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An optimal mixed policy of inspection and burn-in and the optimal production quantity
ARTICLE IN PRESS
Int. J. Production Economics 105 (2007) 483–491 www.elsevier.com/locate/ijpe
An optimal mixed policy of inspection and burn-in and the optimal production quantity Hong-Fwu Yua,, Wen-Ching Yub a
Department of Industrial Management, National Formosa University 64, Wen Hua Road, Hu-Wei, Yunlin 632, Taiwan, ROC b Department of Business Management, National United University 1, Lien Da, Kung-Ching Li, Miao-Li 360, Taiwan, ROC Received 15 September 2003; accepted 20 April 2006 Available online 13 June 2006
Abstract Inspection and burn-in are two techniques that are widely used by the vendor to screen out defective items in a production lot in order that an outgoing batch satisfies the purchaser’s quality requirements. Due to two types of inspection errors and high cost of burn-in, how to make a trade-off between them is a challenge for the vendor. The main purpose of this paper is to deal with the problem of determining the optimal mixed policy of inspection and burn-in, where the average outgoing quality (AOQ) is used as a measure of inspection and burn-in success. More specifically, under the constraint that the outgoing batch meets an AOQ requirement, the following issues are determined to maximize the expected profit that the vendor makes: (a) the total number of items that the vendor needs to produce, (b) the number of items for inspection, the number of items for burn-in, and the number of items that need neither inspection nor burn-in, and (c) the optimal burn-in time if burn-in test is needed. Finally, an example is provided to illustrate the proposed method. r 2006 Elsevier B.V. All rights reserved. Keywords: Mixed policy; Inspection; Burn-in; Average outgoing quality
1. Introduction In order to manufacture a product, some parts of the product need to be purchased from vendors. To guarantee a high-quality level for this product, some quality requirements for each part would be specified in the contract between the manufacturer of this product and the vendor. To achieve the manufacturer’s quality requirements, the vendor must make efforts to screen out defective items in Corresponding author. Tel.: +886 5 6315716; fax: +886 5 6311548. E-mail address: [email protected] (H.-F. Yu).
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. However, the inspection testing fails to be perfect and two types of errors (Types I and II) can occur. Thus, the vendor usually faces two practical problems: (1) false acceptance of defectives resulting in that the outgoing batch may not satisfy the quality requirements and (2) false rejection of nondefectives resulting in the necessity of producing additional items beyond the quantity ordered by the manufacturer. Certainly, this will increase the vendor’s cost. In fact, inspection is not
0925-5273/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2006.04.013
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cost-free, so the need for inspection should be justified in terms of the costs involved. Raz and Thomas (1983) studied sequencing several inspectors with different inspection precision levels to meet a predetermined outgoing conformation rate at minimum cost. Tang and Schneider (1988) proposed a method of determining the optimal inspection precision level based on the tradeoff of inspection cost and the costs incurred by inspection errors. In addition, vast amounts of excellent research on inspection policies for various situations exists (see Britney, 1972; Chen and Lambrecht, 1997; Duffuaa and Roauf, 1990; Eppen and Hurst, 1974; Greenshtein and Rabinowitz, 1997; Hau, 1988; Hau and Rosenblatt, 1987; Jaraiedi et al., 1987; Park et al., 1991; Raouf et al., 1983; Raz, 1986; Raz and Kaspi, 1991; Tang and Tang, 1989; White, 1966; Yum and McDowell, 1981; Liou et al., 1994; Wang, 2005). Another screening method which is also widely used to guarantee a high-quality level for the outgoing batch of items is the burn-in test. In general, if the burn-in time is long enough, then most of defective items would be screened out. Thus, the outgoing batch of items would satisfy the quality requirements. A general background on burn-in can be found in Jensen and Petersen (1982) and Kuo and Kuo (1990), and many references are also given there. Unfortunately, compared with the inspection test, burn-in is usually costly. That is, the longer the burn-in is, the higher the manufacturing cost is. Thus, a practical problem to consider is how long the burn-in procedure should continue. Cozzolino (1970) studied the optimal length of the burnin process for repairable products. Weiss and Dishon (1971) discussed two situations where a specific number of items are required at the end of the burn-in process. Nguyen and Murthy (1982) formulated a model for determining the burn-in time for a product sold with a warranty. Mi (1994) considered the optimal burn-in and maintenance policies for repairable products at minimum cost. For a variety of products (e.g., Logic IC, Memory IC, fan of a power supply, etc.) in semiconductor industries, both inspection and burn-in tests are widely utilized to screen out defective items. In contrast with the burn-in test, a ‘‘pure’’ inspection policy has two shortcomings: (1) more items need to be produced because of false rejection of nondefectives and (2) the quality level among the items remained (after inspection) may not attain to the quality requirements because of false acceptance of
defectives. The former leads directly to the increase in cost and the latter may lead to that the outgoing batch is rejected. A remedy to these two shortcomings, especially the latter, is to employ the burnin test. However, compared to the inspection test, a ‘‘pure’’ burn-in policy is time-consuming and, hence, cost-consuming, although it can guarantee that the outgoing batch satisfies the quality requirements if the burn-in time is long enough. Thus, from both economic and quality’s points of view, the two screening methods cannot take the place of each other completely. Hence, how to make a trade-off between them such that the quality requirement is satisfied with low cost is no doubt an interesting topic for the vendor. Unfortunately, according to the literature, this has not been discussed. To this end, we will investigate the optimal mixed use of inspection and burn-in methods, where the average outgoing quality (AOQ) is utilized as a measure of inspection and burn-in success. More specifically, under the constraint that the outgoing batch meets an AOQ requirement, the following issues are determined to maximize the expected profit for the vendor: (a) the total number of items that the vendor should produce, (b) the number of items for inspection, the number of items for burn-in, and the number of items that need neither inspection or burn-in, and (c) the optimal burn-in time if burn-in testing is adopted. The rest of this paper is organized as follows. Section 2 briefly describes the assumptions on which a mixed policy of inspection and burn-in is based, and presents the corresponding optimization problem. Section 3 presents the optimal plan. Section 4 provides a numerical example to illustrate the proposed method. Finally, a conclusion is drawn in Section 5. 2. Assumptions and the optimization problem This section is devoted to describing the assumptions of a mixed use of inspection and burn-in tests and the optimization problem. 2.1. Assumptions Assume that the manufacturer of a product wants to purchase some kind of part from a vendor. The contract between them is as follows: (i) The quantity that the manufacturer orders is Q0 for each period (e.g., per week or per month).
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(ii) AOQ is set to be p. (iii) The price of each part is C dollars. (iv) The manufacturer will pay the vendor for the Q0 parts after them are shipped to the manufacturer.
manufacturer satisfy the conditions in that contract while accomplishing low cost, the decision problem can be formulated as follows:
Assume that the part is produced in batches or lots (not continuously) from a production process with a defective rate of p0 ð4pÞ on average and that the production quantity is Q for each batch. To guarantee the outgoing batch shipped to the manufacturer would satisfy the AOQ requirement, the vendor decides to adopt both inspection and burn-in techniques to screen out defective parts, where the probabilities of Type I error and Type II error of the inspection are a and b, respectively. For the vendor, to maximize the total profit in each period, he/she faces the following decision problems:
(1)
Maximize
Pv ðQ; Qs ; Qb ; tb Þ
Subject to
d AOQv ¼ N N a pp; N a XQ0 ;
Q; Qs ; Qb 2 N ¼ f1; 2; 3; . . .g; tb X0: In practical situations, the vendor is usually requested to keep promised delivery dates. Otherwise good-will would surely be lost. To achieve this, he/she must control the times of all operations in the production and delivery processes. This indicates that the burn-in time is not unlimited when burn-in is employed. Hence, the following constraint to the burn-in time tb is necessary: tb pt0 ,
1. How many parts (Q) need to be produced? 2. Among the Q parts, how many (Qs ) need to be inspected? How many (Qb ) need to be put into burn-in? And, how many (Qn ) need neither inspection nor burn-in? 3. How long (tb ) need the burn-in procedure continue for those requiring the burn-in test?
where t0 is a tolerance limit prespecified by the vendor.
