52 IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 1, MARCH 2001
Redundancy Optimization for Multi-State System
with Fixed Resource-Requirements and Unreliable
Gregory Levitin, Senior Member, IEEE
Abstract—This paper considers a redundancy optimization number of available versions for elements of MPS
problem for a multi-state system of: 1) elements that consume a availability of element of version belonging to
fixed amount of resources to perform their task, and 2) a number RGS ,
of resource generating subsystems. The algorithm finds the
optimal system-structure, subject to availability constraints, by availability of MPS element of version
choosing system elements from a list of available equipment. Each generating capacity of element of version be-
element is characterized by its productivity, availability, and cost. longing to RGS ,
Elements of the main producing subsystem also have their specific amount of resource required for MPS element of
resource consumption limitations. The objective is to minimize version
the sum of investment costs while satisfying demand, represented
by a cumulative demand curve, with given probability. To solve productivity of MPS element of version
the problem, a genetic algorithm is used for optimization. The version-number of element belonging to RGS ,
procedure, based on the universal generating function, is used to
evaluate the system-availability while assuming that the working number of different possible levels of resource
elements of the main producing subsystem are chosen in such generation
a way that the total system performance rate is maximal under
given resource constraints. discrete r.v. which represents available amount of
Examples demonstrate how to obtain the optimal structures of resource and can have possible values, ,
a simple 2-level system for various availability constraints.
Index Terms—Fixed resource consumption, genetic algorithm,
redundancy optimization, universal generating function. cost of element of version belonging to RGS ,
cost of MPS element of version
number of demand levels considered
GA genetic algorithm , : vector of demand levels
MPS main producing subsystem , : vector of intervals, corre-
PDN preliminarily defined number sponding to demand levels
PRD performance rate distribution discrete r.v representing the entire system produc-
RGS resource generating subsystem tivity
r.v. random variable system availability
-func- universal moment generating function minimal permissible
tion , where is any variable
PDN of genetic cycles in the GA
I. INTRODUCTION population size in the GA
Notation: PDN of crossovers per genetic cycle in the GA.
number of different resources
maximal permissible number of parallel elements in
maximal permissible number of parallel elements in
T THE PROBLEM of total investment-cost minimization,
subject to reliability or availability constraints is well
known as the redundancy-optimization problem. It has been
number of available versions for elements of RGS addressed in many studies, e.g., , where the binary-state
, reliability was considered. When applied to a wide variety of
systems (e.g., production, power generation, data transmission),
reliability is considered to be a measure of the ability of the
Manuscript received June 8, 1998; revised May 20, 2000.
The author is with the Reliability and Equipment Department, PD&T Divi- system to meet the demand, and the outage effect is: 1) different
sion, Israel Electric Corporation Ltd., P.O. Box 10, Haifa 31000 Israel (e-mail: for units with different nominal generating or transmitting
Levitin@iec.co.il). capacity, and 2) depends on demand distribution. Therefore,
Publisher Item Identifier S 0018-9529(01)06804-X.
the capacities of system components should be considered as
1The singular and plural of an acronym are always spelled the same. well as the demand distribution curve. The nonhomogeneous
0018–9529/01$10.00 © 2001 IEEE
LEVITIN: REDUNDANCY OPTIMIZATION FOR MULTI-STATE SYSTEM 53
system containing elements with different capacities can be The problem is to find the minimal-cost RGS and MPS struc-
considered to be a multi-state system, because its components ture which provides the desired level of system-ability to meet
can have different performance levels depending on the state of the demand.
the elements they contain. For such a system, each component In spite of the fact that this paper considers only 2-level
can be characterized by its PRD. The redundancy optimization RGS–MPS hierarchy, the method can easily be expanded to
problem for such a system is a problem of system-structure systems with multilevel hierarchy. When solving the problem
optimization. This problem for a series-parallel multi-state for multi-level systems the entire RGS–MPS system (with PRD
system was formulated in . The algorithm for system defined by its structure) can be considered in its turn as one of
structure optimization, subject to availability constraints, was RGS for higher level MPS.
suggested in ; in this algorithm, the appropriate versions As in , , to solve the combinatorial optimization
of system elements are to be chosen from a list of available problem, a genetic algorithm is used which operates only
products for each type of component as well as number of with values of solution quality and does not require derivative
parallel elements. Each element is characterized by its capacity information. A solution-quality index is comprised of both
(productivity), availability, and cost. The objective is to meet availability and cost estimates.
