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Available online at www.sciencedirect.com Procedia Computer Science 00 (2018) 000–000

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Procedia Computer Science 127 (2018) 170–179

The First International Conference On Intelligent Computing in Data Sciences

An Effective Oriented Genetic Algorithm for solving redundancy allocation problem in multi-state power systems Mustapha Essadqi, Abdellah Idrissi, Amine Amarir Computer Science Laboratory (LRI), Computer Science Department, Faculty of Sciences, Mohammed V University, Rabat, Morocco

Abstract The main goal in the design of industrial systems is to improve their reliability. This can be achieved by reducing the complexity of the system by increasing the reliability of its components (reliability allocation) or hardware redundancy (redundancy allocation) or a combination of these two approaches. In this paper, we are interested in solving the redundancy allocation problems RAP in serial-parallel multi-state systems MSS. Our contribution is the Effective Oriented Genetic Algorithm (EOGA) which aims to improve the genetic algorithm by introducing a new parameter which is the degree of relevance of the component, thus allowing a more oriented search in the generation of the initial population, and also the use of specific operators and fitness function, giving better results. In the following, we will cite some errors in the calculations encountered in previous work. Our goal is to improve the redundancy of the system to satisfy a required demand under the constraint of maximizing the reliability of the system and minimizing its cost. To evaluate the reliability, we will hybridize our algorithm with the universal generating function UGF. A comparison with the literature will be discussed at the end of this work showing the effectiveness of our approach, and the significant gap between our results and previous work. © 2018 The Authors. Published by Elsevier B.V. © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and This is an open access article under the CC BY-NC-ND (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection peer-review under responsibility of International Neural license Network Society Morocco Regional Chapter. and peer-review under responsibility of International Neural Network Society Morocco Regional Chapter. Keywords : Redundancy allocation problem (RAP); Reliability; Multi-State Systems (MSS); Universal Generating Function (UGF); Genetic algorithm (GA); Effective Oriented Genetic Algorithm (EOGA).

1. Introduction A parallel serial multi-state system (MSS) usually consists of several subsystems connected in series and each subsystem consists of several components in parallel. This structure acquires a certain level of redundancy from the system which enables it to be available in the event of failure of one or other components. The system and its components then have a performance level ranging from nominal operation to total failure. These components are 1877-0509© 2018 The Authors. Published by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of International Neural Network Society Morocco Regional Chapter. 1877-0509 © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of International Neural Network Society Morocco Regional Chapter. 10.1016/j.procs.2018.01.112

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chosen from a list on the market and are characterized by their performance G, availability A and cost C [1]. The redundancy allocation problem (RAP) consists to reduce the complexity of the system by increasing hardware redundancy in order to achieve an optimal topology. The system thus constituted must satisfy a demand threshold under the constraint of maximizing its reliability and of minimizing its cost [2]. The reliability of a system represents the probability that a system is operational in a time interval [0, t]. As for availability, it represents the probability that a system is operational at a given time t. This very broad definition, and generalizing the notion of reliability, leads us to take an interest in the notion of availability of systems. In general, the evaluation methods of the latter are based on four different approaches: the structure function approach, the Markov stochastic process, the Monte-Carlo simulation technique and the universal generator function (UGF) approach. The RAP is a complex combinatorial optimization problem and the exhaustive examination of the huge number of possible solutions is not simple enough given the time limitation. Solving this problem leads to solutions that are in the form of possible configurations. The calculation within its configurations must give results respecting the constraints which are defined by mathematical equations. However, several errors are encountered in the literature affecting the cost function and reflect the incompatibility of the configuration with the cost of the components. In [3], a comparison between these four approaches emphasizes that the UGF approach is fast enough to be used in optimization problems in the case where the search space is important. However, several errors are encountered in the literature affecting the cost function and reflect the incompatibility of the configuration with the cost of the corresponding components. These errors will be cited in detail in Section 3. Our contribution is to remedy this problem and improve the results found. We will show improvements, and more relevant results through the use of a new algorithm, EOGA. Our work will be organized as follow: A formulation of the problem and UGF technique in section 2. Quotation of some errors encountered in the literature in 3. Introduce and apply our approach in 4. In section 5, we will test our algorithm while showing its efficiency through the errors cited in 3 and at the end a conclusion and our perspectives. 2. Problem formulation and Universal Generating Function technique Redundancy is a technique used to improve the reliability of systems. Optimization of the latter plays a key role in system design engineering. In this section we will try to mathematically model our problem after formulating it. Nomenclature: Let ⊗𝑓𝑓 be a multiplicative operator, this operator acts differently on the function 𝑢𝑢𝑖𝑖 𝑧𝑧 ; on one hand, it performs a multiplication of the components of the vector p i, on the other hand, the function 𝑓𝑓 acts on the components of the vector xi. The function 𝑢𝑢𝑖𝑖 𝑧𝑧 checks the following strong properties: i ii iii iv

