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

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Problems 153

5.3. Find the system performance measures of CT

, WIP

, and throughput for a pure

serial system consisting of three single capacity workstations. The arrival rate to the

system is 3 jobs per hour, with the inter-arrival time being exponentially distributed.

The processing time data:

Workstation iE[T

] C

]

10.25hr4

20.29hr3

30.30hr2

5.4. Resolve Problem 3 under the assumption that the machine in each workstation

is subject to breakdowns. The necessary data the processor at each workstation are

given in the following table (reference Section 4.2).

Workstation # Availability E[R] C

[R]

1 0.85 1 hr 1.50

2 0.90 1 hr 1.75

3 0.95 1 hr 2.00

5.5. Find the system performance measures of CT

, WIP

, and throughput for a

three-workstation pure serial system. The mean arrival rate to the system is one

job every two hours with an SCV of 2.0. The processing time data for the three

single-capacity workstations are given below. Assume that the machines are avail-

able 100% of the time.

Workstation iE[T

] C

]

1 1.6 hr 0.75

2 1.5 hr 1.50

3 1.7 hr 2.00

5.6. Find the system performance measures of CT

, WIP

, and throughput for a three

workstation pure serial system. The arrival rate to the system is one job every two

hours with an SCV of 2.0. The machine data for the three single-capacity worksta-

tions are given below.

Workstation iE[T

] C

] Availability E[ R] C

[R]

1 1.6 hr 0.75 0.85 2.0 hr 1.30

2 1.5 hr 1.50 0.90 2.5 hr 1.50

3 1.7 hr 2.00 0.90 3.0 hr 1.75

5.7. Develop a spreadsheet model to solve Problem 5.5.

5.8. Develop a spreadsheet model to solve Problem 5.6.

5.9. Consider again the serial ﬂow factory of Problem 5.5. Management expects

there to be a slow increase in demand (i.e., arrival rates) over the next few years.

(a) In order to help management plan for the future, ﬁnd the system performance

measures (CT

sys

and WIP

sys

) for arrival rates of 0.51/hr, 0.53/hr, and 0.55/hr.

154 5 Multiple-Stage Single-Product Factory Models

Fig. 5.6 Diagram for Prob-

lem 5.11

u = 0.8

= 3

= 1.5

(1)

(2)

1-α

λ = 4

(b) The engineering department is considering some machine changes that will re-

duce the processing time variance for the bottleneck machine (i.e., the machine with

the largest cycle time). Assuming the arrival rate increases to 0.51/hr, what percent-

age reduction in the C

for the bottleneck machine is necessary so that the average

system cycle time remains the same as for the original system with an arrival rate of

0.5/hr?

is possible to reduce the mean service time while the coefﬁcient of variation remains

according to the original system. Assuming that the arrival rate increases to 0.55/hr,

what mean service rate is necessary for the bottleneck machine so that the average

system cycle time remains the same as for the original system with an arrival rate of

0.5/hr?

(d) If the mean service time of the bottleneck machine is reduced enough, the bot-

tleneck will “move” to a different machine. With an arrival rate of 0.55/hr, what is

the mean service time of the current bottleneck machine that is required so that two

workstations become “tied” for the bottleneck location?

5.10. Consider a three-workstation serial system, with one machine in workstations

one and three and two machines available in workstation two. The external ﬂow

enters workstation one, with parameters of

= 4 jobs per hour and C

(1)=0.75,

and proceeds sequentially through workstation two and then workstation three (i.e.,

a serial system). The processing time data for the three workstations are given below.

Find the system performance measures of CT

, WIP

, and throughput for this system.

To accomplish this, you need to compute, for each workstation i, C

(i), CT (i), and

WIP

(i).

Workstation iE[T

] C

]

112min2.0

218min0.7

3 13.2 min 1.0

5.11. Solve the spitting branch problem for the unknowns (C

(1),C

(2))

for three different values of branching probabilities

=(1/3, 1/2,3/4) as shown

in Fig. 5.6.

