Multicriteria Decision Making

Abstract

The models and methods in this chapter all consider problems, in which the consequences of a decision are no longer one-dimensional, while, in contrast to the scenario in decision analysis, the outcomes are again deterministic. In other words, if we were to, say, change the composition of a soft drink we manufacture or change the controls on a television, we do not just deal with profit as a result of this decision, but face changing customer acceptance (and demand) for the product, different costs, changing market share, customer satisfaction, and other factors, all of which will influence short- and long-term viability of the firm. The models discussed in this chapter are similar to those in the chapter on multiobjective optimization, except that these are discrete: we face only a finite (and usually fairly small) number of choices.

Exercises

Problem 1 (Generic method)

Consider the problem of purchasing a vehicle. The decision maker has narrowed down the problem to three choices and three criteria. The evaluations of the vehicles are summarized in Table 10.6.

Table 10.6 Evaluations for problem 1

It can reasonably be assumed that the decision maker has utility functions

$$ u(x) = 1 - \frac{{2,778}}{{{{10}^{{10}}}}}{x^2} $$

for a purchase price of x, and

$$ u(y) = 1 - 50{y^{{ - 2}}} $$

for a gas mileage of y. For the qualitative scores regarding the comfort, the decision maker uses the so-called 5-point Likert scale: terrible – poor – medium – good – excellent in increments of 0.25.

• (a) Set up the utility matrix.

• (b) Given that gas mileage and comfort combined are deemed equally important and that combined, they are assessed as important as the purchase price, determine the decision maker’s weights of the criteria.

• (c) What are the values (utilities) of the individual decisions and how would you advise the decision maker?

• (d) Assume now that the decision maker decided to sell the car in 3 years’ time, after having driven for 20,000 miles per year, which would cost him $4 per gallon. For simplicity, we will ignore interest and inflation. The resale values of the three vehicles have been estimated at 40 %, 60 %, and 70 % of their present purchase price. The decision maker will then only consider two criteria, viz., the cost of operating the vehicle for 3 years, and the comfort the vehicle provides. The utility of the costs is expressed as $u(z){ } = 1 - 2{e^{{ - \frac{{40}}{z}}}}$, where z denotes the cost (in $1,000) of the vehicle during the 3 years. The decision maker considers costs three times as important as comfort. Set up the modified utility matrix and use the generic method to make a recommendation to the decision maker.

Solution

• (a) Given the utility functions, the utility matrix is

  $$ {\mathbf{U}} = \left[ {\begin{array}{*{20}{c}} {.3055} & {.6528} & {.5} \\ {.8889} & {.92} & {.25} \\ 0 & {.8457} & 1 \\ \end{array} } \right]. $$

• (b) ${w_{{1}}} =.{5},{w_{{2}}} =.{25},{w_{{3}}} =.{25}$.

• (c) ${\mathbf{v}}(d) = \left( {.{44}0{95},{ }.{73695},{ }.{461425}} \right)$ and the analyst would recommend that the decision maker purchase the Ford Ranger.

• (d) The loss of value of the three vehicles is 50,000 − 20,000 = $30,000, 20,000 − 12,000 = $8,000, and 60,000 − 42,000 = $18,000, respectively. The gas consumptions of the three vehicles during the 3 years are $\frac{1}{{12}}60,000 = 5,000$, $\frac{1}{{25}}60,000 = 2,400$, and $\frac{1}{{18}}60,000 = 3,333.33$ gallons, respectively, for costs of $20,000, $9,600, and $13,333.33, respectively. The combined costs for the three vehicles are then $50,000, $17,600, and $31,333.33, so that the utilities with the function provided above are.1649,.7061, and.4767. The evaluation matrix with columns for cost and comfort is then

$$ \left[ {\begin{array}{*{20}{c}} {.1013} & {.5} \\ {.7939} & {.25} \\ {.4420} & {.1} \\ \end{array} } \right]. $$

The weights specified by the decision maker are w ₁ = .75 and w ₂ = .25, so that the weighted average utilities of the three automobiles are.2010,.6579, and.3565, which gives a large advantage to the Ford Ranger.

Problem 2 (AHP, normalization technique)

An investor may invest his available funds in blue-chip stocks, real estate, and bonds. These investments are to be evaluated on two criteria, viz., short-term profits and long-term viability, with the former being evaluated to be 1.5 times as important as the latter. Pairwise preference statements have been recorded as follows:

Criterion 1: ${{\mathbf{C}}^1} = \left[ {\begin{array}{*{20}{c}} 1 & 3 & 3 \\ { \frac{1}{3} } & 1 & 2 \\ { \frac{1}{3} } & \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} & 1 \\ \end{array} } \right]$, criterion 2: ${{\mathbf{C}}^2} = \left[ {\begin{array}{*{20}{c}} 1 & \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} & {\tfrac{1}{5}} \\ 2 & 1 & \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \\ 5 & 2 & 1 \\ \end{array} } \right]$, and the relation between the two criteria is expressed in the matrix ${\mathbf{C}} = \left[ {\begin{array}{*{20}{c}} 1 & {\tfrac{3}{2}} \\ { \frac{2}{3} } & 1 \\ \end{array} } \right]$.

• (a) Use the Analytic Hierarchy Method to determine a ranking of the three decisions.

• (b) Examine the consistency in the matrices C ¹ and C ², using a quantitative criterion. What are your conclusions?

Solution

• (a) The normalized matrices are

  $$ \begin{array}{*{20}{c}} {{{{\mathbf{\tilde{C}}}}^1} = \left[ {\begin{array}{*{20}{c}} {.6000} & {.6667} & {.5000} \\ {.2000} & {.2222} & {.3333} \\ {.2000} & {.1111} & {.1667} \\ \end{array} } \right],} & {{{{\mathbf{\tilde{C}}}}^2} = \left[ {\begin{array}{*{20}{c}} {.1250} & {.1429} & {.1176} \\ {.2500} & {.2857} & {.2941} \\ {.6250} & {.5714} & {.5882} \\ \end{array} } \right]\,\,and} & {{\mathbf{\tilde{C}}} = \left[ {\begin{array}{*{20}{c}} {.6} & {.6} \\ {.4} & {.4} \\ \end{array} } \right].} \\ \end{array} $$

  The row averages of these normalized matrices are the columns of the utility matrix U and the weight vector w, respectively, so that

  $$ {\mathbf{U}} = \left[ {\begin{array}{*{20}{c}} {.5889} & {.1285} \\ {.2518} & {.2766} \\ {.1593} & {.5949} \\ \end{array} } \right]\,\,\,\,{\text{and}}\,\,\,\,{\mathbf{w}} = { }\left[ {.{6},{ }.{4}} \right]. $$

  We then obtain v(d) = [.4047,.2617,.3335], so that we recommend the first decision (investment in blue-chip stocks) to the decision maker.

• (b) As far as consistency is concerned, we compute coefficients of variation of.20156,.2316, and.2302 for matrix ${{\mathbf{\tilde{C}}}^1}$ i.e. an average variation of more than 20 %. This would give us sufficient reason to return the evaluation matrix C ¹ to the decision maker for reevaluation.

Consider now the matrix ${{\mathbf{\tilde{C}}}^2}$. The three rows have coefficients of variation of.1432,.1197, and.0652 with an average of 10.94 %, which appears fairly high, but more or less acceptable. The matrix C has zero variability and it is consistent.