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

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Problems 193

Fig. 6.6a Process routing for

Product 1 for Problem 6.9

1/4

4/5

1/5

3/4

2/10

3/10

5/6

1/6

5/10

Fig. 6.6b Process routing for

Product 2 for Problem 6.9

(f) Write the equation (give explicit numbers whenever possible) for the average

product arrival SCV into Workstation 2, C

(2). The external arrivals are assumed to

be Poisson processes. Evaluate C

(2) given the other C

(i)’s.

Workstation 1 2 3 4

(i) 1.046 ? 1.380 1.615

(g) Complete the workstation performance measures:

Workstation CT

(k) CT (k) WIP(k)

1 0.625 hr 0.753 hr 10.021

2 0.939 hr 1.018 hr 10.828

3???

4 2.509 hr 2.635 hr 19.098

(h) What are the values of the factory system performance measures of cycle time

(CT

), work-in-process (WIP

) and throughput?

(i) What are the values of the individual product system performance measures of

throughput and cycle time (CT

6.9. This problem is designed to encompasses all of the basic components of build-

ing a multiple product model. To help reduce the time to solve the problem, many

of the numerical values are given so that the entire problem need not be worked out.

There are nine parts to the problem with several tables of numerical values being

given; however, many of the tables will contain incomplete information, so those

places in the table should be ﬁlled in.

A company is developing a factory to produce two different products. The ﬁrst

products use four distinct machining processes; whereas, the second product uses

only the ﬁrst three workstations used by Product 1. Company management would

194 6 Multiple Product Factory Models

like to know several things about the factory before it is built. To support this analy-

sis engineering has developed estimates for the necessary product processing infor-

mation. This information is listed below, and Figs. 6.6a and 6.6b depict the product

processing routings. Answer the questions and ﬁll in the missing data.

Mean processing times by product and workstation.

Product/WS 1 2 3 4

1 7.2 min 6 min 6 min 9 min

2 6 min 2.1 min 3.6 min

SCV of the processing times by product and workstation.

Product/WS 1 2 3 4

1 1.0 0.9 0.8 0.7

2 0.8 0.9 1.0

Machine availabilities and repair time characteristics.

Workstation 1 2 3 4

availability 0.90 0.95 0.93 0.95

E[R] 1hr 1hr 1hr 1hr

[R] 1.75 2.00 1.50 5/3

Answer the following questions and ﬁll in the missing information.

(a) Write the system of equations needed to ﬁnd the mean ﬂow rates into each work-

station for Product 1. These equations would yield the following mean arrival rates

for the two products into t he four workstations:

Product/WS 1 2 3 4

1 7.869/hr 6.295/hr 4.721/hr 4.800/hr

2 3.433/hr 4.044/hr 4.326/hr

Total 11.301/hr 10.339/hr 9.047/hr 4.800/hr

(b) Composite product mean and SCV processing time data by workstation, this

data is not adjusted for downtime and repairs:

Workstation 1 2 3 4

E[S] 6.84 min ? 4.86 min 9 min

[S] 0.966 ? 0.963 0.700

(exponential time between breakdowns):

Workstation 1 2 3 4

E[ S] ? 4.74 min 5.22 min 9.48 min

[S] ? 3.155 2.975 1.544

(d) Average product branching probability by workstation is:

Problems 195

From/To 1 2 3 4

1 0 0.800 0.200 0

2 0.183 0 0.656 0.122

3 ? 0.143 0 0.391

4 0.167 0 0 0

(e) The minimum number of machines required for each workstation:

Workstation 1 2 3 4

Workload ? ? ? ?

Min Machines ? ? ? ?

(f) Write the equation (give explicit numbers whenever possible) for the average

product arrival SCV into Workstation 1, C

(1). The external arrivals are assumed to

be Poisson processes. Evaluate C

(1) given the other C

(k)’s.

Workstation 1 2 3 4

(i) ? 1.602 1.841 1.497

(g) Complete the workstation performance measures:

Workstation CT

(i) CT(i) WIP(i)

1 17.40 min 24.96 min 4.703

2 48.24 min 52.98 min 9.127

3???

4 45.12 min 54.60 min 4.366

(h) What are the values of the factory system performance measures of cycle time

(CT

), work-in-process (WIP

) and throughput?

