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Example ¹3. Transportation Problem by the Least Cost Method
(Unbalanced problem. Fictitious consumer)

This solution was made using the calculator presented on the site.
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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
3
5
4
   20  
A 2
6
3
1
   40  
A 3
3
2
7
   30  
  Customer  
needs 		30 35 20
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: 20 + 40 + 30 = 90 units.
The total needs of consumers: 30 + 35 + 20 = 85 units.
The difference is 5 units.
Let's enter a fictitious consumer B4 into use. Let customer needs B4 is 5 units.
Let the cost of delivery from all suppliers to the consumer B4 is zero (see the table below).
Now 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.
Therefore, if we have a situation where it is necessary to exclude a column and a row at the same time, we will exclude one thing.
First of all we will use the routes with the lowest cost of delivery.
Routes from all suppliers to the fictitious consumer B4 will be used last.
Perhaps this will help us reduce the total cost of delivery for the initial solution.
Supplier Consumer   Supply  
B 1 B 2 B 3 B 4
A 1
3
5
4
0
   20  
A 2
6
3
?
1
0
   40  
A 3
3
2
7
0
   30  
  Customer  
needs 		30 35 20 5
20 = min { 20, 40 }
Supplier Consumer   Supply  
B 1 B 2 B 3 B 4
A 1
3
5
4
0
   20  
A 2
6
3
20
1
0
   40   20  
A 3
3
?
2
7
0
   30  
  Customer  
needs 		30 35 20
No		 5
30 = min { 35, 30 }
Supplier Consumer   Supply  
B 1 B 2 B 3 B 4
A 1
?
3
5
4
0
   20  
A 2
6
3
20
1
0
   40   20  
A 3
3
30
2
7
0
   30   No  
  Customer  
needs 		30 35
5		 20
No		 5
20 = min { 30, 20 }
Supplier Consumer   Supply  
B 1 B 2 B 3 B 4
A 1
20
3
5
4
0
   20   No  
A 2
6
?
3
20
1
0
   40   20  
A 3
3
30
2
7
0
   30   No  
  Customer  
needs 		30
10		 35
5 		20
No		 5
5 = min { 5, 20 }
Supplier Consumer   Supply  
B 1 B 2 B 3 B 4
A 1
20
3
5
4
0
   20   No  
A 2
?
6
5
3
20
1
0
   40   20   15  
A 3
3
30
2
7
0
   30   No  
  Customer  
needs 		30
10 		35
5
No		 20
No		 5
10 = min { 10, 15 }
Supplier Consumer   Supply  
B 1 B 2 B 3 B 4
A 1
20
3
5
4
0
   20   No  
A 2
10
6
5
3
20
1
?
0
   40   20   15   5  
A 3
3
30
2
7
0
   30   No  
  Customer  
needs 		30
10
No		 35
5
No		 20
No		 5
5 = min { 5, 5 }
Supplier Consumer   Supply  
B 1 B 2 B 3 B 4
A 1
20
3
5
4
0
   20   No  
A 2
10
6
5
3
20
1
5
0
   40   20   15   5   No  
A 3
3
30
2
7
0
   30   No  
  Customer  
needs 		30
10
No		 35
5
No		 20
No		 5
No
Let's calculate the total cost of delivery for the initial solution.

