Reliability and Redundancy Allocation in ParallelSeries and SeriesParallel Systems

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Reliability and Redundancy Allocation in
Parallel-Series and Series-Parallel Systems
Chengbin CHU Institut des Sciences et de
Technologies de linformation de Troyes
(istit) Université de Technologie de Troyes (UTT)
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Plan de présentation

• Introduction
• Parallel-series systems (PS)
• Series-parallel systems (SP)
• Conclusions and perspectives

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Plan de présentation

• Introduction
• Parallel-series systems (PS)
• Series-parallel systems (SP)
• Conclusions and perspectives

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Introduction

Management of resources
Possession of reliable resources
Competitiveness
Improvement of the product design
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Introduction

• Reliability allocation
• Goal improve system reliability (Rs), tradeoff
  between reliability and cost, respect constraints
• Means Reduction of complexity, increasing ri
  (reliability allocation), introduction of
  redundancy (redundancy allocation), maintenace
  policies,
• Approaches
• Weighting (Kececioglu 91, Elegbede et Adjallah
  98)
• Optimization maximize Rs, under cost or volume
  constraints, (Misra 86, Kuo et al. 01)

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• Introduction
• Parallel-series systems (PS)
• Series-parallel systems (SP)
• Conclusions and perspectives

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• Introduction
• Parallel-series systems (PS)
• Series-parallel systems (SP)
• Conclusions and perspectives

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PS Description

• PS system s sub-systems in series
• Subsystem i (1is) ki components in active
  redundancy

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PS Description

• In practice ni types of components available
  in the market with the same functionality
• discrete reliability
• component of type k reliability pik and cost
  cik 1kni, 1is
• Goal Develop reliability and redundancy
  allocation
• Minimize Cost subject to relaibity requirement
• Maximize Reliability subject to budget constraint

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PS Description

• State of the art
• Complexity allocation of redundancy NP-hard
  (Chern 92)
• Many types of redondancy GA for PS systems
  (Coit et Smith 96), BAB for any systems (Prasad
  et Kuo 00)
• Kuo et al. 01 no method for discrete
  reliabilities, PS system, combined allocation

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Modelling Continuous reliability

• Decision variables rij and ki (i1,,N et
  j1,ki)

Mixed Non Linear Programming
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Algorithme ECAY

• yijln(1-rij) et fi(rij)hi(yij)
• Fonctions hi are assumed to be strictly convex.
• General scheme
• Sub-system i ki et Ri (reliability of subsystem
  i) assumed to be given.
• Expression of rij (j1,,ki) according to ki et
  Ri

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Algorithme ECAY

• Expression of the cost Ci(ki,Ri)
• Sub-system i Ri assumed to be given.
• Expression of the cost
• ? allows to obtain ki according to Ri but
  fonction non differentiable.
• Global system Rmin given computation of Ri
  (i1,,N) according to Rmin. The problem is

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Algorithme ECAY

• Ris are solutions of the following system
  (agt0), according to Rmin

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Algorithme D-ECAY

• Initialisation of reliability R(n)(R1(n),
  R2(n),,RN(n)) with n0 and fix e
• Solve to
  obtain ki(n) (i1,N) and compute
• For ngt0, if stop, otherwise go
  to 4.
• Solve the previous system to obtain R(n).
• nn1 and go to 2.

Method converges to optimum
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PS Discrete Reliabilities

• Mathematical Formulation

Integer non linear programming
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PS Discrete Reliabilities

• Solving sub-system i
• Model

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PS Discrete reliabilities

• Variable susbstitution zikui-xik
• Equivalent problem one-dimensional knapsack ni
  types of objects, maximize le total profit of
  objets selected to put into a knapsack with
  limited capacity

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PS Discrete Reliabilities
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PS Discrete Reliabilities

• Recursive equation
• L resolution 0 ? V ? L Vi,max
• Global problem
• Ri discrete values between 0 and 1
• Þ Impossible to determine Ri (1i s)
• Bounds over Ri

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PS Discrete Reliabilities

• Two-step approaches
• Determine Ni couples (Rij, Cij) (1is) such
  that
• Solve the global problem
• Model
• Knapsack (
  )

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PS Discrete Reliabilities

• DP subsystem i choose a reliability among Ni
  possibles
• Algorithm YCC
• 1is compute and look for Ni
  feasible solutions by dynamic programming (DP)
• Global problem DP, select among those solutions
  such that Rs Rmin, a solution with the least
  cost.

