Penalty method for fuzzy linear programming with trapezoidal numbers

In this paper we shall propose an algorithm for solving fuzzy linear programming problems with trapezoidal numbers using a penalty method. We will transform the problem of maximizing a function having trapezoidal fuzzy number values under some constraints into a deterministic multi-objective programming problem by penalizing the objective function for possible constraint violation. Furthermore, the obtained deterministic problem will have only unavoidable inequalities between trapezoidal fuzzy numbers parameters as constraints

Lexicographic Methods for Fuzzy Linear Programming

Fuzzy Linear Programming (FLP) has addressed the increasing complexity of real-world decision-making problems that arise in uncertain and ever-changing environments since its introduction in the 1970s. Built upon the Fuzzy Sets theory and classical Linear Programming (LP) theory, FLP encompasses an extensive area of theoretical research and algorithmic development. Unlike classical LP, there is not a unique model for the FLP problem, since fuzziness can appear in the model components in different ways. Hence, despite fifty years of research, new formulations of FLP problems and solution methods are still being proposed. Among the existing formulations, those using fuzzy numbers (FNs) as parameters and/or decision variables for handling inexactness and vagueness in data have experienced a remarkable development in recent years. Here, a long-standing issue has been how to deal with FN-valued objective functions and with constraints whose left- and right-hand sides are FNs. The main objective of this paper is to present an updated review of advances in this particular area. Consequently, the paper briefly examines well-known models and methods for FLP, and expands on methods for fuzzy single- and multi-objective LP that use lexicographic criteria for ranking FNs. A lexicographic approach to the fuzzy linear assignment (FLA) problem is discussed in detail due to the theoretical and practical relevance. For this case, computer codes are provided that can be used to reproduce results presented in the paper and for practical applications. The paper demonstrates that FLP that is focused on lexicographic methods is an active area with promising research lines and practical implications.Spanish Ministry of Economy and CompetitivenessEuropean Union (EU) TIN2017-86647-

Developing an Overbooking Fuzzy-Based Mathematical Optimization Model for Multi-Leg Flights

Overbooking is one of the most vital revenue management practices that is used in the airline industry. Identification of an overbooking level is a challenging task due to the uncertainties associated with external factors, such as demand for tickets, and inappropriate overbooking levels which may cause revenue losses as well as loss of reputation and customer loyalty. Therefore, the aim of this paper is to propose a fuzzy linear programming model and Genetic Algorithms (GAs) to maximize the overall revenue of a large-scale multi-leg flight network by minimizing the number of empty seats and the number of denied passengers. A fuzzy logic technique is used for modeling the fuzzy demand on overbooking flight tickets and a metaheuristics-based GA technique is adopted to solve large-scale multi-leg flights problem. As part of model verification, the proposed GA is applied to solve a small multi-leg flight linear programming model with a fuzzified demand factor. In addition, experimentation with large-scale problems with different input parameters’ settings such as penalty rate, show-up rate and demand level are also conducted to understand the behavior of the developed model. The validation results show that the proposed GA produces almost identical results to those in a small-scale multi-leg flight problem. In addition, the performance of the large-scale multi-leg flight network represented by a number of KPIs including total booking, denied passengers and net-overbooking profit towards changing these input parameters will also be revealed

Fuzzy linear programming problems : models and solutions

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

A Novel Technique for Solving Multiobjective Fuzzy Linear Programming Problems

This study considers multiobjective fuzzy linear programming (MFLP) problems in which the coefficients in the objective functions are triangular fuzzy numbers. The study proposing a new technique to transform MFLP problems into the equivalent single fuzzy linear programming problem and then solving it via linear ranking function using the simplex method, supported by numerical example

Computational intelligence techniques in asset risk analysis

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The problem of asset risk analysis is positioned within the computational intelligence paradigm. We suggest an algorithm for reformulating asset pricing, which involves incorporating imprecise information into the pricing factors through fuzzy variables as well as a calibration procedure for their possibility distributions. Then fuzzy mathematics is used to process the imprecise factors and obtain an asset evaluation. This evaluation is further automated using neural networks with sign restrictions on their weights. While such type of networks has been only used for up to two network inputs and hypothetical data, here we apply thirty-six inputs and empirical data. To achieve successful training, we modify the Levenberg-Marquart backpropagation algorithm. The intermediate result achieved is that the fuzzy asset evaluation inherits features of the factor imprecision and provides the basis for risk analysis. Next, we formulate a risk measure and a risk robustness measure based on the fuzzy asset evaluation under different characteristics of the pricing factors as well as different calibrations. Our database, extracted from DataStream, includes thirty-five companies traded on the London Stock Exchange. For each company, the risk and robustness measures are evaluated and an asset risk analysis is carried out through these values, indicating the implications they have on company performance. A comparative company risk analysis is also provided. Then, we employ both risk measures to formulate a two-step asset ranking method. The assets are initially rated according to the investors' risk preference. In addition, an algorithm is suggested to incorporate the asset robustness information and refine further the ranking benefiting market analysts. The rationale provided by the ranking technique serves as a point of departure in designing an asset risk classifier. We identify the fuzzy neural network structure of the classifier and develop an evolutionary training algorithm. The algorithm starts with suggesting preliminary heuristics in constructing a sufficient training set of assets with various characteristics revealed by the values of the pricing factors and the asset risk values. Then, the training algorithm works at two levels, the inner level targets weight optimization, while the outer level efficiently guides the exploration of the search space. The latter is achieved by automatically decomposing the training set into subsets of decreasing complexity and then incrementing backward the corresponding subpopulations of partially trained networks. The empirical results prove that the developed algorithm is capable of training the identified fuzzy network structure. This is a problem of such complexity that prevents single-level evolution from attaining meaningful results. The final outcome is an automatic asset classifier, based on the investors’ perceptions of acceptable risk. All the steps described above constitute our approach to reformulating asset risk analysis within the approximate reasoning framework through the fusion of various computational intelligence techniques

