There are production facilities in Battle Creek, Cherry Creek, and Dee Creek wit

Question

There are production facilities in Battle Creek, Cherry Creek, and Dee Creek with annual capacities of 500 units, 400 units, and 600 units, respectively. The annual demands at warehouses in Worchester, Dorchester, and Rochester are 300 units, 700 units, and 400 units, respectively. The table below gives the unit transportation costs between the production facilities and the warehouses.

How much of the demand at each of the warehouses must be met by each of the production facilities?

This problem can be modeled as a linear programming model as follows:

Decision Variables

Xbw = # of units to be transported from Battle Creek to Worchester

Xcw = # of units to be transported from Cherry Creek to Worchester

Xdw = # of units to be transported from Dee Creek to Worchester

Xbd = # of units to be transported from Battle Creek to Dorchester

Xcd = # of units to be transported from Cherry Creek to Dorchester

Xdd = # of units to be transported from Dee Creek to Dorchester

Xbr = # of units to be transported from Battle Creek to Rochester

Xcr = # of units to be transported from Cherry Creek to Rochester

Xdr = # of units to be transported from Dee Creek to Rochester

Objective Function

Minimize total annual transportation cost ($):

= 20*Xbw + 10*Xcw + 15*Xdw + 30*Xbd + 5*Xcd + 12*Xdd + 13*Xbr + 17*Xcr + 45*Xdr

Constraints

Demand Constraints

Xbw + Xcw + Xdw 300 (demand at Worchester)

Xbd + Xcd + Xdd 700 (demand at Dorchester)

Xbr + Xcr + Xdr 400 (demand at Rochester)

Capacity Constraints

Xbw + Xbd + Xbr 500 (capacity at Battle Creek)

Xcw + Xcd + Xcr 400 (capacity at Cherry Creek)

Xdw + Xdd + Xdr 600 (capacity at Dee Creek)

Non-Negativity Constraints

Xbw, Xcw, Xdw, Xbd, Xcd, Xdd, Xbr, Xcr, and Xdr are 0

Integer Constraints

Xbw, Xcw, Xdw, Xbd, Xcd, Xdd, Xbr, Xcr, and Xdr are integers

The above model can be solved using the Microsoft Excel Solver tool.

MUST BE DONE WITH EXCEL SOLVER

Worcester Dorchester Rochester Battle Creek $20/unit $30/unit $13/unit Cherry Creek $10/unit $5/unit $17/unit Dee Creek $15/unit $12/unit $45/ unit

Explanation / Answer

The following Table depicts the way in which the said problem can be plotted in Excel Solver. And the table also depicts the fulfillment of demand by each warehouse.

The above Table clearly shows Battle Creek should send 400 units to Rochester.

Cherry Creek should send 400 units to Dorchester

Dee Creek should send 300 units each to Worcester and Dorchester. This shall enable the most cost effective solution to fulfill all the market requirements while being in their maximum capacities.

Demand Supply Constraints C1 C2 C3 C4
=(C1+C2+C3) C5
(C4<=C5) Worcester Dorchester Rochester Supplies of Capacity Constraints R1 Battle Creek 0 0 400 400 500 R2 Cherry Creek 0 400 0 400 400 R3 Dee Creek 300 300 0 600 600 R4 = (R1+R2+R3) 300 700 400 1400 R5: R5>=R4 Demand 300 700 400 Cost Matrix Worcester Dorchester Rochester Battle Creek 20 30 13 Cherry Creek 10 5 17 Dee Creek 15 12 45 Total Cost 15300

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