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Economic design of a Vp X̄ chart
Int. J. Production Economics 74 (2001) 191}200
Economic design of a Vp XM chart Maysa S. De Magalha es *, Eugenio K. Epprecht , Antonio F.B. Costa Department of Industrial Engineering, Pontifn& cia Universidade Cato& lica do Rio de Janeiro, Rio de Janeiro, RJ 22453-900, Brazil Department of Production, FEG-UNESP, Guaratingueta& , SP 12500-000, Brazil
Abstract We develop an economic model for XM control charts having all design parameters varying in an adaptive way, that is, in real time considering current sample information. In the proposed model, each of the design parameters can assume two values as a function of the most recent process information. The cost function is derived and it provides a device for optimal selection of the design parameters. Through a numerical example one can foresee the savings that the developed model possibly provides. 2001 Elsevier Science B.V. All rights reserved. Keywords: Control charts; Process control; Variable design parameters; Expected cost
1. Introduction Control charts are widely used to maintain and establish statistical control of a process. Control charts were introduced by W. Shewhart, a statistician of Bell Laboratories, in the 1920s. Since then many improvements have been made in the Shewhart control charts. The design and operation of Shewhart control charts require the determination of three design parameters: the sample size (n), the sampling interval (h), and the width coe$cient of control limits (k) (number of standard deviations of the sample statistic separating each control limit from the center line). The usual practice is to keep the design parameters constant during the production process, independent of whether the design adopted for the control chart is statistical or economic. These kinds
* Corresponding author. Tel.: #55-24-528-8144; fax: #5524-523-3779. E-mail address: [email protected] (M. S. De Magalha es).
of charts are referred to in the literature as "xed parameters control charts (Fp control charts). However, this classical control scheme is, usually, slow in signaling small to moderate shifts in the process parameter being controlled. Consequently, several alternatives have been proposed to improve the performance of Fp control charts. Also, the technological evolution motivated the development of more #exible, more elaborated Shewhart control charts, in which one or more design parameters are allowed to vary in an adaptive manner, that is, in real time based on current sample information, during production. The models proposed up to now have shown that control charts with one or more variable design parameters are more e$cient than Fp control charts in detecting shifts of the process parameter being controlled. The usual approaches to control chart design are statistical or economic, no matter if the design parameters are kept "xed or not. In the statistical design, the design parameters are chosen based on statistical considerations, such as probabilities of type I and type II errors. In the economic design,
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Nomenclature 1/
n n h h
w w k k ;=¸ G ¸=¸ G ;C¸ G
the mean time that the process is in control shift in the process mean to $ indicator variable (1, if production continues during searches, 0 otherwise) indicator variable (1, if production continues during repair, 0 otherwise) the smaller sample size the larger sample size the length of the larger sampling interval the length of the smaller sampling interval the wider width of the warning limits the narrower width of the warning limits the wider width of the control limits the narrower width of the control limits upper warning limit when the design parameters (n , h , k ) are being used; G G G i"1, 2 lower warning limit when the design parameters (n , h , k ) are being used; G G G i"1, 2 upper control limit when the design parameters (n , h , k ) are being used; G G G i"1, 2
the choice of design parameters are based on a cost function. In the statistical design approach, the work of Reynolds et al. [1] introduced the idea of varying the XM chart sampling interval as a function of what is observed from the process. A variable sampling interval (VSI) was also considered by Runger and Pignatiello [2], Amin and Miller [3], Runger and Montgomery [4], Reynolds et al. [5] and Reynolds [6]. The variable sampling interval feature was extended to Cusum and EWMA charts (see [7,8]).
lower control limit when the design parameters (n , h , k ) are being used; G G G i"1, 2 C cost per unit time of production when the process is in control C cost per unit time of production when the process is out of control > cost per false alarm a "xed cost of sampling per sample b variable cost of sampling per unit sampled = cost of "nding and repairing an assignable cause ¹ expected search time associated with a false alarm ¹ expected time to discover the assignH able cause ¹ expected time to eliminate the assignHH able cause G mean time to take a sample and chart it s("E(N)) the average number of samples drawn while the process is in control s the average number of samples drawn while the process is out of control n the average sample size while the process is in control n the average sample size while the process is out of control h the average time between samples while the process is o! target AA¹S the adjusted average time to signal
¸C¸ G
Recently, Baxley [9] presented an application of EWMA chart with VSI for Monsanto's nylon "ber plant in Pensacola, Florida. The size of the samples was the second design parameter to be considered variable (see [10,11]). Subsequently, both parameters (sample size and sampling interval) were made variable (see [12,13]). Rendtel [14] considered Cusum schemes with variable sampling intervals and sample sizes. Finally, all design parameters were considered variable (see [15]).
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The economic design of XM chart with variable sample size (VSS) was studied by Flaig [16] and Park and Reynolds [17]. Park and Reynolds proposed an economic model for XM chart with VSS when the process is subject to the occurrence of several assignable causes. Recently, Das et al. [18] developed a cost model for optimal dual sampling interval (DSI) policies with and without run rules. Also, Das and Jain [19] proposed a further generalization of VSI policy for XM charts in which the sampling intervals are treated as random variables and the sample sizes are considered a function of the sampling intervals. In this paper, we propose an economic model for XM control charts when all design parameters are variable (Vp XM chart). The design parameters are allowed to vary in real time based on current sample information. A model is developed, providing a cost function which represents the cost per time unit of controlling the quality of a process through a Vp XM chart. As the cost function is a function of the design parameters of the control chart, it provides a device for optimal selection of the design parameters. Recently, Costa [15] considered the statistical design of Vp XM charts and showed that considerable performance improvements (in terms of ATS) can be achieved over other XM control charts. That study motivated us to consider an economic model for XM chart having all design parameters variable. The paper is organized as follows. Section 2 presents the description of a Vp XM chart. The process model and the cost function are presented in Section 3. Section 4 contains the expressions of the policydependent variables. Section 5 presents a numerical example. Conclusions are placed in Section 6. 2. Description of the XM chart with variable parameters Suppose that an XM control chart allowing all design parameters to vary is employed to monitor a process whose quality characteristic of interest (say, X) is normally distributed with mean and variance (which are assumed to be known). The target value of the process mean is represented by
193
, that is, when the process is in control " . To use a control chart the user should specify: the sample size (n), the sampling interval (h) and the width coe$cient of the control (or action) limits (k). These parameters are called design parameters of a control chart. In the Fp XM chart, we denote the design parameters by n , h and k and the XM values from the samples are plotted on an XM chart with upper and lower control limits given by $k /(n and centerline . The Vp XM chart is a modi"cation of the Fp XM chart. The design parameters of the Vp XM chart can, in the general case, be di!erent for each sampled value of XM . However, in the model developed here, each of these parameters can assume two values as function of the most recent process information. The position of each sample point on the chart establishes the size of the next sample, the instant of its sampling and the width coe$cient of the control limits. See Fig. 1. We denote the sample size values by n , n , the sampling interval values by h , h and the coe$cient values used in determining the width of the control limits by k , k . For an XM chart with Vp, suppose that ¸C¸ and ;C¸ represent, respectively, the lower and upper control limits. The interval (¸C¸, ;C¸) is partitioned into three distinct intervals or regions: (¸C¸, ¸=¸); (¸=¸, ;=¸); (;=¸, ;C¸); with ¸C¸(¸=¸(;=¸( ;C¸. Here ¸=¸ and ;=¸ represent, respectively, the lower and upper warning limits of the XM chart. The region de"ned by (¸C¸, ¸=¸) and (;=¸, ;C¸) is called warning region of the chart while the region (¸=¸, ;=¸) is the central region. The region above ;C¸ and below ¸C¸ is denominated by action region of the chart. As in the case of control charts with Fp, a signal is produced when a point falls outside the control limits, that is, in the action region. Note that for each sample point xN , two possibilities will be provided for the G warning and control limits (that is, $w /(n G G and $k /(n , i"1, 2, respectively) because G G each design parameters can assume two values. Here, w and w denote the width coe$cients of the warning limits. Since the control limits are di!erent, then to avoid the use of two control charts, one for the design parameters (n , h , w , k ) and other for the design parameters (n , h , w , k ), one can
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Fig. 1. Vp XM chart.
