Lot sizing and quality investment with quality cost analyses for imperfect production and inspection processes with commercial return

Int. J. Production Economics 140 (2012) 922–933

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Lot sizing and quality investment with quality cost analyses for imperfect production and inspection processes with commercial return Seung Ho Yoo a, DaeSoo Kim b,n, Myung-Sub Park b a b

Sun Moon University, Tangjeong-myeon, Asan, Chungnam 336-708, Republic of Korea Korea University, Business School, Anam-dong, Seongbuk-gu, Seoul 136-701, Republic of Korea

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 September 2010 Accepted 14 June 2012 Available online 27 July 2012

This study examines an imperfect production and inspection system with customer return and defective disposal. We develop an optimal lot sizing model with production and inspection quality investment, incorporating all the quality costs. We find the optimal lot size, rework frequency, defective proportion, and Type I and Type II inspection error proportions which minimize the total quality cost and maximize the total profit. We further analyze the solutions for no, partial, sequential and joint investment decisions on production and/or inspection processes in terms of quality costs using numerical analyses. The result provides important managerial insights into practice. & 2012 Elsevier B.V. All rights reserved.

Keywords: Inventory Imperfect production Inspection Return Quality cost

1. Introduction In a closed-loop supply chain, we frequently discover firms producing defective items and passing some to customers due to imperfect quality in both production and inspection processes. This imperfect process reliability then causes customers to return defective items for exchange or refund. For example, the average return rate in the apparel and the consumer electronics industry amounts to 19.44% and 8.46%, of which 49.45% and 35.71% are due to defects, respectively (Reverse Logistics Executive Council, 1999). The annual value of product returns is estimated at $100 billion (Stock et al., 2002; Blackburn et al., 2004) and commercial returns in online retailing and e-commerce are recorded up to 25% of sales (Krikke et al., 2004). In addition, imperfect inspection incurs an opportunity loss by falsely screening out non-defective items and disposing of them as defectives. Thus, it is important for firms to understand both internal and external effects of defective production and inspection failure on lot sizing, inventory, quality costs and profit. There have been a vast number of studies that deal with imperfect production, inspection and reverse logistics issues. Most studies focused mainly on developing cost-minimizing models for either internal effects of defective production and process quality improvement, or recycling and rework of reusable item returns. The imperfect production and inspection system in practice, however, incurs not only internal failure costs related to

n

Corresponding author. Tel.: þ82 232 902 817; fax: þ82 2922 7220. E-mail address: [email protected] (D. Kim).

0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2012.07.014

rework, salvage and scrap, but also external failure costs from defective item return, reverse logistics, resolution of customer quality problems, and refund or exchange, negatively affecting the firm’s profit. Thus, to reduce these negative effects, the firm often invests in prevention activities to improve process capability, worker skills, inspection and test equipment design, as well as appraisal activities to determine the degree of conformance and internally screen out defectives. Consequently, it is imperative for a firm to carefully consider the tradeoff relationships among the internal and external aspects of quality costs (i.e., prevention, appraisal, internal failure and external failure) in the optimal investment decision along with lot sizing (Juran and Gryna, 1988). In this light, this study explores internal and external issues of defective production, inspection failure (Type I and Type II errors), and related quality improvement investment for a firm in a closed-loop supply chain with commercial return. Specifically, we develop a profit-maximizing model that jointly determines the optimal production lot size, rework frequency, and defective production and inspection error proportions related to respective investment. We solve the model optimally using differential calculus and nonlinear programming. We also analyze our model in terms of four quality cost components to discover their tradeoff relationships in lot sizing and quality investment decision making. Moreover, we investigate the solutions and quality cost structure of different decisions in practice with no, partial, sequential and joint investment in production and inspection process reliability, in order to provide important managerial insights and help managers make well-informed decisions. The significance of this study may lie in building a more practical model, quantifying and analyzing quality costs and comparing

S.H. Yoo et al. / Int. J. Production Economics 140 (2012) 922–933

different investment decision-making situations, thereby extending the current body of knowledge on imperfect-quality inventory and quality management. The rest of the paper is organized as literature review and analysis, basic imperfect-quality inventory model and solution approach, extended quality investment model and quality cost decomposition, and optimal solution and quality cost analysis for different investment decisions, followed by conclusions.

2. Literature review and analysis Most previous research of an imperfect production system has focused on developing cost-minimization models for either deteriorating or stable production processes with different inspection methods (see Yoo et al., 2009, 2012 for a review). The studies of a deteriorating process since Porteus (1986) and Rosenblatt and Lee (1986) typically assumed that the production process goes from an in-control state to an out-of-control state as a run cycle progresses, mostly with inspection based on regular intervals and instantaneous rework of defective items (see Yano and Lee, 1995 for a review). Besides the determination of lot size or production cycle, these studies also dealt with investment in production process quality (Porteus, 1986; Ouyang et al., 2002), inspection schedule or policy (Lee and Rosenblatt, 1987, 1989; Lee and Park, 1991; Liou et al., 1994; Rahim, 1994; Kim et al., 2001; Rahim and Ben-Daya, 2001; Wang and Sheu, 2001), inspection errors (Lee and Park, 1991; Liou et al., 1994; Rahim, 1994; Rahim and Ben-Daya, 2001; Wang and Sheu, 2003; Yeh and Chen, 2006), inspection size (Vickson, 1998; Yeh and Chen, 2006), delivery of defective items to customers (Lee and Park, 1991; Liou et al., 1994; Yeh and Chen, 2006), and finite multiple lot sizing (Guu and Zhang, 2003). More directly related to the present study are the studies of a stable production process since Deming (1982). These studies typically assumed that the production process follows a Bernoulli process generating binomial yields, primarily with inspection by entire lot screening and scrap of defective items at no cost (Schwaller, 1988; Cheng, 1989, 1991a, 1991b; Anily, 1995; Lee et al., 1996; Agnihothri et al., 2000; Otake and Min, 2001; Affisco et al., 2002; Elhafsi, 2002; Tripathy et al., 2003; Grosfeld-Nir, 2005; Grosfeld-Nir et al., 2006; Leung, 2007; Chiu et al., 2007). Some exceptions include Zhang and Gerchak (1990) examining inspection of a fractional lot, Salameh and Jaber (2000) dealing with salvage, and Yoo et al. (2009) investigating two-way inspection errors with rework and salvage of sales returns. Some studies also considered investment in production process reliability with lot sizing (Cheng, 1989, 1991a, 1991b; Lee et al., 1996; Otake and Min, 2001; Affisco et al., 2002; Tripathy et al., 2003; Leung, 2007; Yoo et al., 2012). In contrast to most of the studies above dealing with internal issues related to defective production, reverse logistics modeling studies have investigated external issues of recycling or reusable item returns with mixed disposal methods of rework and scrap (Schrady, 1967; Richter, 1996a, 1996b; Teunter, 2001; Koh et al., 2002; Dobos and Richter, 2004, 2006). Their focus was primarily on finding cost-minimizing production and recycling (repair or remanufacturing) lot sizes to satisfy market demand, not related to commercial returns (see Stock et al., 2002; Blackburn et al., 2004 for return issues). Through the literature review, we discover some crucial points which serve as underpinning of the present study. First, most imperfect production system studies focused on developing costminimization models that reflect only internal effects of defective production, not considering an imperfect inspection process and external reverse logistics issues of commercial returns of defective

