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A mixed inspection policy for CSP-1 and precise inspection under inspection errors and return cost
Computers & Industrial Engineering 57 (2009) 652–659
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A mixed inspection policy for CSP-1 and precise inspection under inspection errors and return cost q Hong-Fwu Yu a,*, Wen-Ching Yu b, Wen Ping Wu c a
Graduate Institute of Commerce, National Kaohsiung University of Applied Sciences, 415 Chien Kung Road, Kaohsiung 807, Taiwan, ROC Department of Business Management, National United University, 1, Lien Da, Kung-Ching Li, Miao-Li 360, Taiwan, ROC c Department of Industrial Management, National Formosa University, 64, Wen Hua Road, Hu-Wei, Yunlin 632, Taiwan, ROC b
a r t i c l e
i n f o
Article history: Received 2 October 2007 Received in revised form 3 April 2008 Accepted 27 December 2008 Available online 10 January 2009 Keywords: CSP-1 Inspection errors Return cost Optimal sampling policy
a b s t r a c t This paper is to deal with a mixed policy between precise inspection and CSP-1 with inspection errors (Types I and II) and return cost. First, the concept of a renewal reward process is employed to obtain the long-term net profit of the mixed inspection policy. Then, with respect to non-repairable and repairable products, we determine the following four decision variables such that the unit net profit is maximal: (1) the optimal clearance number, (2) the optimal sampling frequency, and (3) the proportions which should be taken precise inspection for the non-inspected items in the procedure of CSP-1 and the non-defectives identified by CSP-1. Overall, the analytical results indicate that depending on seven parameters (Type I error, Type II error, the selling price of an item, the unit repair cost, the unit return cost, the unit cost of precise inspection, and the process defective fraction), there are three possible optimal inspection policies for CSP-1: ‘‘Do not inspect”, ‘‘Do 100% inspection”, or any setting for (i, s); for both of the non-defectives identified in CSP-1 and the non-inspected items, there are three possible proportions which should be performed a precise inspection: all, none, or any proportion. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, consumers are paying more and more attention to merchandise quality and reliability. However, due to technology limits, worn-down mechanical parts, or human negligence, the production of defective items is inevitable. Hence, in order to satisfy customers’ requirement, the manufacturer must make efforts to screen out defective items in the outgoing batch. Tang and Tang (1994) surveyed the literature on the approaches of designing screening procedures. Among these screening procedures, inspection for product quality continues to be an important means. For a continuous production process, CSP-1 is the earliest method used to control the quality of products. It consists of 100% inspection and random partial inspection (Dodge, 1943). In CSP-1, if the product defective fraction is known, average fraction inspected (AFI), average outgoing quality (AOQ), and average outgoing quality limit (AOQL) can be calculated (Dodge, 1943). There are many related researches on CSP-1, such as Resnikoff (1960) proposed in minimum average inspection fraction for a continuous sampling plan. Chen and Chou (2002) studied the inspection method under linear inspection cost model to decide the minimum total expected cost. Govindaraju and Kandasamy (2000) proposed the
q
This manuscript was processed by Area Editor E.A. Elsayed. * Corresponding author. Tel.: +886 7 3814526x6706. E-mail address: [email protected] (H.-F. Yu).
0360-8352/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.12.008
modified CSP-1 plan, which is based on single level continuous sampling plan with the acceptance number. The modified plan is similar to CSP-1 but differs in inspection process; it has an acceptance number of defectives and will shift into 100% inspection when exceeds this limit. The advantage of this plan is to achieve a reduction in the average fraction inspected at good quality levels. Chen and Chou (2004) considered the minimum average fraction inspected (AFI) for Govindaraju and Kandasamy’s (2000) modified CSP-1 plan. They developed a solution procedure to find the CSP-1 parameters that will meet the AOQL requirement, while also minimizing the AFI for the modified CSP-1 plan when the process average is known. Two basic assumptions are involved in CSP-1: (i) inspections are assumed to be perfect and (ii) the production process is not perfect. It has a constant defective rate. Also, some defectives are permitted for the products shipped to the consumer under the protection of a pre-specified AOQL. Most studies mentioned earlier have been operated under these two assumptions. However, in real applications, normal inspection may have errors that can decrease the outgoing quality. In fact, there are two types of inspection errors: Type I and II. Type I error occurs when the good products get classified as defectives that will be destroyed or reworked. This error can lead to increase in the production cost. Type II error happens when the defectives are classified as non-defectives and are allowed for shipment. This type of error can affect customer’s loyalty. Presence of inspection errors in CSP-1 plan was first discussed in Case, Bennett, and Schmidt (1973) that mentioned
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the need of an alternate sampling method. Case et al. (1973) developed a compensating sampling plan, considering inspection errors, which yield the desired actual AOQL. Endres (1979) then discussed the minimum AFI plan in case of inspection error. Wang and Chen (1997) further modified Endres’s (1979) method and formulated a minimum average fraction inspection model under inspection errors. In real situations, some machinery parts are prohibited to have defectives, such as vehicle brake system, radar homing system, air defense system, etc. For such parts, a great fine or a high return cost will be put on the purchase contract. Haji and Haji (2004) employed a renewal reward process on CSP-1 to develop optimal policies in terms of the clearance number. Their results showed that depending on the defective fraction of the production process, the unit rework cost, the return cost, and the unit inspection cost, there are only three different optimal policies: (1) the number of consecutive inspection is zero (partial inspection), (2) the number of consecutive inspection is infinity (100% inspection) in CSP-1, or (3) the number of consecutive inspection can be any value from zero to infinity. It is worth noting that due to the defective rate inherent in the production process and Type II error, the non-inspected portion in CSP-1 may still contain defectives. Additionally, one would normally discard defectives found in a production; however, because of the presence of Type I error, such action may cause great loss in opportunity cost and production cost. One of the effective methods to screen out the defectives is a precise inspection. It can almost 100% screen out the defectives among the lot shipped to the consumer. Also, precise inspection can pick up the good items among the apparent defectives identified in the procedure of CSP-1 and thus can prevent the additional production resulting from Type I error. However, precise inspection is usually costly. An example of the ‘‘precise inspection” can be seen in finding faults on ball grid array (BGA). BGA is a packaging for surface-mounted integrated electronic circuit and is sometimes very difficult to look for faults. For such a difficulty, X-ray can almost 100% screen out the defectives; while, it is costly. Obviously, the costliness of precise inspection is in conflict with the requirement of high quality of the shipped lot. The purpose of this paper is to determine the optimal mixed policy of precise inspection and CSP-1 with inspection errors and return cost. More specifically, with respect to nonrepairable and repairable products, by adopting a precise inspection to screen out the defectives for the products behind the procedure of CSP-1, we determine the following four decision variables such that the unit net profit is maximal: (1) the optimal clearance number, (2) the optimal sampling frequency, and (3) the proportions which should be taken precise inspection for the non-inspected items in the procedure of CSP-1 and the non-defectives identified by CSP-1. This paper is organized as follows. Section 2 briefly introduces the procedure of the mixed inspection policy and the corresponding optimization model. Section 3 shows the derivation of the optimal net profit model. Section 4 gives a numerical example to demonstrate the optimal policy. Finally, we give some concluding remarks in Section 5.
2. Inspection procedure and the optimization model In this section, we will introduce the procedure of the mixed inspection policy and the corresponding optimization model. The inspection procedure of the mixed policy considered in this paper is as follows: 1. At the outset, inspect 100% of the units consecutively as produced and continue such inspection until i units in succession are found to be non-defective.