(i) What are the values of the individual product system performance measures of

throughput and cycle time (CT

6.10. Using a spreadsheet program such as Excel, solve Problem 6.1.

6.11. Using a spreadsheet program such as Excel, solve Problem 6.2.

6.12. A factory consists of ﬁve workstations and produces two products. Develop

the product and factory performance measures of throughput, cycle time and work-

in-process. Job Type 1 arrives according to a Poisson process with a mean rate of

5 per hour and Job Type 2 arrives with a mean rate of 3 per hour and a squared

coefﬁcient of variation of the inter-arrival times of 2. There are two machines at

Workstation 2; all other workstations have only one machine. The process ﬂow,

mean processing times in hours, and squared coefﬁcient of variation of the process-

ing times are as follows:

Workstation Mean Service Time SCV Service Time

by Step # by Step # by Step #

Job Type 1 2 3 1 2 3 1 2 3

1 1 2 3 0.16 0.17 0.18 2.0 1.3 1.00

2 4 2 5 0.30 0.30 0.28 1.0 1.5 0.75

196 6 Multiple Product Factory Models

6.13. For the factory of Problem 6.12, the factory can be segmented into two cellular

lines - one product manufactured in each line. Assuming that the mean processing

times for Machine 2 can be reduced by 15% when the products are not processed

on the same machine. That is, the workstation can specialize its setup operation

by product type. Compute the product and total factory performance measures and

compare these with the composite factory organization computed in Problem 6.12.

6.14. Consider a workstation that processes three products with each going to a

speciﬁed and different workstation upon leaving this workstation. Figure 6.2 illus-

trates this situation. The individual arrival stream information and necessary pro-

cessing time parameters are:

Product

(i) E[S

] C

(i)

1 1/hr 1.50 9 min 1.5

2 2/hr 2.50 9 min 1.5

3 3/hr 0.75 9 min 1.5

Compute the arrival SCV, C

, at the next workstation for each product using the

three different estimators: probabilistic routing (Eq. 6.9), Poisson intervening units

assumption (Eq. 6.14), and the asymptotic method (Eq. 6.15).

References

1. Askin, R.G. and Standridge, C.R. (1993). Modeling and Analysis of Manufacturing Systems.

John Wiley & Sons, New York.

2. Bitran, G.R., and Tirupati, D. (1988). Multiproduct Queueing Networks with Deterministic

Routings: Decomposition Approach and the Notion of Interference. Management Science,

34:75–100.

3. Caldentey, R. (2001). Approximations for Multi-Class Departure Processes. Queueing Systems,

38:205–212.

4. Deuermeyer, B.L., Curry, G.L., and Feldman, R.M. (1993). An Automatic Modeling Approach

to the Strategic Analysis of Semiconductor Fabrication Facilities. Production and Operations

Management, 2:195–220.

5. Groover, M.P. (2001). Automation, Production Systems, and Computer-Integrated Manufactur-

ing (Second Ed.) Prentice-Hall, Upper Saddle River, NJ.

6. Whitt, W. (1994). Towards Better Multi-class Parametric-Decomposition Approximations for

Open Queueing Networks. Ann. Oper. Res., 48:221–248.

Chapter 7

Models of Various Forms of Batching

Grouping individual jobs into sets, called batches, is a strategy frequently used in

industry. One cause of batching is for the purpose of transportation between work-

stations. For instance, workers may require mechanical help for moving heavy items

between two machines. If the mechanical help is a large machine such as a forklift,

then a pallet might be loaded ﬁrst before a forklift truck is requested. Another form

of batching occurs when items are batched by type for the purpose of sharing a

machine setup step even though the items are actually processed individually. By

batching like items, only one setup need be performed for the whole set. And ﬁ-

nally, a frequently encountered batch service process is that of a multiple service

capacity resource such as an oven. Due to the slow processing rates of some heat-

treatment or plating processes, large capacity machines have been developed that

can process several units of an item simultaneously.

The batching phenomenon is motivated by a perceived beneﬁcial effect of group-

ing. However, the impact on downstream processing stations can be signiﬁcant. To

illustrate, consider the batch move concept where, say k, items are grouped together

for the convenience of moving them to a subsequent single unit processing station.

Items will arrive at the next workstation k at a time, so the workstation might be idle

for a while and then instantaneously have a queue of waiting units. The variability

of a batch arrival process when the batch is broken back into individuals (frequently

caused by processing items simultaneously) is much greater than the inter-arrival

variability of the individual items and, the workstation queueing behavior will be

exacerbated. This leads to increased cycle times and larger WIP levels at the down-

stream workstation. In addition, the batch process itself causes an increased delay

because units must wait for the completion of other units before they can be grouped

and continue processing.

In this chapter, models are developed for various forms of batching and so that

the beneﬁts and costs of the grouping process under consideration can be quanti-

ﬁed. For the setup sharing situation, there will be a trade-off between the cycle time

increase and the setup time savings due to batching. The chapter is concluded with

a discussion of network models that include a batch (oven-type) processing work-

station. The term “job” can be confusing because in some contexts a job my refer

G.L. Curry, R.M. Feldman, Manufacturing Systems Modeling and Analysis, 2nd ed., 197

DOI 10.1007/978-3-642-16618-1 7,

 Springer-Verlag Berlin Heidelberg 2011

198 7 Models of Various Forms of Batching

Single Unit Server

Batch Forming

Queued Batches

Single Unit Server

Holding Area

Fig. 7.1 The batch move model structure: batches are formed after single unit processing and are

transported to the next workstation; batches wait in the queue until service on individual items

within the batch commences; ﬁnally, items leave as individuals as soon their processing has been

completed

to an individual item and in other contexts it may refer to the entire batch. To avoid

confusion, the term “item” will always be used for an individual job and never to

the entire batch.

