Form rincian soal ujian mata kuliah CO OR.doc

7/28/2019 Form rincian soal ujian mata kuliah CO OR.doc

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Session 18 Layout Design

Production Line Balancing.

.

D 0 8 5 4Supply Chain : Manufacturing and Warehousing


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Bina Nusantara University

2


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Bina Nusantara University

3


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Assembly line balancing is traditionally thought of as afacilities design and layout problem

There are a variety of factors that contribute to the difficulty of the problem Precedence constrains: some tasks may have to be completed in a

particular sequence Zoning restriction: Some tasks cannot be performed at the same

workstation

Let t 1, t 2, , t n be the time required to complete the respectivetasks

Assembly Line Balancing


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The total work content (time) associated with the production of anitem, say T , is given by


==n

iit T 1

For a cycle time of C, the minimum number of workstations possibleis [ T /C], where the brackets indicate that the value of T /C is to berounded to the next larger integer

Ranked positional weight technique: the method places a weight oneach task based on the total time required by all of the succeedingtasks. Tasks are assigned sequentially to stations based on theseweights


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Example 8.11 The Final assembly of Noname personal computers, a generic mail-order PCclone, requires a total of 12 tasks. The assembly is done at the Lubbock,

Texas, plant using various components imported from the Far East. Thenetwork representation of this particular problem is given in the following

figure.


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Precondition The job times and precedence relationships for thisproblem are summarized in the table below.

Task Immediate Predecessors Time1 _ 12

2 1 63 2 64 2 25 2 26 2 127 3, 4 78 7 5

9 5 110 9, 6 411 8, 10 6

12 11 7


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Assembly Line Balancing:Assembly Line Balancing:Helgeson and Birnie Heuristic (1961)Helgeson and Birnie Heuristic (1961)

Ranked positional weight technique

The solution precedence requires determining the positional weight of each task.

The positional weight of task i is defined as the time required toperform task i plus the times required to perform alltasks having task i as a predecessor.

t3 + t 7 + t 8 + t 11 + t 12 = 31Task Time Positional

Weight

1 12 702 6 58

3 6 31

4 2 27

5 2 20

6 12 29

7 7 25

8 5 18

9 1 18

10 4 17

11 6 1312 7 7

The ranking: 1, 2, 3, 6, 4, 7, 5, 8, 9, 10, 11, 12

ti = 70, and the production rate is a unit per 15 minutes;The minimum number of workstations = [70 / 15] = 5


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Assembly Line Balancing:Helgeson and Birnie Heuristic (1961)

The ranking: 1, 2, 3, 6, 4, 7, 5, 8, 9, 10, 11,

12

Task ImmediatePredecessors

Time

1 _ 12

2 1 63 2 64 2 25 2 2

6 2 127 3, 4 78 7 59 5 1

10 9, 6 411 8, 10 612 11 7

Station 1 2 3 4 5 6

Tasks 1 2, 3, 4 5, 6, 9 7, 8 10, 11 12

Processing time 12 14 15 12 10 7

Idle time 3 1 0 3 5 8

C=15


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T2=6

C=15

Station 1 2 3 4 5 6

Tasks 1 2,3,4 5,6,9 7,8 10,11 12


Idle time 3 1 0 3 5 8

Cycle Time=15

T1=12

T2=6 T3=6 T4=2

T5=2 T6=12 T9=1

T5=2

T8=5T7=7 T10=4

T10=4 T11=6 T12=7

T12=7

15 Evaluate the balancing results bythe efficiency ti/NC

The efficiencies for C=15 is 77.7%,C=16 is 87.5%, andC=13 is 89.7% isthe best one

Helgeson and Birnie Heuristic (1961)


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C=15

Station 1 2 3 4 5 6

Tasks 1 2,3,4 5,6,9 7,8 10,11 12


Idle time 3 1 0 3 5 8

Helgeson and Birnie Heuristic (1961)Helgeson and Birnie Heuristic (1961)

Station 1 2 3 4 5

Tasks 1 2,3,4,5

6,9 7,8,10 11,12

Idle time 4 0 3 0 3

C=16Increasing the cycletime from 15 to 16,the total idle timehas been cut downfrom 20 min/units to10 improvement in

balancing rate.

