1. Chapter 15 Short-term Scheduling

2. Outline Introduction Loading Jobs: Assignment Method Sequencing Jobs
One-machine sequencing
Two-machine sequencing

3. INTRODUCTION

4. Introduction
Scheduling is the process by which the production or service plan is implemented.
Scheduling determines where and when a job is to be processed.
A scheduling is a timetable for performing activities, utilizing resources, or allocating facilities.

5. Introduction Two issues
Loading: to which machine a job should be assigned
Sequencing: in what order the jobs assigned to a machine should be processed

6. Introduction
Five printing jobs (1, 2, 3, 4, and 5) are received from local customers and each of them may be done on either of the two high-speed copiers (A and B) in the shop. Scheduling the five jobs calls for a two-step procedure.
Determine to which copier each job should be assigned (i.e., loading).
Determine the order in which the jobs assigned to each copier should be processed (i.e., sequencing).
job
copier
Loading: assign jobs 1, 2 and 4 to copier A;
assign jobs 3 and 5 to copier B 1 A 2
Sequencing: 2 then 1 then 4 on A;
5 then 3 on B 3 4 B 5

7. Importance of Short-Term Scheduling
Effective and efficient scheduling can be a competitive advantage
Faster movement of goods through a facility means better use of assets and lower costs
Additional capacity resulting from faster throughput improves customer service through faster delivery
Good schedules result in more dependable deliveries

8. LOADING JOBS
Assignment Method

9. Loading Assignment method:
Quantitative technique for identifying the optimal job-machine pairings
Goal is to allocate one and only one job to one and only one machine such that the total cost is as low as possible
Arrange information into a square matrix called the assignment matrix

10. Loading Procedure
For each row in the original matrix, subtract the smallest number from each of the numbers
For each column in the new matrix, subtract the smallest number from each of the numbers
Determine the minimum number of (horizontal or vertical) lines needed to cover all the zeros in the existing matrix. If this is equal to the number of rows (or columns) of the square matrix, go to Step (5); otherwise, go to Step (4)
Two sub-steps
Subtract the smallest uncovered number from each of the uncovered numbers in the matrix
Add the same smallest uncovered number to every number where two covering lines intersect. Go to Step (3)
Start with any row or column containing only one zero (a tie can be broken arbitrarily), match the "job" and the "machine" associated with the zero and then exclude them from further consideration by drawing a horizontal line and a vertical line. Repeat the same process until the complete assignments have been made.

11. Example 1
The XYZ Corporation produces four products (A, B, C, and D), which can be manufactured by any of the four machines (1, 2, 3, and 4) in the Production Department. Because of expensive set-up costs, each product is usually produced by one and only one machine. Given the following cost data (in dollars), determine the product-machine assignments such that the total cost is minimized. What is the minimum total cost?
Assignment Matrix 1 2 3 4 A 30 50 40 B 20 10 C D

12. Step 1: For each row in the original matrix, subtract the smallest number from each of the numbers.
Step 2: For each column in the new matrix, subtract the smallest number from each of the numbers
Solution
Assignment Matrix 1 2 3 4 A 30 50 40 B 20 10 C D 1 2 3 4 A B C D 1 2 3 4 A 20 10 B C D 20 10 10 20 10 10 20 10 10
Step 3: Determine the minimum number of (horizontal or vertical) lines needed to cover all the zeros. If this is equal to the number of rows (or columns) of the square matrix, go to Step (5); otherwise, go to Step (4)
Step 5: Starting with any row or column containing only one zero, match the "job" and the "machine" associated with the zero and then exclude them from further consideration. Repeat the same process until the complete assignments have been made 1 2 3 4 A 20 10 B C D 1 2 3 4 A 20 10 B C D
4 lines to cover all zeros
A  1
Matrix includes 4 lines
B  2
Thus, 4 = 4
D  3
C  4
Total cost = = $90

13. Example 2
At Valley Hospital in Tyler, TX, nurses beginning a new shift report to a central area to receive their primary patient assignments. Not every nurse is as efficient as another with particular kinds of patients. Given the following patient roster, list of nurses, and estimated task times (in minutes), which nurse should take care of which patient so that the total amount of time needed to complete the routine tasks on this shift will be as small as possible? What is the minimum total time requirement?
Assignment Matrix
Nurse 1
Nurse 2
Nurse 3
Nurse 4
Jones 70
110
115
130
Hawkins 10
105
135 50
Becker 30 90 75 35
Sweeney 55 25 15 65

