Duality-Motivating Example Moscow industrial Ghetto (MIG) manufactures small and

Question

Duality-Motivating Example Moscow industrial Ghetto (MIG) manufactures small and large iron dragons. The large dragons are used for garden ornaments, the small ones for automobile ornaments. Each week, they can purchase up to 1000 pounds of iron at $0.25. For small dragons, it takes 3 Ibs of iron, 2 hours of labor and 1.5 hours of machine time. A large dragon uses 125 lbs of iron, 12 hours of labor, and 9 hours of machine time. MIG employs 5 workers, each of which works 30 hours per week. The worker's salary is fixed and does not depend on productivity. The machinery needed to make the dragons is availably for up to 300 hours/week. The company can sell a small dragon for $35, and a large dragon for $300. Let x, the number of small dragons produced, and x2- the number of large dragons produced. Then we can summarize the information in a table Small dragon Large dragon Availability of resources Iron per dragon lb] Labor per dragon hr Machine time [hr 125 12 3-50 = 150 s 300 s 1000 Selling price I$ Cost per dragon [S 1.5 35 0.75 300 31.25 268.75 Profit per dragon IS34.25 Set up the associated LP.

Explanation / Answer

By simplex method ( x1 = x & x2 = y )

Tableau #1
x y s1 s2 s3 p
3 125 1 0 0 0 1000   
2 12 0 1 0 0 150
1.5 9 0 0 1 0 300
-34.25 -268.75 0 0 0 1 0

Tableau #2
x y s1 s2 s3 p
0.024 1 0.008 0 0 0 8   
1.712 0 -0.096 1 0 0 54
1.284 0 -0.072 0 1 0 228   
-27.8 0 2.15 0 0 1 2150

Tableau #3
x y s1 s2 s3 p
0 1 0.00934579 -0.0140187 0 0 7.24299   
1 0 -0.0560748 0.584112 0 0 31.5421   
0 0 0 -0.75 1 0 187.5   
0 0 0.591121 16.2383 0 1 3026.87   

Optimal Solution: Profit (p) = 3026.87$; Small dragon produced aprox (x1) = 31.5421,Large dragon (x2) = 7.24299

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