2. When i units in succession are found to be non-defective, discontinue 100% inspection, and inspect only a fraction f of the units, selecting individual sample units one at a time from the flow of product, in a random manner to assure an unbiased sample. Continue this partial inspection until a sample unit is found defective and then revert immediately to the 100% inspection of succeeding units, as in step 1. 3. Precision inspections are taken on a proportion r1 of the nondefectives identified in Steps 1–2 and on a proportion r2 of items not inspected in Step 2, respectively. 4. As to the defectives screened out in Steps 1–3, they will be repaired if the products are repairable; otherwise will be discarded. Note that Step 1 and Step 2 are the process of CSP-1. The CSP-1 process goes from Step 1 to Step 2 and then goes back to Step 1. For ease exposition, we call such process a cycle. Let R and TC be the corresponding expected revenue and expected total cost, respectively. Then the corresponding expected profit P is
P ¼ R TC Next, let Q represent the expected production quantity in a cycle. Thus, the net profit in a cycle can be written as:
NP ¼
P R TC ¼ Q Q
ð1Þ
Let W be the number of products produced until time t. Since there are defectives and inspection errors in CSP-1, the number of actually sold products may be less than W. If the profit earned is P(W), then the net unit profit for the W products is:
NPðWÞ ¼
PðWÞ W
Therefore, from the property of a renewal process, we can obtain (see Ross, 1993)
PðWÞ W E½profit earned in a cycle ¼ NP ¼ E½production quantity in a cycle
lim NPðWÞ ¼ lim NPðWÞ ¼ lim
t!1
W!1
W!1
It is quite obvious that different arrangements with i, f, r1, and r2 will influence the cost, profit, and shipment quantity. Our goal is to decide the optimal combination of i, f, r1, and r2 for CSP-1 in order to achieve the maximal net unit profit. Let N0 = {0, 1, 2, 3, . . .} and I = [0, 1]. Then the decision problem can be expressed as the following optimization model:
Maximize
NP
Subject to i 2 N0 f ; r1 ; r2 2 I 3. Optiaml inspection policy In this section, we will establish the expressions for the expected revenue and the expected total cost and then determine the optimal inspection policy. Let p be the process defective fraction (non-defective fraction q = 1 p), U be the expected number of items manufactured in Step 1, and V be the expected number of items manufactured in Step 2. Then U = Ug + Ud and V = Vg + Vd + Vn, where Ud and Vd are the expected numbers of defectives found in Steps 1 and 2, respectively; Ug and Vg are the expected numbers of non-defectives identified in Steps 1 and 2, respectively; Vn is the expected number of
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non-inspected products in Step 2. Obviously, Vd = 1 and, from Dodge (1943), we know:
1 qi pqi 1 V¼ fp 1 qi Ud ¼ qi 1 Vg ¼ 1 p U¼
ð2Þ
sqiid qiid þ 1 pid qiid 1 qiid 1 1 Ug ¼ pid qiid
ð3Þ
and
ð4Þ
Vn ¼
ð5Þ
sqi qi þ 1 pqi 1 qi 1 1 Ug ¼ U Ud ¼ p qi Q ¼UþV ¼
ð6Þ ð7Þ
and
1 1 ðs 1Þ V n ¼ V ðV g þ V d Þ ¼ 1 ¼ p f p
ð8Þ
where s = 1/f. Thus, the non-defectives and defectives in Vn can be:
Vn q ¼
q 1 q 1 ¼ ðs 1Þ p f p
ð70 Þ
1 1 ðs 1Þ 1 ¼ pid f pid
ð80 Þ
and
Vn p ¼
So,
ð60 Þ
Q¼
pðs 1Þ pid
ð90 Þ
Thus, in Step 1, the expected number of identified non-defectives which are truly defectives is
1 qiid ð1 p2 Þ U g ¼ ð1 p2 Þ qiid
! 1 1 pid
Also, in Step 2, the expected number of identified non-defectives which are truly defectives is
ð1 p2 Þ V g ¼ ð1 p2 Þ
1 1 pid
Hence, the total expected number of the identified non-defectives which are truly defectives in a cycle is
ð1 p2 Þ U g þ ð1 p2 Þ V g ¼ ð1 p2 Þ
and
Vn p ¼
1 1¼s1 f
ð10Þ
ð9Þ 3.1. Non-repairable products (Case 1)
Suppose that Type I error (non-defectives are classified as defectives) and Type II error (defectives are classified as non-defectives) occurring in the inspections taken in CSP-1 are a and b, respectively. Define the following four events: Eid: an item is identified as a defective Etd: an item is truly a defective Eig: an item is identified as a non-defective Etg: an item is truly a non-defective Let pid = P(Eid), qid = P(Eig), p1 = P(Etd|Eid), and p2 = P(Etg|Eig). Then it can be easily shown that
pid ¼ að1 pÞ þ ð1 bÞp qid ¼ 1 pid ¼ ð1 aÞð1 pÞ þ bp p1 ¼
1 pid : pid qiid
In this case, because the defectives identified in the procedure of CSP-1 will go through a precise inspection to pick up all of the misidentified items, there will be a precise inspection cost. Next, since r1 and r2 proportions for the non-defectives and non-inspected products will have a precise inspection, this will result in a precise inspection cost; while, because the remaining portions which do not have the precise inspection may contain defectives, a return cost may be incurred. Overall, the expected total cost consists of the following four segments: (a) expected precise inspection cost, (b) expected return cost, (c) expected inspection cost, and (d) expected production cost. These four costs are discussed below: (a) Expected precise inspection cost: The cost consists of three parts. (a-1) From Eq. (40 ), we know the expected number of defectives identified in the procedure of CSP-1 is U d þ V d ¼ q1i . So, the expected number of precise inspection
ð1 bÞp
að1 pÞ þ ð1 bÞp
id
for this portion is
and
1 . qiid
(a-2) From Eqs. (50 ) and (70 ), we know
the expected number of the non-defectives identified in
ð1 aÞð1 pÞ p2 ¼ ð1 aÞð1 pÞ þ bp
the procedure of CSP-1 is
U¼
1 qiid pid qiid
ð2 Þ
V¼
1 fpid
ð30 Þ
1 qiid qiid 1 1 Vg ¼ pid So,
and if we perform a precise
inspection on a r1 fraction of these, then the expected num-
Thus, under inspection errors, Eqs. (2)–(9) should be modified as follows:
Ud ¼
1pid , pid qiid
0
ð40 Þ ð50 Þ
id ber of precise inspection is r1 1p . (a-3) The expected nump qi id id
ber of the non-inspected items is
s1 , pid
and if we conduct a
precise inspection on a r 2 fraction of these products, then the expected number of precise inspection for this portion is r2 s1 . Overall, aggregating (a-1), (a-2), and (a-3), the p id
expected total precise inspection cost is
! 1 1 pid s1 Cp þ r1 þ r2 pid qiid pid qiid where Cp denotes the unit precise inspection cost.
ð11Þ
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(b) Expected return cost: The cost comprises two segments. (b1) According to Eq. (10), the expected number of the identified non-defectives which are truly defectives in a cycle is id . Since a ð1 r 1 Þ portion of the non-defectives ð1 p2 Þ 1p p qi
in CSP-1. Thus, the expected overall revenue can be expressed as follows:
Rð1Þ ¼ C Q e ¼C
id id
identified in CSP-1 does not have a precise inspection, the expected number of true defectives that will be shipped to id . (b-2) Since a the consumer is ð1 r 1 Þ ð1 p2 Þ 1p p qi
"
1 1 pid Q p1 i r1 ð1 p2 Þ qid pid qiid
!
pðs 1Þ r2 pid
id id
ð1 r2 Þ portion of the non-inspected items does not have a precise inspection, from Eq. (90 ), the expected number of defectives which are shipped to the consumer is ð1 r2 Þ pðs1Þ . Overall, collecting (b-1) and (b-2), the p id
expected total return cost is
! # 1 pid pðs 1Þ þ ð1 r2 Þ C t ð1 r 1 Þ ð1 p2 Þ pid pid qiid
ð16Þ 0
Substituting Eqs. (6 ), (15) and (16) into Eq. (1) gives the expected net profit as follows:
Rð1Þ TC ð1Þ ¼ Q (
(
ð1 p2 Þ ðC C t Þ C p sqiid qiid þ 1 !) 1 þ r 2 ½p ðC C t Þ C p 1 i sqid qiid þ 1
NPð1Þ ¼
"
ð12Þ
)
r 1 ð1 pid Þ
þ fC pC t C d
where Ct denotes the unit return cost. (c) Expected inspection cost: From Eqs. (20 ) and (50 ), we know the expected inspection number in CSP-1 process is 1qiid 1 . So, the expected inspection cost is þ p p qi
C pid p1 C t ð1 pid Þð1 p2 Þ þ C t p C s C p pid þ sqiid qiid þ 1 ð1Þ
ð1Þ
)
ð1Þ
¼ NP 1 þ NP2 þ NP 3 where
id
id id
Cs
#
1 qiid 1 þ pid pid qiid
ð1Þ
NP 1 ¼ r 1 ð1 pid Þ
! ¼ Cs
1 pid qiid
ð13Þ ð1Þ NP 2
where Cs denotes the unit inspection cost.
ð1 p2 Þ ðC C t Þ C p sqiid qiid þ 1
! 1 ¼ r 2 ½p ðC C t Þ C p 1 i sqid qiid þ 1
and (d) Expected production cost: based on Eq. (60 ), the expected production cost is
Cd Q ¼ Cd
sqiid
qiid þ pid qiid
þ 1
ð14Þ ð1Þ
where Cd denotes the unit manufacturing cost. Summing up the costs in (a)–(d), we can obtain the expected total cost as follows:
" TC ð1Þ ¼ C p
1 1 pid þ r1 qiid pid qiid
ð1Þ
NP3 ¼ C pC t C d
! þ r2
þ Cd Q þ Ct "
1 pid pid qiid
ð1 r 1 Þ ð1 p2 Þ
# s1 1 þ Cs pid pid qiid
! þ ð1 r 2 Þ
pðs 1Þ pid
#
ð15Þ As to the expected income Rð1Þ , it is the product of the selling price (C) and the expected number of the products shipped to the consumer (Qe). Note that
Q e ¼ Q Q d1 Q d2 Q d3 Q d4 1qi is the expected number of the identified where Q d1 ¼ p1 qi id id
defectives which are truly defective in step 1, Q d2 ð¼ p1 1Þ is the expected number of the identified defective which is truly defective id is the expected number of in step 2, Q d3 ¼ r 1 ð1 p2 Þ 1p i p q id id
defectives contained in the r1 fraction of the non-defectives identi fied in CSP-1, and Q d4 ¼ r2 pðs1Þ is the expected number of the p id
defectives contained in the r2 fraction of the non-inspected items
C pid p1 C t ð1 pid Þð1 p2 Þ þ C t p C s C p pid sqiid qiid þ 1 ð1Þ
Let D1 ¼ ð1 p2 Þ ðC C t Þ C p ; D2 ¼ p ðC C t Þ C p and Dð1Þ In 3 ¼ C pid p1 C t ð1 pid Þð1 p2 Þ þ C t p C s C p pid . this scenario, r1, r2, i and s are determined by first determining i ð1Þ ð1Þ ð1Þ and s from NP 3 and then substituting i and s into NP2 and NP1 to determine r2 and r1. The derivation steps are shown below: ð1Þ (A) When D3 < 0: ð1Þ increases as Since sqiid qiid þ 1 > 0, the value for NP 3 i i sqid qid þ 1 increases. So, if we take s = 1 and i 2 N 0 , then ð1Þ sqiid qiid þ 1 ¼ 1 and NP 3 has the maximum value. Substituting ð1Þ ð1Þ ð1Þ ð1Þ i i sqid qid þ 1 ¼ 1 into NP 2 can get NP 2 ¼ r2 D2 1 ¼ r2 D2 . ð1Þ Thus, the value of r2 can be determined depending on D2 to give ð1Þ the maximum value for NP 2 as follows:
8 ð1Þ > if D2 > 0 > < 1; ð1Þ r 2 ¼ 0; if D2 < 0 > > : ð1Þ any value between 0 and 1; if D2 ¼ 0
ð1Þ
Similarly, substituting sqiid qiid þ 1 ¼ 1 into NP1 can get ð1Þ NP1 ¼ r 1 0 ¼ 0. Thus, r1 can take any value between 0 and 1 (i.e., r 1 2 I). Note that the inspection policy s = 1 and i 2 N 0 for CSP-1 implies that after the sampling plan goes Step 2, no inspection can be conducted any longer, since the fraction of the units which will 1 ¼ 0. Here, for simplicity, we call such an be inspected is f ¼ 1s ¼ 1 inspection policy ‘‘Do not inspect”. As to the non-defectives identified in CSP-1 and the non-inspected items, any proportion of the former can be performed a precise inspection; there are three possible proportions for the latter which should be performed a precise inspection: all, none, or any proposition.
