Using the Big M method, construct the complete first simplex tableau for the simplex method and identify the corresponding initial (artificial) BF solution. Also identify the initial entering basic variable and the leaving basic variable.

1. Follow the instructions of for the following problem. Minimize Z _ 5,000×1 _ 7,000×2, subject to
_2x1 _ x2 _ 1

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_ x1 _ 2×2 _ 1
and
x1 _ 0, x2 _ 0.
2. Consider the following problem. Maximize Z _ 2×1 _ 5×2 _ 3×3, subject to
x1 _ 2×2 _ x3 _ 20
2×1 _ 4×2 _ x3 _ 50
and
x1 _ 0, x2 _ 0, x3 _ 0.
(a) Using the Big M method, construct the complete first simplex tableau for the simplex method and identify the corresponding initial (artificial) BF solution. Also identify the initial entering basic variable and the leaving basic variable.
(b) Work through the simplex method step by step to solve the problem.
(c) Using the two-phase method, construct the complete first simplex tableau for phase 1 and identify the corresponding
initial (artificial) BF solution. Also identify the initial entering basic variable and the leaving basic variable.
(d) Work through phase 1 step by step.
(e) Construct the complete first simplex tableau for phase 2.
(f ) Work through phase 2 step by step to solve the problem.
(g) Compare the sequence of BF solutions obtained in part (b) with that in parts (d) and ( f ). Which of these solutions are feasible only for the artificial problem obtained by introducing artificial variables and which are actually feasible for the real problem?
(h) Use a software package based on the simplex method to solve the problem.