20*3 + 10*6 + 5*3 + 20*1 + 5*0 + 30*2 = 215

Is the initial solution optimal?
Let's check this using the MODI method (UV method).
To each supplier A i we associate a some number U i
To each consumer B j we associate a some number V j
For the routes used: U + V = cost of delivery.
Let's find U i and V j step by step.
To do this, we need to enter the value of one of them. Let u2 = 0.
A2B1 :   v1 + u2 = 6     v1 = 6 - 0 = 6
A2B2 :   v2 + u2 = 3     v2 = 3 - 0 = 3
A2B3 :   v3 + u2 = 1     v3 = 1 - 0 = 1
A2B4 :   v4 + u2 = 0     v4 = 0 - 0 = 0
A3B2 :   v2 + u3 = 2     u3 = 2 - 3 = -1
A1B1 :   v1 + u1 = 3     u1 = 3 - 6 = -3
More info about finding UV
  Supplier   Consumer   U  
B 1 B 2 B 3 B 4
A 1
20
3
5
4
0
   u1 = -3  
A 2
10
6
5
3
20
1
5
0
   u2 = 0  
A 3
3
30
2
7
0
   u3 = -1  
  V   v1 = 6 v2 = 3 v3 = 1 v4 = 0
Let's find evaluations of unused routes (cij - cost of delivery). ?
A1B2 :   Δ12 = c12 - ( u1 + v2 ) = 5 - ( -3 + 3 ) = 5
A1B3 :   Δ13 = c13 - ( u1 + v3 ) = 4 - ( -3 + 1 ) = 6
A1B4 :   Δ14 = c14 - ( u1 + v4 ) = 0 - ( -3 + 0 ) = 3
A3B1 :   Δ31 = c31 - ( u3 + v1 ) = 3 - ( -1 + 6 ) = -2
A3B3 :   Δ33 = c33 - ( u3 + v3 ) = 7 - ( -1 + 1 ) = 7
A3B4 :   Δ34 = c34 - ( u3 + v4 ) = 0 - ( -1 + 0 ) = 1
There is negative evaluation. Therefore it is possible to reduce the total cost of delivery.
STEP ¹1.
The cell A3B1 (unused route) is selected because its evaluation is negative.
Please, move the mouse cursor to the selected cell A3B1. Only horizontal and vertical cursor movements can be used.
Connect the filled cells with a continuous line so that you return to the cell A3B1.
The cells located at the vertices of the plotted line form a loop for the selected cell (see highlighted cells in the table below).
There is only one loop for cell A3B1.
Supplier Consumer   Supply  
B 1 B 2 B 3 B 4
A 1
20
3
5
4
0
   20  
A 2
10
6
5
3
20
1
5
0
   40  
A 3
-2
3
30
2
7
0
   30  
  Customer  
needs   		  30     35     20     5  
10 = min { 30, 10 } ?
Supplier Consumer   Supply  
B 1 B 2 B 3 B 4
A 1
20
3
5
4
0
   20  
A 2
10
6
5
3
20
1
5
0
   40  
A 3
-2
3
30
2
7
0
   30  
  Customer  
needs   		  30     35     20     5  
This transformation will not change the balance.
But this transformation will change the total cost of delivery by:
3 * 10 - 2 * 10 + 3 * 10 - 6 * 10 = ( 3 - 2 + 3 - 6 ) * 10 = -2 * 10
You correctly noted that   -2 * 10 = Δ31 * 10 ?
Supplier Consumer   Supply  
B 1 B 2 B 3 B 4
A 1
20
3
5
4
0
   20  
A 2
10 - 10
6
5 + 10
3
20
1
5
0
   40  
A 3
+10
-2
3
30 - 10
2
7
0
   30  
  Customer  
needs   		  30     35     20     5  
We have a new solution. ?
Supplier Consumer   Supply  
B 1 B 2 B 3 B 4
A 1
20
3
5
4
0
   20  
A 2
6
15
3
20
1
5
0
   40  
A 3
10
3
20
2
7
0
   30  
  Customer  
needs   		  30     35     20     5  
We will calculate the total cost of delivery of the new solution.

S = 215 + Δ31 * 10 = 215 -2 * 10 = 195

Is the new solution optimal?
Let's check this using the MODI method (UV method).
To each supplier A i we associate a some number U i
To each consumer B j we associate a some number V j
For the routes used: U + V = cost of delivery.
Let's find U i and V j step by step.
To do this, we need to enter the value of one of them. Let u2 = 0.
A2B2 :   v2 + u2 = 3     v2 = 3 - 0 = 3
A2B3 :   v3 + u2 = 1     v3 = 1 - 0 = 1
A2B4 :   v4 + u2 = 0     v4 = 0 - 0 = 0
A3B2 :   v2 + u3 = 2     u3 = 2 - 3 = -1
A3B1 :   v1 + u3 = 3     v1 = 3 - (-1) = 4
A1B1 :   v1 + u1 = 3     u1 = 3 - 4 = -1
More info about finding UV
  Supplier   Consumer   U  
B 1 B 2 B 3 B 4
A 1
20
3
5
4
0
   u1 = -1  
A 2
6
15
3
20
1
5
0
   u2 = 0  
A 3
10
3
20
2
7
0
   u3 = -1  
  V   v1 = 4 v2 = 3 v3 = 1 v4 = 0
Let's find evaluations of unused routes (cij - cost of delivery). ?
A1B2 :   Δ12 = c12 - ( u1 + v2 ) = 5 - ( -1 + 3 ) = 3
A1B3 :   Δ13 = c13 - ( u1 + v3 ) = 4 - ( -1 + 1 ) = 4
A1B4 :   Δ14 = c14 - ( u1 + v4 ) = 0 - ( -1 + 0 ) = 1
A2B1 :   Δ21 = c21 - ( u2 + v1 ) = 6 - ( 0 + 4 ) = 2
A3B3 :   Δ33 = c33 - ( u3 + v3 ) = 7 - ( -1 + 1 ) = 7
A3B4 :   Δ34 = c34 - ( u3 + v4 ) = 0 - ( -1 + 0 ) = 1
There are no negative evaluations. Therefore it is not possible to reduce the total cost of delivery.
Result:

X opt =

Çíàê ñèñòåìû 20 0 0 0Çíàê ñèñòåìû
0 15 20 5
10 20 0 0

Smin = 195


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