uces units of 15. Combining raw materials Mi and M2 and labor, a company prod fo

Question

uces units of 15. Combining raw materials Mi and M2 and labor, a company prod for the A, B, and C. The requirements and profit (excluding the cost of labor) production and sale of a unit of each are as follows: Profit (S) 105 165 60 Labor (hr) M2 (lb) 16 25 7 Mi (lb) B 12 For the next month, the company has available 1 ton of Mi , 2.5 tons of M, 500 hr of labor at $18/hr, and up to an additional 120 hr of overtime at $24/hr. (The company pays only for the labor used.) To determine an optimal production schedule, the company manager defines .x x2, and xj to be the number of A's, B's, and C's to be produced and x4 to be the number of hours of overtime to be used and formulates the following model: Maximize z = 105xi + 165r2 + 60r3-18 (2x1 + 3x2 + x3)-6x4 = 69n + 1 1 1x2 + 42x3-6x4 subject to 6xi + 12r2 + 4x3 2000 16x + 25x2 + 7x3 5000 x4 120 Adding four slack variables and applying the simplex algorithm, the red tableaux resolution (initial and final tableaux only) are shown in Table 5.2. uced (a) What is the optimal production schedule, and what profit does it yield? (b) Write out the dual problem and determine an optimal solution point. (c) Several employees offer to work additional hours of overtime (at the same $24/hr rate). Should the manager accept their offer? (d) Suppose additional pounds of Mi could be purchased, at a cost of $7.75i1b over what the company now pays for the raw material. Should more be purchased? 16. In Fra 6. The company in Problem 15 of Section 5.1 could also make D's, with a unit D requiring 8 Ib of Mi, 10 lb of M2, and 1.5 hr of labor. What minimum profit, excluding labor costs (i.e., selling price less cost of raw materials), is necessary before the company would produce and sell D's? 7. You, the production sunei

Explanation / Answer

So solving the above with 0 as profit:

we get the profit as:

The quantity produced is the first row

Now setting the profit more than >= 26005.71

Using trial and error in excel, we get:

X5 as 125 units (Which is D)

SO the value of X5 coeff is 86.

Hence if the value is above 86 (as profit) D should be produced.

X1 X2 X3 X5 X4 0 0 428.5714 0 120 <= 120 6 12 4 8 0 1714.286 <= 2000 16 25 7 10 0 3000 <= 3000 2 3 1 1.5 -1 308.5714 <= 500 105 165 60 0 -6 26005.71

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