Redundancy Optimization for Multi-State System with Fixed Resource-Requirements and Unreliable Sources

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					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
             RGS ,
             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., [1], 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                                                                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 [2]. 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 [3]; in this algorithm, the appropriate versions            As in [3], [4], 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 [4] 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 [1]) 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
                                                                    version       ],


                                                                    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
                                                             (3)    capacity:

   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
[5], 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 [3]. 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
producing elements.
  In terms of entire system output, the -function of RGS is:
                                                                       subject to
                                                                                       for                                     (21)
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 [6], [7]. Appli-
and consumes         of each resource                 .                  cation of GA in reliability optimization are reported in [3], [4],
   The PRD of the system can be defined by evaluating all pos-           [7]–[26].
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 [27]. As reported in [28] 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 [29].
to obtain          is                 . In practice, the number of                The same double-point crossover technique as in [3],
programs to be solved can be drastically reduced using the set                 [4] 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 [3], [4]. 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)
                                                                                to         :
   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-
                                                                        (24)    ments,

Each solution is represented by string                              , where                                                                 (29)
for each
                                                                                     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

                                                              TABLE I
                                                PARAMETERS OF THE MPS UNITS AVAILABLE

                           TABLE II

                           TABLE III

                          TABLE IV

                                                                     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,
                                                                     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

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                                                                                         661–663, 1997.
                                                                                    [19] 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
 [14] 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
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      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.

Description: This paper considers a redundancy optimization problem for a multi-state system of: 1) elements that consume a fixed amount of resources to perform their task, and 2) a number of resource generating subsystems. The algorithm finds the optimal system-structure, subject to availability constraints, by choosing system elements from a list of available equipment. Each element is characterized by its productivity, availability, and cost. Elements of the main producing subsystem also have their specific resource consumption limitations. The objective is to minimize the sum of investment costs while satisfying demand, represented by a cumulative demand curve, with given probability. To solve the problem, a genetic algorithm is used for optimization. The procedure, based on the universal generating function, is used to evaluate the system-availability while assuming that the working elements of the main producing subsystem are chosen in such a way that the total system performance rate is maximal under given resource constraints. Examples demonstrate how to obtain the optimal structures of a simple 2-level system for various availability constraints.