11.4 Solving Unbalanced Models
To solve an unbalanced transportation model, the first step is to convert it into a balanced model by introducing a dummy supply or demand point, depending on the nature of the imbalance.
Example
For example, to balance the unbalanced model on the previous page, where the total supply exceeds the total demand by 10 units, a dummy demand point is added with a demand equal to the surplus, 10 units in this case. This artificial demand point, often labelled as Market 4 (M4), does not represent a real market but serves to absorb the excess supply.
Since M4 is not an actual destination, the transportation cost associated with shipping to this dummy point is set to zero. This ensures that the model remains cost-neutral while satisfying the mathematical requirement of balance.
The resulting transportation tableau below includes this dummy column, allowing the model to be solved using standard optimization techniques. The inclusion of the dummy point ensures that all supply and demand constraints are met without distorting the cost structure of the real distribution network.
Least Cost Method
Step 1: Problem Set Up
| D1 | D2 | D3 | D4 | Supply | |
|---|---|---|---|---|---|
| S1 | 2 | 4 | 3 | 0 | 50 |
| S2 | 3 | 1 | 5 | 0 | 70 |
| S3 | 4 | 2 | 6 | 0 | 60 |
| Demand | 60 | 80 | 30 | 10 |
There is a 10-unit surplus. To balance the model, a dummy demand point (D4) is created, and the cost of transporting to D4 is set to 0.
Step 2: Find the cell with the lowest cost
- The cell with the lowest cost is S2/D2.
- Allocate 70 units from S2, and adjust the remaining demand in D2 (80-70=10) and the remaining supply in S2 (70-70=0)
| D1 | D2 | D3 | D4 | Supply | |
|---|---|---|---|---|---|
| S1 | 2 | 4 | 3 | 0 | 50 |
| S2 | 3 | (70) 1 | 5 | 0 | |
| S3 | 4 | 2 | 6 | 0 | 60 |
| Demand | 60 | 30 | 10 |
Step 3: Find the cell with the next least cost
- The cell with the next least cost is S3/D2.
- Allocate 10 units (the remaining demand from D2) and adjust the supply in S3 and the demand in D2.
| D1 | D2 | D3 | D4 | Supply | |
|---|---|---|---|---|---|
| S1 | 2 | 4 | 3 | 0 | 50 |
| S2 | 3 | (70) 1 | 5 | 0 | 0 |
| S3 | 4 | (10) 2 | 6 | 0 | |
| Demand | 60 | 30 | 10 |
Step 4: Move to the next least cost
- The cell with the next least cost is S1/D1.
- Allocate 50 units from S1 and adjust the demand in D1.
| D1 | D2 | D3 | D4 | Supply | |
|---|---|---|---|---|---|
| S1 | (50) 2 | 4 | 3 | 0 | |
| S2 | 3 | (70) 1 | 5 | 0 | 0 |
| S3 | 4 | (10) 2 | 6 | 0 | 50 |
| Demand | 0 | 30 | 10 |
- The cell with the next least cost is S3/D1.
- Allocate 10 units from S3 (the remaining demand in D1), and adjust the numbers for supply and demand in S3 and D1.
| D1 | D2 | D3 | D4 | Supply | |
|---|---|---|---|---|---|
| S1 | (50) 2 | 4 | 3 | 0 | 0 |
| S2 | 3 | (70) 1 | 5 | 0 | 0 |
| S3 | (10) 4 | (10) 2 | 6 | 0 | |
| Demand | 0 | 30 | 10 |
- The cell with the next least cost is S3/D3.
- Allocate 30 units from S3 (the remaining demand in D3), and adjust the numbers for supply and demand in S3 and D1.
| D1 | D2 | D3 | D4 | Supply | |
|---|---|---|---|---|---|
| S1 | (50) 2 | 4 | 3 | 0 | 0 |
| S2 | 3 | (70) 1 | 5 | 0 | 0 |
| S3 | (10) 4 | (10) 2 | (30) 6 | 0 | |
| Demand | 0 | 0 | 10 |
Step 5: Moving to the next least cost
- The last allocation is the remaining units in S3 to the dummy demand (D4).
| D1 | D2 | D3 | D4 | Supply | |
|---|---|---|---|---|---|
| S1 | (50) 2 | 4 | 3 | 0 | 0 |
| S2 | 3 | (70) 1 | 5 | 0 | 0 |
| S3 | (10) 4 | (10) 2 | (30) 6 | (10) 0 | |
| Demand | 0 | 0 | 0 | 0 |
Step 6: Total Transportation Cost Calculation
Use the cost multiplied by the allocation values.
[latex]\begin{align*} \text{Total Cost}&= (\small\text{S1/D1} \times 2) + (\small\text{S2/D2} \times 1) + (\small\text{S3/D1} \times 4)+ (\small\text{S3/D2} \times 2) + (\small\text{S2/D3} \times 6) + (\small\text{S3/D4} \times 0) \\ \text{Total Cost}&= (50 \times 2) + (70 \times 1) + (10 \times 4) + (10 \times 2) + (30 \times 6) + (10 \times 0)\\ \text{Total Cost}&= 100 + 70 + 40 + 20 + 180 + 0 \\ \text{Total Cost}&= 410 \end{align*}[/latex]
