Question

Brach Chemicals Inc. needs to build warehouses from where it can distribute its fertilizers to farmers. Three potential locations are considered although the management has not decided how many warehouses to build. The following table provides information about the fixed cost of operating each warehouse (including warehouse depreciation and operating cost for each planning period), the cost of shipping a ton of fertilizer from a potential warehouse to each of the 4 farmers, the planned capacity of each potential warehouse and the demand of each farmer.

Potential Location	Fixed Cost	Farmers				Capacity
		1	2	3	4
1	35000	23	20	18	28	24000
2	42000	22	15	24	16	35500
3	38000	17	31	26	21	32000
Demand		9500	6800	7200	5400

You are required to formulate an integer programming model to determine if a warehouse should be build at each of the potential locations and to determine the quantity to ship from a potential warehouse to a farmer so as to minimize the total cost.

Answer 1

Integer programming model is formulated as under:

Decision variables:

Let Wi = 1, if a warehouse is to be built at location i, otherwise Wi = 0

Xij = quantity to be shipped from warehouse i to farmer j

Objective function:

Minimize Z = 35000W1+42000W2+38000W3+23X11+20X12+18X13+28X14+22X21+15X22+24X23+16X24+17X31+31X32+26X33+21X34

Constraints:

X11+X12+X13+X14-24000W1 <= 0

X21+X22+X23+X24-35500W2 <= 0

X31+X32+X33+X34-32000W3 <= 0

X11+X21+X31=9500

X12+X22+X32=6800

X13+X23+X33=7200

X14+X24+X34=5400

Xij >= 0

Wi = {0,1}

Solution of the linear program using LINGO is as follows:

Optimal solution:

W1=1

W2=1

W3=1

Therefore, warehouse should be built at each of the three location 1, 2 and 3

X13=7200

X22=6800

X24=5400

X31=9500

7200 units should be shipped from warehouse 1 to Farmer 3,

6800 units should be shipped from warehouse 2 to Farmer 2

5400 units should be shipped from warehouse 2 to Farmer 4

9500 units should be shipped from warehouse 3 to Farmer 1

Total cost = $ 594,500