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H.-F. Yu et al. / Computers & Industrial Engineering 57 (2009) 652–659 ð1Þ
(B) When D3 > 0: ð1Þ Similarly, because sqiid qiid þ 1 1, the NP 3 value will increase as sqiid qiid þ 1 decreases. So, if we take ðs ¼ 1; i 2 N 0 Þ or ðs 2 R0 ; i ¼ 1Þ where R0 ¼ ð1; 1Þ, then we have sqiid qiid þ 1 ¼ 1 ð1Þ and NP 3 will has the maximum value. Substituting sqiid qiid þ 1 ¼ 1 ð1Þ ð1Þ ð1Þ ð1Þ into NP2 can get NP 2 ¼ r2 D2 0 ¼ 0. The zero value of NP 2 indicates that r2 can take any value between 0 and 1 (i.e., r 2 2 I). Subseð1Þ quently, substituting sqiid qiid þ 1 ¼ 1 into NP 1 can get: ð1Þ
NP 1 ¼ r 1 ð1 pid Þ
Dð1Þ 1 1
ð1Þ
¼ r 1 ð1 pid Þ D1
ð1Þ
Thus, the value of r1 can be determined depending on D1 to give ð1Þ the maximum value for NP 1 as follows:
8 ð1Þ > if D1 > 0 > < 1; ð1Þ r 1 ¼ 0; if D1 < 0 > > : ð1Þ any value between 0 and 1; if D1 ¼ 0 Note that it is easily seen that both of the inspection policies ðs ¼ 1; i 2 N0 Þ and ðs 2 R0 ; i ¼ 1Þ mean that CSP-1 should perform 100% inspection of the units consecutively. Here, for simplicity, we call such inspection policies ‘‘Do 100% inspection”. As to the non-defectives identified in CSP-1 and the non-inspected items, there are three possible proportions for the former which should be performed a precise inspection: all, none, or any proposition; any proportion of the latter can be performed a precise inspection. ð1Þ (C) When D3 ¼ 0: It is easily seen that
ð1Þ
ð1Þ
value of NP 3 . In such a situation, we will determine the values of r1, r2, i and s via ð1Þ NP1
ð1Þ NP 1
þ
ð1Þ NP 2
þ
ð1Þ NP 2
þ
1 , sqi qi þ1
Let k ¼
id
then 0 k 1. Thus,
id
can be re-written as follows:
ð1Þ
That is, NP1 þ NP 2 is a linear combination of r 1 ð1 pid ÞD1 and ð1Þ r 2 D2 . Accordingly, let ð1Þ M1
n
¼ Max r 1 ð1 0r 1 1
ð1Þ pid ÞD1
o
then it can be easily shown that if
and
ð1Þ M2
ð1Þ NP 1
þ
n o ð1Þ ¼ Max r 2 D2 0r 2 1
ð1Þ NP2
ð1Þ
ð1Þ
M1
ð1Þ
id
number of the non-defectives identified in the procedure of CSP1pid , pid qi
and if we perform a precise inspection on a r1 fraction
of these, then the expected number of true defectives that can be id . identified by the precise inspection will be r 1 ð1 p2 Þ 1p i p q id id
Thus,
expected repair cost for this portion is id . (e-3) As mentioned in (a-3) of Case 1, C r r1 ð1 p2 Þ 1p i p q
the
the expected number of the non-inspected items is
8 ð1Þ ð1Þ > > < ð1 pid ÞD1 ðhere r 1 ¼ 1Þ; if D1 > 0 ð1Þ ¼ 0 ðhere r 1 ¼ 0Þ; if D1 < 0 > > : ð1Þ 0 ðhere r 1 2 IÞ; if D1 ¼ 0 8 ð1Þ ð1Þ > > < D2 ðhere r2 ¼ 1Þ; if D2 > 0 ð1Þ ¼ 0ðhere r 2 ¼ 0Þ; if D2 < 0 > > : ð1Þ 0ðhere r 2 2 IÞ; if D2 ¼ 0
The results associated with this case indicate that (i) depending on ð1Þ ð1Þ M1 and M 2 , there are three possible optimal inspection policies for CSP-1: ‘‘Do not inspect”, ‘‘Do 100% inspection”, or any setting
s1 , pid
and if we
conduct a precise inspection on a r 2 fraction of these products, then the expected number of true defectives that can be identified by the precise inspection will be r 2 pðs1Þ . Thus, the expected repair p id
cost for this portion is C r r 2 pðs1Þ . Overall, aggregating (e-1), (ep id
2), and (e-3), the expected repair cost is:
"
1 1 pid C r p1 i þ r 1 ð1 p2 Þ qid pid qiid
!
pðs 1Þ þ r2 pid
# ð17Þ
Summing up the costs in (a)–(e), we can obtain the expected total cost as follows:
"
ð1Þ
and M2 , it can be easily obtained as
and
M2
id
is C r p1 q1i . (e-2) As mentioned in (a-2) of Case 1, the expected
is maximal if and only
8 ð1Þ ð1Þ 0 0 > if M1 > M 2 > < 1 ði:e:;ðs ¼ 1; i 2 N Þ or ðs 2 R ; i ¼ 1ÞÞ; ð1Þ ð1Þ k ¼ 0 ði:e:;s ¼ 1 and i 2 N 0 Þ; if M1 < M 2 > > : ð1Þ ð1Þ any value between 0 and 1 ði:e:;ði;sÞ any valueÞ; if M1 ¼ M 2 As for the values of M1 follows:
id
defectives is p1 q1i . Thus, the expected repair cost for this portion
id id
¼ k ½r1 ð1 pid ÞD1 þ ð1 kÞ ðr 2 D2 Þ
ð1Þ
In this case, because the defectives identified in the procedure of CSP-1 will go through a precise inspection to pick up all of the mis-identified items, there will be a precise inspection cost. Next, since r1 and r2 proportions for the non-defectives and non-inspected products will have a precise inspection, this will result in a precise inspection cost; while, because the remaining portions which do not have the precise inspection may contain defectives, a return cost may be incurred. Finally, since the defectives identified by precise inspection will be repaired, a repair cost will be incurred. Overall, the expected total cost consists of the following five segments: (a) expected precise inspection cost, (b) expected return cost, (c) expected inspection cost, (d) expected production cost, and (e) expected repair cost. It is easily seen that the first four costs are the same as the ones in (a), (b), (c), and (d) of Case 1, respectively. As to the expected repair cost, it consists of three parts. (e-1) As in (a) of Case 1, the expected number of defectives found by CSP-1 is q1i . Then, of these, the expected number of true
id
That is, NP 3 is a constant. This means that ði; sÞ does not affect the ð1Þ NP 2 .
ð1Þ
3.2. Repairable products (Case 2)
1 is
ð1Þ
NP 3 ¼ C pC t C d þ 0 ¼ C pC t C d
ð1Þ NP 1
ð1Þ
for (i, s); and (ii) depending on D1 and D2 , there are three possible proportions for both of the non-defectives identified in CSP-1 and the non-inspected items which should be performed a precise inspection: all, none, or any proportion.
TC
ð2Þ
! # p1 1 pid pðs 1Þ þ r2 ¼ C r i þ r 1 ð1 p2 Þ pid qid pid qiid " ! # 1 1 pid s1 þ r2 þ Cd Q þ C p i þ r1 pid pid qiid qid " ! # 1 pid pðs 1Þ þ ð1 r þ C t ð1 r1 Þ ð1 p2 Þ Þ 2 pid pid qiid þ Cs
1 pid qiid
ð18Þ
As to the expected revenue Rð2Þ , it is the product of the selling price (C) and the expected number of the products shipped to the consumer (Qe). Note that because the defectives identified in the procedure of CSP-1 will go through a precise inspection to pick
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H.-F. Yu et al. / Computers & Industrial Engineering 57 (2009) 652–659
up all of the mis-identified items and the remaining defectives will be repaired, all of the production quantity will be shipped to the consumer. Hence,
Qe ¼ Q Thus, the expected overall revenue is
Rð2Þ ¼ C Q
ð19Þ
Substituting Eqs. (60 ), (18) and (19) into Eq. (1) gives the expected net profit as follows:
NP ð2Þ ¼ ( ¼ (
Rð2Þ TC ð2Þ Q
ð1 p2 Þ ðC r C t Þ C p r 1 ð1 pid Þ sqiid qiid þ 1
)
ð2Þ
(C) When D3 ¼ 0: Let
þ fC pC t C d C r pid p1 C t ð1 pid Þð1 p2 Þ þ C t p C s C p pid þ sqiid qiid þ 1 ¼
ð2Þ NP 1
þ
ð2Þ NP2
þ
)
ð2Þ NP 3
where ð2Þ
NP1 ¼ r 1 ð1 pid Þ
ð1 p2 Þ ðC r C t Þ C p sqiid qiid þ 1
ð2Þ NP2 ¼ r 2 p ðC r C t Þ C p 1
As to the value of r 1 , it can take any value between 0 and 1 (i.e., r1 2 I). ð2Þ (B) When D3 > 0: Similarly, in this scenario, we take ðs ¼ 1; i 2 N 0 Þ or ðs 2 R0 ; i ¼ 1Þ. As to the value of r 2 , it can take any value between 0 and 1 (i.e., r2 2 I). Finally, the value of r 1 can be determined as follows:
8 ð2Þ > if D1 > 0 > < 1; ð2Þ r 1 ¼ 0; if D1 < 0 > > : ð2Þ any value between 0 and 1; if D1 ¼ 0
!) 1 r2 ½p ðC r C t Þ C p 1 i sqid qiid þ 1
þ
8 ð2Þ > if D2 > 0 > < 1; ð2Þ r 2 ¼ 0; if D2 < 0 > > : ð2Þ any value between 0 and 1; if D2 ¼ 0
n o n o ð2Þ ð2Þ ð2Þ ð2Þ and M 2 ¼ Max r 2 D2 M1 ¼ Max r1 ð1 pid ÞD1 0r 1 1
then
8 ð2Þ ð2Þ 0 0 > > < ðs ¼ 1; i 2 N Þ or ðs 2 R ; i ¼ 1Þ; if M 1 > M2 ð2Þ ð2Þ 0 s ¼ 1 and i 2 N ; if M 1 < M2 > > : ð2Þ ð2Þ ði; sÞ any value; if M 1 ¼ M2 ð2Þ
1 sqiid qiid þ 1
!
