An analysis of distributions of priority values from alternative comparison scales within AHP

The analytic hierarchy process (AHP) introduced by T.L. Saaty is a well known and popular method of multi-criteria decision making. Central to this method are the pairwise comparisons between criteria (and decision alternatives) made using a 9-unit scale. The appropriateness of Saaty's original one-to-nine (1–9) scale has been the subject of much debate and cause for concern. This paper contrasts the appropriateness of the 1–9 scale with other alternative 9-unit scales also used in AHP, by looking at the probability distributions of the associated priority values. For large problems, estimated probability distributions are found for the priority values through using the method of Parzen Windows.     

Ordinal judgments in multiattribute decision analysis

The article discusses the contradiction between the ambiguity of human judgment in a multicriterion environment and the exactness of the assessments required in the majority of the decision-making methods. Preferential information from the decision makers in the ordinal form (e.g., “more preferable”, “less preferable”, etc.) is argued to be more stable and more reliable than cardinal input. Ways of obtaining and using ordinal judgments for rank ordering of multiattribute alternatives are discussed. The effectiveness of the step-wise procedure of using ordinal tradeoffs for comparison of alternatives is evaluated. We introduce the notion of ordinal tradeoffs, presentation of ordinal tradeoffs as a flexible three-stage process, a paired joint ordinal scale (PJOS), and evaluation of the effectiveness of the three-stage process. Simulation results examine the sensitivity of the number of pairwise comparisons required for given numbers of criteria and categories within criteria, as well as the number of alternatives analyzed. This simulation shows that ordinal pairwise comparisons provide sufficient power to discriminate between 75% and 80% of the alternatives compared. While the proportional number of pairwise comparisons relative to the maximum possible decreases with the number of criteria and categories, the method is relatively insensitive to the number of alternatives considered.     

A method for stochastic multiple criteria decision making based on pairwise comparisons of alternatives with random evaluations

This paper proposes a method for solving stochastic multiple criteria decision making (MCDM) problems, where evaluations of alternatives on considered criteria are random variables with known probability density functions or probability mass functions. Probabilities on all possible results of pairwise comparisons of alternatives are first calculated using Probability Theory. Then, all possible results of pairwise comparisons are classified into superior, indifferent and inferior ones using a predefined identification rule. Consequently, the probabilities on all possible results of pairwise comparisons are partitioned into superior, indifferent and inferior probabilities. Furthermore, based on the derived probabilities, an algorithm is developed to rank the alternatives. Finally, a numerical example is used to illustrate the feasibility and validity of the proposed method.     

Modeling subjective evaluation for fuzzy group multicriteria decision making

This paper presents a new fuzzy multicriteria decision making (MCDM) approach for evaluating decision alternatives involving subjective judgements made by a group of decision makers. A pairwise comparison process is used to help individual decision makers make comparative judgements, and a linguistic rating method is used for making absolute judgements. A hierarchical weighting method is developed to assess the weights of a large number of evaluation criteria by pairwise comparisons. To reflect the inherent imprecision of subjective judgements, individual assessments are aggregated as a group assessment using triangular fuzzy numbers. To obtain a cardinal preference value for each decision alternative, a new fuzzy MCDM algorithm is developed by extending the concept of the degree of optimality to incorporate criteria weights in the distance measurement. An empirical study of aircraft selection is presented to illustrate the effectiveness of the approach.     

Multi-criteria decision making with fuzzy linguistic preference relations

Although the analytic hierarchy process (AHP) and the extent analysis method (EAM) of fuzzy AHP are extensively adopted in diverse fields, inconsistency increases as hierarchies of criteria or alternatives increase because AHP and EAM require rather complicated pairwise comparisons amongst elements (attributes or alternatives). Additionally, decision makers normally find that assigning linguistic variables to judgments is simpler and more intuitive than to fixed value judgments. Hence, Wang and Chen proposed fuzzy linguistic preference relations (Fuzzy LinPreRa) to address the above problem. This study adopts Fuzzy LinPreRa to re-examine three numerical examples. The re-examination is intended to compare our results with those obtained in earlier works and to demonstrate the advantages of Fuzzy LinPreRa. This study demonstrates that, in addition to reducing the number of pairwise comparisons, Fuzzy LinPreRa also increases decision making efficiency and accuracy.     

