Transcript Formulation and solution of the drum-buffer

Application of the TOC
FAL, IE, Seoul National University
1999. 11. 4
Eoksu Sim([email protected])
Contents
• General concepts and terms in TOC
• Recent domestic researches
• Formulation and solution of the drum-buffer-rope
constraint scheduling problem(DBRCSP)
• Comment
• Reference list
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General concepts and terms in TOC(1/3)
• What is the TOC?
– A production control methodology that maximizes profits in a
plant that has a demonstrated bottleneck.
– A management philosophy developed by Eliyahu M.Goldratt
which is useful in identifying core problems of an organization
• The TOC provides 5 step
–
–
–
–
–
Identifying the constraints
Deciding how to exploit the constraint
Subordinating all other activities to the constraint
Elevating the constraint
Continuous improvement step of admonishing against managerial
inertia
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General concepts and terms in TOC(2/3)
Theory of Constraints
Problem solving/
thinking process
Logistics
Five-Step
Scheduling V-A-T
focusing
process
analysis
process
DBR
ECE
diagrams
ECE
audit
Buffer
Performance system
management
Throughput
Product
Inventory
Operating expense mix
Cloud
diagrams
Five-Step
focusing
process
Throughput
dollar days
Inventory
dollar days
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General concepts and terms in TOC(3/3)
• Performance Measure under TOC
– Throughput : the rate at which the system generates money through
sales
– Inventory: All the money that the system invests in purchasing
things the system intends to sell
– Operation expense : All the money the system spends in turning
inventory into the throughput
• Several additional supporting measurement
– Throughput = selling price - raw material
– Net profit from production line = total throughput - the additional
operation expense
– ROI = net profit divided by the inventory
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Recent domestic researches
• In IE conference on 30th, Oct. 1999
– Development of the TOC-based MPS component
• Using UML, COM like Mr. Yoon
– Present conditions of the TOC
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Formulation and solution of the drumbuffer-rope constraint scheduling
problem(DBRCSP)
Int. J. Prod. Res., 1996, Vol. 34, No. 9, 2405-2420
J. V. Simons, Jr., W. P. Simpson, III, B. J. Carlson, S. W. James, C. A. Lettiere and B.
A. Mediate, Jr.
Graduate School of Logistics and Acquisition Management, Air Force Institute of
Technology, Wright-Patterson AFB, OH 45433-7765, USA
Constraint scheduling in DBR systems
• Constraints
– Bottleneck, temporary(wandering) bottleneck
– In the TOC, a capacity constraint resource(CCR) is defined to be
any resource which restricts throughput
• TOC’s five focusing steps and the idea of DBR production
systems are typically focused on two primary problems
– product mix : maximizes the overall system value
– product flow : a schedule for each constraint
• This paper will focus on product flow, as determined
constraint scheduling
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• The multiple constraint schedules cannot be generated in
isolation from each other. - interactive constraints
• Goldratt introduced the notion of rods to deal with both
interactive constraints and multiple constraint operations
for a single job
– Rods - required time lags b/w constraint operations
• time rods : rods b/w operations on different constraints
• batch rods : rods b/w operations on same constraints
– The rod’s placement in the constraint schedule
• function of two factors: transfer batch size and the relative magnitude
of the per unit processing time
– Fig. 1 and Fig. 2
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10/21
Problem formulation - general job shop problem
Minimize
 xik
(1)
i
xik  tijk  0
(i, j  1, h)  (i, j , k )
x pk  xik  H (1  yipk )  t pqk
 1  i , p  n, 1  k  m
(3)
xik  x pk  Hyipk  tijk
 1  i , p  n, 1  k  m
(4)
xik  0
 i, k
(5)
yipk  0 or 1
i, p, k
(6)
where:
tijk  processing tim e for operation j of j ob i, on m achinek
t pqk  processing tim e for operation q of j ob p, on m achinek
xik  com pletion tim e of j ob i on m achinek
yipk  indicator variable which takes on a value of 1 if j ob i precedesj ob p
ki
(directlyor indirectly) on m ahcinek, or 0 otherwise
 m achineat which the last operation of j ob i is scheduled
H  a very large num ber
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• Process batch vs. transfer batch
– The net effect is a partial relaxation of the general job shop
precedence constraint (2)
• The general job shop problem produces schedules for each
machine.
– DBR systems simplify the problem by requiring schedules only for
the constraint(s).
• In the DBRCSP, rods(time lags) may be placed b/w
operations on either the first part in a batch or the last,
depending on whether the first constraint operation is
faster or slower that the second.
