The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making

QUALIFLEX, a generalization of Jacquet-Lagreze’s permutation method, is a useful outranking method in decision analysis because of its flexibility with respect to cardinal and ordinal information. This paper develops an extended QUALIFLEX method for handling multiple criteria decision-making problems in the context of interval type-2 fuzzy sets. Interval type-2 fuzzy sets contain membership values that are crisp intervals, which are the most widely used of the higher order fuzzy sets because of their relative simplicity. Using the linguistic rating system converted into interval type-2 trapezoidal fuzzy numbers, the extended QUALIFLEX method investigates all possible permutations of the alternatives with respect to the level of concordance of the complete preference order. Based on a signed distance-based approach, this paper proposes the concordance/discordance index, the weighted concordance/discordance index, and the comprehensive concordance/discordance index as evaluative criteria of the chosen hypothesis for ranking the alternatives. The feasibility and applicability of the proposed methods are illustrated by a medical decision-making problem concerning acute inflammatory demyelinating disease, and a comparative analysis with another outranking approach is conducted to validate the effectiveness of the proposed methodology.     

Least-squares estimates in fuzzy regression analysis

Regression is a very powerful methodology for forecasting, which is considered as an essential component of successful OR applications. In this paper an idea stemmed from the classical least squares is proposed to handle fuzzy observations in regression analysis. Based on the extension principle, the membership function of the sum of squared errors is constructed. The fuzzy sum of squared errors is a function of the regression coefficients to be determined, which can be minimized via a nonlinear program formulated under the structure of the Chen–Klein method for ranking fuzzy numbers. To illustrate how the proposed method is applied, three cases, one crisp input-fuzzy output, one fuzzy input-fuzzy output, and one non-triangular fuzzy observations, are exemplified. The results show that the least-squares method of this paper is able to determine the regression coefficients with better explanatory power. Most important, it works for all types of fuzzy observations, not restricted to the triangular one.     

Time-cost trade-off analysis of project networks in fuzzy environments

This paper proposes a novel approach for time-cost trade-off analysis of a project network in fuzzy environments. Different from the results of previous studies, in this paper the membership function of the fuzzy minimum total crash cost is constructed based on Zadeh’s extension principle and fuzzy solutions are provided. A pair of two-level mathematical programs parameterized by possibility level α is formulated to calculate the lower and upper bounds of the fuzzy minimum total crash cost at α. By enumerating different values of α, the membership function of the fuzzy minimum total crash cost is constructed, and the corresponding optimal activity time for each activity is also obtained at the same time. An example of time-cost trade-off problem with several fuzzy parameters is solved successfully to demonstrate the validity of the proposed approach. Since the minimum total crash cost is expressed by a membership function rather than by a crisp value, the fuzziness of parameters is conserved completely, and more information is provided for time-cost trade-off analysis in project management. The proposed approach also can be applied to time-cost trade-off problems with other characteristics.     

Parametric nonlinear programming approach to fuzzy queues with bulk service

This paper proposes a procedure to construct the membership functions of the performance measures in bulk service queuing systems with the arrival rate and service rate are fuzzy numbers. The basic idea is to transform a fuzzy queue with bulk service to a family of conventional crisp queues with bulk service by applying the α-cut approach. On the basis of α-cut representation and the extension principle, a pair of parametric nonlinear programs is formulated to describe that family of crisp bulk service queues, via which the membership functions of the performance measures are derived. To demonstrate the validity of the proposed procedure, two fuzzy queues often encountered in transportation management are exemplified. Since the performance measures are expressed by membership functions rather than by crisp values, they completely conserve the fuzziness of input information when some data of bulk-service queuing systems are ambiguous. Thus the proposed approach for vague systems can represent the system more accurately, and more information is provided for designing queuing systems in real life. By extending to fuzzy environment, the bulk service queuing models would have wider applications.     

The diameter of a fuzzy set

In pattern recognition one often wants to measure geometric properties of imprecisely defined subsets of an image. This paper proposes definitions of intrinsic and extrinsic diameter for fuzzy subsets which reduce to the ordinary definitions when the subsets are crisp. We also define height and width for a fuzzy subset and show how they relate to the area (i.e., integral of membership). For convex fuzzy subsets the intrinsic diameter cannot exceed the extrinsic diameter, but it can be smaller. Finally, for piecewise constant convex fuzzy subsets the intrinsic diameter cannot exceed half the fuzzy perimeter, but this need not be true in the nonconvex case.     

Measuring performances of multiple-channel queueing systems with imprecise data: a membership function approach

This paper proposes a mixed integer nonlinear programming (MINLP) approach to measure the system performances of multiple-channel queueing models with imprecise data. The main idea is to transform a multiple-channel queue with imprecise data to a family of conventional crisp multiple-channel queues by applying the α-cut approach in fuzzy theory. On the basis of α-cut representation and the extension principle, two pairs of parametric MINLP are formulated to describe the family of crisp multiple-channel queues, via which the membership functions of the performance measures are derived. To demonstrate the validity of the proposed procedure, a real-world case of multiple-channel fuzzy queue is investigated successfully. Since the performance measures are expressed by membership functions rather than by crisp values, the fuzziness of input information is completed conserved. Thus, the results obtained from the proposed approach can represent the system more accurately, and more information is provided for system design in practice.     

