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

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1 JOURAL OF SYSTEMS EGIEERIG Vol.30 o.3 Jun ,2,3, 1,2,3, 1,2,3 (1., ; 2., ; 3., ) : X.,., ;,. : ; ; ; : F253.3 : A : (2015) doi: /j.cnki.jse 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 , China; 2. State Key Laboratory for Manufacturing Systems Engineering, Xi an , China; 3. The Key Lab of the Ministry of Education for Process Control & Efficiency Engineering, Xi an , 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,.,,,.,,,,. : ; : : ( ; ).

2 298 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 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 = β 1 β. (4) G (0,0),(0,0) G (0,0),(0,1) G (0,0),(1,0) G (0,0),(1,1) 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

4 ,,,., : 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) = ]. 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,

5 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.

6 ) 1, U,,. 2) 2,, U,,. 3),,. 1 3 : 1),.,,. 2),.,,,.,. C n = 1 n = 2 n = 5 (n = 1) (n = 2) (n = 5) T 1 Fig. 1 C T Cost per unit-production duration diagram C n = 1 n = 2 n = 5 (n = 1) (n = 2) (n =5) % 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% T 3 Q T Fig. 3 Production duration-defective percentage diagram

7 3 : ] :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 % % % ε = 1 : (100,1,4) : (100,1,4) % % % p = ε = 2 : (24,4,1.2) : (32,5,1.1) % % % ε = 3 : (14,2,1.7) : (19,5,1.2) % % % ε = 1 : (100,5,3.9) : (13,5,3.9) % % % p = 0.01 ε = 2 : (8,5,1.1) : (12,5,1) % % % ε = 3 : (6,4,1.2) : (6,5,1.1) % % % ε = 1 : (5,5,3.9) : (5,5,4) % / / / % p = 0.1 ε = 2 : (5,5,4) / % / / / % 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,,,,.

8 304 30,. 2(a) Table 2(a) Results of sensitivity analysis C i C a C n (b) Table 2(b) Results of sensitivity analysis L h i h a :,. 6,,,.,, X.,,,,.,.,,,.,,,,,.,.,,,. : 1] Balm J G, alina S. Project management with time, cost and quality considerationsj]. European Journal of Operation Research, 1998, 47(2): ] Khang D B, Myint Y M. Time, cost and quality trade-off in project management: A case studyj]. International Journal of Project Management, 1999, 13(9): ],,. J]., 2006, 26(7): Yang Yaohong, Wang Yingluo, Wang engmin. Fuzzy tradeoff optimization of time-cost-quality in construction projectj]. Systems Engineering: Theory & Practice, 2006, 26(7): (in Chinese)

9 3 : 305 4] Bakir M A, Altunkaynak B. The optimization with the genetic algorithm approach of the multi-objective, joint economical design of the X and R control chartsj]. Journal of Applied Statistics, 2004, 31(7): ] Shamsuzzaman M, Wu Z. Control chart design for minimizing the proportion of defective unitsj]. Journal of Manufacturing Systems, 2006, 25(4): ] Montgomery D C, Heikes R G, Mance J F. Economic design of fraction defective control chartsj]. Management Science, 1975, 21(11): ] Shamsuzzaman M, Wu Z. Design of EWMA control chart for minimizing the proportion of defective unitsj]. The International Journal of Quality & Reliability Management, 2012, 29(8): ] Chen Y K, Liao H C. Multi-criteria design of an X control chartj]. Computers & Industrial Engineering, 2004, 46(4): ] Makis V. Multivariate Bayesian control chartj]. Operations Research, 2008, 56(2): ] Ho L L, Anderson G T. Economic design of an X chart for short-run productionj]. International Journal of Production Economics, 2009, 120(2): ] Hillier F S. Chart control limits based on a small number of subgroupsj]. Industrial Quality Control, 1964, 20(8): ] enes G, Tagaras G. The economically designed CUSUM chart for monitoring short production runsj]. International Journal of Production Research, 2006, 44(8): ] Makis V. Multivariate Bayesian process control for a finite production runj]. European Journal of Operational Research, 2009, 194(3): ] Montgomery D C. Introduction to Statistical Quality ControlM]. ew York: John Wiley & Sons, ] Yang M, Wu Z, Lee K M, et al. The X control chart for monitoring process shifts in mean and variancej]. International Journal of Production Research, 2012, 50(3): ] enes G, Tagaras G. The economically designed two-sided Bayesian X control chartj]. European Journal of Operational Research, 2007, 183(1): ],,. J]., 2012, 27(1): Zhang Pengwei, Su Qin, Liu Weiyan. Economic design of a Bayesian chart for short-run productionj]. Journal of Systems Engineering, 2012, 27(1): (in Chinese) 18],,. J]., 2013, 28(3): Zhang Pengwei, Su Qin, Li Chenglong. Economically designed multivariate Bayesian control chart for short-run productionj]. Journal of Systems Engineering, 2013, 28(3): (in Chinese) 19] Duncan A J. The economic design of X charts when there is a multiplicity of assignable causesj]. Journal of the American Statistical Association, 1971, 66(333): ] Chiu W K. Economic design of np charts for processes subject to a multiplicity of assignable causesj]. Management Science, 1976, 23(4): ],. J]., 1999, 14(3): Da Qingli, Liu Xinwang. A weighted aggregate method for the fuzzy decision making of multiobjective linear programmingj]. Journal of Systems Engineering, 1999, 14(3): (in Chinese) 22] Taguchi G, Elsayed E A, Hsiang T. Quality Engineering in Production SystemsM]. ew York: McGraw-Hill, ],,. J]., 2004, 19(2): Wang Jian, Liu Erlie, Luo Gang. Alalysis of time-cost-quality tradeoff optimization in construction project managementj]. Journal of Systems Engineering, 2004, 19(2): (in Chinese) : (1989 ),,,, :, Liassy@stu.xjtu.edu.cn; (1963 ),,,,, :,, qinsu@mail.xjtu.edu.cn; (1985 ),,,, :, kimi.2004@stu.xjtu.edu.cn. η (u,w) γ (u,w) n > 1 η (u,w) γ (u,w) : η (0,0) m n, ( 318 )

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