Factory Physics

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Factory Physics
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Definition
The term factory physics denotes those
manufacturing relationships that can be described
mathematically, much as the term physics refer to
physical relationships that can be framed in
mathematical terms.
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Other Definitions
Workstation Collection of one or more machines
or workers performing identical functions. Raw
Material parts purchased from outside. Component
s individual pieces that are assembled into more
complex products. Subassemblies assembly of
several components. End items a part that is
directly sold to the customer.
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Other Definitions
Bill of Materials list of parts that go into a
product. Consumables items used in production
but are not part of a product. Routing
description of the sequence of workstations
passed through by a part. Orders request from a
customer for a particular part number. Purchase
order a collection of one or more orders.
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Other Definitions
Jobs a set of physical materials that traverse
routings together with the logical
information. Throughput (TH) the average
output of a production process. Capacity of a
station upper limit in its throughput. Crib
inventory intermediate inventory location
usually at the end of a routing. Finished goods
inventory place where the end items are held
prior to shipping.
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Other Definitions
Work in process (WIP) the inventory between the
start and the end points of a production routing.
All the products between, but not including the
stock points. Cycle time (CT) average time from
release of a job at the beginning of a routing to
until it reaches an inventory point at the end of
the routing. I.e., the time the part spends as
WIP. It is more difficult to define this for the
entire product.
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Other Definitions
Lead time time allotted for production of a part
on the routing. Service level Pcycle time lt
lead time) Fill rate fraction of orders that
are filled from stock. Bottleneck rate (rb)
rate of the process center with the least long
term capacity.
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Other Definitions
Raw process time (T0) sum of the long-term
average process times of each workstation in the
line. Congestion coefficient (?) a
dimensionless coefficient that measures
congestion in the line.
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Factory 1
Consider a production line composed of four
machines. Identical jobs go through machines 1-4
repetitively. After each machine processes a job
it becomes immediately available at the next
station for processing. The line runs 24 hours a
day with breaks and lunches covered by spare
operators. For the sake of completeness we assume
that there is unlimited demand for the product.
Also assume that each machine takes 2 hours to
process a single job. From the information given
above, the capacity or rate of each machine is
the same and equals one unit every two hours or
1/2 parts per hour. Therefore, from the
definition of bottleneck, any of the four
machines can be regarded as a bottleneck,
and rb 0.5 units/hour or 12 units per day. In
assembly line terminology this is a balanced line
since all station have the same capacity. The
total amount of processing time for one unit
through this line is T0 24 8
hours. Therefore, the critical WIP level is given
by W0 rbT0 0.58 4 units
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Factory 2
This is the same four step operations as in
Factory 1but now each operation has differing
number of machines available and their processing
times are different as shown in table below.
The capacity of the line with multi-machine
stations is still defined by the rate of the
bottleneck, or slowest station in the line. i
Factory2 the bottleneck is station 2,
therefore rb 0.4 jobs per hour.
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Factory 2
The complete processing time of the line is still
the sum of the processing times for one product.
Thus T0 20 hours Regardless of whether the
line has single- or multiple-machine stations,
the critical WIP level is always defined the same
way thus W0 rbT0 0.420 8 units.
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Best Case Performance for Factory 1
To understand the behavior of Factory 1 we will
simulate several scenarios. The first scenario
is the case where we allow only one job into the
line at a time. The first job enters the line at
time zero and exits the line 8 hours later at
which time the second job enters the line, etc.
Obviously, the throughput of this line under this
scenario is 1/8 jobs per hour. Also note that the
cycle time is still T0 8 hours and the
throughput is one fourth of the bottleneck rate,
rb 0.5 jobs per hour.
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Best Case Performance for Factory 1
The second scenario is as follows. We introduce
two successive jobs into the line immediately
after the first machine completes the processing
of the first job it start working on the second
job. This process of second job following the
first job will continue down to the last
operation. Except the initial waiting of the
second job in front of the first machine while it
is processing the first job, the second job never
waits for processing ever again. Moreover, since
2 jobs exit the line in every 8 hours, the
throughput increased to 2/8 units per hour Double
that when WIP level was one and 50 of the line
capacity (rb 0.5.)
