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a. Production Rate
b. Plant Capacity
c. Utilization and Availability (Reliability)
d. Manufacturing Lead Time
e. WorkinProcess

*PRODUCTION CONCEPTS AND
MATHEMATICAL MODELS*

A number of production concepts are
quantitative, or they require a quantitative approach to measure them. The
purpose of this section is to define some of these concepts. In subsequent chapters,
we refer back to these production concepts in our discussion of specific topics
in automation and production systems. The models developed in this section are
ideal, in the sense that they neglect some of the realities and complications
that are present in the factory. For example, our models do not include the
effect of scrap rates. In some manufacturing operations, the percentage of
scrap produced is high enough to adversely affect production rate, plant
capacity, and product costs. Most of these issues are considered in later
chapters as we focus on specific types of production systems.

**1 ****Production Rate**

The production rate for an
individual processing or assembly operation is usually expressed as an hourly
rate, that is, parts or products per hour. Let us consider how this rate is
determined for the three types of production: job shop production, batch
production, and mass production.

For any production operation,
the *operation cycle time T*_{c}
is defined as the time that one work unit spends being processed or assembled.
It is the time between when one work unit begins processing (or assembly) and
when the next unit begins. *T*_{c}
is the time an individual part spends at the machine, but not all of this time
is productive (recall the Merchant study, Section 2.2.2). In a typical
processing operation, such as machining, *T*_{c}
consists of: (1) actual machining operation time, (2) workpart handling time,
and (3) tool handling time per workpiece. As an equation, this can be
expressed:

where *T*_{c}=operation cycle time (min pc), *T*_{o}=time of the actual processing or assembly operation
(min pc), *T*_{h}=handling time
(min pc), and *T*_{th}=tool
handling time (min pc). The tool handling time consists of time spent changing
tools when they wear out, time changing from one tool to the next, tool
indexing time for indexable inserts or for tools on a turret lathe or turret
drill, tool repositioning for a next pass, and so on. Some of these tool
handling activities do not occur every cycle; therefore, they must be spread over
the number of parts between their occurrences to obtain an average time per
workpiece.

where *T*_{b}=batch processing time (min), *T*_{su}=setup time to prepare for the batch( min), Q=batch
quantity (pc), and *T*_{c}=operation
cycle time per work unit (min cycle). We assume that one work unit is completed
each cycle and so *T*_{c} also
has units of min pc. If more than one part is produced each cycle, then Eq.
(2.9) must be adjusted accordingly. Dividing batch time by batch quantity, we
have the average production time per work unit *T*_{p} for the given machine:

The average production rate
for the machine is simply the reciprocal of production time. It is usually
expressed as an hourly rate:

where *R*_{p}=hourly
production rate (pc hr), *T*_{p}=average
production time per minute (min pc), and the constant 60 converts minutes to
hours.

For *job shop production* when quantity Q=1, the production time per work
unit is the sum of setup and operation cycle times:

For job shop production when
the quantity is greater than one, then this reverts to the batch production
case discussed above.

For *quantity type mass production*, we can say that the production rate
equals the cycle rate of the machine (reciprocal of operation cycle time) after
production is underway and the effects of setup time become insignificant. That
is, as Q becomes very large, A *T*_{su} QB ** S**
0 and

where *R*_{c}=operation
cycle rate of the machine (pc hr), and *T*_{c}=operation
cycle time (min pc).

For *flow line mass production*, the production rate approximates the
cycle rate of the production line, again neglecting setup time. However, the
operation of production lines is complicated by the interdependence of the
workstations on the line. One complication is that it is usually impossible to
divide the total work equally among all of the workstations on the line;
therefore, one station ends up with the longest operation time, and this
station sets the pace for the entire line. The term *bottleneck station* is sometimes used to refer to this station. Also
included in the cycle time is the time to move parts from one station to the
next at the end of each operation. In many production lines, all work units on
the line are moved simultaneously, each to its respective next station. Taking
these factors into account, the cycle time of a production line is the sum of
the longest processing (or assembly) time plus the time to transfer work units
between stations. This can be expressed:

where *T*_{c}=cycle time of the production line (min cycle), *T*_{r}=time to transfer work
units between stations each cycle (min pc), and Max *T*_{o}=operation time at the bottleneck station (the maximum
of the operation times for all stations on the line, min cycle). Theoretically,
the production rate can be determined by taking the reciprocal of *T*_{c} as follows:

where *R*_{c}=theoretical
or ideal production rate, but let us call it the cycle rate to be more precise
(cycles hr), and *T*_{c}=ideal
cycle time from Eq. (2.14) (min cycle).

