Α-cut Approach Research Articles

Multi-Objective Production Planning Problem: A Case Study for Optimal Production

In this paper, we have formulated a multi-objective production planning model for a hardware firm. This firm produces different types of hardware locks and other items in their production run. The objectives of the firm are to minimise the production cost, minimise the inventory holding cost and maximise the net profit subject to the set of realistic constraints. The production planning problem of a similar type in the past formulated under the certain environment where the input information precisely known to the decision maker (DM). However, in most of the situations, the input information is not precisely known. In such situations, fuzzy set theory plays a vital role in modelling of the problem where the input data has some vagueness. The proposed model of production planning also been formulated under fuzzy environment. Both triangular and trapezoidal fuzzy numbers used to present the vagueness in the input information. The equivalent crisp form of the fuzzy model obtained by two different defuzzification approaches namely ranking function and α-cut approach. Henceforth, the formulated models under the certain and fuzzy environment have been solved by the fuzzy goal programming approach.

Multi Criteria Decision Making Based on Classical α-Cut Approach

Multi Criteria Decision Making (MCDM) is the process to find the best alternatives among a set of available alternatives in presence of various conflicting criteria. In this paper MCDM method is presented using α-cut approach and it is observed that α-cut approach is general enough to deal with MCDM methods. Here first fuzzy numbers have been assigned to quantify different qualitative criteria of alternatives and then their α-cut values are evaluated and then performed some arithmetic operation to order ranking of the alternatives. Finally, a decision making problem is carried out with different case studies by using different shapes of fuzzy numbers. Keywords: Multi criteria decision making, fuzzy numbers, α-cut of fuzzy number.

Stochastic Fuzzy Multi-level Multi-objective Fractional Programming Problem: A FGP Approach

In this paper, a fuzzy goal programming (FGP) approach is considered for solving stochastic fuzzy multi-level multi-objective fractional programming (ML-MOFP) problem. In the developed stochastic fuzzy ML-MOFP model the fractional objective function coefficients and scalars are represented by fuzzy parameters. Moreover, in the constraints, the right-hand sides are independent random variable with known distribution function while both the left-hand side coefficients and the tolerance measures are considered to be fuzzy parameters. Therefore, the chance-constrained approach with dominance possibility criteria and the α-cut approach are utilized to transform the stochastic fuzzy ML-MOFP problem to its equivalent deterministic-crisp problem. Then, the membership functions for the defined fuzzy goals are setup. Also, in the proposed FGP model, a linearization procedures for the membership goals of the objective functions is developed. Hence, the FGP approach is used to achieve the highest degree of each of the membership goals by minimizing the sum of the negative deviational variables. Finally, an algorithm to clarify the developed FGP approach, as well as Illustrative numerical example, are presented.

One-unit repairable systems with active and standby redundancy and fuzzy parameters

PurposeThe purpose of this paper is to propose a procedure to construct the membership functions for a one-unit repairable system, which has both active and standby redundancy. The coverage factor is the same for the operating and standby unit failure.Design/methodology/approachThe α-cut approach is used to extract a family of conventional crisp intervals from the fuzzy repairable system for the desired system characteristics. This can be determined with a set of non-linear parametric programing using the membership functions.FindingsWhen system characteristics are governed by the membership functions, more information is provided to use by management. On the other hand, fuzzy theory is applied for the redundant system; therefore, the results are more useful for designers and practitioners.Originality/valueDifferent from other studies, the authors’ model provides more accurate estimation compared to uncertain environments based on fuzzy theory. The research would help managers and manufactures to make a better decision in order to have the optimal maintenance strategy based on the desired mean time to failure and availability of the systems.

A single server fuzzy queues with priority and unequal service rates

This article presents to develop a new mathematical model to construct membership functions of the fuzzy single channel queueing model F(M1, M2)/F(M1, M2)/1/PR-NP with two class of non-preemptive priority with unequal service rates. The main idea of this article is to merge between mathematical programming technique and Yager's ranking method by using α-cut approach and Zadeh extension principle to transform fuzzy queues into a family of conventional crisp (distinct) queues in this context. A numerical example is given to explain the validity of this new approach to obtain real exact values under different levels of α in the system.

