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![ion 5 Logic constraints and free points. 120 points) n can manufacture three kinds of clothing: shirts, shorts, and patively ing requires kind of cloth pants. Making unlimited capacity, but cannot be used o 160 hours of labor and 170 yards of cloth per renting a particular kind of machine, which has effectively The following table describes the produc 70yards d for any other kind of clothing. Gandy can obtain up to week. tion proc Cost of Machine Hours Cloth p $/Week) Per Unit Per Unit Labor Yards of Revenue Variable per Unit Cost $6 $4 $8 per Unit Shirts $1803 Shorts $150 Pants $100 2 6 4. 3 4 $12 $8 $15 Note: you do not need to solve this problem. a. [5 points] Define the decision variables b. [5 points] Formulate the objective function c.[10 points) Formulate the LUB constraints if there are any. And explain the logic.](https://media.cheggcdn.com/media%2Fb0d%2Fb0db3856-c228-4fa8-a2e7-a0a7aa140a43%2Fimage)
Expert Answer
Let, x1 = number of shirts produced each week
x2 = number of shorts produced each week
x3 = number of pants produced each week
Weekly sales revenue = 12×1 + 18×2 + 15×3
weekly variable costs = 6×1 + 4×2 + 8×3
weekly cost of renting the machine = 180y1 + 150y2 + 100y3
Thus, weekly profits = (weekly sales revenue) – (weekly variable costs) – (weekly cost of renting the machinery)
= (12×1 + 8×2 + 15×3) – (6×1 + 4×2 + 8×3) – (180y1 + 150y2 + 100y3)
= 6×1 + 4×2 +7×3 – 180y1 – 150y2 – 100y3
Since its supply of labor and cloth is limited, company faces the following two constraints:
Constraint 1 At most, 160 hours of labor can be used each week
Constraint 2 At most, 170 sq yds of cloth can be used each week
Constraint 1 can be expressed as –
3×1 + 2×2 + 6×3 < 160 (labor constraint)
Constraint 2 is expressed as –
4×1 + 3×2 + 4×3 < 170 (cloth constraint)
max z = 6×1 + 4×2 + 7×3 – 180y1 – 150y2 – 100y3 ——— 1
s.t. 3×1 + 2×2 + 6×3 < 160 ———– 2
4×1 + 3×2 + 4×3 < 170 ———— 3
x1, x2, x3 > 0;
y1, y2, y3 = 0 or 1
There is no optimum solution to the problem. Because, x1 = 20, x2 = 20, x3 = 10 is not satisfying both equation 2 and 3.
