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A study on negative binomial inspection for imperfect production systems
Computers & Industrial Engineering 65 (2013) 605–613
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A study on negative binomial inspection for imperfect production systems q Yeu-Shiang Huang a,⇑, Yan-Jun Lin a, Jyh-Wen Ho b a b
Department of Industrial and Information Management, National Cheng Kung University, Taiwan Department of Industrial Management and Enterprise Information, Aletheia University, Taiwan
a r t i c l e
i n f o
Article history: Received 13 September 2012 Received in revised form 20 May 2013 Accepted 25 May 2013 Available online 3 June 2013 Keywords: Imperfect production system Inspection Negative binomial distribution
a b s t r a c t Production systems continuously deteriorate with age and usage due to corrosion, fatigue, and cumulative wear in production processes, resulting in an increasing possibility of producing defective products. To prevent selling defective products, inspection is usually carried out to ensure that the performance of a sold product satisfies the customer requirements. Nevertheless, some defective products may still be sold in practice. In such a case, warranties are essential in marketing products and can improve the unfavorable image by applying higher product quality and better customer service. The purpose of this paper is to provide manufacturers with an effective inspection strategy in which the task of quality management is performed under the considerations of related costs for production, sampling, inventory, and warranty. A Weibull power law process is used to describe the imperfection of the production system, and a negative binomial sampling is adopted to learn the operational states of the production process. A free replacement warranty policy is assumed in this paper, and the reworking of defective products before shipment is also discussed. A numerical application is employed to demonstrate the usefulness of the proposed approach, and sensitivity analyses are performed to study the various effects of some influential factors. Ó 2013 Published by Elsevier Ltd.
1. Introduction The performance of facilities has been substantially improved due to the significant enhancement in technology. Nevertheless, it is possible that some highly reliable production systems may still produce inferior products which cannot be detected by ordinal control charts using quantitative measures. Inspection before the distribution of products to the market turns out to be a common way of ensuring that the quality of a sold product can satisfy customer requirements. However, in practice, a full inspection is unfeasible due to high inspection cost and long inspection time. Thus, a random sampling inspection is usually adopted to save both inspection time and cost which has to preset an upper limit of the number of defective items for inspection, and the entire lot is sent out as long as the number of defective items found in the randomly selected sample does not reach the upper limit. This may result in an undesirable situation in which a considerable number of defective products have been sold to customers, and unsatisfied customers might take certain actions that would have
q
This manuscript was processed by Area Editor Min Xie.
⇑ Corresponding author. Address: Department of Industrial and Information Management, National Cheng Kung University, 1 University Rd., Tainan 701, Taiwan. Tel.: +886 6 2757575x53723; fax: +886 6 2362162. E-mail address: [email protected] (Y.-S. Huang). 0360-8352/$ - see front matter Ó 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.cie.2013.05.017
unfavorable impacts on the future sales of the product. Since selling defective products is unavoidable with a random sampling inspection, warranties thus play a crucial role after a product has been sold. The deterioration of production systems is unavoidable because of component corrosion, material fatigue, and cumulative wear in production processes. Several researchers explored the deterioration with the deliberation of the inventory. For instance, Sett, Sarkar, and Goswami (2012) proposed an inventory model by assuming the product deterioration varies with time in order to minimize the system cost. Sarkar (2013) employed different types of deterioration function to develop an inventory model for investigating the optimal lot size. Sarkar (2012c) explored the order model for an imperfect production system, whose deterioration rate depends on time, to maximize the product profit. Furthermore, the deteriorating production system is actually an imperfect production system that has a threshold to separate the system into in-control and out-of-control states. Recent years have seen increased attention to develop more realistic models for dealing with such imperfect production systems (Chung & Hou, 2003; Salameh & Jaber, 2000; Sarkar, 2012a, 2012b; Wang & Sheu, 2000, 2003; Yeh, Ho, & Tseng, 2000), but research on imperfect production systems has generally assumed that the elapsed time of the production system has an exponential distribution, with the aim of simplifying the model construction.
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For a reliable highly automatic production system whose rate of producing defective items is usually fairly small, performing a product inspection without considering the system operation status will lead to an unnecessary increase of inspection cost. It turns out that the defective items might need to be reworked and the product profit would be diminished. Some of the related issues can be seen in the literature. For instance, Cárdenas-Barrón, Sarkar, and Treviño-Garza (2013) proposed a model regarding the defective items being reworked with an aim to investigate the optimal replenishment lot size. Sarkar, Saren, and Wee (2013) developed a quantitative model to deliberate the profit maximization under the limited production rate with the consideration of the defective items. Additionally, on account of the deterioration, the performance of a production system decreases as the production time increases, warranties are of special importance in raising product sales and increasing customer satisfaction. However, the trade-off between higher warranty cost and greater sales revenue is always a crucial decision for manufacturers. Murthy and Djamaludina (2002) stated that warranties play different roles from the perspectives of manufacturers and customers, respectively. For customers, warranties are a safeguard with which customers are able to regain the functionality of a broken product without bearing any expense. It is thus of special interest for the manufacturer to determine an appropriate warranty policy in which the warranty cost and product sales can be optimally managed to maximize the expected profit. However, due to variations of the production process, defective items inevitably increase the warranty cost. The discussion of issues with various warranty policies is widely studied in literature (Jung & Park, 2003; Mi, 1999; Nancy & Lutz, 1998). Since warranty terms usually last more than 1 year in practice, time discounting is essential in evaluating the expected warranty cost. Based on the aforementioned discussion, warranties can increase customer confidence of the product, enhancing its competitiveness. Manufacturers must focus more on quality management to ensure product quality and to prevent excessive warranty cost. To prevent defective or nonconforming products from being sold to customers, which would increase the warranty cost and damage the company reputation, an appropriate sampling inspection scheme is necessary. Inspection ensures better product reliability, and monitors whether the status of the production system is in the in-control state, since the defect rate increases when the production system deteriorates and gradually switches into the outof-control state (Wang & Sheu, 2000). Inspection can assist the manufacturer in evaluating the current status of the production system and cut down the warranty cost but it incurs a sampling and testing cost. The primary concern of an inspection strategy is the trade-off between the high inspection cost from a full inspection and the high warranty cost from a random inspection. Dradjad (1996) stated that traditional product inspection is performed after a production cycle has completed, which implies that the nonconforming items have been produced before the inspection. However, recently, due to technology improvements in automation and small-size production, a full inspection is feasible and affordable for the manufacturer. In addition, since both the production system and the product deteriorate, a reasonable inspection program is beneficial for the manufacturer. Various sampling plans have been proposed to assist management in determining an optimal inspection program (Hsu & Kuo, 1995; Mohandas, Dipak, & Rao, 1992). In summary, an effective inspection plan has to consider related costs from three aspects: the production system, the product inspection, and the post-sale warranty. Huang, Lo, and Ho (2008) proposed an inspection scheme in which the optimal sampling number is determined by considering the trading off among the costs of production, inspection, and warranty. However, since the evaluation of inventory cost is not based on time, and the product demand along with the system and product inspection times as
well as the reworking of defective items are not considered, their model seems insufficient for a practical implementation. Consequently, in this study, the model developed in Huang et al. (2008) is extended. The proposed inspection strategy provides management with an effective and efficient inspection mechanism to deal with possible imperfections in production systems with the aim of minimizing the expected total cost by determining the optimal inspection number and reworking time. The rest of this paper is organized as follows: Section 2 introduces the problem of product inspection for imperfect production systems. Section 3 shows the development of the proposed inspection scheme for imperfect production systems. Section 4 demonstrates the usefulness of the proposed inspection scheme using a practical numerical case. Sensitivity analyses are also carried out to investigate the factors which affect the optimal sampling scheme. Finally, Section 5 contains the concluding remarks. 2. Product inspection for imperfect production systems The deterioration of an imperfect production system usually results in two system statuses during the production process: the incontrol state and the out-of-control state. The system starts in the in-control state in which the production process is stable with a low defect rate and is capable of producing reliable products. As time goes by, the system may switch to the out-of-control state in which the production process is undependable and the defect rate gets worse. Such a state might be due to either the improper management during the manufacture or primarily the deterioration of the machine function. It turns out that an appropriate program for system repair and product inspection would be beneficial to prevent costly reworking and to uphold the product image, especially for the application to improve the imperfect production system under the mass manufacture. For example, Wang and Tsai (2012) proposed a heuristic inspection policy to determine the optimal production lot size according to the quality degree and unit nonconforming cost of the input material. Tirkel and Rabinowitz (2012) devised the inspection policy in a decision support tool to prevent the production yield from declining. Inman, Blumenfeld, Huang, and Li (2013) integrated the quality inspection process with the production system in order to diminish the influence of improper system on the quality. Rezaei and Salimi (2012) conducted product inspection with considering the relationship between buyer and supplier to determine the economic order quantity of the buyer. Chung (2013) developed an inspection plan with extra consideration of the warranty and inventory policies for effectively reducing the quality cost. Furthermore, in this study, a negative binomial inspection is proposed to improve the manufacturing efficiency under the imperfect production system. 2.1. Negative binomial inspection Suppose that an imperfect production system whose elapsed time from the in-control state to the out-of-control state can be modeled by a Weibull distribution with a scale factor a and a shape factor b. It is assumed that, for every production cycle T, the system starts in the in-control state and keeps functioning until the end of the production cycle. At the end of each production cycle, the production system has to be set up again and inspected to examine the system status and to prepare it for the next run of production. If the out-of-control state is identified by the system inspection, a repair action is undertaken to restore the system to the in-control state and a product inspection is then carried out to ensure the quality of the products to be sold. As for the preventive maintenance (PM), Wong, Chan, and Chung (2013) proposed a scheduling way to perform the PM in terms of inspection and reliability regarding
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production with an aim to minimize the makespan, whereas in this study, the maintenance toward the production system is taken into account with the concern of the optimal reworking time, which is also searched by Yoo, Kim, and Park (2012) through a lot sizing model for cost minimization. Accordingly, the system maintenance related costs under consideration include the system inspection cost Cs, the set up cost Ck, and the system repair cost Cb. Suppose that the production rate is p, the demand rate is d, the unit production cost is Cm, the system inspection time is tr, the product inspection time is tI per unit, and the time of reworking a defective product is ts. In addition, it is assumed that, after a production cycle, the produced products stay in a temporary buffer area near the production line and wait for further processing which depends on the result of the system inspection. If the system inspection indicates the in-control state, the entire lot is sent to the warehouse. Otherwise, if the out-of-control state is detected, a product inspection process is undertaken in the buffer. After the product inspection process, the inspected conforming items and the unexamined items are all sent to the warehouse and the inspected nonconforming items are sent to the reworking station for the further processing. Note that the defective products after reworking are treated as conforming products and sent to the warehouse for sale, and the inventory related costs are the warehouse holding cost per unit Ch and the temporary buffer holding cost per unit Ca. Furthermore, Fig. 1 depicts the outline for the process of negative binomial inspection in this study. The production system is assumed to be highly reliable to such an extent that the defect rate of production is extremely small and can be neglected when the system is functioning in the in-control state, while there is a reasonably fair variability in producing defective products in the out-of-control state. The product inspection is undertaken only when the system inspection indicates that the production system is in the out-of-control sate to prevent costly inspection and long inspection times. When the system inspection reveals the out-of-control state after a production run, the production system is believed to be relatively unreliable. The product inspection is used to investigate the quality of the production lot. The defect rate in the out-of-control state of production is assumed to be h, and the breakdown rates for products produced in the in-control state and the out-of-control state are h1(t) and h2(t), respectively. Note that h1(t) < h2(t) for any specific t after a product has been sold. Since the free replacement warranty (FRW) policy is adopted in the study, the repair cost for each breakdown of the
At the end of production cycle
Examine production system in buffer area
In-control state?
