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                        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




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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.



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    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.


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                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




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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

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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.




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