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# Chapter 7

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Submitted By naveen838
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7 - 14)
Let A = Number of air conditioners to be produced F = Number of large fans to be produced. Department | Air conditioners | Fans | Total number of hours available | Wiring | 3 Hours | 2 Hours | 240 | Drilling | 2 Hours | 1 Hour | 140 | Profit per unit | \$ 25 per unit | \$ 15 per unit | | Maximum profit P = (\$25) * (A) + (\$15) * (F)
Hours of wiring time used is ≤ 240 ==> 3A + 2F ≤ 240
Hours of drilling time used is ≤ 140 ==> 2A + 1F ≤ 140
Both ( A and F ≥ 0 )
Draw the graph for
3A + 2F = 240 ==> Substitute A as zero we will get point as (0, 120) Substitute F as zero we will get point as ( 80, 0)
2A + 1F = 140 ==> Substitute A as zero we will get point as (0, 140) Substitute F as zero we will get point as (70, 0)

The optimal solution is air conditioners A = 40 and large fans F = 60
Maximum Profit P = (\$25 * 40) + (\$15 * 60) = 1000+900 = \$1,900 . OR
Solving for A and F from above equations
3A+2F = 240 -----------------------equation1
2A+1F = 140 -----------------------equation2 from equation2
F = 140 - 2A ---------------------equation3 substitute equation3 in equation1 for finding A
3A+2(140-2A) = 240 ==> 3A + 280 - 4A =240 ==> A = 40
Substitute value of A in equation 2 to find F
2(40) + 1F = 140 ==> F = 60
The optimal solution of A is 40 and F is 60
Maximum profit P = (\$25 * 40) + (\$15 * 60) = 1000+900 = \$1,900 .

7 - 18)
Let UG = Number of undergraduate courses to offer G = Number of graduate courses to offer
Based on the students demand at least 30 undergraduate and 20 graduate courses need to be offered
At least 30 undergraduate courses i.e., UG ≥ 30
At least 20 undergraduate courses i.e., G ≥ 20
Both ( UG and G) ≥ 0
Based on the faculty contract at least 60 courses need to be offered
UG + G ≥ 60
S = (\$2,500 * UG) + (\$3,000 * G)
Based on the students demand minimum 30UG + 20G courses need to be offered i.e., total 50 courses
Based on the faculty demand minimum of 60 courses need to be offered.
Comparing to graduate course fee undergraduate course fee is less so to make it a total 60 courses with minimum cost we need to add 10 additional under graduate courses. so total undergraduate courses is 40 and graduate courses is 20.
S = (\$2,500 * 40) + (\$3,000 * 20) = 1,00,000 + 60,000 = \$1,60,000.

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