2. Find the initial basic feasible solution for the following transportation problem using least cost method (LCM) and test for optimality using modified distribution (MODI) method with their respective total transportation costs in birr (10 points). }(c) TO From & DA & DB & DC & SUPPLY F1 & 6 & 8 & 10 & 150 F2 & 7 & 11 & 11 & 175 F3 & 4 & 5 & 12 & 275 DEMAND & 200 & 100 & 300 & 600

## Step 1: Identify the Least Cost Method (LCM)### The Least Cost Method involves selecting the cell with the lowest transportation cost and allocating as much as possible to that cell, then adjusting the supply and demand accordingly. Repeat this process until all supplies and demands are satisfied.## Step 2: Allocate Supplies and Demands### Start with the cell having the least cost:- **F3 to DA**: Cost is 4. Allocate 200 units (since DA's demand is 200). Update F3's supply to 75 (275 - 200) and DA's demand to 0.- **F3 to DB**: Cost is 5. Allocate 75 units (remaining supply of F3). Update F3's supply to 0 and DB's demand to 25 (100 - 75).- **F1 to DB**: Cost is 8. Allocate 25 units (remaining demand of DB). Update F1's supply to 125 (150 - 25) and DB's demand to 0.- **F1 to DC**: Cost is 10. Allocate 125 units (remaining supply of F1). Update F1's supply to 0 and DC's demand to 175 (300 - 125).- **F2 to DC**: Cost is 11. Allocate 175 units (remaining demand of DC). Update F2's supply to 0 and DC's demand to 0.## Step 3: Calculate Total Transportation Cost### Compute the total cost based on allocations:- - - - - ### Total Cost = ## Step 4: Test for Optimality using MODI Method### The Modified Distribution Method (MODI) checks if the current solution is optimal by calculating opportunity costs for unallocated cells. If all opportunity costs are non-negative, the solution is optimal.### Calculate opportunity costs for unallocated cells:- For cell F2 to DA: Opportunity cost = \( 7 - (u_2 + v_A) \)- For cell F2 to DB: Opportunity cost = \( 11 - (u_2 + v_B) \)- For cell F3 to DC: Opportunity cost = \( 12 - (u_3 + v_C) \)### Assign potentials and such that the occupied cells have zero opportunity cost:- - - - - ### Solve these equations to find and :- From and , we get .- From and , we get .- From , solve for .### Check opportunity costs:- All opportunity costs should be non-negative for optimality.### Since all opportunity costs are non-negative, the solution is optimal.