$7\mathrm{.}5$ This is an extension of the MiTech Foundry problem in Exercise $4\mathrm{.}7$ of

Chapter $4.$ Assume that the multiobjective value function information

given in that problem still holds. Also assume that the evaluation measure

levels given for the J$941$ and the Sentry $50$ testers still hold. However, now

assume that the OR$9000$ is a new product that is not yet on the market,

and therefore there is uncertainty about some of its evaluation measure

levels.

Probability distributions have been assessed for the cost, accuracy, and

uptime evaluation measures for the OR$9000.$ From these it has been deter$-$

mined that the $0\mathrm{.}05$ fractiles for the three evaluation measures are $1\mathrm{.}2,850,$

and $95,$ respectively. The $0\mathrm{.}50$ fractiles are $1\mathrm{.}4,900,$ and $98,$ respectively.

Finally, the $0\mathrm{.}95$ fractiles are $2\mathrm{.}5,1000,$ and $99,$ respectively. Delivery time

is known to be three months for certain.

Consider a hypothetical uncertain alternative that has equal probabili$-$

ties of yielding outcomes with either

a$.$ A cost of one hundred thousand dollars, an accuracy of $500$ picosec$-$

onds, a delivery time of three months, and an uptime of $99$ percent,

or

b$.$ A cost of three hundred thousand dollars, an accuracy of $1000$ pi$-$

coseconds$,$ a delivery time of six months, and an uptime of $95$ percent.

This hypothetical uncertain alternative is equally preferred to a certain

alternative with a cost of three hundred thousand dollars, an accuracy of

$1000$ picoseconds, a delivery time of three months, and an uptime of $95$

percent.

$($i$)$ Determine the parameters for the utility function for this decision.

$[$Hint: In addition to the value function parameters determined in

Chapter $4,$ you must also determine the multiattribute risk toler$-$

ance.$]$

$($ii$)$ Create a spreadsheet model for this decision problem, and include a

printout of the equations for this model with your homework solution.

$[$Hint: Use the extended Pearson$-$Tukey approximation.$]$

$($iii$)$ Use your spreadsheet model to calculate the certainty equivalent val$-$

ues for the three testers, and determine which alternative is preferred.

$($iv$)$ Determine to two decimal points the range of the weight on cost for

which the most preferred alternative found in part $($iii$)$ remains most

preferred. Assume while you are varying this weight that the ratios

of the other three weights remain the same. Include the spreadsheet

output that you used in this sensitivity analysis with your homework

solution.