Mathematical modeling !!!!! An IT company sells smart phones in two sizes. One h
ID: 1169683 • Letter: M
Question
Mathematical modeling !!!!!
An IT company sells smart phones in two sizes. One has a 4 inch screen and cost $20 to produce. The 6-inch version costs $35 to produce. It is estimated that the average selling price of the 4-inch model is $200 minus $0.01 for each 4 inch sold and minus $0.002 for each 6-inch model sold. The average selling price of the 6 inch is predicted to be $350 minus $0.01 for each 6-inch version sold and minus $0.003 for each 4-inch version sold. The company can produce at most 15000 phones per month. Using constrained optimization, find the number of phones needed to maximize profits.
Explanation / Answer
Let 4 inch screen phone be A
Let 6-inch version phone be B
Given :
4 inch screen phone cost $20 to produce.
6 inch screen phone cost $35 to produce.
selling price of the 4-inch model = 200-0.01A-0.002B
selling price of the 6 inch is = 350-0.01B-0.003A
Profit= SP-Cost
Profit A = 180-0.01A-0.002B
Profit B = 315-0.01B-0.003A
Objective Function is to maximize profit :
Z=Profit A+Profit B
Z=180-0.01A-0.002B+315-0.01B-0.003A
Contraint : A+B<=15000
A>=0,B>=0
Putting values of A and B to 0 we get corner points i.e.
B=15000-A
(0,15000),(15000,0)
Substituting the values of corner points in profit function we get maximum profit :
taking (0,15000)
Z=180-0.01A-0.002B+315-0.01B-0.003A
=180-0.002*(15000)+315-0.01*(15000) =315
Similarly taking (15000,0)
Z=180-0.01A-0.002B+315-0.01B-0.003A
=180-0.01*(15000)+315-0.003*(15000)=300
Thus profit is maximium at 315 i.e at (0,15000)
180-0.01A-0.002B+315-0.01B-0.003A = 315
180-0.002B+315-0.01B=315
495-0.012B=315
180=0.012B
B=15000