5.12. Solve the merging branch problem illustrated in Fig. 5.7 for the unknowns.

5.13. Obtain the mean ﬂow rates for the system illustrated in Fig. 5.8.

5.14. Obtain the mean ﬂow rates for the system illustrated in Fig. 5.9.

Problems 155

Fig. 5.7 Diagram for Prob-

lem 5.12

(merged) = ?

= 1.365

= 2.095

Merge

= 1

= 3

λ = ?

Fig. 5.8 Diagram for Prob-

lem 5.13

Fig. 5.9 Diagram for Prob-

lem 5.14

2/3

4/5

1/5

1/3

Fig. 5.10 Diagram for Prob-

lem 5.15

2/3

4/5

1/5

1/3

5.15. Obtain the mean ﬂow r ates for the system illustrated in Fig. 5.10.

5.16. For the network illustrated in Fig. 5.11, ﬁnd the total inﬂows (arrival) rates

for each workstation and terminator (B and G). Terminator G represents good jobs

and Terminator B represents bad product. What is the probability that a job ends up

good?

5.17. Using a spreadsheet program such as Excel, solve Problem 5.15.

5.18. Reconsider Problem 5.13 using the following service time data for each single-

server workstation and assuming that the squared coefﬁcient of variation of the inter-

arrival times for the jobs arriving from an external source is 1.5.

(a) Compute the system performance measures of throughput, cycle time and work-

in-process for this network.

156 5 Multiple-Stage Single-Product Factory Models

Fig. 5.11 Diagram for Prob-

lem 5.16

1/10

4/10

1/10

1/3

1/6

1/2

2/10

6/10

2/10

1/6

4/6

1/6

Workstation iE[T

(i)] C

(i)

1 0.086 1.3521

2 0.110 0.8264

3 0.080 1.5625

(b) Compute the system performance measures of throughput, cycle time and work-

in-process for this network given that the machines have breakdowns. The availabil-

ity data and parameters for the repair time, random variable R, by workstation are

given in the following table.

Workstation # Availability E[R] C

[R]

1 0.95 0.2 1

2 0.93 0.3 1

3 0.87 0.4 1

5.19. Reconsider Problem 5.15 using the following service time data for each work-

station and assuming that the squared coefﬁcient of variation of the inter-arrival

times for the jobs arriving from an external source is 1.5.

(a) Compute the system performance measures of throughput, cycle time and work-

in-process for this network.

Workstation iE[T

(i)] C

(i) c

1 0.086 1.3521 2

2 0.110 0.8264 2

3 0.080 1.5625 2

(b) Compute the system performance measures of throughput, cycle time and work-

in-process for this network given that the machines have breakdowns. The availabil-

ity data and parameters for the repair time, random variable R, by workstation are

given in the following table.

Workstation # Availability E[R] C

[R]

1 0.95 0.2 1

2 0.93 0.3 1

3 0.87 0.4 1

References 157

Fig. 5.12 Diagram for Prob-

lem 5.20

5.20. Consider the factory model illustrated in Fig. 5.12 with C

(0,1)=2 and the

workstation service time data displayed below. Compute the system performance

measures of throughput, cycle time and system work-in-progress assuming that

there is only one machine at each workstation.

Workstation iE[T

(i)] V [T

(i)]

1 0.13 0.02

2 0.13 0.03

3 0.20 0.04

4 0.08 0.01

5.21. Using a spreadsheet program such as Excel, solve Problem 5.18(b).

5.22. Using a spreadsheet program such as Excel, solve Problem 5.19(b).

5.23. Using a spreadsheet program such as Excel, solve Problem 5.20.

References

1. Albin, S.L. (1984). Approximating a Point Process by a Renewal Process, II: Superposition

Arrival Processes to Queues. Operations Research, 30:1133–1162.

2. Burke, P.J. (1968). The Output Process of a Stationary M/M/s Queueing System. Annuals

Math. Stat., 39:1144–1152.

3. Buzacott, J.A., and Shanthikumar, J.G. (1963). Stochastic Models of Manufacturing Systems.

Prentice Hall, Englewood Cliffs, NJ.

