Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods

A key step in solving minimax distributionally robust optimization (DRO) problems is to reformulate the inner maximization w.r.t. probability measure as a semiinfinite programming problem through Lagrange dual. Slater type conditions have been widely used for strong duality (zero dual gap) when the ambiguity set is defined through moments. In this paper, we investigate effective ways for verifying the Slater type conditions and introduce other conditions which are based on lower semicontinuity of the optimal value function of the inner maximization problem. Moreover, we propose two discretization schemes for solving the DRO with one for the dualized DRO and the other directly through the ambiguity set of the DRO. In the absence of strong duality, the discretization scheme via Lagrange duality may provide an upper bound for the optimal value of the DRO whereas the direct discretization approach provides a lower bound. Two cutting plane schemes are consequently proposed: one for the discretized dualized DRO and the other for the minimax DRO with discretized ambiguity set. Convergence analysis is presented for the approximation schemes in terms of the optimal value, optimal solutions and stationary points. Comparative numerical results are reported for the resulting algorithms.

     

Distributionally robust discrete optimization with Entropic Value-at-Risk

We study the discrete optimization problem under the distributionally robust framework. We optimize the Entropic Value-at-Risk, which is a coherent risk measure and is also known as Bernstein approximation for the chance constraint. We propose an efficient approximation algorithm to resolve the problem via solving a sequence of nominal problems. The computational results show that the number of nominal problems required to be solved is small under various distributional information sets.     

Sourcing decision under disruption risk with supply and demand uncertainty: A newsvendor approach

Multi-sourcing is considered as a common practice to hedge against supply disruption risk. In this context, this paper proposes two models for optimal order allocation in newsvendor setting, where both supply and demand are uncertain. The first model considers a risk neutral decision maker who maximizes the total expected profit under disruption risk. The second one is for a risk averse decision maker who does so under service level constraints. Analytical closed form solutions for both the models are derived. To overcome the computational complexity of the exact optimal solution, two algorithms are developed to generate optimal order quantity and the corresponding set of suppliers. The solutions with exact optimization algorithms and the proposed ones are illustrated and compared with numerical examples. The results show that the proposed algorithms give the exact optimal solution while being tractable. Finally, a case study is used to illustrate the applicability of the proposed model.     

Improving the computational efficiency in a global formulation (GLIDE) for interactive multiobjective optimization

In this paper, we present a new general formulation for multiobjective optimization that can accommodate several interactive methods of different types (regarding various types of preference information required from the decision maker). This formulation provides a comfortable implementation framework for a general interactive system and allows the decision maker to conveniently apply several interactive methods in one solution process. In other words, the decision maker can at each iteration of the solution process choose how to give preference information to direct the interactive solution process, and the formulation enables changing the type of preferences, that is, the method used, whenever desired. The first general formulation, GLIDE, included eight interactive methods utilizing four types of preferences. Here we present an improved version where we pay special attention to the computational efficiency (especially significant for large and complex problems), by eliminating some constraints and parameters of the original formulation. To be more specific, we propose two new formulations, depending on whether the multiobjective optimization problem to be considered is differentiable or not. Some computational tests are reported showing improvements in all cases. The generality of the new improved formulations is supported by the fact that they can accommodate six interactive methods more, that is, a total of fourteen interactive methods, just by adjusting parameter values.     

On consistency measures of linguistic preference relations

Inspired by the concept of deviation measure between two linguistic preference relations, this paper further defines the deviation measure of a linguistic preference relation to the set of consistent linguistic preference relations. Based on this, we present a consistency index of linguistic preference relations and develop a consistency measure method for linguistic preference relations. This method is performed to ensure that the decision maker is being neither random nor illogical in his or her pairwise comparisons using the linguistic label set. Using this consistency measure, we discuss how to deal with inconsistency in linguistic preference relations, and also investigate the consistency properties of collective linguistic preference relations. These results are of vital importance for group decision making with linguistic preference relations.     

Multicriteria decision support using rules that represent rough-graded preference relations

The approach described in this paper aims to support multicriteria choice and ranking of actions when the input preference information acquired from the decision maker is a graded comprehensive pairwise comparison (or ranking) of reference actions. It is based on decision-rule preference model induced from a rough approximation of the graded comprehensive preference relation among the reference actions. The set of decision rules applied to a new set of actions provides a graded fuzzy preference relation, which can be exploited by weighted-fuzzy net flow score or lexicographic-fuzzy net flow score procedure to obtain a final recommendation in terms of the best choice or of the ranking.     

An interactive fuzzy satisficing method for multiobjective stochastic linear programming problems through an expectation model

For decision making problems involving uncertainty, both stochastic programming as an optimization method based on the theory of probability and fuzzy programming representing the ambiguity by fuzzy concept have been developing in various ways. In this paper, we focus on multiobjective linear programming problems with random variable coefficients in objective functions and/or constraints. For such problems, as a fusion of these two approaches, after incorporating fuzzy goals of the decision maker for the objective functions, we propose an interactive fuzzy satisficing method for the expectation model to derive a satisficing solution for the decision maker. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.     

