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FINAL EXAMINATION SAMPLE TEST 01

QUANTITATIVE METHODS FOR BUSINESS Duration: 120 minutes

Student ID: Name:

Date: 2013 Summer

PART I: MULTIPLE CHOICE QUESTIONS: (30%) 1. Which of the following is not true about slack variables in a simplex tableau?

a. They are used to convert constraint inequalities to equations. b. They require the addition of an artificial variable. c. They represent unused resources. d. They may represent machine time, labor hours, or warehouse space.

2. How should the constraint, 5X - 2Y 6, be converted into simplex tableau form?

a. 5X - 2Y - S + A = 6 b. 5X - 2Y + S = 6

c. 5X - 2Y + A = 6 d. 5X - 2Y + S - A = 6

3. Using the simplex method, we know we have an optimal solution when

a. all basic variables are negative. b. when all the artificial variables have a positive value. c. all slack variables have a zero value. d. none of the above

4. If, in the optimal tableau of a linear programming problem, an artificial variable is present in the solution

mix, this implies a. unboundedness. b. degeneracy.

c. alternate optimal solutions. d. infeasibility.

5. If, in the final optimal simplex tableau, the Cj - Zj value for a nonbasic variable is zero, this implies

a. alternate optimal solutions. b. unboundedness.

c. infeasibility. d. degeneracy.

Using the following data for next 3 questions The following represents a project with known activity times. All times are in weeks.

Activity Immediate Processor Time A - 4 B - 3

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C A 2 D B 7 E C,D 4 F B 5

6. What is the minimum possible time required for completing the project? a. 25 b. 14 c. 10

d. 8 e. none of the above

7. What is the critical path:

a. A-C-E b. B-F c. B-D-E

d. A-C-F e. None of the above

8. Compute the slack time for activity F.

a. 5 b. 3 c. 0

d. 6 e. none of the above

9. Slack time in a network is the ________

a. shortest amount of time that could be required to complete the activity. b. amount of time that you would expect it would take to complete the activity. c. amount of time that an activity would take assuming very unfavorable conditions. d. amount of time that an activity can be delayed without delaying the entire project.

10. Consider a project that has an expected completion time of 60 weeks and a standard deviation of 5 weeks. What is the probability that the project is finished in 70 weeks or fewer? (Round to two decimals.) a. 0.02 b. 0.48 c. 0.80

d. 0.98 e. none of the above

11. Given the following key factors and their importance weights, determine a total weighted evaluation for

A3.

Key Factor Importance Weights A1 A2 A3

K1 0.2 0.2 0.9 0.7 K2 0.7 0.8 0.5 0.4 K3 0.1 0.9 0.6 0.8

a. 0.50 b. 0.74

c. 0.69 d. none of the above

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12. In AHP method, higher values of the consistent ratio (CR) indicate a. an indication of low consistency of the pairwise comparisons. b. the greater the level of consistency of the pairwise comparisons. c. a moderate consistency of the pairwise comparisons. d. none of the above

13. Which of the following matrices is not a correct pairwise comparison matrix?

a. A b. B

c. C d. D

Using the following data for next 3 questions Each table and each bookcase produced by ABC company yields a profit of $9 and $12 respectively. During any given production period (of one week), 10 gallons of varnish and 12 length of high quality redwood are available. Each table requires 1 gallon of varnish and 1 length of redwood. Each bookcase takes 1 gallon of varnish and 2 lengths of wood. T = number of tables produced each week B = number of bookcases produced each week 14. if we were to frame this as a linear programming problem, the objective function would be

a. maximize 12T + 10B b. maximize 9B + 12T

c. maximize 9T + 12B d. none of the above

15. Which of the following constraints would be used?

a. 1T + 2B 12 b. 1T + 2B 12 c. 1T + 1B 10

d. a and b e. b and c

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PART II: WRITING: (70%) Question 1: (20 pts) The following table contains the activities of a project, the predecessor activities and activity time estimate:

Activity Predecessor Time Estimate (in day) Optimistic Most Likely Pessimistic A - 6 10 17 B A 4 6 9 C A,B 11 14 20 D B 7 16 22 E B,C 5 8 10 F C 3 12 22 G D,E 1 4 8 H F,G 9 15 20

a. (4pts) Draw the PERT diagram for this project. b. (10pts) Determine earliest start and finish times, latest start and finish times, and slack for each

activity. Indicate the critical path. c. (3pts) Compute the expected value and variance of project completion time. d. (3pts) Determine the probability that the finished time will be less than or equal 50 days. Assume

that the project completion time follows normal distribution.

Question 2: (15 pts) A manufacturing company makes two categories of products including product X1 and product X2. A pair of product X1 requires 2 man-hours for cutting, 1 man-hour for shaping and 3 man-hours for finishing while a pair of product X2 requires 2 man-hours for cutting, 2 man-hours for shaping and 1 man-hour for finishing. Each day the company has available 140 man-hours for cutting, 120 man-hours for shaping and 150 man-hours for finishing. How many numbers of each categories of product should the company manufacture each day in order to maximize profit if a pair of product X1 yields a profit of $10 and a pair of product X2 yields a profit of $8? Question 3: (20 pts) A parent is finding a school for their son. They are considering and going to choose one of three schools: Hoa Binh (HB), Dinh Tien Hoang (DTH) and Ky Dong (KD). The main factors that affect their choice are the fame, the location and the facilities of school.

a. (5pts) Draw the Hierarchy structure of this problem. b. (10pts) Give the pairwise comparison matrix of the factors as following:

Factors Fame Location Facilities Fame 1 3 2

Location 1 0.5 Facilities 1

Determine the priority vector of these factors. Is the priority result consistent? c. (5pts) Give the priority vectors of alternatives (schools) for each factor as following:

Fame Scores Location Scores Facilities Scores HB 0.347 HB 0.234 HB 0.429

DTH 0.540 DTH 0.129 DTH 0.429 KD 0.113 KD 0.637 KD 0.142

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Which school is the best choice for their son? Question 4: (15 pts) Solve the following LP problem by using Simplex method

:Subject to34 21 XXZMaximize

0,1414621032

21

21

21

21

XXXX

XXXX

a. (5pts) Transform the original LP model above to standard LP form (simplex form). b. (7pts) Develop the initial simplex table and 2nd simplex table for this problem. c. (3pts) Give the final simplex table as following:

4 3 0 0 -M -M Cj Sol. X1 X2 s1 s2 A1 A2 Qty. 4 X1 1 0 3 0 2 -3 2 3 X2 0 1 -2 0 -1 2 2 0 s2 0 0 -10 1 -7 10 4 Zj 4 3 6 0 5 -6 14 Cj-Zj 0 0 -6 0 -M-5 -M+6

Determine the ranges of coefficient of X1 and the RHS of constraint 1414 21 XX that the solution is held.

GOOD LUCK!