construct the Vp XM chart with two scales, one on the left side for samples of size n and other on the right side for samples of size n , in such a way that warning and action limits coincide on the two sides, as illustrated on Fig. 1. Vp policy. The design parameters should be chosen such that h 'h , n (n , k 'k and w 'w . If the sample point falls in the central region, then the next sample should be small; that is, with size n , it should be sampled after a long time interval (h ) and the factor used in determin ing the width of the action limits should be wide (k ). On the other hand, if the sample point falls in the warning region, then: the next sample should be large (n ), it should be sampled after a short time interval (h ) and the factor used in determining the width of the action limits should be narrow (k ). Finally, if the sample statistic falls in the action region an investigation should be initiated to verify if, actually, the process is out of control (that is, if the mean has changed from " to " $, '0) or has occurred just a false alarm (that is, the process is still in control). If, in fact, the process is out of control, a corrective action should be undertaken. Size of the xrst sample. The size of the "rst sample, that is taken from the process when it is just starting or after a false alarm, is chosen at random. If the sample was chosen to be large (small) it should be sampled after a short (long) time interval. Probability p . During the in-control period all samples, including the "rst one, have probability p of being small and (1!p ) of being large,
where p "P(Z(w Z(k )"P(Z(w Z(k ) and Z&N(0, 1) (1)
3. The process model and the cost function An economic modelling approach for the design of a Vp XM chart is described in the rest of the paper. The hourly cost function is derived. This function provides a device for the optimal economic selection of the design parameters n , n , h , h , w , w , k , and k . In the development of the proposed model, assumptions about the production process are made and some important concepts are de"ned. After that the cost function formulation is considered. 3.1. The process model Assumptions. These assumptions characterize the class of production process to be analyzed. We assume that the samples are independent, and that the process starts in a state of statistical control with mean " and standard deviation . Also, the occurrence of the assignable cause results in a shift in the process mean from to $. The length of time the process stays in control is exponentially distributed with mean 1/. The process is not self-corrective. During the search for an assignable cause the process can continue in operation or not. This possibility is represented by the indicator
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variable ( "1, if production continues during search and "0, otherwise). During repair the process can continue functioning or not. The indicator variable expresses this possibility ( "1, if production continues during repair and "0, otherwise). Finally, the parameters , , and are assumed to be known and the parameters to be determined are n , n , h , h , w , w , k , and k . Costs. Several costs are associated to the quality control of a production process. Five costs are considered: Costs of sampling and inspection (C ), the cost of investigating false alarms (C ), the cost of "nding and repairing the assignable cause (C ), costs of producing nonconformities while the process is operating in control (C ) and out of control (C ). These are, in general, random variables. Process cycle. The production cycle consists of "ve periods. The in control period (¹ ) is the period that starts from the beginning of the production process until occurrence of the assignable cause, including interruptions for false alarms. The out of control period (¹ ), is the time since process shift until a signal. The period of analysis (¹ ), is the time to analyze a sample and chart the result. The investigation period (¹ ), is the time to "nd an assign able cause when it exists. The repairing period (¹ ), is the time to repair the process. The cycle is the sum of these periods. Hence the process cycle is de"ned as the time since the beginning of production until elimination of the assignable cause. 3.2. The cost model The expected cost per time unit (EC¹;) is the ratio of the expected cost per cycle (E(C)) to the expected cycle time (E(¹)), that is EC¹;"E(C)/E(¹).
(2)
In the computation of E(C) and E(¹), the expressions for some variables are policy-dependent (that is, they depend on the Vp policy adopted), and they are placed in the next section. The expected cost per cycle. E(C) consists of the sum of the costs incurred while the process is in control and out of control. Hence, the components of E(C) are obtained as follows:
195
(i) The expected cost per cycle due to nonconformities while the process is in control (E(C )) and out of control (E(C )) is given by 1 E(C )#E(C )" C #C [AA¹S#E(¹ ) # ¹ # ¹ ]. (3) H HH Here, C and C represent the costs per hour due to nonconformities produced while the process is in control and out of control, respectively. The mean time that the process stays in control is equal to 1/. The adjusted average time to signal (AA¹S) is the mean time interval since a process shift until an alarm occurs. E(¹ ) is equal to nG, where G is the time to take a sample of size 1 and chart it, and n represents the average sample size while the process is out of control. The expressions for AA¹S and n are in Section 4. ¹ and ¹ are the expected times H HH to discover the assignable cause and to repair the process, respectively. (ii) The expected cost due to false alarms (E(C )) is given by E(C )">E(F). (4) Here > represents the cost per false alarm and E(F) is the expected number of false alarms. E(F) is policy-dependent and then its expression is given in Section 4. (iii) The expected cost of "nding and repairing an assignable cause when one exists (E(C )): E(C )"=. (5) This quantity includes any downtime that is appropriate and is assumed policy independent. (iv) The expected cost of sampling and inspection (E(C )) is given by E(C )"(a#bn)s#(a#bn)s, (6) where a is the "xed cost per sample and b is the variable cost per unit sampled. Here, n (n) represents the average sample size while the process is in control (out of control). And s (s) represents the average number of samples drawn while the process is in control (out of control). The expressions for n, n, s, and s are given in Section 4 due to their dependence on the Vp policy.