To facilitate the analysis that follows, we use symbols and notations to summarize the vendor’s mixed policy as follows:
1. Compute N a , N d , and AOQv . 2. Characterize Pv ðQ; Qs ; Qb ; tb Þ.
AOQv average outgoing quality of the outgoing batch, Na expected number of items both correctly and incorrectly accepted after inspection and burn-in tests, Nd expected number of defective parts contained in N a . In the following sections, we will propose an optimal plan to determine Q, Qs , Qb , Qn , and tb simultaneously. 2.2. The optimization problem Let Pv ¼ Pv ðQ; Qs ; Qb ; tb Þ denote the expected profit that the vendor makes in each period. Since AOQ is a ratio of the number of undetected defective items to the total number of items that are accepted, we can obtain AOQv ¼ N d =N a . Then, to make the outgoing batch shipped to the
3. The optimal plan The framework for solving the optimization model consists of two major steps:
3.1. The computation of N a , N d , and AOQv First, as to Qs items, we calculate the expected number of items remaining after the inspection and the defective percentage in this lot. Since there exist two types of errors, the Qs items can be divided into four categories, which are listed in the first row of Table 1, after the inspection is carried out. Next, let us consider the Qb items that are allocated for burn-in testing. Let F ðtÞ denote the cumulative distribution function (CDF) of the defective part’s lifetime distribution and let GðtÞ denote the CDF of nondefective parts. Then, after the burn-in test, the Qb items can be divided into four categories, which are listed in the second row of Table 1. Finally, as to the items Qn that no actions are taken, they, of course, comprise two categories: defective and nondefective items, which are listed in the third row of Table 1. Thus, after the inspection
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Table 1 Ten categories of items after inspection and burn-in tests Qs
Qs p0 b Qs ð1 bÞ p0 Qs ð1 p0 Þ ð1 aÞ Qs a ð1 p0 Þ
expected expected expected expected
number number number number
of of of of
false acceptance of defectives correct classification of defectives correct classification of nondefectives false rejection of nondefectives
Qb
Qb p0 F ðtb Þ Qb p0 ½1 F ðtb Þ Qb ð1 p0 Þ Gðtb Þ Qb ð1 p0 Þ ½1 Gðtb Þ
expected expected expected expected
number number number number
of of of of
correct classification of defectives false acceptance of defectives false rejection of nondefectives correct classification of nondefectives
Qn
Qn p0 Qn ð1 p0 Þ
expected number of defectives expected number of nondefectives
and the burn-in tests, Q items can be divided into 10 categories. Note that, for tb which is not very large, since almost all nondefective parts will survive the burnin procedure, it is reasonable to assume Gðtb Þ 0.
(2)
Thus, from Eq. (2) and Table 1, N d and N a can be expressed as follows: N d ¼ Qs p0 b þ Qb p0 ½1 F ðtb Þ þ ðQ Qs Qb Þp0 and N a ¼ Q ð1 bÞQs p0 Qs að1 p0 Þ Qb p0 F ðtb Þ. Then, according to the preceding two equations, we can obtain AOQv ¼
Qs p0 b þ Qb p0 ½1 F ðtb Þ þ ðQ Qs Qb Þp0 . Q ð1 bÞQs p0 Qs að1 p0 Þ Qb p0 F ðtb Þ
3.2. The characterization of Pv ðQ; Qs ; Qb ; tb Þ The profit that the vendor realizes in each period can be expressed as follows: Pv ðQ; Qs ; Qb ; tb Þ ¼ revenue total cost. The revenue of the vendor, what the purchaser pays for the Q0 items, is CQ0 , where C is the price of a nondefective item if it is sold. The total cost consists of the following four segments:
It can be expressed as C s Qs , where C s is the unit cost of inspection. 3. The cost of burn-in: it can be expressed as C b tb Qb , where C b is the unit cost of burn-in per unit of time. 4. The cost due to false rejection of nondefective items: it can be expressed as C r að1 p0 Þ Qs , where C r is the unit cost due to false rejection of a nondefective item. As to the loss due to false rejection of a nondefective item C r , it is basically equal to the value or price of a nondefective item if it is sold. Hence, it is reasonable to set C r ¼ C. It is worth noting that when inspection error is present, losses are incurred by rejecting nondefective items and accepting defective items (see Tang and Tang, 1994). Several studies take such costs resulting from Type I and Type II errors into account in their optimization models (e.g., Tang and Schneider, 1988; Tang and Schneider, 1990; Badia et al., 2002; Duffuaa and Khan, 2005; etc.). In this paper, because false acceptance of defective items is allowed subject to the restriction on Assumption (ii) in Section 2, only the cost due to false rejection is included in the expected total profit function. Thus, by combining the five parts, we have Pv ¼ CQ0 ½C d Q þ Cað1 p0 ÞQs þ C s Qs þ C b tb Qb .
ð3Þ
3.3. The optimization model 1. The manufacturing cost of Q items: it can be expressed as C d Q, where C d is the manufacturing cost of each item. 2. The cost of inspection: it may include expenses of testing materials, labor, equipment, and so forth.
Since the term CQ0 in Eq. (3) is a constant, maximizing Pv is equivalent to minimizing the second term of the right-hand side in Eq. (3). Hence, synthesizing the results above, the optimization
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model (1) can be rewritten as follows: Minimize C d Q þ Cað1 p0 ÞQs þ C s Qs þ C b tb Qb Subject to
Qs p0 b þ Qb p0 ½1 F ðtb Þ þ ðQ Qs Qb Þp0 pp, Q ð1 bÞQs p0 Qs að1 p0 Þ Qb p0 F ðtb Þ Q ð1 bÞQs p0 Qs að1 p0 Þ
ð4Þ
Qb p0 F ðtb ÞXQ0 , Qs þ Qb pQ, tb pt0 Qs ; Qb 2 N [ f0g,
Note that the choice of the value of m in Step 1 depends on the approximation precision that the decision maker requires. Algorithm 2: In contrast to the integer programming in Algorithm 1, another rather practical method to solve the optimal solution ðQ ; Qb ; Qs ; tb Þ is to convert the optimization model (4) to a linear programming for a given tb (or F ðtb Þ) by utilizing the following changes of variables: Q ¼ rq Q 0 ;
Q 2 N,
Qs ¼ rs Q0 ;
tb X0.
Algorithm 1: Due to the complexity of this optimization model, an efficient method to obtain the optimal solution is missing. However, a closer examination of the objective function and the constraints reveals that, for a given tb (or F ðtb Þ), the optimization model can be expressed as an integer programming problem of Q, Qs , and Qb . Thus, an approximate solution ðQ ; Qb ; Qs ; tb Þ can be determined by the following steps: Step 1. Partition the interval ð0; t0 Þ equally into m (say, m ¼ 1000) subintervals and set tb ðkÞ ¼ k=m t0 , k ¼ 0; 1; 2; . . . ; m. Step 2. Fix k, the corresponding optimal value ðQðkÞ; Qb ðkÞ; Qs ðkÞÞ of ðQ; Qb ; Qs Þ can be obtained by solving the following integer programming: Minimize
C d Q þ ½Cað1 p0 Þ þ C s Qs
Subject to
þC b tb ðkÞQb ½p0 b p0 þ ð1 bÞpp0 þapð1 p0 ÞQs þ ½p0 pF ðtb ðkÞÞ p0 F ðtb ðkÞÞQb pðp p0 ÞQ; Q ð1 bÞp0 Qs að1 p0 ÞQs p0 F ðtb ðkÞÞQb XQ0 ; Qs þ Qb pQ; Qs ; Qb 2 N [ f0g; Q 2 N:
Step 3. Finally, an approximate optimal solution ðQ ; Qb ; Qs ; tb Þ can be determined if tb satisfies Pv ðQ ; Qs ; Qb ; tb Þ ¼ min Pv ðQðkÞ; Qs ðkÞ, 0pkpm
Qb ðkÞ; tb ðkÞÞ.
487
and
Q b ¼ rb Q 0 . (5)
The linear programming can be easily obtained as follows: Minimize
Q0 fC d rq þ ½Cað1 p0 Þ þ C s rs þ C b tb rb g
Subject to
ð6Þ
½p0 b p0 þ ð1 bÞpp0 þ apð1 p0 Þrs þ ½p0 pF ðtb Þ p0 F ðtb Þrb pðp p0 Þrq , rq ½ð1 bÞp0 þ að1 p0 Þrs p0 F ðtb Þrb X1, rs þ rb prq , tb pt0 , rq ; rs ; rb ; tb X0.
