An algorithm for projecting a reference direction onto thenondominated set of given points

We consider the problem of searching nondominated alternatives in a discrete multiple criteria problem. The search procedure is based on the use of a reference direction. A reference direction reflects the desire of the decision maker (DM) to specify a search direction. To find a set of given alternatives related somehow to the reference direction specified by the DRI, the reference direction has to be projected onto the set of nondominated alternatives. Our purpose is to develop an efficient algorithm for making this projection. The projection of each given reference direction determines a nondominated ordered subset. The set is provided to a decision maker for evaluation. The decision maker will choose the most preferred alternative from this subset and continues the search from this alternative with a new reference direction. The search will end when no direction of improvement is found. A critical point in the procedure is the efficiency of the projection operation. This efficiency of our algorithm is considered theoretically and numerically. The projection is made by parametrizing an achievement scalarizing function originally proposed by Wierzbicki (1980) to project any single point onto the nondominated set     

Ordered weighted enhancement of preference modeling in the reference point method for multiple criteria optimization

The reference point method is an interactive technique for multiple criteria optimization problems. It is based on the optimization of the scalarizing achievement function built as the augmented max–min aggregation of individual outcomes with respect to the given reference levels. Actually, the worst individual achievement is optimized, but regularized with the term representing the average achievement. In order to avoid inconsistencies caused by the regularization, we apply the ordered weighted averages (OWA) with monotonic weights to combine all the individual achievements. Further, following the concept of the weighted OWA (WOWA), we incorporate the importance weighting of several achievements into the RPM. We show that the resulting WOWA RPM can be quite effectively implemented as an extension of the original constraints and criteria with simple linear inequalities.     

A CHIM-based interactive Tchebycheff procedure for multiple objective decision making

This study presents a new solution procedure for multiple objective programming (MOP). It applies the concept of the normal boundary intersection (NBI) within the framework of the interactive weighted Tchebycheff procedure (IWTP). The proposed procedure is a collaborative approach to overcome the weak points inherent in both the NBI method and IWTP. In order to control the solution procedure of pinpointing a final solution properly, we parameterized the Pareto-frontier via a set of reference point vectors based on the convex hull of individual maxima (CHIM) instead of using the varying weights of each objective. Using well-distributed reference point vectors, we could identify well-distributed Pareto-optimal solutions, thereby eliminating the IWTP filtering stages and reducing the chance of missing the best compromise solution from the decision maker (DM)'s utility point of view. Moreover, by working with a sequence of progressively smaller subsets of reference point vectors, the DM can identify a final solution at earlier stages than with the IWTP.     

Normal vector identification and interactive tradeoff analysis using minimax formulation in multiobjective optimization

In multiobjective optimization, tradeoff analysis plays an important role in determining the best search direction to reach a most preferred solution. This paper presents a new explicit interactive tradeoff analysis method based on the identification of normal vectors on a noninferior frontier. The interactive process is implemented using a weighted minimax formulation by regulating the relative weights of objectives in a systematic manner. It is proved under a mild condition that a normal vector can be identified using the weights and Kuhn-Tucker (K-T) multipliers in the minimax formulation. Utility gradients can be estimated using local preference information such as marginal rates of substitution. The projection of a utility gradient onto a tangent plane of the noninferior frontier provides a descent direction of disutility and thereby a desirable tradeoff direction, along which tradeoff step sizes can be decided by the decision maker using an explicit tradeoff table. Necessary optimality conditions are established in terms of normal vectors and utility gradients, which can be used to guide the elicitation of local preferences and also to terminate an interactive process in a rigorous yet flexible way. This method is applicable to both linear and nonlinear (either convex or nonconvex) multiobjective optimization problems. Numerical examples are provided to illustrate the theoretical results of the paper and the implementation of the proposed interactive decision analysis process.     

Improving convergence of evolutionary multi-objective optimization with local search: a concurrent-hybrid algorithm

A local search method is often introduced in an evolutionary optimization algorithm, to enhance its speed and accuracy of convergence to optimal solutions. In multi-objective optimization problems, the implementation of local search is a non-trivial task, as determining a goal for local search in presence of multiple conflicting objectives becomes a difficult task. In this paper, we borrow a multiple criteria decision making concept of employing a reference point based approach of minimizing an achievement scalarizing function and integrate it as a search operator with a concurrent approach in an evolutionary multi-objective algorithm. Simulation results of the new concurrent-hybrid algorithm on several two to four-objective problems compared to a serial approach, clearly show the importance of local search in aiding a computationally faster and accurate convergence to the Pareto optimal front.     

Behavioral ordered weighted averaging operator and the application in multiattribute decision making

This paper introduces a new type of behavioral ordered weighted averaging (BOWA) operator, to incorporate decision maker’s gains and losses behavior tendency into the information aggregation process. The main characteristic of this BOWA operator is that it considers behavioral weights and ordered weights in the same formulation. We further provide a calculation method of the behavioral weights, in which various psychological preferences of different attribute types of the decision maker can be expressed intuitively. In addition, we discuss some particular cases of BOWA operator and its main properties. Finally, a numerical example is used to illustrate the use of the proposed method.     

