## Parametric Programming

Definition

The values of P1, P2, P3, represent the supply sources

The values of T1, T2,…, T13 represent the destinations

xij represent the amount of to be transported from the i-th supply source to the j-th demand destination ( where i = 1, …, 3; j = 1, …, 13);

cij represent the transport cost transportation per unit of goods transported.

ai represent the available quantity at the i-th origin (i = 1 ,…, 3) for supply.

bj – amount of goods demanded at the j-th demand destination) (j = 1 ,…, 13)

we wish to find the optimal values of the variables xij (i=1,…,3; j=1,…,13) this refers to the optimum amount transported at minimum cost.

Optimize the equation

 m n z = ∑ ∑cijxij i=1 j=1

Obtain the basic initial solution and optimizing

Using the Vogel’s method

Step 1

We obtain the row and column difference

That is the values of ui to represent the row difference and the values of vj to represent the column difference.

The differences can be labelled as 1st, 2nd

The firs difference is obtained by finding the difference between the least 2 column or row values.

The table above shows the values of u’ and v’ calculated by finding the difference between the minimum values of cost.

Step 2

u’ = min values between the least two Cij along the rows

v’ = min values between the least two Cij across the columns

The u’ and v’ are then ranked to obtain the maximum

u’+ v’ = c’ij

u’’ + v’’ = c’’ij

The min value between the demand and supply capacities is then allocated to the cell with the least cost.

I our case, 28 is the highest penalty, we allocate 3211 (value of demand since demand is less than supply) to column Minna, and row Sureja since 20 is the least cost in that column.

We then subtract the demand (the allocated) from the supply.

We then look for u’’ …… and v’’ …. Consecutively until the allocation inside the cells is complete.

The end result of the process is

The initial basic solution using this method is also the optimal solution.

The optimal transport cost is obtained by the formula.

 3 13 z = ∑ ∑cijxij i=1 j=1

Step 3

The calculations of the u’ and v’ leads to costs c’ij . This values are creates linear equations.

Cij = c’ij +λc’’ij

Yy= αij+ λ βij

Where

When βij < 0

On the other hand, when

When  βij  < 0

When βij < 0

This can be used to estimate the limits of coefficients

 Lower Objective Upper Objective Limit Result Limit Result 2067 546884.4 2067 546884.4 1515 546884.4 1515 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 817 590593.9 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 672 546884.4 672 546884.4 0 546884.4 0 546884.4 208 546884.4 208 546884.4 1318 546884.4 1318 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 219 546884.4 219 546884.4 641 546884.4 641 546884.4 0 546884.4 70 552029.4 1264 546884.4 1264 546884.4 1236 546884.4 1236 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 564 546884.4 564 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 480 546884.4 480 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 3211 546884.4 3211 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4 1 546884.4 1 546884.4 0 546884.4 0 546884.4 0 546884.4 0 546884.4