Thus, similar to Algorithm 1, an approximate solution ðQ ; Qb ; Qs ; tb Þ can be determined by the following steps: Step 0. For convenience, let Pv ðrq ; rs ; rb ; tb Þ ¼ Pv ðQ; Qs ; Qb ; tb Þ. Step 1. Partition the interval ð0; t0 Þ equally into m (say, m ¼ 1000) subintervals and set tb ðkÞ ¼ k=m t0 , k ¼ 0; 1; 2; . . . ; m. Step 2. For a fixed k, the corresponding optimal value ðrq ðkÞ; rb ðkÞ; rs ðkÞÞ of ðrq ; rb ; rs Þ, can be obtained by solving the following linear programming: Minimize
C d rq þ ½Cað1 p0 Þ þ C s rs þC b tb ðkÞrb
Subject to
½p0 b p0 þ ð1 bÞpp0 þapð1 p0 Þrs þ ½p0 pF ðtb ðkÞÞ p0 F ðtb ðkÞÞrb pðp p0 Þrq ; rq ½ð1 bÞp0 þ að1 p0 Þrs p0 F ðtb ðkÞÞrb X1;
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rs þ rb prq ; tb pt0 ; rq ; rs ; rb ; tb X0: Step 3. An approximate optimal solution ðrq ; rb ; rs ; tb Þ can be determined if tb satisfies Pv ðrq ; rs ; rb ; tb Þ ¼ min Pv ðrq ðkÞ; rs ðkÞ, 0pkpm
rb ðkÞ; tb ðkÞÞ. Step 4. Finally, by Eq. (5), ðQ ; Qb ; Qs Þ can be obtained as follows: Q ¼ rq Q0 ; and
Qs ¼ rs Q0 ,
Qb ¼ rb Q0 .
Similarly, the choice of the value of m in Step 1 depends on the approximation precision that the decision maker wants. Note that because rq , rs , and rb usually assume fractional values, to satisfy the integral constraints on Q, Qb , and Qs , we may round the optimal solutions Q , Qb , and Qs to the closest integer values. Thus, some integer rounding errors will be inevitable in computing the optimal solution ðQ ; Qb ; Qs Þ by using Algorithm 2. Fortunately, the rounding errors will tend to small if the order quantity Q0 is large enough. Compared with Algorithm 1, one advantage of Algorithm 2 is that it does not need to be recalculated for different order quantities. That is, a great deal of work can be saved in computing the optimal solution by Algorithm 2. In the next section, we will provide a numerical example solved by Algorithm 2 to illustrate this procedure. 4. A numerical example Assume that a product’s manufacturer wants to purchase a kind of part from a vendor to assemble his product. The manufacturer’s demand for this part is 3000 per month. It is well known that the lognormal distribution is one of the most appropriate lifetime distributions for electronic products (see Nelson, 1990). Hence, we assume that the lifetimes td and tg of defective and nondefective items follow lognormal distributions with location parameters md and mg (md 5mg ) and scale parameter s, respectively. For illustrative purposes, we set Q0 ¼ 3000, C ¼ 3.0 dollars per item, C s ¼ 0.1 dollars per inspection, C d ¼ 1.0 dollar per item, t0 ¼ 300 h, and ðmd ; sÞ ¼ ð4:2; 1:0Þ. Then, by Algorithm 2 with
m ¼ 1000 given in Section 3, the optimal test plans for 64 combinations of p0 , p, a, b, and C b are listed in Table 2, where all of p0 , a, b, and C b are set at two levels, and p is set at 0.02, 0.04, 0.06, and 0.08. In addition, C b means that the burn-in cost per hour is C b dollars. Now, we take two numerical results in Table 2 for example to justify the assertion that inspection and burn-in play a complementary role of each other in Section 1. Result 1: For ðp0 ; p; a; b; C b Þ ¼ ð0:10; 0:02; 0:05; 0:10; 0:002Þ, it is easily seen, from Table 2, that the optimal production quantity is Q ¼ 3394 items; the optimal quantity for inspection is Qs ¼ 2546 items; the optimal quantity for burn-in is Qb ¼ 848 items; the number of the remainder is Qn ¼ Q Qs Qb ¼ 0 items; the optimal burn-in time is tb ¼ 84:3 h; while the profit is Pv ðQ ; Qs ; Qb ; tb Þ ¼ 4864:750 dollars. To facilitate the understanding of how superior the optimal policy is over the individual ‘‘pure’’ ones, the corresponding optimal test plans for inspection and burn-in tests are computed as follows: Test
ðQ ; Qs ; Qb ; tb Þ
Pv ðQ ; Qs ; Qb ; tb Þ
Inspection (3469, 3469, 0, 0) 4715.785 Burn-in (3267, 0, 3267, 164.3) 4659.464
Result 2: For ðp0 ; p; a; b; C b Þ ¼ ð0:15; 0:02; 0:10; 0:15; 0:005Þ, it is seen, from Table 2, that the optimal production quantity is Q ¼ 3628 items; the optimal quantity for inspection is Qs ¼ 1691 items; the optimal quantity for burn-in is Qb ¼ 1937 items; the number of the remainder is Qn ¼ Q Qs Qb ¼ 0 items; the optimal burn-in time is tb ¼ 280:2 h; while the profit is Pv ðQ ; Qs ; Qb ; tb Þ ¼ 2057:878 dollars. Similarly, to highlight the superiority of the mixed policy over the individual ‘‘pure’’ ones, the corresponding optimal test plans for ‘‘pure’’ inspection and ‘‘pure’’ burn-in tests are listed as follows: Test
ðQ ; Qs ; Qb ; tb Þ
Pv ðQ ; Qs ; Qb ; tb Þ
Inspection Burn-in
No solution (3459, 0, 3459, 220.75)
– 32637:71
In this case, a ‘‘pure’’ inspection test results in AOQv ¼ 0:02905 which is greater than
ARTICLE IN PRESS H.-F. Yu, W.-C. Yu / Int. J. Production Economics 105 (2007) 483–491 Table 2 The optimal policies ðQ ; Qs ; Qb ; tb Þ and the profits Pv ðQ ; Qs ; Qb ; tb Þ under various combinations of ðp0 ; p; a; b; C b Þ ðp0 ; p; a; b; C b Þ
ðQ ; Qs ; Qb ; tb Þ
Pv ðQ ; Qs ; Qb ; tb Þ
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.10, 0.10, 0.10, 0.10,
0.002) 0.002) 0.002) 0.002)
(3394, (3217, (3133, (3067,
2546, 848, 84.3) 347, 2870, 84.6) 0, 3121, 55.5) 0, 1752, 49.2)
4864.750 5215.458 5520.194 5760.947
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.10, 0.10, 0.10, 0.10,
0.005) 0.005) 0.005) 0.005)
(3424, (3318, (3212, (3106,
3137, 0, 0) 2353, 0, 0) 1569, 0, 0) 784, 0, 0)
4839.222 5129.416 5419.616 5709.811
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.15, 0.15, 0.15, 0.15,
0.002) 0.002) 0.002) 0.002)
(3415, (3201, (3133, (3067,
2963, 452, 99.6) 13, 3188, 91.5) 0, 3121, 55.5) 0, 1752, 49.2)
4798.907 5212.905 5520.188 5760.945
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.15, 0.15, 0.15, 0.15,
0.005) 0.005) 0.005) 0.005)
(3433, (3325, (3217, (3108,
3333, 0, 0) 2500, 0, 0) 1667, 0, 0) 833, 0, 0)
4783.415 5087.502 5391.670 5695.838
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.10, 0.10, 0.10, 0.10,
0.002) 0.002) 0.002) 0.002)
(3267, (3200, (3133, (3067,
0, 0, 0, 0,
4659.548 5212.829 5520.193 5760.941
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.10, 0.10, 0.10, 0.10,
0.005) 0.005) 0.005) 0.005)
(3600, (3450, (3300, (3150,
3333, 0, 0) 2500, 0, 0) 1667, 0, 0) 833, 0, 0)
4167.291 4625.004 5083.344 5541.674
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.15, 0.15, 0.15, 0.15,
0.002) 0.002) 0.002) 0.002)
(3267, (3200, (3133, (3067,
0, 0, 0, 0,
4659.548 5212.829 5520.193 5760.940
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.15, 0.15, 0.15, 0.15,
0.005) 0.005) 0.005) 0.005)
(3622, (3467, (3311, (3156,
3556, 0, 0) 2667, 0, 0) 1778, 0, 0) 889, 0, 0)
4062.406 4546.673 5031.119 5515.561
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.10, 0.10, 0.10, 0.10,
0.002) 0.002) 0.002) 0.002)
(3635, (3494, (3353, (3247,
3523, 112, 80.4) 2111, 1383, 80.4) 699, 2654, 80.4) 6, 3242, 67.8)
4545.546 4803.641 5061.733 5311.801
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.10, 0.10, 0.10, 0.10,
0.005) 0.005) 0.005) 0.005)
(3639, (3540, (3442, (3344,
3599, 3045, 2491, 1938,
0) 0) 0) 0)
4542.726 4766.784 4991.007 5215.229
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.15, 0.15, 0.15, 0.15,
0.002) 0.002) 0.002) 0.002)
(3530, (3499, (3320, (3247,
1427, 2103, 257.4) 2219, 1280, 94.2) 54, 3267, 94.2) 7, 3241, 67.8)
4062.523 4754.820 5052.014 5311.674
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.15, 0.15, 0.15, 0.15,
0.005) 0.005) 0.005) 0.005)
(3557, (3550, (3450, (3350,
1961, 3235, 2647, 2059,
2603.373 4713.975 4947.796 5181.624
(0.15, 0.02, 0.10, 0.10, 0.002) (0.15, 0.04, 0.10, 0.10, 0.002)
3266, 3198, 3121, 1752,
3266, 3198, 3121, 1763,
0, 0, 0, 0,
164.4) 91.8) 55.5) 49.2)
164.4) 91.8) 55.5) 48.9)
1596, 300.0) 0, 0) 0, 0) 0, 0)
