MultiAttribute Utility Theory
MultiAttribute Utility Theory or MAUT is a method of MultiCriteria Decision Aid
Utility Functions
Suppose we have a global utility function $U(a)$ that aggregates all criteria into one value:
 $U(a) = U(g_1(a), ..., g_k(a))$
Additive Utility Functions
$U(a) = \sum_{j=1}^k u_j(g_j(a))$
The evaluation table:

Price 
Comfort

$a$

300 
Medium

$b$

350 
Good

$c$

400 
Good

$d$

450 
Very Good


The utility functions:
$u_1$ 
$u_2$

8.5 
4

8 
7

6 
7

5 
10


With weights $k_1 = 7$ and $k_2 = 3$ we establish the ranking based on the following values:
 $u(a) = 7 \cdot 8.5 + 3 \cdot 4 = 71.5$
 $u(b) = 7 \cdot 8 + 3 \cdot 7 = 77$
 $u(c) = 7 \cdot 6 + 3 \cdot 7 = 63$
 $u(d) = 7 \cdot 5 + 3 \cdot 10 = 65$
 $b > a > d > c$
Marginal Utilities
At first we transform the evaluation into two "marginal utilities":
 we transform evaluation of all criteria into a scale $[0, 1]$
So with the weighted sum we have
 $U(a) = \sum{k=1}{n} w_k u_k(a)$
 we want $u_k \in [0, 1]$
Suppose you want to buy a car
 the prices range from 15k to 50k
 so first step is assigning 1 to 15k and 0 to 50k
 we've got a linear utility function
 but this is only just the first approximation of our preferences

 we know that we're willing to spend somewhere around 20k
 we want $u(20k) = 0.5$
 so we modify the function by adding an additional point  now there're two linear function
 this is the second approximation of our preferences

 can repeat this for both left and right sides to get the 3rd approximation
 and define $u(p_1) = 0.75$ and $u(p_2) = 0.25$

 and so on
Examples
Consider an additive model and the following evaluation table

$g_1$ 
$g_2$

$a_1$

1 
1

$a_2$

1 
3

$a_3$

1 
5

$a_4$

2 
1

$a_5$

2 
3

$a_6$

2 
5

$a_7$

3 
1

$a_8$

3 
3

$a_9$

3 
5

Based on this table a Decision Maker gives his preferences:
 $a_9 \ P \ a_6 \ P \ a_8 \ P \ a_5 \ P \ a_3 \ I \ a_7 \ P \ a_2 \ I \ a_4 \ P \ a_1$
This ordering satisfied the Preferential Independence criteria.
Definition: $\exists a,b,c,d \in A$, and set of criteria $J \cup \overline{J} = G$
 $g_i(a) = g_i(b), \forall i \not \in J$
 $g_i(c) = g_i(d), \forall i \not \in J$
 $g_i(a) = g_i(a), \forall i \in J$
 $g_i(b) = g_i(d), \forall i \in J$
In this case $J = \{g_1\}, \overline{J} = \{g_2\}$
 under $J$: $a_1 = a_2 = a_3; a_4 = a_5 = a_6; a_7 = a_8 = a_8$
 under $\overline{J}$: $a_1 = a_4 = a_7; a_2 = a_5 = a_8; a_3 = a_7 = a_9$
Need to check if this principle is satisfied for all possible combinations
For example, $a_1, a_4, a_5, a_5$:
 $a_4 \ P \ a_1 \iff a_5 \ P \ a_2$
 this indeed holds
Now we want to check how the decision maker obtained this ranking
 can we model it with an utility function?
 note that $a_3 \ I \ a_7$
 then under the utility model it should be true that $U(a_3) = U(a_7)$
 $u_1(g_1(a_3)) + u_2(g_2(a_3)) = u_1(g_1(a_7)) + u_2(g_2(a_7)) = ...$
 $... = u_1(1) + u_2(5) = u_1(3) + u_2(1) \ \ \ (*)$
 also $a_2 \ I \ a_4$
 then $U(a_2) = U(a_4)$
 or $u_1(1) + u_2(3) = u_1(2) + u_2(1) \ \ \ (**)$
 let's try to find the utility function
 check $(*)  (**)$
 $u_2(5)  u_2(3) = u_1(3)  u_1(2) \Rightarrow u_1(2) + u_2(5) = u_1(3) + u_2(3)$
 we see that $a_6$ is evaluated to $(2, 5)$ and $a_8$ is evaluated to $(3, 3)$
 but it means that we should also have $a_6 \ I \ a_8$  which is not the case
 thus it's not possible to establish
Sources