the demand (represented by a demand distribution curve) Examples are presented in which the optimal structures of
with the desired level of system availability while minimizing simple 2-level systems are obtained for various availability con-
system cost. This approach allows the reliability engineer to straints.
solve practical problems in which a variety of products exist
and in which analytic dependencies are unavailable for the cost II. PROBLEM FORMULATION AND DESCRIPTION OF SYSTEM
of system components. MODEL
Extending the algorithm in  solves the system structure Consider a system consisting of MPS and different RGS,
optimization problem without limiting the diversity of versions as in Fig. 1. The MPS can have up to different elements
of elements connected in parallel; hence both series-parallel and connected in parallel. Each producing element consumes re-
parallel-series systems (according to classification in ) can be sources supplied by RGS, and produces a final product. To dis-
optimized. tinguish among elements with different characteristics, the no-
While the algorithms mentioned here cover a wide range of tion of element version is introduced. There are versions of
series-parallel systems, they are restricted to systems with con- producing-elements available. Each version ,
tinuous flow. Systems with continuous flow are made of ele- is characterized by its , , , and ,
ments that can process any piece of product (resource) within its . MPS element of version works only if it re-
capacity (productivity) limits. In this case, the minimal amount ceives the amount of each resource defined by .
of product which can proceed through the system is not limited. Each resource is generated by the corresponding RGS
which can contain up to parallel resource-generating
In practice, there are technical elements that work only if the
elements of different versions. Each version of an element of
amount of some resources is not lower than specified limits.
RGS supplying resource , is characterized by its availability,
If this requirement is not met, the element fails to work. Ex-
productivity, and cost.
ample #1 is a control system which stops the controlled process
All the properties of element of subsystem can be ob-
if a decrease in its computational resources does not allow nec-
tained from a list of MPS and RGS elements available in the
essary information to be processed within required cycle time.
market according to a chosen for this element. The struc-
Example #2 is a metal-working machine which cannot perform
ture of subsystem is defined by numbers of versions of ele-
its task if the flow of supplied coolant is less than required. In
ments , chosen for this subsystem. The
both examples the amount of resource necessary to provide the
, , defines the en-
usual operation of a given composition of main producing units
tire system structure. To allow the number of elements in each
(controlled processes or machines) is fixed. Any deficit of the
subsystem to vary, use “dummy” elements of version 0. Such
resource makes it impossible for all the units from the compo-
elements have productivity 0 and cost 0. Therefore, all the
sition to operate together (in parallel) because any unit can not
elements of can vary in the range .
reduce the amount of resource it consumes. Therefore any re-
For a given ,
source deficit leads to turning off some producing units.
This paper considers systems containing producing elements
with fixed resource consumption. The systems consist of sev- (1)
eral RGS that supply different resources to the MPS. Each sub-
system consists of different elements connected in parallel. Each The is used as a measure of the entire system availability.
element of MPS can perform only by consuming a fixed amount If the operation period is divided into intervals, each with
of resources. If, following failures in RGS, there are not enough duration and demand level , , then
resources to allow all the available producing elements to work,
some of these elements should be turned off. We assume that
the choice of the working MPS elements is made in such a way (2)
as to maximize the total performance rate of MPS under given
resources constraints. and define the cumulative demand curve.
54 IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 1, MARCH 2001
This can be done using
True False (7)
Consider the single elements with total failures. Since each
element belonging to RGS has nominal performance
and availability [corresponding to chosen
The -function of such an element has only 2 terms and
The -function of a “dummy” element with performance 0 does
not depend on its availability and is .
Fig. 1. Structure of simple RGS–MPS system. The for the entire subsystem represents the prob-
abilistic distribution of the (amount of resource ) which
Formulate the problem of system structure optimization as can be supplied to the MPS: each value of can be supplied
follows: find the minimal cost system configuration that pro- with probability .
vides the required : If subsystem contains 1 element with availability and
capacity , the amount of resource can have 2 levels. The
distribution of is the same as the distribution of the element
III. SYSTEM-AVAILABILITY ESTIMATION
The procedure in this paper for evaluating system availability
is based on the universal moment-generating function technique If subsystem contains elements connected in parallel, its
, and was proven to be very effective for high-dimension total capacity in each moment is the sum of the capacities of
combinatorial problems. The detailed description of the uni- its elements. For example, if element #1 has capacity with
versal -transform applied to system-availability estimation is probability and element #2 has capacity with proba-
in . A brief introduction to this technique is given here. bility , the total capacity of the component containing these
The universal moment generating function ( -transform) of two elements is with probability , which
a discrete variable is defined as a polynomial corresponds to term in the -function
representing the entire component capacity distribution. In the
general case, the -function of elements connected in parallel
(4) can be defined using the operator:
the discrete r.v. has possible values; .
