2 A multiechelon distribution system consists of two factori

2. A multiechelon distribution system consists of two factories, three regional warehouses, and five wholesale distribution points. Production costs at factories 1 and 2 are $3 and $4 per unit, respectively. Costs of shipping from factories to the regional warehouses are given in table (c), and costs of shipping from the warehouses to the wholesale distribution points arc given in Table (d). Factory 1 can produce up to 2100 units per year while factory 2 can produce up to 3000 units per year. Annual demands at the five distribution points are 1400, 500, 1200, 1300 and 600, respectively. It must be decided how much to produce and ship to each regional warehouse from each factory, and how much to ship from each regional warehouse to each distribution point. Formulate this problem as a cost-minimizing linear programming.

Solution

here let,

supply of factory 1 to three warehouses be u1, u2, u3 respectively.

supply of factory 2 to three warehouses be v1, v2, v3 respectively.

supply of warehouse 1 to five distribution points be x1, x2, x3, x4, x5 respectively.

supply of warehouse 2 to five distribution points be y1, y2, y3, y4, y5 respectively.

supply of warehouse 3 to five distribution points be z1, z2, z3, z4, z5 respectively.

thus the lp is to minimize cost price p as,

Minimize p = (3.5)u1 + (3.51)u2 + (3.49)u3 + (4.45)v1 + (4.43)v2 + (4.44)v3 + (0.15)x1 + (0.20)x2 + (0.25)x3 + (0.14)x4 + (0.22)x5 + (0.17)y1 + (0.23)y2 + (0.24)y3 + (0.17)y4 + (0.23)y5 + (0.16)z1 + (0.22)z2 + (0.23)z3 + (0.18)z4 + (0.22)z5
subject to
u1 + u2 + u3 <= 2100
v1 + v2 + v3 <= 3000
x1 + y1 + z1 = 1400
x2 + y2 + z2 = 500
x3 + y3 + z3 = 1200
x4 + y4 + z4 = 1300
x5 + y5 + z5 = 600
x1 + x2 + x3 + x4 + x5 = u1 + v1
y1 + y2 + y3 + y4 + y5 = u2 + v2
z1 + z2 + z3 + z4 + z5 = u3 + v3