A consistency and consensus-based method to group decision making with interval linguistic preference relations

Preference relations are a powerful tool to address decision-making problems. In some situations, because of the complexity of decision-making problems and the inherent uncertainty, the decision makers cannot express their preferences by using numerical values. Interval linguistic preference relations, which are more reliable and informative for the decision-makers’ preferences, are a good choice to cope with this issue. Just as with the other types of preference relations, the consistency and consensus analysis is very importance to ensure the reasonable ranking order by using interval linguistic preference relations. Considering this situation, this paper introduces a consistency concept for interval linguistic preference relations. To measure the consistency of interval linguistic preference relations, a consistency measure is defined. Then, a consistency-based programming model is built, by which the consistent linguistic preference relations with respect to each object can be obtained. To cope with the inconsistency case, two models for deriving the adjusted consistent linguistic preference relations are constructed. Then, a consistency-based programming model to estimate the missing values is built. After that, we present a group consensus index and present some of its desirable properties. Furthermore, a group consensus-based model to determine the weights of the decision makers with respect to each object is established. Finally, an approach to group decision making with interval linguistic preference relations is developed, which is based on the consistency and consensus analysis. Meanwhile, the associated numerical examples are offered to illustrate the application of the procedure.     

Fuzzy Pattern Recognition Approach to Construction Contractor Selection

Contractor selection is a complex process crucial to ensuring the success of construction projects. Existing methods by which owners select a suitable contractor have been inadequate because it is difficult for decision-makers to evaluate contractor bids against inexact qualitative criteria. The purpose of this paper is to propose a Multiple-layer Fuzzy Pattern Recognition (MFPR) approach to solve contractor selection problem. Integrating judgments, experience and preferences of decision-makers, this paper uses the paired comparison method to decide relative membership degrees of qualitative criteria as well as weights of the criteria set, a case study for a channel construction project was used to demonstrate the feasibility of this approach. The findings clearly indicate that the techniques may successfully harmonize different opinions and reach a group consensus.     

A statistical approach to the analytic hierarchy process with interval judgements. (I). Distributions on feasible regions

This paper addresses the problem of extracting preferences for alternatives from interval judgement matrices in the Analytic Hierarchy Process (AHP). The method of Arbel for extracting such preferences, which is based on the assumption that the interval judgements specify a feasible region in the weight space of the alternatives, is critically appraised from a statistical perspective and some new ideas emanating from this approach are developed and discussed. In particular it is proposed that a distribution for the weights on the feasible region, which is both tractable and meaningful, be adopted. The mean of the distribution can then be used as an assessment of the overall ranking of the alternatives and quantities of interest, such as probabilities and marginal distributions, can immediately be quantified. Two specific distributions on the feasible region, the uniform distribution and the distribution of random convex combinations with coefficients which are uniform spacings, are examined in some detail and the ideas which emerge are illustrated by means of selected examples.     

Stochastic multiobjective acceptability analysis for the Choquet integral preference model and the scale construction problem

The Choquet integral preference model is adopted in Multiple Criteria Decision Aiding (MCDA) to deal with interactions between criteria, while the Stochastic Multiobjective Acceptability Analysis (SMAA) is an MCDA methodology considered to take into account uncertainty or imprecision on the considered data and preference parameters. In this paper, we propose to combine the Choquet integral preference model with the SMAA methodology in order to get robust recommendations taking into account all parameters compatible with the preference information provided by the Decision Maker (DM). In case the criteria are on a common scale, one has to elicit only a set of non-additive weights, technically a capacity, compatible with the DM’s preference information. Instead, if the criteria are on different scales, besides the capacity, one has to elicit also a common scale compatible with the preferences given by the DM. Our approach permits to explore the whole space of capacities and common scales compatible with the DM’s preference information.     

Transitive calibration of the AHP verbal scale

One of the strengths of the Analytic Hierarchy Process (AHP) is that it allows decision-makers to specify their preferences using a verbal scale. Yet, as is well known, a strict reliance on the corresponding Saaty 1–9 numeric scale can induce some inconsistency. Hence we argue that, in certain situations, it may be appropriate to calibrate the verbal scale. The result of this calibration is a geometric scale based on a single parameter. We present some limited evidence that this geometric scale marginally outperforms the Saaty 1–9 scale. Moreover, we suggest that this calibration can be used to do a simple sensitivity analysis in cases where judgements are uncertain.     

