time limits. [ABSTRACT FROM AUTHOR]

One of the special cases of type-2 fuzzy sets is the interval type-2 fuzzy sets, which are less complicated and easier to understand than T2FSs. In this study, we explore the interval type-2 fuzzy linear programming problem with the resources vector that has imprecision of the vagueness type. These types of vagueness are expressed via membership functions. First, we review the three available methods, including the Figueroa and Sarani methods. Then, using the three ideas of Verdegay, Werners, and Guu and Wu for solving fuzzy linear programming problems with vagueness in the resources vector, we propose three new methods for solving interval type-2 fuzzy linear programming problems with vagueness in the resources vector. Finally, we demonstrate the effectiveness of our proposed methods by solving an example and comparing the results obtained with each other and with those of previous methods. [ABSTRACT FROM AUTHOR]

In contemporary decision-making, reliance on information is paramount, yet much of it is fraught with uncertainty, making logical decision-making challenging. To tackle this uncertainty, various methods, including the use of fuzzy numbers, have been employed. This paper specifically delves into addressing linear programming (LP) problems characterized by fuzzy coefficients in the objective function, fuzzy values in the right-hand side, and fuzzy coefficients of constraints. The proposed approach involves employing linear ranking functions such as Maleki, Campos, Yager’s F1 and Yager linear ranking functions to solve these fuzzy linear programming (FLP) problems and attain optimal solutions. Furthermore, the paper elucidates the resolution steps through the presentation of numerical examples, in this study, a comprehensive methodology is presented for effectively addressing a wide range of linear programming problems.

International law is mainly the law between countries, which is the code of conduct for countries to conduct international interactions, and this paper researches the role of international law in international relations interactions. First of all, this paper combines the fuzzy set on the basis of the classical linear regression model algorithm, establishes the fuzzy linear programming model, and explores the fuzzy correlation analysis. Then, taking EU law as an example, the role of EU law in EU relations is analyzed through the theory of EU law, EU and the role of EU law in the EU embodied. Finally, based on the role of EU law, the importance of international law in international relations is studied through a fuzzy linear programming model, and the role of strengthening international law is considered. The results show that the countries consider the legitimacy of EU law to be above 0.9, and the countries’ participation in EU law also ranges from 0.9 to 1.0, and the countries’ self-identification increases by about 0.1 to 0.2 after the enactment of EU law. The correlation coefficients of international law on the construction of the identities of the 10 countries are all greater than 0.75, and the interests of the countries have increased by about 2.3, and international law promotes the order of international relations.

There are several approaches to address fuzzy linear programming problems (FLPP). However, due to using standard interval arithmetic (SIA), these methods have some limitations and are not complete solutions. This article establishes a new approach to fuzzy linear programming via the theory of horizontal membership functions and the multidimensional relative-distance-measure fuzzy interval arithmetic. Furthermore, we propose a multidimensional solution based on the primal Simplex approach that satisfies any equivalent form of FLPP. The new solutions of FLPP are also compared with the results of existing methods. Some numerical examples have been illustrated to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

An alternative new method for solving an epistemic uncertainty such as fuzzy-based linear programming problem with equality constraints has been proposed here. In this approach, the optimal solution of the considered fuzzy problem has been achieved by solving a related crisp problem. Moreover, the objective function of the crisp problem has been obtained by taking the linear combination of the fuzzified version of the original uncertain objective function. The coefficients of the linear combination are considered as strictly positive real numbers. And the constraints of the crisp problem are obtained equivalently from the equality constraints of the fuzzy problem. Triangular fuzzy numbers are considered for this analysis. Comparisons have been made with the results of Kumar et al. (Appl Math Model 35:817–823, 2011), Najafi and Edalatpanah (Appl Math Model 37:7865–7867, 2013), Ezzati et al. (Appl Math Model 39:3183-3193, 2015) and Ghoushchi et al. (Mathematics 9:2937, 2021. https://doi.org/10.3390/math9222937). [ABSTRACT FROM AUTHOR]

There are several approaches to address fuzzy linear programming problems (FLPP). However, due to using standard interval arithmetic (SIA), these methods have some limitations and are not complete solutions. This article establishes a new approach to fuzzy linear programming via the theory of horizontal membership functions and the multidimensional relative-distance-measure fuzzy interval arithmetic. Furthermore, we propose a multidimensional solution based on the primal Simplex approach that satisfies any equivalent form of FLPP. The new solutions of FLPP are also compared with the results of existing methods. Some numerical examples have been illustrated to show the efficiency of the proposed method.