The polynomial can define PRD: it represents all possible
states of the system (or element) by relating the with of the operator is a product of polynomials representing the in-
the system in this state. dividual -functions.
To evaluate , the coefficients of should be Consider, for example, subsystem consisting of 2 elements
summed for every term with : with and , and and respectively. Having
the -functions, representing capacity distribution for individual
LEVITIN: REDUNDANCY OPTIMIZATION FOR MULTI-STATE SYSTEM 55
the -function of the subsystem is The RGS which can provide the work of minimal number of
producing units, becomes the system bottleneck. Therefore, this
RGS defines the total system capacity. To calculate the -func-
tion for a system containing 2 different RGS, the operator
should be used. This operator for a pair of RGS is:
which corresponds to the distribution of :
Successively applying the operator using the rule
The -function which represents PRD of RGS containing
elements with their individual -functions , one can obtain -function for all the resource generating sub-
is: systems which represents the entire system PRD for an
unlimited number of producing elements.
The system productivity equals the minimum of total theo-
retical productivity which can be achieved using available re-
sources and total productivity of available producing elements.
To obtain the system PRD, accounting for the availability of
MPS elements, the same operator should be applied:
number of different levels of resource generation.
The same operator can be used to obtain the -function B. PRD of System Containing Different Elements in MPS
representing maximal PRD of MPS, . is number of
Let MPS have different elements; then there are pos-
different levels of MPS productivity.
sible states of element-availability composition. Each state can
The represents the distribution of system productivity
be characterized by set of available elements.
defined only by MPS elements’ availability. This distribution
The probability of state is
corresponds to situations in which there are no limitations on
required resources. (20)
A. PRD of System Containing Identical Elements in MPS
If a producing subsystem contains only identical elements, The maximal possible productivity of MPS, and corresponding
the number of the elements that can work in parallel, when resources consumption, in state are, respectively:
the available amount of resource is , is
which corresponds to total system productivity
The amount of resources generated by RGS is defined by their
productivity of a single element of MPS
PRD. It is not always enough to provide maximal possible pro-
amount of resource required for this element ductivity of MPS at state . To define maximum possible pro-
for ductivity, , of MPS under resource constraints, solve the in-
teger linear programming problem:
represents the total theoretical productivity, which can be
achieved using available resource by an unlimited number of
In terms of entire system output, the -function of RGS is:
56 IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 1, MARCH 2001
available amount of resource , [element by a principle of evolution. The comprehensive description of
works]; if element works it produces units of main product GA theory and application in engineering is in , . Appli-
and consumes of each resource . cation of GA in reliability optimization are reported in , ,
The PRD of the system can be defined by evaluating all pos- –.
sible combinations of available resources generated by RGS and Unlike constructive optimization algorithms which use so-
states of MPS. For each combination, a solution of (21) defines phisticated methods to obtain a good single solution, the GA
the system productivity. The -function representing PRD of the deals with a set of solutions (population) and tends to manip-
entire system is: ulate each solution in the simplest way. “Chromosomal” rep-
resentation requires the solution to be coded as a finite-length
string. The steady-state version of GA used in this paper was de-
veloped by . As reported in  this version, named GEN-
ITOR, outperforms the basic “generational” GA. The structure
of steady state GA is:
1) Generate an initial population of randomly con-
structed solutions (strings) and evaluate their fitness.2
To evaluate for the system with its PRD, the probability that 2) Select 2 solutions randomly, and produce a new solution
the total productivity of the system is not less than a specified (offspring) using a crossover procedure providing inheri-
must be calculated: tance of some basic properties of the parent strings in the
offspring. The probability of selecting the solution as a
parent is proportional to the rank of this solution.3 Unlike
To obtain the system PRD, its productivity should be deter- the fitness-based parent selection scheme, the rank-based
mined for each unique combination of available resources and scheme reduces GA dependence on the fitness function
for each unique state of MPS. Equation (22) shows that, in gen- structure, which is especially important when constrained
eral, the total number of integer linear programs to be solved optimization problems are considered .
to obtain is . In practice, the number of The same double-point crossover technique as in ,
programs to be solved can be drastically reduced using the set  is used here: the fragment of the string is randomly
of 4 rules: chosen as a set of adjacent positions. All the elements
1) If for the given vector and for allocated within the fragment are copied into the child
the given set of MPS elements , there exists for solution string from its first parent and the rest of the
which , then system productivity elements are copied from the second one.
