The Stamper company make two varieties of stapler. Model H is of high quality an

Question

The Stamper company make two varieties of stapler. Model H is of high quality and they make a profit of $1 per stapler sold with a market size of 2500 per week while Model S is of medium quality and they make a $0.6 per stapler sold with a market size of 5000 per week. The factory uses the same equipment to make the two models. If only S staplers were made, then the stamping section could handle 5000 per week, the assembly section 6000 per week and the paint section 8000 per week. The H staplers take 25 % more stamping time, 50% more assembly time and 150% more printing time than S staplers. At least 25% of the staplers made must be Model S.

Explanation / Answer

suppose x1 = amount of S sold profit = 1$
x2 = amount of H sold profit =0.6$
so total profit by selling = x1 + 0.6x2 .... hence this is the objective function to be maximized

production of X1 of S needs x1/5000 week for the stamping section, x1/6000 for the assembly section, x1/8000 week for paint section

production of X2 of S needs 1.25*x2/5000 week for the stamping section, 1.5*x2/6000 for the assembly section, 2.5*x2/8000 week for paint section
x1 < 2500 , x2 < 5000 (because of the market size)
hence in a week,
x1/5000 + 1.25* x2/5000 <=1
x1/6000 + 1.5* x2/6000 <=1
x1/8000 + 2.5* x2/8000 <=1
and x1 > 3x2 (since atleast 25% of them should be of model S)

finally the problem is
Max x1 + 0.6x2 (profit function)
s.t
0 < x1 <2500 , 0 < x2 < 5000
x1 + 1.25* x2 <=5000

x1 + 1.5* x2 <=6000
x1 + 2.5* x2 <=8000
x1 - 3*x2 >= 0

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