Date post: | 15-Dec-2015 |
Category: |
Documents |
Upload: | deshaun-sills |
View: | 215 times |
Download: | 0 times |
This Program is Presented This Program is Presented
as Oral Bulletin as Oral Bulletin
at XXV. National Conference at XXV. National Conference
on Operational Research/Industrial on Operational Research/Industrial Engineering Engineering
(OR/IE)(OR/IE)
( Koç University, July 4-6, 2005 )( Koç University, July 4-6, 2005 )
ARİF GÜRSOY, BELGİN VATANSEVERSCIENCE FAC., MATHS DEPART. COMP. SCIENCE,
MASTER STU.
PROF. DR. URFAT NURİYEVSCIENCE FAC., MATHS DEPART. COMP. SCIENCE,
ASSIST. PROF. MÜCELLA GÜNER
ENGINEERING FAC., TEXTILE ENGINEERING DEPART.
RESEARCH ASSIST. MURAT ERŞEN BERBERLERSCIENCE FAC., MATHS DEPART. COMP. SCIENCE
WITH LEAN MANUFACTURING COMPONENTSWITH LEAN MANUFACTURING COMPONENTS
PREPARING A MATHEMATICAL MODEL AND PREPARING A MATHEMATICAL MODEL AND A COMPUTER PROGRAM FOR LINE A COMPUTER PROGRAM FOR LINE
BALANCING PROBLEMBALANCING PROBLEM
THE AIM:THE AIM:
For line balancing in manufacturing,in order to
• Provide regular work flow,Provide regular work flow,
• Make idle times that may occur at workstations and Make idle times that may occur at workstations and cannot be removed , be at leastcannot be removed , be at least
determining the ideal production quantity.determining the ideal production quantity.
Line BalancingLine Balancing
Line balancing is the problem of assigning approximately the same Line balancing is the problem of assigning approximately the same amount of workload to each workstatition and reducing the idle time amount of workload to each workstatition and reducing the idle time to a minimal degree for smooth production.to a minimal degree for smooth production.
Lean ProductionLean Production
• Just-In-Time ProductionJust-In-Time Production
• Lean Manufacturing ComponentsLean Manufacturing Components– WorkerWorker– Cellular ProductionCellular Production– Synchronization Between Machines and WorkshopsSynchronization Between Machines and Workshops
PROCESS CHARTPROCESS CHART
Process chart is a diagram on which all Process chart is a diagram on which all tasks,control flow and material entry points in tasks,control flow and material entry points in process are indicated.process are indicated.
MANUFACTURING OPERATIONS OF TROUSERS-SKIRT MANUFACTURING OPERATIONS OF TROUSERS-SKIRT
PROCESS CHART FOR TROUSERS-SKIRTPROCESS CHART FOR TROUSERS-SKIRT
BIN and PACK NOTIONSBIN and PACK NOTIONS
Bin notion: Daily working time of a worker ( min. ) e.g. , 540 (+ overtime ) min.
Pack size notion: Required time for any production quantity
of any operation
For example;
1st pack’s size =Standart unit time of 1st operation * production
quantity
2nd pack’s size =Standart unit time of 2nd operation * production
quantity
Pack size of any operation changes according to production quantity; on the other hand,size difference between two packs at the same quantity
changes according to the standart unit times of operations which compose the packs.
GROUP CONCEPTGROUP CONCEPT
Group is the bin/bins that is filled by pack or packs.
If we consider a bin as a worker who has
daily working time + overtime;
a group consists of one or more workers.
If the group consists of two or more workers,required time for each operation is distributed equally to these
workers.
THE MATERIALTHE MATERIAL
Sewing department of garment company Sewing department of garment company
and trousers-skirt modal have chosen as material and trousers-skirt modal have chosen as material
of the problemof the problem..
THE METHODTHE METHOD• Line balancing is a planning problem and Line balancing is a planning problem and
most of the planning problems are in most of the planning problems are in NP-complete class.Bin Packing Problem NP-complete class.Bin Packing Problem (BPP) is also in NP-complete class.(BPP) is also in NP-complete class.
• BPPs are,in fact, the form of planning BPPs are,in fact, the form of planning problem obtained by reversing them.problem obtained by reversing them.
• BPP’s solution principles has considered for BPP’s solution principles has considered for the solution of line balancing problem.the solution of line balancing problem.
MATHEMATICAL MODAL OF THE MATHEMATICAL MODAL OF THE PROBLEMPROBLEM
and
( 1 )
operations’
required times ( std. unit time ); p is the production quantity and (T+t) is the maximum daily working time for a worker.
are
,
personltTltptTl ij )()()1( )(
)()(2
)(1 )(,...,, q
n
qqqttt )()(
2)(
1 )(,...,, q
n
qqqaaa
)()(3
)(2
)(1
)()(2
)(1
)()(2
)(1 )(...,,...,... q
n
qr
qr
qr
qr
qk
qk
qk
qqqaaaaaaaaaa
s ,...,, 21 )()(2
)(1 )(,...,, q
n
qqq qaaa
)( l
( 2 )
• In this case,the goal is to minimize total In this case,the goal is to minimize total space in bins.space in bins.
• Operations have partial priority according to Operations have partial priority according to (1) ordering.(1) ordering.
• Another point is that capacities are flexible Another point is that capacities are flexible because of t variable.because of t variable.
• Let’s call BPP, defined like this, as Prior Let’s call BPP, defined like this, as Prior Group Bin Packing Problem (PGBPP).Group Bin Packing Problem (PGBPP).
( 3 )
( 4 )
( 5 )
.,...,2,1,,...,2,1,),...,,( )()()(2
)(1
)()(
in
qi
qi
qi misqxxxX q
otherwise
groupitoassignedisaoperationifx
qqjq
ij,0
.,1 )()()(
)(
1
)()()(
)(
qij
n
j
qij
qi
qi xtXT
q
)()()()()( qi
qi
qi
qi
qi XTltTXT
,,...,2,1,),...,,( )()(2
)(1
)()( sqXXXX q
m
qqqq
,),...,,( )()2()1( sXXXX
( 6 )
( 7 )
( 8 )
( 9 )
s
q
qq XTXT1
)()()(
)()(
,1max qq
sq
XMTXMT
)()()()(
,1
)()( max)(
qi
qi
qi
qi
mi
qq XlXTXMTq
)(
1
)()()()(
qm
i
qi
qi
qq XTXT
( 10 )
( 11 )
( 12 )
( 13 )
( 14 )
)(
1
)()()()( ,...,2,1,,...,2,1,)(
qn
j
qqi
qij
qj sqmiltTxtp
)(
1
)()( ,...,2,1,,...,2,1,1qm
i
qqij sqnjx
)(
1
)()( ,...,2,1,,...,2,1,1qn
j
qqij sqmix
)( )(
1
)(
1
)(
q qm
i
qn
j
qij nx
sqnjmix qqqij ,...,2,1,,...,2,1,,...,2,1,10 )()()(
)(0 XTTEnkX
With these notations,mathematical modal for PGBPP becomes as follows;
( 15 )
In order to generate a synchronization between different
cells ; the most ideal value for all cells is chosen
from interval.
For this value, is determined, then it is written
in (5) and (10) formulas instead of . Thus,total idle time for determined assignment is assured.
Here ; is a variable, is the suggested plan,
is the flexibility constant and is the optimal solution.
pPppP ~~ **
)(XMT
)( tT
*P
P̂
p
p~ P̂
APPLICATION OF APPLICATION OF THE PROGRAMTHE PROGRAM