Cybernetics and Systems Analysis, Vol. 44, No. 3, 2008
RELIABILITY OPTIMIZATION OF A COMPLEX SYSTEM BY THE STOCHASTIC BRANCH AND BOUND METHOD V. I. Norkina and B. O. Onishchenkob
UDC 519.248
An optimal redundancy problem is considered as a stochastic optimization problem. The mean lifetime of a network is maximized by the stochastic branch and bound algorithm. To obtain (stochastic) estimates of branches, use is made of stochastic tangent minorants and majorants of the objective functional, interchange relaxation (permutation of maximization and expectation operations), and multiple solution of auxiliary dynamic programming problems. Keywords: reliability optimization, optimal redundancy, minorants, majorants, stochastic minorants, stochastic programming, discrete optimization, stochastic branch and bound algorithm, permutable relaxation, dynamic programming. INTRODUCTION A reliability optimization problem and, in particular, optimal redundancy problem is usually formulated as a minimization problem for system failure probability under constraints imposed on the cost of resources (redundant elements) intended to increase the reliability [1–3]. If subsystems or elements fail independently, given the topology of a system and the laws governing its operation, the failure probability for the whole system can be calculated as a nonlinear and nonconvex function of the failure probabilities of elements. The failure probability for each element is a function of the amount of resources invested to increase the reliability of this element, and this may be a continuous or a discrete dependence, in the latter case, for example, on the number of redundant elements. Therefore, the optimal redundancy problem is a complex problem of nonlinear continuous or discrete optimization. To solve it, methods of nonlinear programming and discrete optimization are widely used, such as the dynamic programming method, branch and bound algorithm, the method of sequential analysis of alternatives, various heuristic procedures. For special problems, special methods are developed to reduce the enumeration. It is usually impossible to represent explicitly the failure probability of the whole system as a function of failure probabilities of isolated elements in the case of dependent failures of subsystems; therefore, the practical formulation of the problem is a challenge. The reliability index should be represented as a function of continuous or discrete parameters being optimized. However, the reliability optimization problem can be reformulated in terms of the lifetime of the system and its subsystems. It appears that the random lifetime of the whole system can easily be represented in terms of the random lifetimes of the elements, which, in turn, can be expressed in terms of the numbers of redundant elements, and their lifetimes. Then the reliability optimization problem can be presented as a problem of maximizing the mean lifetime of the system under constraint imposed on the cost of the redundant elements [1, 2]. The objective function in this problem is, as a rule, not computed in explicit form since it is the expectation (a multivariate integral) of a complex integrand, and parameters of this function (lifetimes of elements) are dependent random variables. Such an objective function can only be estimated statistically. Therefore, this problem is classed among nonlinear discrete stochastic programming problems. In case of so-called highly reliable systems, where the time of practical use of a system is small as compared with its lifetime, the latter is not an adequate parameter to estimate the reliability. Then the probability that the system fails on a given a
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine,
[email protected]. bCherkassy National University, Cherkassy, Ukraine. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 129–141, May–June 2008. Original article submitted February 7, 2007. 418
1060-0396/08/4403-00418
©
2008 Springer Science+Business Media, Inc.
time interval is used [4, 5], i.e., the probability that a random lifetime of the system is less than a given boundary. It is generally impossible to derive an analytic expression for this probability as a function of system parameters; therefore, significant efforts of contributors go into deriving deterministic and stochastic estimates of failure probability [4, 5]. Optimization of highly reliable systems is also classed among stochastic programming problems but with probability objective functions or constraints [4, 6]. Note that there are other probabilistic and nonprobabilistic approaches to the reliability optimization of random networks [7, 8]. We consider here the problem of optimal redundancy of unreliable elements in a complex system with dependent failures of elements in terms of the lifetimes of the system and subsystems. We maximize the expectation of the system