4. Marshall, K.T. (1968). Some Inequalities in Queueing, Operations Research, 16:651–665.

5. Whitt, W. (1982). Approximating a Point Process by a Renewal Process, I: Two Basic Methods.

Operations Research, 30:125–147.

6. Whitt, W. (1983). The Queueing Network Analyzer, The Bell System Technical Journal,

62:2779–2814.

Chapter 6

Multiple Product Factory Models

Most manufacturing facilities are setup to produce more than a single product. Even

in the case of single product facilities, if the product visits a workstation more

than once with different processing times at each visit, then the workstation sees

the equivalent of multiple products. Such revisiting production schemes, called re-

entrant ﬂow systems, are prevalent in the semiconductor industry where it is not

unusual for a product to be routed to the same machine group for distinct process-

ing20ormoretimes.

Modeling multiple product facilities is not signiﬁcantly more difﬁcult than single

product models. There are two basic principles to keep in mind. First, the workload

on a workstation is, as before, the sum of all the visits multiplied by the processing

time per visit. This concept was introduced in the previous chapter (see p. 143) and

since we use it in a more general setting here, we give a formal deﬁnition.

Deﬁnition 6.1. The offered workload (or simply the workload)ofaworkstationis

the total amount of work that is required of a workstation per unit of time. The

workload is determined by the sum of the total arrival rate (per hour) for each prod-

uct type multiplied by its associated mean processing time (in hours). For purposes

of determining workload, when a speciﬁc product type revisits a workstation, it is

considered as a separate product type.

The s econd basic principle is that job ﬂow needs to be maintained by product

type. That is, the number of visits to each workstation by product class is needed.

Different products can have different probabilistic ﬂows through the production fa-

cility as well as different processing time characteristics. Hence, the number of visits

to each workstation by product needs to be developed. This analysis requires the so-

lution of a network ﬂow system of equations by product. Here again as was done in

the preceding chapter, the processing time is assumed to follow the same distribution

for each product on each visit to a given workstation (of course due to randomness,

the actual processing times will vary even though the distribution is the same). The

re-entrant ﬂow situation with different processing distributions per visit requires a

different modeling paradigm that is taken up in Sect. 6.5.

G.L. Curry, R.M. Feldman, Manufacturing Systems Modeling and Analysis, 2nd ed., 159

DOI 10.1007/978-3-642-16618-1 6,

 Springer-Verlag Berlin Heidelberg 2011

160 6 Multiple Product Factory Models

6.1 Product Flow Rates

To compute the workload on the workstation, the number of visits to the workstation

by each product is computed ﬁrst. This requires an analysis for each product similar

to that performed in Sect. 5.4.1 for a single product. A method of distinguishing

between products visiting the same workstation is required. Previously a subscript

was used to denote the workstation visited by a product so that

denoted the ar-

rival rate of jobs to Workstation k. Two subscripts will now be used to distinguish

among the various product types; thus,

i,k

is the arrival rate of Product Type i to

Workstation k. Since a single subscript refers to a workstation, we will use a super-

script when a single index refers to a product type; thus,

is a vector giving the

total arrival rates of Product Type i into each workstation so that the k

component

of the vector

i,k

Arrivals from an external source are denoted as before by

so that

i,k

is the exter-

nal arrival rate of Product i into Workstation k. Additionally a workstation branching

probability matrix for each product type will be needed. Since it is standard to al-

ready use two subscripts for this matrix of probabilities, the product type will be

denoted by a s uperscript such as p

meaning the probability that an individual item

of Product i leaving Workstation j goes to Workstation k. The matrix of these prob-

abilities for Product i is denoted as P

With the above notation, we can rewrite Property 5.7 so that it applies to more

than one product type.

Property 6.1. Consider a factory of n workstations where Product Type i fol-

lows the switching rule deﬁned by the routing matrix P

and assume that the

sum of at least one row of P

is strictly less than one (i.e., jobs exit the net-

work from at least one workstation). Let

i,1

,··· ,

i,n

) denote a vector

consisting of the mean arrival rate of Type i jobs from an external source to

the workstations. Both P

and

are known. Let

i,1

,···,

i,n

) be the

(unknown) vector denoting mean arrival rate of all Type i jobs to the worksta-

tions. The vector

is given by



I −(P

)



−1

where I is an n ×n identity matrix and (P

)

is the transpose of P

Once the arrival rates for the various product types have been determined, the total

arrival rate of jobs to Workstation k is given by the sum of the different product

types; that is

∑

i=1

i,k

where m is the total number of product types within the factory.

