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Chapter 3 Manufacturing Models and Metrics
A variety of metrics help manage the operations ofmanufacturing companies. Manufacturing metrics aredivided into two basic categories:
1. Production performance measuresProduction rate, plant capacity, proportion uptime onequipment (reliability measure), manufacturing lead time
2. Manufacturing costs
Labor and material costs, the costs of producing theproducts, the costs of operating a given piece ofequipment
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3.1 MATHEMATICAL MODELS OF PRODUCTION PERFORMANCE
3.1.1 Production Rate
Cycle time Tc: 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 nextunit begins.
Typical cycle time for a production operation:Tc= To + Th + Tth
where Tc= cycle time (min/pc), To = processing time for theoperation (min/pc), Th = handling time(min/pc), (e.g., loadingand unloading the production machine), and Tth = toolhandling time (min/pc), (e.g., time to change tools).
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3.1.1 Production Rate
Batch production: batch time Tb = Tsu + QTc
Average production time per work unit Tp = Tb/Q
Production rate (pc/min) Rp= 1/Tp or Production rate (pc/hr)Rp= 60/Tp
where Tb = batch processing time (min), Tsu = setup time
to prepare for the batch (min), Q = batch quantity (pc),Tc= cycle time per work unit (min/cycle).
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3.1.1 Production Rate
Job shop production:
For quantity Q = 1, Tp = Tsu + Tc
For job shop production when Q is greater than one, the
production rate is determined as in batch production case.
where Tp = production time per work unit, Tsu = setup time,
Tc= cycle time.
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3.1.1 Production Rate
Forhigh quantity production ormass production, the
production rate equals the cycle rate of the machine
(reciprocal of operation cycle time) because the effects
of setup time become insignificant, i.e., as Q becomes
very large, Tsu/Q 0.
Rp = Rc= 60/Tc
where Rc= operation cycle rate of the machine (pc/hr),
Tc= operation cycle time (min/pc).
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3.1.1 Production Rate
Forflow line production, the production rate approximates
the cycle rate of the production line, i.e., neglecting setup
time.
Tc= Tr+ Max Toand Rc= 60/Tc
where Tc= cycle time of the production line (min/cycle),Tr= time to transfer work units between stations eachcycle (min/cycle), Max T
o
= operation time at thebottleneck station (the maximum of the operation times forall stations on the line, min/cycle),Rc= theoretical or idealproduction rate or cycle rate (cycle/hr).
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3.1.2 Production Capacity
Definition
The maximum rate of output that a production facility (orproduction line, work center, or group of work centers)
is able to produce under a given set of a assumed
operation conditions. It usually refers to a plant or factory
so that it is also termedplant capacity.
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3.1.2 Production Capacity
Plant capacity for facility in which parts are made in oneoperation (no = 1):
PCw= nSwHshRpwhere PCw= weekly plant capacity (unit/wk), n = number
of machines or work centers in the facility, Sw= number of
shifts per week, Hsh = number of hours per shift, Rp = hourlyproduction rate of each work center (unit/hr).
Plant capacity for facility in which parts require multipleoperations (no > 1):
PCw=o
pshw
n
RHnS
where no
= number of operations in the routing.
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3.1.2 Production Capacity
Example 3.1
Production Capacity
The turret lathe section has six machines, all devoted to theproduction of the same part. The section operates 10 shift/wk.The number of hours per shift averages 8. Average production
rate of each machine is 17 unit/hr. Determine the weekly
production capacity of the turret lathe section.
Solution
PC= 6 10 8 17 = 8160 output unit/wk
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3.1.2 Production Capacity
For short term:
Change the number of shifts per week Sw to affect plant capacity
Change the number of hours worked per shift Hsh to affect plant
capacity
For long term:
Increase the number of work centers n in the shop to increase plantcapacity
Reduce the number of operations no required per work unit by usingcombined operations, simultaneous operations, or integration ofoperations to increase plant capacity
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3.1.3 Utilization and Availability
Utilization UIt refers to the amount of output of a production facility relative to itscapacity.
