Synthesis criterion

1. Introduction

The multicriteria principle of synthesis criterion aims at building a function $g$ synthesizing all criteria:

Such approach allows for

• Pre-order on the set of alternatives $A$
• Compare alternatives
• Choose among them
• Rank them
• Assign them to classes

Nonetheless, building $g$ is often difficult and requires a big amount of preference information from the decision-maker.

In particular, we should focus on its two aspects:

1. Which properties should the decision-maker preferences possess so as to be representable by a $g$ function?
2. How to build $g$ and set the values to the parameters involved in the chosen parametric form?

1.1. Weighted sum

It is the simplest and the most used analytic form, The function g takes the following analytic form:

and the corresponding structure is:

1.2. Multi-attribute value model

This weighted sum can be generalized to non-linear operations for the preferences on a criteria. It is the role of the Multi-Attribute Value Theory, which takes the following form:

$\begin{cases} a \succeq b \Leftrightarrow u(a) > u(b)\\ a = b \Leftrightarrow u(a) = u(b)\\ \end{cases}$ with $u(a) = f(g_{1}(a),g_{2}(a),…,g_{n}(a))$.

A frequently encountered case is the additive form of the MAVT:

with $u_{i}(g_{i}^{\text{min}}) = 0$, $u_{i}(g_{i}^{\text{max}}) = 1$ and normalized weights.

2. Constructing value functions

To specify an additive MAVT model, one should elicit marginal value functions $u_{i}$, $\forall i \in F$ and “weights” $w_{i}$, $\forall i \in F$. In particular, we are interested in eliciting:

1. Value function $u_{i}$ (for each criterion).
2. Weight vector $w_{i}$.

2.1. Elicitation of value function $u_{i}$

Method 1: convenient when the evaluation scale $E_{i}$ is finite.

1. Rank elements of $E_{i}$
2. Rank differences between consecutive elements in the preceding ranking
3. Assign values compatible with the information obtained at step 1. and 2.

2.2. Eliciting importance coefficients $w_{i}$

Consider $b_{j}$ an alternative such that $g_{i}(b_{j}) = g_{i}^{\text{min}}$ , $\forall i \neq j$ and $g_{j}(b_{j}) = g_{j}^{\text{max}}$.

1. Rank $b_{j}$ , $j \in F$ by order of preference,
2. Reorder criteria such that $b_{n} \succ . . . \succ b_{1}$, we induce that $w_{n} \geq …\geq w_{1}$

Now consider $b_{j}^{n}$ such that $g_{i}(b_{j}^{n}) = g_{i}^{min}\ \forall i \neq n$ and $g_{i}(b_{j}^{n}) = x$.

3 Determine g_{n}(b_{n}(j)) such that $b_{1} \sim b_{n}^{j}$ to obtain: $\sum_{i=1}^{n} u_{i} (b_{1}) = \sum_{i=1}^{n} u_{i}(b_{n}^{j})$

1. Finally, we obtain: $\frac{w_{1}}{w_{n}} = u_{n}(x)$ or $u_{n}(g_{n}(x))$

2. Repeat for all $g_{2}, …, g_{n-1}$

Hence we obtain the ratios $\frac{w_{i}}{w_{n}},\ i = 1, …, n−1$

TODO: Revoir notations! Indices tout ca!

2.3. UTA

TODO: Add image

The goal of UTA is to:

1. Define a ranking on a subset of alternatives $(a_{2} \succ a_{1} \succ a_{6} \succ a_{8})$.
2. Define a linear program which minimizes an error function and infers an additive value model compatible with the stated ranking.
3. Based on this inferred additive value model, rank order all alternatives.

3. Inferring a value function

Searched model: $u(a) = \sum_{i=1}^{n} u_{i} (g_{i} (a))$ with $u_{i}$ piecewise linear marginal value function.

If $z ∈[g_{i}^{l}, g_{i}^{l+1}]$ then:

$A’$ is a subset of alternative ranked by the decision-maker Inferred value $u’$: $u’(a) = u(a) + \sigma(a), \forall a ∈ A’$ where $\sigma(a)$ is the error in the estimation of $u(a)$.

TODO: Custom program