A single server fuzzy queues with priority and unequal service rates

This article presents to develop a new mathematical model to construct membership functions of the fuzzy single channel queueing model F(M1, M2)/F(M1, M2)/1/PR-NP with two class of non-preemptive priority with unequal service rates. The main idea of this article is to merge between mathematical programming technique and Yager's ranking method by using α-cut approach and Zadeh extension principle to transform fuzzy queues into a family of conventional crisp (distinct) queues in this context. A numerical example is given to explain the validity of this new approach to obtain real exact values under different levels of α in the system.

Z-number DEA: A new possibilistic DEA in the context of Z-numbers

Data envelopment analysis (DEA) is a methodology that uses multiple inputs and outputs for measuring the efficiencies of a set of decision making units (DMUs). When data are crisp, conventional DEA models are used. However the values of inputs and outputs in many cases are imprecise and vague. In addition, most of these data are expert-based. Thus, taking into account expert’s reliability is quite important. In this paper we propose a Z-number version of the CCR (named after Charnes, Cooper, and Rhodes) and BCC (named after Banker, Charnes and Coopers) DEA models. The proposed method can be converted into the fuzzy DEA model when experts are confident about their opinions. Also, it can be converted into the conventional DEA models when the inputs and outputs are crisp numbers. In this study, the Z-number DEA model is transformed into possible linear programming and then by applying an alternative α-cut approach, a crisp linear programming model is obtained. Furthermore, the proposed model is applied to a portfolio selection problem in IS/IT (Information Systems/Information Technology) project to tackle uncertainties, interactions between projects and reliabilities. To the best of our knowledge, this is the first study that presents a unique Z-number DEA model.

An improved α-cut approach to transforming fuzzy membership function into basic belief assignment

In practical applications, pieces of evidence originated from different sources might be modeled by different uncertainty theories. To implement the evidence combination under the Dempster–Shafer evidence theory (DST) framework, transformations from the other type of uncertainty representation into the basic belief assignment are needed. α-Cut is an important approach to transforming a fuzzy membership function into a basic belief assignment, which provides a bridge between the fuzzy set theory and the DST. Some drawbacks of the traditional α-cut approach caused by its normalization step are pointed out in this paper. An improved α-cut approach is proposed, which can counteract the drawbacks of the traditional α-cut approach and has good properties. Illustrative examples, experiments and related analyses are provided to show the rationality of the improved α-cut approach.

Optimistic and pessimistic performance and congestion analysis in fuzzy data envelopment analysis

The notion of input congestion in data envelopment analysis (DEA) is analogous to the 'law of diminishing returns' in the classical economic theory of production which states that if a single input is increased while other inputs are held constant, the marginal product of the variable input diminishes. Congestion has been an under-researched topic in economic theory especially when there is a need for augmenting inputs to serve important objectives besides output maximisation. We propose a fuzzy DEA model and represent the imprecise and ambiguous input and output data with fuzzy numbers. We solve the model with an α-cut approach and obtain the value of input congestion for the optimistic and pessimistic cases. The fundamental idea in this paper is to transform the fuzzy DEA model into a crisp linear programming model using the α-cut approach. Two auxiliary crisp models are solved to obtain optimistic and pessimistic values of congestion for evaluating the decision-making units (DMUs). We use a numerical example from the literature to demonstrate the applicability of the proposed method and exhibit the efficacy of the procedures and algorithms.

Estimating reliability of a repairable system with imperfect coverage and fuzzy parameters using parametric non-linear programming approach

In this paper, we consider the problem of evaluating system characteristics (MTTF, reliability) using Markov modeling approach, in which times to failure and times to repair of the operating units are, assumed to follow fuzzified exponential distribution. A method has been developed to construct a fuzzy set as an estimator for unknown parameters in the proposed statistical model. Using the α-cut approach the fuzzy repairable system is extracted from the conventional crisp intervals for the desired system characteristics, which are determined with a set of parametric nonlinear programs using their membership functions. With the proposed approach, explicit closed-form expressions of the system characteristics are obtained by inverting the interval limits of α-cuts of membership functions.

