Curry G.L., Feldman R.M. Manufacturing Systems Modeling and Analysis

Подождите немного. Документ загружается.

2.1 The Basics 53

Fig. 2.5 A representation of the number of jobs in a simulated factory

WIP

(1 −

)

, given that

Note that the condition,

, establishes the existence of a steady-state for this

system. Using Little’s Law (Deﬁnition 2.1), the expected time in the system or cycle

time, CT

, becomes

−

The goal of our modeling efforts in future chapters will be to develop equations

such as the above. Often, the long-run distribution will be derived and then the mean

measures will be obtained from the distributions. The next chapter addresses single

workstation models and the associated queueing theory mechanics for their devel-

opment and approximation.

• Suggestion: Do Problems 2.1–2.2.

54 2 Introduction to Factory Models

Fig. 2.6 A four machine

serial ﬂow production factory

with constant service times

and a constant WIP

level

new

out

2.2 Introduction to Factory Performance

In this section a factory consisting of four machines in series with deterministic pro-

cessing times is analyzed. The purpose of the model is to illustrate several issues and

properties of manufacturing systems that will be studied in this text. This analysis

is patterned after the “penny fab” model in [3]. Through this modeling and analysis

exercise, several terms will become more meaningful. In particular, long-term or

steady-state performance measures, the validity and robustness of Little’s Law for

these performance measures, and the impact of a bottleneck or throughput limiting

machine will be illustrated.

Consider a factory that makes only one type of product. The processing require-

ments for this product consists of four processing steps that must be performed in

sequence. Each processing operation is performed on a separate machine. These

machines can process only one unit of the product at a time (called a job). The pro-

cessing times for the four operations are constant. These processing times are 1, 2,

1 and 1 hour(s) on each of the four machines, respectively. This idealized factory

has no machine downtimes, no product unit losses due to faulty production, and

operates continuously. The factory is operated using a constant number of jobs in

process (i.e., WIP

(t) is constant for all t). When a job has completed its four pro-

cessing steps, it is immediately removed from the factory and a new job is started

at Machine 1 to keep the total factory WIP

at the speciﬁed level. This process is

depicted in Fig. 2.6.

Since the processing times at each machine are not identical, the factory inven-

tory will not necessarily be the same at each machine. The factory has ample storage

space and the factory management policy is to move a job to the next machine area

as soon as it completes processing on each machine. Thus, no machine will set idle

if there is a part that is ready to be processed on that machine.

This factory is running smoothly at the current time. Management has set a con-

stant WIP

level at 10 jobs. This accomplishes a throughput rate of th = 0.5/hr jobs

(leaving the factory). That is, the factory produces one ﬁnished job every two hours

on the average. This is the maximum throughput rate for this factory because its

slowest processing step (at Machine 2) takes two hours per job. Thus, jobs can be

completed no faster than this single machine completes its own processing because

of the single unit machines and the serial nature of the production process.

Management is quite pleased with the throughput of the factory since it is at its

maximum capacity. However, management is somewhat concerned with the total

time that it takes a job from release to ﬁnish in the factory (the cycle time). This

cycle time is currently running at 20 hours per job. Management feels like this is

2.2 Introduction to Factory Performance 55

high since it only takes 5 hours of processing to complete each job. The ratio of the

cycle time to the processing time is a standard industry measure that will be called

the x-factor.

Deﬁnition 2.6. The x-factor for a factory is the ratio of CT

to the average total

processing time per job.

The average for this industry is currently running at 2.6 as reported in a recent

publication by the industry’s professional journal. With this factory’s x-factor being

4, management is worried about their ability to keep customers when the industry

on average produces the same product with a considerably shorter lead-time f rom

order placement to receipt.

To address the cycle time problem, management has been considering a large

capital outlay to purchase a 25% faster machine (1.5 hours) for processing step two.

This purchase would be made expressly for the purpose of reducing the x-factor

for the factory to be more in line with the industry average. The company selling

the machine says that this investment will bring the x-factor down to 3.33 and the

additional throughput of 0.166 units per hour would pay for the cost of the new

machine in three years.

Management has decided that this investment is not worthwhile just based on

increased throughput because the funds needed for the large capital outlay to buy

the machine are sorely needed in other aspects of the company. The life blood of

the company has been its ability to keep pace with the competition in new product

development. This level of expenditure would decimate the company’s investment

in research and development of new products.

