SCIENCE Discussion(NO PLAGIARISM, A++ WORK, QUALITY, ON TIME)

On an IC wire bonding machine production–inventory problem with time value of money

Dah-Chuan Gonga*, Jia-Lun Kangb, Gary C. Linc and T. C. Houb

aDepartment of Industrial and Business Management, Chang Gung University, Guishan, Taiwan, ROC; bDepartment of Industrial and Systems Engineering, Chung Yuan Christian University, Chungli, Taiwan, ROC; cDepartment of Industrial & Manufacturing

Engineering & Technology, Bradley University, Peoria, IL, USA

(Received 1 June 2015; accepted 7 July 2016)

The semiconductor industry in Taiwan has received excellent performance ratings in the past. SinoPac’s statistics in January, 2016 found that two of the top four global packaging and testing companies are from Taiwan, including ASE Group and SPIL. In the IC packaging process, the wire bonding machine requires careful attention. It is also costly at approximately 50% of the equipment investment. Thus, this paper focuses on wire bonding machines’ production prob- lems. According to the on-site interviews, three key control modules are identified, which may change a machine to an out-of-control state and are responsible for 90% of the defective products. A mathematical model is developed to deter- mine the optimal production time of an imperfect production process. Taking the time value of money, the objective is to minimise the total of the set-up cost, inventory cost and the defect cost. Besides applying MacLaurin Series, a math property and an effective solution range are derived to help obtain near-optimal solutions. For the justification of solution quality, the bisection method working on practical data is used. Finally, managerial insights are explored from observed outputs through the changes of various parameters. Moreover, these explorations are confirmed by experts in this field.

Keywords: imperfect production system; production time; time value; IC packaging industry; wire bonding machine

1. Introduction

With the advancement of science and technology, as well as the demand for information, people make more frequent use of electronic products. The main technology for processing and executing those products relies on the integrated cir- cuit (IC) module developed by the semiconductor industry. The semiconductor industry can be divided into front-end and back-end production companies. The front-end contains IC design and wafer manufacturing, and this stage determi- nes the performance of the manufactured IC module. The back-end contains IC packaging and IC testing, and this stage aims to use packaging technology to ensure the easy and safe storage of the IC modules.

According to information released by ASE Group, an international packaging factory, the wire bonding machine in the IC packaging process not only creates a bottleneck in the overall process, but also accounts for the highest invest- ment portion among all other equipment used in the process (about 50%). The wire bonding process is set-up to connect the signal point on the die to the inner pin of the lead frame via metal wire (e.g. gold, aluminium and copper) so that the IC signal can be sent through this connection. Therefore, the quality of wire bonding is the key to the normal opera- tions of the IC. Wire bonding is achieved by extending the wire through the capillary tip, and uses a discharging rod to release sparks, which melts the tip of the wire into a sphere, as shown in Figure 1(a). Then, the ball is pressed onto the chip pad, making the first bond. An ultrasonic vibration generates a eutectic bond between the ball and the pad, as shown in Figure 1(b). Afterwards, the wire is pulled up and then pressed down to the pin inside the lead frame, making the second bond. In the same way, the wire and pin inside the lead frame are connected. Finally, the wire is cut off and pulled up, and the overall wire bonding process is complete as shown in Figure 1(c).

There are multiple causes for defects from a wire bonding machine, including human error, equipment, materials and method. In this paper, we analyse the four aspects and identify the factors that contribute to nonconforming parts. We then divide the factors into two types, namely human control and non-human control.

There are many human control factors including whether the materials of wire and pad/pin inside the lead frame are likely to result in eutectic bonding, the operator’s proficiency, the number of machines to be simultaneously under the control of one operator, the wire bonding mode, the wire cutting mode after eutectic bonds are formed and the machine settings before mass production. Although all of these factors may cause defects, rigorous personnel training and

*Corresponding author. Email: [email protected]

© 2016 Informa UK Limited, trading as Taylor & Francis Group

International Journal of Production Research, 2017 Vol. 55, No. 9, 2431–2453, http://dx.doi.org/10.1080/00207543.2016.1213914

multiple tests by the R&D personnel can effectively improve the manufacturing procedures. Hence, those causes are classified as human control factors. Non-human control factors mostly occur on the wire bonding machine, which become the main focus in the research scope of this paper.

Figure 2 shows a cause and effect diagram based on the discussion with ASE experts and analysis on the past equip- ment failure records. The selected key factors account for 90% of the machine fault rate, including the thermosonic

Figure 1. Wire bonding process (Source: ASM Taiwan, www.asm.com).

What reason make the machine to

produce defect item?

The thermosonic bonding power control module

Power

Time

Low/High

Breakdown Long

Short

Loader/Indexer/Unloader Transportation System

Unloader/Loader

Indexer

Good

Breakdown Good

Breakdown

The lead frame temperature

control module

Temperature

Time

Long

Low/High

Short

Breakdown

The capillary tip force

control module

Force

Time

Long

Small/Large

Short

Breakdown

Figure 2. Cause and effect diagram of a wire bonding machine.

2432 D.-C. Gong et al.

bonding power control module (PCM), the capillary tip force control module (FCM), the lead frame temperature control module (TCM) and the Loader/Indexer/Unloader transportation system (TS). In the case of a TS fault, the entire process needs to be immediately stopped for repair, and the production process is not resumed until the TS is repaired. In terms of machinery management, it can be considered as a downtime maintenance analysis model.

The other three control modules share a similar property that allows shifts within a predetermined range. They are denoted as Key Control Modules (KCMs). An imperfect production process dictated by these three KCMs are more dif- ficult to deal with than with one or two KCMs. Hence, this paper studies a production–inventory problem for a wire bonding machine with the three KCMs of Power, Force and Temperature. In addition, according to the literature review, many researchers also defined the parameters of the above three modules as key parameters in their articles of dis- cussing a wire bonding machine, such as Prasad (2004), Xu et al. (2010a, 2010b, 2010c), Nan et al. (2011), Zulkifli et al. (2012, 2013), Wang and Han (2013a, 2013b), Yong et al. (2013), and Lim et al. (2014).

The three KCMs described above have respective module controllers and sensors. The sensor instantly captures environmental data, and feeds the data back to the module controller in order to monitor whether they exceed the preset range of parameters. The optimum range of parameters of the machine is provided by the equipment manufacturer. The allowable values of the sensor are determined by the packaging factory. Those allowable ranges are usually wide. FCM and PCM production conditions are different from the continuous operation mode of TCM. The two modules are actu- ated when the wire and pad/pin inside the lead frame are connected, and there is a demand for a move between solder joints. Compared to the processing time, the moving time between two solder joints is very short. Hence, if the move delay time is omitted, FCM and PCM can be regarded as continuous actuation.

Taking the TCM of the ASM AB559 wire bonding machine as an example, the optimum temperature range of the equipment is 140~160 °C, and the preset range for the sensor is 135~165 °C. When the lead frame preheating tempera- ture is lower than 140 °C, the sensor sends a signal to inform the module controller to actuate the heating unit. When the sensor detects the lead frame temperature is 160 °C, it immediately informs the module controller to switch off the heating unit. Under normal conditions, the module controller can accurately maintain the temperature within the opti- mum range (140~160 °C). However, when the module controller has a shift due to fatigue after a period of operation, namely the control sensitivity of the module controller declines, the lead frame temperature changes to 135~140 °C or 160~165 °C. This buffer stage is not the optimum production temperature range, and there is a certain probability for generating defective products. However, as the shutdown standard is not met, the machine continues to run.

As the semiconductor industry progresses to become a worldwide supply chain where the parties involved may be from different countries, the time value of money that contains inflation rates needs to be considered in the study of a production–inventory model. Therefore, in addition to the proportion of non-confirming products due to the machine’s shift, this paper also considers the influence of the time value of money that contains the inflation rate on the total cost.

Past studies on the wire bonding machine mostly focus on the technical aspect. Although there are few discussions on the production–inventory problem, the emphases are on the supply chain (Lendermann et al. 2003), dynamic scheduling (Li et al. 2000), the stochastic economic lot scheduling problem (Löhndorf and Minner 2013) or the internal production flow of the entire plant (Lee, Chen, and Chien 2014). Those studies often used simulation tools to construct models, and applied different business strategies and cooperative schemes in simulated situations. They also compared the differences between various strategies or schemes, in order to determine the best decision combination.

