Inspection scheduling for imperfect production processes under free repair warranty contract

European Journal of Operational Research 183 (2007) 238–252 www.elsevier.com/locate/ejor

Stochastics and Statistics

Inspection scheduling for imperfect production processes under free repair warranty contract B.C. Giri, T. Dohi

*

Department of Information Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, Japan Received 26 August 2004; accepted 26 September 2006 Available online 14 December 2006

Abstract The paper considers scheduling of inspections for imperfect production processes where the process shift time from an ‘in-control’ state to an ‘out-of-control’ state is assumed to follow an arbitrary probability distribution with an increasing failure (hazard) rate and the products are sold with a free repair warranty (FRW) contract. During each production run, the process is monitored through inspections to assess its state. If at any inspection the process is found in ‘out-of-control’ state, then restoration is performed. The model is formulated under two different inspection policies: (i) no action is taken during a production run unless the system is discovered in an ‘out-of-control’ state by inspection and (ii) preventive repair action is undertaken once the ‘in-control’ state of the process is detected by inspection. The expected sum of pre-sale and post-sale costs per unit item is taken as a criterion of optimality. We propose a computational algorithm to determine the optimal inspection policy numerically, as it is quite hard to derive analytically. To ease the computational difficulties, we further employ an approximate method which determines a suboptimal inspection policy. A comparison between the optimal and suboptimal inspection policies is made and the impact of FRW on the optimal inspection policy is investigated in a numerical example.  2006 Elsevier B.V. All rights reserved. Keywords: EMQ; Imperfect production processes; Inspection; Sequential policy; Warranty

1. Introduction The classical economic manufacturing quantity (EMQ) model assumes that the production process always remains in ‘in-control’ state and produces items of acceptable quality. This assumption may not be true in general. In many practical situations, a random process shift from an ‘in-control’ state to an ‘out-of-control’ state is frequently observed due to equipment uncertainty or aging and as a result, substandard items are produced. Maintenance by inspection is a useful means of controlling the quality of the products. Recently, a great deal of research efforts have been devoted to extend the classical EMQ model in order to address the strategic issues such as quality and maintenance along with production quantity. *

Corresponding author. Fax: +81 824 22 7195. E-mail address: [email protected] (T. Dohi).

0377-2217/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.09.062

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Initial researches had been carried out to study the impact of process deterioration on the optimal production policy by Rosenblatt and Lee (1986) and Porteus (1986), the impact of machine breakdown on lot sizing decisions by Groenevelt et al. (1992) and the joint effect of process deterioration and machine breakdown by Makis and Fung (1998). Since this paper focuses on process deterioration, we confine ourselves in reviewing the following related works. In an EMQ modeling framework, Rosenblatt and Lee (1986) assumed an exponential process shift distribution whereas Porteus (1986) assumed that the process goes to an ‘out-of-control’ state with a given probability at each time when it produces an item. In their models, monitoring the state of the production process during a production run was not considered. Later, Lee and Rosenblatt (1987), Porteus (1990) and Lee and Park (1991), among others, modeled from different view points the situation where the state of the production process is identified through inspections. Lee and Rosenblatt (1987) assumed a periodic (constant) inspection to detect the state of the production process and the cost of a defective item regardless of whether it is reworked before sale or the cost incurred for reworking after sale (i.e., warranty cost) is the same. They determined the optimal lot size and inspection schedule when the ‘in-control’ periods are exponentially distributed and showed that the optimal lot size is smaller than that of the classical EMQ model. Lee and Park (1991) addressed the problem of joint determination of production cycle and process inspection intervals by considering the difference between reworking cost before sale and warranty cost after sale when the process shifts to an ‘out-of-control’ state after an exponentially distributed production time. Porteus (1990) studied the impact of time delay in obtaining the results of each inspection on lot sizing. Rahim and Banerjee (1993) determined jointly the optimal design parameters on an x control chart and preventive replacement time for a production system with an increasing failure rate (IFR). For a deteriorating production process with IFR, Rahim (1994) investigated an integrated model for production quantity, inspection schedule and control chart design for a typical process which is subject to non-Markovian random shock. He approximately derived the length of inspection schedule under the assumption that the cumulative hazard over each interval is equivalent. Makis (1998) derived the optimal lot sizing and inspection policy for an imperfect production process assuming that the ‘in-control’ periods are generally distributed and inspections are imperfect i.e., the true state of the process is not necessarily revealed through inspections. Tseng (1996) incorporated a preventive maintenance policy to enhance the system reliability instead of an inspection policy into an imperfect EMQ model. Process inspection or preventive maintenance can only reduce the number of defective items in a deteriorating production system but cannot make the system absolutely perfect. Because of this deficiency, manufacturers often offer the buyers a warranty with the sale of the products, that is, if their products do not perform satisfactorily within the specified warranty period then they will be responsible to either repair or replace the items with free of charge. Thus, offering a warranty results in additional costs to the manufacturer due to servicing of any item that arises during the warranty period. Djamaludin et al. (1994) studied the effect of product warranty on the optimal production lot size assuming that the production system can shift to an ‘out-of-control’ state with a given probability at each time when an item is produced. Describing the production process as a two-state discrete time Markov chain, they developed a cost model under free repair warranty (FRW) policy to determine the optimal lot size by minimizing the expected cost per unit item. Yeh and Lo (1998) studied the effects of a FRW policy on the optimal production lot size and optimal burn-in time. Recently, Yeh et al. (2000) reformulated Djamaludin et al.’s (1994) model considering exponentially distributed process shift distribution. Wang (2004) and Wang and Sheu (2003) further extended Yeh et al.’s (2000) model by incorporating continuous and discrete time general shift distributions, respectively. Wang (2004, 2005) developed EMQ models without considering inspection and maintenance during a production run. At the end of a production lot, the system is inspected once to detect the state of the production process. If the system is in ‘outof-control’ state then restoration is done with some additional cost. Otherwise, preventive maintenance is carried out to return back the system to the original ‘in-control’ state. The purpose of this paper is to incorporate sequential inspections during a production run in an imperfect EMQ model where the process shift time from an ‘in-control’ state to an ‘out-of-control’ state follows a general probability distribution with increasing failure rate (IFR). The produced items are repairable and are sold with a free repair warranty (FRW). The objective of this study is to determine jointly the optimal production run length, the optimal number of inspections during a production run and the inspection schedule which minimize the expected cost per unit item where the resulting inspection sequence is not always constant.