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items. In practice, however, the inspection process is often not perfect or error-free, let alone the production process, thereby resulting in Type II inspection error of falsely not screening out some proportion of defectives even with entire lot screening and thus passing them to customers. This then subsequently causes customers’ defective return for exchange or refund and incurs lost sales and additional costs (Reverse Logistics Executive Council, 1999; Stock et al., 2002; Blackburn et al., 2004). Second, those studies dealing with process reliability investment were mainly concerned with production processes. In practice, however, planning and control of appraisal processes including inspection and test, training of employees in an inspection line, design of an inspection process and introduction of better inspection equipments are also important prevention activities of quality management in reducing the delivery of defective items to customers and subsequent return and loyalty problems (Juran and Gryna, 1988). Third, besides the external failure issues due to Type II inspection error, an imperfect inspection process also involves Type I inspection error, which falsely screens out some non-defective items and regards them as defectives. Thus, this yields an opportunity loss in practice by not being able to sell those falsely screened nondefective items to customers (Liou et al., 1994; Yoo et al., 2009). Fourth, many previous studies have dealt with only one defective disposal option. In practice, however, firms in consumer electronics, apparel, automotive parts industries, etc. use multiple disposal options of rework, salvage and scrap together (Cyber Atlas, 2000; Yoo et al., 2012). And fifth, previous studies have not examined all the quality cost components comprehensively. In practice, however, identifying and measuring quality costs (i.e., prevention, appraisal, internal failure and external failure costs) are crucial for firms in quantifying the size of the quality problem, thereby helping justify control and improvement efforts, guide their development and track progress in quality management activities (Juran and Godfrey, 1998). Therefore, to close the gap between practice and academia, we extend extant studies, considering all the above issues previously unexplored. The next two sections develop basic and extended inventory and/or quality investment models.

3. Basic imperfect-quality inventory model and solution approach 3.1. Problem description and assumptions We investigate a practice in which a manufacturer’s production and inspection processes are stable and non-deteriorating but not perfectly reliable. Thus the two imperfect processes result in producing defective items, and not only delivering some defective items to customers leading to the customers’ return but also yielding an opportunity loss by falsely screening out some non-defectives as defectives. In modeling, we extend particularly previous two studies, Yoo et al. (2012) dealing with continuous improvement for imperfect production and inspection processes with Type II inspection error and Yoo et al. (2009) considering both Type I and Type II inspection errors without quality investment. Further, different from these studies, the present study focuses on detailed quality cost decomposition analyses in addition to building a more comprehensive practical model. Fig. 1 describes the forward and reverse flow of inventories and transactions (see notations in Table 1 henceforth). The imperfect production system produces lot size Q in cycle T. Given that the production follows a stable Bernoulli process generating binomial yields with defective proportion p from its probability density function (pdf), f(p), the process yields defective quantity pQ among Q. Then, the entire lot Q is inspected

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Sales and Exchanges

(᧧)

Demand [D]

Lot size Q

Screened items ((1-)+(1-))Q [F]

Production [M]

Inspection [I]

Non-defective items (1-)(1-)Q

Note

Defective items Q

Stock or Activity qty. [rate]

Defective returns from customers Q

Returned defective items Q

Customers’ choice

Items for disposition ((1-)+)Q

Rework 1((1-)+)Q [W]

Disposition method

(᧧)

Net depletion [D’]

(᧧)

Serviceable Items ((1-)(1-)+)Q

Exchange  Q [E] Refund (1-)Q

(᧩)

Investment in production process reliability 1-

Sales/exchange of reworked items 1((1-)+)Q [R]

Investment in inspection process reliability 1- and 1-

(᧩)

Salvage 2( (1-)+)Q [V]

Scrap 3( (1-)+)Q

* 1+ 2+ 3=1

Fig. 1. Inventory flow involving imperfect production and inspection processes.

Table 1 Notation for mathematical models. D D0 E F R V M W Q T Tp

p a b l lIF lEF

g 1g

Demand rate (unit/unit time) Net depletion rate in production line (unit/unit time), D0 ¼ D þE þF  R  V Exchange rate (unit/unit time), E¼ glEFD0 Screening rate by inspection (unit/unit time), F¼ lIFD0 Accumulation rate of items being reworked or depletion rate of items reworked (unit/unit time), R ¼ d1lD0 Accumulation and depletion rate of salvage items (unit/unit time), V ¼ d2lD0 Production rate (unit/unit time), M 4D0 Rework rate (unit/unit time), W4R Lot size (unit/cycle; decision variable) Cycle time of a production lot, T¼ Q/D0 Production run time, Tp ¼ Q/M Proportion of defective items during production, 0r p r 1 (decision variable) Proportion of Type I inspection error ¼P(items screened as defective items 9 nondefective items), 0 r a r 1 (decision variable) Proportion of Type II inspection error ¼P(items not screened as defective items 9 defective items), 0 r b r 1 (decision variable) Total failure proportion, l ¼ lIF þ lEF ¼ a(1  p) þ p Internal failure proportion, lIF ¼ a(1  p) þ(1  b)p External failure proportion, lEF ¼ bp Proportion of exchange among returns, 0 r g r 1 Proportion of refund among returns

continuously at an inspection rate I, which is assumed equal to a net depletion rate D0 so as not to incur unnecessary inventory holding from different rates. The inspection process is also assumed to follow a stable binomial process with Type I and Type II inspection errors, given their respective proportions of a ¼P (items screened as defective items9non-defective items) and b ¼P (items not screened as defective items9defective items) (0o a, b o1) following pdf’s of f(a) and f(b). We assume a and b are independent of defective proportion p, as in Liou et al. (1994) and Yoo et al. (2009). Then, the internal failure proportion is defined as lIF ¼ a(1 p)þ(1 b)p, including falsely screened non-defective proportion a(1 p) due to Type I inspection error and screened defective proportion (1 b)p. The external failure proportion is defined as lEF ¼ bp, the proportion falsely not screened and regarded as non-defectives due to Type II inspection error. Therefore, the total failure proportion becomes l ¼ lIF þ lEF ¼(a(1 p)þ(1 b)p)þ bp ¼ a(1 p)þ p. The serviceable

d1,2,3 n Qr Tr Tw K Kr h H u i p v r w g l Y YP YI

Proportion of rework, salvage, scrap among defectives, respectively, 0r d1,2,3 r 1, where d1 þ d2 þ d3 ¼1 Rework (setup) frequency per cycle time T, n40 (decision variable) Rework lot size (unit), Qr ¼ d1lQ/n Cycle time of rework items, Tr ¼T/n ¼ Q/(nD0 ) Rework run time, Tw ¼ Qr/W ¼ d1lQ/(nW) Setup cost per production run ($/cycle) Rework setup cost per rework run ($/rework cycle) Inventory holding cost rate fraction (fraction/unit time) Inventory holding cost rate ($/unit/unit time) Production cost per unit ($/unit) Inspection cost per unit ($/unit) Selling price per unit ($/unit) Salvage price per unit ($/unit) Return cost per unit ($/unit) Rework cost per unit ($/unit) Scrap cost per unit ($/unit) Penalty cost per unit from customers’ quality dissatisfaction ($/unit) Total interest and depreciation cost ($/unit time), Y ¼YP þYI Interest and depreciation cost for a production process ($/unit time), YP ¼ap-b where a, b40 Interest and depreciation cost for an inspection process ($/unit time), YI ¼Ya þ Yb ¼ ca  d þ eb  f where c, d, e, f 40

