A Comarative Study of Various Loss Functions in the Economic Tolerance Design Jeh-Nan Pan Deartment of Statistics National Chen-Kung University, Tainan, Taiwan 700, ROC Jianbiao Pan Deartment of Industrial and Manufacturing Engineering California Polytechnic State University, San Luis Obiso, CA 93407, USA Abstract - Engineering tolerance design lays an imortant role in modern manufacturing. In this aer, the Kaur s model was modified so that the economic secification limits for both symmetric and asymmetric losses can be established. Three different loss functions ( Taguchi s quadratic loss function ( Inverted Normal Loss Function (3 Revised Inverted Normal Loss Function are comared in the economic tolerance design. The relationshis between the three loss functions and rocess caability indices for symmetric tolerance are established. The results suggest that the revised inverted normal loss function be used in determination of economic secification limits. I. INTRODUCTION Product quality is highly regarded in today s business environment. Generally seaking, quality characteristics can be divided into three tyes: nominal the best, larger the better and smaller the better. In the traditional concet of the quality evaluation system, a roduct is determined to be nonconforming if the quality characteristic of a roduct fails to meet the engineering secification limits and then a certain amount of quality loss is incurred. Reference [9], on the other hand, believed that a oorly designed roduct causes society to incur losses from the initial design stage to the roduct usage. Therefore, he defined the loss function as the deviation from the target/nominal quality characteristic. In other words, the Taguchi s quality loss is incurred when quality characteristics of a roduct deviates from its target value regardless how small the deviation is. Since then, the quality loss concet has been shifted from defined by secification limits to defined by user and Taguchi s loss function has been extensively used for determining the engineering tolerance. The estimation of total quality losses lays an imortant role in the economic tolerance design. Reference [] and [] roosed an otimization method to develo the secification limits. The secification limits were determined on the basis of minimizing total cost or loss to the customer as well as to the roducer. In the model, the scra cost and rework cost were assumed to be the same. However, the scra cost and rework cost may not be equal in many real manufacturing cases. For examle, in metal cutting or machining rocesses, if a art is greater than the uer secification limit (USL, it still can be reworked to the secified length/thickness. However, if a art is less than the lower secification limit (LSL, it can no longer be used and has to be scraed. The objective of this study is to develo a new model of determining the economic secification limits. First, various loss functions were reviewed. The Kaur s model was then modified to relax the limitation that the scra cost and the rework cost have to be same. The exected total losses including scra, rework and insection costs were estimated using three different loss functions: ( Taguchi s quadratic loss function, ( Inverted Normal Loss Function (INLF, and (3 Revised Inverted Normal Loss Function (RINLF. To decide which loss function is the best in the economic tolerance design, the relationshis between rocess caability indices and exected loss er unit under normal distribution are derived. The results suggest RINLF be the most aroriate loss function in the economic tolerance design. II. LOSS FUNCTION REVIEW Taguchi [9] defined the quadratic loss function as L ( y = k ( y T ( where y is the quality characteristics, k is the coefficient of quality loss. Taguchi s loss function has been extensively used for determining the engineering tolerance ([]; []; [3]. The drawbacks of Taguchi s quality loss function are that it is unbounded and symmetrical ([4], [0]. In many manufacturing rocesses, it is unrealistic to assume the quality loss is unbounded even if the material, labor and other administrative costs are included. Asymmetric quality loss is also common in cases such as the scra cost is different from the rework cost. To overcome the unbounded loss in Taguchi's loss function, Reference [6] roosed a loss function as below: if y T K B( y T B L( y= ( K if y T > K B where K is the maximum value of quality loss, B reresents the coefficient of quality loss within the secification limits. In order to better describe the quality loss for the "Nominal the better" case of engineering secification, Reference [7] -444-048-8/06/$0.00 c 006 IEEE 783
