OPERATIONS SCHEDULING
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SUPPLEMENT J
OPERATIONS SCHEDULING
LEARNING GOALS
After reading this supplement, you should
be able to:
1. Define new performance measures (beyond
flow time and past due) for evaluating a
schedule.
2. Describe the decision rules (beyond FCFS
and EDD) to sequence jobs.
3. Determine schedules for single and
multiple workstations.
T
his supplement focuses on operations
scheduling, which involves assigning jobs to
workstations or employees to jobs for specified time periods. Effective scheduling helps managers achieve the full potential of their supply
chains. Chapter 14, “Operations Planning and
Scheduling,” covers the basics of scheduling—Gantt
charts, workforce scheduling, two rules (FCFS and
EDD) for sequencing work at a single workstation,
and two commonly used performance measures
(flow time and past due). Here we deepen your
understanding with additional performance measures and priority sequencing rules, a discussion of
scheduling multiple workstations, and a discussion of
scheduling a two-station flow shop.
myomlab and the Companion Website at
www.pearsonhighered.com contain many tools,
activities, and resources designed for this supplement.
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SUPPLEMENT J
OPERATIONS SCHEDULING
operations scheduling
A type of scheduling in which jobs are
assigned to workstations or employees
are assigned to jobs for specified time
periods.
Scheduling Service and Manufacturing
Processes
The scheduling techniques we discuss in this supplement cut across the various process
types found in services and manufacturing. Many service firms are characterized by a
front-office process with high customer contact, divergent work flows, customization,
and, consequently, a complex scheduling environment. Often customer demands are
difficult to predict, which puts a high premium on scheduling employees to handle the
varied needs of customers. At the other extreme in the service industry, a back-office
process has low customer involvement, uses more line work flows, and provides standardized services. Inanimate objects are processed; these processes take on the appearance of manufacturing processes.
Manufacturing processes also benefit from operations scheduling techniques. Our
discussion of the operations scheduling techniques in this supplement has application
for job, batch, and line processes in services as well as in manufacturing. Schedules for
continuous processes can be developed with linear programming (see Supplement E,
“Linear Programming”). Although the scheduling techniques in this chapter provide
some structure to the selection of good schedules, many alternatives typically need to be
evaluated. We begin by looking at the performance measures managers use to select
good schedules.
Performance Measures
We already covered two important performance measures in Chapter 14, “Operations
Planning and Scheduling.” Flow time is the time a job spends in the service or manufacturing system, and past due (tardiness) is the amount of time by which a job missed its due
date. In this regard, a job is the object receiving service or being manufactured. For example,
a job may be a customer waiting for service at a state licensing bureau or it may be a batch
of pistons waiting for a manufacturing process. These two performance measures can be
insufficient, depending on the competitive priorities of a process. Additional performance
measures follow:
makespan
The total amount of time required to
complete a group of jobs.
total inventory
The sum of scheduled receipts and onhand inventories.
쐍 Makespan. The total amount of time required to complete a group of jobs is called
makespan. Minimizing makespan supports the competitive priorities of cost (lower
inventory) and time (delivery speed).
Makespan = Time of completion of last job - Starting time of the first job
쐍 Total Inventory. This performance measure is used to measure the effectivness of
schedules for manufacturing processes. The sum of scheduled receipts and on-hand
inventories is the total inventory.
Total inventory = Scheduled receipts for all items + On-hand inventories of all items
Minimizing total inventory supports the competitive priority of cost (inventory holding costs).
쐍 Utilization. The degree to which equipment, space, or the workforce is currently being
used, measured as the ratio of the average output rate to maximum capacity.
Maximizing the utilization of a process supports the competitive priority of cost (slack
capacity).
These performance measures often are interrelated. For example, minimizing
the average flow time tends to increase utilization. Minimizing the makespan for a group
of jobs tends to increase utilization. Understanding how flow time, makespan, past due,
and utilization interact can make the selection of good schedules easier.
Sequencing Jobs
Operations schedules are short-term plans designed to implement the sales and operations plan. Often, several jobs must be processed at one or more workstations. Typically,
a variety of tasks can be performed at each workstation. If schedules are not carefully
planned to avoid bottlenecks, waiting lines may develop. For example, Figure J.1 depicts
OPERATIONS SCHEDULING
the complexity of scheduling a manufacturing process. When a job order is received for
a part, the raw materials are collected and the batch is moved to its first operation. The
colored arrows show that jobs follow different routes through the manufacturing
process, depending on the product being made. At each workstation, the next job to
process is a decision because the arrival rate of jobs at a workstation often differs from
the processing rate of the jobs at a workstation, thereby creating a waiting line. In addition, new jobs can enter the process at any time, thereby creating a dynamic environment. Such complexity puts pressure on managers to develop scheduling procedures
that will handle the workload efficiently.
In this section, we focus on scheduling approaches used in two environments: (1) divergent flow processes and (2) line flow processes. A manufacturer's operation with divergent
flows is often called a job shop, which specializes in low- to medium-volume production
and utilizes job or batch processes. The front office would be the equivalent for a service
provider. Jobs in divergent flow processes are difficult to schedule because of the variability
in job routings and the continual introduction of new jobs to be processed. Figure J.1 depicts
a manufacturer’s job shop. A manufacturer's operation with line flows is often called a flow
shop, which specializes in medium- to high-volume production and utilizes line or
continuous flow processes. The back office would be the equivalent for a service provider.
Tasks are easier to schedule because the jobs have a common flow pattern through the system. Nonetheless, scheduling mistakes can be costly in either situation.
Job Shop Sequencing
SUPPLEMENT J
job shop
A manufacturer's operation that specializes
in low- to medium-volume production and
utilizes job or batch processes.
flow shop
A manufacturer's operation that
specializes in medium- to high-volume
production and utilizes line or continuous
flow processes.
Just as many schedules are feasible for a specific group of jobs at a particular set of workstations,
numerous methods can be used to generate schedules. They range from straightforward
manual methods, such as manipulating Gantt charts, to sophisticated computer models for
developing optimal schedules. One way to generate schedules in job shops is by using
priority sequencing rules, which allows the schedule for a workstation to evolve over a period
of time. The decision about which job to process next is made with simple priority rules
whenever the workstation becomes available for further processing. One advantage of this
method is that last-minute information on operating conditions can be incorporated into
the schedule as it evolves.