The production ratehas to be reduced fromone unit/15 minutes toone unit/16minute;

Station 1 2 3 4 5 6

Tasks 1 2,3 6 4,5,7,9 8,10 11,12

Idle time 1 1 1 1 4 0

C=13


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C=15

Station 1 2 3 4 5 6

Tasks 1 2,3,4 5,6,9 7,8 10,11 12


Idle time 3 1 0 3 5 8

Helgeson and Birnie Heuristic (1961)

Station 1 2 3 4 5

Tasks 1 2,3,4,5

6,9 7,8,10 11,12

Idle time 4 0 3 0 3

C=16

Station 1 2 3 4 5 6

Tasks 1 2,3 6 4,5,7,9 8,10 11,12

Idle time 1 1 1 1 4 0

C=13

13 minutes appear to be the minimumcycle time with sixstation balance.

Increasing thenumber of stationsfrom 5 to 6 resultsin a greatimprovement in

production rate;


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Stochastic Scheduling: Static Case

Single machine case: Suppose that processing times are randomvariables. If the objective is to minimize average weighted flow time,

jobs are sequenced according to expected weighted SPT. That is, if jobtimes are t 1, t 2, . . ., and the respective weights are u 1, u 2, . . . then job i precedes job i+1 if

E(t i )/u i < E(t i+1 )/u i+1

Multiple Machines: Requires the assumption that the distribution of jobtimes is exponential, (memoryless property). Assume parallelprocessing of n jobs on two machines. Then the optimal sequence is toto schedule the jobs according to LEPT (longest expected processingtime first).

Johnsons algorithm for scheduling n jobs on two machines in thedeterministic case has a natural extension to the stochastic case aslong as the job times are exponentially distributed.


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Stochastic Scheduling: Queueing Theory

A typical queueing process

The basic phenomenon of queueing arises whenever a shared facility needs tobe accessed for service by a large number of jobs or customers. (Bose)

The study of the waiting times, lengths, and other properties of queues.(Mathworld)

Applications:

Telecommunications Health servicesTraffic control Predicting computer performance

Airport traffic, airline ticket sales Layout of manufacturing systemsDetermining the sequence of computer operations

Service FacilityCustomers arriving Served customers leaving

Discouraged customers leaving


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Examples of Queueing Theory

http://www.bsbpa.umkc.edu/classes/ashley/Chaptr14/sld006.htm


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Stochastic Scheduling: Dynamic Analysis

View network as collections of queues

FIFO data-structures

Queuing theory provides probabilistic analysis of these queues

Typical operating characteristics of interest include: Lq = Average number of units in line waiting for service

L = Average number of units in the system (in line waiting for service andbeing serviced)

Wq = Average time a unit spends in line waiting for service W = Average time a unit spends in the system Pw = Probability that an arriving unit has to wait for service Pn = Probability of having exactly n units in the system P0 = Probability of having no units in the system (idle time) U = Utilization factor, % of time that all servers are busy


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Characteristics of Queueing Processes

Arrival pattern of customers

Service pattern of servers

Queue discipline

System capacity

Number of service channels

Number of service stages


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Arrival pattern of customers Probability distribution describing the times between successive

customer arrivals Time independent Stationary arrival patterns Time dependent Non-stationary

Batch or Bulk customer arrivals Probability distribution describing the size of the batch

Customers behavior while waiting Wait no matter how long the queue becomes

If the queue is too long, customer may choose not to enterinto the system

Enter, wait, and choose to leave without being serviced If there is more than one waiting line, customer may

switch jockey


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Arrival pattern of customers

Service pattern of servers Single or Batch May depend on the number of customers waiting state dependent Stationary or Non-stationary

