THE OPTIMIZATION OF SYSTEM RELIABILITY
- COMBINE REVISED PATTERN SEARCH AND DYNAMIC PROGRAMMING
劉家熙 周宣光
國防大學國防管理學院企管系 國立政治大學資訊管理系
ABSTRACT
The application of reliability optimization in the system design and development phase affects system availability, maintainability, logistics support requirement and related cost control greatly. In this research, we consider the system reliability optimization problem under different design constraints such as weight, volume, system and subsystem reliability requirements, etc. System reliability can be improved by enhancing component reliability level and/or adding redundant components. Of course, both component reliability and redundancy enhancement will definitely spend a lot of costs and resources. Therefore, there exists a trade-off between these two alternatives for resource-constrained reliability optimization.
Many optimization techniques have been used to allocate component reliability level and the number of redundancy individually. Few of them can solve reliability-redundancy
allocation problems simultaneously except for some kind of combinations. The proposed model developed in this paper is based on revised H-J pattern search combined with dynamic programming approach using the concept of dominating sequences. We try to solve the system reliability-redundancy allocation problem which can handle more general system configurations which include series- parallel system, element stand-by system, and n-stage mixed series system etc.. Besides general resource capacity constraints, some specific system and subsystem performance constraints such as reliability will also be considered. Through our approach, we can get the optimal allocation of component redundancy number and reliability level to minimize the total system design and development cost
A couple of numerical examples have been demonstrated for different system
configurations. The information we obtain from those is very helpful to facilitate the system design and development process. The performance comparison between this proposed approach and other reliability- redundancy allocation methods are also discussed and it turns out to be very good result. We show that this proposed approach can fit various problem types and is more realistic, flexible, even can get better results in some aspect than some traditional reliability optimization approaches.
Keywords: Reliability Optimization, Reliability-Redundancy Allocation, Revised H-J Pattern Search, Dynamic Programming, Dominating Sequences
1. INTRODUCTION
For many systems, the costs related to design and development, construction, and production, etc., are relatively well known. However, the costs occurred with the utilization, maintenance and support of the system throughout its planned life cycle are somewhat hidden. Blanchard (1998) had mentioned that a large percentage of the total life cycle cost for a given system is attributed to operating and maintenance activities and a significant part of this cost depends on the decisions made during the
phases of initial design and development. In the initial system design and
development period, reliability optimization plays a key role to affect system availability, maintenance activities,logistics requirements and related costs during its planned life cycle. In the past decades, for solving reliability optimization problems, a lot of research efforts have been put on how to find the component reliability and the number of redundancy for each stage which maximize the system reliability under some resource constraints, such as weight, volume, budget, etc.. In the recent review paper, Kuo & Prasad (2000) classified papers on reliability optimization by system structure and solution method since 1977. For reliability-redundancy allocation problems, a couple of famous methods have been published, such as Tillman et al (1977) combined the H-J Pattern Search (Hooke & Jeeves, 1961) and AGM Heuristic Approach (Aggarwal et al, 1975), Kuo et al (1987) used Lagrange Multiplier and Branch and Bound Technique (LMBB), and Xu et al (1990) combined the sequential search and XKL Heuristic Approach to solve these
problems. In addition, Hikita et al (1992) used a dynamic programming approach to solve single-constraint surrogate problems. Coit&Smith(1996) developed a problem- specific genetic algorithm to analyze series- parallel system and to select components and redundancy-levels to optimize some objective function, given system-level constraints on reliability, cost and weight. Reliability-redundancy allocation problems can also be applied in software reliability optimization. Chi et al (1990) formulated mixed integer nonlinear programming problems for reliability-redundancy allocation in software systems with common-cause failures and systems involving both software and hardware. Most of these reliability-redundancy optimization methods have their own specific applied area and limitations depending on different system configurations, objective functions and resource constraints.
