# MATHEMATICS

1. Consider the following linear programming problem:

Max 1 2 x 2x

s.t. 3 1 2 x x

x1 2×2 0

1 2 x

, 0 1 2 x x

a. Write the problem in standard form. Identify slack/surplus variables. (5 points)

b. Identify the feasible region (5 points)

c. Find the optimal solution. (5 points)

d. What are the values of the slack and surplus variables at the optimal solution? (5 points)

2. Thomas Industries and Washburn Corporation supply three firms (Zrox, Hewes, Rockwright) with customized shelving for its offices. They both order shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc. Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are given below:

Thomas Washburn

Arnold 5 8

Supershelf 7 4

The chart below gives the cost to install the shelving at the various locations:

Zrox Hewes Rockwright

Thomas 1 5 8

Washburn 3 4 4Currently weekly demand by the user are 50 for Zorx, 60 for Hewes, and 40 for Rockwright. Both Arnold and Supershelf can supply at most 75 units to its customers.

a. Draw a network representation for this problem. (6 points)

b. Formulate this problem as a LP model. (9 points)

3. Bart’s Barometer Business (BBB) is a retail outlet that deals exclusively with weather equipment. Currently BBB is trying to decide on an inventory and reorder policy for home barometers. These cost BBB $50 each and demand is about 500 per year distributed fairly evenly throughout the year. Reordering costs are $80 per order and holding costs are figured at 20% of the cost of the item.

BBB is open 300 days a year (6 days a week and closed two weeks in August).

Lead time is 60 working days.

a. What is the optimal reorder quantity? (5 points)

b. What is the cycle time? (5 points)

c. What is the reorder policy for BBB? (5 points)

d. What total annual cost does the model give? (5 points)

4. Consider the following linear programming problem:

Min 3 1 4 2 x x

s.t. 3 6 1 2 x x

x1 x2 4

, 0 1 2 x x

a. Solve this problem using Excel Solver. Provide answer report and sensitivity report. (5 points)

b. If the objective function coefficient for 1x decreases from 3 to 2 and the objective function coefficient for 2 x increases from 4 to 6, will the current optimal solution still remain the same? Why? (8 points)

c. If you would change the right-hand side of one constraint by one unit, which one would you choose? How would you make the change? Why? (8 points)5. Given the following network with activities and times estimated in days,

D

E

F

G

H J

I K Finish

A

B

C

StartActivity Optimistic

Most

Probable Pessimistic

A 2 5 6

B 1 3 7

C 6 7 10

D 5 12 14

E 3 4 5

F 8 9 12

G 4 6 8

H 3 6 8

I 5 7 12

J 12 13 14

K 1 3 4

a. Fill in all the blanks in the following table. (13 points)

Note:

i) To reduce the computation load, please round the expected activity time (t) up to an integer if the computed expected activity time is not an integer. For example, if tA=4.15, round it up to 5.

ii) For variance (σ

2) in activity time, keep two decimal places.

Activity

Precedence

Activities

Expected

Time t (days)

Varianc

e σ2 ES LS EF LF Slack

Critical

Path?

A

B

C

D

E

F

G

H

I

J

K

b. What is the expected time and variance to complete the project? (5 points)

c. What is the probability the project will take more than 28 days to complete? (6 points)