Question

The ABC Company is one of the largest producers of power tools in the United States. The company is preparing to replace its current product line with the next generation of products: specifically, three exciting new power tools with the latest state-of-the-art features. Because of the limited amount of capital available, management needs to make some difficult choices about how much to invest in each of these products. Another concern is the effect of these decisions on the company’s ability to maintain a relatively stable employment level. In addressing these decisions, management wants primary consideration given to three factors: total profit, stability in the workforce, and the level of capital investment needed to launch these products.

Goal 1: Achieve a total profit (NPV) of at least $250 million.

Goal 2: Maintain the current employment level of 8,000 employees. Goal 3: Hold the capital investment down to no more than $110 million.

All goals are important, but by small margin their order of importance is: Priority 1: Goal 1,

Priority 2: Goal 3,

Priority 3: Part of Goal 2 (avoid increasing the employment level), Priority 4: Part of Goal 2 (avoid decreasing the employment level)

The company estimated contributions per unit of each product along with all the necessary information as follows:

Factor	P1	P2	P3	Goal
Total profit ($mil)	12	9	15	≥ 250
Employee level (00s)	5	3	4	= 80
Capital investment ($mil)	5	7	8	≤ 110

Goal	Factor	Penalty Weight for Missing Goal
1	Total profit	50 (per $1 mil under the goal)
2	Employment level	20 (per 100 employees under the goal 30 (per 100 employees over the goal)
3	Capital investment	40 (per $1 mil over the goal)

1. Formulate the above problem into Goal Programming (GP).
2. Find the optimal solution (Attach Excel file).

Answer 1

Solution:

Goal Formulation

Let x, y and z be the number of units of P1, P2 and P3 produced respectively.

Goal 1: 12 x + 9 y + 15 z >= 250

Goal 2 5 x + 7 y + 8 z <= 110

Goal 3 5 x + 3 y + 4 z = 80

Penalty weights:

Penalty 1 : If ((250 > 12x +9 y + 15 z), then (250 - (12 x + 9 y + 15 z)) * 7, otherwise 0)

Penalty 2 : If ((110 < 5x+7y+8z). then ((5 x + 7y + 8z ) - 110) * 5, otherwise 0)

Penalty 3 & 4 : If(80 > 5 x + 3y + 4 z , then (80 - (5x+3y +4z))*4. otherwise (( 5x + 3y + 4 z) -80) * 2)

x, y, z = 0

x,y,z = Integers (no fractions allowed)

Using Solver and using trial and error method, the following seems to be the optimal solution:

P1 qty = x = 6 units

P2 qty = y = 3 units

P3 qty = z = 10 units

And the resultant Profit = 249, Capital Expenditure = 131 and Labor = 79

Solver output is shown below:

		Qty
	x	6			Goal 1 Profit	249
	y	3			Goal 2 Capex	131
	z	10			Goal 3 Labor	79
			If Profit < 250	Penalty 1	7
			If Capex > 110	Penalty 2	105
			If labour<8000	Penalty3	4
			If labour > 8000	Penalty 4	0
					Total Penalty	116