Experiments with a new selection criterion in a fast interval optimization algorithm

Usually, interval global optimization algorithms use local search methods to obtain a good upper (lower) bound of the solution. These local methods are based on point evaluations. This paper investigates a new local search method based on interval analysis information and on a new selection criterion to direct the search. When this new method is used alone, the guarantee to obtain a global solution is lost. To maintain this guarantee, the new local search method can be incorporated to a standard interval GO algorithm, not only to find a good upper bound of the solution, but also to simultaneously carry out part of the work of the interval B&B algorithm. Moreover, the new method permits improvement of the guaranteed upper bound of the solution with the memory requirements established by the user. Thus, the user can avoid the possible memory problems arising in interval GO algorithms, mainly when derivative information is not used. The chance of reaching the global solution with this algorithm may depend on the established memory limitations. The algorithm has been evaluated numerically using a wide set of test functions which includes easy and hard problems. The numerical results show that it is possible to obtain accurate solutions for all the easy functions and also for the investigated hard problems.     

A surrogate based multi-fidelity approach for robust design optimization

Robust design optimization (RDO) is a field of optimization in which certain measure of robustness is sought against uncertainty. Unlike conventional optimization, the number of function evaluations in RDO is significantly more which often renders it time consuming and computationally cumbersome. This paper presents two new methods for solving the RDO problems. The proposed methods couple differential evolution algorithm (DEA) with polynomial correlated function expansion (PCFE). While DEA is utilized for solving the optimization problem, PCFE is utilized for calculating the statistical moments. Three examples have been presented to illustrate the performance of the proposed approaches. Results obtained indicate that the proposed approaches provide accurate and computationally efficient estimates of the RDO problems. Moreover, the proposed approaches outperforms popular RDO techniques such as tensor product quadrature, Taylor’s series and Kriging. Finally, the proposed approaches have been utilized for robust hydroelectric flow optimization, demonstrating its capability in solving large scale problems.     

基于前景理论和熵权法的交叉效率集结方法

传统的交叉效率集结过程通常采用算术平均方法,不仅会低估自评的重要性,而且未考虑决策者的风险偏好。针对上述问题,提出一种基于前景理论和熵权法的交叉效率集结方法。首先,求解交叉效率矩阵,运用熵权法确定他评过程中评价单元的指标权重。然后,引入前景理论以考虑决策者在交叉效率集结过程中的风险偏好,利用TOPSIS方法识别正负参考点,进而构造总体效用函数,得到前景交叉效率矩阵。随后,构建最大化前景价值模型,求解集结权重。该方法既考虑到交叉效率集结的相对重要性权重,又将决策者的风险偏好纳入到效率评价中,从而实现决策单元的全排序。最后,结合实例验证方法的有效性。     

Bi-objective optimisation with multiple decision-makers: a convex approach to attain majority solutions

An interactive approach for solving bi-objective optimisation problems with multiple decision-makers in a context where the decisions are made via the strict majority voting rule is proposed. An adequate use of projection and decomposition techniques leads to a hierarchical algorithm in which the upper level is represented by a relaxed version of the bi-objective problem in the objective space. The feasibility of the partial consensus obtained in the upper (decision) level by the strict majority voting rule is tested in the lower (analysis) level of the algorithm. Some properties of the relaxed bi-objective problem allow that different preference structures of the decision-makers can be represented by appropriate multiobjective methods. The paper includes numerical examples that illustrate the characteristics of the approach proposed.     

