Problem 1: Consider the partial simplex tableau shown below:

a) What is the current solution given in the tableau?

X1 = ____, X2 = ____, X3 = ____, S1 = ____, S2 = ____, S3 = ____, Z = ____,

b) Is the given solution optimal? If not, what is the optimal solution?

X1 = ____, X2 = ____, X3 = ____, S1 = ____, S2 = ____, S3 = ____, Z = ____,

c) In the above tableau, what are the non basic variables?

d) In the optimal tableau, what are the basic variables?

Problem 2: The manager of a very busy bank branch has estimated the requirements for bank tellers between 10 and 18 tellers, depending on the time of the day. The branch opens between 9:00 AM and 5:00 PM. At present the branch employs 12 full-time tellers who are allowed one hour lunch break (Half of the full-timers eat at 11 AM, and the other half at noon.) The Table below indicates the estimated number of tellers needed at various hours of the day.

The branch is considering hiring part time tellers who must work exactly four hours per day and they can start any hour between 9:00 AM and 1:00 PM. The bank policy limits total part time hours to a maximum of 50% of the day's total teller requirement (i.e. 112.) The part-timers are paid $8:00 per hour or $32:00 per day on average, and full-timers earns $100:00 per day, on average. The bank seeks an advise to minimize the tellers cost. It will release one or more of its full time tellers if it is profitable to do so. Formulate the model to help solving this problem.

Hint: Let F = total available full-time tellers

P1 = part-timers starting at 9:00 AM and leaving at 1:00 PM

P2 = part-timers starting at 10:00 AM and leaving at 2:00 PM

P3 = part-timers starting at 11:00 AM and leaving at 3:00 PM

P4 = part-timers starting at Noon and leaving at 4:00 PM

P5 = part-timers starting at 1:00 PM and leaving at 5:00 PM

Problem 3: Consider the following linear program and the graph for the constraints set as equalities:

Maximize Z = 8X1 + 5X2

Subject to

2X1+ 1X2 ≤ 12 = I

1X1 + 1X2 ≤ 10 = II

X2 ≥ 4 = III

X1, X2 ≥ 0

Adjustable Cells

Constraints

Find the missing values in the above Table.

Problem 4: A firm produces two types of steel hammers with wooden handles. The small one sells for $6 and consists of 0.50 pounds of steel; the big one sells for $15 and consists of 1 pound of steel. For each hammer, the cost of labor, wooden handle and packaging add up to $1. The steel costs the firm $4 per pound.

Let X1 = Number of the small type hammer to produce.

X2 = Number of the big type hammer to produce.

a) Write down an appropriate objective function that maximizes the profit for this firm?

b) Given that the shadow price for steel is $2, which of the following statements is correct:

1) The firm should not buy more steel.

2) The firm should buy all the steel it can only if it can purchase it for less than $2 per pound.

3) The firm should buy at least as much as the "ALLOWABLE INCREASE", but only if it can be purchased for less than $2 per pound.

4) The firm should buy at least as much as the "ALLOWABLE INCREASE", but only if it can be purchased for less than $6 per pound.

c) If the firm has a budget of $1000 for steel, write a constraint to reflect this limitation.

d) If there is a requirement by the marketing research group in the firm to produce at least twice as much from the smaller hammer as the big one, write a constraint to show this requirement.

Problem 5: Consider the following ILP problem:

Maximize Z = 3X1 + 5X2

Subject to

2X1 + 6X2 ≤ 27

X2 ≥ 2

3X1 + X2 ≤ 19

X1 , X2 ≥ 0 and integer

a) The optimal LP relaxation solution to this problem:

X1 = ____, X2 = ____, X3 = ____, s1 = ____, s2 = ____, s3 = ____, z = ____,

b) The best rounding feasible solution to this problem:

X1 = ____, X2 = ____, X3 = ____, z = ____,

c) The integer optimal feasible solution to this problem:

X1 = ____, X2 = ____, s1 = ____, s2 = ____, s3 = ____, s = ____,

Problem 6: What LP problem will have the following dual?

Minimize 2Y₁ + Y₃ - 4Y₄

Subject to

2Y₁ + Y₂ + Y₄ ≤ 15

2Y₁ + Y₂ + Y₃ + Y₄ ≥ 8

Y₁ ≥ 0, Y₂ ≤ 0,

Y₃ unconstraint in sign, Y₄ ≤ 0