Using target costing concept in loss function and ... - Springer Link

Int J Adv Manuf Technol (2004) 24: 206–213 DOI 10.1007/s00170-003-1547-8

O R I GI N A L A R T IC L E

Hsin-Hung Wu

Using target costing concept in loss function and process capability indices to set up goal control limits

Received: 30 September 2002 / Accepted: 30 September 2002 / Published online: 30 June 2004
Springer-Verlag London Limited 2004

Abstract This paper depicts the relationship among the loss function, process capability indices and control charts to establish goal control limits by extending the target costing concept. The specification limits derived from the reflected normal loss function is linked through the Cpm value, computed either directly from the raw data or given by management or engineers, to conventional control charts to obtain goal control limits. The target value can be taken into consideration directly. The advantages of applying the target costing philosophy are also discussed. This paper explains, from a quantitative approach, that reducing process variation is not enough to solve quality problems. In fact, reducing process variation should be used along with bringing the process mean to the target value. Keywords Reflected normal loss function Æ Target costing concept Æ Process capability index Æ Goal control limits

rT: r^: ^0 : r n: d2: D 4, D 3, and A2: c4: B 4 , B 3, and A3: L(y): L1(y): h: f(y):

A list of symbols K: The maximum-loss parameter in the reflected normal loss function c: The shape parameter in the reflected normal loss function, D/4 T: The target value D: The distance from the target value to the point where K first occurs (tolerance or specification limit) E(L(y)): The expected loss associated with the reflected normal loss function l: The average value (mean) of a population r: The standard deviation of a population H.-H. Wu Department of Business Administration, National Changhua University of Education, No. 2 Shda Road, Changhua City, Changhua, Taiwan 500 E-mail: [email protected] Tel.: +886-4-7232105 ext. 7405 Fax: +886-4-7211162

The standard deviation from the target value of a population The estimated standard deviation ofr The new process standard deviation when 2D’ are applied The sample size of the subgroup
2, ^ ¼ R=d The parameter used to estimate r determined by n
The parameters in Xand Rcontrol charts, determined by n
^ ¼ S
c4 , the parameter used to estimate r determined by n The parameters in X
and Scontrol charts, determined by n The general loss function The general loss function when the quality improvement is implemented The parameter used to determine L1(y), where L1(y)=hK The probability density function

1 Introduction Loss functions are often used to measure the loss of a product characteristic deviated from a desired target value, largely due to the Taguchi philosophy. Taguchi [1] has used a modified quadratic loss function to assess the loss associated with deviations of a product characteristic from the target value. As the process departs from the target value, a loss is incurred. Moreover, the loss increases as the process is further away from the target value. Therefore, the purpose of loss functions is to reflect the economic loss associated with variation in, and deviations from, the process target or the target value of a product characteristic [2]. The loss function has been widely used in many areas. For instance, Chen [3] has applied the Taguchi loss function towards setting up specification limit(s) when the product characteristic follows an exponential distri-

207

bution. In addition, Chen [4] developed a specification limit under the membership function by implementing the Taguchi loss function. Sauers [5] also applied the Taguchi loss function along with process capability indices and control charts to set up goal control limits by considering the target costing philosophy. Spiring and Yeung [2], on the other hand, have developed a general class of loss functions with industrial examples to demonstrate the advantage of using loss functions. The traditional loss function has been criticised by practitioners and researchers because it fails to provide a quantifiable maximum loss and magnitude of losses associated with extreme deviations from the target value [6, 7]. On the other hand, Spiring [8] has pointed out that things such as production resources, cost of identification, scrap or rework and liability generally have a maximum loss. As a result, the traditional loss function with an infinite maximum loss is inadequate to describe the loss associated with a product characteristic. For an organization’s viewpoints, it is always economical to reduce unit-to-unit performance variation around the target value even if the products are within specification limits [9]. Continuous quality improvement and cost reduction are necessary for an organisation to stay in a competitive economy. Besides, quality improvement requires the continuous reduction of variation in product and/or process performance around the target value. Therefore, the purpose of this paper is to apply target costing philosophy discussed by Sauers [5] along with Spiring’s loss function with finite maximum loss in goal control limits to reduce common causes of variation. The paper is organised as follows: first, the loss function developed by Spiring [8] and Spiring and Yeung [2] is reviewed. Then, control charts and process capability indices and the target costing philosophy are discussed. The framework based upon the target costing philosophy is then developed. Finally, an example and conclusions are provided.