Reformulation versus cutting-planes for robust optimization

Robust optimization (RO) is a tractable method to address uncertainty in optimization problems where uncertain parameters are modeled as belonging to uncertainty sets that are commonly polyhedral or ellipsoidal. The two most frequently described methods in the literature for solving RO problems are reformulation to a deterministic optimization problem or an iterative cutting-plane method. There has been limited comparison of the two methods in the literature, and there is no guidance for when one method should be selected over the other. In this paper we perform a comprehensive computational study on a variety of problem instances for both robust linear optimization (RLO) and robust mixed-integer optimization (RMIO) problems using both methods and both polyhedral and ellipsoidal uncertainty sets. We consider multiple variants of the methods and characterize the various implementation decisions that must be made. We measure performance with multiple metrics and use statistical techniques to quantify certainty in the results. We find for polyhedral uncertainty sets that neither method dominates the other, in contrast to previous results in the literature. For ellipsoidal uncertainty sets we find that the reformulation is better for RLO problems, but there is no dominant method for RMIO problems. Given that there is no clearly dominant method, we describe a hybrid method that solves, in parallel, an instance with both the reformulation method and the cutting-plane method. We find that this hybrid approach can reduce runtimes to 50–75 % of the runtime for any one method and suggest ways that this result can be achieved and further improved on.     

A Bayesian approach for multiple criteria decision making with applications in Design for Six Sigma

Linking end-customer preferences with variables controlled at a manufacturing plant is a main idea behind popular Design for Six Sigma management techniques. Multiple criteria decision making (MCDM) approaches can be used for such purposes, but in these techniques the decision-maker's (DM) utility function, if modelled explicitly, is considered known with certainty once assessed. Here, a new algorithm is proposed to solve a MCDM problem with applications to Design for Six Sigma based on a Bayesian methodology. At a first stage, it is assumed that there are process responses that are functions of certain controllable factors or regressors. This relation is modelled based on experimental data. At a second stage, the utility function of one or more DMs or customers is described in a statistical model as a function of the process responses, based on surveys. This step considers the uncertainty in the utility function(s) explicitly. The methodology presented then maximizes the probability that the DM's or customer's utility is greater than some given lower bound with respect to the controllable factors of the first stage. Both stages are modelled with Bayesian regression techniques. The advantages of using the Bayesian approach as opposed to traditional methods are highlighted.     

Uncertainty analysis for structures with hybrid random and interval parameters using mathematical programming approach

A novel computational method, namely the unified perturbation mathematical programming (UPMP) approach, for hybrid uncertainty analysis of engineering structures is proposed in this paper. The presented study considers a mixture of random and interval system parameters which are frequently encountered in engineering applications. Within the UPMP approach, matrix perturbation theory is adopted in combination with the mathematical programming approach. The proposed computational method provides a non-simulative hybrid uncertainty analysis framework, which is competent to offer the extreme bounds of the statistical characteristics (i.e., mean and variance) of any concerned structural responses in computationally tractable fashion. In order to thoroughly explore various intricate aspects of the engineering system involving hybrid uncertainties, systematic numerical experiments have also been conducted. Diverse statistical analyses are implemented to identify the bounded probability profile of the uncertain structural responses. Both academic and practical engineering structures are investigated to justify the applicability, accuracy and efficiency of the proposed UPMP approach.     

基于物理规划的工程项目综合均衡优化

质量、工期和成本是工程项目三大主要控制目标,对于工程项目中质量-工期-成本综合均衡优化问题,传统的基于权重的决策方法存在各目标权重难以合理确定的问题.为此引入物理规划方法建立工程项目质量-工期-成本综合均衡优化模型.决策者只需设定各目标的偏好,即可通过该均衡优化模型获得符合决策者偏好的优化方案,使决策过程更加符合工程实际,避免了确定无实际物理意义的各目标权重的问题.通过桥梁工程实例验证了该方法的有效性和实用性.     

Aggregating individual judgments and priorities with the analytic hierarchy process

The Analytic Hierarchy Process (AHP) is often used in group settings where group members either engage in discussion to achieve a consensus or express their own preferences. Individual judgments can be aggregated in different ways. Two of the methods that have been found to be most useful are the aggregation of individual judgments (AIJ) and the aggregation of individual priorities (AIP). We propose that the choice of method depends on whether the group is assumed to act together as a unit or as separate individuals and explain why AIJ is appropriate for the former while AIP is appropriate for the latter. We also address the relationships between the choice of method, the applicability of the Pareto principle, and the use of arithmetic or geometric means in aggregation. Finally, we discuss Ramanathan and Ganesh's method to derive priorities for individual decision-makers that can be used when aggregate group preferences of individuals whose judgments are not all equally weighted. We conclude that while this method can be useful, it is applicable only in special circumstances.     