⊗ 𝑝𝑝𝑍𝑍 𝑎𝑎 = 𝑝𝑝𝑍𝑍 𝑎𝑎 ⊗𝑓𝑓 𝑝𝑝𝑍𝑍 𝑎𝑎 𝑞𝑞𝑍𝑍 𝑏𝑏 = 𝑝𝑝𝑝𝑝𝑍𝑍𝑓𝑓 𝑎𝑎 𝑏𝑏 where f is defined depending on the nature of the MSS ⊗ 𝑢𝑢1 𝑧𝑧 … 𝑢𝑢𝑘𝑘 𝑧𝑧 𝑢𝑢𝑘𝑘+1 𝑧𝑧 … 𝑢𝑢𝑛𝑛 𝑧𝑧 = ψ ψ 𝑢𝑢1 𝑧𝑧 … 𝑢𝑢𝑘𝑘 𝑧𝑧 ; ψ 𝑢𝑢𝑘𝑘+1 𝑧𝑧 … 𝑢𝑢𝑛𝑛 𝑧𝑧 ∀ . ⊗ 𝑢𝑢1 𝑧𝑧 … 𝑢𝑢𝑘𝑘 𝑧𝑧 𝑢𝑢𝑘𝑘+1 𝑧𝑧 … 𝑢𝑢𝑛𝑛 𝑧𝑧 = ψ 𝑢𝑢1 𝑧𝑧 … 𝑢𝑢𝑘𝑘 𝑧𝑧 𝑢𝑢𝑘𝑘+1 𝑧𝑧 … 𝑢𝑢𝑛𝑛 𝑧𝑧 ∀ .

2.1 Mathematical formulation of the problem

Let us consider a MSS that consists of n subsystems. Those subsystems are in series and each subsystem . For each version (1 ), the component contains components in parallel, each, with a version , performance and cost . The structure of each subsystem i is defined by is characterized by its availability for each version [6, 7], as shown below in Figure 1. the number of parallel components

Corresponding author. Tel:+212600777517;

E-mail address:[email protected]; [email protected].

Es-sadqi Mustapha/ Procedia Computer Science 00 (2018) 000–000 Mustapha Essadqi et al. / Procedia Computer Science 127 (2018) 170–179

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Fig. 1. A series-parallel system consisting of n parallel subsystems, each with

3

components

Let us consider the structure of our system as the one described in 2.1, the entire system will be defined by the vector

{

2.1.1. Cost.

} such that:{

The cost function is linear, so, the cost of the entire system is the sum of costs of each component. Given the set of ={ 1 … }, the cost of the entire system will be given by the formula: vectors (1) = ∑ 1∑ 1

