Solving multi-criteria decision making with incomplete linguistic preference relations

When there are n criteria or alternatives in a decision matrix, a pairwise comparison methodology of analytic hierarchy process (AHP) with the time of n(n ? 1)/2 is frequently used to select, evaluate or rank the neighboring alternatives. But while the number of criteria or comparison level increase, the efficiency and consistency of a decision matrix decrease. To solve such problems, this study therefore uses horizontal, vertical and oblique pairwise comparisons algorithm to construct multi-criteria decision making with incomplete linguistic preference relations model (InLinPreRa). The use of pairwise comparisons will not produce the inconsistency, even allows every decision maker to choose an explicit criterion or alternative for index unrestrictedly. When there are n criteria, only n ? 1 pairwise comparisons need to be carried out, then one can rest on incomplete linguistic preference relations to obtain the priority value of alternative for the decision maker’s reference. The decision making assessment model that constructed by this study can be extensively applied to every field of decision science and serves as the reference basis for the future research.     

Multi-class classification using hybrid soft decision model for agriculture crop selection

A hybrid soft decision model has been developed in this paper to take decision on agriculture crop that can be cultivated in a given experimental land by integrating few soft computing techniques. The proposed model comprises of three parts, namely weight calculation, classification and prediction. Twenty-seven input criteria were categorized into seven broad criteria, namely soil (11 sub-criteria), water (2 sub-criteria), season (no sub-criterion), input (6 sub-criteria), support (2 sub-criteria), facilities (3 sub-criteria) and risk (2 sub-criteria). In the proposed model, relative weights of main criteria were calculated using Shannon’s Entropy method and relative weights of sub-criteria in each main criterion were calculated using rough set approach. As VIKOR method is effective in sorting the alternatives, it is used to determine the ranking index of main criteria in this study. A soft decision system was constructed from the results of rough set method, VIKOR method and Shannon’s Entropy method. Classification rules were generated for five agriculture crops, namely paddy, groundnut, sugarcane, cumbu and ragi based on the soft decision system using bijective soft set approach. The developed model predicts each site in the validation dataset into one of the five crops. The performance of the proposed model has been sanity checked by agriculture experts.

     

An intuitionistic fuzzy group decision making method using entropy and association coefficient

Group decision making is a process in which experts rank and choose the most desirable alternatives based on some accepted criteria. The aim of this paper was to introduce a method to solve group decision making problems with Atanassov’s intuitionistic fuzzy sets. First, the weight of each criterion is calculated using intuitionistic fuzzy entropy. Then, the total criteria weight vector is calculated by aggregating the calculated weights. Using the obtained weight vector, the alternatives are ranked based on the association coefficient of the performance of alternatives related to each criterion and the positive ideal intuitionistic fuzzy set value and the negative ideal intuitionistic fuzzy set value. Finally, to show the application of the proposed method, it is implemented in software vendor selection.     

A family of dominance rules for multiattribute decision making under uncertainty

Multiattribute decision-making involves choosing from a set of alternatives each of which is evaluated along multiple criteria that reflect the dimensions of interest to the goals and values of the decision-maker. Dominance-based decision-making narrows down the focus of the decision to the Pareto optimal set. The elimination of dominated alternatives is a compelling principle of rationality since each dominated alternative is logically inferior to its dominating alternative, given the criteria of evaluation. One kind of uncertainty in multiattribute decision making arises out of noisy or inaccurate criteria evaluations. The application of the principle of dominance is not quite rational if the criteria evaluations are known to be noisy. In this paper, we see how dominance-based decision-making can be applied to multiattribute decision-making problems with uncertainty due to noisy criteria values. In particular it will be shown that, for bounded uncertainty it is possible to produce the smallest sufficient subset that is guaranteed to contain all of the nondominated alternatives, and the largest necessary subset that contains only nondominated alternatives. For unbounded uncertainty, we will see how these notions of sufficiency and necessity can be adapted to varying degrees of probabilistic assurances desired by the decision-maker, and that the varying degrees of user assurance map naturally to a family of dominance rules.     

A DES-based group decision model for group decision making with large-scale alternatives

To solve group decision making problems with large-scale alternatives, this paper proposes a dynamic ensemble selection (DES) based group decision model by using historical decision data. The historical decision data of a group of experts are collected from the same multi-criteria decision framework and are mixed to train a set of base classifiers (BCs) to learn group preferences. For each new alternative, the predictions derived from BCs are used to determine its similar historical alternatives from historical data, and the BC with the highest accuracy in predicting the similar historical alternatives is identified as the best individual BC for the new alternative. By iteratively comparing the accuracy of an ensemble of randomly selected BCs and the best individual BC in predicting the similar historical alternatives of the new alternative, a novel DES method is developed to select a competent subset of BCs for the new alternative. The developed DES method effectively avoids the error-independence assumption to a certain extent. Based on the similar historical alternatives determined by the ensemble of selected BCs, a group decision optimization model is developed to learn criterion weights from their assessments on criteria and ensemble predictions derived from the selected BCs. With the learned criterion weights, the understandable group decision result is generated for the new alternative. Case study validates the superiority of the proposed model in diagnosing thyroid nodules using group capabilities. Empirical comparisons on thirty real datasets examine the competence of the proposed DES method against five representative DES methods.