Simulation and optimization for crop water allocation based on crop water production functions and climate factor under uncertainty

In this paper, the uncertainty methods of interval and functional interval are introduced in the research of the uncertainty of crop water production function itself and optimal allocation of water resources in the irrigation area. The crop water production functions in the whole growth period under uncertainty and the optimal allocation of water resources model in the irrigation area under uncertainty are established, and the meteorological factor is considered in the model. It can promote the practical application of the uncertain methods, reflect the complexity and uncertainty of the actual situation, and provide more reliable scientific basis for using water resources economically, fully improving irrigation efficiency, and keeping the sustainable development of the irrigated area. This approach has important value on theoretical and practical for the optimal irrigation schedule, and has very broad prospects for research and development to other related agriculture water management.     

An Education in Robustness

A case study is presented of the application of robustness analysis to the choice of examination subjects to be made by a 14-year-old girl. The analysis is conducted in terms of the desirable future career options which may be preserved by different combinations of subjects, and also of the balance with topics selected for their intrinsic interest. The approach, which employs very limited technical apparatus, is justified on a number of grounds-in particular, the crucial importance of uncertainty, and the need for understanding and acceptance by a "client" very unlike the corporate manager. Some tentative conclusions are reached on the extension of such methods to other cases of decision-makers with little, if any, resources to control except their own lives.     

Direct versus indirect credit scoring classifications

We introduce a new approach to assigning bank account holders to ‘good’ or ‘bad’ classes based on their future behaviour. Traditional methods simply treat the classes as qualitatively distinct, and seek to predict them directly, using statistical techniques such as logistic regression or discriminant analysis based on application data or observations of previous behaviour. We note, however, that the ‘good’ and ‘bad’ classes are defined in terms of variables such as the amount overdrawn at the time at which the classification is required. This permits an alternative, ‘indirect’, form of classification model in which, first, the variables defining the classes are predicted, for example using regression, and then the class membership is derived deterministically from these predicted values. We compare traditional direct methods with these new indirect methods using both real bank data and simulated data. The new methods appear to perform very similarly to the traditional methods, and we discuss why this might be. Finally, we note that the indirect methods also have certain other advantages over the traditional direct methods.     

Robust location of new housing developments using a choice model

We consider the issue of choosing a subset of locations to construct new housing developments maximizing the satisfaction of potential buyers, which has not been previously studied in the literature. The allocation of demands to the selected locations is modeled by a choice model, based on the distance to the location, real-estate prices and incomes. We study two robust counterparts of the optimal location problem, where uncertainty lies on demand volumes for the first one, and on customer preferences for the second one. In both cases, the parameters subject to uncertainty appear both in the objective function and constraints. The second robust model combines a scenario-based approach with nominal, price-centric and distance-centric scenarios on customers preferences, and an uncertainty budget approach that limits the number of customers that can deviate from the nominal scenario. We show that the subproblem of finding the worst-case deviation of parameters subject to uncertainty is tractable and leads to linear formulations of the robust problem. Computational experiments conducted on instances of the Paris region show that the average loss of value of the robust solution is reasonably low when compared to the optimal solution of deviated instances. We also derive insights for the new housing development issue.     

Accelerating data uncertainty quantification by solving linear systems with multiple right-hand sides

The subject of this work is accelerating data uncertainty quantification. In particular, we are interested in expediting the stochastic estimation of the diagonal of the inverse covariance (precision) matrix that holds a wealth of information concerning the quality of data collections, especially when the matrices are symmetric positive definite and dense. Schemes built on direct methods incur a prohibitive cubic cost. Recently proposed iterative methods can remedy this but the overall cost is raised again as the convergence of stochastic estimators can be slow. The motivation behind our approach stems from the fact that the computational bottleneck in stochastic estimation is the application of the precision matrix on a set of appropriately selected vectors. The proposed method combines block conjugate gradient with a block-seed approach for multiple right-hand sides, taking advantage of the nature of the right-hand sides and the fact that the diagonal is not sought to high accuracy. Our method is applicable if the matrix is only known implicitly and also produces a matrix-free diagonal preconditioner that can be applied to further accelerate the method. Numerical experiments confirm that the approach is promising and helps contain the overall cost of diagonal estimation as the number of samples grows.