d e of question number 1

1. A factory manufactures three products. Three resources technical services, labor, and administration are required to produce these products. The following table gives the requirements on each of the resources for the three products: Resources (hours) Unit Profit Product Technical Service Labor Administration (8) 10 2 6 There are 100 hours of technical services available, 600 hours of labor, and 300 hours of administration. In order to determine the optimal product mix which will maximize the total profit, the following linear program was solved: Maximize: Z=10x, +6x2 + 4x3 Subject to: X, + x2 + x3 5100 (technical) 10x, +4x2 + 5x600 (labor) 2x + 2x, +6x, S 300 (administration) *, X2, 4320 where x, Xy, x, are the number of product 1, product 2, and product 3 produced. The optimal solution is given by the following tableau, where xq, X5, X, are the slack variables: 0 Constants Basis 5/6 10/6 400/6 -1/6 1/6 -476 1/6 200/6 100 - 16/6 -20/6 0 -4/6 0 Z-4400/6 Row 0 X5 0 26 0 Using sensitivity analysis, answer the following with respect to the above optimal tableau: ya) Determine the shadow prices of all the resources. Camlar b) What should be the profit of product 3 before it becomes worthwhile to manufacture? Find the most profitable product mix if the profit on product 3 were increased to $50/6. c) What is the range on the profit of product 1 so that the current solution is still optimal? d) The manufacturing department comes up with a proposal to produce a new product requiring 1 hour of technical service, 4 hours of labor, and 3 hours of administration. The marketing and sales department predicts that the product can be sold at a unit profit of $8. What should be the management's decision? Suppose the company decides to produce at least 10 units of product 3; determine the optimal product mix. It is believed that the estimates of the available hours of technical services might be wrong. The correct estimate is 100 + 10 2 where is some unknown parameter. Find the range of values of 2 within which the given product mix is still optimal. e) 2. The following tableau gives an optimal solution to a standard LP: Maximize: Z CX, subject to: AX=b, X 20, where C=(2,3,1,0,0)