- Email: [email protected]
Multiobjective economic design of an X control chart
o o n ~ p u ~ 4 i r o &~ i~riml
PERGAMON
Computers& Industrial Engineering 37 (1999) 129-132
MULTIOBJECTIVE ECONOMIC DESIGN OF AN X CONTROL CHART Celano G. and Fichera S. Istituto di Macchine, University of Catania, Italy. v.le A. Doria n° 6 95100 Catania - Italy e-mail [email protected] ABSTRACT The prevention of defective products is a fundamental principle of total quality management and control charts are a powerful statistical tool to reach this objective, but they are expensive and may increase the cost of production. For this reason an appropriate design is necessary before the chart is used. In this paper a new approach, based on an evolutionary algorithm, to solve this problem is proposed. The design of the chart has been developed considering the optimisation of the cost of the chart and at the same time the statistical proprieties. The proposed multiobjective approach has been compared to some well-known heuristics; the obtained results show the effectiveness of the evolutionary algorithm. © 1999 Elsevier Science Ltd. All rights reserved. KEYWORDS Quality control, X control chart, multiobjective, average run length, genetic algorithms. INTRODUCTION The control chart developed by W. A. Shewhart (1931) is a useful tool to evaluate if the process is in a state of statistical control or if there is pre~mt a system of assignable causes. A model to determine the costs associated with the implementation of X chart has been developed by Duncan (1956). It permits one to define a cost function which depends on the sample size n, the width of control limits k, and the time interval between samples h. Many authors proposed different heuristic approaches to minimise the cost function and to find the optimal design of the chart. But, as pointed out by Woodall (1986), control charts based on economically optimal design generally have poor statistical properties; in fact the values of ARL0 and ARLi are obtained by the chart parameters and they are not optimised accordingly to a statistical point of view. In order to resolve this problem Saniga (1989) proposed a constrained model, while Del Castillo et al. (1996) considered an interactive multiobjective algorithm on the basis of a simplified procedure. In this paper an evolutionary algorithm which considers both the economic and statistical objective has been developed. It handles directly the feasible solutions of the problem and permits one to consider the cost function without any simplification. The performance of the proposed algorithm has been tested by computer simulation utilising the examples reported in literature. 0360-8352/99 - see front matter © 1999 Elsevier Science Ltd. All rights reserved. PII: S0360-8352(99)00038-8
130
Proceedings of the 24th International Conference on Computers and Industrial Engineering
ECONOMIC MODEL The economic design of X control charts is fulfilled by searching the values of sample size n, the width of control limits k and the interval between samples h, which allows one to minimise the costs associated with the implementation of the chart defined by Duncan's model (Duncan, 1956). In this model the process is divided into cycles constituted by the following phases: production, monitoring and adjustment; each cycle begins with the production process in the "in control state" and continues until an "out of control signal" appears on the control chart. Following an adjustment in which the process returns to the in control state, a new cycle begins. The expected length of a cycle E(T) is. E ( T ) = I + h . A R L - ~ + g . n +D
(1)
where ~ is the number of occurrences per hour of an assignable cause, g is the time required to take a single item and intercept the results and D is the time required to find an assignable cause after a point plotted outside the control limits. The average run length with the process in out of control state is