We already covered two important sequencing rules in Chapter 14, “Operations Planning
and Scheduling.” The first-come, first-served (FCFS) rule gives the job arriving at the workstation
first the highest priority. The earliest due date (EDD) rule gives the job with the earliest due date
based on assigned due dates the highest priority. Such rules can be applied by a worker or
왗 FIGURE J.1
Raw materials
Shipping department
Diagram of a Manufacturing Job
Shop Process
Legend:
Batch of parts
Workstation
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SUPPLEMENT J
OPERATIONS SCHEDULING
incorporated into a computerized scheduling system that generates a dispatch list of jobs and
priorities for each workstation. Additional priority sequencing rules follow:
critical ratio (CR)
A ratio that is calculated by dividing the
time remaining until a job’s due date by the
total shop time remaining for the job, which
is defined as the setup, processing, move,
and expected waiting times of all remaining
operations, including the operation being
scheduled.
shortest processing time (SPT)
A priority sequencing rule that specifies that
the job requiring the shortest processing
time is the next job to be processed.
slack per remaining
operations (S/RO)
A priority sequencing rule that
determines priority by dividing the slack
by the number of operations that remain,
including the one being scheduled.
쐍 Critical Ratio. The critical ratio (CR) is calculated by dividing the time remaining until
a job’s due date by the total shop time remaining for the job, which is defined as the
setup, processing, move, and expected waiting times of all remaining operations,
including the operation being scheduled. The formula is
CR =
Due date - Today’s date
Total shop time remaining
The difference between the due date and today’s date must be in the same time units as
the total shop time remaining. A ratio less than 1.0 implies that the job is behind schedule, and a ratio greater than 1.0 implies that the job is ahead of schedule. The job with
the lowest CR is scheduled next.
쐍 Shortest Processing Time. The job requiring the shortest processing time (SPT) at the
workstation is processed next.
쐍 Slack per Remaining Operations. Slack is the difference between the time remaining
until a job’s due date and the total shop time remaining, including that of the operation
being scheduled. A job’s priority is determined by dividing the slack by the number of
operations that remain, including the one being scheduled, to arrive at the slack per
remaining operations (S/RO).
(Due date - Today’s date) - Total shop time remaining
S/RO =
Number of operations remaining
The job with the lowest S/RO is scheduled next. Ties are broken in a variety of ways if
two or more jobs have the same priority. One way is to arbitrarily choose one of the tied
jobs for processing next.
Although the priority sequencing rules seem simple, the actual task of scheduling hundreds of jobs through hundreds of workstations requires intensive data gathering and manipulation. The scheduler needs information on each job’s processing requirements: the job’s
due date; its routing; the standard setup, processing, and expected waiting times at each
operation; whether alternative workstations could be used at each operation; and the inputs
from internal or external suppliers at each operation. In addition, the scheduler needs to
know the job’s current status: its location (waiting in line for a workstation or being processed
at a workstation), how much of the operation has been completed, the actual arrival and
departure times at each operation or waiting line, and the actual processing and setup times.
The scheduler or software uses the priority sequencing rules to determine the processing
sequence of jobs at a workstation and the remaining information for estimating job arrival
times at the next workstation, as well as determining whether an alternative workstation
should be used when the primary one is busy. Because this information may change throughout the day, computers are needed to track the data and to maintain valid priorities.
Sequencing Jobs for One Workstation
Any priority sequencing rule can be used to schedule any number of workstations. For
the purpose of illustrating the rules, however, we focus on scheduling several jobs at a
single workstation. We divide the rules into two categories: (1) single-dimension rules
and (2) multiple-dimension rules.
single-dimension rules
Single-Dimension Rules Some priority sequencing rules (e.g., FCFS, EDD, and SPT)
A set of rules that bases the priority of a
job on a single aspect of the job, such as
arrival time at the workstation, the due
date, or the processing time.
base a job’s priority assignment only on information about the jobs waiting for processing at
the individual workstation. We call these rules single-dimension rules because they determine priority based on a single aspect of the job, such as arrival time at the workstation, the
due date, or the processing time. We begin with an example of single-dimension rules.
EXAMPLE J.1
Comparing the EDD and SPT Rules
Tutor J.1 in myomlab provides a new
example to practice EDD and SPT rules.
The Taylor Machine Shop rebores engine blocks. Currently, five engine blocks are waiting for processing. At any
time, the company has only one engine expert on duty who can do this type of work. The engine problems have
been diagnosed, and the processing times for the jobs have been estimated. Expected completion times have
been agreed upon with the shop’s customers. The accompanying table shows the current situation. Because the
OPERATIONS SCHEDULING
Taylor Machine Shop is open from 8:00 A.M. until 5:00 P.M. each weekday, plus weekend hours as needed, the
customer pickup times are measured in business hours from the current time. Determine the schedule for the
engine expert by using (a) the EDD rule and (b) the SPT rule. For each rule, calculate the average flow time, average hours early, and average hours past due. If average past due is most important, which rule should be chosen?
Active Model J.1 in myomlab provides
additional insight on the use of singledimension rules.
Business Hours
Until Due Date
(customer pickup time)
Business Hours
Since Order Arrived
Processing Time,
Including Setup (hours)
Ranger
12
8
10
Explorer
10
6
12
Bronco
1
15
20
Econoline 150
3
3
18
Thunderbird
0
12
22
Engine Block
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SUPPLEMENT J
SOLUTION
a.
The EDD rule states that the first engine block in the sequence is the one with the closest due date.
Consequently, the Ranger engine block is processed first. The Thunderbird engine block, with its due
date furthest in the future, is processed last. The sequence is shown in the following table, along with
the flow times, the hours early, and the hours past due.