6.1 Product Flow Rates 161

After the product ﬂow rates have been computed, it is straight-forward to obtain

the expected number of visits to each workstation per product type. For example, if

a given product type arrives from an external source at a rate of 5 per hour, but the

calculated total arrival rate to a workstation is 10 per hour, it follows that each job

visit the workstation an average of two times. This reasoning leads to the following

property.

Property 6.2. Consider a factory of n workstations with m different job types,

and let the arrival rate of Job Type i from an external source be given by

∑

k=1

i,k

. Then the expected number of visits to Workstation k by Job Type i is

i,k

∑

j=1

i, j

, where

i,k

is the arrival rate as determined by Property 6.1.

Example 6.1. To demonstrate Property 6.1, we take advantage of two examples from

the previous chapter. Consider a four workstation facility that processes two prod-

ucts with each product arriving to the ﬁrst workstation according to individual Pois-

son arrival streams, each at a rate of 5 per hour. Product 1 uses only the ﬁrst three

workstations with the routing structure displayed in Fig. 5.4 (p. 141). Product 2 uses

all four workstations with the routing structure displayed in Fig. 5.5 (p. 144). To de-

termine the mean arrival rate to each workstation of Type 1 jobs is simply to repeat

the steps of Example 5.6 yielding

=(5.690,4.267,5.263,0) .

The calculations necessary to give the mean arrival rates for Type 2 jobs are con-

tained in Step 1 of Example 5.7 and are

=(6.25,6.667,5.833, 12.5) .

The total rate into each workstation is merely the sum of the individual product

inﬂows; namely

∑

i=1

, and is given as

=(11.940,10.934,11.096,12.5) .

The average number of visits of Job Type 1 to Workstation 1 is 1.138, but the average

number of visits of Job Type 1 to the second workstation is slightly less than 1

(actually it equals 0.8534) implying that some jobs bypass Workstation 2 completely

and some jobs visit the workstation more than once. The most visited workstation

by a single product type is the fourth workstation that has each Job Type 2 visiting

it an average of 2.5 times. 

162 6 Multiple Product Factory Models

6.2 Workstation Workloads

Once the workstation arrival rates by product type have been determined, the work-

load for each workstation can be computed. Again, the previously used notation for

the service time is extended for product types by including a second index to denote

the t ype; namely, the mean and squared coefﬁcient of variation of the processing

time of Product i at Workstation k are denoted as E[T

(i,k)] and C

(i,k), respec-

tively. By Deﬁnition 6.1, the workload at Workstation k, WL

, is computed as the

sum of the product visits multiplied by their respectively mean processing times;

that is,

∑

i=1

E[ T

(i,k)] , (6.1)

where m is the total number of product types within the factory.

The utilization factor, u

, for Workstation k is then the workload divided by the

available capacity; thus,

∑

i=1

i,k

E[ T

(i,k)]

, (6.2)

where c

is the number of identical processors available at Workstation k to handle

the workload.

Example 6.2. We return to Example 6.1 and assume that there is one machine at each

workstation and that the processing time data for the two products are as given in

Table 6.1. Since there is one machine per workstation, the workload and utilization

Table 6.1 Processing time characteristics for Example 6.2

Workstation kE[T

(1,k)] C

(1,k) E[T

(2,k)] C

(2,k)

1 1/14 hr 0.8 1/15 hr 1.33

2 1/10 hr 1.2 1/18 hr 2.00

3 1/15 hr 1.5 1/12 hr 1.50

4 — — 0.06 hr 0.75

factors are the same at each workstation so that

u =(0.8231,0.7971,0.8369, 0.75) .

With utilization factors all less than 1.0, the factory can achieve steady-state and

further analysis is possible. 