In this work, we propose a 3D dynamic optimization model that enables the design of an underground mine ore pass system with uncertainties. Ore transportation costs and ore pass development costs are quantified by triangular fuzzy numbers. Transportation costs are treated as production costs, and they vary over the duration of mining operation, while development costs of ore passes are treated as an investment, and they are treated as constant. The developed model belongs to the class of fuzzy 0–1 linear programming models, where the fuzzy objective cost function achieves a minimum value, with respect to given set of techno-dynamic constraints. Searching for optimal value in the fuzzy environment is a hard task, and because of that, we developed a new ranking function which transforms the fuzzy optimization model into a crisp one. A triangular fuzzy number can be presented as a triangular graph G(V,E) composed of vertices and edges. The x-coordinate of the Torricelli point of a triangular graph presents the crisp value of a triangular fuzzy number. The use of this model lets us know the optimal number of ore passes, optimal location of ore passes, and optimal dynamic ore transportation plan. [ABSTRACT FROM AUTHOR]

Electrical impedance tomography (EIT) is an imaging technique that realizes the image reconstruction of conductivity distribution in the field. The existing EIT algorithms ignore the hidden fuzzy features during imaging, making the EIT technique exhibit a high degree of uncertainty, imprecision, incompleteness, and inconsistency in the actual use process, resulting in a low spatial resolution of the reconstructed images. In order to solve this problem, we introduce fuzzy linear programming into EIT imaging. On the basis of analyzing the fuzzy features of EIT in detail, a new model of fuzzy optimization is built up, whose optimal solution is obtained by using constrained coefficient fuzzy linear programming. To this end, we devise two types of simulation experiments to verify the performance of the optimization algorithm. Experimental results prove that compared with the traditional Tikhonov regularization algorithm, the correlation coefficient of the reconstructed image of the proposed algorithm is higher, the relative error value is smaller.

The necessarily optimal solution is known as the most reasonable solution to linear programming problems with interval/fuzzy objective function coefficients. As it remains optimal against the certain fluctuations of objective function coefficients, the necessarily optimal solution can be seen as a robust optimal solution. In this paper, we demonstrate that the necessary optimality degree of a non-degenerate basic feasible solution can be obtained easily by utilizing the tolerance approach. The necessary optimality degree evaluates to what extent the solution remains optimal against the fluctuations of objective function coefficients. Several types of fuzzy subsets showing the possible range of the objective function coefficient vector are considered. For each type of fuzzy subset, an efficient calculation method of necessary optimality degree is proposed. Numerical examples are given to illustrate the proposed methods. [ABSTRACT FROM AUTHOR]

The best allocation of limited resources to activities with the aim of accomplishing the desired goal, such as maximization of profit or minimization of cost, is the focus of linear programming. The relationships between activities in linear programming models satisfy the proportionality and additivity requirements since they are linear interactions. This feature of linear programming is extended to the tourism industry which is one of the major industries in the global economy with respect to invested capital and earnings of foreign currencies. In todays era of design thinking, automation, and met averse, there is a very close margin between solutions to similar real-world problems. This is where the need and demand for fuzzy and imprecise linear programming arises. The membership functions provide the model developer the freedom to grade the imprecision as per his/her preference, thereby enabling a unique solution for each problem. While considering the tourism problem a number of factors like natural resources, people, history, culture, security, accommodation, entertainment, political stability, cost of services, tour operator, tour information, and advertisement play an important role in enhancing the sector to a large extent. Combined and classified they come under the following categories like leisure tourism, therapeutic/spa tourism, conference tourism, political tourism, sports/recreational tourism, cultural tourism, social tourism, conference tourism, recreational tourism, sports tourism, religious tourism, health tourism, etc. Each of these sectors requires careful investment on the part of the Government and other stake holders for development purpose. i.e. only a proper marketing mix will ensure a better return for the state. Taking this aspect into consideration the authors decided to introduce a hypothetical LP maximization model for the tourism industry in North East. Cost and space allocation are the constraints in the model. Among the various categories, three tourism forms (heritage, eco-tourism, and pilgrimage tourism) are considered which are used to optimize the allocation of the States marketing budget in tourism in such a way that the appropriate sector provides the greatest likelihood of producing the strongest return on investment. The maximization of profit has been done using various methods, like Werner's, Verdegay's, and Zimmermans. The authors have concluded the results based on the sensitivity analysis that has been done in the process of maximizing profit along with a maximal satisfaction level. [ABSTRACT FROM AUTHOR]

