Multi-objective design of control chart for short-run production

30 3 2015 6 JOURAL OF SYSTEMS EGIEERIG Vol.30 o.3 Jun. 2015 1,2,3, 1,2,3, 1,2,3 (1., 710049; 2., 710049; 3., 710049) : X.,., ;,. : ; ; ; : F253.3 : A : 1000 5781(2015)03 0297 09 doi: 10.13383/j.cnki.jse.2015.03.002 Multi-objective design of control chart for short-run production Li Chenglong 1,2,3, Su Qin 1,2,3, Zhang Pengwei 1,2,3 (1. School of Management, Xi an Jiaotong University, Xi an 710049, China; 2. State Key Laboratory for Manufacturing Systems Engineering, Xi an 710049, China; 3. The Key Lab of the Ministry of Education for Process Control & Efficiency Engineering, Xi an 710049, China) Abstract: Based on Markov chain this paper constructs and studies an X control chart model for short-run production which bases on Markov chain. A synthesis optimization is built considering cost per unit, production duration and defective percentage, while the existing relevant research only focuses on cost objective. A real production case study is carried out and indicates that multi-objective control chart is superior to economic control chart in most cases. Besides, the application of economic control chart easily induces the enterprise s short-term behavior under certain conditions, while the use of synthesis optimization can maximally avoid it. Key words: multi-objective; synthesis optimization; short-run; control chart design 1,.,,,.,,,,. : 2013 01 31; : 2013 10 10. : (70872091; 71371151).

298 30,.,,,.,, 1 3],., 4,5].,,,. Montgomery 6],.,,. Shamsuzzaman 7] X EWMA. Chen 8],,.,,,,, 9,10],,.,,. Hillier 11],,., 12,13]. Montgomery 14] X R, 4 6. Yang 15],X n = 1., enes 16] 17,18],.,,, (n 5), n = 1,2,...,5.,,,.,. X.,,,,,. 2,, T T U, Q Q U. m, n (0 n m), d,. : 1) ( ) X, X (µ,σ 2 ), µ {0,ε}. ( I, µ 1 = 0); 2) 1 ; 3) p, q = 1 p.,, ε, µ 2 = µ 1 + ε( II, )., 1 Duncan 19] Chiu 20],,.

3 : 299 ; 4), ; 5) +, i X i >, ; 6),,. 3, Ω = {(0,0),(0,1),(1, 0), (1,1),(2,0), (2, 1)}, (u,w), u : u = 0,., u = 0,u = 1, ( ), ; u = 2,,. w, w = 0, w = 1., ( i ) i 1, i 1., G (0,0),(0,0) G (0,0),(0,1) G (0,0),(2,0) G (0,0),(2,1) G (0,1),(0,0) G (0,1),(0,1) G (0,1),(2,0) G (0,1),(2,1) G m =......., (1) G (2,0),(0,0) G (2,0),(0,1) G (2,0),(2,0) G (2,0),(2,1) G (2,1),(0,0) G (2,1),(0,1) G (2,1),(2,0) G (2,1),(2,1) G m 6 6, m m.g (a),(b) a b,, G (0,0),(0,0) G (0,0),(1,0). G (0,0),(0,0) m, G (0,0),(0,0) = q m (1 α), (2) G (0,0),(1,0),, n G (0,0),(1,0) = (1 q m n )β+ τ i pq m n+i 1, (3) ( τ i = Φ (d (n i + 1)ε/n) / ) ( σ 2 /n Φ ( d (n i + 1)ε/n) / ) σ 2 /n, α β.. G (0,0),(0,0) G (0,0),(0,1) G (0,0),(1,0) G (0,0),(1,1) 0 0 G (0,0),(0,0) G (0,0),(0,1) G (0,0),(1,0) G (0,0),(1,1) 0 0 G m = 0 0 0 0 β 1 β. (4) G (0,0),(0,0) G (0,0),(0,1) G (0,0),(1,0) G (0,0),(1,1) 0 0 0 0 0 0 β 1 β G (0,0),(0,0) G (0,0),(0,1) G (0,0),(1,0) G (0,0),(1,1) 0 0