Hours
Engine Block Since Order
Arrived
Sequence
Ranger
12
Explorer
Begin
Work
Scheduled
Actual
Customer
Finish
Flow Time Customer
Pickup Time Pickup Time
Time (hr)
(hr)
Processing
Time (hr)
Hours
Past
Due
—
0
+
8
=
8
20
10
10
2
10
8
+
6
=
14
24
12
14
—
2
Econoline 150
3
14
+
3
=
17
20
18
18
1
—
Bronco
1
17
+
15
=
32
33
20
32
—
12
Thunderbird
0
32
+
12
=
44
44
22
44
—
22
The flow time for each job is its finish time, plus the time since the job arrived.1 For example, the Explorer
engine block’s finish time will be 14 hours from now (8 hours waiting time before the engine expert started
to work on it plus 6 hours processing). Adding the 10 hours since the order arrived at this workstation
(before the processing of this group of orders began) results in a flow time of 24 hours. You might think of
the sum of flow times as the total job hours spent by the engine blocks since their orders arrived at the
workstation until they were processed.
The performance measures for the EDD schedule for the five engine blocks are
20 + 24 + 20 + 33 + 44
= 28.2 hrs
Average flow time =
5
2 + 0 + 1 + 0 + 0
= 0.6 hrs
Average hours early =
5
0 + 2 + 0 + 12 + 22
Average hours past due =
= 7.2 hrs
5
Flow time, as a performance measure in its traditional use, does not count the time a job spends “outside the system under our control.” Our “system” in this supplement is the single workstation (or two
workstations in the case of Johnson’s rule in the next section). Arrival time here relates to when the job
was first available for processing at the workstation. Adding the time since the order arrived at the workstation to the job’s finish time departs from conventions used in early research on static problems,
which assumed that no jobs arrive during the time span covered by the resulting schedule. With traditional assumptions, a job’s finish time and flow time are identical and SPT will always have the best flow
time performance. With our definition of flow time, the SPT rules do not necessarily produce the best
flow time performance, such as when the job with the shortest processing time arrived at the workstation
well before the other jobs.
1
Hours
Early
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SUPPLEMENT J
OPERATIONS SCHEDULING
b.
Hours
Engine Block Since Order
Arrived
Sequence
Econoline 150
Begin
Work
Under the SPT rule, the sequence starts with the engine block that has the shortest processing time,
the Econoline 150, and it ends with the engine block that has the longest processing time, the Bronco.
The sequence, along with the flow times, early hours, and past due hours, is contained in the following table:
Scheduled
Actual
Customer
Finish
Flow Time Customer
Pickup Time Pickup Time
Time (hr)
(hr)
Processing
Time (hr)
Hours
Early
Hours
Past
Due
3
0
+
3
=
3
6
18
18
15
—
Explorer
10
3
+
6
=
9
19
12
12
3
—
Ranger
12
9
+
8
=
17
29
10
17
—
7
Thunderbird
0
17
+
12
=
29
29
22
29
—
7
Bronco
1
29
+
15
=
44
45
20
44
—
24
The performance measures are
6 + 19 + 29 + 29 + 45
= 25.6 hrs
5
15 + 3 + 0 + 0 + 0
= 3.6 hrs
Average hours early =
5
0 + 0 + 7 + 7 + 24
Average hours past due =
= 7.6 hrs
5
Average flow time =
DECISION POINT
The EDD rule is better than the SPT rule with respect to average past due (keeping promises to customers),
but worse with respect to average flow time for the set of jobs in this example. Management’s choice
depends on which performance measure it values the most. More experimentation should be conducted
before a final choice is made.
As the solution of Example J.1 shows, the EDD schedule gave better customer service, as
measured by the average hours past due, and a lower maximum hours past due (22 versus 24).
However, the SPT schedule provided a lower average flow time. In general, the SPT priority rule
will push most jobs through the system to completion more quickly than will the other rules.
Speed can be an advantage—but only if jobs can be delivered sooner than promised and revenue
collected earlier. If they cannot, the completed job must stay in finished inventory. Consequently,
the priority rule chosen can help or hinder the firm in meeting its competitive priorities.
Researchers have studied the implications of the single-dimension rules for various
performance measures. In most of these studies, all jobs were considered to be independent
(in contrast to the parent-component dependencies in MRP environments), and the
assumption was made that sufficient capacity generally was available. These studies found
that the EDD rule performs well with respect to the percentage of jobs past due and the variance of hours past due. For any set of jobs to be processed at a single workstation, it minimizes the maximum of the past due hours of any job in the set. The EDD rule is popular with
firms that are sensitive to achieving due dates, which usually are the basis for setting priorities using MRP systems.
Often referred to as the world champion, the SPT rule tends to minimize the mean flow
time (assuming time since arrival is 0 for all jobs) and the percentage of jobs past due. It also
tends to maximize shop utilization. For the single-workstation case, the SPT rule always will
provide the lowest mean finish time. However, it could increase total inventory because it
tends to push all work to the finished state. In addition, it tends to produce a large variance
in past due hours because the larger jobs might have to wait a long time for processing. Also,
it provides no opportunity to adjust schedules when due dates change. The advantage of this
rule over others diminishes as the load on the shop increases.
Finally, though the FCFS rule is considered fair to the jobs (or customers), it performs
poorly with respect to all performance measures. This result is to be expected because FCFS
does not acknowledge any job (or customer) characteristics. However, FCFS usually is the
only acceptable choice for service processes where the customer is present and demand leveling options such as appointments or reservations are not used.
OPERATIONS SCHEDULING
SUPPLEMENT J
J-7
Multiple-Dimension Rules Priority rules, such as CR and S/RO, incorporate information
about the remaining workstations at which the job must be processed, in addition to the
processing time at the present workstation or the due date considered by single-dimension
rules. We call these rules multiple-dimension rules because they apply to more than one
aspect of the job. Example J.2 demonstrates their use for sequencing jobs.
The first five columns of the following table contain information about a set of four jobs that just arrived (end of
hour 0 or beginning of hour 1) at an engine lathe. They are the only ones now waiting to be processed. Several
operations, including the one at the engine lathe, remain to be done on each job. Determine the schedule by using
(a) the CR rule and (b) the S/RO rule. Compare these schedules to those generated by FCFS, SPT, and EDD.
Processing
Time at Engine
Lathe (hours)
Time
Remaining
Until Due
Date (days)
Number of
Operations
Remaining
Shop Time
Remaining
(days)
CR
S/RO
1
2.3
15
10
6.1
2.46
0.89
2
10.5
10
2
7.8
1.28
1.10
3
6.2
20
12
14.5
1.38
0.46
4
15.6
8
5
10.2
0.78
- 0.44
SOLUTION
a.
