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Integration of production quantity and control chart design in automotive manufacturing
Computers & Industrial Engineering xxx (2016) xxx–xxx
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Integration of production quantity and control chart design in automotive manufacturing E.E. Gunay ⇑, U. Kula Department of Industrial Engineering, Sakarya University, Sakarya, Turkey
a r t i c l e
i n f o
Article history: Available online xxxx Keywords: Attribute chart (p-chart) design Stochastic models Sample average approximation Quality management
a b s t r a c t This study presents a two-stage stochastic programming model for the determination of control limits in p-charts when a production process produces above a certain quantity. Consideration of production quantity needed along with control limit determination is important for the following competing two reasons: (1) Wider control limits make it difficult to detect the changes in the process, therefore producing excessive number of cars with paint defects. (2) Narrower control limits, on the other hand, increase the number of unnecessary interventions even if there is no deterioration in the process so that inspection costs increase. In both cases, quantity produced reduces due to defective products and unnecessary interventions. Therefore, it is important to design a control chart for proportion of defects that takes production quantity requirements into account. We consider the problem in an automotive manufacturing setting in which the cars are inspected for paint defects after paint operations. We formulate the problem as a two-stage stochastic programming model. In the first stage, control limit parameter k is decided for the p-chart and in the second stage, production quantity is determined that minimizes total quality-related and production costs. We solve the model by sample average approximation algorithm (SAA). In a numerical study, we investigate the effect of various factors on control limit parameter k and the total cost. Our numerical study shows that (i) an increase on the mean defect rate increases both the total cost and the total production quantity, (ii) effect of an increasing process variance to the control limit parameter k is significantly small, (iii) frequency of special cause occurrences affects the total cost significantly and (iv) all the experiments show that the commonly used 3r control limits in practice are wider than required. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In automobile manufacturing, after body shop operations, welded bodies are directed to the paint shop. Vehicle painting operation includes several steps such as cleaning, primer coat painting, top coating and polishing. Once painting is completed, the vehicles are inspected for paint defects before they are released to the Final Assembly (FA) department. Since paint defects may be as much as 40%, some manufacturers use control charts to monitor the paint process and determine whether the current painting process is out-of-control. An out-of-control situation means that there is a special cause increasing the proportion of paint defects. In vehicle painting operations, defects are categorized as either minor or major ones. Minor defects just need small touchups, usually done on the line, which does not cause any delays in assembly
⇑ Corresponding author. E-mail addresses: [email protected] (E.E. Gunay), [email protected] (U. Kula).
operations. On the other hand, vehicles with major defects must be taken off the line and sent back to paint shop for re-painting. Since minor defects do not add any significant cost and do not disturb the vehicle flows in the line, the scope of this study is confined to major paint defects. Fig. 1 shows the simplified inspection procedure used in the paint shop of a major car manufacturer’s plant located in Turkey. A sample of size n is drawn in a sampling interval of h hours at the end of epoch t from the batch size of X t vehicles. Number of the defective vehicles in the sample is counted and estimated aver^, is calculated. The painting process age defect rate of the process p ^, is above is deemed out-of-control if the defect rate of the sample p the upper control limit (UCL). When the p-chart signals an out-ofcontrol situation, the engineering team searches for special causes that might increase the proportion of paint defects. If a cause for the defect rate increase is found, the painting process is restored to the in-control state at some cost and all the newly painted vehicles in the sampling interval are inspected. Since no action is taken if the defect rate of the sample is below UCL, determination of lower control limit in p-charts is of no practical importance.
http://dx.doi.org/10.1016/j.cie.2016.06.016 0360-8352/Ó 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Gunay, E. E., & Kula, U. Integration of production quantity and control chart design in automotive manufacturing. Computers & Industrial Engineering (2016), http://dx.doi.org/10.1016/j.cie.2016.06.016
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Randomly take sample of size n from a batch of Xt vehicles at the end of production period t
t
t 1
Does the chart signal an out-ofcontrol in the tth production period?
Yes
Inspect all the newly painted vehicles in the tth production period
No
Send defective vehicles for re-painting.
Fig. 1. Simplified procedure of paint control process.
Setting an optimal UCL is important to reduce the production and quality costs: A widely set UCL decreases the sensitivity of a control chart to detect the special cause occurrences. On the other hand, a narrow UCL increases the number of false out-of-control signals in the process even if there is no deterioration in the process. As Eq. (1) suggests, setting an optimal UCL involves finding the optimal k that would minimize quality related costs. Determination of optimal control limits along with optimal sampling size and sampling frequency that minimize quality related costs is known as economical design of chart parameters and is commonly studied in the literature.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p0 ð1 p0 Þ UCL ¼ p0 þ k n
ð1Þ
Ladany (1973) is the first paper that considers economical design of p-chart parameters. His model includes cost of sampling, cost of not detecting a change in the process (Type 2 error cost), cost of false indication of change (Type 1 error cost) and cost of readjusting detected change. Montgomery, Heikes, and Mance (1975) consider the same problem when there are several out-of-control states. Chiu (1975a) presents a model which minimizes loss cost function in an np-control chart and Chiu (1975b) investigate the effects of variation in cost factors by drawing loss cost surface as contour plots. Chiu (1976) considers the case where there are several out-ofcontrol states in an np-control chart. More recent studies on economical design of chart parameters consider variable sample size, variable sampling interval and both variable sample size/sampling interval and present the advantages of these models rather than traditional p-chart design (Aslam, Azam, Khan, & Jun, 2015; Inghilleri, Lupo, & Passannanti, 2015; Kooli & Limam, 2011; Wu & Luo, 2004). Recognizing the effect of the monitoring policy on the production capacity, Lee and Rosenblatt (1987) develop a model addressing the problem of joint determination of optimal production run time, number of inspections to minimize quality-related costs in an X chart. Lee and Park (1991) consider the same problem by focusing on the difference between rework cost before sale and warranty cost after sale. Rahim (1994) considers the same joint problem by developing a non-Markovian shock model under production setup, inventory holding and maintenance cost. Rahim and Ben-Daya (1998) extend this model to consider the case in which the production is halted not only if there is true alarm but also there is a false alarm. Ben-Daya and Rahim (2000) consider
the joint-optimization of preventive maintenance actions and X chart parameters when in-control state follows a general probability distribution with increasing hazard rate. Pan, Jin, Wang, and Cang (2012) develop a model to minimize the total expected production costs while jointly determining the optimal parameters of control chart and the maintenance decision policy whereas Bouslah, Gharbi, and Pellerin (2015) consider joint design of production, quality and maintenance control policy problem in cchart. In vehicle painting process, since defects in the paint shop occur randomly, it is not possible to know before how many vehicles to paint in order to meet the FA demand. Therefore, a control chart policy should take random paint defect occurrences into account to meet assembly line demand and to minimize quality related costs simultaneously. Economic design problem of control charts, including p-chart design, involve determination of three important parameters: Control limit width, sampling interval, and sample size. In a p-chart, narrowly determined UCL increases the number of false alarms and when a p-chart reports an alarm, all the vehicles painted in between previous and the current sampling epochs are inspected one by one, which increases the unnecessary cost of inspection. This inspection cost depends on the number of vehicles painted in the batch. Therefore, if the number of painted vehicles is more than the optimal number needed, extra inspection costs occur. On the other hand widely determined UCL makes difficult to detect the shift in the process so defective vehicles will be sent to the FA department. In this paper, we develop a two-stage mathematical model that jointly determines the optimal upper control limit in a p-chart and the paint batch size X t . In the first stage, the control limit parameter k is set and after observing the random paint defects, the paint batch size X t is decided. The two-stage decision making framework optimizes the first stage decision k given the fact that vehicles are repeatedly painted and paint defects are experienced over and over again. In fact, in statistical process control applications, the control chart parameters are determined first and random process shifts occur repeatedly as process is monitored over time. Therefore, two-stage stochastic programming provides an appropriate modeling framework for incorporating various constraints such as demand, service level, and maximum inventory level extra into control chart design problem. Bouslah, Gharbi, and Pellerin (2013) built a stochastic mathematical model and use a simulation based optimization approach for joint determination of the production quantity, hedging level and the sample size in X charts.