2.1.2. Availability and demand. According to the standard (EN 13306) [8], the reliability of a given system is defined as the ability of a property to perform a required functioning under given conditions and for a given time . According to the standard (NF X 60-500) [9], it is defined as a characteristic of a property that is well expressed by the probability that it ful fills a required function under given conditions and for a given time. This is expressed mathematically by: = As our system belongs to the repairable systems, we can no longer speak about reliability but about availability , which, according to the standard (EN 13306) [10], is specified as the ability of a property to be able to accomplish a required function under given conditions at a given time, or during a given time interval, and assuming that the necessary external means are provided. This result in: = At each time t, the system has a performance . However, the demand is predicted by a cumulative load curve by periods , those periods can be, for example, a distribution according to the four seasons of the year. In this case, availability is measured by the probability that the performance of the system is greater than or equal to a given demand [11]. If is the instantaneous availability of the system and is its performance and W0 the . required demand during a period , we can write using (3) that: = In fact, the system during its functioning, goes through a distribution of constant failures and acquires a stationary , this is true to the extent that the external factors (climate, shock, ...) and long production state when = periods are neglected. In this case, we’re talking about the stationary state (equilibrium) and the probability of reaching this state tends to the stationary probability .This state is reached after a sufficient time and the = = distribution of probability states can be written as [12]: = ∑ and using (4): This allows writing, for a required The demand described in (6), allows dividing the period of functioning T in M time intervals of duration , the availability of MSS will be written as follows: =



1



1

In Engineering and for a given system , the availability is related to the load loss probability index (LOLP) and is defined by [13, 14]: = . This allows to write: = In this study, the case of components with total failure each component is characterized by its nominal performance and its availability . Thus, we can write:

4

Mustapha Essadqi et al. / Procedia Computer Science 127 (2018) 170–179 Es-sadqi Mustapha/ Procedia Computer Science 00 (2018) 000–000

=

[

{

173

]=

[ = ]= To evaluate the availability related to a given performance, we will use the UGF detailed in the next subsection. 2.2. Universal Generating Function UGF were introduced by I. Ushakov in 1986 [15], many researchers, such as G. Levitin and A. Lisnianski [16, 17, 18], have proved the efficiency of the method. This is a fairly rapid technique; it is based on simple algebraic operations and helps to deter-mine the level of performance and availability of an overall system based on the characteristics of its components. 2.2.1. Mathematical background. Let be the number of independent discrete random variables can be represented by vectors

and

The 𝒵𝒵 transform of the variable

=(

, such as: {

=(

1

1



)



)

1



and let’s assume that each variable = {

=

}

=

is defined by the distribution function in the polynomial form: =∑

1



1



To obtain the 𝒵𝒵 transform of the n functions distribution function, we replace the polynomial product by a composition operator ⊗ for the individual 𝒵𝒵 transforms of each function, by applying the property (2) we obtain: ⊗ ∑

=∑

1

…∑



1

This technique, merging the 𝒵𝒵 transform and the composition operator ⊗ , is called the Universal Generating Function (UGF). The 𝒵𝒵 transform of a variable is called the 𝒰𝒰-function of the function f ( ) (12), and is denoted (11), while the 𝒰𝒰-function of the function f ( ) (12) is denoted as such that: … =⊗ 1 To obtain the UGF of a subset with a number of components, we use the composition operators. These operators determine the resulting function U(z) of a series or parallel components group based on simple algebraic operations. If the random variable is identical to a given performance, which is the case of our study, according to formula (12), the 𝒰𝒰-function resulting from the combination of a set of m components is: ⊗ (

1



)=∑

1



1

…∑

1



1



.

Note that the function represents the equivalent productivity of components. When these components = … … are connected in series, the function takes the following form: 1 1 And in the case where the m components are connected in parallel, the function becomes: To satisfy a requested state of performance

1



=∑

1

, we introduce a satisfaction operator =

defined as follows:

={

2.2.2. Application of UGF on MSS. To evaluate the availability of a MSS, two operators are used; the operator 𝒮𝒮 is introduced for the series composition and the operator 𝒫𝒫 for the parallel one. These operators determine the polynomial for a group of elements. i.

Parallel components.

When the measurement of the performance is linked to the system capacity, the overall performance of the of a parallel system is the sum of performances according to the equation (15). Therefore the 𝒰𝒰-function component containing elements in parallel can be obtained by using the operator 𝒫𝒫: = 𝒫𝒫 1 … Where 𝒫𝒫{ 1 … }=∑ 1 Thus, using the property 2 of 𝒰𝒰-functions and using (11), (13) and (15) we find for two parallel components:

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174

𝒫𝒫(

+

) = ∑∑

1

1

5

1

We can notice that the operator 𝒫𝒫 performs a simple multiplication between individual 𝒰𝒰-functions as follows: ii.

