an LPP problem asked in aktu mba exam in 2008 (operations research) formulated, and solved using simplex method

1. LPP Numerical
2008

2. The ABC Company has been a producer of picture tubes for TV sets and
certain printed circuits for radios. The company has just expanded into full
scale production and marketing of AM-FM radios. It has built a new plant
that can operate 48 hours per week. Production of an AM radio in the new
plant will require 2 hours and production of AM-FM radio requires 3 hours.
Each AM radio will contribute Rs.40 to profits while an AM-FM radio will
contribute Rs.80 to profits. The marketing department, after extensive
research, has determined that a maximum of 15 AM radios and 10 AM-FM
radios can be sold each week. Formulate a linear programming model to
determine the optimum production mix of AM and AM-FM radios that will
maximize profits. Solve it using the graphical method and the simplex
method. (2008)

3. 𝑀𝑎𝑥𝑖𝑚𝑖𝑠𝑒 𝑧 = 40𝑥1 + 80𝑥2
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑠:
2𝑥1 + 3𝑥2 ≤ 48
0 ≤ 𝑥1 ≤ 15
0 ≤ 𝑥2 ≤ 10

4. 𝐼𝑛𝑡𝑟𝑜𝑑𝑢𝑐𝑖𝑛𝑔 𝑠𝑙𝑎𝑐𝑘 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠:
𝑀𝑎𝑥𝑖𝑚𝑖𝑠𝑒 𝑧 = 40𝑥1 + 80𝑥2 + 0𝑆1 + 0𝑆2 + 0𝑆3
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑠:
2𝑥1 + 3𝑥2 + 𝑆1 = 48
𝑥1 + 𝑆2 = 15
𝑥2 + 𝑆3 = 10

5. 𝐶𝑗 40 80 0 0 0
𝐶 𝑏 𝐵𝑉 𝑆𝑉 𝑥1 𝑥2 𝑆1 𝑆2 𝑆3
0 𝑆1 48 2 3 1 0 0
48
3
= 16
0 𝑆2 15 1 0 0 1 0
15
0
0 𝑆3 10 0 1 0 0 1
10
1
= 10 
𝑍𝑗 0 0 0 0 0 0
𝐶𝑗 − 𝑍𝑖 40 80 0 0 0


6. 𝐶𝑗 40 80 0 0 0
𝐶 𝑏 𝐵𝑉 𝑆𝑉 𝑥1 𝑥2 𝑆1 𝑆2 𝑆3
0 𝑆1 18 2 0 1 0 −3
𝑅1
= 𝑅1 − 3𝑅3
18
2
= 9
0 𝑆2 15 1 0 0 1 0
𝑅2
= 𝑅2 − 0𝑅3
15
1
= 15
80 𝑥2 10 0 1 0 0 1 𝑅3 = 𝑅3
10
0
𝑍𝑗 800 0 80 0 0 80
𝐶𝑗
− 𝑍𝑖
40 0 0 0 −80


7. 𝐶𝑗 40 80 0 0 0
𝐶 𝑏 𝐵𝑉 𝑆𝑉 𝑥1 𝑥2 𝑆1 𝑆2 𝑆3
40 𝑥1 9 1 0
1
2
0 −
3
2
𝑅1 =
𝑅1
2
0 𝑆2 6 0 0 −
1
2
1
3
2
𝑅2 = 𝑅2 − 𝑅1
80 𝑥2 10 0 1 0 0 1 𝑅3 = 𝑅3 − 0𝑅1
𝑍𝑗 1160 40 80 20 0 20
𝐶𝑗 − 𝑍𝑖 0 0 −20 0 −20