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# Linear programming using two-phase simplex and graphical method

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a) When solving linear programming problems a number of problem cases can arise. Explain, with the aid of diagrams where appropriate, how you would identify each of the following cases when solving a two-variable problem using the graphical method and when using the two-phase Simplex method.

i) A non-unique solution
ii) An infeasible problem.
iii) An unbounded problem.
iv) A degenerate solution.
v) Describe the usual consequence of degeneracy and explain briefly how degeneracy can be avoided.

b) Explain how the two-phase Revised Simplex method indicates that a linear programming problem is (i) infeasible, (ii) unbounded and (iii) has infinitely many solutions.

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#### Solution Preview

QUESTION 2
Two-phase Simplex method
A non-unique solution
If Dz=0, then we know that there is an alternate optima or the solution is non-unique. Refer to the inner productive rule for the derivation and formula for Dz.
An infeasible problem
If w'>0 in the Phase 1 LP, then the problem is infeasible
An unbounded problem
If maxin Z = +Â¥, then the problem is unbounded. In other words, if the optimum value, which is represented by Z in the LP, approaches infinity, then the problem is unbounded.
A degenerate solution
If w'=0 in the Phase 1 LP and one or more of the basic variables are equal to 0, then the problem has a feasible solution, but it is degenerate

Graphical method
A non-unique solution
When the obkective function line is moved parallel to the farthest or nearest boundary of the LP ...

#### Solution Summary

The expert describes the usual consequence of degeneracy and explain briefly how degeneracy can be avoided. Identify non-unique solution, degenerate solution, infeasible problem, unbounded problem using graphical method and two-phase simplex method.

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