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The paper is organized as follows. The assumptions and notation of the proposed model are given in the next section. The basic model and two proposed inspection policies – I and II are described in Section 3. In Section 4, we formulate and analyse the model under inspection policy I. A computational algorithm is proposed to find the optimal number of inspections and optimal inspection schedule numerically, as it is quite hard to derive them analytically. To ease the computational difficulties, we also employ an approximate method which determines only a suboptimal inspection policy. Section 5 treats the model with inspection policy II. A numerical example is presented in Section 6 to illustrate the developed model and investigate the impact of FRW on the optimal inspection policy. Finally, the paper is concluded in Section 7. 2. Notations and assumptions The following notations are used throughout the paper: f(Æ), F(Æ) probability density function, distribution function of the time to process shift F ðÞ the survivor function i.e., F ðÞ ¼ 1  F ðÞ d (>0) demand rate p (>d) production rate cs (>0) setup cost cm (>0) manufacturing cost per unit product ch (>0) inventory holding cost per unit product per unit time cr (>0) cost of each minimal repair v0 (>0) inspection cost v1 (>0) preventive maintenance cost w (>0) free repair warranty period n (P1) number of inspections undertaken during each production run Ti elapsed time from the beginning of the production run until the ith inspection takes place, "i = 1, 2, . . . , n {T1, T2, . . . , Tn} inspection time sequence T (=Tn) production run length ti the ith inspection interval, i.e., ti = Ti  Ti1, "i = 1, 2, . . . , n RðsÞ a0 + a1s, the linear restoration cost where a0 > 0, a1 P 0 and s, the detection delay time h1 probability that the produced item in the ‘in-control’ state is non-conforming h2 (>h1) probability that the produced item in the ‘out-of-control’ state is non-conforming r1(t) hazard rate of a conforming item r2(t) hazard rate of a non-conforming item; r2(t)  r1(t), 0 6 t < 1 The proposed model is based on the following assumptions: (1) The production facility starts always in an ‘in-control’ state. It may shift to an ‘out-of-control’ state at any random time but never breaks down. (2) The duration of ‘in-control’ period follows an arbitrary probability distribution with an increasing failure (hazard) rate. (3) Inspections are undertaken during a production run where each inspection is perfect. (4) If any inspection finds that the process is in ‘out-of-control’ state then restoration is done with negligible time and that the restoration cost is a function of detection delay. Here restoration means minor repair or replacement that returns the system to ‘good-as-new’ condition, that is, the failure rate after repair is significantly lower than the failure rate at the time of failure. Such a repair policy is particularly appropriate for a simple system where repairing or replacement of its mechanical components such as rotating shaft, valve or cam, etc. affect the failure rate of the system. (5) Non-conforming items are produced not only in ‘out-of-control’ state but also in ‘in-control’ state. In this paper, we assume that the non-conforming items are operational when put into use (Type B nonconforming items, see Djamaludin et al. (1994)) but they have performance characteristics inferior to

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241

those of conforming items. These non-conforming items can be detected only through time-testing for some significant time period in contrast to the other type (Type A, see Djamaludin et al. (1994)) of non-conforming items which can be detected either by inspection or testing for a very short time duration. (6) The produced items are repairable and are sold with a free minimal repair warranty. 3. Model description Consider a single-unit, single-item production system which starts in an ‘in-control’ state to produce items at a rate p (>0) to meet a demand rate d (
Stock level : Inspection time

p-d

T0 = 0

T1

T2

.