=∏

Series components.

1

In the case where elements are in series, the functionality is provided by the lower performance element. The 𝒰𝒰in this case is given using the operator 𝒮𝒮. Considering as the cardinal of levels, (12) and (13) function give: = 𝒮𝒮 1 … = Where 𝒮𝒮 1 … … 1 So for two elements in series using equations (11), (13) and (14): 𝒮𝒮(

iii.

Series-parallel components.

) = ∑∑

1

1

1

The 𝒰𝒰-function of the whole parallel-series system is given by using consecutively the operators, 𝒫𝒫 and 𝒮𝒮. = = According to (9), the 𝒰𝒰-function of the component is then: Therefore, for a system consisting of subsystems in series, for each subsystem and for each version of , the subsystem is modeled as follows: components 𝑣𝑣 with 𝑣𝑣 = 𝒫𝒫 (

The whole system contains Once using (16):

1

(

)

1



)

subsystems, thus, by introducing the operator 𝒮𝒮 we obtain: = 𝒮𝒮[

1



]

s determined, we will need to evaluate the probability of meeting a given demand .

=

, to do this,

2.2.3. Formulation of the problem. The problem can be summarized as the following: -

Maximize the availability resulting of {⊗

Minimize the Cost

=∑

1∑

1

〈∑

〈∑







(23)



(22) while respecting the demand

3. Calculation errors found in the literature As indicated in 1, these errors are linked to the cost function C and the allocation of components in the optimal configuration. Indeed, the solutions found by an algorithm are in the form of optimal configurations. Each configuration contains a number of components that must comply with the constraints outlined in 2.2.6. 3.1. Errors related to the function cost The cost function given by equation (1) is linear, which means that the total cost of the system is equal to the sum of the costs of the components of the optimal configuration. Let us consider the example in [19], the study presented deals with a hybrid system: Wind energy / gas. The authors used the method of Gravitational Search (GS) for its optimization. Table 1 presents data that the authors have used for tests. Table 1. Data example (Detailed electrical power system) [19]

Subsystem

Device Number

Availability A (%)

Power Units

1 2

0,992 0,986

Cost C (mln $) 7,735 6,475

Performance (MW) 5 5

6

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HT Transformers

HT Lines

HT/MT Transformers

MT Lines

3 4 5 6 7 8 9 10 1 2 3 4 5 1 2 3 4 1 2 3 4 5 6 7 8 1 2 3 4 5

0,994 0,988 0,980 0,991 0,992 0,984 0,993 0,985 0,994 0,990 0,997 0,991 0,998 0,971 0,997 0,991 0,976 0,978 0,986 0,978 0,983 0,981 0,971 0,985 0,973 0,984 0,993 0,989 0,981 0,986

6,698 6,290 6,146 6,945 6,984 6,465 125 150 2,805 2,272 2,594 2,569 1,857 1,985 1,983 1,842 1,318 0,842 0,875 0,745 0,654 0,625 0,608 0,492 0,415 0,456 0,432 0,364 0,283 0,242

175

5 6 6 6 6 6 80 120 18 18 12 12 8 16 14 14 12 16 16 14 12 12 12 10 10 14 12 12 8 8

Table 2. Parameter of cumulative load [19]

Load (MW) 200 160 120 80 Duration (h) 1080 2640 2880 2160 This system consists of five subsystems in series, and each subsystem consists of different components in parallel. In Table 3, the authors presented the results corresponding to two different demands: 200MW and 160MW. Thus two configurations, each gives a corresponding cost C (M $) and its availability (%). For the cost, the authors found: VT =280,15M$ for the first configuration and VT =185,12M$ for the second. Table 3. Optimal solution obtained by GS (Gravitational Search Algorithm)

Constraints M$

Topology

Subsystem 1 200

200

160

Cost, Reliability and Performance Method

MW

300

Optimal Topology

Subsystem 2 Subsystem 3 Subsystem 4 Subsystem 5 Subsystem 1 Subsystem 2 Subsystem 3 Subsystem 4 Subsystem 5