In this paper, new concepts of (ˆ-feasibility and ã-efficiency of solutions for fuzzy mathematical programming problems are used, where ã is a vector of distinct satisfaction degrees. Recently, a special kind of fuzzy mathematical programming entitled Fuzzy Flexible Linear programming (FFLP) is attracted much interest. Using the mentioned concepts, we propose a two-phase approach to solve FFLP. In the first phase, the original FFLP problem converts to a Multi-Objective Linear Programming (MOLP) problem, and then in phase II a weighting technique for the reduced program is introduced. We saw that it was observed that using this concept as a generalization of the parametric approach in linear programming provides a more appropriate tool for modeling real problems and improving the solving process. Thus, by using this two-phase approach, we achieve better utilization of available resources. Further, the solution resulting from these two approaches is always an ã-efficient solution. Finally, an example in the real world is described to express this approach. [ABSTRACT FROM AUTHOR]

Demand forecasting is of crucial importance for make-to-stock companies because product demand is uncertain and it changes with time. Fuzzy linear programming (FLP) can be an optimum approach for such uncertain situations. In this study, the FLP model was used for more accurate demand forecasting in a make-to-stock company. Demand forecasting study was carried out according to the FLP method and linear programming (LP) method. Solutions of FLP and LP were compared in terms of imputed shortage cost, inventory carrying cost, and net profit. Results show that the applied FLP method is more advantageous than LP as it provides a 67% decrease in costs and a 15% increase in net profit. [ABSTRACT FROM AUTHOR]

One of the various optimization methods that addresses optimization under uncertainty is fuzzy linear programming. This model can be used when there is ambiguity in the situation because it is not precisely specified or when the problem does not require an exact value. With fuzzy linear programming, there is a range of grey between the two extremes as opposed to binary models, where an event may only be either black or white. As a result, it broadens the range of potential applications because most scenarios involve a spectrum of values rather than a bipolar state. In this article, a new FLP-based method is developed using a single MF, called modified logistics MF. The modified MF logistics and its modifications taking into account the characteristics of the parameter are from the analysis process. This MF was tested for useful performance by modeling using FLP. The developed version of FLP provides confidence in the existing IPPP application. This approach to resolving the IPPP can get feedback from the decision maker, the implementer and the analyst. In this case, this process can be called FLP interaction. FS self-assembly for MPS problems can be developed to find satisfactory solutions. The decision maker, researcher and practitioner can apply their knowledge and experience to get the best results. [ABSTRACT FROM AUTHOR]

Electrical impedance tomography (EIT) is an imaging technique that realizes the image reconstruction of conductivity distribution in the field. The existing EIT algorithms ignore the hidden fuzzy features during imaging, making the EIT technique exhibit a high degree of uncertainty, imprecision, incompleteness, and inconsistency in the actual use process, resulting in a low spatial resolution of the reconstructed images. In order to solve this problem, we introduce fuzzy linear programming into EIT imaging. On the basis of analyzing the fuzzy features of EIT in detail, a new model of fuzzy optimization is built up, whose optimal solution is obtained by using constrained coefficient fuzzy linear programming. To this end, we devise two types of simulation experiments to verify the performance of the optimization algorithm. Experimental results prove that compared with the traditional Tikhonov regularization algorithm, the correlation coefficient of the reconstructed image of the proposed algorithm is higher, the relative error value is smaller. [ABSTRACT FROM AUTHOR]

When some coefficients of the constraints are uncertain with only their possible ranges being given, a conventional linear programming (LP) problem can be generalised to the one with set-inclusive constraints. We consider the case where the possible ranges are given by fuzzy sets in this paper. The set-inclusive constraints with fuzzy coefficients have been treated by a necessity measure. However, the usual necessity measure cannot express well the decision-maker's requirement about the trade-off between the robustness level and the satisfaction level of the constraints. We extend the necessity measure to incorporate the trade-off between the robustness level and the satisfaction level of the constraints. We apply the extended necessity measure to LP problems with fuzzy coefficients. After the problem is formulated and reduced to the conventional programming problem, we propose a solution algorithm. Numerical examples are given to illustrate the proposed approach. [ABSTRACT FROM AUTHOR]

In this manuscript, three contributions are proposed. First contribution is proposing a good evaluation between the fuzzy and neutrosophic approaches using a novel fuzzy-neutrosophic transfer. Second contribution is introducing a general framework for solving the neutrosphic linear programming problems using the advantages of the method of Abdel-Basset et al. and the advantages of Singh et al.'s method. Third contribution is proposing a new neutrosophic exterior point simplex algorithm NEPSA and its fuzzy version FEPSA. NEPSA has two paths to get optimal solutions. One path consists of basic not feasible solutions, but the other path is feasible. Finally, the numerical examples and results analysis show that NEPSA more than accurate FEPSA.