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Note that the "rst and second terms of Eq. (6) gives, respectively, the expected cost of sampling while the process is in control and out of control. Then adding Eqs. (3)}(6), the expected cost per cycle is obtained: 1 E(C)" C #C [AA¹S#nG# ¹ # ¹ ] H HH #>E(F)#=#(a#bn)s#(a#bn)s. (7) The expected cycle time. The expected cycle time is the sum of the expected time of each period of cycle, then: E(¹)"E(¹ )#E(¹ )#E(¹ )#E(¹ )#E(¹ ) 1 " #(1! )E(¹ )#AA¹S #nG#¹ #¹ . (8) H HH The expression (1! )E(¹ ) is the part of E(¹ ) due to the fact that the in control period can include interruptions for false alarms. The indicator variable takes into account this possibility. The expected time searching for false alarms is equal to the expected search time associated with a false alarm (¹ ) times the expected number of false alarms (E(F)). As the expression for E(¹ ) is depen dent of the Vp policy, it is placed in Section 4. The AA¹S counts for the mean time since process shift until an alarm is given. nG counts the time to analyze a sample and chart the result. The expected time to "nd an assignable cause (¹ ) and repair it H (¹ ) are not dependent of whether the production HH process stops or not, then they always count to the expected cycle length. From Eqs. (2), (7) and (8), we get the expression for EC¹;:
random variable F depends on the number of samples (N) taken before the process shift. The expected number of false alarms (E(F)) is given by E(F)"( p # (1!p ))s, where "P(Z'k ) is the probability of type G G I error when the chart being used has design parameters n , h , k , for i"1, 2. Here, p is the probG G G ability of a sample to be small during the in control period, and (1!p ) is the probability of a sample to be large during the in control period. Also, s("E(N)) represents the average number of samples taken while the process is in control. Average number of samples taken while the process is in control. Let N be the number of samples taken before the process shift. Suppose that the assignable cause has occurred between the jth and the (j#1)th samples, that is, the process mean has shifted from to $ between these two samples. Then j samples were taken during the in control period, that is, N"j. To have j samples before the shift is equivalent to the assignable cause occurring between the sampling times ¹ and H ¹ , then: H> E(N)" jP(N"j)" jP(¹ (¹(¹ ). H H> H H Note that the time that the process stays in control (¹) has exponential distribution with parameter . Taking into account the memoryless property of the exponential distribution and that the sampling interval (H) is a random variable with probability distribution given by P(H"h )"p and P(H"h )"1!p , we get e\HF p #e\HF (1!p ) . E(N)"s" 1!e\HF p !e\HF (1!p ) Average time since occurrence of an assignable cause until an alarm occurs. This average time is,
(1/)C #C [AA¹S#nG# ¹ # ¹ ]#>E(F)#=#(a#bn)s#(a#bn)s H HH EC¹;" . (1/)#(1! )E(¹ )#AA¹S#nG#¹ #¹ H HH 4. Expressions of the policy-dependent variables Expected number of false alarms. Let F be the number of false alarms incurred in a cycle. The
usually, called adjusted average time to signal (AA¹S). Let ¹ be the time since occurrence of an assignable cause until an alarm is given and let A be a random variable representing the length of the
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197
Fig. 2. The relationship between ¹ , A, ¹, R, S.
interval in which the shift occurs. Also, let R be the time from the process shift until the "rst sample after the shift, S the time since the "rst sample after the shift until an alarm and let ¹ represent the time from last sample before the shift until the process shift. The relationship between ¹ , A, ¹, R, and S is shown in Fig. 2. From Fig. 2, we see that ¹ "R#S and then AA¹S"E(¹ )"E(R)#E(S). Determination of E(R). As Reynolds et al. [1], we assume that P(A"h ) is proportional to the length G of the interval (A) times the probability of an interval of this length occurring. This means that p h , P(A"h )" p h #(1!p )h (1!p )h P(A"h )" . p h #(1!p )h Since E(R) can be written as E(R)"E(E(RA)) and E(E(RA))"E(E((h !¹ )A)) G " E((h !¹)A)P(A"h ), G G G we, therefore, get
1!e\HF (1#h ) P(A"h ) E(R)" h ! (1!e\HF )
1!e\HF (1#h ) P(A"h ). # h ! (1!e\HF ) The expression above gives the expected time from the process shift until the "rst sample after the shift.
Determination of E(S). The expected value of S depends on the position of the "rst sample point after shift (B). The probability of plotting this point in the central region (B"B ), warning region (B"B ), or action region (B"B ), depends on the length of the interval in which the shift occurs. This leads to P(B"B )"P(B"B A"h )P(A"h ) # P(B"B A"h )P(A"h ) "p P(A"h )#p P(A"h ), P(B"B )"p P(A"h )#p P(A"h ) and P(B"B )"1!P(B"B )!P(B"B ), where p "P(;(w ;&N((n , 1)), p "P(w (;(k ;&N((n , 1)), p "P(;(w ;&N((n , 1)), p "P(w (;(k ;&N((n , 1)). When the "rst sample point after the shift falls in the central region (warning region) let (SB"B )"¹ (let (SB"B )"¹ ). Therefore, E(S)"E(¹ )P(B"B )#E(¹ )P(B"B ), where ¹ (¹ ) is the time since the "rst sample after the shift, when the "rst sample after the shift falls in the central region (when the "rst sample after the shift falls in the warning region), until an alarm occurs.
