A factory can produce four products denoted hy P1, P2, P3 and P4. Each product m

Question

A factory can produce four products denoted hy P1, P2, P3 and P4. Each product must be processed In each of two workshops. The processing times (in hours per urnt produced) are given in the following table. 400 hours of labour are available in each workshop. The profit margins are 4, 6. 10, arul 9 TL per unit of P1, P2, P3, and P4 produced, respectively, liverything that is produced can be sold. Thus, maximising profits. the following linear program can be used. Introducing slack variables s1 and s2 in Rows 1 anil 2. respectively, and applyiugg the simplex method, we get the final tableau: How many units of P1, P2, P3 and P4 should be produced in order to maximise profits? Assume that 20 units of P3 have been produced by mistake. What is the resulting decrease in profit? In what range ran the profit margin per unit of P, vary without changing the optimal basis? In what range can the profit margin per unit of P2 vary without changing the optimal basis? What is the marginal value of increasing the production capacity of Workshop 1? In what range can the capacity of Workshop], vary without changing the optimal basis? Management is considering the production of a new product ft. that would require 2 hours iu Workshopl and ten hours in Workshop2. What is the minimum profit margin needed on this new product to make it worth producing?

Explanation / Answer

a) From the final tableau, we read that x2 = 100 is basic and x1 = x3 = x4 = 0 are nonbasic. So, 100 units of P2 should be produced and none of P1, P3 and P4. The resuting profit is $600 and that is the maximum possible, given the constraints.

(b) The reduced cost for x3 is 2 (found in Row 0 of the final tableau). Thus, the effect on profit of producing x3 units of P3 is 2x3. If 20 units of P3 have been produced by mistake, then the profit will be 2*20 = $40 lower than the maximum stated in (a).

(c) Let 4 + d be the profit margin on P1. The reduced cost remains non negative in the final tableau if 0.5 - d >= 0. That is d <= 0.5. Therefore, as long as the profit margin on P1 is less than 4.5, the optimal basis remains unchanged.

(d) Let 6 + d be the profit margin on P2. Since x2 is basic, we need to restore a correct basis. This is done by adding d times Row 1 to Row 0. This effects the reduced costs of the nonbasic variables, namely x1, x3, x4 and s1. All these reduced costs must be nonnegative. This implies:
0.5 + 0.75d >= 0
2 + 2d >= 0
0 + 1.5d >= 0
1.5 + 0.25d >= 0.
Combining all these inequalities, we get d >= 0. So, as long as the profit margin on P2 is 6 or greater, the optimal basis remains unchanged.

(e) The marginal value of increasing capacity in Workshop 1 is lambda1 = 1.5.

(f) Let 400 + d be the capacity of Workshop 1. The resulting RHS in the final tableau will be:
100 + 0.25d in Row 1, and
200 - 0.5d in Row 2.
The optimal basis remains unchanged as long as these two quantities are nonnegative. Namely, -400 <= d <= 400. So, the optimal basis remains unchanged as long as the capacity of Workshop 1 is in the range 0 to 800.

(g) The effect on the optimum profit of producing x5 units of P5 would be

lambda1*(2x5) + lambda2*(10x5) = 1.5(2x5) + 0(10x5) = 3x5.

If the profit margin on P5 is sufficient to offset this, then P5 should be produced. That is, we should produce P5 if its profit margin is at least 3.

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