Joint Chance-Constrained Reliability Optimization with General Form of Distributions

 Probabilistic or stochastic programming is a framework for modelling optimization problems that involve uncertainty, undoubtedly supporting many decision problems in business and management. Stochastic programming models arise as reformulations or extensions of reliability optimization problems with random parameters. Moreover, the resource elements vary and it is reasonable to consider them as stochastic variables. In this paper, we describe the chance-constrained reliability stochastic optimization (CCRSO) problem for which the objective is to maximize the system reliability for the given joint chance constraints where only the resource variables are random in nature and which follow different general form of distributions. Few numerical examples are also presented to illustrate the applicability of the methodology.

Reliability is defined as the probability that a device or system is able to perform its intended functions satisfactorily under specified conditions for a specified period of time.However, traditional reliability assumes that a system and its components can be in either a completely working or a completely failed state only (Birnbaum, Esary, & Saunders, 1961), i.e., no intermediate states are allowed.A reliability-based methodology for the robust optimal design of uncertain linear structural systems subjected to stochastic dynamic loads was also presented by Papadimitriou, Katafygiotis, and Siu (1997) and Papadimitriou and Ntotsios (2004).
Solution methods in the literature for reliability optimization of complex systems are mainly heuristic methods.In recent years, metaheuristic algorithms such as genetic algorithm (Gen & Cheng, 1997), simulated annealing (Ravi, Murty, & Reddy, 1997), and tabu search (Glover & Laguna, 1993) have also been applied to reliability optimization of complex systems.A comprehensive review of heuristic and metaheuristic algorithms for reliability optimization can be found in a ..., .
It is given that the th i random variable i b has three known parameters   , and Now, for the above pdf the joint probabilistic constraints in System (1) can be written as: After integration, we have: Hence, for the given random variable, the joint chance constraints in System (2) are converted into joint deterministic constraints as follows:   The deterministic constraints may get the information from the following distributions when the latter follow the parameters   , , , : Hence, in this case, the deterministic form of the chance-constrained programming problem for n-stage series with m-joint chance constraints is given by: Charles, Ansari, and Khodabakhshi: Joint Chance-Constrained Reliability Optimization with General Form of Distributions IJOR Vol. 12, No. 3, 057−068 (2015) After inserting the particular value of the parameters   , , ,   in the above deterministic constraints, we get different deterministic constraints for different distributions.
The th i random variable i b is acknowledged to have three known parameters Now, the joint chance constraints in System (2) for the above pdf can be written as: The integration leads to:

  
As such, for the given random variable, the following joint deterministic constraints can be obtained from the conversion of the joint chance constraints in System (2): The deterministic constraints may obtain the information from the distributions listed below when the latter follow the parameters   , , , : Now, the below represent the joint chance constraints for the above pdf: The following is obtained after integration: The below joint deterministic constraints are then derived from the conversion of the joint chance constraints in System (2) for the given random variable: When the subsequent specified distributions follow the parameters

 
,, A B h b   , the deterministic constraints may contract the information from these distributions:

    
, and Burr Type XII distribution In consequence, the chance-constrained programming problem for n-stage series with m-chance constraints can be found in its deterministic form, defined as follows: Finally, different deterministic constraints for different distributions can be obtained after inserting the particular value of the parameters

 
,, where In order to obtain the reliability of the system, the above 3-stage series with 3-chance constraints problem is solved using the LINGO software.The results show that the reliability of the system is s R = 0.9998, at

Example 2: (for case 2)
This second example builds upon the following stochastic model: where, in this case, We now have a 3-stage series but with 4-chance constraints problem that we solve by means of employing the LINGO software and we obtain the reliability of the system, that is, s R = 0.9983, at The LINGO software is used once again to solve the above 5-stage series with 4-chance constraints problem and the reliability of the system is obtained, that is, s R = 0.9979, at x =4, and 5 x =3.

CONCLUSION
In this paper, we formulate the chance-constrained reliability stochastic optimization problem for optimal solution to an n-stage series system with m-joint chance constraints in which only resource variables are random in nature.Various cases have been discussed with different general form of distributions when resource variables are random in nature and have different general form of distributions.After formulating the problem, we solved it using the LINGO software.One of the limitations of the study is that the current approach to tackle the problem assumes that only the right-hand side of the constraints is random in nature; simultaneously studying the case in which one can introduce randomness on the left-hand side of the joint chance constraints, as well as in the objective function, separately or combined, which is used to measure system performances, such as mean system-life time,  -system lifetime, and system reliability; many real life management problems actually do have multiple objectives, i.e., minimizing the cost, maximizing the performance, maximizing the reliability, and so on, subject to satisfying several requirements.Taking the lead from this, and in line with Charles and Udhayakumar (2012) and Charles, Udhayakumar, and Rhymend Uthariaraj (2010), the present work may be extended to multi-objective reliability optimization problems with constraints having the finite probability being violated, as well as it may be extended to solve the proposed systems using hybrid algorithms.

1 :
the following form: Charles, Ansari, and Khodabakhshi: Joint Chance-Constrained Reliability Optimization with General Form of Distributions IJOR Vol. 12, No. 3, 057−068 (In System (1), let i b follow a general form of distributions    

FF
  , and   i hb is a monotonic, continuous, and differentiable function of i b in the interval   , ii  .In this context, the pdf of the random variable i b is given by:

Case 3 :FF
 .Charles, Ansari, and Khodabakhshi: Joint Chance-Constrained Reliability Optimization with General Form of Distributions IJOR Vol. 12, No. 3, 057−068 (h b   in the above deterministic constraints, we then obtain different deterministic constraints for different distributions.In System (1), let i b follow a general form of distributions    .Moreover, in the interval   , ii  ,   i hb is a monotonic, continuous, and differentiable function of i b .The pdf of the random variable i b is then defined by:

1 b 3 b 4 b
follows a Pareto distribution with parameters 10follows a Burr type IV distribution with parameters 6follows a Burr type IX distribution with parameters 1 20, 1 3 k   .In this case, the deterministic model of the above stochastic problem becomes as follows: and

Example 3 : (for case 3 )
For our final example, we have the following stochastic problem: Charles, Ansari, and Khodabakhshi: Joint Chance-Constrained Reliability Optimization with General Form of Distributions IJOR Vol. 12, No. 3, 057−068 ( h b   in the above deterministic constraints, we get different deterministic constraints for the various distributions listed below: Power Function distribution 0.5, , tanh ii kb , Burr Type VIII distribution

Proposition 2 :
System (A2) is a complement of System (h b   in the above deterministic constraints, we get different deterministic constraints for the various distributions listed below: Inverse Weibull distribution Joint Chance-Constrained Reliability Optimization with General Form of Distributions in the above deterministic constraints.Charles, Ansari, and Khodabakhshi:

Charles, Ansari, and Khodabakhshi:
Joint Chance-Constrained Reliability Optimization with General Form of Distributions