9. Monthly profits are maximized for the company described in Problem 15 d Secti
ID: 361330 • Letter: 9
Question
9. Monthly profits are maximized for the company described in Problem 15 d Section 5.1 by the making and selling of 240 A's and 140 Cs (a) Using the data from Table S.2, determine by how mach the proit on the sale af a unit of A (the S140) can vary before the optiral pod ton sild b) By how much do we need to increase the profit on the B's for them to be the prott on the sale of a unit of Che S60, vary before (d) Suppose labor requests a pay raise, from Si Shr to $(18 + )h, with ovet- would change part of the optimal production schedule? the optimal production schedule would change? time pay to be $(24 hr. How large can be belorethe optimal produc- (c) By how much can tion schedule would change? units of 15. Combining raw materials M, and M2 and labor, a company prodacos A. B, and C. The requirements and profit (escluding the cost of labon) to production and sale of a unit of each are as follows: M2 (D) 16 25 Preyle (5) 105 AtB) Labor (hr, 8 12 For the next month, the company has available I ton of M.2.5 tons of M, 50 Ir of labor at $1shr. and upto an allitional 120 hr ofovertime at company pays only for the labor used.) To determine an optimal production schodule, the company manager defines x2, and x) to be the number of AS, B's, and C's to be produced and 4 to bethe number of bhours of overtime to be used and formulates the following model Maimize z-105r1 + 165r2 + 60r-18( 2n + 3+1)-6r4 6911 + 111x2 +420-6x4 subject to 6x1 + 12n + 43 2000 Iáty + 25x+7x S 5000 2 + 3x2+ x3 50+14 x S 120 Adding four slack variables and applying the simplex algorithm, the tableaux resolution (initial and final tableaux only) are shown in Table 52 a) What is the optimal production schedule, and what profit does it yield? (b) Write out the dual problem and determine an optimal solution (c) Several employees offer to work additional hours of overtime (at the same $24Phr rate). Should the manager accept their offer? (d) Suppose additional pounds of M, could be parchasod, at a cost of $1.3Sib over what the company now pays for the raw material. Should more be Table 5.2 12 4 1O02000 25700100 162 31-1 01500 0 1 0 0 01 120 69-l11 -42 600 0 0 4 0 11-11-11 180 0 I O 1 120 1000 0 221 210 3 1 01 0 -3 -3140 0 15 0 0 7 0 12 6 21,730 ryExplanation / Answer
(a) Current profit on sale of product A is $ 105 (in the question it is incorrectly mentioned as the $ 140 )
Range of profit on product A to sustain current Optimal solution is 105-3 = 102 and 105+15 = 120
(b) Profit on B needs to increase by $ 15 (refer corresponding reduced cost of x2 in the last row of table 5.2)
(c) Profit on product C needs to decrease by 5 before the optimal solution changes
(d) The maximum allowable decrease is 6. Therefore lambda can increase by a maximum of 6 before optimal production schedule changes.
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