Question

Chen is president of Chen cabinets Inc., a firm that manufactures two types of metal file cabinets.  Chen has a weekly labor capacity of 1,300 hours, with each smaller cabinet taking 1 hour to produce and the larger cabinet requiring 2 hours each. One wooden plank is used for each smaller cabinet and 1.5 planks are used for each larger cabinet.  Chen can get a supply of a maximum of 1000 planks each week.  Each two-drawer model sold yields a $10 profit, and the profit for the larger model is $25.

Chen has the following goals (1). Maximize profit, (2). Maximize number of cabinets produced.

1. Formulate this Goal programming problem.
2. Solve the goal programming problem to minimize the maximum % deviation from each goal. Show the solution, target values and % deviation from each goal.
3. What will be new solution if Goal # 1 was twice as important as Goal # 2?

Answer 1

What if all the resources were consumed for the smaller cabinet, possible production given 1300 hours of labor capacity = 1300/1 = 1300 unit; possible production given 1000 planks = 1000/1 = 1000 units. So, maximum possible production = min(1300, 1000) = 1000

What if all the resources were consumed for the larger cabinet, possible production given 1300 hours of labor capacity = 1300/2 = 650 unit; possible production given 1000 planks = 1000/1.5 = 667 units. So, maximum possible production = min(650, 667) = 650

So, we can take the upper bound of profit as 1000*$10 + 650*$25 = $26,250

Similarly, we can take upper bound for the total number fo cabinets = 1000+650 = 1650

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(a)

Let x₁ and x₂ be the number of small and large cabinets to be produced. Also, let d_j^- and d_j⁺ be the under- and over-achievement of the goal-j for j=1,2

Min (p₁d₁^-, p₂d₂^-)

Subject to,

1 x₁ + 2 x₂ <= 1300 (labor hrs.)

1 x₁ + 1.5 x₂ <= 1000 (wood)

10 x₁ + 25 x₂ + d₁^- - d₁⁺ = 26250 (goal-1)

x₁ + x₂ + d₂^- - d₂⁺ = 1650 (goal-2)

x₁, x₂, d_j^-, d_j⁺ >= 0 for j=1,2

(b)

We will solve this in a non-preemptive manner i.e. taking all goals at a time with equal weights to their % deviation as follows:

Solver inputs:

Solution:

	x1	x2	d1+	d1-	d2+	d2-
Value of	100	600	0	10250	0	950
Z2				3.80952E-05		0.000606061	0.966233766
s.t.
Labor	1	2					1300	<=	1300
Wood	1	1.5					1000	<=	1000
Goal-1	10	25	-1	1			26250	=	26250
Goal-2	1	1			-1	1	1650	=	1650

x1 = 100
x2 = 600
Target value for goal-1 = 26250
% deviation from goal-1 = 10250 / 26250 = 39%
Target value for goal-2 = 1650
% deviation from goal-2 = 950 / 1650 = 58%

(c)

Here, we will put twice weight to the first goal's % deviation as follows:

The solver inputs will be the same:

Solution (same as before):

	x1	x2	d1+	d1-	d2+	d2-
Value of	100	600	0	10250	0	950
Z2				3.80952E-05		0.000606061	0.966233766
s.t.
Labor	1	2					1300	<=	1300
Wood	1	1.5					1000	<=	1000
Goal-1	10	25	-1	1			26250	=	26250
Goal-2	1	1			-1	1	1650	=	1650

x1 = 100
x2 = 600
Target value for goal-1 = 26250
% deviation from goal-1 = 10250 / 26250 = 39%
Target value for goal-2 = 1650
% deviation from goal-2 = 950 / 1650 = 58%