8 represents the number of standard deviations cr in the shit~ of the mean po of a quality characteristic and the expected time of occurrence of the cause within the interval between two samples is j~ff')hX. e-Xt( t - jh)dt x=
~;+'>h~. e-X,dt
1_ (1 + Xh).e-~ -
(3)
7~.(1- e -~ )
The expected net income per cycle E(C) has the following expression: E ( C ) = V 0 . 1 + VI" ( 1-----~ h -x+g'n+D
) -a3
ot.a'3-e 1 - e ~-~
(a I +aE.n) E(T)h
(4)
where Vo and V~ are respectively the net income per hour of operation in the in and out control state, the ai values are the unitary cost components and finally, the reciprocal of the in control average run length with the process in control state is a - ARL-----~ The expected net income per hour E(A) can be determined by dividing E(C) by E(T): rearranging the terms of this expression, it may be written as: E(A) = Vo - E(L) (6) where the expected loss per hour E(L) is equal to: E(L)= a4 • [h/(1-13)- x + g ' n +D]+a3 +a'3 " a ' e - ~ / ( 1 - e-~).l a 1 + a 2 . n (7) 1/L +h/(1-fJ)-x +g.n + D h This objective function must be minimised in order to pursue the economic goal, whereas the statistical objectives are reached by minimising the ct value and maximising the 1-13 value. The variables of the function are n, h and k. EVOLUTIONARY ECONOMIC DESIGN ALGORITHM The proposed evolutionary algorithin simulates the evolution of a population of Ns individuals, based on the rule "the survival of the finest" for a given environment. A genetic alsorithrn for a
Proceedings of the 24th International Conference on Computers and Industrial Engineering
131
control chart economic design problem needs customisation of coding individuals and in selec,~ing andworking out proper genetic operators (Michalewicz, 1994). A decimal encoding of individuals has been adopted: each individual is a decimal string of three elements, n, h and k, which represent a possible solution of the chart design problem. The fitness value of each individual is evaluated in two ways depending on the configuration of the algorithm. When a single objective is considered the fitness value coincide with the expected loss per hour Fs =E(L) (8) wl~reas in the multiobjective configuration the fitness is the expected loss per hour multiplied by a coefficient function of the weighed sum of or and 1-13. Fm= E(L)* (W~ +W 2 * a - W 3 *(1-13)) (9) where WI W2 W3 are the coefficients which values ranging from 0.1 and 2. Point crossover and mutation operators (Fig. 1), operating on a random selected set of genes, have been chosen avoiding feasibility problems in the creation of new individuals. Crossover is applied at each iteration, the total iteration being N~. A pair of parents is chosen on a fitness basis, i.e., assigning a higher reproduction probability to the fittest individuals; the new individuals replace the parents only if they better fit the goal. Mutation is randomly applied during iterations with probability P,,, a chromosome is casually chosen on a fitness basis, and substituted with a new one generated by the mutation operator. nl kl hl Parmzt I n2 k2 h2 Pa~-e~6.2 nl kl hi P~u-e~t 1 [0 [ 1 [ Selected point 0 1 0 Selectedpoint ni Ik2 Ih~ [ Off~ring 1 nl k2 h, O mg n2 [k~ ]h2 [ Offspring 2 Point crossover Mutation Fig. 1 Genetic operators Moreover, an elitist strategy has been pursued during the evolutionary process, which consists in preserving the best chromosome from disruption caused by the application of the mutation operator. In order to avoid premature convergence, due to a quick increase in the number of copies of the fittest individuals, a population diversity control technique, which, over a given number Dmx mutates the duplicates of chromosomes in the actual population, has also been embedded in the developed algorithm.
