Problem $3($Textbook problem $7-3\mathrm{.}6)$

$(-$ pts$.)$

David, LaDeana, and Lydia are the sole partners and workers in a company which produces fine clocks. David and LaDeana each are available to work a maximum of $40$ hours per week at the company, while Lydia is available to work a maximum of $20$ hours per week. $q,$

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The company makes two different types of clocks: a grandfather clock and a wall clock. To make a clock, David $($a mechanical engineer$)$ assembles the inside mechanical parts of the clock while LaDeana $($a woodworker$)$ produces the hand$-$carved wood casings. Lydia is responsible for taking orders and shipping the clocks. The amount of time required for each of these tasks is shown below.

$\backslash$table$[[$Task$,$Time required $($hours$)],[$Grandfather clock,Wall clock$],[$Assemble clock mechanism,$6,4],[$Carve wood casting,$8,4],[$Shipping$,3,3]]$

Each grandfather clock built and shipped yields a profit of $$300,$ while each wall clock yields a profit of $$200.$ The three partners now want to determine how many clocks of each type should be produced per week to maximize the total profit.

Formulate a linear programming model for this problem.

Use AMPL to solve this problem and generate sensitivity analysis information.

Use this sensitivity analysis information to determine whether the optimal solution must remain optimal if the estimate of the unit profit for grandfather clocks is changed from $300$ to $375($with no other changes in the model$).$

To increase the total profit, the three partners have agreed that one of them will slightly increase the maximum number of hours available to work per week. The choice of which one will be based on which one would increase the total profit the most. Use the sensitivity analysis information to make this choice. $($Assume no change in the original estimates of the unit profits.$)$

Explain why one of the shadow prices is equal to zero.

Can the shadow prices given in the sensitivity analysis information be validly used to determine the effect if Lydia were to change her maximum number of hours available to work per week from $20$ to $25?$ If so$,$ what would be the increase in the total profit?