An interactive procedure for linguistic multiple attribute decision making with incomplete weight information

In this paper, we consider the multiple attribute decision making (MADM) problems, in which the information about attribute weights is partly known and the attribute values are expressed in linguistic labels. We first define the concepts of linguistic positive ideal point, linguistic negative ideal point, and satisfactory degree of alternative. Based on these concepts, we then establish some linear programming models, through which the decision maker interacts with the analyst. Furthermore, we establish a practical interactive procedure for solving the MADM problems considered in this paper. The interactive process can be realized by giving and revising the satisfactory degrees of alternatives till an optimum satisfactory solution is achieved. Finally, a practical example is given to illustrate the developed procedure.     

Approximate solutions for a class of first order nonlinear difference equations

In this paper, a measure-theoretical approach to find the approximate solutions for a class of first order nonlinear difference equations is introduced. In this method the problem is transformed to an equivalent optimization problem. Then, by considering it as a calculus of variations problem, some concepts in measure theory are used to approximate the solution. The procedure of constructing approximate solution in form of an algorithm is shown. Finally a numerical example is given.     

Aspiration level approach in stochastic MCDM problems

The paper considers a discrete stochastic multiple criteria decision making problem. This problem is defined by a finite set of actions A, a set of attributes X and a set of evaluations of actions with respect to attributes E. In stochastic case the evaluation of each action with respect to each attribute takes form of a probability distribution. Thus, the comparison of two actions leads to the comparison of two vectors of probability distributions. In the paper a new procedure for solving this problem is proposed. It is based on three concepts: stochastic dominance, interactive approach, and preference threshold. The idea of the procedure comes from the interactive multiple objective goal programming approach. The set of actions is progressively reduced as the decision maker specifies additional requirements. At the beginning the decision maker is asked to define preference threshold for each attribute. Next, at each iteration the decision maker is confronted with the set of considered actions. If the decision maker is able to make a final choice then the procedure ends, otherwise he/she is asked to specify aspiration level. A didactical example is presented to illustrate the proposed technique.     

Evaluation of rankings with regard to the possible number of agreements and conflicts

An interesting problem in group decision analysis is how many different agreements can occur, or conversely disagreements may exist, between two or more different rankings of a set of alternatives. In this paper it is assumed that a reference ranking has been established for the set of alternatives. This reference ranking may represent the ranking of a high authority decision maker or be just a virtual ranking to be used in determining the discrepancy between pairs of rankings. Then, the problem examined here is to evaluate the number of possible rankings when the ranking method is the number of agreements with some reference ranking. The analysis presented here illustrates that this problem is not trivial and moreover, its simple context conceals complexity in its depth. The purpose of this paper is to provide an evaluation of the number of possible agreements in rankings given to a set of concepts, alternatives or ideas, by two or more decision makers. The number of possible agreements takes on the values 0, 1, 2,…, n − 2, or n when n concepts are compared. This paper develops a recursive closed form formula for calculating the frequencies for the various numbers of agreements.     

On Computing the Size of a Lattice and Related Decision Problems

The problem of determining the size of a finite concept lattice is shown to be #P-complete. Since any finite lattice can be represented as a concept lattice, the problem of determining the size of a lattice given by the ordered sets of its irreducibles is also #P-complete. Some results about NP-completeness or polynomial tractability of decision problems related to concepts with bounded extent, intent, and the sum of both are given. These problems can be reformulated as decision problems about lattice elements generated by a certain amount of irreducibles.     

Bilevel programming approach applied to the flow shop scheduling problem under fuzziness

This paper presents a fuzzy bilevel programming approach to solve the flow shop scheduling problem. The problem considered here differs from the standard form in that operators are assigned to the machines and imposing a hierarchy of two decision makers with fuzzy processing times. The shop owner considered higher level and assigns the jobs to the machines in order to minimize the flow time while the customer is the lower level and decides on a job schedule in order to minimize the makespan. In this paper, we use the concepts of tolerance membership function at each level to define a fuzzy decision model for generating optimal (satisfactory) solution for bilevel flow shop scheduling problem. A solution algorithm for solving this problem is given. Mathematics Subject Classification: 90C70, 90B36, 90C99     

INSDECM—an interactive procedure for stochastic multicriteria decision problems

A new interactive technique for a discrete stochastic multiattribute decision making problem is proposed in this paper. It is assumed that performance probability distribution for each action on each attribute is known. Two concepts are combined in the procedure: stochastic dominance and interactive approach. The first one is employed for generating efficient actions and constructing rankings of actions with respect to attributes. The second concept is used when the communication between the DM and the model is conducted. It is assumed that decision maker’s restrictions are defined by specifying minimal or maximal values of scalar criteria measuring either expected outcome or variability of outcomes. As such restrictions are, in general, not consistent with stochastic dominance rules, we suggest verifying this consistency and asking the decision maker to redefine inconsistent restrictions.     