In an effort to seek a lower cost solution to the x-factor performance measure

for the factory, a consulting team from the manufacturing engineering department

of a local university was hired to perform a short term factory ﬂow analysis study.

The ﬁrst activity of the consulting team was to devise a method of predicting the

long-term factory performance measures of cycle time and throughput.

2.2.1 The Modeling Method

The consulting team accomplished the performance estimation task rather quickly

devising a hand simulation procedure of the factory ﬂow. They started with the

speciﬁed number of 10 jobs in the factory, all placed at Machine 1, and made hourly

updates to each job’s status. Each job that was on a machine was allocated one hour

of processing time and if this completed their requirements on that machine, the job

was moved to the next machine. Empty machines were loaded with the ﬁrst job in

the machine queue, for those with a queue, and the next hourly update was started.

The jobs soon distributed themselves throughout the factory and after a short period

of time a two-hour cyclic pattern emerged. Every cycle of this pattern produced one

completed job and the factory returned to the identical state for each machine and

associated queue. This set of conditions is referred to as the factory status.

56 2 Introduction to Factory Models

Once the team had the model in the cyclic behavior pattern, they would mark

the time that each job entered the factory and again when it exited the factory. The

difference in these times is the job cycle time. All of these cycle times had identical

values after the system reached the cyclic behavior pattern. Thus, the job cycle time

was determined and agreed with the company’s actual cycle time of 20 hours.

The consulting team also computed the number of jobs that were completed dur-

ing the marked job’s residence in the model factory. When the marked job emerged

this completion total was always 10 (including the marked job). In retrospect this is

not surprising since a constant number of jobs is kept in the factory and, thus, when

the marked job entered the factory there were 9 other jobs ahead of it in the factory.

When the marked job emerged, all 9 of these jobs plus the marked job had been

completed. Thus, the total throughput was 10 jobs over the cycle time of 20 hours

or 0.5 j obs per hour. This modeling process exactly predicts the long-term factory

performance.

The simulation study is detailed in Table 2.1, where the factory status at the start

of each hour is displayed. The ﬁrst entry is the initial factory setup at time 0. Notice

that after hour 15 the factory status repeats every two hours; thus, the factory status

at the start of hours 15, 17, 19, 21, etc. are identical. Note also that the even hours

from time 16 on are also identical. In other words, this factory has reached a cyclic

behavior pattern at the start of time 15. Hours 0 through 14 represent the transient

phase of the simulation, and after hour 15, the limiting behavior is established.

Consider the system status at beginning with hour 15. There is a new job that has

just entered (no processing has occurred) into Machine 1. There are 8 jobs at Ma-

chine 2 with no processing completed on the job in the machine. There is one job that

just entered Machine 3 and Machine 4 is empty. This factory status is represented

by four pairs of numbers, one for each machine. The ﬁrst number in a machine pair

is the number of jobs at the machine, including the job being processed, and the

second number is the hours of processing at this machine already completed on the

job. The last entry is the cumulative number of completed jobs through this point in

time. The hour 15 the factory status entry in the table is

15 : (1,0),(8,0),(1,0),(0, 0) :6

After an additional hour of processing the factory status is

16 : (0,0),(9,1),(0,0),(1, 0) :6

which shows that the job in Machine 1 was completed and moved to Machine 2.

The job processing in Machine 2 needs an additional hour before being completed

since it requires a total of two hours for processing, and the job in Machine 3 was

completed and moved to Machine 4 to begin processing.

After one more hour of processing, the job in Machine 4 is completed and re-

moved from the factory and a new job is, therefore, entered into Machine 1. The job

processing on Machine 2 is completed and moved to Machine 3. Thus, the system

status at the end of time 17 is identical to that of time 15, except that one additional

job is completed.

2.2 Introduction to Factory Performance 57

Table 2.1 Factory simulation with WIP = 10, four single-machine workstations, and processing

times of (1,2,1,1) for one 24-hour day using a time step of one hour; data pairs under each work-

station are the number of jobs at the workstation and the elapsed processing time for the job being

processed

Time WS #1 WS #2 WS #3 WS #4 Cum. Thru.

