Question

Oranges are grown, picked, and then stored in warehouses in Tampa, Miami, and Fresno, These warehouses supply oranges to markets in New York, Philadelphia, Chicago, and Boston. The following table shows the shipping costs per truckload (in hundreds of dollars), supply, and demand. Because of an agreement between distributors, shipments are prohibited from Miami to Chicago:

		To (Cost in $100's)
From	New York	Philadelphia	Chicago	Boston	Supply
Tampa	$9	$14	$12	$17	200
Miami	11	10	6	10	200
Fresno	12	8	15	7	200
Demand	130	170	100	150

Formulate this problem as a linear programming model by hand, and then solve it using the computer.

Answer 1

Let, Xij = number of truckloads from supply i to Demand centre j

So, X11 = number of truckloads from Tampa to Newyork

X12 = number of truckloads from Tampa to Philadelphia

X13 = number of truckloads from Tampa to Chicago

X14 = number of truckloads from Tampa to Boston

X21 = number of truckloads from Miami to Newyork

X22 = number of truckloads from Miami to Philadelphia

X23 = number of truckloads from Miami to Chicago

X24 = number of truckloads from Miami to Boston

X31 = number of truckloads from Fresno to Newyork

X32 = number of truckloads from Fresno to Philadelphia

X33 = number of truckloads from Fresno to Chicago

X34 = number of truckloads from Fresno to Boston

Here, X23 = 0 as from Miami to Chicago shipments are prohibited

objective function would be to minimize the cost.

Min Z = 9X11+14X12+12X13+17X14+11X21+10X22+10X24+12X31+8X32+15X33+7X34

Subject to,

X11+X12+X13+X14 <= 200

X21+X22+X24 <=200

X31+X32+X33+X34 <=200

X11+X21+X31 = 130

X12+X22+X32 = 170

X13+X33 = 100

X14+X24+X34 = 150

Xij >=0

Solving in solver, we get, Minimum cost = 5080

Decision variables,

X11	X12	X13	X14	X21	X22	X24	X31	X32	X33	X34
100	0	100	0	30	120	0	0	50	0	150