4. A product is manufactured by four factories A, B, C and D. The unit production costs in them are ETB 2, ETB 3, ETB 1 and ETB 5 respectively. Their production capacities are 50, 70, 30 and 50 units respectively. These factories supply the product to four stories, demands of which are 25, 35, 105, and 20 units respectively. Unit transportation cost in ETB for each factory to each store is given in the table belo
w:
| | Store 1 | Store 2 | Store 3 | Store 4 |
|---|---------|---------|---------|---------|
| A | 2 | 3 | 5 | 4 |
| B | 4 | 2 | 6 | 5 |
| C | 3 | 4 | 7 | 6 |
| D | 5 | 6 | 8 | 7 |
To minimize the total cost of production and transportation, we need to determine how much to produce at each factory and how much to transport to each store. This can be formulated as a linear programming problem. Let:
- \( x_{ij} \) = number of units produced at factory \( i \) and sent to store \( j \)
- \( y_{i} \) = 1 if factory \( i \) is operational, 0 otherwise
The objective function to minimize is:
\[ Z = \sum_{i=1}^{4} \sum_{j=1}^{4} (c_{ij}x_{ij}) \]
Subject to the constraints:
1. Factory capacity constraints:
\[ \sum_{j=1}^{4} x_{ij} \leq y_{i}c_{i} \quad \forall i \in \{1,2,3,4\} \]
\[ y_{i} \in \{0,1\} \quad \forall i \in \{1,2,3,4\}\]
2. Store demand constraints:
\[ \sum_{i=1}^{4} x_{ij} = d_{j} \quad \forall j \in \{1,2,3,4\} \]
Where:
- \( c_{ij} \) = unit transportation cost from factory \( i \) to store \( j \)
- \( c_{i} \) = production cost at factory \( i \)
- \( d_{j} \) = demand at store \( j \)
By solving this linear programming problem, we can find the optimal production quantities at each factory and the optimal transportation quantities to each store in order to minimize the total cost.