roosed inverted normal loss function (INLF from a standoint of normal robability density function (df. The INLF can be written as: (y T L( y = K ex (3 σ L where K is the maximum loss if the characteristic deviated from the target, σ L is the arameter for controlling the shae of loss function deending on the realistic loss. In addition, Reference [7] and [8] roosed an asymmetric INLF as below: (y T K ex if y < T σ y = (4 (y T K ex if y T σ L L( Reference [5] roosed a revised inverted normal loss function (RINLF to measure the quality loss for roduct interference. Due to the fact that quality loss will not incur when the quality characteristic falls within the neighborhood of target value from the customer s oint of view, it would be more reasonable for a customer or manufacturer to secify an accetable range (L, U in which no quality loss would be incurred. Therefore, the Siring s inverted normal loss function can be modified as (5. (y L K { ex( } σ y < L L(y = 0 L y U (5 K { ex( (y U y > U } σ L where (L, U is the accetable range of a quality characteristic; K is the maximum loss if the characteristic deviates from the target and exceeds the LSL; K the maximum loss if the characteristic deviates from the target and exceeds the USL; σ L and σ L are the arameters for controlling the shae of function deending on the realistic loss. III. ECONOMIC TOLERANCE DESIGN To minimize total loss to the customer as well as to the manufacturer, Reference [] and [] roosed otimization models for determination of secification limits. Three costs were considered: insection costs, scra/rework costs, and loss due to variation. Assuming that both the target of a roduct T and the quality characteristic Y follow a normal distribution, Kaur s economic model of otimization can be written as: TC = L Q + ( q SC + IC (6 where TC is the total exected losses er unit roduct, L Q is the exected loss er unit roduct shied to the customer, SC is the scra cost er unit, IC is the insection cost er unit and q is the fraction of good roducts actually shied to the customer. In the model, the scra cost and rework cost were assumed to be the same. Considering that the quality losses above the uer or lower secification limit may not be equal, we revise Kaur s economic tolerance design model as TC = L + q SC + q RC + IC (7 where TC is total exected losses er unit roduct, SC is the scra cost er unit, RC is the rework cost er unit, IC is the insection cost er unit, L is the exected loss er unit roduct shied to the customer using the above-mentioned loss functions, q is the fraction of good roducts actually shied to the customer and q, q denotes the robability of scra or rework resectively. IV. ESTIMATION OF EXPECTED LOSS Next we will estimate the total exected loss er unit roduct based on the revised Kaur s economic tolerance design model. Assuming the quality characteristics Y follows a normal distribution, i.e.y~n( µ, σ, the exected loss er unit roduct is derived for revised Taguchi s quadratic, INIF, and RINLF loss functions. As described in the introduction section, quality characteristics can be divided into three tyes: the nominal the best, the larger the best, and the small the best. The formulae of the total exected losses er unit roduct under two tyes of quality characteristics, i.e. the nominal the best (bilateral secification and the smaller the better (unilateral secification are derived. When the quality characteristics are the larger the better (unilateral secification, the derivation of total exected losses er unit roduct is very similar to the case of the smaller the better case. When the quality characteristics are the nominal the best (bilateral secification, we assume the lower secification limit LSL µ η σ ; the uer secification limit = USL =µ+ησ ; the roduct will be scraed if Y exceeds the LSL and it can be reworked if Y exceeds the USL. A. Exected loss estimation using revised Taguchi s loss function Case : no insection is erformed. The total exected losses er unit roduct is the exected loss er unit roduct shied to the customer, which can be written as L Q = E L Q ( Y = L Q ( y f ( y dy = k σ + {µ T} T µ T µ + (k k σ( µ T φ ( +[(µ T + σ ][ Φ ( ] σ σ where k, k reresent the coefficient of two different 784 006 IEEE International Conference on Management of Innovation and Technology
quality losses. Case : 00% insection is erformed. The exected loss er unit roduct shied to the customer can be written as: L µ + η σ = E L Q Y Q ( = LQ ( y f T ( y dy µ η σ = k ( + V Y T T} T { E( Y k k T µ + [(µ T + σ ][Φ(η Φ( ] q σ T µ η +σ ( µ T φ ( σ [( µ T + ] φ( η σ where E[ Y ] T = µ + σ φ ( η φ( η and q η η φ( η φ( η V[ YT ] = σ φ( η φ( η q q q q= Φ( η q= Φ( η q =Φη ( + Φ( η φ ( and Φ ( denote the standard normal robability density function and the cumulative distribution. Note that (µ T σ η µ σ and η ( T µ σ Considering the scra, rework, and insection costs, the total exected loss er unit roduct can be calculated according to (7. B. Exected loss estimation using revised INLF Case : no insection is erformed. The total exected losses er unit roduct can be written as: L = E INLF L INLF ( y = L y f y dy INLF ( ( T µ σ (µ T = K Φ ex σ σ + σ (σ + σ σ (T µ T µ Φ + K Φ σ σ + σ σ σ (µ T σ (T µ Φ σ + σ L (σ + σ L σ σ L L ex + σ L Case : 00% insection is erformed. The exected loss er unit roduct shied to the customer can be written as: µ + η σ L INLF = E L INLF ( Y = L INLF ( y f T ( y dy µ η σ K T µ σ (µ T = Φ( + Φ( η {ex[ ]} σ q σ σ + σ ( + σ σ ( T µ σ ( µ T η ( σ + σ Φ ( Φ( σ σ + σ σ σ + σ L K T µ σ (µ T + Φ ( η Φ( {ex[ ]} q σ σ + σ (σ + σ L L σ ( µ T + η ( σ + σ σ L L (T µ Φ( Φ( σ σ + σ σ σ + σ L L L Considering the scra, rework, and insection costs, the total exected loss er unit roduct can be calculated according to (7. C. Exected loss estimation using revised RINLF Case : no insection is erformed. The total exected losses er unit roduct can be written as L RINLF = E L RINLF ( y = L RINLF ( y f y dy ( L µ σ = K Φ σ σ + σ (µ L σ (L µ ex Φ (σ + σ σ σ + σ U µ σ +K Φ L σ σ + σ L (µ U σ L (U µ ex Φ (σ + σ σ σ + σ L L Case : 00% insection is erformed. The exected loss er unit roduct shied to the customer can be written as: µ η σ L + RINLF = E L RINLF ( Y = L y f y dy µ η σ RINLF ( T ( K L µ σ (µ L = Φ( + Φ(η {ex[ ]} q σ σ + σ (σ + σ σ (L µ σ ( µ L η ( σ + σ Φ ( Φ( σ σ + σ σ σ + σ L σ σ + σ (σ + σ L L K U µ σ (µ U + Φ( η Φ( {ex[ ]} q σ ( µ U +η ( σ + σ σ ( U µ L L Φ ( Φ( σ L σ + σ L σ σ + σ L where η (µ L σ and η ( U µ σ Considering the scra, rework, and insection costs, the total exected loss er unit roduct can be calculated according to (7. Estimation of the exected losses for unilateral secification can be derived similarly. V. RELATIONSHIP BETWEEN LOSS FUNCTION AND PROCESS CAPABILITY INDICES We have described exected loss estimation using Taguchi s quadratic, INLF, and RINLF loss functions. To answer which loss function is aroriate for determining the economic secification limits, it is necessary to exlore the relationshi between loss function and rocess caability indices. Assuming that a quality characteristics Y follows a normal distribution, the rocess caability indices for a bilateral secification are: 006 IEEE International Conference on Management of Innovation and Technology 785
USL LSL C = = Φ (3 C 6 φ ( 3 C C 6σ 3σ L = (K + K Q 8[ Φ(3 C ] C µ LSL USL µ µ M C k = min {C l,c u } = min, = Considering the scra, rework and insection costs, the 3σ 3σ 3σ total exected loss er unit roduct is USL LSL C m = = TC = L Q + Φ(3C 3 σ + 6{ EY ( T } (SC + RC + IC (µ T To exlore the relationshi between INLF and rocess caability indices and to achieve INLF s maximum loss at USL + LSL where M = is the center of a bilateral secification secification limits, we first set the arameters σ USL LSL = σ L = 4 = [(3 σ 4] C according to the rule and = denotes half of the secification width. roosed by [7]. Then, the exected loss er unit roduct To exlore the relationshis between loss functions and shied to the customer is rocess caability indices for symmetric tolerance, we first (K + K L INLF = Φ(3C assume that quality characteristic of a symmetric tolerance (T Φ (3 C = M, as shown in Fig., follows a normal distribution. If the rocess caability indices C and C k are known, then one can 3C [Φ( 6 + 9C obtain µ M = 3 σ (C C k. Thus, ] 6 + 9C LSL = µ ησ η = [ ( T µ ] σ = [3σ C ( T µ ] σ The total exected losses er unit roduct is USL = µ + η σ η = [3σ C + ( T µ ] σ TC = L INLF + Φ(3C (SC + RC + IC When the rocess average µ is equal to the target value T, To exlore the relationshi between RINLF and rocess then C = C k and η =η =η=3c. Note that only the caability indices and to achieve RINLF s maximum loss at secification limits, we first set the arameters: asymmetric loss (SC RC is lotted in Fig. since the L LSL 3 σ ( mc USL L 3 σ ( nc symmetric loss (SC=RC can be considered as a secial case of σ = = σ = = asymmetric loss. By utilizing C, one can derive the 4 4 4 4 relationshis between three different loss functions and rocess according to the rule roosed by [7]. Then the exected loss caability indices. er unit roduct shied to the customer can be written as To exlore the relationshi between the revised Taguchi s loss function and rocess caability indices, we first assume L RINLF = Φ( 3mC + Φ (3C ex Φ (3 C 6 + 9( m C the quality losses exceed the uer or lower secification limit are not equal and k, k reresent the coefficient of 3 ( mc 9m ( mc 6 9 ( mc Φ Φ two different quality losses, namely, 6 + 9 ( m C 6 + 9 ( m C 6 + 9 ( m C k = K = K (3σ C ; k = K = K (3σ C, where K denotes the maximum loss if the quality K 7n C + Φ 3C Φ 3nC ex characteristic deviates from the target and exceeds the LSL; Φ (3C { ( ( 6 + ( 9 n C and K denotes the maximum loss if the characteristic deviates from the target and exceeds the USL. The exected loss er 3 ( nc 6 9 ( nc 9n ( nc Φ Φ unit roduct shied to the customer is 6 + 9 ( n C 6 + 9 ( n C 6 + 9 ( n C K 7m C Fig.. Comarison of three loss functions for symmetric tolerance where m = T L denotes the difference between the target and the lower limit for the accetable range, in which no quality loss will be incurred, divided by half of the U T secification width; n = denotes the difference between the target and the uer limit for the accetable range, divided by half of the secification width. The total exected loss er unit roduct is TC = L + Φ(3C RINLF (SC + RC + IC When a rocess average µ is greater or less than the target value T, the derivation of the relationshi between rocess 786 006 IEEE International Conference on Management of Innovation and Technology
