Case Study $4\mathrm{.}1$ rescaling or normalization step is usually based on the minimum and maximum

performance of the alternatives on each criterion:

$f_j^{\text{'}}(a_i)=\frac{f_j(a_i)-\text{m}\text{i}\text{n}(f_j)}{\text{m}\text{a}\text{x}(f_j)-\text{m}\text{i}\text{n}(f_j)}$

when maximizing the criterion, or

$f_j^{\text{'}}(a_i)=1+(\frac{\text{m}\text{i}\text{n}(f_j)-f_j(a_i)}{\text{m}\text{a}\text{x}(f_j)-\text{m}\text{i}\text{n}(f_j)})$

when minimizing the criterion. Table $4\mathrm{.}2$ represents the rescaled performances of

Case Study $4\mathrm{.}1.$ Let us remark that this rescaling step might be performed after

applying the marginal utility function in order to ensure utility scores of between

$0$ and $1.$

Table $4\mathrm{.}2$ Rescaled performance table.

Figure $4\mathrm{.}3$ Positive exponential marginal utility function. Figure $4\mathrm{.}4$ Negative exponential marginal utility function.

Suppose that the marginal utility functions of the screen size and storage size

criteria are linear. On the other hand, consider an exponential marginal utility

function for customer review and price as depicted in Figure $4\mathrm{.}3$ and Figure $4\mathrm{.}4$

with an exponent of $3$ and $2,$ respectively. We remark that $(4\mathrm{.}8)$ uses the rescaled

values $($with $(4\mathrm{.}6)$ and $(4\mathrm{.}7)),$ which explains the decreasing utility function for

the price utility function $($Figure $4\mathrm{.}4)$ despite the value of $2$ for its exponent. The

corresponding utility values are given in Table $4\mathrm{.}3.$

Table $4\mathrm{.}3$ Marginal utility values for the alternatives.

The exponential marginal utility score for the price criterion is computed as

follows:

$U_1(a_j)=\frac{\text{e}\text{x}\text{p}(f_j^{\text{'}}(a_i{)}^2)-1}{1\mathrm{.}71}$

exp$(1)=2\mathrm{.}71,$ which explains the subtraction of $1$ in the nominator as well as the division by $1\mathrm{.}71$

in order to obtain $1$ for the best alternative. Central to the MAUT method are the marginal utility function and weights, which

reflect the decision maker's preferences for each criterion. These parameters need

As illustrative example, consider the ranking of five fictitious smartphones

$($SP$1,.,$ SP$5),$ which are evaluated on the following criteria:

$(1)$ price $($to be minimized$)$;

$(2)$ customer reviews $($to be maximized$)$;

$(3)$ screen size $($to be maximized$)$;

$(4)$ storage size $($to be maximized$).$

The performance of the five smartphones on those criteria.

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