The value of setup cost reduction and process

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European Journal of Operational Research xxx (2005) xxx–xxx www.elsevier.com/locate/dsw

Production, Manufacturing and Logistics

The value of setup cost reduction and process improvement for the economic production quantity model with defects Michael Freimer a, Douglas Thomas a

a,*

, John Tyworth

a

509 Business Administration Building, Smeal College of Business Administration, The Pennsylvania State University, University Park, PA 16802, USA Received 19 January 2004; accepted 29 November 2004

Abstract This paper investigates the effect of imperfect yield on economic production quantity decisions. The production system is assumed to produce some time-varying proportion of defective parts which can be repaired at some unit cost. For a general defect rate function, we develop results that characterize the optimal run length and expected total cost and how these objects are affected by the cost parameters. Two kinds of investments in process improvements are considered: (i) reducing setup costs and (ii) improving process quality. We develop expressions for the marginal value of both setup cost reduction and process improvement and discuss the relationship between these marginal values and problem cost parameters. We show that any investment in setup cost reduction will result in a reduction in the number of defects produced, and the total number of defects can increase or decrease with an investment in quality improvement. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Manufacturing; Quality management; Economic production quantity

1. Introduction An enormous body of research has stemmed from the original economic production quantity (EPQ) model. The basic model determines the optimal lot size when demand and production

*

Corresponding author. E-mail addresses: [email protected] (M. Freimer), [email protected] (D. Thomas), [email protected] (J. Tyworth).

rates have known, deterministic values. The manufacturing process is assumed to be perfect; production occurs without defects. One branch of the literature has accounted for imperfections in the manufacturing process. Much of this work has focused on the benefits of smaller lot sizes in imperfect systems, and the relationship between quality improvement (improving yield) and reducing setup costs. Wright and Mehrez [17] provide a taxonomy and survey of the work relating inventory and quality issues.

0377-2217/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.11.024

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Perhaps the first paper to relate quality and lot size was written by Porteus [11]. Porteus describes a system that begins each production run in control (i.e. producing only good units). As each unit is produced, there is a probability p that the system goes out of control, at which time all subsequent units (until the end of the production run) are defective. The time until the process goes out of control therefore follows a geometric distribution. Porteus used this model to study the optimal setup investment in relation to reducing the probability p of the process going out of control. Other researchers have used this model to investigate the economic benefits of reducing setup costs (see [6,5]). More recently, Affisco et al. [1] investigated investment in quality improvement and setup cost reduction in a joint supplier–customer system with defects produced at a known constant rate. Rosenblatt and Lee [14] present a model where the process goes out of control after some exponentially distributed time. Once out of control, the process produces units at a constant defect rate a. We extend prior work in this area by providing a general method for modeling production yields. Several papers (see [2,1]) model systems in which defects are produced at a known constant rate. Others, such as Porteus, model the production of defects as a random process. Yano and Lee [18] provide a representative survey of research involving lot sizing with random yields, together with a useful description of the modeling issues involved. They list several methods for modeling random yield that are represented in the literature, which include: 1. modeling defects as a Bernoulli process, where each unit is defective with probability p (see [4,19]); 2. describing a distribution for the overall fraction of defective units (see [4,15,9]); 3. specifying a distribution for the time in which the process remains in control (i.e. producing good units), after which the process is out of control (a fraction of the units are defective). In the first two methods, the fraction of defective units is stochastically constant with respect to the length of the production run. In the third

case it is stochastically increasing with the duration of the run (see [4,11,13]). Yano and Lee [18] also point out that expected cost is the dominant performance measure among models described by the literature. In this context, our approach generalizes the three methods described above, as well as the case in which defects are produced at a known rate which may vary over time. That is, we allow any random yield function as long as all defective units are repaired instantly at some per unit cost. If all units are not repaired instantly, then one must allow for the possibility of lost sales or backorders, or impose conditions on the random yield function such that the effective production rate always exceeds the demand rate. It should be noted that while our approach generalizes several prior methods, some recent research has addressed other aspects of the EPQ problem such as stochastic demand [16], coordination between buyer and seller [3,7] and choosing an inspection schedule [8,12]. We focus on systems in which the process deteriorates over time, although we point out where our results may be generalized (in the opposite direction) for a process that improves over time. We develop properties of the optimal production quantity and optimal cost and consider the options of investing in reducing setup cost and improving process quality. Expressions for the marginal value of setup cost reduction and process improvement are also developed. This paper is organized as follows. In Section 2 we present our model formulation and establish results regarding the optimal solution. In Section 3 we investigate the percentage cost penalty of using the standard economic production quantity even when the process is imperfect. In Sections 4 and 5 we explore optimizing investment in setup cost reduction and process improvement. Concluding remarks are offered in Section 6.

