$4\mathrm{.}7-3.$ This chapter has described the simplex method as applied to linear programming

problems where the objective function is to be maximized. Section $4\mathrm{.}6$ then described how

to convert a minimization problem to an equivalent maximization problem for applying

the simplex method. Another option with minimization problems is to make a few

modifications in the instructions for the simplex method given in the chapter in order to

apply the algorithm directly.

$($a$)$ Describe what these modifications would need to be$.$

$($b$)$ Using the Big $M$ method, apply the modified algorithm developed in part $($a$)$ to solve

the following problem directly by hand. $($Do not use your OR Courseware.$)$

Minimize $,Z=3x_1+8x_2+5x_3.$

subject to

$3x_2+4x_{3}70$

$3x_1+5x_2+2x_{3}70$

and

$x_{1}0,x_{2}0,x_{3}0.$