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Last Name _______________________ First Name ________________________ ID ___________________________ Operations Management II 73-431 Summer 2007 Odette School of Business University of Windsor Midterm Exam Solution Thursday, July 19, 11:30 am – 12:50 pm Instructor: Mohammed Fazle Baki Aids Permitted: Calculator, straightedge, and a one-sided formula sheet Time available: 1 hour 20 min Instructions: This exam has 10 pages including this cover page and 2 blank pages. Please be sure to put your name and student ID number on each page. Do not return tables the formula sheets. Show your results up to four decimal places. Show your work. Grading: Question Marks: 1 /10 2 /15 3 /12 4 /8 5 /20 Total: /65 Last Name _______________________ First Name ________________________ ID ___________________________ Question 1: (10 points) Circle the most appropriate answer 1.1 Which of the following is false? a. Expedite critical activities. b. Do not expedite non-critical activities. c. Expedite the least cost alternative. d. If there are many critical paths, expedite one of them. 1.2 Consider an assembly line. Suppose that a single shift of 8 hours is used and the demand is 48 units per day. Then, the required cycle time is a. 6 hours b. 10 minutes c. 6 minutes d. 10 seconds 1.3 Consider two-job scheduling problem. If a 45-degree line has a horizontal component of 5 min, its length is a. 5 min b. 5 2 min c. 10 min d. unknown 1.4 Consider a single machine stochastic scheduling problem. The EEDD rule minimizes a. maximum probability that a job is late b. maximum probability that a job is completed c. maximum lateness d. maximum expected lateness 1.5 Which of the following is not a network construction rule? a. There must be a single start node. b. There must be a single finish node. c. Every pair of nodes is connected by an arc. d. More than one arc must not represent an activity. 1.6 Consider Johnson’s rule. If minimum processing time is on Machine 2 for Job E a. assign Job E in the beginning b. assign Job E in the end c. assign Job E to position 1 d. a and c 1.7 In which of the following cases, it is most desirable to find a schedule that minimizes number of tardy jobs? a. Cost of utilities is rising b. Customers are complaining about the congestion and long waiting time c. Buyers are asking for faster delivery d. The fashion designer decides to start taking orders for wedding gowns 2 Last Name _______________________ First Name ________________________ ID ___________________________ 1.8 Which of the following objectives are equivalent? a. minimizing makespan, mean completion time, average lateness and maximizing space utilization b. maximizing space utilization, efficiency, machine utilization and minimizing inventory c. minimizing inventory, total completion time, average lateness and mean completion time d. minimizing maximum lateness, maximum completion time and number of tardy jobs 1.9 If a facility start at 8:30 am, one job completes at 10:30 am and another job completes at 11:30 am, what is the total completion time? a. 2 hour b. 3 hour c. 4 hour d. 5 hour 1.10 A flow shop is suitable for a. high volume, high customization b. high volume, low customization c. low volume, high customization d. low volume, low customization Question 2: (15 points) Consider the following single-machine scheduling problem: Job Arrival Time Processing Time (min) Due Date (min) A 0 13 15 B 0 10 20 a. (2 points) If the jobs are processed according to the FCFS order, compute total completion time. Job Starting time Processing time Completion time A 0 13 13 B 13 10 23 Total Completion time 36 b. (1 point) Which sequence minimizes total completion time A-B or B-A? SPT, B-A. c. (4 points) Find the sequence that follows from the CR rule. If the jobs are processed according to the CR rule, compute maximum lateness. CR = (Due date – Today’s date)/ Processing time Today’s date = 0. For Job A, CRA = (15-0)/13 = 1.1539 (less, so assign A before B) For Job B, CRB = (20-0)/10 = 2 Hence the CR sequence is A-B. 3 Last Name _______________________ First Name ________________________ ID ___________________________ Job j Starting Processing Completion Due Date, Lateness, Time Time Time, C j dj A 0 13 13 15 -2 B 13 10 23 20 3 Maximum