7.1 Batch Moves

Consider a situation where individual items are grouped together into ﬁxed batches

of size k at the completion of processing at a workstation that processes single units.

Items wait in the incomplete batch until the proper quantity has accrued and then the

full batch is transported to the next workstation. (A basic assumption used through-

out this text is that transportation time is negligible and, therefore, is not explicitly

considered. If transportation time is signiﬁcant, it can often be approximated in the

model by considering the transporter as a separate workstation.) An additional as-

sumption is made that the receiving workstation processes items individually, hence,

the batch is merely a convenient transportation tool. The modeling of the batch move

situation is a building block for more complex models. In addition, we will demon-

strate that batch moves add to the cycle time in comparison to a system where items

are moved individually. Figure 7.1 illustrates a batch move system.

To model the batch move, several aspects of the problem will have to be con-

sidered. First, the batch forming time as it contributes to each individual item, or

the average item delay within a partial batch, needs to be computed. (The batch

forming time is added to the cycle time of the workstation receiving the batch even

though the batch forming actually takes place at the workstation sending the batch.)

Then the arrival stream characteristics for the batch receiving workstation need be

developed; that is, the mean arrival rate for batches and the squared coefﬁcient of

variation of the inter-arrival batch times must be computed. And ﬁnally, the model-

ing approach for developing the second workstation cycle time is different than our

previous analyses. The cycle time model is separated into the standard components

of the queue time and the service time. The queue time, however, is developed from

7.1 Batch Moves 199

the batch point of view. The individual item’s service time is the average time for

processing individuals. These individuals are assumed to be released immediately

after processing to move on to their next workstation. But since there are k items in

the batch, the items have different delays while awaiting their turn at service. The

ﬁrst item served from a batch has no additional delay due to waiting for others from

the same batch, while the second item serviced from the batch waits for the ﬁrst

item, the third item waits for the ﬁrst two selected items, and so on. The average

delay is then taken over the composite delay for all items in the batch. These cycle

time component analyses are addressed one at a time below.

7.1.1 Batch Forming Time

A batch is formed by grouping k individually arriving items together. Let T

be the

random variable denoting the time between departures from the source workstation

that are to be batched for transportation to the destination workstation. Note that

the arrival rate of individuals

(I), in the absence of batching, to the destination

workstation is given by

(I)=

E[T

]

The random variable T (B) is the inter-arrival time of batches. This random variable

is the sum of k individual inter-departure times

T (B)=T

d,1

+ T

d,2

+ ···+T

d,k

The individual inter-departure times T

d,i

for i = 1,··· ,k are independent and iden-

tically distributed (i.i.d.) random variables; thus, the expected value of the batch

inter-arrival time is

E[ T (B)] = kE[T

] .

Hence, the arrival r ate of batches to the destination workstation,

(B),is

(B)=

E[T (B)]

kE[T

]

(I)

. (7.1)

The squared coefﬁcient of variation, C

[T (B)], of the batch inter-arrival times at

the destination workstation is obtained from

[T (B)] =

V [T (B)]

E[T (B)]

where the variance is computed from

V [T (B)] = V [T

d,1

+ T

d,2

+ ···+ T

d,k

]=kV [T

] ,

200 7 Models of Various Forms of Batching

since the inter-departure random variables are assumed independent (see Prop-

erty 1.6 or Eq. 1.27 ). Thus, the squared coefﬁcient of variation of the batch inter-

arrival times can be computed from the squared coefﬁcient of variation of the indi-

vidual inter-arrival times by

[T (B)] =

V [T (B)]

E[ T (B)]

kV [T

]

(kE[T

])

]

. (7.2)

The delay that an individual item encounters when being placed into a batch

depends on where it is among the k batched items. The ﬁrst departing item used to

start the formation of a new batch must wait for k −1 more items to depart before the

batch has been formed and released for transportation to the destination workstation.

Denote this delay by the random variable D

where

= T

d,2

+ ···+T

d,k

The second item forming the new batch has to wait for k −2 succeeding departures

and its waiting time is the random variable D

given by

= T

d,3

+ ···+T

d,k

The other items in the batch have delay times similarly developed with the last

item encountering no delay (i.e., D

= 0). The last item’s arrival signals the batch is

complete and the batch is instantaneously transported to the destination workstation.