06r 2 61
ð2Þ
where M1 and M2 can be obtained as follows: ð2Þ
M1
and
8 ð2Þ ð2Þ > > < ð1 pid ÞD1 ðhere r 1 ¼ 1Þ; if D1 > 0 ð2Þ ¼ 0 ðhere r 1 ¼ 0Þ; if D1 < 0 > > : ð2Þ 0 ðhere r 1 2 IÞ; if D1 ¼ 0
ð2Þ
NP 3 ¼ C pC t C d þ
and
C r pid p1 C t ð1 pid Þð1 p2 Þ þ C t p C s C p pid sqiid qiid þ 1
Dð2Þ 1
ð2Þ C p ; D2
Let ¼ ð1 p2 Þ ðC r C t Þ ¼ p ðC r C t Þ C p and Dð2Þ 3 ¼ C r pid p1 C t ð1 pid Þð1 p2 Þ þ C t p C s C p pid . ð2Þ ð2Þ ð2Þ ð1Þ ð1Þ ð1Þ Note that D1 ; D2 , and D3 are the same as D1 ; D2 , and D3 , respectively except that Cr should be replaced by C. Hence, the optimal combinations of r1, r2, i and s for Case 2 can be easily obtained by duplicating the results for Case 1 obtained above. The results are summarized as follow: ð2Þ (A) When D3 < 0: In this scenario, we take s ¼ 1 and i 2 N 0 . Next, the value of r2 can be determined as follows:
ð2Þ
M2
8 ð2Þ ð2Þ > > < D2 ðhere r 2 ¼ 1Þ; if D2 > 0 ð2Þ ¼ 0 ðhere r 2 ¼ 0Þ; if D2 < 0 > > : ð2Þ 0 ðhere r 2 2 IÞ; if D2 ¼ 0
In the next section, due to the similarity in the structures of Case 1 and Case 2, we will take Case 1 as an example to demonstrate the procedure of the proposed method. 4. Numerical example To simplify the calculation, we set Cd = 1. For illustrative purpose, Table 1 lists six combinations of p, a, b, C, C t , C s , and C p which
Table 1 The optimal inspection policies corresponding to six combinations of p, a, b, C, Ct, Cs, and Cp. (p, a, b, C, Ct, Cs, Cp)
The proposed model ð1Þ
(0.15, 0.025, 0.05, 5, 10, 0.01, 0.1) (0.15, 0.15, 0.30, 2.5, 20, 0.01, 0.1) (0.20, 0.10, 0.20, 2.5, 15, 0.01, 0.1) (0.20, 0.10, 0.20, 2.5, 2.3, 0.01, 0.1) (0.20, 0.10, 0.20, 2.0875, 2.3, 0.01, 0.1) (0.05, 0.01, 0.02, 2, 2, 0.2, 0.07)
ð1Þ
ð1Þ
HHM
ðD1 ; D2 ; D3 Þ ði; s; r1 ; r2 Þ
NP
(0.05516, 0.6500, 0.6861) ð1; R0 ; 0; IÞ or ðN 0 ; 1; 0; IÞ (0.9261, 2.525, 1.8043) 1; R0 ; 1; IÞ or ðN 0 ; 1; 1; IÞ (0.5579, 2.4, 1.966) ð1; R0 ; 1; IÞ or ðN 0 ; 1; 1; IÞ (0.1105, 0.14, 0.066) ðN0 ; 1; I; 1Þ (0.08882, 0.0575, 0) (i, s): any setting (r1, r2) = (0, 0) (0.07, 0.07, 0.2041) (N0, 1, I, 1)
3.1861
Remark: X ¼ pid ðC d C t Þ þ C s (see Haji and Haji, 2004).
X
NPHHM
Inspection policy
1.015 0.89 0.90 0.6275 0.83
1.4638 100% Inspection 4.4075 100% Inspection 3.35 100% Inspection 0.302 100% Inspection 0.302 100% Inspection 0.0115 Partial inspection
3.0963 0.00875 0.29 1.04 0.6275 0.9
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H.-F. Yu et al. / Computers & Industrial Engineering 57 (2009) 652–659 ð1Þ
ð1Þ
are set to generate the scenarios associated with D3 < 0, D3 ¼ 0, ð1Þ ð1Þ ð1Þ ð1Þ and D3 > 0. The corresponding ðD1 ; D2 ; D3 Þ s are also given in ð1Þ ð1Þ ð1Þ Table 1. Based on the values of ðD1 ; D2 ; D3 Þ, the corresponding optimal inspection policies can be easily obtained by the procedures of Section 3. The results are listed in the second column of Table 1. For example, for (p, a, b, C, Ct, Cs, Cp) = (0.20, 0.10, 0.20, 2.0875, 2.3, ð1Þ 0.01, 0.1), because the corresponding D3 ¼ 0, according to the procedure of Section 3, the optimal inspection policy ði; s; r 1 ; r 2 Þ deð1Þ ð1Þ ð1Þ pends on the values of D1 and D2 . Since D1 ¼ 0:08882 and ð1Þ ð1Þ D2 ¼ 0:0575, by the definitions of M1 and Mð1Þ 2 , we can obtain ð1Þ ð1Þ M 1 ¼ 0 (here, we take r1 ¼ 0) and M2 ¼ 0 (here, we take ð1Þ ð1Þ r2 ¼ 0). Finally, since M 1 ¼ M2 ; we have that ði; sÞ can be set to be any value. At the end of this section, we make a comparison of the proposed model with that of Haji and Haji (2004) (called HHM). To make the comparison possible, the concept ‘‘the net unit profit” is selected as the comparison standard. In the following, we will build up the net unit profit associated with HHM. First, since the expected number of defectives found in the procedure of CSP-1 is q1i and the defectives are all replaced with perfect id
units, the quantity shipped to the customer in a cycle is Q e ¼ Q . So, the revenue is RHHM ¼ C Q and the production cost is
1 Cd Q þ i qid
!
Next, since precise inspection is not adopted, the expected number of defectives contained in the quantity shipped to the customer is
pðs 1Þ 1 pid þ ð1 p2 Þ pid pid qiid This results in a return cost
Ct
" # pðs 1Þ 1 pid þ ð1 p2 Þ pid pid qiid
Finally, as the result (c) of Case 1, the inspection cost for HHM is C s p 1qi . Thus, the total cost is id id
TC HHM
1 ¼ Cd Q þ i qid þ Cs
!
" # pðs 1Þ 1 pid þ Ct þ ð1 p2 Þ pid pid qiid
1 pid qiid
Hence, the net unit profit associated with HHM is
NP HHM ¼
RHHM TC HHM Q
¼ ðC pC t C d Þ
C d pid þ C t ½ð1 p2 Þð1 pid Þ p þ C s sqiid qiid þ 1
The optimal inspection policies and the net unit profits of HHM corresponding to the five combinations of p, a, b, C, and Ct in Table 1 are listed in the third column of Table 1. As expected, it is seen, from Table 1, that adopting precise inspection is profitable to the manufacturer if the inspection errors are large and/or the return cost is relatively large to the selling price.
ucts behind the procedure of CSP-1. The main purpose of this paper is to investigate the mixed inspection policy of CSP-1 and precise inspection under inspection errors and return cost. First, the concept of a renewal reward process is employed to obtain the longterm net profit of the mixed inspection policy. Then, by using the criterion of minimizing the long-term net profit, the following four decision variables for non-repairable and repairable products are determined: (1) the optimal clearance number, (2) the optimal sampling frequency, and (3) the proportions which should be taken precise inspection for the non-inspected items in the procedure of CSP-1 and the non-defectives identified by CSP-1. The analytical results indicate that depending on seven parameters (Type I error, Type II error, the selling price of an item, the unit repair cost, the unit return cost, the unit cost of precise inspection, and the process defective fraction), there are three possible optimal inspection policies for CSP-1: ‘‘Do not inspect”, ‘‘Do 100% inspection”, or any setting for (i, s); for both of the non-defectives identified in CSP-1 and the non-inspected items, there are also three possible proportions which should be performed a precise inspection: all, none, or any proportion. At the end of this section, some concluding remarks are addressed as follows: 1. Except CSP-1, the US Military standard No. MIL-STD-1235B announced in December 12, 1982 contains four types of single-level continuous sampling inspection plans (CSP-2, CSP-3, CSP-F, and CSP-V) and a multilevel sampling plan CSP-T (Banzhoof & Brugger, 1970; Dodge & Torrey, 1951). The continuous inspection plans listed above are all the modified versions of CSP-1. However, in this paper, we only focused on the optimal policy of CSP-1 with inspection errors and return cost while employing precise inspection. It is no doubt an interesting topic to research into the other plans listed above under the same conditions set in the present paper. 2. In practical applications, the decision-maker usually needs to face the problem that the true values of cost parameters are unknown in advance. A feasible method of dissolving such a problem is to estimate the parameters by past experience or data obtained from the accounting and/or the marketing departments of the company, and then assume that the parameters can be approximated well by these estimates through some deterministic equivalent. Besides, it might be feasible in dealing with the problem of parameter uncertainties to fuzzify the parameters, treat them as random variables, or make a sensitivity analysis for the optimal solution. No doubt, this is an interesting topic for the future study. 3. It is worth noting that an assumption of the present paper is that the production is in control. However, after a period of production time, the process may deteriorate and, then, shift to an out-of-control state, producing items with a higher defective percentage. Hence, the manufacturer usually arranges an inspection schedule for the production process to keep the production process in control and to prevent the defectives. Obviously, it is an interesting topic for the future study to investigate a joint policy between the mixed inspection policy of the paper and an inspection schedule for the production process. References
5. Concluding remarks In practice, inspection testing usually fails to be perfect and defectives are strictly prohibited for some products and a great return cost will be fined for each defective. Hence, in this study, we adopt a precise inspection to screen out the defectives for the prod-
Banzhoof, R. A., & Brugger, R. M. (1970). Reviews of standards and specificationsMIL-STD-1235 (ORD), single and multi-level continuous sampling procedures and tables for inspection by attributes. Journal of Quality Technology, 2(1), 41–53. Case, K. E., Bennett, G. K., & Schmidt, J. W. (1973). The Dodge CSP-1 continuous sampling plan under inspection error. AIIE Transactions, 5, 193–202.
H.-F. Yu et al. / Computers & Industrial Engineering 57 (2009) 652–659 Chen, C. H., & Chou, C. Y. (2002). Economic design of continuous sampling plan under linear inspection cost. Journal of Applied Statistics, 29, 1003–1009. Chen, C. H., & Chou, C. Y. (2004). Minimum average fraction inspected for modified CSP-1 plan. Pan-Pacific Management Review, 7, 35–43. Dodge, H. F. (1943). A sampling inspection plan for continuous production. Annals of Mathematical Statistics, 14, 264–279. Dodge, H. F., & Torrey, M. N. (1951). Additional continuous sampling inspection plans. Industrial Quality Control, 7, 7–12. Endres, A. C. (1979). Minimum AFI plan designs for inspection error. In ASQC technical conference transaction–Houston, pp. 414–416. Govindaraju, K., & Kandasamy, C. (2000). Design of generalized CSP-C continuous sampling plan. Journal of Applied Statistics, 27, 829–841.