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There are also some other studies that both decide on quality and maintenance decisions to both lower the cost of quality and maintenance. These models aim to both improve quality of items produced also lower maintenance costs (Chan & Wu, 2009; Mehdi, Nidhal, & Anis, 2010; Mehrafrooz & Noorossana, 2011; Pandey, Kulkarni, & Vrat, 2011). After developing the two-stage stochastic model, we perform a numerical study that investigates the effect of several cost components and random disturbances on first stage decision variable k and the total cost. Lam and Rahim (2002) present an extensive sensitivity analysis similar to ours to investigate the effects of cost and inspection parameters on production and quality costs in X charts. Chakraborty, Giri, and Chaudhuri (2009) study the joint effect of process shift, machine breakdown and inspection decisions on production quantity for deteriorating production system where p-chart is used to control the process. To our knowledge, our study is the first that models determination of p-chart parameter k when a certain production quantity needs to be produced in a planning horizon of length T with the objective of minimizing total production and quality-related costs. In the first stage two-stage stochastic model, we decide on the p-chart control limit parameter k which determines the UCL for p-chart. In the second stage of the model, the production quantities X t are determined for each period t for a planning horizon of length T. In addition to the demand constraint, a maximum inventory level constraint that guarantees the total production quantity does not exceed a certain percentage of the quantity produced, i.e. safety stock level SSL, is added to the second stage model. This paper is organized as follows. Section 2 formulates the problem and presents the two-stage stochastic model. In Section 3 we give the details of the sample average approximation (SAA) algorithm used to solve the developed model. A numerical study is performed to offer managerial insights in Section 4. Section 5 concludes the paper and explores possible future research directions. 2. Two stage stochastic model
pout ¼ 1 ek
where pout is the probability that the painting process defect rate would shift from p0 to p1 . Due to sampling error, it is possible to get a false out-of-control signal even if the process defect rate has not shifted from p0 to p1 . This error is called as Type 1 error and ^ P UCLjp1 ¼ p0 Þ. is denoted by Pðp Since number of defectives D in a sample of size n is a binomial random variable with mean np0 , Type 1 error probability, the probability of having more than n(UCL) defects in the sample given that the process mean stays as p0 , a ¼ PðD P nUCLjp1 ¼ p0 Þ is given as follows:
a¼
n X nd n d p0 1 pd0 d dPnðUCLÞ
ð3Þ
which can be approximated by normal distribution. Therefore we have,
a¼P ZP
nUCL np0
r
Z
¼ PðZ P kÞ ¼ k
1
1 1 xl 2 pffiffiffiffiffiffiffi e2ð r Þ dx; r 2p
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r is equal to np0 ð1 p0 Þ and l equals to np0 . When a Type 1 error occurs, an unnecessary inspection cost is incurred since the true mean of the process is still p0 . Total expected Type 1 error cost in a production period of length T is given in Eq. (4), where c1 shows the Type 1 error cost per vehicle and Xt shows the number of vehicles painted in production period t. T X X t c1 ð1 pout Þa
ð4Þ
t¼1
When a special cause occurs and the defect rate of the process shifts to p1 , the probability of not detecting this shift is, ^ 6 UCLjp1 > p0 Þ, called as Type 2 error. Type 2 error occurs b ¼ Pðp when the defect rate of the process shifts to p1 but the sample ^ is less than the UCL so the deterioration in the process defect rate p is not detected. By Normal approximation to binomial distribution, we can calculate the Type 2 error probability b as follows:
Z Consider a painting process that is controlled by a p-chart and let the upper control limit of the p-chart is set at pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UCL ¼ p0 þ k p0 ð1 p0 Þ=n, where p0 is the in-control defect rate of the process and k is the sigma level of the process. Once the value of k is decided, the painting process would be monitored at each discrete time epoch t. At each epoch a sample of size of n vehicles are inspected for paint defects and proportion of defectives in ^ is calculated. If p ^ is below UCL, the paint nozzles are the sample p purged and the next batch of vehicles are painted until the next ^ is above the UCL, it means that the pinspection epoch t þ 1. If p chart signals a shift in process defect rate from p0 to p1 , where p1 > p0 . ^ is above the UCL, a search for the special cause is perWhen p formed. If a special cause is found resulting in a higher than acceptable defect rate UCL, the process is brought back to in-control state at some cost and no cessation in the production. We assume that the special causes in the painting process occur according to a Poisson Process with rate k vehicles per inspection interval, ½t; t þ 1. Without loss of generality, we assume that inspection period length is one time units. Therefore, time between special cause occurrences follows an exponential distribution with mean 1=k. Modeling special cause occurrences as a Poisson Process is a reasonable and a common assumption since multiple sources contribute to special cause occurrences. (Lam & Rahim, 2002; Lee & Rosenblatt, 1987; Montgomery et al., 1975). Since the process shifts to an out-of-control state when there is one or more special cause occurrences, the probability that there is one or more special cause occurrences in each period ½t; t þ 1 is,
ð2Þ
b¼
UCL
1
1 1 pffiffiffiffiffiffiffi e 2 r p1 2p
xp1 rp1
2
dx
ð5Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where l is equal to p1 and r equals to p1 ð1 p1 Þ=n. When the process defect rate shifts to p1 , a p-chart will detect this shift with probability ð1 bÞ and all the newly painted vehicles are inspected one by one. Since this inspection is different than the inspection of sample at the end of each production period t and requires inspection of all the vehicles painted in that production period, we call it as re-inspection cost. Let c0 shows the cost of re-inspection per vehicle, then the expected total cost of reinspection is, T X X t c0 pout ð1 bÞ
ð6Þ
t¼1
On the other hand, when the process defect rate shifts to p1 , a pchart will not detect this shift with probability b. If the shift in the average defect rate of the process is not detected, the defective vehicles will be sent to following department, FA. In this case FA workers will be busy to take the vehicles off the line which may cause delays and line stoppages. In some cases, defective vehicles may not be noticed at the final inspection after the FA and may be shipped to the customer. In practice, usually this cost is much higher than the cost of false alarm, Type 1 cost. Let c2 presents the cost of Type 2 error per vehicle. Then the total cost of Type 2 error is, T X X t c2 pout b
ð7Þ
t¼1
Please cite this article in press as: Gunay, E. E., & Kula, U. Integration of production quantity and control chart design in automotive manufacturing. Computers & Industrial Engineering (2016), http://dx.doi.org/10.1016/j.cie.2016.06.016
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Once the value of k is decided, i.e., once UCL is set, the defective number of vehicles in the sample at epoch t is observed, which depends on whether the process is in-control or out-of-control. Let the random variable It be an in-control or in an out-ofcontrol state,
It ¼
1; if an out-of-control signal is not reported 0;
if an out-of-control signal is reported
For example if production period is limited with 6 periods, T ¼ 6 and the realization of the sequence of random variables It is as follows nðxÞ ¼ ð1; 0; 1; 1; 0; 1Þ. Then out-of-control signal is reported in production period 2 and 5 while an out-of-control signal is not reported in other periods 1, 3, 4 and 6. If out-of-control ^ 6 UCL, number of defective vehicles signal is not reported, i.e. p in the sample ytin , are sent back to paint shop for re-painting, where ytin is a binomial random variable with mean np0 and variance np0 ð1 p0 Þ. When an out-of-control signal is reported, i.e. ^ P UCL, then all the vehicles painted in that production period p the are inspected and number of defects out of all the vehicles painted in period t, ytout are also sent back to paint for repainting, where ytout is a binomial random variable with mean X t p1 and variance X t p1 ð1 p1 Þ. However since ytout depends on the decision variable X t , we replace ytout by its expected value X t p1 and obtain a fat solution. Then the expected cost due to repainting can be written as,
" # T T X X cp ytin ðIt Þ þ cp E X t p1 ð1 It Þ t¼1
ð8Þ
t¼1
where cp is cost of re-painting a vehicle. Note that if It ¼ 1, i.e., if pchart does not signal an out-of-control situation, the second cost component vanishes, and if It ¼ 0, i.e., if p-chart signals an out-ofcontrol situation, the first cost component vanishes. The probability distribution of It , which is given by
PðIt ¼ 0Þ ¼ PðIt ¼ 0jp1 > p0 ÞPðp1 > p0 Þ þ PðIt ¼ 0jp1 ¼ p0 ÞPðp1 ¼ p0 Þ : PðIt ¼ 1Þ ¼ PðIt ¼ 1jp1 > p0 ÞPðp1 > p0 Þ þ PðIt ¼ 1jp1 ¼ p0 ÞPðp1 ¼ p0 Þ Since Pðp1 > p0 Þ ¼ pout ¼ ek , we have
^ > UCLjp1 > p0 Þek þ Pðp ^ > UCLjp1 ¼ p0 Þð1 ek Þ PðIt ¼ 0Þ ¼ Pðp k ^ 6 UCLjp1 > p0 Þe þ Pðp ^ 6 UCLjp1 ¼ p0 Þð1 ek Þ PðIt ¼ 1Þ ¼ Pðp Since the costs given in Eq. (8) is due to repainting at the end of each inspection epoch is a random variable, a realization of (8) may be calculated by obtaining random drawing from the probability distribution of It by the following routine: (1) first generate an uniform random variable RN in ½0; 1, (2) If RN 6 pout ¼ ek , then the process is out-of-control, i.e. p1 > p0 . If RN > pout ¼ ek , then the process is in-control, i.e. p1 ¼ p0 . (3) If p1 > p0 , decide on the magnitude of shift d by drawing a uniform random number RN in ½0; 1. Let c is a unit cost of painting a vehicle. Then the total cost of painting vehicles is, T X cX t
ð9Þ
t¼1
In our two-stage stochastic programming model, the first stage decision is to decide on control limit parameter, k. Control limit parameter k is decided before status of the process It as in-control or out-of-control is known. The optimal k is one that minimizes the expected total cost E½Q ðk; nÞ, respect to predetermined limit B, and the solution of the first stage (FS) problem. Objective function of the FS problem E½Q ðk; nÞ in Eq. (10) is the average sum of all the costs in Eqs. (4) and (6)–(9).
8 z ¼ min fE½Q ðk; nÞg > < k2K ðFSÞ k 6 B > : kP0
ð10Þ
In Eq. (10), E½Q ðk; nÞ is taken with respect to random vector n, which is a vector of size T, whose ith element shows whether ith production period is in-control or out-of-control. First stage (FS) decision is usually associated with a cost coefficient cT, and the objective function in Eq. (10) is written as minfcT k þ E½Q ðk; nÞg. In our model since no cost is associated with FS decision k, the cost coefficient of the k is zero. The second stage (SS) model determines minimum total cost Q ðk; nÞ, i.e. sum of all the costs in Eqs. (4) and (6)–(9), according to the FS decision k and random realization nðxÞ of the random vector n. Once the status of the processes nðxÞ are observed, then the SS decisions on production quantity for the next production period X tþ1 is decided at the end of production period t so the FA demand should be met. Production quantity X t shows the number of vehicles to be painted in production periods between t and t + 1. Before, introducing the second stage (SS) model, we list the parameters, indices, variables, subscripts and decision expressions used in the model. Input parameters n Sample size p0 Average defect rate of the process when in-control ^ Average defect rate of the sample p p1 The average defect rate of the process after special cause occurrence k Rate of the occurrence of the special cause c Painting cost of each vehicle cp Re-painting cost of a vehicle Inspection cost c0 c1 Type 1 error cost c2 Type 2 error cost D Re-painting time of defective vehicles FD Final assembly demand Cap Capacity of paint process for each production period BM Very big integer number SSL Safety stock level in percentage Indices t Length of production period also sampling interval, t ¼ 1; . . . ; T Decision variables The distance of the control limit from the process k average defect rate, control limit parameter Xt Number of vehicles painted in period t Decision expression Probability of exceeding UCL given the process is incontrol b Probability of not exceeding UCL given the process is out-of-control z Standard difference between UCL and p1 , z-statistic value 1; If variable z is negative bv ¼ 0; Otherwise
a
Once the first stage decision k is determined and the random vector n is observed, the following SS is solved in order to minimize the total cost Q ðk; nÞ. Note that the production quantity X t and the status of the process It depend on the random defect occurrences. To show this dependency, they should be typed as X t ðnÞ and It ðnÞ.
Please cite this article in press as: Gunay, E. E., & Kula, U. Integration of production quantity and control chart design in automotive manufacturing. Computers & Industrial Engineering (2016), http://dx.doi.org/10.1016/j.cie.2016.06.016
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However, to simplify the presentation of the SS model, we drop n from X t and It .
zero, variable bv in Eq. (11f) and (11g) takes a value of 1 and Type 2 error probability equals to,
8 " # T T T T T T X X X X X X > > t > min Qðk; nÞ ¼ X c ð1 p Þ a þ X c p ð1 bÞ þ X c p b þ c y ðI Þ þ c E X p ð1 I Þ þ cX t > t 1 t 0 t 2 p t p t t out out out 1 in > > > t¼1 t¼1 t¼1 t¼1 t¼1 t¼1 > > > > T > >X > > X t 6 FDð1 þ SSLÞ > > > t¼1 > > > > D1g maxf0;tD1g > > X t þ ymaxf0;t ðImaxf0;tD1g Þ þ yout ð1 ðImaxf0;tD1g Þ 6 Cap; 8t ¼ 1; . . . ; T > in > > > 8 > 2 > a ¼ ð1 þ 0:09979271k þ 0:04432014k þ 0:0096992k3 0:00009862k4 þ 0:0058155k5 Þ =2 > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > p0 ð1p0 Þ p1 ð1p1 Þ > > p1 > z ¼ p0 þ k n n > > > > > > < z 6 BM ð1 bv Þ SS ¼ z 6 BM bv > > > 8 > > 1 ð1 þ 0:09979271jzj þ 0:04432014jzj2 þ 0:0096992jzj3 0:00009862jzj4 þ 0:0058155jzj5 Þ =2 b 6 BM bv > > > > 8 > > > ð1 þ 0:09979271jzj þ 0:04432014jzj2 þ 0:0096992jzj3 0:00009862jzj4 þ 0:0058155jzj5 Þ =2 b 6 BM ð1 bv Þ > > > þ > > > > X t 2 Z ; 8t ¼ 1; . . . ; T > > > 06a61 > > > > > 06b61 > > > > > k 2 Rþ > > > > > bv 2 f0; 1g > > > > > > y0in ¼ 0 > > : 0 yout ¼ 0
Objective function of the model in Eq. (11a) minimizes the total cost. Total cost is the sum of costs as presented in Eqs. (4) and (6)– (9). Constraint (11b) ensures that the difference between total production and FA demand must not exceed the safety stock level, SSL. Constraint (11c) guarantees that number of vehicles painted and repainted must not exceed the production capacity, Cap. We use normal approximation to binomial distribution for the calculation of Type 1 error in Eq. (3). There are several polynomial approximation methods to calculate the area under the normal curve that is greater than k, i.e. PðZ P kÞ, easily (Patel & Read, 1982). One of the polynomial rational methods which has good 5
approximation with sufficiently small error, i.e. jeðkÞj < 2 10 , is presented in Eq. (12). Approximation given in Eq. (12) is written as Eq. (11d) in the SS problem.