     

一种信息不完全确定的多准则语言群决策方法

针对权系数信息不完全确定且方案的准则值为确定语言等级或位于两个连续语言等级之间,甚至缺失的群决策问题,提出一种群体语言决策方法．该方法利用证据推理算法得到方案属于各语言等级的信任度,利用二元语义对方案进行语言集结;然后结合决策者和准则权重的不完全确定信息及方案与理想方案的二元语义问的距离构建非线性规划模型．利用遗传算法求解所得模型,计算得到各方案的排序．实例计算表明了该方法的可行性和有效性．     

Improved method of multicriteria fuzzy decision-making based on vague sets

An improved method is presented, which provides improved score functions to measure the degree of suitability of each of a set of alternatives, with respect to a set of criteria presented with vague values. The improved algorithm for score functions is introduced by taking into account the effect of an unknown degree (hesitancy degree) of the vague values on the degree of suitability to which each alternative satisfies the decision-maker’s requirement. The meaning of the proposed function is more transparent than that of other existing functions, which are not reasonable in some cases. The proposed function illustrates that it has stronger discrimination in comparison with previous functions. The applicability of this improved multicriteria fuzzy decision-making approach is also demonstrated by means of examples. The improved method can be used to rank the decision alternatives according to the decision criteria. The functions proposed in this paper can provide a more useful technique than previous functions, in order to efficiently help the decision-maker.     

TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment

A single-valued neutrosophic set is a special case of neutrosophic set. It has been proposed as a generalization of crisp sets, fuzzy sets, and intuitionistic fuzzy sets in order to deal with incomplete information. In this paper, a new approach for multi-attribute group decision-making problems is proposed by extending the technique for order preference by similarity to ideal solution to single-valued neutrosophic environment. Ratings of alternative with respect to each attribute are considered as single-valued neutrosophic set that reflect the decision makers’ opinion based on the provided information. Neutrosophic set characterized by three independent degrees namely truth-membership degree (T), indeterminacy-membership degree (I), and falsity-membership degree (F) is more capable to catch up incomplete information. Single-valued neutrosophic set-based weighted averaging operator is used to aggregate all the individual decision maker’s opinion into one common opinion for rating the importance of criteria and alternatives. Finally, an illustrative example is provided in order to demonstrate its applicability and effectiveness of the proposed approach.     

Clustering and selection of multiple criteria alternatives using unsupervised and supervised neural networks

There are decision-making problems that involve grouping and selecting a set of alternatives. Traditional decision-making approaches treat different sets of alternatives with the same method of analysis and selection. In this paper, we propose clustering alternatives into different sets so that different methods of analysis, selection, and implementation for each set can be applied. We consider multiple criteria decision-making alternatives where the decision-maker is faced with several conflicting and non-commensurate objectives (or criteria). For example, consider buying a set of computers for a company that vary in terms of their functions, prices, and computing powers. In this paper, we develop theories and procedures for clustering and selecting discrete multiple criteria alternatives. The sets of alternatives clustered are mutually exclusive and are based on (1) similar features among alternatives, and (2) preferential structure of the decision-maker. The decision-making process can be broken down into three steps: (1) generating alternatives; (2) grouping or clustering alternatives based on similarity of their features; and (3) choosing one or more alternatives from each cluster of alternatives. We utilize unsupervised learning clustering artificial neural networks (ANN) with variable weights for clustering of alternatives, and we use feedforward ANN for the selection of the best alternatives for each cluster of alternatives. The decision-maker is interactively involved by comparing and contrasting alternatives within each group so that the best alternative can be selected from each group. For the learning mechanism of ANN, we proposed using a generalized Euclidean distance where by changing its coefficients new formation of clusters of alternatives can be achieved. The algorithm is interactive and the results are independent of the initial set-up information. Some examples and computational results are presented.     