300 30 4 4.1,,,., : C i, ; C f,, ; C a,, ; C c,, ; C n,, ; L,,.,h p,h i, h f, h a ( ), h c. 4.2,,,,, ;,,,., C (u,w) = nc i + kh i L + η (u,w) C n + γ (u,w) (C c + h c L) + ( Ca + h a L (C a + h a L C f h f L)I (0) (u) ) I (1) (w), (5) T (u,w) = mh p + kh i + γ (u,w) h c + ( h a (h a h f )I (0) (u) ) I (1) (w), (6) k, k = 1.η (u,w) γ (u,w),η (u,w), γ (u,w)., γ (u,w),.,, C m = (C (0,0),C (0,1),C (1,0),C (1,1),C (2,0),C (2,1) ) T, T m = (T (0,0),T (0,1),T (1,0),T (1,1),T (2,0),T (2,1) ) T, η m = (η (0,0),η (0,1),η (1,0),η (1,1),η (2,0),η (2,1) ) T. m, m m.,,. m < m, n < m < m, n,, 0 < m n. π (0),, π (0) = 1 0 0 0 0 0]. π (i) = π (i 1) G m., π (i) C m, m = 0 C = π (i) C m + π C m, 0 < m n (7) π (i) C m + π G m C m, n < m < m,

3 : 301 π (i) η m, m = 0 Q = π (i) η m + π η m, 0 < m n (8) π (i) η m + π G m η m, n < m < m, π (i) T m + m h p, 0 m n T = π (i) T m + π G m T m, n < m < m, ] m m, C m η m. 4.3, (m,n,d),,. (C s,t s,q s ), Min {v C F(C) + v T F(T) + v Q F(Q)} s.t. (10) T T U Q Q U, F(C),F(T) F(Q) C,T Q ( ), v C,v T v Q, 21], v C + v T + v Q = 1.,,. (C e,t e,q e ) ( v C = 1,v T = v Q = 0), (C e,t e,q e ) = arg min {C}. (11) C, T Q; (10) (11),. (9) 5. 22], 5]..,, = 100, T 600, Q 2%, = 2.5. p = 0.01, ε = 2. C i = 0.2, C f = 1.5, C a = 50, C c = 4 C n = 12, L = 0.5, h p = 5, h i = 1, h f = 3, h a = 20, h c = 3.,. 5.1,. C, T Q.

302 30 1) 1, U,,. 2) 2,, U,,. 3),,. 1 3 : 1),.,,. 2),.,,,.,. C 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 n = 1 n = 2 n = 5 (n = 1) (n = 2) (n = 5) 0.9 500 520 540 560 580 600 620 640 T 1 Fig. 1 C T Cost per unit-production duration diagram C 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 n = 1 n = 2 n = 5 (n = 1) (n = 2) (n =5) 0.9 0 2% 4% 6% 8% 10% 12% Q Fig. 2 2 C Q Cost per unit-defective percentage diagram 14% 12% 10% n = 1 n = 2 n = 5 (n = 1) (n = 2) (n = 5) 8% Q 6% 4% 2% 0 500 520 540 560 580 600 620 640 T 3 Q T Fig. 3 Production duration-defective percentage diagram

3 : 303 5.2 23] :v C = 0.4,v T = 0.3, v Q = 0.3.,,. 1. : m(1 100), n(1 5), d(0 4). Table 1 1 Comparative results between economic optimization and synthesis optimization p ε C e T e Q e C s T s Q s C o T o Q o 0.187 0 501.1 1.49% 0.187 0 501.1 1.49% 0.181 1 500 1.51% ε = 1 : (100,1,4) : (100,1,4) 0.288 1 507.0 1.35% 0.291 5 505.9 1.44% 0.322 7 500 2.69% p = 0.001 ε = 2 : (24,4,1.2) : (32,5,1.1) 0.320 0 510.2 1.49% 0.327 5 508.8 1.36% 0.547 3 500 4.56% ε = 3 : (14,2,1.7) : (19,5,1.2) 0.392 9 501.7 3.04% 0.4232 512.1 1.89% 0.393 1 500 3.28% ε = 1 : (100,5,3.9) : (13,5,3.9) 0.944 0 536.2 0.97% 0.950 2 531.2 1.71% 1.472 2 500 12.27% p = 0.01 ε = 2 : (8,5,1.1) : (12,5,1) 1.054 5 543.8 1.08% 1.055 2 545.3 0.59% 3.183 3 500 26.53% ε = 3 : (6,4,1.2) : (6,5,1.1) 0.641 7 538.6 0.00% 0.641 7 538.5 0.00% 0.745 5 500 6.21% ε = 1 : (5,5,3.9) : (5,5,4) 1.850 4 604.6 0.00% / / / 3.382 7 500 28.19% p = 0.1 ε = 2 : (5,5,4) / 3.835 1 710.0 0.00% / / / 7.564 2 500 63.04% ε = 3 : (5,5,4) / : (10), /.,,.,. 33.3%(3/9 ) (T 600) (Q 2%),.,,,., 1, : 1) ( p = 0.001,ε = 1),, ; 2) ( p = 0.1,ε = 3),.,,,. 5.3,. 2(a) 2(b). 2,,,,.

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