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If the xrst sample after the shift falls in the central region, the time from the "rst sample after the shift until an alarm occurs is given by + ¹ " >. G G The expected value of ¹ is + E(¹ )"E > G G + "E E > M G G "E(M )E(<). Here, the random variable M represents the number of points that falls in the central region until an alarm occurs. The variable M is distrib uted geometrically with parameter (1!p ), where p is the probability to obtain a point in the central region. This probability is given by
p "p #p pG\p , G where the p s were given above. GH The expected value of M is given by 1 , E(M )" 1!p where p is given above. The random variables > s are independent and G identically distributed as <. The random variable < represents the length of time to obtain a point outside the warning region since the last point that fell in the central region. The probability function of < is given by P(<"h )"p #p "1!p P(<"h #ih )"p pG\p #p pG\p "p pG\(1!p ) where i"1, 2,2 . Calculating the expected value of < we obtain that p . E(<)"h #h 1!p
Then, substituting E(M ) and E(<) in E(¹ ), we get [h (1!p )#h p ] . E(¹ )" 1!p !p #p p !p p In a similar way, if the xrst sample after the shift falls in the warning region, we get [h (1!p )#h p ] E(¹ )" . 1!p !p #p p !p p The determination of n, n, h, and s. Calculations of n, n, h, and s are straightforward: n"n p #n (1!p ), n"n p ()#n (1!p ()), and h"h p ()#h (1!p ()), where p was given before by Eq. (1), and p ()"P(!w !(n (Z(w !(n !k G G G G G ! (n (Z(k !(n ), G G G for i"1, 2. Finally, expected cycle length while the process is off target s" average time between samples while the process is off target
AA¹S#nG# ¹ # ¹ H HH . " h 5. Numerical example An application of the developed economic model is demonstrated for the numerical example of Lorenzen and Vance [20]. It is an example of foundry operations, where periodic samples of molten iron are taken to monitor the carbon}silicate content of the casting. A high carbon}silicate content results in casting of low tensile strength. The values of the input variables are G"¹ "¹ " hours; H ¹ " hours; 1/"50; C "$114.24/hour; HH C "$949.20/hour; >"="$977.40; a"0; b"$4.22; "1; "0. In order to accomplish the optimization of the unit cost function, the following constraints were considered: n )n ; n *5; n *1; 0.1)h )h ; h *1; w )w ; w and w *0.1; k )k ; k and k *1. A nonlinear constrained optimization algo rithm (MATLAB toolbox software) was applied to
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Table 1 Comparison between the economic performance of Vp and Fp XM control charts Shift
Optimum parameters for Vp (n , n , h , h , w , w , k , k )
0.5
Opt. ECTU for Vp
Opt. ECTU and opt. par. for Fp (n , h , k )
236.70
248.77 (17, 2.77, 1.87)
203.32
216.27 (12, 2.28, 2.20)
185.88
197.65 (9, 1.98, 2.43)
175.44
185.75 (7, 1.77, 2.60)
169.41
177.47 (5, 1.45, 2.71)
165.62
171.88 (5, 1.54, 2.88)
162.99
168.66 (5, 1.59, 3.06)
(11, 15, 2.15, 0.1, 1.15, 1.08, 2.62, 1.97) 0.75 (7, 9, 1.70, 0.1, 1.23, 1.18, 3.03, 2.22) 1.0 (5, 6, 1.43, 0.1, 1.30, 1.26, 3.26, 2.38) 1.25 (3, 5, 1.25, 0.1, 1.43, 1.39, 3.44, 2.47) 1.5 (3, 5, 1.11, 0.1, 1.71, 1.6, 3.62, 2.44) 1.75 (2, 5, 1, 0.1, 1.9, 1.8, 3.65, 2.42) 2 (2, 5, 1, 0.1, 2.06, 1.9, 3.56, 2.42)
the cost function. We considered several shifts of the mean: "0.5; 0.75; 1.0; 1.25; 1.5 and 2. Di!erent starting vectors were used, in the optimization process, to "nd the minimum value of the ECTU and the corresponding optimal design parameters. Since, for each speci"c shift, all of the iterations converged to the same solution, independent of the given starting vectors, we have evidence that, perhaps, the global mimimum was attained. We have, nevertheless, obtained a better control chart, as will be seen next. In order to compare the results, we computed also the optimal design parameters and the ECTU for the economic model for the Fp XM chart. Table 1 presents the results. From this table, we can observe that, for small to moderate shifts, there is a decrease of more than 5% on ECTU for the economic model with variable parameters when compared to the Fp economic model. In other words, if an economic model with Vp is used, then for shifts from 0.5 to 1.25 a saving of more than $20 000 per year is obtained and for shifts varying from 1.5 to 2.0 a saving of more than $10 000 per year is obtained, supposing that the process is operating 8 hours per day during 5 days per week.
6. Conclusions A quality cost model for an XM chart with all design parameters varying in an adaptive way was developed. This model makes possible the optimal economic selection of the design parameters of the chart. We could check, through a numerical example, that the economic model for a XM chart with Vp reduces the quality cost of a process. The economic model also provides a way to analyze for sensitivity of some parameters and to have information on how the cost is a!ected. Also, it allows for comparisons of several types of control charts. References [1] M.R. Reynolds Jr., R.W. Amin, J.C. Arnold, J.A. Nachlas, XM Charts with variable sampling intervals, Technometrics 30 (1988) 181}192. [2] G.C. Runger, J.J. Pignatiello Jr., Adaptive sampling for process control, Journal of Quality Technology 23 (1991) 135}155. [3] R.W. Amin, R.W. Miller, A robustness study of XM charts with variable sampling intervals, Journal of Quality Technology 25 (1993) 36}44.
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[4] G.C. Runger, D.C. Montgomery, Adaptive sampling enhancements for Shewhart control charts, IIE Transactions 25 (1993) 41}51. [5] M.R. Reynolds Jr., J.C. Arnold, J.W. Baik, Variable sampling interval XM charts in the presence of correlation, Journal of Quality Technology 28 (1996) 1}28. [6] M.R. Reynolds Jr., Shewhart and EWMA variable sampling interval control charts with sampling at "xed times, Journal of Quality Technology 28 (1996) 199}212. [7] M.R. Reynolds Jr., R.W. Amin, J.C. Arnold, Cusum charts with variable sampling intervals, Technometrics 32 (1990) 371}384. [8] M.S. Saccuci, R.W. Amin, J.M. Lucas, Exponentially weighted moving average control schemes with variable sampling intervals, Communication in Statistics } Simulation and Computation 21 (1992) 627}657. [9] R.V. Baxley Jr., An application of variable sampling interval control charts, Journal of Quality Technology 27 (1995) 275}282. [10] S.S. Prabhu, G.C. Runger, J.B. Keats, An adaptive sample size XM chart, International Journal of Production Research 31 (1993) 2895}2909. [11] A.F.B. Costa, XM charts with variable sample size, Journal of Quality Technology 26 (1994) 155}163.
