Using Genetic Algorithm to Find the Optimal Shopping Policy for 1-out-of-n Active-redundancy Series Systems under Budget Constraint

The mathematical model to find the optimal shopping policy from many available manufacturers for 1-out-of-n active redundancy series systems under budget constraint was formulated and tested using GA. The study showed that the number of possible combinations for this problem can be very high and the majority of those combinations are infeasible. This renders the enumeration technique ineffective or even impossible in practice, the matter that calls for a solution through GA. The results showed that the proposed genetic algorithm has high degree of robustness. Moreover, the results showed that the proposed algorithm is superior to the enumeration technique in terms of both computational time and quality of solution. Furthermore, the results showed that the convergence of the algorithm to the optimal solution is high.


Introduction
System reliability can be defined as the probability that the system will conduct its intended functions satisfactorily at least for a given period of time when operated under normal operating conditions.Redundancy, in its both configurations: high and low redundancy level, can increase the reliability of the system.This increase in the system reliability is normally accompanied with increase in the system cost.Redundancy allocation problem (RAP) is considered an optimization problem involving the selection of components and their layout in the system to maximize the system reliability under certain constraints.
The problem of how to choose and mix between different manufacturers that supply similar components with different reliabilities and costs for a series system can be treated as a special case of series-parallel RAP.
The problem of optimal shopping policy considered in this study is as follows: Consider a system that consists of k different series subsystems such that each subsystem consists of a single different component than the other subsystems.For each component, there are number of different manufacturers available in the market for that component with different reliabilities and costs.The optimal shopping problem is to determine the optimal configuration of the system within the budget, i.e., the configuration that will have the best reliability within the given budget.The configuration of the system includes the determination of the manufacturer(s) along with the number of components that will be used from each manufacturer in each subsystem such that the best reliability possible is reached and the budget of the system is not exceeded.

Related work
The RAP problem was proved to be an NP-hard problem.The redundancy allocation problem had been studied extensively in the literature.Fyffe et al. (1968) developed goal programming to solve system reliability allocation problem.You and Chen (2005) proposed an efficient heuristic to solve the RAP.Tian and Zuo (2006) used multi-objective optimization to solve RAP where the model maximized the system performance and minimized the system cost and weight simultaneously.Kulturel and Coit (2008) used objective prioritization in a multi-objective frame work to optimize the RAP.Coit (2001) optimized the RAP for non-repairable systems using cold standby strategy for the subsystems.Taboada and Coit (2012)  ive is es the given on of milar e, the those s, the m is a rating ystem ms Consider a system consists of K different series components (subsystems) with n k different manufacturers for each k th subsystem.Furthermore, assume that each subsystem's manufacturer has a different cost and reliability for the subsystem denoted by , , and , , , respectively, where , denotes the i th manufacturer for the k th subsystem.The design variables for the model is the number of components used from each manufacturer in each subsystem, , , to reach the optimal system reliability, , within the available budget, .The mathematical model for the problem can be expressed as follows: , Equation ( 1) is the objective function for this model which denotes the overall reliability of the 1-out-of-n active redundancy series system with K subsystems.Equation ( 2) guarantees that the overall cost of the system will not exceed the budget.Equation ( 3) limits the number of components for any subsystem that can be used from any manufacturer in the system and also guarantees that at least one component must be used in any subsystem, i.e., guarantees that the system must consist of a total of K subsystems.Equation ( 4) limits the number of components used in any subsystem for a certain manufacturer, and finally, equation ( 5) guarantees that only whole components can be used in the subsystems.
This model is clearly a non-linear integer model that belongs to the NP-hard class, therefore solving it with exact methods is mathematically intractable.Evolutionary algorithms can be used to solve such problem efficiently.Hence, in this study, a genetic algorithm will be proposed and used to solve this problem.Figure 2, shows a schematic diagram for the problem in hand.
The proposed genetic algorithm will be explained in the next section.