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Problem
f  FS
formulation - f  0
'
ij
i
Minimize
 fj dj
 fn
(7)
(i, j )  H '
(9)
1
the general production
scheduling problem(GPSP)
(8)
fi  di  gi
i
(10)
fi  hi
i
(11)
k , t  1,2,, f n
(12)
 rik  akt
iS t
where:
FSij'  m ax{  s j  (n j  1) * p j ; s j  (ni  1) * pi ; si  ni * pi }
si
 sequence- independent setup tim e for activity i, i  1,2,...n
ni
 batch size for activity i, i  1,2,..., n
pi
 processing tim e per unit for activity i, i  1,2,..., n
(13)
H '  set of pairs of activities with finish - start relations with a tim e lag of FSij'
fi
 finish tim e of activity i, i  1,2,..., n
di
 duration of activity i, i  1,2,..., n
gi
 ready tim eof activity i, i  1,2,..., n
hi
 due date of activity i, i  1,2,..., n
rik  am ountof renewableresourcety pe k (k  1,2,..., K) requiredby activity i
akt  availability of renewableresourcety pe k during period [t - 1, t]
St  set of activities in processin tim e interval[t - 1, t]  {i | fi - di13/21
 t  fi }
• Equation 13 addresses the advantage gained by the use of
single unit transfer batches
• Consideration of all three relationships in equation 13 is
necessary to ensure the improvement in the schedule is
achievable
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Problem
formulation - f  FS
'
ij
i
The drum-buffer-rope
constraint scheduling
problem(DBRCSP)
Min(max{m ax{(fi  hi ),0}})
iC
 fj dj
(i, j )  H '
f1  0
(14)
(15)
(16)
f j  fi  (1  yij )  d j
 (i, j )  M k , k  1,2,..., K (17)
fi  f j  yij  di
 (i, j )  M k , k  1,2,..., K (18)
where:
FSij'  max{  s j  (n j  1) * p j  0.5 * bij ; s j  (ni  1) * pi  0.5 * bij }
si
 sequence- independent setup tim e for activity i, i  1,2,...n
ni
 batch size for activity i, i  1,2,..., n
pi
 processing tim e per unit for activity i, i  1,2,..., n
(19)
H '  set of pairs of activities with finish - start relations with a tim e lag of FSij'
fi
 finish tim e of activity i, i  1,2,..., n
di
 duration of activity i, i  1,2,..., n
hi
 due date of activity i, i  1,2,..., n
C  set of activities representing the final operation in a job
  a very large number
yij  {1 if activity i precedesactivity j, or 0 otherwise}
M k  set of all activities requiring processing on machinek
bij  buffersize betweenactivities i and j (0.5 * bij representsthe 15/21
rod length )
An evaluation of Goal System solutions
• Optimized Production Technology(OPT) - 1979,
proprietary
– 1990 AGI - DISASTER
– 1994 TOC Center - the Goal System
• The GS is heuristic in nature and is organized consistently
with the five focusing steps
– Essentially, GS begins by assuming that job order due dates are the
only constraints to be met
• Two aspects of the GS process are particularly noteworthy
– whenever constraint schedules are produced, rods are inserted to
protect either previously developed constraint schedule or
intervening non-constraint operations
– GS builds schedules for multiple constraints sequentially
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• Construct benchmark test problems(108 problems)
– Resource criticality factor as a measure of the severity of the load
placed on a resource
– ten resources types, two resources(overloaded)
– A plant, V plant, T plant
• Obtain multiple GS solutions
– Total days late(TDL) for both the best and worst GS solution
– Maximum tardy days(MTD) for both the best and worst GS
solution
– %difference b/w best and worst GS solutions for both TDL and
MTD
• Obtain optimal solutions
– we sought to evaluate the quality of GS solutions compared to
solutions which optimally minimize maximum tardiness(B&B) 17/21
18/21
Conclusion
• The sequence in which multiple constraints are scheduled
can make a substantial difference in the quality of GS
solutions
• Following the GS recommendation concerning which
constraint to schedule first usually produces the best
solution
• GS solution appear to do a good job of minimizing
maximum tardiness, especially when GS is run more than
once(scheduling different constraints first)and the best
solution used
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Comments
• TOC is not a really new concept.
• There are some areas that TOC can be applied.
• If you want to understand the Goldratt’s theory more easily,
you can read the book “The Goal” which is located in our
lab.
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Reference Lists
• Demeulemeester, E. L. and Herroelen, W. S., 1992, A
branch-and-bound procedure for the multiple resourceconstrained project scheduling problem. Management
Science, 38(12), 1803-1818
• Plenert, G., 1993, Optimizing theory of constraints when
multiple constrained resources exist, EJOR, 70, 126-133
• www.goldratt.com
• www.tocc.com
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