lifetime as a reliability characteristic, i.e., we do not consider the above case of highly reliable systems. To solve the problem, we apply the stochastic branch and bound algorithm, which takes into account the specific nature of the problem. Namely, to estimate branches, the method of interchange relaxation [9, 10] (permutation of maximization and expectation) and stochastic minorants and majorants of the objective function [11] are used. This allows reducing the branch estimation to a multiple solution of some special deterministic dynamic programming problems. At the same time, the present paper is an illustration of applying the minorant-majorant method of solving discrete optimization problems. To solve continuous global optimization problems, this method was justified in [12] and then rediscovered and used in many studies. Despite the fact that the method was mainly used to solve continuous problems, it is obvious that it is also applicable to solve discrete problems. The key concept of the minorant-majorant optimization method is tangent minorants (majorants) of a function, which are a source of global information on the function being optimized. Therefore, the number of studies develop the calculus of tangent minorants of composite functions [11–15], in particular, the papers [11, 15] specify a way of constructing tangent minorants of the functions of minimum, functions of expectation, and probability. This opens up the possibility of applying the method of minorants to solve stochastic global optimization problems [11] and, in particular, as shown here, reliability optimization problems. 1. MODELING AND MAXIMIZING RELIABILITY (MEAN LIFETIME) OF A NETWORK Let us consider the reliability of a network (system) from the standpoint of its lifetime. A network consists of a set of nodes connected by arcs. Messages arrive at the network at some input nodes and pass along the network to some destination nodes. Arcs and nodes may fail, so that a message cannot pass through them. A network is assumed efficient if there is a path from each input node to each destination node. It is obvious that only failures of arcs can be considered. Indeed, failures of nodes can be represented as failures of all arcs adjacent to them. Each arc is assumed to have a random lifetime. Then the reliability of a network is expressed in terms of the lifetimes of the arcs. If there is only one input and one output, then a random lifetime is f = maxi Î I min j Î i f ij , where the maximum is taken with respect to paths i Î I that connect the input and output, and the minimum with respect to the lifetimes of arcs j appearing in the path i. If there are several input-output pairs, then to obtain the lifetime of the whole system, the minimization should be with respect to all the pairs. There can be several types of arc failures, for example, disconnections and short-circuits. We also distinguish arc and node failures. Failures in different arcs and different failures in one arc may be dependent. Of interest is the reliability optimization with respect to parameters x = {x ij }; therefore, it is necessary to compose a model that reveal a relationship between the lifetime and parameters of the solution. We assume that all the lifetimes f ij depend on the vector x ij and are random; thus, they are functions of a corresponding abstract random variable w ij or of a common random variable w defined on an individual or common probabilistic space, i.e., on (W ij , S ij , Pij ) or (W , S, P ) . Introducing redundant arcs is usually limited by the resources available, which can be expressed by the inequality å (i , j ) Î R cij xij £ C, where R denotes the set of arcs that can be reserved. Denote by F ( x ) the expected lifetime of the network, for example,
é F ( x ) = E ê f ( x, w ) = maxi Î I min ë
j Îi
ù f ij ( x ij , w )ú . û
419
Then a typical optimal redundancy problem has the form F ( x ) ® maxx Î {0 , 1}n under the constraints
å (i , j ) Î R cij xij
£ C.
Note that the objective function here is the expectation with the operations max / min under the sign of expectation, and constraints may contain several types of resources. 1.1. Reliability of a Redundant Element. Denote by t ( w ) a random lifetime of an arc (element). Increasing the reliability of the corresponding connection by a redundant (idle) arc with the lifetime t ( w ) makes the lifetime of this connection to be equal to t ( w ) + t ( w ) . If we use a variable xÎ{0, 1} to denote the presence (x = 1) or absence (x = 0) of a redundant arc, the lifetime is f ( x, w ) = t ( w ) + xt ( w ) . If we use several types of redundant elements k = 1, . . . , n in a cold redundancy (as the main device and redundant elements involved earlier fail, redundant elements are included) with the lifetimes t k ( w ) in numbers of x k Î{0, 1}, then the corresponding lifetime of the arc is represented by f ( x, w ) = t ( w ) + å k = 1 x k t k ( w ) , n