Single Model Approach To Evaluate Optimistic And Pessimistic Congestion In Fuzzy Data Envelopment Analysis

Data Envelopment Analysis (DEA) is an almost new data oriented approach for measuring the performance of series of homogenous entities named Decision Making Unit (DMU) with the ability to alter multiple inputs into multiple outputs. This article investigates the development of congestion model with regard to fuzzy input and output. Also, a new concept called as optimistic and pessimistic are introduced for congestion model. To create this concept, the upper and lower bound of efficiency, outputs and inputs are required. This concept is provided with α-cut approach in fuzzy data envelopment analysis (FDEA). In order to evaluate the efficiency of given models, data of some universities of Iran are applied. The results reveal the proper validity of proposed model.

A fuzzy DEA model with undesirable fuzzy outputs and its application to the banking sector in India

Data envelopment analysis (DEA) is a widely used technique for measuring the relative efficiencies of decision making units (DMUs) with multiple inputs and multiple outputs. However, in real life applications, undesirable outputs may be present in the production process which needs to be minimized. The present study endeavors to propose a DEA model with undesirable outputs and further to extend it in fuzzy environment in view of the fact that input/output data are not always available in exact form in real life problems. We propose a fuzzy DEA model with undesirable fuzzy outputs which can be solved as crisp linear program for each α in (0,1] using α-cut approach. Further, cross-efficiency technique is applied to increase the discrimination power of the proposed models and to rank the efficient DMUs at every α in (0,1]. Moreover, for better understanding of the proposed methodology, we present a numerical illustration followed by an application to the banking sector in India. This is the first study which attempts to measure the performance of public sector banks (PuSBs) in India using fuzzy input/output data for the period 2009–2011. The results obtained from the proposed methodology not only depict the impact of undesirable output on the performance of PuSBs but also analyze efficiently the influence of the presence of uncertainty in the data over the efficiency results. The findings show that the efficiency results of many PuSBs vary with the variation in α during the selected period.

Efficiency Measure by Fuzzy Data Envelopment Analysis Model

Data envelopment analysis (DEA) is a non-parametric technique to measure the relative efficiencies of a set of decision making units (DMUs) with common crisp inputs and outputs. Input and output data of DMUs often fluctuate. These fluctuating data can be represented as linguistic variable characterized by fuzzy numbers. This paper attempts to extend the traditional DEA model to a fuzzy framework, thus proposing a fuzzy DEA model based on α-cut approach to deal with the efficiency measuring and ranking problem with the given fuzzy input and output data. Finally, a numerical example is presented to illustrate the fuzzy DEA model. Since the efficiency measures are expressed by membership functions rather than by crisp values, more information is provided for management. By extending to fuzzy environment, the DEA approach is made more powerful for application.

Generalized fuzzy data envelopment analysis methods

The conventional data envelopment analysis (DEA) measures the relative efficiencies of a set of decision making units (DMUs) with exact values of inputs and outputs. In real-world problems, however, inputs and outputs typically have some levels of fuzziness. To analyze a DMU with fuzzy input/output data, previous studies provided the fuzzy DEA (FDEA) model and proposed an associated evaluating approach. Nonetheless, numerous deficiencies must be improved, including the α-cut approaches, types of fuzzy numbers, and ranking techniques. Moreover, a fuzzy sample DMU (SDMU) still cannot be evaluated for the FDEA model. Therefore, the present paper proposes a generalized FDEA model which can evaluate SDMU and the traditional FDEA model. Five evaluation methods are provided and these methods not only improve the types of FDEA model, the types of fuzzy number, the α-cut approach but also firstly propose a new evaluation method based on vector. At last related algorithm and ranking methods are provided to test our new methods. A numerical experiment is used to demonstrate and compare the results with those obtained using alternative approaches.

A genre-based fuzzy inference approach for effective filtering of movies

Recommendation techniques are utilized in electronic commerce because of their potential commercial value. Many e-commerce sites employ collaborative filtering techniques to provide recommendations to customers based on the preferences of similar users. However, as the number of customers and the range of products increase, the prediction accuracy of memory-based collaborative filtering algorithms declines because of sparse ratings. In addition, the time complexity of such algorithms is quite high in the prediction phase. To resolve these issues, we propose a genre-based fuzzy inference filtering approach for predicting movie preferences. We use content-based and collaborative filtering algorithms as baseline methods to evaluate the performance of our approach. The results of experiments demonstrate that the hybrid approach exploits the strengths of the content-based and collaborative filtering algorithms to achieve effective filtering in terms of precision. Moreover, the computation time can be reduced by using the α-cut approach. The findings have implications for the design of an interactive movie recommendation system for the World Wide Web.