2. Model formulation and notation We consider the production system of a single item on a single machine which produces some (time-varying) fraction of defective items. Specifi-

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cally, we assume that the fraction of defective goods produced is described by u(x) at any time x, with 0 6 u(x) 6 1. As noted above, we focus on the situation in which the process deteriorates over time, u 0 (x) P 0. All defective units are repaired instantaneously at a per unit cost s. If repair takes some constant time then the analysis either remains the same if units are considered to count in inventory when they are first produced, or the inventory expression is slightly modified if units are considered to count in inventory when the final, non-defective unit is produced. In addition, we define the following notation: R D h h* v h0 i h K

production rate per unit time, demand rate per unit time, actual production run time, optimal production run time, unit production cost, non-financial holding cost per unit per unit time, opportunity cost of capital per unit time, h0 + iv, holding cost per unit time, setup cost for each production run.

The separation of the per unit holding cost into financial and non-financial pieces facilitates the discussion of investment in setup cost and process improvements in later sections. The total cost per unit time as a function of h is given by KD hðR  DÞh sD CðhÞ ¼ þ þ Rh 2 h

Z

h

uðxÞ dx:

ð1Þ

0

Recall that our approach generalizes the three methods from Yano and Lee mentioned above. The first approach (modeling defects as a Bernoulli process with parameter p) corresponds to a constant defect rate u(x) = p. The second approach (describing the constant fraction of defective items with random variable Y) is also equivalent to the constant defect rate case (u(x) = E[Y]) since we do not incorporate the variance of defective units into our model. Finally, for the third method (specifying a random time until the process goes out of control) consider the model in Rosenblatt and Lee [14] where the process goes out of control

3

after an exponential time T with mean b1 after which defects are produced at the constant fraction a. Let N(h) be the number of defective units produced for a run length of h, N ðhÞ ¼ aR maxðh  T ; 0Þ; E½N ðhÞ ¼

Z

ð2Þ

h

aRðh  tÞbebt dt

ð3Þ

  a 1  ebt dt:

ð4Þ

0

¼R

Z

h

0

This is equivalent to our model with u(x) = a(1  ebx). As Yano and Lee point out, the geometric time to failure in PorteusÕ model can be viewed as the discrete approximation to the Rosenblatt and Lee model. For notational convenience, define a = KD/R, b = h(R  D)/2, c = sD. With these substitutions a c CðhÞ ¼ þ bh þ h h

Z

h

uðxÞ dx;

  Z h a c C ðhÞ ¼ 2 þ b þ 2 huðhÞ  uðxÞ dx : h h 0 0

ð5Þ

0

ð6Þ

Note that the first two terms of Eq. (5) form the standard EPQ cost function, which pffiffiffiffiffiffiffiffiis convex and b has a unique minimizer at h a=b with optimal pffiffiffiffiffi cost per unit time 2 ab. Our first result establishes that with a deteriorating process, the optimal run length is shorter than the run length suggested by the classical EPQ. Furthermore, the faster a process deteriorates, the shorter the optimal run length. While this is intuitive, we show (a) that the result holds for our generalized model, (b) how to obtain this run length and (c) that the cost penalty for using the standard EPQ model can be significant. Proposition 1 1. There is a unique optimal run length h* that is less than (greater than) b h for defect rate u(x) increasing (decreasing) over time. 2. The faster a process deteriorates (improves), the shorter (longer) the optimal run length. 3. The optimal cost is given by C(h*) = 2bh* + cu(h*).