VI. caability indices and loss functions is similar to the case of rocess average µ = T as described above. The relationshi between loss function and rocess caability indices for asymmetric tolerance can be derived similarly. SELECTION OF THE APPROPRIATE LOSS FUNCTION FOR ECONOMIC TOLERANCE DESIGN Based on the relationshi between rocess caability indices and the total exected losses er unit roduct for various loss functions, the total exected losses er unit roduct for three loss functions under various C can be comared. We have estimated the total exected losses er unit roduct under various C for both the symmetric and asymmetric tolerances when a rocess average µ equals to T. Fig. shows the total exected losses er unit roduct decline as the rocess caability index C imroves regardless which loss function is used (Here the case of symmetric tolerance and symmetric loss is illustrated as an examle. Notice that the failure/defect rate is 66 m (art er million for C =.33 or 4 sigma rocess, 0.54 m for C =.67 or 5 sigma rocess, and 0.00 m for C = or 6 sigma rocess. When C =, the total exected losses er unit roduct calculated using three different loss functions are $0.4 (K + K for the revised Taguchi s quadratic loss function, $0.084 (K + K for the INLF loss function, and $0.00004 (K + K for the RINLF, resectively. It aears that the exected loss er unit roduct calculated using RINLF is the most close to the failure/defect rate. Since engineering secification lays a key imact in the calculation of C or C k and the loss estimation of RINLF is more consistent with the failure rate than that of the revised Taguchi loss function and INLF, it is suggested that RINLF be used in the determination of economic secification limits. VII. SUMMARY In the manufacturing rocess of an industrial roduct, the determination of engineering secification and the selection of loss function have a significant imact on the estimation of quality loss. In this aer, we roose a new method for determining the economic secification limits for the manufacturing rocesses with symmetric or asymmetric losses. By exloring the relationshi between loss functions and rocess caability indices for symmetric and asymmetric tolerances and then comaring the unit total exected loss, we have shown that RINLF is the most reasonable one to reflect the actual quality loss/failure rate among three loss functions. Therefore, it is suggested that RINLF be used in the determination of new economic secification limits. Total unit loss $(K +K 5.0 4.5 4.0 3.5 3.0.5.0.5.0 0.5 0.0 TaguchiLoss [] K.C. Kaur and C.J. Wang, Economic Design of Secifications Based on Taguchi's Concet of Quality Loss Function, Quality Design, Planning, and Control, The American Society of Mechanical Engineers, Boston,. 3-36, 987. [] K.C. Kaur, An Aroach for Develoment of Secifications for Quality Imrovement, Quality Engineering, vol.,. 63-77, 988. [3] K.C. Kaur and B.R. Cho, Economic Design of the Secification Region for Multile Quality Characteristics, IIE Transactions, vol. 8,. 37-48, 994. [4] R. V. Leon and C. F. Wu, A Theory of Performance Measures in Parameter Design, Statistic Sinica, vol.,. 335~358, 99. [5] J. N. Pan and J. H. Wang, A Study of Loss Functions for Product Interference Analysis, Industrial Engineering Research, vol. (,. 80-00, 000. [6] T. P. Ryan, Statistical Methods for Quality Imrovement, Wiley, New York, 989. [7] F. A. Sring, The Reflected Normal Loss Function, The Canadian Journal of Statistics, vol,. 3-330, 993. [8] F. A. Sring and A. S. Yueng, A General Class of Loss Functions with Industrial Alications, Journal of Quality Technology, vol. 30(,. 5-6, 998. [9] G. Taguchi, Introduction to Quality Engineering: Designing Quality Into Products and Processes, Asian Productivity Organization, Tokyo, 986. [0] M. Tribus and G. Szonyi, An alternative view of the Taguchi aroach, Quality Progress, vol. (5,. 46-5, 989. INLF RINLF 0. 0. 0.3 0.33 0.4 0.5 0.6 0.67 0.7 0.8 0.9...3.33.4.5.6.67.7.8.9 Fig.. Comarison of the total exected losses er unit roduct for various loss functions under different rocess caability indices for the symmetric tolerance and symmetric loss case C ACKNOWLEDGMENT The first author would like to gratefully acknowledge financial suort from the National Science Council of Taiwan on this research roject. REFERENCES 006 IEEE International Conference on Management of Innovation and Technology 787