Int J Adv Manuf Technol (2002) 20:660–663 Ownership and Copyright 2002 Springer-Verlag London Limited
Bayesian Analysis on Deming’s Model with Consideration of Inspection Errors C.-C. Chyu1 and F.-C. Wu1,2 1
Department of Industrial Engineering and Management, Yuan-Ze University, Chung-Li, Taiwan; and 2Department of Industrial Management, Van-Nung Institute of Technology, Chung-Li, Taiwan
This paper studies an extension to Deming’s model for inspection sampling. This new model is more useful in practice since it takes into account the two inspection errors (i.e. accepting out of tolerance components or rejecting in tolerance components) with the goal of minimising the expected total cost. Mathematical formulae corresponding to the model are derived based on the Bayesian approach. A computer program of this model is available for industry. Some numerical analysis of the effects due to model factors, such as inspection errors, component quality, and k2/k1 ratio, is also presented. As a result, we conclude that a model that involves the two types of inspection error will lead to a looser inspection policy and higher expected total cost than a model without inspection errors. This method can be used to select the best suppliers when the total cost of the product is the main concern of the producer. Keywords: Bayesian decision analysis; Inspection sampling plans
1.
Introduction
Very few industrial firms manufacture all the major components of their own products. One or more types of component must be purchased from outside. When purchasing components, there is often a choice among several suppliers whose quality, prices, and services vary. Deming [1, chap. 13] discusses in detail, inspection sampling relative to two different cost set-ups (Fig. 1). The first is the cost per unit to inspect an item. This cost, k1, includes all labour, material, and amortisation costs associated with inspection. The second cost, k2, is the cost per unit of a non-conforming item placed in an assembly that fails. The Deming decision rule in this model is based on a classical statistical approach. The rule states that if the probability that an item is non-conforming is less than k1/k2, no inspection is Correspondence and offprint requests to: Dr C.-C. Chyu, Department of Industrial Engineering and Management, Yuan-Ze University, Chung-Li, Taiwan 320. E-mail:
[email protected]
Fig. 1. Deming’s inspection model.
always best. Otherwise, 100% inspection is more economical. An (n, c) acceptance sampling plan has never been the best under the Deming model using a classical statistical approach. Many authors including Hald [2], Case and Keats [3], Papadakis [4], Vardeman [5], Sower et al. [6], Fitzsimmons [7], and Milligan [8] have cited the work of Deming. Burke et al. [9] devised the (k1, k2) game to provide histograms of the total costs for use in acceptance sampling, 100% inspection, and zero inspection. Kaminsky et al. [10] have derived the probability distributions and variances of the three inspection plans. Inspection planning using a Bayesian approach provides another good means of supplier selection because this method takes into account the cost effects due to the uncertainty of the component quality. Barlow et al. [11,12] discussed Deming’s models using a Bayesian approach and came to a different conclusion. In the experimental analysis using  priors for the component quality, they found that an (n, c) sampling plan is frequently the best choice when the degree of the uncertainty for the component quality increases or, in other words, when the variance of the distribution is large. In this paper, we focus our study on Deming’s model with consideration of inspection errors by a Bayesian approach in which a finished assembly cannot be repaired by replacing defective units. Deming describes this model as “value added to the substrate”.
Baysian Analysis on Deming’s Model
Fig. 2. Inspection sampling model.
In the paper, Section 2 describes the model and its mathematical formulation. Section 3 presents some numerical analysis of the model.
2.