Production lines are of two
basic types: (1) manual and (2) automated. In the operation of automated
production lines, another complicating factor is reliability. Poor reliability
reduces the available production time on the line. This results from the interdependence
of workstations in an automated line, in which the entire line is forced to
stop when one station breaks down. The actual average production rate *R*_{p} is reduced to a value that
is often substantially below the ideal *R*_{c}
given by Eq. (2.15). We discuss reliability and some of its terminology in
Section 2.4.3. The effect of reliability on automated production lines is
examined in Chapters 18 and 19.

It is important to design the
manufacturing method to be consistent with the pace at which the customer is
demanding the part or product, sometimes referred to as the *takt* time (a German word for cadence or
pace). The takt time is the reciprocal of demand rate, but adjusted for the
available shift time in the factory. For example, if 100 product units were
demanded from a customer each day, and the factory operated one shift day, with
400 min of time available per shift, then the takt time would be 400 min 100
units=4.0 min work unit.

** 2 Production
Capacity**

We mentioned production capacity in our
discussion of manufacturing capabilities (Section 2.3.3). *Production capacity* is defined as the maximum rate of output that a
production facility (or production line, work center, or group of work centers)
is able to produce under a given set of assumed operating conditions. The
production facility usually refers to a plant or factory, and so the term *plant capacity* is often used for this
measure. As mentioned before, the assumed operating conditions refer to the
number of shifts per day (one, two, or three), number of days in the week (or
month) that the plant operates, employment levels, and so forth.

The number of hours of plant
operation per week is a critical issue in defining plant capacity. For
continuous chemical production in which the reactions occur at elevated
temperatures, the plant is usually operated 24 hr day, 7 day wk. For an
automobile assembly plant, capacity is typically defined as one or two shifts.
In the manufacture of discrete parts and products, a growing trend is to define
plant capacity for the full 7day week, 24 hr day. This is the maximum time
available (168 hr wk), and if the plant operates fewer hours than the maximum,
then its maximum possible capacity is not being fully utilized.

Quantitative measures of
plant capacity can be developed based on the production rate models derived
earlier. Let PC=the production capacity of a given facility under
consideration. Let the measure of capacity=the number of units produced per
week. Let n=the number of machines or work centers in the facility. A *work center* is a manufacturing system in
the plant typically consisting of one worker and one machine. It might also be
one automated machine with no worker, or multiple workers working together on a
production line. It is capable of producing at a rate *R*_{p} unit hr, as defined in Section 2.4.1. Each work
center operates for H hr shift. Provision for setup time is included in *R*_{p} , according to Eq. (2.11).
Let S denote the number of shifts per week. These parameters can be combined to
calculate the production capacity of the facility:

where PC=production capacity of the facility
(output units wk), n=number of work centers producing in the facility, S=number
of shifts per period (shift wk), H=hr shift (hr), and *R*_{p}=hourly production rate of each work center (output
units hr). Although we have used a week as the time period of interest, Eq.
(2.16) can easily be revised to adopt other periods (months, years, etc.). As
in previous equations, our assumption is that the units processed through the
group of work centers are homogeneous, and therefore the value of *R*_{p} is the same for all units
produced.

**EXAMPLE 2.3** **Production Capacity**

The turret lathe section has
six machines, all devoted to the production of the same part. The section
operates 10 shift wk. The number of hours per shift averages 8.0. Average
production rate of each machine is 17 unit hr. Determine the weekly production
capacity of the turret lathe section.

If we include the possibility
that each work unit is routed through n_{o} operations, with each
operation requiring a new setup on either the same or different machine, then
the plant capacity equation must be amended as follows:

where n_{o}=number of distinct
operations through which work units are routed, and the other terms have the
same meaning as before.

Eq. (2.17) indicates the
operating parameters that affect plant capacity. Changes that can be made to
increase or decrease plant capacity over the short term are:

1. Change the number of shifts per week (S). For
example, Saturday shifts might be authorized to temporarily increase capacity.