0 (10,0) (0,0) (0,0) (0,0) 0

1 (9,0) (1,0) (0,0) (0,0) 0

2 (8,0) (2,1) (0,0) (0,0) 0

3 (7,0) (2,0) (1,0) (0,0) 0

4 (6,0) (3,1) (0,0) (1,0) 0

5 (6,0) (3,0) (1,0) (0,0) 1

6 (5,0) (4,1) (0,0) (1,0) 1

7 (5,0) (4,0) (1,0) (0,0) 2

8 (4,0) (5,1) (0,0) (1,0) 2

9 (4,0) (5,0) (1,0) (0,0) 3

10 (3,0) (6,1) (0,0) (1,0) 3

11 (3,0) (6,0) (1,0) (0,0) 4

12 (2,0) (7,1) (0,0) (1,0) 4

13 (2,0) (7,0) (1,0) (0,0) 5

14 (1,0) (8,1) (0,0) (1,0) 5

15 (1,0) (8,0) (1,0) (0,0) 6

16 (0,0) (9,1) (0,0) (1,0) 6

17 (1,0) (8,0) (1,0) (0,0) 7

18 (0,0) (9,1) (0,0) (1,0) 7

19 (1,0) (8,0) (1,0) (0,0) 8

20 (0,0) (9,1) (0,0) (1,0) 8

21 (1,0) (8,0) (1,0) (0,0) 9

22 (0,0) (9,1) (0,0) (1,0) 9

23 (1,0) (8,0) (1,0) (0,0) 10

24 (0,0) (9,1) (0,0) (1,0) 10

Computing the cycle time for a job consists of starting with a new job release

into the factory and following through 10 subsequent job completions. This release

occurs at the end of the given period that coincides with the start of the next time

period. It is convenient to place the new job into its location in the Machine 1 list

before recording the factory status so that the system maintains the required 10

jobs. Consider the job that just enters the factory at the end of time period 15 (the

beginning of time period 16), this job leaves the factory at the end of time period 35

(that is, actually equal to time 36). The time in the system for this job is 36-16 = 20

hours.

The consulting team also modeled the factory under the assumption of a new

Machine 2 with a constant processing time of 1.5 hours. To model this situation,

the consulting team used 1/2 hour time increments for the model time step and,

thus, the associated processing time requirements at the machines were (2, 3,2,2)

in terms of the number of time steps needed to complete a job. Again the results

obtained for this situation agreed with those proposed by the company trying to sell

the new machine. These results were a cycle time of 30 time increments (15 hours)

and a throughput rate of 2/3 jobs per hour (10 jobs every 15 hours).

58 2 Introduction to Factory Models

2.2.2 Model Usage

The consulting team, recognizing that their modeling approach was general and be-

ing familiar with Little’s Law (WIP equals throughput multiplied by CT), decided

to estimate the x-factor, the ratio of cycle time to total processing time, for vari-

ous numbers of jobs in the system (WIP). Letting CT represent cycle time and th

represent throughput, then Little’s Law (Property 2.1) yields

CT = WIP/th.

Using a throughput rate of 1/2 jobs per hour, then cycle time is given by

CT = 2 ×WIP .

Since the total processing time is 5 time units, then the ratio of the cycle time to the

processing time called the x-factor for the factory would be

x =

WIP

2.5

Notice that as long as the processing speeds of the machines do not change, the

maximum throughput rate for this factor is 1/2 per hour due to the speed of the

second workstation. Thus, the above formula gives shows the relationship between

the x-factor and WIP provided that WIP does not get too small so as to “starve”

Machine 2. Therefore, using the above formula, notice that if WIP = 6.5, the x-

factor will equal the desired level of 2.6.

Since 6.5 is a non-integer, the constant WIP level should be set to 6 or 7. A ﬁxed

WIP level of 6 should yield an x-factor lower than 2.6 and a ﬁxed WIP level of 7

should yield an x-factor slightly higher than 2.6 as long as the throughput rate of 1/2

per hour can be maintained.