Model Description
A producer makes a certain product requiring an assembly process of a certain key item. It is assumed that a failed assembly unit cannot be disassembled and reassembled. The failed product will either be discarded or sold for its residual or scrap value. It is also assumed that the items in the lot are produced, one at a time and independently, by a process in which the probability of a non-conforming item remains constant from item to item. This probability constant is unknown, but, from a Bayesian point of view, it has a probability distribution. The producer can estimate this distribution based on surveyed data and past experience. Berger [3] provides several methods for quantifying prior information as a distribution. Before the lot is put into assembly, the producer can take a sample of size n. Whenever an item in the lot is reported “non-conforming” during inspection, items in a spare lot are drawn and inspected until a reported “conforming” one is found. The producer pays for all such extra cost. Each inspection may be imperfect. A reported “non-conforming” item in the inspection may be conforming, and a reported “conforming” item may cause a failed assembly. The former mistake is often referred to as the type error and the latter as the type error. Both mistakes will create a loss for the producer. The notation and definitions of random variables used are as follows: P = probability of a conforming item, which is a random variable Q1 = type I error, a random variable Q2 = type II error, a random variable
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n = sample size T1 = number of reported “non-conforming” items in the inspection sample T2 = number of reported “non-conforming” items in the remaining (N ⫺ n ) items of the lot k1 = inspection cost per unit k2 = product failure cost per unit Mj = number of inspections made in the spare lot due to the jth reported “non-conforming” in the inspection sampling Z = number of non-conforming items in the remaining (N ⫺ n) items of the lot Zn = number of failed assemblies due to n reported “conforming” items ZN⫺n = number of failed assemblies due to (N ⫺ n) reported “conforming” items Xj = the functional state of the jth item in the lot. Xj = 1 if the item is “conforming” and Xj = 0 otherwise Wj = the inspection result of the jth item in the lot. Wj = 1 if the item is reported “conforming” and Wj = 0 otherwise Model assumptions: 1. Components in the purchased lot and in the spare lot have the same quality. 2. (X1,X2,$,Xn)兩P are i.i.d. Bernoulli trials. 3. (P,X1,X2,$,Xn) and (Q1, Q2) are probabilistically independent. The inspection errors have nothing to do with the supplier’s item quality. 4. (X1,W1), (X2,W2),$,(XN,WN)兩P,Q1Q2 are i.i.d. random pairs. 5. No testing errors for the assembled products. Thus, according to the above assumptions, we obtain Pr{Wj=0兩P=p,Q1=q1,Q2=q2} =Pr{Wj=0,Xj=1兩P,q1,q2}+Pr{Wj=0,Xj=0兩p,q1,q2} =q1p+(1 ⫺ q2)(1 ⫺ p)
(1)
Let U = Q1(1 ⫺ P) + (1 ⫺ Q2)P, which is the probability that an item is reported “conforming” when inspected. Under the above assumptions, it can be shown that the probability that the number of items reported as “conforming” in a sample size of n, given that U = u, is a binomial distribution with parameters n and u. Therefore, Pr{T1=tn}=E[Pr{T1=t1兩n,U}] =
冘冤 冥 u
n n⫺t u 1(1 ⫺ u)t1Pr{U=u} t1
(2)
On the other hand, the probability that an item is nonconforming given that it is reported as “conforming” is Pr{Xj=0兩Wj=1}=E[Pr{Xj=0兩Wj=1,P,Q1,Q2}兩Wj=1]
冤
=E
Q2(1 ⫺ P) (1 ⫺ Q1)P + Q2(1 ⫺ P)
冥
(3)
The model has a two-stage decision requirement (Fig. 2). The first is to determine the optimum sample size, and the second
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C.-C. Chyu and F.-C. Wu
Fig. 3. Decision tree for this model.
decides what to do for the remaining components of the lot, i.e. either to stop inspection or to continue to inspect all the components. Let symbol s denote “stop inspection” and d denote “inspect all”. Given that the sampling outcome is (n, t1), the cost due to action s is
冘 t1
C(s兩n,t1)=nk1+
Mjk1+Zk2+Znk2
(4)
j=1
冘
Table 1. The distributions of inspection errors Q1 and Q2.
Pr{Qi Pr{Qi Pr{Qi Pr{Qi
= = = =
0.99} 0.95} 0.90} 0.80}
= = = =
(Qi = I) E(Qi) = 0.99
(Qi = II) E(Qi) = 0.946
(Qi = III) E(Qi) = 0.879
1.00 0.00 0.00 0.00
0.40 0.40 0.10 0.10
0.10 0.20 0.30 0.40
t1
C(d兩n,t1)=nk1+
Mjk1+Znk2+(N ⫺ n)k1
Table 2. The effects due to k2/k1 ratio and (Q1, Q2) with N = 500 and P 苲  (4.5, 0.5).
j=1
冘
t1+T2
+
Mjk1+ZN⫺nk2
(5)
k2/k1
Q1
Q2
(n*,c)
Cost
Q1
Q2
(n*,c) Cost
7
I II III
I I I
(17,3) (17,5) (0,0)
309.2 329.8 333.9
I I I
I II III
(17,3) 309.2 (16,3) 317.7 (15,3) 329.4
9
I II III
I I I
(24,3) (24,5) (31,10)
348.1 381.2 402.1
I I I
I II III
(24,3) 348.1 (23,3) 363.0 (23,3) 384.3
12
I II III
I I I
(34,3) (34,5) (45,12)
388.1 433.9 468.3
I I I
I II III
(34,3) 388.1 (33,3) 412.0 (32,3) 447.0
j=t1+1
冘
t1+T2
=N$k1+
Mjk1+ZNk2
j=1
The action s is better if E[C(s兩n,t1)] ⱕ E[C(d兩n,t1)]. Otherwise, the action d is better. Further algebraic operations give the following.