There are numerous studies concerning the imperfect production problem; however, no discussion has been made regarding the wire bonding machine. The economic order quantity (EOQ) and the economic production quantity (EPQ) models are the most representative production inventory models in the literature. However, the traditional EPQ model assumes the production process is perfect during a production run and no defective products will be produced, which does not reflect reality. Porteus (1986) proposed an EPQ model with an imperfect process, and suggested that the pro- duction of each product has a certain probability of shift over the control limit, thus there is a significant relationship between production yield and production lot size. Rosenblatt and Lee (1986), and Porteus (1986) discussed the deterio- ration of the process of a single stage production system over time. However, the difference between the two is that Por- teus assumed the process would be out of control after the shift, producing 100% defects. Rosenblatt and Lee assumed that the time-to-shift is a random variable following an exponential distribution, and a proportion of defects would be produced after a shift. Some studies changed the probability distribution of the shift time or applied the fuzzy theory in the generation of the machine shift time point (Kim, Hong, and Chang 2001; Chiu 2003; Halim, Giri, and Chaudhuri 2009); however, the focus of discussion was on production time determination.

In order to avoid delivering non-conforming products to customers, previous studies have adopted two methods, periodical inspection of the products and periodical inspection of the machines. When a defect is found in either method, coping strategies will be carried out. If the machine is found to have defects, it will be shut down for repair,

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which will delay the product delivery schedule. Thus, there should be a balance among machine maintenance, delivery date and quality (Lee and Rosenblatt 1987; Groenevelt, Pintelon, and Seidmann 1992; Chung 1997).

Regarding product inspection, previous studies have discussed the requirement for rework, the rework costs and the quality after rework in case of defects (Schwaller 1988; Makis and Fung 1998; Lin and Lin 2007). If the non-conform- ing products are discarded without rework, an extra cost will incur. Therefore, some researchers have divided the non- conforming products into sub-quality products and defective products. The sub-quality products are discounted and sold in batches in order to recover partial production costs (Salameh and Jaber 2000; Rezaei and Davoodi 2008). Further- more, the inspection can be error-prone due to different sources. Detailed discussion on this issue including three prob- lem-solving procedures are provided in (Karimi-Nasab and Sabri-Laghaie 2014).

Besides modifying the model, some studies have developed new solutions. Early studies often used heuristic algo- rithms for computation. Using MacLaurin Series to approximate any exponential functions appearing in an equation was another approach often adopted to derive an quadratic equation so as to determine the value of a decision variable. Recent studies have used Cauchy inequality to obtain the optimal solution. (Teng 2009; Ouyang and Chang 2013; Wee et al. 2013).

The topics of incorporating inflation and discount have attracted wide attention since the 1980s, due to the globalisa- tion of transactions. First, studies have discussed the influences of scarcity, linear demand, deteriorating products, and value-added products in the ordering model and under the impact of time value of money rate (Dohi, Kaio, and Osaki 1992; Hariga and Ben-Daya 1996; Moon, Giri, and Ko 2005; Mishra et al. 2011). Other studies have discussed the correlation between the interest rate and production time for imperfect processes, such as Sana and Chaudhuri (2003), Sarkar and Moon (2011), Gong et al. (2012), and Sarkar (2012).

Recent studies have been more concerned about practicality. Taking the production inventory research as an exam- ple, researchers have explored problems within the industry, and used the inventory models to construct mathematical models to solve problems. The actual data are used in the model to better reflect the situation of the production line. Managerial insights are also obtained from sensitivity analysis. These results can provide references for enterprise decision-makers. The range of industry categories covered extends from the lumber industry to the high-tech industry (Björk 2012; Dhouib, Gharbi, and Ben Aziza 2012; Pan et al. 2012; Tsao, Chen, and Zhang 2013; Jiang, Chen, and Zhou 2015).

Based on the above discussion, one can see that a production–inventory model is able to express the actual state of a production line such as a wire bonding machine. The remainder of this paper is organised as follows: In Section 2, we describe the causes of assumed conditions. Then a model is developed in Section 3. While in Section 4, we derive mathematically an appropriate computing range. A solution procedure of the proposed model is presented in Section 5. Section 6 gives the application of actual data. In Section 7, the solution quality is assessed and justified. Section 8 shows a sensitivity analysis. Section 9 conducts managerial insights based on obtained analytical results. Finally, some concluding remarks are provided in Section 10.

2. Model assumptions and notation

2.1 Assumptions

This paper considers a production–inventory model where a wire bonding machine parameters may shift during the semiconductor packaging process. The assumptions are moderately proposed in order to focus on the problem, and the rationality of the assumptions is discussed as follows:

A trial is conducted before each production run, and the non-conforming products are immediately detected, thus, the status of the production machine after trial is determined as stable and normal. In other words, the machine returns to the normal state between two productions.

The wire bonding machine is a high-precision machine equipment; thus, the operating reliability of the machine is high, and the machine is regularly maintained. Therefore, in the actual production line, the probability of having machine failures is slim. Thus, machine breakdowns are neglected in this study. Only the production scenario of having parameter shifts on the three KCMs of the bonding machine is considered.

The automobile air bag starting IC is the selected product. As this product’s order source is usually stable and of regular production, the IC design is unlikely to be changed. In addition, the product value is high. Therefore, it is assumed that the production rate is greater than the demand rate and no shortage will happen.

A wire bonding machine can apply a mixed production policy to produce different products of various batches. However, from the view of a single product, the machine can be regarded as intermittently producing this product. Thus, the next production will not be implemented until the product inventory is exhausted.

2434 D.-C. Gong et al.

The order source of a precision IC component is stable and continuous, not seasonal, and as it is a product category of preferential scheduling, the number of production cycles can be regarded as infinite.

The probability distribution of the time-to-shift of a KCM is assumed to be exponentially distributed. The proportion of defects is not fixed; however, in the long run, the probability of defects can be reasonably assumed to be a fixed value.

In practice, when a wire bonding machine produces non-conforming products, those products are gathered first, and then sent to the recovery station after the process is finished, in order to recover the gold wire. However, as this recov- ery operation is independent of the original production process, the defect handling method can be assumed to be directly discarded with a certain cost.

2.2 Notation

The following notation will be used throughout the remainder of this paper:

3. Model development

Discount and inflation factors are included in the mathematical model developed in this paper in order to obtain the minimum total production inventory cost. This cost is composed of the inventory holding cost, the set-up cost, and the defect cost, multiplied by the time function, and are calculated, as follows:

3.1 Inventory holding cost

Referring to Figure 3, the quantity of inventory is multiplied by the inventory holding cost in unit time and time func- tion in order to obtain the inventory holding cost per cycle. The mathematical expression is shown as follows:

hIðsÞ ¼ h Rs 0 ðp � dÞt expð�rtÞdt þ h

Rps=d s d

p d s � t

� � expð�rtÞdt

¼ h p�d�p expð�rsÞþd expð �psr d Þ

r2

h i (1)

3.2 Set-up cost

At the start of each production cycle, the machine should be set-up. Certain activities must be executed including (1) change fixture, mould, clamp, gold wire and capillary tip, (2) clean the work environment in order to avoid dust from affecting the production quality and (3) produce products on a trial basis to ensure the normal function of the machine. The above work content is the basis of machine set-up cost (K).

Figure 3. Production inventory model with shifts on three key modules.

International Journal of Production Research 2435

3.3 Defect cost

The three KCMs of a wire bonding machine may shift from an in-control to an out-of-control state after a period of operation. The time-to-shift is a random variable following an exponential distribution. It is assumed that after a specific key module has shifted, a fixed proportion of defective items may be produced by the machine. We assume these time- to-shift random variables are dependent. Therefore, a multivariate exponential distribution (MVE), proposed by Marshall and Olkin (1967) and Barlow and Proschan (1981), n = 3 is applied to model the behaviour of the production process. The complement of cumulative distribution function of the MVE distribution is given in Equation (2).

�FXYZðx; y; zÞ ¼ P½X [ x; Y [ y; Z [ z� ¼ exp½�k1x � k2y � k3z � k4maxðx; yÞ � k5maxðx; zÞ

�k6maxðy; zÞ � k7maxðx; y; zÞ� (2)

According to the calculation method for the joint probability density function, as mentioned by Lindgren (1976), the probability density function of MVE can be written as follows:

fXYZðx; y; zÞ ¼

k3ðk2 þ k6Þðk1 þ k4 þ k5 þ k7Þ�FXYZðx; y; zÞ for x [ y [ z k2ðk3 þ k6Þðk1 þ k4 þ k5 þ k7Þ�FXYZðx; y; zÞ for x [ z [ y k3ðk1 þ k5Þðk2 þ k4 þ k6 þ k7Þ�FXYZðx; y; zÞ for y [ x [ z k1ðk3 þ k5Þðk2 þ k4 þ k6 þ k7Þ�FXYZðx; y; zÞ for y [ z [ x k2ðk1 þ k4Þðk3 þ k5 þ k6 þ k7Þ�FXYZðx; y; zÞ for z [ x [ y k1ðk2 þ k4Þðk3 þ k5 þ k6 þ k7Þ�FXYZðx; y; zÞ for z [ y [ x

8>>>>>>< >>>>>>:

(3)

Let X, Y and Z axes represent the time variables of the shift of temperature, force and power KCM, respectively. According to the order combination of the shift of three KCMs, the time segmentation corresponding to different changes in the machine can be drawn in Figure 4.

