Multiattribute utility functions,partial information on coefficients,and efficient choice

Summary The expected utility analysis of decision under risk needs information on the alternatives and on the decision maker's preferences which in many practical situations are difficult to obtain. This paper presents a procedure for choosing between multiattribute risky alternatives when the probabilities of outcomes are known, the utility function is general multilinear (i.e., can be decomposed into sums and products of univariate utility functions), and there is some partial information on univariate utilities (viz. increasingness) and arbitrary partial information on the scaling coefficients. Pairwise comparisons in the set of alternatives yield a subset which is efficient under the given partial information. Additive and multiplicative utility functions are special cases of the multilinear one. The paper gives particular attention to linear partial information (LPI) on coefficients, which is obtained by standard assessment procedures. The approach can be combined with dominance procedures which use other partial information as LPI on probabilities.     

Solving dominance and potential optimality in imprecise multi-attribute additive problems

We consider the multicriteria decision-making problem where there is partial information on decision maker preferences, represented by means of an imprecise multiattribute additive utility function, and where the consequences of the alternatives or strategies are also possibly imprecise. Under these circumstances we consider how useful problem-solving concepts, namely nondominated, potentially optimal, adjacent potentially optimal alternatives, can be analytically computed. Thus, the problem can be solved much more efficiently using the classical methodology of linear programming.     

Bayesian decision analysis and reliability certification

Reliability certification is set as a problem of Bayesian Decision Analysis. Uncertainties about the system reliability are quantified by assuming the parameters of the models describing the stochastic behavior of components as random variables. A utility function quantifies the relative value of each possible level of system reliability after having been accepted or the opportunity loss of the same level if the system has been rejected. A decision about accepting or rejecting the system can be made either on the basis of the existing prior assessment of uncertainties or after obtaining further information through testing of the components or the system at a cost. The concepts of value of perfect information, expected value of sample information and the expected net gain of sampling are specialized to the reliability of a multicomponent system to determine the optimum component testing scheme prior to deciding on the system's certification. A component importance ranking is proposed on the basis of the expected value of perfect information about the reliability of each component. The proposed approach is demonstrated on a single component system failing according to a Poisson random process and with natural conjugate prior probability density functions (pdf) for the failure rate and for a multicomponent system under general assumptions.     

Using Interactive Multiobjective Optimization in Continuous Casting of Steel

We discuss some pros and cons of using different types of multiobjective optimization methods for demanding real-life problems like continuous casting of steel. In particular, we compare evolutionary approaches that are used for approximating the set of Pareto-optimal solutions to interactive methods where a decision maker actively takes part and can direct the solution process to such Pareto-optimal solutions that are interesting to her/him. Among the latter type of methods, we describe an interactive classification-based multiobjective optimization method: NIMBUS. NIMBUS converts the original objective functions together with preference information coming from the decision maker into scalar-valued optimization problems. These problems can be solved using any appropriate underlying solvers, like evolutionary algorithms. We also introduce an implementation of NIMBUS, called IND-NIMBUS, for solving demanding multiobjective optimization problems defined with different modelling and simulation tools. We apply NIMBUS and IND-NIMBUS in an optimal control problem related to the secondary cooling process in the continuous casting of steel. As an underlying solver we use a real-coded genetic algorithm. The aim in our problem is to find a control resulting with steel of the best possible quality, that is, minimizing the defects in the final product. Since the constraints describing technological and metallurgical requirements are so conflicting that they form an empty feasible set, we formulate the problem as a multiobjective optimization problem where constraint violations are minimized.     

A methodology for evaluating appropriateness of new energy resources in rural applications

In this paper a method to determine the appropriateness of energy resources in rural applications is discussed. Feasible energy resources are comparatively evaluated using eight attributes representing the criteria of appropriateness. ‘Appropriateness’ is defined as a linear combination of attribute weights multiplied by attribute attainment levels which have been mapped into utility for decision-makers. The uncertainty in data is handled using Monte-Carlo techniques. Sample results indicate a set of dominant energy resources for a particular task. This method can be applied in real-life decision making concerning energy resources for rural applications.     