where the vector x = ( x1 , . . . , x n ) has 0-1 components. In this model, the lifetime of an arc linearly depends on parameters of the solution x = ( x1 , . . . , x n ) Î{0, 1}n . The optimal redundancy problem with multitype independent redundant elements was considered in [16]. The model of the dependence of the lifetime of an element on the resources invested can be introduced in a different way. Assume that the lifetime f ( x ) of an arc has an exponential distribution, P { f ( x ) ³ t} = e
- t / l( x )
,
where the mean lifetime l( x ) of the arc depends on the investment of some resource in the quantity of x into the reliability of the arc. Let us introduce a random variable w uniformly distributed on [ 0, 1]. Then the random lifetime of the arc can be expressed as 1 f ( x, w ) = l ( x ) ln . 1- w 1.2. Reliability of a Renewable Element. Assume that a primary unreliable but renewable element has a redundant auxiliary but also unreliable element. If the primary element is broken, it is temporarily replaced with the auxiliary one. After renewal, the primary element starts operating, and the redundant one goes over into the standby mode. Such renewal cycles are repeated until the redundant element fails earlier than the primary element is renewed. Let us model the lifetime of a redundant element. Let t k and t k be random operating and renewal times of the primary element at the kth renewal cycle, T k be a random time of nonfailure operation of the redundant element at the kth renewal cycle. A random lifetime of a redundant element t can easily be modeled by sequential simulation of random variables {t k , t k , T k }, k = 1, 2, . . . . Let us introduce a binary variable xÎ{0, 1} such that x = 1 if a redundant element exists, otherwise x = 0 . Then a random lifetime of the renewal element as a function of x can be specified by the expression f ( x ) = (1 - x )t 1 + x æç è
k*
åk = 1 (t k
+ tk ) + T k
*
+1ö
÷, ø
ü ì where k * = supí k :T i > t i " i Î[1, . . . , k ]ý is the number of successful renewal cycles. Thus, f ( x ) is a linear function of þ î the binary variable x with random coefficients. Formally, denote these indices of the random variable w so that the lifetime of the redundant element be equal to f ( x, w ) . 1.3. Reliability of a Parallel/Series Circuit. There are n devices with random lifetimes t i ( w ) connected in parallel and thus guaranteeing that the lifetime of the whole circuit is max1 £ i £ n t i ( w ) . Assume that each failed device can potentially be replaced with a redundant device with a random lifetime t i ( w ) and cost c i . Let us introduce a binary variable x i : x i = 1denotes that the corresponding device has a redundancy, otherwise x i = 0. Then the lifetime f ( x, w ), x = ( x1 , . . . , x n ) , in the parallel circuit with redundancy is f par ( x, w ) = max1£ i £ n ( t i ( w ) + x i t i ( w )) . 420
The optimal redundancy plan x = ( x1 , . . . , x n ) under a cost constraint is a solution of the problem Fpar ( x ) = Ef par ( x, w ) = E max1£ i £ n ( t i ( w ) + x i t i ( w )) ® maxx Î {0 , 1}n under the constraints cx = å i = 1 c i x i £ C. n
The expected lifetime of the series circuit can be optimized similarly: é ù Fseq ( x ) = E ê f seq ( x, w ): = min 1£ i £ n ( t i ( w ) + x i t i ( w ))ú ® maxx Î {0 , 1}n : cx £ C . ë û In case of discrete distribution of the probability { p w } of w, the last problem can be solved with relative ease. Namely, introduce new variables z w , w ÎW; then this problem is equivalent to the problem
å w ÎW p w z w ® maxx, {zw} under the constraints z w £ t i ( w ) + x i t i ( w ), w ÎW , i = 1, . . . , n; cx £ C; x Î{0, 1}n . In case of a continuous distribution of w, the expectation can be approximated by discrete observed mean, but the corresponding mixed approximation of the integer problem may be excessively unwieldy. 1.4. Reliability of a Parallel-Series Circuit. Let us consider a parallel-series circuit consisting of m series circuits (paths) connected in parallel, the ith chain containing n i elements. Note that any circuit with one input node and one output node can be represented as a parallel-series circuit (with dependent failures). Denote by t ij ( w ) the lifetime of the jth element in the ith chain. Then the lifetime of the whole system is ì f ( w ) = max1£ i £ m í min 1£ î
j £ ni
ü t ij ( w )ý . þ
Assume that for each ( i, j )th element of the circuit, we can use a redundant element with the lifetime t ij ( w ) and cost c ij . Let us introduce a binary variable x ij ; if x ij = 1, the ( i, j )th element has a redundant device, otherwise x ij = 0. The redundancy circuit has the lifetime f ( x, w ) = max1£ i £ m f i ( x i , w ) , f i ( x i , w ) = min 1£
j £ ni
n
f ij ( x ij , w ), x i = {x ij } j i = 1 ,
f ij ( x ij , w ) = t ij ( w ) + x ij t ij ( w ), x = {x ij }. The lifetime optimization problem has the form F ( x ) = Ef ( x, w ) ® maxx Î X , where
