Question

In the furniture store that sells tables and chairs, business is good and so the demand for tables is up to 500 pieces per week and for chairs up to 300 pieces. The production has a weekly capacity of 1200 hours, with a table taking 40 minutes to produce and a chair 80 minutes. Each table nets a profit of $7/piece but a chair’s profit is now $10. The manager’s goals are to attain a profit of $10,000; to avoid underutilization of the production department; and to sell as many tables and chairs as demand permits. Help the manager achieve her goals

Answer 1

Let

T = No. of tables to be produced
C = No. of chairs to be produced
d_j^- = Underachievement of goal-j for j=1,2,3
d_j⁺ = Overachievement of goal-j for j=1,2,3

Min Z = d₁^- + d₂^- + d_3T^- + d_3C^-

Subject to,

7 T + 10 C + d₁^- - d₁⁺ = 10000 (Goal-1)

40 T + 80 C + d₂^- - d₂⁺ = 72000 (Goal-2)

T + d_3T^- = 500 (Goal-3 for tables)

C + d_3C^- = 300 (Goal-3 for chairs)

T, C, d₁^-, d₁⁺, d₂^-, d₂⁺, d_3T^-, d_3C^- >= 0

Note: The above is a non-preemptive goal programming formulation where we have taken equal weights to all the goals. We now solve this using Excel solver as follows:

Solution:

	T	C	d1-	d1+	d2-	d2+	d3T-	d3C-
Value of	500	300	3500	0	28000	0	0	0
Z	0	0	1	0	1	0	1	1	31500
s.t.
Goal-1	7	10	1	-1					10000	10000
Goal-2	40	80			1	-1			72000	72000
Goal-3(table)	1						1		500	500
Goal-3(chair)		1						1	300	300

Note that Goal-1 and 2 were not achieved.

Is it possible to achieve Goal-1? Of course not. Because the maximum production possible is 500 and 300 for T and C, the max profit can be 500*7 + 300*10 = 6500.

Is it possible to achieve Goal-2? Of course not. Because the maximum production possible is 500 and 300 for T and C, the max utilization possible is 500*40 + 300*80 = 44000. So, the underutilization can not be made zero.