2. The ABC Appliance Company requires sheet metal for its appliances. The compan

Question

2. The ABC Appliance Company requires sheet metal for its appliances. The company can purchase long coils of sheet metal in two different widths: 65 inches and 40 inches. The company must purchase the coils by linear foot of length: $1.20 per foot for a 64-inch coils and $1.00 per foot for a 40-inch coil. (This implies that a square foot, say, of the wider coil is less expensive.) Up to 4000 feet of the 65-inch coil is available, and up to 6000 feet of the 40 inch coil is available. There are manufacturing requirements for six different widths: 50, 45, 40, 35, 20, and 10 inches. ABC's requirements are expressed as lengths of the various widths The company requires 1000 feet of 50-inch width, 2500 feet of 45-inch width, 3000 feet of 40 inch width, 2300 feet of 35-inch width, 1300 feet of 20-inch width, and 2000 feet of 10-inch width. Develop a mathematical model to determine how much of each width coil ABC should purchase to meet its requirements at minimal cost

Explanation / Answer

Following are the sets or patterns that can be cut from the coils of given widths considering minimum wastages.

Let Xj be the feets of pattern-j to be cut. j=1,2,..,12

Minimize Z = total cost = 1.2 (X1+X2+X3+X4+X5+X6+X7) + 1.0 (X8+X9+X10+X11+X12)

Subject to,

Demand constraints

X1 >= 1000
X2 >= 2500
X3 + X8 >= 3000
X4 + X9 >= 2300
X2 + X3 + X4 + 3 X5 + 2 X6 + 2 X10 + X12 >= 1300
X1 + X4 + 2 X6 + 6 X7 + 4 X11 + 2 X12 >= 2000

Supply constraints

X1 + X2 + X3 + X4 + X5 + X6 + X7 <= 4000
X8 + X9 + X10 + X11 + X12 <= 6000

Xj >= 0

From 65" coil Pattern 50 45 40 35 20 10 Use Waste 1 1 1 60 5 2 1 1 65 0 3 1 1 60 5 4 1 1 1 65 0 5 3 60 5 6 2 2 60 5 7 6 60 5 From 40" coil Pattern 50 45 40 35 20 10 Use Waste 8 1 40 0 9 1 35 5 10 2 40 0 11 4 40 0 12 1 2 40 0

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