ìï m X = í x ij Î{0, 1}: å i = 1 îï
å ji= 1 cij xij n
üï £ C ý. þï
1.5. Reliability of a Circuit with Element Failures of two Types. Assume in the assumptions in Sec. 1.4 that there may be two types of failures of each element: break and short-circuit. Assume also that the lifetime of the ( ij )th element to break is f ij b ( w ) , and the lifetime to a short-circuit is f ijs ( w ) . The circuit is viable if there is a path from operating elements and there is no path consisting of short-circuited elements. This circuit can be modeled by a circuit with only one type of failure (break). An equivalent circuit consists of m + 1series-connected sections: first there is the original circuit with errors, the next m components correspond to parallel paths of the original circuit, each consisting of corresponding elements connected in parallel but with short-circuits only. The lifetime of such a circuit is ì f = min í max1£ i £ m min 1£ î
b s j £ ni min ( f ij , f ij
); max1£
s s ü j £ n1 f1 j , . . . , max1£ j £ nm f mj ý ,
þ
421
where f ijb ( x ij , w ) = t ijb ( w ) + x ij t bij ( w ), f ijs ( x ij , w ) = t ijs ( w ) + x ij t ijs ( w ), t ijb ( w ) and t ijs ( w ) are the lifetimes of the ( ij )th element, t bij ( w ) and t sij ( w ) are the lifetimes of the ( ij )th redundant element, to break and to a short-circuit, respectively. 1.6. Reliability of Parallel-Series Circuit Consisting of n Identical Elements (or at the Maximum of n Identical Elements). Assume in the parallel-series circuit that m = n and n i = n. For all i = 1, K , n , x ij = 1 if the jth element is present in the ith chain, otherwise x ij = 0 ,
å i , j = 1 xij n
= n ( £ n ) . The lifetime of this circuit is
f ( x, w ) = max1£ i £ n min 1£
j £n
(T (1 - x ij ) + t ij ( w ij ) x ij ) ,
where T is a very large number (plus infinity), t ij ( w ij ) is the lifetime of the ( ij )th element, depending on random eigenfactors w ij , x = {x ij }, w = {w ij }. A similar model but with two types of failures and in terms of failure probabilities of elements and the whole circuit was analyzed in [1, 3, 17]. 2. BRANCH AND BOUND ALGORITHM TO MAXIMIZE MEAN LIFETIME OF THE NETWORK In the branch and bound algorithm, the original optimization problem is divided into subproblems by sequentially fixing some components of the vector x on their possible values 0 or 1, the remaining components being free (may take both values, 0 and 1). With iterations, each subproblem can be divided in the same way into subproblems of lower dimension by fixing some free variables of this subproblem on their possible values 0 or 1. The subproblems generated in this process are also called branches. A feasible set of some branch may appear empty; such a branch is certainly eliminated from the further consideration. Thus, at each iteration of the method, some variables i Î I k have the fixed value of x i , the remaining i Î I k = I \ I k are free. Denote x k = {x i , i Î I k }Î{0, 1}nk . Then the following subproblems are solved to optimize the reliability of the parallel circuit by the branch and bound algorithm: ì ü Fk ( x k ) = E maxí max ( t i ( w ) + x i t i ( w )), max ( t i ( w ) + x i t i ( w ))ý ® maxxk Î {0 , 1} nk i ÎI k îi ÎIk þ under the constraints å i Î I ci xi £ C - å i Î I ci xi . k
(1)
(2)
k
Let us rearrange (1), (2) as
é ù F ( x k ) = E ê f k ( x k , w ) = maxi Î {I k , 0} j i ( x i , w )ú ® maxxk Î {0 , 1}nk êë úû
under the constraints
å i Î I k ci xi
£ Ck ,
(3)
(4)
where j i ( x i , w ) = t i ( w ) + x i t i ( w ), j 0 ( x 0 , w ) = maxi Î I k ( t i ( w ) + x i t i ( w )), C k = C - å i Î I c i x i , x 0 k variable introduced for convenience. Similarly, the corresponding subproblems for the series circuit have the form é ù . F ( x k ) = E ê f k ( x k , w ) = min i Î {I k , 0} j i ( x i , w )ú ® maxxk Î {0 , 1}nk : å i Î Ik ci xi £ C k ë û
is a dummy
(5)
In the branch and bound algorithm, upper- and lower-bound estimates of the optimal values of subproblems are also used to reduce the enumeration (cut-off of unpromising subproblems/branches). 2.1. Upper-Bound Estimate for the Parallel Circuit. Denote by Fk* the optimal value in (3), (4); by virtue of the interchange relaxation, it is majorized by Ef k* ( w ) , where f k* ( w ) is the optimal value in the problem f k ( x k , w ) = maxi Î {I k , 0} j i ( x i , w ) ® maxxk = {x
i
under the constraints
422
åi Î I k
ci xi £ C k .
: i Î I k}
The problem can be solved by exhaustive search for low dimensions, and by the dynamic programming method for high dimensions. Let us consider a problem j( y ) = max1 £ i £ m j i ( y i ) ® max y under the constraints
å i = 1 ci yi m
F i ( z ) = max{ y j : 1£
£ C, where j i ( y i ) = t i + y i t i . Denote ì
j £ i } í max1£ j £ i
î
ü i j j ( y j ) | å j = 1 c j y j £ z ý, þ
ìï j1 ( 0), z < c1 , F1 ( z ) = í ïî j1 (1), z ³ c1 ,
0£ z £ C.
