First, let$$s acquire a general idea about what a transportation problem is$.$ fgbThe transportation problem seeks to determine the transportation schedule of shipping a single product from a number of sources $($plants$)$ to a number of destinations $($warehouses$)$ so that the total transportation cost is minimized.

The parameters of the transportation problem include:

Level of supply at each source; i$.$e$.,$ the capacity of each plant $($denoted by Si$).$

Amount of demand at each destination $($denoted by Dj$).$

The cost per unit of transporting the product from each source to each destination, or warehouse to plant, represented by Cij.

Well, we need to understand that the transportation problem can be modelled as a linear programming problem. In fact, it is considered as a special type of linear programming problem where all constraints coefficients are only $0$ or $1.$ Although the transportation problem can be solved using the linear programming method, we will later show that it can be solved in a more efficient manner using the transportation method.

To illustrate how we can formulate a transportation problem as a linear programming problem, we consider the following example.

A company produces a product at each of its plants, X$,$ Y$,$ and W$.$ The weekly capacities of the three plants are $6,8,$ and $10$ truckloads, respectively. The weekly demands at the three distribution centers, warehouses A$,$ B$,$ and C$,$ are $7,12,$ and $5$ truckloads, respectively. The transportation costs per truckload $($in $$00)$ between the plants and distribution centers are as follows:

task:

Use POM$-$QM to find the optimal transportation schedule by solving the above LP problem.