14. Solution Step 1 Step 2 Assignment Matrix 40 45 60 95 125 40 60 45 5 40J 70
110
115
130 H 10
105
135 50 B 30 90 75 35 S 55 25 15 65 N1 N2 N3 N4 J H B S N1 N2 N3 N4 J 30 45 55 H 85
125 35 B 50 S 40 40 45 60 95
125 40 60 45 5 40 10 50
Go back Step 3
Step 4(a): Subtract the smallest uncovered number from each of the uncovered numbers in the matrix N1 N2 N3 N4 J 15 25 H 55 95 5 B 30 50 45 S 70
Step 3
Step 4(b): Add the same smallest uncovered number to every number where two covering lines intersect. Go to Step (3). N1 N2 N3 N4 J 30 45 55 H 85
125 35 B 50 S 40
4 = 4 N1 N2 N3 N4 J H B S
Step 5 15 25
B  N4 N1 N2 N3 N4 J 15 25 H 55 95 5 B 30 50 45 S 70
3 lines to cover all zeros
S  N3 55 95 5
Matrix includes 4 lines
J  N2 30 50 45
H  N1
Thus, 3 ≠ 4 70 45
Total time = = 170

15. EX 1 in class
Coach Terry Young is putting together a relay team for a 400-meter relay. The four swimmers are Gary Hall (GH), Mark Spitz (MS), Jim Montgomery (JM), and Chet Jastremski (CJ) and each of them must swim 100 meters of breaststroke (BR), backstroke (BA), butterfly (BU), or freestyle (FR). Based on the past records, the coach believes that each swimmer will attain the times (in seconds) given in the following table: BR BA BU FR GH 55 52 54 MS 53 58 JM 51 56 CJ 57
To minimize the team’s total time for the race so that it has the best chance to win, which swimmier should swim which stroke? What is the minimum total time for the team to complete the relay
Repeat (1) by assuming that it takes Gary Hall and Mark Spitz, respectively, 52 seconds and 54 seconds, to swim 100 meters of freestyle. By how many seconds does the minimum total swimming time differ from the one in (1) in this case?

16. Loading Summary of traditional loading problem
The number of jobs is equal to the number of machines
Each of the jobs is processed by any of the machines
The goal is to minimize the total cost
Three special loading problems in reality
Case I: unequal numbers of jobs and machines
Case II: prohibited job-machine combination
Case III: maximization objective

17. Loading (special case I)
Case I: Unequal numbers of jobs and machines
Solution
A dummy row (job) or column (machine) of zero costs should be introduced to make the assignment matrix a square one
The standard method discussed before should then be applied in the usual manner.
Be careful in interpreting the final optimal solution obtained since the dummy job or machine added does not exist.

18. Example 3
Four professors are responsible for three projects funded by the National Science Foundation. The table presented below shows the time (in weeks) needed for each professor to finish each project. Determine the optimal assignments as well as the minimum total time requirement.
Assignment Matrix 1 2 3
Thompson 50 60 40
Johnson 62 78 36
Nelson 57 74 38
Parkinson 42 66 45

19. Solution Dummy machine Assignment Matrix Step 1 Step 2 Step 3 Step 54 T 50 60 40 J 62 78 36 N 57 74 38 P 42 66 45 1 2 3 4 T 50 60 40 J 62 78 36 N 57 74 38 P 42 66 45 1 2 3 4 T 8 J 20 18 N 15 14 P 6 9
Step 3
Step 5 1 2 3 4 T 8 J 20 18 N 15 14 P 6 9 1 2 3 4 T 8 J 20 18 N 15 14 P 6 9
P  1
T  2
J  3
4 = 4
Minimum total time = = 138 weeks
Note that Professor Nelson will not be responsible for any project since "4" is a dummy project

20. Loading (special case II)
Case II: prohibited job-machine combination
A job cannot be processed by a machine for some reason
Solution
Replace the cost corresponding to the prohibited job-machine pair in the matrix with an exceedingly large positive number (e.g., 9,999)
The standard method discussed before should then be applied in the usual manner.

21. Example 4
Three managers (1, 2, and 3) are to be assigned to three fast-food outlets (A, B, and C). The following table indicates the expense (in dollars) to be incurred for each manager to run each outlet. Suppose that Manager 3 is not willing to work at outlet B due to travel distance. What are the optimal personnel assignments? What is the minimum total expense?
Assignment Matrix A B C 1
4,000
3,500
3,400 2
4,800
5,000
4,700 3
4,100
3,200
4,400

22. Solution
Manager 3 is not willing to work at outlet B due to travel distance, thus, replace 3,200 with 99,999.
Assignment Matrix
Revised Assignment Matrix
Step 1 A B C 1
4,000
3,500
3,400 2
4,800
5,000
4,700 3
4,100
3,200
4,400 A B C 1
4,000
3,500
3,400 2
4,800
5,000
4,700 3
4,100
99,999
4,400 A B C 1
600
100 2
300 3
95,899
Step 2
Step 3
Step 5 A B C 1
600 2
100
200 3
95,799
300 A B C 1
600 2
100
200 3
95,799
300 A B C 1
600 2
100
200 3
95,799
300
3 = 3
3 A
1 B
Total cost = 4, , ,700 = $12,300
2 C

23. Loading (special case III)
Case III: maximization objective
Solution
Replace every number with the difference between the largest number in the entire matrix and the number under consideration.
The standard method discussed before should then be applied in the usual manner.

