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Q$1.$ Implement the following LP problem in a spreadsheet. Use Solver to solve the problem and create a Sensitivity Report. Use this information to answer the following questions:

a$.$ What range of values can the objective function coefficient for variable X$1$ assume without changing the optimal solution?

b$.$ Is the optimal solution to this problem unique, or are there alternate optimal solutions?

c$.$ How much does the objective function coefficient for variable X$2$ have to increase before it enters the optimal solution at a strictly positive level?

d$.$ What is the optimal objective function value if X$2$ equals $1?$

e$.$ What is the optimal objective function value if the RHS value for the second constraint changes from $15$ to $25?$

f$.$ Is the current solution still optimal if the coefficient for X$2$ in the second constraint changes from $5$ to $1?$ Explain.

Q$2.$ The CitruSun Corporation ships frozen orange juice concentrate from processing plants in Eustis and Clermont to distributors in Miami, Orlando, and Tallahassee. Each plant can produce $20$ tons of concentrate each week. The company has just received orders of $10$ tons from Miami for the coming week, $15$ tons for Orlando, and $10$ tons for Tallahassee. The cost per ton for supplying each of the distributors from each of the processing plants is shown in the following table.

Miami Orlando Tallahassee

Eustis $$260$ $$220$ $$310$

Clermont $$230$ $$240$ $$290$

The company wants to determine the least costly plan for filling their orders for the coming week.

a$.$ Formulate an LP model for this problem.

b$.$ Implement the model in a spreadsheet and solve it$.$

c$.$ What is the optimal solution?

d$.$ Is the optimal solution degenerate?

e$.$ Is the optimal solution unique? If not, identify an alternate optimal solution for the problem.

f$.$ How would the solution change if the plant in Clermont is forced to shut for one day resulting in a loss of four tons of production capacity?

g$.$ What would the optimal objective function value be if the processing capacity in Eustis was reduced by five tons?

h$.$ Interpret the reduced cost for shipping from Eustis to Miami.

Q$3.$Consider the following LP problem:

a$.$ Use slack variables to rewrite this problem so that all its constraints are equal$-$to constraints.

b$.$ Identify the different sets of basic variables that might be used to obtain a solution to the problem.

c$.$ Of the possible sets of basic variables, which lead to feasible solutions and what are the values for all the variables at each of these solutions?

d$.$ Graph the feasible region for this problem and indicate which basic feasible solution corresponds to each of the extreme points of the feasible region.

e$.$ What is the value of the objective function at each of the basic feasible solutions?

f$.$ What is the optimal solution to the problem?

g$.$ Which constraints are binding at the optimal solution?

Q$4.$ Consider the following constraint, where S is a slack variable:

a$.$ What was the original constraint before the slack variable was included?

b$.$ What value of S is associated with each of the following points:

i$)$ X$1=5,$ X$2=0$

ii$)$ X$1=2,$ X$2=2$

iii$)$ X$1=7,$ X$2=1$

iv$)$ X$1=4,$ X$2=0$