Service for Solving Linear Programming Problems

Example 4. Transportation Problem by North-West Corner Method (Balanced problem)

This solution is done by the program presented on the site.
Problem:
The cost of delivery of a unit of production from the supplier to the consumer is located in the lower right corner of the cell.
Supplier Consumer   Supply  
B 1 B 2 B 3
A 1
5
3
1
  10  
A 2
3
2
4
  20  
A 3
4
1
2
  30  
  Customer  
needs
15 20 25
It is necessary to find a transport plan in which the total cost of delivery will be the lowest.
Solution:
This is a necessary condition for solving the problem:
the total supply of suppliers should be equal to the total needs of consumers.

Let's check it.
The total supply of suppliers: 10 + 20 + 30 = 60 units.
The total needs of consumers: 15 + 20 + 25 = 60 units.
The total supply of suppliers equals the total needs of consumers.
This is a necessary condition for solving the problem:
number of used routes = number of suppliers + number of consumers - 1.

So if there is a situation where it is necessary to exclude a column and a row at the same time, we will exclude one thing.
We will start filling out the table from the upper left corner and gradually move to the lower right corner.
From the North-West to South-East.
Supplier Consumer   Supply  
B 1 B 2 B 3
A 1
?
5
3
1
  10  
A 2
3
2
4
  20  
A 3
4
1
2
  30  
  Customer  
needs
15 20 25
10 = min { 15, 10 }
Supplier Consumer   Supply  
B 1 B 2 B 3
A 1
10
5
3
1
  10   no  
A 2
?
3
2
4
  20  
A 3
4
1
2
  30  
  Customer  
needs
15
5
20 25
5 = min { 5, 20 }
Supplier Consumer   Supply  
B 1 B 2 B 3
A 1
10
5
3
1
  10   no  
A 2
5
3
?
2
4
  20   15  
A 3
4
1
2
  30  
  Customer  
needs
15
5
no
20 25
15 = min { 20, 15 }
Supplier Consumer   Supply  
B 1 B 2 B 3
A 1
10
5
3
1
  10   no  
A 2
5
3
15
2
4
  20   15   no  
A 3
4
?
1
2
  30  
  Customer  
needs
15
5
no
20
5
25
5 = min { 5, 30 }
Supplier Consumer   Supply  
B 1 B 2 B 3
A 1
10
5
3
1
  10   no  
A 2
5
3
15
2
4
  20   15   no  
A 3
4
5
1
?
2
  30   25  
  Customer  
needs
15
5
no
20
5
no
25
25 = min { 25, 25 }
Supplier Consumer   Supply  
B 1 B 2 B 3
A 1
10
5
3
1
  10   no  
A 2
5
3
15
2
4
  20   15   no  
A 3
4
5
1
25
2
  30   25   no  
  Customer  
needs
15
5
no
20
5
no
25
no
Let's calculate the total cost of delivery for the initial solution.

