Question

A craftsman named William Barnes builds two kinds of ​birdhouses, one for wrens and a second for bluebirds. Each wren birdhouse takes 44 hours of labor and 44 units of lumber. Each bluebird house requires 22 hours of labor and  1212 units of lumber. The craftsman has available 6060 hours of labor and 120120 units of lumber. Wren houses yield a profit of $ 8$8 each and bluebird houses yield a profit of $ 15$15 each. The aim of the objective function for William should be to ▼ Minimize Maximize the objective value. Decision​ variables: X​ = number of wren houses to be produced Y​ = number of bluebird houses to be produced ​a) Objective​ function: Zequals= nothingXplus+nothingY Subject​ to: 44Xplus+22Y ▼ greater than or equals≥ less than or equals≤ 6060 ​(Upper C 1C1​) 44Xplus+1212Y ▼ greater than or equals≥ less than or equals≤ 120120 ​(Upper C 2C2​) ​X, Y greater than or equals≥0 ​b) On the graph on​ right, constraints C1 and C2 have been plotted.   Using the point drawing tool​, plot all the corner points for the feasible area. The optimum solution​ is: X​ = nothing ​(round your response to two decimal​ places). Y​ = nothing ​(round your response to two decimal​ places). Optimal solution value​ 'Z' ​ = nothing ​(round your response to two decimal​ places).

Answer 1

If X quantity of wren and Y quantity of bluebird houses are produced, then

Profit = 8X+15Y, which needs to be miximised.

OF = Max [ 8X+15Y]

such that

4X+2Y<=60

4X+12Y<=120

and X,Y=>0

The points of optimality are ( 12,6) (0.10) and ( 15,0), as given in the graph.The feasible area has also been given in graph.

Value of OF at three points

At ( 0,10) = 150,

At ( 12,6) =186

AT ( 15,0) = 120.

The max value is obtained at (12,6) which is the most feasible option. The associated profits is 186.

Hence company should produce 12 Wrem and 6 Bluebird units