Pseudo-criteria versus linear utility function in stochastic multi-criteria acceptability analysis

Stochastic multi-criteria acceptability analysis (SMAA) is a multi-criteria decision support method for multiple decision-makers (DMs) in discrete problems. SMAA does not require explicit or implicit preference information from the DMs. Instead, the method is based on exploring the weight space in order to describe the valuations that would make each alternative the preferred one. Partial preference information can be represented in the weight space analysis through weight distributions. In this paper we compare two variants of the SMAA method using randomly generated test problems with 2–12 criteria and 4–12 alternatives. In the original SMAA, a utility or value function models the DMs' preference structure, and the inaccuracy or uncertainty of the criteria is represented by probability distributions. In SMAA-3, ELECTRE III-type pseudo-criteria are used instead. Both methods compute for each alternative an acceptability index measuring the variety of different valuations that supports this alternative, and a central weight vector representing the typical valuations resulting in this decision. We seek answers to three questions: (1) how similar are the results provided by the decision models, (2) what kind of systematic differences exists between the models, and (3) how could one select indifference and preference thresholds of the pseudo-criteria model to match a utility model with given probability distributions?     

一种基于证据推理的信息不完全的多准则决策方法

针对权系数信息不完全、准则值不确定且不完全的多准则决策问题,提出了一种基于证据推理的方法.该方法通过证据推理算法构造方案的目标函数,结合不完全信息的权系数建立非线性规划模型,使用遗传算法求解模型得到效用值的区间数,从而得到整个方案集的排序.最后以实例表明该方法的有效性和可行性.     

Convergence properties and practical estimation of the probability of rank reversal in pairwise comparisons for multi-criteria decision making problems

In this paper, we address the impact of uncertainty introduced when the experts complete pairwise comparison matrices, in the context of multi-criteria decision making. We first discuss how uncertainty can be quantified and modeled and then show how the probability of rank reversal scales with the number of experts. We consider the impact of various aspects which may affect the estimation of probability of rank reversal in the context of pairwise comparisons, such as the uncertainty level, alternative preference scales and different weight estimation methods. We also consider the case where the comparisons are carried out in a fuzzy manner. It is shown that in most circumstances, augmenting the size of the expert group beyond 15 produces a small change in the probability of rank reversal. We next address the issue of how this probability can be estimated in practice, from information gathered simply from the comparison matrices of a single expert group. We propose and validate a scheme which yields an estimate for the probability of rank reversal and test the applicability of this scheme under various conditions. The framework discussed in the paper can allow decision makers to correctly choose the number of experts participating in a pairwise comparison and obtain an estimate of the credibility of the outcome.     

Comparing non-structural fuzzy decision support system and analytical hierarchy process in decision-making for construction problems

Analytical Hierarchy Process (AHP) is a commonly used decision-aiding tool for resolving multi-criteria decision problems. However, users sometimes find it difficult to ensure a consistent pairwise comparison between voluminous decisions. The cause of which is that the Consistency Ratio (CR) is produced after the evaluation process and its global acceptance criteria is limited. When the derived ratio reports some inconsistency, it requires a long process to locate and rectify the problem. The major aim of this study is to look for an alternative decision-aiding tool to AHP, helping to avoid the above problem. The alternative approach proposed in this study is the Non-Structural Fuzzy Decision Support System (NSFDSS). The application of the system is illustrated with a worked example. The results generated by NSFDSS are compared against those generated by the conventional AHP that shows the effectiveness and some unique advantages of the proposed tool over AHP.     

Probabilistic judgments specified partially in the analytic hierarchy process

The Analytic Hierarchy Process (AHP) is a decision-making tool which yields priorities for decision alternatives. This paper proposes a new approach to elicit and synthesize expert assessments for the group decision process in the AHP. These new elicitations are given as partial probabilistic specifications of the entries of pairwise comparisons matrices. For a particular entry of the matrix, the partial probabilistic elicitations could arise in the form of either probability assignments regarding the chance of that entry falling in specified intervals or selected quantiles for that entry. A new class of models is introduced to provide methods for processing this partial probabilistic information. One advantage of this approach is that it allows to generate as many pairwise comparison matrices of the decision alternatives as one desires. This, in turn, allows us to determine the statistical significance of the priorities of decision alternatives.     

A method for aggregating ordinal assessments by a majority decision rule

A new method for aggregating ordinal assessments of individuals is proposed to obtain a collective preference ordering of alternatives. The method converts the ordinal assessments to collective pairwise comparisons of the alternatives and incorporates a deduction process to yield transitively consistent pairwise comparisons with respect to majority principle, and hence yields a linear order of the alternatives. Its algorithm is presented and its properties are studied. The method is applied to an opinion survey concerning quality of life, and a collective ordering of items pertaining to the quality of life is obtained with respect to their importance.     

