New method for solving Fuzzy transportation problems with LR flat fuzzy numbers

A new method to determine the Fermatean fuzzy optimal solution of transportation problems

 Transportation Problems (TP) have multiple applications in supply chain management to reduce costs. Efficient methods have been developed to address TP when all factors, including supply, demand, and unit transportation costs, are precisely known. However, due to uncertainty in practical applications, it is necessary to study TP in an uncertain environment. In this paper, we define the Trapezoidal Fermatean Fuzzy Number (TrFFN) and its arithmetic operations. Then we introduce a new approach to solve TP, where transportation cost, supply, and demand are treated as TrFFN, and we call it Fermatean Fuzzy TP (FFTP). We illustrate the feasibility and superiority of this method with two application examples, and compare the performance of this method with existing methods. Furthermore, the advantages of the proposed method over existing methods are described to address TP in uncertain environments.

Fuzzy efficiency evaluation in relational network data envelopment analysis: application in gas refineries

In contrast to classical data envelopment analysis (DEA), network DEA has attention to the internal structure of a production system and reveals the relationship between the efficiency of system and efficiencies of the processes. However, the flexibility of weights and the need for crisp input and output data in the evaluation process are two major shortcomings of classical network DEA models. This paper presents a common weights approach for a relational network DEA model in a fuzzy environment to measure the efficiencies of the system and the component processes. The proposed approach first finds upper bounds on input and output weights for a given cut level and then it determines a common set of weights (CSW) for all decision-making units (DMUs). Hence, the fuzzy efficiencies of all processes and systems for all DMUs are obtained based on the resulting CSW. The developed fuzzy relational network DEA and the proposed common weights approach are illustrated with a numerical example. The obtained results confirm that the fuzzy data affects over the efficiency scores and complete ranking of DMUs. The applicability of the proposed network model is illustrated by performance evaluation of gas refineries in Iran.

Dynamic programming approach for fuzzy linear programming problems FLPs and its application to optimal resource allocation problems in education system

This research is about the development of a dynamic programming model for solving fuzzy linear programming problems. Initially, fuzzy dynamic linear programming model FDLP is developed. This research revises the established dynamic programming model for solving linear programming problems in a crisp environment. The mentioned approach is upgraded to address the problem in an uncertain environment. Dynamic programming model can either be passing forward or backward. In the proposed approach backward dynamic programming approach is adopted to address the problem. It is then followed by implementing the proposed method on the education system of Pakistan. The education system of Pakistan comprises of the Primary, Middle, Secondary, and Tertiary education stages. The problem is to maximize the efficiency of the education system while achieving the targets with minimum usage of the constrained resources. Likewise the model tries to maximize the enrollment in the Primary, Middle, Secondary and Tertiary educational categories, subject to the total available resources in a fuzzy uncertain environment. The solution proposes that the enrollment can be increased by an amount 9997130, by increasing the enrollment in the Middle and Tertiary educational categories. Thus the proposed method contributes to increase the objective function value by 30%. Moreover, the proposed solutions violate none of the constraints. In other words, the problem of resources allocation in education system is efficiently managed to increase efficiency while remaining in the available constrained resources. The motivation behind using the dynamic programming methodology is that it always possesses a numerical solution, unlike the other approaches having no solution at certain times. The proposed fuzzy model takes into account uncertainty in the linear programming modeling process and is more robust, flexible and practicable.

Fuzzy fractional mathematical model of COVID-19 epidemic

In this paper, we develop a mathematical model with a Caputo fractional derivative under fuzzy sense for the prediction of COVID-19. We present numerical results of the mathematical model for COVID-19 of most three infected countries such as the USA, India and Italy. Using the proposed model, we estimate predicting future outbreaks, the effectiveness of preventive measures and potential control strategies of the infection. We provide a comparative study of the proposed model with Ahmadian’s fuzzy fractional mathematical model. The results demonstrate that our proposed fuzzy fractional model gives a nearer forecast to the actual data. The present study can confirm the efficiency and applicability of the fractional derivative under uncertainty conditions to mathematical epidemiology.

Modeling of relative collision risk based on the ships group situation

The collision risk of ships is a fuzzy concept, which is the measurement of the likelihood of a collision between ships. Most of existed studies on the risk of multi-ship collision are based on the assessment of two-ship collision risk, and collision risk between the target ship and each interfering ship is calculated respectively, to determine the key avoidance ship. This method is far from the actual situation and has some defects. In open waters, it is of certain reference value when there are fewer ships, but in busy waters, it cannot well represent the risk degree of the target ship, since it lacks the assessment of the overall risk of the perceived area of the target ship. Based on analysis of complexity of ships group situation, the concept of relative domain was put forward and the model was constructed. On this basis, the relative collision risk was proposed, and the corresponding model was obtained, so as to realize risk assessment. Through the combination of real ship and simulation experiments, the variation trend, stability and sensitivity of the model were verified. The results showed that risk degree of the environment of ships in open and busy waters could be well assessed, and good references for decision-making process of ships collision avoidance could be provided.

