3. A company has two production facilities, #1 and #2 and the capacities of faci

Question

3. A company has two production facilities, #1 and #2 and the capacities of facilities at #1 and #2 are Ki and respectively. Assume the total number of markets is 5, the cost of producing and shipping one unit product from facility i (i = 1,2) to market(j = 1, 2, 3, 4, 5) is cii The demand from all markets are given as Di (js 1, 2, 3, 4, 5) The company's goal is to allocate the demand from different markets to the two plants so that the cost is minimized. Formulate an optimization model; and identify decision variables, objective function and constraints. (15 points)

Explanation / Answer

The Flow Diagram has been attached below. The following are the colour codes-

Green - Plant Capacity (Ki)

Black - Cost for production and transportation through that route (Cij)

Red - Quantity of product transported through that route (Xij)

Yellow - Demand of each market (Di)

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Our objective is to minimize the cost while fulfilling the orders. Cost through each node is Quantity transported multiplied by transport cost/unit on that route. Summation of cost through all routes will be the total cost and thus we have to minimize the total cost.

Objective Function -

Min(C11*X11+C12*X12+C13*X13+C14*X14+C15*X15+C21*X21+C22*X22+C23*X23+C24*X24+C25*X25)

Thus we can say that the quantity transported through all routes decided the minimization. Hence Xij are the decision variables.

Constraints-

From Capacity.Side (assuming full capacity is not utilized)

K1 = X11 +X12 +X13 + X14 + X15

K2 = X21 +X22 +X23 + X24 + X25

From Demand Side (assuming all demands are fulfilled)

D1 = X11 + X21

D2 = X12 +X22

D3 = X13 + X23

D4 = X14 + X24

D5 = X15 + X25

As the Quantity transported cant be negative-

Xij >= 0

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