2. Change the number of hours worked per shift
(H). For example, overtime on each regular shift might be authorized to
increase capacity.

Over the intermediate or longer term, the
following changes can be made to increase plant capacity:

3. Increase the number of work centers, n, in the
shop. This might be done by using equipment that was formerly not in use and
hiring new workers. Over the long term, new machines might be acquired.
Decreasing capacity is easier, except for the social and economic impact:
Workers must be laid off and machines decommissioned.

4. Increase the production rate, *R*_{p} by making improvements in
methods or process technology.

5. Reduce the number of operations n_{o}
required per work unit by using combined operations, simultaneous operations,
or integration of operations (Section 1.5.2: strategies 2, 3, and 4).

This capacity model assumes
that all n machines are producing 100% of the time, and there are no bottleneck
operations due to variations in process routings to inhibit smooth flow of work
through the plant. In real batch production machine shops where each product
has a different operation sequence, it is unlikely that the work distribution
among the productive resources (machines) can be perfectly balanced.
Consequently, there are some operations that are fully utilized while other
operations occasionally stand idle waiting for work. Let us examine the effect
of utilization.

**3 Utilization and Availability**

*Utilization *refers
to the amount of output of a production facility relative to its capacity.* *Expressing this as an equation,

where U=utilization of the facility, Q=actual
quantity produced by the facility during a given time period (i.e., pc wk), and
PC=production capacity for the same period (pc wk).

Utilization can be assessed
for an entire plant, a single machine in the plant, or any other productive
resource (i.e., labor). For convenience, it is often defined as the proportion
of time that the facility is operating relative to the time available under the
definition of capacity. Utilization is usually expressed as a percentage.

**EXAMPLE 2.4** **Utilization**

A production machine operates 80 hr wk (two
shifts, 5 days) at full capacity. Its production rate is 20 unit hr. During a
certain week, the machine produced 1000 parts and was idle the remaining time.
(a) Determine the production capacity of the machine. (b) What was the
utilization of the machine during the week under consideration?

** Solution:** (a)
The capacity of the machine can be determined using the assumed 80hr week as
follows:

PC=80(20)=1600 unit wk

(b) tilization can be determined as the ratio
of the number of parts made by the machine relative to its capacity.

U=1000 1600=*0.625* *(62.5*%)

The alternative way of
assessing utilization is by the time during the week that the machine was
actually used. To produce 1000 units, the machine was operated

*Availability *is a
common measure of reliability for equipment. It is especially appropriate for
automated production equipment. Availability is defined using two other
reliability terms, *mean time between failure*
(MTBF) and *mean time to repair*
(MTTR). The MTBF indicates the average length of time the piece of equipment
runs between breakdowns. The MTTR indicates the average time required to
service the equipment and put it back into operation when a breakdown occurs.
Availability is defined as follows:

where A=availability,
MTBF=mean time between failures (hr), and MTTR=mean time to repair (hr).
Availability is typically expressed as a percentage. When a piece of equipment
is brand new (and being debugged), and later when it begins to age, its
availability tends to be lower.

**EXAMPLE 2.5** **Effect of Utilization and Availability on
Plant Capacity**

Consider previous Example 2.3. Suppose the same
data from that example were applicable, but that the availability of the
machines A=90%, and the utilization of the machines U=80%. Given this
additional data, compute the expected plant output.

** Solution:** Previous Eq. (2.16) can
be altered to include availability and utilization as follows

where A=availability and U= utilization.
Combining the previous and new data, we have

Q=0.90(0.80)(6)(10)(8.0)(17)=5875
output unit wk

**4 ****Manufacturing Lead Time**

In the competitive environment of modern
business, the ability of a manufacturing firm to deliver a product to the
customer in the shortest possible time often wins the order. This time is
referred to as the manufacturing lead time. Specifically, we define *manufacturing lead* *time *(MLT) as the total time required to process a given part or
product through the plant.* *Let us
examine the components of MLT.