Multi-criteria location planning for public facilities in tsunami-prone coastal areas

We present a model for multi-objective decision analysis with respect to the location of public facilities as schools in areas near to coasts, taking risks of inundation by tsunamis into account. A mathematical programming formulation with three objective functions is given. The first objective function is a weighted mean of a minisum and a maximum coverage criterion. The second objective function expresses risk by possible tsunami events; for quantifying this risk, a statistical model for tsunami occurrences by Kaistrenko and Pinegina is applied. The third criterion represents costs. For the solution of the multi-objective optimization problem, we propose a heuristic approach based on the NSGA-II algorithm and compare it with a decomposition technique where the region under consideration is partitioned into smaller sub-regions, and the problem is solved for each separate subregion either exactly or heuristically. Both approaches are tested on two real-life instances from southern Sri Lanka.     

Sensitivity analysis scheme of boundary value problem of 2D Poisson equation by using Trefftz method

This paper describes the sensitivity analysis of the boundary value problem of two-dimensional Poisson equation by using Trefftz method. A non-homogeneous term of two-dimensional Poisson equation is approximated with a polynomial function in the Cartesian coordinates to derive a particular solution. The unknown function of the boundary value problem is approximate with the superposition of the T-complete functions of Laplace equation and the derived particular solution with unknown parameters. The parameters are determined so that the approximate solution satisfies boundary conditions. Since the T-complete functions and the particular solution are regular, direct differentiation of the expression results leads to the sensitivity expressions. The boundary-specified value and the shape parameter are taken as the variables of the sensitivity analysis to formulate the sensitivity analysis methods. The present scheme is applied to some numerical examples in order to confirm the validity of the present algorithm.     

A bi-objective model for design and analysis of sustainable intermodal transportation systems: a case study of Turkey

This paper presents a mixed-integer linear optimisation model to analyse the intermodal transportation systems in the Turkish transportation industry. The solution approach includes mathematical modelling, data analysis from real-life cases and solving the resulting mathematical programming problem to minimise total transportation cost and carbon dioxide emissions by using two different exact solution methods in order to find the optimal solutions. The novel approach of this paper generates Pareto solutions quickly and allows the decision makers to identify sustainable solutions by using a newly developed solution methodology for bi-objective mixed-integer linear problems in real-life cases.     

Preferential reducts and constructs in robust multiple criteria ranking and sorting

We revisit the multiple criteria ranking and sorting methods based on ordinal regression, which accept preference information in the form of, respectively, pairwise comparisons or assignment examples for some reference alternatives. Robust ordinal regression methods consider the whole set of value functions reproducing these holistic statements provided at the input. Its impact on the recommendation is expressed in terms of the necessary and possible preference relations or assignments. We propose methods for generating explanations of this impact, showing pieces of preference information provided by the decision maker (DM), which led to the observed outcomes. In particular, the minimal set of preference information pieces, called preferential reduct, is identified to justify some result observable for the whole set of compatible value functions (e.g., the truth of the necessary relation for some pair of alternatives). Further, the maximal set of preference information pieces, called preferential construct, is discovered to reveal the conditions under which some result non-observable for the whole set of compatible value functions (e.g., the falsity of the possible relation for some pair of alternatives) is possible. Knowing such explanations, the DM can better understand the impact of each piece of preference information on the result and, in consequence, get conviction about the obtained recommendation.     

Maintenance optimisation from a decision theoretical point of view

Maintenance optimisation is rarely discussed from a decision theoretical point of view. It is believed that maintenance programmes may benefit from using decision theory in a more formal manner. In decision theory there is a sharp line of demarcation between establishing requirements and preferences on one side, and methods for seeking an optimal solution in accordance with the requirements and preferences on the other side. We discuss requirements and preferences concerning maintenance, and how to model these by value and utility functions. Next we discuss how to choose the optimum set of maintenance actions. Influence diagrams are introduced to visualise the relation between maintenance actions, system characteristics and value functions. Finally an illustrative example is given.     

Multiple objective facility layout: a heuristic to generate efficient alternatives

Most multiple-objective approaches to solving the facility layout problem generate different layouts by varying the weights of the objectives using heuristic methods. It is possible, however, to generate the same layout using different weights. It is also possible that many efficient layouts will not be generated because it is not clear which weights would generate them. Furthermore, there may be some efficient layouts that cannot be generated regardless of what weights are used. In this paper we propose a heuristic algorithm for generating efficient layouts through pairwise interchange of departments. We show that if the decision maker's preferences for the selection of the best layout cannot be expressed as a linear utility function, it is essential to use other techniques such as the one developed in this paper, rather than weighting techniques, to generate efficient layouts. An example and some experiments are discussed.     

