Question

2. The state of NJ has three major power companies (A, B, and C). During the months of peak demand, NJ Power Authority authorizes these companies to pool their excess supply and to distribute it to smaller independent power companies that do not have generators large enough to handle the demand. Excess supply is distributed on the basis of cost per kilowatt hour transmitted. The following table shows the demand and excess supply in million of kilowatt hours and the cost per kilowatt hour of transmitting electric power to four small companies, W, X, Y, and Z: To From W X Y Z Excess Supply A $0.12 $0.03 $0.08 $0.05 55 B $0.02 $0.05 $0.05 $0.07 60 C $0.08 $0.11 $0.03 $0.06 35 Demand 35 40 25 50 a) Develop a network representation of this problem. b) Formulate a linear programming model for this problem. (Write the complete model for the problem. Make sure to give clear definitions of your decision variables). c) Solve the problem by using Excel Solver (Hand-in the one-page value and one-page formulas printouts for the problem).

Answer 1

Network Representation:

LPP Formulation:

Let,

C_i = cost per kilowatt hour transmitted from plant i to small companies j

Decision Variable:

X_ij be the kilowatt hour to be transmitted from power plants i to small Companiesy j,

Where,

i = a, b, and c for A, B, and C respectively

j = 1,2,3,4 for W, X, Y, and Z respectively

Objective Function:

Objective is to minimize the total production and transportation cost:

Min. Z = (C_ij x X_ij)

Min. Z = $0.12X_aw + $0.03X_ax + $0.08X_ay + $0.05X_az + $0.02X_bw + $0.05X_bx + $0.05X_by + $0.07X_bz

+ $0.08X_cw + $0.11X_cx + $0.03X_cy + $0.06X_cz

Subject to:

Supply Constraint:

Power Plant A: X_aw + X_ax + X_ay + X_az <= 55

Power Plant B: X_bw + X_bx + X_by + X_bz <= 60

Power Plant C: X_cw + X_cx + X_cy + X_cz <= 35

Demand constraint:

Small Company W: X_aw + X_bw + X_cw = 35

Small Company X: X_ax + X_bx + X_cx = 40

Small Company Y: X_ay + X_by + X_cy = 25

Small Company Z: X_az + X_bz + X_cz = 50

Nonnegative Constraint: X_ij>= 0

Excel Formulation: