Question: Consider the following LP problem:

\[\begin{aligned} & \max \text{ }0.8{{x}_{1}}+0.4{{x}_{2}}+1.2{{x}_{3}}-0.1{{x}_{4}} \\ & \text{s}\text{.t}\text{. }{{x}_{1}}+2{{x}_{2}}+{{x}_{3}}+5{{x}_{4}}\le 150 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{x}_{2}}-4{{x}_{3}}+8{{x}_{4}}=70 \\ & \,\,\,\,\,\,\,\,\,\,\,\,6{{x}_{1}}+7{{x}_{2}}+2{{x}_{3}}-{{x}_{4}}\ge 120 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}\ge 0 \\ \end{aligned}\]

a. Convert the constraints to equalities by adding the appropriate slack, surplus or artificial variable. Also, add the new variables into the problem’s objective function.

b. Set up the complete initial simplex tableau for this problem.

c. Give the values for all variables in this initial solution.