PðZ P kÞ ¼ 1 UðkÞ; 2
3
4
5 8
1 UðkÞ ¼ ½ð1 þ a1 k þ a2 k þ a3 k a4 k þ a5 k Þ =2 þ 2eðkÞ; jeðkÞj < 2 105 ;
kP0
a1 ¼ 0:09979271; a2 ¼ 0:04432014; a3 ¼ 0:0096992; a4 ¼ 0:00009862; a5 ¼ 0:00581551
ð12Þ
For the polynomial approximation to Type 2 error probability in Eq. (5), some modifications to Eq. (12) is required: If the new defect rate of the process p1 is less than UCL, Type 2 error is greater than 0.5 and less than 0.5 vice versa. Since the approximation in Eq. (12) is defined for the calculation of the right hand tail of the normal curve and maximum value of this approximation equal to 0.5, we introduce a binary indicator variable bv to calculate Type 2 accurately. The standard difference between UCL and p1 is calculated with variable z, in Eq. (11e). If the variable z in Eq. (11e) is less than
ðaÞ ðbÞ ðcÞ ðdÞ ðeÞ ðfÞ ðgÞ
ð11Þ
ðhÞ ðiÞ ðjÞ ðkÞ ðlÞ ðmÞ ðnÞ ðoÞ ðpÞ
PðZ 6 zÞ ¼ UðzÞ; 8
UðzÞ ¼ ½ð1 þ a1 jzj þ a2 jz2 j þ a3 jz3 j a4 jz4 j þ a5 jz5 jÞ =2 þ 2eðzÞ; jeðzÞj < 2 105 ;
z60 ð13Þ
When the variable z in Eq. (11e) is greater than zero, variable bv in Eq. (11f) and (11g) takes a value of 0 and Type 2 error probability is calculated as in Eq. (14),
PðZ 6 zÞ ¼ UðzÞ; 8
UðzÞ ¼ 1 ½ð1 þ a1 jzj þ a2 jz2 j þ a3 jz3 j a4 jz4 j þ a5 jz5 jÞ =2 þ 2eðzÞ; jeðzÞj < 2 105 ; z P 0
ð14Þ
According to the values of variable bv, constraint (11h) and (11i) calculates the Type 2 error probability accurately as presented in Eqs. (13) and (14). The types of the variables used in the model are defined in Eq. (11j)–(11n). Constraints (11o) and (11p) satisfy that at the beginning of the production i.e., t ¼ 0, no defective vehicles from previous day is observed.
3. SAA algorithm The main difficulty of solving stochastic optimization problems by conventional techniques is the number of possible realizations (scenarios) of the random variables. If the number of possible realizations of the random variable considered in the problem is large, evaluation of the expectation E½Q ðk; nÞ, becomes extremely difficult sometimes not possible. For example, in our problem
QT PnðUCLÞ n the number of scenarios jXj is It þ t¼1 d¼0 d
Please cite this article in press as: Gunay, E. E., & Kula, U. Integration of production quantity and control chart design in automotive manufacturing. Computers & Industrial Engineering (2016), http://dx.doi.org/10.1016/j.cie.2016.06.016
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PX t
Xt ð1 It Þ . Since the calculation of the exact ytout expected value of the total costs E½Q ðk; nÞ, in FS problem in Eq. (15) is impossible due to the increase on number of scenarios. The sample average approximation (SAA) which is a simulation based method developed by Kleywegt, Shapiro, and Homemde-Mello (2001) is used to solve two-stage stochastic optimization problems.
ytout ¼nðUCLÞ
E½Q ðk; nÞ ¼
jXj X pf Q ðk; nf Þ
ð15Þ
f ¼1
The idea of the SAA algorithm is to approximate the expectation in (15). A sample size R of n is taken from the scenario space X and P the expectation E½Q ðk; nÞ is replaced by 1=R Rr¼1 Q ðk; nr Þ and an approximate first stage problem (AFS) is developed. It is shown by Kleywegt et al. (2001), that the optimal objective value ^z and ^ to AFS, converge to the true optimal objective optimal solution k
value z and true optimal solution k of the stochastic program as the size of the scenario R increases. In our model, we assume that there is no cost associated with k so the cost coefficient cT of the first stage decision variable all zero. Therefore, we can write the AFS problem correspond to Eq. (10) as below,
8 > > > > < ^z ¼ mink ðAFSÞ > > k6B > > : kP0
1 R
R X
!
Q ðk; nr Þ ð16Þ
r¼1
Let AFSm be the mth solution to AFS, where m ¼ 1; 2; . . . ; M and ^m be the optimal total cost and optimal control limit let ^zm and k parameter to AFSm , then ^z ¼ minm¼1;2;...;M f^zm g is a point estimate for the optimal value z of the true total cost in FS problem, and ^ ¼ arg min^m f^zm g, i.e. k ^ such that zðk ^ Þ ¼ minm¼1;2;...;M f^zm ðk ^m Þg, k k
is an estimated control limit parameter k of the true FS problem. It is shown that average of M problems of AFSm problems, P zM ¼ M1 M ^m is an estimated lower bound on the optimal averm¼1 z
age total cost z (Kleywegt et al., 2001). An estimated upper bound on z is calculated by fixing any ^ , the control feasible solution to AFS. A usual practice is to fix k
limit parameter that satisfy minimum expected total cost out of m solutions, to solve the SS problem given in Eqs. (11a)–(11p). An estimated upper bound on minimum average total cost z of the FS problem is calculated by solving the SS problem for R0 ^ Þ ¼ different scenarios of n where R0 > R as follows, ^zR0 ðk PR0 r 1 ^ Q ðk ; n Þ. 0 R
r¼1
^ Þ and zM . The estimated gap for z is the difference between ^zR0 ðk Note that the estimated gap could be negative due to sampling ^ Þ < zM in some iterations error since it is possible to obtain ^zR0 ðk of the algorithm. Therefore, the estimated gap is set ^ Þ zM ; 0g. The algorithm is stopped when the gap is sufmaxf^zR0 ðk ficiently small. The steps of the algorithm are detailed below: Step 1: Generate M independent samples, each of size R, i.e., ðnR1 ; . . . ; nRm Þ for each instance of AFSm . For each m ¼ 1; 2; . . . ; M, solve the AFSm problem. Let ^zm be the optimal objective value ^m be the optimal control limit parameter (FS (total cost) and k decision) for the mth problem. Step 2: Calculate the average total cost zM of M problems solved P ^ ^m in step 1: zM ¼ M1 M m¼1 z and find the minimum of solutions, k which is the estimated optimal control limit parameter that ^ ¼ arg min^m f^zm g. minimizes the average total cost, i.e., k k
Step 3: Increase the number of samples R to R0 , where R0 R. ^ , solve the second stage problem Fix the first stage decision to k ^ Þ correspond to k ^ and calculate the expected total cost ^zR0 ðk 0
R X ^ Þ ¼ 1 ^ ; nr Þ: ^zR0 ðk Q ðk 0 R r¼1
^ Þ zM and Step 4: Calculate the optimality gap estimate, ^zR0 ðk 2 ^ 2 ^ þ r ^ 2z ^ ^ ¼r the estimated variance of the gap, r ^z ðk Þz ^z ðk Þ M R0
M
R0
where,
r^ ^2z 0 ðk^ Þ ¼ R
r^ 2zM ¼
R0 X 1 ^ ; nr Þ ^z 0 ðk ^ ÞÞ2 ðQðk R R ðR 1Þ r¼1 0
0
and
M X 1 2 ð^zm zM Þ : MðM 1Þ m¼1
^ Þ zM is less than 1% of the costs, then stop. Step 5: If gap ^zR0 ðk Otherwise go to Step 1. 4. Numerical analysis In this part of study, we present a set of numerical examples to provide insights on how the magnitude of the shift, special cause occurrence rate, k and ratio of Type 1 error cost to Type 2 error cost affect total inspection and production costs, i.e. total cost. We solve the problem in GAMS 23.5.2 (version 18.08.2010) with KNITRO solver. The numerical examples are run on an Intel(R) Core(TM)2 Duo 2.67 GHz CPU PC, with 2 GB of memory. It takes approximately 20 min to solve the problem for a production horizon consisting of 24 production periods, which correspond to a day’s production. The automobile company that we worked with in this project randomly inspects 10 vehicles at the end of each hour. The average defect rate of the process p0 , is 0.3 and rp0 is 0.15. The values of the parameters involved in the model are as follows: painting cost of each vehicle c = 200 $, re-painting cost of a defective vehicle cp = 600 $, inspection cost of vehicles c0 = 10 $, Type 1error cost c1 = 1500 $. Type 2 error cost is considered in six different levels (1500 $, 3000 $, 4500 $, 7500 $, 15,000 $, 30,000 $) in the experiments. Re-paining time of defective vehicles takes 4 h. Capacity of the paint process is 80 vehicles per hour. At the end of each day, 1200 painted vehicles should be ready to meet FA demand. The safety stock level allowed by the company is 10% of the daily demand. In the numerical study, we assume that the average defect rate of the process is same for every color. If the defect rate of the process highly depends on the color, it is also possible to design new charts for different colors. We solve the two stage stochastic model by SAA. The parameters used in SAA algorithm are: M ¼ 10; R ¼ 100 and R0 ¼ 200. We model the magnitude of the shift, d as a random variable in our model, and it represents the percent change in the process average defect rate when the process runs out-of-control: p1 ¼ p0 þ d. We assume that magnitude of shift d is normally distributed with mean ld and variance r2d . Hence the new defect rate of the process is p1 ¼ p0 þ drp0 . In our numerical study we consider effect of four different parameters on control limit parameter k, Type 1 and Type 2 error probability and total cost. These four parameters are (i) mean of the shift ld , (ii) variance of the shift r2d , (iii) occurrence of the special cause rate k and (iv) cost ratio of Type 2 error cost to Type 1 error cost. 4.1. Effect of the mean shift on k and costs Table 1 shows the effect of the mean shift, ld . Mean shift ld , is considered at five different levels and these levels are reported in
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E.E. Gunay, U. Kula / Computers & Industrial Engineering xxx (2016) xxx–xxx Table 1 Effect of mean shift on k, Type 1 and Type 2 probability.