(3459, 0, 3459, 220.8) (3389, 3, 3385, 136.8)
4013.838 4684.071
489
ARTICLE IN PRESS H.-F. Yu, W.-C. Yu / Int. J. Production Economics 105 (2007) 483–491
490 Table 2 (continued ) ðp0 ; p; a; b; C b Þ
ðQ ; Qs ; Qb ; tb Þ
Pv ðQ ; Qs ; Qb ; tb Þ
(0.15, 0.06, 0.10, 0.10, 0.002) (0.15, 0.08, 0.10, 0.10, 0.002)
(3318, 0, 3316, 95.1) (3247, 0, 3240, 68.1)
5051.718 5311.665
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.10, 0.10, 0.10, 0.10,
0.005) 0.005) 0.005) 0.005)
(3841, (3712, (3582, (3453,
3824, 3235, 2647, 2059,
0) 0) 0) 0)
3801.507 4139.707 4477.943 4816.187
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.15, 0.15, 0.15, 0.15,
0.002) 0.002) 0.002) 0.002)
(3459, (3389, (3318, (3247,
0, 6, 0, 0,
220.8) 136.8) 95.1) 68.1)
4013.838 4683.535 5051.718 5311.659
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.15, 0.15, 0.15, 0.15,
0.005) 0.005) 0.005) 0.005)
(3628, (3733, (3600, (3467,
1691, 3451, 2824, 2197,
1937, 280.2) 0, 0) 0, 0) 0, 0)
2057.878 4041.836 4397.651 4753.732
0, 0, 0, 0,
3459, 3383, 3316, 3240,
AOQ ¼ 0:02. That is, an outgoing batch will not satisfy the quality requirement if only inspection test is employed. Hence, no solution can be obtained. On the other hand, if only burn-in test is employed to guarantee that the AOQ requirement is satisfied, then the vendor will lose much money in the trade. In addition, from the results in Table 2, it is seen that (a) The larger p0 is, the larger Q is. (b) The larger p is, the smaller Q is. (c) If p is small, then the burn-in test is profitable for the vendor. (d) Basically, if the unit cost of burn-in C b is relatively less than the unit cost of inspection C s , then the burn-in test is profitable to the vendor. Conversely, if C b is relatively greater than C s , then the inspection test is profitable for the vendor. (e) Basically, if a and b are small, then the inspection is profitable for the vendor. However, if a and b are large, then the burn-in test is profitable for the vendor.
satisfies the purchaser’s AOQ requirement while remaining low cost is a challenge for the vendor. In this paper, we deal with the problem of determining the optimal mixed policy of inspection and burn-in. More specifically, under the constraint that the outgoing batch satisfies the purchaser’s AOQ requirement, the following issues are determined to maximize the expected profit that the vendor makes: (a) the total number of items that the vendor needs to produce, (b) the number of items for inspection, the number of items for burn-in, and the number of items that need neither inspection nor burn-in, and (c) the optimal burn-in time for those needing burn-in testing. In practice, a product often requires inspection on more than one characteristic. It is no doubt an interesting topic to extend our proposed method to such a situation. Besides, in real situations, the buyer usually gains a discount from the vendor if the contract between them is a long-term one. Hence, it is also an important topic for future research to investigate the optimal mixed policy of inspection and burn-in and the optimal production quantity under infinite horizon with a discount factor.
5. Conclusion
References
Inspection and burn-in are two techniques that are widely used by vendors to ensure the outgoing product’s quality. Due to inspection errors and high cost of burn-in, how to make a trade-off between these two techniques such that the outgoing batch
Badia, F.G., Berrade, M.D., Campos, C.A., 2002. Optimal inspection and preventive maintenance of units with revealed and unrevealed failures. Reliability Engineering & System Safety 78, 157–163. Britney, R.R., 1972. Optimal screening plans for non-serial production systems. Management Science 18, 550–559.
ARTICLE IN PRESS H.-F. Yu, W.-C. Yu / Int. J. Production Economics 105 (2007) 483–491 Chen, S.X., Lambrecht, M., 1997. The optimal frequency and sequencing of tests in the inspection of multicharacteristics components. IIE Transactions 29, 1039–1049. Cozzolino, J.M., 1970. Optimal burn-in testing of repairable equipments. Naval Research Logistics Quarterly 17, 167–182. Duffuaa, S.O., Khan, M., 2005. Impact of inspection on the performance measures of a general repeat inspection plan. International Journal of Production Research 43, 4945–4967. Duffuaa, S.O., Roauf, A., 1990. On optimal sequencing of multicharacteristic inspection. Journal of Optimization Theory and Applications 67, 79–86. Eppen, G.D., Hurst, E.G., 1974. Optimal location of inspection stations in a multistage production process. Management Science 20, 1194–1200. Greenshtein, E., Rabinowitz, G., 1997. Double-stage inspection for screening multi-characteristic items. IIE Transactions 29, 1057–1061. Hau, H.L., 1988. On the optimality of a simplified multicharacteristic component inspection model. IIE Transactions 20, 393–398. Hau, H.L., Rosenblatt, M.J., 1987. Simultaneous determination of production cycle and inspection schedules in a production system. Management Science 33, 1125–1136. Jaraiedi, M., Kochhar, D.S., Jaisingh, S.C., 1987. Multiple inspections to meet desired outgoing quality. Journal of Quality Technology 19, 46–51. Jenson, F., Petersen, N.E., 1982. Burn-In. Wiley, New York. Kuo, W., Kuo, Y., 1990. Facing the headaches of early failures: a state-of-the-art review of burn-in decisions. Proceedings of the IEEE 71, 1257–1266. Liou, M.J., Tseng, S.T., Lin, T.M., 1994. Optimal sequence of partial inspections subject to errors. International Journal of Production Economics 33, 109–119. Mi, J., 1994. Burn-in and maintenance policies. Advance in Applied Probability 26, 207–221. Nelson, W., 1990. Accelerated Testing: Statistical Models, Test Plans, and Data Analysis. Wiley, New York. Nguyen, D.D., Murthy, D.N.P., 1982. Optimal burn-in time to minimize cost for products sold under warranty. IIE Transactions 14, 167–174.
491
Park, J.S., Peter, M.H., Tang, K., 1991. Optimal inspection policy in sequential screening. Management Science 37, 1058–1061. Raouf, A., Jain, J.K., Sathe, P.T., 1983. A cost-minimization model for multicharacteristic component inspection. IIE Transactions 15, 187–194. Raz, T., 1986. A survey of models for allocating inspection effort in multistage production systems. Journal of Quality Technology 18, 239–247. Raz, T., Kaspi, M., 1991. Location and sequencing of imperfect inspection operations in a serial multi-stage production system. International Journal of Production Research 29, 1645–1659. Raz, T., Thomas, M.U., 1983. A method for sequencing inspection activities subject to errors. IIE Transactions 15, 12–18. Tang, K., Schneider, H., 1988. Selection of the optimal inspection precision level for a complete inspection plan. Journal of Quality Technology 20, 153–156. Tang, K., Schneider, H., 1990. Cost effectiveness of using a corrected variable in a complete inspection plan when inspection error is present. Naval Research Logistics 37, 893–904. Tang, K., Tang, J., 1989. Design of product specifications for multi-characteristic inspection. Management Science 35, 743–756. Tang, K., Tang, J., 1994. Design of screening procedures: a review. Journal of Quality Technology 26, 209–226. Wang, C.H., 2005. Integrated production and product inspection policy for a deteriorating system. International Journal of Production Economics 95, 123–134. Weiss, G.H., Dishon, M., 1971. Some economic problems related to burn-in programs. IEEE Transactions on Reliability 20, 190–195. White, L.S., 1966. An analysis of a simple class of multistage inspection plans. Management Science 12, 685–693. Yum, B.J., McDowell, E.D., 1981. The optimal allocation of inspection efforts in a class of non-serial production system. AIIE Transactions 13, 193–285.
