Parametric Programming

Parametric Linear Programming is the extension of sensitivity analysis procedures.

We can investigate the effect on the optimal solution of varying several parameters simultaneously. When we vary just  b_i  parameters, we express the new value b`_i  in terms of the original value  b_i  as follows:

                                                     b`_i   =   b_i + α_iӨ                             for   i = 1, 2, 3..... , m,
                                          

where the  α_i are input constants specifying the desired rate of increase (positive or negative) of the corresponding right hand side as θ is increased.

Similarly, we can have

                                                     c`_j   =   c_j + β_iθ                               for   j = 1, 2, 3,..... , n,

The general idea of a parametric analysis is to start with the optimal solution at  θ = 0. Then, using the optimality and feasibility conditions of the Simplex Method, we determine the range  0 ≤ θ ≤ θ₁      for which the solution at  θ  =  0 remains optimal. In this case θ₁ is known as critical value.

The process continues by determining successive critical values and corresponding optimal feasible solutions and will terminate at  θ  =  θ_r   , When there is indication that either the last solution will remain unchanged for θ  > θ_r   or that no feasible solution exists beyond that point.

Applying new parameters we get the new value of Z as

                                               Z`  =  Z +  γθ                                          where  γ  is any constant,

However if  Z` decreases as  θ  increases from 0 indicates that the best choice for   is  θ  = 0, so none of the possible shifting in the model parameters should be done and in real practice it should be avoided.