An interactive fuzzy satisficing method for structured multiobjective linear fractional programs with fuzzy numbers

In this paper, by considering the experts' vague or fuzzy understanding of the nature of the parameters in the problem formulation process, multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated. Using the a-level sets of fuzzy numbers, the corresponding nonfuzzy a-multiobjective linear fractional programming problem is introduced. The fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions including nonlinear ones. Through the introduction of extended Pareto optimality concepts, if the decision maker specifies the degree a and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the minimax problems for which the Dantzig-Wolfe decomposition method and Ritter's partitioning procedure are applicable. Then a linear programming-based interactive fuzzy satisficing method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an extended Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.     

Preference relations on a set of fuzzy utilities as a basis for decision making

In fuzzy decision problems, we often encounter situations of choosing among alternatives which are assigned fuzzy utilities. These problems have been approached using fuzzy implications or direct comparisons among fuzzy utilities. In the literature, however, there are few attempts to investigate the issues addressing reasonable choice or reasonable ordering using fuzzy sets theory. This paper first introduces some fundamental properties of fuzzy binary relations and certain conditions of reasonable orderings of fuzzy utilities. Then a method for constructing a fuzzy preference relation on a given set of fuzzy utilities is proposed for the sake of rational decision making. This procedure employs the concepts of the extended minimum and the Hamming distance between the greatest upper sets or the greatest lower sets of fuzzy utilities. Finally it is shown that the proposed fuzzy preference relations have reasonable properties as fuzzy orderings for decision making.     

几何分布时间序贯检验的贝叶斯推断

设有统计模型｛ｘ,Ｂｘ,Ｐθ｝,θ∈（０,１）,其中Ｐθ为几何分布：Ｐθ（Ｘ＝ｋ）＝（１－θ）θ＾ｋ－１ｋ＝１,２,…。考虑检验问题：θ＝θｏ ｖｓ． θ＝θ１（０〈θ０〈θ１〈１）本文对一种依次试验的时间序贯样本,给出了上述检验问题的贝叶斯停止判决法则,其中损失函数为试验费用和误判损失之和,贝叶斯停止判决法则由后验概率的两组界（上界和下界）所给出。     

Deterministic solution approach for some classes of nonlinear multilevel programs with multiple followers

In this paper we investigate multilevel programming problems with multiple followers in each hierarchical decision level. It is known that such type of problems are highly non-convex and hard to solve. A solution algorithm have been proposed by reformulating the given multilevel program with multiple followers at each level that share common resources into its equivalent multilevel program having single follower at each decision level. Even though, the reformulated multilevel optimization problem may contain non-convex terms at the objective functions at each level of the decision hierarchy, we applied multi-parametric branch-and-bound algorithm to solve the resulting problem that has polyhedral constraints. The solution procedure is implemented and tested for a variety of illustrative examples.     

Satisfactory solution concepts and their relations for Stochastic Multiobjective Programming problems

This work deals with the concept of satisfactory solution for Stochastic Multiobjective Programming (SMP) problems. Based on previous literature, we will introduce different concepts of satisfactory solutions for SMP problems, define a new concept of solution (where the decision maker (DM) sets his/her preferences in terms of two aspiration levels for the stochastic objective and two probabilities to reach those levels), and establish some relationship between these concepts. The results will aim at featuring these concepts and determine the differences between them. Moreover, the paper proposes a new step by step procedure to exchange information between the analyst and DM prior to solving the problem. Thus, the DM will be able to choose the transformation criterion for each stochastic objective and the aspiration level.     