Int. J. Production Economics 105 (2007) 483–491 www.elsevier.com/locate/ijpe
An optimal mixed policy of inspection and burn-in and the optimal production quantity Hong-Fwu Yua,, Wen-Ching Yub a
Department of Industrial Management, National Formosa University 64, Wen Hua Road, Hu-Wei, Yunlin 632, Taiwan, ROC b Department of Business Management, National United University 1, Lien Da, Kung-Ching Li, Miao-Li 360, Taiwan, ROC Received 15 September 2003; accepted 20 April 2006 Available online 13 June 2006
Abstract Inspection and burn-in are two techniques that are widely used by the vendor to screen out defective items in a production lot in order that an outgoing batch satisfies the purchaser’s quality requirements. Due to two types of inspection errors and high cost of burn-in, how to make a trade-off between them is a challenge for the vendor. The main purpose of this paper is to deal with the problem of determining the optimal mixed policy of inspection and burn-in, where the average outgoing quality (AOQ) is used as a measure of inspection and burn-in success. More specifically, under the constraint that the outgoing batch meets an AOQ requirement, the following issues are determined to maximize the expected profit that the vendor makes: (a) the total number of items that the vendor needs to produce, (b) the number of items for inspection, the number of items for burn-in, and the number of items that need neither inspection nor burn-in, and (c) the optimal burn-in time if burn-in test is needed. Finally, an example is provided to illustrate the proposed method. r 2006 Elsevier B.V. All rights reserved. Keywords: Mixed policy; Inspection; Burn-in; Average outgoing quality
1. Introduction In order to manufacture a product, some parts of the product need to be purchased from vendors. To guarantee a high-quality level for this product, some quality requirements for each part would be specified in the contract between the manufacturer of this product and the vendor. To achieve the manufacturer’s quality requirements, the vendor must make efforts to screen out defective items in Corresponding author. Tel.: +886 5 6315716; fax: +886 5 6311548. E-mail address: [email protected] (H.-F. Yu).
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. However, the inspection testing fails to be perfect and two types of errors (Types I and II) can occur. Thus, the vendor usually faces two practical problems: (1) false acceptance of defectives resulting in that the outgoing batch may not satisfy the quality requirements and (2) false rejection of nondefectives resulting in the necessity of producing additional items beyond the quantity ordered by the manufacturer. Certainly, this will increase the vendor’s cost. In fact, inspection is not
0925-5273/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2006.04.013
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cost-free, so the need for inspection should be justified in terms of the costs involved. Raz and Thomas (1983) studied sequencing several inspectors with different inspection precision levels to meet a predetermined outgoing conformation rate at minimum cost. Tang and Schneider (1988) proposed a method of determining the optimal inspection precision level based on the tradeoff of inspection cost and the costs incurred by inspection errors. In addition, vast amounts of excellent research on inspection policies for various situations exists (see Britney, 1972; Chen and Lambrecht, 1997; Duffuaa and Roauf, 1990; Eppen and Hurst, 1974; Greenshtein and Rabinowitz, 1997; Hau, 1988; Hau and Rosenblatt, 1987; Jaraiedi et al., 1987; Park et al., 1991; Raouf et al., 1983; Raz, 1986; Raz and Kaspi, 1991; Tang and Tang, 1989; White, 1966; Yum and McDowell, 1981; Liou et al., 1994; Wang, 2005). Another screening method which is also widely used to guarantee a high-quality level for the outgoing batch of items is the burn-in test. In general, if the burn-in time is long enough, then most of defective items would be screened out. Thus, the outgoing batch of items would satisfy the quality requirements. A general background on burn-in can be found in Jensen and Petersen (1982) and Kuo and Kuo (1990), and many references are also given there. Unfortunately, compared with the inspection test, burn-in is usually costly. That is, the longer the burn-in is, the higher the manufacturing cost is. Thus, a practical problem to consider is how long the burn-in procedure should continue. Cozzolino (1970) studied the optimal length of the burnin process for repairable products. Weiss and Dishon (1971) discussed two situations where a specific number of items are required at the end of the burn-in process. Nguyen and Murthy (1982) formulated a model for determining the burn-in time for a product sold with a warranty. Mi (1994) considered the optimal burn-in and maintenance policies for repairable products at minimum cost. For a variety of products (e.g., Logic IC, Memory IC, fan of a power supply, etc.) in semiconductor industries, both inspection and burn-in tests are widely utilized to screen out defective items. In contrast with the burn-in test, a ‘‘pure’’ inspection policy has two shortcomings: (1) more items need to be produced because of false rejection of nondefectives and (2) the quality level among the items remained (after inspection) may not attain to the quality requirements because of false acceptance of
defectives. The former leads directly to the increase in cost and the latter may lead to that the outgoing batch is rejected. A remedy to these two shortcomings, especially the latter, is to employ the burnin test. However, compared to the inspection test, a ‘‘pure’’ burn-in policy is time-consuming and, hence, cost-consuming, although it can guarantee that the outgoing batch satisfies the quality requirements if the burn-in time is long enough. Thus, from both economic and quality’s points of view, the two screening methods cannot take the place of each other completely. Hence, how to make a trade-off between them such that the quality requirement is satisfied with low cost is no doubt an interesting topic for the vendor. Unfortunately, according to the literature, this has not been discussed. To this end, we will investigate the optimal mixed use of inspection and burn-in methods, where the average outgoing quality (AOQ) is utilized as a measure of inspection and burn-in success. More specifically, under the constraint that the outgoing batch meets an AOQ requirement, the following issues are determined to maximize the expected profit for the vendor: (a) the total number of items that the vendor should produce, (b) the number of items for inspection, the number of items for burn-in, and the number of items that need neither inspection or burn-in, and (c) the optimal burn-in time if burn-in testing is adopted. The rest of this paper is organized as follows. Section 2 briefly describes the assumptions on which a mixed policy of inspection and burn-in is based, and presents the corresponding optimization problem. Section 3 presents the optimal plan. Section 4 provides a numerical example to illustrate the proposed method. Finally, a conclusion is drawn in Section 5. 2. Assumptions and the optimization problem This section is devoted to describing the assumptions of a mixed use of inspection and burn-in tests and the optimization problem. 2.1. Assumptions Assume that the manufacturer of a product wants to purchase some kind of part from a vendor. The contract between them is as follows: (i) The quantity that the manufacturer orders is Q0 for each period (e.g., per week or per month).
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485
(ii) AOQ is set to be p. (iii) The price of each part is C dollars. (iv) The manufacturer will pay the vendor for the Q0 parts after them are shipped to the manufacturer.
manufacturer satisfy the conditions in that contract while accomplishing low cost, the decision problem can be formulated as follows:
Assume that the part is produced in batches or lots (not continuously) from a production process with a defective rate of p0 ð4pÞ on average and that the production quantity is Q for each batch. To guarantee the outgoing batch shipped to the manufacturer would satisfy the AOQ requirement, the vendor decides to adopt both inspection and burn-in techniques to screen out defective parts, where the probabilities of Type I error and Type II error of the inspection are a and b, respectively. For the vendor, to maximize the total profit in each period, he/she faces the following decision problems:
(1)
Maximize
Pv ðQ; Qs ; Qb ; tb Þ
Subject to
d AOQv ¼ N N a pp; N a XQ0 ;
Q; Qs ; Qb 2 N ¼ f1; 2; 3; . . .g; tb X0: In practical situations, the vendor is usually requested to keep promised delivery dates. Otherwise good-will would surely be lost. To achieve this, he/she must control the times of all operations in the production and delivery processes. This indicates that the burn-in time is not unlimited when burn-in is employed. Hence, the following constraint to the burn-in time tb is necessary: tb pt0 ,
1. How many parts (Q) need to be produced? 2. Among the Q parts, how many (Qs ) need to be inspected? How many (Qb ) need to be put into burn-in? And, how many (Qn ) need neither inspection nor burn-in? 3. How long (tb ) need the burn-in procedure continue for those requiring the burn-in test?
where t0 is a tolerance limit prespecified by the vendor.