24. Example 5
The following table shows the expected sales volumes (in units) if four salespersons are assigned to four different marketing territories. Determine the optimal assignment plan and the maximum total sales.
Assignment Matrix 1 2 3 4
Mary
1,500
1,400
1,600
Jim
1,450
Lori
1,550
1,350
Tom
1,650

25. Solution Assignment Matrix Revised Assignment Matrix Step 1 Step 2 & 34 M
1,500
1,400
1,600 J
1,450 L
1,550
1,350 T
1,650 1 2 3 4 M
150
250 50 J
200 L
100
300 T 1 2 3 4 M
100
200 J
150 L T
Step 2 & 3
Step 4 & 3
Step 5 1 2 3 4 M
100 J
150 L
200 T 50 1 2 3 4 M J L T 1 2 3 4 M
200 J 50 L
100 T
200
200 50 50
100 50
100 50
3 ≠ 4
4 = 4
M  4
L  3
Total sales = 1, , , ,600 = 6,200 units
J  2
T  1

26. EX 2 In Class
The following table contains information on the cost to run three jobs on four available machines. Determine an assignment plan that will minimize costs
Machine A B C D 1 12 16 14 10
Job 2 9 8 13 7 3 15 11

27. SEQUENCING JOBS
One-machine sequencing
Two-machine sequencing

28. Sequencing
To determine the order in which the jobs assigned to a machine are to be processed
One-machine sequencing
Two-machine sequencing

29. One-machine sequencing
Each of the jobs needs to go through only one machine in a certain order
Applications
Books ("jobs") borrowed by students need to be checked out at the circulation desk ("machine") of a university library.
Patients ("jobs") are treated by a doctor ("machine") in the local hospital.
Passengers ("jobs") check in at an airline's ticket counter ("machine") in the airport.
Research manuscripts ("jobs") are typed by a secretary ("machine").
Tasks ("jobs") are performed by a worker ("machine") in a manufacturing plant

30. One-machine sequencing
Objectives – minimize
m: total time needed to process the set of jobs
ACT: average completion time (or mean flow time)
AJL: average job lateness
AWT: average waiting time
NLJ: number of late jobs

31. One-machine sequencing
Dispatching (or priority) rules
FCFS: First come, first served, i.e., the job that arrives first is processed first
EDD: The job with the earliest due date is processed first
SPT: The job with the shortest processing time is processed first

32. Example 6
Five machines in a shop have failed and will be repaired by a maintenance mechanic. The estimated repair times (in days) along with the due times (in days) for the machines are given below. Assume that the machines were down in the order shown in the table. Apply each of the FCFS, EDD, and SPT rules to dispatch the jobs and then compute m, ACT, AJL, AWT, and NLJ for each of the resulting sequences
Machine A B C D E
Repair time 4 7 2 6 3
Due date 15 16 8 23 9

33. Solution Machine A B C D E Repair time 4 7 2 6 3 Due date 15 16 8 23 9
(1) FCFS: First come, first served
waiting time plus its processing time
ci = wi + pi
Sum of the processing time of preceding jobs
wi = p1 + p pi-1
Li = max {ci - di, 0}
Job
(i)
Processing time for job i (Pi)
Waiting time of job i
(Wi)
Completion time of job i (Ci)
Due time for job i (di)
Lateness of job i (Li)
total A B C D E 4 7 2 6 3 4 15 16 8 23 9 4
7 + 4 = 11
4 + 7 =11
= 13
13 – 8 = 5
=13
= 19
=19
= 22
22 – 9 = 13 22 47 69 18
m = 22 days
AWT = 47/5 = 9.4 days
ACT = 69/5 = 13.8 days
NLJ = 2 machines
AJL = 18/5 = 3.6 days

34. Processing time for job i (Pi) Completion time of job i (Ci)
Solution
Machine A B C D E
Repair time 4 7 2 6 3
Due date 15 16 8 23 9
(2) EDD: The job with the earliest due date is processed first
Order the due date from the smallest to the largest: 8 (C), 9 (E), 15 (A), 16 (B) and 23 (D)
Job
(i)
Processing time for job i (Pi)
Waiting time of job i
(Wi)
Completion time of job i (Ci)
Due time for job i (di)
Lateness of job i (Li)
total C E A B D 2 3 4 7 6
2 + 0 = 2 8 9 15 16 23 2
3 + 2 = 5
2 + 3 = 5
4 + 5 = 9
= 9
7 + 9 = 16
= 16
= 22 22 32 54
m = 22 days
AWT = 32/5 = 6.4 days
ACT = 54/5 = 10.8 days
NLJ = 0 machines
AJL = 0/5 = 0.0 days