In this work, we propose a 3D dynamic optimization model that enables the design of an underground mine ore pass system with uncertainties. Ore transportation costs and ore pass development costs are quantified by triangular fuzzy numbers. Transportation costs are treated as production costs, and they vary over the duration of mining operation, while development costs of ore passes are treated as an investment, and they are treated as constant. The developed model belongs to the class of fuzzy 0–1 linear programming models, where the fuzzy objective cost function achieves a minimum value, with respect to given set of techno-dynamic constraints. Searching for optimal value in the fuzzy environment is a hard task, and because of that, we developed a new ranking function which transforms the fuzzy optimization model into a crisp one. A triangular fuzzy number can be presented as a triangular graph G(V,E) composed of vertices and edges. The x-coordinate of the Torricelli point of a triangular graph presents the crisp value of a triangular fuzzy number. The use of this model lets us know the optimal number of ore passes, optimal location of ore passes, and optimal dynamic ore transportation plan.

A selection of suitable sustainability management systems (SMS) is a major part of supply chain strategies to create a competitive advantage, reduce total costs, and manage long-term sustainability. A framework and method for prioritizing supply chain SMSs are presented in this research. Analytic hierarchy process (AHP) is the most common method for alternative selection in multi-criteria decision-making (MCDM). However, complex information is mixed with ambiguity and uncertainty, which makes decision makers unable to use precise or crisp numbers, so fuzzy numbers are presented to remedy this difficulty. Therefore, this research proposes a fuzzy additive preference programming (FAPP) to select the optimum SMS for a supply chain. FAPP method can produce the unique normalized optimal priority vector of fuzzy pairwise comparison matrices for SMS selection effectively with linear constraints. The additive linear constraints can eliminate the weaknesses of existing methods and equalize the upper and lower triangular fuzzy judgments. In addition, the proposed method can identify abnormal pairs of fuzzy judgments that cause inconsistency. The proposed methodology can prioritize the key criteria which lead to the selection of the most appropriate SMS. An example of SMS selection in a rubber factory demonstrates the feasibility and validity of the proposed method. [ABSTRACT FROM AUTHOR]

Nowadays, the intensification of a competitive environment in markets in conjunction with sustainability issues has forced organizations to concentrate on designing sustainable closed-loop supply chains. In this study, a sustainable closed-loop supply chain network is configured under uncertain conditions based on fuzzy theory. The proposed network is a multi-product multi-period problem which is formulated by a bi-objective mixed-integer linear programming model with fuzzy demand and return rate. The objectives are to maximize the supply chain profit and customer satisfaction at the same time. Moreover, the carbon footprint is included in the first objective function in terms of cost (tax) to affect the total profit and treat the environmental aspect. Fuzzy linear programming and Lp-metric method are then applied to deal with the uncertainty and bi-objectiveness of the model, respectively. In order to validate the methodology, a case study problem in the dairy industry is investigated where the proposed Lp-metric is also compared to goal attainment method. The obtained results demonstrate the superiority of Lp-metric against goal attainment method as well as the applicability and efficiency of the proposed methodology to treat a real case study problem. Furthermore, from the management perspective, outsourcing the production during high-demand periods is highly recommended as an efficient solution. [ABSTRACT FROM AUTHOR]

Resource-Constrained Multi-Project Scheduling Problem (RCMPSP) is considered a significant topic in project management studies and many types of research have been carried out in this field in which numerous approaches have been proposed so far. However, in most of these approaches, the viewpoints of clients and contractors, i.e., two important stakeholders of the project, are not directly considered. The current research attempts to introduce a new approach, namely RCMPSP, to schedule the project portfolio and allocate the budget as a limited resource simultaneously. In doing so, first, clients and contractors' budget satisfaction is defined. Then, a number of budget allocation models have been proposed to maximize the satisfaction of clients and contractors. These models consider such constraints as minimum cost required for each project, maximum budget for each period, and exibility of the start date of each project. To illustrate the viability of the proposed models, a real case of the project portfolio is considered. [ABSTRACT FROM AUTHOR]

The aim of this research is focused on the problem of allocation and selection of the construction equipment when construction firms bid for construction projects. The main objective of the selection of construction equipment is to ensure the timely completion of all considered projects, while at the same time the planned budget is fulfilled. In this paper fuzzy linear programming is applied with the fuzzy objective function and fuzzy constrains. The proposed mathematical model is applied for optimal selection and allocation of equipment for five projects with different quantities of concrete works and general budget.