To facilitate the analysis that follows, we use symbols and notations to summarize the vendor’s mixed policy as follows:
1. Compute N a , N d , and AOQv . 2. Characterize Pv ðQ; Qs ; Qb ; tb Þ.
AOQv average outgoing quality of the outgoing batch, Na expected number of items both correctly and incorrectly accepted after inspection and burn-in tests, Nd expected number of defective parts contained in N a . In the following sections, we will propose an optimal plan to determine Q, Qs , Qb , Qn , and tb simultaneously. 2.2. The optimization problem Let Pv ¼ Pv ðQ; Qs ; Qb ; tb Þ denote the expected profit that the vendor makes in each period. Since AOQ is a ratio of the number of undetected defective items to the total number of items that are accepted, we can obtain AOQv ¼ N d =N a . Then, to make the outgoing batch shipped to the
3. The optimal plan The framework for solving the optimization model consists of two major steps:
3.1. The computation of N a , N d , and AOQv First, as to Qs items, we calculate the expected number of items remaining after the inspection and the defective percentage in this lot. Since there exist two types of errors, the Qs items can be divided into four categories, which are listed in the first row of Table 1, after the inspection is carried out. Next, let us consider the Qb items that are allocated for burn-in testing. Let F ðtÞ denote the cumulative distribution function (CDF) of the defective part’s lifetime distribution and let GðtÞ denote the CDF of nondefective parts. Then, after the burn-in test, the Qb items can be divided into four categories, which are listed in the second row of Table 1. Finally, as to the items Qn that no actions are taken, they, of course, comprise two categories: defective and nondefective items, which are listed in the third row of Table 1. Thus, after the inspection
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Table 1 Ten categories of items after inspection and burn-in tests Qs
Qs p0 b Qs ð1 bÞ p0 Qs ð1 p0 Þ ð1 aÞ Qs a ð1 p0 Þ
expected expected expected expected
number number number number
of of of of
false acceptance of defectives correct classification of defectives correct classification of nondefectives false rejection of nondefectives
Qb
Qb p0 F ðtb Þ Qb p0 ½1 F ðtb Þ Qb ð1 p0 Þ Gðtb Þ Qb ð1 p0 Þ ½1 Gðtb Þ
expected expected expected expected
number number number number
of of of of
correct classification of defectives false acceptance of defectives false rejection of nondefectives correct classification of nondefectives
Qn
Qn p0 Qn ð1 p0 Þ
expected number of defectives expected number of nondefectives
and the burn-in tests, Q items can be divided into 10 categories. Note that, for tb which is not very large, since almost all nondefective parts will survive the burnin procedure, it is reasonable to assume Gðtb Þ 0.
(2)
Thus, from Eq. (2) and Table 1, N d and N a can be expressed as follows: N d ¼ Qs p0 b þ Qb p0 ½1 F ðtb Þ þ ðQ Qs Qb Þp0 and N a ¼ Q ð1 bÞQs p0 Qs að1 p0 Þ Qb p0 F ðtb Þ. Then, according to the preceding two equations, we can obtain AOQv ¼
Qs p0 b þ Qb p0 ½1 F ðtb Þ þ ðQ Qs Qb Þp0 . Q ð1 bÞQs p0 Qs að1 p0 Þ Qb p0 F ðtb Þ
3.2. The characterization of Pv ðQ; Qs ; Qb ; tb Þ The profit that the vendor realizes in each period can be expressed as follows: Pv ðQ; Qs ; Qb ; tb Þ ¼ revenue total cost. The revenue of the vendor, what the purchaser pays for the Q0 items, is CQ0 , where C is the price of a nondefective item if it is sold. The total cost consists of the following four segments:
It can be expressed as C s Qs , where C s is the unit cost of inspection. 3. The cost of burn-in: it can be expressed as C b tb Qb , where C b is the unit cost of burn-in per unit of time. 4. The cost due to false rejection of nondefective items: it can be expressed as C r að1 p0 Þ Qs , where C r is the unit cost due to false rejection of a nondefective item. As to the loss due to false rejection of a nondefective item C r , it is basically equal to the value or price of a nondefective item if it is sold. Hence, it is reasonable to set C r ¼ C. It is worth noting that when inspection error is present, losses are incurred by rejecting nondefective items and accepting defective items (see Tang and Tang, 1994). Several studies take such costs resulting from Type I and Type II errors into account in their optimization models (e.g., Tang and Schneider, 1988; Tang and Schneider, 1990; Badia et al., 2002; Duffuaa and Khan, 2005; etc.). In this paper, because false acceptance of defective items is allowed subject to the restriction on Assumption (ii) in Section 2, only the cost due to false rejection is included in the expected total profit function. Thus, by combining the five parts, we have Pv ¼ CQ0 ½C d Q þ Cað1 p0 ÞQs þ C s Qs þ C b tb Qb .
ð3Þ
3.3. The optimization model 1. The manufacturing cost of Q items: it can be expressed as C d Q, where C d is the manufacturing cost of each item. 2. The cost of inspection: it may include expenses of testing materials, labor, equipment, and so forth.
Since the term CQ0 in Eq. (3) is a constant, maximizing Pv is equivalent to minimizing the second term of the right-hand side in Eq. (3). Hence, synthesizing the results above, the optimization
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model (1) can be rewritten as follows: Minimize C d Q þ Cað1 p0 ÞQs þ C s Qs þ C b tb Qb Subject to
Qs p0 b þ Qb p0 ½1 F ðtb Þ þ ðQ Qs Qb Þp0 pp, Q ð1 bÞQs p0 Qs að1 p0 Þ Qb p0 F ðtb Þ Q ð1 bÞQs p0 Qs að1 p0 Þ
ð4Þ
Qb p0 F ðtb ÞXQ0 , Qs þ Qb pQ, tb pt0 Qs ; Qb 2 N [ f0g,
Note that the choice of the value of m in Step 1 depends on the approximation precision that the decision maker requires. Algorithm 2: In contrast to the integer programming in Algorithm 1, another rather practical method to solve the optimal solution ðQ ; Qb ; Qs ; tb Þ is to convert the optimization model (4) to a linear programming for a given tb (or F ðtb Þ) by utilizing the following changes of variables: Q ¼ rq Q 0 ;
Q 2 N,
Qs ¼ rs Q0 ;
tb X0.
Algorithm 1: Due to the complexity of this optimization model, an efficient method to obtain the optimal solution is missing. However, a closer examination of the objective function and the constraints reveals that, for a given tb (or F ðtb Þ), the optimization model can be expressed as an integer programming problem of Q, Qs , and Qb . Thus, an approximate solution ðQ ; Qb ; Qs ; tb Þ can be determined by the following steps: Step 1. Partition the interval ð0; t0 Þ equally into m (say, m ¼ 1000) subintervals and set tb ðkÞ ¼ k=m t0 , k ¼ 0; 1; 2; . . . ; m. Step 2. Fix k, the corresponding optimal value ðQðkÞ; Qb ðkÞ; Qs ðkÞÞ of ðQ; Qb ; Qs Þ can be obtained by solving the following integer programming: Minimize
C d Q þ ½Cað1 p0 Þ þ C s Qs
Subject to
þC b tb ðkÞQb ½p0 b p0 þ ð1 bÞpp0 þapð1 p0 ÞQs þ ½p0 pF ðtb ðkÞÞ p0 F ðtb ðkÞÞQb pðp p0 ÞQ; Q ð1 bÞp0 Qs að1 p0 ÞQs p0 F ðtb ðkÞÞQb XQ0 ; Qs þ Qb pQ; Qs ; Qb 2 N [ f0g; Q 2 N:
Step 3. Finally, an approximate optimal solution ðQ ; Qb ; Qs ; tb Þ can be determined if tb satisfies Pv ðQ ; Qs ; Qb ; tb Þ ¼ min Pv ðQðkÞ; Qs ðkÞ, 0pkpm
Qb ðkÞ; tb ðkÞÞ.
487
and
Q b ¼ rb Q 0 . (5)
The linear programming can be easily obtained as follows: Minimize
Q0 fC d rq þ ½Cað1 p0 Þ þ C s rs þ C b tb rb g
Subject to
ð6Þ
½p0 b p0 þ ð1 bÞpp0 þ apð1 p0 Þrs þ ½p0 pF ðtb Þ p0 F ðtb Þrb pðp p0 Þrq , rq ½ð1 bÞp0 þ að1 p0 Þrs p0 F ðtb Þrb X1, rs þ rb prq , tb pt0 , rq ; rs ; rb ; tb X0.