ld
r2d
k
Cost ratio
k
UCL
Type 1 error a
Type 2 error b
1.00 1.40 1.60 1.80 2.20 2.30 2.40 2.50 2.60
0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
2 2 2 2 2 2 2 2 2
2.110 1.960 1.660 1.880 1.881 1.889 1.901 1.927 1.933
0.617 0.594 0.549 0.582 0.582 0.583 0.585 0.589 0.589
0.015 0.023 0.060 0.030 0.030 0.030 0.030 0.026 0.026
0.866 0.697 0.620 0.504 0.343 0.339 0.323 0.306 0.278
the first column of Table 1. In experiments in Table 1, we assume that r2d ¼ 0:03 and Type 2 to Type 1 error cost ratio is two. Table 1 shows mean shift of the random variable d increases k initially. As mean of d keeps increasing k starts to decrease in order to balance out Type 1 and Type 2 error costs. Fig. 2(a) shows that total cost increases as the magnitude of the average shift ld increases. The reason for this is that an increase on average defect rate increases the number of non-conforming units therefore more vehicles needed to be painted to meet the FA demand, which increases painting costs. The increase on number of vehicles painted increases the painting cost. Also, since all the newly painted vehicles are inspected when an out-of-control signal is reported, the increase on the total number of vehicles painted increase re-inspection cost. In addition to these two cost components, the increase on the number of defects increases re-painting cost. Fig. 2(b) shows that since Type 2 error cost is twice the Type 1 error cost, the p-chart parameter k is optimized to give a priority to reduce Type 2 error cost over Type 1 error cost. Both Table 1 and Fig. 2(b) shows that as ld increases to 1.6, Type 1 error cost increases since UCL initially increases to reduce the k, Type 2 error probability and its cost. As the mean shift ld continue to increase above 1.6, both Type 1 and Type 2 error costs decrease since both Type 1 and Type 2 error probabilities decrease.
(a)
Fig. 2(a) shows that even though Type 1 and Type 2 error costs decrease, the total cost function increases due to the increase on average defect rate. The increase on average defect rate ld increases the re-painting and defective production costs which cannot be compensated by the decrease in Type 1 and Type 2 error costs. 4.2. Effect of the shift variance on k and the costs In Table 2, we investigate the effect of the increase in variance of the shift d. Table 2 shows that as shift variance r2d increases, variation in process defect rate also increases. Therefore lower k values hence lower UCL, are determined to detect the special cause occurrences, which in turn decreases Type 2 error probability and increases Type 1 error probability. As Fig. 3(a) shows, total cost increases as the variance of the shift increases as expected. Fig. 3(b) shows that as shift variance r2d increases, Type 1 error cost is increased in order to minimize the Type 2 error cost. In experiments given in Table 3, we changed the mean and the variance of the shift simultaneously to see their effect on chart parameters and the total cost. Table 3 shows that as the mean and the variance of the shift d increase, k decreases to 1.80 and the corresponding UCL decreases to 0.57. As the mean and the variance of the shift continue to increase, higher k values are determined to decrease Type 1 error probability and its associated costs.
Table 2 Effect of shift variance on k, Type 1 and Type 2 probability.
r2d
ld
k
Cost ratio
k
UCL
Type 1 error a
Type 2 error b
0.003 0.03 0.12 0.24 0.33
1 1 1 1 1
0.05 0.05 0.05 0.05 0.05
2 2 2 2 2
2.110 2.040 1.937 1.890 1.870
0.617 0.606 0.591 0.584 0.581
0.015 0.018 0.024 0.030 0.030
0.866 0.803 0.654 0.500 0.432
(a) 410000
400000 395000
Total Cost
405000
Total Cost
400000 395000 390000
385000 380000 375000 365000 360000
380000 1
1.4
1.6
1.8
2.2
2.3
2.4
2.5
0.003
2.6
Mean shift
0.03
0.12
0.24
0.33
Shift variance
(b) 80000
65000 60000 55000 50000 45000 40000 35000 30000 25000 20000 15000
70000 60000
Cost
Cost
390000
370000
385000
(b)
405000
50000 40000 30000 20000
1
1.4
1.6
1.8
2.2
2.3
2.4
2.5
2.6
Mean shift Type1 cost Type2 cost Fig. 2. Effect of mean shift on total cost, Type 1 and Type 2 cost.
0.003
0.03
0.12
0.24
0.33
Shift variance Type1 cost Type2 cost Fig. 3. Effect of shift variance of on total cost, Type 1 and Type 2 cost.
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Table 3 Effect of changing mean and variance of shift on k, Type 1, Type 2 probability and total cost. k
Cost ratio
k
UCL
Type 1 error a
Type 2 error b
Total cost
0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2 2.1 2.2 2.3
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
2 2 2 2 2 2 2 2 2 2 2 2
2.50 2.27 2.07 2.04 2.02 1.99 1.80 2.00 2.11 2.27 2.32 2.43
0.68 0.64 0.61 0.61 0.60 0.60 0.57 0.60 0.62 0.64 0.65 0.66
0.0007 0.0020 0.0070 0.0075 0.0080 0.0090 0.0200 0.0090 0.0050 0.0030 0.0020 0.0010
0.900 0.851 0.780 0.730 0.672 0.613 0.606 0.590 0.570 0.560 0.550 0.530
394,205 396,828 405,570 408,755 411,064 421,267 426,146 430,574 434,681 436,086 439,305 440,343
Total Cost
r2d
580000 530000 480000 430000 380000 0.05
0.07
0.1
0.15
0.2
Special cause rate
(b)
160000 140000 120000
Cost
ld
(a)
100000 80000 60000 40000 20000 0.05
440000
0.07
0.1
0.15
0.2
Special cause rate Type1 cost Type2 cost
Total cost 420000
Fig. 5. Effect of special cause rate k on total cost, Type 1 and Type 2 cost. 0,6
400000 0,4 0,5
1,0
0,2 1,5
Variance of shift
2,0
Mean of shift
Fig. 4. Effect of changing mean and variance of shift on total cost.
Fig. 4 plots the change in total cost as the mean and the variance of the shift increase: As mean and the variance of the shift increase, the number of non-conforming vehicles and hence the number of re-painted vehicles, therefore total painting and re-painting cost increases.
Table 5 Effect of changing the cost ratio of Type 2 error cost to Type 1 error cost. Cost ratio
ld
r2d
k
k
UCL
Type 1 error a
Type 2 error b
1 2 3 5 10 20
1 1 1 1 1 1
0.03 0.03 0.03 0.03 0.03 0.03
0.05 0.05 0.05 0.05 0.05 0.05
2.350 2.110 2.090 1.990 1.931 1.629
0.652 0.617 0.614 0.599 0.590 0.544
0.001 0.015 0.018 0.023 0.026 0.072
0.918 0.866 0.860 0.838 0.727 0.642
(a) 800000
Table 4 Effect of special cause rate on k, Type 1 and Type 2 probability.
700000 600000 500000 400000 300000 1
2
3
5
10
20
10
20
Cost ratio
(b) 401700 351700 301700
Cost
In experiments given in Table 4, we investigate the effect of special cause rate k. Experiments in Table 4 show that as k increases, the control limit parameter k decreases. The reason for this decrease in k can be explained as follows: As the probability of the process running out-of-control increases with k, the lower values of k must be set to detect the shift, hence to decrease Type 2 error probability. For instance when k is 0.05, control limit parameter k is set to 2.11 and Type 2 error probability is 0.866. As k increases to 0.07, the control limit parameter k decreases to 1.816 and Type 2 error probability decreases to 0.76. Fig. 5(a) shows that as k increases, total cost increases since the number of times the process runs out-of-control state increases on average. Fig. 5(b) shows that as k increase both Type 1 and Type 2 error probabilities increase.
Total Cost
4.3. Effect of special cause rate k on k and the costs
251700 201700 151700 101700
k
ld
r2d
Cost ratio
k
UCL
Type 1 error a
Type 2 error b
0.05 0.07 0.10 0.15 0.20
1 1 1 1 1
0.03 0.03 0.03 0.03 0.03
2 2 2 2 2
2.110 1.816 1.688 1.560 1.460
0.617 0.572 0.553 0.534 0.519
0.015 0.035 0.044 0.060 0.070
0.866 0.760 0.738 0.710 0.670
51700 1700 1
2
3
5
Cost ratio Type1 cost Type2 cost Fig. 6. Effect of cost ratio on total cost, Type 1 and Type 2 costs.
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4.4. Change on the cost ratio of Type 2 error cost to Type 1 error cost, cost ratio Table 5 shows how the ratio of Type 2 error cost to Type 1 error cost affects the chart parameter k and the costs. As the Type 2 error cost increase, lower k values are selected to decrease Type 2 error cost. Since usually Type 2 error cost is higher than Type 1 error cost, our examples show that commonly used 3r control limits are quite far from being optimal. Exponential increase on cost ratio exponentially increase total cost and Type 2 error cost as seen in Fig. 6(a) and (b). 5. Conclusion In our study we present a two-stage stochastic model to determine control chart parameter, upper control limit (UCL) and production quantity of vehicles painted for each sampling period Xt. The model is solved by sample average approximation (SAA) algorithm. A numerical study is performed to investigate the behavior of the model according to various parameters: mean of the shift, variance of the shift, occurrence rate of the special cause, cost ratio of Type 2 error to Type 1 error. Our numerical study provides the following insights: (i) An increase on the mean defect rate increases both the total cost and the total production quantity. (ii) Effect of an increasing process variance to the control limit parameter k is insignificant. (iii) Frequency of special cause occurrences significantly affects the total cost. (iv) All the experiments show that the commonly used 3r control limits in practice are wider than required, which increases Type 1 error risk. As a future study, determination of optimal sample size n and sampling interval h along with parameter k may be an interesting research direction. Another research direction is to study effect of several types of special cause occurrences on the control chart parameter k and the total quality and production related costs. Also, since total cost and total production quantity are very sensitive to the special cause occurrence rates, effect of different probability distributions of special cause occurrences may be explored. References Aslam, M., Azam, M., Khan, N., & Jun, C.-H. (2015). A mixed control chart to monitor the process. International Journal of Production Research, 53(15), 4684–4693. http://dx.doi.org/10.1080/00207543.2015.1031354. Ben-Daya, M., & Rahim, M. A. (2000). Effect of maintenance on the economic design of x-control chart. European Journal of Operational Research, 120(1), 131–143. http://dx.doi.org/10.1016/S0377-2217(98)00379-8. Bouslah, B., Gharbi, A., & Pellerin, R. (2013). Joint production and quality control of unreliable batch manufacturing systems with rectifying inspection. International Journal of Production Research (April), 1–15. http://dx.doi.org/ 10.1080/00207543.2012.746481. Bouslah, B., Gharbi, A., & Pellerin, R. (2015). Integrated production, sampling quality control and maintenance of deteriorating production systems with AOQL constraint. Omega. http://dx.doi.org/10.1016/j.omega.2015.07.012.