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Haji, A., & Haji, R. (2004). The optimal policy for a sampling plan in continuous in terms of the clearance number. Computers & Industrial Engineering, 47, 141–147. Resnikoff, G. J. (1960). Minimum average fraction inspected for a continuous sampling plan. Journal of Industrial Engineering, 11, 208–209. Ross, S. M. (1993). Introduction to probability models (5th ed.). New York: Academic Press, Inc. Tang, K., & Tang, J. (1994). Design of screening procedures: A review. Journal of Quality Technology, 26, 209–226. Wang, R. C., & Chen, C. H. (1997). Minimum average fraction inspected for continuous sampling plan CSP-1 under inspection error. Journal of Applied Statistics, 24, 539–548.
Contents lists available at ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
A mixed inspection policy for CSP-1 and precise inspection under inspection errors and return cost q Hong-Fwu Yu a,*, Wen-Ching Yu b, Wen Ping Wu c a
Graduate Institute of Commerce, National Kaohsiung University of Applied Sciences, 415 Chien Kung Road, Kaohsiung 807, Taiwan, ROC Department of Business Management, National United University, 1, Lien Da, Kung-Ching Li, Miao-Li 360, Taiwan, ROC c Department of Industrial Management, National Formosa University, 64, Wen Hua Road, Hu-Wei, Yunlin 632, Taiwan, ROC b
a r t i c l e
i n f o
Article history: Received 2 October 2007 Received in revised form 3 April 2008 Accepted 27 December 2008 Available online 10 January 2009 Keywords: CSP-1 Inspection errors Return cost Optimal sampling policy
a b s t r a c t This paper is to deal with a mixed policy between precise inspection and CSP-1 with inspection errors (Types I and II) and return cost. First, the concept of a renewal reward process is employed to obtain the long-term net profit of the mixed inspection policy. Then, with respect to non-repairable and repairable products, we determine the following four decision variables such that the unit net profit is maximal: (1) the optimal clearance number, (2) the optimal sampling frequency, and (3) the proportions which should be taken precise inspection for the non-inspected items in the procedure of CSP-1 and the non-defectives identified by CSP-1. Overall, the analytical results indicate that depending on seven parameters (Type I error, Type II error, the selling price of an item, the unit repair cost, the unit return cost, the unit cost of precise inspection, and the process defective fraction), there are three possible optimal inspection policies for CSP-1: ‘‘Do not inspect”, ‘‘Do 100% inspection”, or any setting for (i, s); for both of the non-defectives identified in CSP-1 and the non-inspected items, there are three possible proportions which should be performed a precise inspection: all, none, or any proportion. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, consumers are paying more and more attention to merchandise quality and reliability. However, due to technology limits, worn-down mechanical parts, or human negligence, the production of defective items is inevitable. Hence, in order to satisfy customers’ requirement, the manufacturer must make efforts to screen out defective items in the outgoing batch. Tang and Tang (1994) surveyed the literature on the approaches of designing screening procedures. Among these screening procedures, inspection for product quality continues to be an important means. For a continuous production process, CSP-1 is the earliest method used to control the quality of products. It consists of 100% inspection and random partial inspection (Dodge, 1943). In CSP-1, if the product defective fraction is known, average fraction inspected (AFI), average outgoing quality (AOQ), and average outgoing quality limit (AOQL) can be calculated (Dodge, 1943). There are many related researches on CSP-1, such as Resnikoff (1960) proposed in minimum average inspection fraction for a continuous sampling plan. Chen and Chou (2002) studied the inspection method under linear inspection cost model to decide the minimum total expected cost. Govindaraju and Kandasamy (2000) proposed the
q
This manuscript was processed by Area Editor E.A. Elsayed. * Corresponding author. Tel.: +886 7 3814526x6706. E-mail address: [email protected] (H.-F. Yu).
0360-8352/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.12.008
modified CSP-1 plan, which is based on single level continuous sampling plan with the acceptance number. The modified plan is similar to CSP-1 but differs in inspection process; it has an acceptance number of defectives and will shift into 100% inspection when exceeds this limit. The advantage of this plan is to achieve a reduction in the average fraction inspected at good quality levels. Chen and Chou (2004) considered the minimum average fraction inspected (AFI) for Govindaraju and Kandasamy’s (2000) modified CSP-1 plan. They developed a solution procedure to find the CSP-1 parameters that will meet the AOQL requirement, while also minimizing the AFI for the modified CSP-1 plan when the process average is known. Two basic assumptions are involved in CSP-1: (i) inspections are assumed to be perfect and (ii) the production process is not perfect. It has a constant defective rate. Also, some defectives are permitted for the products shipped to the consumer under the protection of a pre-specified AOQL. Most studies mentioned earlier have been operated under these two assumptions. However, in real applications, normal inspection may have errors that can decrease the outgoing quality. In fact, there are two types of inspection errors: Type I and II. Type I error occurs when the good products get classified as defectives that will be destroyed or reworked. This error can lead to increase in the production cost. Type II error happens when the defectives are classified as non-defectives and are allowed for shipment. This type of error can affect customer’s loyalty. Presence of inspection errors in CSP-1 plan was first discussed in Case, Bennett, and Schmidt (1973) that mentioned
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H.-F. Yu et al. / Computers & Industrial Engineering 57 (2009) 652–659
the need of an alternate sampling method. Case et al. (1973) developed a compensating sampling plan, considering inspection errors, which yield the desired actual AOQL. Endres (1979) then discussed the minimum AFI plan in case of inspection error. Wang and Chen (1997) further modified Endres’s (1979) method and formulated a minimum average fraction inspection model under inspection errors. In real situations, some machinery parts are prohibited to have defectives, such as vehicle brake system, radar homing system, air defense system, etc. For such parts, a great fine or a high return cost will be put on the purchase contract. Haji and Haji (2004) employed a renewal reward process on CSP-1 to develop optimal policies in terms of the clearance number. Their results showed that depending on the defective fraction of the production process, the unit rework cost, the return cost, and the unit inspection cost, there are only three different optimal policies: (1) the number of consecutive inspection is zero (partial inspection), (2) the number of consecutive inspection is infinity (100% inspection) in CSP-1, or (3) the number of consecutive inspection can be any value from zero to infinity. It is worth noting that due to the defective rate inherent in the production process and Type II error, the non-inspected portion in CSP-1 may still contain defectives. Additionally, one would normally discard defectives found in a production; however, because of the presence of Type I error, such action may cause great loss in opportunity cost and production cost. One of the effective methods to screen out the defectives is a precise inspection. It can almost 100% screen out the defectives among the lot shipped to the consumer. Also, precise inspection can pick up the good items among the apparent defectives identified in the procedure of CSP-1 and thus can prevent the additional production resulting from Type I error. However, precise inspection is usually costly. An example of the ‘‘precise inspection” can be seen in finding faults on ball grid array (BGA). BGA is a packaging for surface-mounted integrated electronic circuit and is sometimes very difficult to look for faults. For such a difficulty, X-ray can almost 100% screen out the defectives; while, it is costly. Obviously, the costliness of precise inspection is in conflict with the requirement of high quality of the shipped lot. The purpose of this paper is to determine the optimal mixed policy of precise inspection and CSP-1 with inspection errors and return cost. More specifically, with respect to nonrepairable and repairable products, by adopting a precise inspection to screen out the defectives for the products behind the procedure of CSP-1, we determine the following four decision variables such that the unit net profit is maximal: (1) the optimal clearance number, (2) the optimal sampling frequency, and (3) the proportions which should be taken precise inspection for the non-inspected items in the procedure of CSP-1 and the non-defectives identified by CSP-1. This paper is organized as follows. Section 2 briefly introduces the procedure of the mixed inspection policy and the corresponding optimization model. Section 3 shows the derivation of the optimal net profit model. Section 4 gives a numerical example to demonstrate the optimal policy. Finally, we give some concluding remarks in Section 5.
2. Inspection procedure and the optimization model In this section, we will introduce the procedure of the mixed inspection policy and the corresponding optimization model. The inspection procedure of the mixed policy considered in this paper is as follows: 1. At the outset, inspect 100% of the units consecutively as produced and continue such inspection until i units in succession are found to be non-defective.