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Best Case Performance for Factory 1
Now in the third scenario we start with three
jobs immediately available in front of station 1
and they follow each other as in the second
scenario. The cycle time is still 8 hours but
throughput increased to 3/8 units per hour or 75
of rb.
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Best Case Performance for Factory 1
Finally in the fourth scenario we add the fourth
job in front of machine 1 and repeat the process.
We now see that all the stations stay busy all
the time once the system is in steady state.
Because there is no waiting at the stations, the
cycle time is still 8 hours and it produces one
penny in every two hours, thus the throughput is
now 1/2 units per hour which equals the line
capacity rb. This very special behavior, in
which the cycle time is T0 (its minimum value)
and throughput is rb (its maximum value) is only
achieved when the WIP level is set at the
critical WIP level, which for Factory1 was
evaluated as W0 rbT0 0.58 4 units.
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Best Case Performance for Factory 1
The fifth scenario is the case where we now add
the fifth job in front of the line and process 5
units at a time. Because there are only four
machines one job will always wait in front of the
first machine. Since the measure of cycle time is
the time between a job is introduced to the line
until the time it leaves the line, it now becomes
10 hours due to an extra 2 hour wait in front of
machine 1. Thus for the first time now the cycle
time is larger than the minimum cycle time of 8
hours. However, since all the stations are busy,
the throughput remains at rb 0.5. At this point
the conclusion is that each job we add increases
the cycle time by two hours with no increase in
throughput.
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Best Case Performance for Factory 1
The following table summarizes 10 different
scenarios and their associated performance
measures.
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Littles Law
Law 1. (Little's law)
TH WIP/CT

• Queue length calculations
• Since the law applies to individual stations
  then we can calculate the expected queue length
  at each station as well as station utilization.
  For example, in Factory 2, suppose it was running
  at the bottleneck rate of 0.4 jobs per hour. From
  Lttle's Law we calculate the WIP to be (for
  station 1)
• 0.4 jobs/hour 2 hours 0.8 jobs.
• Since there is one machine in station 1,
  then it is utilized by 80 of the
• time. Similarly, at station 3 the Little's
  Law predicts an average WIP of 4
• jobs. Since there are 6 machines at this
  station, the average machine
• utilization will be (4/6) 100 66.7. Note
  that the same ratio may be
• obtained by the ratio of the rate of
  bottleneck to the rate of station
• (0.4/0.6).

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Littles Law
2. Cycle time reduction From CT WIP/TH
we deduce that reducing cycle time implies
reducing WIP provided that the
throughput remains the same. Hence large queues
are a sign of opportunity to improve cycle
time as well as WIP. 3. Measure of Cycle Time
It is difficult to measure the cycle time
since we must keep track of each unit in
the system. Instead, throughput and WIP is
tracked routinely. Thus we can then use the
Little's Law to estimate CT given the other two
quantities. 4. Planned Inventory
Particularly as a just in time supplier, we may
have to maintain some finished good
inventories to supply our customers on time. This
may be expressed as the number of days of
sales inventory storage. If n days worth of
finished goods are to be stocked, then the
amount of finished goods inventory (FGI) is
given by FGI n TH
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Best Case Performance
Law 2 (Best case performance) The minimum cycle
time (CTbest) for a given WIP level, w, is given
by T0 , if w lt W0 CTbest
w/rb , otherwise The maximum
throughput (THbest) for a given WIP level, w, is
given by w/T0 , if w lt
W0 THbest rb , otherwise
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Best Case Performance
Consider now a real scenario. Four pallets are
recycled through the system with one additional
information. Suppose that instead of each job
requiring exactly two hours at each station, only
the job on pallet 1 requires 8 hours at each
station while pallets 2, 3, and 4 require zero
hours. The average processing time at each
station is (8 0 0 0)/4 2 hours. as
before and we still have rb 0.5 jobs/hour and
T0 8 hours. However, every time pallet 4
reaches a station it finds itself behind pallets
1, 2, and 3. This is absolute maximum amount of
wait each pallet can have and thus represents the
worst case. The cycle time for this system is 8
8 8 8 32 hours or 4T0 and since 4 jobs
are output each time pallet 1 finishes on station
4, the throughput is 4/32 1/8 jobs per
hour. or 1/T0 jobs per hour. Also notice that the
product of cycle time and throughput is 32 1/8
4, which is the WIP level, so as always, the
Little's Law holds.