. This example illustrates the crossover procedure in
2) If for each element from , there exists for which the offspring is obtained from the 2 parent
which , then the system productivity strings , of length 8 (the fragment is between posi-
. tions 3 and 7):
3) If there exists element for which , for
some , then in (21) must be set to 0. The dimension
of the integer program can be reduced by removing all
such elements from .
4) If for the given vector and for
the given set , the solution of (21) determines subset 3) Allow the offspring to mutate. Mutation results in slight
of turned-on MPS elements if , changes in the offspring structure and maintains diver-
the same solution must be optimal for the MPS states sity of solutions. This procedure avoids premature con-
characterized by any set : . This allows vergence to a local optimum, and facilitates jumps in the
one to avoid solving many integer programs by assigning solution space. In this GA, the mutation procedure just
to all the . swaps elements initially located in 2 randomly chosen po-
For systems with many elements and/or resources, the sitions of the string.
required computation for solving redundancy optimization 4) Decode offspring to obtain the objective function (fitness)
problem can be unaffordable even when using this computa- values. These values are a measure of quality, which is
tion-complexity reduction technique. In this case, the use of used to compare various solutions.
fast heuristics for solving integer programs is recommended 5) Apply a selection procedure that compares the new off-
instead of exact algorithms. spring with the worst solution in the population and se-
2Unlike the “generational” GA, the steady-state GA performs the evolution
IV. OPTIMIZATION TECHNIQUE search within the same population, improving its average fitness by replacing
worst solutions with better ones.
To solve the optimization problem in (3), use the same ap- 3All the solutions in the population are ranked in order of their fitness in-
proach as in , . It is based on a GA, a technique inspired crease.
LEVITIN: REDUNDANCY OPTIMIZATION FOR MULTI-STATE SYSTEM 57
lects one that is better. The better solution joins the pop- For different MPS elements, the algorithm forms a set of ver-
ulation, and the worse one is discarded. If the population sions of elements included into MPS:
contains equivalent solutions after the selection process,
redundancies are eliminated and, as a result, the popula-
tion size decreases.
and for all its different subsets , , ,
6) Generate new randomly-constructed solutions to re-
corresponding to MPS states, the state probability is determined
plenish the population after repeating steps 2–5
using (20). Then it solves optimization problems (21) for each
times (or until the population contains a single solution
composition of available resources and, finally, obtains -func-
or solutions with equal quality). Run new genetic cycle
tion using (22).
(return to step 2).
To obtain the probability that the system output rate exceeds
7) Terminate the GA after genetic cycles.
given , the decoding procedure applies the operator (6), (7)
The final population contains: a) the best solution achieved,
and b) different near-optimal solutions which might be of in- (27)
terest in decision-making.
The for all the demand levels is calculated using (2).
A. Solution Representation and Decoding in GA To let GA look for the solution with minimal total cost and
To apply this GA to a specific problem, one has to define the with , the solution quality (fitness) is evaluated:
solution representation as well as the decoding procedure. This
GA deals with length integer strings; maximal number (28)
of elements that the entire system may contain:
for any , which corresponds to the “dummy” ele-
Each solution is represented by string , where (29)
a sufficiently large penalty.
(25) For solutions meeting the requirement , the “fitness
of solution” equals its total cost.
number of version chosen for element of subsystem ; V. EXAMPLE
corresponds to MPS and to RGS .
To vary the number of elements included in subsystems by A. Description of the System to be Optimized
using “dummy” elements, the solution generation procedure is The main producing component of the system can have up
designed as: for each , the random value from the to 6 parallel producing elements (chemical reactors) working
range is assigned to element of the in parallel. To perform their task, producing elements require 3
string with probability , and a value of 0 is assigned with prob- different resources:
ability .4 1) Power, generated by energy supply subsystem (group of
To allow each randomly generated string to represent a converters),
feasible solution, the decoding procedure obtains the version- 2) Computational resource, provided by control subsystem
number chosen for element of subsystem using: (group of controllers),
3) Cooling water, provided by water supply subsystem
(26) (group of pumps).
Each of these resource generating subsystems can have up
is calculated using (25). to 5 parallel elements. Both “producing units” and “resource
To obtain the -function of RGS represented by elements generating units” can be chosen from the list of products avail-
of string with position numbers from able in the market. Each producing unit is characterized by its
availability, productivity, cost and amount of resources required
to for its work. Table I shows the characteristics of available pro-
ducing units. The resource generating units are characterized by
their availability, generating capacity (productivity), and cost.
the decoding algorithm uses (15). For identical MPS elements it Table II shows the characteristics of available resource gener-
also obtains MPS using the same expression, transforms ating units. Each element of the system is considered to be a
-functions of RGS to the form (16), and obtains the system unit with total failures.