U=
where Q = quantity actually produced, PC= plant capacity
Utilization can be assessed for an entire plant, a single machine in the
plant, or any other productive resources, i.e., labor. It is also defined asthe proportion of time that the facility is operating relative to the timeavailable under the definition of capacity.
PC
Q
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3.1.3 Utilization and Availability
Example 3.2
Utilization
A production machine operates 80 hr/wk (2 shifts, 5 days) atfull 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 duringthe week under consideration?
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3.1.3 Utilization and Availability
Solution(a) PC= 80 20 = 1600 unit/wk
(b) The ratio of the number of parts made by the machine
relative to its capacity
U= 1000/1600 = 0.625 = 62.5% or
The hours required to produce the given output
hr
His the time during the week that the machine was actually used.
5020
1000H
%5.62625.080
50U
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3.1.3 Utilization and Availability
AvailabilityA
It is a common measure of reliability for equipment.
It is especially appropriate for automated
production. Mean time between failure (MTBF)
Average length of time the piece of equipment runs betweenbreakdowns
Mean time to repair (MTTR)Average time required to service the equipment
and put it back into operation when a breakdown
occurs.
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3.1.3 Utilization and Availability
Availability:A =MTBF
MTTRMTBF
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3.1.3 Utilization and Availability
Example 3.3
Effect of Utilization and Availability on Plant 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 ofhours per shift averages 8. Average production rate of each machine is
17 unit/hr. The availability of the machine is 90% and the utilization of
the machine is 80%. Determine the weekly production capacity of the
turret lathe section.
SolutionQ =AU(nSwHshRp) = 0.9 0.8 6 10 8 17 = 5875 output unit/wk
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3.1.4 Manufacturing Lead Time
Definition: Manufacturing lead time (MLT) is the total time
required to process a given part or product
through the plant, including any lost time due to
delays, time spent in storage, reliabilityproblems, and so on.
Activities of production: Operation Performed on a work unit when it is in the production
machine Nonoperation Handling, temporary storage, inspection, and other
sources of delay when the work unit is not in the
production machine
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3.1.4 Manufacturing Lead Time
where MLTj= manufacturing lead time forPart orProductj(min), Tsuji= setup time for operation i(min),Qj= quantity of part or productjin the Batch beingprocessed (pc), Tcji= operation cycle time for
operation i(min/pc), and Tnoji= nonoperation timeassociated with operation i(min), and iindicates theoperation sequence in the processing; i= 1, 2,, noj,noj= number of operations on partj.
ojn
i
nojicjijsu jij TTQTMLT1
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3.1.4 Manufacturing Lead Time
Simplified model:All setup times, operation cycle times, and
nonoperation times are equal for the nojmachines. Batch quantities
of all parts or products processed through the plant are equal and
they are processed through the same number of machines, i.e.,noj= no.
MLT= no (Tsu + QTc+ Tno)
where MLT= manufacturing lead time for a part or product,no = number of operations, Tsu = setup time, Q = batch
quantity, Tccycle time per part, and Tno = nonoperation time
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3.1.4 Manufacturing Lead Time
Averaging procedure
In an actual batch production factory, the terms no, Q, Tsu,
Tc, and Tno would vary by product and by operation. These
variations can be accounted for by using properly weighted
average vales of the various terms.
The average procedure is explained in Appendix.
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3.1.4 Manufacturing Lead Time
Example 3.4
Manufacturing Lead Time
A certain part is produced in a batch size of 100 units. The batch must be
routed through 5 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 nonoperation time due to handling, delays, inspections,
etc., is 7 hr for each operation. Determine how many days it will take
to complete the batch, assuming that the plant runs one 8 hr shift/day.
Solution
MLT= no (Tsu + QTc+ Tno)
= 5 (3 + 100 0.1 + 7) = 100 hr = 100/8 days = 12.5 days
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3.1.4 Manufacturing Lead Time
For job shop production, Q = 1
For mass production, Q term in the MLT equation is verylarge and dominates the other terms. In the case ofno = 1,the MLT simply becomes the operation cycle time.
MLT= no (Tsu + Tc+ Tno)
MLT= QTc
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3.1.4 Manufacturing Lead Time
ForFlow Line mass production, the entire production lineis set up in advance. Also, the nonoperation time betweenprocessing steps is simply the transfer time Tr to move thepart or product from one workstation to the next.
