introducing buffer inventories in the rbd

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INTRODUCING BUFFER INVENTORIES INTHE RBD ANALYSIS OF PROCESS

PRODUCTION SYSTEMS

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CONTENT

Introduction

Objective

Throughput analysis of process production lines by means of

RBDs

Proposition of RBD models for buffered production lines

Validation of proposed method.

Conclusion

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INTRODUCTION

Modeling a system by using RBDs is used to make reliability and availability

analysis.

Presence of buffer inventory plays a role on the propagation of the effect of

failure along the entire systems.

Buffer inventory level should provide proper isolation time before material

starvation and blocking of production.

The buffer inventory level between two production stages.

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OBJECTIVE

To develop RBDs capability to analyze material buffering

To improve RBDs potentials toward including analysis of material starvation and

blocking of production.

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THROUGHPUT ANALYSIS OF PROCESS PRODUCTION LINES BY

MEANS OF RBDS

RBDs allows to reach a multileveled hierarchical model by using a simple procedure.RBDs can deal with complex production structures.

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PROPOSITION OF RBD MODELS FOR BUFFERED

PRODUCTION LINES

Three different RBD models are proposed for analyzing the behavior of

buffered production lines

First RBD model does not consider the presence of buffers

Second model is based on the assumption that buffers perfectly isolate

the production stages of the line.

Third model considers a non perfect isolation of buffers.

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Model 1.

The subsystems are kept as two coupled components so that when a failure occurs

at a subsystem, the other also stops.

Series logic of production system7

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RBD model of the production line.8

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Availability and throughput analysis

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Model 2.

The buffer perfectly isolates the two subsystems.

The buffer inventory is high enough to avoid a propagation of the failure

downward, when the failure occurs upwards.

The buffer capacity is high enough to avoid propagation upward, when the

failure occurs downward.

Buffered line with perfect isolation10

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where:

K = 1, 2(1 = upward and 2 = downward subsystem);

Ak = subsystem availability;

Thknom= nominal throughput of the subsystem k;

Thbest= throughput of the line (best case).11

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Model 3.

Its distinctive feature is the non perfect isolation attitude of the buffer.

Time losses can be propagated along the line in conjunction with blocking or

starvation effects.

RBD logics can be used to identify the state of the subsystems with respect to the

production flow degradation

(i) PFD corresponds to a totalreduction of production flow,in case the failure occurs

in a subsystem behaving as a series reliability structure;

(ii) PFD corresponds to a partial reduction of production flow, in case the failure

occurs in a MSS reliability structure (the impact factor allows to determine the

reduction rate);(iii) PFD corresponds to a noreduction of production flow,in case the failure occurs

in a standby or, in general, in a parallel structure .

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Modeling the dynamics of the buffer inventory level

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W1,kno PFD happens after failureW2,kwhen a partial PFD happens after failure

W3,k- when a total PFD happens after the failure



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Analytical solution of the model

Identification of the bottleneck and the non-bottleneck subsystems.

State space analysis of the non-bottleneck subsystem.

Joint availability analysis of non-bottleneck subsystem and the buffer

inventory.

Availability and throughput analysis of the line.

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Probability of PFDs

Considering the degraded states wi,1of the non-bottleneck subsystem (i.e.,

subsystem upward, so k is fixed to 1).

where:

0,i,1is the failure rate leading to a transition from the nominal state (indexedwith 0) of the non-bottleneck subsystem (indexed with 1)to any ith degraded

state having a partial or total PFD;

I is total number of degraded states having a partial or total PFD.15



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Probability that PFDs cause a time loss.

where:

ILb,t1is the buffer inventory level at time t1;

t1is the time when a machine failure occurs;t2,iis the time when the buffer becomes empty;

t2,it1is the isolation time toward material starvation.

M(t) is the maintainability of the machine under repair 16



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Joint availability

The expected time loss per failure of the non-bottleneck can be calculated by,

Total number of failures at the non-bottleneck can be,

where T is the time when the line is open for production.

Equivalent subsystem availability, expressing the joint

availability provided by the non-bottleneck subsystem andthe buffer inventory is

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where:

Ak = subsystem availability;

Thknom= nominal throughput of the subsystem k;

Threal= throughput of the line

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VALIDATION OF THE PROPOSED METHOD

The case of a production line taken out from the mining industry is carried on in orderto demonstrate proposed method.

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Design of simulation experiments

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Three data sets were considered sufficient for the validation along a range of

operating conditions, adequately ample to cover real situations as well as to tackle

border cases.

Input data sets

A first data set was selected in order to test the models under conditions resembling

the real industrial case (data set 1).

a condition of quasi perfectisolation(data set 2)

a condition of quasi perfect coupling between the upward and downward

subsystems (data set 3).

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TH1nom=1.1 units/h;

TH2nom=1.0 units/h.

BC = 4.8 units;

Ilmin = 0.5xBC = 2.4 units.

Data set 1

Data set 2

Data set 3

TH1nom=1.1 units/h;

TH2nom

=1.0 units/h.

TH1nom=1.1 units/h;TH2

nom=1.0 units/h.

BC = 15 units;

Ilmin = 0.5xBC = 7.5 units.

BC = 1units;Ilmin = 0.5xBC = 0.5units.

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Performance analysis

Model 3.

Model 1.

Aworst = 91.90%

Thworst = 0.92 unit/h

Model 2.

Abest = 96.93%Thbest = 0.97 unitlh

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CONCLUSION

The main result of this paper is the demonstration of a new modeling approach to

introduce buffer inventories in the RBD analysis of a production line with complexreliability structures.

The method is based on

oIntegration of the RBD availability analysis with a state space analysis of the

PFDs and related buffer dynamics.

oThe analytical modeling of the joint availability resulting from the combinedaction of the non-bottleneck subsystem and the buffer.

APPLICATION

When BC is low model 1 should be used.

Greater the size of BC model 2 and model 3 are advisable.The quasi perfect isolation by model 2 can achieved only in specific conditions.

Model 1 &3 are used where machines have repair time exceeding isolation time

guaranteed by the buffer.

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REFERENCES

Marco Macchi Introducing buffer inventories in the RBD analysis of process

production systems Reliability Engineering and System Safety 104 (2012) 8495

Barabady J, Kumar U. Reliability analysis of mining equipment: a case study of a

crushing plant at Jajarm Bauxite Mine in Iran. Reliability Engineering & System Safety

2008;93:64753.

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introducing buffer inventories in the rbd

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