Production usually consists
of a series of individual processing and assembly operations. Between the
operations are material handling, storage, inspections, and other nonproductive
activities. Let us therefore divide the activities of production into two main
categories, operations and non-operation elements. An operation is performed on
a work unit when it is in the production machine. The non-operation elements
include handling, temporary storage, inspections, and other sources of delay
when the work unit is not in the machine. Let *T*_{c}=the operation cycle time at a given machine or
workstation, and *T*_{no}=the
non-operation time associated with the same machine. Further, let us suppose* *that the number of separate operations
(machines) through which the work unit must be routed to be completely
processed=n_{o} . If we assume batch production, then there are Q work
units in the batch. A setup is generally required to prepare each production
machine for the particular product, which requires a time=*T*_{su} . Given these terms, we can define manufacturing
lead time as:

where MLT_{j}=manufacturing lead time
for part or product j (min), *T*_{suji}=setup
time for operation i (min), *Q*_{j}=
quantity of part or product j in the batch being processed (pc),

*T*_{cji}=operation cycle time for operation* *i* *(min pc),* T*_{noji}=* *non operation time associated with operation i (min), and i
indicates the operation sequence in the processing;

i=1, 2, ** p** n

To simplify and generalize
our model, let us assume that all setup times, operation cycle times, and
nonoperation times are equal for the n_{oj} machines. Further, let us
suppose that the batch quantities of all parts or products processed through
the plant are equal and that they are all processed through the same number of
machines, so that n_{oj}=n_{o}. With these simplifications, Eq.
(2.21) becomes:

where MLT=average manufacturing lead time for a
part or product (min).

In an actual batch production
factory, which this equation is intended to represent, the terms n_{o},
Q, *T*_{su}, *T*_{c}, and *T*_{no} would vary by product and by operation. These
variations can be accounted for by using properly weighted average values of
the various terms. The averaging procedure is explained in the Appendix at the
end of this chapter.

**EXAMPLE 2.6** **Manufacturing Lead Time**

A certain part is produced in a batch size of
100 units. The batch must be routed through five operations to complete the
processing of the parts. Average setup time is 3 hr operation, and average
operation time is 6 min (0.1 hr). Average non operation time due to handling,
delays, inspections, etc., is 7 hours for each operation. Determine how many
days it will take to complete the batch, assuming the plant runs one 8hr shift
day.

** Solution:** The
manufacturing lead time is computed from Eq. (2.22)

MLT=5(3+100*0.1+7)=*100 hours*

At 8 hr day, this amounts to
100 8=*12.5 days.*

Eq. (2.22) can be adapted for
job shop production and mass production by making adjustments in the parameter
values. For a job shop in which the batch size is one (Q=1), Eq. (2.22) becomes

For mass production, the Q
term in Eq. (2.22) is very large and dominates the other terms. In the case of
quantity type mass production in which a large number of units are made on a
single machine A n_{o}=1B , the MLT simply becomes the operation cycle
time for the machine after the setup has been completed and production begins.

For flow line mass
production, the entire production line is set up in advance. Also, the
nonoperation time between processing steps is simply the transfer time *T*_{r} to move the part or
product from one workstation to the next. If the workstations are integrated so
that all stations are processing their own respective work units, then the time
to accomplish all of the operations is the time it takes each work unit to progress
through all of the stations on the line. The station with the longest operation
time sets the pace for all stations.

where MLT=time between start
and completion of a given work unit on the line (min), n_{o}=number of
operations on the line; *T*_{r}=transfer
time (min), Max *T*_{o}=operation
time at the bottleneck station (min) and *T*_{c}=cycle
time of the production line (min pc).

where the symbols have the same meaning as
above, and we have substituted n (number of workstations or machines) for number
of operations n_{o} .

**5 ****Work-in-Process**

*Work-in-process
*(WIP) is the quantity of parts or products
currently located in the factory that are either being processed or are between
processing operations. WIP is inventory that is in the state of being transformed
from raw material to finished product. An approximate measure of work-in-process
can be obtained from the following, using terms previously defined:

where WIP=workinprocess in the facility (pc),
A=availability, U=utilization, PC=production capacity of the facility (pc wk),
MLT=manufacturing lead time, (wk), S=number of shifts per week (shift wk), and
H=hours per shift (hr shift). Eq. (2.26) states that the level of WIP equals
the rate at which parts flow through the factory multiplied by the length of
time the parts spend in the factory. The units for (PC) SH (e.g., pc wk) must
be consistent with the units for MLT (e.g., weeks).

Work-in-process represents an
investment by the firm, but one that cannot be turned into revenue until all processing
has been completed. Many manufacturing companies sustain major costs because
work remains in-process in the factory too long.

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