-function using (17)–(19). The demand for final product varies with time. Table III
4It was experimentally found that p = 0 8 provides the fastest GA conver-
shows the demand distribution in the form of a cumulative-de-
gence to the best solution. mand curve.
58 IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 1, MARCH 2001
PARAMETERS OF THE MPS UNITS AVAILABLE
PARAMETERS OF THE AVAILABLE RGS UNITS
PARAMETERS OF THE CUMULATIVE DEMAND CURVE
PARAMETERS OF THE OPTIMAL SOLUTIONS
Fig. 2. Coefficient of variation of best-in-population solution fitness as
function of number of crossovers. For 5 search processes.
TABLE V rithm for solving the problem for different MPS elements,
PARAMETERS OF THE OPTIMAL SOLUTIONS
which requires much greater computational effort, usually
yields better solutions then one for -identical elements.
C. Computational Effort and Algorithm Consistency
The language realization of the algorithm was tested on a
DEC station 5000/240. The parameters of GA were chosen as:
, , .
B. Optimization Results For the time-consuming optimization problem in which MPS
Table IV contains minimal-cost solutions for required levels can have different elements, the time to obtain the best-in-popu-
of . The structure of each subsystem is presented by the list of lation solution (time of the last modification of the best solution
numbers of versions of the elements included in the subsystem. obtained) did not exceed 45 minutes. The average time for ar-
The actual estimated availability of the system and its total cost riving at the best solutions for the solved problems of this type
are in the table for each solution. was 27 minutes. The corresponding time for the problems with
Table V contains (for comparison) the solutions of the identical MPS elements was less than 1 minute.
system-structure optimization problem when the main pro- To demonstrate the consistency of the algorithm, GA was
ducing subsystem can contain only identical elements. When repeated 5 times with different starting solutions (initial pop-
MPS is composed of elements of different types, the same ulation) for the system structure optimization problems with
system availability can be achieved at much lower cost. Using and . The coefficient of variation was
elements with different availability and capacity (productivity) calculated for fitness values of best-in-population solutions ob-
provides much greater flexibility for optimizing the entire tained during the genetic search by different GA search pro-
system performance in different states. Therefore, the algo- cesses. Fig. 2 shows the variation of this index during the GA
LEVITIN: REDUNDANCY OPTIMIZATION FOR MULTI-STATE SYSTEM 59
proceeding. All the processes converge to the same solution fit-  V. Ramachandran and V. Sivakumar et al., “Genetic based redundancy
ness values. optimization,” Microelectronics and Reliability, vol. 37, no. 4, pp.
 A. Lisnianski, G. Levitin, H. Ben-Haim, and D. Elmakis, “Power system
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Systems, vol. 7, no. 5, pp. 681–694, 1995. Gregory Levitin received the M.S. (1982, with honors) in electrical engineering
 L. Painton and J. Campbell, “Genetic algorithm in optimization of from Kharkov (Ukraine) Politechnical Institute, the B.S. (1986) in mathematics
system reliability,” IEEE Trans. Reliability, vol. 44, no. 2, pp. 172–178, from Kharkov State University and Ph.D. (1989) in industrial automation from
June 1995. Moscow Research Institute of Metalworking Machines. From 1982 to 1990 he
 D. Coit and A. Smith, “Reliability optimization of series-parallel sys- was a software engineer and research associate in industrial automation at the
tems using genetic algorithm,” IEEE Trans. Reliability, vol. 45, no. 2, Ukrainian and Russian Research Institutes. From 1991 to 1993 worked at the
pp. 254–266, June 1996. Technion-Israel Institute of Technology as a post-doctoral fellow at the faculty
 , “Penalty guided genetic search for reliability design optimiza- of Industrial Engineering and Management. In 1993 Dr. Levitin joined the R&D
tion,” Computers Ind. ?? Engineering, vol. 30, no. 4, pp. 895–904, 1996. Division of The Israel Electric Corporation, and is an engineer-expert in the
 , “Solving the redundancy allocation problem using a combined Reliability Department and an adjunct lecturer at the Technion. His current re-
neural network/genetic algorithm approach,” Computers Operations Re- search is in artificial-intelligence and operations-research application in relia-
search, vol. 23, no. 6, pp. 515–526, 1996. bility and power engineering. He is a Senior Member of IEEE.