Question

15. Combining raw materials M and M2 and labor, a company produces units of A. B, and C. The requirements and profit (excluding the cost of labor) for the production and sale of a unit of each are as follows: Mi (lb) M2 (lb) Labor (hr) Profit (S) 105 165 60 12 For the next month, the company has available 1 ton of Mi. 2.5 tons of M2, 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 xi, x2, and x3 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: Maximizez 105x1+1 65X2+60X3 18(2x11 3x2lxs) 6x4 69x1+111x2 +42x3 -6x subject to 6xi 12x2 + 4x3 2000 16X1 + 25x2 + 7x3-5000 2xi 1 3x2+ X3 500+x4 14120 Adding four slack variables and applying the simplex algorithm, the reduced tableaux resolution initial and final tableaux only) are shown in Table 5.2 (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 $7.75/lb over what the company now pays for the raw material. Should more be purchased? CHAPTER 5. SENSITIVITY ptimal solution points for thei 172 d) Show that these solutions are, in fact, optimal solution spective problems. e) Referring to the last paragraph of to the last paragraph of Section 2.3 on page 26, assist t of the original What does the marginal value for the first const (0 ponding to the vice president's questions manager in res A, B, and C. The requirements and profit (excluding teces uni production and sale of a unit of each are as follows: suggest about the use of overtime? 15 Combining raw materials Mi and Mz and labor, a company prod M2 (lb) 16 Labor (hr) -Profit($) MI (lb) 105 165 60 B 12 25 For the next month, the company has available 1 ton of Mi, 2.5 tons ot hr of labor at SI 8 hr, and up to an additional 120 hr of overtime at $24ん2.50 company pays only for the labor used.) re So To determine an optimal production schedule, the company manage t2, and x3 to be the number of A's, B's, and C's to be produced and number of hours of overtime to be used and formulates the following model x4 to be the no 105rı + 165x2 + 60r3-18 (2n + 3x2 + x3)-6x4 - 69x1 +111x2 +42x3-6x4 Maximize z subject to ór +12x2 4x3 2000 16x +25x2 +7x3 5000 2x + 3x2 x500+x4 x4 120 Pou Sen Adding four slack variables and applying the simplex algorithm, the reduceal tableaux resolution (initial and final tableaux only) are shown in Table 5.2. (o) What is the optimal production schedule, and what profit does it yield? (b) Write out the dual problem and determine an optimal solution po (c) Several employees offer to work additional hours of overtime int $24/hr rate). Should the manager accept their offer? (d) Suppose additional pounds of M, could be purchased, at a cost ol .noree over what the company now pays for the raw material. Should purchased? (at the same In Example 5.1.1 it was determined that (1,2) is the optimal souf the dual of the poultry producer's diet problem as l ong as the rato 9. Monthly profits are maximized for the company described in Problem 15 of Section 5.1 by the making and selling of 240 A's and 140 C's (a) Using the data from Table 5.2, determine by how much the profit on the sale of a unit of A (the S140) can vary before the optimal production schedule (b) By how much do we need to increase the profit on the B's for them to be (c) By how much can the profit on the sale of a unit of C (the $60) vary before (d) Suppose labor requests a pay raise, from $18/hr to $(18+A)/hr, with over- would change part of the optimal production schedule? the optimal production schedule would change? time pay to be S(24A)hr. How large can A be before the optimal produc tion schedule would change?

Answer 1

Siven the uontity 28 6 12 16 25 1 | 10 | | 000 |七| 3000 35 306 500 2. 10S 165 6 06

Nou etting the Profit move tha 26005 0 | | 2S0 | 125 | 180 | 6 1 16 | 180 0 |2000 | | 2000 10 | O |300 |仁 18000 26032 5 o, the ValuzCeee 6 Hence If the Volut U above %6.