[12] S.S. Prabhu, D.C. Montgomery, G.C. Runger, A combined adaptive sample size and sampling interval XM control scheme, Journal of Quality Technology 26 (1994) 164}176. [13] A.F.B. Costa, XM with variable sample size and sampling intervals, Journal of Quality Technology 29 (1997) 197}204. [14] U. Rendtel, Cusum-schemes with variable sampling intervals and sample sizes, Statistical Papers 31 (1990) 103}118. [15] A.F.B. Costa, XM Charts with variable parameters, Journal of Quality Technology 31 (1999) 408}416. [16] J.J. Flaig, Adaptive control charts, in: J.B. Keats and D.C. Montgomery (Eds.), Statistical Process Control in Manufacturing, Marcel Dekker, New York, 1991, pp. 111}122. [17] C. Park, M.R. Reynolds Jr., Economic design of a variable sample size XM -chart, Communications in Statistics-Simulation and Computation 23 (1994) 467}483. [18] T.K. Das, V. Jain, A. Gosavi, Economic design of dualsampling-interval policies for XM charts with and without run rules, IIE Transactions 29 (1997) 497}506. [19] T.K. Das, V. Jain, An economic design model for XM charts with random sampling policies, IIE Transactions 29 (1997) 507}518. [20] T.J. Lorenzen, L.C. Vance, The economic design of control charts: A uni"ed approach, Technometrics 28 (1986) 3}10.
Economic design of a Vp XM chart Maysa S. De Magalha es *, Eugenio K. Epprecht , Antonio F.B. Costa Department of Industrial Engineering, Pontifn& cia Universidade Cato& lica do Rio de Janeiro, Rio de Janeiro, RJ 22453-900, Brazil Department of Production, FEG-UNESP, Guaratingueta& , SP 12500-000, Brazil
Abstract We develop an economic model for XM control charts having all design parameters varying in an adaptive way, that is, in real time considering current sample information. In the proposed model, each of the design parameters can assume two values as a function of the most recent process information. The cost function is derived and it provides a device for optimal selection of the design parameters. Through a numerical example one can foresee the savings that the developed model possibly provides. 2001 Elsevier Science B.V. All rights reserved. Keywords: Control charts; Process control; Variable design parameters; Expected cost
1. Introduction Control charts are widely used to maintain and establish statistical control of a process. Control charts were introduced by W. Shewhart, a statistician of Bell Laboratories, in the 1920s. Since then many improvements have been made in the Shewhart control charts. The design and operation of Shewhart control charts require the determination of three design parameters: the sample size (n), the sampling interval (h), and the width coe$cient of control limits (k) (number of standard deviations of the sample statistic separating each control limit from the center line). The usual practice is to keep the design parameters constant during the production process, independent of whether the design adopted for the control chart is statistical or economic. These kinds
* Corresponding author. Tel.: #55-24-528-8144; fax: #5524-523-3779. E-mail address: [email protected] (M. S. De Magalha es).
of charts are referred to in the literature as "xed parameters control charts (Fp control charts). However, this classical control scheme is, usually, slow in signaling small to moderate shifts in the process parameter being controlled. Consequently, several alternatives have been proposed to improve the performance of Fp control charts. Also, the technological evolution motivated the development of more #exible, more elaborated Shewhart control charts, in which one or more design parameters are allowed to vary in an adaptive manner, that is, in real time based on current sample information, during production. The models proposed up to now have shown that control charts with one or more variable design parameters are more e$cient than Fp control charts in detecting shifts of the process parameter being controlled. The usual approaches to control chart design are statistical or economic, no matter if the design parameters are kept "xed or not. In the statistical design, the design parameters are chosen based on statistical considerations, such as probabilities of type I and type II errors. In the economic design,
0925-5273/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 1 ) 0 0 1 2 6 - 8
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Nomenclature 1/
n n h h
w w k k ;=¸ G ¸=¸ G ;C¸ G
the mean time that the process is in control shift in the process mean to $ indicator variable (1, if production continues during searches, 0 otherwise) indicator variable (1, if production continues during repair, 0 otherwise) the smaller sample size the larger sample size the length of the larger sampling interval the length of the smaller sampling interval the wider width of the warning limits the narrower width of the warning limits the wider width of the control limits the narrower width of the control limits upper warning limit when the design parameters (n , h , k ) are being used; G G G i"1, 2 lower warning limit when the design parameters (n , h , k ) are being used; G G G i"1, 2 upper control limit when the design parameters (n , h , k ) are being used; G G G i"1, 2
the choice of design parameters are based on a cost function. In the statistical design approach, the work of Reynolds et al. [1] introduced the idea of varying the XM chart sampling interval as a function of what is observed from the process. A variable sampling interval (VSI) was also considered by Runger and Pignatiello [2], Amin and Miller [3], Runger and Montgomery [4], Reynolds et al. [5] and Reynolds [6]. The variable sampling interval feature was extended to Cusum and EWMA charts (see [7,8]).
lower control limit when the design parameters (n , h , k ) are being used; G G G i"1, 2 C cost per unit time of production when the process is in control C cost per unit time of production when the process is out of control > cost per false alarm a "xed cost of sampling per sample b variable cost of sampling per unit sampled = cost of "nding and repairing an assignable cause ¹ expected search time associated with a false alarm ¹ expected time to discover the assignH able cause ¹ expected time to eliminate the assignHH able cause G mean time to take a sample and chart it s("E(N)) the average number of samples drawn while the process is in control s the average number of samples drawn while the process is out of control n the average sample size while the process is in control n the average sample size while the process is out of control h the average time between samples while the process is o! target AA¹S the adjusted average time to signal
¸C¸ G
Recently, Baxley [9] presented an application of EWMA chart with VSI for Monsanto's nylon "ber plant in Pensacola, Florida. The size of the samples was the second design parameter to be considered variable (see [10,11]). Subsequently, both parameters (sample size and sampling interval) were made variable (see [12,13]). Rendtel [14] considered Cusum schemes with variable sampling intervals and sample sizes. Finally, all design parameters were considered variable (see [15]).
