The value of additional information in multicriteria decision making choice problems with information imperfections

Abstract

Processing information is a key ingredient for decision making. In most decision-making cases, information is distributed across various sources that may differ in reliability and accuracy. Various sources and kinds of uncertainty are encountered in the same decision situation. “Information imperfections” is a general term that encompasses all kinds of “deficiencies” (such as uncertainty, imprecision, ambiguity, incompleteness) that may affect the quality of information at hand. In discrete multicriteria decision making, where several alternatives are assessed according to heterogeneous and conflicting criteria, information used to assess such alternatives can also be imperfect. It is rather natural, in such a context, to seek additional information to reduce these imperfections. This paper aims at extending the Bayesian model for assessing the value of additional information to multicriteria decision analysis in a context of imperfect information. A unified procedure for processing additional information has been proposed in a previous work. It leads to prior and posterior global preference relational systems. It will be extended here to include pre-posterior analysis where concepts such as the expected value of perfect information and the expected value of imperfect information are adapted to multicriteria decision making choice problems.

Appendices

Appendix 1: Concepts from evidence and possibility theories

Let $\varOmega$ be the universe of discernment. The theory of evidence (Shafer 1976), is based on the concept of belief mass, that is, a mapping m, called “Basic Belief Assignment”, BBA, from $P(\varOmega )$ the set of subsets of $\varOmega$ , to [0, 1], such that:

$$\begin{aligned} \sum _{B\in P(\varOmega )}{m(B)=1} \end{aligned}$$

(10)

For $B \subseteq \varOmega$, m(B), called “Basic Belief Mass” BBM, is the part of belief that supports B. Any non empty part F of $\varOmega$ such that $m(F) \ne 0$, is called a focal element.

The plausibility function of any subset B of $\varOmega$ is the sum of all the masses of the subsets D that intersect B:

$$\begin{aligned} P \ell (B)=\sum _{B \cap D \ne \emptyset }{m(D)} \end{aligned}$$

(11)

Let $\varOmega$ be a universe of discourse. $\varOmega$ is a finite set. Let $P(\varOmega )$ be the set of subsets of $\varOmega$. A possibility measure is a function $\prod : P(\varOmega )\rightarrow {}[0, 1]$ such that:

• $\prod (\emptyset )=0$, $\prod (\varOmega )=1$

• $\forall {} B_i \in P(\varOmega )$ disjoint subsets of $\varOmega , i=1,2,\ldots$ $\prod (\cup _{i=1,2\ldots }\, B_i) = sup_{i=1,2\ldots } \prod (B_i)$

The possibility distribution $\pi$ is a mapping from $\varOmega$ to [0, 1], such that:

$$\begin{aligned} \forall {} B \in P(\varOmega ), \prod (B)=sup_{\omega \in B} \pi (\omega ). \end{aligned}$$

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1.1 Transformation rules

For evidential criteria we use the pignistic transformation justified by Smets (1993):

$$\begin{aligned} BetP(\omega _h^j)=\sum _{B^j_{b'}: \omega ^j_h \in B^j_{h'} \subseteq P(\varOmega ^j)} \frac{m(B^j_{h'})}{|B^j_{h'}| (1-m(\phi ))}, \;\; \forall \; \omega _h^j \in \; \varOmega ^j, m(\phi ) \ne 1. \end{aligned}$$

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where $|B^j_{h'}|$ is the cardinality of $B^j_{h'}$ in $P(\varOmega )$ and $BetP(\omega _h^j)$ is the pignistic probability of $\omega _h^j$.

For possibilistic criteria, as mentioned in Sect. 2, possibility measures are a special case of plausibility functions when focal elements are embedded, meaning that focal elements can be viewed as a sequence:

$B_1 = \{\omega _1\}, B_2 = \{\omega _1, \omega _2\}, B_h = \{\omega _1, \omega _2,\ldots , \omega _h\}, B_H = \varOmega$ such that $\forall B \ne B_h,\; m(B) = 0$. It is possible though for a certain h, to have $m(B_h) = 0$.

