$4\mathrm{.}7$ MiToch Foundry plans to purchnse a new tester for use in its manufacturing

processes for very large scale integrated circeits. Three possible testers are

under consideration: the OR$9000,$ the J$941,$ and the Sentry $50.$ There

are four evaluation measures: cost in hundreds of thousands of dollars,

accuracy in picoseonds, delivery time in months, and uptime in percent.

For the following analysis, assume a range of one to three hundred thou$-$

sand dollars for cost, $500$ to $1000$ piccoseconds for sccuracy, three to six

months for delivery time, and $95$ to $99$ percent for uptime. Preferences

are monotonically incressing $("$more is better"$)$ for uptime, and monoton$-$

ically decreasing $("$more is worse"$)$ for cost, accurscy, and delivery time.

The single dimensional value fusetioe over cost is linear $($that is$,$ this fune$-$

tion is piecewise linear with only one straight line sogment$).$ The single

dimensional value function over accuracy is exponential with a midvalue

of $900.$

The single dimensional value functions over delivery time and uptime are

pieorwise linear. The value function for delivery time has two segments,

which join at a delivery time of $4$ months. The value increment from

decreasing delivery time from four months to three months is three times

as great as the value increment from decreasing delivery time from six

months to four months. The value function for uptime has three segments,

one from $95$ to $96$ percent, one from $96$ to $97$ percent, and one from $97$

to $99$ percent. The value increment from increasing uptime from $96$ to $97$

percent is the same as the value increment from increasing uptime from

$97$ to $99$ percent. The value increment from increasing uptime from $95$ to

$96$ percent is twice as great as the value increment from incressing uptime

from $96$ to $97$ percent.

The value increment obtained by decreaxing accuracy from $1000$ pioneec$-$

onds to $500$ picoecoonds is three times as great as the value increment

obtained by decreasing cost from three hundred thousand dollars to one

husdred thousand dollars. The value increment obtained by decreasing do$-$

livery time from six months to three months is equal to the value incresent

obtained by increaving accuracy from $95$ percent to $99$ peroent. The value

increment obtained by increasing uptime from $95$ percent to $99$ percent

is twice as great as the vilue increment obtained by decreasing cost from

three hundred thousand dollars to one hundred thousand dollars.

The evaluation measure scores $($levels$)$ for the three alteraatives are given

in the following table:

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

This requires fiading $(1)$ the exponential conitant for any exponen$-$

tial single dimensional value functions, $(2)$ the values for the points

at which the segrnents of piecewise linear single dimensional value

functions join, and $(3)$ the evaluation measure weights.

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

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

$($iii$)$ Use your spreadshoct model to calculate the values for the three

testers, and deternine which alternative is preferred.

$($iv$)$ Determine to two decimal poirts 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.