Stability analysis of efficient solutions in multiobjective integer programming: A case study in load management

This paper presents a multiobjective linear integer programming model for supporting the choice of remote load control strategies in electric distribution network management. The model takes into account the main concerns in load management, considering three objective functions: minimization of the peak demand as perceived by the distribution network dispatch center, maximization of the utility profit associated with the energy services delivered by the controlled loads and minimization of the discomfort caused to consumers. The problem was analyzed using an interactive reference point method for multiobjective integer (and mixed-integer) linear programming. This approach exploits the use of the branch-and-bound algorithm for solving the reference point scalarizing programs through which efficient solutions are computed. Post-optimality techniques enable a stability analysis of the efficient solutions by means of computing and displaying graphically sets of reference points that correspond to the same solution.     

Multi-period lot-sizing with supplier selection using achievement scalarizing functions

In this paper, an integration of Analytic Network Process (ANP) and achievement scalarizing functions is proposed to choose the best suppliers and define the optimum quantities among the selected suppliers by considering tangible–intangible criteria and time horizon. To reflect the decision maker’s (DM’s) preferences more accurate, an additive achievement function is defined consist of several components. In this additive function while unwanted deviations from periodic budget and aggregate quality goals are balanced by Minmax Goal Programming (MGP), and unwanted deviations from total cost, total value of purchasing (TVP) and aggregate quality are minimized by Achimedean Goal Programming (AGP) to provide more acceptable solutions. This multi-period model enables us to reflect DM’s preferences more flexible than the other traditional models that use only one type of achievement function. The sensitivity analysis was also performed for different levels of periodic demands. It is also possible to enlarge the sensitivity analyses for other parameters such as different levels of capacity, and different weights of components.     

An interactive fuzzy satisficing method for multiobjectivenonconvex programming problems with fuzzy numbers through coevolutionarygenetic algorithms

In this paper, by considering the experts' fuzzy understanding of the nature of the parameters in the problem-formulation process, multiobjective nonconvex nonlinear programming problems with fuzzy numbers are formulated and an interactive fuzzy satisficing method through coevolutionary genetic algorithms is presented. Using the alpha-level sets of fuzzy numbers, the corresponding nonfuzzy alpha-programming problem is introduced. After determining the fuzzy goals of the decision maker, if the decision maker specifies the degree alpha and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the augmented minimax problems for which the coevolutionary genetic algorithm, called GENOCOP III, is applicable. In order to overcome the drawbacks of GENOCOP III, the revised GENOCOP III is proposed by introducing a method for generating an initial feasible point and a bisection method for generating a new feasible point efficiently. Then an interactive fuzzy satisficing method for deriving a satisficing solution for the decision maker efficiently from an extended Pareto optimal solution set is presented together with an illustrative numerical example.     

区间Pythagorean模糊交互式多准则决策模型

研究了权重信息部分已知,评价信息为区间Pythagorean模糊数的交互式多准则决策问题。利用区间Pythagorean模糊数得分函数,计算各方案的加权得分向量在Pythagorean模糊正理想点和Pythagorean模糊负理想点上的投影,构建基于方案满意度最大的非线性规划准则权重确定模型。根据决策者的主观偏好并结合现有客观信息建立单目标规划模型,通过对方案满意度的给定与修正来实现交互决策。通过算例说明模型及方法的可行性和有效性。     

An interactive rectangle elimination method for biobjective decision making

This paper presents a new man-machine interactive method for biobjective decision making. It is specifically designed to cope with both the ill-defined nature of the decision problem and the high cost of computation points in the tradeoff (Pareto optimal) set. With this method, the decision maker may efficiently approximate the tradeoff set and/or estimate his preferred objective value. First, the notion of a rectangle representation of the tradeoff set by a set of points, called experiments, and a set of rectangles, defined by the experiments, is introduced. Next, a special class of decision makers is considered. For a decision maker in this special class, the rectangle representation of the tradeoff set defines a rectangle of uncertainty that contains the decision maker's preferred objective value. A measure of the worst ease uncertainity is formulated and minimized to yield an optimal strategy for interactively selecting experiments. Finally, this strategy is employed in a general interactive algorithm that works under minimal assumptions on the tradeoff set and on the decision maker.     

An entropy target-point approach to multiobjective programming

In this paper, we present a method for solving multiple objective programming problems. The method can be interpreted as a ‘distance’ method, i.e. the method minimizes the ‘distance’ from a target point specified by the decision maker. The auxiliary ‘distance’ objective we use in our method is the entropy function. With this choice of auxiliary objective, we obtain a computationally efficient method. This algorithmic efficiency is especially emphasized when the method is to be used in an interactive scheme where the auxiliary problem has to be solved repeatedly for a number of different target points. Another attractive feature of the choice of an entropy auxiliary objective function is that it generates stable solutions.     

