An action learning approach for assessing the consistency of pairwise comparison data

Pairwise comparison data are used in various contexts including the generation of weight vectors for multiple criteria decision making problems. If this data is not sufficiently consistent, then the resulting weight vector cannot be considered to be a reliable reflection of the evaluator’s opinion. Hence, it is necessary to measure its level of inconsistency. Different approaches have been proposed to measuring the level of inconsistency, but they are often based on ‘rules of thumb” and/or randomly generated matrices, and are not interpretable. In this paper we present an action learning approach for assessing the consistency of the input pairwise comparison data that offer interpretable consistency measures.     

Ranking by pairwise comparisons for Swiss-system tournaments

Pairwise comparison matrices are widely used in multicriteria decision making. This article applies incomplete pairwise comparison matrices in the area of sport tournaments, namely proposing alternative rankings for the 2010 Chess Olympiad Open tournament. It is shown that results are robust regarding scaling technique. In order to compare different rankings, a distance function is introduced with the aim of taking into account the subjective nature of human perception. Analysis of the weight vectors implies that methods based on pairwise comparisons have common roots. Visualization of the results is provided by multidimensional scaling on the basis of the defined distance. The proposed rankings give in some cases intuitively better outcome than currently used lexicographical orders.     

Inferring consensus weights from pairwise comparison matrices without suitable properties

Pairwise comparison is a popular method for establishing the relative importance of n objects. Its main purpose is to get a set of weights (priority vector) associated with the objects. When the information gathered from the decision maker does not verify some rational properties, it is not easy to search the priority vector. Goal programming is a flexible tool for addressing this type of problem. In this paper, we focus on a group decision-making scenario. Thus, we analyze different methodologies for getting a collective priority vector. The first method is to aggregate general pairwise comparison matrices (i.e., matrices without suitable properties) and then get the priority vector from the consensus matrix. The second method proposes to get the collective priority vector by formulating an optimization problem without determining the consensus pairwise comparison matrix beforehand.     

Decomposition of generalized vector variational inequalities

A multicriteria optimization problem is called Pareto reducible if its weakly efficient solutions actually are efficient solutions for the problem itself or for at least one subproblem obtained from it by selecting certain criteria. The aim of this paper is to investigate a similar property within a special class of generalized vector variational inequalities, under appropriate generalized convexity assumptions.     

An intelligent decomposition of pairwise comparison matrices for large-scale decisions

A Pairwise Comparison Matrix (PCM) has been used to compute for relative priorities of elements and are integral components in widely applied decision making tools: the Analytic Hierarchy Process (AHP) and its generalized form, the Analytic Network Process (ANP). However, PCMs suffer from several issues limiting their applications to large-scale decision problems. These limitations can be attributed to the curse of dimensionality, that is, a large number of pairwise comparisons need to be elicited from a decision maker. This issue results to inconsistent preferences due to the limited cognitive powers of decision makers. To address these limitations, this research proposes a PCM decomposition methodology that reduces the elicited pairwise comparisons. A binary integer program is proposed to intelligently decompose a PCM into several smaller subsets using interdependence scores among elements. Since the subsets are disjoint, the most independent pivot element is identified to connect all subsets to derive the global weights of the elements from the original PCM. As a result, the number of pairwise comparison is reduced and consistency is of the comparisons is improved. The proposed decomposition methodology is applied to both AHP and ANP to demonstrate its advantages.     

A Goal Programming Method for Generating Priority Vectors

The generation of priority vectors from pairwise comparison information is an integral part of the Analytic Hierarchy Process (AHP). Traditionally, either the right eigenvector method or the logarithmic least squares method have been used. In this paper, a goal programming method (GPM) is presented that has, as its objective, the generation of the priority vector whose associated comparison values are, on average, the closest to the pairwise comparison information provided by the evaluator. The GPM possesses the properties of correctness in the consistent case, comparison order invariance, smoothness, and power invariance. Unlike other methods, it also possesses the additional property that the presence of a single outlier cannot prevent the identification of the correct priority vector. The GPM also has a pair of naturally meaningful consistency indicators that offer the opportunity for empowering the decision maker. The GPM is thus an attractive alternative to other proposed methods.     