Design of the Proposed Genetic Algorithm
Genetic algorithm is one type of heuristic optimization search based on Darwin's natural selection theory.Genetic algorithm simulate the natural systems in solving problems.It enhances the initial population (initial solutions) through continuous evolution over generations through successive application of exploration and exploitation operators to reach a superior population that contains solutions (chromosomes) superior to the initial population.The genetic algorithm starts by encoding the solutions (phenotype) into chromosomes (genotype) using certain vector string template.This template is used to generate the initial population.The fitness value for each individual in the population is calculated, using a fitness function, to determine the next generation's parents through selection operator.Then, exploration (crossover) operator is applied on the parents to produce the offspring after which the exploitation (mutation) operator is applied on the offspring to generate the individuals of that generation.This evolutionary process, reproduction and selection, continues until certain level of convergence or a predetermined termination criterion is satisfied. www.ccsen The geneti

Chrom
The     Computer an osome in the c will be evaluat carry the sam ne means that er of the subsy he offspring w ll have their va tor.Table 3. Shopping policy for the system It is worth to mention here that the number of different configurations (feasible and infeasible) for the system is enormous as the number of combinations that can be used in any subsystem is huge, the matter that makes the enumeration of different configurations intractable.Considering the constraint given by equation ( 4), the total number of configuration combinations can be calculated by the following equation: In equation ( 6), 1 was added to count for the case of having zero components from the prospective manufacturer.
For the large problem, the number of possible combinations is calculated using equation ( 6) as: .This enormous number renders the enumeration technique intractable.
To assess the effectiveness of the proposed GA, ten millions random chromosomes where generated and the percentage of the feasible chromosomes was less than 0.0002%.This indicates clearly that the problem is fairly constrained to the extent that only about 2 in a million randomly generated chromosomes are feasible (the budget is tight).The average reliability of the 19 feasible random chromosomes found from the ten millions randomly generated chromosomes was 0.4232 which is significantly lower than the worst average reliability found by the proposed GA (population size of 50 and generation number of 1000).Also the S.D. for these randomly feasible chromosomes was 0.1904 with a coefficient of variation of 0.4500 compared to 0.082 and 0.1088 for the worst case found by the proposed GA respectively.Moreover, the time for generating and evaluating the ten millions random chromosomes was 383720 seconds (using a machine with the following specifications: Manufacturer HP, Model HPE-500f, Processor AMD phenon (tn) IIX6 1045T processor 2.70GHz, RAM 8.0 GB, system 64-bit operating system).compared with 93 seconds per replication for the GA on the same machine.This clearly indicates that the proposed GA is very effective in terms of solution quality and computational time compared to enumeration.
Table 2 clearly indicates that as the number of generations and the size of the population increases, the average reliability increases and the S.D decreases and hence the coefficient of variance decreases.Furthermore, the table also shows that the population size is more important than the number of generations for the performance of the proposed GA.For example, using 200 chromosomes and 1000 generations gives better results in terms of average reliability and coefficient of variation than using 100 chromosomes and 2000 generations.In addition, the table also indicates that at the same population size, as the number of generations increases the average reliability and coefficient of variations deceases.The number of possible combinations for this problem evaluated by equation ( 6) was 5.7 10 .The problem was solved by GA using population size of 50 and 1000 generations with 100 replications.The results obtained for the reliability of the system by enumeration was 0.9409 while for the average 100 replications of the proposed GA was 0.9405.The S.D for the 100 replications was 0.0017 with coefficients of variance of 0.0018.Out of the 100 replications, GA found the optimal solution 88 times.This means that GA was capable to converge to the optimal value 88% of the time in this problem.
Figure 8 shows the shopping policy (chromosome) corresponds to the optimal solution found for this problem.

Conclusions
The problem of selecting the optimal shopping policy of products from many available manufacturers for 1-out-of-n active redundancy series systems under budget constraint was formulated and tested using GA.The study showed that the number of possible combinations for this problem can be very high from which the majority of the possible solutions are infeasible, the matter that renders the enumeration technique ineffective or even practically impossible.
The results showed that the proposed algorithm has high degree of robustness.Moreover, the results showed that the proposed algorithm is superior to the enumeration technique in terms of both computational time and quality of solution.Furthermore, the results showed that convergence of the algorithm to the optimal solution is high.

Figure 8 .
Figure 8. Optimal shopping policy for the small problem

Table 4 .
Relevant information for the small problem