KONSEP RELIABILITY Rt 1 Ft dimana Ft Probalitity

KONSEP RELIABILITY R(t) = 1 – F(t) dimana F(t) = Probalitity of Failure R(t) = Probalitity of Survive 10/31/2021

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

Contoh R(t) = 1 / (1+0. 001 t) F(t)=1 - (1+0. 001 t)-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) = P(A). P(B) (Conditional) (Independent) Bila ada n unit harus bekerja simultant dan independent R(t)1 R(t)2 R(t)3 R(t)n P(R 1. R 2. R 3…. Rn) = P(R 1). P(R 2). P(R 3)… P(Rn) Reliability system R(t)s = [R(t)]n

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

Jumlah unit Rusak Laju Kegagalan (Failure Rate) 10 1 Hours 10 10 00 50 0 0

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(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) = F(t+ t) – F(t) = f(t) t f(t) h(t) = R(t) Hazard Function Atau Failure Rate

Cumulative Hazard Function Cumulative hazard dalam selang t 1 -t 2 atau F(t) = 1 – e-H(t)

Average Failure Rate (AFR)

Bila waktu dari 0 hinga time period ( 0 – T), maka : H(T) AFR(T) = T = - ln R(T) T Probabilitas failure selang time period T F(t) = 1 – e-Tx. AFR(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(T 1. T 2) Failure rate in FIT = 109 x h(t) = 104 x Failure rate in %/K AFR in FIT = 109 x AFR (T 1. T 2)

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

Armada Taxi Contoh 1 2 3 4 5 6 x x x 7 8 x x x 9 10 0 100 x 200 300 = Pertama kali Fail 400 500 x 600 700 = Fail ke kali sekian time

Pertanyaan 1. 2. 3. 4. 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)

Armada Taxi 1 2 3 4 5 6 x x x 7 8 F(500) = 5/10 9 R(500) = 5/10 x x F(550) = 7/10 R(550) = 3/10 100 200 300 f(500 550) =(2/10)/50 hr = 1/(5 x 50) x x h(500 550) =(1/(5 x 50) /x(5/10) = 10 / (5 x 5 x 50) = (2/5) / 50 10 0 x x x x 400 500 600 700 time r(500 550) = (4/10) / 50 hr