In this paper, we try to solve reliability- redundancy allocation problem for more general system configurations such as series-parallel system, element stand-by system, and n-stage mixed series system etc. Also, the problem has been changed from different aspects compared to the template problem referred in a lot of previous studies that the system reliability and some subsystem reliability requirements were put into the constraints and the cost function was placed as objective function. It means we try to find the minimum cost solution under some constraints which include one to require the minimum system reliability and some others on subsystem reliability. The argument is, in reality, we need to meet some specific system and subsystem
performance requirements at the minimum development cost as possible as we can, when a new system is designed.
In section 2, we will first describe the problem statement for this paper and do the modeling work. In section 3, an optimization procedure is developed. The Dynamic Programming and Dominating Sequences will be applied to find the optimal solution in each step of revised H-J Pattern Search modified by our model, which leads to exact optimal solution on components reliability and number of redundancies at each stage. We show some numerical examples for this proposed model in section 4 and compare performance with other reliability and redundancy optimization approaches in section 5. Finally some conclusions will be presented in section 6.
Notation
xj,rj number of components and component reliability at stage j
xUj ,xLj upper bound and lower bound of xj n, m number of stage in the system,
number of resource constraints R,R0 component reliability vector, initial
component reliability vector
X vector of number of components used at each stage ) X , R (
Cs system cost function )
x , r (
Cj j j cost function of stage j )
x , r (
Gij j j the resource i required by stage j )
X , R (
Gi the resource i required by system bi,Sj the available quantity of resource i,
state variable for state j
Rs,RLs system reliability, lower bound of system reliability
Qs,QsUsystem unreliability, upper bound of system unreliability ) x , r (
Rj j j reliability function of stage j RLj lower bound of reliability of stage j C set of subsystems which has lower
bound in reliability
X*(R) the optimum redundancy allocation policy when reliability vector is R Wj, W weight of one component at stage j,
limitation on weight
Pj product of the weight and volume per element at stage j
P limitation on product of the weight and volume
αj,βj constants representing the inherent reliability characteristics of each component at stage j,
2. PROBLEM STATEMENT AND MODELING
The traditional template problem applied in many studies for reliability-redundancy optimization methods is to maximize system reliability function by conforming to different resource and related cost constraints. On the contrary, in this model we use the cost function as the objective function and try to minimize it by conforming to those constraints such as system configuration, resource capacity, system and subsystem reliability requirement etc.. The systems we consider in this model include series-parallel, element stand-by and n-stage mixed series system, etc.. It means that all subsystems of this proposed system are connected in series, and these subsystems can be either or both of parallel and stand-by configurations. The number of components xj and the component reliability rj at the stage j are the decision variables to be determined. Therefore, the template problem in this
model is defined as follows: Min ( , ) ( , ) 1C r x X R C j j n j j s = ∑ = (1) s.t. b ) x , r ( G j j i n 1 j ij ≤ ∑ = , for i = 1,…,m (2) ∏ ≥ = n j L s j j j r x R R 1 ( , ) (3) R ) x , r ( Rj j j ≥ Lj , for some j in C (4) In the above equations, equation (1) is the objective function which will be focused on system design and development cost, equation (2) is the general form of system configuration or resource capacity constraint, and equation (3), (4) are system and subsystem reliability requirement constraints. As usual, the reliability parameters: R, RLs , RLj are all real numbers between 0 and 1; X are all positive integers.
In this proposed model, the reliability constraint on equation (3) and (4) has two different forms. For the series-parallel
system,Rj(rj,xj)becomes[1−(1−rj)xj] (5) and for the element stand-by system,
) x , r ( Rj j j becomes ] )! 1 x ( ) r (ln ) 1 ( ! 2 ) r (ln r ln 1 [ r j 1 j 1 j 2 j j j x x j − − + − + − − − L (6)
To set up the problem, several assumptions need to be made:
1. All stages is connected in series, ie., 1-out-of-N: F configuration.
2. For a parallel stage, it is a 1-out-of-xj: G configuration and for a stand-by stage, we assume each switch from component to component is instant and successful.
3. All components in the same stage have the same probability of failure.
4. All the stages and all the components
used at each stage are S-independent.
