4. Canning Transport is to move goods from three factories to three distribution centers.
Information about the move is given below. Give the network model and the linear
programming model for this problem.Canning Transport is to move goods from three factories to three distribution centers.
Information about the move is given below. Give the network model and the linear
programming model for this problem.
Source Supply Destination Demand
A 200 X 50
B 100 Y 125
C 150 Z 125
Shipping costs are:
Destination
Source X Y Z
A 3 2 5
B 9 10 --
C 5 6 4
(Source B cannot ship to destination
To create a network model for this problem, we can represent the factories and distribution centers as nodes in a graph, and the shipping paths between them as edges. The supply and demand values can be represented as the capacities on the nodes.
First, let's create a graph with nodes representing the factories (A, B, C) and the distribution centers (X, Y, Z).
Next, we can add edges between the nodes to represent the shipping paths. The shipping costs can be represented as the weights on these edges.
Based on the given information, the network model would look like this:
```
X(50)
/ | \
3 / | \
A(200) C(150)
2 / 4\ /
/ \ /
B(100) Z(125)
| 6 |
10 5
| |
Y(125) (Demand)
```
Now, let's create the linear programming model for this problem.
1. Decision Variables:
Let's define the decision variable as Xij, representing the amount of goods shipped from factory i to distribution center j.
2. Objective Function:
Our objective is to minimize the total shipping cost. To do this, we can define the objective function as the sum of the products of the shipping costs and the decision variables:
minimize Z = 3XAX + 9XBX + 2XAY + 10XBY + 5XAZ + 6XCZ + 4XCO
3. Constraints:
We need to satisfy the supply and demand constraints:
- For Factory A: XAX + XAY + XAZ = 200 (Supply constraint)
- For Factory B: XBX + XBY = 100 (Supply constraint)
- For Factory C: XCZ + XCO = 150 (Supply constraint)
- For Distribution Center X: XAX + XBX = 50 (Demand constraint)
- For Distribution Center Y: XAY + XBY = 125 (Demand constraint)
- For Distribution Center Z: XAZ + XCZ + XCO = 125 (Demand constraint)
Also, we need to ensure that the decision variables are non-negative:
Xij ≥ 0 for all i and j.
This linear programming model can be solved using various optimization techniques to determine the optimal shipping quantities and minimize the total shipping cost.