III
SIMULATION RESULTS Two configurations of evolutionary algorithm have been considered: the single, SEEDA, and the multiobjective, MEEDA, with the algorithm structure being the same, the only difference is the calculation of the fitness. In order to assess the effectiveness of the proposed technique, the algorithms have been tested on a set of 20 problems reported in literature, where the first five, indicated with M, have been elaborated by Montgomery (1982), and the remaining fifteen by Duncan (1956), indicated with D. The following setting of control parameters has been employed: NsffiS0, Pm=O.01, Dmx=2, N~--'-50000, W1= 1.3, W2 =1, W3= 0.5. The results of the two algorithms, together with the ones obtained by two of the best conventional optimisation algorithms, Duncan (1956) and Montgomery (1982), (each one for the reference example) are resumed in Table 1. The Fs average of SEEDA compared with the conventional algorithms
132
Proceedings of the 24th International Conference on Computers and Industrial Engineering
shows a significant improvement, all examples reaches a cost reduction up to a maximum of 11% in the example D6. In order to evaluate the effectiveness of the MEEDA the fitness function Fm has also been calculated on the basis of the values obtained with the conventional algorithms. The average Fm values show that the evolutionary hybrid approach outperforms the other conventional algorithms tested here. All examples show a little increase in cost, but at the same time a significant improvement of statistical proprieties. Table 1 Simulation results Montgomery-Duncan SEEDA MEEDA 1-~ Fm Fs (x 1-13 Fm Fs Fs ¢x M1 10,38 0,0028 0,931 8,69 10,37 10,48 0,0018 0,985 8,48 M2 13,88 0,0028 0,931 11,62 13,86 14,04 0,0019 0,985 11,36 M3 12,59 0,0091 0,871 11,00 12T56 12,77 0,0088 0,948 10,66 M4 13,95 0,0011 0,948 11,54 13,93 14,01 0,0014 0,982 11,36 M5 11,43 0,0024 0,969 9,35 11,41 11,47 0,0022 0,987 9,28 331t98 D1 402,13 0,0014 0,8983 3 4 2 , 7 0 4 0 1 , 2 8 405,94 0,0024 0,969 D2 696,55 0,0014 0,8983 593,60 694,6 703,86 0,0026 0,971 575,26 D3 962,39 0,0014 0,8983 8 2 0 , 1 5 9 5 9 , 2 4 974,44 0,0028 0,972 795,97 D4 416,49 0,0014 0,8983 3 5 4 , 9 3 4 1 5 , 2 6 419,78 0,0025 0,97 343,19 D5 2756,7 0,0014 0,8983 2349123 2697,82 2774,58 0,0032 0,974 2264,06 D6 23683 0,0014 0,8983 20182,42 22880,7 23701,7 0,0074 0,963 19570,49 D7 608,86 0,0014 0,8983 5 1 8 , 8 7 5 4 0 , 0 6 574,74 0,0074 0 , 9 0 7 490,89 D8 1840,8 0,0014 0,8983 1568,71 1837,15 1852,65 0,0016 0,994 1490,92 D9 362,14 0,0164 0,9452 3 0 5 , 5 7 3 6 0 , 8 7 367,88 0,0108 0,973 303,30 D10 659,37 0,0001 0,8641 5 7 2 , 3 9 6 3 6 , 6 9 644,54 0,0003 0 , 9 7 8 523,04 Dll 2834,2 0,0000 0,8137 2531,40 2828,58 2850,48 0 0,996 2285,51 D12 589,86 0,0014 0,9818 4 7 8 , 0 8 5 8 6 , 6 9 588,93 0,0029 0,99 475,91 D13 594,13 0,0093 0,5903 6 0 2 , 5 4 5 6 3 , 1 3 571,34 0,0135 0 , 9 3 7 482,78 D14 1010,7 0,1615 0,9234 1010,53 9 8 7 , 3 4 1101,85 0,0385 0 , 9 1 8 968~86 D15 3241,7 0,0093 0,5903 3287,53 3174,99 3234,97 0,0147 0,941 2731,61 CONCLUSION The present paper deals with X control chart design. A multiobjective approach is proposed to pursue contemporaneously an economic obiective and a statistical objective. The results obtained with the developed method show a notable Improvement in front of classical heuristics. REFERENCE Del Castillo, E., Mackin, P. and Montgomery, D. C. (1996). Multiple-criteria optimal design ofx control charts. IIE Transactions 28 467-474. Duncan, A. J. (1956). The economic design of x charts used to maintain current control of a process. Journal of American Statistical Association, 51, 228-242. Michaiewicz, Z. (1994). Genetic Algorithms + Data Structures = Evolution Program, SpringerVerlag, New York. Montgomery, D. C. (1982). Economic designs of an x control chart. Journal of Quality Technology, 14 (1). 40-43. Saniga, E. M. (i989). Economic statistical control-chart designs with application to x and r charts. Technometrics, 31(3), 313-320. Shewhart, W. A. (1931). Economic Control of Quality of Manufactured Product. Van Nostrand, New York. Woodall, W. H. (1986). Weaknesses of the economic design of control charts. Technometrics, 28(4), 408-409.