An exact solution to a robust portfolio choice problem with multiple risk measures under ambiguous distribution

This paper proposes a unified framework to solve distributionally robust mean-risk optimization problem that simultaneously uses variance, value-at-risk (VaR) and conditional value-at-risk (CVaR) as a triple-risk measure. It provides investors with more flexibility to find portfolios in the sense that it allows investors to optimize a return-risk profile in the presence of estimation error. We derive a closed-form expression for the optimal portfolio strategy to the robust mean-multiple risk portfolio selection model under distribution and mean return ambiguity (RMP). Specially, the robust mean-variance, robust maximum return, robust minimum VaR and robust minimum CVaR efficient portfolios are all special instances of RMP portfolios. We analytically and numerically show that the resulting portfolio weight converges to the minimum variance portfolio when the level of ambiguity aversion is in a high value. Using numerical experiment with simulated data, we demonstrate that our robust portfolios under ambiguity are more stable over time than the non-robust portfolios.     

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.     

A decision theory for partially consonant belief functions

Partially consonant belief functions (pcb), studied by Walley, are the only class of Dempster-Shafer belief functions that are consistent with the likelihood principle of statistics. Structurally, the set of foci of a pcb is partitioned into non-overlapping groups and within each group, foci are nested. The pcb class includes both probability function and Zadeh’s possibility function as special cases. This paper studies decision making under uncertainty described by pcb. We prove a representation theorem for preference relation over pcb lotteries to satisfy an axiomatic system that is similar in spirit to von Neumann and Morgenstern’s axioms of the linear utility theory. The closed-form expression of utility of a pcb lottery is a combination of linear utility for probabilistic lottery and two-component (binary) utility for possibilistic lottery. In our model, the uncertainty information, risk attitude and ambiguity attitude are separately represented. A tractable technique to extract ambiguity attitude from a decision maker behavior is also discussed.     

A new distributionally robust p-hub median problem with uncertain carbon emissions and its tractable approximation method

The p-hub median problem is to determine the optimal location for p hubs and assign the remaining nodes to hubs so as to minimize the total transportation costs. Under the carbon cap-and-trade policy, we study this problem by addressing the uncertain carbon emissions from the transportation, where the probability distributions of the uncertain carbon emissions are only partially available. A novel distributionally robust optimization model with the ambiguous chance constraint is developed for the uncapacitated single allocation p-hub median problem. The proposed distributionally robust optimization problem is a semi-infinite chance-constrained optimization model, which is computationally intractable for general ambiguity sets. To solve this hard optimization model, we discuss the safe approximation to the ambiguous chance constraint in the following two types of ambiguity sets. The first ambiguity set includes the probability distributions with the bounded perturbations with zero means. In this case, we can turn the ambiguous chance constraint into its computable form based on tractable approximation method. The second ambiguity set is the family of Gaussian perturbations with partial knowledge of expectations and variances. Under this situation, we obtain the deterministic equivalent form of the ambiguous chance constraint. Finally, we validate the proposed optimization model via a case study from Southeast Asia and CAB data set. The numerical experiments indicate that the optimal solutions depend heavily on the distribution information of carbon emissions. In addition, the comparison with the classical robust optimization method shows that the proposed distributionally robust optimization method can avoid over-conservative solutions by incorporating partial probability distribution information. Compared with the stochastic optimization method, the proposed method pays a small price to depict the uncertainty of probability distribution. Compared with the deterministic model, the proposed method generates the new robust optimal solution under uncertain carbon emissions.     

考虑风险偏好的动态生产库存问题的鲁棒优化模型

不同阶段需求不确定情况下,决策者的风险偏好和生产过程中的废品处理影响着供应链生产库存管理和供应链整体效益。本文考虑决策者风险偏好下,构建了包含I个生产者企业,一个库存点和一个废物处理基地的T阶段动态供应链生产库存框架,建立了椭球型需求不确定集下,以追求整体收益最大化为目标的不确定优化模型,并应用鲁棒优化理论得到了数据确定性线性鲁棒对应模型,讨论了模型解的可靠性和有效性。最后的算例表明,只有当决策者风险偏好参数在一定范围内时,才会存在满足条件且具有较高可靠性的鲁棒决策,验证了该鲁棒优化模型的合理性。     

Capturing decision maker preference: Experimental comparison of decision analysis and MCDM techniques

This paper describes the results of a laboratory study which investigates preference in decision making under certainty with multiple, conflicting objectives and continuous decision variables. Techniques for solving such problems are taken from the fields of decision analysis and optimization: the SMART technique for the former and both the NAIVE and Zionts-Wallenius techniques for the latter. The purpose of the experiment is to determine the ability of each technique to correctly capture decision maker preference. In addition, the relative preference of the decision maker for each technique was obtained. The experiment was conducted on a random sample of business school undergraduates and involved a decision with three criteria. The results give insight into the use of several techniques when confronted with decisions with multiple criteria.     

