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Unbalanced Transportation Problem

An unbalanced transportation problem occurs when the total supply is not equal to the total demand in a transportation problem. In other words, the sum of the supplies at all sources does not match the sum of the demands at all destinations.

  • If total supply exceeds total demand, a "dummy" destination is added to absorb the excess supply. The transportation cost to this dummy destination is set to zero.
  • If total demand exceeds total supply, a "dummy" source is added to cover the shortfall in supply. The transportation cost from this dummy source is set to zero.

These adjustments help convert the problem into a balanced form so that standard transportation algorithms can be applied.

Example of an Unbalanced Transportation Problem

Suppose we have three factories (A, B, C) supplying goods to three warehouses (X, Y, Z), with the following supply and demand:

X Y Z Supply
A 4 6 8 50
B 2 5 7 60
C 3 4 6 40
Demand 30 70 20

Here, the total supply is 50 + 60 + 40 = 150, and the total demand is 30 + 70 + 20 = 120. Since supply exceeds demand, this is an unbalanced problem.

Step 1: Balancing the Problem

To balance the problem, we need to add a dummy destination to absorb the excess supply. The excess supply is calculated as:

[ 150 \text{ (total supply)} - 120 \text{ (total demand)} = 30 ]

We add a dummy column (let's call it D) with a demand of 30. The transportation cost to this dummy destination is set to zero.

The modified transportation table becomes:

X Y Z D (Dummy) Supply
A 4 6 8 0 50
B 2 5 7 0 60
C 3 4 6 0 40
Demand 30 70 20 30

Now, the total supply (150) equals the total demand (150), making the problem balanced.

Step 2: Solving the Balanced Problem

Now that the problem is balanced, we can solve it using any of the standard methods, such as the North-West Corner Rule, Least Cost Method, or Vogel's Approximation Method.

Example Solution using the Least Cost Method

  1. Select the cell with the lowest cost: B-X (cost = 2).

    • Allocate 30 units to B-X (min of 60 supply at B and 30 demand at X).
    • Update the remaining supply at B to 30 and cross out column X (demand met).

  2. Select the next lowest cost: B-Y (cost = 5).

    • Allocate 30 units to B-Y (min of 30 supply at B and 70 demand at Y).
    • Update the remaining supply at B to 0 and cross out row B (supply exhausted).

  3. Select the next lowest cost: C-Y (cost = 4).

    • Allocate 40 units to C-Y (min of 40 supply at C and 40 remaining demand at Y).
    • Update the remaining supply at C to 0 and cross out row C (supply exhausted).

  4. Allocate the remaining units to the dummy destination: A-D (cost = 0).

    • Allocate 30 units to A-D (remaining supply at A).
    • Update the supply at A to 0 and cross out row A (supply exhausted).

Final Allocation

The final allocation table is:

X Y Z D (Dummy)
A 0 0 20 30
B 30 30 0 0
C 0 40 0 0

Summary

  • Unbalanced transportation problems occur when total supply does not equal total demand.
  • To balance the problem, add a dummy source or destination with the required supply or demand.
  • The dummy row or column is used to absorb the excess supply or demand, with transportation costs set to zero.
  • Once balanced, the problem can be solved using standard transportation methods.
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