TENTISSÄ KÄYTETTÄVÄ KAAVAKOKOELMA KURSSILLE Tilastollinen laadunvalvonta

Transkriptio

1 TENTISSÄ KÄYTETTÄVÄ KAAVAKOKOELMA KURSSILLE Tilastollie laauvalvota Shewharti muuttujakartat ARL I = α ARL II = β x-kartta x = x + + x Ex =µ ja Vx = µ ± k Φx = π x e t t α = Φk β =Φk Φ k S-kartta S = x i x gx = X = S x 3 Γ e x, ku x>0 i

2 ii Γx = ES = Γ Γ Γ VS = Γ 0 s x e s s = c 4, ES = c 5 = c 4 B 5 = max0,c 4 kc 5 B 6 =c 4 + kc 5 = c 5 α = PB 5 <S<B 6 = P B 5 <X< B 6 B5 β =PB 5 <S<B 6 =P + <X< B6 + R-kartta R = maxx,...,x mix,...,x f R r = F t F t r ftft r t, fx = e x µ x µ, Fx =Φ π ER = π ER = π 0 s 0 e u +u s Φu Φu s us = s e u +u s Φu Φu s us VR =ER ER = 3 D = max0, k 3 D = + k 3

3 iii α = π e u Φu Φu D Φu Φu D u β = π e u Φu Φ u D Φu Φ u D u + + Karttoje käyistys x-kartta: m m x i = x, m m S i = S, m m R i = R S-kartta: x ± x ± k S = x ± A 3 S, A 3 = k c 4 c 4 k R = x ± A R, A = k max 0, kc 5 S = B 3 S c 4 + kc 5 c 4 S = B 4 S

4 iv R-kartta: max 0, k 3 R = D 3 R + k 3 R = D 4 R Yksittäisarvokartat MR-kartta: MR = y i y i = maxy i,y i miy i,y i =, 3 = 4 π π Käyistys: MR i = y i y i i =,...,m, MR = m π π A = k, D 3/ 3 = max 0, k m MR i i= π, D 4 =+k Shewharti attribuuttikartat p-kartta p = D Ep = ED =θ, Vp = θ θ max 0,θ k θ θ mi,θ+ k θ θ α = Pθ k θ θ <D<θ+ k θ θ, PD = = θ θ

5 v Käyistys: β =Pθ k θ θ <D<θ+ k θ θ, PD = = θ + θ θ θ m m p i = p max 0, p k p p mi, p + k p p c-kartta Pc = i = i i! e i =0,,... Ec =, Vc = max0, k + k α = P k <c<+ k Käyistys: β =P k <c<+ k, + i Pc = i = e i! α = Γ, k +Γ, + k β =Γ +, k Γ +, + k m c i = c m max0, c k c c + k c

6 vi u-kartta ja epämeriittikartta u-kartta: u = D + + D = D Eu =, Vu = VD = max 0, k + k Epämeriittikartta: α = P k <D<+ k, PD = i = i e i! β =P k < D <+ k, = PD = i = + i e + i! L j= ω j D j i = L j= ω j D j i E = j= L ω j j, V = j= L j= L max 0, ω j j k L ωj j j= j= L ω j j + k L ωj j j= ω j j

7 vii Liukumakartat CUSUM-kartta Ey =m, Vy =v C + r C r = max0,y r k + + C + r = mi0,y r k + C r, C + 0 = C 0 =0 PC + s =F s + k + z s 0 RL + z = mi C + r h + r, C + 0 = z p + j z =PRL+ z = j p + z =PC + h + = F h + + k + z p + j h+ z = p + j xfx + k+ z x + p + j 0F k+ z j =, 3,... 0 ARL ± I = ARL ± II = j= j= jp ± j jp ± j 0, Ey =m 0, Ey =m + δ± EWMA-kartta Ey =m, Vy =v z 0 = m z i = y i + z i i =,,... i z i = j y i j + i z 0 j=0 Ez i =m, Vz i = i v

