Question

1. You are in charge of scheduling shipment from the 4 regional warehouses to 4 customers. The unit shipment costs are as given below:

From Warehouse	To Customer
	1	2	3	4
1	25	21	11	21
2	23	16	17	21
3	20	20	14	25
4	17	25	16	12

Each of the warehouses have a capacity of 2,500 units, and the demand levels at the four customers are as follows: 1,200 units at Customer 1; 800 units at Customer 2; 5,250 units at Customer 3; and 1,850 at Customer 4. Surely, your main objective is to figure out the shipment plan in order to minimize total transportation costs.

Formulate this problem as a linear program. By following the 4 steps, write out the linear model.

Answer 1

We will first formulate the linear program model:

Decision variables

Xij= number of units shipped from Warehouse i to Customer j

X11 = number of units shipped from Warehouse 1 to Customer 1

X12 = number of units shipped from Warehouse 1 to Customer 2

X13 = number of units shipped from Warehouse 1 to Customer 3

X14 = number of units shipped from Warehouse 1 to Customer 4

X21 = number of units shipped from Warehouse 2 to Customer 1

X22 = number of units shipped from Warehouse 2 to Customer 2

X23 = number of units shipped from Warehouse 2 to Customer 3

X24 = number of units shipped from Warehouse 2 to Customer 4

X31 = number of units shipped from Warehouse 3 to Customer 1

X32 = number of units shipped from Warehouse 3 to Customer 2

X33 = number of units shipped from Warehouse 3 to Customer 3

X34 = number of units shipped from Warehouse 3 to Customer 4

X41 = number of units shipped from Warehouse 4 to Customer 1

X42 = number of units shipped from Warehouse 4 to Customer 2

X43 = number of units shipped from Warehouse 4 to Customer 3

X44 = number of units shipped from Warehouse 4 to Customer 4

Objective function

The objective is to minimize total transportation costs.

MIN 25X11+21X12+11X13+21X14+23X21+16X22+17X23+21X24+20X31+20X32+14X33+25X34+17X41+25X42+16X43+12X44

Constraints

Demand constraints:

X11+X21+X31+X41=1200

X12+X22+32+X42=800

X13+X23+33+X43=5250

X14+X24+34+X44=1850

Supply constraints:

X11+X12+X13+X14<= 2500

X21+X22+X23+X24<= 2500

X31+X32+X33+X34<= 2500

X41+X42+X43+X44<= 2500

Xij >=0 (non negativity constraints)

Solving using Excel -> Solver

Below excel screenshot show the approach for solving using Excel -> Solver

Setup Solve table with decision variables, objective function and constraints

Click on Solve to get the optimal solution

Final optimal shipment schedule is :

From Warehouse	To Customer
1	2	3	4
1	0	0	2500	0
2	0	800	800	0
3	550	0	1950	0
4	650	0	0	1850

Total minimum transportation cost is 125,450

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