5. A short circuit failure will not be considered, that is, only a single mode of failure is assumed.
6. The stage costs, resources, unreliability are additive.
7. Rj should be large enough, such that the higher order terms can be neglected.
8. Assume all components have the characteristics of electrical component ( i.e. R(t)=e−λt )
To facilitate the modeling procedure, according to assumption 7, we can simplify equation (3) for the typical series-parallel
system from ∏ − − ≥ = n 1 j L s j j) ] R r 1 ( 1 [ x to ) R 1 Q ( Q ) r 1 ( L s L s L s j n 1 j j x ≤ = − ∑ − = (7)
and for standby system, the equation (3) is simplified from R ] )! 1 x ( ) r (ln ) 1 ( ! 2 ) r (ln r ln 1 [ r Ls j 1 j j 1 j 2 j n 1 j j j x x ≥ − − + − + ∏ − − − = L to Q ] )! 1 x ( ) r (ln ) 1 ( ! 2 ) r (ln r ln 1 [ r 1 [ Ls n 1 j j 1 j j 1 j 2 j j j x x ≤ ∑ − − + − + − − = − − L (8)
Then the stage variable vector of dynamic programming used in this model can be defined as: ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ∏ ∑ ∑ = = = = k 1 j j j j j j k 1 j mj j j k 1 j 1j k ) x , r ( R ) x , r ( G ) x , r ( G S M (9)
Base on the above transformation which approximates the system unreliability by the addition of unreliability of all stages, the Sk is changed to
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ∑ − ∑ ∑ = = = = k 1 j j j j j j k 1 j mj j j k 1 j 1j k )] x , r ( R 1 [ ) x , r ( G ) x , r ( G S M (10)
The recurrent formula in Dynamic Programming becomes )] , ( [ ) ( 1 1 1 1 1 1 1 1 x r G Min x x x S F U L≤ ≤ = (11) n j for S F x r G Min S F j j j j j U j j L j j j x x x , , 2 )] ( ) , ( [ ) ( = + −1 −1 = K ≤ ≤
In the following section, we will apply dominating sequences concept in our dynamic programming model to find the optimal redundancy allocation under every specific component reliability level decided by revised H-J Pattern Search approach. Therefore, the above recurrent formula will be modified to only keep the dominating solutions for every step calculation.
3. AN OPTIMIZATION PROCEDURE
The combination of the revised Hooks and Jeeves pattern search and Dynamic Programming accompanied with dominating sequences is used to solve the template problem in section 2.
The main structure of this model is to apply revised H-J pattern search to find the appropriate component reliability level, and at that component reliability level, Dynamic Programming with dominating sequences is applied to get the optimal redundancy number to minimize system cost at that step. The result obtained from the computation procedure of Dynamic Programming with dominating sequences goes back to the procedure of H-J pattern search to decide if it is the optimal solution for whole model.
The descriptive flow diagram is shown in Figure 1
The computation procedure of Dynamic Programming with dominating sequences for evaluating the function value of system cost, Cs( R , X ) at any given point in Figure 1 are
1. Calculate xU j
(a) Assume xk=1, k=1,2,..., n and k≠j
(b) xU j =min{max{xj|Gi(R,(1, ,xj, ,1)) bi} i K K ≤ } m , , 1 i for = K 2. Calculate xL
j which satisfies (a) and (b) below: (a) For stage j in C,
)} R , R ( max ) x , r ( R | x min{ xLj = j j j j ≥ sL Lj
(b) For other stage j not in C, } R ) x , r ( R | x min{ xLj = j j j j ≥ Ls
3. Set up a matrix for the combination of stage1 and stage2
(1) The number of components, the stage cost, G
, ,
G1 KK m, the stage unreliability for stages 1 and 2 are presented as the column to the left of the matrix and the row above the matrix, respectively.
(2) Each entry of the matrix is a vector, which shows the system cost,G1,KK,Gm, the system unreliability which are results of the combination of stages 1 and 2.
4. Obtain the dominating sequence by eliminating entries of matrix which are dominated by others:
(1) When any element of the entries in the matrix exceeds the constrained available resource or unreliability limit, this entry is eliminated.
(2) The dominating sequence will then be determined as follows:
a. Consider the entry having the lowest system cost, which is always one term of dominating sequences no matter
what other elements of this entry are. Now compare the elements of all other entries with the elements of this entry.