Large sample properties of estimates of a discrete grade of membership model

Increasingly, fuzzy partitions are being used in multivariate classification problems as an alternative to the crisp classification procedures commonly used. One such fuzzy partition, the grade of membership model, partitions individuals into fuzzy sets using multivariate categorical data. Although the statistical methods used to estimate fuzzy membership for this model are based on maximum likelihood methods, large sample properties of the estimation procedure are problematic for two reasons. First, the number of incidental parameters increases with the size of the sample. Second, estimated parameters fall on the boundary of the parameter space with non-zero probability. This paper examines the consistency of the likelihood approach when estimating the components of a particular probability model that gives rise to a fuzzy partition. The results of the consistency proof are used to determine the large sample distribution of the estimates. Common methods of classifying individuals based on multivariate observations attempt to place each individual into crisply defined sets. The fuzzy partition allows for individual to individual heterogeneity, beyond simply errors in measurement, by defining a set of pure type characteristics and determining each individual's distance from these pure types. Both the profiles of the pure types and the heterogeneity of the individuals must be estimated from data. These estimates empirically define the fuzzy partition. In the current paper, this data is assumed to be categorical data. Because of the large number of parameters to be estimated and the limitations of categorical data, one may be concerned about whether or not the fuzzy partition can be estimated consistently. This paper shows that if heterogeneity is measured with respect to a fixed number of moments of the grade of membership scores of each individual, the estimated fuzzy partition is consistent.     

Topological relations between fuzzy regions: derivation of verbal terms

A method is presented to generate verbal terms about topological relations between fuzzy regions. The methodology relies on the fuzzy 4-intersection, which is a generalization of the crisp 4-intersection of Egenhofer and co-workers. The computation of the similarity between the fuzzy- and the crisp 4-intersection enables the verbal term, i.e., the linguistic variable, to be derived. The linguistic variable contains a semantic part which gives an immediate association to a crisp relation and a quantifier which indicates the strength of the relationship. Since the derivation of the linguistic variable depends on the definition of the boundary of the fuzzy regions, a method is presented to compute fuzzy boundaries. The approach here defines fuzzy boundaries so that each point in the fuzzy region is associated a partial membership in both the interior and the boundary of the region. This view is different from the boundary definition in crisp topology, but it agrees with the fuzzy set idea that elements can have partial membership in different sets. A simulation experiment demonstrates the properties of the proposed methodology, and it shows how the linguistic variable relates to an inclusion index. An example illustrates how some level of action can be associated to the linguistic variable, which is applicable in the course control of moving crafts, in military applications or in other kinds of operations where the level of warning or action depends on the topological relation between the fuzzy regions.The findings in this article are applicable to geographical information systems (GIS), the modelling of objects with indeterminate boundaries, in the reasoning about relations between geographical objects, or the evaluation of database queries. If the ideas in the present article are implemented in GIS, this will provide an enhanced user interface compared to most GIS today.     

Solving fuzzy queueing decision problems via a parametric mixed integer nonlinear programming method

This paper proposes a mathematical programming method to construct the membership functions of the fuzzy objective value of the cost-based queueing decision problem with the cost coefficients and the arrival rate being fuzzy numbers. On the basis of Zadeh’s extension principle, three pairs of mixed integer nonlinear programs (MINLP) parameterized by the possibility level α are formulated to calculate the lower and upper bounds of the minimal expected total cost per unit time at α, through which the membership function of the minimal expected total cost per unit time of the fuzzy objective value is constructed. To provide a suitable optimal service rate for designing queueing systems, the Yager’s ranking index method is adopted. Two numerical examples are solved successfully to demonstrate the validity of the proposed method. Since the objective value is completely expressed by a membership function rather than by a crisp value, it conserves the fuzziness of the input information, thus more information is provided for designing queueing systems. The successful extension of queueing decision models to fuzzy environments permits queueing decision models to have wider applications in practice.     

带模糊参数的系统模糊可靠度分析

含模糊参数系统的可靠性理论研究具有广泛的实际应用背景,但由于模糊数运算的隶属函数表达困难,影响和制约着模糊参数系统的模糊可靠性理论与应用的研究。本文利用模糊数的结构元表示,给出了模糊表达式隶属函数确定的两种方法,进而得到了具有模糊参数的不可修复串联和并联系统模糊可靠度的隶属函数表达式。     

灰色关联分析方法在机场风险评价中的应用

本文通过对机场风险因素的分析,建立了一套合理的机场风险评价指标体系,并运用模糊层次分析法确定各评价指标的权重,然后结合灰色关联分析,建立了机场风险评价模型.最后将此方法运用到具体的实例中,说明了该方法的有效性.     