items of ((1 a)(1 p)þ lEF)Q are then sold to customers, including non-defectives (1 a)(1 p)Q and falsely unscreened defectives lEFQ. After sales, the customers who bought defective items detect defect and return them due to quality dissatisfaction. The defective returns lEFQ are assumed to occur continuously like demand requests, for either exchange (with proportion g following its pdf, f(g)) or refund (with proportion 1 g), as evidenced by the BizRate study (Cyber Atlas, 2000). This return choice is also assumed to follow a binomial process with exchange quantity glEFQ and refund quantity (1  g)lEFQ where lEF ¼ bp. In each cycle, the firm handles the total failure items lQ ( ¼(lIF þ lEF)Q¼(a(1 p)þ p)Q) including screened and returned items, using three main defective disposal options, i.e., rework, salvage and scrap (Reverse Logistics Executive Council, 1999). These options are assumed to follow a multinomial process with the expected quantity of d1lQ,

S.H. Yoo et al. / Int. J. Production Economics 140 (2012) 922–933

To rework = δ1λQ

R

Rework and salvage = (δ1+δ2)λQ

Time Reworked = δ1λQ

D+F+E Q

W

Qr

Screened and exchange = (λIF +γλEF)Q

925

R

Qr

Tw Tr

D′=D+F+E–R–V

M

W

T D Salvage = δ2λQ

Tp

V

V

Time T

T

Time

Fig. 2. Inventory behavior of (a) production line, (b) rework line and (c) salvage process.

d2lQ and d3lQ, given their respective proportions d1, d2 and d3 following their pdf0 s, f(  ) where d1 þ d2 þ d3 ¼1. These proportions can be obtained in practice by analyzing historical disposal data of defective items. To satisfy customer demand D along with exchange request E, the firm runs two separate main production line and rework line. Fig. 2 illustrates the inventory behavior of regular items produced and depleted in the main production line, items to rework and reworked in the rework line, and salvage items. As depicted in Fig. 2a, the main line produces lot Q at a rate M during its run cycle Tp in each cycle T ( ¼Q/D’), as in the typical EPQ model. The finished goods inventories are depleted at a net inventory depletion rate D0 below in T, which is positively affected by demand rate D, screening rate by inspection F ( ¼ lIFD0 ) and exchange rate E ( ¼ glEFD0 ), and negatively influenced by reworked item depletion rate R ( ¼ d1lD0 ) and salvage item depletion rate V (¼ d2lD0 ). From D0 ¼ Dþ FþE  R V, we obtain D0 ¼ D0 ðp, a, bÞ ¼ D=ð12d3 lIF 2ðg2d1 2d2 ÞlEF Þ:

ð1Þ

where lIF ¼ a(1 p) þ(1 b)p, lEF ¼ bp and d1 þ d2 þ d3 ¼1. Observe that D0 increases when a increases while the effect of either p or b depends on the interactions among customers’ return choice and disposal options, and that D0 ¼D when p ¼0 and a ¼0. Also note that D comprises demand for both regular and salvage items, instead of independent demand in two separate markets, as found for many consumer products including apparel, consumer electronics and furniture. As shown in Fig. 2b, items to rework among defective items are accumulated and depleted at the same rate R to avoid unnecessary inventory holding from different rates. The separate rework line produces rework lot size Qr ( ¼ d1lQ/n) at a continuous and constant rate W during its rework production cycle Tw ( ¼ d1lQ/(nW)) in the rework cycle Tr (¼ T/n¼Q/(nD0 )) with rework setup frequency n in T. To ensure no shortages, we assume M 4 D0 and

ð2Þ

W 4R ¼ d1 lD0 :

ð3Þ

The reworked items are assumed to be perfectly repaired and used to satisfy customers’ D and E like the items produced in the main line. Further, in practice where rework involves simple repairs of items (e.g., small electronic appliances) which do not require major rework setups, the model can be easily modified by excluding them, thereby not incurring rework setup and holding costs, given the depletion of reworked items right upon completion.

As in Fig. 2c, the items to salvage as they are (d2lQ) to the second market among defective items are accumulated and depleted at the same continuous and constant rate V in T, again to avoid unnecessary inventory holding from different rates. These sales are assumed to be final without returns as many second market resellers do in practice. Finally, those defective items not suitable for rework or salvage are scrapped instantaneously at variable costs. 3.2. Mathematical formulation The objective of the model is to simultaneously determine lot size Q and rework frequency n that maximize total profit per unit time (TPU). In each cycle T, the firm sells at a unit selling price p serviceable items (i.e., non-defectives (1 a)(1 p)Q and unscreened defectives lEFQ) and reworked items d1lQ, which include items for exchange (glEFQ) at no charge, generating sales revenue of p(1(1 d1)l þ(1 g)lEF)Q. It also obtains salvage sales revenue of nd2lQ. However, refund requests at p incur a revenue loss of p(1 g)lEFQ. Thus, total revenue per cycle (TR) becomes TR ¼ pð12ð12d1 Þl þ ð12gÞlEF ÞQ þ vd2 lQ 2pð12gÞlEF Q ¼ pð12ð12d1 ÞlÞQ þ vd2 lQ :

ð4Þ

where l ¼ lIF þ lEF ¼ a(1 p)þ p. Total cost per cycle consists of fixed and variable costs. Fixed costs comprises production line and rework setup costs of K and nKr in cycle T. And variable costs include costs of production (uQ), inspection (iQ), return (rlEFQ), rework (wd1lQ), scrap (gd3lQ), and penalty (l(1 g)lEFQ) of returned defective items refunded, where l is incurred due to the loss of goodwill or adjustment of customers’ quality complaints. Further, given h as inventory holding cost rate fraction, inventory holding cost rate H becomes hu for items produced in the production line, h(u þiþr(lEF/l)) for items to rework, and h(u þiþr(lEF/l)þ w) for reworked items, since on average there are returned items with external failure proportion lEF among total failure proportion l in the rework line due to Type II inspection error. From Fig. 2a and b, we obtain average inventory per cycle in production and rework lines. Note that salvage items do not incur inventory holding cost (see Fig. 2c). Thus, total cost per cycle (TC) becomes TC ¼ ðK þnK r Þ þ ðu þi þ r lEF þ ðwd1 þ g d3 Þl þlð1gÞlEF ÞQ     hu 1 1 hð2u þ 2i þ 2rðlEF =lÞ þ wÞ 1 d1 l d1 lQ 2 Q2 þ : þ 0 0 2 D M 2 W n D ð5Þ