Eliminate all entries which have higher system costs, higher consumptions of all resources and higher unreliability.
b. Choose the entry having the next lower system cost to be the other term of dominating sequence. Compare the elements of all other entries with the elements of this entry. Eliminate all entries which have higher system cost, higher consumptions of all resources and higher unreliability.
c. Repeat step b until the dominating sequence for the combination of stage 1and stage 2 has been finally reached.
5. Set up a matrix for the combination of stage1-2 (the dominating sequence) and stage 3 as step 3 and repeat step 4 to obtain the dominating sequence of stage 1-2-3.
6. Repeat step 5 until the matrix for the combination of stage 1-2-……-(n-1) and stage n is set up and the final dominating sequence is reached.
7. Choose the entry in the final dominating sequence which have the lowest system cost to be the optimal solution. Go back to main model (revised H-J Pattern Search model).
4. NUMERICAL EXAMPLES
One research team is assigned to develop a new system and need to meet some specific logistics requirements. It is noted that reliability optimization in the initial system design period plays a critical role to affect system availability, maintenance activities, logistics requirements and life cycle cost during system’s service life. For simplifying the situation, we assume that they need to find the optimal component reliability level and redundancy number in each subsystem to minimize the system design and development cost by conforming to the constraints for two design configurations (weight, volume), system and subsystem reliability requirements. In the following we demonstrate three numerical examples which respectively represent different system configurations or require different constraints. Before proceeding our numerical examples, table 1 lists all the parameters or related information we need as follows.
Table 1: Constant used in Examples
j αj(10−5) Pj Wj 1 2.33 1 7 2 1.45 2 8 3 0.541 3 8 4 8.05 4 6 5 1.95 2 9 βj=1.5, j=1,2,3,4,5 t=1000
Start
Assume initial base point R0
Combine Dynamic Programming and Dominationg Sequences to find the optimal allocation policy
and calculate ) R ( X* 0 )) R ( X , R ( 0 * 0 s C
Start at base point
Make exploratory moves with respect to at each move to find by Dynamic programming with Domonationg sequences and calculate
Is present functional value lower than the
functional value at base point
Set new base point
yes
Make pattern moves with respect to at each move to find
by Dynamic programming with Domonationg sequences and see if lower than the value of base point
Make exploratory moves with respect to at each move to find by Dynamic programming with Domonationg sequences and calculate yes R0 ) R ( X* )) R ( X , R ( * s C )) R ( X , R ( * s C R R X(R ) new * )) R ( X , R ( new new s * C Rnew ) R ( X new1 * )) R ( X , R ( 1 new * 1 new s C
Is step size with respect to small enough?R
Is present functional value lower than the
functional value at base point
no
Reach optimal solution
Reduce step size with respect to yes R )) R ( X , R ( 1 new * 1 new s C yes no Figure 1. Descriptive Flow Diagram for Combining of Revised H-J Pattern Search and
Dynamic Programming with Dominating Sequences Approach
Example 1: The typical series-parallel
system is considered. There are totally 5 subsystems in this system. The limitation on product of the weight and volume is 110. The limitation on weight is 200. The low bound of system reliability is 0.9.
)] 4 x exp( x ][ ) r ln t ( [ ) X , R ( C Min j j j j n 1 j j s = ∑ α − β + = r r s.t ∑ ≤ = n 1 j 2 j) P x ( pj ∑ ∗ ≤ = n 1 j wjxj exp(xj 4) W (12) R ] ) r 1 ( 1 [ L S j j n 1 j x ≥ − − Π = ( (1 r) 1 RL S j j n 1 j x ≤ − − ∑ = )
The results are shown in the following table 2.