Then for dynamic programming, the recurrence has the form F i ( z ) = max{ y j : 1£ ì = maxí max{ y j : 1£ î max{ y j : 1£
ì
j £ i } í max1 £ j £ i
î
ì
ì
î
î
ü i j j ( y j ) | å j = 1 c j y j £ zý þ
j £ i - 1} í maxí j i ( 0 ), max1£ j £ i - 1
ïì
ïì ïî
j £ i - 1} í maxí j i (1), max1£ j £ i - 1
ïî
ü ü i -1 j j ( y j )ý | å j = 1 c j y j £ z ý , þ þ
i -1 ïü ïü üï j j ( y j )ý | å j = 1 c j y j £ z - c i ý ý ïþ ïþ þï
ìï max{j i ( 0), j i (1), F i - 1 ( z ), F i - 1 ( z - c i )}, z - c i ³ 0, =í ïî max{j i ( 0), F i - 1 ( z )}, z - c i < 0. Since j i ( 0) £ j i (1) , and F i - 1 ( z ) increases monotonically in z, the formula becomes ìï max{ j i (1), F i - 1 ( z )}, c i £ z £ C , Fi ( z ) = í ïî max{j i ( 0), F i - 1 ( z )}, 0 £ z < c i. Observing the values of {t i , t i , i = 1, . . . , m} results in the values of {j i ( 0) = t i , j i (1) = t i + t i , i = 1, . . . , m}. The resultant recurrence allows us to find carefully and quickly the values of F i ( z ), 0 £ z £ C, by improving the values of F i - 1 ( z ) on a part of the domain of definition z Î[ 0, C ] , and to find eventually the optimal value of the problem F m (C ) under consideration. 2.2. Upper-Bound Estimate for a Series Circuit. Let us consider the problem j( y ) = min 1 £ i £ m j i ( y i ) ® max y under the constraints
å i = 1 ci m
y i £ C, where j i ( y i ) = t i + y i t i .
First, this problem can be reduced to the mixed integer programming problem z ® max y, z ³ 0 under the constraints z £ t i ( w ) + y i t i ( w ), i = 1, . . . , m;
å i = 1 ci yi m
£ C; y Î{0, 1}m ,
and thus it can be solved by any appropriate integer optimization method. Second, the original problem can be solved also by the dynamic programming method. In case of a series circuit, the corresponding recurrence has a similar form: F i ( z ) = max y j : 1 £
ì
j £ i í min 1 £ j £ i
î
ü i j j ( y j ) | å j = 1 c j y j £ zý þ
423
ìï = maxí max{ y j : 1 £ ïî max{ y j : 1 £
ìï
ìï ïî
j £ i - 1} í min í j i ( 0 ), min 1 £ j £ i - 1
ïì
ïî
ïì ïî
j £ i -1} í min í j i (1), min 1 £ j £ i -1
ïî
ïü j j ( y j )ý | ïþ
üï üï i -1 j j ( y j )ý | å j = 1 c j y j £ z ý , ïþ þï i -1
å j =1c j y j
ïü üï £ z - ci , z ³ ci ý ý ïþ þï
ì min {j i ( 0), F i - 1 ( z )}, 0 £ z < c i , ï =í max min{j i ( 0), F i - 1 ( z )}, min {j i (1), F i - 1 ( z - c i )} , c i £ z £ C , îï
{
}
ì j ( 0), z < c1 , F1 ( z ) = í 1 î j1 (1), z ³ c1. One more upper-bound estimate for the series circuit follows from Jensen’s inequality maxxk :
ì
å i Î Ik
³ maxxk :
ü min i Î I k ( Et i ( w ) + x i Et i ( w ))ý î þ
ci xi £ C k , xi Î {0 , 1} í
å i Î Ik
ci xi £ C k , xi Î {0 , 1}
E min i Î I k ( t i ( w ) + x i t i ( w )) .