Lateness 3 Hence the maximum lateness is 3. d. (1 point) Which sequence minimizes maximum lateness A-B or B-A? EDD, A-B. e. (4 points) Find the optimal sequence, if the objective is to minimize the number of tardy jobs. What is the optimal number of tardy jobs? Arrange the jobs in the EDD order Job j Starting Processing Completion Due Date, Lateness, Tardy? Time Time Time, C j dj A 0 13 13 15 -2 No B 13 10 23 20 3 Yes The first job tardy is B. Between A and B, A is longer. So, remove A. When assigned in the beginning, Job B is not tardy. Append A. Optimal sequence, B-A, one job tardy. f. (3 points) Suppose each job requires a setup time of 3 minutes which is not included in the processing time shown above. Redo part e, using the setup time information. Arrange the jobs in the EDD order Job j Starting Processing Completion Due Date, Lateness, Tardy? Time Time Time, C j dj A 0 16 16 15 1 Yes B 16 13 29 20 9 Yes The first job tardy is A. Remove A. When assigned in the beginning, Job B is not tardy. Append A. Optimal sequence, B-A, one job tardy. Question 3: (12 points) Sharon and Donna plan to attend a social function. Each requires hair styling and fitting for a gown. Assume that the fittings are done before the styling and that there is only a single hair stylist and a single seamstress available. Sharon Donna Activity Time (min) Activity Time (min) Fitting 15 Fitting 10 Styling 18 Styling 16 a. (8 points) Determine how the activities should be scheduled in order to minimize the makespan. Use the graphical solution procedure for two-job problem. 4 Last Name _______________________ First Name ________________________ ID ___________________________ Sharon’s Time Start Process End Fitting 0 15 15 30 Styling 15 18 33 Path 1 44 (33,26) Donna's tme 20 49 Styling Donna’s Time 10 Start Process End Fitting Path 2 Fitting 0 10 10 0 Styling 10 16 26 10 20 30 40 Sharon's time Coordinates of Blocks Blocks Lower-left corner (start, start) Upper-right corner (end, end) History (0,0) (15,10) Science (15,10) (33,26) Interpretation of Path 1 Clock Clock Line Sharon’s Donna’s Activity Start End Type Activity 0 10 Vertical Idle Fitting 10 25 450 Fitting Styling 25 26 Vertical Idle Styling 26 44 Horizontal Styling Idle b. (4 points) Draw the Gantt chart for the optimal schedule obtained in a. 10 25 Fitting 10 26 44 Styling 0 10 20 30 40 50 Time 5 Last Name _______________________ First Name ________________________ ID ___________________________ Question 4: (8 points) Three jobs are to be scheduled on two machines M1 and M2. Assume that every job is first processed on M1 and then on M2. The processing times are independent exponentially distributed random variables with the mean times as stated below: Job M1 M2 J1 10 12 J2 5 8 J3 4 5 Find a schedule that minimizes expected makespan. Show your work. M1 M2 Mean Processing Mean Processing Job Processing Rate Processing Rate a j b j Time Time 1 1 (2 points) a j bj Aj Aj Bj Bj J1 10 0.1 12 0.0833 -0.0166 J2 5 0.2 8 0.125 0.075 J3 4 0.25 5 0.2 0.05 Arrange the jobs in the descending order of a j b j . Hence, the sequence is J2, J3, J1 Question 4: (20 points) Information on a project network is given below: Activity Immediate Predecessor Processing time (days) A --- 8 B --- 5 C B 4 D --- 6 E D 3 a. (5 points) Construct the project network. Use the Activity on Arc method. Use no dummy arc. 2 D E B C 1 3 4 A 6 Last Name _______________________ First Name ________________________ ID ___________________________ b. (5 points) What are the slack times of each activity? Activity ES EF LS LF Slack time =LS-ES =LF-EF A 0 8 1 9 1 B 0 5 0 5 0 C 5 9 5 9 0 D 0 6 0 6 0 E 6 9 6 9 0 c. (2 points) Which activities are critical? B,C,D,E d. (2 points) What is the minimum time required to complete the project? 9 days e. (6 points) Suppose that the activities can be expedited at some additional costs. The cost of expediting per day is $100, $200, $300, $400 and $500 respectively for activities A, B, C, D and E. What is the minimum cost of expediting the project by 2 days? There are two critical paths, B-C and D-E. Expedite B and D each by 1 day, for a total cost of 200+400=$600 Next, there are three critical paths, A, B-C and D-E Expedite A, B and D each by 1 day for a total cost of 100+200+400=$700 Hence, the minimum cost = $1,300. 7