The transformation rule between possibility, belief masses and probability measures Dubois et al. (1993) applies as follows:

Let $m_h$ the belief mass $m(B_h), h = 1,\ldots , H$ and $\Pi _h = \Pi (\{\omega _h\})$ where $\Pi$ is the possibility measure defined using the belief masses m. We have:

$$\begin{aligned} \left. \begin{array}{ll} \Pi _1=m_1+m_2+\cdots +m_H=1 &{} (when \;\; m(\phi )=0)\\ \Pi _2=m_2+\cdots +m_H &{}\\ \ldots &{}\\ \Pi _H=m_H &{}\\ \end{array} \right. \end{aligned}$$

In a more compact way:

• $\forall h=1,\ldots ,H, \;\; \Pi _h=\sum _{t=h}^{H} m_t \;$ and inversely

• $\forall h=1,\ldots ,H, \;\; m_h=\Pi _h - \Pi _{h+1} \;$ with $\; \Pi _{H+1}=0$ and $\Pi _1=1$.

Let $p_h$ the probability of $\omega _h$ obtained as the pignistic transformation of $m_h$. We have:

$$\begin{aligned} p_h=\sum _{t=h}^H \frac{1}{t} \cdot m_t =\sum _{t=h}^H \frac{1}{t} (\Pi _t - \Pi _{t+1}). \end{aligned}$$

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For this relation to be applicable, we proceed with necessary $\alpha$-cuts in order to describe the given possibility distribution. We then assign appropriate $\Pi _h$ to compute required probabilities. For the possibilistic criterion in our example, $\varOmega = \{ \omega _1^3, \omega _2^3, \omega _3^3\}$ and the possibility distribution $\pi$ is such that $\pi (\omega _1^3) = 0.8, \; \pi (\omega _2^3) = 0.6$ and $\pi (\omega _3^3) = 1$. It can be represented by the following sequence of $\alpha$-cuts:

• For $\alpha = 1, \; B_1 = \{\omega _3^3\}$ and $\Pi _1 = \Pi (\{\omega _3^3\})=1$;

• For $\alpha = 0.8, \; B_2 = \{\omega _3^3, \omega _1^3\}$ and $\Pi _2 = \Pi (\{\omega _1^3\})= 0.8$;

• For $\alpha = 0.6, \; B_3 = \{\omega _3^3, \omega _1^3, \omega _2^3\}= \varOmega$ and $\Pi _3 = \Pi (\{\omega _2^3\})= 0.6$;

• $\Pi _4= 0$.

Probability distribution using the transformation relation $p_h=\sum _{t=h}^H \frac{\Pi _t - \Pi _{t+1}}{t}$, leads to:

• $p_1=P(\omega _3^3)=\sum _{t=1}^3 \frac{\Pi _t - \Pi _{t+1}}{t}= \frac{1-0.80}{1}+\frac{0.80-0.60}{2}+\frac{0.60-0}{3}=0.50$

• $p_2=P(\omega _1^3)=\sum _{t=2}^3 \frac{\Pi _t - \Pi _{t+1}}{t}=\frac{0.80-0.60}{2}+\frac{0.60-0}{3}=0.30$ and

• $p_3=P(\omega _2^3)=\sum _{t=3}^3 \frac{\Pi _t - \Pi _{t+1}}{t}=\frac{0.60-0}{3}=0.20$.

Let us note that we can obtain belief masses for this possibility distribution using the relation $m_h = \Pi _h - \Pi _{h+1}$:

• $m_1=m(B_1)=\Pi _1-\Pi _2=0.20$

• $m_2=m(B_2)=\Pi _2-\Pi _3=0.20$

• $m_3=m(B_3)=\Pi _3-\Pi _4=0.60$

These results are displayed in Table 6. Let us note that applying the pignistic probability transformation to the resulting belief masses leads to the same probability distribution obtained above.

1.2 Revision rules

Revision rules presented in Table 3 are here described. Let’s consider the additional information $y^j_{\ell '} \in Y^j$, $\ell '=1,\ldots ,{L_j}$, and $Y^j=\{y_1^j,\ldots ,y^j_\ell ,\ldots , y_{L_j}^j\}$. This information is introduced through a conditional information (or likelihood) $f(y^j_{\ell '} \mid B^j_{h'})$ , $h'=1,\ldots ,2^{H_j}$, f is a “conditional measure” which can be a conditional plausibility, a conditional probability or a conditional possibility measure.