Using CR to schedule the machine, we divide the time remaining until the due date by the shop time
remaining to get the priority index for each job. For job 1,
CR =
Time remaining until the due date
15
=
= 2.46
Shop time remaining
6.1
By arranging the jobs in sequence with the lowest critical ratio first, we determine that the sequence
of jobs to be processed by the engine lathe is 4, 2, 3, and finally 1, assuming that no other jobs arrive
in the meantime.
b.
Using S/RO, we divide the difference between the time remaining until the due date and the shop time
remaining by the number of remaining operations. For job 1,
S/RO =
A set of rules that apply to more than one
aspect of a job.
Sequencing with the CR and S/RO Rules
EXAMPLE J.2
Job
multiple-dimension rules
Time remaining until the due date - Shop time remaining
15 - 6.1
=
= 0.89
Number of operations remaining
10
Arranging the jobs by starting with the lowest S/RO yields a 4, 3, 1, 2 sequence of jobs.
DECISION POINT
Note that the application of the two priority rules gives two different schedules. Moreover, the SPT sequence,
based on processing times (measured in hours) at the engine lathe only, is 1, 3, 2, and 4. No preference is given
to job 4 in the SPT schedule, even though it may not be finished by its due date. The EDD sequence is 4, 2, 1,
and 3. For illustration purposes, we assume that the FCFS sequence is 1, 2, 3, and 4. All four jobs arrived at the
workstation at the end of hour 0, so the finish times and flow times are identical for all five rules. The following
table shows the comparative performance of the five priority sequencing rules at the engine lathe:
Priority Rule Summary
FCFS
SPT
EDD
CR
S/RO
Average flow time
17.175
16.100
26.175
27.150
24.025
Average early time
3.425
6.050
Average past due
7.350
8.900
0
12.925
0
13.900
0
10.775
Tutor J.2 in myomlab provides a
new example to practice the CR and
S/RO rules.
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SUPPLEMENT J
OPERATIONS SCHEDULING
The S/RO rule is better than the EDD rule and the CR rule, but it is much worse than the SPT rule and the
FCFS rule for this example. However, EDD, CR, and S/RO all have the advantage of allowing schedule
changes when due dates change. These results cannot be generalized to other situations because only four
jobs are being processed.
Research studies have shown that S/RO is better than EDD with respect to the percentage of jobs past due but worse than SPT and EDD with respect to average flow times.
These studies also indicate that CR results in longer flow times than SPT, but CR also
results in less variance in the distribution of past due hours. Consequently, even though
the use of the multiple-dimension rules requires more information, no choice is clearly
best. Each rule should be tested in the environment for which it is intended.
Scheduling Jobs for Multiple Workstations
Priority sequencing rules can be used to schedule more than one operation. Each operation
is treated independently. When a workstation becomes idle, the priority rule is applied to
the jobs waiting for that operation, and the job with the highest priority is selected. When
that operation is finished, the job is moved to the next operation in its routing, where it
waits until it again has the highest priority. At any workstation, the jobs in the waiting line
change over a period of time, so the choice of a priority rule can make quite a difference in
the processing sequence. Schedules can be evaluated with the performance measures
already discussed.
Identifying the best priority rule to use at a particular operation in a process is a complex problem because the output from one operation becomes the input to another. The priority rule at a workstation determines the sequence of work the workstation will perform,
which in turn determines the arrival of work at the next workstation downstream. Computer
simulation models are effective tools to determine which priority rules work best in a given
situation. Once the current process is modeled, the analyst can make changes to the priority rules at various operations and measure the impact on performance measures, such as
past due, flow time, and utilization.
Scheduling Jobs for a Two-Station Flow Shop
Suppose that a flow shop has several jobs ready for processing at two workstations
and that the routings of all jobs are identical. In the scheduling of two or more workstations
in a flow shop, the makespan varies according to the sequence chosen. Determining a production sequence for a group of jobs to minimize the makespan has two
advantages:
1. The group of jobs is completed in minimum time.
2. The utilization of the two-station flow shop is maximized. Utilizing the first workstation
continuously until it processes the last job minimizes the idle time on the second
workstation.
Johnson’s rule
A procedure that minimizes makespan
when scheduling a group of jobs on two
workstations.
Johnson’s rule is a procedure that minimizes makespan when scheduling a group of
jobs on two workstations. S. M. Johnson showed that the sequence of jobs at the two stations
should be identical and that the priority assigned to a job should, therefore, be the same at
both. The procedure is based on the assumption of a known set of jobs, each with a known
processing time and available to begin processing on the first workstation. The procedure is
as follows.
Step 1. Scan the processing times at each workstation and find the shortest processing
time among the jobs not yet scheduled. If two or more jobs are tied, choose one job
arbitrarily.
Step 2. If the shortest processing time is on workstation 1, schedule the corresponding job
as early as possible. If the shortest processing time is on workstation 2, schedule the corresponding job as late as possible.
Step 3. Eliminate the last job scheduled from further consideration. Repeat steps 1 and
2 until all jobs have been scheduled.
OPERATIONS SCHEDULING
SUPPLEMENT J
J-9
Scheduling a Group of Jobs on Two Workstations
EXAMPLE J.3
The Morris Machine Company just received an order to refurbish five motors for materials handling equipment
that were damaged in a fire. The motors have been delivered and are available for processing. The motors will be
repaired at two workstations in the following manner.
Workstation 1: Dismantle the motor and clean the parts.
Workstation 2: Replace the parts as necessary, test the motor, and make adjustments.
Tutor J.3 in myomlab provides a new
example to practice Johnson’s rule.
The customer’s shop will be inoperable until all the motors have been repaired, so the plant manager is
interested in developing a schedule that minimizes the makespan and has authorized around-the-clock operations
until the motors have been repaired. The estimated time to repair each motor is shown in the following table:
Time (hr)
Motor
Workstation 1
Workstation 2
M1
12
22
M2
4
5
M3
5
3
M4
15
16
M5
10
8
SOLUTION
The logic for the optimal sequence is shown in the following table:
Establishing a Job Sequence
Iteration
Job Sequence
1
Comments
M3 The shortest processing time is 3 hours for M3 at workstation 2. Therefore, M3 is
scheduled as late as possible.
2
M2
M3 Eliminate M3 from the table of estimated times. The next shortest processing time
is 4 hours for M2 at workstation 1. M2 is therefore scheduled first.
3
M2
4
M2
5
M2 M1 M4 M5 M3 The last motor to be scheduled is M4. It is placed in the last remaining position, in
the middle of the schedule.
M5 M3 Eliminate M2 from the table. The next shortest processing time is 8 hours for M5
at workstation 2. Therefore, M5 is scheduled as late as possible.
M1 M5 M3 Eliminate M5 from the table. The next shortest processing time is 12 hours for M1
at workstation 1. M1 is scheduled as early as possible.
DECISION POINT
No other sequence of jobs will produce a shorter makespan. To determine the makespan, we can draw a Gantt
chart, as shown in Figure J.2. In this case, refurbishing and reinstalling all five motors will take 65 hours. This
schedule minimizes the idle time of workstation 2 and gives the fastest repair time for all five motors. Note that
the schedule recognizes that a job cannot begin at workstation 2 until it has been completed at workstation 1.
왔 FIGURE J.2
Gantt Chart for the Morris Machine
Company Repair Schedule
Workstation
1
M2
(4)
2
Idle
0
M1
(12)
M2
(5)
5
M4
(15)
M3
(5)
M1
(22)
Idle
10
M5
(10)
15
20
25
Idle—available
for further work
M4
(16)
30
Hour
35
40
45
M5
(8)
50
55
M3
(3)
60
65
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OPERATIONS SCHEDULING
Labor-Limited Environments
labor-limited environment
An environment in which the resource
constraint is the amount of labor
available, not the number of machines or
workstations.
Thus far, we have assumed that a job never has to wait for lack of a worker. The limiting
resource has been the number of machines or workstations available. More typical, however,
is a labor-limited environment in which the resource constraint is the amount of labor
available, not the number of machines or workstations. In this case, workers are trained to
work on a variety of machines or tasks to increase the flexibility of operations.
In a labor-limited environment, the scheduler not only must decide which job to
process next at a particular workstation but also must assign workers to their next workstations. The scheduler can use priority rules to make these decisions, as we used them to
schedule engine blocks in Example J.1. In labor-limited environments, the labor-assignment
policies, as well as the priority sequencing rules, affect performance. The following examples
provide some labor-assignment rules.
쐍 Assign personnel to the workstation with the job that has been in the system longest.
쐍 Assign personnel to the workstation with the most jobs waiting for processing.
쐍 Assign personnel to the workstation with the largest standard work content.
쐍 Assign personnel to the workstation with the job that has the earliest due date.
The manufacturing scheduling process is a key element of an integrated supply chain.
Advanced planning and scheduling (APS) systems attempt to link the scheduling process to
demand data and forecasts, supply chain facility and inventory decisions, and the capability
of suppliers so that the entire chain can operate as efficiently as possible. A firm’s ability to
change its schedules quickly and still keep the supply chain flowing smoothly provides a
competitive edge.
Internet Resources
myomlab and the Companion Website at www.pearsonhighered.com contain many tools, activities, and resources designed for
this supplement.
Key Equations
1. Performance measures:
Flow time = Finish time + Time since the job arrived at the workstation
Past due = Time by which a job missed its due date
Makespan = Time of completion of last job - Starting time of the first job
Total inventory = Scheduled receipts for all items + On-hand inventories of all items
2. Critical ratio:
CR =
Due date - Today’s date
Total shop time remaining
3. Slack per remaining operations:
S/RO =
(Due date - Today’s date) - Total shop time remaining
Number of operations remaining
OPERATIONS SCHEDULING
Solved Problem 1
The Neptune’s Den Machine Shop specializes in overhauling outboard marine engines.
Some engines require replacement of broken parts, whereas others need a complete overhaul. Currently, five engines with varying problems are awaiting service. The best estimates
for the labor times involved and the promise dates (in number of days from today) are shown
in the following table. Customers usually do not pick up their engines early.
Time Since Order
Arrived (days)
Processing Time,
Including Setup (days)
Promise Date (days
from now)
50-hp Evinrude
4
5
8
7-hp Johnson
6
4
15
100-hp Mercury
8
10
12
1
1
20
15
3
10
Engine
50-hp Honda
75-hp Nautique
a. Develop separate schedules by using the SPT and EDD rules.
b. Compare the two schedules on the basis of average flow time, percentage of past due
jobs, and maximum past due days for any engine.
SOLUTION
a. Using the SPT rule, we obtain the following schedule:
Days Since
Order
Processing
Arrived
Time
Repair Sequence
50-hp Honda
Finish
Time
Flow
Time
Actual
Promise Pickup
Date
Date
Days
Early
Days
Past Due
1
1
1
2
20
20
19
—
15
3
4
19
10
10
6
—
7-hp Johnson
6
4
8
14
15
15
7
—
50-hp Evinrude
4
5
13
17
8
13
—
5
100-hp Mercury
8
10
23
31
12
23
—
11
Days
Early
Days
Past Due
3
—
75-hp Nautique
Total
83
Using the EDD we obtain this schedule:
Days Since
Order
Processing
Arrived
Time
Repair Sequence
50-hp Evinrude
4
75-hp Nautique
15
100-hp Mercury
8
7-hp Johnson
6
50-hp Honda
1
Finish
Time
5
Flow
Time
Actual
Promise Pickup
Date
Date
5
9
8
8
3
8
23
10
10
2
—
10
18
26
12
18
—
6
4
22
28
15
22
—
7
1
23
24
20
23
—
3
Total
110
b. Performance measures are as follows:
Average flow time is 16.6 (or 83/5) days for SPT and 22.0 (or 110/5) days for EDD. The
percentage of past due jobs is 40 percent (2/5) for SPT and 60 percent (3/5) for EDD.
For this set of jobs, the EDD schedule minimizes the maximum days past due but has
a greater flow time and causes more jobs to be past due.
SUPPLEMENT J
J-11
J-12
SUPPLEMENT J
OPERATIONS SCHEDULING
Solved Problem 2
The following data were reported by the shop floor control system for order processing at the
edge grinder. The current date is day 150. The number of remaining operations and the total
work remaining include the operation at the edge grinder. All orders are available for processing, and none have been started yet. Assume the jobs were available for processing at
the same time.
Processing
Time (hr)
Due Date
(day)
Remaining
Operations
Shop Time
Remaining (days)
A101
10
162
10
9
B272
7
158
9
6
Current Order
C105
15
152
1
1
D707
4
170
8
18
E555
8
154
5
8
a. Specify the priorities for each job if the shop floor control system uses slack per
remaining operations (S/RO) or critical ratio (CR).
b. For each priority rule, calculate the average flow time per job at the edge grinder.
SOLUTION
a. We specify the priorities for each job using the two sequencing rules.
(Due date - Today’s date) - Shop time remaining
Number of operations remaining
(154 - 150) - 8
E555:S/RO =
= - 0.80 [1]
5
(158 - 150) - 6
B272:S/RO =
= 0.22 [2]
9
(170 - 150) - 18
D707:S/RO =
= 0.25 [3]
8
(162 - 150) - 9
A101:S/RO =
= 0.30 [4]
10
(152 - 150) - 1
C105:S/RO =
= 1.00 [5]
1
S/RO =
The sequence of production for S/RO is shown in the preceding brackets.
CR =
E555:CR =
D707:CR =
B272:CR =
A101:CR =
C105:CR =
Due date - Today’s date
Shop time remaining
154 - 150
8
170 - 150
18
158 - 150
6
162 - 150
9
152 - 150
1
= 0.50 [1]
= 1.11 [2]
= 1.33 [3]
= 1.33 [4]
= 2.00 [5]
The sequence of production for CR is shown in the preceding brackets.
b. We are sequencing a set of jobs at a single machine, so each job’s finish time equals the
finish time of the job just prior to it in sequence plus its own processing time. Further,
OPERATIONS SCHEDULING
all jobs were available for processing at the same time, so each job’s finish time equals
its flow time. Consequently, the average flow times at this single machine are
8 + 15 + 19 + 29 + 44
= 23.30 hours
5
8 + 12 + 19 + 29 + 44
CR:
= 22.4 hours
5
In this example, the average flow time per job is lower for the CR rule, which is not
always the case. For example, the critical ratios for B272 and A101 are tied at 1.33. If we
arbitrarily assigned A101 before B272, the average flow time would increase to
(8 + 12 + 22 + 29 + 44)/5 = 23.0 hours.
S/RO:
Solved Problem 3
The Rocky Mountain Arsenal, formerly a chemical warfare manufacturing site, is said to be
one of the most polluted locations in the United States. Cleanup of chemical waste storage
basins will involve two operations.
Operation 1: Drain and dredge basin.
Operation 2: Incinerate materials.
Management estimates that each operation will require the following amounts of time (in days):
Storage Basin
A
B
C
D
E
F
G
H
I
J
Dredge
3
4
3
6
1
3
2
1
8
4
Incinerate
1
4
2
1
2
6
4
1
2
8
Management’s objective is to minimize the makespan of the cleanup operations. All
storage basins are available for processing right now. First, find a schedule that minimizes
the makespan. Then calculate the average flow time of a storage basin through the two operations. What is the total elapsed time for cleaning all 10 basins? Display the schedule in a
Gantt machine chart.
SOLUTION
We can use Johnson’s rule to find the schedule that minimizes the total makespan. Four
jobs are tied for the shortest process time: A, D, E, and H. E and H are tied for first place,
while A and D are tied for last place. We arbitrarily choose to start with basin E, the first
on the list for the drain and dredge operation. The 10 steps used to arrive at a sequence
are as follows:
1. Select basin E first (tied with basin H);
put it at the front.
E
—
—
—
—
—
—
—
—
—
2. Select basin H next; put it toward
the front.
E
H
—
—
—
—
—
—
—
—
3. Select basin A next (tied with basin D);
put it at the end.
E
H
—
—
—
—
—
—
—
A
4. Put basin D toward the end.
E
H
—
—
—
—
—
—
D
A
5. Put basin G toward the front.
E
H
G
—
—
—
—
—
D
A
6. Put basin C toward the end.
E
H
G
—
—
—
—
C
D
A
7. Put basin I toward the end.
E
H
G
—
—
—
I
C
D
A
8. Put basin F toward the front.
E
H
G
F
—
—
I
C
D
A
9. Put basin B toward the front.
E
H
G
F
B
—
I
C
D
A
E
H
G
F
B
J
I
C
D
A
10. Put basin J in the remaining space.
SUPPLEMENT J
J-13
J-14
SUPPLEMENT J
OPERATIONS SCHEDULING
Several optimal solutions are available to this problem because of the ties at the start of
the scheduling procedure. However, all have the same makespan. The schedule would be as
follows:
Operation 1
Basin
Start
Operation 2
Finish
Start
Finish
E
0
1
1
3
H
1
2
3
4
G
2
4
4
8
F
4
7
8
14
B
7
11
14
18
J
11
15
18
26
I
15
23
26
28
C
23
26
28
30
D
26
32
32
33
A
32
35
35
36
Total 200
The makespan is 36 days. The average flow time is the sum of incineration finish times
divided by 10, or 200/10 = 20 days. The Gantt machine chart for this schedule is given in
Figure J.3.
Storage basin
Dredge
E H
Incinerate
G
E
F
H
B
G
J
I
F
C
B
D
J
I
C
A
D
A
FIGURE J.3 왖
Discussion Question
1. Suppose that two alternative approaches for determining
workstation schedules are available. One is an optimizing
approach that can be run once a week on the computer.
The other approach utilizes priority sequencing rules to
determine the schedule as it evolves. Discuss the advantages and disadvantages of each approach and the conditions under which each approach is likely to be better.
Problems
Software, such as OM Explorer, Active Models, and POM for
Windows, is available in myomlab. Check with your instructor
on how best to use it. In many cases, the instructor wants you to
understand how to do the calculations by hand. At most, the
software provides a check on your calculations. When calculations are particularly complex and the goal is interpreting the
results in making decisions, the software replaces entirely the
manual calculations.
1. The Hickory Company manufactures wooden desks.
Management schedules overtime every weekend to
reduce the backlog on the most popular models. The
automatic routing machine is used to cut certain types of
edges on the desktops. The following orders need to be
scheduled for the routing machine:
Order
Time Since Order
Estimated
Arrived (hr)
ProcessingTime (hr)
10
Due Date
(hr from now)
1
12
12
2
10
3
8
3
7
15
18
4
3
9
20
5
1
7
21
OPERATIONS SCHEDULING
The due dates reflect the need for the order to be at its next
operation.
a. Develop separate schedules by using the FCFS, SPT,
and EDD rules.
J-15
SUPPLEMENT J
scheduling procedures that would reduce inventory and
increase customer service in the shop. Assume that at
9:00 A.M. on Monday the NC welding machine was idle.
Also assume that job “arrival times” are the “release
times” to the workstation.
b. Compare the schedules on the basis of average flow
time, the average early time, and average past due
hours for any order.
a. Develop schedules for SPT and EDD priority rules,
and draw a Gantt machine chart for each schedule.
c. Comment on the performance of the two rules relative
to these measures.
b. For each schedule in part (a), calculate the average
flow time per job and the average past due hours per job.
2. The drill press is a bottleneck operation. Currently, five
jobs are waiting to be processed. Following are the available operations data. Assume that the number of remaining operations and the shop time remaining include the
processing at the drill press.
Job
Time Since
Time to
Order
Processing Due Date
Arrived (hr) Time (hr)
(wk)
AA
24
4
10
Shop Time
Operations Remaining
(wk)
Remaining
3
4
BB
16
8
16
4
6
CC
14
13
21
10
9
DD
12
6
23
3
12
EE
10
2
12
5
3
a. Specify the priority for each job if the shop floor control system uses each of the following priority rules:
SPT, S/RO, EDD, and CR.
b. For each priority rule, calculate the average flow time
per job at the drill press.
c. Which of these priority rules would work best for priority planning with an MRP system? Why?
3. The machine shop at Bycraft Enterprises operates 24 hours
a day and uses a numerically controlled (NC) welding
machine. The load on the machine is monitored, and no
more than 24 hours of work is released to the welding
operators in one day. The data for a typical set of jobs are
shown in Table J.1. Management has been investigating
TABLE J.1
MANUFACTURING DATA
Processing
Time
(hr/unit)
Setup
Time
(hr)
Release
Time
Lot
Size
1
9:00 A.M.
Monday
50
0.06
4
9:00 P.M.
Monday
2
10:00 A.M.
Monday
120
0.05
3
10:00 P.M.
Monday
3
11:00 A.M.
Monday
260
0.03
5
11:00 P.M.
Monday
4
12:00 P.M.
Monday
200
0.04
2
2:00 A.M.
Tuesday
Job
Due
Date
4. Refer to the Gantt machine chart in Figure J.4.
a. Suppose that a routing requirement is that each job
must be processed on machine A first. Can the
makespan be improved? If so, draw a Gantt chart with
the improved schedule. If not, state why.
b. Suppose that the machine sequence has no routing
restriction; in other words, jobs can be processed in
any sequence on the machines. Can the makespan in
the chart be improved in this case? If so, draw a Gantt
chart with your schedule. If not, state why.
Machine
A
Job
1
Job
2
B
Idle
Job
1
0
1
2
Job
3
Idle
Job
2
3
4
Job
3
5
6
7
8
9
왖 FIGURE J.4
5. A manufacturer of sails for small boats has a group of
custom sails awaiting the last two processing operations
before the sails are sent to the customers. Operation 1
must be performed before operation 2, and the jobs have
different time requirements for each operation. The
hours required are as follows:
Job
1
2
3
4
5
6
7
8
9
10
Operation 1
1
5
8
3
9
4
7
2
4
9
Operation 2
8
3
1
2
8
6
7
2
4
1
a. Use Johnson’s rule to determine the optimal sequence.
b. Draw a Gantt chart for each operation.
6. McGee Parts Company is under tremendous pressure to
complete a government contract for six orders in 31 working days. The orders are for spare parts for highway maintenance equipment. According to the government contract, a late penalty of $1,000 is imposed each day the
order is late. Owing to a nationwide increase in highway
construction, McGee Parts has received many orders for
J-16
SUPPLEMENT J
OPERATIONS SCHEDULING
spare parts replacement and the shop has been
extremely busy. To complete the government contract,
the parts must be deburred and heat treated. The production control manager has suggested the following
schedule:
Debur
his inventory low and is adamant about processing the
jobs through his department on the basis of shortest processing time. Pat Mooney, supervisor for department 22,
pointed out that if Mangano were more flexible the orders
could be finished and shipped earlier. The processing
times (in days) for each job in each department follow:
Heat Treat
Job
Job
Start
Finish
Start
Finish
1
0
2
2
8
2
2
5
8
13
3
5
12
13
17
4
12
15
17
25
5
15
16
25
30
6
16
24
30
32
a. Use Johnson’s rule to determine the optimal sequence.
b. Draw a Gantt chart for each operation.
7. Carolyn Roberts is the operations manager of the
machine shop of Reliable Manufacturing. She has to
schedule eight jobs that are to be sent to final assembly
for an important customer order. Currently, all eight jobs
are in department 12 and must be routed to department 22
next. All jobs arrived at the same time. Jason Mangano,
supervisor for department 12, is concerned about keeping
1
2
3
4
5
6
7
8
Department 12
2
4
7
5
4
10
8
2
Department 22
3
6
3
8
2
6
6
5
a. Determine a schedule for the operation in each
department. Use SPT for department 12 and the same
sequence for department 22. What is the average flow
time for department 12? What is the makespan
through both departments? What is the total number
of job-days spent in the system?
b. Find a schedule that will minimize the makespan
through both departments, and then calculate the
average flow time for department 12. What is the total
number of job-days spent in the system?
c. Discuss the trade-offs represented by these two schedules. What implications do they have for centralized
scheduling?
ADVANCED PROBLEMS
8. The repair manager at Standard Components needs to
develop a schedule for repairing eight Dell PCs. Each job
requires analysis using the same diagnostic system.
Furthermore, each job will require additional processing
after the diagnostic evaluation. The manager does not
expect any rescheduling delays, and the jobs are to move
directly to the next process after the diagnostic work has
been completed. The manager has collected the following
processing time and scheduling data for each repair job:
Time Since Processing Due Date Shop Time
Order Arrived
Time
(days
Remaining Operations
(days)
(days)
from now)
(days)
Remaining
Job
1
10
1.25
6
2.5
5
2
9
2.75
5
3.5
7
3
7
2.50
7
4.0
9
4
6
3.00
6
4.5
12
5
5
2.50
5
3.0
8
6
4
1.75
8
2.5
6
7
3
2.25
7
3.0
9
8
1
2.00
5
2.5
3
a. Compare the relative performance of the FCFS, SPT,
EDD, S/RO, and CR rules in terms of the percent of
jobs past due, average days past due, and maximum
days of past due. (Hint: The time since an order was
placed is needed just to establish the sequence for the
FCFS rule, because all performance measures deal
with past due statistics.)
b. Discuss the selection of one of the rules for this company. What criteria do you consider most important in
the selection of a rule in this situation?
9. Penultimate Support Systems makes fairly good speaker
and equipment support stands for music groups. The
assembly process involves two operations: (1) fabrication,
or cutting aluminum tubing to the correct lengths, and
(2) assembly, with purchased fasteners and injectionmolded plastic parts. Setup time for assembly is negligible.
Fabrication setup time and run time per unit, assembly run
time per unit, and the production schedule for next week
follow. All jobs arrived at the same time. Organize the work
to minimize makespan, and create a Gantt chart. Can this
work be accomplished within two 40-hour shifts?
Fabrication
Assembly
Model
Quantity
Setup (hr)
Run Time
(hr/unit)
Run Time
(hr/unit)
A
200
2
0.050
0.04
B
300
3
0.070
0.10
C
100
1
0.050
0.12
D
250
2
0.064
0.60
OPERATIONS SCHEDULING
10. Eight jobs must be processed on three machines in the
sequence M1, M2, and M3. The processing times (in
hours) are as follows:
SUPPLEMENT J
J-17
processing on M3. Other jobs require processing on M2
before M3. Currently, six jobs are waiting at M1 and four
jobs are waiting at M2. The following data have been supplied by the shop floor control system:
Job
Processing Time (hr)
1
2
3
4
5
6
7
8
Machine 1
2
5
2
3
1
2
4
2
Machine 2
4
1
3
5
5
6
2
Machine 3
6
4
5
2
3
2
6
Job
M1
M2
M3
1
1
6
—
4
13
2
2
2
—
1
18
3
4
—
7
22
4
5
—
3
16
5
7
—
4
30
6
3
—
1
29
7
—
4
6
42
8
—
2
10
31
9
—
6
9
48
10
—
8
2
40
Machine M2 is a bottleneck, and management wants to
maximize its use. Consequently, the schedule for the
eight jobs, through the three machines, was based on the
SPT rule on M2. The proposed schedule is 2, 8, 7, 3, 1, 4,
5, and 6.
a. It is now 4:00 P.M. on Monday. Suppose that processing
on M2 is to begin at 7:00 A.M. on Tuesday. Use the proposed schedule to determine the schedules for M1 and
M3 so that job 2 begins processing on M2 at 7:00 A.M. on
Tuesday. Draw Gantt charts for M1, M2, and M3. What
is the makespan for the eight jobs?
b. Find a schedule that utilizes M2 better and yields a
shorter makespan.
11. The last few steps of a production process require two
operations. Some jobs require processing on M1 before
Due Date (hr from now)
a. Schedule this shop by using the following rules: SPT,
EDD, S/RO, and CR.
b. Discuss the operating implications of each of the
schedules you developed in part (a). Assume all jobs
arrived at the same time.
Active Model Exercise
This Active Model appears in myomlab. It allows you to evaluate the application of single-dimension priority rules for
scheduling jobs at one workstation.
QUESTIONS
1. Which rule minimizes the average job flow time in the
system for this example?
2. Use the scroll bars to change the five processing times
and the five due dates. Does the same rule always minimize the average flow time and average past due?
3. Which rule minimizes the average hours past due for this
example?
4. Use the scroll bar to change the processing time for the
Thunderbird and to modify the due date for the
Thunderbird. Does the same rule always minimize the
average hours past due?
5. Which rule minimizes the average hours early for this
example?
6. Use the scroll bar to change the processing time for the
Econoline and to modify the due date for the Econoline.
Does the same rule always minimize the average hours
past due?
J-18
SUPPLEMENT J
OPERATIONS SCHEDULING
Job Shop Scheduling Using Data from Example J.1
Selected References
Baker, K. R. Elements of Sequencing and Scheduling. Hanover,
NH: Baker Press, 2002.
Hartvigsen, David. SimQuick: Process Simulation with Excel,
2nd ed. Upper Saddle River, NJ: Prentice Hall, 2004.
Johnson, S. M. “Optimal Two Stage and Three Stage Production
Schedules with Setup Times Included.” Naval Logistics
Quarterly, vol. 1, no. 1 (1954), pp. 61–68.
Kiran, Ali S., and Thomas H. Willingham. “Simulation: Help for
Your Scheduling Problems.” APICS—The Performance
Advantage (August 1992), pp. 26–28.
LaForge, R. Lawrence, and Christopher W. Craighead.
“Computer-Based Scheduling in Manufacturing Firms: Some
Indicators of Successful Practice.” Production and Inventory
Management Journal (First Quarter 2000), pp. 29–34.
Metters, Richard, and Vincente Vargas. “A Comparison of
Production Scheduling Policies on Costs, Service Levels,
and Schedule Changes.” Production and Operations
Management, vol. 17, no. 3 (1999), pp. 76–91.
Pinedo, Michael. Scheduling: Theory, Algorithms, and Systems,
2nd ed. Upper Saddle River, NJ: Prentice Hall, 2002.
Pinedo, M., and X. Chao. Operations Scheduling with
Applications in Manufacturing and Services. Boston:
McGraw-Hill/Irwin, 1998.
Suresh, V., and D. Chaudhuri. “Dynamic Scheduling-A Survey
of Research.” International Journal of Production
Economics, vol. 32 (1993), pp. 52–63.
Vollmann, Thomas E., William Berry, D. Clay Whybark, and
Robert Jacobs. Manufacturing Planning and Control
Systems for Supply Chain Management, 5th ed. New York:
McGraw-Hill/Irwin, 2005.