$5-22.$ Consider the following final simplex tableau for a linear programming problem:

Simplex Tableau: $1$

a$.$ Identify the optimal solution.

b$.$ What is the marginal value of an additional unit of resource in the first constraint? What is the range

of $b_1$ within which the marginal value is valid $(b_1=120)?$

c$.$ Is there an alternate optimal solution? If so$,$ identify the new solution.

d$.$ Suppose that the firm can obtain $10$ additional units of resource for the fourth constraint at the price

of $$10$ each $(b_4=20).$ Should it obtain these additional units? Why or why not?

e$.$ Suppose that the firm can obtain additional resources for the second constraint at a price of $$1$ per unit

more than the current price $(b_2=160).$ Should it purchase these additional resources? If so$,$ how many

units?

f$.$ Suppose that the firm signed a trade agreement with its competitor, and the agreement resulted in a

new constraint, $x_{1}8.$ Derive the new solution with this additional constraint.