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. [ABSTRACT FROM AUTHOR]

Network planning is an important design-step wherein various network resources are provisioned at different network locations. Traffic demand for all node pairs in the network is an important determining factor for network planning, and typically, a static traffic demand matrix consisting of long-term average traffic values for all node pairs is employed. However, assessment of traffic demand matrix may not be accurate (precise). In this paper, we propose an optimization framework to minimize traffic congestion in a wavelength-routed wavelength-division multiplexing WDM optical backbone network using such approximate traffic demands. Modeling such a network is constrained by limited network resources, and can be formulated in general as a fuzzy optimization problem. Moreover, decision maker might want to relax traffic congestion goal by a small margin in presence of approximate traffic demand matrix, so as to reach some aspiration level, rather missing the crisp target by a very small amount. We use the network planning model to minimize traffic congestion so as to achieve an aspiration level of traffic congestion. We use Zimmermann's fuzzy programming technique to explore mixed integer linear programing based optimization model incorporating fuzzified constraints and objective function. All related constraints, such as traffic and lightpath routing, wavelength selection, restriction due to average propagation delay, lightpath degree and maximum hop count constraints are taken into consideration. Following the proposed model, under fuzzy environment, we implement reliable and efficient network planning and traffic provisioning, and estimate traffic congestion more accurately. [ABSTRACT FROM AUTHOR]

In this study, we discussed a fuzzy programming approach to bi-level linear programming problems and their application. Bi-level linear programming is characterized as mathematical programming to solve decentralized problems with two decision-makers in the hierarchal organization. They become more important for the contemporary decentralized organization where each unit seeks to optimize its own objective. In addition to this, we have considered Bi-Level Linear Programming (BLPP) and applied the Fuzzy Mathematical Programming (FMP) approach to get the solution of the system. We have suggested the FMP method for the minimization of the objectives in terms of the linear membership functions. FMP is a supervised search procedure (supervised by the upper Decision Maker (DM)). The upper-level decision-maker provides the preferred values of decision variables under his control (to enable the lower level DM to search for his optimum in a wider feasible space) and the bounds of his objective function (to direct the lower level DM to search for his solutions in the right direction).

Many problems in operations research including problems from management, production planning and scheduling, transportation, location, and many others necessitate decision making in the presence of uncertainty. Therefore, many theories and methodologies have been developed to deal with optimization problems under uncertainty in general. To understand uncertain data envelopment analysis models, the process was started by introducing the definition of uncertain variables, uncertain vectors, fuzzy linear programming (FLP). Selection criteria of an incoming vector in improving uncertain linear programming is established along with its unbounded nature. This paper based on the established theoretical framework proposes an alternative linear programming model that can include some uncertainty information. Finally, input-oriented CCR model with fuzzy variables is developed and an effective approach for measuring efficiency is demonstrated with a numerical example.

This paper presents a literature review and analysis of fuzzy linear programming models and methods. This review traces back to the linear programming with fuzzy linear constraints models proposed by Hans Jürgen Zimmermann during 1974–76 which have become the seminal works for scholars and practitioners in this field due to its ease to implement and understand. To do so, we analyze the timeline of publication of the most important works in the field, evaluate the influence of relevant papers using a time-weighted index based on the amount of citations in order to provide a perspective about the future directions in fuzzy linear programming and to recommend selected books, journals and some recent contributions to readers.

Supplier evaluation and selection (SES) problems have long been studied, leading to the development of a wide range of individual and hybrid models for solving them. However, the lack of widespread diffusion of existing SES models in the industry points to a need for simpler models that can systematically evaluate both qualitative and quantitative attributes of potential suppliers while enhancing the flexibility decision-makers need to account for relevant situational factors. Furthermore, empirical validations of existing models in SES have been few and far between. With a view to addressing these issues, this paper proposes an integrated solution framework that can be used to evaluate both tangible and intangible attributes of potential suppliers. The proposed framework combines three individual methods, namely the fuzzy analytic hierarchy process, fuzzy complex proportional assessment and fuzzy linear programming. The framework is validated through application in a Turkish textile company. The results generated using the proposed framework is compared with the actual historical data collected from the company. Additionally, a feasibility assessment is conducted on the sample supplier selection criteria employed, as well as assessment of the results generated using the proposed model. [ABSTRACT FROM AUTHOR]

In today's competitive environment, it is essential to design a flexible-responsive manufacturing system with automatic material handling systems. In this article, a fuzzy mixed integer linear programming model is designed for cell formation problems including the scheduling of parts within cells in a cellular manufacturing system (CMS) where several automated guided vehicles (AGVs) are in charge of transferring the exceptional parts. Notably, using these AGVs in CMS can be challenging from the perspective of mathematical modeling due to consideration of AGVs’ collision as well as parts pickup/delivery. This article investigates the role of AGVs and human factors as indispensable components of automation systems in the cell formation and scheduling of parts under fuzzy processing time. The proposed objective function includes minimizing the makespan and intercellular movements of parts. Due to the NP-hardness of the problem, a hybrid genetic algorithm (GA/heuristic) and a whale optimization algorithm (WOA) are developed. The experimental results reveal that our proposed algorithms have a high performance compared to CPLEX and the other two well-known algorithms, i.e., particle swarm optimization and ant colony optimization, in terms of computational efficiency and accuracy. Finally, WOA stands out as the best algorithm to solve the problem. [ABSTRACT FROM AUTHOR]

