u. of thessaly5th int'l conf. on manuf. syst.1 analysis of a bufferless, paced, automatic...

U. of Thessaly 5th Int'l Conf. on Manuf. Syst. 1

Analysis of a Bufferless, Paced, Automatic Transfer Line with Massive Scrapping of Material

during Long Failures

George Liberopoulos, George Kozanidis, Panagiotis Tsarouhas

University of Thessaly, Greece


Outline

• Introduction• Model description• Model analysis

– Case 1: Material has NO memory of damage during previous stoppages

– Case 2: Material has memory of damage during previous stoppages

• Effect of parameters on performance• Comparison with model in which workstation

downtimes do not have memoryless distributions


Introduction

• Manufacturing Setting– Continuous or semi-continuous processing manufacturing– High-speed, paced transfer lines with NO buffers in

between workstations

• Examples– metallurgical products– nonmetallic mineral products (e.g., ceramics, glass, and

cement)– basic chemicals– food and beverage products– paper products


Introduction

• Motivation– When a failure occurs, the section of the line upstream of

the failure stops.– This causes a gap in production downstream of the failure

(loss of productivity).– The quality of the material that is trapped in the stopped

section of the line deteriorates with time.– If the stoppage lasts long enough, the trapped material may

have to be scrapped because its quality becomes unacceptable.

– RESULT: (1) Havoc, (2) wasted material, and (3) an additional significant gap in production upstream of the failure (additional loss in productivity).


Introduction

• Our own experience:– In a pizza processing line, approximately half of the 10% drop in

efficiency of the line was due to the gap in production caused by failures while the other half was due to the gap caused by scrapping of material during long failures (Liberopoulos & Tsarouhas, 2005).

• Many other real-life situations (material solidification, too much exposure to heat, humidity, acidity, etc.)

• The problem is important from a manufacturing systems practitioner’s point of view but has not been studied from a manufacturing systems engineering perspective.


Introduction

• Literature on transfer lines and scrapping– Okamura & Yamashina (1977), Shanthikumar & Tien (1983), Jafari &

Shanthikumar (1987), Buzacott & Shanthikumar (1993), Altiok (1996), Dogan & Altiok (1998)

• Literature on transfer lines and scrap/rework– Pourbabai (1990), Yu & Bricker (1993), Gopalan & Kannan (1994),

Helber (2000), Li (2004)

• In all of the above works, wherever scrapping is involved, it is assumed that when a failure occurs at a workstation, a single part – that which is on the workstation – is either always scrapped or scrapped with a given stationary probability, independently of the failure time.


Model description

• Assumptions– M workstations in series with NO buffers in between– Space and time are discretized– Ni = Number of discrete positions of workstation i– Processing time at every position is 1 time unit (cycle)– Inexhaustible supply of raw parts upstream of the first workstation– Unlimited storage area for finished parts downstream of the last

workstation– Uptime of workstation i geometrically distributed with mean 1/pi – Downtime of workstation i geometrically distributed with mean 1/ri

Workstation 1

2 … 1 N1 2 … j 1 Ni …

Workstation i Workstation M

2 … 1 NM

… …


Model analysis

• Efficiency (availability) of workstation i in isolation

• Efficiency (availability) of workstation i in the system

• Operating time of workstation i geometrically distributed with mean

• Stoppage time of workstation i geometrically distributed with mean

ii

i i

re

r p

Mdi j

j i

E e

1 d

ip

1 dir

1 1M

di j

j i

p p

1

dd d i

i i di

Er p

E


Model analysis• Case 1: Material with NO memory of damage during

previous stoppages– Ri = stoppage time of workstation i ~ Geom ( )

– ni = maximum allowable stoppage time of workstation i

– qi = conditional probability that a part will not be scrapped from a particular position of workstation i, given that it has entered this position

– li = conditional expected time that a part spends in any position of workstation i, given that it has moved into this position

1

1 1 1i

di i i i i i i i i i

dndi

idi

l p E R R n P R n n P R n

pr

r

dir

1 1 1ind d d d

i i i i i i iq p p P R n p r


Model analysis– qi,j = conditional probability that a part will enter position j of

workstation i, given that it has entered workstation i

– li,j = conditional expected time that a part spends in position j of workstation i, given that it has entered workstation i