2. When i units in succession are found to be non-defective, discontinue 100% inspection, and inspect only a fraction f of the units, selecting individual sample units one at a time from the flow of product, in a random manner to assure an unbiased sample. Continue this partial inspection until a sample unit is found defective and then revert immediately to the 100% inspection of succeeding units, as in step 1. 3. Precision inspections are taken on a proportion r1 of the nondefectives identified in Steps 1–2 and on a proportion r2 of items not inspected in Step 2, respectively. 4. As to the defectives screened out in Steps 1–3, they will be repaired if the products are repairable; otherwise will be discarded. Note that Step 1 and Step 2 are the process of CSP-1. The CSP-1 process goes from Step 1 to Step 2 and then goes back to Step 1. For ease exposition, we call such process a cycle. Let R and TC be the corresponding expected revenue and expected total cost, respectively. Then the corresponding expected profit P is
P ¼ R TC Next, let Q represent the expected production quantity in a cycle. Thus, the net profit in a cycle can be written as:
NP ¼
P R TC ¼ Q Q
ð1Þ
Let W be the number of products produced until time t. Since there are defectives and inspection errors in CSP-1, the number of actually sold products may be less than W. If the profit earned is P(W), then the net unit profit for the W products is:
NPðWÞ ¼
PðWÞ W
Therefore, from the property of a renewal process, we can obtain (see Ross, 1993)
PðWÞ W E½profit earned in a cycle ¼ NP ¼ E½production quantity in a cycle
lim NPðWÞ ¼ lim NPðWÞ ¼ lim
t!1
W!1
W!1
It is quite obvious that different arrangements with i, f, r1, and r2 will influence the cost, profit, and shipment quantity. Our goal is to decide the optimal combination of i, f, r1, and r2 for CSP-1 in order to achieve the maximal net unit profit. Let N0 = {0, 1, 2, 3, . . .} and I = [0, 1]. Then the decision problem can be expressed as the following optimization model:
Maximize
NP
Subject to i 2 N0 f ; r1 ; r2 2 I 3. Optiaml inspection policy In this section, we will establish the expressions for the expected revenue and the expected total cost and then determine the optimal inspection policy. Let p be the process defective fraction (non-defective fraction q = 1 p), U be the expected number of items manufactured in Step 1, and V be the expected number of items manufactured in Step 2. Then U = Ug + Ud and V = Vg + Vd + Vn, where Ud and Vd are the expected numbers of defectives found in Steps 1 and 2, respectively; Ug and Vg are the expected numbers of non-defectives identified in Steps 1 and 2, respectively; Vn is the expected number of
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H.-F. Yu et al. / Computers & Industrial Engineering 57 (2009) 652–659
non-inspected products in Step 2. Obviously, Vd = 1 and, from Dodge (1943), we know:
1 qi pqi 1 V¼ fp 1 qi Ud ¼ qi 1 Vg ¼ 1 p U¼
ð2Þ
sqiid qiid þ 1 pid qiid 1 qiid 1 1 Ug ¼ pid qiid
ð3Þ
and
ð4Þ
Vn ¼
ð5Þ
sqi qi þ 1 pqi 1 qi 1 1 Ug ¼ U Ud ¼ p qi Q ¼UþV ¼
ð6Þ ð7Þ
and
1 1 ðs 1Þ V n ¼ V ðV g þ V d Þ ¼ 1 ¼ p f p
ð8Þ
where s = 1/f. Thus, the non-defectives and defectives in Vn can be:
Vn q ¼
q 1 q 1 ¼ ðs 1Þ p f p
ð70 Þ
1 1 ðs 1Þ 1 ¼ pid f pid
ð80 Þ
and
Vn p ¼
So,
ð60 Þ
Q¼
pðs 1Þ pid
ð90 Þ
Thus, in Step 1, the expected number of identified non-defectives which are truly defectives is
1 qiid ð1 p2 Þ U g ¼ ð1 p2 Þ qiid
! 1 1 pid
Also, in Step 2, the expected number of identified non-defectives which are truly defectives is
ð1 p2 Þ V g ¼ ð1 p2 Þ
1 1 pid
Hence, the total expected number of the identified non-defectives which are truly defectives in a cycle is
ð1 p2 Þ U g þ ð1 p2 Þ V g ¼ ð1 p2 Þ
and
Vn p ¼
1 1¼s1 f
ð10Þ
ð9Þ 3.1. Non-repairable products (Case 1)
Suppose that Type I error (non-defectives are classified as defectives) and Type II error (defectives are classified as non-defectives) occurring in the inspections taken in CSP-1 are a and b, respectively. Define the following four events: Eid: an item is identified as a defective Etd: an item is truly a defective Eig: an item is identified as a non-defective Etg: an item is truly a non-defective Let pid = P(Eid), qid = P(Eig), p1 = P(Etd|Eid), and p2 = P(Etg|Eig). Then it can be easily shown that
pid ¼ að1 pÞ þ ð1 bÞp qid ¼ 1 pid ¼ ð1 aÞð1 pÞ þ bp p1 ¼
1 pid : pid qiid
In this case, because the defectives identified in the procedure of CSP-1 will go through a precise inspection to pick up all of the misidentified items, there will be a precise inspection cost. Next, since r1 and r2 proportions for the non-defectives and non-inspected products will have a precise inspection, this will result in a precise inspection cost; while, because the remaining portions which do not have the precise inspection may contain defectives, a return cost may be incurred. Overall, the expected total cost consists of the following four segments: (a) expected precise inspection cost, (b) expected return cost, (c) expected inspection cost, and (d) expected production cost. These four costs are discussed below: (a) Expected precise inspection cost: The cost consists of three parts. (a-1) From Eq. (40 ), we know the expected number of defectives identified in the procedure of CSP-1 is U d þ V d ¼ q1i . So, the expected number of precise inspection
ð1 bÞp
að1 pÞ þ ð1 bÞp
id
for this portion is
and
1 . qiid
(a-2) From Eqs. (50 ) and (70 ), we know
the expected number of the non-defectives identified in
ð1 aÞð1 pÞ p2 ¼ ð1 aÞð1 pÞ þ bp
the procedure of CSP-1 is
U¼
1 qiid pid qiid
ð2 Þ
V¼
1 fpid
ð30 Þ
1 qiid qiid 1 1 Vg ¼ pid So,
and if we perform a precise
inspection on a r1 fraction of these, then the expected num-
Thus, under inspection errors, Eqs. (2)–(9) should be modified as follows:
Ud ¼
1pid , pid qiid
0
ð40 Þ ð50 Þ
id ber of precise inspection is r1 1p . (a-3) The expected nump qi id id
ber of the non-inspected items is
s1 , pid
and if we conduct a
precise inspection on a r 2 fraction of these products, then the expected number of precise inspection for this portion is r2 s1 . Overall, aggregating (a-1), (a-2), and (a-3), the p id
expected total precise inspection cost is
! 1 1 pid s1 Cp þ r1 þ r2 pid qiid pid qiid where Cp denotes the unit precise inspection cost.
ð11Þ
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H.-F. Yu et al. / Computers & Industrial Engineering 57 (2009) 652–659
(b) Expected return cost: The cost comprises two segments. (b1) According to Eq. (10), the expected number of the identified non-defectives which are truly defectives in a cycle is id . Since a ð1 r 1 Þ portion of the non-defectives ð1 p2 Þ 1p p qi
in CSP-1. Thus, the expected overall revenue can be expressed as follows:
Rð1Þ ¼ C Q e ¼C
id id
identified in CSP-1 does not have a precise inspection, the expected number of true defectives that will be shipped to id . (b-2) Since a the consumer is ð1 r 1 Þ ð1 p2 Þ 1p p qi
"
1 1 pid Q p1 i r1 ð1 p2 Þ qid pid qiid
!
pðs 1Þ r2 pid
id id
ð1 r2 Þ portion of the non-inspected items does not have a precise inspection, from Eq. (90 ), the expected number of defectives which are shipped to the consumer is ð1 r2 Þ pðs1Þ . Overall, collecting (b-1) and (b-2), the p id
expected total return cost is
! # 1 pid pðs 1Þ þ ð1 r2 Þ C t ð1 r 1 Þ ð1 p2 Þ pid pid qiid
ð16Þ 0
Substituting Eqs. (6 ), (15) and (16) into Eq. (1) gives the expected net profit as follows:
Rð1Þ TC ð1Þ ¼ Q (
(
ð1 p2 Þ ðC C t Þ C p sqiid qiid þ 1 !) 1 þ r 2 ½p ðC C t Þ C p 1 i sqid qiid þ 1
NPð1Þ ¼
"
ð12Þ
)
r 1 ð1 pid Þ
þ fC pC t C d
where Ct denotes the unit return cost. (c) Expected inspection cost: From Eqs. (20 ) and (50 ), we know the expected inspection number in CSP-1 process is 1qiid 1 . So, the expected inspection cost is þ p p qi
C pid p1 C t ð1 pid Þð1 p2 Þ þ C t p C s C p pid þ sqiid qiid þ 1 ð1Þ
ð1Þ
)
ð1Þ
¼ NP 1 þ NP2 þ NP 3 where
id
id id
Cs
#
1 qiid 1 þ pid pid qiid
ð1Þ
NP 1 ¼ r 1 ð1 pid Þ
! ¼ Cs
1 pid qiid
ð13Þ ð1Þ NP 2
where Cs denotes the unit inspection cost.
ð1 p2 Þ ðC C t Þ C p sqiid qiid þ 1
! 1 ¼ r 2 ½p ðC C t Þ C p 1 i sqid qiid þ 1
and (d) Expected production cost: based on Eq. (60 ), the expected production cost is
Cd Q ¼ Cd
sqiid
qiid þ pid qiid
þ 1
ð14Þ ð1Þ
where Cd denotes the unit manufacturing cost. Summing up the costs in (a)–(d), we can obtain the expected total cost as follows:
" TC ð1Þ ¼ C p
1 1 pid þ r1 qiid pid qiid
ð1Þ
NP3 ¼ C pC t C d
! þ r2
þ Cd Q þ Ct "
1 pid pid qiid
ð1 r 1 Þ ð1 p2 Þ
# s1 1 þ Cs pid pid qiid
! þ ð1 r 2 Þ
pðs 1Þ pid
#
ð15Þ As to the expected income Rð1Þ , it is the product of the selling price (C) and the expected number of the products shipped to the consumer (Qe). Note that
Q e ¼ Q Q d1 Q d2 Q d3 Q d4 1qi is the expected number of the identified where Q d1 ¼ p1 qi id id
defectives which are truly defective in step 1, Q d2 ð¼ p1 1Þ is the expected number of the identified defective which is truly defective id is the expected number of in step 2, Q d3 ¼ r 1 ð1 p2 Þ 1p i p q id id
defectives contained in the r1 fraction of the non-defectives identi fied in CSP-1, and Q d4 ¼ r2 pðs1Þ is the expected number of the p id
defectives contained in the r2 fraction of the non-inspected items
C pid p1 C t ð1 pid Þð1 p2 Þ þ C t p C s C p pid sqiid qiid þ 1 ð1Þ
Let D1 ¼ ð1 p2 Þ ðC C t Þ C p ; D2 ¼ p ðC C t Þ C p and Dð1Þ In 3 ¼ C pid p1 C t ð1 pid Þð1 p2 Þ þ C t p C s C p pid . this scenario, r1, r2, i and s are determined by first determining i ð1Þ ð1Þ ð1Þ and s from NP 3 and then substituting i and s into NP2 and NP1 to determine r2 and r1. The derivation steps are shown below: ð1Þ (A) When D3 < 0: ð1Þ increases as Since sqiid qiid þ 1 > 0, the value for NP 3 i i sqid qid þ 1 increases. So, if we take s = 1 and i 2 N 0 , then ð1Þ sqiid qiid þ 1 ¼ 1 and NP 3 has the maximum value. Substituting ð1Þ ð1Þ ð1Þ ð1Þ i i sqid qid þ 1 ¼ 1 into NP 2 can get NP 2 ¼ r2 D2 1 ¼ r2 D2 . ð1Þ Thus, the value of r2 can be determined depending on D2 to give ð1Þ the maximum value for NP 2 as follows:
8 ð1Þ > if D2 > 0 > < 1; ð1Þ r 2 ¼ 0; if D2 < 0 > > : ð1Þ any value between 0 and 1; if D2 ¼ 0
ð1Þ
Similarly, substituting sqiid qiid þ 1 ¼ 1 into NP1 can get ð1Þ NP1 ¼ r 1 0 ¼ 0. Thus, r1 can take any value between 0 and 1 (i.e., r 1 2 I). Note that the inspection policy s = 1 and i 2 N 0 for CSP-1 implies that after the sampling plan goes Step 2, no inspection can be conducted any longer, since the fraction of the units which will 1 ¼ 0. Here, for simplicity, we call such an be inspected is f ¼ 1s ¼ 1 inspection policy ‘‘Do not inspect”. As to the non-defectives identified in CSP-1 and the non-inspected items, any proportion of the former can be performed a precise inspection; there are three possible proportions for the latter which should be performed a precise inspection: all, none, or any proposition.