As shown in Figure 4, defective items produced during a production run, NiðsÞ, can be expressed as follows (Lin and Gong 2011):

N1ðsÞ ¼ ps1ðmin Y; Z; sf g � XÞ; if ðX ; Y; ZÞ 2 0 � X � Y; Z; sf gf g N2ðsÞ ¼ ps2ðmin X ; Z; sf g � YÞ; if ðX ; Y; ZÞ 2 0 � Y � X ; Z; sf gf g N3ðsÞ ¼ ps3ðmin X ; Y; sf g � ZÞ; if ðX ; Y; ZÞ 2 0 � Z � X ; Y; sf gf g N4ðsÞ ¼ ps4ðmin Z; sf g � max X ; Yf gÞ; if ðX ; Y; ZÞ 2 0 � X ; Yf g � Z; sf gf g N5ðsÞ ¼ ps5ðmin Y; sf g � max X ; Zf gÞ; if ðX ; Y; ZÞ 2 0 � X ; Zf g � Y; sf gf g N6ðsÞ ¼ ps6ðmin X ; sf g � max Y; Zf gÞ; if ðX ; Y; ZÞ 2 0 � Y; Zf g � X ; sf gf g N7ðsÞ ¼ ps7ðs � max X ; Y; Zf gÞ; if ðX ; Y; ZÞ 2 0 � X ; Y; Zf g � sf g

8>>>>>>>>< >>>>>>>>:

(4)

The fXYZ(x, y, z) is substituted into the probability distribution of Ni(τ), and the expected number of defective items, E NiðsÞ½ � produced during a cycle is obtained by integration. However, since the mathematical expression is very

X Y

Z

τ τ

τ

Figure 4. Time segmentation.

2436 D.-C. Gong et al.

complex, it can be simplified by an algebraic process. Let A ¼ 1 1 0 1 0 1 0 1 1

0 @

1 A, and I3 ¼ 1 0 00 1 0

0 0 1

0 @

1 A. The algebraic

correspondence is expressed as follows:

xk ¼

A � k1 k2 k3½ �T; if k ¼ 1; 2; 3 A � k1 k2 k3½ �TþI3 � kk; if k ¼ 4; 5; 6 I3 � k1 k2 k3½ �TþA � k4 k5 k6½ �TþI3 � k7; if k ¼ 7; 8; 9 A � k1 k2 k3½ �TþI3 �

P6 i¼4 ki þ I3 � k7; if k ¼ 10; 11; 12P7

i¼1 ki; if k ¼ 13

8>>>>< >>>>:

(5)

The E NiðsÞ½ � after the aforementioned algebraic simplification is expressed as follows:

E½N1ðsÞ� ¼ ps1 ðx6 þ k6Þx7

x26x13

� þ exp �x13s½ �

x13 � exp½�x6s�

x6

�k6 x7

þ x6 þ k6 x6

þ k6s � ��

(6)

E½N2ðsÞ� ¼ ps2 ðx5 þ k5Þx8

x25x13 þ

� exp �x13s½ �

x13 � exp½�x5s�

x5

�k5 x8

þ �

x5 þ k5 x5

þ k5s ��

(7)

E½N3ðsÞ� ¼ ps3 ðx4 þ k4Þx9

x24x13 þ exp �x13s½ �

x13

� � exp½�x4s�

x4

�k4 x9

þ x4 þ k4 x4

þ k4s � ��

(8)

E½N4ðsÞ� ¼ ps4 exp½�x6s��1½ �x6 n

þ exp½�x5s��1½ �x5 � x10þk4þk7ð Þ expð�k3sÞ

k3x10

� ðx1þk5þk6Þ exp �x13s½ �x10x13 þ x13þk3þk4þk7ð Þ

k3x13

o (9) E½N5ðsÞ� ¼ ps5 exp½�x6s��1½ �x6

n þ exp½�ðx4Þs��1½ �x4 �

x11þk5þk7ð Þ expð�k2sÞ k2x11

� ðx2þk4þk6Þ exp �x13s½ �x11x13 þ x13þk2þk5þk7ð Þ

k2x13

o (10) E½N6ðsÞ� ¼ ps6 exp½�x5s��1½ �x5

n þ exp½�x4s��1½ �x4 �

x12þk6þk7ð Þ expð�k1sÞ k1x12

� ðx3þk4þk5Þ exp �x13s½ �x12x13 þ x13þk1þk6þk7ð Þ

k1x13

o (11) E½N7ðsÞ� ¼ ps7 x13þk6þk7ð Þ expð�k1sÞ�1½ �k1x12 þ

x13þk5þk7ð Þ expð�k2sÞ�1½ � k2x11

n þ x13þk4þk7ð Þ expð�k3sÞ�1½ �k3x10 þ

sðx6þk6Þ x6

þ sðx4þk4Þx4 þ sðx5þk5Þ

x5

þ exp �x13s½ ��1½ � x2

13

k2ðx2þk4þk6Þ x11

h þ k3ðx1þk5þk6Þx10 þ

k1ðx3þk4þk5Þ x12

i � 2s x13�k7ð Þx13 þ

x2 1�exp½�x5s�½ � x2

5 þ x3 1�exp½�x6s�½ �

x2 6

þ x1 1�exp½�x4s�½ � x2

4

o (12)

E NiðsÞ½ � is multiplied by πi and summed up to obtain the expected defective cost of one cycle.

3.4 Total cost

Finally, the total cost incorporating time value of money can be calculated according to the method proposed by Silver, Pyke, and Peterson (1998) and Sarkar and Moon (2011). The expected total cost of the first cycle, Z1(τ), which is the sum of the set-up cost, inventory cost and defect cost, is expressed as follows:

Z1ðsÞ ¼ K þ hIðsÞ þ X7 i

piE½NiðsÞ� !

exp � rp d s

� (13)

As shown in Figure 3, the same production cycle is repeated over the planning horizon and each cycle has a total cost of Z1(τ). Referring to Sarkar and Moon (2011), exp(−rpτ/d) is the time value discount for a cycle. The total production cost Z∞(τ), which is the sum of every single cycle’s present value, can be written as follows:

International Journal of Production Research 2437

Z1ðsÞ ¼ P1 j¼0

Z1ðsÞ exp �j rpd s � � �

¼ Z1ðsÞ P1 j¼0

exp �j rp d s

� � ¼ KþhIðsÞþ

P7 i¼1 piE½NiðsÞ�

� � exp �rpd sð Þ

1�exp �rpd sð Þ (14)

4. Convexity of the total production cost function

The expression of Z∞(τ) is very complex. Hence, referring to Rosenblatt and Lee (1986), when the conditions rs � 1; prs=d � 1 and ks � 1 are met, the exponential function can be approximated by the MacLaurin Series as fol- lows:

expð�rsÞ � 1 � rs þ 1 2

rsð Þ2� 1 6

rsð Þ3 (15)

expð� rps d Þ � 1 � rps

d þ 1 2

rps d

� 2 � 1 6

rps d

� 3 (16)

expð�ksÞ � 1 � ks þ 12 ksð Þ 2� 16 ksð Þ

3

k 2 k1 þ rpd � �

; k2 þ rpd � �

; k3 þ rpd � �

; x4 þ rpd � ��

; x5 þ rpd � �

; x6 þ rpd � �

; x13 þ rpd � � (17)

After substituting Equations (15), (16) and (17) in Equation (14), where the coefficients of the partial complex math- ematical terms are replaced by parameters A, B and C, the total production cost ~Z1ðsÞ, is written as follows:

Z1ðsÞ � 6d3 As4 þ Bs3 þ Cs2 þ Kð Þ sprðs2p2r2 � 3spdr þ 6d2Þ ¼

~Z1ðsÞ (18)

The detailed expressions of A, B and C are provided in Appendix 1. The convexity of the simplified objective equation in Equation (18) must be shown before the optimum production

time can be determined, and then the minimum total production cost can be calculated. The second derivative of ~Z1ðsÞ with respect to τ must be obtained. If the second derivative is greater than 0, it is a convex function. We can show that when 0 < τ < θ, an optimum production time can be found, and θ value is expressed as follows:

h ¼ d pr ; when B � 0

min d pr ; ffiffi ½

p 3��K

4B

n o ; when B\0

( (19)

The second derivative of ~Z1ðsÞ is obtained as follows:

g00 sð Þ ¼ d2 ds2

eZ1ðsÞh i ¼ 12d3

s3pr s2p2r2�3sdprþ6d2ð Þ3 3Ad 2p2r2 þ 3Bdp3r3 þ Cp4r4ð Þ½ s6

� 54Ad3pr þ 18Bd2p2r2ð Þs5 þ 45Kd2p2r2s2 � 54Kd3prs þ 108Ad4 � 18Cd2p2r2 þ 6Kp4r4ð Þs4 þ 36Kd4 þ 36Bd4 þ 18Cd3pr � 24Kdp3r3ð Þs3�

(20)

Next, we present several lemmas before providing a property showing the objective function becomes convex for a range of τ.