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Now by subtracting TC in (5) from TR in (4) and dividing it by cycle time T ( ¼Q/D0 ), we obtain total profit per unit time (TPU) as a function of Q and n below: TPU ¼ ðpð1ð1d1 ÞlÞ þ ðvd2 wd1 g d3 Þluiðr þ lð1gÞÞlEF ÞD0   ðK þ nK r ÞD0 hu D0 1  Q  2 Q M   hð2u þ 2i þ 2rðlEF =lÞ þ wÞ d1 lD0 d1 lQ 1 , ð6Þ  2 n W where D0 ¼D0 (p, a, b) in (1), l ¼ a(1 p)þ p and lEF ¼ bp. Since p, a, b, g, d1, d2 and d3 are random variables with their known pdf0 s, TPU has its expected value, ETPU. For simplicity, we assume that g, d1, d2 and d3 are their fixed long-run average values. Then, ETPU becomes   K þ nK r huQ DUE½C ETPU ¼ ðpuiÞ þ 2M Q      pð1d1 Þðvd2 wd1 g d3 Þ DUE lC   hð2u þ 2i þwÞd21 Q D h 2 i UE l C ðr þ lð1gÞÞDUE lEF C þ 2nW 2  hð2u þ 2iþ wÞd1 Q   hrd1 Q D  UE lEF lC  UE l þ nW 2n hrd1 Q   hu UE lEF  Q, ð7Þ  n 2 where E[  ] is an expected value, C ¼ 1/(1 d3lIF (g  d1  d2)lEF) and CD ¼D0 in (1) with lIF ¼ a(1  p)þ(1  b)p, lEF ¼ bp and l ¼ lIF þ lEF. 3.3. Solution approach To obtain the optimal solution of Q and n that maximize ETPU, we use differential calculus. From the first-order necessary optimality condition for ETPU in (7), we obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KDUE½C Qn ¼ , ð8Þ huð1ðD=MÞUE½CÞ

nn ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 K d1 fð2u þ 2iþ wÞðE½lðd1 D=WÞUE½l CÞ þ 2rðE½lEF ðd1 D=WÞUE½lEF lCÞg , K r uð1ðD=MÞUE½CÞ

ð9Þ where C ¼1/(1 d3lIF  (g  d1  d2)lEF) with lIF ¼ a(1 p)þ(1 b)p, lEF ¼ bp and l ¼ lIF þ lEF. Now from the second-order sufficient optimality conditions of ETPU in (7), we prove the following principal minors of the Hessian with respect to Q and n for global optimality: 9H1 9 ¼ ETPU Q Q ¼ 

2ðK þ nK r ÞD Q3

UE½C o 0

and

9H2 9 ¼ ETPU Q Q UETPU nn ETPU 2Q n      d1 D h 2 i 2Khd1 D UE l C ¼ ð2u þ 2iþ wÞ E l  2 W n3 Q     d1 D   UE lEF lC UE½C 4 0, þ 2r E lEF  W

ð10Þ

ð11Þ

since W4R¼ d1lD0 in (3). 3.4. Illustrative example Now we show numerically the solution of the basic model using the parameters below. Example parameters: unit time¼ year, demand rate D¼20,000 units/year, production rate M¼40,000 units/year, rework rate W¼30,000 units/year, production setup cost K¼$500/setup, rework setup cost Kr ¼$50/rework setup, unit production cost u¼$100/unit, unit inspection cost i¼$5/unit, unit selling price p¼$200/unit, unit

salvage price v¼$80/unit, inventory holding cost rate fraction h¼0.3/unit/year, unit return cost r¼$5/unit, unit rework cost w¼$30/unit, unit penalty cost l¼ $50/unit, and unit scrap cost g¼$3/unit. Additionally, we assume that defective proportion p, Type I inspection error proportion a and Type II inspection error proportion b are uniformly distributed with their pdf0 s below: 25, 0 r p r0:04 10, 0:1r a r0:2 f ðpÞ ¼ , f ðaÞ ¼ , 0 otherwise 0 otherwise 10, 0:05r b r 0:15 : and f ðbÞ ¼ 0 otherwise Also, exchange proportion g ¼0.4 and proportions of rework, salvage and scrap (d1, d2, d3)¼ (0.5, 0.2, 0.3). Now by putting the above parameter values into Qn and nn in (8–9) and ETPU in (7), we obtain the optimal solution (Qn, nn)¼(1,215.79 units/cycle, 1.99/cycle) with its maximum profit ETPUn ¼$1,624,208/year, using MAPLE 12. 3.5. Extreme cases When b ¼1 and i¼0 in (7–9), the model simply reduces to the one with no inspection, which entails the external effect of defective production. On the other hand, when a ¼0 and b ¼0 in (7–9), the model becomes the one with perfect inspection (i.e., entire lot screening without inspection errors) as in many previous studies. Also when p ¼0 and a ¼ 0 (i.e., the production and inspection processes are perfect without defect and Type I inspection error), D0 ¼D and thus (8) and (9) reduce to the classic EPQ, i.e., Qn ¼(2KMD/(hu(M D))1/2 and nn ¼0.

4. Extended quality investment model and quality cost decomposition 4.1. Inventory model with investment in production and inspection quality In this section, we extend the prior basic (Q, n) model by treating defective proportion p and inspection error proportions a and b as decision variables from capital investment in production and inspection process reliability. For simplicity, we assume that the processes involving defective production, two types of inspection errors, customers’ return choice and defective disposal option choice are all stable, so all the related proportions are regarded as fixed long-run average values. Therefore, we consider TPU in (6) as a function of p, a and b instead of ETPU in (7) in this investment problem. Then Qn and nn in (8) and (9) become sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KMD0 and ð12Þ Q ðp, a, bÞ ¼ huðMD0 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kð2u þ2i þ 2rðlEF =lÞ þ wÞd1 lð1d1 lD0 =WÞ , nðp, a, bÞ ¼ K r uð1D0 =MÞ

ð13Þ

where D0 ¼D0 (p, a, b) in (1), lEF ¼ bp and l ¼ a(1  p) þ p. Further, we include in TPU in (6) interest and depreciation costs YP and YI from investment in production and inspection process reliability, respectively. YI involves investment for both Type I and Type II inspection errors, Ya and Yb. These costs are assumed to take power function forms below, widely accepted in the literature related to quality problems (Cheng, 1989, 1991a, 1991b; Lee et al., 1996; Tripathy et al., 2003; Leung, 2007). Y P ¼ f ðpÞ ¼ apb

and

ð14Þ f

Y I ¼ f ða, bÞ ¼ Y a þY b ¼ cad þeb ,

ð15Þ

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where a, b, c, d, e and f40 and are real numbers chosen in practice as the best fit of the estimated cost functions based on the industry, firm and product characteristics. Note that we adopt YP and YI as costs per unit time, rather than per cycle (Cheng, 1989; Leung, 2007), to avoid an effect of the non-fixed cycle time T on the decision of p, a and b. Therefore, the modified TPU in the extended quality investment model becomes TPU ¼ TPU in ð6ÞðY P þ Y I Þ in ð14Þ and ð15Þ ¼ ðpð1ð1d1 ÞlÞ þðvd2 wd1 g d3 Þluiðr þ lð1gÞÞlEF ÞD0   ðK þ nK r ÞD0 hu D0 hð2u þ 2i þ 2rðlEF =lÞ þ wÞ 1  Q  2 2 Q M   d1 lD0 d1 lQ f apb cad eb , ð16Þ  1 n W where D0 ¼D0 (p, a, b) in (1), l ¼ a(1 p)þ p and lEF ¼ bp. In solution, finding the closed-form optimal solutions of Q, n, p, a and b is impossible due to the mathematical complexity from the non-fixed D0 in (1) and YP and YI in (14) and (15) with Q(p, a, b) and n(p, a, b) in (12) and (13). Thus, we develop the nonlinear TPU maximization model below: Maximize TPU subject to

in (16)