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Worst Case Performance
Law 3 (Worst Case Performance) The worst case
cycle time for a given WIP level, w, is given
by CTworst wT0 The worst case throughput
for a given WIP level, w, is given by THworst
1/T0
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Practical Worst Case Performance
No system ever performs under best case or the
worst case scenarios. In order to have a
practical worst case we will introduce some
randomness to the system. First we need a
definition for "system state". Simply put, a
state is a description of location of jobs at
stations. We will provide the possible states of
our system with four machines and three jobs in
the table below
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Practical Worst Case Performance
The practical worst case (PWC) cycle time for a
given WIP level, w, is given by, CTpwc T0
(w-1)/rb The PWC throughput for a given WIP
level, w, is given by, THpwc w/(W0 w
-1)rb
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Measuring Congestion
We will first state that the congestion in a
production line is a function of its WIP level,
w, indicating the average cycle time that is CT
c(w). We can then use the ratio of the actual
cycle time at the critical WIP level, c(W0) to
the best possible cycle time, T0, as an indicator
of congestion, i.e.,
a c(w)/T0. The logic is
that a congested line will have a long cycle time
and hence a large ratio. The least congested
system, the best case will have c(W0) T0., so
the ratio is 1. To force to a range between 0 and
1 we normalize this measure. We use the measure
c(w)/T0 -1, instead. Second, to make a 1 for
PWC, notice that, c(W0)
T0 (W0 - 1)/rb for the PWC, so that for this
case
c(W0)/T0 -1
(W0 - 1)/W0
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Measuring Congestion
Therefore, if we further adjust the modified
ratio by multiplying by W0/(W0 - 1), it will
equal 1 for PWC. We therefore define a to be
a W0/(W0 - 1)c(w)/T0 - 1 Now, since a 0
for the best case and a 1 for the practical
worst case, since c(W0) W0T0 (by the Worst Case
Law), it turns out that a W0 for the worst
case. Now, with this modification, if a is
between 0 and 1 then the line is a fairly good
line. If, on the other hand, a is above 1 then
the line is on the "bad" side and therefore a
candidate for improvement.
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Measuring Congestion
To see this consider Factory 2. Suppose all
processes have zero variability. Then it is easy
to simulate the line to find that the WIP is
equal to the critical WIP, W0 8 jobs. The cycle
time remains at T0 20 hours. and so, the
congestion coefficient is a 0. Next, suppose
that all processing times are exponentially
distributed. In order to find the cycle time with
WIP set at W0, we must simulate the line. After
performing such a simulation we get an average
cycle time of 25.79 minutes, implying that
a W0/(W0 - 1)c(W0)/T0 - 1
8/(8-1)25.79/20 - 1 0.33. This is
substantially less than 1 indicating that the
system performs significantly better than the
PWC. The reason, as noted before is the imbalance
and the parallel machine stations. The net effect
is to reduce cycle time by excess capacity.
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Bottleneck Rates and the Cycle Time
The bottleneck rate rb plays an important role in
improving production lines since it establishes
the capacity of the line. First of all, if we
are operating a good line ( value less than 1),
then the cycle time will be very close to w/rb ,
where w is the WIP level. Hence, increasing the
bottleneck rate rb, will reduce cycle time for
any given WIP level.
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Variability

• The most prevalent sources of variability in
  manufacturing environment are
• Natural variability, which includes minor
  fluctuations in process time
• due to differences in operators, machines
  and material
• Random outages
• Setups
• Operator availability
• Recycle