Gulf Coast Electronics is ready to award contracts to suppliers for providing reservoir capacitors for use in its electronic devices. For the past several years, Gulf Coast Electronics has relied on two suppliers for its reservoir capacitors: Able Controls and Lyshenko Industries. A new firm, Boston Components, has inquired into the possibility of providing a portion of the reservoir capacitors needed by Gulf Coast.

The quality of products provided by Lyshenko Industries has been extremely high; in fact, only $0\mathrm{.}5\%$ of the capacitors provided by Lyshenko had to be discarded because of quality problems. Able Controls has also had a high quality level historically, producing an average of only $1\%$ unacceptable capacitors. Because Gulf Coast Electronics has had no experience with Boston Components, it estimated Boston Components' defective rate to be $10\%.$

Gulf Coast would like to determine how many reservoir capacitors should be ordered from each firm to obtain $75,000$ acceptable$-$quality capacitors to use in its electronic devices. To ensure that Boston Components will receive some of the contract, management specified that the volume of reservoir capacitors awarded to Boston Components must be at least $10\%$ of the volume given to Able Controls.

In addition, the total volume assigned to Boston Components, Able Controls, and Lyshenko Industries should not exceed $30,000,50,000,$ and $50,000$ capacitors, respectively. Because of Gulf Coast's long$-$term relationship with Lyshenko Industries, management also specified that at least $30,000$ capacitors should be ordered from Lyshenko.

The cost per capacitor is $$2\mathrm{.}45$ for Boston Components, $$2\mathrm{.}50$ for Able Controls, and $$2\mathrm{.}75$ for Lyshenko Industries.

$($a$)$

Formulate a linear program for determining how many reservoir capacitors should be ordered from each supplier to minimize the total cost of obtaining $75,000$ acceptable$-$quality reservoir capacitors. $($Let B $=$ number of capacitors ordered from Boston Components, A $=$ number of capacitors ordered from Able Controls, and L $=$ number of capacitors ordered from Lyshenko Industries.$)$

Min

Correct: Your answer is correct.

s$.$t$.$

volume for Boston

volume for Able

volume for Lyshenko

# useful capacitors

Incorrect: Your answer is incorrect.

Boston relative to Able

Lyshenko minimum

B$,$ A$,$ L $>=0$

Solve the linear program to determine how many reservoir capacitors should be ordered from each supplier. $($Round your answers to the nearest integer.$)$

Boston Components

capacitors

Able Controls

$50000$

Incorrect: Your answer is incorrect.

capacitors

Lyshenko Industries

capacitors

$($b$)$

Suppose that the quality level for reservoir capacitors supplied by Boston Components is much better than estimated, with a defective rate of $2\%.$ What effect, if any, would this quality level have?

The optimal solution would change, with $30,000$ reservoir capacitors assigned to Boston Components.

The optimal solution would change, with Lyshenko Industries being assigned the maximum of $50,000$ reservoir capacitors.

The optimal solution would change, with $7,526$ reservoir capacitors assigned to Boston Components.

The optimal solution would remain the same.

Correct: Your answer is correct.

$($c$)$

Suppose that management is willing to reconsider their requirement that at least $30,000$ capacitors must be ordered from Lyshenko Industries. What effect, if any, would this consideration have on the solution in part $($a$)?$

The total cost will decrease about $$2\mathrm{.}54$ for every unit the required minimum is reduced by$,$ with no lower limit on how much the required minimum can be reduced.

The total cost will decrease about $$0\mathrm{.}22$ for every unit the required minimum is reduced by$,$ for at least $21,106$ capacitors.

The total cost will decrease about $$0\mathrm{.}16$ for every unit the required minimum is reduced by$,$ for at least $25,440$ capacitors

Reducing the requirement will not change the total cost of obtaining the reservoir capacitors.