Digital Controls, Inc. $($DCI$),$ manufactures two models of a radar gun used by police to monitor the speed of automobiles. Model A has an accuracy of plus or minus $1$ mile per hour, whereas the smaller model B has an accuracy of plus or minus $3$ miles per hour.A$($a$)$ Interpret the ranges of optimality for the objective function coefficients.

If a single change to the injection molding time or assembly time for either model case is within the allowable range, the optimal solution will not change.

If a single change to the manufacturing cost or purchase cost for either model case is within the allowable range, the optimal solution will not change.

If multiple changes are made to the manufacturing or purchase costs for either model case within their respective allowable ranges, the optimal solution will not change.

If multiple changes are made to the injection molding time or assembly time for either model case within their respective allowable ranges, the optimal solution will not change.

$($b$)$ Suppose that the manufacturing cost increases to $$11\mathrm{.}50$ per case for model $A.$ Would the optimal solution change?

Yes, it is necessary to solve the model again.

No$,$ t

For the next week, the company has orders for $100$ units of model A and $150$ units of model B$.$ Although DCI purchases all the electronic components used in both models, the plastic cases for both models are manufactured at a DCI plant in Newark, New Jersey. Each model A case requires $4$ minutes of injection$-$molding time and $6$ minutes of assembly time. Each model B case requires $3$ minutes of injection$-$molding time and $8$ minutes of assembly time. For next week, the Newark plant has $600$ minutes of injection$-$molding time available and $1,080$ minutes of assembly time available. The manufacturing cost is $$10$ per case for model A and $$6$ per case for model B$.$ Depending upon demand and the time available at the Newark plant, DCI occasionally purchases cases for one or both models from an outside supplier in order to fill customer orders that could not be filled otherwise. The purchase cost is $$14$ for each model A case and $$9$ for each model B case.

Management wants to develop a minimum cost plan that will determine how many cases of each model should be produced at the Newark plant and how many cases of each model should be purchased. The following decision variables were used to formulate a linear programming model for this problem:

AM $=$ number of cases of model A manufactured

BM $=$ number of cases of model B manufactured

AP $=$ number of cases of model A purchased

BP $=$ number of cases of model B purchased

The linear programming model that can be used to solve this problem is as follows:

Min $10$AM $+6$BM $+14$AP $+9$BP

s$.$t$.$

$1$AM $++1$AP $+=100$ Demand for model A

$1$BM $+1$BP $=150$ Demand for model B

$4$AM $+3$BM Injection molding time

$6$AM $+8$BM Assembly time

AM$,$ BM$,$ AP$,$ BP $>=0$

Refer to the computer solution below.

Optimal Objective Value $=2170\mathrm{.}00000$

Variable Value Reduced Cost

AM $100\mathrm{.}000000\mathrm{.}00000$

BM $60\mathrm{.}000000\mathrm{.}00000$

AP $0\mathrm{.}000001\mathrm{.}75000$

BP $90\mathrm{.}000000\mathrm{.}00000$

Constraint Slack$/$Surplus Dual Value

$10\mathrm{.}0000012\mathrm{.}25000$

$20\mathrm{.}000009\mathrm{.}00000$

$320\mathrm{.}000000\mathrm{.}00000$

$40\mathrm{.}000000\mathrm{.}37500$

Variable Objective

Coefficient Allowable

Increase Allowable

Decrease

AM $10\mathrm{.}000001\mathrm{.}75000$ Infinite

BM $6\mathrm{.}000003\mathrm{.}000002\mathrm{.}33333$

AP $14\mathrm{.}00000$ Infinite $1\mathrm{.}75000$

BP $9\mathrm{.}000002\mathrm{.}333333\mathrm{.}00000$

Constraint RHS

Value Allowable

Increase Allowable

Decrease

$1100\mathrm{.}0000011\mathrm{.}42857100\mathrm{.}00000$

$2150\mathrm{.}00000$ Infinite $90\mathrm{.}00000$

$3600\mathrm{.}00000$ Infinite $20\mathrm{.}00000$

$41,080\mathrm{.}0000053\mathrm{.}33333480\mathrm{.}00000$

$($a$)$

Interpret the ranges of optimality for the objective function coefficients.

If a single change to the injection molding time or assembly time for either model case is within the allowable range, the optimal solution will not change.

If a single change to the manufacturing cost or purchase cost for either model case is within the allowable range, the optimal solution will not change.

If multiple changes are made to the manufacturing or purchase costs for either model case within their respective allowable ranges, the optimal solution will not change.

If multiple changes are made to the injection molding time or assembly time for either model case within their respective allowable ranges, the optimal solution will not change.

$($b$)$

Suppose that the manufacturing cost increases to $$11\mathrm{.}50$ per case for model A$.$ Would the optimal solution change?

Yes, it is necessary to solve the model again.

No$,$ the optimal solution remains the same.

What is the new optimal solution?

at $($AM$,$ BM$,$ AP$,$ BP$)=$

Suppose that the manufacturing cost increases to $$11\mathrm{.}50$ per case for model A and the manufacturing cost for model B decreases to $$4$ per unit. Would the optimal solution change?

Yes, it is necessary to solve the model again.

No$,$ the optimal solution remains the same.

What is the new optimal solution?

at $($AM$,$ BM$,$ AP$,$ BP$)=$