8 viii i LCL i = m kv i UCL i = m + kv LCL = m kv UCL = m + kv ARL z =+ UCL LCL u z ARL u f u RL z = mi r, z 0 = z z r LCL tai z r UCL p j z =PRL z = j UCL z LCL z p z = F + F p j z = UCL LCL ARL I = ARL II = u z p j uf u jp j m, Ey =m j= jp j m, Ey =m + δ j= Moimuuttujakartat x = x + + x S = S = x i xx i x T x i µx i µ T Ex =µ, Vx = Σ

9 ix x = m x + + x m, S = m S + + S m fx = π p/ etσ e x µt Σ x µ χ -kartta X = x µ T Σ x µ EX =p, VX =p Hotelligi kartat T M = x µ T S x µ UCL = p p F p, p, α T = x x T S x x UCL i = pi i i p + F p,i i p+, α, UCL = χ p, α W -kartta: W = p l e Alti kartat l et S etσ + traceσ S S -kartta: et S etσ UCL = χ pp+/, α = p ets etσ χ,,..., p EetS = b etσ, VetS = b etσ b =! p! p, b = max0, b k b etσ b + k b etσ! +! p! p +! p b

10 x Kykyieksit m = USL + LSL, = USL LSL Ieksit Kae ieksit: Taguchi ieksi: C P = 3 C PL = µ LSL 3 C PU = USL µ 3 C PK = mic PL,C PU = µ m 3 Pear Kotz Johso-ieksi: C PM = 3 +µ T Y L = 00 Φ Y = 00 Φ C A = µ T LSL µ USL µ %, Y U = 00Φ USL µ LSL µ Φ % Y L = 00Φ3C PL %, Y U = 00Φ3C PU % % Boylesi ieksi: 00Φ3C PK % Y<00Φ3C PK % C PS = 3 Φ Φ USL µ Y = 00Φ3C PS % + µ LSL Φ

11 xi Ieksie estimoiti ja testaus ˆµ = x, ˆ = S b = S C P : testaus: Ĉ P = 3S b, Ĉ PK = Ĉ PL = x LSL 3S b, Ĉ PM = x m 3S b 3 Sb +x T Ĉ PU = USL x, Ĉ A = x T 3S b X = S b = 9 Ĉ P C A : testaus: CA a PĈA a = Φ C PL : testaus: C PU : testaus: C PM : testaus: = C P Ĉ P CA +a +Φ X = S b, U = x µ /, = µ LSL / =3 C PL 3 ĈPL = U +3 C PL X X = S b, U = x µ /, = µ USL / = 3 C PU 3 ĈPU = U 3 C PU X X = S b, U = x µ /, δ = µ T / C PK : testaus: 9 Ĉ PM = X +U + δ X = S b, U = x µ /, ν = δ = 3C PK = ν 3 C PK, δ = µ m /

12 xii 3ĈPK = x m S b = ν U + δ X F y = Γ by 0 x 3 e x Φν y x + δ Φ ν + y x + δ x, ν, ku y>0 by = y, ku y 0 Tarkastusotata: attribuuttiotata Yksikertaie otata PD = = M N M =0,..., N TN T N c P AA T =, T = M N N =0 c fθ,, c =P AB θ =P A θ = θ θ =0 fθ,, c + fθ,, c = θ c+ θ c c + fθ, +,c fθ,, c = θ c+ θ c c fθ,, c = θ c θ c θ c ATIθ =P A θ+n P A θ AOQθ =θp A θ tai AOQθ = θp A θ N

13 xiii Kaksikertaie otata P A θ = c =0 θ θ + ASNθ = + c t=c + =0 c 3 t c =c + t θ θ θ t+ θ + t Tarkastusotata: muuttujaotata Ala- ja yläpuolie tarkastus µ L P A µ =Φ / x0 µ θ =Φ U µ, P A µ =Φ /. µ x0, θ =Φ µ = x 0 Φ θ, µ = x 0 + Φ θ µ = x 0 Φ p, µ = x 0 + Φ p µ = x 0 Φ p, µ = x 0 + Φ p = Φ α+φ β Φ p Φ p L = µ + Φ α =x 0 Φ p + Φ α U = µ Φ α =x 0 + Φ p Φ α Kaksipuolie tarkastus U µ L µ P A µ =Φ / Φ / x µ µ x θ =Φ +Φ