1,2,3,4,4,5,6,8,9,1 0 1,2,3,4,5 1,2,2,4 1,2,3,4,4,6,7,8 1,2,3,4,5 1,2,2,3,5,5,5,8,8,9 1,2,2,3 2,2,3,4 2,3,3,4,5,5,6,7 1,1,1,2,3

GS

GS

C(M$)

C=280,15

C=185,12

A (%)

W (MW)

0,998

200

0,96

160

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75

160

Subsystem 1 Subsystem 2 Subsystem 3 Subsystem 4 Subsystem 5

2,3,3,4,5,6,6,7,8,9 2,2,3,4,5,5,7 2,2,3,3,3 2,2,3,3,4,5,7,7 2,2,3,3,4

GS

C=84,159

0,99999

7

160

Let us now check the linearity of the cost function for these two configurations: ∑ 1 ∑ 1 . ∑ ∑ With C0=300M$ and G0=200MW, for the whole system the total cost is:  and we find: 1 1 C=354,472M$ instead of 280,15M$ indicated on the paper. It can be seen that only the cost of the first subsystem largely exceeds C0 which is set at 300M$, which means that the calculation method does not respect constraint on the cost. The same remark as for the second configuration. We have done the calculation and found that the cost that corresponds to it is 209,263M$ instead of 185,12M$. Again, we went beyond the C0 that equals 160M$. We will talk about the configuration 3 in 4.2 that reflects an unauthorized allocation error. Several cost-type errors have been encountered in our research and will be cited in [20,21,22,23,24,25,26]. 3.2. Errors related to the allocation of components In the architecture of a MSS, each subsystem performs a well-defined task. The component of a subsystem cannot be allocated to another subsystem. For example, in electrical systems, a wind turbine cannot be replaced by a transformer. We shall cite here three errors encountered in [19, 20, 21]. In [19] (Table 3) and on configuration 3 let's = i.e. look at the allocated component 7 in the second configuration. This allocation is not allowed because there are only five types of components in the subsystem 2. Consequently, this configuration is not allowed. The same error was found in [30] when the authors applied the improved bat algorithm (IBA) on an electrical system. Table 4. Extract of the result obtained in [30]

0,990

Sub-system1 Sub-system2 Sub-system3 Sub-system4 Sub-system5

6,6,6,6,6,6,6 4,3,1,5 ,4 1,5,5,5 6,8,9,7,9,8,2,4,9 3,4,4,4

0,97

VT=11,828

IBA [30]

4. Effective Oriented Genetic Algorithm The genetic algorithm (GA) was first introduced in [27], GA maintains a population of individual solutions. In literature, there are several encoding schemes for representing an electrical system. In our case, each chromosome is represented by a finite string of symbols encoding a possible solution in a given problem space. This space contains all possible solutions to the problem. GA tries to emulate Darwin’s Evolutionary process, it exploits the idea of the “survival of the fittest” and an interbreeding population to create a new innovative search strategy. It iteratively creates new solutions from the old ones by selecting the strings using the Roulette wheel selection which gives the fittest individuals a greater chance of survival, and interbreeding them. So in each generation, the GA creates a set of string from the bits and pieces of the previous strings (crossover operator), and adding random new data (mutation operator). The end result is a search strategy that is tailored for vast, complex; multi-modal search spaces. Several authors as in [28,29,30,31] used the GA and [32] used CSP and Forward Checking to deal with RAP in MSS having as objectives: either the cost or availability, or both of them. In our study, we apply an enhanced version of GA hybridized with the UGF. We call it Effective Oriented Genetic Algorithm on a MSS with the objectives of minimizing the cost and maximizing the availability of the given system. 4.1. Encoding of solutions.

