4.6-11. Label each of the following statements as true or false, and then justif

Question

4.6-11. Label each of the following statements as true or false, and then justify your answer. (a) When a linear programming model has an equality constraint, an artificial variable is introduced into this constraint in order to start the simplex method with an obvious initial basic solu- tion that is feasible for the original model. (b) When an artificial problem is created by introducing artificial variables and using the Big M method, if all artificial variables in an optimal solution for the artificial problem are equal to zero, then the real problem has no feasible solutions. (c) The two-phase method is commonly used in practice because it usually requires fewer iterations to reach an optimal solution than the Big M method does.

Explanation / Answer

Given :

(a) True .

Explanation : In a Linear Programming Model : An Artifical Varaibale is introduced when the equation does not contains a slack variable. In such cases, Simplex method is only possible when Artifical variables are introduced. We have to work out the problem using the 2-Phase Method. This makes the equations similar to simplex method by adding the artifical variables.

(b) False .

Explanation : If the Artificial Variable in the solution are equal to 0 (Zero), then we have a intial basic feasible solution to the original problem. If the Artificial variables are positive then the original problem will not have any basic feasible solutions.

(c) False .

Explanation : Practically, Big M Method contains fewer Iterations than Two-Phase Method. Two -Phase Method is commonly used because it can solves the problem in 2 different phases. First phase is used to solve the problem by intrducing the artificial variable , if the artificial variables are positive . Then the original solution has no basic feasible solution. If the first phase is success,that is, all the artificial variables are equal to zero then we can proceed to second phase in finding the solution.

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