Table 2
Optimal Solution for Example 1 (RS =0.9, P=110, w=200) L Stage 1 2 3 4 5 Starting point, R0 0.8 0.8 0.8 0.8 0.8 Optimal point R*0.83125 0.84375 0.83906 0.7 0.74844 Optimal point X* 2 2 3 3 3 System Cost 151.014378 System Reliability 0.900021 Used P 93 Used W 195.534639 Initial step size 0.05 Final step size 0.000195
Example 2: The typical element stand-by
system is considered. There are totally 5 subsystems in this system. The limitation on product of the weight and volume is 110. The limitation on weight is 200. The low bound of system reliability is 0.9. Therefore,
the whole problem is the same with example 1 except the third constraint (reliability constraint) need to be modified as follows:
R ] )! 1 x ( ) r (ln ) 1 ( ! 2 ) r (ln r ln 1 [ r SL j 1 j x j 1 j 2 j j j 5 1 j x ≥ − ∗ − + + + − Π − − = LL => R 1 ]} )! 1 x ( ) r (ln * ) 1 ( ! 2 ) r (ln r ln 1 [ r 1 { L S j 1 j x j 1 j 2 j j j n 1 j x ≤ − − − + + + − − ∑ − − = L
The results are shown in the following table 3.
Table 3
Optimal Solution for Example 2 (RS =0.9, P=110, w=200) L Stage 1 2 3 4 5 Starting point, R0 0.8 0.8 0.8 0.8 0.8 Optimal point R* 0.63438 0.63672 0.69688 0.7 0.8 Optimal point X* 3 3 3 2 2 System Cost 88.998256 System Reliability 0.9000 Used P 78 Used W 195.534639 Initial step size 0.05 Final step size 0.000195
Example 3: The 5-stage mixed series
system with stage 2 stand-by and other four stages are parallel configurations. We also limit the reliability of subsystem 3 greater or equal to 0.95. The problem can be modified by changing reliability constraint and adding one subsystem reliability constraint as follows: ] ) r 1 ( 1 [ x1 1 − − − + +L+ ! 2 ) r (ln r ln 1 [ r [ 2 2 2 2 R ] ) r 1 ( 1 [ ]] )! 1 x ( ) r (ln ) 1 ( L S j j 5 3 j 2 1 2 x 2 1 2 x x Π − − ≥ − ∗ − = − − + + + − − + − ⇒ L ! 2 ) r (ln r ln 1 [ r 1 [ ) r 1 ( 2 2 2 2 1 1 x R 1 ) r 1 ( ]] )! 1 x ( ) r (ln ) 1 ( L S j j 5 3 j 2 1 2 x 2 1 2 x x + ∑ − ≥ − − ∗ − = − −
and [1 (1 r )x3] 0.95
3 ≥
− −
The results are shown in the following table 4.
Table 4
Optimal Solution for Example 3 (RS =0.9, P=110, w=200) L Stage 1 2 3 4 5 Starting point, R0 0.8 0.8 0.8 0.8 0.8 Optimal point R* 0.776562 0.8 0.7975 0.78 0.79 Optimal point X* 3 2 3 2 3 System Cost 154.129294 System Reliability 0.900008 Used P 78 Used W 198.588197 Initial step size 0.05 Final step size 0.000195
According to the above results of table 2, table3 and table 4, the design and development cost is 151.014 for series- parallel system, 88.998 for element stand-by system, and 154.129294 for n-stage mixed series system. It is very clear that the lowest cost is incurred in element stand-by system. The reason for this is because the component reliability level of all subsystems for element stand-by system which is (0.63438, 0.63672, 0.69688, 0.7, 0.8) are almost lower than the one for series-parallel system which is (0.83125, 0.84375, 0.83906, 0.7, 0.74844), and the redundancy number for all subsystems are pretty much the same for both systems which is (3,3,3,2,2) compared to (2,2,3,3,3). The research team can easily achieve system reliability requirement (0.9) by using lower reliability component in all stand-by subsystems when they design a new system. It can save a lot of money compared to typical series-parallel system. The highest cost is incurred in
n-stage mixed series system of example 3. The reason for this is intuitive that there exists a limit in subsystem 3 which reliability needs to be equal to or greater than 0.95. Therefore, the system cost for example 3 is a little higher than example 1.
5. PERFORMANCE COMPARISONS
In this section we compare the system reliability performance of this proposed method with other 11 reliability-redundancy allocation methods by conforming to the constraints for two design configurations (weight, volume), system design and development budget. In order to facilitate comparison, we apply the following template problem referred in a lot of previous studies to evaluate the performance for all compared methods.