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Integration of production quantity and control chart design in automotive manufacturing E.E. Gunay ⇑, U. Kula Department of Industrial Engineering, Sakarya University, Sakarya, Turkey
a r t i c l e
i n f o
Article history: Available online xxxx Keywords: Attribute chart (p-chart) design Stochastic models Sample average approximation Quality management
a b s t r a c t This study presents a two-stage stochastic programming model for the determination of control limits in p-charts when a production process produces above a certain quantity. Consideration of production quantity needed along with control limit determination is important for the following competing two reasons: (1) Wider control limits make it difficult to detect the changes in the process, therefore producing excessive number of cars with paint defects. (2) Narrower control limits, on the other hand, increase the number of unnecessary interventions even if there is no deterioration in the process so that inspection costs increase. In both cases, quantity produced reduces due to defective products and unnecessary interventions. Therefore, it is important to design a control chart for proportion of defects that takes production quantity requirements into account. We consider the problem in an automotive manufacturing setting in which the cars are inspected for paint defects after paint operations. We formulate the problem as a two-stage stochastic programming model. In the first stage, control limit parameter k is decided for the p-chart and in the second stage, production quantity is determined that minimizes total quality-related and production costs. We solve the model by sample average approximation algorithm (SAA). In a numerical study, we investigate the effect of various factors on control limit parameter k and the total cost. Our numerical study shows that (i) an increase on the mean defect rate increases both the total cost and the total production quantity, (ii) effect of an increasing process variance to the control limit parameter k is significantly small, (iii) frequency of special cause occurrences affects the total cost significantly and (iv) all the experiments show that the commonly used 3r control limits in practice are wider than required. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In automobile manufacturing, after body shop operations, welded bodies are directed to the paint shop. Vehicle painting operation includes several steps such as cleaning, primer coat painting, top coating and polishing. Once painting is completed, the vehicles are inspected for paint defects before they are released to the Final Assembly (FA) department. Since paint defects may be as much as 40%, some manufacturers use control charts to monitor the paint process and determine whether the current painting process is out-of-control. An out-of-control situation means that there is a special cause increasing the proportion of paint defects. In vehicle painting operations, defects are categorized as either minor or major ones. Minor defects just need small touchups, usually done on the line, which does not cause any delays in assembly
⇑ Corresponding author. E-mail addresses: [email protected] (E.E. Gunay), [email protected] (U. Kula).
operations. On the other hand, vehicles with major defects must be taken off the line and sent back to paint shop for re-painting. Since minor defects do not add any significant cost and do not disturb the vehicle flows in the line, the scope of this study is confined to major paint defects. Fig. 1 shows the simplified inspection procedure used in the paint shop of a major car manufacturer’s plant located in Turkey. A sample of size n is drawn in a sampling interval of h hours at the end of epoch t from the batch size of X t vehicles. Number of the defective vehicles in the sample is counted and estimated aver^, is calculated. The painting process age defect rate of the process p ^, is above is deemed out-of-control if the defect rate of the sample p the upper control limit (UCL). When the p-chart signals an out-ofcontrol situation, the engineering team searches for special causes that might increase the proportion of paint defects. If a cause for the defect rate increase is found, the painting process is restored to the in-control state at some cost and all the newly painted vehicles in the sampling interval are inspected. Since no action is taken if the defect rate of the sample is below UCL, determination of lower control limit in p-charts is of no practical importance.
http://dx.doi.org/10.1016/j.cie.2016.06.016 0360-8352/Ó 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Gunay, E. E., & Kula, U. Integration of production quantity and control chart design in automotive manufacturing. Computers & Industrial Engineering (2016), http://dx.doi.org/10.1016/j.cie.2016.06.016
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Randomly take sample of size n from a batch of Xt vehicles at the end of production period t
t
t 1
Does the chart signal an out-ofcontrol in the tth production period?
Yes
Inspect all the newly painted vehicles in the tth production period
No
Send defective vehicles for re-painting.
Fig. 1. Simplified procedure of paint control process.
Setting an optimal UCL is important to reduce the production and quality costs: A widely set UCL decreases the sensitivity of a control chart to detect the special cause occurrences. On the other hand, a narrow UCL increases the number of false out-of-control signals in the process even if there is no deterioration in the process. As Eq. (1) suggests, setting an optimal UCL involves finding the optimal k that would minimize quality related costs. Determination of optimal control limits along with optimal sampling size and sampling frequency that minimize quality related costs is known as economical design of chart parameters and is commonly studied in the literature.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p0 ð1 p0 Þ UCL ¼ p0 þ k n
ð1Þ
Ladany (1973) is the first paper that considers economical design of p-chart parameters. His model includes cost of sampling, cost of not detecting a change in the process (Type 2 error cost), cost of false indication of change (Type 1 error cost) and cost of readjusting detected change. Montgomery, Heikes, and Mance (1975) consider the same problem when there are several out-of-control states. Chiu (1975a) presents a model which minimizes loss cost function in an np-control chart and Chiu (1975b) investigate the effects of variation in cost factors by drawing loss cost surface as contour plots. Chiu (1976) considers the case where there are several out-ofcontrol states in an np-control chart. More recent studies on economical design of chart parameters consider variable sample size, variable sampling interval and both variable sample size/sampling interval and present the advantages of these models rather than traditional p-chart design (Aslam, Azam, Khan, & Jun, 2015; Inghilleri, Lupo, & Passannanti, 2015; Kooli & Limam, 2011; Wu & Luo, 2004). Recognizing the effect of the monitoring policy on the production capacity, Lee and Rosenblatt (1987) develop a model addressing the problem of joint determination of optimal production run time, number of inspections to minimize quality-related costs in an X chart. Lee and Park (1991) consider the same problem by focusing on the difference between rework cost before sale and warranty cost after sale. Rahim (1994) considers the same joint problem by developing a non-Markovian shock model under production setup, inventory holding and maintenance cost. Rahim and Ben-Daya (1998) extend this model to consider the case in which the production is halted not only if there is true alarm but also there is a false alarm. Ben-Daya and Rahim (2000) consider
the joint-optimization of preventive maintenance actions and X chart parameters when in-control state follows a general probability distribution with increasing hazard rate. Pan, Jin, Wang, and Cang (2012) develop a model to minimize the total expected production costs while jointly determining the optimal parameters of control chart and the maintenance decision policy whereas Bouslah, Gharbi, and Pellerin (2015) consider joint design of production, quality and maintenance control policy problem in cchart. In vehicle painting process, since defects in the paint shop occur randomly, it is not possible to know before how many vehicles to paint in order to meet the FA demand. Therefore, a control chart policy should take random paint defect occurrences into account to meet assembly line demand and to minimize quality related costs simultaneously. Economic design problem of control charts, including p-chart design, involve determination of three important parameters: Control limit width, sampling interval, and sample size. In a p-chart, narrowly determined UCL increases the number of false alarms and when a p-chart reports an alarm, all the vehicles painted in between previous and the current sampling epochs are inspected one by one, which increases the unnecessary cost of inspection. This inspection cost depends on the number of vehicles painted in the batch. Therefore, if the number of painted vehicles is more than the optimal number needed, extra inspection costs occur. On the other hand widely determined UCL makes difficult to detect the shift in the process so defective vehicles will be sent to the FA department. In this paper, we develop a two-stage mathematical model that jointly determines the optimal upper control limit in a p-chart and the paint batch size X t . In the first stage, the control limit parameter k is set and after observing the random paint defects, the paint batch size X t is decided. The two-stage decision making framework optimizes the first stage decision k given the fact that vehicles are repeatedly painted and paint defects are experienced over and over again. In fact, in statistical process control applications, the control chart parameters are determined first and random process shifts occur repeatedly as process is monitored over time. Therefore, two-stage stochastic programming provides an appropriate modeling framework for incorporating various constraints such as demand, service level, and maximum inventory level extra into control chart design problem. Bouslah, Gharbi, and Pellerin (2013) built a stochastic mathematical model and use a simulation based optimization approach for joint determination of the production quantity, hedging level and the sample size in X charts.