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PS Discrete reliabilities

• Complexity pseudopolynomial
• Numerical experiments (C, Pentium 4)
• s?28, li?12, ui?37, ni?28, Rmin?0.6
  0.99, rij?0.50.99, random choice of cost
  functions
• Convergence to optimum according to L
• s8 optimum reached with L10000

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• Introduction
• Parallel-series systems (PS)
• Series-parallel systems (SP)
• Conclusions and perspectives

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• Introduction
• Parallel-series systems (PS)
• Series-parallel systems (SP)
• Conclusions and perspectives

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SP Description

• SP system k sub-systems (technologies) in active
  redundancy
• sub-system i (1 i k) ni components in
  series
• Two cases rij discrete or continuous
• Objectives Reliability allocation to minimize
  cost under reliability constraint

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SP State of the art

• Few wrok on SP systems
• Jensen (70) Redundancy allocation, identical
  technologies, DP
• Marquez et Coit (04) Redundancy allocation,
  multi-states, heuristics
• Kuo et al.(01) No method dedicated to
• SP system (different technologies)
• Reliability allocation
• Continuous or discrete reliability

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SP Continuous reliability

• Notations and assumptions
• Cost function cijfij(rij)
• yijln(rij) and hij(yij)fij(rij) hij
  strictly convex
• Formulation

Decision variables
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SP continuous reliability

• Solving subsystem i

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SP Continuous reliability

• rijs are solution of (S2) which can be solved by
  dichotomy (Agt0)
• Solving global problem determination of Ri
• Yiln(1-Ri) i1,k
• Cost function Hi(Yi)

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SP continuous reliability

• Formulation
• Hi(Yi) convex if
• with
• but too restrictive

A condition of convexity
Hi convex?
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SP continuous reliability

• Heuristic determination of Ri
• Ziln(Ri) i1,k
• Cost function Gi(Zi)
• Formulation

Non convex functions and Gi(Zi) unknown

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SP continuous reliability

• The least square approximation des
  Gi(Zi)
• Approximate model
• Lagrangian relaxation

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SP continuous reliability

• Local minimum if
• Equivalent to system (Bgt0) (dichotomy)

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SP continuous reliability

• Heuristic
• 1 Generation of functions
• 2 - Solve (S3) Ri
• - Solve (S2) pour i1,k rij
• Numerical Results (C, Pentium 4)
• Function de Truelove ( )
  
• Computation time lt 0.4 s CPU for s5

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SP discrete reliabilities

• Minimize C, subject to Rmin
• The same approach as for PS systems
• Analogy to Knapsack problems
• Dynamic Programming
• Bounds on the reliability of sub-systems
• Algorithm YCC-SP pseudopolynomial

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• Introduction
• Parallel-series systems (PS)
• Series-parallel systems (SP)
• Conclusions and perspectives

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• Introduction
• Parallel-series systems (PS)
• Series-parallel systems (SP)
• Conclusions and perspectives

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Conclusions, perspectives PS systems

• Analogy between combined allocation and knapsack
  problem (discrete case)
• Algorithms which converges to optimum (ECAY, YCC)
• Use other recent methods to efficiently solve the
  knapsack problem (discrete case)
• Other constraints (weight, volume, )

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Conclusions, perspectives SP systems

• continuous reliability theoretical results and a
  new heuristic (RESS)
• Discrete reliabilities YCC-SP (convergence to
  optimum)
• Continuous case improve the heuristic and adapt
  it to different cost functions
• Discrete case recent method to solve the
  knapsack problems
• Include other constraints

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Perspectives

• Generalize to complex systems
• Consider new constraints and criteria
• Identify new problems other parameters and
  constraints related to production / maintenance
• Integrate the approaches into the (re-)design
  process