1 Rod Ent packages and ships cheese trays to four different

1. Rod Ent packages and ships cheese trays to four different retailers. He has operations on both the East and West Coasts and in the Midwest. The cost of packaging is $3.10 per tray on the West Coast, $3.25 per tray on the East Coast, and $3.15 per tray in the Midwest. Rod\'s forecasts for demand indicate that shipments must be as indicated in Table (a). Packaging capacity on the west Coast is 8000 trays, on the East Coast 7000 trays, and 10, 000 trays in the Midwest. The shipping costs from the three locations to the four retailers are given in Table (b). Formulate this LP problem.

Solution

Here the total packaging capacity(25000 trays) is equal to the total demand(25000 trays) thus the packaging cost does not come into the problem as all 3 locations has to produce maximum trays,

thus we minimize shipping price p

let,

west coast supply to retailer 1,2,3,4 be x1, x2, x3, x4 respectively.

east coast supply to retailer 1,2,3,4 be y1, y2, y3, y4 respectively.

midwest coast supply to retailer 1,2,3,4 be z1, z2, z3, z4 respectively.

thus the lp is,

Minimize p = (0.02)x1 + (0.03)x2 + (0.02)x3 + (0.05)x4 + (0.04)y1 + (0.04)y2 + (0.03)y3 + (0.04)y4 + (0.03)z1 + (0.02)z2 + (0.02)z3 + (0.05)z4
subject to
x1 + x2 + x3 + x4 = 8000
y1 + y2 + y3 + y4 = 7000
z1 + z2 + z3 + z4 = 10000
x1 + y1 + z1 = 5000
x2 + y2 + z2 = 7000
x3 + y3 + z3 = 6000
x4 + y4 + z4 = 7000

and the solution will be,

Optimal Solution: p = 640; x1 = 5000, x2 = 0, x3 = 3000, x4 = 0, y1 = 0, y2 = 0, y3 = 0, y4 = 7000, z1 = 0, z2 = 7000, z3 = 3000, z4 = 0