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Two Phase Method

Given that Linear Programming has multiple methods to be followed, it is important that detailed knowledge of the various methods be understood by a student. Given that Big M method has certain negatives associated with it, it is important that the two-phase method is understood in detail.

As one of the most important methods that have dual facets associated with it, this process can be understood in two phases.

Phase 1:

Here before beginning this process, certain conditions are to be kept. They are:

  • The constraints associated with this phase are to be expressed in the standard format.
  • Those terms that are placed on the right-hand side of the table have to be positive. In case of any negative element being present there, it has to be turned into a positive one.
  • Artificial variables are to be added in both type and equality constraints.
  • A new objective function (W) is to be considered that is taken to be the summation of artificial variables.
  • Also, it is to be noted that the function W has to be taken at a minimal level that is subjected to constraints of the original given problem and by doing so, the basic solution that is both feasible and optimum is found.

As a result of this phase, any of the 3 given conditions may arise as per the latest association.

  • It can so happen that the minimum value of W can be equal to zero and there is at least 1 artificial variable available in the column at the zero level. It may so happen that the optimal feasible solution is not equal to the original feasible solution that is given as per linear programming method.
  • It can also be placed that minimum value of W is greater than zero and minimal one artificial variable is present at a positive level. In this case, there is no acceptable or feasible solution that is there in regards to the original linear programming curve, hence no final answer can be derived from this.
  • Where W is taken to be equal to zero, and there is no artificial variable that is present, in such a scenario, an acceptable solution can be said to have been found and therefore, one can proceed to the level of phase 2 to carry on with this process in details.

Phase 2:

In this case, it is taken initially that the acceptable solution of Phase 1 that is feasible is started off as the base point. In this case,simplex method is taken, and with help of this optimal basic feasible solution is to be understood and obtained.

Also, another very important point that is to be noted is that,the objective function that is associated with W is to be always of the minimization type, and it does not depend on whether the original linear programming is of the maximization or minimization types.

Thus, it needs to be understood that with help of this method, a better understanding of the concept of linear programming can be obtained.

 

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