A risk assessment method based on Pythagorean fuzzy set and artificial-neuron-like evaluation node

Taking concepts from supply management, we developed a specification-assessment-compliance approach to obtain a transparent multi-criteria decision-making method. We designed an artificial-neuron-like node that allows the implementation of networks to represent hierarchies of evaluation criteria. A new graphical model based on functions in the unit segment uses the concept of Pythagorean fuzzy set (PFS). The specification PFSs’ entropies modulate the widths of one-sided triangular fuzzy numbers (TFNs) with positive slopes that become the evaluation nodes’ activation functions. All the specifications refer to the same point to facilitate the evaluation and ensure coherence. One-sided TFNs with negative slopes biunivocally represent the assessment PFSs at the input layer of the network. A risk case study on the options for the outsourcing of an information technology development project shows the proposed method’s implementation. We compare the results with those of the application of two other previous methods.

Optimizing a fully rough interval integer solid transportation problems

An innovative method, namely modified slice-sum method using the principle of zero point method is proposed for finding an optimal solution to fully rough interval integer solid transportation problems (FRIISTP). The proposed method yields an optimal solution to the fully rough interval integer solid transportation problem directly. In this method, there is no necessity to find an initial basic feasible solution to FRIISTP and also need not to use the existing MODI and stepping stone methods for testing the optimality to improve the basic feasible solution to the FRIISTP but directly obtain an optimal solution to the given FRIISTP by using the proposed method. The optimal values of decision variables and the objective function of the fully rough interval integer solid transportation problems provided by the proposed method are rough interval integers. The advantages of the proposed method over existing method are discussed in the context of an application example. The modified slice-sum method has been applied to calculate the optimal compromise solutions of FRIISTP, and then it was solved by using TORA software. The proposed method can be served as an appropriate tool for the decision makers when they are handling logistic models of real life situations involving three items with rough interval integer parameters.

A new algorithm for solving uncapacitated transportation problem with interval-defined demands and suppliers capacities

The uncapacitated transportation problem (UTP) deals with minimizing the transportation costs related to the delivery of a homogeneous product from multi-suppliers to multi-consumers. The application of the UTP can be extended to other areas of operations research, including inventory control, personnel assignment, signature matching, product distribution with uncertainty, multi-period production and inventory planning, employment scheduling, and cash management. Such a UTP with interval-defined demands and suppliers capacities (UTPIDS) is investigated in this paper. In UTPIDS, the demands and suppliers capacities may not be known exactly but vary within an interval due to variation in the economic conditions of the global economy. Following the variation, the minimal total cost of the transportation can also be varied within an interval and thus, the cost bounds can be obtained. Here, although the lower bound solution can be attained methodologically, the correct estimation of the worst case realization (the exact upper bound) on the minimal total transportation cost of the UTPIDS is an NP-hard problem. So, the decision-makers seek for minimizing the transportation costs and they are interested in the estimation of the worst case realization on these minimal costs for better decision making especially, for proper investment and return. In literature very few approaches are available to find this estimation of the worst case realization with some shortcomings. First, we demonstrate that the available heuristic methods fail to obtain the correct estimation of the worst case realization always. In this situation, development of a better heuristic method to find the better near optimal estimation of the worst case realization on the minimal total costs of the UTPIDS is desirable. Then this paper provides a new polynomial time algorithm that runs in O (N2) time (N, higher of the numbers of source and destination nodes) for better estimation. A comparative assessment on solutions of available benchmark instances, some randomly generated numerical example problems and a real-world application shows promising performance of the current technique. So, our new finding would definitely be benefited to practitioners, academics and decision makers who deal with such type of decision making instances.

On derivatives of fuzzy multi-dimensional mappings and applications under generalized differentiability

This article identifies and presents the generalized difference (g-difference) of fuzzy numbers, Fréchet and Gâteaux generalized differentiability (g-differentiability) for fuzzy multi-dimensional mapping which consists of a new concept, fuzzy g-(continuous linear) function; Moreover, the relationship between Fréchet and Gâteaux g-differentiability is studied and shown. The concepts of directional and partial g-differentiability are further framed and the relationship of which will the aforementioned concepts are also explored. Furthermore, characterization is pointed out for Fréchet and Gâteaux g-differentiability; based on level-set and through differentiability of endpoints real-valued functions a characterization is also offered and explored for directional and partial g-differentiability. The sufficient condition for Fréchet and Gâteaux g-differentiability, directional and partial g-differentiability based on level-set and through employing level-wise gH-differentiability (LgH-differentiability) is expressed. Finally, to illustrate the ability and reliability of the aforementioned concepts we have solved some application examples.

Modified artificial bee colony algorithm for solving mixed interval-valued fuzzy shortest path problem

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

Multi-objective fully intuitionistic fuzzy fixed-charge solid transportation problem

During past few decades, fuzzy decision is an important attention in the areas of science, engineering, economic system, business, etc. To solve day-to-day problem, researchers use fuzzy data in transportation problem for presenting the uncontrollable factors; and most of multi-objective transportation problems are solved using goal programming. However, when the problem contains interval-valued data, then the obtained solution was provided by goal programming may not satisfy by all decision-makers. In such condition, we consider a fixed-charge solid transportation problem in multi-objective environment where all the data are intuitionistic fuzzy numbers with membership and non-membership function. The intuitionistic fuzzy transportation problem transforms into interval-valued problem using $$(\alpha ,\beta )$$ ( α , β ) -cut, and thereafter, it reduces into a deterministic problem using accuracy function. Also the optimum value of alternative corresponds to the optimum value of accuracy function. A numerical example is included to illustrate the usefulness of our proposed model. Finally, conclusions and future works with the study are described.