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Rh Proof. Define f ðhÞ ¼ ðc=hÞ 0 uðxÞdx. This is the cost of repairing defective units per unit time. When u(x) is non-decreasing, f(h) Ris non-decreash ing sinceR f 0 ðhÞ ¼ ðc=h2 ÞðhuðhÞ  0 uðxÞdxÞ and h huðhÞ P 0 uðxÞdx. Furthermore, since 0 6 u(x) 6 1, 0 6 f(h) 6 c. Now, the total cost function, C(h), is the sum of a convex function and a bounded, non-decreasing function, implying that there exists a unique minimizer, h 6 b h. Similarly, when u(x) is non-increasing, f is bounded and non-increasing, implying that C(h) has a unique minimizer h P b h. Next, consider two defect rate functions u1, u2 with u01 ðxÞ P u02 ðxÞ P 0 for all x P 0. Define ~ u1 ðxÞ ¼ u1 ðxÞ  u1 ð0Þ with ~ u2 similarly defined. Let C1, C2 represent cost functions associated with defect rate functions u1, u2 respectively. C 01 ðhÞ

  Z h a c ~ ¼ 2 þ b þ 2 h~ u1 ðhÞ  u1 ðxÞ dx ; h h 0 ð7Þ

C 01 ðhÞ

  Z h a c ~ ¼ 2 þ b þ 2 h~ u2 ðhÞ  u2 ðxÞ dx : h h 0

~01 ðxÞ P ~ u02 ðxÞ P 0, C 01 ðhÞ P C 02 ðhÞ. SimiSince u 0 larly, if ~ u1 ðxÞ 6 ~ u02 ðxÞ 6 0, C 01 ðhÞ 6 C 02 ðhÞ. Finally, if the first order condition is satisfied at h*, setting (6) equal to zero yields: Z h c a uðxÞ dx ¼ bh þ cuðh Þ   ; ð9Þ  h 0 h ð10Þ



Now we derive expressions for the changes in optimal run length h* and optimal cost C* C(h*) with respect to the cost parameters a, b and c. These expressions will be useful in establishing later results. The first order condition will be satisfied at h*, implying " 2





a þ bh þ c h uðh Þ 

Z

h

oh oh þ ch u0 ðh Þ ¼ 0: oa oa

ð12Þ

Recall that the optimal cost is given by Cðh Þ ¼ 2bh þ cuðh Þ; ð13Þ so after differentiating and substituting (12), oC  oh oh 1 ¼ 2b þ cu0 ðh Þ ¼ : h oa oa oa

ð14Þ

Differentiating (11) with respect to b yields: h2 þ 2bh

oh oh þ ch u0 ðh Þ ¼ 0; ob ob

ð15Þ

implying oC  oh oh ¼ 2h þ 2b þ cu0 ðh Þ ¼ h : ob ob ob

ð16Þ

Differentiating (11) with respect to c yields: " # Z h   oh   þ h uðh Þ  uðxÞ dx 2bh oc 0 þ ch u0 ðh Þ

oh oc

¼ 0; ð8Þ

Cðh Þ ¼ 2bh þ cuðh Þ:

1 þ 2bh

implying R h uðxÞ dx oC  ¼ 0  : h oc

ð17Þ

ð18Þ

Rearranging terms in Eq. (12): 

oh 1 ¼ P 0: 2bh þ ch u0 ðh Þ oa

ð19Þ

Similarly, rearranging terms in Eqs. (15) and (17) establishes: oh h2 ¼ 60  ob 2bh þ ch u0 ðh Þ

ð20Þ

and

h i R h   oh  h uðh Þ  0 uðxÞ dx ¼ 6 0: 2bh þ ch u0 ðh Þ oc

ð21Þ

# uðxÞ dx ¼ 0:

ð11Þ

0

Differentiating this expression with respect to a gives:

3. Cost penalty of using the EPQ Despite the strong underlying assumptions, the EPQ can often provide a reasonable solution for

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more complicated settings (see [16]). We now examine the percentage cost penalty of using the EPQ when the production process is imperfect. Define pffiffiffiffiffi c Z bh b b uðxÞ dx ð22Þ C Cð hÞ ¼ 2 ab þ b h 0 and C  Cðh Þ ¼ 2bh þ cuðh Þ:

ð23Þ

The percentage cost penalty if using the EPQ rather than the optimal quantity is given by b  C C D : ð24Þ C Analyzing D for general u(x) appears to be difficult. We first present general expressions for the change in cost at EPQ and then analyze the special case where the production process deteriorates linearly: 2 3 rffiffiffi Z bh b oC b c ob h 4b b ¼ huð hÞ  uðxÞ dx5 ð25Þ þ 2 oa a b 0 h oa 2 0 13 Z bh 14 c @b b 1 þ pffiffiffiffiffi huð hÞ  ¼ uðxÞ dxA5; b 0 h 2b h ab 2 3 rffiffiffi Z bh b b oC a c o h 4b b ¼ huð hÞ  uðxÞ dx5 þ 2 ob b b 0 h ob 2 0 13 Z bh c @b huðb hÞ  ¼b h 41  uðxÞ dxA5; 2 b 0 2b h

ð26Þ ð27Þ

ð28Þ

Z a c h ða þ bxÞ dx CðhÞ ¼ þ bh þ h h 0 a ¼ þ ðb þ cb=2Þh þ ca; h a 0 C ðhÞ ¼ 2 þ ðb þ cb=2Þ; h rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a h ¼ ; b þ cb=2

ð30Þ ð31Þ ð32Þ ð33Þ

and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cb Cðh Þ ¼ 2 aðb þ Þ þ ca: 2 pffiffiffiffiffiffiffiffi b Since h ¼ a=b, rffiffiffi pffiffiffiffiffi cb a b Cð hÞ ¼ 2 ab þ þ ca 2 b 

ð34Þ

ð35Þ

and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffi 2 ab þ cb2 ab  2 aðb þ cb2 Þ pffiffiffiffiffi D¼ : pffiffi 2 ab þ cb2 ab þ ca

ð36Þ

The analysis of the percentage cost penalty is complicated by the constant term ca. We momentarily drop this constant term by defining b  ca, C  C   ca. b0 C C 0 Proposition 2. For the linear deterioration case with no constant term a (u(x) = bx), the percentage cost penalty of using the EPQ is: (1) not affected by the ordering cost K, (2) decreasing in the holding cost h and (3) increasing in the repair cost s. Proof.

and b oC ¼ oc

5

R bh 0

uðxÞ dx : b h

ð29Þ

b b Note that o C=oc P oC  =oc, and that o C=ob can be negative. That is, if a firm is going to use the EPQ, overall cost could be reduced by increasing the holding cost h.

b0 C 1 C 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b cb 1 þ  1: ¼ b þ cb=2 4 bðb þ cb=2Þ

D0

ð37Þ

Part (1) is obvious from inspection of D0.

3.1. Linearly increasing defect rate

oD0 c2 b2 ¼ ½bðb þ cb=2Þ3=2 6 0; ob 16

For illustrative purposes, we consider the special case of a process that deteriorates linearly, u(x) = a + bx with a, b P 0. From (5) we have:

oD0 bcb2 3=2 ¼ ½bðb þ cb=2Þ P 0: oc 16

ð38Þ



ð39Þ

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Proposition 3. For the linear deterioration case, the percentage cost penalty of using the EPQ is: (1) increasing in the ordering cost K, (2) decreasing in the holding cost h (3) increasing in the repair cost s and (4) increasing in the rate of deterioration b. Proof. If we differentiate D with respect to any one of the three cost parameters, we have D0 ¼

b 0  ðC b 0 þ caÞC 0 ðC 0 þ caÞ C 0 0 ðC 0 þ caÞ

ð40Þ

2

0

¼

b C b 0 C 0 þ cað C b 0  C 0 Þ C 0 C 0 0 0 0 2

ðC 0 þ caÞ

ð41Þ

:

Note that the denominator is always positive, and we are only interested in the sign of this expression. From the proof of Proposition 2, the sum of the first two terms in the numerator is zero, negative and positive for derivatives with respect to a, b and c respectively. It is therefore sufficient to show

Differentiating (35), pffiffiffi pffiffiffi b 2 b þ cb=ð2 bÞ b þ cb=4 oC pffiffiffi ¼ ¼ pffiffiffiffiffi : oa 2 a ab Differentiating (34), rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oC  b þ cb=2 ¼ a oa rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bðb þ cb=2Þ ¼ ab sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 b þ bcb=2 þ ðcb=4Þ  ðcb=4Þ ¼ ab sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ðb þ cb=4Þ2  ðcb=4Þ2 o C : ¼ 6 oa ab