Lemma 1. If s\ 2d pr , then s3pr s2p2r2 � 3sdpr þ 6d2ð Þ3 [ 0.

Lemma 2. If 0\s\ 6d pr , then 3Bdp3r3τ6 − 18Bdp2r2τ5 > 0.

Lemma 3. If s\ 2d pr

or s [ 16d pr , then 3Ad2p2r2τ6 − 54Ad3prτ5 + 96Ad4τ4 > 0.

Lemma 4. If s\ �d pr

or s [ 0, then −18Cd2p2r2τ4 + 18Cd3prτ3 > 0. Lemma 5. If s\ d

pr or s [ 3d

pr , then 6Kp4r4τ4 − 24Kdp3r3τ3 + 18Kd2p2r2τ2 > 0.

Lemma 6. If s3 [ �K 4B , then 36Bd4τ3 + 9Kd4 > 0.

Proofs of Lemma 1 through Lemma 6 can be found in Appendix 2. Property 1. If 0 < τ < θ, then the ~Z1ðsÞ is a convex function. Proof By combining the results of Lemma 1 to Lemma 5, all the intervals of τ are plotted in Figure 5 to determine the

interval of the intersection, However, Lemma 6 only gives a range for the B value. Thus, if B ≥ 0, θ = d/pr; if B < 0, h ¼ minfd=pr;

ffiffi ½

p 3��K

4Bg. This implies that 0 < τ < θ is the solution interval of ~Z1ðsÞ.

2438 D.-C. Gong et al.

5. Solution procedure

The paper proposes two solution methods to rapidly obtain approximate solutions for minimising the objective function described by Equation (18). To validate the solution’s quality, the approximate solutions will be compared with the results obtained by applying a bisection algorithm to be developed in Section 7, which can be applied to find a solution for the original objective function not an approximate solution.

5.1 Method I

According to the method used by Lin and Lin (2007), we solve d ds

~Z1ðsÞ ¼ 0 by omitting the high-order terms and keeping only the quadratic term of the equation to obtain a positive root of the equation. Consequently, the following result is obtained:

d ds eZ1ðsÞ ¼ 6d3s2pr s2p2r2�3dprsþ6d2ð Þ2 Ap2r2s6 � 6Adprs5 þ 18Ad2 � 3Bdprð½

�Cp2r2Þs4 þ 12Bd2s3 þ 6Cd2 � 3Kp2r2ð Þs2 þ 6Kdprs � 6Kd2� One of the conditions of using MacLaurin series is rτ ≪ 1, which suggests that the high-order terms in the denomi-

nator can be ignored because they are too small to have any significant impact. Thus, in order to simplify the calcula- tion, the cubic and higher order terms in the denominator are deleted. Then, d

ds ~Z1ðsÞ becomes:

�Kd prs2

� � þ K

s þ Cd

pr � Kpr

2d

� � þ 2Bds

pr þ 3Ad

pr � B

2 � Cpr

6d

� � s2 � As3 þ Aprs

4

6d (21)

If we further delete the high-order terms and set the remaining terms to zero, we obtain the following quadratic equation:

g02nd sð Þ ¼ Cd

pr � Kpr

2d

� � s2 þ Ks þ �Kd

pr

� � ¼ 0 (22)

Let s 1 denote the positive root of Equation (22), it can be written as

s 1 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4d4KC

K2p4r4 � 4Kp2r2Cd2 þ 4d4C2

s (23)

5.2 Method II

In order to avoid a probable oversimplification of Method I, Cardano’s method is used to obtain a solution in this paper. Again, the first derivative of the objective function with respect to the decision variable τ is obtained. The denominator’s high-order terms are omitted, but the cubic and all other lower order terms are retained. After setting the result equal to zero, we obtain

d

pr

– d

pr

2d

pr

3d

pr

6d

pr

16d

pr 0

Lemma 2.

Lemma 1.

Lemma 3. Lemma 3.

Lemma 4. Lemma 4.

Lemma 5. Lemma 5.

Figure 5. Interval of production time.

International Journal of Production Research 2439

g03rdðsÞ ¼ 2Bds3

pr þ Cd

pr � Kpr

2d

� � s2 þ Ks þ �Kd

pr

� � ¼ 0 (24)

For ease of further explanation, Equation (24) is algebraically replaced by aτ3 + bτ2 + cτ + d = 0, where a ¼ 2Bd=pr; b ¼ Cd=pr � Kpr=2d; c ¼ K; d ¼ ð�KdÞ=pr. In fact, a, b, c and d are real numbers and a ≠ 0.

Let m ¼ 3ac�b2 9a2

and n ¼ 9abc�27a2d�2b3 54a3

. Define Δ = m3 + n2. There are two cases for Δ

Case 1: When Δ > 0, then the equation (m3 + n2 − Δ = 0) has one real root and two non-real roots. Define s ¼

ffiffi ½

p 3�n þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m3 þ n2

p and t ¼

ffiffi ½

p 3�n �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m3 þ n2

p .

Case 2: When Δ ≤ 0, then the equation has three distinct real roots. Let q ¼

ffiffiffiffiffiffiffiffiffi �q3

p and h ¼ cos�1 r=qð Þ, then define s ¼

ffiffi ½

p 3�qexp ih=3ð Þ and t ¼

ffiffi ½

p 3�qexp �ih=3ð Þ.

According to these two cases, we calculate the s and t, substitute them into Equation (25), and then find an optimum production run time s 2 with the following equations:

s 2 ¼ s þ t � b

3a

s 2 ¼ � 1

2 s þ tð Þ � b

3a þ

ffiffiffi 3

p

2 s � tð Þi (25)

s 2 ¼ � 1

2 s þ tð Þ � b

3a �

ffiffiffi 3

p

2 s � tð Þi

(Note that negative number and imaginary roots are invalid.) Disregarding the time value rate, if the machine does not have any shift, the above results can be shown to become

the same with those obtained for the traditional EPQ model. If there is only one KCM, the result has consistent mathe- matical expression with the imperfect EPQ model proposed by Rosenblatt and Lee (1986). This shows that our model includes the basic EPQ model and the model of Rosenblatt and Lee (1986) as special cases.

6. Numerical example

In this section, a real-world example is obtained to illustrate the solution procedure. The data collected from a produc- tion line of the ASE Group are shown in Table 1. Detailed descriptions of these data are given as follows:

6.1 Productivity (p)

The optimum production rate of a wire bonding machine is 14,000 gold wires per hour, excluding the delay and han- dling time of changing the IC component. If the delay and handling time is included in the production rate, the time should be multiplied by 0.88 according to an empirical rule. In terms of the starting IC component for the automobile air bag, one component must bond about 40 gold wires. The production line of a wire bonding machine is run for 24 h a day. However, after subtracting the time spent on changing a tool module and machine maintainance activity, the effective operating time becomes about 22 h. Therefore, the production rate can be computed as 14,000*0.88/ 40*22 = 6787 pieces/day. For ease of further calculation, the production rate in this paper is set as 6800 pieces/day.

Table 1. Example parameters (doller unit: TWD, 1 USD = 30TWD).

p d K h r 6800 unit/day 500 unit/day 300/order 0.2 unit/day 0.02775

s1 s2 s3 s4 s5 s6 s7 0.25 0.3 0.2 0.7 0.75 0.8 0.95 π1 π2 π3 π4 π5 π6 π7

40/unit 40/unit 40/unit 42.5/unit 42.5/unit 42.5/unit 45/unit λ1 λ2 λ3 λ4 λ5 λ6 λ7

0.004 0.0055 0.0045 0.00095 0.001 0.0009 0.0005

2440 D.-C. Gong et al.

6.2 Demand rate (d)

A typical purchase order of the starting IC is for about 90,000~100,000 IC components. Depending on the delivery date set for an order, the total throughput per day is usually run between 3000 and 4500 units, which exceeds the capacity of one machine. As a result, an order is usually fulfilled by running 5 to 9 wire bonding machines together. Therefore, the demand rate of a single wire bonding machine is set as 500 pieces/day in this paper.