Thus, similar to Algorithm 1, an approximate solution ðQ ; Qb ; Qs ; tb Þ can be determined by the following steps: Step 0. For convenience, let Pv ðrq ; rs ; rb ; tb Þ ¼ Pv ðQ; Qs ; Qb ; tb Þ. Step 1. Partition the interval ð0; t0 Þ equally into m (say, m ¼ 1000) subintervals and set tb ðkÞ ¼ k=m t0 , k ¼ 0; 1; 2; . . . ; m. Step 2. For a fixed k, the corresponding optimal value ðrq ðkÞ; rb ðkÞ; rs ðkÞÞ of ðrq ; rb ; rs Þ, can be obtained by solving the following linear programming: Minimize
C d rq þ ½Cað1 p0 Þ þ C s rs þC b tb ðkÞrb
Subject to
½p0 b p0 þ ð1 bÞpp0 þapð1 p0 Þrs þ ½p0 pF ðtb ðkÞÞ p0 F ðtb ðkÞÞrb pðp p0 Þrq ; rq ½ð1 bÞp0 þ að1 p0 Þrs p0 F ðtb ðkÞÞrb X1;
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rs þ rb prq ; tb pt0 ; rq ; rs ; rb ; tb X0: Step 3. An approximate optimal solution ðrq ; rb ; rs ; tb Þ can be determined if tb satisfies Pv ðrq ; rs ; rb ; tb Þ ¼ min Pv ðrq ðkÞ; rs ðkÞ, 0pkpm
rb ðkÞ; tb ðkÞÞ. Step 4. Finally, by Eq. (5), ðQ ; Qb ; Qs Þ can be obtained as follows: Q ¼ rq Q0 ; and
Qs ¼ rs Q0 ,
Qb ¼ rb Q0 .
Similarly, the choice of the value of m in Step 1 depends on the approximation precision that the decision maker wants. Note that because rq , rs , and rb usually assume fractional values, to satisfy the integral constraints on Q, Qb , and Qs , we may round the optimal solutions Q , Qb , and Qs to the closest integer values. Thus, some integer rounding errors will be inevitable in computing the optimal solution ðQ ; Qb ; Qs Þ by using Algorithm 2. Fortunately, the rounding errors will tend to small if the order quantity Q0 is large enough. Compared with Algorithm 1, one advantage of Algorithm 2 is that it does not need to be recalculated for different order quantities. That is, a great deal of work can be saved in computing the optimal solution by Algorithm 2. In the next section, we will provide a numerical example solved by Algorithm 2 to illustrate this procedure. 4. A numerical example Assume that a product’s manufacturer wants to purchase a kind of part from a vendor to assemble his product. The manufacturer’s demand for this part is 3000 per month. It is well known that the lognormal distribution is one of the most appropriate lifetime distributions for electronic products (see Nelson, 1990). Hence, we assume that the lifetimes td and tg of defective and nondefective items follow lognormal distributions with location parameters md and mg (md 5mg ) and scale parameter s, respectively. For illustrative purposes, we set Q0 ¼ 3000, C ¼ 3.0 dollars per item, C s ¼ 0.1 dollars per inspection, C d ¼ 1.0 dollar per item, t0 ¼ 300 h, and ðmd ; sÞ ¼ ð4:2; 1:0Þ. Then, by Algorithm 2 with
m ¼ 1000 given in Section 3, the optimal test plans for 64 combinations of p0 , p, a, b, and C b are listed in Table 2, where all of p0 , a, b, and C b are set at two levels, and p is set at 0.02, 0.04, 0.06, and 0.08. In addition, C b means that the burn-in cost per hour is C b dollars. Now, we take two numerical results in Table 2 for example to justify the assertion that inspection and burn-in play a complementary role of each other in Section 1. Result 1: For ðp0 ; p; a; b; C b Þ ¼ ð0:10; 0:02; 0:05; 0:10; 0:002Þ, it is easily seen, from Table 2, that the optimal production quantity is Q ¼ 3394 items; the optimal quantity for inspection is Qs ¼ 2546 items; the optimal quantity for burn-in is Qb ¼ 848 items; the number of the remainder is Qn ¼ Q Qs Qb ¼ 0 items; the optimal burn-in time is tb ¼ 84:3 h; while the profit is Pv ðQ ; Qs ; Qb ; tb Þ ¼ 4864:750 dollars. To facilitate the understanding of how superior the optimal policy is over the individual ‘‘pure’’ ones, the corresponding optimal test plans for inspection and burn-in tests are computed as follows: Test
ðQ ; Qs ; Qb ; tb Þ
Pv ðQ ; Qs ; Qb ; tb Þ
Inspection (3469, 3469, 0, 0) 4715.785 Burn-in (3267, 0, 3267, 164.3) 4659.464
Result 2: For ðp0 ; p; a; b; C b Þ ¼ ð0:15; 0:02; 0:10; 0:15; 0:005Þ, it is seen, from Table 2, that the optimal production quantity is Q ¼ 3628 items; the optimal quantity for inspection is Qs ¼ 1691 items; the optimal quantity for burn-in is Qb ¼ 1937 items; the number of the remainder is Qn ¼ Q Qs Qb ¼ 0 items; the optimal burn-in time is tb ¼ 280:2 h; while the profit is Pv ðQ ; Qs ; Qb ; tb Þ ¼ 2057:878 dollars. Similarly, to highlight the superiority of the mixed policy over the individual ‘‘pure’’ ones, the corresponding optimal test plans for ‘‘pure’’ inspection and ‘‘pure’’ burn-in tests are listed as follows: Test
ðQ ; Qs ; Qb ; tb Þ
Pv ðQ ; Qs ; Qb ; tb Þ
Inspection Burn-in
No solution (3459, 0, 3459, 220.75)
– 32637:71
In this case, a ‘‘pure’’ inspection test results in AOQv ¼ 0:02905 which is greater than
ARTICLE IN PRESS H.-F. Yu, W.-C. Yu / Int. J. Production Economics 105 (2007) 483–491 Table 2 The optimal policies ðQ ; Qs ; Qb ; tb Þ and the profits Pv ðQ ; Qs ; Qb ; tb Þ under various combinations of ðp0 ; p; a; b; C b Þ ðp0 ; p; a; b; C b Þ
ðQ ; Qs ; Qb ; tb Þ
Pv ðQ ; Qs ; Qb ; tb Þ
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.10, 0.10, 0.10, 0.10,
0.002) 0.002) 0.002) 0.002)
(3394, (3217, (3133, (3067,
2546, 848, 84.3) 347, 2870, 84.6) 0, 3121, 55.5) 0, 1752, 49.2)
4864.750 5215.458 5520.194 5760.947
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.10, 0.10, 0.10, 0.10,
0.005) 0.005) 0.005) 0.005)
(3424, (3318, (3212, (3106,
3137, 0, 0) 2353, 0, 0) 1569, 0, 0) 784, 0, 0)
4839.222 5129.416 5419.616 5709.811
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.15, 0.15, 0.15, 0.15,
0.002) 0.002) 0.002) 0.002)
(3415, (3201, (3133, (3067,
2963, 452, 99.6) 13, 3188, 91.5) 0, 3121, 55.5) 0, 1752, 49.2)
4798.907 5212.905 5520.188 5760.945
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.15, 0.15, 0.15, 0.15,
0.005) 0.005) 0.005) 0.005)
(3433, (3325, (3217, (3108,
3333, 0, 0) 2500, 0, 0) 1667, 0, 0) 833, 0, 0)
4783.415 5087.502 5391.670 5695.838
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.10, 0.10, 0.10, 0.10,
0.002) 0.002) 0.002) 0.002)
(3267, (3200, (3133, (3067,
0, 0, 0, 0,
4659.548 5212.829 5520.193 5760.941
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.10, 0.10, 0.10, 0.10,
0.005) 0.005) 0.005) 0.005)
(3600, (3450, (3300, (3150,
3333, 0, 0) 2500, 0, 0) 1667, 0, 0) 833, 0, 0)
4167.291 4625.004 5083.344 5541.674
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.15, 0.15, 0.15, 0.15,
0.002) 0.002) 0.002) 0.002)
(3267, (3200, (3133, (3067,
0, 0, 0, 0,
4659.548 5212.829 5520.193 5760.940
(0.10, (0.10, (0.10, (0.10,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.15, 0.15, 0.15, 0.15,
0.005) 0.005) 0.005) 0.005)
(3622, (3467, (3311, (3156,
3556, 0, 0) 2667, 0, 0) 1778, 0, 0) 889, 0, 0)
4062.406 4546.673 5031.119 5515.561
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.10, 0.10, 0.10, 0.10,
0.002) 0.002) 0.002) 0.002)
(3635, (3494, (3353, (3247,
3523, 112, 80.4) 2111, 1383, 80.4) 699, 2654, 80.4) 6, 3242, 67.8)
4545.546 4803.641 5061.733 5311.801
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.10, 0.10, 0.10, 0.10,
0.005) 0.005) 0.005) 0.005)
(3639, (3540, (3442, (3344,
3599, 3045, 2491, 1938,
0) 0) 0) 0)
4542.726 4766.784 4991.007 5215.229
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.15, 0.15, 0.15, 0.15,
0.002) 0.002) 0.002) 0.002)
(3530, (3499, (3320, (3247,
1427, 2103, 257.4) 2219, 1280, 94.2) 54, 3267, 94.2) 7, 3241, 67.8)
4062.523 4754.820 5052.014 5311.674
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.05, 0.05, 0.05, 0.05,
0.15, 0.15, 0.15, 0.15,
0.005) 0.005) 0.005) 0.005)
(3557, (3550, (3450, (3350,
1961, 3235, 2647, 2059,
2603.373 4713.975 4947.796 5181.624
(0.15, 0.02, 0.10, 0.10, 0.002) (0.15, 0.04, 0.10, 0.10, 0.002)
3266, 3198, 3121, 1752,
3266, 3198, 3121, 1763,
0, 0, 0, 0,
164.4) 91.8) 55.5) 49.2)
164.4) 91.8) 55.5) 48.9)
