The utility efficient set and its interactive reduction

We assume a decision situation under risk with incomplete information on preferences modelled as a vector utility function. We consider an additive aggregation of its components and partial information on the scaling constants. We develop the concept of utility efficiency to identify efficient strategies in discrete problems when the information about the scaling constants of the decision maker is in the form of a polyhedral cone. A characterization of the utility efficient set provides a practical way to compute such efficient strategies. We then discuss an interactive method based on the assessment of the scaling constants via an interactive paired comparison with its convergence. The method is complemented by a procedure to reduce the utility efficient set to aid in the process of reaching a final strategy.     

An interactive method for multiple-objective mathematical programming problems

An interactive method is developed for solving the general nonlinear multiple objective mathematical programming problems. The method asks the decision maker to provide partial information (local tradeoff ratios) about his utility (preference) function at each iteration. Using the information, the method generates an efficient solution and presents it to the decision maker. In so doing, the best compromise solution is sought in a finite number of iterations. This method differs from the existing feasible direction methods in that (i) it allows the decision maker to consider only efficient solutions throughout, (ii) the requirement of line search is optional, and (iii) it solves the problems with linear objective functions and linear utility function in one iteration. Using various problems selected from the literature, five line search variations of the method are tested and compared to one another. The nonexisting decision maker is simulated using three different recognition levels, and their impact on the method is also investigated.     

Multiple objectives decomposition–coordination methods for hierarchical organizations

In this paper, the integration of goal programming models and hierarchical programming models is analyzed. The systems under study are assumed to consist of interconnected subsystems with multiple goals in each. Three possible cases regarding the number of decision makers will be considered: (1) one decision maker for the overall goals and one decision maker for each subsystem, (2) conflicting decision makers who are interested in their subsystems, and (3) just one decision maker for the overall system. Next, conditions are stated under which the problem of obtaining satisfying solutions for problems (1) and (3) can be reduced to the problem of obtaining satisfying solutions for the case (2). In order to determine such solutions, hierarchical techniques which exploit the structure of a decomposable system are analyzed. The empirical implementation of the two algorithms proposed shows their efficiency in terms of processing time.     

An interactive decision procedure with multiple attributes under risk

Consider a finite set of alternatives under risk which have multiple attributes. MARPI is an interactive computer-based procedure to find an efficient choice in the sense of linear expected utility. The choice is based on incomplete information about the decision maker's preferences which is elicited and processed in a sequential way. The information includes qualitative properties of the multivariate utility function such as monotonicity, risk aversion, and separability. Further, in case of an additively separable utility function, bounds on the scaling constants are elicited, and preferences (not necessarily indifferences) between sure amounts and lotteries are asked from the decision maker. The lotteries are Bernoulli lotteries generated by MARPI using special strategies. At every stage of the procedure the efficient set of alternatives is determined with respect to the information elicited so far.The procedure has been fully implemented on a PC. The paper exhibits the basic ideas of MARPI and some details of its implementation.     

Utility efficiency and its approximation

Summary We assume a multi-attribute decision making problem under uncertainty with partial information on the decision maker's preferences, by a vector utility function with two components and imprecision over their scaling constants. We propose an approximation set whose determination may be easier than the one of the utility efficient set and we consider an interactive procedure which uses such approximation to decision aid. We study some nesting and convergence properties based on the interactive reduction of the approximation set. Finally, we illustrate the procedure with a numerical example.     

Information and preference reversals in lotteries

Several approaches have been proposed for evaluating information in expected utility theory. Among the most popular approaches are the expected utility increase, the selling price and the buying price. While the expected utility increase and the selling price always agree in ranking information alternatives, Hazen and Sounderpandian [11] have demonstrated that the buying price may not always agree with the other two. That is, in some cases, where the expected utility increase would value information A more highly than information B, the buying price may reverse these preferences. In this paper, we discuss the conditions under which all these approaches agree in a generic decision environment where the decision maker may choose to acquire arbitrary information bundles.     

A new PCCA method: IDRA

IDRA (Intercriteria Decision Rule Approach), a new MCDA (Multiple Criteria Decision Aid) method which adopts the PCCA (Pairwise Criterion Comparison Approach) methodolgoy, is presented. It is based on two principal hypotheses about intercriteria information furnished by the decision maker: the mixed utility function, i.e. it is assumed that in the decision process both tradeoff and importance intercriteria information are considered; and bounded consistency, i.e. no constraint is imposed on the consistency of the intercriteria information.     