Ranking multicritena fuzzy alternatives in the decision-maker's preference subspace

This paper offers a new procedure for ranking multicritena fuzzy alternatives when the decision-maker subscribes to the notion of ‘the larger, the better’. For each alternative a joint membership function captures possible interactions among ratings for each criterion. The ranking procedure first orthogonally projects the joint membership functions from the multicritena decision space to the one-dimensional preference subspace, and then the fuzzy projections are ranked in that subspace. A method for generating joint membership functions is introduced, and a numerical example is presented.     

基于前景理论的具有指标期望的多指标决策方法

针对决策者给出单一与组合指标期望情形的多指标决策问题, 提出一种基于前景理论的决策分析方法. 首先, 依据前景理论将决策者给出的指标期望视为参照点, 分别计算各方案针对各单一与各组合指标期望的前景价值,并构建各方案的综合前景价值向量; 然后, 将原始决策问题转化为相应的广义优序模糊约束满意问题(GPFCSP), 进而计算各方案针对相应推理准则的总体满意度, 并据此对各方案进行排序; 最后, 通过算例表明了该方法的可行性.     

Hierarchical semi-numeric method for pairwise fuzzy group decision making

Gradual improvements to a single-level semi-numeric method, i.e., linguistic labels preference representation by fuzzy sets computation for pairwise fuzzy group decision making are summarized. The method is extended to solve multiple criteria hierarchical structure pairwise fuzzy group decision-making problems. The problems are hierarchically structured into focus, criteria, and alternatives. Decision makers express their evaluations of criteria and alternatives based on each criterion by using linguistic labels. The labels are converted into and processed in triangular fuzzy numbers (TFNs). Evaluations of criteria yield relative criteria weights. Evaluations of the alternatives, based on each criterion, yield a degree of preference for each alternative or a degree of satisfaction for each preference value. By using a neat ordered weighted average (OWA) or a fuzzy weighted average operator, solutions obtained based on each criterion are aggregated into final solutions. The hierarchical semi-numeric method is suitable for solving a larger and more complex pairwise fuzzy group decision-making problem. The proposed method has been verified and applied to solve some real cases and is compared to Saaty's (1996) analytic hierarchy process (AHP) method.     

Multiple criteria classification with an application in water resources planning

A new kind of multiple criteria decision aid (MCDA) problem, multiple criteria classification (MCC), is studied in this paper. Traditional classification methods in MCDA focus on sorting alternatives into groups ordered by preference. MCC is the classification of alternatives into nominal groups, structured by the decision maker (DM), who specifies multiple characteristics for each group. Starting with illustrative examples, the features, definition and structures of MCC are presented, emphasizing criterion and alternative flexibility. Then an analysis procedure is proposed to solve MCC problems systematically. Assuming additive value functions, an optimization model with constraints that incorporate various classification strategies is constructed to solve MCC problems. An application of MCC in water resources planning is carried out and some future extensions are suggested.     

准则关联的直觉模糊多准则决策方法

针对准则值为直觉三角模糊数,准则间相互关联的多准则决策问题,提出基于Choquet分的决策方法．该方法首先利用偏好函数定义方案在各准则下的优序关系,若模糊测度已知,则直接利用Choquet积分进行求解;若准则集上的模糊测度未知,则利用部分决策信息和最小方差法建立二次规划模型,求解模糊测度,再利用Choquet分进行决策．最后通过实例表明了该方法的有效性和可行性．     

Clustering and group selection of multiple criteria alternatives with application to space-based networks

In many real-world problems, the range of consequences of different alternatives are considerably different. In addition, sometimes, selection of a group of alternatives (instead of only one best alternative) is necessary. Traditional decision making approaches treat the set of alternatives with the same method of analysis and selection. In this paper, we propose clustering alternatives into different groups so that different methods of analysis, selection, and implementation for each group can be applied. As an example, consider the selection of a group of functions (or tasks) to be processed by a group of processors. The set of tasks can be grouped according to their similar criteria, and hence, each cluster of tasks to be processed by a processor. The selection of the best alternative for each clustered group can be performed using existing methods; however, the process of selecting groups is different than the process of selecting alternatives within a group. We develop theories and procedures for clustering discrete multiple criteria alternatives. We also demonstrate how the set of alternatives is clustered into mutually exclusive groups based on 1) similar features among alternatives; 2) ideal (or most representative) alternatives given by the decision maker; and 3) other preferential information of the decision maker. The clustering of multiple criteria alternatives also has the following advantages. 1) It decreases the set of alternatives to be considered by the decision maker (for example, different decision makers are assigned to different groups of alternatives). 2) It decreases the number of criteria. 3) It may provide a different approach for analyzing multiple decision makers problems. Each decision maker may cluster alternatives differently, and hence, clustering of alternatives may provide a basis for negotiation. The developed approach is applicable for solving a class of telecommunication networks problems where a set of objects (such as routers, processors, or intelligent autonomous vehicles) are to be clustered into similar groups. Objects are clustered based on several criteria and the decision maker's preferences.     