Q ¼ Q ðp, a, bÞ n ¼ nðp, a, bÞ M 4 D0 W 4R 0 r p, a, b r1,

in in in in

ð12Þ, ð13Þ, ð2Þ, ð3Þ, ð17Þ

4.2. Decomposition of quality costs By substituting Q(p, a, b) and n(p, a, b) in (12) and (13), TPU in (16) reduces to  TPUðp, a, bÞ ¼ pð1ð1d1 ÞlÞ þ ðvd2 wd1 g d3 Þl sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    D0 uiðr þ lð1gÞÞlEF D0  2huKD0 1 M sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   d1 lD0  2hð2u þ 2i þ2rðlEF =lÞ þ wÞK r d1 lD0 1 W ðY P þ Y I Þ:

ð18Þ

Then by subtracting TPU(p, a, b) in (18) from non-quality related components TPUT (equivalent to the profit function of the classic EPQ model without quality problems and activities), we obtain total quality cost TQC(p, a, b) below, which consists of prevention cost (QCP), appraisal cost (QCA), internal failure cost (QCIF) and external failure cost (QCEF) (Juran and Gryna, 1988). TQ Cðp, a, bÞ ¼ TPU T TPUðp, a, bÞ ¼ Q C P þQ C A þQ C IF þ Q C EF ¼ ðY P þ Y I Þ þuðD0 DÞ þ iD0 þ ðr þ ðp þ lÞð1gÞÞlEF D0 þ ðwd1 þ g d3 ÞlD0 þ ðpvÞd2 lD0  pffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2huK D0 ð1D0 =MÞ Dð1D=MÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   d1 lD0 : þ 2hð2u þ2i þ 2rðlEF =lÞ þ wÞK r D0 d1 l 1 W

Q C A ¼ iD0 :

927

ð21Þ

Regarding internal and external failure costs (QCIF, QCEF), first as seen from Fig. 2a, quality problems in the production line affect production and inventory (holding and setup) costs not only internally by screened items with proportion d3lIF (¼(1  d1– d2)lIF) from the production of those scrapped, but also externally by returned defectives with proportion (g  d1  d2)lEF due to the production for customers’ exchange request, reuse of reworked items and salvage. Thus, the quality-related production cost, u(D0  D), is divided into QCIF component of ud3lIFD0 and QCEF component of u(g  d1  d2)lEFD0 where D0 ¼D0 (p, a, b) in (1). And the quality-related inventory cost, (2huK)1/2[(D0 (1 D0 /M))1/2  (D(1  D/M))1/2], is obtained by subtracting inventory costs in TPU(p, a, b) in (18) from inventory costs only related to customers’ original demand request in TPUT in (19). Then, as in the production cost, the quality-related inventory cost in the production line is divided into QCIF component with proportion d3lIF and QCEF component with proportion (g  d1  d2)lEF. Second, the terms only related to customers’ defective return, such as return cost rlEFD0 , penalty cost l(1 g)lEFD0 and revenue loss from refund p(1  g)lEFD0 , are obviously included only in QCEF. Third, the total revenue difference between TPUT in (19) and TPU(p, a, b) in (18), excluding the revenue loss from refund p(1 g)lEFD0 (already included in QCEF), becomes (p v)d2lD0 , indicating the revenue loss from salvage instead of regular sales. This is divided into QCIF with proportion lIF and QCEF with proportion lEF, since salvage items consist of screened and returned defective portions. And fourth, similarly, rework cost wd1lD0 and scrap cost gd3lD0 are divided into QCIF with proportion lIF and QCEF with proportion lEF. However, for inventory (setup and holding) costs of items to rework and reworked in the rework line, we need to consider different holding cost rates based on the type of items. While 2hr(lEF/l) is obviously only for returned defectives, h(2uþ2iþw) consists of internally screened items and returned defectives with their respective proportions lIF and lEF among total failure proportion l. Now, putting all the components together, we get QCIF and QCEF: Q C IF ¼ ðwd1 þðpvÞd2 þ ðu þ gÞd3 ÞlIF D0 "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi  ffi#  ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d3 lIF 2huK D0 D 0  D 1 þ D 1 M d3 lIF þ ðgd1 d2 ÞlEF M sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ð2u þ2i þ wÞlIF 2hK r d1 lD0 d1 lD0 1 þ l 2u þ2iþ 2rðlEF =lÞ þw W ð22Þ and Q DEF ¼ ðuðgd1 d2 Þ þ wd1 þ ðpvÞd2 þg d3 þ r þðp þ lÞð1gÞÞlEF D0 pffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi#  ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðgd1 d2 ÞlEF 2huK D0 D 0  D 1 þ D 1 M d3 lIF þ ðgd1 d2 ÞlEF M sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ð2u þ2i þ 2r þ wÞlEF 2hK r d1 lD0 d1 lD0 1 : þ l 2u þ 2iþ 2rðlEF =lÞ þw W ð23Þ

ð19Þ where TPUT ¼pD uD KD/QT  hu/2(1 D/M)QT with QT ¼(2KMD/ (hu(M–D)))1/2. In detail, prevention cost (QCP) consists of the firm’s interest and depreciation costs from investment in production and inspection process reliability, and appraisal cost (QCA) comprises item inspection cost. b

Q C P ¼ Y P þY I ¼ ap

d

þca

þeb

f

and

ð20Þ

5. Optimal solution and quality cost analysis for different investment decisions 5.1. Optimal solutions for various investment decisions This section illustrates the global optimal solution of the extended nonlinear maximization model in (17) for joint production and inspection process reliability investment using the

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previous example in Section 3.4 additionally with YP ¼10,000p  1, Ya ¼1000a  1 and Yb ¼3000b  1. We assume that the firm currently yields defective proportion p0 ¼ 20% and Type I and Type II inspection error proportions a0 and b0 ¼20%. In practice, it is not easy for a firm to reach the joint optimum of all the quality attributes at once since it has always a diversified portfolio of projects or is not capable of investing a huge capital at one time. Especially when the firm has a myopic view, quality improvement activities are often conducted partially or sequentially. Therefore, we need to consider practical situations where the firm enhances its quality level under different investment decisions. We demonstrate 14 cases in order to investigate how optimal solutions and quality cost structure can be different. The cases include:





 Case (a): one basic case without investment (i.e., (p, a, b)¼(p0, a0, b0)).

 Cases (b1–g1): six partial investment cases, investing in only one or two failure proportion(s) among p, a and b.