Ranking generalized fuzzy numbers in fuzzy decision making based on the left and right transfer coefficients and areas

Although a number of recent studies have proposed ranking fuzzy numbers based on the deviation degree, most of them have exhibited several shortcomings associated with non-discriminative and counter-intuitive problems. In fact, none of the existing deviation degree methods has guaranteed consistencies between the ranking of fuzzy numbers and that of their images under all situations. They have also ignored decision maker’s attitude toward risk, which significantly influences final ranking result. To overcome the above-mentioned drawbacks, this study proposes a new approach for ranking fuzzy numbers that ensures full consideration for all information of fuzzy numbers. Accordingly, an overall ranking index is obtained by the integration of the information from the left and the right (LR) areas between fuzzy numbers, the centroid points of fuzzy numbers and the decision maker’s attitude toward risk. This new method is efficient for evaluating generalized fuzzy numbers and distinguishing symmetric fuzzy numbers. It also overcomes the shortcomings of the existing approaches based on deviation degree. Several numerical examples are provided to illustrate the superiority of the proposed approach. Lastly, a new fuzzy MCDM approach for generalized fuzzy numbers is proposed based on the proposed ranking approach and the concept of generalized fuzzy numbers. The proposed fuzzy MCDM approach does not require the normalization process and thus avoids the loss of information results from transforming generalized fuzzy numbers to normal form.     

Loss-averse newsvendor problem with supply risk

Modeling the manufacturer as a newsvendor, in this paper we study the ordering decisions of a loss-averse newsvendor with supply and demand uncertainties. Using the stylized newsvendor models, we analyse several key issues, including the effect of the newsvendor’s loss aversion, the effect of demand uncertainty, and the effect of supply uncertainty on the decision maker’s optimal decision under the procurement model, in which the decision maker only pays for the actual quantity received. Through our analysis, we find the following facts: the optimal order quantity decreases with respect to the degree of loss-aversion; the supply uncertainty induces the decision maker to order more than that in a deterministic environment; a stochastically larger demand always results in a larger order quantity and a larger expected utility; the optimal expected utility decreases in the demand volatility while the optimal order quantity may increase or decrease. Moreover, with numerical experiments, we demonstrate that the supply risk negatively affects the utility more than the demand risk does.     

Newsvendor problem under complete uncertainty: a case of innovative products

The paper presents a new scenario-based decision rule for the classical version of the newsvendor problem (NP) under complete uncertainty (i.e. uncertainty with unknown probabilities). So far, NP has been analyzed under uncertainty with known probabilities or under uncertainty with partial information (probabilities known incompletely). The novel approach is designed for the sale of new, innovative products, where it is quite complicated to define probabilities or even probability-like quantities, because there are no data available for forecasting the upcoming demand via statistical analysis. The new procedure described in the contribution is based on a hybrid of Hurwicz and Bayes decision rules. It takes into account the decision maker’s attitude towards risk (measured by coefficients of optimism and pessimism) and the dispersion (asymmetry, range, frequency of extremes values) of payoffs connected with particular order quantities. It does not require any information about the probability distribution.

     

Interactive scheme for a MOLP problem given two partial orders: One on variables and one on objectives

An interactive approach for solving a multiple objective linear programming (MOLP) problem is presented. This approach assumes that the decision maker (in contrast to the analyst) can establish a (preferential) partial order P_v on the decision variables and a partial order P_o on the objective functions. The solution approach presented in this paper requires minimal additional information in its interactive stage. Results are presented for a number of tests: both conceptual development and applicability in solving specific problems have been successfully tested.     

A simple Tabu Search method to solve the mixed-integer linear bilevel programming problem

Multilevel programming is characterized as mathematical programming to solve decentralized planning problems. The models partition control over decision variables among ordered levels within a hierarchical planning structure of which the linear bilevel form is a special case of a multilevel programming problem. In a system with such a hierarchical structure, the high-level decision making situations generally require inclusion of zero-one variables representing ‘yes-no’ decisions. We provide a mixed-integer linear bilevel programming formulation in which zero-one decision variables are controlled by a high-level decision maker and real-value decision variables are controlled by a low-level decision maker. An algorithm based on the short term memory component of Tabu Search, called Simple Tabu Search, is developed to solve the problem, and two supplementary procedures are proposed that provide variations of the algorithm. Computational results disclose that our approach is effective in terms of both solution quality and efficiency.     

A Radon-Nikodym approach to measure information

We consider a decision maker facing uncertainty which behaves as a subjective expected utility maximizer. The value of information is traditionally captured as a greater expected utility the decision maker can achieve by selecting a best strategy as information arrives. We deal with the limit process of being better informed, and introduce an information density function depending solely on the states that gives an exact least upper bound to being more informed. This information density function is given by a Radon-Nikodym-type theorem for set functions and is explicitly computed for the countable case.     