An efficient procedure for finding best compromise solutions to the multi-objective assignment problem

In this paper, we consider the problem of determining a best compromise solution for the multi-objective assignment problem. Such a solution minimizes a scalarizing function, such as the weighted Tchebychev norm or reference point achievement functions. To solve this problem, we resort to a ranking (or k-best) algorithm which enumerates feasible solutions according to an appropriate weighted sum until a condition, ensuring that an optimal solution has been found, is met. The ranking algorithm is based on a branch and bound scheme. We study how to implement efficiently this procedure by considering different algorithmic variants within the procedure: choice of the weighted sum, branching and bounding schemes. We present an experimental analysis that enables us to point out the best variants, and we provide experimental results showing the remarkable efficiency of the procedure, even for large size instances.     

Solving scalarized multi‐objective network flow problems using an interior point method

In this paper, we present a primal‐dual interior‐point algorithm to solve a class of multi‐objective network flow problems. More precisely, our algorithm is an extension of the single‐objective primal infeasible dual feasible inexact interior point method for multi‐objective linear network flow problems. Our algorithm is contrasted with standard interior point methods and experimental results on bi‐objective instances are reported. The multi‐objective instances are converted into single objective problems with the aid of an achievement function, which is particularly adequate for interactive decision‐making methods.     

基于方案达成度和综合度的交互式多属性决策法

定义了方案达成度和方案综合度，对于属性权重信息不能完全确知且对方案有偏好的多属性决策问题，提出一种基于方案达成度和综合度的交互式决策方法，该方法既能充分利用已知的客观信息，又有最大限度地发挥决策的主观能动性，并通过对方案达成度和综合度的给定和修正来实现人机交互决策，应用实例说明了该方法的有效性和实用性。     

Methods and theory for off-line machine learning

Many problems in machine learning can be abstracted to the sequential design task of finding the minimum of an unknown erratic and possibly discontinuous function on the basis of noisy measurements. In the present work, it is presumed that there is no penalty for bad choices during the experimental stage, and at some time, not known to the decision maker, or under his control, the experimentation will be terminated, and the decision maker will need to specify the point considered best, on the basis of the experimentation. In this paper, we seek the best trade-off between: i) acquiring new test points, and ii) retesting at points previously selected so as to improve the estimates of relative performance. The algorithm is shown to achieve a performance standard described herein. This decision setting would seem natural for function minimization in a simulation contest or for tuning up a production process prior to putting it into service     

基于前景理论的犹豫模糊多属性群决策方法

针对属性指标值为犹豫模糊信息且属性权重完全未知的多属性群决策问题,提出一种基于新的决策参考点和前景理论的多属性群决策方法。将犹豫模糊决策矩阵转变为区间模糊决策矩阵,并结合t-分布估计方法,构建期望得分函数值,将其作为决策参考点;基于属性值得分函数与决策参考点之间的差异,确定价值函数,进而得到前景价值综合矩阵;利用同一属性下的前景价值方差计算属性的权重,并基于前景理论计算各方案的加权前景价值,进而对决策方案进行优劣排序;通过对云计算产品的选择实例验证提出的决策方法的可行性与有效性。     

Multi-criteria inventory classification with reference items

The ABC method is a well-known approach to classify inventory items into ordered categories, such as A, B and C. As emphasized in the literature, it is reasonable to evaluate the inventory classification problem in the multi-criteria context. From this point of view, it corresponds to a sorting problem where categories are ordered. Here, one important issue is that the weights of the criteria and categorization preferences can change from industry to industry. This requires the analysis of the problem in a specific framework where the decision maker (expert)’s preferences are considered. In this study, the preferences of the decision maker are incorporated into the decision making process in terms of reference items into each class. We apply two utility functions based sorting methods to the problem. We perform an experiment and compare results with other algorithms from the literature.     

An Interactive Reference Direction Algorithm For Solving Multi-Objective Convex Nonlinear Integer Programming Problems

We present a learning-oriented interactive reference direction algorithm for solving multi-objective convex nonlinear integer programming problems. At each iteration the decision-maker (DM) sets his/her preferences as aspiration levels of the objective functions. The modified aspiration point and the solution found at the previous iteration define the reference direction. Based on the reference direction, we formulate a mixed-integer scalarizing problem with specific properties. By solving this problem approximately, we find one or more integer solutions located close to the efficient surface. At some iteration (usually at the last iteration), the DM may want to solve the scalarizing problem to obtain an exact (weak) efficient solution. Based on the proposed algorithm, we have developed a research-decision support system that includes one exact and one heuristic algorithm. Using this system, we illustrate the proposed algorithm with an example, and report some computational results.