2.3. Lower-Bound Estimates for Optimal Values of Subproblems. Note that the optimal value of the maximization problem is always minorized by the value of the objective function at any feasible point. This point should be selected heuristically so that the value of the objective function at this point is as maximum as possible. To estimate optimal values of subproblems in the branch and bound algorithm, it is possible to use stochastic tangent minorants (and majorants) of the objective function. Definition 1 [13]. Let X be a topological space, functions F ( x ), x Î X , and j( x, y ), x Î X , y Î X , be related by the following conditions: (a) F ( x ) ³ j( x, y ) for all x Î X , y Î X ; (b) F ( y ) = j( y, y ) for all y Î X ; (c) the function j( x, y ) is continuous in x equipotentially in y. Then the functions { j( × , y ), y Î X } are called tangent (at points y) minorants for F ( x ) . A tangent majorant can be defined similarly. Note that the space X in Definition 1 may also be discrete. For example, let f ( x ) = inf y ( x, z ) and functions y( × , z ) for all z Î Z admit (concave) tangent minorants f( x, y, z ) z ÎZ
at points y. Then the function j( x, y ) = inf z Î Z f( x, y, z ) is a (concave) tangent minorant for f ( x ) at y [15]. Definition 2 [11]. Functions {f( × , y, q )}, y Î X , q Î Q} , where Q is a carrier of a probabilistic space ( Q, S, P ) , are called stochastic tangent minorants for F ( x ) if the functions f( x, y, q ) are measurable in q and the expectations j( x, y ) = Ef( x, y, q ) are finite and are tangent minorants at the point y for F ( x ) for each y Î X (in the sense of Definition 1). The lemma below shows that it is possible to take tangent minorants of the integrand f ( x, q ) as stochastic tangent minorants of the function of expectation F ( x ) = Ef ( x, q ) . The situation is similar to computing stochastic gradients of the function of expectation [7]. It may cause difficulties to find deterministic gradients, as well as deterministic minorants of the functions of expectation; however, it is quite possible to compute and use stochastic quasigradients and stochastic minorants. LEMMA 1 [11]. Assume that the function f ( × , q ) admits tangent minorants f( x, y, q ) at points y Î X , i.e., the following conditions are satisfied for almost all q: (i) f ( x, q ) ³ f( x, y, q ) for all x Î X , y Î X ; (ii) f ( y, q ) = f( y, y, q ) for all y Î X ; (iii) the function f( x, y, q ) is continuous in ( x, y ) for almost all q; (iv) f( x, y, q ) is measurable in q for any x, y Î X ; (v) | f( x, y, q )| £ M ( q ) for all x, y Î X with an integrable function M ( q ) . Then functions j( x, y ) = Ef( x, y, q ) are continuous and are tangent minorants for the function of expectation F ( x ) = Ef ( x, q ) . 424
Let us construct a lower-bound minorant estimate for the optimal value in the problem (1), (2) or (3), (4) for the parallel circuit. Specify a fixed point y k = { y i , i Î I k È 0} satisfying constraints (2). Denote i k ( y k , w ) = arg maxi Î {I k È 0} j i ( y i , w ) , ì 1, i = i k ( y k , w ), ï 1( i = i k ( y , w )) = í ïî 0, i ¹ i k ( y k , w ). k
The function Y( x k , y k ) = Ej i
k(
yk , w )
( x k , w ) = å i Î I E1( i = i k ( y k , w )) j i k
k(
yk , w )
(x k , w )
is a tangent (at the point y k ) minorant for the objective function Fk ( x k ) of the problem (1), (2). In case of linear components j i ( y i , w ) , the function Y( x k , y k ) is also linear. Then ì ü F *k = maxxk í Y ( x k , y k ) | å i Î I c i x i £ C k ý k î þ is a lower-bound estimate for the optimal value Fk* and can be found elementarily. This lower-bound estimate can be improved by repeating the above-mentioned steps of constructing new tangent minorants and of maximizing minorant functions. 2.4. Boundaries of Optimal Values for a Parallel-Series Circuit. Let us fix some feasible plans x = {x ij }, x = {x ij } and find a critical path i * ( x ) = arg max1£ i £ m f i ( x i , w ) and tangent minorant f ( x, w ) = f i* ( x) ( x i* ( x) , w ) = min 1 £
j £n*
i ( x)
f i* ( x ) j ( x i* ( x ) j , w )
for the lifetime of the whole circuit. Obviously, f ( x, w ) = f ( x, w ) and f ( x, w ) £ f ( x, w ) for all x. Similarly, for x, we find a critical element in the ith path j * ( i, x ) = arg min 1 £ j £ ni f ij ( x ij , w ) and tangent majorant f i ( x i , w ) = f ij* ( i , x ) ( x ij* ( i , x ) , w ) for the lifetime of the ith path. Obviously, f i ( x i , w ) £ f i ( x i , w ) for all x and f i ( x i , w ) = f i ( x i , w ) . The function f ( x, w ) = max1 £ i £ m f i ( x, w ) is a tangent majorant for f ( x, w ) . Then for each feasible x, min 1 £ j £ n * f i* ( x) j ( x i* ( x) j , w ) = f ( x, w ) £ f ( x, w ) £ f ( x, w ) = max1£ i £ m f ij* ( i , x ) ( x ij* ( i , x ) , w) i ( x)
and, therefore, E f ( x, w ) £ F ( x ) £ Ef ( x, w ) . Thus, the lifetime of the parallel-series circuit can be minorized by the lifetime of a series circuit and majorized by the lifetime of a parallel circuit. Denote n* ì ü X = í x j = x i* ( x) j Î{0, 1}: å i (x) c i* ( x) j x i* ( x) j £ C ý , j =1 î þ ì ü m X = í x i = x ij* ( i , x ) Î{0, 1}: å i = 1 c ij* ( i , x ) x ij* ( i , x ) £ C ý . þ î We get maxx Î X Emin 1 £
j £n* i
æ ö ç t * (w ) + x * ÷£F* t ( w ) * i ( x) j i ( x) ÷ ( x) ç i ( x ) è ø