冤冘 k1
E
冥 冤
Mj兩(n,t1) =t1E
j=1
1 兩n,t 1⫺U 1
冥
(6)
E[Z兩n,t1]=E[E[Z兩n,t1,P]兩n,t1]=(N ⫺ n)(1 ⫺ E[P兩n,t1]) (7) E[Zn兩n,t1]=nPr{Xj=0兩Wj=1,n,t1} =nE
冤
| 冥 | 冥
Q2(1 ⫺ P) n,t (1 ⫺ Q1)P+Q2(1 ⫺ P) 1
冤
(8)
Q2(1 ⫺ P) E[ZN⫺n兩n,t1]=(N ⫺ n)E n,t (1 ⫺ Q1)P+Q2(1 ⫺ P) 1
冤冘 | t1+T2
E
冥
Mj n,t1,U=u =(N ⫺ n)
J=t1+1
冤 冘 | 冥 冤 冤 冘 | 冥| 冥 冤 | 冥 t1+T2
E
u 1⫺u
(Q1,Q2)
(I, I) (I, II) (II, I) (II, II)
 (3.5,0.5) E(P) = 7/8
 (4.5,0.5) E(P) = 9/10
 (7.5,0.5)  (9.5,0.5) E(P) = 15/16 E(P) = 19/20
(n*,c) Cost
(n*,c) Cost
(n*,c) Cost
(n*,c) Cost
(35,4) (33,6) (35,4) (33,6)
(34,4) (27,5) (27,3) (26,5)
(23,3) (0, 0) (23,3) (0, 0)
(15,2) (0, 0) (0, 0) (0, 0)
405.0 445.3 429.1 469.7
冢 冤
Mj n,t1,U n,t1
J=t1+1
u =(N ⫺ n)E n,t 1⫺U 1
Thus, E[C(s兩n,t1)] ⱕ E[C(d兩n,t1)] implies that
363.6 401.4 381.2 419.0
(1 ⫺ E[P兩n,t1])k2 ⱕ 1+E
t1+T2
Mj n,t1 =E E
J=t1+1
(10)
Table 3. The effects of quality and inspection errors under N = 500 and k2/k1 = 10.
+E (11)
冤
280.4 304.8 288.5 304.8
| 冥冣 | 冥
U n,t 1⫺U 1
242.8 245.1 245.1 245.1
k1
Q2(1 ⫺ P) n,t k (1 ⫺ Q1)P+Q2(1 ⫺ P) 1 2
(12)
By applying Bayesian decision theory, we use the folding back method (Fig. 3) to decide the optimal sampling size
Baysian Analysis on Deming’s Model
n* and obtain the mathematical formulation of the model as follows.
冦冘 冦冘 冧冧
Table 4. The effects of the ratio and quality variance under N = 500 and E(P) = 9/10. k2/k1
[Min{E[C(s兩n,t1)],E[C(d兩n,t1)]}]
Min
0ⱕnⱕN
(Q1,Q2)
k1
n
Pr
Wj=k1
(13)
j=1
7 9
3.