An interactive fuzzy programming system

An interactive system is introduced which supports a decision maker in solving programming models with crisp or fuzzy constraints and crisp or fuzzy goals. One part of the system is the determination of membership functions representing goals. To this purpose fuzzy extreme solutions are computed and are presented to the decision maker. These and each of the proposed compromise solutions are fuzzy-efficient.     

基于贝塞尔曲线理论的备件需求模糊隶属度函数构建模型

用模糊理论描述备件需求是一种科学适用的方法,针对现有模糊变量隶属度函数构建方法的不足, 设计了基于贝塞尔曲线理论的备件需求模糊隶属度函数构建方法,给出了隶属度求解算法,分析了使拟合误差最小的控制点选择方法.同时通过实例验证以及与最小二乘法的对比分析,验证了贝塞尔曲线方法在构建备件需求模糊隶属度函数方面的有效性.此方法无需事先假设隶属度函数的形态,简单易用、使用灵活并且精度较高.     

一类大系统模糊随机多目标决策的建模与求解方法

本文提出和建立了一类大系统模糊随机多目标决策（ＦＳＭＯＤＭ）模型，讨论了模糊事件的模糊概率的若干基本性质．在此基础上，将ＦＳＭＯＤＭ模型转化为清晰的多目标决策模型，并给出了递阶分解协调求解方法．把该模型应用到大连市水资源与经济发展规划研究中取得了令人满意的结果．     

模糊向量真值修正模糊化方法

模糊化是将物理空间的实测精确量影射为模糊推理空间上的模糊集合。在模糊控制中，不同的模糊推理方法要求不同的模糊化方法，不同的模糊化方法对模糊控制性能影响很大。本文首先系统地总结了现有的模糊化方法，然后提出了模糊向量真值修正模糊化方法，最后，针对常用的CRI法，完成了不同模糊化方法的一阶惯性时滞定常系统的模糊控制仿真，结果表明，该方法能够提高CRI法的模糊控制性能，消除稳态误差。     

基于模糊一致矩阵的模糊层次分析法的排序

分析基于模糊一致矩阵的模糊层次分析法 (FAHP) ,改正文 [4 ]中存在的两个将会误导决策的错误 ,给出并证明模糊一致矩阵的一些重要性质 ,获得模糊互补判断矩阵下的一个简明排序计算公式 ,完善模糊层次分析法的排序原理。     

Analysis of critical paths in a project network with fuzzy activity times

This paper proposes an approach to critical path analysis for a project network with activity times being fuzzy numbers, in that the membership function of the fuzzy total duration time is constructed. The basic idea is based on the extension principle and linear programming formulation. A pair of linear programs parameterized by possibility level α is formulated to calculate the lower and upper bounds of the fuzzy total duration time at α. By enumerating different values of α, the membership function of the fuzzy total duration time is constructed, and the fuzzy critical paths are identified at the same time. Moreover, by applying the Yager ranking method, definitions of the most critical path and the relative degree of criticality of paths are developed; and these definitions are theoretically sound and easy to use in practice. Two examples with activity times being fuzzy numbers of L-R and L-L types discussed in previous studies are solved successfully to demonstrate the validity of the proposed approach. Since the total duration time is completely expressed by a membership function rather than by a crisp value, the fuzziness of activity times is conserved completely, and more information is provided for critical path analysis.     

A fuzzy MCDM approach for stock selection

A fuzzy MCDM approach is applied to the stock selection problem, where the proposed approach can deal with qualitative information in addition to quantitative information. A hierarchy of major–sub criteria is then established to reduce the dependence between criteria. The ratings of alternatives versus qualitative sub-criteria and the weights of major- and sub-criteria are assessed in linguistic terms represented by fuzzy numbers. Each sub-criterion is in a benefit, cost, or balanced nature. New standardization methods for fuzzy numbers in the cost and balanced nature are presented. The algorithms of membership functions of the final aggregation are completely developed instead of approximation. The final aggregations in fuzzy numbers are then defuzzified to crisp values in order to rank the performance of alternatives. Moreover, the ratio of market price to performance (PP) is suggested to filter the over/under-pricing of alternatives. A set of buying/selling strategies are recommended according to the performance and PP. An empirical example then demonstrates the processing of the proposed approach.     

A redundant repairable system with imperfect coverage and fuzzy parameters

This paper proposes a procedure to construct the membership functions of the system characteristics of a redundant repairable system with two primary units and one standby in which the coverage factor is the same for an operating unit failure as that for a standby unit failure. Times to failure and times to repair of the operating and standby units are assumed to follow fuzzified exponential distributions. The α-cut approach is used to extract from the fuzzy repairable system a family of conventional crisp intervals for the desired system characteristics, determined with a set of parametric nonlinear programs using their membership functions. A numerical example is solved successfully to illustrate the practicality of the proposed approach. Because the system characteristics are governed by the membership functions, more information is provided for use by management, and because the redundant system is extended to the fuzzy environment, general repairable systems are represented more accurately and the analytic results are more useful for designers and practitioners.