Question

Consider a small company with three sources of supply: Toronto, Montreal, and Halifax. These may either be manufacturing facilities or warehouses for a foreign firm doing business in Canada. The capacities of each of these locations are 500, 400 and 300 units respectively. These locations need to supply markets in four other cities: London, Ottawa, Kingston and Quebec City. The requirements of each of these locations are respectively: 450, 350, 250 and 150 units. The major costs are from the transportation of the units. The following costs per unit have been estimated for each of the routes.

To	London	Ottawa	Kingston	Quebec City
From
Toronto	$6	9	8	13
Montreal	12	17	6	7
Halifax	1	14	13	10

Write the formulation for this problem as a transportation problem

Answer 1

Solution:

Let the decision variables be as shown in the table below:

To	London	Ottawa	Kingston	Quebec City
From
Toronto	T1	T2	T3	T4
Montreal	M1	M2	M3	M4
Halifax	H1	H2	H3	H4

This means that T1 units are shipped from Toronto to London, T2 from Toronto to Ottawa and so on.

Objective function is the total cost = 6T1+9T2+8T3+13T4+12M1+17M2+6M3+7M4+1H1+14H2+13H3+10H4

This has to be minimized.

Constraints are:

1. T1+T2+T3+T4<=500 (capacity constraint of Toronto)

2. M1+M2+M3+M4<=400 (capacity constraint of Montreal)

3. H1+H2+H3+H4<=300 (capacity constraint of Halifax)

4. T1+M1+H1 = 450 (requirement of London)

5. T2+M2+H2 = 350 (requirement of Ottawa)

6. T3+M3+H3 = 250 (requirement of Kingston)

7. T4+M4+H4 = 150 (requirement of Quebec)

8. All variables >=0 (non negativity)

Solving using excel's solver the following solution is obtained:

Units shipped is shown in the first table below:

To	London	Ottawa	Kingston	Quebec City
From
Toronto	150	350	0	0
Montreal	0	0	250	150
Halifax	300	0	0	0
To	London	Ottawa	Kingston	Quebec City
From
Toronto	6	9	8	13
Montreal	12	17	6	7
Halifax	1	14	13	10
Total cost	6900
Constraints				Formula
500	<=	500		T1+T2+T3+T4<=500
400	<=	400		M1+M2+M3+M4<=400
300	<=	300		H1+H2+H3+H4<=300
450	=	450		T1+M1+H1 = 450
350	=	350		T2+M2+H2 = 350
250	=	250		T3+M3+H3 = 250
150	=	150		T4+M4+H4 = 150

Image of solver: