OPERATIONS SCHEDULING

OPERATIONS SCHEDULING

J

SUPPLEMENT J


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.

J-1

J-2

SUPPLEMENT J


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


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

J-3

J-4

SUPPLEMENT J


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


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

J-5

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

J-6

SUPPLEMENT J


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)


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.


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.

J-8

SUPPLEMENT J


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.


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

J-10

SUPPLEMENT J


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


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


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.


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,


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


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


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


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


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


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.