6.3 Set-up cost (K)

According to the historical data of the production line, the set-up cost includes the labour rates of operators and techni- cians, and machine parts changed in each set-up. This set-up cost is about 300 Taiwan dollars (TWD) each time.

6.4 Holding cost (h)

The selling price of one starting IC for the automobile air bag is about 10 US dollars (USD). Depending on the exchange rate, one USD is roughly equivalent to 30 TWD. The computing equation for the inventory holding cost per unit time is h = P*ratio, and the ratio is generally 20~25%, thus, the holding cost is about 0.2/piece/day (h = 30*10*0.25/365 ≒ 0.2055).

6.5 Time value of money rate (r)

The discount rate of Taiwan was 1.875% in 2013, the inflation factor of May 2013 was −0.9%, as compared with the previous May, thus, the time value rate of Taiwan in May 2013 = 2.775% (cnYES, 2014). The value of r in Equation (14) is assumed to be 2.775%.

6.6 Defective cost (πi)

The non-conforming products from the wire bonding process are gathered first, and then processed by other stations. The incurred cost includes non-recoverable material cost and the processing cost of the recovery operation. Thus, the corresponding πi is estimated based on historical data as: π1 = 40, π2 = 40, π3 = 40, π4 = 42.5, π5 = 42.5, π6 = 42.5, π7 = 45.

6.7 Time-to-shift parameter value (λi) and defective rate (si) after shift

Based on our interviews with an expert, the yield rate of the present semiconductor IC packaging process is about 99.9%, meaning there is only one non-conforming product among 1000 products. About 90% of the defects result from the shift of one wire bonding machine occurred in the three key modules. In summary, in the wire bonding process of IC packaging, the probability value of defects produced by the three key modules of a wire bonding machine is about 0.09%, where k1; k2; k3 [ k4; k5; k6 [ k7 can be reasonably assumed; and the condition of s1, s2, s3 < s4, s5, s6 < s7 is met. As the number of shifts of a single module is usually larger than the number of simultaneous shifts of two or three modules, the probability of generating defective products resulting from the shift of a single module must be smaller than the probability of defects resulting from the simultaneous shift of two modules, and smaller than the probability value of defects resulting from simultaneous shift of three modules.

Figure 6 illustrates the relationship between cost and time in this group of parameters (Table 1). The inventory holding cost begins to fall on about Day 4, as the exponential term of the holding cost (HCa) was approximated by the MacLaurin Series. However, since it does not meet the condition (prτ/d ≪ 1), the posterior curve of HCa becomes concave.

Figure 6 also shows the variation between the original total cost curve and the approximated total cost curve. They are very close to each other when τ is very small; however, they are farther from each other with time. This phe- nomenon is the motive for this paper to seek an optimum solution interval. The approximate solutions of s , which are calculated by the two solution methods proposed in Section 5, i.e. s 1s

1 and s

2, are within the interval of 0 < τ < θ.

The parameters in Table 1 are used with Method I to obtain a solution of s1 ¼ 0:1978 day. According to ASE Group’s provision, at most two production lots are performed per machine per an eight-hour shift. One reason is to pre- vent from generating too many defectives when the machine goes wrong. Another reason is to match with just-in-time (JIT) demands of various product types required by customers. The obtained value of s1 (=0.1978 day = 4.7472 h) is

International Journal of Production Research 2441

reasonable. We then calculate the number of defects ( P

E Ni sð Þ½ � ¼ 1:2154Þ, the products produced per cycle (pτ = 1345.04), and thus the defective rate is about 9:04 10�4, which is in compliance with the specification of only one non-conforming product among 1000 products.

7. Justification of solutions

Section 5 described a simplified process of the objective function, and proposed two solution methods to generate two approximate solutions denoted as s 1 and s

2. To justify the approximate solutions’ quality, a bisection search procedure

is applied for the purpose in this paper. There are 20 probability combinations of (λi, si) randomly generated, as based on the parameters in Table 1, to determine the difference between s 1, s

2 and the solution ðs b) as obtained by the bisec-

tion search. The VBA programme language of EXCEL is used to develop a bisection method, which solves the original objective function without MacLaurin series approximations. The optimum production time range 0\s\hð Þ, as found in Section 5, is used as the upper and lower bounds. The range is gradually narrowed to obtain an optimum production time (s b). The bisection method is briefly described as follows:

7.1 Bisection search method

Step 1. Set tolerance = 0.0000001, τR = θ and τL = 0.0000001. Then let f 0 sð Þ ¼ d

ds Z1 sð Þ, and calculate f 0 sRð Þ and f 0 sLð Þ:

Step 2. Let τold = τR. Step 3. Let smid ¼ sRþsL2 . Step 4. If smid�soldsmid

��� ���\tolerance, then go to Step 6. Otherwise, let τold = τmid, and go to Step 5. Step 5. Check whether f 0 smidð Þ and f 0 sLð Þ are equal, i.e. f

0 smidð Þ�f 0 soldð Þ f 0 smidð Þ

��� ���\tolerance. If yes, let τL = τmid. If not, let τR = τmid, go to Step3.

Step 6. When τopt = τmid, stop. When using MacLaurin Series to obtain approximate solutions, if the condition (ks � 1) is not met, the result is

likely to change in quality. Therefore, the ki values on 0; 1�ð 0; 0:1�ð are tested first, and then substituted into the equa- tions of Method I and Method II for computation. The results are mostly negative or imaginary, and even if a small part has solutions, they are different from s b.

0 2 4 6 8 0

5 104×

1 105×

1.5 105×

2 105× TCa DCa SCa HCa TCo DCo SCo HCo

Production Time

C os

t

Figure 6. Variations of approximated set-up cost, holding cost, defect cost and total cost.

2442 D.-C. Gong et al.

Therefore, we suggest that ki is in the interval of 0; 0:01�ð , and si is in the interval of 0; 1�ð , and the relationship between ki and si shall meet the constraints as specified in Section 6. Twenty probability combinations are randomly selected for computation. The obtained s 1 and s

2 are compared with s

b, and the results are shown in Table 2. The devia-

tion percentage used in Table 2 is calculated as follows:

Deviation Percentage of sð%Þ ¼ s i � s b s b

���� ���� � 100%; i ¼ 1; 2

Deviation Percentage of Total Costð%Þ ¼ TC i � TC b TC b

���� ���� � 100%; i ¼ 1; 2

Figures 7 and 8 illustrate the data of Table 2, where the deviation percentage of s 1 and s b is lower than 3.48%, the devi-

ation percentage of s 2 and s b is lower than 1.69% and the deviation percentage of total cost is lower than 6%.

8. Sensitivity analysis

For enterprises, the time value of money rate is unable to control. This section focuses on the influence of the variance in the time value rate on the total cost, and determine the corresponding strategies. It contains two parts. First, the time value rate is adjusted, and the variance of the total production cost is observed. Second, let the time value rate be 2.775%, while the other parameter values are changed, in order to observe the variation in the total production cost. The τ values mentioned in this section are obtained by Method I.

Before going through all the analyses, the effect in considering the time value factor is first examined. We resolve the problem by assuming the time value rate equal to zero (r = 0). The obtained optimal s b by the bisection method is 0.17966. This number is further substituted in the objective function (Equation 18) to calculate the total cost, which is 9084.7239. It is 12.33% higher than the total cost when τ equals 0.1978 (note: τ = 0.1978 is the solution when the time value of money is considered).