The average delay encountered by an item in the batch is then the expected value of

the sum of all these delays divided by the batch size k ; that is,

E[D]=

E[D

+ D

+ ···+D

k−1

+ D

]

(k −1)E[T

]+(k −2)E[T

]+···+1E[T

]+0E[T

]

((k −1)k/2)E[T

]

(k −1)

E[ T

] . (7.3)

Thus, the average delay encountered by an individual item when waiting for a batch

of size k to form is (k −1)E[T

]/2.

One should recognize that the term E[T

] in the expected batch waiting time per

individual is the time between arrivals for the particular situation that was used to

motivate this analysis. In this situation, batching after job completion at a worksta-

tion is done for the purpose of t ransporting the items to the next workstation. The

batching operation could occur at the front of a batch service workstation such as

an oven. In this case, the average delay would follow the same form as the result

just derived, but the individual item’s inter-arrival time would then be denoted as

7.1 Batch Moves 201

E[T

]. This is the expected time between arrivals to the workstation and, therefore,

the inter-arrival time for batch items.

The batching form that has been analyzed is for a single product or for indis-

criminate grouping of multiple products. It is very likely that batching is restricted

to jobs of the same type. In this situation, then multiple batch types can be forming

simultaneously and the wait associated with batch formation of a given item type

would be a function of the inter-arrival time to the workstation of that job type.

7.1.2 Batch Queue Cycle Time

Modeling the cycle time for the recipient workstation for the batch move situation

has two distinct components: the queue time and the service time. The queueing

delay is modeled from the batch units point of view. The items within a batch see

this queueing phenomenon as batches waiting and then moving up in the line based

on batches being served. The arrival rate to this queue is

(B) with corresponding

squared coefﬁcient of variation C

[T (B)]. It may be clearer to denote t he random

variable T (B) as T

(B) to distinguish and relate this random variable with the indi-

vidual inter-arrival time random variable T

(I).

The service time that these batches observe while they wait in the queue is for

batches (they move forward one location in the queue whenever a batch has been

completely served). The service time for these batches is the time it takes the server

to process all of the items within the batch being served, which is the random vari-

able T

(B) given by

(B)=T

s,1

(I)+T

s,2

(I)+···+ T

s,k

(I) ,

where the (B) and (I) notation again stands for batches and individuals, respectively.

Note that since this is a single item service facility (items are processed one at a

time), the processing times T

s,i

(I) are independent and identically distributed ran-

dom variables with known mean E[T

(I)] and known squared coefﬁcient of variation

(I)]. Thus, the (perceived) batch service time characteristics can be computed

from these known individual item data as

E[T

(B)] = kE[ T

(I)] ,

(B)] =

(I)]

The G/G/1 cycle time approximation model is used to compute the cycle time

in the queue for the waiting batches. The utilization factor for the workstation must

also be computed for batches. This computation is

202 7 Models of Various Forms of Batching

u(B)=

(B)E[T

(B)]

(I)

(kE[T

(I)])

(I)E[T

(I)] = u (I) .

Thus, the utilization factor for the workstation in the batch service mode as per-

ceived by the waiting move batches is the same as the normal single unit processing

utilization factor for the workstation. The queue time estimate is given by

(B)=



(B)] +C

(B)]



u(B)

1 −u(B)



E[ T

(B)] (7.4)



(I)]/k)+(C

(I)]/k)



u(I)

1 −u(I)



kE[T

(I)]



(I)] +C

(I)]



u(I)

1 −u(I)



E[ T

(I)] = CT

(I) .

So the expected cycle time in the queue for individuals in a move batch is iden-

tical to the cycle time in the queue for the workstation operating in a single item

mode. Therefore, utilizing move batches does not change the expected queueing de-

lay time. (It should be noted here that this is an approximation. The expected cycle

time in the queue for individuals in a move batch is actually smaller than the ex-

pected cycle time in the queue for the workstation operating in a single item mode,

but the Kingman approximations for t he two situations are the same.) It does, how-

ever, affect the processing time delay as seen by individuals within a batch as is

developed in the next section.

7.1.3 Batch Move Processing Time Delays

The ﬁnal element in the cycle time computation for batch moves between successive

workstations is the processing time delay for batched items. The individual item

processing time has not been altered by the batching process. The cycle time in the

workstation has been separated into the queue waiting time for the batch and the

processing time for the batch. Hence, the previous section developed the time that

the batch waits until the server is working on items within the batch. There is another

element to the actual waiting time until a particular item is served that consists of

the delay encountered while the item is waiting its turn for service with respect to

the other items within the batch. This delay is analogous to the batch forming delay

analyzed above. That is, the ﬁrst item selected from the batch for processing sees no

delay at the processor due to having been part of a move batch. However, the second

and subsequent items must wait their turn for service. Thus, items are delayed for

their own processing time plus the processing times of all items that were part of