In this manuscript, three contributions are proposed. First contribution is proposing a good evaluation between the fuzzy and neutrosophic approaches using a novel fuzzy-neutrosophic transfer. Second contribution is introducing a general framework for solving the neutrosphic linear programming problems using the advantages of the method of Abdel-Basset et al. and the advantages of Singh et al.'s method. Third contribution is proposing a new neutrosophic exterior point simplex algorithm NEPSA and its fuzzy version FEPSA. NEPSA has two paths to get optimal solutions. One path consists of basic not feasible solutions but the other path is feasible. Finally, the numerical examples and results analysis show that NEPSA more than accurate FEPSA. [ABSTRACT FROM AUTHOR]

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.

Environmental concerns are on the rise around the world, as carbon emissions are increasing rapidly and which is one of the leading causes of global warming. These emissions come from different sectors but the biggest contribution comes from the transport sector. As countermeasures, countries are shifting toward electric vehicles, which are much greener modes of transportation. Currently, different types of electric vehicles are available in the market. The selection of a viable type of electric vehicle for a developing country like Pakistan needs proper evaluation. In this study, two techniques are used. Firstly, fuzzy Strength, Weaknesses, Opportunities, and Threats (SWOT) approach is used for the selection of a suitable vehicle for the future market of Pakistan. Secondly, fuzzy linear programming is used to check the viability of electric vehicles in the current market scenario. Using these techniques, hydrogen fuel cell vehicle is selected as the most advantageous vehicle for Pakistan's future market and the hybrid electric vehicle comes out as the best alternative for the current market. The techniques and the case study of Pakistan are the main novelty of this study, as no literature exists on electric vehicles with these methods on Pakistan's perspective. Based on the results, different recommendations are given to the policymakers of Pakistan. The recommended policies could make electric vehicles a prime competitor to the combustion vehicles in the current and future markets. [ABSTRACT FROM AUTHOR]

The growth of DNA databases used to store large number of biological sequence data, has stimulated the importance of alignment of sequences for phylogenetics. Most of the phylogenetic methods based on alignment of sequences consume long time to provide the results. In this regard, a new alignment free measure, based on frequency of occurrence of different nucleotides in sequences has been reported. The Euclidean distance metric has been used over these frequencies of nucleotides to obtain the dissimilarities among DNA sequences. These distances are then used to construct the phylogenetic tree among sequences. In addition, a fuzzy linear programming model has been developed here to construct the phylogenetic network which is considered as the generalized form of phylogenetic tree. As an application, the proposed method is applied over the data set of beta - globin gene of nine species and is validated by comparing the obtained results with the already existing method. The results obtained are more promising over the available method and can be applied over any length of input data sequences.

Recently, Ghanbari et al. (IEEE Transactions on Fuzzy Systems 27:1286–1294, 2019) have proposed modified Kerre's method for comparison of LR fuzzy numbers. Here, we make use of the modified Kerre's method to solve fuzzy linear programming problems with LR coefficients. In an approach to solve a fuzzy linear program with fuzzy LR coefficients, a bi-objective optimization problem is formulated. For the associated bi-objective optimization problem, we present a time variant multi-objective particle swarm optimization (TV-MOPSO) algorithm to compute the Pareto front, a set containing a large number of solutions. Contrary to methods that change the fuzzy optimization problem to a crisp problem by use of a ranking function, using modified Kerre's method, the fuzzy optimization problem is solved directly, with no need for changing it to a crisp program. A comparative investigation using illustrative examples with triangular fuzzy coefficients show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Remanufacturing can bring used or end of life products back to like-new products. The production planning of inventory control policies in a hybrid manufacturing/ remanufacturing system with different prioritizations (manufacturing vs. remanufacturing) is investigated in this study. For this production planning problem, the imprecision of customer demand, related operating costs, number of returned used components and all lead time and production times are uncertain. This uncertainty complicates the planning of the production and inventory control. The fuzzy set theory is employed due to the presence of the imprecise information, and the Fuzzy Linear Programming (FLP) is applied to optimize the model under these uncertainties. The proposed approach maximizes the most likely value of the profit, minimizes the risk of obtaining a lower profit, and maximizes the possibility of obtaining a higher profit for each production planning policy. The results show that the Priority-To-Remanufacturing (PTR) policy shows a higher profit than the policy of Priority-To-Manufacturing (PTM) and FLP can help decision makers to be aware of the risks and effects of uncertainties in their plans. As a result, they can prepare in advance for such scenarios. [ABSTRACT FROM AUTHOR]

In this paper we try to define a percentage form of LR fuzzy numbers which is a useful form of fuzzy numbers and its’ arithmetics. So, we show how the maximum variation range of optimal value of fuzzy objective function can be predicted by using this form of fuzzy numbers. Since fuzzy problems are generally solved through a complicated manner, the purpose of this study is releasing a kind of prediction for the final solution in the way that the manager can access to an outlook to optimal solution (Z∗) without solving the problem. Finally, optimal value of fuzzy objective function on fuzzy linear programming is predicted whenmaximum variation range of fuzzy variable have been predetermined.