Solving fuzzy multi-objective shortest path problem based on data envelopment analysis approach

The shortest path problem (SPP) is a special network structured linear programming problem that appears in a wide range of applications. Classical SPPs consider only one objective in the networks while some or all of the multiple, conflicting and incommensurate objectives such as optimization of cost, profit, time, distance, risk, and quality of service may arise together in real-world applications. These types of SPPs are known as the multi-objective shortest path problem (MOSPP) and can be solved with the existing various approaches. This paper develops a Data Envelopment Analysis (DEA)-based approach to solve the MOSPP with fuzzy parameters (FMOSPP) to account for real situations where input–output data include uncertainty of triangular membership form. This approach to make a connection between the MOSPP and DEA is more flexible to deal with real practical applications. To this end, each arc in a FMOSPP is considered as a decision-making unit with multiple fuzzy inputs and outputs. Then two fuzzy efficiency scores are obtained corresponding to each arc. These fuzzy efficiency scores are combined to define a unique fuzzy relative efficiency. Hence, the FMOSPP is converted into a single objective Fuzzy Shortest Path Problem (FSPP) that can be solved using existing FSPP algorithms.

A common-weights approach for efficiency evaluation in fuzzy data envelopment analysis with undesirable outputs: Application in banking industry

Data Envelopment Analysis (DEA) is a non-parametric approach based on linear programming for evaluating the performance of decision making units (DMUs) with multiple inputs and multiple outputs. The lack of the ability to generate the actual weights, not considering the impact of undesirable outputs in the evaluation process and the measuring of efficiencies of DMUs based upon precise observations are three main drawbacks of the conventional DEA models. This paper proposes a novel approach for finding the common set of weights (CSW) to compute efficiencies in DEA model with undesirable outputs when the data are represented by fuzzy numbers. The proposed approach is based on fuzzy arithmetic which formulates the fuzzy additive DEA model as a linear programing problem and gives fuzzy efficiencies of all DMUs based on resulting CSW. We demonstrate the applicability of the proposed model with a simple numerical example. Finally, in the context of performance management, an application of banking industry in Iran is presented for analyzing the influence of fuzzy data and depicting the impact of undesirable outputs over the efficiency results.

Solving the fully fuzzy multi-objective transportation problem based on the common set of weights in DEA

A transportation problem basically deals with the problem which aims to minimize the total transportation cost or maximize the total transportation profit of distributing a product from a number of sources or origins to a number of destinations. While, in general, most of the real life applications are modeled as a transportation problem (TP) with the multiple, conflicting and incommensurate objective functions. On the other hand, for some reason such as shortage of information, insufficient data or lack of evidence, the data of the mentioned problem are not always exact but can be fuzzy. This type of problem is called fuzzy multi-objective transportation problem (FMOTP). There are a few approaches to solve the FMOTPs. In this paper, a new fuzzy DEA based approach is developed to solve the Fully Fuzzy MOTPs (FFMOTPs) in which, in addition to parameters of the MOTPs, all of the variables are considered fuzzy. This approach considers each arc in a FFMOTP as a decision making unit which produces multiple fuzzy outputs using the multiple fuzzy inputs. Then, by using the concept of the common set of weights (CSW) in DEA, a unique fuzzy relative efficiency is defined for each arc. In the following, the unique fuzzy relative efficiency is considered as the only attribute for the arcs. In this way, a single objective fully fuzzy TP (FFTP) is obtained that can be solved using the existing standard algorithms for solving this kind of TPs. A numerical example is provided to illustrate the developed approach.

Linear Programming and Fuzzy Optimization to Substantiate Investment Decisions in Tangible Assets

This paper studies the problem of tangible assets acquisition within the company by proposing a new hybrid model that uses linear programming and fuzzy numbers. Regarding linear programming, two methods were implemented in the model, namely: the graphical method and the primal simplex algorithm. This hybrid model is proposed for solving investment decision problems, based on decision variables, objective function coefficients, and a matrix of constraints, all of them presented in the form of triangular fuzzy numbers. Solving the primal simplex algorithm using fuzzy numbers and coefficients, allowed the results of the linear programming problem to also be in the form of fuzzy variables. The fuzzy variables compared to the crisp variables allow the determination of optimal intervals for which the objective function has values depending on the fuzzy variables. The major advantage of this model is that the results are presented as value ranges that intervene in the decision-making process. Thus, the company's decision makers can select any of the result values as they satisfy two basic requirements namely: minimizing/maximizing the objective function and satisfying the basic requirements regarding the constraints resulting from the company's activity. The paper is accompanied by a practical example.