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The economic design of XM chart with variable sample size (VSS) was studied by Flaig [16] and Park and Reynolds [17]. Park and Reynolds proposed an economic model for XM chart with VSS when the process is subject to the occurrence of several assignable causes. Recently, Das et al. [18] developed a cost model for optimal dual sampling interval (DSI) policies with and without run rules. Also, Das and Jain [19] proposed a further generalization of VSI policy for XM charts in which the sampling intervals are treated as random variables and the sample sizes are considered a function of the sampling intervals. In this paper, we propose an economic model for XM control charts when all design parameters are variable (Vp XM chart). The design parameters are allowed to vary in real time based on current sample information. A model is developed, providing a cost function which represents the cost per time unit of controlling the quality of a process through a Vp XM chart. As the cost function is a function of the design parameters of the control chart, it provides a device for optimal selection of the design parameters. Recently, Costa [15] considered the statistical design of Vp XM charts and showed that considerable performance improvements (in terms of ATS) can be achieved over other XM control charts. That study motivated us to consider an economic model for XM chart having all design parameters variable. The paper is organized as follows. Section 2 presents the description of a Vp XM chart. The process model and the cost function are presented in Section 3. Section 4 contains the expressions of the policydependent variables. Section 5 presents a numerical example. Conclusions are placed in Section 6. 2. Description of the XM chart with variable parameters Suppose that an XM control chart allowing all design parameters to vary is employed to monitor a process whose quality characteristic of interest (say, X) is normally distributed with mean and variance (which are assumed to be known). The target value of the process mean is represented by
193
, that is, when the process is in control " . To use a control chart the user should specify: the sample size (n), the sampling interval (h) and the width coe$cient of the control (or action) limits (k). These parameters are called design parameters of a control chart. In the Fp XM chart, we denote the design parameters by n , h and k and the XM values from the samples are plotted on an XM chart with upper and lower control limits given by $k /(n and centerline . The Vp XM chart is a modi"cation of the Fp XM chart. The design parameters of the Vp XM chart can, in the general case, be di!erent for each sampled value of XM . However, in the model developed here, each of these parameters can assume two values as function of the most recent process information. The position of each sample point on the chart establishes the size of the next sample, the instant of its sampling and the width coe$cient of the control limits. See Fig. 1. We denote the sample size values by n , n , the sampling interval values by h , h and the coe$cient values used in determining the width of the control limits by k , k . For an XM chart with Vp, suppose that ¸C¸ and ;C¸ represent, respectively, the lower and upper control limits. The interval (¸C¸, ;C¸) is partitioned into three distinct intervals or regions: (¸C¸, ¸=¸); (¸=¸, ;=¸); (;=¸, ;C¸); with ¸C¸(¸=¸(;=¸( ;C¸. Here ¸=¸ and ;=¸ represent, respectively, the lower and upper warning limits of the XM chart. The region de"ned by (¸C¸, ¸=¸) and (;=¸, ;C¸) is called warning region of the chart while the region (¸=¸, ;=¸) is the central region. The region above ;C¸ and below ¸C¸ is denominated by action region of the chart. As in the case of control charts with Fp, a signal is produced when a point falls outside the control limits, that is, in the action region. Note that for each sample point xN , two possibilities will be provided for the G warning and control limits (that is, $w /(n G G and $k /(n , i"1, 2, respectively) because G G each design parameters can assume two values. Here, w and w denote the width coe$cients of the warning limits. Since the control limits are di!erent, then to avoid the use of two control charts, one for the design parameters (n , h , w , k ) and other for the design parameters (n , h , w , k ), one can
194
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Fig. 1. Vp XM chart.
construct the Vp XM chart with two scales, one on the left side for samples of size n and other on the right side for samples of size n , in such a way that warning and action limits coincide on the two sides, as illustrated on Fig. 1. Vp policy. The design parameters should be chosen such that h 'h , n (n , k 'k and w 'w . If the sample point falls in the central region, then the next sample should be small; that is, with size n , it should be sampled after a long time interval (h ) and the factor used in determin ing the width of the action limits should be wide (k ). On the other hand, if the sample point falls in the warning region, then: the next sample should be large (n ), it should be sampled after a short time interval (h ) and the factor used in determining the width of the action limits should be narrow (k ). Finally, if the sample statistic falls in the action region an investigation should be initiated to verify if, actually, the process is out of control (that is, if the mean has changed from " to " $, '0) or has occurred just a false alarm (that is, the process is still in control). If, in fact, the process is out of control, a corrective action should be undertaken. Size of the xrst sample. The size of the "rst sample, that is taken from the process when it is just starting or after a false alarm, is chosen at random. If the sample was chosen to be large (small) it should be sampled after a short (long) time interval. Probability p . During the in-control period all samples, including the "rst one, have probability p of being small and (1!p ) of being large,
where p "P(Z(w Z(k )"P(Z(w Z(k ) and Z&N(0, 1) (1)
3. The process model and the cost function An economic modelling approach for the design of a Vp XM chart is described in the rest of the paper. The hourly cost function is derived. This function provides a device for the optimal economic selection of the design parameters n , n , h , h , w , w , k , and k . In the development of the proposed model, assumptions about the production process are made and some important concepts are de"ned. After that the cost function formulation is considered. 3.1. The process model Assumptions. These assumptions characterize the class of production process to be analyzed. We assume that the samples are independent, and that the process starts in a state of statistical control with mean " and standard deviation . Also, the occurrence of the assignable cause results in a shift in the process mean from to $. The length of time the process stays in control is exponentially distributed with mean 1/. The process is not self-corrective. During the search for an assignable cause the process can continue in operation or not. This possibility is represented by the indicator
M.S. De MagalhaJ es et al. / Int. J. Production Economics 74 (2001) 191}200
variable ( "1, if production continues during search and "0, otherwise). During repair the process can continue functioning or not. The indicator variable expresses this possibility ( "1, if production continues during repair and "0, otherwise). Finally, the parameters , , and are assumed to be known and the parameters to be determined are n , n , h , h , w , w , k , and k . Costs. Several costs are associated to the quality control of a production process. Five costs are considered: Costs of sampling and inspection (C ), the cost of investigating false alarms (C ), the cost of "nding and repairing the assignable cause (C ), costs of producing nonconformities while the process is operating in control (C ) and out of control (C ). These are, in general, random variables. Process cycle. The production cycle consists of "ve periods. The in control period (¹ ) is the period that starts from the beginning of the production process until occurrence of the assignable cause, including interruptions for false alarms. The out of control period (¹ ), is the time since process shift until a signal. The period of analysis (¹ ), is the time to analyze a sample and chart the result. The investigation period (¹ ), is the time to "nd an assign able cause when it exists. The repairing period (¹ ), is the time to repair the process. The cycle is the sum of these periods. Hence the process cycle is de"ned as the time since the beginning of production until elimination of the assignable cause. 3.2. The cost model The expected cost per time unit (EC¹;) is the ratio of the expected cost per cycle (E(C)) to the expected cycle time (E(¹)), that is EC¹;"E(C)/E(¹).