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H.-F. Yu et al. / Computers & Industrial Engineering 57 (2009) 652–659 ð1Þ
(B) When D3 > 0: ð1Þ Similarly, because sqiid qiid þ 1 1, the NP 3 value will increase as sqiid qiid þ 1 decreases. So, if we take ðs ¼ 1; i 2 N 0 Þ or ðs 2 R0 ; i ¼ 1Þ where R0 ¼ ð1; 1Þ, then we have sqiid qiid þ 1 ¼ 1 ð1Þ and NP 3 will has the maximum value. Substituting sqiid qiid þ 1 ¼ 1 ð1Þ ð1Þ ð1Þ ð1Þ into NP2 can get NP 2 ¼ r2 D2 0 ¼ 0. The zero value of NP 2 indicates that r2 can take any value between 0 and 1 (i.e., r 2 2 I). Subseð1Þ quently, substituting sqiid qiid þ 1 ¼ 1 into NP 1 can get: ð1Þ
NP 1 ¼ r 1 ð1 pid Þ
Dð1Þ 1 1
ð1Þ
¼ r 1 ð1 pid Þ D1
ð1Þ
Thus, the value of r1 can be determined depending on D1 to give ð1Þ the maximum value for NP 1 as follows:
8 ð1Þ > if D1 > 0 > < 1; ð1Þ r 1 ¼ 0; if D1 < 0 > > : ð1Þ any value between 0 and 1; if D1 ¼ 0 Note that it is easily seen that both of the inspection policies ðs ¼ 1; i 2 N0 Þ and ðs 2 R0 ; i ¼ 1Þ mean that CSP-1 should perform 100% inspection of the units consecutively. Here, for simplicity, we call such inspection policies ‘‘Do 100% inspection”. As to the non-defectives identified in CSP-1 and the non-inspected items, there are three possible proportions for the former which should be performed a precise inspection: all, none, or any proposition; any proportion of the latter can be performed a precise inspection. ð1Þ (C) When D3 ¼ 0: It is easily seen that
ð1Þ
ð1Þ
value of NP 3 . In such a situation, we will determine the values of r1, r2, i and s via ð1Þ NP1
ð1Þ NP 1
þ
ð1Þ NP 2
þ
ð1Þ NP 2
þ
1 , sqi qi þ1
Let k ¼
id
then 0 k 1. Thus,
id
can be re-written as follows:
ð1Þ
That is, NP1 þ NP 2 is a linear combination of r 1 ð1 pid ÞD1 and ð1Þ r 2 D2 . Accordingly, let ð1Þ M1
n
¼ Max r 1 ð1 0r 1 1
ð1Þ pid ÞD1
o
then it can be easily shown that if
and
ð1Þ M2
ð1Þ NP 1
þ
n o ð1Þ ¼ Max r 2 D2 0r 2 1
ð1Þ NP2
ð1Þ
ð1Þ
M1
ð1Þ
id
number of the non-defectives identified in the procedure of CSP1pid , pid qi
and if we perform a precise inspection on a r1 fraction
of these, then the expected number of true defectives that can be id . identified by the precise inspection will be r 1 ð1 p2 Þ 1p i p q id id
Thus,
expected repair cost for this portion is id . (e-3) As mentioned in (a-3) of Case 1, C r r1 ð1 p2 Þ 1p i p q
the
the expected number of the non-inspected items is
8 ð1Þ ð1Þ > > < ð1 pid ÞD1 ðhere r 1 ¼ 1Þ; if D1 > 0 ð1Þ ¼ 0 ðhere r 1 ¼ 0Þ; if D1 < 0 > > : ð1Þ 0 ðhere r 1 2 IÞ; if D1 ¼ 0 8 ð1Þ ð1Þ > > < D2 ðhere r2 ¼ 1Þ; if D2 > 0 ð1Þ ¼ 0ðhere r 2 ¼ 0Þ; if D2 < 0 > > : ð1Þ 0ðhere r 2 2 IÞ; if D2 ¼ 0
The results associated with this case indicate that (i) depending on ð1Þ ð1Þ M1 and M 2 , there are three possible optimal inspection policies for CSP-1: ‘‘Do not inspect”, ‘‘Do 100% inspection”, or any setting
s1 , pid
and if we
conduct a precise inspection on a r 2 fraction of these products, then the expected number of true defectives that can be identified by the precise inspection will be r 2 pðs1Þ . Thus, the expected repair p id
cost for this portion is C r r 2 pðs1Þ . Overall, aggregating (e-1), (ep id
2), and (e-3), the expected repair cost is:
"
1 1 pid C r p1 i þ r 1 ð1 p2 Þ qid pid qiid
!
pðs 1Þ þ r2 pid
# ð17Þ
Summing up the costs in (a)–(e), we can obtain the expected total cost as follows:
"
ð1Þ
and M2 , it can be easily obtained as
and
M2
id
is C r p1 q1i . (e-2) As mentioned in (a-2) of Case 1, the expected
is maximal if and only
8 ð1Þ ð1Þ 0 0 > if M1 > M 2 > < 1 ði:e:;ðs ¼ 1; i 2 N Þ or ðs 2 R ; i ¼ 1ÞÞ; ð1Þ ð1Þ k ¼ 0 ði:e:;s ¼ 1 and i 2 N 0 Þ; if M1 < M 2 > > : ð1Þ ð1Þ any value between 0 and 1 ði:e:;ði;sÞ any valueÞ; if M1 ¼ M 2 As for the values of M1 follows:
id
defectives is p1 q1i . Thus, the expected repair cost for this portion
id id
¼ k ½r1 ð1 pid ÞD1 þ ð1 kÞ ðr 2 D2 Þ
ð1Þ
In this case, because the defectives identified in the procedure of CSP-1 will go through a precise inspection to pick up all of the mis-identified items, there will be a precise inspection cost. Next, since r1 and r2 proportions for the non-defectives and non-inspected products will have a precise inspection, this will result in a precise inspection cost; while, because the remaining portions which do not have the precise inspection may contain defectives, a return cost may be incurred. Finally, since the defectives identified by precise inspection will be repaired, a repair cost will be incurred. Overall, the expected total cost consists of the following five segments: (a) expected precise inspection cost, (b) expected return cost, (c) expected inspection cost, (d) expected production cost, and (e) expected repair cost. It is easily seen that the first four costs are the same as the ones in (a), (b), (c), and (d) of Case 1, respectively. As to the expected repair cost, it consists of three parts. (e-1) As in (a) of Case 1, the expected number of defectives found by CSP-1 is q1i . Then, of these, the expected number of true
id
That is, NP 3 is a constant. This means that ði; sÞ does not affect the ð1Þ NP 2 .
ð1Þ
3.2. Repairable products (Case 2)
1 is
ð1Þ
NP 3 ¼ C pC t C d þ 0 ¼ C pC t C d
ð1Þ NP 1
ð1Þ
for (i, s); and (ii) depending on D1 and D2 , there are three possible proportions for both of the non-defectives identified in CSP-1 and the non-inspected items which should be performed a precise inspection: all, none, or any proportion.
TC
ð2Þ
! # p1 1 pid pðs 1Þ þ r2 ¼ C r i þ r 1 ð1 p2 Þ pid qid pid qiid " ! # 1 1 pid s1 þ r2 þ Cd Q þ C p i þ r1 pid pid qiid qid " ! # 1 pid pðs 1Þ þ ð1 r þ C t ð1 r1 Þ ð1 p2 Þ Þ 2 pid pid qiid þ Cs
1 pid qiid
ð18Þ
As to the expected revenue Rð2Þ , it is the product of the selling price (C) and the expected number of the products shipped to the consumer (Qe). Note that because the defectives identified in the procedure of CSP-1 will go through a precise inspection to pick
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H.-F. Yu et al. / Computers & Industrial Engineering 57 (2009) 652–659
up all of the mis-identified items and the remaining defectives will be repaired, all of the production quantity will be shipped to the consumer. Hence,
Qe ¼ Q Thus, the expected overall revenue is
Rð2Þ ¼ C Q
ð19Þ
Substituting Eqs. (60 ), (18) and (19) into Eq. (1) gives the expected net profit as follows:
NP ð2Þ ¼ ( ¼ (
Rð2Þ TC ð2Þ Q
ð1 p2 Þ ðC r C t Þ C p r 1 ð1 pid Þ sqiid qiid þ 1
)
ð2Þ
(C) When D3 ¼ 0: Let
þ fC pC t C d C r pid p1 C t ð1 pid Þð1 p2 Þ þ C t p C s C p pid þ sqiid qiid þ 1 ¼
ð2Þ NP 1
þ
ð2Þ NP2
þ
)
ð2Þ NP 3
where ð2Þ
NP1 ¼ r 1 ð1 pid Þ
ð1 p2 Þ ðC r C t Þ C p sqiid qiid þ 1
ð2Þ NP2 ¼ r 2 p ðC r C t Þ C p 1
As to the value of r 1 , it can take any value between 0 and 1 (i.e., r1 2 I). ð2Þ (B) When D3 > 0: Similarly, in this scenario, we take ðs ¼ 1; i 2 N 0 Þ or ðs 2 R0 ; i ¼ 1Þ. As to the value of r 2 , it can take any value between 0 and 1 (i.e., r2 2 I). Finally, the value of r 1 can be determined as follows:
8 ð2Þ > if D1 > 0 > < 1; ð2Þ r 1 ¼ 0; if D1 < 0 > > : ð2Þ any value between 0 and 1; if D1 ¼ 0
!) 1 r2 ½p ðC r C t Þ C p 1 i sqid qiid þ 1
þ
8 ð2Þ > if D2 > 0 > < 1; ð2Þ r 2 ¼ 0; if D2 < 0 > > : ð2Þ any value between 0 and 1; if D2 ¼ 0
n o n o ð2Þ ð2Þ ð2Þ ð2Þ and M 2 ¼ Max r 2 D2 M1 ¼ Max r1 ð1 pid ÞD1 0r 1 1
then
8 ð2Þ ð2Þ 0 0 > > < ðs ¼ 1; i 2 N Þ or ðs 2 R ; i ¼ 1Þ; if M 1 > M2 ð2Þ ð2Þ 0 s ¼ 1 and i 2 N ; if M 1 < M2 > > : ð2Þ ð2Þ ði; sÞ any value; if M 1 ¼ M2 ð2Þ
1 sqiid qiid þ 1
!