MLT= no (Tr+ MAX To) = noTc or
MLT= n (Tr+ MAX To) = nTc
where MLT= time between start and completion of a given work unit on
the line, no = number of operations on the line, n = number of stationson the line, no = n, because each station completes one operation,
Tr= transfer time, Max To = operation time at the bottleneck station,
Tc= cycle time of the production line.
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3.1.5 Work-in-Process
WIP=
where WIP= work-in-process (pc),A = availability, U= utilization,PC= ideal plant capacity (pc/wk), MLT= manufacturing lead time(hr), Sw= shifts per week, Hsh = hours per shift (hr/shift).
w sh
AU PC MLT
S H
Definition:Work-In-Process (WIP) is the quantity of parts or products
currently located in the factory that either are being
processed or are between processing operations. It is
inventory that is in the state of being transformation from raw
material to finished product.
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3.2 MANUFACTURING COSTS
3.2.1 Fixed and Variable Costs
Two major categories of manufacturing costs:
1. Fixed costs - Remain constant for any output
level, cost of the factory building and productionequipment, insurance, property taxes
2. Variable costs - Vary in proportion to productionoutput level; increase with output, direct labor, raw
materials, electric power to operate the productionequipment
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3.2.1 Fixed and Variable Costs
Adding fixed and variable costs
TC= FC+ VC(Q)
where TC= total costs, FC= fixed costs (e.g., building,equipment, taxes), VC= variable costs (e.g., labor,materials, utilities), Q = output level.
When comparing automated and manual production
methods, it is typical that the fixed cost of the automatedmethod is high relative to the manual method, and thevariable cost of automation is low relative to the manualmethod.
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3.2.1 Fixed and Variable CostsFixed and variable costs as a function of production output
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3.2.2 Direct Labor, Material, and Overhead
Alternative classification of manufacturing costs:
1. Direct labor - Wages and benefits paid to workers
2. Materials - Costs of raw materials
3. Overhead - All of the other expenses associated with
running the manufacturing firm
Factory overhead
Corporate overhead
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3.2.2 Direct Labor, Material, and Overhead
Typical factory overhead expenses
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3.2.2 Direct Labor, Material, and Overhead
Typical corporate overhead expenses
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3.2.2 Direct Labor, Material, and OverheadCosts for a manufactured product
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3.2.2 Direct Labor, Material, and Overhead
Allocation of overhead costs:
Direct labor cost
Material cost
Direct labor hours
Space
Most common in industry is direct labor cost, which will
be used to illustrate how overheads are allocated andsubsequently used to compute factors such as sellingprice of the product.
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3.2.2 Direct Labor, Material, and OverheadOverhead rates
Factory overhead rate (FOHR):
FOHR=
Corporate overhead rate (COHR):
COHR=
where DLC= annual direct labor costs, FOHC= annual factoryoverhead costs, COHC= annual corporate overhead costs.
If material cost were used as the allocation basis, then material costwould be used as the denominator in both ratios.
DLC
FOHC
DLC
COHC
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3.2.2 Direct Labor, Material, and Overhead
Example 3.5
Determining Overhead Rates
Suppose that all costs have been compiled for a certain
manufacturing firm for last year. The summary is shown in
the table below. The company operates two different
manufacturing plants plus a corporate headquarters.
Determine (a) the factory overhead rate for each plant, and(b) the corporate overhead rate. These rates will be used
by the firm to predict the following years expenses.
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3.2.2 Direct Labor, Material, and Overhead
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3.2.2 Direct Labor, Material, and Overhead
Solution
(a) For plant 1,
For plant 2,
(b)
%2505.2108
1025
6
1
1
1
DLC
FOHCFOHR
%27575.2104
101.15
6
2
2
2
DLC
FOHCFOHR
%6006102.1
102.76
6
DLC
COHCCOHR
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3.2.2 Direct Labor, Material, and Overhead
Example 3.6
Estimating Manufacturing Costs and Establishing Selling
Price
A customer order of 50 parts is to be processed through plant of
Example 3.5. Raw materials and tooling are supplied by the
customer. The total time for processing the parts (including setup and
other direct lobor) is 100 hr. Direct labor cost is $ 10/hr. The factory
overhead rate is 250% and the corporate overhead rate is 600%.(a) Compute the cost of the job. (b) What price should be quoted to a
potential customer if the company uses a 10% markup.
