Unnormalized vs Normalized

Consider a simple model with two objectives (O₁ and O₂) and two alternatives (a₁ and a₂).
Let’s assume that we provided judgments for a₁ and a₂ wrt O₁ and O₂, and those judgments are in unnormalized mode (for instance, we used ratings), so that priorities of a₁ and a₂ do not add up to 1 wrt O₁ and O₂.

                  Goal
                 /    \
          p = 0.6      p = 0.4
              O₁           O₂
             /  \         /   \
          0.8    0.7   0.6    0.5
           a₁     a₂    a₁     a₂

Here:

For O₁: p(a₁) + p(a₂) = 0.8 + 0.7 ≠ 1
For O₂: p(a₁) + p(a₂) = 0.6 + 0.5 ≠ 1

Now, let’s see how we calculate the resulting (global) priorities of a₁ and a₂ in normalized and unnormalized modes.

1. Normalized mode:

First, we normalize priorities of a₁ and a₂ wrt each covering objective:

$p (a_{1} wrt O_{1}) = \frac{0.8}{(0.8 + 0.7)} = 0.53$
$p (a_{2} wrt O_{1}) = \frac{0.7}{(0.8 + 0.7)} = 0.47$
$\sum = 1$
$p (a_{1} wrt O_{2}) = \frac{0.6}{(0.6 + 0.5)} = 0.55$
$p (a_{2} wrt O_{2}) = \frac{0.5}{(0.6 + 0.5)} = 0.45$
$\sum = 1$

After that, we perform regular synthesis:

$p (a_{1}) = 0.53 \cdot 0.6 + 0.55 \cdot 0.4 = 0.538$
$0.47 \cdot 0.6 + 0.45 \cdot 0.4 = 0.462$

Since all clusters were normalized, global priorities are also normalized: 0.538 + 0.462 = $1$

2. Unnormalized mode:

In this case, we skip the normalization step and go straight to synthesis:

$p (a_{1}) = 0.8 \cdot 0.6 + 0.6 \cdot 0.55 = 0.72$
$p (a_{2})$

As we can see, the sum of global priorities does not add up to 1: $0.72 + 0.62 =$

If we normalize global unnormalized priorities, we will get the following values:

$p (a_{1}) = 0.537$
$p (a_{2}) = 0.463$

Results are close, but not quite the same.

Let’s check the general case and see if the results should or should not match up.

                Goal
                 |
     -------------------------
     |                       |
    O₁ (p₁)                 O₂ (p₂)
   /   \                   /     \
 v₁₁   v₂₁              v₁₂     v₂₂
 a₁     a₂              a₁       a₂

In normalized mode:

$p^{N} (a_{1}) = p_{1} \cdot \frac{v_{11}}{v_{11} + v_{21}} + p_{2} \cdot \frac{v_{12}}{v_{12} + v_{22}}$

(normalized priority wrt O₁ and wrt O₂)

In unnormalized mode after normalization:

$p^{UN} (a_{1}) = \frac{p_{1} v_{11} + p_{2} v_{12}}{p_{1} v_{11} + p_{2} v_{12} + p_{1} v_{21} + p_{2} v_{22}}$

(denominator is normalization)

If results are the same in normalized mode and unnormalized after normalization, then:

$p^{N} (a_{1}) = p^{UN} (a_{1})$

That is:

$p_{1} \cdot \frac{v_{11}}{v_{11} + v_{21}} + p_{2} \cdot \frac{v_{12}}{v_{12} + v_{22}} = \frac{p_{1} v_{11} + p_{2} v_{12}}{p_{1} v_{11} + p_{2} v_{12} + p_{1} v_{21} + p_{2} v_{22}}$

Let’s assume $p_{1} = p_{2} = 0.5$ to simplify calculations:

$\frac{v_{11}}{v_{11} + v_{21}} + \frac{v_{12}}{v_{12} + v_{22}} = \frac{v_{11} + v_{12}}{v_{11} + v_{12} + v_{21} + v_{22}}$
$\frac{v_{11} (v_{12} + v_{22}) + v_{12} (v_{11} + v_{21})}{(v_{11} + v_{21}) (v_{12} + v_{22})} = \frac{v_{11} + v_{12}}{v_{11} + v_{12} + v_{21} + v_{22}}$
$(v_{11} v_{12} + v_{11} v_{22} + v_{12} v_{11} + v_{12} v_{21}) (v_{11} + v_{12} + v_{21} + v_{22})$ $= (v_{11} + v_{12}) (v_{11} v_{22} + v_{12} v_{21} + v_{21} v_{22})$ $v_{12} v_{21}^{2} + v_{11} v_{22}^{2} + 2 v_{11} v_{12}^{2} + 2 v_{11} v_{12} v_{22} + 2 v_{11} v_{12} v_{21} + 2 v_{11} v_{21} v_{22} + 2 v_{12} v_{21} v_{22} - v_{11}^{2} v_{12} - v_{12}^{2} v_{11} = 0$ $v_{12} v_{21}^{2} + v_{11} v_{22}^{2} + v_{11}^{2} v_{12} + v_{12}^{2} v_{11} + 2 v_{11} v_{12} v_{21} + 2 v_{11} v_{12} v_{22} = 0$

This is true only if:

$v_{11} = v_{12} = v_{21} = v_{22} = 0$

Conclusion: We should NOT expect results in normalized mode to match results in unnormalized mode after normalization.

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