Question

The distribution system for the Herman Company consists of three plants, three warehouses, and four customers. Plant capacities and shipping costs per unit (in $) from each plant to each warehouse are as follows:

		Warehouse
Plant	Capacity	1	2	3
1	450	200	500	300
2	600	50	250	200
3	380	100	400	250

Shipping costs per unit (in $) from each warehouse to each customer are as follows:

	Customer
Warehouse	1	2	3	4
1	100	50	75	100
2	50	75	125	100
3	75	150	50	150

Customer demand is as follows:

	Customer
	1	2	3	4
Total Demand	300	300	300	400

Develop a network representation of this problem, then, using excel solver, solve the linear program to determine the optimal shipping plan.

Answer 1

Formulation:

Decision Variables:

Let X_ij = represents units to be transported form plant i to the warehouse j.

X_jk = represents units to be transported form warehouse j to customer k

Objective Function:

Objective is to minimize the total cost of transportation from manufacture plants to Distribution center to market.

Cij = units transportation cost form plant i to warehouse j

Cjk = units transportation cost form warehouse j to customer k

Min Z = Σ(C_ij)(X_ij) + Σ(C_jk)(X_jk)

Subject to:

Capacity constraint:

Total units transported from Plant i <= 1000:

ΣX_ij <= capacity

Demand constraint:

Total units shipped to customer k = demand

ΣX_jk = demand

Transshipment balance constraint at warehouse

Total shipped to customer j = total shipped from plant j

ΣX_ij = ΣX_jk

Or X_ij – X_jk = 0

Nonnegative constraint:

All ΣX_ij and ΣX_jk >= 0     

Excel Model:

Optimal Solution:

Total Cost = $239,500

From/To	WH 1	WH 2	WH 3	C1	C2	C3	C4	Shipped
Plant 1	320	0	0					320
Plant 2	600	0	0					600
Plant 3	380	0	0					380
WH 1				300	300	300	400	1300
WH 2				0	0	0	0	0
WH 3				0	0	0	0	0
Received	1300	0	0	300	300	300	400