KONSEP RELIABILITY R(t) = 1 – F(t) dimana F(t) = Probalitity of Failure R(t) = Probalitity of Survive 11/20/2018

Karakteristik Relibility R(t)=e-t 1 Contoh R(t) = (1+g(t))-1 R(t) = e-g(t) R(t) t (hours) Probability Density Function (PDF) dan Cumulative Distribution Function (CDF)

Contoh R(t) = 1 / (1+0.001t) F(t)=1 - (1+0.001t)-1 Berapa probalilitas kerusakan produk baru setelah dioperasikan : - 1000 Jam - 4000 Jam - antara 1000 jam hingga 4000 jam Jawab F(1000) = 1-(1+1)-1= 1-1/2 = 0.5 -F(4000) = 1-(1+4)-1= 1-1/5 = 0.8 -F(4000)-F(1000) = 0.3

Contoh 150 armada taxi (baru) beroperasi. Berapa taxi yang gagal beroperasi setelah 1000 jam? dan 3000 jam berikutnya? Jawab F(1000) = 0.5 x 150 = 75 mobil F(4000) = 0.8 x 150 = 120 mobil

Ingat ! P(AB) = P(A).P(B|A) (Conditional) = P(A).P(B) (Independent) Bila ada n unit harus bekerja simultant dan independent R(t)1 R(t)2 R(t)3 R(t)n P(R1. R2. R3…. Rn) = P(R1). P(R2). P(R3)… P(Rn) Reliability system R(t)s = [R(t)]n

Probabilitas Kerusakan pada sistem tersebut : Fs(t) = 1 –[R(t)]n

Laju Kegagalan (Failure Rate) 10 Jumlah unit Rusak 1 Hours 10000 10500

Teorema Failure Rate (Hazard Function) Event A : Survive hingga selang waktu t => P(A) Event B : Failure selang t setelah t => P(B) P(AB) = P(A).P(B|A) P(AB) P(B|A) = P(A) P(A & B keduanya terjadi) = P(B|A) P(survive)

F(t+ t) – F(t) P(gagal selang t|survive hingga selang t) = R(t) Untuk t -> 0 F(t+ t) – F(t) F’(t) = = f(t) t f(t) Hazard Function Atau Failure Rate h(t) = R(t)

Cumulative Hazard Function Cumulative hazard dalam selang t1-t2 atau F(t) = 1 – e-H(t)

Average Failure Rate (AFR)

Bila waktu dari 0 hinga time period ( 0 – T), maka : H(T) - ln R(T) = AFR(T) = T T Probabilitas failure selang time period T F(t) = 1 – e-TxAFR(T)

Satuan Failure rate = percent / 1000 hours = %/K Failure rate = parts per million / 1000 hours = ppm/K = FIT (fails in time) Konversi Failure rate in %/K = 105 x h(t) AFR in %/K = 105 x AFR(T1.T2) Failure rate in FIT = 109 x h(t) = 104 x Failure rate in %/K AFR in FIT = 109 x AFR (T1.T2)

Bathtub Curve Renewal Failure Rate Time Item Life Early Failure Period Stable Failure Period Wear out Failure Period Time

Contoh x x x x x x x x x x x x x x x time x x Armada Taxi 1 x x x 2 x x 3 x 4 5 x x x Armada Taxi 6 x x 7 x 8 x x 9 10 100 200 300 400 500 600 700 time x x = Pertama kali Fail = Fail ke kali sekian

Pertanyaan Estimasi CDF dan Reliability Function pada 500 & 550 jam Estimasi f(500-550) Estimasi Failure Rate atau Hazard rate, h(500-550), Estimasi Renewal Rate, r(500-550)

x x x x x x x x x x x x F(500) = 5/10 R(500) = 5/10 x x 2 x x 3 x 4 5 x x x Armada Taxi 6 x x 7 F(500) = 5/10 R(500) = 5/10 x x 8 h(500550) =(1/(5x50) / (5/10) = 10 / (5x5x50) = (2/5) / 50 x 9 10 F(550) = 7/10 R(550) = 3/10 100 200 300 400 500 600 700 f(500550) =(2/10)/50 hr = 1/(5x50) r(500550) = (4/10) / 50 hr time

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