Consider the following problem.

Maximize $Z=2x_1+3x_2$

subject $to$

$x_1+2x_{2}4$

$x_1+x_2=3$

and $x_{1}0,x_{2}0,x_{3}0,{\stackrel{}{x}}_{4}0.$

Work through the simplex method step $by$ step $to$ demonstrate that this problem does not possess any

feasible solutions. Use $of$ artificial variables and the Big $M$ method where appropriate. Describe the

resulting adjustments $in$ the fundamental insight. Identify the numbers $in$ the final simplex tableau. Show

your calculations.

$(a)$ Solve this problem graphically.

$(b)$ Introduce $an$ artificial variable $to$ reformulate this problem $as$ a convenient artificial problem for

preparing $to$ apply the simplex method.

$(c)$ 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.

$(d)$ Continue from part $(c)to$ work through the simplex method step $by$ step $to$ solve the problem.