] ) r 1 ( 1 [ ) X , R ( R Max j xj n 1 j s =Π − − = (13) s.t 175 )] 4 x exp( x ][ ) r ln t ( [ j j j j n 1 j j + ≤ − ∑ α β = ∑ ≤ = n 1 j 2 j) 110 x ( pj ∑ ∗ ≤ = n 1 j wjxj exp(xj 4) 200
According to the above equations, the objective function in our model needs to be changed to system reliability function. To facilitate algorithm computation process in our model, the additive property of objective and constraint function for all subsystems is required. By taking the logarithm to the above objective function, it can be modified as ∑ − − = = 5 1 j j j s(R,X) ln[1 (1 r) ] R ln Max x
table 5.
Table 5
System. Unused Cost Unused P Unused W Rel.
The Proposed Method 0.931054 0.02655 27 7.518918 XKL Method (Xu et al) 0.93167 0.014 27 7.519 LMBB(Kuo et al) 0.92975 0.00001 27 10.572 H-J& 0.91494 0.03373 28 1.4118 A-G-M(Tillman et al) H-J(Hooke&Jeeves)& 0.91068 0.014 32 1.412 S-V (Sharma&Venkateswaran) H-J(Hooke&Jeeves)& 0.91068 0.014 32 1.412 G-A-G(Gopal et al, 1980) H-J(Hooke&Jeeves)& 0.89829 0.036 47 25.846 N-N(Nakagawa&Nakashima) H-J(Hooke&Jeeves)& 0.93102 0.047 27 7.519 K-I (Kohda&Inoue) G-A-G(Gupal et al, 0.89821 0.013 32 4.465 1978) & S-V(Sharma&Venkateswaran) G-A-G(Gupal et al, 0.90446 0.025 32 4.465 1978) &G-A-G(Gopal et al, 1980) G-A-G(Gupal et al, 0.90376 0.020 27 7.519 1978) &N-N(Nakagawa&Nakashima) G-A-G(Gupal et al, 0.90373 0.004 27 4.465 1978) &K-I(Kohda&Inoue)
From the results of the above table, it shows that our proposed approach gets better system reliability performance than other methods except for XKL method. But, from the previous studies, our approach can take advantage of more general objective and constraint function than XKL method, which usually uses derivatives to define sensitivity function in its algorithm. Also, this proposed approach performs normally in spending of cost and all resources compared to other combination methods.
6. CONCLUDING REMARKS
In this paper we have developed a specific reliability-redundancy optimization approach combining the revised H-J pattern
search and dynamic programming with dominating sequences. Three examples have been demonstrated for different system configurations including series-parallel, element stand-by, and 5-stage mixed series system. The results show that the lowest cost is incurred in element stand-by system, then is the series-parallel system, and the highest cost is for n-stage mixed series system. This information is very helpful when system design and development process is initiated. We also compare the performance of this proposed method with other 11 reliability-redundancy allocation methods by conforming to different constraints. It clearly shows that our proposed approach gets better system reliability performance than other methods except for XKL method.
Besides the above numerical results and comparisons, in this model the dynamic programming with dominating sequences applied at each step on the revised H-J pattern search technique seems better than traditional heuristic approach or Genetic Algorithm. It is because that dynamic programming gives more exact optimal solution. Also, by applying dominating sequence concept on dynamic programming to limit the range of redundancy, it makes our approach more efficient compared to traditional dynamic programming method and Genetic Algorithm. In addition, Genetic Algorithm seems not to be popular and efficient to solve the allocation problem of the number of redundancy and arbitrary component reliability level simultaneously. It can be applied in reliability optimization problem by selecting appropriate “commercial-off-the shelf (COTS)” items
from a number of different sources of supply, though.
The reliability constraints applied in some subsystems give more tools in design process to conform to performance requirement threshold or technique bottleneck. We can also easily modify our model to include maintenance consideration to meet availability requirement. All of these show that this proposed approach is not only an efficient way to solve the reliability- redundancy optimization problem, but also more realistic and flexible than a lot of conventional approaches.
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