Perform negative binomial inspection
N
product Cw within the warranty term w is borne by the product producer. In regard to an imperfect production system, Hsu and Hsu (2013) developed an economic production quantity model with the deliberation of Type I and Type II inspection errors subject to the specific probability density function, whereas in this study, a negative binomial sampling is employed to start inspecting inversely from the last product of the corresponding production lot in turn with unit inspection cost CI, and stops when a predetermined number of conforming products r has been collected. Note that the inspection number r and the defect rate of production h in the outof-control state are the two parameters for the negative binomial sampling process. Furthermore, both inspected conforming and nonconforming products reworked with reworking cost per unit Cr after the product inspection process can be accepted. The uninspected items in the production lot after the product inspection process are treated as conforming products produced in the incontrol state and are all accepted. In addition, the cost of company image damage CR is also considered if a defective product is sold. It is assumed that the deteriorating manner of all products after sale is subject to a Weibull power law process, i.e., c hi ðtÞ ¼ ki i ci t ci1 1 ; t > 0, (i = 1, 2), where ki and ci are the scale parameter and the shape parameter of the failure intensities for the products produced in the two states, respectively. 2.2. Assumptions and notations The following assumptions are made in this study: (1) Only two system states are considered: the in-control and the out-of-control states. The system always starts in the in-control state, and possibly switches to the out-of-control state with a Weibull distributed elapsed time. The product inspection is performed only when the out-of-control state is detected. The system in the in-control state produces more reliable products and has a negligible defect rate. (2) The policy of FRW is adopted in the study. (3) The system starts in the in-control state and keeps functioning until the end of the production cycle. (4) After a production cycle, the produced products stay in a temporary buffer area near the production line and wait for further processing which depends on the result of the system inspection.
Inspect inversely from the last product
Number of conforming items = r* ?
Conforming items & unexamined items Y
Warehouse for sale Rework nonconforming items with ts* in reworking station
Fig. 1. Process of negative binomial inspection.
Nonconforming item
N
Conforming item
Defective ?
Y
N
Y
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(5) The costs of maintaining company reputation and implementing warranty obligations should be higher than the inventory holding cost. (6) Delayed delivery is not allowed, and the product inspection is performed in the temporary buffer area. (7) The warranty cost and the company image damage cost are relatively high for the product in supporting of the use of negative binomial sampling. (8) The product after sale deteriorates according to a Weibull power law deteriorating process. (9) The system has been well-established for a fair time period, and the related parameters can be properly estimated by the field experts using sufficient historical data. Moreover, the notations used throughout this paper are as follows:
a, b: The scale and shape factors of the Weibull distribution for the system elapsed time. T: The elapsed time of a production cycle. th: The waiting time for sale for products in the warehouse. tI: The product inspection time per unit. tr: The time for performing a system inspection. ts: The reworking time per unit. cm: The unit production cost. ch: The holding cost per unit in the warehouse. ca: The holding cost per unit in the temporary buffer area. ck: The set-up cost. cs: The system inspection cost. cb: The cost for restoring the production system to the in-control state. cI: The product inspection cost per unit. cr: The reworking cost per unit under normal working conditions. cw: The repair cost for each product breakdown. cR: The cost of damaging company image per unit. h1(t): The breakdown rate of a product produced in the in-control state. h2(t): The failure rate of a product produced in the out-of-control state, where h2(t) > h1(t). p, d: The production rate and demand rate. h: The defect rate of a production lot produced in the out-ofcontrol state. r: The number of conforming products for inspection. n: The number of total items for inspection. w: The warranty period. N: The lot size of a production run (N = pT).
restore the system back to the initial in-control state for the next production cycle. The set up cost SC can be expressed as SC = ck. In addition, with negative binomial sampling, having collected the optimal number of conforming products r, if the number of total inspected products is n, then n r nonconforming products have to be sent to the reworking station. As the number of nonconforming products increases, in order to prevent late delivery, the reworking time per unit has to be shortened by utilizing more resources, resulting in higher total reworking cost. Suppose that the total reworking cost is the reciprocal of the reworking time for each nonconforming item, ts, and since the reworking occurs only when the system is in the out-of-control state, which happens with b a probability of 1 eðT=aÞ , the total cost of reworking nonconforming items RWC can be expressed as:
RWC ¼
b cr ðn rÞ 1 eðt=aÞ : ts
ð3:1Þ
For the inventory holding cost, both the system and the product inspections are considered. If the in-control system state is detected, the entire batch of products is transferred to the warehouse without any product inspection performing. In such a case, the inventory holding cost includes the costs incurred from both the temporary buffer area and the final goods warehouse. Fig. 2 shows the changes of inventory levels during the time considered for the in-control state. The inventory holding cost for the of in-control system (HCI) is thus given by:
HC I ¼ ca
pT 2 þ pTt r 2
! þ ch
t2h d : 2
ð3:2Þ
On the other hand, if the out-of-control state is detected, the extra inspection time for inspecting products one by one, and the reworking time of the inspected nonconforming products have to be considered in evaluating the inventory holding cost. Fig. 3 shows the changes of inventory levels during the time considered for the out-of-control state. As can be seen in Fig. 3, once the system is detected as being in the out-of-control state, n out of N products is inspected one by one until r conforming products have been collected; i.e., (n r) nonconforming products have to be reworked. Since the reworking time for each nonconforming product is ts, the holding cost during h dÞ the reworking time is ch ðnrÞts ðpT2nþ2rþt , and the holding cost after 2 receiving the reworked batch of (n r) nonconforming products is
t2 d
ch h2 . Therefore, the inventory holding cost for the out-of-control system (HCO) is given by:
3. Optimal sampling number of conforming items and reworking time In this section, the three costs involved in the inspection scheme are first described: production cost, inspection cost, and warranty cost. The determination of the optimal sampling number of conformed products and reworking time of the negative binomial product inspection process is then discussed.
In-Control State Inventory Level
Buffer System Inspection
3.1. Production cost The production cost consists of the manufacturing cost of the products, the setup cost of the production system, the reworking cost of defective products, and the inventory holding cost. Since the unit production cost is assumed to be cm, the manufacturing cost MC can be obtained by multiplying cm by the production lot size N = pT, given by MC = cmpT. Furthermore, upon each production cycle termination, the production system has to be set up to
Production
tr
Demand (Warehouse)
th T
Fig. 2. Changes of inventory levels for the in-control state.
Y.-S. Huang et al. / Computers & Industrial Engineering 65 (2013) 605–613
breakdown rates of products produced by the production system, which may be in the in-control or the out-of-control state after the inspection, are different, the evaluation of warranty cost should be carefully considered for the two cases.
Out of Control State Inventory Level
Buffer
tr Production
Receipt of Reworking Products
ti
3.3.1. Warranty cost if the in-control state is detected We assume that a product produced in the in-control state is quite reliable with deterioration subject to a Weibull power law process with a breakdown rate h1(t). The warranty cost is the product of the maintenance cost per item cw, the production lot size Rw N = pT, and the expected number of breakdowns 0 h1 ðtÞdt; i.e., Rw Ncw 0 h1 ðtÞdt. Therefore, the warranty cost for the in-control state (IWC) is given by:
Product Inspection
System Inspection
Warehouse
(n-r)ts
T th
Fig. 3. Changes of inventory levels for the out-of-control state. 2
IWC ¼ Ncw
!
pT HC O ¼ ca þ pTtr þ pTnt i 2 ðn rÞt s ðpT 2n þ 2r þ t h dÞ t2h d : þ þ ch 2 2
609
Z
w
h1 ðtÞdt ¼ Ncw k1 wc1 ;
ð3:6Þ
0
c
where h1 ðtÞ ¼ k11 c1 tc1 1 ; t > 0.