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. [ABSTRACT FROM AUTHOR]

The growing awareness about the adverse effects of fossil fuels and the ensuing pollution has encouraged and promoted the use of renewable energies throughout the world. This in turn has encouraged governments and investors to show more interest in generation of renewable energy through construction of facilities such as wind farms. The novelty of this research work is the budget constraints consideration for the first time in wind energy projects. This work is done for the Khorasan provinces of Iran. This study is based on regional and municipal data pertaining to the year 2016, and uses the fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method to deal with the uncertainty of the data. The fuzzy TOPSIS method is used to evaluate and rank the candidate cities, and the results are then used to develop a budget-constrained model by fuzzy linear programming with the Maxmin objective function. The resulting mathematical model is optimised with GAMS software. The modelling results show that in the presence of budget constraints, of 10 Khorasan cities considered as candidate locations, Afriz, Rudab, and Fadashk exhibit the best conditions for the construction of wind farms.

A major problem connected with planning the organization of trains on a railway network is the optimization of the scheme of movement, which determines the routing and the number of trains. In this paper, an integrated approach of fuzzy linear programming method and multi-criteria analysis including three steps is proposed. In the first step, we defined the schemes of transportation of intercity trains and optimized each scheme in terms of direct operating costs by taking into account the uncertainty of passenger flows and utilization of train capacity using the fuzzy linear programming method. In the second step we determined the additional technological criteria to assess the variant schemes. The Fuzzy AHP method was applied to determine the weights of criteria. Using the results obtained from Fuzzy AHP, we prioritized the variant schemes of transportation by applying the PROMETHEE method. The third step presents the optimal choice of transportation of trains on a railway network based on minimum ratio of normalized costs and normalized PROMETHEE net outranking flow. In this step, the model uses the results obtained in the first and second steps. The practicability of the integrated approach is demonstrated through the case study of Bulgaria’s railway network, and nine schemes were investigated. The model results and the real situation were compared. It was found out that the optimal scheme of intercity train transportation improves the service and reduces direct operating costs.

One of the most important operational issues in urban drinking water production and distribution systems is to assign a plan for running hours of water supplying electric pumps. The cost of consuming electricity in these pumps allocates most of water and wastewater companies operational costs to itself which is dependent to their running hours. In this paper, meanwhile having a field study in Sistan rural water and wastewater company, the constraints for specifying electric pumps operational time in water supplying resources such as restrictions in fulfilling demand, supply potable water with suitable quality and uselessness of electric pumps have been identified. Due to uncertainty and fuzziness of the constraints, a linear programming model with fuzzy restrictions for determining electric pumps running hours per day is submitted with the aim to minimize electricity consumption and cost. After collecting and using required data for model, it proved that using the proposed model could reduce the costs of electrical energy and increase productivity up to 23 percent per month. The proposed mathematical fuzzy programming is able to specify electric pumps scheduling plan for water supply resources with the aim to reduce the costs of consuming energy.

In this study, a multi-element wind turbine blade that consists of NACA 6411 and NACA 4412 leading-edge slat design is investigated computationally. Optimum design parameters of the slatted wind turbine blade leading to maximum value of CL/CD related to the turbine power are obtained. In the optimization process, a new fuzzy logic linear programming methodology integrating with fuzzy linear regression and 2D CFD analysis is proposed. The aerodynamic characteristics of the slatted blade are computed by using Incompressible Navier–Stokes equations and k-omega turbulence modeling. Results are compared with the results of linear programming method and direct search optimization method. The computational results reveal that the proposed methodology for performance optimization is more effective than other methods to obtain high-performance value of the CL/CD. The maximum value of the CL/CD is obtained as 25.1 leading the maximum efficiency of 0.52. [ABSTRACT FROM AUTHOR]

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, α -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. [ABSTRACT FROM AUTHOR]

