MODELLING AND OPTIMISING OF AN INTEGRATED RELIABILITY REDUNDANT SYSTEM

The reliability of a system is generally treated as a function of cost; but in many real-life situations reliability will depend on a variety of factors. It is therefore interesting to probe the hidden impact of constraints apart from cost – such as weight, volume, and space. This paper attempts to study the impact of multiple constraints on system reliability. For the purposes of analysis, an integrated redundant reliability system is considered, modelled and solved by applying a Lagrangian multiplier that gives a real valued solution for the number of components, for its reliability at each stage, and for the system. The problem is further studied by using a heuristic algorithm and an integer programming method, and is validated by sensitivity analysis to present an integer solution.


INTRODUCTION
The reliability of a system can be increased by keeping redundant units, or by using components of greater reliability, or by employing both methods simultaneously [3,4].Either of them consumes additional resources.Optimising system reliability, subject to resource availability such as cost, weight, and volume, is considered.Generally, reliability is treated as a function of cost; but when applied to real-life problems, the hidden impact of other constraints like weight, volume, etc, will have a definite impact on optimising reliability.The novel application of a redundant reliability model with multiple constraints is considered to optimise the proposed system.

Figure 1: Series-parallel system
The problem considers the unknowns -that is, the number of components (x j ), the component reliabilities (r j ), and the stage reliability (R j ) at each stage for a given multiple of constraints to maximise the system's reliability, which is called an integrated reliability model (IRM) [9].In the literature, integrated reliability models are optimised using cost constraints where there is an established relation between cost and reliability.The novelty of the proposed work is its consideration of weight and volume as additional constraints, along with cost, to design and optimise the redundant reliability system for a series-parallel system configuration.

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All the components in each stage are assumed to be identical -i.e., all the components have the same reliability.

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The components are assumed to be statistically independent -i.e., the failure of a component does not affect the performance of other components in any system.

MATHEMATICAL MODEL
The objective function and the constraints of the model subject to the constraints non-negative restriction that x j is an integer and r j , R j >0

MATHEMATICAL FUNCTION
To establish the mathematical model, the most commonly-used function is considered for the purpose of reliability design and analysis.The proposed mathematical function where a j , b j are constants System reliability for the given function The number of components at each stage X j is given through the relation The problem under consideration is
The number of components in each stage (x j ), the optimum component reliability (r j ), the stage reliability (R j ), and the system reliability (R s ) are derived by using the Lagrangian method.The method provides a real valued solution with reference to cost, weight, and volume.

Total volume 5000
Table 3: Volume constraint details System reliability = R s = 0.7860

HEURISTIC METHOD
Since the Lagrangian method gives a real valued solution, a heuristic approach is applied to derive an approximate integer solution.

Heuristic algorithm
Step 1: Initialise and enter the values of required input parameters.
Step 2: Enter the maximum number of components (n).
Step 3: Initialise the number of components, set to 1 for the 1 st stage, and calculate the values of system cost (C), weight (W), volume (V) for the 1 st stage.
Step 4: Initialise the number of components, set to 1 for the 2 nd stage, and calculate the values of system cost(C), weight(W), volume(V) for the 2 nd stage.
Step 5: (i) Initialise the number of components, set to 1 for the 3 rd stage, and calculate the values of system cost (C), weight (W), volume (V) for the 3 rd stage.(ii) Sum up all the values C, W, V for all three stages.(iii) Calculate the system reliability (R s ).(iv) Check the constraints.
Step 6: (i) If the constraints are satisfied, print the corresponding values of number of components and system reliability (R s ).(ii) If the constraints are not satisfied, increment the number of components of stage three by 1 and go to Step 5. (iii) Repeat the above step until the number of components in all three stages reaches the value less than or equal to the maximum number of components (n).

R s =
System reliability R j = Reliability of stage j, 0< R j < 1 r j = Reliability of each component in stage j, 0< rj < 1 x j = Number of components in stage j c j = Cost coefficient of each component in stage j w j = Weight coefficient of each component in stage j v j = Volume coefficient of each component in stage j C o = Maximum allowable system cost W o = Maximum allowable system weight V o = Maximum allowable system volume a j = Constant b j = Constant

Table 4 :
Reliability design relating to costVariation in total cost = 10.28%

Table 5 : Reliability design relating to weight
Variation in total weight = 12.96%