Robot selection using an integrated approach based on quality function deployment and fuzzy regression

The decision-makers have been experiencing difficulties in determining the most suitable robot alternative due to the increase in number of robots and the diversity in their application areas. A robust decision framework for robot selection should consider multiple and conflicting criteria and the dependencies among them. This paper introduces a decision model for robot selection based on quality function deployment (QFD) and fuzzy linear regression. The proposed approach benefits from the fact that QFD focuses on delivering value by taking into account the customer requirements and then by deploying this information throughout the development process, and applies this perspective to robot selection. Fuzzy linear regression is considered as an alternative decision aid for robot selection problems where imprecise relationships among system parameters exist. An example robot selection problem is presented to illustrate the proposed decision-making approach.     

潜在市场需求不确定下的风险厌恶供应链决策

考虑新产品投入市场时潜在市场需求不确定下的供应链风险厌恶,构造了一个考虑制造商和零售商的效用函数,建立了基于广告费用、服务水平以及销售价格的最优短期决策模型,分析了需求风险和供应链成员风险容忍水平对供应链最优短期决策的影响,采用数值仿真分析了供应链最优长期决策。研究结果表明：最优广告费用分别是需求风险和风险容忍水平的单调递增和递减函数,最优销售价格和服务水平则因行业等因素不同与需求风险和风险容忍水平呈U型或倒U型关系;当市场环境变化时,销售价格和服务水平在不同市场条件下应做出不同调整,而广告费用调整与市场特征无关;从长期来看制造商可以达到效用最大化,而零售商效用只能次优。a     

A multiple criteria decision making approach for the assembly line balancing problem

In practice, measuring total profit for a given assembly line balancing (ALB) problem is an involved process that is sometimes impossible because of much uncertainty and unavailability of data. In this paper, ALB is formulated as a multiple criteria problem where several easily quantifiable criteria (objectives) and constraints are defined. Objective functions include number of stations, cycle time, and operations cost, all to be minimized. After a discussion of applications and an overview of multiple criteria decision making (MCDM) approaches for ALB, the MCDM-ALB problem is formulated. Basic definitions and properties of MCDM for ALB are outlined and then an interactive MCDM approach is developed for solving the MCDM-ALB problem. To solve the problem, the decision maker (DM) interactively responds to paired comparisons of multicriteria alternatives. Through a limited number of interactions with the DM, the most preferred alternative is obtained. Many unexplored alternatives are eliminated by using a one-dimensional multiple criteria search. To present the DM's preference, we use the most flexible and general class of utility functions; namely, either quasi-concave or quasi-convex utility functions. An example is solved and computational experiments are reported. It is demonstrated that the bicriteria ALB, cycle time versus number of stations, can be easily solved by using the developed procedure. For the case that there are different criteria, an improved goal programming is developed to solve the MCDM-ALB problem. The motivation for development of the method, based on a case study of a lamp-making plant of the General Electric Company, is discussed.     

Decisiones en incertidumbre con multiatributos

This paper gives a formalization of the relation between the Debreu’s value function and the Von Neumann’s utility function, with a generalization of this result for their respective vectorial functions. Finally the problem of incorporating complementary information is consisdered.

     

Dynamic portfolio choice: a simulation-and-regression approach

Simulation-and-regression algorithms have become a standard tool for solving dynamic programs in many areas, in particular financial engineering and computational economics. In virtually all cases, the regression is performed on the state variables, for example on current market prices. However, it is possible to regress on decision variables as well, and this opens up new possibilities. We present numerical evidence of the performance of such an algorithm, in the context of dynamic portfolio choices in discrete-time (and thus incomplete) markets. The problem is fundamentally the one considered in some recent papers that also use simulations and/or regressions: discrete time, multi-period reallocation, and maximization of terminal utility. In contrast to that literature, we regress on decision variables and we do not rely on Taylor expansions nor derivatives of the utility function. Only basic tools are used, bundled in a dynamic programming framework: simulations—which can be black-boxed—as a representation of exogenous state variable dynamics; regression surfaces, as non-anticipative representations of expected future utility; and nonlinear or quadratic optimization, to identify the best portfolio choice at each time step. The resulting approach is simple, highly flexible and offers good performance in time and precision.     