Experimental Analysis
In this section, we present some numerical analysis on the model. From our experiments, we conclude that the (n, c) inspection sampling policy is likely to be the optimum policy when the ratio k1/k2 does not deviate far from the expected value of the component quality E(P), and uncertainty of the quality P exists. Table 1 presents the effects of type I and type II errors on the inspection sampling policy and the expected total cost. In the following three tables, Q1 and Q2 have three levels. Table 1 explains the distributions and expectations with respect to the notation (Qi= j), j = I, II, and III. Table 2 shows the effects of the ratio k2/k1 and inspection errors (Q1, Q2) on the inspection policy and expected total cost. Increasing the likelihood of a type I or type II inspection error, while holding the other parameters fixed, will lead to a cost increase and a more loose (n*, c) inspection policy, or in other words, an increase of the ratio c/n*. If E(Q1) or E(Q2) is bad, the inspection policy will suggest inspecting no items in order to avoid the greater additional cost due to inspection errors. As the ratio k2/k1 increases, inspection policy becomes more restrictive, and the expected total cost will increase while the other parameters are held unchanged. However, the cost effect due to type errors is more severe than for type II errors. Table 3 shows the effects of component quality and the two inspection errors on the inspection policy and the expected total cost. As E(P) increases, the expected total cost decreases and the ratio c/n* increases, and the inspection policy becomes more loose. As the inspection errors become severe, the (n, c) policy becomes looser, and no inspection tends to be the best policy when the component quality is judged to be “good.” Table 4 presents the effects of the degree of uncertainty of the component quality on the inspection policy and expected total cost. In the case where 1 ⫺ E(P) ⬍ k1/k2, the inspection policy tends to be loose as the degree of uncertainty, Var(P), decreases. If Var(P) is very small, the optimum inspection policy would be no inspection at all, which is the same as stated by Deming. On the other hand, in the case where 1 ⫺ E(P) ⬎ k1/k2, the inspection policy will suggest inspecting all items if the value of P is certain.
4.
Conclusion
This paper studies an extension of Deming’s model. This new model takes into account the two inspection errors, which
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12
(I, I) (II, II) (I, I) (II, II) (I, I) (II, II)
 (4.5,0.5) Var(P) = 0.015
 (9,1) Var(P) = 0.0082
 (45,5) Var(P) = 0.0018
(n*,c) Cost
(n*,c) Cost
(n*,c) Cost
(17,3) (0, 0) (24,3) (23,5) (34,3) (32,5)
(20,4) (0, 0) (30,4) (22,5) (44,4) (42,6)
(0, 0) (0, 0) (22,4) (0, 0) (96,9) (70,9)
309.2 333.9 348.1 395.4 388.1 458.3
339.0 341.3 391.6 434.0 441.5 508.7
349.6 349.6 449.5 449.5 523.4 579.9
occur very often in practice. Mathematical formulae using a Bayesian approach with respect to this model are derived and a computer program is reported that can be used in industry. Some numerical analysis of the effects on the model due to inspection errors, component quality, and k2/k1 ratio are also presented.
References 1. W. E. Deming, Quality, Productivity, and Competitive Position, Massachusetts Institute of Technology, Center for Advanced Engineering Study, Cambridge, MA, 1982. 2. A. Hald, “The compound hypergeometric distribution and a system of single sampling inspection plans based on prior distributions and costs”, Technometrics, 2, pp. 275–340, 1960. 3. K. E. Case J. B. Keats, “On the selection of a prior distribution in Bayesian acceptance sampling”, Journal of Quality Technology, 14, pp. 10–18, 1982. 4. E. P. Papadakis, “The Deming inspection criterion for choosing zero of 100 percent inspection”, Journal of Quality Technology, 17, pp. 121–127, 1985. 5. S. B. Vardeman, “The legitimate role of inspection in modern SQC”, American Statistician, 40, pp. 325–328, 1986. 6. V. E. Sower, J. Motwani and M. J. Savole, “Are acceptance sampling and SPC complementary of incompatible?”, Quality Progress, 26(9), pp. 85–89, 1993. 7. D. K. Fitzsimmons, “Gaining acceptance for acceptance sampling”, Quality Progress, 22(4), pp. 46–48, 1989. 8. G. W. Milligan, “Is sampling really dead?”, Quality Progress, 24(2), pp. 77–81, 1991. 9. R. J. Burke, R. D. Davis and F. C. Kaminsky, “The (k1,k2) game”, Quality Progress 26(2), pp. 49–53, 1993. 10. F. C. Kaminsky and K. R. Haberle, “Probability distributions for the Deming cost models”, Journal of Quality Technology, 27(5), pp. 355–362, 1995. 11. R. E. Barlow and X. Zhang, “A critique of Deming’s discussion of acceptance sampling procedures”, in: A. P. Basu (ed.), Reliability and Quality Control, pp. 21–32, North-Holland, Amsterdam, 1986. 12. R. E. Barlow and X. Zhang, “Bayesian analysis of inspection sampling procedures discussed by Deming”, Journal of Statistical Planning and Inference 16, pp. 285–296, 1987. 13. J. Q. Berger, Statistical Decision Theory and Bayesian Analysis, Springer-Verlag, 1985.