ð3:3Þ
Since the elapsed time of the deteriorating production system follows a Weibull distribution with a scale factor a and a shape factor b, the total inventory holding cost (HC) can be obtained by multiplying the two holding costs shown in Eqs. (3.2) and (3.3) with their corresponding probabilities and summing them up, i.e., HC ¼ b b eðT=aÞ HC I þ 1 eðT=aÞ ÞHC O . As a result, the total production cost (PC) is given by:
b cr PC ¼ MC þ SC þ RWC þ HC ¼ cm pT þ ck þ ðn rÞ 1 eðT=aÞ ts ! 2 2 b b pT t d þ 1 eðT=aÞ þ pTtr þ ch h þ eðT=aÞ ca 2 2 " ! 2 pT þ pTt r þ pTnti ca 2 ðn rÞts ðpT 2n þ 2r þ t h dÞ t 2h d þ ch : ð3:4Þ þ 2 2 3.2. Inspection cost The inspection cost includes costs related to inspections of both the production system and products, which consist of the costs of inspection and possible restoration of the production system, and the inspection costs for the products. Since it is assumed that the production system is examined after each production run, the inspection cost for the production system (EC) is given by EC = cs. Once the out-of-control state is detected, the cost cb is defrayed so that the system can be restored to the in-control state for the next production run. Since the restoration cost (RC) occurs only when the system is in the out-of-control state, which happens with b b a probability of 1 eðT=aÞ , RC is given by RC ¼ cb ð1 eðT=aÞ Þ. The product inspection cost cI per product has to be defrayed as well for the total number of inspected products n. Since the inspection cost for products (CC) also occurs only when the system is in the out-ofb control state, which happens with a probability of 1 eðT=aÞ , CC is b given by CC ¼ cI nð1 eðT=aÞ Þ. As a result, the total inspection cost IC is given by:
b b IC ¼ EC þ RC þ CC ¼ cs þ cb 1 eðT=aÞ þ cI n ð1 eðT=aÞ : ð3:5Þ 3.3. Warranty cost The warranty cost includes the repair and maintenance costs of the product within the term of the warranty after sale. Since the
3.3.2. Warranty cost if the out-of-control state is detected The evaluation of warranty cost under the out-of-control state considers the costs involving unchecked items, conforming items, and reworked items. In fact, the (N n) unchecked products in the production lot would probably be produced in the in-control state or in the out-of-control state. According to the discussion in Huang et al. (2008), l ¼ Nn is used to denote that l and 1 l are the estimated percentages of conforming products produced in the in-control and the out-of-control states, respectively. As assumed in the in-control state, a Weibull power law intensity with scale parameter k2 and shape paramec ter c2, i.e., h2 ðtÞ ¼ k22 c2 tc2 1 , h2(t) > h1(t), is employed to denote the breakdown rate of conforming items produced in the outof-control state. Accordingly, the R wwarranty cost of unchecked Rw items (WN) is W N ¼ cw ðN nÞ l 0 h1 ðtÞdt þ ð1 lÞ 0 h2 ðtÞdt, c and which results as W N ¼ cw ðN nÞ½lk1 w 1 þ ð1 lÞk2 wc2 . Moreover, the defect rate of products produced in the out-ofcontrol state is assumed to be h, and the cost of selling a defective product is cr, which is the loss of damaging the company image. Therefore, the cost of losing the company’s reputation (WL) is given by WL = cRh(N n)(1 l). It is reasonable to believe that r inspected conforming products are produced in the out-of-control state with a breakdown rate h2(t). In such a case, the warranty cost for conforming products Rw (WG) is given by W G ¼ cw r 0 h2 ðtÞdt ¼ cw rk2 wc2 . In addition, since the (n r) nonconforming items are also produced in the out-ofcontrol state and will be sold after a reworking process, they are assumed to have a breakdown rate h2(t) as well. Therefore the warranty cost for reworked products (WR) can be given by Rw W R ¼ cw ðn rÞ 0 h2 ðtÞdt ¼ cw ðn rÞk2 wc2 . As a result, the warranty cost for the out-of-control state (OWC) is given by:
OWC ¼ W N þ W L þ W R þ W G ¼ cw ðN nÞ½lk1 wc1 þ ð1 lÞk2 wc2 þ cR ðN nÞð1 lÞh þ cw nk2 wc2 :
ð3:7Þ
Therefore, the total warranty cost (WC) can be obtained by multiplying the warranty costs for the in-control and out-of-control states shown in Eqs. (3.6) and (3.7) with their corresponding probabilities and summing them up, which is thus given by:
b b WC ¼ eðT=aÞ Ncw k1 wc1 þ 1 eðT=aÞ fcw ðN nÞ½lk1 wc1 þ ð1 lÞk2 wc2 þ cR ðN nÞð1 lÞh þ cw nk2 wc2 g:
ð3:8Þ
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Consequently, the expected total cost (TC) is given as:
TC ¼ PC þ IC þ WC b NT t2 d ¼ eðT=aÞ cm N þ ck þ cs þ cw Nk1 wc1 þ ca þ Ntr þ ch h 2 2 b c r þ 1 eðT=aÞ cm N þ ck þ ðn rÞ þ cs þ cb tr þ cI n þ cw ðN nÞ½lk1 wc1 þ ð1 lÞk2 wc2 þ cR ðN nÞð1 lÞh NT þ Ntr þ Nnt i þ cw nk2 wc2 þ ca 2
ðN 2n 2r þ t h dÞðn rÞts t2h d þch : þ 2 2
maintaining company reputation and implementing warranty obligations should be higher than the inventory holding cost, the second order derivative of E[TC] with respect to r, o2E[TC]/or2, is thus greater than zero. Moreover, because the time and cost for reworking are both greater than zero, the second order derivative of E[TC] with respect to ts, @ 2 E½TC=@t2s , is greater than zero, too. However, to ensure that the determinant of the matrix H, which 2 ½cR h þ cw ð1 hÞ k2 wc2 k1 wc1 can be calculated as jHj ¼ Nð1hÞ2 t3s h i2 2 ch Nh2 t s ch rh 2 , is greater than zero, Eq. (3.11) has to be ð1hÞ
satisfied. h
ð3:9Þ 4. Numerical application 3.4. Product inspection strategy According to the empirical investigation performed by Huang et al. (2008), as long as the optimal number r is not very close to N, the difference between the values of moments of n calculated from either theoretical method or empirical approach does not affect whether the negative binomial distribution is truncated or not. Moreover, for case of both h 6 0:2 and r/N < 0.7 seems to be a good criteria for ignoring the truncated negative binomial distribution for further analysis. In this study, it is assumed that h is relatively small while the product inspection cost is somewhat larger due to the assumption that the production system is quite reliable, since a low product inspection cost might result in a full inspection. Under the aforementioned assumptions, r is not close to N, and therefore r rhþr 2 we can employ both E½n ¼ 1h and E½n2 ¼ ð1hÞ 2 for further analytical processes with valid conclusions. Proposition. With the total expected cost shown in Eq. (3.9), the optimal number of conforming items that have to be collected r and the optimal reworking time per product ts can be obtained as follows: h 8 c c2 w k1 w 1 > r ¼ Nð1 hÞ cNð1hÞ þ 2cw ð1 hÞk2 wc2 þ 2hcR ctrsh cI cw k1 wc1 cw hkN2 w > > > < i c Nht t dht cR h2 Nð1hÞ ca Nt i h 2 s h 2 s ð2cw ½k2 wc2 k1 wc1 þ 2hcR 2Nh2 t s Þ1 ; > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2ð1hÞ : t ¼ ; s c ½ðNþdt Þð1h2r cr hÞ h
h
ð3:10Þ
" #2 2 c h r h2 2 c2 c1 if ½cR h þ cw ð1 hÞk2 w k1 w ch Nh t s > : Nts 3 ð1 hÞ ð3:11Þ
Proof. By setting the first order derivatives of E[TC] with respect to r and ts to be zero, respectively, Eq. (3.10) is obtained. However, the optimal values of r and t s are valid only under the condition that the second order derivative of E[TC] with respect to r and ts are both greater than zero, and the determinant of the Hessian matrix of the problem, which is given by:
" H¼
@ 2 E½TC=@r2
@ 2 E½TC=@r@ts
@ 2 E½TC=@ts @r
@ 2 E½TC=@t2s c2
# ; c1
2
k1 w 2ch Nh ts r where @ 2 E½TC=@r 2 ¼ 2hcR þ2cw ½k2 wNð1hÞ , @ 2 E½TC=@t 2s ¼ 2c , and t 3s 2 th hd cr h Nh 2rh @ 2 E½TC=@r@t s ¼ @ 2 E½TC=@t s @r ¼ ch 2ð1hÞ , has þ 2ð1hÞ ð1hÞ 2 t 2 ð1hÞ s
to be greater than zero as well. Since k2 wc2 k1 wc1 > 0 because of h2(t) > h1(t), and in this study, the production system is assumed to be highly reliable which results that the defect rate is small and the costs of
In this section, the data used in Huang et al. (2008) are utilized to perform numerical analyses in which some adequate values for the remaining parameters are also adopted. It is assumed that an FRW policy is offered to the customers, and a Weibull distribution, b1 FðtÞ ¼ 1 eðatÞ , with a = 8 and b = 2.5 can be used to describe the deterioration of the imperfect production system. In addition, the lifetime distributions of conforming and nonconforming products are Weibull distributed with breakdown rates c c h1 ðtÞ ¼ k11 c1 tc1 1 and h2 ðtÞ ¼ k22 c2 tc2 1 , where k1 = 1/36 and c1 = 2, and k2 = 1/12 and c2 = 2, respectively. Note that, presumably, the production system has been well-established for a fairly long period of time, and the related parameters can be properly estimated by the field experts using sufficient historical data and experiences. Table 1 lists the numerical values of the parameters. The prior conditions for obtaining the optimal values of sampling number of conforming products and reworking time per product have to be examined first, which are o2E[TC]/or2 = 0.03, @ 2 E½TC=@t 2s ¼ 106744:57, and |H| = 3200.88. Since they are all greater than zero, the optimal values of interest can be calculated by substituting the values listed in Table 1 into Eq. (3.10) and solving it. The optimal number of conforming items that have to be collected r is 620.597, and the optimal reworking time per product is 0.02659 h, which is about 1.6 min. However, since r should be an integer in practice, the expected total costs for cases where r = 620 and r = 621 are investigated. The case with the smaller expected Table 1 Numerical values of parameters. Parameter
Value
a
8 2.5 4 500 2000 5 150 0.8 150 200 5 15 1 5 1 1/36 2 1/12 2 0.05 18 400 0.01 0.01
b T p N(N = pT) cm ck ch cs cb cI cw ca cR cr k1
c1 k2
c2 h w d tr tI
Y.-S. Huang et al. / Computers & Industrial Engineering 65 (2013) 605–613 Table 2 The results of the analysis. Item
Value
Optimal sampling number of conforming products Optimal reworking time per product Optimal total sampling number Optimal sampling percentage Total expected cost
620 0.02659 652 32.6% $33351.9
Table 3 Relationship of the optimal values and the parameters. p
d
h
w
ch
ca
cR
ts
r
E[TC]
300 400 500 600 700
400 400 400 400 400
0.05 0.05 0.05 0.05 0.05
18 18 18 18 18
0.8 0.8 0.8 0.8 0.8
1 1 1 1 1
5 5 5 5 5
0.030100 0.028180 0.026595 0.025241 0.024073
395 512 620 722 817
$21087.1 $27207.4 $33351.9 $39518.9 $45707.0
500 500 500 500 500
200 300 400 500 600
0.05 0.05 0.05 0.05 0.05
18 18 18 18 18
0.8 0.8 0.8 0.8 0.8
1 1 1 1 1
5 5 5 5 5
0.030244 0.028244 0.026595 0.025205 0.024012
635 627 620 614 608
$32023.4 $32688.5 $33351.9 $34013.7 $34674.2
500 500 500 500 500
400 400 400 400 400
0.01 0.025 0.05 0.075 0.1
18 18 18 18 18
0.8 0.8 0.8 0.8 0.8
1 1 1 1 1
5 5 5 5 5
0.026407 0.026483 0.026595 0.026687 0.026761
736 691 620 554 491
$32984.8 $33127.7 $33351.9 $33559.4 $33751.1
500 500 500 500 500
400 400 400 400 400
0.05 0.05 0.05 0.05 0.05
18 24 30 36 42
0.8 0.8 0.8 0.8 0.8
1 1 1 1 1
5 5 5 5 5
0.026595 0.026652 0.026678 0.026693 0.026702
620 764 831 867 889
$33351.9 $44984.3 $59835.2 $77946.1 $99331.3
500 500 500 500 500
400 400 400 400 400
0.05 0.05 0.05 0.05 0.05
18 18 18 18 18
0.2 0.4 0.6 0.8 1
1 1 1 1 1
5 5 5 5 5
0.053236 0.037630 0.030716 0.026595 0.023788
679 655 636 620 607
$31223.4 $31952.0 $32657.9 $33351.9 $34038.4
500 500 500 500 500
400 400 400 400 400
0.05 0.05 0.05 0.05 0.05
18 18 18 18 18
0.8 0.8 0.8 0.8 0.8
0.4 0.8 1 1.2 1.6
5 5 5 5 5
0.026610 0.026600 0.026595 0.026590 0.026580
658 633 620 608 583
$30700.9 $32469.1 $33351.9 $34233.8 $35995.1
500 500 500 500 500
400 400 400 400 400
0.05 0.05 0.05 0.05 0.05
18 18 18 18 18
0.8 0.8 0.8 0.8 0.8
1 1 1 1 1
1 2.5 5 7.5 10
0.026591 0.026593 0.026595 0.026597 0.026599
612 615 620 626 631
$33322.3 $33333.5 $33351.9 $33370.2 $33388.3
