Problem 1: Set up the initial simplex tableau and work through one iteration (in TABULAR form). Show the values of x₁, x₂ and z after one iteration.

Maximize Z = 3x₁ + 2 x₂,

Subject to x₁ 4

x₁ + 3x₂ 15

2x₁ + x₂ 10

and x₁ 0, x₂ 0.

Problem 2: The following tableau represents a specific simplex iteration. All variables are nonnegative. The tableau is not optimal for either a maximization or a minimization problem. Thus, when a nonbasic variable enters the solution, it can either increase or decrease z or leave it unchanged, depending on the parameters of the nonbasic variable.

Basic	X1	X2	X3	X4	X5	X6	X7	X8	RHS
Z	0	-5	0	4	-1	-10	0	0	620
X8	0	3	0	-2	-3	1	5	1	12
X3	0	1	1	3	1	0	3	0	6
X1	1	-1	0	0	6	-4	0	0	0

1. categorize the variables as basic and nonbasic and provide the current values of all the variables.
2. Suppose that the problem is of the maximization type: identify the nonbasic variables that have the potential to improve the value of z. if such a variable enters the basic solution, determine the associated leaving variable., if any, and the associated change in z. Do not use the row operations.
3. Which nonbasic variable(s) will not cause a change in the value of z when selected to enter the solution? Why?

Problem 3: Consider the graphical representation of the following LP problem in which X1 and X2 are the number of desks and file cabinets produced per week, respectively:

Max z = 3X1 + 2X2

S.t. 2X1 + X2 < 100 (cutting operation constraint)

X1 + X2 < 80 (welding operation constraint)

X1 < 35 (woodwork constraint)

X1, X2 > 0

a) Using the graphical representation, Find the optimal solution, and answer the following questions:

b) Let c2 be the contribution to profit by each file cabinet. Currently c2 =2. For what range of c2 values does the current solution remains optimal?

c) Let b3 be the number of available woodwork hours. Currently b3 = 35. For what range of b3 values does the current solution remains optimal?