基于贝叶斯信息更新的风险规避库存策略研究

在贝叶斯库存控制研究中一个著名的结论是：当缺货需求不能被观测到时,最优贝叶斯库存水平总会高于短视策略库存水平,原因是决策者需要通过多订货来获取对需求分布的认识. 这是基于风险中性的研究,然后现实中决策者都期望规避风险. 基于贝叶斯信息更新研究了风险规避背景下需求部分可观测的多周期报童问题,决策者的周期内效用函数满足独立可加性公理. 通过引入非正规化概率,研究发现,对风险规避的决策者,当其效用函数具有不变绝对风险规避特征时,最优贝叶斯库存水平也会高于短视策略库存水平. 非正规化概率简化了动态规划方程与结果的证明.     

Decision making under uncertainty with unknown utility function and rank-ordered probabilities

We consider the ranking of decision alternatives in decision analysis problems under uncertainty, under very weak assumptions about the type of utility function and information about the probabilities of the states of nature. Namely, the following two assumptions are required for the suggested method: the utility function is in the class of increasing continuous functions, and the probabilities of the states of nature are rank-ordered. We develop a simple analytical method for the partial ranking of decision alternatives under the stated assumptions. This method does not require solving optimization programs and is free of the rounding errors.     

Optimal portfolio choice for an insurer with loss aversion

The problem of optimal investment for an insurance company attracts more attention in recent years. In general, the investment decision maker of the insurance company is assumed to be rational and risk averse. This is inconsistent with non fully rational decision-making way in the real world. In this paper we investigate an optimal portfolio selection problem for the insurer. The investment decision maker is assumed to be loss averse. The surplus process of the insurer is modeled by a Lévy process. The insurer aims to maximize the expected utility when terminal wealth exceeds his aspiration level. With the help of martingale method, we translate the dynamic maximization problem into an equivalent static optimization problem. By solving the static optimization problem, we derive explicit expressions of the optimal portfolio and the optimal wealth process.     

A posterior preference articulation approach to multiresponse surface optimization

In multiresponse surface optimization (MRSO), responses are often in conflict. To obtain a satisfactory compromise, the preference information of a decision maker (DM) on the tradeoffs among the responses should be incorporated into the problem. In most existing work, the DM expresses a subjective judgment on the responses through a preference parameter before the problem-solving process, after which a single solution is obtained. In this study, we propose a posterior preference articulation approach to MRSO. The approach initially finds a set of nondominated solutions without the DM’s preference information, and then allows the DM to select the best solution from among the nondominated solutions. An interactive selection method based on pairwise comparisons made by the DM is adopted in our method to facilitate the DM’s selection process. The proposed method does not require that the preference information be specified in advance. It is easy and effective in that a satisfactory compromise can be obtained through a series of pairwise comparisons, regardless of the type of the DM’s utility function.     

Multiobjective stochastic programming for feed formulation

The minimum cost linear programming model used traditionally for feed formulation does not take account of variability of nutrients in feed ingredients. Therefore, it may be that the nutrient requirements of the animal are not adequately met. In this paper, we show how a multiobjective stochastic model that permits confronting the cost of the ration with the probabilities of meeting the nutrient requirements of the animal can enhance the process of animal diet formulation. The model presented here does not require any a priori information from the decision maker, eliciting his preferences through an interactive process. This is the main advantage in relation to other models found in the literature for treating the problem of nutrient variability, which introduce stochastic constraints in the single objective minimum cost model requiring fixing the level of probability desired for each one of the nutrients in advance.     

Sensitivity analysis incorporating fuzzy evaluation for scaling constants of multiattribute utility functions

A multiattribute utility function can be represented by a function of single-attribute utility functions if the decision maker’s preference satisfies additive independence or mutually utility independence. Additive independence is a preference condition stronger than mutually utility independence, and the multiattribute utility function is in the additive form if the former condition is satisfied, otherwise it is in the multiplicative form. In this paper, we propose a method for sensitivity analysis of multiattribute utility functions in multiplicative form, taking into account the imprecision of the decision maker’s judgment in the procedures for determining scaling constants (attribute weights).     

Performance bounds for hierarchical systems under partial dynamic information

This paper discusses a general approach to obtain optimum performance bounds for (N+1)-person deterministic decision problems,N+1>2, with several levels of hierarchy and under partial dynamic information. Both cooperative and noncooperative modes of decision making are considered at the lower levels of hierarchy; in each case, it is shown that the optimum performance of the decision maker at the top of the hierarchy can be obtained by solving a sequence of open-loop (static) optimization problems. A numerical example included in the paper illustrates the general approach.