PERGAMON
Computers& Industrial Engineering 37 (1999) 129-132
MULTIOBJECTIVE ECONOMIC DESIGN OF AN X CONTROL CHART Celano G. and Fichera S. Istituto di Macchine, University of Catania, Italy. v.le A. Doria n° 6 95100 Catania - Italy e-mail [email protected] ABSTRACT The prevention of defective products is a fundamental principle of total quality management and control charts are a powerful statistical tool to reach this objective, but they are expensive and may increase the cost of production. For this reason an appropriate design is necessary before the chart is used. In this paper a new approach, based on an evolutionary algorithm, to solve this problem is proposed. The design of the chart has been developed considering the optimisation of the cost of the chart and at the same time the statistical proprieties. The proposed multiobjective approach has been compared to some well-known heuristics; the obtained results show the effectiveness of the evolutionary algorithm. © 1999 Elsevier Science Ltd. All rights reserved. KEYWORDS Quality control, X control chart, multiobjective, average run length, genetic algorithms. INTRODUCTION The control chart developed by W. A. Shewhart (1931) is a useful tool to evaluate if the process is in a state of statistical control or if there is pre~mt a system of assignable causes. A model to determine the costs associated with the implementation of X chart has been developed by Duncan (1956). It permits one to define a cost function which depends on the sample size n, the width of control limits k, and the time interval between samples h. Many authors proposed different heuristic approaches to minimise the cost function and to find the optimal design of the chart. But, as pointed out by Woodall (1986), control charts based on economically optimal design generally have poor statistical properties; in fact the values of ARL0 and ARLi are obtained by the chart parameters and they are not optimised accordingly to a statistical point of view. In order to resolve this problem Saniga (1989) proposed a constrained model, while Del Castillo et al. (1996) considered an interactive multiobjective algorithm on the basis of a simplified procedure. In this paper an evolutionary algorithm which considers both the economic and statistical objective has been developed. It handles directly the feasible solutions of the problem and permits one to consider the cost function without any simplification. The performance of the proposed algorithm has been tested by computer simulation utilising the examples reported in literature. 0360-8352/99 - see front matter © 1999 Elsevier Science Ltd. All rights reserved. PII: S0360-8352(99)00038-8
130
Proceedings of the 24th International Conference on Computers and Industrial Engineering
ECONOMIC MODEL The economic design of X control charts is fulfilled by searching the values of sample size n, the width of control limits k and the interval between samples h, which allows one to minimise the costs associated with the implementation of the chart defined by Duncan's model (Duncan, 1956). In this model the process is divided into cycles constituted by the following phases: production, monitoring and adjustment; each cycle begins with the production process in the "in control state" and continues until an "out of control signal" appears on the control chart. Following an adjustment in which the process returns to the in control state, a new cycle begins. The expected length of a cycle E(T) is. E ( T ) = I + h . A R L - ~ + g . n +D
(1)
where ~ is the number of occurrences per hour of an assignable cause, g is the time required to take a single item and intercept the results and D is the time required to find an assignable cause after a point plotted outside the control limits. The average run length with the process in out of control state is