The consulting team recognized that Little’s Law is a relationship between three

factory performance measures and that two of these measures must be known be-

fore the third can be obtained. The issue of concern is whether or not the throughput

would stay at 1/2 when the total factory WIP was reduced below 10 jobs. For in-

stance if only one job is allowed in the factory, the throughput rate is one job every

5 hours or 1/5. It certainly is obvious that at some job level, the factory throughput

would drop below 1/2. So the consulting team decided to perform a study of the fac-

tory performance for all ﬁxed job levels from 1 to 10 using their performance anal-

ysis modeling approach. These results are displayed in Table 2.2. They found that

the WIP level in the factory can be reduced all the way down to 3 jobs while main-

taining the factory throughput rate of 1/2. The cycle time reduces to 2 ×WIP = 6

hours with an x-factor of 1.2. Thus at no expense, the factory can maintain its cur-

rent throughput rate and reduce its cycle time from 20 to 6 hours. The cycle time and

throughput performance measures for this factory as a function of the ﬁxed factory

WIP level are displayed in Figs. 2.7 and 2.8, respectively.

2.2 Introduction to Factory Performance 59

Fig. 2.7 Average cycle time

for the simple factory model

as a function of the constant

WIP level

810

Constant WIP Level

Cycle Time

Fig. 2.8 Average throughput

rate for the simple factory

model as a function of the

constant WIP level

810

Constant WIP Level

Throughput

0.2

0.4

0.6

2.2.3 Model Conclusions

Detailed consideration has been given to the factory performance measures of

throughput, cycle time and work-in-process for a simple factory model. Little’s

Law for long-term system behavior is valid for both deterministic and stochastic

factory models. Little’s Law applies to individual workstations and to the system as

Table 2.2 Factory performance measures as a function of the WIP level

WIP Throughput Cycle Time x-factor

1 0.2 5 1.0

2 0.4 5 1.0

3 0.5 6 1.2

4 0.5 8 1.6

5 0.5 10 2.0

6 0.5 12 2.4

7 0.5 14 2.8

8 0.5 16 3.2

9 0.5 18 3.6

10 0.5 20 4.0

60 2 Introduction to Factory Models

a whole. For serial systems, the factory performance is controlled by the bottleneck

workstation (herein, the slowest machine). When there is enough WIPin the system

the maximum throughput rate is reached and is equal to the bottleneck workstation

(machine). As WIP increases beyond the minimum needed to reach the maximal

throughput rate, factory cycle time performance degrades proportionally.

• Suggestion: Do Problems 2.3–2.14.

2.3 Deterministic vs Stochastic Models

The simple throughput analysis of a serial factory with deterministic processing

times of the last section was used to illustrate several system performance mea-

sures and their inter-relationships (i.e., Little’s Law). The modeling approach was

developed speciﬁcally for deterministic processing times. This approach does not

necessarily yield accurate results when processing times are random. If the mean

processing time for a stochastic system is used in the above deterministic model-

ing approach, the results can be misleading and the wrong decisions can be drawn.

This problem is illustrated below with a system similar to the above example. The

key point to be made here is that for the evaluation of stochastic systems, stochas-

tic methodologies should be employed. How one models stochastic production and

manufacturing systems is the purpose of this book.

Consider the four-step production system represented by Fig. 2.6. Now instead

of the constant processing time of two hours at workstation 2, let us assume that

this time actually varies between two values: 1 hour and 3 hours. If these times

occur with equal probability, then the system has a mean processing time of 2 hours

and using this time one would draw the conclusions of the previous section. Recall

that the principle problem was to determine the constant WIP level that yields a

maximal throughput rate while maintaining a cycle time that is as small as possible.

The decision arrived at using the deterministic analysis was that a WIP level of 3

jobs in the system at all times yields the maximum throughput rate of 0.5 jobs per

hour with the minimal cycle time of 6 hours.

This stochastic system is now analyzed more thoroughly. One (incorrect) ap-

proach would be to develop the system performance measures using the determinis-

tic model but recognizing that the processing times at Machine 2 are not 2 hours but

1 hour and 3 hours, with equal frequency. Thus, one approach would be to model the

system using constant processing times of 1 hour and 3 hours and then average these

results since these times occur with equal frequencies. This leads to the results in

Table 2.3. These average results indicate that the decision to limit the constant WIP

level to 3 jobs is incorrect and that 4 jobs would yield the maximum throughput of

2/3 jobs per hour. Thus, this slightly more involved (but still not proper) methodol-

ogy, would indicate that the throughput level is up by 33% over the previous estimate

of 0.5 jobs per hour.