2 Spiring’s loss function Spiring [8] has developed the reflected normal loss function based upon the Taguchi philosophy. The general equation and figure of this reflected normal loss function are shown in Eq. 1 and Fig. 1:

n  o Þ2 Lð y Þ ¼ K 1
exp
ðy
T 2 2c n  o 8ðy
T Þ2 ; ¼ K 1
exp
D2

ð1Þ

where y represents the quality measurement, K is the maximum-loss parameter, T is the target value and c is a shape parameter. The target value, shape and maximumloss parameters allow a customisation of the loss function to meet practitioners’ requirements. In addition, c is defined as D/4, where D is the distance from the target value to the point where K first occurs. The reflected normal loss function is asymptotic to the maximum loss and will attain this value only at ±¥. The equation of c=D/4 ensures that the loss function at the points T±D will be 0.9997K, which can be considered to be K. The expected loss associated with the reflected normal loss function defined by Spiring [8] is: ! Z ðy
T Þ2 EðLð y ÞÞ ¼ K
K exp
f ð y Þdy ; ð2Þ 2c2 where f(y) is the associated probability density function. If the quality characteristic follows a normal distribution with a mean of l and a standard deviation of r, the expected loss becomes: n  o 2 R 1 ðy
lÞ2 1 ðy
T Þ ELð y Þ ¼ K
K pffiffiffiffi dy exp
þ 2 2 c2 2pr   r   ð3Þ ðl
T Þ2 c ¼ K 1
pffiffiffiffiffiffiffiffiffi exp
; 2 2 2ðr þc Þ 2 2 r þc

where the minimum happens at l=T. Spiring and Yeung [2] stated that using this type of loss function provides the optimal setting for the process from an economic consideration. When the process deviates from the target value, the loss increases. In addition, the loss function can be used as a control chart to monitor the loss from a process. Though the control charts and charts developed by the loss function might be similar, the standard deviation of the economic loss is usually larger than those of control charts. Moreover, the subgroup averages for the economic loss and measurements are different, which provide practitioners with alternatives for assessing process performance.

3 Control charts and process capability indices Statistical process control (SPC) charts serve three purposes [5]: The first purpose is to ensure that the process is in statistical control. The second purpose is to provide alarms when the process shows out-of-control signals. Finally, SPC charts also provide the prerequisite information needed for process capability analysis. Typically, X
-R and X
-S control charts are the two most commonly used control charts. For X
and R control charts, the formulas are described as follows:
; UCLð RÞ ¼ D4 R

Fig. 1 The reflected normal loss function

ð4Þ

208


; CLð RÞ ¼ R

ð5Þ


; LCLð RÞ ¼ D3 R

ð6Þ


^ r R
; UCLðX
Þ ¼ X
þ 3 pffiffiffi ¼ X
þ 3 pffiffiffi ¼ X
þ A2 R n d2 n

ð7Þ

CLðX
Þ ¼ X
;

ð8Þ

ð16Þ

Cpk

ð17Þ

and Cpm ¼

and
^ r R
; LCLðX
Þ ¼ X

3 pffiffiffi ¼ X

3 pffiffiffi ¼ X

A2 R n d2 n

USL
LSL D ¼ ; 6r 3r   USL
l l
LSL ; ; ¼ min 3r 3r

Cp ¼

USL
LSL USL
LSL D ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 6rT 6 r 2 þ ðl
T Þ2 3 r 2 þ ðl
T Þ2 ð18Þ

ð9Þ

UCLðS Þ ¼ B4 S
;

ð10Þ

CLðS Þ ¼ S
;

ð11Þ

LCLðS Þ ¼ B3 S
;

ð12Þ

^ r S
UCLðX
Þ ¼ X
þ 3 pffiffiffi ¼ X
þ 3 pffiffiffi ¼ X
þ A3 S
; n c4 n

ð13Þ

where USL and LSL are the upper specification limit and lower specification limit, respectively, and 2D is the distance between USL and LSL. In Eqs. 16, 17, and 18, l is the mean which is the sum of the numerical values of the measurement divided by the number of items examined, r is the standard deviation which is the square root of the average squared deviates from the mean and rT is the standard deviation from the target value which is the square root of the average squared deviation from the target value. The values of l, r and rT are usually unknown, and estimations from the sample data are required. If control charts, such as X
-R and X
-S control charts, are implemented prior to the process capability analysis, Eqs. 16, 17 and 18 can be revised as follows:

CLðX
Þ ¼ X
;