– Li = conditional expected flow time of a part at workstation i, given that it has entered this workstation

– Qi = conditional probability that a part will move from workstation i to workstation i + 1, given that it has entered workstation i, i.e. yield of workstation i

1,

ji j iq q

1, ,

ji j i i j i il l q l q

1,

1 1

1

1

ii i NN Nj i

i i j i i ij j i

qL l l q l

q

0 1Q , for 1, , .iNi iQ q i M


Model analysis– = unconditional probability that a part will move from workstation

i to workstation i + 1, given that it has entered the system

Note: = yield of the entire line

= line availability line yield = line efficiency

– = unconditional expected flow time of a part at workstation i

– = total unconditional expected flow time of a part in the line

îQ

0

î

i jj

Q Q

îL

1ˆˆ

i i iL L Q ˆ

TOTL

1

ˆ ˆM

TOT ii

L L

ˆMQ

1ˆd

ME Q


Model analysis

– Bi = average number of parts in workstation i

– BTOT = average number of parts in the entire line

1

M

TOT ii

B B

1 1 1ˆ ˆd d

i i i iB E L Q E L


Model analysis• Case 2: Material WITH memory of damage during

previous stoppages– Si,j = cumulative time that workstation i is stopped from the moment

that a part enters position 1 of workstation i until it exits position j of workstation i

– Q: P{Si,j = k}=?

A: For k =0,

For k > 0, consider the event that in its trajectory from position 1 to position j, a part does not stop in m out of the j positions and stops in the remaining j – m positions, and that the cumulative time that the part is stopped is k, where k > 0 . The probability of this event is

, 0 1jd

i j iP S p

( )11 1

1

m j m k j m j md d d di i i i

j kp p r r

m j m


Model analysis

is the probability that the part does not stop in m positions and stops in the remaining j – m positions (binomial).

is the probability that the cumulative time until the (j – m)th resumption of operation of workstation i following a stoppage is equal to k (Pascal or negative binomial).

To find P{Si,j = k}, add over all possible values of m:

1m j md d

i i

jp p

m

( )11

1

k j m j md di i

kr r

j m

1( )

,max 0,

11 1 ,

1

1,2, .

jm j m k j m j md d d d

i j i i i im j k

j kP S k p p r r

m j m

k


Model analysis– Each time a part enters a new position j of workstation i, where j = 2,

…, Ni, its remaining maximum allowable standstill time is ni – Si,j-1 instead of ni, as was the situation in the NO memory case.

– Conditional expected flow time of a part in position j of workstation i, given that the part has entered workstation i:

– Yield of workstation i:

– The rest of the expressions are the same as in the NO memory case.

,1 1 1 1i

dndi

i idi

pl r

r

, , 10

1 1 1 , 2, ,i

in d

n kdii j i i j id

k i

pl r P S k j N

r

, ,0

, for 1, ,i

i i

n

i i N i i Nk

Q P S n P S k i M

0 1Q


Effect of parameters on performance

• Problem instance with M = 6 identical workstations with parameters: Ni = 10, pi = 0.1, ri = 0.8, and ni = 10, i = 1,…, M.

• How do pi, ri, ni affect , , BTOT?ˆ

MQ ˆTOTL



0.8

0.85

0.9

0.95

1

0.3 0.4 0.5 0.6 0.7 0.8 0.9

i = 1

i = 2

i = 3

i = 4

i = 5

i = 6



37

40

43

46

0.3 0.4 0.5 0.6 0.7 0.8 0.9

i = 1

i = 2

i = 3

i = 4

i = 5

i = 6



89

91

93

95

97

0.3 0.4 0.5 0.6 0.7 0.8 0.9

i = 1

i = 2

i = 3

i = 4

i = 5

i = 6



0

0.2

0.40.6

0.8

1

0 5 10 15



• Other observations:– , , and BTOT are higher in the NO memory case than

in the memory case (there is less scrapping in the NO memory case).