 Cases (b2–g2): six sequential investment cases, investing in the 

other failure proportion(s) after partial investment in Cases (b1–g1). Case (h): one joint investment case, yielding the optimal p, a and b at once.

Tables 2 and 3 summarize the optimal solution results of the above 14 cases under the different investment structure, using LINGO 13. We obtain several important managerial implications below from the result summarized in Tables 2 and 3.

 As shown, all the investment decisions (i.e., partial, sequential

 



and joint) decrease p, a and/or b (or improve (1  p), (1 a) and/or (1 b)) from those in the basic state (Case (a)). Moreover, they differently affect the control variables in the production and rework lines, including lot size Q and rework frequency n, besides the corresponding YP, Ya and/or Yb. The net depletion rate D ( ¼D/(1  d3lIF (g  d1  d2)lEF)) in (1) varies by the decision of p, a and b, different from most previous studies with fixed demand or item depletion rate. The optimal failure proportions p, a and b from partial and sequential investment deviate from the result of joint investment in Case (h) which yields the global optimum. Therefore, the partial and sequential investment cases (Cases (b1–g1) in Table 2 and Cases (b2–g2) in Table 3) incur too much or less YP, Ya and/or Yb due to their myopic decisions. Especially when first investing in Type II inspection error proportion b without investment in the production process (Cases (d1) and (g1) in Table 2, and subsequently Cases (d2) and (g2) in Table 3), b shows much deviation from the global optimum in Case (h). This is mainly due to the control of external failure proportion lEF ( ¼ bp) even while not enhancing production process reliability (1 p).

The difference in profit (TPU) solutions solely comes from the difference in total quality cost TQC given the same non-qualityrelated TPUT as shown in Tables 2 and 3. As for the quality costs of the different investment cases, we obtain the below implications:

Overall, all partial and sequential investment cases show inferior TPU and TQC results to the joint investment case. This is due to the deviation in each quality cost, QCP, QCA, QCIF or QCEF. Thus, in practice where quality improvement activities are often conducted partially or sequentially, it is imperative for managers to understand and prudently examine how these affect each quality cost component and result in deviations from the optimal balance in TQC. It cannot be overemphasized to consider quality management and control activities jointly and integrate related investment decisions. The next subsection investigates the equilibrium behaviors of failure proportions and quality cost structure based on p, a and b in joint investment.

5.2. Equilibrium behaviors and quality cost analysis In this section, we investigate how a change in each failure proportion affects the decision of the other failure proportions and quality cost structure. We illustrate the equilibrium behaviors of defective production and inspection error proportions p, a and b, and internal and external failure proportions lIF and lEF in Fig. 3 with respect to p, a or b in the joint investment case. Fig. 4 also depicts the changes in total quality cost TQC and each quality cost QCP, QCA, QCIF and QCEF, given the same example in the previous section. From the results in Fig. 3, we excavate some meaningful managerial implications about the relationship among failure proportions below:

 Interestingly, defective proportion p and Type I inspection

 All partial investment cases (Cases (b1–g1) in Table 2) improve



TQC (or TPU) from the basic state (Case (a)). Specifically, while partial investment in p, a and/or b increases prevention cost QCP, it decreases all the other quality costs, i.e., appraisal, internal failure and external failure costs (QCA, QCIF and QCEF) in all cases except Case (d1). When investing partially only in b as in Case (d1), it decreases only QCEF while increasing QCA and QCIF as well as QCP. This is

because Type II inspection error proportion b increases internal failure proportion lIF ( ¼ a(1  p)þ(1  b)p) when there is no improvement in defective proportion p. Note that total failure proportion l (¼ lIF þ lEF ¼ a(1  p)þ p) stays the same as in Case (a) while b increases lIF and decreases lEF by the same magnitude. Most partial investment cases show much deviation in TQC from the joint investment case. Among all, Case (e1) provides relatively better performance due to the significant improvement in both lIF and lEF through the smallest deviation in p and a from joint investment. As shown in Table 3, all sequential investment cases (Cases (b2–g2)) reduce TQC from their prior partial investment cases (Cases (b1–g1) in Table 2) despite higher prevention cost QCP. Also, most cases do not show much deviation from the joint investment case (Case (h)). Yet we need to note that each quality cost component deviates from the best-balanced quality cost structure in Case (h). Cases (d2) and (g2) yield relatively inferior results among the sequential investment cases due to the relatively lower b, resulting from their prior partial investment in Cases (d1) and (g1). This, in turn, results in deviations in quality costs from the best-balanced joint investment case, incurring higher QCP, QCA and QCIF and lower QCEF.



error proportion a change in the same direction as shown in Fig. 3(a1) and (b1). Conversely, Type II inspection error proportion b behaves in the opposite direction of p or a, as p or a affects the optimum of b inversely, or vice versa, as shown in Fig. 3(a2), (b2), (c1) and (c2). These changes, in turn, jointly affect the equilibrium behaviors of lIF and lEF. As for the effect of the change in p, a or b on lIF and lEF in the joint investment case, when p or a increases, it increases both lIF and lEF as shown in Figs. 3(a3), (a4), (b3) and (b4). On the other hand, when b increases, it decreases lIF and increases lEF as shown in Fig. 3(c3) and (c4).

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929

Table 2 Optimal solutions of basic, partial and joint investment in production and/or inspection processes. (a) Basic (p0,a0,b0 ¼20%)

Partial investment in

(h) Joint investment

(b1) p (a0, b0 ¼20%)

(c1) a (p0, b0 ¼20%)

(d1) b (p0, a0 ¼20%)

(e1) p and a (b0 ¼ 20%)

(f1) p and b (a0 ¼ 20%)

(g1) a and b (p0 ¼20%)

b l lIF lEF

20.0000% 20.0000% 20.0000% 36.0000% 32.0000% 4.0000%

7.7050 20.0000 20.0000 26.1640 24.6230 1.5410

20.0000 2.7710 20.0000 22.2168 18.2168 4.0000

20.0000 20.0000 9.5841 36.0000 34.0832 1.9168

7.3761 2.6618 20.0000 9.8415 8.3663 1.4752

7.9005 20.0000 14.9207 26.3204 25.1416 1.1788

20.0000 2.7622 9.3621 22.2098 20.3373 1.8724

7.5551 2.6619 14.9322 10.0159 8.8878 1.1282

Depreciation and interest cost YP Ya Yb

$50,000/yr $5000/yr $15,000/yr

129,786 5000 15,000

50,000 36,088 15,000

50,000 5000 31,302

135,574 37,569 15,000

126,575 5000 20,106

50,000 36,203 32,044

132,360 37,567 20,091

Net depletion rate and other rates D0 D F E R V

21,834 units/yr 20,000 units/yr 6987 units/yr 349 units/yr 3930 units/yr 1572 units/yr