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There are also some other studies that both decide on quality and maintenance decisions to both lower the cost of quality and maintenance. These models aim to both improve quality of items produced also lower maintenance costs (Chan & Wu, 2009; Mehdi, Nidhal, & Anis, 2010; Mehrafrooz & Noorossana, 2011; Pandey, Kulkarni, & Vrat, 2011). After developing the two-stage stochastic model, we perform a numerical study that investigates the effect of several cost components and random disturbances on first stage decision variable k and the total cost. Lam and Rahim (2002) present an extensive sensitivity analysis similar to ours to investigate the effects of cost and inspection parameters on production and quality costs in X charts. Chakraborty, Giri, and Chaudhuri (2009) study the joint effect of process shift, machine breakdown and inspection decisions on production quantity for deteriorating production system where p-chart is used to control the process. To our knowledge, our study is the first that models determination of p-chart parameter k when a certain production quantity needs to be produced in a planning horizon of length T with the objective of minimizing total production and quality-related costs. In the first stage two-stage stochastic model, we decide on the p-chart control limit parameter k which determines the UCL for p-chart. In the second stage of the model, the production quantities X t are determined for each period t for a planning horizon of length T. In addition to the demand constraint, a maximum inventory level constraint that guarantees the total production quantity does not exceed a certain percentage of the quantity produced, i.e. safety stock level SSL, is added to the second stage model. This paper is organized as follows. Section 2 formulates the problem and presents the two-stage stochastic model. In Section 3 we give the details of the sample average approximation (SAA) algorithm used to solve the developed model. A numerical study is performed to offer managerial insights in Section 4. Section 5 concludes the paper and explores possible future research directions. 2. Two stage stochastic model
pout ¼ 1 ek
where pout is the probability that the painting process defect rate would shift from p0 to p1 . Due to sampling error, it is possible to get a false out-of-control signal even if the process defect rate has not shifted from p0 to p1 . This error is called as Type 1 error and ^ P UCLjp1 ¼ p0 Þ. is denoted by Pðp Since number of defectives D in a sample of size n is a binomial random variable with mean np0 , Type 1 error probability, the probability of having more than n(UCL) defects in the sample given that the process mean stays as p0 , a ¼ PðD P nUCLjp1 ¼ p0 Þ is given as follows:
a¼
n X nd n d p0 1 pd0 d dPnðUCLÞ
ð3Þ
which can be approximated by normal distribution. Therefore we have,
a¼P ZP
nUCL np0
r
Z
¼ PðZ P kÞ ¼ k
1
1 1 xl 2 pffiffiffiffiffiffiffi e2ð r Þ dx; r 2p
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r is equal to np0 ð1 p0 Þ and l equals to np0 . When a Type 1 error occurs, an unnecessary inspection cost is incurred since the true mean of the process is still p0 . Total expected Type 1 error cost in a production period of length T is given in Eq. (4), where c1 shows the Type 1 error cost per vehicle and Xt shows the number of vehicles painted in production period t. T X X t c1 ð1 pout Þa
ð4Þ
t¼1
When a special cause occurs and the defect rate of the process shifts to p1 , the probability of not detecting this shift is, ^ 6 UCLjp1 > p0 Þ, called as Type 2 error. Type 2 error occurs b ¼ Pðp when the defect rate of the process shifts to p1 but the sample ^ is less than the UCL so the deterioration in the process defect rate p is not detected. By Normal approximation to binomial distribution, we can calculate the Type 2 error probability b as follows:
Z Consider a painting process that is controlled by a p-chart and let the upper control limit of the p-chart is set at pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UCL ¼ p0 þ k p0 ð1 p0 Þ=n, where p0 is the in-control defect rate of the process and k is the sigma level of the process. Once the value of k is decided, the painting process would be monitored at each discrete time epoch t. At each epoch a sample of size of n vehicles are inspected for paint defects and proportion of defectives in ^ is calculated. If p ^ is below UCL, the paint nozzles are the sample p purged and the next batch of vehicles are painted until the next ^ is above the UCL, it means that the pinspection epoch t þ 1. If p chart signals a shift in process defect rate from p0 to p1 , where p1 > p0 . ^ is above the UCL, a search for the special cause is perWhen p formed. If a special cause is found resulting in a higher than acceptable defect rate UCL, the process is brought back to in-control state at some cost and no cessation in the production. We assume that the special causes in the painting process occur according to a Poisson Process with rate k vehicles per inspection interval, ½t; t þ 1. Without loss of generality, we assume that inspection period length is one time units. Therefore, time between special cause occurrences follows an exponential distribution with mean 1=k. Modeling special cause occurrences as a Poisson Process is a reasonable and a common assumption since multiple sources contribute to special cause occurrences. (Lam & Rahim, 2002; Lee & Rosenblatt, 1987; Montgomery et al., 1975). Since the process shifts to an out-of-control state when there is one or more special cause occurrences, the probability that there is one or more special cause occurrences in each period ½t; t þ 1 is,
ð2Þ
b¼
UCL
1
1 1 pffiffiffiffiffiffiffi e 2 r p1 2p
xp1 rp1
2
dx
ð5Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where l is equal to p1 and r equals to p1 ð1 p1 Þ=n. When the process defect rate shifts to p1 , a p-chart will detect this shift with probability ð1 bÞ and all the newly painted vehicles are inspected one by one. Since this inspection is different than the inspection of sample at the end of each production period t and requires inspection of all the vehicles painted in that production period, we call it as re-inspection cost. Let c0 shows the cost of re-inspection per vehicle, then the expected total cost of reinspection is, T X X t c0 pout ð1 bÞ
ð6Þ
t¼1
On the other hand, when the process defect rate shifts to p1 , a pchart will not detect this shift with probability b. If the shift in the average defect rate of the process is not detected, the defective vehicles will be sent to following department, FA. In this case FA workers will be busy to take the vehicles off the line which may cause delays and line stoppages. In some cases, defective vehicles may not be noticed at the final inspection after the FA and may be shipped to the customer. In practice, usually this cost is much higher than the cost of false alarm, Type 1 cost. Let c2 presents the cost of Type 2 error per vehicle. Then the total cost of Type 2 error is, T X X t c2 pout b
ð7Þ
t¼1
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Once the value of k is decided, i.e., once UCL is set, the defective number of vehicles in the sample at epoch t is observed, which depends on whether the process is in-control or out-of-control. Let the random variable It be an in-control or in an out-ofcontrol state,
It ¼
1; if an out-of-control signal is not reported 0;
if an out-of-control signal is reported
For example if production period is limited with 6 periods, T ¼ 6 and the realization of the sequence of random variables It is as follows nðxÞ ¼ ð1; 0; 1; 1; 0; 1Þ. Then out-of-control signal is reported in production period 2 and 5 while an out-of-control signal is not reported in other periods 1, 3, 4 and 6. If out-of-control ^ 6 UCL, number of defective vehicles signal is not reported, i.e. p in the sample ytin , are sent back to paint shop for re-painting, where ytin is a binomial random variable with mean np0 and variance np0 ð1 p0 Þ. When an out-of-control signal is reported, i.e. ^ P UCL, then all the vehicles painted in that production period p the are inspected and number of defects out of all the vehicles painted in period t, ytout are also sent back to paint for repainting, where ytout is a binomial random variable with mean X t p1 and variance X t p1 ð1 p1 Þ. However since ytout depends on the decision variable X t , we replace ytout by its expected value X t p1 and obtain a fat solution. Then the expected cost due to repainting can be written as,
" # T T X X cp ytin ðIt Þ þ cp E X t p1 ð1 It Þ t¼1
ð8Þ
t¼1
where cp is cost of re-painting a vehicle. Note that if It ¼ 1, i.e., if pchart does not signal an out-of-control situation, the second cost component vanishes, and if It ¼ 0, i.e., if p-chart signals an out-ofcontrol situation, the first cost component vanishes. The probability distribution of It , which is given by
PðIt ¼ 0Þ ¼ PðIt ¼ 0jp1 > p0 ÞPðp1 > p0 Þ þ PðIt ¼ 0jp1 ¼ p0 ÞPðp1 ¼ p0 Þ : PðIt ¼ 1Þ ¼ PðIt ¼ 1jp1 > p0 ÞPðp1 > p0 Þ þ PðIt ¼ 1jp1 ¼ p0 ÞPðp1 ¼ p0 Þ Since Pðp1 > p0 Þ ¼ pout ¼ ek , we have
^ > UCLjp1 > p0 Þek þ Pðp ^ > UCLjp1 ¼ p0 Þð1 ek Þ PðIt ¼ 0Þ ¼ Pðp k ^ 6 UCLjp1 > p0 Þe þ Pðp ^ 6 UCLjp1 ¼ p0 Þð1 ek Þ PðIt ¼ 1Þ ¼ Pðp Since the costs given in Eq. (8) is due to repainting at the end of each inspection epoch is a random variable, a realization of (8) may be calculated by obtaining random drawing from the probability distribution of It by the following routine: (1) first generate an uniform random variable RN in ½0; 1, (2) If RN 6 pout ¼ ek , then the process is out-of-control, i.e. p1 > p0 . If RN > pout ¼ ek , then the process is in-control, i.e. p1 ¼ p0 . (3) If p1 > p0 , decide on the magnitude of shift d by drawing a uniform random number RN in ½0; 1. Let c is a unit cost of painting a vehicle. Then the total cost of painting vehicles is, T X cX t
ð9Þ
t¼1
In our two-stage stochastic programming model, the first stage decision is to decide on control limit parameter, k. Control limit parameter k is decided before status of the process It as in-control or out-of-control is known. The optimal k is one that minimizes the expected total cost E½Q ðk; nÞ, respect to predetermined limit B, and the solution of the first stage (FS) problem. Objective function of the FS problem E½Q ðk; nÞ in Eq. (10) is the average sum of all the costs in Eqs. (4) and (6)–(9).
8 z ¼ min fE½Q ðk; nÞg > < k2K ðFSÞ k 6 B > : kP0
ð10Þ
In Eq. (10), E½Q ðk; nÞ is taken with respect to random vector n, which is a vector of size T, whose ith element shows whether ith production period is in-control or out-of-control. First stage (FS) decision is usually associated with a cost coefficient cT, and the objective function in Eq. (10) is written as minfcT k þ E½Q ðk; nÞg. In our model since no cost is associated with FS decision k, the cost coefficient of the k is zero. The second stage (SS) model determines minimum total cost Q ðk; nÞ, i.e. sum of all the costs in Eqs. (4) and (6)–(9), according to the FS decision k and random realization nðxÞ of the random vector n. Once the status of the processes nðxÞ are observed, then the SS decisions on production quantity for the next production period X tþ1 is decided at the end of production period t so the FA demand should be met. Production quantity X t shows the number of vehicles to be painted in production periods between t and t + 1. Before, introducing the second stage (SS) model, we list the parameters, indices, variables, subscripts and decision expressions used in the model. Input parameters n Sample size p0 Average defect rate of the process when in-control ^ Average defect rate of the sample p p1 The average defect rate of the process after special cause occurrence k Rate of the occurrence of the special cause c Painting cost of each vehicle cp Re-painting cost of a vehicle Inspection cost c0 c1 Type 1 error cost c2 Type 2 error cost D Re-painting time of defective vehicles FD Final assembly demand Cap Capacity of paint process for each production period BM Very big integer number SSL Safety stock level in percentage Indices t Length of production period also sampling interval, t ¼ 1; . . . ; T Decision variables The distance of the control limit from the process k average defect rate, control limit parameter Xt Number of vehicles painted in period t Decision expression Probability of exceeding UCL given the process is incontrol b Probability of not exceeding UCL given the process is out-of-control z Standard difference between UCL and p1 , z-statistic value 1; If variable z is negative bv ¼ 0; Otherwise
a
Once the first stage decision k is determined and the random vector n is observed, the following SS is solved in order to minimize the total cost Q ðk; nÞ. Note that the production quantity X t and the status of the process It depend on the random defect occurrences. To show this dependency, they should be typed as X t ðnÞ and It ðnÞ.
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However, to simplify the presentation of the SS model, we drop n from X t and It .
zero, variable bv in Eq. (11f) and (11g) takes a value of 1 and Type 2 error probability equals to,
8 " # T T T T T T X X X X X X > > t > min Qðk; nÞ ¼ X c ð1 p Þ a þ X c p ð1 bÞ þ X c p b þ c y ðI Þ þ c E X p ð1 I Þ þ cX t > t 1 t 0 t 2 p t p t t out out out 1 in > > > t¼1 t¼1 t¼1 t¼1 t¼1 t¼1 > > > > T > >X > > X t 6 FDð1 þ SSLÞ > > > t¼1 > > > > D1g maxf0;tD1g > > X t þ ymaxf0;t ðImaxf0;tD1g Þ þ yout ð1 ðImaxf0;tD1g Þ 6 Cap; 8t ¼ 1; . . . ; T > in > > > 8 > 2 > a ¼ ð1 þ 0:09979271k þ 0:04432014k þ 0:0096992k3 0:00009862k4 þ 0:0058155k5 Þ =2 > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > p0 ð1p0 Þ p1 ð1p1 Þ > > p1 > z ¼ p0 þ k n n > > > > > > < z 6 BM ð1 bv Þ SS ¼ z 6 BM bv > > > 8 > > 1 ð1 þ 0:09979271jzj þ 0:04432014jzj2 þ 0:0096992jzj3 0:00009862jzj4 þ 0:0058155jzj5 Þ =2 b 6 BM bv > > > > 8 > > > ð1 þ 0:09979271jzj þ 0:04432014jzj2 þ 0:0096992jzj3 0:00009862jzj4 þ 0:0058155jzj5 Þ =2 b 6 BM ð1 bv Þ > > > þ > > > > X t 2 Z ; 8t ¼ 1; . . . ; T > > > 06a61 > > > > > 06b61 > > > > > k 2 Rþ > > > > > bv 2 f0; 1g > > > > > > y0in ¼ 0 > > : 0 yout ¼ 0
Objective function of the model in Eq. (11a) minimizes the total cost. Total cost is the sum of costs as presented in Eqs. (4) and (6)– (9). Constraint (11b) ensures that the difference between total production and FA demand must not exceed the safety stock level, SSL. Constraint (11c) guarantees that number of vehicles painted and repainted must not exceed the production capacity, Cap. We use normal approximation to binomial distribution for the calculation of Type 1 error in Eq. (3). There are several polynomial approximation methods to calculate the area under the normal curve that is greater than k, i.e. PðZ P kÞ, easily (Patel & Read, 1982). One of the polynomial rational methods which has good 5
approximation with sufficiently small error, i.e. jeðkÞj < 2 10 , is presented in Eq. (12). Approximation given in Eq. (12) is written as Eq. (11d) in the SS problem.