2.3 Deterministic vs Stochastic Models 61

Table 2.3 Weighed average throughput rate results for the factory of Fig. 2.6 with Workstation 2

processing times of 1 and 3 hours, and constant WIP levels of 3, 4 and 5

Processing Times

WIP 1 hour 3 hours Average

3 3/4 1/3 13/24

41 1/3 2/3

51 1/3 2/3

The reason for the averaged deterministic results not yielding the correct stochas-

tic result is that the factory throughput is not an instantaneous function of the pro-

cessing rate of Machine 2. This processing rate has an impact on the number of

jobs allowed into downstream machines and, hence, there is a longer term impact

on system performance. The length of this impact is also such that the system might

re-enter this rate status more than once while a job is in the system. Hence, com-

plex and longer term impacts cannot be properly estimated by merely performing

a weighted average of the constant processing time results. To illustrate this idea,

the t hroughput gain for the average results is obtained from the system when the

processing time is only one hour. This situation corresponds to a throughput rate of

1 job per hour (for a WIP level of at least 4 jobs). This high level of throughput is

balanced by the lower throughput rate (1/3 jobs per hour) when the system has a

3 hours processing time at Machine 2. These situations occur at the machine with

equal probability for a given job. However, the proportion of the time that the sys-

tem is operating in the slow state is 75%. Thus, one would expect a more accurate

throughput rate estimate to be





(1)=

This is the expected throughput rate for the stochastic system if the WIP level is at

least the minimum of 4 jobs. If there are only 3 jobs allowed in the system simul-

taneously, then the throughput rate reduces to around the 0.47 jobs per hour level.

Notice the detrimental effect of the variability in the processing time; namely, a

necessary increase in WIP and CT to maintain the same throughput rate. In general,

variability in workplace parameters always is detrimental in that it increases average

work-in-process and cycle times!

The calculation of throughput rates in our stochastic system can be obtained by

simulation or by the analytical decomposition method of Chap. 8. The bottom line

is that stochastic systems are much more difﬁcult to evaluate than deterministic

systems and the purpose of this textbook is to expose the reader to some of the

analytical approaches available for stochastic modeling of manufacturing systems.

62 2 Introduction to Factory Models

Appendix

In this appendix, Microsoft Excel will be used to present a discrete simulation model

of the factory given in Fig. 2.6 with a generalization that the processing time at

Workstation 2 is random as discussed in Sect. 2.3. In the next chapter, we will

present a more general simulation methodology (an event driven simulation) that

can better handle continuous time. For now, we shall limit ourselves to discrete

time. (For practice in developing similar models, see Problem 2.15.) We also sug-

gest that the understanding of this material is best accomplished by reading the

appendix while Excel is available so that the reader can build the spreadsheet as it

is presented below.

Simulation is a very important tool, especially for testing the validity of the mod-

els and approximations developed in these chapters. Simulation modeling is gener-

ally robust with respect to modeling distributional assumptions and allows for more

realistic modeling of system interactions. The price that one pays with simulation

is the time requirement for obtaining accurate estimates of system performance pa-

rameters. With analytical models, the system response can often be characterized by

studying the mathematical structure; while this must be accomplished in the simu-

lation environment by experimentation that again adds another dimension to the

already time consuming computational burden.

Before building the spreadsheet simulation model, it is important to understand

ﬁve Excel functions. The Excel function

RAND()

generates random numbers that are uniformly distributed between 0.0 and 1.0. Note

that the RAND function has no parameter, although the parentheses are used. The

Excel function

IF(boolean

expression, true value, false value)

evaluates the boolean expression and returns the value contained in the second pa-

rameter if the boolean expression is true and returns the value contained in the

third parameter if the boolean expression is false. The Excel IF() function can

act similar to an If — ElseIf structure by replacing either the true value or

false value with another IF() function. The above two functions can be used

together to create the random law mentioned previously; namely, the function

IF(RAND()<0.5,1,3) will yield a value of one 50% of the time and a value

of three 50% of the time. The

OFFSET(cell ref, number rows offset, number cols offset)

function allows for the referencing of a cell relative to another cell. For example,

OFFSET(A1,3,0) references the A4 cell. The function

MATCH(value, array reference, 0)