(2)
In the computation of E(C) and E(¹), the expressions for some variables are policy-dependent (that is, they depend on the Vp policy adopted), and they are placed in the next section. The expected cost per cycle. E(C) consists of the sum of the costs incurred while the process is in control and out of control. Hence, the components of E(C) are obtained as follows:
195
(i) The expected cost per cycle due to nonconformities while the process is in control (E(C )) and out of control (E(C )) is given by 1 E(C )#E(C )" C #C [AA¹S#E(¹ ) # ¹ # ¹ ]. (3) H HH Here, C and C represent the costs per hour due to nonconformities produced while the process is in control and out of control, respectively. The mean time that the process stays in control is equal to 1/. The adjusted average time to signal (AA¹S) is the mean time interval since a process shift until an alarm occurs. E(¹ ) is equal to nG, where G is the time to take a sample of size 1 and chart it, and n represents the average sample size while the process is out of control. The expressions for AA¹S and n are in Section 4. ¹ and ¹ are the expected times H HH to discover the assignable cause and to repair the process, respectively. (ii) The expected cost due to false alarms (E(C )) is given by E(C )">E(F). (4) Here > represents the cost per false alarm and E(F) is the expected number of false alarms. E(F) is policy-dependent and then its expression is given in Section 4. (iii) The expected cost of "nding and repairing an assignable cause when one exists (E(C )): E(C )"=. (5) This quantity includes any downtime that is appropriate and is assumed policy independent. (iv) The expected cost of sampling and inspection (E(C )) is given by E(C )"(a#bn)s#(a#bn)s, (6) where a is the "xed cost per sample and b is the variable cost per unit sampled. Here, n (n) represents the average sample size while the process is in control (out of control). And s (s) represents the average number of samples drawn while the process is in control (out of control). The expressions for n, n, s, and s are given in Section 4 due to their dependence on the Vp policy.
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Note that the "rst and second terms of Eq. (6) gives, respectively, the expected cost of sampling while the process is in control and out of control. Then adding Eqs. (3)}(6), the expected cost per cycle is obtained: 1 E(C)" C #C [AA¹S#nG# ¹ # ¹ ] H HH #>E(F)#=#(a#bn)s#(a#bn)s. (7) The expected cycle time. The expected cycle time is the sum of the expected time of each period of cycle, then: E(¹)"E(¹ )#E(¹ )#E(¹ )#E(¹ )#E(¹ ) 1 " #(1! )E(¹ )#AA¹S #nG#¹ #¹ . (8) H HH The expression (1! )E(¹ ) is the part of E(¹ ) due to the fact that the in control period can include interruptions for false alarms. The indicator variable takes into account this possibility. The expected time searching for false alarms is equal to the expected search time associated with a false alarm (¹ ) times the expected number of false alarms (E(F)). As the expression for E(¹ ) is depen dent of the Vp policy, it is placed in Section 4. The AA¹S counts for the mean time since process shift until an alarm is given. nG counts the time to analyze a sample and chart the result. The expected time to "nd an assignable cause (¹ ) and repair it H (¹ ) are not dependent of whether the production HH process stops or not, then they always count to the expected cycle length. From Eqs. (2), (7) and (8), we get the expression for EC¹;:
random variable F depends on the number of samples (N) taken before the process shift. The expected number of false alarms (E(F)) is given by E(F)"( p # (1!p ))s, where "P(Z'k ) is the probability of type G G I error when the chart being used has design parameters n , h , k , for i"1, 2. Here, p is the probG G G ability of a sample to be small during the in control period, and (1!p ) is the probability of a sample to be large during the in control period. Also, s("E(N)) represents the average number of samples taken while the process is in control. Average number of samples taken while the process is in control. Let N be the number of samples taken before the process shift. Suppose that the assignable cause has occurred between the jth and the (j#1)th samples, that is, the process mean has shifted from to $ between these two samples. Then j samples were taken during the in control period, that is, N"j. To have j samples before the shift is equivalent to the assignable cause occurring between the sampling times ¹ and H ¹ , then: H> E(N)" jP(N"j)" jP(¹ (¹(¹ ). H H> H H Note that the time that the process stays in control (¹) has exponential distribution with parameter . Taking into account the memoryless property of the exponential distribution and that the sampling interval (H) is a random variable with probability distribution given by P(H"h )"p and P(H"h )"1!p , we get e\HF p #e\HF (1!p ) . E(N)"s" 1!e\HF p !e\HF (1!p ) Average time since occurrence of an assignable cause until an alarm occurs. This average time is,
(1/)C #C [AA¹S#nG# ¹ # ¹ ]#>E(F)#=#(a#bn)s#(a#bn)s H HH EC¹;" . (1/)#(1! )E(¹ )#AA¹S#nG#¹ #¹ H HH 4. Expressions of the policy-dependent variables Expected number of false alarms. Let F be the number of false alarms incurred in a cycle. The
usually, called adjusted average time to signal (AA¹S). Let ¹ be the time since occurrence of an assignable cause until an alarm is given and let A be a random variable representing the length of the
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197
Fig. 2. The relationship between ¹ , A, ¹, R, S.
interval in which the shift occurs. Also, let R be the time from the process shift until the "rst sample after the shift, S the time since the "rst sample after the shift until an alarm and let ¹ represent the time from last sample before the shift until the process shift. The relationship between ¹ , A, ¹, R, and S is shown in Fig. 2. From Fig. 2, we see that ¹ "R#S and then AA¹S"E(¹ )"E(R)#E(S). Determination of E(R). As Reynolds et al. [1], we assume that P(A"h ) is proportional to the length G of the interval (A) times the probability of an interval of this length occurring. This means that p h , P(A"h )" p h #(1!p )h (1!p )h P(A"h )" . p h #(1!p )h Since E(R) can be written as E(R)"E(E(RA)) and E(E(RA))"E(E((h !¹ )A)) G " E((h !¹)A)P(A"h ), G G G we, therefore, get
1!e\HF (1#h ) P(A"h ) E(R)" h ! (1!e\HF )
1!e\HF (1#h ) P(A"h ). # h ! (1!e\HF ) The expression above gives the expected time from the process shift until the "rst sample after the shift.