total cost is chosen, which is the case of r = 620; i.e., the products are inversely inspected from the last product of the corresponding production lot in turn until 620 conforming products have been collected. In addition, the expected number of inspected products can be r 620 derived from r and h, which is given as E½n ¼ ð1hÞ ¼ 10:05 ffi 652. Therefore, the required percentage of inspection (POI) is n 652 ¼ 2000 ¼ 0:326 ¼ 32:6%, which indicates that the product inspecN tion process can be stopped after inspecting 32.6% of the lot size. The expected total cost for the production lot can also be calculated as $33351.9. Table 2 shows the results of the analysis. The effects of the corresponding parameters on the optimal values of interest are essential to decision makers, and are investigated in this study. The results are shown in Table 3. From Table 3, the production rate is inversely correlated to the reworking time per product. This can be explained by the fact that an increase of production rate increases the production lot size, which results in an increase of defective products. Therefore, in order to prevent late delivery, the reworking time per product has to
611
be reduced accordingly. Moreover, the sampling number of conforming products increases as the production rate increases. This can also be explained by an increase of lot size, which may need more product inspections to ensure product quality to lower the warranty cost after product sale. It is observed that the total cost increases as the production rate increases because more money is needed to produce more products. The demand rate is inversely correlated to the reworking time per product. This can be explained by the fact that the reworking time per product has to be reduced to prevent increased demand which may cause late delivery because of long reworking time. Moreover, since more conforming products are collected, more defective products are detected and reworked, which results in a long reworking process and an inventory shortage. Therefore, the sampling number of conforming products has to decrease as the demand rate increases. It is also interesting to find that when demand is increased, the reworking time per unit and the sampling number of conforming product have to decrease. This would result in the possible condition in which the reworking cost may increase because extra operation shifts will be needed, and the warranty cost may increase because more uninspected defective products may be sold in the market. Consequently, the total cost would increase as the demand rate increases. As can be seen in Table 3, the sampling number of conforming products decreases as the defective rate increases. Since when the defect rate is relatively high, collecting more conforming products for sampling results in detecting more defective products. This increases the inspection and reworking costs which may not be offset by the possible saved warranty cost in the case. In addition, the reworking time per unit is also affected, becoming longer, because that the sampling number of conforming products has to be decreased to cut down the total cost, and thus there are fewer detected defective products that require reworking. The total cost increases as the defect rate increases because a higher defect rate increases costs corresponding to reworking and company image damage. The sampling number of conforming products increases as the warranty term increases. This can be explained by the fact that for a longer warranty term, the product quality has to be higher to ensure that the warranty cost will be tolerable within a certain range; this can be achieved using a stricter inspection process. The warranty term is not significantly correlated to the reworking time per unit; however, it affects the total cost since the increase of the inspection and reworking cost are incapable of offsetting the warranty cost in this case, and the total cost increases as the warranty term increases. As can be seen in Table 3, the holding cost per unit in the warehouse is inversely correlated to the reworking time per unit. This is due to the fact that as the holding cost increases, the reworking time has to be reduced to shorten the period that the products are stored in the inventory to cut down the total inventory cost. A small number of conforming products collected in the inspection process also cuts down the total inventory cost. It is obvious that the total cost increases as the holding cost per unit increases. Similar effects can be seen in Table 3 for the holding cost per unit in the buffer; only the effect on the reworking time per unit is not very significant in the case. The sampling number of conforming products increases as the image damage cost increases. This can be explained by the fact that for a higher image damage cost, the product quality has to be higher to ensure that the image will not be damaged seriously when too many defective products are sold; this can be achieved using a stricter inspection process. The image damage cost is not significantly correlated to the reworking time per unit; however, it affects the total cost. The total cost increases as the image damage cost increases. Figs. 4–6 show the impacts of the parameters on ts, r, and E[TC], respectively.
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Y.-S. Huang et al. / Computers & Industrial Engineering 65 (2013) 605–613 Table 4 The evaluated results of impacts for the parameters.
0.06 0.055
p
0.05
θ
0.045
d
0.04
ch
ts
ca
0.035
cR 0.03
w
0.025 0.02 0.015 -100% -80% -60% -40% -20% 0%
20% 40% 60% 80% 100%
p d h w ch ca cR
r
ts
TC
+(2) (3) – +(1) – – –
(2) +(3)
+(2) + + +(1) + +(3) +
(1)
Note: The Arabic numerals in the parenthesis denote the rank of influence. + Denotes positively correlated. Denotes inversely correlated.
the reworking time per unit the most in the case. Decision makers can refer to the results to determine how their limited resources can be invested to enhance their decision making process.
Percentage of Change Fig. 4. The impacts of parameters on ts.
5. Conclusion 1000 900 800 700
p
600
θ
500
r
d
400
ch
300
ca
200
cR
100
w
0 -100 -200 -100% -80% -60% -40% -20% 0%
20% 40% 60% 80% 100%
Percentage of Change Fig. 5. The impacts of parameters on r.
58500 53500 48500 43500
p
38500
θ d
E [TC] 33500
ch 28500
ca
23500
cR
18500
w
13500 8500 -100% -80% -60% -40% -20% 0% 20% 40% 60% 80% 100%
Percentage of Change
In this study, we consider an imperfect production system whose elapsed time is distributed as a two-parameter Weibull distribution for developing a reasonable and economic inspection strategy with an aim to provide manufacturers with an effective decision-making on quality examination. Negative binomial sampling is employed in the production inspection scheme to detect the possible switching point from the in-control state to the outof-control state of the production system. The products are inversely inspected from the last item of the corresponding production lot in turn and the inspection terminates after the optimal number of conforming items has been collected. In addition, the nonconforming products detected in the inspection process are reworked and then transferred to the warehouse for sale. Since late delivery is not allowed in this study, the decision of critical interest is to simultaneously determine the optimal sampling number of conforming products and the optimal reworking time per unit. By using this inspection strategy, manufacturers can effectively manage both the inspection and the reworking processes to ensure an acceptable level of product quality for maintaining a good company reputation and thus achieve the goal of cost minimization. Further research would be performed by allowing shortage to occur to investigate its influence on the decision making process, which may lead to a significant change of the inventory holding cost evaluated in the proposed model. Additionally, it is of interest to further consider that the inspection time and the reworking time, instead of being constants, can be considered to be random variables, and the resultant costs may modify the structure of the elements in the model. Accordingly, such a stochastic problem needs a considerable extension of the model elements and therefore becomes a somewhat complicated problem to be addressed. Finally, the warranty cost in this research is deliberated under an FRW policy, thus dissimilar warranty policies might be employed to revise the proposed product inspection strategy, such as the application of pro-rata warranty (PRW) in the future extension of the proposed model.
Fig. 6. The impacts of parameters on E[TC].
References As can be seen in Figs. 4–6, the relative importance of each parameter can be evaluated, and the results are shown in Table 4. As can be seen in Table 4, the warranty term plays an important role in determining the sampling number of conforming products and has the greatest impact on the expected cost in the case. The production rate is important for the three decision variables of interest, and the holding cost per unit in the warehouse influences
Cárdenas-Barrón, L. E., Sarkar, B., & Treviño-Garza, G. (2013). An improved solution to the replenishment policy for the EMQ model with rework and multiple shipments. Applied Mathematical Modelling, 37(7), 5549–5554. Chung, C. J. (2013). Investigating imperfect process and demand effects on inspection scheduling and supply chain replenishment policy. Computers and Industrial Engineering, 64(1), 31–44. Chung, K. J., & Hou, K. L. (2003). An optimal production run time with imperfect production processes and allowable shortages. Computers and Operations Research, 30, 483–490.
Y.-S. Huang et al. / Computers & Industrial Engineering 65 (2013) 605–613 Dradjad, I. (1996). Inspection and correction policies in setting economic product tolerance. International Journal of Industrial Ergonomics, 46–47, 587–593. Hsu, J. T., & Hsu, L. F. (2013). Two EPQ models with imperfect production processes, inspection errors, planned backorders, and sales returns. Computers and Industrial Engineering, 64(1), 389–402. Hsu, L. F., & Kuo, S. (1995). Design of optimal maintenance policies based on on-line sampling plans. European Journal of Operational Research, 86, 345–357. Huang, Y. S., Lo, H. C., & Ho, J. W. (2008). Design of effective inspection schema for imperfect production systems. International Journal of Production Research, 46(16), 4537–4551. Inman, Robert R., Blumenfeld, Dennis E., Huang, N., & Li, J. (2013). Survey of recent advances on the interface between production system design and quality. IIE Transactions, 45(6), 557–574. Jung, G. M., & Park, D. H. (2003). Optimal maintenance policies during the postwarranty period. Reliability Engineering and System Safety, 82, 173–185. Mi, J. (1999). Comparisons of renewable warranties. Naval Research Logistics, 46(1), 91–106. Mohandas, K., Dipak, C., & Rao, B. V. A. (1992). Optimal periodic replacement for a deteriorating production system with inspection and minimal repair. Reliability Engineering and System Safety, 37(1), 73–77. Murthy, D. N. P., & Djamaludina, I. (2002). New product warranty: A literature review. International Journal of Production Economics, 79, 231–260. Nancy, A., & Lutz, V. P. (1998). Warranties, extended warranties, and product quality. International Journal of Industrial Organization, 16, 463–493. Rezaei, J., & Salimi, N. (2012). Economic order quantity and purchasing price for items with imperfect quality when inspection shifts from buyer to supplier. International Journal of Production Economics, 137(1), 11–18. Salameh, M. K., & Jaber, M. Y. (2000). Economic production quantity model for items with imperfect quality. International Journal of Production Economics, 64, 59–64. Sarkar, B. (2012a). An inventory model with reliability in an imperfect production process. Applied Mathematics and Computation, 218(9), 4881–4891. Sarkar, B. (2012b). An EOQ model with delay in payments and stock dependent demand in the presence of imperfect production. Applied Mathematics and Computation, 218(17), 8295–8308.