PðZ P kÞ ¼ 1 UðkÞ; 2
3
4
5 8
1 UðkÞ ¼ ½ð1 þ a1 k þ a2 k þ a3 k a4 k þ a5 k Þ =2 þ 2eðkÞ; jeðkÞj < 2 105 ;
kP0
a1 ¼ 0:09979271; a2 ¼ 0:04432014; a3 ¼ 0:0096992; a4 ¼ 0:00009862; a5 ¼ 0:00581551
ð12Þ
For the polynomial approximation to Type 2 error probability in Eq. (5), some modifications to Eq. (12) is required: If the new defect rate of the process p1 is less than UCL, Type 2 error is greater than 0.5 and less than 0.5 vice versa. Since the approximation in Eq. (12) is defined for the calculation of the right hand tail of the normal curve and maximum value of this approximation equal to 0.5, we introduce a binary indicator variable bv to calculate Type 2 accurately. The standard difference between UCL and p1 is calculated with variable z, in Eq. (11e). If the variable z in Eq. (11e) is less than
ðaÞ ðbÞ ðcÞ ðdÞ ðeÞ ðfÞ ðgÞ
ð11Þ
ðhÞ ðiÞ ðjÞ ðkÞ ðlÞ ðmÞ ðnÞ ðoÞ ðpÞ
PðZ 6 zÞ ¼ UðzÞ; 8
UðzÞ ¼ ½ð1 þ a1 jzj þ a2 jz2 j þ a3 jz3 j a4 jz4 j þ a5 jz5 jÞ =2 þ 2eðzÞ; jeðzÞj < 2 105 ;
z60 ð13Þ
When the variable z in Eq. (11e) is greater than zero, variable bv in Eq. (11f) and (11g) takes a value of 0 and Type 2 error probability is calculated as in Eq. (14),
PðZ 6 zÞ ¼ UðzÞ; 8
UðzÞ ¼ 1 ½ð1 þ a1 jzj þ a2 jz2 j þ a3 jz3 j a4 jz4 j þ a5 jz5 jÞ =2 þ 2eðzÞ; jeðzÞj < 2 105 ; z P 0
ð14Þ
According to the values of variable bv, constraint (11h) and (11i) calculates the Type 2 error probability accurately as presented in Eqs. (13) and (14). The types of the variables used in the model are defined in Eq. (11j)–(11n). Constraints (11o) and (11p) satisfy that at the beginning of the production i.e., t ¼ 0, no defective vehicles from previous day is observed.
3. SAA algorithm The main difficulty of solving stochastic optimization problems by conventional techniques is the number of possible realizations (scenarios) of the random variables. If the number of possible realizations of the random variable considered in the problem is large, evaluation of the expectation E½Q ðk; nÞ, becomes extremely difficult sometimes not possible. For example, in our problem
QT PnðUCLÞ n the number of scenarios jXj is It þ t¼1 d¼0 d
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PX t
Xt ð1 It Þ . Since the calculation of the exact ytout expected value of the total costs E½Q ðk; nÞ, in FS problem in Eq. (15) is impossible due to the increase on number of scenarios. The sample average approximation (SAA) which is a simulation based method developed by Kleywegt, Shapiro, and Homemde-Mello (2001) is used to solve two-stage stochastic optimization problems.
ytout ¼nðUCLÞ
E½Q ðk; nÞ ¼
jXj X pf Q ðk; nf Þ
ð15Þ
f ¼1
The idea of the SAA algorithm is to approximate the expectation in (15). A sample size R of n is taken from the scenario space X and P the expectation E½Q ðk; nÞ is replaced by 1=R Rr¼1 Q ðk; nr Þ and an approximate first stage problem (AFS) is developed. It is shown by Kleywegt et al. (2001), that the optimal objective value ^z and ^ to AFS, converge to the true optimal objective optimal solution k
value z and true optimal solution k of the stochastic program as the size of the scenario R increases. In our model, we assume that there is no cost associated with k so the cost coefficient cT of the first stage decision variable all zero. Therefore, we can write the AFS problem correspond to Eq. (10) as below,
8 > > > > < ^z ¼ mink ðAFSÞ > > k6B > > : kP0
1 R
R X
!
Q ðk; nr Þ ð16Þ
r¼1
Let AFSm be the mth solution to AFS, where m ¼ 1; 2; . . . ; M and ^m be the optimal total cost and optimal control limit let ^zm and k parameter to AFSm , then ^z ¼ minm¼1;2;...;M f^zm g is a point estimate for the optimal value z of the true total cost in FS problem, and ^ ¼ arg min^m f^zm g, i.e. k ^ such that zðk ^ Þ ¼ minm¼1;2;...;M f^zm ðk ^m Þg, k k
is an estimated control limit parameter k of the true FS problem. It is shown that average of M problems of AFSm problems, P zM ¼ M1 M ^m is an estimated lower bound on the optimal averm¼1 z
age total cost z (Kleywegt et al., 2001). An estimated upper bound on z is calculated by fixing any ^ , the control feasible solution to AFS. A usual practice is to fix k
limit parameter that satisfy minimum expected total cost out of m solutions, to solve the SS problem given in Eqs. (11a)–(11p). An estimated upper bound on minimum average total cost z of the FS problem is calculated by solving the SS problem for R0 ^ Þ ¼ different scenarios of n where R0 > R as follows, ^zR0 ðk PR0 r 1 ^ Q ðk ; n Þ. 0 R
r¼1
^ Þ and zM . The estimated gap for z is the difference between ^zR0 ðk Note that the estimated gap could be negative due to sampling ^ Þ < zM in some iterations error since it is possible to obtain ^zR0 ðk of the algorithm. Therefore, the estimated gap is set ^ Þ zM ; 0g. The algorithm is stopped when the gap is sufmaxf^zR0 ðk ficiently small. The steps of the algorithm are detailed below: Step 1: Generate M independent samples, each of size R, i.e., ðnR1 ; . . . ; nRm Þ for each instance of AFSm . For each m ¼ 1; 2; . . . ; M, solve the AFSm problem. Let ^zm be the optimal objective value ^m be the optimal control limit parameter (FS (total cost) and k decision) for the mth problem. Step 2: Calculate the average total cost zM of M problems solved P ^ ^m in step 1: zM ¼ M1 M m¼1 z and find the minimum of solutions, k which is the estimated optimal control limit parameter that ^ ¼ arg min^m f^zm g. minimizes the average total cost, i.e., k k
Step 3: Increase the number of samples R to R0 , where R0 R. ^ , solve the second stage problem Fix the first stage decision to k ^ Þ correspond to k ^ and calculate the expected total cost ^zR0 ðk 0
R X ^ Þ ¼ 1 ^ ; nr Þ: ^zR0 ðk Q ðk 0 R r¼1
^ Þ zM and Step 4: Calculate the optimality gap estimate, ^zR0 ðk 2 ^ 2 ^ þ r ^ 2z ^ ^ ¼r the estimated variance of the gap, r ^z ðk Þz ^z ðk Þ M R0
M
R0
where,
r^ ^2z 0 ðk^ Þ ¼ R
r^ 2zM ¼
R0 X 1 ^ ; nr Þ ^z 0 ðk ^ ÞÞ2 ðQðk R R ðR 1Þ r¼1 0
0
and
M X 1 2 ð^zm zM Þ : MðM 1Þ m¼1
^ Þ zM is less than 1% of the costs, then stop. Step 5: If gap ^zR0 ðk Otherwise go to Step 1. 4. Numerical analysis In this part of study, we present a set of numerical examples to provide insights on how the magnitude of the shift, special cause occurrence rate, k and ratio of Type 1 error cost to Type 2 error cost affect total inspection and production costs, i.e. total cost. We solve the problem in GAMS 23.5.2 (version 18.08.2010) with KNITRO solver. The numerical examples are run on an Intel(R) Core(TM)2 Duo 2.67 GHz CPU PC, with 2 GB of memory. It takes approximately 20 min to solve the problem for a production horizon consisting of 24 production periods, which correspond to a day’s production. The automobile company that we worked with in this project randomly inspects 10 vehicles at the end of each hour. The average defect rate of the process p0 , is 0.3 and rp0 is 0.15. The values of the parameters involved in the model are as follows: painting cost of each vehicle c = 200 $, re-painting cost of a defective vehicle cp = 600 $, inspection cost of vehicles c0 = 10 $, Type 1error cost c1 = 1500 $. Type 2 error cost is considered in six different levels (1500 $, 3000 $, 4500 $, 7500 $, 15,000 $, 30,000 $) in the experiments. Re-paining time of defective vehicles takes 4 h. Capacity of the paint process is 80 vehicles per hour. At the end of each day, 1200 painted vehicles should be ready to meet FA demand. The safety stock level allowed by the company is 10% of the daily demand. In the numerical study, we assume that the average defect rate of the process is same for every color. If the defect rate of the process highly depends on the color, it is also possible to design new charts for different colors. We solve the two stage stochastic model by SAA. The parameters used in SAA algorithm are: M ¼ 10; R ¼ 100 and R0 ¼ 200. We model the magnitude of the shift, d as a random variable in our model, and it represents the percent change in the process average defect rate when the process runs out-of-control: p1 ¼ p0 þ d. We assume that magnitude of shift d is normally distributed with mean ld and variance r2d . Hence the new defect rate of the process is p1 ¼ p0 þ drp0 . In our numerical study we consider effect of four different parameters on control limit parameter k, Type 1 and Type 2 error probability and total cost. These four parameters are (i) mean of the shift ld , (ii) variance of the shift r2d , (iii) occurrence of the special cause rate k and (iv) cost ratio of Type 2 error cost to Type 1 error cost. 4.1. Effect of the mean shift on k and costs Table 1 shows the effect of the mean shift, ld . Mean shift ld , is considered at five different levels and these levels are reported in
Please cite this article in press as: Gunay, E. E., & Kula, U. Integration of production quantity and control chart design in automotive manufacturing. Computers & Industrial Engineering (2016), http://dx.doi.org/10.1016/j.cie.2016.06.016
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E.E. Gunay, U. Kula / Computers & Industrial Engineering xxx (2016) xxx–xxx Table 1 Effect of mean shift on k, Type 1 and Type 2 probability.
