This case study involves a transportation model where four plants (P1, P2, P3, P4) must distribute their automobile production to nine markets (C1 to C9). The matrix in the image provides the transportation costs (in terms of distance) between the plants and markets. The goal is to allocate the plant capacities to meet the market demands while minimizing the transportation cost.

Problem setup:

• Plants: P1, P2, P3, P4 with respective capacities:

  • P1: 8500 units
  • P2: 8000 units
  • P3: 6000 units
  • P4: 9500 units
  • Total supply: 32,000 units.

• Markets: C1 to C9 with respective demands:

  • C1: 4000 units
  • C2: 6000 units
  • C3: 3500 units
  • C4: 1000 units
  • C5: 4000 units
  • C6: 1500 units
  • C7: 5000 units
  • C8: 6500 units
  • C9: 500 units
  • Total demand: 33,000 units.

The cost of transporting one unit per kilometer is 30 UM. The distances (or costs per unit) between plants and markets are given in the matrix.

Steps to solve the transportation problem:

1. Formulate the problem as a linear programming model:

   • Define variables $x_{ij}$ as the number of units shipped from plant $i$ to market $j$.
   • The objective function will be to minimize the total transportation cost: $\text{Minimize} \sum_{i=1}^{4} \sum_{j=1}^{9} x_{ij} \cdot d_{ij} \cdot 30$ Where $d_{ij}$ is the distance (cost) between plant $i$ and market $j$.

2. Constraints:

   • Supply constraints: The total units shipped from each plant should not exceed the plant's capacity: $\sum_{j=1}^{9} x_{ij} \leq \text{capacity of plant } i$
   • Demand constraints: The total units received by each market must meet the market's demand: $\sum_{i=1}^{4} x_{ij} = \text{demand of market } j$
   • Non-negativity constraints: $x_{ij} \geq 0$ for all $i, j$.

3. Balancing supply and demand: Since the total demand (33,000) exceeds the total supply (32,000), this is an unbalanced transportation problem. One option is to introduce a dummy supply node with a supply of 1,000 units and zero transportation cost to absorb the excess demand.

4. Solution:

   • Use methods like the North-West Corner Rule, Least Cost Method, or Vogel's Approximation Method to find an initial feasible solution.
   • Apply optimization methods like the Stepping-Stone Method or MODI Method to improve the solution and minimize costs.

Would you like to proceed with the solution using a specific method or tool (e.g., a linear programming solver)?

Related questions:

1. What are the basic methods to obtain an initial feasible solution in transportation problems?
2. How does the unbalanced nature of supply and demand affect the solution?
3. How do you apply the Stepping-Stone Method to optimize the transportation cost?
4. How can you implement the Vogel’s Approximation Method for this case study?
5. What are the advantages of using a linear programming approach to solve transportation problems?

Tip:

In unbalanced transportation problems, introducing a dummy supply or demand node helps in making the problem solvable without violating constraints.