£ E maxx Î X max1£ i £ m f ij* ( i , x ) ( x ij* ( i , x ) , w ) . Thus, to obtain lower-bound estimates, we replace the original circuit with a series one that consists of the initial elements {( i º i * ( x ), j Î[1, n i* ( x) ])}, and to compute upper-bound estimates, we replace the original circuit with a parallel one that consists of the components { i Î[1, m], j * ( i, x )} of the initial parallel-series circuit. 425
Fig. 1. Parallel-series network. TABLE 2. Node Characteristics of ParallelSeries Network
TABLE 1. Node Characteristics of Parallel and Series Networks Node No.
lt
lt
c
Node No.
lt
lt
c
1 2 3 4 5 6 7 8 9 10
20 10 20 20 30 20 40 20 10 10
40 10 10 10 20 10 20 20 30 30
30 20 10 40 30 20 50 20 30 10
11 12 13 14 15 16 17 18 19 20
20 10 20 20 30 20 40 20 10 10
40 10 10 10 20 10 20 20 30 30
30 20 10 40 30 20 50 20 30 10
Node No.
lt
lt
c
Node No.
lt
lt
c
1 2 3 4 5 6 7 8 9 10 11
20 10 20 20 30 20 40 20 10 10 20
40 10 10 10 20 10 20 20 30 30 10
30 20 10 40 30 20 50 20 30 10 20
12 13 14 15 16 17 18 19 20 21 22
10 10 10 20 10 30 40 20 40 20 10
20 10 10 30 30 30 30 30 20 20 30
30 20 10 30 20 10 20 30 50 20 30
TABLE 3. Operation Results of the Branch and Bound Algorithm and Exhaustive Search Algorithm
Network type Iterations
Exhaustive search Run time
Iterations
Run time
Parallel
4146
13 sec
1048576
2 min 9 sec
Series
508
1 min 6 sec
1048576
2 min 9 sec
Parallel-series
7072
1 min 50 sec
4194304
8 min 29 sec
3. NUMERICAL EXPERIMENTS The method of branches can be illustrated using reliability (mean lifetime) optimization problems for three types of networks: (i) a network of parallel-connected elements; (ii) a network of series-connected elements; (iii) a parallel-series network. The results and operating time of the branch and bound algorithm are compared with the exhaustive search. The parallel and series networks consist of 20 nodes, each being characterized by the following: — a parameter determining the random lifetime of the primary element ( l t ) ; — a parameter determining the random lifetime of the redundant element (l t ); — the cost (ñ) of introducing a redundant element. Table 1 presents characteristics of the networks. The parallel-series network has the topology shown in Fig. 1 and consist of 22 nodes with the characteristics represented in Table 2. The numerical experiments were performed on a Celeron 1.1 GHz, 256 MB RAM PC. The algorithms for numerical experiments were implemented in the Borland Delphi 6 programming environment. In particular, a version of the branch and
426
bound algorithm was implemented, with the upper-bound estimates described in Secs. 2.1 (parallel circuit), 2.2 (series circuit), 2.4 (parallel-series network), and branching by choosing a current free component of the vector of solutions with the minimum price of redundancy of the corresponding element. To estimate the expectations of the lifetime, its lower and upper bounds, 100 independent realizations of a vector random parameter with independent realizations of components were used. The results are presented in Table 3. As follows from the experimental results, the mean lifetime optimization algorithm substantially reduces the enumeration of alternatives. CONCLUSIONS In the paper, we understand a reliability optimization problem for a complex system as a mean lifetime (serviceability) optimization problem for a corresponding network of unreliable and probably dependent elements. The lifetime of an element is represented as a function of resources invested, in particular, the number of redundant elements. The random lifetime of the whole network can be represented as the maximin function of the lifetimes of its elements. To optimize the reliability, the expectation of the lifetime of the network is maximized under limited resources. This is a (discrete) stochastic programming problem with the objective functional not calculated in explicit form. The problem is solved by a specialized stochastic branch and bound algorithm since the estimates of branches are, generally speaking, random variables. To obtain estimates of branches, stochastic minorants and majorants of the objective functional as sources of the global information on the objective function are used. The serviceability of the algorithm is illustrated using optimization problems for parallel-series networks with 22 redundant elements. Note that this approach can also be applied to optimize the reliability of systems whose failures can be modeled using fault trees [19]. In this case, the lifetime of a system can be presented using a fault tree as the maximin function of the lifetimes of elements (leaves of the tree). A technique of constructing stochastic tangent minorants and majorants is developed in [11] for such functions, and we use it in the present paper to develop a technique for estimating the reliability of various types of networks. The problem of optimizing highly reliable systems differs from the formulation considered since it optimizes the probability of system failure up to a certain instant of time, which is much shorter than the mean lifetime of the system. The estimate of probability as a function of parameters being optimized is a challenge; the studies [4, 5] are devoted to point estimates and probability sensitivity estimates. However, conceptually, this is also a (discrete or continuous) stochastic programming problem with the probability as a specific objective functional. Basically, it is possible to optimize such a probability by discrete stochastic optimization methods.
