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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

    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 = +¥, 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.