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3.2.2 Direct Labor, Material, and Overhead
Solution
(a) The direct labor cost for the job is 100 10 = $1000.
The allocated factory overhead charge, at 250% of direct labor, is
1000 2.5 = $2500.
The total factory cost of the job, including allocated factory overhead
is 1000 + 2500 = $3500.
The allocated corporate overhead charge, at 600% of direct labor, is
1000 6 = $6000.
The total cost of the job including corporate overhead is$3500 + $6000 = $9500.
(b) If the company uses a 10% markup, the price quoted to the customerwould be 1.1 9500 = $10450.
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3.2.3 Cost of Equipment Usage
Deficiency of overhead rates
Based on labor cost only
Equipment factor neglected
Therefore, it is appropriate to divide the cost of a worker
running a machine into two components:
Direct labor
Machine
These costs apply not to the entire factory operations, but to
individual work centers.
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3.2.3 Cost of Equipment Usage
A work center can be any of the following:
One worker and one machine
One worker and several machines
Several workers operating one machine
Several workers and machines
The direct labor cost consists of the wages and benefits paidto operate the work center.
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3.2.3 Cost of Equipment Usage
Applicable factory overhead expenses allocated to direct
Labor cost include
State taxes
Certain fringe benefits
Line supervision
The machine annual cost is the initial cost of the machine
apportioned over the life of the asset at the appropriate rateof return used by the firm.
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3.2.3 Cost of Equipment Usage
Equivalent uniform annual cost is expressed as
niPAICUAC ,,/where IC= initial cost of the machine, (A/P, i, n) = capital recovery
factor that converts initial cost at year zero into a series of equivalent
uniform annual year-end values, where i= annual interest rate and
n = number of years in the service life of the equipment.
11
1,,/
n
n
i
iiniPA
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3.2.3 Cost of Equipment Usage
The uniform annual cost can be expressed as an hourlyrate by dividing the annual cost by the number of annualhours of equipment use.
The machine overhead rate is based on those factoryexpenses that are directly assignable to the machine,including power to drive the machine, floor space,maintenance and repair expenses, and so on.
The total cost rate for the work center is the sum of laborand machine costs.
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3.2.3 Cost of Equipment Usage
For a work center consisting of one worker and one machine,
hourly cost of worker-machine system:
Co = CL(1 + FOHRL) + Cm(1 + FOHRm)
where Co = hourly rate to operate the work center,
CL = direct labor wage rate, FOHRL = factory overhead rate
for labor, Cm = machine hourly rate, FOHRm = factory
overhead rate applicable to the machine.
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3.2.3 Cost of Equipment Usage
Example 3.7
Hourly Cost of a Work Center
The following data are given for a work center consisting of one
worker and one machine: direct labor rate = $10/hr, applicable
factory overhead rate on labor = 60%, capital investment in
machine = $105, service life of the machine = 8 yr,
rate of return = 20%, salvage value in 8 yr = 0, and applicable
factory overhead rate on machine = 50%. The work center will be
operated one 8 hr shift, 250 day/yr. Determine the appropriate
hourly rate for the work center.
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3.2.3 Cost of Equipment Usage
Solution
Labor cost per hour is
CL(1 + FOHRL) = 10 (1 + 0.6) = $16/hr
The number of hours per year = 8 250 = 2000 hr/yr.
/hr
2606.012.01
2.012.011
1,,/8
8
n
n
i
iiniPA
26060$2606.010,,/ 5 niPAICUAC
03.13$2000
26060mC
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3.2.3 Cost of Equipment Usage
Machine cost per hour is
Cm(1 + FOHRm) = 13.03 (1 + 0.5) = $19.55/hr
Total cost rate for the work center is
Co = CL(1 + FOHRL) + Cm(1 + FOHRm)
= 16 + 19.55
= $35.55/hr