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Sarkar, B. (2012c). An EOQ model with delay in payments and time varying deterioration rate. Mathematical and Computer Modelling, 55(3–4), 367–377. Sarkar, B. (2013). A production–inventory model with probabilistic deterioration in two-echelon supply chain management. Applied Mathematical Modelling, 37(5), 3138–3151. Sarkar, B., Saren, S., & Wee, H. M. (2013). An inventory model with variable demand, component cost and selling price for deteriorating items. Economic Modelling, 30, 306–310. Sett, B. K., Sarkar, B., & Goswami, A. (2012). A two-warehouse inventory model with increasing demand and time varying deterioration. Scientia Iranica, 19(6), 1969–1977. Tirkel, I., & Rabinowitz, G. (2012). The relationship between yield and flow time in a production system under inspection. International Journal of Production Research, 50(14), 3686–3697. Wang, C. H., & Sheu, S. H. (2000). Fast approach to the optimal production/PM policy. Computers and Mathematics with Applications, 40, 1297–1314. Wang, C. H., & Sheu, S. H. (2003). Determining the optimal production–maintenance policy with inspection errors using a Markov chain. Computers and Operations Research, 30, 1–17. Wang, C. H., & Tsai, W. C. (2012). Determining the production lot size with a heuristic inspection policy for controlling the quality of input materials and products. International Journal of Systems Science, 43(11), 2030–2039. Wong, C. S., Chan, F. T. S., & Chung, S. H. (2013). A joint production scheduling approach considering multiple resources and preventive maintenance tasks. International Journal of Production Research, 51(3), 883–896. Yeh, R. H., Ho, W. T., & Tseng, S. T. (2000). Optimal production run length for products sold with warranty. European Journal of Operational Research, 120, 575–582. Yoo, S. H., Kim, D., & Park, M. S. (2012). Lot sizing and quality investment with quality cost analyses for imperfect production and inspection processes with commercial return. International Journal of Production Economics, 140(2), 922–933.
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Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
A study on negative binomial inspection for imperfect production systems q Yeu-Shiang Huang a,⇑, Yan-Jun Lin a, Jyh-Wen Ho b a b
Department of Industrial and Information Management, National Cheng Kung University, Taiwan Department of Industrial Management and Enterprise Information, Aletheia University, Taiwan
a r t i c l e
i n f o
Article history: Received 13 September 2012 Received in revised form 20 May 2013 Accepted 25 May 2013 Available online 3 June 2013 Keywords: Imperfect production system Inspection Negative binomial distribution
a b s t r a c t Production systems continuously deteriorate with age and usage due to corrosion, fatigue, and cumulative wear in production processes, resulting in an increasing possibility of producing defective products. To prevent selling defective products, inspection is usually carried out to ensure that the performance of a sold product satisfies the customer requirements. Nevertheless, some defective products may still be sold in practice. In such a case, warranties are essential in marketing products and can improve the unfavorable image by applying higher product quality and better customer service. The purpose of this paper is to provide manufacturers with an effective inspection strategy in which the task of quality management is performed under the considerations of related costs for production, sampling, inventory, and warranty. A Weibull power law process is used to describe the imperfection of the production system, and a negative binomial sampling is adopted to learn the operational states of the production process. A free replacement warranty policy is assumed in this paper, and the reworking of defective products before shipment is also discussed. A numerical application is employed to demonstrate the usefulness of the proposed approach, and sensitivity analyses are performed to study the various effects of some influential factors. Ó 2013 Published by Elsevier Ltd.
1. Introduction The performance of facilities has been substantially improved due to the significant enhancement in technology. Nevertheless, it is possible that some highly reliable production systems may still produce inferior products which cannot be detected by ordinal control charts using quantitative measures. Inspection before the distribution of products to the market turns out to be a common way of ensuring that the quality of a sold product can satisfy customer requirements. However, in practice, a full inspection is unfeasible due to high inspection cost and long inspection time. Thus, a random sampling inspection is usually adopted to save both inspection time and cost which has to preset an upper limit of the number of defective items for inspection, and the entire lot is sent out as long as the number of defective items found in the randomly selected sample does not reach the upper limit. This may result in an undesirable situation in which a considerable number of defective products have been sold to customers, and unsatisfied customers might take certain actions that would have
q
This manuscript was processed by Area Editor Min Xie.
⇑ Corresponding author. Address: Department of Industrial and Information Management, National Cheng Kung University, 1 University Rd., Tainan 701, Taiwan. Tel.: +886 6 2757575x53723; fax: +886 6 2362162. E-mail address: [email protected] (Y.-S. Huang). 0360-8352/$ - see front matter Ó 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.cie.2013.05.017
unfavorable impacts on the future sales of the product. Since selling defective products is unavoidable with a random sampling inspection, warranties thus play a crucial role after a product has been sold. The deterioration of production systems is unavoidable because of component corrosion, material fatigue, and cumulative wear in production processes. Several researchers explored the deterioration with the deliberation of the inventory. For instance, Sett, Sarkar, and Goswami (2012) proposed an inventory model by assuming the product deterioration varies with time in order to minimize the system cost. Sarkar (2013) employed different types of deterioration function to develop an inventory model for investigating the optimal lot size. Sarkar (2012c) explored the order model for an imperfect production system, whose deterioration rate depends on time, to maximize the product profit. Furthermore, the deteriorating production system is actually an imperfect production system that has a threshold to separate the system into in-control and out-of-control states. Recent years have seen increased attention to develop more realistic models for dealing with such imperfect production systems (Chung & Hou, 2003; Salameh & Jaber, 2000; Sarkar, 2012a, 2012b; Wang & Sheu, 2000, 2003; Yeh, Ho, & Tseng, 2000), but research on imperfect production systems has generally assumed that the elapsed time of the production system has an exponential distribution, with the aim of simplifying the model construction.
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For a reliable highly automatic production system whose rate of producing defective items is usually fairly small, performing a product inspection without considering the system operation status will lead to an unnecessary increase of inspection cost. It turns out that the defective items might need to be reworked and the product profit would be diminished. Some of the related issues can be seen in the literature. For instance, Cárdenas-Barrón, Sarkar, and Treviño-Garza (2013) proposed a model regarding the defective items being reworked with an aim to investigate the optimal replenishment lot size. Sarkar, Saren, and Wee (2013) developed a quantitative model to deliberate the profit maximization under the limited production rate with the consideration of the defective items. Additionally, on account of the deterioration, the performance of a production system decreases as the production time increases, warranties are of special importance in raising product sales and increasing customer satisfaction. However, the trade-off between higher warranty cost and greater sales revenue is always a crucial decision for manufacturers. Murthy and Djamaludina (2002) stated that warranties play different roles from the perspectives of manufacturers and customers, respectively. For customers, warranties are a safeguard with which customers are able to regain the functionality of a broken product without bearing any expense. It is thus of special interest for the manufacturer to determine an appropriate warranty policy in which the warranty cost and product sales can be optimally managed to maximize the expected profit. However, due to variations of the production process, defective items inevitably increase the warranty cost. The discussion of issues with various warranty policies is widely studied in literature (Jung & Park, 2003; Mi, 1999; Nancy & Lutz, 1998). Since warranty terms usually last more than 1 year in practice, time discounting is essential in evaluating the expected warranty cost. Based on the aforementioned discussion, warranties can increase customer confidence of the product, enhancing its competitiveness. Manufacturers must focus more on quality management to ensure product quality and to prevent excessive warranty cost. To prevent defective or nonconforming products from being sold to customers, which would increase the warranty cost and damage the company reputation, an appropriate sampling inspection scheme is necessary. Inspection ensures better product reliability, and monitors whether the status of the production system is in the in-control state, since the defect rate increases when the production system deteriorates and gradually switches into the outof-control state (Wang & Sheu, 2000). Inspection can assist the manufacturer in evaluating the current status of the production system and cut down the warranty cost but it incurs a sampling and testing cost. The primary concern of an inspection strategy is the trade-off between the high inspection cost from a full inspection and the high warranty cost from a random inspection. Dradjad (1996) stated that traditional product inspection is performed after a production cycle has completed, which implies that the nonconforming items have been produced before the inspection. However, recently, due to technology improvements in automation and small-size production, a full inspection is feasible and affordable for the manufacturer. In addition, since both the production system and the product deteriorate, a reasonable inspection program is beneficial for the manufacturer. Various sampling plans have been proposed to assist management in determining an optimal inspection program (Hsu & Kuo, 1995; Mohandas, Dipak, & Rao, 1992). In summary, an effective inspection plan has to consider related costs from three aspects: the production system, the product inspection, and the post-sale warranty. Huang, Lo, and Ho (2008) proposed an inspection scheme in which the optimal sampling number is determined by considering the trading off among the costs of production, inspection, and warranty. However, since the evaluation of inventory cost is not based on time, and the product demand along with the system and product inspection times as
well as the reworking of defective items are not considered, their model seems insufficient for a practical implementation. Consequently, in this study, the model developed in Huang et al. (2008) is extended. The proposed inspection strategy provides management with an effective and efficient inspection mechanism to deal with possible imperfections in production systems with the aim of minimizing the expected total cost by determining the optimal inspection number and reworking time. The rest of this paper is organized as follows: Section 2 introduces the problem of product inspection for imperfect production systems. Section 3 shows the development of the proposed inspection scheme for imperfect production systems. Section 4 demonstrates the usefulness of the proposed inspection scheme using a practical numerical case. Sensitivity analyses are also carried out to investigate the factors which affect the optimal sampling scheme. Finally, Section 5 contains the concluding remarks. 2. Product inspection for imperfect production systems The deterioration of an imperfect production system usually results in two system statuses during the production process: the incontrol state and the out-of-control state. The system starts in the in-control state in which the production process is stable with a low defect rate and is capable of producing reliable products. As time goes by, the system may switch to the out-of-control state in which the production process is undependable and the defect rate gets worse. Such a state might be due to either the improper management during the manufacture or primarily the deterioration of the machine function. It turns out that an appropriate program for system repair and product inspection would be beneficial to prevent costly reworking and to uphold the product image, especially for the application to improve the imperfect production system under the mass manufacture. For example, Wang and Tsai (2012) proposed a heuristic inspection policy to determine the optimal production lot size according to the quality degree and unit nonconforming cost of the input material. Tirkel and Rabinowitz (2012) devised the inspection policy in a decision support tool to prevent the production yield from declining. Inman, Blumenfeld, Huang, and Li (2013) integrated the quality inspection process with the production system in order to diminish the influence of improper system on the quality. Rezaei and Salimi (2012) conducted product inspection with considering the relationship between buyer and supplier to determine the economic order quantity of the buyer. Chung (2013) developed an inspection plan with extra consideration of the warranty and inventory policies for effectively reducing the quality cost. Furthermore, in this study, a negative binomial inspection is proposed to improve the manufacturing efficiency under the imperfect production system. 2.1. Negative binomial inspection Suppose that an imperfect production system whose elapsed time from the in-control state to the out-of-control state can be modeled by a Weibull distribution with a scale factor a and a shape factor b. It is assumed that, for every production cycle T, the system starts in the in-control state and keeps functioning until the end of the production cycle. At the end of each production cycle, the production system has to be set up again and inspected to examine the system status and to prepare it for the next run of production. If the out-of-control state is identified by the system inspection, a repair action is undertaken to restore the system to the in-control state and a product inspection is then carried out to ensure the quality of the products to be sold. As for the preventive maintenance (PM), Wong, Chan, and Chung (2013) proposed a scheduling way to perform the PM in terms of inspection and reliability regarding