A geometric averaging procedure for constructing supertransitive approximation to binary comparison matrices

The usefulness of encoding the fuzzy evaluations of alternatives and the importance weights of criteria, in a multiple objective decision problem through binary comparison matrices (or pairwise judgment matrices) is receiving considerable attention. The methodology for identifying the best alternative in a given decision problem involves the computation of the principal eigenvectors of the binary comparison matrices. The eigenvectors transform the fuzzy evaluations of the importance of the criteria and the ratings of the alternatives into a ratio scale. A difficulty that is often experienced in using this approach in practice, is the inconsistency of the binary evaluations. This paper proposes a simple averaging procedure to construct a supertransitive approximation to a binary comparison matrix, where inconsistency is a problem. It is further suggested that such an adjustment might be necessary to more closely reflect the inherent fuzziness of the evaluations contained in a binary comparison matrix. The procedure is illustrated by means of examples.     

A fuzzy ranking method with range reduction techniques

For ranking alternatives based on pairwise comparisons, current analytic hierarchy process (AHP) methods are difficult to use to generate useful information to assist decision makers in specifying their preferences. This study proposes a novel method incorporating fuzzy preferences and range reduction techniques. Modified from the concept of data envelopment analysis (DEA), the proposed approach is not only capable of treating incomplete preference matrices but also provides reasonable ranges to help decision makers to rank decision alternatives confidently.     

Numerical scaling of ratio scale utilities in multi-criteria decision analysis with geometric model

The ratio scale measurement of utilities in multi-criteria decision analysis has been criticised due to theoretical and practical interpretation of the required assessments. Compared to the ratio scale models, for example, the ‘0–1-scale’ models based on interval scale have been more generally accepted. On the other hand, under some conditions it has been shown that the results of multi-criteria decision support based on ratio scale comparisons are actually independent of the numerical magnitude of the ratio scale assessments. This paper reviews and interprets these results and proposes new solutions for the remaining problems, for which the numerical magnitude of the ratio scale utilities is required. Moreover, we interpret the scale-independent ratio scale models with respect to some other multi-criteria decision support models. The theoretical analysis is carried out by statistical models and illustrated by numerical examples.     

Using tropical optimization techniques in bi-criteria decision problems

Computational Management Science - We consider decision problems of rating alternatives based on their pairwise comparisons according to two criteria. Given pairwise comparison matrices for each...     

基于新精确函数的区间直觉模糊多属性决策方法

基于区间直觉模糊数隶属度和非隶属度构成的二维几何图形特征给出区间直觉模糊数精确函数的新定义,并将其作为区间直觉模糊数的排序指标,区间直觉模糊数的精确函数值越大,则区间直觉模糊数就越大,进而提出一种权重信息不完全确定的区间直觉模糊多属性决策方法.通过算例分析说明所提出排序指标的有效性和决策方法的可行性.     

Incremental analysis for MCDM with an application to group TOPSIS

The study aims to exploit incremental analysis or marginal analysis to overcome the drawbacks of ratio scales utilized in various multi-criteria or multi-attribute decision making (MCDM/MADM) techniques. In the proposed 11-step procedure, multiple criteria of alternatives are first reorganized as two categories – benefits and costs – and decision information will be manipulated separately. The performances of alternatives are then evaluated on their incremental benefit–cost ratio, and the rank can be obtained by applying the group TOPSIS (technique for order preference by similarity to ideal solution) model (Shih et al., 2007). Two representations of cost, i.e., a cost index and utility index, are proposed in the model to better-fit real-world situations. In addition, some considerations on costs and input–output relations are also discussed in order to understand the essentials of incremental analysis. In the final part, a case of robot selection demonstrates the suggested model to be both robust and efficient in a group decision-making environment.     

Inconsistency indices for pairwise comparison matrices: a numerical study

Evaluating the level of inconsistency of pairwise comparisons is often a crucial step in multi criteria decision analysis. Several inconsistency indices have been proposed in the literature to estimate the deviation of expert’s judgments from a situation of full consistency. This paper surveys and analyzes ten indices from the numerical point of view. Specifically, we investigate degrees of agreement between them to check how similar they are. Results show a wide range of behaviors, ranging from very strong to very weak degrees of agreement.     

Pruning of decision alternatives in multiple criteria decision making,based on the Uta method for estimating utilities

A new interactive method for the progressive elimination of elements from a finite set of decision alternatives is proposed. A sequence of alternatives is presented to the decision maker, who places each new alternative presented in rank order relative to the earlier alternatives evaluated. This ranking of elements in a subset of the decision space is used to eliminate other alternatives from further consideration. The approach is broadly based on the UTA method for utility assessment. The emphasis is not, however, on assessing a utility function as such; instead the class of utility functions is used to eliminate elements of the decision space, when the optimality of such elements is inconsistent with the presumed properties of the utility function and the rank orderings given. In this way, the decision maker need only evaluate a relatively small subset of the decision space, before all remaining alternatives are eliminated. The new procedure is illustrated by two numerical examples.