Next, the time value rate is adjusted, where the r values are 1.5, 2, 2.5, 2.775, 3, 3.5 and 4%, respectively, to observe the changes in the total cost (~Z1 sð Þ) and production time (τ). According to Figure 9, the optimal production time decreases as r increases, and the total cost curve approaches the origin. The result validates a managerial phe- nomenon that

Table 2. Comparisons between s 1, s 2 and s

b as well their total costs.

n s b TC b s

1 TC

1 s

2 TC

2

s b vs. s 1 (%)

TC b vs. TC 1 (%)

s b vs. s 2 (%)

TC b vs. TC 2 (%)

1 0.23718472436 6674.67708 0.239986324 6909.378298 0.236343556 6898.284295 1.18 3.52 0.35 3.35 2 0.20757043694 7714.563593 0.204909552 7843.38218 0.207464778 7847.64765 1.28 1.67 0.05 1.73 3 0.20420159081 7849.672323 0.201160348 7963.564274 0.204148039 7967.844243 1.49 1.45 0.03 1.51 4 0.21336653198 7640.788321 0.211419499 7637.016427 0.213163902 7640.362121 0.91 0.05 0.09 0.01 5 0.19647088450 8174.950149 0.192731906 8257.590239 0.196513123 8261.002002 1.90 1.01 0.02 1.05 6 0.20617107900 7769.564889 0.203381951 7884.460025 0.206082593 7888.396403 1.35 1.48 0.04 1.53 7 0.23907864396 6616.411511 0.242338877 6883.615081 0.23817801 6869.334757 1.36 4.04 0.38 3.82 8 0.21629945380 7382.263777 0.214815567 7530.394633 0.216033546 7532.785159 0.69 2.01 0.12 2.04 9 0.23194511195 6842.004511 0.233439987 7053.673396 0.231277936 7047.651509 0.64 3.09 0.29 3.01 10 0.23250370012 6942.309824 0.229529402 7151.896336 0.228582951 7149.274742 1.28 3.02 1.69 2.98 11 0.20955837225 7636.908875 0.207128115 7762.227736 0.209421335 7765.945246 1.16 1.64 0.07 1.69 12 0.24645439894 6394.194913 0.25193045 6699.928912 0.245263189 6674.50475 2.22 4.78 0.48 4.38 13 0.25621835123 6117.521102 0.265136317 6481.934914 0.254587507 6436.210447 3.48 5.96 0.64 5.21 14 0.20729747992 7725.523789 0.204595366 7852.49254 0.207197603 7856.760917 1.30 1.64 0.05 1.70 15 0.18683195460 8615.171931 0.18246103 8689.37122 0.186962354 8692.475563 2.34 0.86 0.07 0.90 16 0.22372555397 7313.486409 0.223453222 7314.763274 0.223296457 7314.352809 0.12 0.02 0.19 0.01 17 0.21967120433 7178.069116 0.218726012 7348.021687 0.221612393 7354.105534 0.43 2.37 0.88 2.45 18 0.22827288985 6966.785132 0.2286917 7169.611815 0.227754048 7167.099724 0.18 2.91 0.23 2.88 19 0.20555191390 7795.064736 0.202656744 7944.494702 0.205478428 7950.07565 1.41 1.92 0.04 1.99 20 0.19416715053 8276.405682 0.190257967 8374.043639 0.194232935 8378.53993 2.01 1.18 0.03 1.23

International Journal of Production Research 2443

the effect of the discount rate increases with the time value parameter, thus increasing the exchange rate, increasing the national currency value, decreasing the purchase price of imported raw material, and decreasing the total production cost.

However, the effect of adjusting the discount rate on controlling the exchange rate is limited, and even if the r continu- ously increases, the reduction in the total production cost only decreases gradually.

In order to further analyse the effect of the time value rate on the total production cost, Figure 10 is created. Observe that when the time value parameter is greater than 5%, the effect of time value on the total production cost gradually becomes flat. This phenomenon can be used by decision makers to evaluate whether or not to set-up factories in other countries. When the time value parameter of a country is above 5%, long-term factory set-up investment is

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Group Number(n)

Deviation Percentage of Production Time(τ)

τb v.s τ1 τb v.s τ2

Figure 7. Deviation percentage of production time.

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Group Number (n)

Deviation Percentage of Total Cost

TCb v.s TC1

TCb v.s TC2

Figure 8. Deviation percentage of total cost.

2444 D.-C. Gong et al.

feasible; however, if it is smaller than 5%, it is noteworthy that the inflation benefit of the country will neutralise the profit of production, which causes the producer’s profit to decrease. This phenomenon matches the expected benefit of inflation, as mentioned in the MBA Library webpage.

Since the time value rate is an uncontrollable factor for enterprises, the existing production parameters should be adaptable, if possible, in order to reduce the total production cost. Therefore, in addition to changing the exponential distribution parameter for the shift of a KCM, the other parameters are considered together for sensitivity analysis. The purpose is to observe the variation in the optimum production time and the total production cost when the parameters change to ± 40 and ± 80%. Figure 11(a) and (c) show the changes in τ when the parameters are changed. Figure 11(b) and (d) show the changes in total production cost when the parameters are changed. The percentage error is calculated using Equations (26) and (27).

Percentage change of Production Timeð%Þ ¼ schange � soriginal soriginal

���� ���� � 100% (26)

Percentage change of Total Costð%Þ ¼ schange � soriginal soriginal

���� ���� � 100% (27)

The variations in TC with ki, si and πi in Figure 11(d) show smaller than the variations due to other parameters; how- ever, it remains in a positive relationship. As TC has no obvious quantitative change, it may be because of the probabil- ity of the machine shifting into out-of-control state is already low.

According to Figure 11(b), the total production cost decreases with the set-up cost (K) and the inventory holding cost (h). Observe in Figure 11(a) that the inventory holding cost is inversely proportional to the production time. When

5000

10000

15000

20000

25000

30000

35000

40000

0.04 0.09 0.14 0.19 0.24 0.29 0.34 0.39 τ

r

r=1.5%

r=2%

r=2.5%

r=2.775%

r=3%

r=3.5%

r=4%

T ot

al C

os t

Figure 9. Curves of total cost vs. τ at different time value rates.

0

10000

20000

30000

40000

50000

1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 9.0% 10.0%

Time Value (r)

T ot

al C

os t

r

TC

2.775

Figure 10. Relationship between total cost and τ when time value rate = 2.775%.

International Journal of Production Research 2445

the inventory holding cost increases, the production time and the throughput should be reduced to decrease the inventory in order to reduce the total production cost outlay for inventory. The set-up cost is proportional to the produc- tion time. When the set-up cost is high, the manufacturer plans to prolong the time of each production to reduce the set-up cost as much as possible while meeting the demand. Since the throughput per cycle is increased, the number of machine set-ups will be reduced.

Observe in Figure 11(b) and (d) that the total production cost increases when p increases or d decreases. Therefore, this paper probes into the correlation among set-up cost (K), inventory holding cost (h), production rate (p) and demand rate (d) in details. Obtained observations and related managerial insights are analysed in the following section.

9. Managerial insight analysis

As shown in Figures 12(a)–(c), when different values of the set-up cost K (150, 200, 250, 300, 350, 400 and 450), inventory holding cost h (0.15, 0.17, 0.19, 0.21, 0.23, 0.25 and 0.27.), production rate p and demand rate d in the form of ðp � dÞ=p (5%, 10%, … , 95%.) are used in the calculation of all cost components, significant effects can be observed in the corresponding total cost or the production time τ. In Figure 12(c), the πi values provided in Table 1 are used in calculating every cost component. In order to know the effect of πi on each cost component, the value of πi in the range 1 < πi < 1.5 are used in the calculation. As a result, Figure 12(d) is created. Our observations on these figures are summarised as follows:

• When the set-up cost increases, the total cost curve moves upward. However, the increasing amplitude of the total production cost can be moderately reduced by prolonging the production time.

• When the holding cost h increases, the total production cost curve also moves upward. However, the increment in total production cost can be reduced moderately by shortening the production time.

• For the lower production total cost, p�d p

value should approach to 1. • When the πi value is small, the

p�d p

value approaches 0 or 1, and the total production cost is low.

