Rasool et al. _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ଵ ൌ ߤ ହ ߠ ହ ܲ Now use normalization condition i.e the sum of all the probabilities is equal to one ሾσܲ ଵ ୀ ൌͳሿ . By solving the equation (15) and (16) and substituting the value of ܲ ଵ in equation (3), we get ܲ ൌ ଵା ଵ ഋ ഇఱ ఱ (17) Availability ( ܣ ௩ ሻൌܲ ൌ ଵା ଵ ഀఱഁ ఱ ൌ ͲǤͻͷͷͺͺʹ͵ͷ͵ Similarly, Reliability (ܴ ሺ ݐ ሻሻൌ݁ି ఓ ఱ ௧ ൌ݁ି Ǥଷ௧ , Maintainability ሺ ܯ ሺ ݐ ሻሻൌͳെ݁ି Ǥହଵ௧ , ܨܤ ܶܯ ൌ ఓ ଵ ఱ ൌ ͵͵Ǥ͵͵͵͵͵͵͵݄ , ܴܶܶܯ ൌ ͳǤͷ͵ͺͶͳͷ݄ , ݁ܦ ܾ݈݁݊݀ܽ݅݅ ݕݐ ሺ݀ ሻ ൌ ʹͳǤʹ , ܦ = ͳെቀ ௗି ଵ ଵ ቁቀ݁ି ష భ െ݁ି ష భ ቁ ൌ ͲǤͲǤͺͶͺʹͺͻͷͷ Finally, the RAMD measures of stock preparation unit are derived by using the probabilistic argument as follows: System Reliability ܴ ௦௬௦ ሺ ݐ ሻൌܴ ௦௦ଵ ሺ ݐ ሻൈܴ ௦௦ଶ ሺ ݐ ሻൈܴ ௦௦ଷ ሺ ݐ ሻൈܴ ௦௦ସ ሺ ݐ ሻൈܴ ௦௦ହ ሺ ݐ ሻ ܴ֜ ௦௬௦ ሺ ݐ ሻൌ݁ି Ǥଶଷ௧ ൈ݁ି Ǥଶ௧ ൈ݁ି Ǥଵ଼ ௧ ൈ݁ି Ǥଵଵ௧ ൈ݁ି Ǥଷ௧ ൌ݁ି Ǥଶସ௧ System Availability ܣ ௦௬௦ ሺ ݐ ሻൌ ܣ ௌௌଵ ൈ ܣ ௌௌଶ ൈ ܣ ௌௌଷ ൈ ܣ ௌௌସ ൈ ܣ ௌௌହ ൌ ͲǤͻͷͷͳͷͻʹ ൈ ͲǤͻͶͲʹͷͻͶ ൈ ͲǤͻ͵Ͳͳ͵ͺ ൈ ͲǤͻ ൈ ͲǤͻͷͷͺͺʹ͵ͷ͵ ൌ ͲǤͺʹʹͳʹͶͶ System Maintainability ܯ ௦௬௦ ሺ ݐ ሻൌ ܯ ௦௦ଵ ሺ ݐ ሻൈ ܯ ௦௦ଶ ሺ ݐ ሻൈ ܯ ௦௦ଷ ሺ ݐ ሻൈ ܯ ௦௦ସ ሺ ݐ ሻൈ ܯ ௦௦ହ ሺ ݐ ሻ ֜ ሺͳെ݁ି Ǥସଽ௧ ሻൈሺͳെ݁ି Ǥସଽଽଽଽଽଵଶ௧ ሻൈሺͳെ݁ି Ǥସଽଽଽଽ଼଼ ௧ ሻൈሺͳെ݁ି Ǥଽଽ௧ ሻൈሺͳ െ݁ି Ǥହଵ௧ ሻൌሺͳെ݁ି ଽǤଷ଼ ଽଽଽ଼ ଵ௧ ሻ System Dependability ܦ ୫୧୬ሺ௦௬௦ሻ ሺ ݐ ሻൌ ܦ ୫୧୬ሺ ௦௦ଵሻ ൈ ܦ ୫୧୬ሺ ௦௦ଶሻ ൈ ܦ ୫୧୬ሺ ௦௦ଷሻ ൈ ܦ ୫୧୬ሺ ௦௦ସሻ ൈ ܦ ୫୧୬ሺ ௦௦ହሻ ͲǤͺൌͶͺͲʹǤͺͺͶͻͻͷ͵ͷͶ͵Ͳͷʹ ൈ ͲǤͻͲͲͲʹͻͻ ൈ ͲǤͻͳͲͷ͵ͷʹͳ͵ ൈ ͲǤͶͲͲͻͺͲ ൈ =0.437288184 _____________________________________________________________________________________________ 10 Braz. J. Biom. ǡǤ 43 ǡǦͶ͵ʹǡʹͲʹͷǤ Performance modeling and Optimization of Stock Preparation Unit In this section, a mathematical model of stock preparation unit is developed by using Markov birth death process and Chapman-Kolmogorov differential-difference equations derived by considering constant failure and repair rates. The steady state availability expression is derived from derived mathematical model and treated as the objective function for availability optimization. All the failure and repair rates are considered as the decision variables. The nature inspired algorithms genetic algorithm (GA) and particle swarm optimization (PSO) are utilized for optimization of objective function. The mathematical model is developed based on state transition diagram (Figure 7) by using simple probabilistic arguments as follows: ܲ ሺ ݐ ᶭ ݐ ሻൌሺͳെ ߤ ଵ ᶭ ݐ െ ߤ ଶ ᶭ ݐ െʹ ߤ ଷ ᶭ ݐ െ ߤ ସ ᶭ ݐ ߤ ହ ᶭ ݐ ሻܲ ሺ ݐ ሻ ߠ ଵ ᶭ ܲݐ ሺ ݐ ሻ ߠ ଶ ᶭ ଼ܲݐ ሺ ݐ ሻ ߠ ଷ ᶭ ܲݐ ଵ ሺ ݐ ሻ ߠ ସ ᶭ ܲݐ ଽ ሺ ݐ ሻ ߠ ହ ᶭ ܲݐ ଵ ሺ ݐ ሻ From equations (15-16) using (3), we get ܲ
Made with FlippingBook. PDF to flipbook with ease