06r 2 61
ð2Þ
where M1 and M2 can be obtained as follows: ð2Þ
M1
and
8 ð2Þ ð2Þ > > < ð1 pid ÞD1 ðhere r 1 ¼ 1Þ; if D1 > 0 ð2Þ ¼ 0 ðhere r 1 ¼ 0Þ; if D1 < 0 > > : ð2Þ 0 ðhere r 1 2 IÞ; if D1 ¼ 0
ð2Þ
NP 3 ¼ C pC t C d þ
and
C r pid p1 C t ð1 pid Þð1 p2 Þ þ C t p C s C p pid sqiid qiid þ 1
Dð2Þ 1
ð2Þ C p ; D2
Let ¼ ð1 p2 Þ ðC r C t Þ ¼ p ðC r C t Þ C p and Dð2Þ 3 ¼ C r pid p1 C t ð1 pid Þð1 p2 Þ þ C t p C s C p pid . ð2Þ ð2Þ ð2Þ ð1Þ ð1Þ ð1Þ Note that D1 ; D2 , and D3 are the same as D1 ; D2 , and D3 , respectively except that Cr should be replaced by C. Hence, the optimal combinations of r1, r2, i and s for Case 2 can be easily obtained by duplicating the results for Case 1 obtained above. The results are summarized as follow: ð2Þ (A) When D3 < 0: In this scenario, we take s ¼ 1 and i 2 N 0 . Next, the value of r2 can be determined as follows:
ð2Þ
M2
8 ð2Þ ð2Þ > > < D2 ðhere r 2 ¼ 1Þ; if D2 > 0 ð2Þ ¼ 0 ðhere r 2 ¼ 0Þ; if D2 < 0 > > : ð2Þ 0 ðhere r 2 2 IÞ; if D2 ¼ 0
In the next section, due to the similarity in the structures of Case 1 and Case 2, we will take Case 1 as an example to demonstrate the procedure of the proposed method. 4. Numerical example To simplify the calculation, we set Cd = 1. For illustrative purpose, Table 1 lists six combinations of p, a, b, C, C t , C s , and C p which
Table 1 The optimal inspection policies corresponding to six combinations of p, a, b, C, Ct, Cs, and Cp. (p, a, b, C, Ct, Cs, Cp)
The proposed model ð1Þ
(0.15, 0.025, 0.05, 5, 10, 0.01, 0.1) (0.15, 0.15, 0.30, 2.5, 20, 0.01, 0.1) (0.20, 0.10, 0.20, 2.5, 15, 0.01, 0.1) (0.20, 0.10, 0.20, 2.5, 2.3, 0.01, 0.1) (0.20, 0.10, 0.20, 2.0875, 2.3, 0.01, 0.1) (0.05, 0.01, 0.02, 2, 2, 0.2, 0.07)
ð1Þ
ð1Þ
HHM
ðD1 ; D2 ; D3 Þ ði; s; r1 ; r2 Þ
NP
(0.05516, 0.6500, 0.6861) ð1; R0 ; 0; IÞ or ðN 0 ; 1; 0; IÞ (0.9261, 2.525, 1.8043) 1; R0 ; 1; IÞ or ðN 0 ; 1; 1; IÞ (0.5579, 2.4, 1.966) ð1; R0 ; 1; IÞ or ðN 0 ; 1; 1; IÞ (0.1105, 0.14, 0.066) ðN0 ; 1; I; 1Þ (0.08882, 0.0575, 0) (i, s): any setting (r1, r2) = (0, 0) (0.07, 0.07, 0.2041) (N0, 1, I, 1)
3.1861
Remark: X ¼ pid ðC d C t Þ þ C s (see Haji and Haji, 2004).
X
NPHHM
Inspection policy
1.015 0.89 0.90 0.6275 0.83
1.4638 100% Inspection 4.4075 100% Inspection 3.35 100% Inspection 0.302 100% Inspection 0.302 100% Inspection 0.0115 Partial inspection
3.0963 0.00875 0.29 1.04 0.6275 0.9
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H.-F. Yu et al. / Computers & Industrial Engineering 57 (2009) 652–659 ð1Þ
ð1Þ
are set to generate the scenarios associated with D3 < 0, D3 ¼ 0, ð1Þ ð1Þ ð1Þ ð1Þ and D3 > 0. The corresponding ðD1 ; D2 ; D3 Þ s are also given in ð1Þ ð1Þ ð1Þ Table 1. Based on the values of ðD1 ; D2 ; D3 Þ, the corresponding optimal inspection policies can be easily obtained by the procedures of Section 3. The results are listed in the second column of Table 1. For example, for (p, a, b, C, Ct, Cs, Cp) = (0.20, 0.10, 0.20, 2.0875, 2.3, ð1Þ 0.01, 0.1), because the corresponding D3 ¼ 0, according to the procedure of Section 3, the optimal inspection policy ði; s; r 1 ; r 2 Þ deð1Þ ð1Þ ð1Þ pends on the values of D1 and D2 . Since D1 ¼ 0:08882 and ð1Þ ð1Þ D2 ¼ 0:0575, by the definitions of M1 and Mð1Þ 2 , we can obtain ð1Þ ð1Þ M 1 ¼ 0 (here, we take r1 ¼ 0) and M2 ¼ 0 (here, we take ð1Þ ð1Þ r2 ¼ 0). Finally, since M 1 ¼ M2 ; we have that ði; sÞ can be set to be any value. At the end of this section, we make a comparison of the proposed model with that of Haji and Haji (2004) (called HHM). To make the comparison possible, the concept ‘‘the net unit profit” is selected as the comparison standard. In the following, we will build up the net unit profit associated with HHM. First, since the expected number of defectives found in the procedure of CSP-1 is q1i and the defectives are all replaced with perfect id
units, the quantity shipped to the customer in a cycle is Q e ¼ Q . So, the revenue is RHHM ¼ C Q and the production cost is
1 Cd Q þ i qid
!
Next, since precise inspection is not adopted, the expected number of defectives contained in the quantity shipped to the customer is
pðs 1Þ 1 pid þ ð1 p2 Þ pid pid qiid This results in a return cost
Ct
" # pðs 1Þ 1 pid þ ð1 p2 Þ pid pid qiid
Finally, as the result (c) of Case 1, the inspection cost for HHM is C s p 1qi . Thus, the total cost is id id
TC HHM
1 ¼ Cd Q þ i qid þ Cs
!
" # pðs 1Þ 1 pid þ Ct þ ð1 p2 Þ pid pid qiid
1 pid qiid
Hence, the net unit profit associated with HHM is
NP HHM ¼
RHHM TC HHM Q
¼ ðC pC t C d Þ
C d pid þ C t ½ð1 p2 Þð1 pid Þ p þ C s sqiid qiid þ 1
The optimal inspection policies and the net unit profits of HHM corresponding to the five combinations of p, a, b, C, and Ct in Table 1 are listed in the third column of Table 1. As expected, it is seen, from Table 1, that adopting precise inspection is profitable to the manufacturer if the inspection errors are large and/or the return cost is relatively large to the selling price.
ucts behind the procedure of CSP-1. The main purpose of this paper is to investigate the mixed inspection policy of CSP-1 and precise inspection under inspection errors and return cost. First, the concept of a renewal reward process is employed to obtain the longterm net profit of the mixed inspection policy. Then, by using the criterion of minimizing the long-term net profit, the following four decision variables for non-repairable and repairable products are determined: (1) the optimal clearance number, (2) the optimal sampling frequency, and (3) the proportions which should be taken precise inspection for the non-inspected items in the procedure of CSP-1 and the non-defectives identified by CSP-1. The analytical results indicate that depending on seven parameters (Type I error, Type II error, the selling price of an item, the unit repair cost, the unit return cost, the unit cost of precise inspection, and the process defective fraction), there are three possible optimal inspection policies for CSP-1: ‘‘Do not inspect”, ‘‘Do 100% inspection”, or any setting for (i, s); for both of the non-defectives identified in CSP-1 and the non-inspected items, there are also three possible proportions which should be performed a precise inspection: all, none, or any proportion. At the end of this section, some concluding remarks are addressed as follows: 1. Except CSP-1, the US Military standard No. MIL-STD-1235B announced in December 12, 1982 contains four types of single-level continuous sampling inspection plans (CSP-2, CSP-3, CSP-F, and CSP-V) and a multilevel sampling plan CSP-T (Banzhoof & Brugger, 1970; Dodge & Torrey, 1951). The continuous inspection plans listed above are all the modified versions of CSP-1. However, in this paper, we only focused on the optimal policy of CSP-1 with inspection errors and return cost while employing precise inspection. It is no doubt an interesting topic to research into the other plans listed above under the same conditions set in the present paper. 2. In practical applications, the decision-maker usually needs to face the problem that the true values of cost parameters are unknown in advance. A feasible method of dissolving such a problem is to estimate the parameters by past experience or data obtained from the accounting and/or the marketing departments of the company, and then assume that the parameters can be approximated well by these estimates through some deterministic equivalent. Besides, it might be feasible in dealing with the problem of parameter uncertainties to fuzzify the parameters, treat them as random variables, or make a sensitivity analysis for the optimal solution. No doubt, this is an interesting topic for the future study. 3. It is worth noting that an assumption of the present paper is that the production is in control. However, after a period of production time, the process may deteriorate and, then, shift to an out-of-control state, producing items with a higher defective percentage. Hence, the manufacturer usually arranges an inspection schedule for the production process to keep the production process in control and to prevent the defectives. Obviously, it is an interesting topic for the future study to investigate a joint policy between the mixed inspection policy of the paper and an inspection schedule for the production process. References
5. Concluding remarks In practice, inspection testing usually fails to be perfect and defectives are strictly prohibited for some products and a great return cost will be fined for each defective. Hence, in this study, we adopt a precise inspection to screen out the defectives for the prod-
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