35. Processing time for job i (Pi) Completion time of job i (Ci)
Solution
Machine A B C D E
Repair time 4 7 2 6 3
Due date 15 16 8 23 9
(3) SPT: The job with the shortest processing time is processed first
Order the repair time from the smallest to the largest: 2 (C), 3 (E), 4 (A), 6 (D) and 7 (B)
Job
(i)
Processing time for job i (Pi)
Waiting time of job i
(Wi)
Completion time of job i (Ci)
Due time for job i (di)
Lateness of job i (Li)
total C E A D B 2 3 4 6 7
2 + 0 = 2 8 9 15 23 16 2
3 + 2 = 5
2 + 3 = 5
4 + 5 = 9
= 9
6 + 9 = 15
= 15
= 22 6 6 22 31 53
m = 22 days
AWT = 31/5 = 6.2 days
ACT = 53/5 = 10.6 days
NLJ = 1 machine
AJL = 6/5 = 1.2 days

36. Solution
Summary m
ACT
AJL
AWT
NLJ
FCFS 22
13.8
3.6
9.4 2
EDD
10.8
6.4
SPT
10.6
1.2
6.2 1
Applying the SPT rule leads to the sequence with the minimum ACT and AWT
Applying the EDD rule leads to the sequence with the minimum AJL and NLJ
The FCFS rule should be used if fairness is the major concern

37. EX 3 in Class
Processing time (including setup times) and due date for six jobs waiting to be processed at a work center are given in the following table. Determine the sequence of jobs, the average flow time (ACT), average job lateness (AJL), average waiting time (AWT), and the number of late jobs (NLJ) at the work center, for each of these rules: FCFS, SPT and EDD. (assume jobs arrived in the order shown)
Job A B C D E F
Processing time (day) 2 8 4 10 5 12
Due date 7 16 17 15 18

38. Two-Machine Sequencing
Each of the jobs to be processed has to go through two machines in the same order
Applications
Athletes ("jobs") competing in the Olympic Games must be subject to a series of two drug tests ("two machines") after each event
Cars ("jobs") in a garage must be sanded ("first machine") and then painted ("second machine")

39. Two-Machine Sequencing
Objectives
Minimize the total time needed to process the set of jobs
Johnson's rule
Step 1: Select the job that has the shortest processing time among all processing times on both machines. If the time is associated with the first machine, then the job should be processed first. If the time is associated with the second machine, however, then the job should be processed last.
Step 2: Schedule the job identified in Step (1) and then eliminate it along with its processing times on both machines. If all of the jobs have been scheduled, stop; otherwise, go to Step (1) and work toward the center of the sequence

40. Example 7
Hirsch Products, Inc., produces certain computer connectors, which first require a shearing operation and then need a punch press operation. The company currently has orders for five jobs with estimated processing times (in days) given below:
Job P Q R S T
Shearing 3 4 12 7 2
Punch press 5 1 6 10
Questions
Determine the order in which the jobs should be processed to minimize the total time needed to process the set of jobs.
Based on the sequence obtained in Part 1 above, construct a time-phased bar chart (or Gantt chart) for both operations
What is the total idle time on both operations?

41. Solution Job P Q R S T Shearing 3 4 12 7 2 Punch press 5 1 6 10
Determine the order in which the jobs should be processed to minimize the total time needed to process the set of jobs.
Sequencing T P S R Q
Based on the sequence obtained in Part 1 above, construct a time-phased bar chart (or Gantt chart) for both operations 2 5 12 24 28
Shearing T P S R Q
Punch press T P S R Q 2 6 11 12 22 24 30 31
What is the total idle time on both operations?
There is no idle time on "Shearing"
Idle time on "Punch Press" = (2 - 0) + ( ) + ( ) = 5 days
Total idle time = = 5 days.

42. EX 4 In Class
A group of six jobs is to be processed through a two-machine flow shop. The first operation involves cleaning and the second involves painting. Determine a sequence that will minimize the total completion time for this group of jobs. Processing times are as follows:
Job A B C D E F
Work Center 1 6 4 8 2 12
Work Center 2 5 3 9 7 15
Questions
Determine the order in which the jobs should be processed to minimize the total time needed to process the set of jobs.
Based on the sequence obtained in Part 1 above, construct a time-phased bar chart (or Gantt chart) for both operations
What is the total idle time on both operations?