395.15%

66.39%

24.92% 0.00%

-16.63% -28.52%

-44.38% -56.85%

-23.17% -10.85%

0.00% 9.79% 18.79%

34.99%

-54.47%

-21.98% -10.26% 0.00%

9.21% 17.64% 32.74%

89.53%

24.84% 10.35% 0.00% -7.87% -14.13% -23.55%

3.18% 1.55% 0.77% 0.00% -0.75% -1.48% -2.90%1.36% 0.67% 0.34% 0.00% -0.33% -0.66% -1.31%0.79% 0.39% 0.20% 0.00% -0.19% -0.39% -0.77%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%3.18% 1.55% 0.77% 0.00% -0.75% -1.48% -2.90%1.36% 0.67% 0.34% 0.00% -0.33% -0.66% -1.31%0.79% 0.39% 0.20% 0.00% -0.19% -0.39% -0.77%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%3.18% 1.55% 0.77% 0.00% -0.75% -1.48% -2.90%0.95% 0.47% 0.23% 0.00% -0.23% -0.46% -0.92%0.51% 0.26% 0.13% 0.00% -0.13% -0.25% -0.51%0.30% 0.15% 0.07% 0.00% -0.07% -0.15% -0.29%0.26% 0.13% 0.06% 0.00% -0.06% -0.13% -0.25%0.26% 0.13% 0.06% 0.00% -0.06% -0.13% -0.25%0.21% 0.11% 0.05% 0.00% -0.05% -0.11% -0.21%

-100.00%

-50.00%

0.00%

50.00%

100.00%

150.00%

200.00%

250.00%

300.00%

350.00%

400.00%

450.00%

-80% -40% -20% 0% 20% 40% 80%

(a)

(c) (d)

(b)

p d K h

0.21% 0.02% 0.01% 0.00% 0.00% 0.00% -0.01%

-53.12%

-21.78%

-10.22%

0.00%

9.26%

17.79%

33.23%

-55.98%

-23.07%

-10.85%

0.00%

9.88%

19.02%

35.68%

-44.43%

-19.55%

-9.27%

0.00%

8.50%

16.38%

30.73%

-3.43% -1.69% -0.84%

0.00% 0.75% 1.47% 2.89%

-1.28% -0.64% -0.32% 0.00% 0.32% 0.64% 1.27%-0.75% -0.37% -0.19% 0.00% 0.19% 0.37% 0.74%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%-0.44% -0.22% -0.11% 0.00% 0.03% 0.03% 0.03%

-3.43% -1.69% -0.84%

0.00% 0.83% 1.65% 3.27%

-1.28% -0.64% -0.32% 0.00% 0.32% 0.64% 1.27%-0.75% -0.37% -0.19% 0.00% 0.19% 0.37% 0.74%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%-0.44% -0.22% -0.11% 0.00% 0.11% 0.22% 0.44%

-2.92% -1.45% -0.72% 0.00% 0.72%

1.43% 2.85% -0.82% -0.42% -0.21% 0.00% 0.21% 0.43% 0.86%-0.43% -0.22% -0.11% 0.00% 0.11% 0.23% 0.46%-0.31% -0.16% -0.08% 0.00% 0.08% 0.15% 0.31%-0.29% -0.14% -0.07% 0.00% 0.07% 0.14% 0.28%-0.28% -0.14% -0.07% 0.00% 0.07% 0.14% 0.27%-0.22% -0.11% -0.06% 0.00% 0.06% 0.11% 0.22%

-80.00%

-60.00%

-40.00%

-20.00%

0.00%

20.00%

40.00%

60.00%

-80% -40% -20% 0% 20% 40% 80%

d

K

h

3.18%

1.55%

0.77%

-0.75%

-1.48%

-2.90%

3.18%

1.55%

0.77%

-0.75%

-1.48%

-2.90%

3.18%

1.55%

0.77%

-0.75%

-1.48%

-2.90%

- 4.00%

- 3.00%

- 2.00%

- 1.00%

0.00%

1.00%

2.00%

3.00%

4.00%

-80% -40% -20% 0% 20% 40% 80%

si πi λi 0.21% 0.02% 0.01% 0.00% 0.00% -0.01%

-3.43%

-1.69%

-0.84%

0.75%

1.47%

2.89%

-3.43%

-1.69%

-0.84%

0.83%

1.65%

3.27%

-2.92%

-1.45%

-0.72%

0.72%

1.43%

2.85%

-4.00%

-3.00%

-2.00%

-1.00%

0.00%

1.00%

2.00%

3.00%

4.00%

-80% -40% -20% 0% 20% 40% 80%

p si πi λi

Figure 11. Sensitivity analyses of τ and TC with various parameters.

2446 D.-C. Gong et al.

The set-up cost and the inventory cost can be reduced by better product design, shop floor improvement and effec- tive warehouse management, which then reduces the total production costs.

Based on the above analysis, improvement suggestions are proposed according to the set-up and the production inventory costs, from the perspective of production management, as shown in Table 3. The discussion is as follows:

9.1 Scenario I: set-up cost and inventory cost are relatively high

When the two costs are high, the optimum production time can be determined by directly applying the methods pro- posed in Section 5. In addition, if the defect cost is very low, multiple machine work orders of like products can be combined for production on one machine. Thus, the batch size of a single work order approaches to the maximum throughput of the machine. The optimum production time is determined using the model constructed in this paper to obtain the minimum total production cost in this scenario, referring to Figure 12(d).

9.2 Scenario II: only set-up cost is effectively reduced

The production time can be shortened when the set-up cost is low. Even if this action increases the number of set-ups, it remains worthy in comparison to the increase in the production inventory cost resulting from an excessive production. The concept of using rapid die changing technology to reduce set-up costs in the Toyota system is an example.

5000

7000

9000

11000

13000

15000

17000

19000

21000

23000

25000

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

τ

(a) (b)

(c) (d)

K=450

K=400

K=350

K=300

K=250

K=200

K=150

T ot

al C

os t

6000

8000

10000

12000

14000

16000

18000

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

τ

h=0.27

h=0.25

h=0.23

h=0.21

h=0.19

h=0.17

h=0.15

C os

t

0

5000

10000

15000

20000

25000

30000

35000

5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95%

C os

t

(p-d)/p

Total Cost

Setup Cost

Holding Cost

Defect Cost

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95%

C os

t

(p-d)/p

Total Cost

Setup Cost

Holding Cost

Defect Cost

Figure 12. Further analyses of the effects of K, h, p, d and πi to the total cost.

Table 3. Improvement suggestions for the four scenarios.

High inventory cost Low inventory cost

High set-up cost

When πi is very small, combine orders to minimise (p − d)/p

Prolong the production time to increase the throughput in unit cycle time

Low set-up cost

Shorten the production time to reduce the throughput in unit cycle time

Split an order to maximise (p − d)/p

International Journal of Production Research 2447

9.3 Scenario III: only the inventory cost is reduced

The production time is prolonged to increase the throughput per cycle, and the number of set-ups is reduced, thus reduc- ing the total production cost, as shown in Figure 12(a).

9.4 Scenario IV: both the set-up cost and the inventory cost are effectively reduced

The optimum production time can be determined by directly using the proposed methods of this paper. However, the findings show that the original order shall be split into several sub-orders with a size much smaller than the machine capacity. Then, the total production cost can be reduced, see Figure 12(c).

10. Concluding remarks

This paper considers a production–inventory problem of a wire bonding machine. We identify the causes of defects resulting from the shift of a wire bonding machine from an in-control state to an out-of-control state. The causes can be divided into three KCMs: capillary tip FCM, ultrasonic vibration PCM and lead frame preheating TCM. In addition, the time value of money including inflation rate is significant in this type of industry and has attracted attention in recent years. Hence, this paper constructs a mathematical model considering the time value rate in the shift of three KCMs of a single machine. We prove that its effective computing range meets convex function properties. Moreover, this paper proposed two methods for the approximate solution of the optimum production time. In order to validate the quality of the solution, 20 randomly generated numerical examples are compared with the solutions obtained by a bisection method. The error between the τ value obtained by the two methods and the τ value of the bisection method is within 3.48%. The parameter values in Table 1 meet industrial characteristics, and can be compared with an actual production line. The parameters are adjusted for sensitivity analysis, and improvement suggestions are proposed in Section 9 based on our findings which can be referenced by a company in decision-making.

As for the effect in considering the time value factor, we have resolved the problem by assuming the time value rate r = 0. The obtained optimal s b by the bisection method is 0.17966. After substituting it in Equation (18), the corre- sponding total cost is 9084.7239. However, when the time value of money is considered, we have the optimal solution s 1 = 0.1978. Its total cost resulting from Equation (18) shows a 12.33% saving.

Moreover, regarding whether or not the improvement suggestions proposed in Section 9 can benefit IC packaging factories, the following information is obtained after another expert interview:

In terms of combining orders, in an actual production line, when the demand of one order is small but there are mul- tiple orders of the same type, the orders are usually combined in order to reduce the number of machine set-ups. How- ever, the combination of orders is not suggested in the case of gradually unstable machines or high defect costs.

The total production cost is possibly reduced by splitting an order. In an actual production line, the order is split into several work orders with small demand if there is a large emergency order. Meanwhile, multiple wire bonding machines are used for production in order to complete the manufacturing of products within a short time. For a single machine, d > p, and after the order is split, it is changed to a large (p − d)/p value ((p − d)/p → 1), and the total production cost is lower.

To simplify the production scenarios, this paper made reasonable assumptions on single-machine multi-product pro- duction. However, in order to extend our analytical model for real-world problems, the production concept of ELSP (Economic Lot Scheduling Problem) can be investigated in future studies.