1596, 300.0) 0, 0) 0, 0) 0, 0)
(3459, 0, 3459, 220.8) (3389, 3, 3385, 136.8)
4013.838 4684.071
489
ARTICLE IN PRESS H.-F. Yu, W.-C. Yu / Int. J. Production Economics 105 (2007) 483–491
490 Table 2 (continued ) ðp0 ; p; a; b; C b Þ
ðQ ; Qs ; Qb ; tb Þ
Pv ðQ ; Qs ; Qb ; tb Þ
(0.15, 0.06, 0.10, 0.10, 0.002) (0.15, 0.08, 0.10, 0.10, 0.002)
(3318, 0, 3316, 95.1) (3247, 0, 3240, 68.1)
5051.718 5311.665
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.10, 0.10, 0.10, 0.10,
0.005) 0.005) 0.005) 0.005)
(3841, (3712, (3582, (3453,
3824, 3235, 2647, 2059,
0) 0) 0) 0)
3801.507 4139.707 4477.943 4816.187
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.15, 0.15, 0.15, 0.15,
0.002) 0.002) 0.002) 0.002)
(3459, (3389, (3318, (3247,
0, 6, 0, 0,
220.8) 136.8) 95.1) 68.1)
4013.838 4683.535 5051.718 5311.659
(0.15, (0.15, (0.15, (0.15,
0.02, 0.04, 0.06, 0.08,
0.10, 0.10, 0.10, 0.10,
0.15, 0.15, 0.15, 0.15,
0.005) 0.005) 0.005) 0.005)
(3628, (3733, (3600, (3467,
1691, 3451, 2824, 2197,
1937, 280.2) 0, 0) 0, 0) 0, 0)
2057.878 4041.836 4397.651 4753.732
0, 0, 0, 0,
3459, 3383, 3316, 3240,
AOQ ¼ 0:02. That is, an outgoing batch will not satisfy the quality requirement if only inspection test is employed. Hence, no solution can be obtained. On the other hand, if only burn-in test is employed to guarantee that the AOQ requirement is satisfied, then the vendor will lose much money in the trade. In addition, from the results in Table 2, it is seen that (a) The larger p0 is, the larger Q is. (b) The larger p is, the smaller Q is. (c) If p is small, then the burn-in test is profitable for the vendor. (d) Basically, if the unit cost of burn-in C b is relatively less than the unit cost of inspection C s , then the burn-in test is profitable to the vendor. Conversely, if C b is relatively greater than C s , then the inspection test is profitable for the vendor. (e) Basically, if a and b are small, then the inspection is profitable for the vendor. However, if a and b are large, then the burn-in test is profitable for the vendor.
satisfies the purchaser’s AOQ requirement while remaining low cost is a challenge for the vendor. In this paper, we deal with the problem of determining the optimal mixed policy of inspection and burn-in. More specifically, under the constraint that the outgoing batch satisfies the purchaser’s AOQ requirement, the following issues are determined to maximize the expected profit that the vendor makes: (a) the total number of items that the vendor needs to produce, (b) the number of items for inspection, the number of items for burn-in, and the number of items that need neither inspection nor burn-in, and (c) the optimal burn-in time for those needing burn-in testing. In practice, a product often requires inspection on more than one characteristic. It is no doubt an interesting topic to extend our proposed method to such a situation. Besides, in real situations, the buyer usually gains a discount from the vendor if the contract between them is a long-term one. Hence, it is also an important topic for future research to investigate the optimal mixed policy of inspection and burn-in and the optimal production quantity under infinite horizon with a discount factor.
5. Conclusion
References
Inspection and burn-in are two techniques that are widely used by vendors to ensure the outgoing product’s quality. Due to inspection errors and high cost of burn-in, how to make a trade-off between these two techniques such that the outgoing batch
Badia, F.G., Berrade, M.D., Campos, C.A., 2002. Optimal inspection and preventive maintenance of units with revealed and unrevealed failures. Reliability Engineering & System Safety 78, 157–163. Britney, R.R., 1972. Optimal screening plans for non-serial production systems. Management Science 18, 550–559.
ARTICLE IN PRESS H.-F. Yu, W.-C. Yu / Int. J. Production Economics 105 (2007) 483–491 Chen, S.X., Lambrecht, M., 1997. The optimal frequency and sequencing of tests in the inspection of multicharacteristics components. IIE Transactions 29, 1039–1049. Cozzolino, J.M., 1970. Optimal burn-in testing of repairable equipments. Naval Research Logistics Quarterly 17, 167–182. Duffuaa, S.O., Khan, M., 2005. Impact of inspection on the performance measures of a general repeat inspection plan. International Journal of Production Research 43, 4945–4967. Duffuaa, S.O., Roauf, A., 1990. On optimal sequencing of multicharacteristic inspection. Journal of Optimization Theory and Applications 67, 79–86. Eppen, G.D., Hurst, E.G., 1974. Optimal location of inspection stations in a multistage production process. Management Science 20, 1194–1200. Greenshtein, E., Rabinowitz, G., 1997. Double-stage inspection for screening multi-characteristic items. IIE Transactions 29, 1057–1061. Hau, H.L., 1988. On the optimality of a simplified multicharacteristic component inspection model. IIE Transactions 20, 393–398. Hau, H.L., Rosenblatt, M.J., 1987. Simultaneous determination of production cycle and inspection schedules in a production system. Management Science 33, 1125–1136. Jaraiedi, M., Kochhar, D.S., Jaisingh, S.C., 1987. Multiple inspections to meet desired outgoing quality. Journal of Quality Technology 19, 46–51. Jenson, F., Petersen, N.E., 1982. Burn-In. Wiley, New York. Kuo, W., Kuo, Y., 1990. Facing the headaches of early failures: a state-of-the-art review of burn-in decisions. Proceedings of the IEEE 71, 1257–1266. Liou, M.J., Tseng, S.T., Lin, T.M., 1994. Optimal sequence of partial inspections subject to errors. International Journal of Production Economics 33, 109–119. Mi, J., 1994. Burn-in and maintenance policies. Advance in Applied Probability 26, 207–221. Nelson, W., 1990. Accelerated Testing: Statistical Models, Test Plans, and Data Analysis. Wiley, New York. Nguyen, D.D., Murthy, D.N.P., 1982. Optimal burn-in time to minimize cost for products sold under warranty. IIE Transactions 14, 167–174.
491
Park, J.S., Peter, M.H., Tang, K., 1991. Optimal inspection policy in sequential screening. Management Science 37, 1058–1061. Raouf, A., Jain, J.K., Sathe, P.T., 1983. A cost-minimization model for multicharacteristic component inspection. IIE Transactions 15, 187–194. Raz, T., 1986. A survey of models for allocating inspection effort in multistage production systems. Journal of Quality Technology 18, 239–247. Raz, T., Kaspi, M., 1991. Location and sequencing of imperfect inspection operations in a serial multi-stage production system. International Journal of Production Research 29, 1645–1659. Raz, T., Thomas, M.U., 1983. A method for sequencing inspection activities subject to errors. IIE Transactions 15, 12–18. Tang, K., Schneider, H., 1988. Selection of the optimal inspection precision level for a complete inspection plan. Journal of Quality Technology 20, 153–156. Tang, K., Schneider, H., 1990. Cost effectiveness of using a corrected variable in a complete inspection plan when inspection error is present. Naval Research Logistics 37, 893–904. Tang, K., Tang, J., 1989. Design of product specifications for multi-characteristic inspection. Management Science 35, 743–756. Tang, K., Tang, J., 1994. Design of screening procedures: a review. Journal of Quality Technology 26, 209–226. Wang, C.H., 2005. Integrated production and product inspection policy for a deteriorating system. International Journal of Production Economics 95, 123–134. Weiss, G.H., Dishon, M., 1971. Some economic problems related to burn-in programs. IEEE Transactions on Reliability 20, 190–195. White, L.S., 1966. An analysis of a simple class of multistage inspection plans. Management Science 12, 685–693. Yum, B.J., McDowell, E.D., 1981. The optimal allocation of inspection efforts in a class of non-serial production system. AIIE Transactions 13, 193–285.