11.5 Solving Balanced Transportation Models
Once a transportation model is confirmed to be balanced (the total supply equals the total demand), it can be solved using one of several initial feasible solution methods. These methods differ in their approach and efficiency in minimizing total transportation cost. The goal of each is to generate a starting solution that satisfies all supply and demand constraints.
Common Initial Feasible Solution Methods:
- North-West Corner Rule (NWCR):
This is the most basic method. It begins at the top-left (north-west) cell of the transportation tableau and allocates as much as possible to that cell, then moves either right or down based on remaining supply and demand. While simple, it does not consider transportation costs and often results in suboptimal solutions. - Least Cost Method (LCM):
This method improves upon NWCR by selecting the cell with the lowest transportation cost for allocation first. It continues to allocate to the next lowest-cost cell, adjusting supply and demand accordingly. This approach generally yields a better starting solution than NWCR. - Vogel’s Approximation Method (VAM):
VAM introduces the concept of opportunity cost by calculating penalties for not using the least-cost routes. It selects allocations based on the highest penalty, thereby balancing cost efficiency with strategic allocation. VAM typically produces the most cost-effective initial solution among the three methods.
These methods will be illustrated using the following example.
Example: Mega Farms Inc.
Mega Farms Inc. operates three strawberry farms—S1, S2, and S3—which supply four regional markets—D1, D2, D3, and D4. The supply capacities and market demands are as follows:
- Supply Capacities:
- S1: 60 cases
- S2: 80 cases
- S3: 100 cases
- Market Demands:
- D1: 50 cases
- D2: 70 cases
- D3: 80 cases
- D4: 40 cases
The transportation cost per case (in $) from each farm to each market is provided in the following tableau:
| D1 | D2 | D3 | D4 | Supply | |
|---|---|---|---|---|---|
| S1 | $6 | $2 | $14 | $8 | 60 Cases |
| S2 | $4 | $12 | $10 | $18 | 80 Cases |
| S3 | $16 | $6 | $6 | $4 | 100 Cases |
| Demand | 50 Cases | 70 Cases | 80 Cases | 40 Cases |
Mega Farms Inc. wants to determine how many cases should be sent from which farm to which market so that the total cost of transportation is minimized.