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production with an aim to minimize the makespan, whereas in this study, the maintenance toward the production system is taken into account with the concern of the optimal reworking time, which is also searched by Yoo, Kim, and Park (2012) through a lot sizing model for cost minimization. Accordingly, the system maintenance related costs under consideration include the system inspection cost Cs, the set up cost Ck, and the system repair cost Cb. Suppose that the production rate is p, the demand rate is d, the unit production cost is Cm, the system inspection time is tr, the product inspection time is tI per unit, and the time of reworking a defective product is ts. In addition, it is assumed that, after a production cycle, the produced products stay in a temporary buffer area near the production line and wait for further processing which depends on the result of the system inspection. If the system inspection indicates the in-control state, the entire lot is sent to the warehouse. Otherwise, if the out-of-control state is detected, a product inspection process is undertaken in the buffer. After the product inspection process, the inspected conforming items and the unexamined items are all sent to the warehouse and the inspected nonconforming items are sent to the reworking station for the further processing. Note that the defective products after reworking are treated as conforming products and sent to the warehouse for sale, and the inventory related costs are the warehouse holding cost per unit Ch and the temporary buffer holding cost per unit Ca. Furthermore, Fig. 1 depicts the outline for the process of negative binomial inspection in this study. The production system is assumed to be highly reliable to such an extent that the defect rate of production is extremely small and can be neglected when the system is functioning in the in-control state, while there is a reasonably fair variability in producing defective products in the out-of-control state. The product inspection is undertaken only when the system inspection indicates that the production system is in the out-of-control sate to prevent costly inspection and long inspection times. When the system inspection reveals the out-of-control state after a production run, the production system is believed to be relatively unreliable. The product inspection is used to investigate the quality of the production lot. The defect rate in the out-of-control state of production is assumed to be h, and the breakdown rates for products produced in the in-control state and the out-of-control state are h1(t) and h2(t), respectively. Note that h1(t) < h2(t) for any specific t after a product has been sold. Since the free replacement warranty (FRW) policy is adopted in the study, the repair cost for each breakdown of the
At the end of production cycle
Examine production system in buffer area
In-control state?
Perform negative binomial inspection
N
product Cw within the warranty term w is borne by the product producer. In regard to an imperfect production system, Hsu and Hsu (2013) developed an economic production quantity model with the deliberation of Type I and Type II inspection errors subject to the specific probability density function, whereas in this study, a negative binomial sampling is employed to start inspecting inversely from the last product of the corresponding production lot in turn with unit inspection cost CI, and stops when a predetermined number of conforming products r has been collected. Note that the inspection number r and the defect rate of production h in the outof-control state are the two parameters for the negative binomial sampling process. Furthermore, both inspected conforming and nonconforming products reworked with reworking cost per unit Cr after the product inspection process can be accepted. The uninspected items in the production lot after the product inspection process are treated as conforming products produced in the incontrol state and are all accepted. In addition, the cost of company image damage CR is also considered if a defective product is sold. It is assumed that the deteriorating manner of all products after sale is subject to a Weibull power law process, i.e., c hi ðtÞ ¼ ki i ci t ci1 1 ; t > 0, (i = 1, 2), where ki and ci are the scale parameter and the shape parameter of the failure intensities for the products produced in the two states, respectively. 2.2. Assumptions and notations The following assumptions are made in this study: (1) Only two system states are considered: the in-control and the out-of-control states. The system always starts in the in-control state, and possibly switches to the out-of-control state with a Weibull distributed elapsed time. The product inspection is performed only when the out-of-control state is detected. The system in the in-control state produces more reliable products and has a negligible defect rate. (2) The policy of FRW is adopted in the study. (3) The system starts in the in-control state and keeps functioning until the end of the production cycle. (4) After a production cycle, the produced products stay in a temporary buffer area near the production line and wait for further processing which depends on the result of the system inspection.
Inspect inversely from the last product
Number of conforming items = r* ?
Conforming items & unexamined items Y
Warehouse for sale Rework nonconforming items with ts* in reworking station
Fig. 1. Process of negative binomial inspection.
Nonconforming item
N
Conforming item
Defective ?
Y
N
Y
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(5) The costs of maintaining company reputation and implementing warranty obligations should be higher than the inventory holding cost. (6) Delayed delivery is not allowed, and the product inspection is performed in the temporary buffer area. (7) The warranty cost and the company image damage cost are relatively high for the product in supporting of the use of negative binomial sampling. (8) The product after sale deteriorates according to a Weibull power law deteriorating process. (9) The system has been well-established for a fair time period, and the related parameters can be properly estimated by the field experts using sufficient historical data. Moreover, the notations used throughout this paper are as follows:
a, b: The scale and shape factors of the Weibull distribution for the system elapsed time. T: The elapsed time of a production cycle. th: The waiting time for sale for products in the warehouse. tI: The product inspection time per unit. tr: The time for performing a system inspection. ts: The reworking time per unit. cm: The unit production cost. ch: The holding cost per unit in the warehouse. ca: The holding cost per unit in the temporary buffer area. ck: The set-up cost. cs: The system inspection cost. cb: The cost for restoring the production system to the in-control state. cI: The product inspection cost per unit. cr: The reworking cost per unit under normal working conditions. cw: The repair cost for each product breakdown. cR: The cost of damaging company image per unit. h1(t): The breakdown rate of a product produced in the in-control state. h2(t): The failure rate of a product produced in the out-of-control state, where h2(t) > h1(t). p, d: The production rate and demand rate. h: The defect rate of a production lot produced in the out-ofcontrol state. r: The number of conforming products for inspection. n: The number of total items for inspection. w: The warranty period. N: The lot size of a production run (N = pT).
restore the system back to the initial in-control state for the next production cycle. The set up cost SC can be expressed as SC = ck. In addition, with negative binomial sampling, having collected the optimal number of conforming products r, if the number of total inspected products is n, then n r nonconforming products have to be sent to the reworking station. As the number of nonconforming products increases, in order to prevent late delivery, the reworking time per unit has to be shortened by utilizing more resources, resulting in higher total reworking cost. Suppose that the total reworking cost is the reciprocal of the reworking time for each nonconforming item, ts, and since the reworking occurs only when the system is in the out-of-control state, which happens with b a probability of 1 eðT=aÞ , the total cost of reworking nonconforming items RWC can be expressed as:
RWC ¼
b cr ðn rÞ 1 eðt=aÞ : ts
ð3:1Þ
For the inventory holding cost, both the system and the product inspections are considered. If the in-control system state is detected, the entire batch of products is transferred to the warehouse without any product inspection performing. In such a case, the inventory holding cost includes the costs incurred from both the temporary buffer area and the final goods warehouse. Fig. 2 shows the changes of inventory levels during the time considered for the in-control state. The inventory holding cost for the of in-control system (HCI) is thus given by:
HC I ¼ ca
pT 2 þ pTt r 2
! þ ch
t2h d : 2
ð3:2Þ
On the other hand, if the out-of-control state is detected, the extra inspection time for inspecting products one by one, and the reworking time of the inspected nonconforming products have to be considered in evaluating the inventory holding cost. Fig. 3 shows the changes of inventory levels during the time considered for the out-of-control state. As can be seen in Fig. 3, once the system is detected as being in the out-of-control state, n out of N products is inspected one by one until r conforming products have been collected; i.e., (n r) nonconforming products have to be reworked. Since the reworking time for each nonconforming product is ts, the holding cost during h dÞ the reworking time is ch ðnrÞts ðpT2nþ2rþt , and the holding cost after 2 receiving the reworked batch of (n r) nonconforming products is
t2 d
ch h2 . Therefore, the inventory holding cost for the out-of-control system (HCO) is given by:
3. Optimal sampling number of conforming items and reworking time In this section, the three costs involved in the inspection scheme are first described: production cost, inspection cost, and warranty cost. The determination of the optimal sampling number of conformed products and reworking time of the negative binomial product inspection process is then discussed.
In-Control State Inventory Level
Buffer System Inspection
3.1. Production cost The production cost consists of the manufacturing cost of the products, the setup cost of the production system, the reworking cost of defective products, and the inventory holding cost. Since the unit production cost is assumed to be cm, the manufacturing cost MC can be obtained by multiplying cm by the production lot size N = pT, given by MC = cmpT. Furthermore, upon each production cycle termination, the production system has to be set up to
Production
tr
Demand (Warehouse)
th T
Fig. 2. Changes of inventory levels for the in-control state.
Y.-S. Huang et al. / Computers & Industrial Engineering 65 (2013) 605–613
breakdown rates of products produced by the production system, which may be in the in-control or the out-of-control state after the inspection, are different, the evaluation of warranty cost should be carefully considered for the two cases.
Out of Control State Inventory Level
Buffer
tr Production
Receipt of Reworking Products
ti
3.3.1. Warranty cost if the in-control state is detected We assume that a product produced in the in-control state is quite reliable with deterioration subject to a Weibull power law process with a breakdown rate h1(t). The warranty cost is the product of the maintenance cost per item cw, the production lot size Rw N = pT, and the expected number of breakdowns 0 h1 ðtÞdt; i.e., Rw Ncw 0 h1 ðtÞdt. Therefore, the warranty cost for the in-control state (IWC) is given by:
Product Inspection
System Inspection
Warehouse
(n-r)ts
T th
Fig. 3. Changes of inventory levels for the out-of-control state. 2
IWC ¼ Ncw
!
pT HC O ¼ ca þ pTtr þ pTnt i 2 ðn rÞt s ðpT 2n þ 2r þ t h dÞ t2h d : þ þ ch 2 2
609
Z
w
h1 ðtÞdt ¼ Ncw k1 wc1 ;
ð3:6Þ
0
c
where h1 ðtÞ ¼ k11 c1 tc1 1 ; t > 0.