Determination of E(S). The expected value of S depends on the position of the "rst sample point after shift (B). The probability of plotting this point in the central region (B"B ), warning region (B"B ), or action region (B"B ), depends on the length of the interval in which the shift occurs. This leads to P(B"B )"P(B"B A"h )P(A"h ) # P(B"B A"h )P(A"h ) "p P(A"h )#p P(A"h ), P(B"B )"p P(A"h )#p P(A"h ) and P(B"B )"1!P(B"B )!P(B"B ), where p "P(;(w ;&N((n , 1)), p "P(w (;(k ;&N((n , 1)), p "P(;(w ;&N((n , 1)), p "P(w (;(k ;&N((n , 1)). When the "rst sample point after the shift falls in the central region (warning region) let (SB"B )"¹ (let (SB"B )"¹ ). Therefore, E(S)"E(¹ )P(B"B )#E(¹ )P(B"B ), where ¹ (¹ ) is the time since the "rst sample after the shift, when the "rst sample after the shift falls in the central region (when the "rst sample after the shift falls in the warning region), until an alarm occurs.
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If the xrst sample after the shift falls in the central region, the time from the "rst sample after the shift until an alarm occurs is given by + ¹ " >. G G The expected value of ¹ is + E(¹ )"E > G G + "E E > M G G "E(M )E(<). Here, the random variable M represents the number of points that falls in the central region until an alarm occurs. The variable M is distrib uted geometrically with parameter (1!p ), where p is the probability to obtain a point in the central region. This probability is given by
p "p #p pG\p , G where the p s were given above. GH The expected value of M is given by 1 , E(M )" 1!p where p is given above. The random variables > s are independent and G identically distributed as <. The random variable < represents the length of time to obtain a point outside the warning region since the last point that fell in the central region. The probability function of < is given by P(<"h )"p #p "1!p P(<"h #ih )"p pG\p #p pG\p "p pG\(1!p ) where i"1, 2,2 . Calculating the expected value of < we obtain that p . E(<)"h #h 1!p
Then, substituting E(M ) and E(<) in E(¹ ), we get [h (1!p )#h p ] . E(¹ )" 1!p !p #p p !p p In a similar way, if the xrst sample after the shift falls in the warning region, we get [h (1!p )#h p ] E(¹ )" . 1!p !p #p p !p p The determination of n, n, h, and s. Calculations of n, n, h, and s are straightforward: n"n p #n (1!p ), n"n p ()#n (1!p ()), and h"h p ()#h (1!p ()), where p was given before by Eq. (1), and p ()"P(!w !(n (Z(w !(n !k G G G G G ! (n (Z(k !(n ), G G G for i"1, 2. Finally, expected cycle length while the process is off target s" average time between samples while the process is off target
AA¹S#nG# ¹ # ¹ H HH . " h 5. Numerical example An application of the developed economic model is demonstrated for the numerical example of Lorenzen and Vance [20]. It is an example of foundry operations, where periodic samples of molten iron are taken to monitor the carbon}silicate content of the casting. A high carbon}silicate content results in casting of low tensile strength. The values of the input variables are G"¹ "¹ " hours; H ¹ " hours; 1/"50; C "$114.24/hour; HH C "$949.20/hour; >"="$977.40; a"0; b"$4.22; "1; "0. In order to accomplish the optimization of the unit cost function, the following constraints were considered: n )n ; n *5; n *1; 0.1)h )h ; h *1; w )w ; w and w *0.1; k )k ; k and k *1. A nonlinear constrained optimization algo rithm (MATLAB toolbox software) was applied to
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199
Table 1 Comparison between the economic performance of Vp and Fp XM control charts Shift
Optimum parameters for Vp (n , n , h , h , w , w , k , k )
0.5
Opt. ECTU for Vp
Opt. ECTU and opt. par. for Fp (n , h , k )
236.70
248.77 (17, 2.77, 1.87)
203.32
216.27 (12, 2.28, 2.20)
185.88
197.65 (9, 1.98, 2.43)
175.44
185.75 (7, 1.77, 2.60)
169.41
177.47 (5, 1.45, 2.71)
165.62
171.88 (5, 1.54, 2.88)
162.99
168.66 (5, 1.59, 3.06)
(11, 15, 2.15, 0.1, 1.15, 1.08, 2.62, 1.97) 0.75 (7, 9, 1.70, 0.1, 1.23, 1.18, 3.03, 2.22) 1.0 (5, 6, 1.43, 0.1, 1.30, 1.26, 3.26, 2.38) 1.25 (3, 5, 1.25, 0.1, 1.43, 1.39, 3.44, 2.47) 1.5 (3, 5, 1.11, 0.1, 1.71, 1.6, 3.62, 2.44) 1.75 (2, 5, 1, 0.1, 1.9, 1.8, 3.65, 2.42) 2 (2, 5, 1, 0.1, 2.06, 1.9, 3.56, 2.42)
the cost function. We considered several shifts of the mean: "0.5; 0.75; 1.0; 1.25; 1.5 and 2. Di!erent starting vectors were used, in the optimization process, to "nd the minimum value of the ECTU and the corresponding optimal design parameters. Since, for each speci"c shift, all of the iterations converged to the same solution, independent of the given starting vectors, we have evidence that, perhaps, the global mimimum was attained. We have, nevertheless, obtained a better control chart, as will be seen next. In order to compare the results, we computed also the optimal design parameters and the ECTU for the economic model for the Fp XM chart. Table 1 presents the results. From this table, we can observe that, for small to moderate shifts, there is a decrease of more than 5% on ECTU for the economic model with variable parameters when compared to the Fp economic model. In other words, if an economic model with Vp is used, then for shifts from 0.5 to 1.25 a saving of more than $20 000 per year is obtained and for shifts varying from 1.5 to 2.0 a saving of more than $10 000 per year is obtained, supposing that the process is operating 8 hours per day during 5 days per week.
6. Conclusions A quality cost model for an XM chart with all design parameters varying in an adaptive way was developed. This model makes possible the optimal economic selection of the design parameters of the chart. We could check, through a numerical example, that the economic model for a XM chart with Vp reduces the quality cost of a process. The economic model also provides a way to analyze for sensitivity of some parameters and to have information on how the cost is a!ected. Also, it allows for comparisons of several types of control charts. References [1] M.R. Reynolds Jr., R.W. Amin, J.C. Arnold, J.A. Nachlas, XM Charts with variable sampling intervals, Technometrics 30 (1988) 181}192. [2] G.C. Runger, J.J. Pignatiello Jr., Adaptive sampling for process control, Journal of Quality Technology 23 (1991) 135}155. [3] R.W. Amin, R.W. Miller, A robustness study of XM charts with variable sampling intervals, Journal of Quality Technology 25 (1993) 36}44.
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