ð45Þ

ð46Þ ð47Þ ð48Þ ð49Þ

Similarly, oC  ¼ ob

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a b þ cb=2

ð50Þ

rffiffiffi  a cb 1 : b 4b

ð51Þ

and

b0 oC oC  P 0; oa oa

ð42Þ

b0 oC 6 ; ob ob

ð43Þ

oC 0

and b0 oC oC  P 0 oc oc

ð44Þ

b oC ¼ ob

 b Note that when c = 0, oC ¼ oobC . Furthermore, ob 3=2 pffiffiffi  o2 C 0 b a 1 ¼ ð52Þ 4 b þ cb=2 oboc

and pffiffiffi  3=2 b0 o2 C b a 1 ¼ ; 4 b oboc

to establish our first three results.

ð53Þ

Table 1 Production quantities, costs and percentage defective units at both the optimal run length and the run length implied by EPQ

Rh*

Optimal values C*

% Defective

At EPQ Rb h

b C

% Defective

D (%)

Base case

242

$1077

2.01

387

$1170

2.61

8.66

K = 90 K = 110

230 254

$1034 $1117

1.96 2.06

367 406

$1123 $1215

2.53 2.69

8.55 8.75

h=7 h=9

248 236

$1056 $1097

2.03 1.98

414 365

$1164 $1178

2.73 2.52

10.24 7.43

s = 22 s = 28

251 234

$1016 $1136

2.05 1.97

387 387

$1091 $1248

2.61 2.61

7.44 9.85

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establishing b 0 oC  oC 6 0 ob ob

ð54Þ

oC  b ¼aþ 2 oc

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a b þ cb=2

ð55Þ

ð58Þ

Eq. (58) together with Eqs. (19)–(21) and Proposition 1-(2) immediately establish the following results. Proposition 4. For a deteriorating process, the marginal value of setup cost reduction is higher for

and b oC b ¼aþ oc 2

From Eq. (14) and the definition a = KD/R, the marginal change in optimal cost cost as a function of K is oC  oC  oa D ¼ : ¼ oK oa oK Rh

as desired. Next,

7

rffiffiffi a : b

ð56Þ

Finally, to establish part (4), pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi oD c2 ab 2a 2b þ cb þ 2b a  2a 2b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi2 ; pffiffiffi ob 2b þ cb 4 ab þ cb a þ 2ca b ð57Þ which is greater pthan ffiffiffiffiffi or equal to zero since pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2a 2b þ cb P 2a 2b. h Example 1. Consider the numerical example given in [10]: K = 100, v = 50, h0 = 0.5, i = 0.15 and D = 1000. We assume R = 1200, s = 25 and u(x) = 0.01 + 0.1x. The quantity given by the standard EPQ is Rb h ¼ 387:3, resulting in a cost of $1170. The optimal quantity is Rh* = 241.9, resulting in a cost of $1077, a percentage difference of 8.6%. Table 1 shows production quantities, costs, percentage of defective units and the cost penalty for this base case and for cases with small changes in the cost parameters K, h and s.

4. Optimization of setup cost reduction In this section we consider investing in reducing the setup cost. First, we develop an expression for the marginal value of setup cost reduction for general u(x). We analyze how the cost parameters K, h and s affect this marginal value. We then consider the linear deterioration case and a specific functional form for investment in setup cost reduction.

1. 2. 3. 4.

Smaller setup cost K; Larger holding cost h; Larger repair cost s; Faster deteriorating process (larger u 0 (x)).

Note also, that the absolute number of defective units produced is increasing in K, so any investment in setup cost reduction will decrease the number of defective units. 4.1. Linear deterioration Following Porteus [10], we consider a logarithmic investment function, MK(K) = A  B ln (K). That is, an investment of MK(K) is required to reduce the setup cost to K from the current level K0. As described in [10], one could fit this functional form by estimating the investment x, required to reduce setup cost from K0 to a specific K0(1  d) and using the fact that MK(K0) = 0. It is easy to verify that B = x/ln (1  d) and A = B ln (K0). Since we deal with the undiscounted model, an opportunity cost of iMK(K) is charged per unit time. With logarithmic investment function MK and linear deterioration rate u(x) = a + bx the total cost function is gðKÞ ¼ iðA  B lnðKÞÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KDðhðR  DÞ þ sbDÞ þ asD þ R