21,488 20,000 5291 132 2811 1124

20,891 20,000 3806 334 2321 928

22,136 20,000 7545 170 3985 1594

20,422 20,000 1709 121 1005 402

21,549 20,000 5418 102 2836 1134

21,173 20,000 4306 159 2351 940

20,477 20,000 1820 92 1025 410

Production line Q T

1265.92 units 21.16 day

1244.06 21.13

1207.34 21.09

1285.38 21.19

1179.34 21.08

1247.89 21.14

1224.52 21.11

1182.56 21.08

Rework line n Qr Tr

2.88/cycle 79.07 units 7.34 day

2.48 65.55 8.51

2.28 58.88 9.26

2.90 79.80 7.31

1.53 37.88 13.76

2.49 65.89 8.48

2.29 59.42 9.22

1.55 38.31 13.64

Profit TPU TPUT

$1,166,210/yr $1,982,679/yr

1,296,771 1,982,679

1,369,399 1,982,679

1,183,460 1,982,679

1,520,668 1,982,679

1,298,409 1,982,679

1,388,278 1,982,679

1,522,301 1,982,679

Quality costs TQC QCP QCA QCIF QCEF

$816,469/yr $70,000/yr $109,170/yr $492,699/yr $144,601/yr

685,908 149,786 107,440 373,814 54,869

613,281 101,088 104,455 269,202 138,536

799,219 86,302 110,681 531,986 70,251

462,012 188,143 102,111 121,666 50,092

684,270 151,681 107,746 382,752 42,091

594,402 118,247 105,864 304,567 65,723

460,379 190,018 102,383 129,571 38,406

–% –% –% –% –% –% $–/yr $–/yr $–/yr $–/yr $–/yr $–/yr

 12.2950 0.0000 0.0000  9.8360  7.3770  2.4590 130,561  130,561 79,786  1731  118,885  89,732

0.0000  17.2290 0.0000  13.7832  13.7832 0.0000 203,189  203,189 31,088  4715  223,497  6065

0.0000 0.0000  10.4159 0.0000 2.0832  2.0832 17,250  17,250 16,302 1510 39,287  74,349

 12.6239  17.3382 0.0000  26.1585  23.6337  2.5248 354,458  354,458 118,143  7059  371,032  94,509

 12.0995 0.0000  5.0793  9.6796  6.8584  2.8212 132,199  132,199 81,681  1425  109,946  102,509

0.0000  17.2378  10.6379  13.7902  11.6627  2.1276 222,068  222,068 48,247  3306  188,131  78,877

 12.4449  17.3381  5.0678  25.9841  23.1122  2.8718 356,090  356,090 120,018  6787  363,127  106,194

12.4449% 17.3381% 5.0678% 25.9841% 23.1122% 2.8718%  $356,090/yr $356,090/yr  $120,018/yr $6787/yr $363,127/yr $106,194/yr

0.1498 17.3381 5.0678 16.1480 15.7352 0.4128  225,530 225,530  40,232 5,056 244,243 16,462

12.4449 0.1091 5.0678 12.2009 9.3290 2.8718  152,902 152,902  88,930 2,072 139,631 100,129

12.4449 17.3381  5.3481 25.9841 25.1954 0.7887  338,840 338,840  103,716 8,297 402,415 31,845

 0.1791  0.0001 5.0678  0.1744  0.5215 0.3471  1,633 1,633  1,876  272  7,905 11,685

0.3453 17.3381  0.0115 16.3044 16.2538 0.0507  223,891 223,891  38,337 5,362 253,181 3,685

12.4449 0.1003  5.5701 12.1938 11.4496 0.7443  134,023 134,023  71,771 3,481 174,996 27,317

– – – – – – – – – – – –

Failure proportion

p a

Difference from basic state

p a b l lIF lEF TPU TQC QCP QCA QCIF QCEF Deviation from joint investment

p a b l lIF lEF TPU TQC QCP QCA QCIF QCEF

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S.H. Yoo et al. / Int. J. Production Economics 140 (2012) 922–933

Table 3 Optimal solutions of sequential and joint investment in production and/or inspection processes. Sequential investment in (after partial investment in Table 2)

(h) Joint investment

(b2) p, then a and b

(c2) a, then p and b

(d2) b, then p and a

(e2) p and a, then b

(f2) p and b, then a

(g2) a and b, then p

b l lIF lEF

7.7050% 2.6630% 14.7897% 10.1628% 9.0232% 1.1395%

7.5570 2.7710 14.9320 10.1186 8.9902 1.1284

7.7603 2.6621 9.5841 10.2157 9.4720 0.7438

7.3761 2.6618 15.1083 9.8415 8.7271 1.1144

7.9005 2.6645 14.9207 10.3545 9.1757 1.1788

7.7710 2.7622 9.3621 10.3185 9.5910 0.7275

7.5551 2.6619 14.9322 10.0159 8.8878 1.1282

Depreciation and interest cost YP Ya Yb

$129,786/yr $37,552/yr $20,284/yr

132,327 36,088 20,091

128,862 37,565 31,302

135,574 37,569 19,857

126,575 37,530 20,106

128,684 36,203 32,044

132,360 37,567 20,091

Net depletion rate and other rates D0 D F E R V

20,484 units/yr 20,000 units/yr 1848 units/yr 93 units/yr 1041 units/yr 416 units/yr

20,483 20,000 1841 92 1036 415

20,538 20,000 1945 61 1049 420

20,467 20,000 1786 91 1007 403

20,492 20,000 1880 97 1061 424

20,546 20,000 1971 60 1060 424

20,477 20,000 1820 92 1025 410

Production line Q T

1183.02 units 21.08 day

1182.94 21.08

1186.18 21.08

1182.01 21.08

1183.44 21.08

1186.69 21.08

1182.56 21.08

Rework line n Qr Tr

1.56/cycle 38.61 units 13.54 day

1.55 38.52 13.57

1.56 38.80 13.50

1.53 37.96 13.76

1.57 38.99 13.41

1.57 39.01 13.43

1.55 38.31 13.64

Profit TPU TPUT

$1,522,251/yr $1,982,679/yr

1,522,239 1,982,679

1,518,397 1,982,679

1,522,226 1,982,679

1,522,033 1,982,679

1,517,912 1,982,679

1,522,301 1,982,679

Quality costs TQC QCP QCA QCIF QCEF

$460,429/yr $187,623/yr $102,422/yr $131,577/yr $38,807/yr

460,440 188,506 102,416 131,092 38,426

464,282 197,728 102,689 138,471 25,394

460,453 192,999 102,337 127,193 37,924

460,647 184,211 102,458 133,823 40,155

464,768 196,931 102,732 140,256 24,849

460,379 190,018 102,383 129,571 38,406

0.0000%  17.3370%  5.2103%  16.0012%  15.5997%  0.4015% $225,480/yr  $225,480/yr $37,836/yr  $5017/yr  $242,236/yr  $16,062/yr

 12.4430 0.0000  5.0680  12.0982  9.2266  2.8716 152,841  152,841 87,418  2040  138,110  100,110

 12.2397  17.3379 0.0000  25.7843  24.6112  1.1731 334,937  334,937 111,427  7992  393,514  44,858

0.0000 0.0000  4.8917 0.0000 0.3608  0.3608 1,558  1,558 4857 226 5527  12,168

0.0000  17.3355 0.0000  15.9659  15.9659 0.0000 223,624  223,624 32,530  5288  248,930  1937

 12.2290 0.0000 0.0000  11.8912  10.7463  1.1449 129,634  129,634 78,684  3133  164,311  40,874