ð3:3Þ
Since the elapsed time of the deteriorating production system follows a Weibull distribution with a scale factor a and a shape factor b, the total inventory holding cost (HC) can be obtained by multiplying the two holding costs shown in Eqs. (3.2) and (3.3) with their corresponding probabilities and summing them up, i.e., HC ¼ b b eðT=aÞ HC I þ 1 eðT=aÞ ÞHC O . As a result, the total production cost (PC) is given by:
b cr PC ¼ MC þ SC þ RWC þ HC ¼ cm pT þ ck þ ðn rÞ 1 eðT=aÞ ts ! 2 2 b b pT t d þ 1 eðT=aÞ þ pTtr þ ch h þ eðT=aÞ ca 2 2 " ! 2 pT þ pTt r þ pTnti ca 2 ðn rÞts ðpT 2n þ 2r þ t h dÞ t 2h d þ ch : ð3:4Þ þ 2 2 3.2. Inspection cost The inspection cost includes costs related to inspections of both the production system and products, which consist of the costs of inspection and possible restoration of the production system, and the inspection costs for the products. Since it is assumed that the production system is examined after each production run, the inspection cost for the production system (EC) is given by EC = cs. Once the out-of-control state is detected, the cost cb is defrayed so that the system can be restored to the in-control state for the next production run. Since the restoration cost (RC) occurs only when the system is in the out-of-control state, which happens with b b a probability of 1 eðT=aÞ , RC is given by RC ¼ cb ð1 eðT=aÞ Þ. The product inspection cost cI per product has to be defrayed as well for the total number of inspected products n. Since the inspection cost for products (CC) also occurs only when the system is in the out-ofb control state, which happens with a probability of 1 eðT=aÞ , CC is b given by CC ¼ cI nð1 eðT=aÞ Þ. As a result, the total inspection cost IC is given by:
b b IC ¼ EC þ RC þ CC ¼ cs þ cb 1 eðT=aÞ þ cI n ð1 eðT=aÞ : ð3:5Þ 3.3. Warranty cost The warranty cost includes the repair and maintenance costs of the product within the term of the warranty after sale. Since the
3.3.2. Warranty cost if the out-of-control state is detected The evaluation of warranty cost under the out-of-control state considers the costs involving unchecked items, conforming items, and reworked items. In fact, the (N n) unchecked products in the production lot would probably be produced in the in-control state or in the out-of-control state. According to the discussion in Huang et al. (2008), l ¼ Nn is used to denote that l and 1 l are the estimated percentages of conforming products produced in the in-control and the out-of-control states, respectively. As assumed in the in-control state, a Weibull power law intensity with scale parameter k2 and shape paramec ter c2, i.e., h2 ðtÞ ¼ k22 c2 tc2 1 , h2(t) > h1(t), is employed to denote the breakdown rate of conforming items produced in the outof-control state. Accordingly, the R wwarranty cost of unchecked Rw items (WN) is W N ¼ cw ðN nÞ l 0 h1 ðtÞdt þ ð1 lÞ 0 h2 ðtÞdt, c and which results as W N ¼ cw ðN nÞ½lk1 w 1 þ ð1 lÞk2 wc2 . Moreover, the defect rate of products produced in the out-ofcontrol state is assumed to be h, and the cost of selling a defective product is cr, which is the loss of damaging the company image. Therefore, the cost of losing the company’s reputation (WL) is given by WL = cRh(N n)(1 l). It is reasonable to believe that r inspected conforming products are produced in the out-of-control state with a breakdown rate h2(t). In such a case, the warranty cost for conforming products Rw (WG) is given by W G ¼ cw r 0 h2 ðtÞdt ¼ cw rk2 wc2 . In addition, since the (n r) nonconforming items are also produced in the out-ofcontrol state and will be sold after a reworking process, they are assumed to have a breakdown rate h2(t) as well. Therefore the warranty cost for reworked products (WR) can be given by Rw W R ¼ cw ðn rÞ 0 h2 ðtÞdt ¼ cw ðn rÞk2 wc2 . As a result, the warranty cost for the out-of-control state (OWC) is given by:
OWC ¼ W N þ W L þ W R þ W G ¼ cw ðN nÞ½lk1 wc1 þ ð1 lÞk2 wc2 þ cR ðN nÞð1 lÞh þ cw nk2 wc2 :
ð3:7Þ
Therefore, the total warranty cost (WC) can be obtained by multiplying the warranty costs for the in-control and out-of-control states shown in Eqs. (3.6) and (3.7) with their corresponding probabilities and summing them up, which is thus given by:
b b WC ¼ eðT=aÞ Ncw k1 wc1 þ 1 eðT=aÞ fcw ðN nÞ½lk1 wc1 þ ð1 lÞk2 wc2 þ cR ðN nÞð1 lÞh þ cw nk2 wc2 g:
ð3:8Þ
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Consequently, the expected total cost (TC) is given as:
TC ¼ PC þ IC þ WC b NT t2 d ¼ eðT=aÞ cm N þ ck þ cs þ cw Nk1 wc1 þ ca þ Ntr þ ch h 2 2 b c r þ 1 eðT=aÞ cm N þ ck þ ðn rÞ þ cs þ cb tr þ cI n þ cw ðN nÞ½lk1 wc1 þ ð1 lÞk2 wc2 þ cR ðN nÞð1 lÞh NT þ Ntr þ Nnt i þ cw nk2 wc2 þ ca 2
ðN 2n 2r þ t h dÞðn rÞts t2h d þch : þ 2 2
maintaining company reputation and implementing warranty obligations should be higher than the inventory holding cost, the second order derivative of E[TC] with respect to r, o2E[TC]/or2, is thus greater than zero. Moreover, because the time and cost for reworking are both greater than zero, the second order derivative of E[TC] with respect to ts, @ 2 E½TC=@t2s , is greater than zero, too. However, to ensure that the determinant of the matrix H, which 2 ½cR h þ cw ð1 hÞ k2 wc2 k1 wc1 can be calculated as jHj ¼ Nð1hÞ2 t3s h i2 2 ch Nh2 t s ch rh 2 , is greater than zero, Eq. (3.11) has to be ð1hÞ
satisfied. h
ð3:9Þ 4. Numerical application 3.4. Product inspection strategy According to the empirical investigation performed by Huang et al. (2008), as long as the optimal number r is not very close to N, the difference between the values of moments of n calculated from either theoretical method or empirical approach does not affect whether the negative binomial distribution is truncated or not. Moreover, for case of both h 6 0:2 and r/N < 0.7 seems to be a good criteria for ignoring the truncated negative binomial distribution for further analysis. In this study, it is assumed that h is relatively small while the product inspection cost is somewhat larger due to the assumption that the production system is quite reliable, since a low product inspection cost might result in a full inspection. Under the aforementioned assumptions, r is not close to N, and therefore r rhþr 2 we can employ both E½n ¼ 1h and E½n2 ¼ ð1hÞ 2 for further analytical processes with valid conclusions. Proposition. With the total expected cost shown in Eq. (3.9), the optimal number of conforming items that have to be collected r and the optimal reworking time per product ts can be obtained as follows: h 8 c c2 w k1 w 1 > r ¼ Nð1 hÞ cNð1hÞ þ 2cw ð1 hÞk2 wc2 þ 2hcR ctrsh cI cw k1 wc1 cw hkN2 w > > > < i c Nht t dht cR h2 Nð1hÞ ca Nt i h 2 s h 2 s ð2cw ½k2 wc2 k1 wc1 þ 2hcR 2Nh2 t s Þ1 ; > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2ð1hÞ : t ¼ ; s c ½ðNþdt Þð1h2r cr hÞ h
h
ð3:10Þ
" #2 2 c h r h2 2 c2 c1 if ½cR h þ cw ð1 hÞk2 w k1 w ch Nh t s > : Nts 3 ð1 hÞ ð3:11Þ
Proof. By setting the first order derivatives of E[TC] with respect to r and ts to be zero, respectively, Eq. (3.10) is obtained. However, the optimal values of r and t s are valid only under the condition that the second order derivative of E[TC] with respect to r and ts are both greater than zero, and the determinant of the Hessian matrix of the problem, which is given by:
" H¼
@ 2 E½TC=@r2
@ 2 E½TC=@r@ts
@ 2 E½TC=@ts @r
@ 2 E½TC=@t2s c2
# ; c1
2
k1 w 2ch Nh ts r where @ 2 E½TC=@r 2 ¼ 2hcR þ2cw ½k2 wNð1hÞ , @ 2 E½TC=@t 2s ¼ 2c , and t 3s 2 th hd cr h Nh 2rh @ 2 E½TC=@r@t s ¼ @ 2 E½TC=@t s @r ¼ ch 2ð1hÞ , has þ 2ð1hÞ ð1hÞ 2 t 2 ð1hÞ s
to be greater than zero as well. Since k2 wc2 k1 wc1 > 0 because of h2(t) > h1(t), and in this study, the production system is assumed to be highly reliable which results that the defect rate is small and the costs of
In this section, the data used in Huang et al. (2008) are utilized to perform numerical analyses in which some adequate values for the remaining parameters are also adopted. It is assumed that an FRW policy is offered to the customers, and a Weibull distribution, b1 FðtÞ ¼ 1 eðatÞ , with a = 8 and b = 2.5 can be used to describe the deterioration of the imperfect production system. In addition, the lifetime distributions of conforming and nonconforming products are Weibull distributed with breakdown rates c c h1 ðtÞ ¼ k11 c1 tc1 1 and h2 ðtÞ ¼ k22 c2 tc2 1 , where k1 = 1/36 and c1 = 2, and k2 = 1/12 and c2 = 2, respectively. Note that, presumably, the production system has been well-established for a fairly long period of time, and the related parameters can be properly estimated by the field experts using sufficient historical data and experiences. Table 1 lists the numerical values of the parameters. The prior conditions for obtaining the optimal values of sampling number of conforming products and reworking time per product have to be examined first, which are o2E[TC]/or2 = 0.03, @ 2 E½TC=@t 2s ¼ 106744:57, and |H| = 3200.88. Since they are all greater than zero, the optimal values of interest can be calculated by substituting the values listed in Table 1 into Eq. (3.10) and solving it. The optimal number of conforming items that have to be collected r is 620.597, and the optimal reworking time per product is 0.02659 h, which is about 1.6 min. However, since r should be an integer in practice, the expected total costs for cases where r = 620 and r = 621 are investigated. The case with the smaller expected Table 1 Numerical values of parameters. Parameter
Value
a
8 2.5 4 500 2000 5 150 0.8 150 200 5 15 1 5 1 1/36 2 1/12 2 0.05 18 400 0.01 0.01
b T p N(N = pT) cm ck ch cs cb cI cw ca cR cr k1
c1 k2
c2 h w d tr tI
Y.-S. Huang et al. / Computers & Industrial Engineering 65 (2013) 605–613 Table 2 The results of the analysis. Item
Value
Optimal sampling number of conforming products Optimal reworking time per product Optimal total sampling number Optimal sampling percentage Total expected cost
620 0.02659 652 32.6% $33351.9
Table 3 Relationship of the optimal values and the parameters. p
d
h
w
ch
ca
cR
ts
r
E[TC]
300 400 500 600 700
400 400 400 400 400
0.05 0.05 0.05 0.05 0.05
18 18 18 18 18
0.8 0.8 0.8 0.8 0.8
1 1 1 1 1
5 5 5 5 5
0.030100 0.028180 0.026595 0.025241 0.024073
395 512 620 722 817
$21087.1 $27207.4 $33351.9 $39518.9 $45707.0
500 500 500 500 500
200 300 400 500 600
0.05 0.05 0.05 0.05 0.05
18 18 18 18 18
0.8 0.8 0.8 0.8 0.8
1 1 1 1 1
5 5 5 5 5
0.030244 0.028244 0.026595 0.025205 0.024012
635 627 620 614 608