8 represents the number of standard deviations cr in the shit~ of the mean po of a quality characteristic and the expected time of occurrence of the cause within the interval between two samples is j~ff')hX. e-Xt( t - jh)dt x=
~;+'>h~. e-X,dt
1_ (1 + Xh).e-~ -
(3)
7~.(1- e -~ )
The expected net income per cycle E(C) has the following expression: E ( C ) = V 0 . 1 + VI" ( 1-----~ h -x+g'n+D
) -a3
ot.a'3-e 1 - e ~-~
(a I +aE.n) E(T)h
(4)
where Vo and V~ are respectively the net income per hour of operation in the in and out control state, the ai values are the unitary cost components and finally, the reciprocal of the in control average run length with the process in control state is a - ARL-----~ The expected net income per hour E(A) can be determined by dividing E(C) by E(T): rearranging the terms of this expression, it may be written as: E(A) = Vo - E(L) (6) where the expected loss per hour E(L) is equal to: E(L)= a4 • [h/(1-13)- x + g ' n +D]+a3 +a'3 " a ' e - ~ / ( 1 - e-~).l a 1 + a 2 . n (7) 1/L +h/(1-fJ)-x +g.n + D h This objective function must be minimised in order to pursue the economic goal, whereas the statistical objectives are reached by minimising the ct value and maximising the 1-13 value. The variables of the function are n, h and k. EVOLUTIONARY ECONOMIC DESIGN ALGORITHM The proposed evolutionary algorithin simulates the evolution of a population of Ns individuals, based on the rule "the survival of the finest" for a given environment. A genetic alsorithrn for a
Proceedings of the 24th International Conference on Computers and Industrial Engineering
131
control chart economic design problem needs customisation of coding individuals and in selec,~ing andworking out proper genetic operators (Michalewicz, 1994). A decimal encoding of individuals has been adopted: each individual is a decimal string of three elements, n, h and k, which represent a possible solution of the chart design problem. The fitness value of each individual is evaluated in two ways depending on the configuration of the algorithm. When a single objective is considered the fitness value coincide with the expected loss per hour Fs =E(L) (8) wl~reas in the multiobjective configuration the fitness is the expected loss per hour multiplied by a coefficient function of the weighed sum of or and 1-13. Fm= E(L)* (W~ +W 2 * a - W 3 *(1-13)) (9) where WI W2 W3 are the coefficients which values ranging from 0.1 and 2. Point crossover and mutation operators (Fig. 1), operating on a random selected set of genes, have been chosen avoiding feasibility problems in the creation of new individuals. Crossover is applied at each iteration, the total iteration being N~. A pair of parents is chosen on a fitness basis, i.e., assigning a higher reproduction probability to the fittest individuals; the new individuals replace the parents only if they better fit the goal. Mutation is randomly applied during iterations with probability P,,, a chromosome is casually chosen on a fitness basis, and substituted with a new one generated by the mutation operator. nl kl hl Parmzt I n2 k2 h2 Pa~-e~6.2 nl kl hi P~u-e~t 1 [0 [ 1 [ Selected point 0 1 0 Selectedpoint ni Ik2 Ih~ [ Off~ring 1 nl k2 h, O mg n2 [k~ ]h2 [ Offspring 2 Point crossover Mutation Fig. 1 Genetic operators Moreover, an elitist strategy has been pursued during the evolutionary process, which consists in preserving the best chromosome from disruption caused by the application of the mutation operator. In order to avoid premature convergence, due to a quick increase in the number of copies of the fittest individuals, a population diversity control technique, which, over a given number Dmx mutates the duplicates of chromosomes in the actual population, has also been embedded in the developed algorithm.