Queue discipline Manner in with customers are selected to service First Come First Served (FCFS) Last Come First Served (LCLS) Random Selection for Service (RSS) Priority Schemes

Preemptive case Non-preemptive case


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Arrival pattern of customers Service pattern of servers Queue discipline System capacity

Finite queueing situations = Limiting amount of waiting room

Number of service channels Single-channel system Multi-channel system, generally assumed that parallel channels

operate independently of each other

Number of service stages


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Notation Used in Queueing Processes

Full notation: A / B / X / Y / Z Shorthand: A / B / X

A indicates the interarrival-time distribution Assumes: Y is infinity,B the probability distribution for service time Z = FCFSX number of parallel service channels

Y the restriction on system capacityZ the queue discipline (FCFS)

Symbol = Explanation

AB

M = Exponential, D = Deterministic, E k = Erlang type

Hk = Mixture of k exponentials, PH = Phase type, G = General

X Y

1, 2, ... , infinity1, 2, ... , infinity

Z FCFS, LCLS, RSS, PR = priority, GD = general discipline


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Queueing Processes: Littles Formulas

One of the most powerful relationships in queueing theory was developed by

John D.C. Little in the early 1960s.Formulas:

and ,

where is an average rate of customers entering the system, and

W is an expected time customer will spend in the system

W L = qq W L =

1

2

3

Number of customersin system

Time, t

t1 t2 t3 t4 t5 t6 t7 T


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Poisson Process & Exponential Distribution

M: stands for "Markovian", implying exponential distribution for servicetimes or inter-arrival times, that carries the memoryless property

past state of the system does not help to predict next arrival / departure

n+1nn-1


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Calculating ExpectedSystem Measures for M/M/1

The utilization rate: = / P0= 1

P i= i(1 ), for i = 1, 2, 3,

these formulas hold only if < 210

CHARACTERISTIC SYMBOL FORMULA

Utilization / Exp. No. in System L / ( ) = / (1-)Exp. No. in Queue Lq 2/ ( ) = 2 / (1-)Exp. Waiting Time W=L/ 1 / ( ) = / (1-)Exp. Time in Queue W q=L q / / ( ) = 2 / (1-)Prob. System is Empty P 0 1 ( / ) = 1 -


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Calculating ExpectedSystem Measures for M/M/m

http://www.ece.msstate.edu/~hu/courses/spring03/notes/note4.ppt


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Assumption- m servers- all servers have the same service rate - single queue for access to the servers- arrival rate n =

- departure rate

( )( )+=

==

,1,,

1,,2,1,0,

mmnm

mnnn

m-1 m m+1210

m

mm(m-1)32


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( ) ( )( )1

11

00 1!!

=+=

mm

nm

P mm

n

n

( ) 021

1! P

m

m L

mm

q

=

+

m L L q += qq LW = 1+= qW W

m-1 m m+1210

m

mm(m-1)32


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Example

Unisex hair salon runs on a first-come, first-served basis. Customersseem to arrive according to a Poisson process with mean arrival rateof 5/hr. Because of Ms. H.R. Cutts excellent reputation, customersare always willing to wait. Average service time of 10 min isexponentially distributed.

Calculate the average number of customers in the shop and theaverage number of customers waiting for a haircut.

Calculate the percentage of time an arrival can walk right in withouthaving to wait at all.

The waiting room has only 4 seats. What is the probability that acustomer upon arrival rill have to stand?

Calculate average system waiting time, and the line delay.


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Other Systems

M/M/1/K - system with a capacity K eff = effective arrival rate

M/D/1; M/G/1; M/G/

Assignment: download the QTS add-in for Excel software to check the homework problems answers

http://www.geocities.com/qtsplus/DownloadInstructions.htm#DOhttp://www.geocities.com/qtsplus/DownloadInstructions.htm#DOWNLOAD_INSTRUCTIONSWNLOAD_INSTRUCTIONS

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