ð59Þ

from Eq. (34) and the definitions of a, b and c. Setting the first derivative, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iB DðhðR  DÞ þ sbDÞ 0 g ðKÞ ¼  þ ; ð60Þ K 2KR

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equal to zero yields 2

K ¼

2ðiBÞ R : DðhðR  DÞ þ bsDÞ

ð61Þ

It is easily verified that g 0 (K) < ( > )0 for K < ( > )K*, implying that g is pseudo-convex, thus K* is a unique cost minimizer. Note that unless one wants to permit positive compensation for increasing setup time, K* > K0 would imply the optimal investment level is zero. Since h* is increasing in K, the marginal value of setup cost reduction (58) is decreasing in K in the general case as well. This implies that for any investment function that has marginally increasing cost of setup cost reduction, there will be a unique solution. Example 2. Consider the data from Example 1, now with setup cost reduction function A  B ln (K). Suppose a 10% setup cost reduction would cost $200, implying A = 8742 and B = 1898. The optimal setup cost is K* = 47 with Rh* = 267. Table 2 shows production quantities, costs and percentage of defective units for the original optimal solution (K = K0) and when setup cost reduction is considered. Cases with small changes in the cost parameters h and s are also shown.

5. Optimization of process quality improvement In this section we consider investing in improving process quality. We examine two kinds of process improvement: (1) a constant downward shift 

in the defect rate u(x) such that the new process produces defects at a rate ~uðxÞ ¼ uðxÞ  , and (2) a scale factor p such that the new defect rate is ~uðxÞ ¼ puðxÞ. We first develop expressions for the marginal values of each kind of process improvement for general u(x), analyzing how the cost parameters K, h and s affect these marginal values. We then consider the linear deterioration case and a specific functional form for investment in process improvement. Proposition 5 1. Let  be a constant shift in u(x) such that the new defect rate function is ~uðxÞ ¼ uðxÞ  . (i) The optimal run length h* is not affected by the shift, and (ii) the rate of change in optimal cost as a function of the (downward) shift  is sD. 2. Let 0 6 p 6 1 define a scaling of u(x) such that the new defect rate function is ~uðxÞ ¼ puðxÞ. (i) The optimal run length h* is decreasing in p (the optimal run length gets longer as the process is improved), and (ii) the rate of change in optimal cost as a function of p is given by the defect repair cost per unit time: R h oC  sD 0 uðxÞ dx ¼ : ð62Þ h op 3. For a deteriorating process, the marginal value of process improvement (by scaling) is larger for greater K and smaller h. 4. The marginal value can increase or decrease with s, and the total number of defects can increase or decrease with p.

Table 2 Optimal production quantities, costs and percentage defective units with and without investment in setup cost reduction Original values

With setup cost reduction

Rh*

C*

% Defective

K*

Rh*

C*

% Defective

Base case

242

$1077

2.01

47

167

$1032

1.69

h=7 h=9

248 236

$1056 $1097

2.03 1.98

50 45

175 159

$1017 $1045

1.73 1.66

s = 22 s = 28

251 234

$1016 $1136

2.05 1.97

51 44

180 155

$980 $1082

1.75 1.65

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~ðxÞ ¼ uðxÞ  , the Proof. With a new defect rate u original cost function (1) can be written as Z KD hðR  DÞh sD h þ þ uðxÞ dx  sD: CðhÞ ¼ Rh 2 h 0 ð63Þ This final term is a constant with respect to h, so h* is clearly not affected by . Also, oC/o is clearly the constant sD. Next, to show part two of the proposition, consider the scaling ~ uðxÞ ¼ puðxÞ. One can re-write the original total cost function (1) as CðhÞ ¼

KD hðR  DÞh psD þ þ Rh 2 h

Z

h

uðxÞ dx:

ð64Þ

0

From Eq. (21) it is clear that Rh* is decreasing in h p. The expression oC  =op ¼ ðsD 0 uðxÞ dxÞ=h follows from Eq. (18). To show part 3, note that when the defect rate is non-decreasing (u 0 (x) P 0), the defect repair cost at the optimal solution (which is also equal to oC*/ op) is increasing in h* when s and D are constant. From Eqs. (19) and (20) it is clear that increasing K or decreasing h will increase h*, thus increasing o C*/op. Finally, to establish part 4, consider a process with a constant defect rate a. The defect repair cost at any run length is the constant asD which is clearly increasing in s. Next, consider a process that is defect free for some very short initial interval, u(x) = 0, 0 6 x 6 t, with a constant defect rate thereafter. For small enough s, the optimal run length will be greater than t and oC*/ op > 0. As s approaches 1, the optimal run length

9

will become t, and the defect repair cost will drop to zero. Now, note that the total number of defects is equal to oC*/op divided by the constant s. With a constant defect rate, it is clear that the total number of defects is increasing in p. Now, consider a process that produces no defects for 0 6 x 6 t and has constant defect rate pa thereafter. For very large s, moving p from 0 to a positive value will shift the run length from b h to t, decreasing total defects. h 5.1. Linear deterioration We consider the case where process improvement scales u(x) with some constant p. We use a similar logarithmic function /(p) = B ln (p). There is no constant term A in /(p) since we assume that keeping the current defect rate (p = 1) requires no investment, /(1) = B ln (1) = 0. Again, one could fit this functional form by estimating the investment required to scale down the defect rate by a specific p. It is worth noting that given this logarithmic investment form, the percentage of defective units at the optimal value for p does not depend on K and h. We seek to minimize wðpÞ ¼ C  ðpÞ þ /ðpÞ:

ð65Þ

Differentiating, oC  ðpÞ iB  op p R h sD 0 uðxÞ dx iB ¼  : h p

w0 ðpÞ ¼

ð66Þ ð67Þ

Setting w 0 (p*) = 0 implies

Table 3 Optimal production quantities, costs and percentage defective units with and without investment in process improvement Original values

With process improvement

Rh*

C*

% Defective

p*

Rh*

C*

% Defective

Base case

242

$1077

2.01

0.52

288

$1011

1.14

K = 90 K = 110

230 254

$1034 $1117

1.96 2.06

0.53 0.50

271 304

$976 $1045

1.14 1.14

h=7 h=9

248 236

$1056 $1097

2.03 1.98

0.51 0.53

300 277

$987 $1035

1.14 1.14

s = 22 s = 28

251 234

$1016 $1136

2.05 1.97

0.59 0.46

288 288

$975 $1044

1.29 1.02

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p

M. Freimer et al. / European Journal of Operational Research xxx (2005) xxx–xxx

R h 0

uðxÞ dx iB : ¼ sD h

ð68Þ

For example, reducing K would reduce the optimal run length, h*, but the optimal p* would be higher such that the number of defective units produced remained constant. This is illustrated in Table 3. With linear deterioration rate u(x) = a + bx, finding the optimal p involves finding the appropriate root of a cubic equation. As such, we do not present the algebra. Example 3. Consider the data from Example 1 now with process improvement investment function /(p) = B ln (p). Suppose a 10% reduction in defect rate would cost $200, implying B = 1898. Table 3 shows production quantities, costs and percentage of defective units for the original optimal solution (K = K0) and when process improvement is considered. Cases with small changes in the cost parameters K, h and s are also shown.

6. Conclusions In this paper we consider the classical economic production quantity model with defects produced according to some time-varying function u(x). Our results hold for any random yield function as long as units are repaired instantly. With a deteriorating process, the optimal run length is shorter than the run length implied by the EPQ. Furthermore, the faster the process deteriorates the shorter the optimal run length. For the special case of linear deterioration, the cost penalty for using the EPQ instead of the optimal quantity is increasing in the setup cost K, the defect repair cost s and the rate of process deterioration b, and decreasing in the holding cost h. We consider the opportunity to invest in reducing setup cost and improving process quality. For a general time varying defect rate u(x), the marginal value of setup cost reduction is inversely proportional to optimal run length while the marginal value of process improvement (as modeled by a constant scaling p of the defect rate) is increasing in the optimal run length. Any investment in setup

cost reduction will result in a reduction in the number of defects produced. Interestingly, the total number of defects can increase or decrease with the scaling p. For the logarithmic investment function investigated, the number of defects at the optimal run length is unaffected by the holding cost h and the setup cost K. These results may provide guidance for manufacturing managers regarding the relative value of setup cost reduction and process improvement investments.

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