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R. Barlow and F. Proschan, Mathematical Theory of Reliability, Wiley, New York (1965). I. A. Ushakov, “Optimal redundancy problems,” in: Optimal Redundancy and Inventory Control [in Russian], Znanie, Moscow (1979). V. L. Volkovich, A. F. Voloshin, V. A. Zaslavskii, and I. A. Ushakov, Models and Methods of Reliability Optimization of Complex Systems [in Russian], Naukova Dumka, Kyiv (1993). I. N. Kovalenko and A. N. Nakonechnyi, Approximate Computation and Reliability Optimization [in Russian], Naukova Dumka, Kyiv (1989). I. N. Kovalenko and N. Yu. Kuznetsov, Methods of Computing Highly Reliable Systems [in Russian], Radio i Svyaz’, Moscow (1988). A. N. Nakonechnyi, “Optimal-system synthesis with reliability constraints,” Cybern. Syst. Analysis, 29, No. 5, 786–788 (1993). Yu. M. Ermol’ev, Problems and Methods in Stochastic Programming [in Russian], Nauka, Moscow (1976). F. A. Sharifov, “The problem of synthesis of reliable networks,” Cybern. Syst. Analysis, 36, No. 4, 596–604 (2000). V. Norkin, G. Ch. Pflug, and A. Ruszczynski, “A branch and bound method for stochastic global optimization,” Math. Progr., 83, 425–450 (1998).
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V. Norkin, Yu. M. Ermoliev, and A. Ruszczcynski, “On optimal allocation of indivisibles under uncertainty,” Oper. Res., 46, No. 3, 381–395 (1998). V. I. Norkin and B. O. Onishchenko, “Minorant methods of stochastic global optimization,” Cybern. Syst. Analysis, 41, No. 2, 203–214 (2005). S. A. Piyavskii, “An algorithm of searching for absolute minima of functions,” Zh. Vych. Mat. Mat. Fiz., 12, No. 4, 888–896 (1972). V. I. Norkin, “The Piyavskii method for solving the general global optimization problem,” Zh. Vych. Mat. Mat. Fiz., 32, No. 7, 992–1007 (1992). O. Khamisov, “On optimization properties of functions with a concave minorant,” J. Global Optimiz., 14, No. 1, 79–101 (1999). V. I. Norkin and B. O. Onishchenko, “Global minimization of minimum functions by the minorant method,” in Theory of Optimal Solutions [in Russian], Inst. of Cybernetics, NAS of Ukraine, Kyiv, Issue 3 (2004), pp. 56–63. V. L. Volkovich and V. A. Zaslavskii, “Reliability optimization algorithm for compound systems with different backup elements in subsystems,” Cybern. Syst. Analysis, 22, No. 5, 599–609 (1986). V. S. Kirilyuk, “Maximum-reliability parallel-serial structure with two types of component faults,” Cybern. Syst. Analysis, 31, No. 1, 25–36 (1995). V. I. Norkin, “Global stochastic optimization: the branch and probabilistic-bound method,” in: Methods of Control and Decision Making under Risk and Uncertainty [in Russian], Inst. of Cybern., NAS of Ukraine, Kyiv (1993), pp. 3–12. N. Yu. Kuznetsov and K. V. Mikhalevich, “Reliability analysis of systems described by fault trees with efficiency,” Cybern. Syst. Analysis, 39, No. 5, 746–752 (2003).
