1. A manufacturing company produces diesel engines in four factories located in

Question

1. A manufacturing company produces diesel engines in four factories located in Phoenix, Seattle, Baltimore, and Cleveland. Three trucking firms purchase these engines for their plants located in Nashville, Orlando, and Charleston. The supplies and demands, along with the per engine transportation costs in dollars are given below: Plant Nashville Orlando Charleston Supply 880 650 550 620 500 700 472 520 20 30 10 25 1150 Phoenix Seattle Baltimore Cleveland 1050 910 20 Factory Demand 35 25 (a) Formulate a linear programming problem to minimize total cost for this transportation problem Transportation problem with 4 origins and 3 destinations Objective is to minimize total distribution cost Total Supply 20 +30+10+ 25-85 engines Total Demand 5 +20 +25-80 engines Since the supply and demand are not the same, this is an unbalanced problem. The supply is greater than the demand, thus a dummy demand variable will be created. Supply constraints will be

Explanation / Answer

Let X 1 = Be the decision variable for Nashville

X2 = Be the decision variable for   Orlando

X2 = Be the decision variable for   Charleston

X 4 = Be the decision variable for   Dummy Location for balancing

Minimization P = 35 X1 + 20 X2 + 25 X3 + 5 X4
subject to
880 X1 +1150 X2 + 500 X3 + 0 X4 <= 20
650 X1 +1050 X2 + 700 X3 + 0 X4 <= 30
550 X1 +815 X2 + 472 X3 + 0 X4 <= 10
620 X1 +910 X2 + 520 X3 + 0 X4 <= 25

Tableau #1
x1 x2 x3 x4 s1 s2 s3 s4 p   
880 1150 500 0 1 0 0 0 0 20   
650 1050 700 0 0 1 0 0 0 30   
550 815 472 0 0 0 1 0 0 10   
620 910 520 0 0 0 0 1 0 25   
-35 -20 -25 -5 0 0 0 0 1 0

Tableau #2
x1 x2 x3 x4 s1 s2 s3 s4 p   
0 -154 -255.2 0 1 0 -1.6 0 0 4
0 86.8182 142.182 0 0 1 -1.18182 0 0 18.1818
1 1.48182 0.858182 0 0 0 0.00181818 0 0 0.0181818
0 -8.72727 -12.0727 0 0 0 -1.12727 1 0 13.7273
0 31.8636 5.03636 -5 0 0 0.0636364 0 1 0.636364   

Total Minimum Cost = 40720

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