Measure recognition problem

This is a paper in mathematics, specifically in set theory. On the example of the measure recognition problem (MRP), the paper highlights the phenomenon of the utility of a multidisciplinary mathematical approach to a single mathematical problem, in particular, the value of a set-theoretic analysis. MRP asks if for a given Boolean algebra, B, and a property, Phi, of measures, one can recognize by purely combinatorial means if B supports a strictly positive measure with property Phi. The most famous instance of this problem is MRP (countable additivity), and in the first part of the paper, we survey the known results on this and some other problems. We show how these results naturally lead to asking about two other specific instances of the problem MRP, namely MRP (non-atomic) and MRP (separable). Then, we show how our recent work gives an easy solution to the former of these problems and some partial information about the latter. The long-term goal of this line of research is to obtain a structure theory of Boolean algebras that support a finitely additive strictly positive measure, along the lines of Maharam theorem, which gives such a structure theorem for measure algebras.     

Decisions with uncertain alternatives

This paper analyses decision models with an uncertain set of alternatives defined by a deterministic objective function and constraints with uncertain coefficients. Here, in contrast to decisions under risk, the stochastic distribution of uncertain coefficients is not known. On the basis of an uncertain multiobjective decision model, we define efficient alternatives and formulate deterministic surrogate models. Uncertain decision models with recourse are introduced, and we present solution concepts that combine approaches from both multiobjective decision making and decision models with uncertain objective functions. These approaches are discussed in relation to stochastic and fuzzy programming and to models with linear partial information. In addition, ramifications of this particular approach are explored.     

A synchronous reference point-based interactive method for stochastic multiobjective programming

Stochastic multiobjective programming models are highly complex problems, due to the presence of random parameters, together with several conflicting criteria that have to be optimized simultaneously. Even the widely used concept of efficiency has to be redefined for these problems. The use of interactive procedures can somehow ease this complexity, allowing the decision maker to learn about the problem itself, and to look for his most preferred solution. Reference point schemes can be adapted to stochastic problem, by asking the decision maker to provide, not only desirable levels for the objectives, but also the desired probability to achieve these values. In this paper, we analyze the different kinds of achievement scalarizing functions that can be used in this environment, and we study the efficiency (in the stochastic sense) of the different solutions obtained. As a result, a synchronous interactive method is proposed for a class of stochastic multiobjective problems, where only the objective functions are random. Several solutions can be generated by this new method, making use of the same preferential information, using the different achievement scalarizing functions. The preferential information (levels and probabilities for the objectives) is incorporated into the achievement scalarizing functions in a novel way to generate the new solutions. The special case of linear normal problems is addressed separately. The performance of the algorithm is illustrated with a numerical example.     

Trefftz‐based dual reciprocity method for hyperbolic boundary value problems

A new dual reciprocity‐type approach to approximating the solution of non‐homogeneous hyperbolic boundary value problems is presented in this paper. Typical variants of the dual reciprocity method obtain approximate particular solutions of boundary value problems in two steps. In the first step, the source function is approximated, typically using radial basis, trigonometric or polynomial functions. In the second step, the particular solution is obtained by analytically solving the non‐homogeneous equation having the approximation of the source function as the non‐homogeneous term. However, the particular solution trial functions obtained in this way typically have complicated expressions and, in the case of hyperbolic problems, points of singularity. Conversely, the method presented here uses the same trial functions for both source function and particular solution approximations. These functions have simple expressions and need not be singular, unless a singular particular solution is physically justified. The approximation is shown to be highly convergent and robust to mesh distortion. Any boundary method can be used to approximate the complementary solution of the boundary value problem, once its particular solution is known. The option here is to use hybrid‐Trefftz finite elements for this purpose. This option secures a domain integral‐free formulation and endorses the use of super‐sized finite elements as the (hierarchical) Trefftz bases contain relevant physical information on the modeled problem. Copyright © 2015 John Wiley & Sons, Ltd.