Question

LG owns five production plants where 90" high definition televisions are produced. LG can sell up to 25,000 televisions per year at the price of $4000 per television. For each plant, the production capacity, the production cost per television, and the fixed cost of operating a plant for year are given below.

	Production	Plant	Cost per
Plant	Capacity	Fixed Cost	Television
1	10000	$9 million	$1,600
2	8000	$5 million	$2,000
3	9000	$3 million	$2,300
4	7000	$4 million	$2,100
5	6000	$1 million	$2,400

a. Use Solver to determine how LG can maximize its yearly profit from television production.

b. Use SolverTable to determine how the optimal plant locations vary as the number of televisions that can be sold varies from 20,000 to 30,000 (in 1000 unit increments).

Answer 1

Plant

Production Capacity

Fixed cost

Variable cost per TV

Let

1

10000

$9,000,000.00

$1600

X1

Number of TV manufactured in Plant 1

2

8000

$5,000,000.00

$2000

X2

Number of TV manufactured in Plant 2

3

9000

$3,000,000.00

$2300

X3

Number of TV manufactured in Plant 3

4

7000

$4,000,000.00

$2100

X4

Number of TV manufactured in Plant 4

5

6000

$1,000,000.00

$2400

X5

Number of TV manufactured in Plant 5

Y1

Binary Variable (1 if X1>0 else 0)

Y2

Binary Variable (1 if X2>0 else 0)

Y3

Binary Variable (1 if X3>0 else 0)

Y4

Binary Variable (1 if X4>0 else 0)

Y5

Binary Variable (1 if X5>0 else 0)

Objective Function

Profit=

Number of TVs from each plant* Selling Price-Fixed Cost-Number of TVs from Each Plant * Variable cost in that plant location

Profit=

(X1+X2+X3+X4+X5)*4000 -( 9,000,000Y1+5,000,000Y2-3,000,000Y3-4,000,000Y4-1,000,000Y5)-(X1*1600-X2*2000-X3*2300-X4*2100-X5*2400)

Profit=

4000X1+4000X2+4000X3+4000X4+4000X5-1600X1-2000X2-2300X3-2100X4-2400X5-(9,000,000Y1+5,000,000Y2-3,000,000Y3-4,000,000Y4-1,000,000Y5)

Z=

2400X1+2000X2+1700X3+1900X4+1600X5-(9,000,000Y1+5,000,000Y2-3,000,000Y3-4,000,000Y4-1,000,000Y5)

Subject to

Constraint 1

Maximum Sales can be 25000

X1+X2+X3+X4+X5<=25000

Constraint 1

Individual plant capacity constraint

Plant 1: X1<=10000Y1

Plant 2: X2<=8000Y2

Plant 3: X3<=9000Y3

Plant 4: X4<=7000Y4

Plant 5: X5<=6000Y5

Non Negative constraint:

X1,X2,X3,X4,X5>=0

Binary Variables

Y1,Y2,Y3,Y4,Y5 are binary variables

Selling Capacity	X1	X2	X3	X4	X5
20000	0	8000	9000	0	3000
21000	0	8000	9000	0	4000
22000	9000	0	0	7000	6000
23000	10000	0	0	7000	6000
24000	10000	8000	0	0	6000
25000	10000	0	9000	0	6000
26000	9000	8000	9000	0	0
27000	10000	8000	9000	0	0
28000	10000	0	0	7000	6000
29000	0	8000	9000	6000	6000
30000	9000	8000	0	7000	6000

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