The parameters of decision models are not clearly known and they are not exact for most real-life problems. Therefore, this article presents a model of linear programming with grey interval numbers for right-hand-side parameters under an uncertainty case. The purpose of this article is to compare the results of a fuzzy linear programming model and those of four proposed grey linear programming models and to find effective solutions for alternative production planning. The results produced a more effective plan to use as an industrial solution according to a fuzzy linear programming model. The objective function value is about 3% more effective than with the fuzzy linear programming method, and the customer satisfaction of total demand is about 25% more effective for the number of served customers. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of determining the optimal production quantity under uncertainties. The treated problem has a critical effect on the achievement of business goals, which further propagates to the competitive advantage of manufacturing company. The considered problem is stated as the fuzzy linear programming task. The objective function is defined as the maximization of all profit over the time period. The set of constraints consists of constraints deriving from the enterprise (available capacities) and from the market (demand). The fuzzy rating coefficient values of objective function of each pair of the considered products are described by linguistic expressions which are modelled by triangular fuzzy numbers. Handling of uncertainties of the stated fuzzy pair-wise comparison matrix is performed by using the extent analysis. The right-hand-side values of the capacity constraints are determined by decision makers and modelled by the triangular fuzzy numbers. The market constraint values are given according to evidence data. Determination of the optimal quantities of considered products for each time period is based on the concept of equal possibilities. The proposed model is illustrated by an example with real-life data. [ABSTRACT FROM AUTHOR]

Although assembly lines are designed to be highly efficient and very cost effective, they have been criticised for their effects on workers. Assembly workers usually perform repetitive tasks and sit or stand for long periods of time, therefore they are at high risk of work-related injuries such as musculoskeletal disorders. In this study, we present a new assembly line balancing problem that incorporates not only task precedence and processing times but also physical workloads in order to reduce work-related injuries. We consider the physical workload of a task as a fuzzy concept and propose a fuzzy linear programming model. The proposed model is applied to an assembly line balancing problem in a textile company and to some of the benchmark problems in order to illustrate the effects of physical workloads. [ABSTRACT FROM AUTHOR]

To achieve carbon neutrality, the power structure is bound to usher in fundamental transformations. In this study, an energy–carbon–water nexus (F-ECWN) system model is developed to assess trade-offs among different decision-making objectives within a power system under varying policy orientations. The model incorporates forecast data derived from various global climate models, leverages neural network prediction, draws upon multi-criteria decision-making theory, and incorporates fuzzy theory. Moreover, to effectively handle a multitude of fuzzy parameters within the model, a novel dual-interval algorithm for solving fuzzy linear programming is proposed. The F-ECWN model has the capability to derive the fuzzy membership function for each variable within the model ensuring that errors remain below 1 %, reflecting the uncertainty stemming from diverse policy orientations and technology choices. Various scenarios were formulated to gauge the impact of policy orientation on the allocation decisions of regional energy systems. Additionally, sensitivity analysis has been conducted to assess the effects of uncertain parameters on modeling outputs. The results of the applied research in Fujian Province have revealed that the presence of uncertainties in the energy system's parameters can significantly influence model outputs and decision-making processes. Furthermore, the modeling results demonstrate the region's substantial potential for reducing carbon emissions. Under optimal policy guidance and climate conditions, the total carbon emissions can be reduced by 65 %, with a 36.14 % increase in the total system cost. These findings are anticipated to provide valuable support for formulating optimal decisions regarding regional energy-carbon-water nexus system and related environmental policies. [Display omitted] • A new algorithm for solving fuzzy multi-objective programming problems is proposed. • A robust fuzzy number reduction algorithm based on probabilistic integration is proposed. • E -C-W nexus in Fujian Province under different policy orientations is quantified. • Incorporation of CMIP6 forecast data into renewable energy parameters construction. • Carbon and water synergy effects in regional power system optimization are observed. [ABSTRACT FROM AUTHOR]

In this paper, we generalize a linear programming problem with symmetric trapezoidal ‎fuzzy number which is introduced by Ganesan and et al. in [3] to a general kind of trapezoidal fuzzy number. In this way, we first establish a new ‏arithmetic operation for multiplication of two ‏trapezoidal fuzzy numbers. Then in order to ‎preparing a method for solving the fuzzy linear programming as well as the primal simplex ‎algorithm, we use a general linear ranking function as a convenient approach in the literature. In fact, our main contribution in this work is based on 3 items: 1) Extending the current fuzzy linear program to a general kind which is not essentially including the symmetric trapezoidal fuzzy number‎s ‎‏,‏ 2) Defining a new multiplication role of two trapezoidal fuzzy number‎s‎, 3) Establishing a fuzzy primal simplex algorithm for solving the generalized model. We in particular emphasize that this study can be used for establishing fuzzy dual simplex algorithm, fuzzy primal-dual simplex algorithm, fuzzy multi objective linear programming and the other similar methods which are appeared in the literature.