$32023.4 $32688.5 $33351.9 $34013.7 $34674.2
500 500 500 500 500
400 400 400 400 400
0.01 0.025 0.05 0.075 0.1
18 18 18 18 18
0.8 0.8 0.8 0.8 0.8
1 1 1 1 1
5 5 5 5 5
0.026407 0.026483 0.026595 0.026687 0.026761
736 691 620 554 491
$32984.8 $33127.7 $33351.9 $33559.4 $33751.1
500 500 500 500 500
400 400 400 400 400
0.05 0.05 0.05 0.05 0.05
18 24 30 36 42
0.8 0.8 0.8 0.8 0.8
1 1 1 1 1
5 5 5 5 5
0.026595 0.026652 0.026678 0.026693 0.026702
620 764 831 867 889
$33351.9 $44984.3 $59835.2 $77946.1 $99331.3
500 500 500 500 500
400 400 400 400 400
0.05 0.05 0.05 0.05 0.05
18 18 18 18 18
0.2 0.4 0.6 0.8 1
1 1 1 1 1
5 5 5 5 5
0.053236 0.037630 0.030716 0.026595 0.023788
679 655 636 620 607
$31223.4 $31952.0 $32657.9 $33351.9 $34038.4
500 500 500 500 500
400 400 400 400 400
0.05 0.05 0.05 0.05 0.05
18 18 18 18 18
0.8 0.8 0.8 0.8 0.8
0.4 0.8 1 1.2 1.6
5 5 5 5 5
0.026610 0.026600 0.026595 0.026590 0.026580
658 633 620 608 583
$30700.9 $32469.1 $33351.9 $34233.8 $35995.1
500 500 500 500 500
400 400 400 400 400
0.05 0.05 0.05 0.05 0.05
18 18 18 18 18
0.8 0.8 0.8 0.8 0.8
1 1 1 1 1
1 2.5 5 7.5 10
0.026591 0.026593 0.026595 0.026597 0.026599
612 615 620 626 631
$33322.3 $33333.5 $33351.9 $33370.2 $33388.3
total cost is chosen, which is the case of r = 620; i.e., the products are inversely inspected from the last product of the corresponding production lot in turn until 620 conforming products have been collected. In addition, the expected number of inspected products can be r 620 derived from r and h, which is given as E½n ¼ ð1hÞ ¼ 10:05 ffi 652. Therefore, the required percentage of inspection (POI) is n 652 ¼ 2000 ¼ 0:326 ¼ 32:6%, which indicates that the product inspecN tion process can be stopped after inspecting 32.6% of the lot size. The expected total cost for the production lot can also be calculated as $33351.9. Table 2 shows the results of the analysis. The effects of the corresponding parameters on the optimal values of interest are essential to decision makers, and are investigated in this study. The results are shown in Table 3. From Table 3, the production rate is inversely correlated to the reworking time per product. This can be explained by the fact that an increase of production rate increases the production lot size, which results in an increase of defective products. Therefore, in order to prevent late delivery, the reworking time per product has to
611
be reduced accordingly. Moreover, the sampling number of conforming products increases as the production rate increases. This can also be explained by an increase of lot size, which may need more product inspections to ensure product quality to lower the warranty cost after product sale. It is observed that the total cost increases as the production rate increases because more money is needed to produce more products. The demand rate is inversely correlated to the reworking time per product. This can be explained by the fact that the reworking time per product has to be reduced to prevent increased demand which may cause late delivery because of long reworking time. Moreover, since more conforming products are collected, more defective products are detected and reworked, which results in a long reworking process and an inventory shortage. Therefore, the sampling number of conforming products has to decrease as the demand rate increases. It is also interesting to find that when demand is increased, the reworking time per unit and the sampling number of conforming product have to decrease. This would result in the possible condition in which the reworking cost may increase because extra operation shifts will be needed, and the warranty cost may increase because more uninspected defective products may be sold in the market. Consequently, the total cost would increase as the demand rate increases. As can be seen in Table 3, the sampling number of conforming products decreases as the defective rate increases. Since when the defect rate is relatively high, collecting more conforming products for sampling results in detecting more defective products. This increases the inspection and reworking costs which may not be offset by the possible saved warranty cost in the case. In addition, the reworking time per unit is also affected, becoming longer, because that the sampling number of conforming products has to be decreased to cut down the total cost, and thus there are fewer detected defective products that require reworking. The total cost increases as the defect rate increases because a higher defect rate increases costs corresponding to reworking and company image damage. The sampling number of conforming products increases as the warranty term increases. This can be explained by the fact that for a longer warranty term, the product quality has to be higher to ensure that the warranty cost will be tolerable within a certain range; this can be achieved using a stricter inspection process. The warranty term is not significantly correlated to the reworking time per unit; however, it affects the total cost since the increase of the inspection and reworking cost are incapable of offsetting the warranty cost in this case, and the total cost increases as the warranty term increases. As can be seen in Table 3, the holding cost per unit in the warehouse is inversely correlated to the reworking time per unit. This is due to the fact that as the holding cost increases, the reworking time has to be reduced to shorten the period that the products are stored in the inventory to cut down the total inventory cost. A small number of conforming products collected in the inspection process also cuts down the total inventory cost. It is obvious that the total cost increases as the holding cost per unit increases. Similar effects can be seen in Table 3 for the holding cost per unit in the buffer; only the effect on the reworking time per unit is not very significant in the case. The sampling number of conforming products increases as the image damage cost increases. This can be explained by the fact that for a higher image damage cost, the product quality has to be higher to ensure that the image will not be damaged seriously when too many defective products are sold; this can be achieved using a stricter inspection process. The image damage cost is not significantly correlated to the reworking time per unit; however, it affects the total cost. The total cost increases as the image damage cost increases. Figs. 4–6 show the impacts of the parameters on ts, r, and E[TC], respectively.
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Y.-S. Huang et al. / Computers & Industrial Engineering 65 (2013) 605–613 Table 4 The evaluated results of impacts for the parameters.
0.06 0.055
p
0.05
θ
0.045
d
0.04
ch
ts
ca
0.035
cR 0.03
w
0.025 0.02 0.015 -100% -80% -60% -40% -20% 0%
20% 40% 60% 80% 100%
p d h w ch ca cR
r
ts
TC
+(2) (3) – +(1) – – –
(2) +(3)
+(2) + + +(1) + +(3) +
(1)
Note: The Arabic numerals in the parenthesis denote the rank of influence. + Denotes positively correlated. Denotes inversely correlated.
the reworking time per unit the most in the case. Decision makers can refer to the results to determine how their limited resources can be invested to enhance their decision making process.
Percentage of Change Fig. 4. The impacts of parameters on ts.
5. Conclusion 1000 900 800 700
p
600
θ
500
r
d
400
ch
300
ca
200
cR
100
w
0 -100 -200 -100% -80% -60% -40% -20% 0%
20% 40% 60% 80% 100%
Percentage of Change Fig. 5. The impacts of parameters on r.
58500 53500 48500 43500
p
38500
θ d
E [TC] 33500
ch 28500
ca
23500
cR
18500
w
13500 8500 -100% -80% -60% -40% -20% 0% 20% 40% 60% 80% 100%
Percentage of Change
In this study, we consider an imperfect production system whose elapsed time is distributed as a two-parameter Weibull distribution for developing a reasonable and economic inspection strategy with an aim to provide manufacturers with an effective decision-making on quality examination. Negative binomial sampling is employed in the production inspection scheme to detect the possible switching point from the in-control state to the outof-control state of the production system. The products are inversely inspected from the last item of the corresponding production lot in turn and the inspection terminates after the optimal number of conforming items has been collected. In addition, the nonconforming products detected in the inspection process are reworked and then transferred to the warehouse for sale. Since late delivery is not allowed in this study, the decision of critical interest is to simultaneously determine the optimal sampling number of conforming products and the optimal reworking time per unit. By using this inspection strategy, manufacturers can effectively manage both the inspection and the reworking processes to ensure an acceptable level of product quality for maintaining a good company reputation and thus achieve the goal of cost minimization. Further research would be performed by allowing shortage to occur to investigate its influence on the decision making process, which may lead to a significant change of the inventory holding cost evaluated in the proposed model. Additionally, it is of interest to further consider that the inspection time and the reworking time, instead of being constants, can be considered to be random variables, and the resultant costs may modify the structure of the elements in the model. Accordingly, such a stochastic problem needs a considerable extension of the model elements and therefore becomes a somewhat complicated problem to be addressed. Finally, the warranty cost in this research is deliberated under an FRW policy, thus dissimilar warranty policies might be employed to revise the proposed product inspection strategy, such as the application of pro-rata warranty (PRW) in the future extension of the proposed model.
Fig. 6. The impacts of parameters on E[TC].
References As can be seen in Figs. 4–6, the relative importance of each parameter can be evaluated, and the results are shown in Table 4. As can be seen in Table 4, the warranty term plays an important role in determining the sampling number of conforming products and has the greatest impact on the expected cost in the case. The production rate is important for the three decision variables of interest, and the holding cost per unit in the warehouse influences
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