AHP判断矩阵权向量的改进最小二乘求解

提出了基于最小二乘法计算判断矩阵权向量的新方法.固定AHP判断矩阵权向量中的一个值为常量,利用判断矩阵的上三角部分元素,设计了一种计算判断矩阵权向量的新算法,算法简单,计算容易,与特征向量排序方法导出标度相同,并且能够证明存在唯一解.实验表明该算法具有有效性和可行性.     

A cosine maximization method for the priority vector derivation in AHP

The derivation of a priority vector from a pair-wise comparison matrix (PCM) is an important issue in the Analytic Hierarchy Process (AHP). The existing methods for the priority vector derivation from PCM include eigenvector method (EV), weighted least squares method (WLS), additive normalization method (AN), logarithmic least squares method (LLS), etc. The derived priority vector should be as similar to each column vector of the PCM as possible if a pair-wise comparison matrix (PCM) is not perfectly consistent. Therefore, a cosine maximization method (CM) based on similarity measure is proposed, which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a PCM. An optimization model for the CM is proposed to derive the reliable priority vector. Using three numerical examples, the CM is compared with the other prioritization methods based on two performance evaluation criteria: Euclidean distance and minimum violation. The results show that the CM is flexible and efficient.     

Search for Efficient Solutions of Multi-Criterion Problems by Target-Level Method

The target-level method is considered for solving continuous multi-criterion maximization problems. In the first step, the decision-maker specifies a target-level point (the desired criterion values); then in the set of vector evaluations we seek points that are closest to the target point in the Chebyshev metric. The vector evaluations obtained in this way are in general weakly efficient. To identify the efficient evaluations, the second step maximizes the sum of the criteria on the set generated in step 1. We prove the relationship between the evaluations and decisions obtained by the proposed procedure, on the one hand, and the efficient (weakly efficient) evaluations and decisions, on the other hand. If the Edgeworth–Pareto hull of the set of vector evaluations is convex, the set of efficient vector evaluations can be approximated by the proposed method.     

Efficient structure of noisy communication networks

In the canonical network model, the connections model, only three specific network structures are generically efficient: complete, empty, and star networks. This renders many plausible network structures inefficient. We show that requiring robustness with respect to stochastic information transmission failures rehabilitates incomplete, redundant network structures. Specifically, we show that star and complete networks are not generally robust to transmission failures, that circular and quasi-circular networks are efficient at intermediate costs in four-player networks, and that if either of them is efficient, then at least one of them is pairwise stable even without reallocation. Thus, incomplete, redundant networks are efficient and stable at intermediate costs.     

Bitopologies on Totally Ordered Sets

We study the category of ray bispaces, that is, the category whose objects are totally ordered sets with two topologies, each having a subbase of rays and so that the resulting bitopological space is pairwise weakly symmetric, and whose morphisms are the pairwise continuous functions. In contrast with the purely topological results of [5], we show that, (1) such spaces are utterly normal and hence monotonically normal (in the sense of [6]), and (2) (Intermediate Value Theorem) the pairwise continuous image of a pairwise connected bitopological space in a selective ray bispace is an interval. We also obtain conditions for the equality of the de Groot dual (see [4]) and the ray dual (see [5]) of a ray topology and show that a selective ray topology is compact if and only if it is skew compact.     

Pairwise opposite matrix and its cognitive prioritization operators: comparisons with pairwise reciprocal matrix and analytic prioritization operators

The pairwise reciprocal matrix (PRM) of the analytic hierarchy/network process has been investigated by many scholars. However, there are significant queries about the appropriateness of using the PRM to represent the pairwise comparison. This research proposes a pairwise opposite matrix (POM) as the ideal alternative with respect to the human linguistic cognition of the rating scale of the paired comparison. Several cognitive prioritization operators (CPOs) are proposed to derive the individual utility vector (or priority vector) of the POM. Not only are the rigorous mathematical proofs of the new models demonstrated, but solutions of the CPOs are also illustrated by the presentation of graph theory. The comprehensive numerical analyses show how the POM performs better than the PRM. POM and CPOs, which correct the fallacy of the PRM associated with its prioritization operators, should be the ideal solutions for multi-criteria decision-making problems in various fields.     

Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems

We consider multi-objective convex optimal control problems. First we state a relationship between the (weakly or properly) efficient set of the multi-objective problem and the solution of the problem scalarized via a convex combination of objectives through a vector of parameters (or weights). Then we establish that (i) the solution of the scalarized (parametric) problem for any given parameter vector is unique and (weakly or properly) efficient and (ii) for each solution in the (weakly or properly) efficient set, there exists at least one corresponding parameter vector for the scalarized problem yielding the same solution. Therefore the set of all parametric solutions (obtained by solving the scalarized problem) is equal to the efficient set. Next we consider an additional objective over the efficient set. Based on the main result, the new objective can instead be considered over the (parametric) solution set of the scalarized problem. For the purpose of constructing numerical methods, we point to existing solution differentiability results for parametric optimal control problems. We propose numerical methods and give an example application to illustrate our approach.     

An integrated approach for deriving priorities in analytic network process

A multiple objective programming approach for the analytic network process (ANP) is proposed to obtain all local priorities for crisp or interval judgments at one time, even in an inconsistent situation. The weakness of the ANP and fuzzy ANP (FANP) is that the complexity of generating priorities is equal to the number of comparison matrices. In the proposed approach, all sets of crisp priorities for each pairwise comparison matrix can be obtained directly. Moreover, from the outcomes of three examples, the power to reach a limiting supermatrix is less than or equal to the power of the FANP. Thus, the proposed approach can be regarded as an efficient alternative of the fuzzy ANP.     

Solution concepts in vector optimization: a fresh look at an old story

Over the past decades various solution concepts for vector optimization problems have been established and used: among them are efficient, weakly efficient and properly efficient solutions. In contrast to the classical approach, we define a solution to be a set of efficient solutions on which the infimum of the objective function with respect to an appropriate complete lattice (the space of self-infimal sets) is attained. The set of weakly efficient solutions is not considered to be a solution, but weak efficiency is essential in the construction of the complete lattice. In this way, two classic concepts are involved in a common approach. Several different notions of semicontinuity are compared. Using the space of self-infimal sets, we can show that various originally different concepts coincide. A Weierstrass existence result is proved for our solution concept. A slight relaxation of the solution concept yields a relationship to properly efficient solutions.     

On pairwise comparison matrices that can be made consistent by the modification of a few elements

Pairwise comparison matrices are often used in Multi-attribute Decision Making for weighting the attributes or for the evaluation of the alternatives with respect to a criteria. Matrices provided by the decision makers are rarely consistent and it is important to index the degree of inconsistency. In the paper, the minimal number of matrix elements by the modification of which the pairwise comparison matrix can be made consistent is examined. From practical point of view, the modification of 1, 2, or, for larger matrices, 3 elements seems to be relevant. These cases are characterized by using the graph representation of the matrices. Empirical examples illustrate that pairwise comparison matrices that can be made consistent by the modification of a few elements are present in the applications.     

On comparison of approximate solutions in vector optimization problems

The problem of comparison of approximations (approximate solutions to a vector optimization problem) obtained using different numerical methods is considered. In the absence of a priori information about the set of weakly efficient vectors, a scalar function is introduced that enables pair-wise comparison of approximations and establishes a binary preference relation according to which the approximations close (in the sense of the Hausdorff distance) to the set containing all possible efficient vectors are preferable to other approximations.     

Criteria for efficiency in vector optimization

We consider unconstrained finite dimensional multi-criteria optimization problems, where the objective functions are continuously differentiable. Motivated by previous work of Brosowski and da Silva (1994), we suggest a number of tests (TEST 1–4) to detect, whether a certain point is a locally (weakly) efficient solution for the underlying vector optimization problem or not. Our aim is to show: the points, at which none of the TESTs 1–4 can be applied, form a nowhere dense set in the state space. TESTs 1 and 2 are exactly those proposed by Brosowski and da Silva. TEST 3 deals with a local constant behavior of at least one of the objective functions. TEST 4 includes some conditions on the gradients of objective functions satisfied locally around the point of interest. It is formulated as a Conjecture. It is proven under additional assumptions on the objective functions, such as linear independence of the gradients, convexity or directional monotonicity. This work was partially supported by grant 55681 of the CONACyT.     

An additional result of Monsuur's paper about intrinsic consistency threshold for reciprocal matrices

The paper addresses ranking of factors with a use of the pairwise comparison method in sense of the analytic hierarchy process (AHP). The aim of this study is to show that the logarithmic least squares method used to approximate judgement matrix in order to calculate the ranking of factors does not cause the reduction of weight of a new alternative as the eigenvector method does.