MODI Method Examples: Transportation Problem

In the previous section, we provided the steps in MODI method (modified distribution method) to solve a transportation problem.

In this section, we provide an example. Let's solve the following example:

Example: MODI Method (Modified Distribution Method)

Consider the transportation problem presented in the following table.

On small screens, use horizontal scrollbar to view full table

Determine the optimal solution of the above problem.

Solution.

An initial basic feasible solution is obtained by Matrix Minimum Method and is shown in table 1.

Table 1

Initial basic feasible solution

12 X 7 + 70 X 3 + 40 X 7 + 40 X 2 + 10 X 8 + 20 X 8 = Rs. 894.

Calculating u_i and v_j using u_i + v_j = c_ij

Substituting u₁ = 0, we get

u₁ + v₄ = c₁₄ ⇒ 0 + v₄ = 12 or v₄ = 12
u₃ + v₄ = c₃₄ ⇒ u₃ + 12 = 20 or u₃ = 8
u₃ + v₂ = c₃₂ ⇒ 8 + v₂ = 10 or v₂ = 2
u₃ + v₁ = c₃₁ ⇒ 8 + v₁ = 40 or v₁ = 32
u₂ + v₁ = c₂₁ ⇒ u₂ + 32 = 70 or u₂ = 38
u₂ + v₃ = c₂₃ ⇒ 38 + v₃ = 40 or v₃ = 2

Table 2

Calculating opportunity cost using c_ij – ( u_i + v_j )

Table 3

Use Horizontal Scrollbar to View Full Table Calculation

Now choose the smallest (most) negative value from opportunity cost (i.e., –13) and draw a closed path from P1D1. The following table shows the closed path.

Table 4

Choose the smallest value with a negative position on the closed path(i.e., 2), it indicates the number of units that can be shipped to the entering cell. Now add this quantity to all the cells on the corner points of the closed path marked with plus signs and subtract it from those cells marked with minus signs. In this way, an unoccupied cell becomes an occupied cell.

Now again calculate the values for u_i & v_j and opportunity cost. The resulting matrix is shown below.

Table 5

Choose the smallest (most) negative value from opportunity cost (i.e., –23). Now draw a closed path from P2D2 .

Table 6

Now again calculate the values for u_i & v_j and opportunity cost

Table 7: MODI Method

Since all the current opportunity costs are non–negative, this is the optimal solution. The minimum transportation cost is: 19 X 5 + 12 X 2 + 30 X 3 + 40 X 7 + 10 X 5 + 20 X 13 = Rs. 799