Decision Support
Influence diagrams with super value nodes involving imprecise information

https://doi.org/10.1016/j.ejor.2006.03.021Get rights and content

Abstract

The concept of super value nodes was established to allow dynamic programming to be performed within the theory of influence diagrams and to reduce the computational complexity in solving problems by means of influence diagrams. This paper is focused on how influence diagrams with super value nodes are affected by the presence of imprecise information. We analyze how to reduce the complexity when evaluating an influence diagram in this framework by modelling these kinds of nodes and random magnitudes in terms of fuzzy random variables. Finally, an applied example of the theoretical results is developed.

Introduction

Influence diagrams (ID’s) have been applied for several years as a graphical model to formulate and solve decision problems. They were introduced at the beginning of the eighties by Howard and Matheson (1981). Since then thorough studies on ID’s have been developed like those by Olmsted, 1983, Shachter, 1986, Smith, 1989a, Smith, 1989b, Tatman and Shachter, 1990, Smith et al., 1993, Shenoy, 2000.

Basically an ID is an acyclic directed graph, whose nodes are of three different kinds: chance nodes, which are associated with random magnitudes, decision nodes, associated with action spaces, and value nodes, associated with the utility or objective function. These nodes are represented by circles, rectangles and diamond shapes respectively. As we will see later, super value nodes are value nodes constructed by adding or multiplying other value nodes.

Directed arcs represent different influences between nodes they connect. So, an arc to a value node means that the utility function depends explicitly on the action spaces or random magnitudes with an arc from their associated nodes to a value node.

Arcs from either chance nodes or decision nodes to one associated with a random magnitude mean that the probability associated with this may depend on the variable values or considered actions in the former nodes.

An arc from a node to a decision node means that the decision maker knows the variable value or considered action in the former node when he/she chooses an action associated with the latter node.

For any node B in an ID, we will denote by PB the set of all the direct predecessors of B, that is, those nodes from which there exists an arc to B from them. PB will denote the set of all the predecessors of B, that is, those nodes from which there exists a path to B from them. We will denote by SB the set of all the direct successors of B, that is, those nodes having an arc connected directly from node B, and by SB the set of all the successors of B or nodes having a path leading from node B. A node B is said to be a barren node if SB = ∅.

In order to solve a decision problem by means of an ID, we need firstly to represent graphically the problem in terms of an ID, taking into account possible relations between different elements of the problem, that is, we should identify random magnitudes, action spaces, the utility function and possible influences or relations among them.

Secondly, we should perform a sequence of graphical transformations to reduce step by step the ID to just a value node, therefore, we need to remove from the ID chance and decision nodes, and simultaneously we should develop mathematical rules associated with each graphical transformation.

These transformations are basically four: barren (chance or decision) node removal, chance node removal, decision node removal and arc reversal. Mathematical rules associated with them are respectively the elimination of irrelevant information, the computation of a conditional expectation, the maximization of the utility and the application of Bayes’ rule.

Shachter (1986) proves the convergence of the graphical process in the sense that there always exists a node removal transformation which can be performed (perhaps after an arc reversal operation), and so, if a decision problem can be modelled by means of an ID, then it can be solved by applying the mathematical rules associated with the graphical transformations.

Mathematical rules in ID’s with imprecise information were thoroughly studied in Rodríguez-Muñiz et al. (2005). In particular, this paper studies ID’s when both, the random magnitudes and the utility function are modelled by means of fuzzy random variables. However, the analysis of ID’s with super value nodes in this uncertainty context remained open. The main purpose of the present paper is to analyze how to develop transformations with super value nodes when ID’s are affected by the presence of fuzziness.

For the case of super value nodes, whose associated utility functions take real values, we should indicate that Tatman and Shachter (1990) develop an exhaustive and explanatory analysis.

The structure of the paper is as follows: in Section 2 we state some preliminaries concepts, supporting results are developed in Section 3, main transformations involving super value nodes are studied in Section 4, while Section 5 contains an illustrative example of the preceding results.

Section snippets

Preliminaries

In this section we collect some concepts about random sets, fuzzy sets and fuzzy random variables needed for our study. First of all, we start with some set classes.

Given BRd with 1  d < ∞, we will denote by P(B) the class of subsets of B, by K(B) the class of non-empty compact subsets of B, and Kc(B) will stand for the class of non-empty compact convex subsets of B.

Let Fc(R) denote the class of fuzzy subsets A:R[0,1] whose α-level sets are in the class Kc(R) for all α  [0, 1], where the α-level

Supporting results concerning fuzzy random variables

In this section we state several results of fuzzy random variables which will allow us to justify mathematically the effects of incorporating fuzziness in super value nodes and chance nodes in ID’s, namely, utility functions and random magnitudes will be modelled by means of fuzzy random variables.

Proposition 3.1

Let (Ω,A,P) be a probability space, let VK(R) and let X:ΩK(Rd) be an integrably bounded random set. We define the mapping Y:ΩP(Rd) given by Y(ω) = VX(ω), where VX(ω) = {vx : v  V, x  X(ω)}. Then

  • (i)

    Y(ω)K(Rd)

Graphical transformations on influence diagrams

In this section we analyze main transformations in ID’s with super value nodes when they are affected by imprecise information, this being modelled by means of fuzzy random variables. We will divide it into two subsections, one devoted to the utility function in our context and the other one to analyze the transformations on influence diagrams with separable utility functions.

Motivating example

This example is a modified version of the Reactor problem, firstly described by Covaliu and Oliver (1995) and then modified by Bielza and Shenoy (1999). We have used the decomposition of the probability distribution introduced by Bielza and Shenoy (1999) and we have modified the utilities so that they are quantified by imprecise labels.

An electricity company must decide whether to build (we will denote this decision by D2) a reactor of advanced design (this action will be denoted by a), a

Conclusions and final remarks

In this paper we have developed a mathematical model to handle multi-step decision problems by means of influence diagrams when the problem is affected by fuzziness and when the fuzzy utility function can be separated by addition or product in some arguments. We show that the same statistical rules of the classical case can be performed in our context, and the Tatman and Shachter (1990) algorithm to solve influence diagrams with super value nodes holds in the fuzzy case, by taking into account

Acknowledgements

The suggestions and comments of the Editor and two anonymous Referees have led to significant improvements in the original manuscript. The authors are in debt to the Spanish Ministry of Science and Technology for financing this research with Grant MCT-BFM2002-03263 and Grant MTM2005-02254.

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