In the case of a stochastic attribute/criterion, Bayes theorem is a well-known rule to derive posterior information:

$$\begin{aligned} P(\omega _h^j \mid y^j_ \ell )= \frac{P(\omega _h^j)P(y^j_ \ell \mid \omega _h^j)}{\sum _{h'}{P(\omega _{h'}^j)P(y^j_ \ell \mid \omega _{h'}^j)}} \end{aligned}$$

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In the case of additional information on an evidential criterion, posterior information is obtained through the generalized Bayes theorem GBT (Smets 1993). This theorem gives posterior beliefs given a vacuous prior belief over $\varOmega ^j$ (i.e., prior belief describing a state of total ignorance). The GBT leads to the following relations: $\forall {B^j_{h'} \subseteq \varOmega ^j}$, $\forall {y^j_{\ell '} \in Y^j}$, we have :

$$\begin{aligned} m(B^j_{h'} \mid y^j_{\ell '}) = \prod _{\omega ^j_h \in B^j_{h'}}{P \ell (y^j_{\ell '} \mid \omega _h^j)} \prod _{\omega ^j_h \in {\bar{B}^j_{h'}=\varOmega ^j - B^j_{h'}}}{(1 - P \ell (y^j_{\ell '} \mid \omega _h^j))} \end{aligned}$$

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Incorporating a priori information on $\varOmega ^j$ can then be achieved by combining the belief induced by y on $\varOmega ^j$ computed with the GBT, with the prior belief using the conjunctive combination rule. Given prior belief $m_0$ and $\forall {y^j_{\ell '} \in Y^j}$, we have:

$$\begin{aligned} m(y^j_{\ell '})=\sum _{B_h^j \subseteq \varOmega ^j}{m_0 (B_{h'}^j) m(y^j_{\ell '} \mid B_{h'}^j) } \end{aligned}$$

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where $m(y^j_{\ell '} \mid B_{h'}^j)$ is obtained through the disjunctive combination operation (Smets 1993).

Appendix 2: Concepts from fuzzy sets theory for multicriteria evaluation

Membership functions in fuzzy evaluation may be built in two different ways:

• 1. Deductively, with the use of formal models constructed according to specific hypotheses.

• 2. Empirically, with the use of two different methods:

  • 1. interpolating a finite number of degrees of membership,

  • 2. constructing a real model of a membership function and seeking to verify.

In order to overcome some of the limitations typical of fuzzy approaches to multicriteria evaluation, we have applied the transformation proposed by Munda et al. (1995) obtained by rescaling the ordinates of a membership function such that:

$$\begin{aligned} f_{ij}(x_j)=k \mu _{ij}(x_j) \end{aligned}$$

(18)

where k is the coefficient such that

$$\begin{aligned} \int _{-\infty }^{\infty } f_{ij}(x_j)d(x_j)=1 \end{aligned}$$

(19)

as in the stochastic case.

Normal convex trapezoidal fuzzy numbers can be characterised by a 4-tuple (a, b, c, d) where [a, b] is the closed interval on which the membership function is equal to 1, c is the left-hand variation and d the right- hand variation.

$$\begin{aligned} \mu _{ij}(x_j) = \left\{ \begin{array}{ll} 0 &{} if \; x_j \le a-c \\ 1-\frac{a-x_j}{c} &{} if \; a-c<x_j< a \\ 1 &{} if \; a\le x_j \le b \\ 1-\frac{x_j-b}{d} &{} if \; b<x_j< b+d \\ 0 &{} if \; x_j \ge b+d \\ \end{array} \right\} \end{aligned}$$

(20)

The parameter k necessary to calculate the function $f_{ij}(x_j)$ for a given membership function according to condition (19) corresponds to:

$$\begin{aligned} k = \frac{1}{((c+d)/2+(b-a))}. \end{aligned}$$

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The rescaling approach has been applied for the trapezoidal fuzzy numbers used in the numerical example. The expected values for the resulting density functions have been computed as shown in Table 13.