A Fuzzy Set Theory Based Methodology for Analysis of Uncertainties in Stage-Discharge Measurements and Rating Curve

River stage and discharge records are essential for hydrological and hydraulic analyses. While stage is measured directly, discharge value is calculated from measurements of flow velocity, depth and channel cross-section dimensions. The measurements are affected by random and systematic measurement errors and other inaccuracies, such as approximation of velocity distribution and channel geometry with a finite number of measurements. Such errors lead to the uncertainty in both, the stage and the discharge values, which propagates into the rating curve established from the measurements. The relationship between stage and discharge is not strictly single valued, but takes a looped form due to unsteady flow in rivers. In the first part of this research, we use a fuzzy set theory based methodology for consideration of different sources of uncertainty in the stage and discharge measurements and their aggregation into a combined uncertainty. The uncertainty in individual measurements of stage and discharge is represented using triangular fuzzy numbers and their spread is determined according to the ISO – 748 guidelines. The extension principle based fuzzy arithmetic is used for the aggregation of various uncertainties into overall stage discharge measurement uncertainty. In the second part of the research we use fuzzy nonlinear regression for the analysis of the uncertainty in the single valued stage – discharge relationship. The methodology is based upon fuzzy extension principle. All input and output variables as well as the coefficients of the stage - discharge relationship are considered as fuzzy numbers. Two different criteria; the minimum spread and the least absolute deviation are used for the evaluation of output fuzziness. The results of the fuzzy regression analysis lead to a definition of lower and upper uncertainty bounds of the stage – discharge relationship and representation of discharge value as a fuzzy number. The third part of this research considers uncertainties in a looped rating curve with an application of the Jones formula. The Jones formula is based on approximate form of unsteady flow equation, which leads to an additional uncertainty. In order to take into account of the uncertainties due to the use of approximate formula and measurement of discharge values, the parameters of the Jones formula are considered fuzzy numbers. This leads to a fuzzified form of Jones formula. Its spread is determined by a multi-objective genetic algorithm. We used a criterion to minimize the spread of the fuzzified Jones formula so that the measurements points are bounded by the lower and upper bound curves. The study therefore considers individual sources of uncertainty from measurements to the single valued and looped rating curves. The study also shows that the fuzzy set theory provides an appropriate methodology for the analysis of the uncertainties in a nonprobabilistic framework.https://ir.lib.uwo.ca/wrrr/1023/thumbnail.jp

Order Allocation and Purchasing Transportation Planning in the Garment Supply Chain: A Goal-Flexible Planning Approach

The garment supply chain is one of the most common supply chains in the world. In this supply chain, quality and cost are the most important factors that are strongly related to the selection of suppliers and the allocation of orders to them. Accordingly, the purpose of this paper is to integrate decisions for supplier selection, order allocation, and multi- source, multi-mode, multi-product shipping plans with consideration of discounts under uncertainty. For this purpose, a multi-objective mixed-integer mathematical model is presented, including the objectives of minimizing costs and products with delays and maximizing the total purchase value. In this mathematical model, the policy of purchasing materials and determining the number and type of transport equipment are specified. To solve this mathematical model, a goal-flexible programming approach with a utility function is presented. In the solution algorithm, a new possibility-flexible programming method has been developed to deal with the uncertainties in the model, which is based on the expected value method and chance constraint. Finally, using a numerical problem, the establishment of the above model in the garment supply chain is investigated. As indicated by the outcomes, the proposed model was touchy to certain boundaries, including blended leaders’ mentality, a boundary identified with fluffy imperatives, and the degree of certainty characterized by the chief for not exactly equivalent limitations

Using Fuzzy Neural Networks and Analytic Hierarchy Process for Supplier Classification in e-Procurement

Electronic procurement is frequently defined as the sourcing of goods or services via electronic means, usually through the internet. A major process in the e-procurement decision making is that of supplier selection process. In the real world, the criteria and constraints for such a process are subjective in nature. In this study, the criteria for supplier selection, which already have been established empirically, has been adopted and no new criteria for the same has been proposed. These criteria and constraints have been modeled using fuzzy logic into constraints, which further has been modeled as a multi-objective decision making process, by combining neural networks and analytic hierarchy process. Then the suppliers have been classified into suitable suppliers and unsuitable suppliers, from the viewpoint of the firm