Strange: An interactive method for multi-objective linear programming under uncertainty

In the field of investment planning within a time horizon, problems typically involve multiple objectives, and basic data are uncertain. In a large number of cases, these decision problems can be written as linear programming problems in which time dependent uncertainties affect the coefficients and the right hand side of constraints. Given the possibility of defining plausible scenarios on basic data, discrete sets of such coefficients are given, each with its subjective probability of occurrence. The corresponding structure is then characteristic for Multi-Objective Stochastic Linear Programming (MOSLP).In the paper, an interactive procedure is described to obtain a best compromise for such a MOSLP problem. This algorithm, called Strange, extends the Stem method to take the random aspects into account. It involves in particular, the concepts of stochastic programming with recourse. In its interactive steps, the efficiency projection techniques are used to provide the decision-maker with detailed graphical information on efficient solution families.As an illustration of the successive steps, a didactic example is solved in some detail, and the results of a case study in energy planning are given.     

Decision Networks

Decision Networks is a technique for solving problems which involve a sequence of decisions. It is similar in style to critical path analysis in that it consists of arrow diagrams which give a visual representation of the problem and are used as a basis for a simple calculation procedure. The technique can deal with deterministic and stochastic problems and in the latter case is more general than decision trees. The decision network approach meets the need for a method of solution for multi-stage decision problems which is easily understood, helps the user to visualize the nature of the problem and is routine in application.     

多集值信息表中的多粒度模糊决策论粗糙集

多粒度粗糙集和决策论粗糙集是Pawlak粗糙集的重要推广,目前已成为人工智能研究的热点.然而,它们大多处理的都是单值信息系统中的问题.而实际生活中绝大多数都是处理多值问题,为了解决这一问题,在多集值信息表中将多粒粗糙集与模糊决策论粗糙集相结合进行研究,提出了其在乐观,悲观情形下的上下近似,研究了一些相关性质并给出了多集值信息表中的多粒度模糊决策论粗糙集精度、粗度的概念,最后通过一个具体例子验证其有效性.     

Quantifying vagueness in decision models

When modelling specific decision situations the decision maker often feels overstrained when he is asked for precise numerical quotations concerning the objectives or the constraints, whereas qualitative statements are easily given. In the recent past the theory of fuzzy sets has proven to be very useful for representing this type of information. Though it is quite advanced formally, the practical determination of its core elements, i.e. membership functions and operators, has only been explored to a very limited extent. This paper presents results of empirical research which focused on the problem of modelling vagueness, i.e. determining membership functions of fuzzy sets which are considered as quantitative representations of vague concepts such as ‘young man’, ‘long sticks’, ‘high profits’, etc.     

Minimizing Labor Requirements in a Periodic Vehicle Loading Problem

In this paper, we address a logistics problem that a manufacturer of auto parts in the north of Spain described to the authors. The manufacturer stores products in its warehouse until customers retrieve them. The customers and the manufacturer agree upon an order pickup frequency. The problem is to find the best pickup schedule, which consists of the days and times during the day that each customer is expected to retrieve his/her order. For a given planning horizon, the optimization problem is to minimize the labor requirements to load the vehicles that the customers use to pick up their orders. Heuristically, we approach this situation as a decision problem in two levels. At the first level, customers are assigned to a calendar, consisting of a set of days with the required frequency during the planning horizon. Then, for each day, the decision at the second level is to assign each customer to a time slot. The busiest time slot determines the labor requirement for a given day. Therefore, once customers have been assigned to particular days in the planning horizon, the second-level decision is a multiprocessor scheduling problem, where each time slot is the equivalent of a processor, and where the objective is to minimize the makespan. A metaheuristic procedure is developed for the problem of minimizing labor requirements in this periodic vehicle-loading problem and artificial as well as real data are used to assess its performance.     

On the empirical Bayes approach to multiple decision problems with sequential components

The empirical Bayes approach to multiple decision problems with a sequential decision problem as the component is studied. An empirical Bayesm-truncated sequential decision procedure is exhibited for general multiple decision problems. With a sequential component, an empirical Bayes sequential decision procedure selects both a stopping rule function and a terminal decision rule function for use in the component. Asymptotic results are presented for the convergence of the Bayes risk of the empirical Bayes sequential decision procedure.     

Performance benchmarking using interactive data envelopment analysis

This paper presents a procedure for performance benchmarking, that extends the performance measurement technique Data Envelopment Analysis (DEA), to incorporate the interactive decision procedure Interactive Multiple Goal Programming (IMGP). The resulting procedure is called Interactive Data Envelopment Analysis (IDEA). It is a decision support tool that helps decision makers to select performance benchmarks that are both feasible and desirable, and to identify benchmark partners that may be helpful in uncovering ways for achieving the selected performance standards. The IDEA concepts and characteristics are illustrated by means of an example IDEA assessment, using previously reported operating performance data of UK university departments.