1. (30 points) Gangbuk manufacturing company makes four different types of produ

Question

1. (30 points) Gangbuk manufacturing company makes four different types of products. To make the products, three different resources are required. is the number of product i, where i = {1, 2, 3, 4} Consider the following final table, where xs-x, and x, are slack variables for resources 1, 2, and 3, Maximization X1 Xs X7 Basis c 4 6 31 0 0 0RHS (b Ratio (e) 1/20 01 1/2 3/10 0 1/5 125 0 13/4 0 0 1/2 -1/2 1425 25 X4 13/20 1 0 1/21/10 02/5 r(i) | 81/20 | 6 | 3 |92| 3/10 | 0 | 9/5 | 525 | Objective ciHzU)1/200 7/23/10 0 9/ (a) Perform a sensitivity analysis for c and ca, which are the objective coefficient of x: and xa, respectively (b) Perform a sensitivity analysis for ba, which is the right-hand-side value of the second constraint. (hint: you will have to find out the initial simplex table before doing the sensitivity analysis) (c) Suppose a new product (say x) were introduced which uses 3 units of resource 1, 4 units of resource 2, and 4 units of resource 3. What is the minimum profit for the new product to be worthwhile to

Explanation / Answer

(a)

Sensitivity of c2: Allowable increase is 3, Allowable decrease is 1/13

Sensitivity of c1: Allowable increase is 0.05, Allowable decrease is INFINITY

(b)

Travelling back to initial tableau

The second constraint is 6x1 + 4x2 + 3x3 + 3x4 <= 900

With the optimal solution, LHS = 6 x 0 + 4 x 25 + 3 x 125 + 3 x 0 = 475

Therefore slack = 900 - 475 = 425 = allowable decrease

Allowable increase = INFINITY (non-binding)

Dual price = 0 (non-binding)

(c)

There will be a new dual constraint corresponding to the new primal variable. This constraint can be written as -

3y1 + 4y2 + 4y3 = b4 (note the equality sign will hold good because we know that that we are doing this for a positive value of x8 for which this constraint will be binding)

The present values of y1, y2, and y3 can be writeen from the final tableau. They are equal to the shadow prices.

i.e. 3 x 0.3 + 4 x 0 + 4 x 1.8 = b4 = 8.1

So, any profit equal to or greater than 8.1 will be good o introduce the product.

Maximization x1 x2 x3 x4 x5 x6 x7 Basis Cj 4 6 3 1 0 0 0 RHS x3 3 1/20 0/1 1/1 1/2 3/10 0/1 - 1/5 125 x6 0 13/4 0/1 0/1 - 1/2 - 1/2 1/1 - 1/1 425 x2 6 13/20 1/1 0/1 1/2 - 1/10 0/1 2/5 25 Zj 81/20 6/1 3/1 9/2 3/10 0/1 9/5 525 Cj - Zj - 1/20 0/1 0/1 - 7/2 - 3/10 0/1 - 9/5 (Cj - Xj)/x2 - 1/13 0/1 - - 7/1 3/1 - - 9/2

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