Multiattribute utility analysis for policy selection and financing for the preservation of the forest

In this paper, we examine effective policies for financing and activities for the preservation of the forest on Mount Ryuoh in the city of Higashi-Hiroshima by multiattribute utility analysis. In multiattribute utility analysis, we deal with decision making problems with multiple attributes and select the most effective solution among several alternatives by deriving preference of the decision maker. Although in our decision making problem, the decision maker is a representative of a hypothetical nonprofit organization established for the preservation of the forest, the decision maker gives serious consideration to intentions of several groups of people receiving the benefit from the mountain, and then from this viewpoint, our problem can be interpreted as a group decision making problem.     

一种考虑决策者风险偏好的区间数多属性决策方法

针对属性值以区间数形式给出的多属性决策问题,提出了一种决策分析方法。在本文中,首先描述了属性值为区间数形式的多属性决策问题;然后通过引入决策者的风险偏好因子将区间数决策信息映射为实数值决策信息,并依据属性值与属性均值绝对偏差的大小确定了属性的权重,在此基础上依据所得权重给出了基于加权和法的方案排序方法,通过对风险偏好因子的不同取值还可进行方案排序的灵敏度分析。最后,通过一个算例说明了本文给出方法的可行性和有效性。     

Hierarchical multi-objective decision systems for general resource allocation problems

We consider optimization methods for hierarchical power-decentralized systems composed of a coordinating central system and plural semi-autonomous local systems in the lower level, each of which possesses a decision making unit. Such a decentralized system where both central and local systems possess their own objective function and decision variables is a multi-objective system. The central system allocates resources so as to optimize its own objective, while the local systems optimize their own objectives using the given resources. The lower level composes a multi-objective programming problem, where local decision makers minimize a vector objective function in cooperation. Thus, the lower level generates a set of noninferior solutions, parametric with respect to the given resources. The central decision maker, then, parametric with respect to the given resources. The central decision maker, then, chooses an optimal resource allocation and the best corresponding noninferior solution from among a set of resource-parametric noninferior solutions. A computational method is obtained based on parametric nonlinear mathematical programming using directional derivatives. This paper is concerned with a combined theory for the multi-objective decision problem and the general resource allocation problem.The authors are indebted to Professor G. Leitmann for his valuable comments and suggestions.     

Interactive outranking approaches for multicriteria decision-making problems with imprecise information

PROMETHEE is a powerful method, which can solve many multiple criteria decision making (MCDM) problems. It involves sophisticated preference modelling techniques but requires too much a priori precise information about parameter values (such as criterion weights and thresholds). In this paper, we consider a MCDM problem where alternatives are evaluated on several conflicting criteria, and the criterion weights and/or thresholds are imprecise or unknown to the decision maker (DM). We build robust outranking relations among the alternatives in order to help the DM to rank the alternatives and select the best alternative. We propose interactive approaches based on PROMETHEE method. We develop a decision aid tool called INTOUR, which implements the developed approaches.     

基于区间直觉梯形模糊数的改进TOPSIS多属性决策方法

研究了属性权重完全未知的区间直觉梯形模糊数的多属性决策问题,结合TOPSIS方法定义了相对贴近度及总贴近度公式.首先由区间直觉梯形模糊数的Hamming距离给出了每个方案的属性与正负理想解的距离,基于此,给出了相对贴近度矩阵,根据所有决策方案的综合贴近度最小化建立多目标规划模型,从而确定属性的权重值,然后根据区间直觉梯形模糊数的加权算数平均算子求出各决策方案的总贴近度,根据总贴近度的大小对方案进行排序;最后,通过实例分析说明该方法的可行性和有效性.     

Mixed uncertainty sets for robust combinatorial optimization

In robust optimization, the uncertainty set is used to model all possible outcomes of uncertain parameters. In the classic setting, one assumes that this set is provided by the decision maker based on the data available to her. Only recently it has been recognized that the process of building useful uncertainty sets is in itself a challenging task that requires mathematical support. In this paper, we propose an approach to go beyond the classic setting, by assuming multiple uncertainty sets to be prepared, each with a weight showing the degree of belief that the set is a “true” model of uncertainty. We consider theoretical aspects of this approach and show that it is as easy to model as the classic setting. In an extensive computational study using a shortest path problem based on real-world data, we auto-tune uncertainty sets to the available data, and show that with regard to out-of-sample performance, the combination of multiple sets can give better results than each set on its own.

     

Using box indices in supporting comparison in multiobjective optimization

Because of the conflicting nature of criteria or objectives, solving a multiobjective optimization problem typically requires interaction with a decision maker who can specify preference information related to the objectives in the problem in question. Due to the difficulties of dealing with multiple objectives, the way information is presented plays a very important role. Questions posed to the decision maker must be simple enough and information shown must be easy to understand. For this purpose, visualization and graphical representations can be useful and constitute one of the main tools used in the literature. In this paper, we propose to use box indices to represent information related to different solution alternatives of multiobjective optimization problems involving at least three objectives. Box indices are an intelligible and easy to handle way to represent data. They are based on evaluating the solutions in a natural and rough enough scale in order to let the decision maker easily recognize the main characteristics of a solution at a glance and to facilitate comparison of two or more solutions in an easily understandable way.