ld
r2d
k
Cost ratio
k
UCL
Type 1 error a
Type 2 error b
1.00 1.40 1.60 1.80 2.20 2.30 2.40 2.50 2.60
0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
2 2 2 2 2 2 2 2 2
2.110 1.960 1.660 1.880 1.881 1.889 1.901 1.927 1.933
0.617 0.594 0.549 0.582 0.582 0.583 0.585 0.589 0.589
0.015 0.023 0.060 0.030 0.030 0.030 0.030 0.026 0.026
0.866 0.697 0.620 0.504 0.343 0.339 0.323 0.306 0.278
the first column of Table 1. In experiments in Table 1, we assume that r2d ¼ 0:03 and Type 2 to Type 1 error cost ratio is two. Table 1 shows mean shift of the random variable d increases k initially. As mean of d keeps increasing k starts to decrease in order to balance out Type 1 and Type 2 error costs. Fig. 2(a) shows that total cost increases as the magnitude of the average shift ld increases. The reason for this is that an increase on average defect rate increases the number of non-conforming units therefore more vehicles needed to be painted to meet the FA demand, which increases painting costs. The increase on number of vehicles painted increases the painting cost. Also, since all the newly painted vehicles are inspected when an out-of-control signal is reported, the increase on the total number of vehicles painted increase re-inspection cost. In addition to these two cost components, the increase on the number of defects increases re-painting cost. Fig. 2(b) shows that since Type 2 error cost is twice the Type 1 error cost, the p-chart parameter k is optimized to give a priority to reduce Type 2 error cost over Type 1 error cost. Both Table 1 and Fig. 2(b) shows that as ld increases to 1.6, Type 1 error cost increases since UCL initially increases to reduce the k, Type 2 error probability and its cost. As the mean shift ld continue to increase above 1.6, both Type 1 and Type 2 error costs decrease since both Type 1 and Type 2 error probabilities decrease.
(a)
Fig. 2(a) shows that even though Type 1 and Type 2 error costs decrease, the total cost function increases due to the increase on average defect rate. The increase on average defect rate ld increases the re-painting and defective production costs which cannot be compensated by the decrease in Type 1 and Type 2 error costs. 4.2. Effect of the shift variance on k and the costs In Table 2, we investigate the effect of the increase in variance of the shift d. Table 2 shows that as shift variance r2d increases, variation in process defect rate also increases. Therefore lower k values hence lower UCL, are determined to detect the special cause occurrences, which in turn decreases Type 2 error probability and increases Type 1 error probability. As Fig. 3(a) shows, total cost increases as the variance of the shift increases as expected. Fig. 3(b) shows that as shift variance r2d increases, Type 1 error cost is increased in order to minimize the Type 2 error cost. In experiments given in Table 3, we changed the mean and the variance of the shift simultaneously to see their effect on chart parameters and the total cost. Table 3 shows that as the mean and the variance of the shift d increase, k decreases to 1.80 and the corresponding UCL decreases to 0.57. As the mean and the variance of the shift continue to increase, higher k values are determined to decrease Type 1 error probability and its associated costs.
Table 2 Effect of shift variance on k, Type 1 and Type 2 probability.
r2d
ld
k
Cost ratio
k
UCL
Type 1 error a
Type 2 error b
0.003 0.03 0.12 0.24 0.33
1 1 1 1 1
0.05 0.05 0.05 0.05 0.05
2 2 2 2 2
2.110 2.040 1.937 1.890 1.870
0.617 0.606 0.591 0.584 0.581
0.015 0.018 0.024 0.030 0.030
0.866 0.803 0.654 0.500 0.432
(a) 410000
400000 395000
Total Cost
405000
Total Cost
400000 395000 390000
385000 380000 375000 365000 360000
380000 1
1.4
1.6
1.8
2.2
2.3
2.4
2.5
0.003
2.6
Mean shift
0.03
0.12
0.24
0.33
Shift variance
(b) 80000
65000 60000 55000 50000 45000 40000 35000 30000 25000 20000 15000
70000 60000
Cost
Cost
390000
370000
385000
(b)
405000
50000 40000 30000 20000
1
1.4
1.6
1.8
2.2
2.3
2.4
2.5
2.6
Mean shift Type1 cost Type2 cost Fig. 2. Effect of mean shift on total cost, Type 1 and Type 2 cost.
0.003
0.03
0.12
0.24
0.33
Shift variance Type1 cost Type2 cost Fig. 3. Effect of shift variance of on total cost, Type 1 and Type 2 cost.
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Table 3 Effect of changing mean and variance of shift on k, Type 1, Type 2 probability and total cost. k
Cost ratio
k
UCL
Type 1 error a
Type 2 error b
Total cost
0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2 2.1 2.2 2.3
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
2 2 2 2 2 2 2 2 2 2 2 2
2.50 2.27 2.07 2.04 2.02 1.99 1.80 2.00 2.11 2.27 2.32 2.43
0.68 0.64 0.61 0.61 0.60 0.60 0.57 0.60 0.62 0.64 0.65 0.66
0.0007 0.0020 0.0070 0.0075 0.0080 0.0090 0.0200 0.0090 0.0050 0.0030 0.0020 0.0010
0.900 0.851 0.780 0.730 0.672 0.613 0.606 0.590 0.570 0.560 0.550 0.530
394,205 396,828 405,570 408,755 411,064 421,267 426,146 430,574 434,681 436,086 439,305 440,343
Total Cost
r2d
580000 530000 480000 430000 380000 0.05
0.07
0.1
0.15
0.2
Special cause rate
(b)
160000 140000 120000
Cost
ld
(a)
100000 80000 60000 40000 20000 0.05
440000
0.07
0.1
0.15
0.2
Special cause rate Type1 cost Type2 cost
Total cost 420000
Fig. 5. Effect of special cause rate k on total cost, Type 1 and Type 2 cost. 0,6
400000 0,4 0,5
1,0
0,2 1,5
Variance of shift
2,0
Mean of shift
Fig. 4. Effect of changing mean and variance of shift on total cost.
Fig. 4 plots the change in total cost as the mean and the variance of the shift increase: As mean and the variance of the shift increase, the number of non-conforming vehicles and hence the number of re-painted vehicles, therefore total painting and re-painting cost increases.
Table 5 Effect of changing the cost ratio of Type 2 error cost to Type 1 error cost. Cost ratio
ld
r2d
k
k
UCL
Type 1 error a
Type 2 error b
1 2 3 5 10 20
1 1 1 1 1 1
0.03 0.03 0.03 0.03 0.03 0.03
0.05 0.05 0.05 0.05 0.05 0.05
2.350 2.110 2.090 1.990 1.931 1.629
0.652 0.617 0.614 0.599 0.590 0.544
0.001 0.015 0.018 0.023 0.026 0.072
0.918 0.866 0.860 0.838 0.727 0.642
(a) 800000
Table 4 Effect of special cause rate on k, Type 1 and Type 2 probability.
700000 600000 500000 400000 300000 1
2
3
5
10
20
10
20
Cost ratio
(b) 401700 351700 301700
Cost
In experiments given in Table 4, we investigate the effect of special cause rate k. Experiments in Table 4 show that as k increases, the control limit parameter k decreases. The reason for this decrease in k can be explained as follows: As the probability of the process running out-of-control increases with k, the lower values of k must be set to detect the shift, hence to decrease Type 2 error probability. For instance when k is 0.05, control limit parameter k is set to 2.11 and Type 2 error probability is 0.866. As k increases to 0.07, the control limit parameter k decreases to 1.816 and Type 2 error probability decreases to 0.76. Fig. 5(a) shows that as k increases, total cost increases since the number of times the process runs out-of-control state increases on average. Fig. 5(b) shows that as k increase both Type 1 and Type 2 error probabilities increase.
Total Cost
4.3. Effect of special cause rate k on k and the costs
251700 201700 151700 101700
k
ld
r2d
Cost ratio
k
UCL
Type 1 error a
Type 2 error b
0.05 0.07 0.10 0.15 0.20
1 1 1 1 1
0.03 0.03 0.03 0.03 0.03
2 2 2 2 2
2.110 1.816 1.688 1.560 1.460
0.617 0.572 0.553 0.534 0.519
0.015 0.035 0.044 0.060 0.070
0.866 0.760 0.738 0.710 0.670
51700 1700 1
2
3
5
Cost ratio Type1 cost Type2 cost Fig. 6. Effect of cost ratio on total cost, Type 1 and Type 2 costs.
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E.E. Gunay, U. Kula / Computers & Industrial Engineering xxx (2016) xxx–xxx
4.4. Change on the cost ratio of Type 2 error cost to Type 1 error cost, cost ratio Table 5 shows how the ratio of Type 2 error cost to Type 1 error cost affects the chart parameter k and the costs. As the Type 2 error cost increase, lower k values are selected to decrease Type 2 error cost. Since usually Type 2 error cost is higher than Type 1 error cost, our examples show that commonly used 3r control limits are quite far from being optimal. Exponential increase on cost ratio exponentially increase total cost and Type 2 error cost as seen in Fig. 6(a) and (b). 5. Conclusion In our study we present a two-stage stochastic model to determine control chart parameter, upper control limit (UCL) and production quantity of vehicles painted for each sampling period Xt. The model is solved by sample average approximation (SAA) algorithm. A numerical study is performed to investigate the behavior of the model according to various parameters: mean of the shift, variance of the shift, occurrence rate of the special cause, cost ratio of Type 2 error to Type 1 error. Our numerical study provides the following insights: (i) An increase on the mean defect rate increases both the total cost and the total production quantity. (ii) Effect of an increasing process variance to the control limit parameter k is insignificant. (iii) Frequency of special cause occurrences significantly affects the total cost. (iv) All the experiments show that the commonly used 3r control limits in practice are wider than required, which increases Type 1 error risk. As a future study, determination of optimal sample size n and sampling interval h along with parameter k may be an interesting research direction. Another research direction is to study effect of several types of special cause occurrences on the control chart parameter k and the total quality and production related costs. Also, since total cost and total production quantity are very sensitive to the special cause occurrence rates, effect of different probability distributions of special cause occurrences may be explored. References Aslam, M., Azam, M., Khan, N., & Jun, C.-H. (2015). A mixed control chart to monitor the process. International Journal of Production Research, 53(15), 4684–4693. http://dx.doi.org/10.1080/00207543.2015.1031354. Ben-Daya, M., & Rahim, M. A. (2000). Effect of maintenance on the economic design of x-control chart. European Journal of Operational Research, 120(1), 131–143. http://dx.doi.org/10.1016/S0377-2217(98)00379-8. Bouslah, B., Gharbi, A., & Pellerin, R. (2013). Joint production and quality control of unreliable batch manufacturing systems with rectifying inspection. International Journal of Production Research (April), 1–15. http://dx.doi.org/ 10.1080/00207543.2012.746481. Bouslah, B., Gharbi, A., & Pellerin, R. (2015). Integrated production, sampling quality control and maintenance of deteriorating production systems with AOQL constraint. Omega. http://dx.doi.org/10.1016/j.omega.2015.07.012.
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Please cite this article in press as: Gunay, E. E., & Kula, U. Integration of production quantity and control chart design in automotive manufacturing. Computers & Industrial Engineering (2016), http://dx.doi.org/10.1016/j.cie.2016.06.016