基于Hurwicz的概率不确定的灰色随机多准则决策方法

首先定义了扩展灰数的可能度和距离公式;然后针对方案准则值为扩展灰数的不确定多准则决策问题,提出一种基于Hurwicz的概率不确定的灰色随机多准则决策方法。该方法通过使用扩展灰数的可能度和Hurwicz准则求得各方案在各准则下的评价值,经规范化后得到标准效用值决策矩阵;利用扩展灰数的距离和TODIM思想计算决策者对每个方案的损益感知价值及方案优势度,进而计算各方案总体感知价值大小以对方案进行排序。最后,通过算例验证了所提出方法的可行性和有效性。     

The analytic hierarchy process: can wash criteria be ignored?

We define a wash criterion as one where the decision-maker is indifferent among the alternatives when they are compared on that criterion. In view of the Belton–Gear example and other such anomalies associated with the analytic hierarchy process (AHP), we ask whether eliminating a wash criterion will affect the overall ranking of objects. In the case where there is only one level of criteria, the rank-order of objects is unaffected by leaving out a wash criterion. However, in the case where the wash criterion is a subcriterion, the rank order may be affected by leaving it out.Scope and purposeA wash criterion is defined as a criterion where the decision-maker is indifferent among the alternatives when they are compared on that criterion. We would like to think that the overall rank-order of objects would be unaffected in the case where the wash criterion is excluded. We give an example of an AHP hierarchy where this is not the case. In our view this presents another challenge to the AHP methodology.     

Establishing dominance and potential optimality in multi-criteria analysis with imprecise weight and value

The model presented in this paper does not require exact estimations of decision parameters such as attribute weights and values that may often be considerable cognitive burden of human decision makers. Information on the decision parameters is only assumed to be in the form of arbitrary linear inequalities which form constraints in the model. We consider two criteria, dominance and potential optimality, to check whether or not each alternative is outperform for a fixed feasible region denoted by the constraints. In particular, we develop a method to identify potential optimality of alternatives when all (or subsets) of the attribute values as well as weights are imprecisely know. This formulation becomes a nonlinear programming problem hard to be solved generally so that we provide in this paper how this problem is transformed into a linear programming equivalent.Scope and purposeMost managerial decisions involve choosing an optimal alternative from a number of available alternatives. Researchers have proposed a lot of methods to assist decision makers in choice making with a set of, usually conflicting, criteria or attributes. Many of these approaches require exact (or precise) information about either or both attribute values and/or trade-off weights. In some practice, however, it is not easy for decision makers to provide such exact data because, for example, intangible attributes to reflect social and environmental impacts may be included. To cope with such problem, a mathematical programming model-based approach to multi-criteria decision analysis is presented in this paper when both attribute weights and marginal values are imprecisely identified. A weighted additive rule is used to evaluate the performance of alternatives. We then show how to obtain non-dominated and potentially optimal alternatives in order to support choice making.     

A linguistic modeling of consensus in group decision making basedon OWA operators

In this paper, a model for group decision-making is proposed and defined in a linguistic context. A multiperson multicriteria decision problem is considered, in which a group of experts is involved in the evaluation of the performances of a set of alternatives with respect to a predefined set of criteria. The objective is to evaluate a consensual judgement and a consensus degree on each alternative. Both the experts' evaluations of the alternatives and the degree of consensus are expressed linguistically. A “soft” consensus degree referred to a fuzzy majority of the experts is proposed based on the concept of linguistic quantifier. The entire process is defined in a linguistic domain and modeled within fuzzy set theory by ordered weighted average (OWA) operators     

Finding the numerical compensation in multiple criteria decision-making problems under fuzzy environment

In this paper, we have developed a methodology to derive the level of compensation numerically in multiple criteria decision-making (MCDM) problems under fuzzy environment. The degree of compensation is dependent on the tranquility and anxiety level experienced by the decision-maker while taking the decision. Higher tranquility leads to the higher realisation of the compensation whereas the increased level of anxiety reduces the amount of compensation in the decision process. This work determines the level of tranquility (or anxiety) using the concept of fuzzy sets and its various level sets. The concepts of indexing of fuzzy numbers, the risk barriers and the tranquility level of the decision-maker are used to derive his/her risk prone or risk averse attitude of decision-maker in each criterion. The aggregation of the risk levels in each criterion gives us the amount of compensation in the entire MCDM problem. Inclusion of the compensation leads us to model the MCDM problem as binary integer programming problem (BIP). The solution to BIP gives us the compensatory decision to MCDM. The proposed methodology is illustrated through a numerical example.