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Our solutions are in the form of a set of strings which represents the set of the subsystems of our system, and each string is a set of integers where the length of the set is equal to the number of devices of the subsystem, and each integer reflects the id of the device. 4.2. Evaluation of each solution. Each time an individual is created, a fitness value is associated to it. This value is used by the selection process to favor the most suitable solutions, as it reflects the performances of such individual towards our problem. The evaluation function to calculate the fitness, in order to satisfy the objectives is the following: With: -

=

(

is the cost of the individual calculated using equation (1) is the availability of the individual calculated using the UGF (3) is the initial cost where as a constraint

)

4.3. Initial population. The choice of the initial population is based on a random solutions design, generated using the roulette wheel selection based on a new parameter which is the degree of relevance of the device. With those solutions respecting . The degree of relevance of a device is as follow: the constraints =

4.4. Selection.

The selection process defines how many times an individual is involved in the reproduction process, the individuals with the best performances (best fitness values) are selected more often than the others and are used in the next step. 4.5. Reproduction The reproduction or variation helps to find better individuals, by producing individuals based on the best solutions of the current population. It is carried out using two main operators: Crossover and mutation. - The crossover consists of applying procedures on two selected individuals called parents to obtain two children; those children have some similar characteristics from both of their parents. For our problem, the order of integers in the encoding of individuals is important to minimize the respond time when comparing two strings and then in a polynomial time, we add each chromosome which doesn’t exist in the existing population, that way, we maintain the diversification of individuals in our population. Fig. 1. An example of a crossover operation

-

After the crossover, the mutation permits to obtain diversity of individuals for the given population, by replacing randomly the characteristics of its individuals. Those mutations must occur with a low probability . And it uses the Swap mutation, since, as Pm. For EOGA, mutations occur with a probability: mentioned before, changing the order of integers for a system doesn’t change anything. The Swap mutation is done by changing two integers chose randomly by two other random integers.

4.6. Evaluation and selection for replacement

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9

EOGA evaluates the obtained individuals for the population, and selects the weaker individuals (min fitness values) and replace them by the new chromosomes. 4.7. Termination. The termination criterion is the number of generations. Once this number is reached, we get the best solution found. 5. Application of the EOGA algorithm and comparison of results In this section we will implement our approach for comparison with the literature [19]. 5.1. Comparison For this comparison, we chose a hybrid system. The authors in [19] used the GS for optimizing the system. Table 5. Optimal solutions obtained by GS (Gravitational Search Algorithm) and EOGA (Effective Oriented Genetic Algorithm)

Constraints (M$) (MW) 300

200

200

160

300

200

200

160

Topology

Optimal topology

Sub 1 Sub 2 Sub 3 Sub 4 Sub 5 Sub 1 Sub 2 Sub 3 Sub 4 Sub 5 Sub 1 Sub 2 Sub 3 Sub 4 Sub 5 Sub 1 Sub 2 Sub 3 Sub 4 Sub 5

1,2,3,4,4,5,6,8,9,10 1,2,3,4,5 1,2,2,4 1,2,3,4,4,6,7,8 1,2,3,4,5 1,2,2,3,5,5,5,8,8,9 1,2,2,3 2,2,3,4 2,3,3,4,5,5,6,7 1,1,1,2,3 8,8,6,5,4,4,4,4,2,2 5,2,2,1 3,3,2,2 8,7,7,6,5,4,3,1 5,2,2,2,1 8,5,5,5,5,4,4,4,2,2 5,4,2,1 4,2,2,2 8,7,5,5,4,3,3,2 5,3,3,2,1

Method

GS

GS

Cost, Reliability and Performance C (M$) A (%) W (MW) 280,15[19]

185,12[19]

0,998[19]

200

0,967[19]

160

EOGA

87.85402

0.9995594

200

EOGA

86.673004

0.9998094

160

5.2. Discussion The EOGA has shown its ability to find optimal results. Moreover, it is able to make an oriented selection as shown in the latter configuration. Our algorithm EOGA adapts to cases of practical and dynamic realization to use the resources deployed to their limits. The EOGA gave the best results according to the cost and the availability. 6. Conclusion We presented in this paper an effective method for solving optimization problems of redundancy in multi-state serial-parallel systems, our method based on a version of the genetic algorithm with an oriented selection in the generation of the initial population using a new parameter which is the degree of relevance, proved its efficiency and gave the best results in our comparative case, it also allows the verification of the constraints (availability and cost). In the next paper, we will try to treat the problem with the variation of other parameters and additional constraints.

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