– – – – – – – – – – – –

0.1498% 0.0011%  0.1425% 0.1468% 0.1354% 0.0114%  $50/yr $50/yr  $2396/yr $39/yr $2006/yr $400/yr

0.0019 0.1091  0.0002 0.1027 0.1024 0.0003  61 61  1512 32 1521 19

0.2051 0.0002  5.3481 0.1998 0.5842  0.3844  3903 3903 7710 305 8900  13,013

 0.1791  0.0001 0.1761  0.1744  0.1607  0.0138  75 75 2981  46  2378  483

0.3453 0.0026  0.0115 0.3385 0.2879 0.0507  268 268  5807 75 4252 1748

0.2158 0.1003  5.5701 0.3026 0.7032  0.4006  4389 4389 6913 348 10,685  13,558

– – – – – – – – – – – –

Failure proportion

p a

Difference from partial investment in Table 2

p a b l lIF lEF TPU TQC QCP QCA QCIF QCEF Deviation from joint investment

p a b l lIF lEF TPU TQC QCP QCA QCIF QCEF

Overall, it is very important to understand the dynamics among production and inspection process reliability and internal and external failure proportions before initiating quality

improvement activities. This is because these failure proportions directly affect the structure of the quality costs. Next, we analyze the equilibrium behaviors of the quality cost structure in joint

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931

Fig. 3. Equilibrium behaviors of defective proportion p, Type I and Type II inspection error proportions a and b, and internal and external failure proportions lIF and lEF with respect to p, a and b in the joint investment case.

investment with respect to production and inspection process reliability, illustrated in Fig. 4:

when either p or a increases. On the other hand, the optimal TQC is determined by a different tradeoff among decreasing QCP, QCA and QCIF and increasing QCEF, when b increases.

 Total quality cost TQC behaves in a convex manner with an









increase in not only p but also a and b. Further, the magnitude of the change in TQC is much greater with p than a or b (Fig. 4(a1), (b1) and (c1)). It is because the amount of quality failure is primarily determined by defective proportion p rather than inspection error proportion a or b. As shown in Fig. 4(a2), (b2) and (c2), prevention cost QCP decreases as p, a or b increases. The magnitude of the change in QCP is the smallest with b since b enforces the other two proportions p and a to change in the opposite direction as shown in Fig. 3(c1) and (c2), reducing the change magnitude in QCP. Appraisal cost QCA increases as either p or a increases. It is mainly because internal failure proportion lIF increases as p or a increases as shown in Fig. 3(a3) and (b3). This, in turn, increases net depletion rate D0 which corresponds to the inspection rate. On the other hand, QCA decreases as b increases since an increase in b renders lIF to decrease as shown in Fig. 3(c3). The behaviors of internal failure and external failure costs QCIF and QCEF directly correspond to the change directions of internal and external failure proportions lIF and lEF. Both QCIF and QCEF increase as p or a increases since p or a directly increases lIF and lEF as depicted in Fig. 3(a3–4) and (b3–4). When b increases, however, QCIF decreases but QCEF increases since b enforces lIF to decrease and lEF to increase as shown in Fig. 3(c3–4). Finally, total quality cost TQC has its minimum at the point in Case (h) shown in Table 2 or 3 based on the tradeoff relationship among decreasing QCP and increasing QCA, QCIF and QCEF,

It should be noted that it is not possible to analytically show the change in the behavior of the total quality cost (or total profit) and their components based on p, a and b due to the mathematical complexity of the model. Nonetheless, the present study provides profound contribution to theory and practice. The framework of analytically decomposing the profit into non-qualityrelated components, total quality cost and its components, first presented to our best knowledge, enables managers to quantify and measure the size of a quality problem, thereby helping them understand and analyze the effect of quality improvement activities, justify control and improvement efforts, guide their development and track progress in quality management activities.

6. Conclusions In this study, we investigated an imperfect-quality inventory problem in practice in which imperfect production and inspection processes cause a firm to take actions of internal prevention and external reverse logistics of customer defective returns. Most previous studies, however, partially examined cost-minimizing aspects of internal defective production, inspection and/or prevention, or external reverse logistics of used items, without considering imperfect inspection, defective delivery to customers, returns for exchange and refund, and reselling to the second market, which affect the firm’s profit. Moreover, few imperfectquality inventory studies performed a quality cost analysis despite its significance of helping managers understand the effect

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to integrate and jointly consider investment decisions in production and inspection process reliability improvement for quality activities. Fourth, understanding the effect of each failure proportion on firms’ total profit or total quality cost is very crucial for practicing managers. The effects of defective proportion and Type I inspection error on the internal and external failure proportions and the quality cost structure are similar, but Type II inspection error behaves inversely and produces different results. Fifth, when investing in process reliability partially or sequentially as in practice, a firm first needs to consider production process reliability and/or Type I inspection process reliability. Enhancing only Type II inspection process reliability cannot guarantee the balanced quality cost structure. Finally, as the difference in the total profit from different investment decisions comes from that in the total quality cost, it is crucial for managers to measure and quantify the size of quality problems, based on the quality cost analysis presented, to accurately justify control and improvement efforts, guide their development and track progress in quality management activities by clearly understanding the tradeoff relationships and the best balance among quality costs. The significance and contribution of this study may lie in developing a more realistic and comprehensive model by integrating and extending the past research, including internal and external aspects of defective production and two types of inspection errors, and providing richer managerial insights into practices through the decomposition and analysis of the total quality cost. For future research, it is desired to extend further the present quality investment model with lot sizing by investigating a deteriorating production process and by examining more complicated multistage production or supply chain systems along with lead time and capacity issues besides quality to further close the gap between practice and academia. Fig. 4. Equilibrium behaviors of TQC and its components (QCP, QCA, QCIF and QCEF) based on defective proportion p, Type I and Type II inspection error proportions a and b in the joint investment case (–, QCP; ~, QCA;  , QCIF; J, QCEF for (a2), (b2) and (c2)).

of quality improvement activities and determine the optimal investment level. Therefore, we incorporated all the above issues into our inventory model, in which the profit is decomposed into non-quality-related components and total quality cost components (i.e., prevention, appraisal, internal failure and external failure). The objective of the model was to determine production lot size, rework frequency, defective proportion, and Type I and Type II inspection error proportions dependent on the production and inspection process reliability investment, which maximize the total profit and minimize the total quality cost. We examined and compared the optimal solutions among the basic model and the models with various investment decisions (partial, sequential and joint) in production and/or inspection processes. Based on the analytical models along with numerical analyses, we provided important managerial implications to practices. First, the net inventory depletion rate to satisfy customer demand and exchange requests for defective items varies by not only demand rate, but also by defective proportion, Type I and II inspection error proportions, customers’ return choice for exchange, and multiple defective disposal options of rework and salvage. Second, firms need to incorporate not only internal but also external effects of defective items in their production and inspection process reliability investment decisions, by considering reverse logistics aspects of defective returns, subsequent exchange and refund, and various defective disposal options, as they are affected by defective proportion, and Type I and II inspection error proportions. Third, as joint investment yields superior profit performance to partial and sequential investment, it is important

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