– The effects of system parameters on system performance are more intense for downstream workstations than for upstream workstations.

– The effect of pi on system performance seems to be linear, whereas the effect of ri seems to be concave.

ˆMQ ˆ

TOTL


Comparison with modified model in which workstation downtimes do not

have memoryless distributions

• Original model:– M = 6 identical workstations with parameters: Ni = 30, pi =

1/1600, ri = 1/30, and ni = 10, 20, 40, 50, i = 1,…, M. E[downtimei] = 30; Var[downtimei] = 870

• Modified model:– Same as original model except that the workstation

downtimes are distributed as the sum of two iid geometrically distributed rv’s, each with mean 15

E[downtimei] = 30; Var[downtimei] = 420


niMemory

of damage

Model &method BTOT

10

No

Oa 0.894529 0.753069 153.750000 137.534000

Os0.894433

± 0.0000720.754544

± 0.000166153.95673

± 0.016972 137.703999± 0.024633

M0.894433

± 0.0000720.712132

± 0.000185148.977831± 0.020243

133.250201± 0.028329

Yes

Oa 0.894529 0.752624 153.674000 137.466000

Os0.894443

± 0.0000840.754142

± 0.000199153.883898± 0.020196

137.640371± 0.028884

M0.89443

± 0.0000710.711956

± 0.000184148.943598± 0.019288

133.219586± 0.026204

20

No

Oa 0.894529 0.815279 163.595000 146.340000

Os0.894449

± 0.0000750.816681

± 0.000129163.776167± 0.012448

146.48947± 0.021735

M0.894449

± 0.0000750.783508

± 0.000179160.458935± 0.016968

143.515893± 0.025492

Yes

Oa 0.894529 0.814089 163.395000 146.161000

Os0.894423± 0.00009

0.815547± 0.000175

163.580014± 0.016357

146.309703± 0.027879

M0.894408

± 0.0000750.782311

± 0.000169160.243363± 0.016295

143.322955± 0.025306

Comparison with modified model

1dE ˆ

MQ ˆTOTL


niMemory

of damage

Model &method BTOT

40

No

Oa 0.894529 0.899484 176.721000 158.082000

Os0.89441

± 0.0000820.899747

± 0.000014176.712874± 0.011722

158.053721± 0.023132

M0.894449

± 0.0000750.902633

± 0.000111178.217000± 0.010237

159.406268± 0.017173

Yes

Oa 0.894529 0.897274 176.355000 157.754000

Os0.894421

± 0.0000740.897596

± 0.000102176.358859± 0.008358

157.739103± 0.018391

M0.894437

± 0.0000630.898836

± 0.000128177.606764± 0.008757

158.858116± 0.016617

50

No

Oa 0.894529 0.926535 180.897000 161.817000

Os0.894362

± 0.0000790.926279± 0.0001

180.815499± 0.008928

161.714527± 0.018372

M0.894362

± 0.0000790.938845

± 0.000098183.441788± 0.009299

164.073864± 0.016607

Yes

Oa 0.894529 0.924187 180.510000 161.471000

Os0.894387

± 0.0000810.924019

± 0.000115180.440969± 0.009227

161.384062± 0.019852

M0.894385

± 0.0000630.934644

± 0.000098182.78321± 0.009395

163.478532± 0.015617


1dE ˆ

MQ ˆTOTL


• Observations:– , , and BTOT are higher in the NO memory case than in the

memory case for both the original and the modified models (there is less scrapping in the NO memory case).

– As ni increases, , , and BTOT increase (there is less scrapping when the material can remain still for a longer period).

– The workstation downtimes in the modified model have half the variance of the workstation downtimes in the original model.

– When ni > E[downtimei], there is less scrapping in the modified model.– When ni < E[downtimei], there is more scrapping in the modified

model.– The difference in performance between the original and the modified

models is less than 6%.


ˆMQ ˆ

TOTL

ˆMQ ˆ

TOTL


downtime pdf

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

0.0300

0.0350

0 10 20 30 40 50 60 70 80 90 100

original

modified


ni

E[downtime]

P{scrapping} = P{downtime > ni}

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