III
SIMULATION RESULTS Two configurations of evolutionary algorithm have been considered: the single, SEEDA, and the multiobjective, MEEDA, with the algorithm structure being the same, the only difference is the calculation of the fitness. In order to assess the effectiveness of the proposed technique, the algorithms have been tested on a set of 20 problems reported in literature, where the first five, indicated with M, have been elaborated by Montgomery (1982), and the remaining fifteen by Duncan (1956), indicated with D. The following setting of control parameters has been employed: NsffiS0, Pm=O.01, Dmx=2, N~--'-50000, W1= 1.3, W2 =1, W3= 0.5. The results of the two algorithms, together with the ones obtained by two of the best conventional optimisation algorithms, Duncan (1956) and Montgomery (1982), (each one for the reference example) are resumed in Table 1. The Fs average of SEEDA compared with the conventional algorithms
132
Proceedings of the 24th International Conference on Computers and Industrial Engineering
shows a significant improvement, all examples reaches a cost reduction up to a maximum of 11% in the example D6. In order to evaluate the effectiveness of the MEEDA the fitness function Fm has also been calculated on the basis of the values obtained with the conventional algorithms. The average Fm values show that the evolutionary hybrid approach outperforms the other conventional algorithms tested here. All examples show a little increase in cost, but at the same time a significant improvement of statistical proprieties. Table 1 Simulation results Montgomery-Duncan SEEDA MEEDA 1-~ Fm Fs (x 1-13 Fm Fs Fs ¢x M1 10,38 0,0028 0,931 8,69 10,37 10,48 0,0018 0,985 8,48 M2 13,88 0,0028 0,931 11,62 13,86 14,04 0,0019 0,985 11,36 M3 12,59 0,0091 0,871 11,00 12T56 12,77 0,0088 0,948 10,66 M4 13,95 0,0011 0,948 11,54 13,93 14,01 0,0014 0,982 11,36 M5 11,43 0,0024 0,969 9,35 11,41 11,47 0,0022 0,987 9,28 331t98 D1 402,13 0,0014 0,8983 3 4 2 , 7 0 4 0 1 , 2 8 405,94 0,0024 0,969 D2 696,55 0,0014 0,8983 593,60 694,6 703,86 0,0026 0,971 575,26 D3 962,39 0,0014 0,8983 8 2 0 , 1 5 9 5 9 , 2 4 974,44 0,0028 0,972 795,97 D4 416,49 0,0014 0,8983 3 5 4 , 9 3 4 1 5 , 2 6 419,78 0,0025 0,97 343,19 D5 2756,7 0,0014 0,8983 2349123 2697,82 2774,58 0,0032 0,974 2264,06 D6 23683 0,0014 0,8983 20182,42 22880,7 23701,7 0,0074 0,963 19570,49 D7 608,86 0,0014 0,8983 5 1 8 , 8 7 5 4 0 , 0 6 574,74 0,0074 0 , 9 0 7 490,89 D8 1840,8 0,0014 0,8983 1568,71 1837,15 1852,65 0,0016 0,994 1490,92 D9 362,14 0,0164 0,9452 3 0 5 , 5 7 3 6 0 , 8 7 367,88 0,0108 0,973 303,30 D10 659,37 0,0001 0,8641 5 7 2 , 3 9 6 3 6 , 6 9 644,54 0,0003 0 , 9 7 8 523,04 Dll 2834,2 0,0000 0,8137 2531,40 2828,58 2850,48 0 0,996 2285,51 D12 589,86 0,0014 0,9818 4 7 8 , 0 8 5 8 6 , 6 9 588,93 0,0029 0,99 475,91 D13 594,13 0,0093 0,5903 6 0 2 , 5 4 5 6 3 , 1 3 571,34 0,0135 0 , 9 3 7 482,78 D14 1010,7 0,1615 0,9234 1010,53 9 8 7 , 3 4 1101,85 0,0385 0 , 9 1 8 968~86 D15 3241,7 0,0093 0,5903 3287,53 3174,99 3234,97 0,0147 0,941 2731,61 CONCLUSION The present paper deals with X control chart design. A multiobjective approach is proposed to pursue contemporaneously an economic obiective and a statistical objective. The results obtained with the developed method show a notable Improvement in front of classical heuristics. REFERENCE Del Castillo, E., Mackin, P. and Montgomery, D. C. (1996). Multiple-criteria optimal design ofx control charts. IIE Transactions 28 467-474. Duncan, A. J. (1956). The economic design of x charts used to maintain current control of a process. Journal of American Statistical Association, 51, 228-242. Michaiewicz, Z. (1994). Genetic Algorithms + Data Structures = Evolution Program, SpringerVerlag, New York. Montgomery, D. C. (1982). Economic designs of an x control chart. Journal of Quality Technology, 14 (1). 40-43. Saniga, E. M. (i989). Economic statistical control-chart designs with application to x and r charts. Technometrics, 31(3), 313-320. Shewhart, W. A. (1931). Economic Control of Quality of Manufactured Product. Van Nostrand, New York. Woodall, W. H. (1986). Weaknesses of the economic design of control charts. Technometrics, 28(4), 408-409.