Download Special Topics Section 4.5 – Northwest Corner Rule Example

Special Topics Section 4.5 – Northwest Corner Rule Example
Example 1
The Sterling Milk Company has three plants located throughout a state with production capacity 50, 75 and 25
gallons. Each day the firm must furnish its four retail shops R1, R2, R3, & R4 with at least 20, 20, 50, and 60
gallons respectively.
Retail Shops
Plant
P1
I
P2
II
P3
III
Demand
R1
R2
R3
R4
1
2
3
4
Supply
3
5
7
6
2
5
8
2
3
6
9
2
20
20
50
60
50
75
25
150
Here is what the numbers in the tableau mean:
1.
The Roman Numerals name the rows. The regular numerals on top name the columns. Thus, cell III4
contains small number 2.
2.
The little numbers in the upper right corner of each cell are the unit cost Sterling pays to deliver the
milk from the plant to the shop.
3.
The larger numbers bottom and right margins are the subtotals of both what the plants can provide
(Supply) and what the shops need (demand). These are also called Rim Conditions.
4.
The grand total is in the lower right cell. The numbers in the rim conditions, either vertical or
horizontal, add up to the grand total.
The question is: How many gallons of milk should be shipped from each plant to each store to stay within the
demands of the stores and to minimize the costs? This is called a Transportation Problem.
One way to solve this is to use the Northwest Corner Rule. Here is the algorithm:
1.
Locate the top far-left-hand cell and ship this cell along with the costs. Ship via this cell with the
smaller of the two rim cells (call the value “s”) and put a circle around the entry in the tableau.
2.
Cross out the row or column that had rim value “s” and reduce the other rim value for this cell by “s”.
3.
When a single cell remains, there will be a tie for the rim conditions of both the row and column
involved, and this amount is entered into the cell and circled.
Let’s just learn it by doing it!
1
I
II
III
Demand
2
7
6
2
5
8
2
3
6
9
2
20
I
II
III
Demand
I
II
III
Demand
6
8
2
9
2
50
20
50
3
10
75
25
75
25
150
4
Supply
5
7
6
5
8
2
6
9
2
50
50
60
30
75
25
60
3
Supply
7
Supply
5
20
4
4
3
2
3
3
4
II
III
Demand
8
2
9
2
40
60
60
4
4
Supply
2
III
Demand
Supply
25
25
2
II
2
III
Demand
Supply
60
35
25
75
25
As you work, you should be completing the first tableau on the note sheet. When you are finished, you have
completed the algorithm. Now calculate the cost.
Example 2
Luminous lamps have three factories - F1, F2, and F3 with production capacity 30, 50, and 20 units per week
respectively. These units are to be shipped to four warehouses W 1, W2, W3, and W4 with requirement of 20,
40, 30, and 10 units per week respectively. The transportation costs per unit between factories and
warehouses are given below:
Warehouse
Factory
F1
I
F2
II
F3
III
Demand
W1
W2
W3
W4
1
2
3
4
20
Supply
1
2
1
4
3
3
2
1
4
2
5
9
40
30
10
30
50
20
100