Question

Solve using the Simplex Method

You own two gold refining companies, A and B. Company A costs $20,000 per day to operate, and it can produce 400 grams of high-grade gold, 300 grams of medium-grade gold, and 200 grams of low-grade gold each day. Company B costs $25,000 per day to operate, and it can produce 300 grams of high-grade gold, 400 grams of medium-grade gold, and 500 grams of low-grade gold each day. You have orders totaling 25,000 grams of high-grade gold, 27,000 grams of medium-grade gold, and 30,000 grams of low-grade gold. How many days should you run each company to minimize the costs and still refine enough gold to meet its orders? More specifically, please complete the following sentences: 1) Minimum cost is $......... 2) Company A should operate ......... days Company B should operate ......... days 3) Totally, at that cost and during that operation days you will produce ............. grams of high-grid gold; grams of medium-grid gold; .............. grams of low-grid gold; 4) So totally you will produce ......... grams of gold Please use additional pages to show your detailed solution below

Answer 1

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Problem is Min Z = 20000 x1 + 25000 X2 subject to 400 X1 + 300 X2 > 25000 300 X1 + 400 X2 2 27000 200 X1 + 500 X2 2 30000 and X1, X2 2 0;

After introducing surplus,artificial variables Min Z = 20000 x1 + 25000 X2 + 0S1 + 0S2 + 053 + MA1 + MA2 + MA3 subject to 400 X1 + 300 X2 - Si + A = 25000 300 x1 + 400 X2 - S2 + A2 = 27000 200 X1 + 500 X2 - S3 + Az = 30000 and X1, X2, S1, S2, S3, A1, A2, A3 2 0 Iteration-1 C; 20000 25000 0 0 0 M M m MinRatio S2 S3 A XB 25000 M 25000 400 300 0 1 0 83. 333 300 27000 M 27000 300 400 0 0 1 400 = 67.5 30000 M 30000 200 (500) 0 0 -1 0 500 = 60 - Z = 82000M Zi 900M 1200M -M -M -M M мм Z; -C; 900M - 20000 1200M - 25000+ -M -M -M 0

Iteration-2 20000 25000 0 0 0 MM MinRatio x x₂ S S2 S3 A, A₂ IR 7000 280 280 = 25 LA MIW 3000 (140) 140 = 21.4286 25000 60 ON aj- z = 150 NILA -= 10000M + 1500000 2; 420M + 10000 2;-G 420M - 10000 1 25000 -M - M 0 -M-M 7 M .50 M M -50 0 0

Iteration-3 20000 25000 MinRatio CB X3 X1 X2 s 1000 5° = 500 - 360 25000 1 0 -0.0043 1 - 18000 350 12000000 Z = 1000M + z; 20000 25000 -M + 7.1429 M M 2M - M2M- SO-M + 7.14290 27-; 0 -M -M + 7.1429

Iteration-4 20000 25000 0 Si 0 S2 0 Sz 20000 0.0021 25000 -0.0029 Z = 1750000 20000 25000 -28.5714 0 0 -28.5714 Since all Z; -C; so Hence, optimal solution is arrived with value of variables as : 1 = 25, X2 = 50 Min Z = 1750000

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1 minimum cost = 1750000

2.

company A operate 25 days

company B operate 50 days ..

3.

25000 gm high grade

27500 gm medium grade

30000 gm low grade ...

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4 ..total 82500 gm gold