10*5 + 5*3 + 15*2 + 5*1 + 25*2 = 150

Is the initial solution optimal?
Let's check it.
To each supplier A i we associate a some number u i called a potential of the supplier.
To each consumer B j we associate a some number v j called a potential of the consumer.
For used routes: potential of the supplier + potential of the consumer = cost of delivery.
Let's find potentials sequentially.
We must enter the value of one potential. Let u2 = 0.
A2B1 :   v1 + u2 = 3     v1 = 3 - 0 = 3
A2B2 :   v2 + u2 = 2     v2 = 2 - 0 = 2
A3B2 :   v2 + u3 = 1     u3 = 1 - 2 = -1
A3B3 :   v3 + u3 = 2     v3 = 2 - (-1) = 3
A1B1 :   v1 + u1 = 5     u1 = 5 - 3 = 2
  Supplier   Consumer   U  
B 1 B 2 B 3
A 1
10
5
3
1
  u1 = 2  
A 2
5
3
15
2
4
  u2 = 0  
A 3
4
5
1
25
2
  u3 = -1  
  V   v1 = 3 v2 = 2 v3 = 3
Let's find evaluations of unused routes (cij - cost of delivery).
A1B2 :   Δ12 = c12 - ( u1 + v2 ) = 3 - ( 2 + 2 ) = -1  
A1B3 :   Δ13 = c13 - ( u1 + v3 ) = 1 - ( 2 + 3 ) = -4  
A2B3 :   Δ23 = c23 - ( u2 + v3 ) = 4 - ( 0 + 3 ) = 1  
A3B1 :   Δ31 = c31 - ( u3 + v1 ) = 4 - ( -1 + 3 ) = 2  
There are negative evaluations. Therefore it is possible to reduce the total cost of delivery.
STEP 1.
The cell A1B3 (unused route) is selected because its evaluation is negative.
Move the mouse cursor to the selected cell A1B3.
Using the horizontal and vertical cursor movements, connect the filled cells with a continuous line so that you return to the cell A1B3.
The cells located at the vertices of the plotted line form a cycle for the selected cell (see the highlighted cells in the table below).
There is only one cycle for cell A1B3.
Supplier Consumer   Supply  
B 1 B 2 B 3
A 1
10
5
3
-4
1
  10  
A 2
5
3
15
2
4
  20  
A 3
4
5
1
25
2
  30  
  Customer  
needs  
  15     20     25  
10 = min { 10, 15, 25 }
Supplier Consumer   Supply  
B 1 B 2 B 3
A 1
10
5
3
-4
1
  10  
A 2
5
3
15
2
4
  20  
A 3
4
5
1
25
2
  30  
  Customer  
needs  
  15     20     25  
This transformation will not change the balance.
But this transformation will change the total cost of delivery by:
1 * 10 - 5 * 10 + 3 * 10 - 2 * 10 + 1 * 10 - 2 * 10 = ( 1 - 5 + 3 - 2 + 1 - 2 ) * 10 = -4 * 10
You correctly noticed that -4 * 10 = Δ13 * 10
Supplier Consumer   Supply  
B 1 B 2 B 3
A 1
10 - 10
5
3
+10
-4
1
  10  
A 2
5 + 10
3
15 - 10
2
4
  20  
A 3
4
5 + 10
1
25 - 10
2
  30  
  Customer  
needs  
  15     20     25  
We got a new solution.
Supplier Consumer   Supply  
B 1 B 2 B 3
A 1
5
3
10
1
  10  
A 2
15
3
5
2
4
  20  
A 3
4
15
1
15
2
  30  
  Customer  
needs  
  15     20     25  
Let's calculate the total cost of delivery for the new solution.

S = 150 + Δ13 * 10 = 150 -4 * 10 = 110

Is the new solution optimal?
Let's check it.
To each supplier A i we associate a some number u i called a potential of the supplier.
To each consumer B j we associate a some number v j called a potential of the consumer.
For used routes: potential of the supplier + potential of the consumer = cost of delivery.
Let's find potentials sequentially.
We must enter the value of one potential. Let u2 = 0.
A2B1 :   v1 + u2 = 3     v1 = 3 - 0 = 3
A2B2 :   v2 + u2 = 2     v2 = 2 - 0 = 2
A3B2 :   v2 + u3 = 1     u3 = 1 - 2 = -1
A3B3 :   v3 + u3 = 2     v3 = 2 - (-1) = 3
A1B3 :   v3 + u1 = 1     u1 = 1 - 3 = -2
  Supplier   Consumer   U  
B 1 B 2 B 3
A 1
5
3
10
1
  u1 = -2  
A 2
15
3
5
2
4
  u2 = 0  
A 3
4
15
1
15
2
  u3 = -1  
  V   v1 = 3 v2 = 2 v3 = 3
Let's find evaluations of unused routes (cij - cost of delivery).
A1B1 :   Δ11 = c11 - ( u1 + v1 ) = 5 - ( -2 + 3 ) = 4  
A1B2 :   Δ12 = c12 - ( u1 + v2 ) = 3 - ( -2 + 2 ) = 3  
A2B3 :   Δ23 = c23 - ( u2 + v3 ) = 4 - ( 0 + 3 ) = 1  
A3B1 :   Δ31 = c31 - ( u3 + v1 ) = 4 - ( -1 + 3 ) = 2  
There are not any negative evaluations. Therefore it is not possible to reduce the total cost of delivery.
Result:

X opt =

 0 0 10
15 5 0
0 15 15

Smin = 110









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