A fuzzy bulk queue with modified Bernoulli vacation and restricted admissible customers

A single server fuzzy queue with modified Bernoulli vacation is analyzed using a technique which is a fusion of Zadeh's extension principle L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 1978, 3--28, α-cut approach and parametric non-linear programming. Batches of customers arrive at the system according to a compound Poisson process. But all arriving batches are not allowed to enter into the system. The restriction policy depends on availability or otherwise of the server. The system is analyzed in Fuzzy environment. Some special cases are discussed. A numerical study is also carried out.

A Nonlinear Programming Approach for a Fuzzy Queue with an Unreliable Server

The aim of this paper is to develop the membership functions of the system characteristic of a queuing g model with an unreliable server, in which the arrival rate, service rate, breakdown rate and repair rate are all fuzzy numbers. The α-cut approach is used to transform a fuzzy queue with an unreliable server into a family of conventional crisp queues with an unreliable server. By using membership functions, a set of parametric nonlinear programmes are developed to describe the family of crisp queues with an unreliable server. An efficient algorithm is developed to find the optimal solutions at or different possibility level α. Numerical examples are solved successfully. Since the system characteristics being expressed and governed by membership functions, more information is provided for the management.

Reliability-based measures for a system with an uncertain parameter environment

This article proposes a procedure to construct membership functions of the system characteristics of a repairable system with one primary machine, one standby machine and an unreliable service station. Failure and repair times of both the operating and standby machines are assumed to follow fuzzified exponential distributions. In addition, the failure and repair times of the service station also follow fuzzified exponential distributions. The α-cut approach is used to extract a family of conventional crisp intervals for the desired system characteristics from the fuzzy repairable system, determined with a set of parametric nonlinear programs using their membership functions. A numerical example successfully illustrates the practicality of the proposed approach. Because system characteristics are governed by the membership functions, more information is provided for use by designers and practitioners. The successful extension of the parameter spaces to fuzzy environments permits the repairable system to have wider practical applications.

Fuzzy data envelopment analysis: A discrete approach

Data envelopment analysis (DEA) as introduced by Charnes, Cooper, and Rhodes (1978) is a linear programming technique that has widely been used to evaluate the relative efficiency of a set of homogenous decision making units (DMUs). In many real applications, the input–output variables cannot be precisely measured. This is particularly important in assessing efficiency of DMUs using DEA, since the efficiency score of inefficient DMUs are very sensitive to possible data errors. Hence, several approaches have been proposed to deal with imprecise data. Perhaps the most popular fuzzy DEA model is based on α-cut. One drawback of the α-cut approach is that it cannot include all information about uncertainty. This paper aims to introduce an alternative linear programming model that can include some uncertainty information from the intervals within the α-cut approach. We introduce the concept of “local α-level” to develop a multi-objective linear programming to measure the efficiency of DMUs under uncertainty. An example is given to illustrate the use of this method.

Stochastic time–cost optimization using non-dominated archiving ant colony approach

Abstract This article employs a non-dominated archiving ant colony approach to solve the stochastic time–cost trade-off optimization problem. The model searches for non-dominated solutions considering total duration and total cost of the project as two objectives. In order to expect more realistic outcomes for the time–cost trade-off problem, uncertainties in time and cost of the project should be taken into account. Fuzzy sets theory is used to answer for uncertainties in time and cost of the project. The model embeds the α-cut approach to account for accepted risk level of the project manager. Left and right dominance ranking method is used for finding non-dominated solutions. The ranking method employs decision maker's optimism using β concept. The performance of the model is tested according to performance metrics for multi-objective evolutionary algorithms proposed in the literature. The results show that the algorithm is adequately reliable. A case study is solved to show the application of the proposed model for the uncertain time–cost trade-off problem.