Transportation Problem_short theory

what is transportation problem: The transportation problem is a special type of linear programming problem where the objective consists in minimizing transportation cost of a given item from a number of sources or origins to a number of destinations.

 

Balanced Transportation Problems:  where the total supply is equal to the total demand.

Unbalanced Transportation Problems: where the total supply is not equal to the total demand.

      When the supply is higher than the demand, a dummy destination is introduced in the equation to make it equal to the supply (with unit(shipping) costs of 0).

      When the demand is higher than the supply, a dummy source is introduced in the equation to make it equal to the demand.

IBFS (initial basic feasible solution): This involves Initial solution to the given balanced Transportation Problems. This is known as Initial Basic Feasible Solution (IBFS).

It is required to derive an initial feasible solution; the only requirement is that the destination needs be met within source supply. 

Methods of IBFS: 

o  North-West Corner Rule (NWCR)

o  Least cost Method (LCM) 

o  Vogel’s Approximation Method (VAM).

 

     These methods used for get the initial solution. there are two phases for solve the transportation problem. In the first phase, the initial basic feasible solution has to be found and the second phase involves optimization of the initial basic feasible solution.

 

Optimal solution: An optimal solution is one in which there are no other transportation routes that would reduce the total transportation cost. 

Optimality check of IFB using MODI method.

 

MODI method: The modified distribution method, also known as MODI method or (u - v) method provides a minimum cost solution to the transportation problem. 

• ·     Applied MODI method if m+n-1=No. of allocations.

 

Degenerate solution: if number of allocations are less than (m + n – 1), then the solution is degenerate. 

No. of allocations < m+n-1

m = no. of row 

n = no. of column

 

For MODI method, Opportunity cost for unoccupied cell:

   Calculating opportunity cost using c_ij – (u_i + v_j).

 

·     If any opportunity cost > 0 then the solution is not optimal. So, improve the   solution.

·     If all opportunity cost < 0 so, solution is optimal & unique.

·     If all opportunity cost ≤ 0, some cost are zero, so solution is optimal but not unique.