ð14Þ

^ p ¼ USL
LSL ¼ D ; C 6^ r 3^ r

where UCL, CL and LCL stand for upper control limit, centre line and lower control limit, respectively,
2 , n is the sample size of the subgroup, and the ^ ¼ R=d r parameters of D4, D3 and A2 can be found in the work of Gitlow, Oppenheim and Oppenheim [9]. For X
and S control charts, the formulas are as follows:

and ^ r S
ð15Þ LCLðX
Þ ¼ X

3 pffiffiffi ¼ X

3 pffiffiffi ¼ X

A3 S
; n c4 n
^ ¼ S
c4 , n is the sample size of the subgroup, where r and the parameters of B4, B3 and A3 can be found in the work of Gitlow, Oppenheim and Oppenheim [9]. After the process is in statistical control, a process capability analysis can be conducted to further examine if the process is capable of producing high quality products. Typically, Cp, Cpk and Cpm are the well-known process capability indices (PCIs) for process capability analysis. These PCIs are widely used throughout industry to quantify the ability of a process within specification limits when the populations are normally distributed. The Cp index, classified as the first generation, only considers the relationship between the spread and the specification limits. On the other hand, the Cpk index, classified as the second generation, considers the location of a process with the specification limits, while the Cpm index, also viewed as the second generation, further considers the departures of a process mean from the target value. In fact, the Cpm index is more sensitive to determine if the process is centred at the target value. This paper only focuses on normal process data. For non-normality-based PCIs, please refer to [10, 11, 12, 13]. The formulas of these normality-based Cp, Cpk, and Cpm indices are described as follows:

! USL
X
X

LSL ^ ; ; Cpk ¼ min 3^ r 3^ r

ð19Þ ð20Þ

and USL
LSL D ^ pm ¼ USL
LSL ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C    2ffi ; 2 6^ rT

2 2 ^ þ X
T ^ þ X
T 6 r 3 r ð21Þ where X
is available directly from the X
control chart, ^ depends on the X
-R and X
-S control charts. For while r
4 is

2 is used to replace r ^, while S/c the X -R charts, R/d ^ for the X
-S charts [14]. The third used to substitute r approach is to use a sample standard deviation, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . n P 2 S¼ ðXi
X
Þ n
1, and X
from the overall samples i¼1

^ and X
from the overall samples to replace r ^ to replace r
and X in Eqs. 19, 20 and 21.

4 The target costing philosophy The ‘target costing’ philosophy has been applied to set the price of products by many companies [15, 16]. Sauers [5], on the other hand, has discussed how the target costing philosophy should be applied to combine the philosophy

209

of the Taguchi loss function, the process capability index and the X
-R charts and developed the procedure for the implementation in practice. The philosophy is as follows: By starting with the anticipated acceptable market price, the companies subtract the desired profit margin to obtain a target manufacturing cost. Design and manufacturing engineers are responsible to bring the product into being at this cost. By applying this technique, price concerns can be driven down to the process level. Continuous improvement can be carried out by listening to the price concerns of the marketplace [5]. Roth [17] also described a similar concept to estimate scrap and rework costs from the process analysis. Different settings in process means would result in different rework costs and scrap costs, and the total cost is the sum of the rework and scrap costs. Typically, the relationship between the total cost and the process settings can be represented as an increasing function ƒ as a quadratic function with a minimum value at some certain point during process settings. If the setting is far away from the best setting point with the minimum value, the total cost increases. If the process is stable and the function ƒ is known, this technique can be utilised to estimate the total cost based upon the actual production run long before the accounting department begins to generate any data. If the process performance does not meet the management’s expectations, corrected actions are required to bring the total cost to the best setting point with the minimum value. If the target costing concept is applied, the implied tolerance or specification limits can be derived from loss functions. Later, the specification limits can be linked through predetermined process capability index values, calculated from the original process data, to conventional SPC charts to provide goal control limits to reduce common causes of variation [5]. The goal control limits form the foundation for directed, continuous improvement efforts by considering the price concerns from the marketplace. Generally, Sauers [5] only applied the Taguchi loss function along with the Cp index and X
-R control charts to set up goal control limits by applying the target costing philosophy. To further strengthen and broaden the use of the target costing philosophy, this paper takes into account Spiring’s loss function and the Cpm index on both X
-R and X
-S control charts. The goal control limits derived from the target costing concept are always tighter than the traditional con
in Eqs. 4, 5, ^ , S
and R trol limits. That is, the values of r 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15 for the target costing concept are usually smaller compared with those values for traditional control limits. For instance, the points slightly below the upper control limit, which are in control without any special causes of variation in SPC charts, might be indicated as the points outside the upper goal control limit. Taguchi and Deming have expressed a similar idea that continuously reducing process variation in a product and process performance around the nominal value for quality improvement is economical [9]. Therefore, the application of goal control limits is

to force manufacturers to reduce the total variation relentlessly, even when SPC charts illustrate no special causes of variation in the process.

5 The implementation of target costing concept Suppose the target costing concept is applied and a quality improvement program is implemented; the general loss is expected to be reduced to L1(y)=hK, where 0