In addition, the proposed math model was developed for a product of stable order source and regular production. The concerned KCMs, which cause an imperfect production process, have a similar property that allows shifts within a predetermined range. Although these limit the applications of this paper, they actually offer an opportunity for future research.

Disclosure statement No potential conflict of interest was reported by the authors.

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Appendix 1

p Production rate per unit time d Demand rate per unit time K Set-up cost for a production cycle h Inventory holding cost per unit time r Time value rate, a fixed value pi Defect cost per unit when a shift of module i occurs ki Exponential distribution parameter for the time-to-shift of module i si Rate of generating defect items when a shift of module i occurs τ Production run time, where s represents the optimal production time IðsÞ Cumulative inventory quantity in a cycle. NiðsÞ Number of defective items produced after the ith module has shifted Z1ðsÞ The total cost of the first cycle, where Z1ðsÞ is the total cost of the entire infinite cycles (planning horizon) The subscript i of pi; ki; siand Ni sð Þ; i = 1, 2 and 3, denotes a shift of ith module. Modules 1, 2 and 3 represent the TCM, FCM and

PCM, respectively. When both temperature and FCMs have shifted, i is set equal to 4. When the temperature and PCMs shift, i equals 5. When the force and PCMs shift, i equals 6. Finally, i = 7 means all control modules shift.

2450 D.-C. Gong et al.

A ¼ p 6d3

p1s1k6 dx6þpr½ �3 x6

n þ p2s2k5 dx5þpr½ �

3

x5 þ p3s3k4 dx4þpr½ �

3

x4

þ p7s7p2r3 6 � x1x4 h

� x2x5 � x3 x6

� 3 x13�k7ð Þx13 io

B ¼ B1 þ B2 þ B3 þ B4 þ B5 þ B6 þ B7 þ B8

B1 ¼ p1s1p6d3 x6þk6ð Þ x2

6 dx6 þ prð Þ3� p

3r3x7 x13

h in � x7�k6ð Þ dx13þprð Þ

3

x7x13

� k6 dx6þprð Þ 2 d x13þ3x7ð Þþpr½ � x6x7

o B2 ¼ p2s2p6d3

x5þk5ð Þ x2

5 dx5 þ prð Þ3� p

3r3x8 x13

h in � x8�k5ð Þ dx13þprð Þ

3

x8x13

� k5 dx5þprð Þ 2 d x13þ3x8ð Þþpr½ � x5x8

o B3 ¼ p3s3p6d3

x4þk4ð Þ x2 4

dx4 þ prð Þ3� p 3r3x9 x13

h in � x9�k4ð Þ dx13þpr½ �

3

x9x13

� k4 dx4þprð Þ 2 d x13þ3x9ð Þþpr½ � x4x9

o

B4 ¼ p4s4p6d3 x10þk4þk7ð Þ dk3þprð Þ3�p3r3½ �

k3x10

� � dx5þprð Þ

3

x5 � dx6þprð Þ

3

x6

þ p3r3 x1þ2k3þk5þk6ð Þx5x6 þ x1þk5þk6ð Þ dx13þprð Þ3�p3r3½ �

x10x13

�

B5 ¼ p5s5p6d3 x11þk5þk7ð Þ dk2þprð Þ3�p3r3½ �

k2x11

� � dx4þprð Þ

3

x4 � dx6þprð Þ

3

x6

þ p3r3 x2þ2k2þk4þk6ð Þx4x6 þ x2þk4þk6ð Þ dx13þprð Þ3�p3r3½ �

x11x13

�

B6 ¼ p6s6p6d3 x12þk6þk7ð Þ dk1þprð Þ3�p3r3½ �

k1x12 � dx4þprð Þ

3

x4 � dx5þprð Þ

3

x5

� þ p3r3 x3þ2k1þk4þk5ð Þx4x5 þ

x3þk4þk5ð Þ dx13þprð Þ3�p3r3½ � x12x13

�

B7 ¼ p7s7p p 2r2

d2 3 � x1

2x4 � x2

2x5 � x3

2x6 � x13�k7x13

� n þ x1 dx4þprð Þ

3�p3r3½ � 6d3x2

4

þ x2 dx5þprð Þ 3�p3r3½ �

6d3x2 5

þ x3 dx6þprð Þ 3�p3r3½ �

6d3x2 6

þ x12þk6þk7ð Þ p 3r3� dk1þprð Þ3½ �

6d3k1x12

þ x11þk5þk7ð Þ p 3r3� dk2þprð Þ3½ �

6d3k2x11 þ x10þk4þk7ð Þ p

3r3� dk3þprð Þ3½ � 6d3k3x10

þ dx13þpr½ � 3�p3r3½ �

x13 k1 x3þk4þk5ð Þ

x12 þ k2 x2þk4þk6ð Þx11 þ

k3 x1þk5þk6ð Þ x10

h i�

B8 ¼ hprðd2 � p2Þ

6d2

C ¼ C1 þ C2 þ C3 þ C4 þ C5

C1 ¼ p1s1p2d2 p2r2x7 x6þk6ð Þ

x2 6 x13

n � x3 dx6þprð Þ d x7þx13ð Þþpr½ �x6x7

þ x3þ2x7ð Þ dx13þprð Þ 2

x7x13 � x6þk6ð Þ dx6þprð Þ

2

x2 6

� d2x7 o

C2 ¼ p2s2p2d2 p2r2x8 x5þk5ð Þ

x2 5 x13

n � x2 dx5þprð Þ d x8þx13ð Þþpr½ �x5x8

þ x2þ2x8ð Þ dx13þprð Þ 2

x8x13 � x5þk5ð Þ dx5þprð Þ

2

x2 5

� d2x8 o

International Journal of Production Research 2451

C3 ¼ p3s3p2d2 p2r2x9 x4þk4ð Þ

x2 4 x13

n � x1 dx4þprð Þ d x9þx13ð Þþpr½ �x4x9

þ x1þ2x9ð Þ dx13þprð Þ 2

x9x13 � x4þk4ð Þ dx4þprð Þ

2

x2 4

� d2x9 o

C4 ¼ p7s7p2d 4pr x13�k7ð Þ

x10

n þ dk1þ2prð Þ x12þk6þk7ð Þx12 � 2d k1 þ k2 þ k3ð Þ

þ dk2þ2prð Þ x11þk5þk7ð Þx11 þ dk3þ2prð Þ x10þk4þk7ð Þ

x10 � 12pr

þ dx10þ2pr½ �x10 k1 x3þk4þk5ð Þ

x12 þ k2 x2þk4þk6ð Þx11 þ

k3 x1þk5þk6ð Þ x10

h io

C5 ¼ hpðp � dÞ

2d

Appendix 2 Proof of Lemma 1.

If � 3sdpr þ 6d2 [ 0

then s3pr s2p2r2 � 3sdpr þ 6d2 � �3

[ 0

�3sdpr þ 6d2 [ 0

s\ 2d

pr

Proof of Lemma 2.

3Bdp3r3s6 � 18Bd2p2r2s5 [ 0

3Bdp2r2s5 prs � 6dð Þ [ 0 The coefficient of B is negative. To have a positive value on the left side, then we should have s prs � 6dð Þ\0

s prs � 6dð Þ\0

0\s\ 6d

pr

Proof of Lemma 3.

3Ad2p2r2s6 � 54Ad3prs5 þ 96Ad4s4 [ 0

3Ad2s4 p2r2s2 � 18dprs þ 32d2 � �

[ 0

3Ad2s4 prs � 2dð Þ prs � 16dð Þ [ 0

s\ 2d

pr or s [

16d

pr

Proof of Lemma 4.

�18Cd2p2r2s4 þ 18Cd3prs3 [ 0

18Cd2prs3 �prs þ dð Þ [ 0

s �prs þ dð Þ [ 0

2452 D.-C. Gong et al.

s [ �d pr

or s [ 0

Proof of Lemma 5.

6Kp4r4s4 � 24Kdp3r3s3 þ 18Kd2p2r2s2 [ 0

6Kp2r2s2 p2r2s2 � 4dprs þ 3d2 � �

[ 0

6Kp2r2s2 prs � dð Þ prs � 3dð Þ [ 0

s\ d

pr or s [

3d

pr

Proof of Lemma 6.

9d4 4Bs3 þ K � �

[ 0

if B � 0; s3 [ �K 4B

ðright - side is negativeÞ

Let s3 [ 0

s [ 0

if B\0; s3\ �K 4B

ðright - side is positiveÞ

s3\ �K 4B

s\ ffiffi ½

p 3��K 4B

International Journal of Production Research 2453

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