. Ti

-d

.

.

.

.

Tn=T

Production run length Production-inventory cycle Fig. 1. Configuration of the EMQ model with inspections.

Time

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4. Model analysis under inspection policy I We derive the expected sum of pre-sale and post-sale costs per unit item which will be considered as a criterion of optimality. For a production lot pT, the expected holding cost per unit item is ch(p  d)T/(2d) and the expected setup and inspection costs per unit item is (cs + nv0)/(pT). Let Pj is the probability that the production process needs to be restored at time Tj, "j = 1, 2, . . . , n. Then, we have P 1 ¼ F ðT 1 Þ; P 2 ¼ F ðT 2 Þ  F ðT 1 Þ þ P 1 F ðT 2  T 1 Þ and in general, Pj ¼

j1 X

P i fF ðT j  T i Þ  F ðT j1  T i Þg;

j ¼ 1; 2; . . . ; n;

i¼0

where P0 = 1, see Tseng (1996) for detailed derivations. Let Ni denote the number of non-conforming items produced in the time interval (Ti1, Ti], i = 1, 2, . . . , n and sj denote the random elapsed time of the first shift from the ‘in-control’ state to the ‘out-of-control’ state given that the previous restoration was held at time Tj, j = 0, 1, 2, . . . , i  1. Then the number of non-conforming items produced in (0, T1] is given by  N1 ¼

ph1 s0 þ ph2 ðT 1  s0 Þ;

if T 0 < s0 6 T 1 ;

ph1 T 1 ;

otherwise:

Therefore, the expected number of non-conforming items produced in (0, T1] is EðN 1 Þ ¼

Z

T1

fph1 s0 þ ph2 ðT 1  s0 Þg dF ðs0 Þ þ 0

Z

1

ph1 T 1 dF ðs0 Þ ¼ pðh2  h1 Þ

Z

T1

F ðtÞ dt þ ph1 T 1

ð1Þ

0

T1

and the expected restoration cost in the time interval (0, T1] is Z T1 RðT 1  s0 Þ dF ðs0 Þ:

ð2Þ

0

Similarly, the number of non-conforming items produced in the second period (T1, T2] is given by  ph1 ðsj  T 1j Þ þ ph2 ðT 2  T j  sj Þ; if T 1  T j < sj 6 T 2  T j ; N2 ¼ ph1 ðT 2  T 1 Þ; if sj > T 2  T j 8j ¼ 0; 1: Therefore, the expected number of non-conforming items produced in the time interval (T1, T2] is "Z # Z 1 1 T 2 T j X EðN 2 Þ ¼ Pj fph1 ðsj  T 1j Þ þ ph2 ðT 2  T j  sj Þg dF ðsj Þ þ ph1 ðT 2  T 1 Þ dF ðsj Þ j¼0

T 1 T j

and the expected restoration cost in the time interval (T1, T2] is Z T 2 T j 1 X Pj RðT 2  T j  sj Þ dF ðsj Þ: j¼0

ð3Þ

T 2 T j

ð4Þ

T 1 T j

By the same argument, the expected number of non-conforming items produced in the ith period (Ti1, Ti] is given by "Z # i1 T i T j X EðN i Þ ¼ Pj fph1 ðsj  T i1j Þ þ ph2 ðT i  T j  sj Þg dF ðsj Þ þ ph1 ðT i  T i1 ÞF ðT i  T j Þ ð5Þ j¼0

T i1 T j

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243

and the expected restoration cost in the ith period (Ti1, Ti] is Z T i T j i1 X Pj RðT i  T j  sj Þ dF ðsj Þ:

ð6Þ

T i1 T j

j¼0

Therefore, the expected sum of restoration and preventive maintenance costs per cycle is given by "Z # Z T i T j n1 X i1 n1 T n T j X X Pj RðT i  T j  sj Þ dF ðsj Þ þ Pj RðT n  T j  sj Þ dF ðsj Þ þ v1 F ðT n  T j Þ : i¼1

j¼0

T i1 T j

T n1 T j

j¼0

Hence, the expected pre-sale cost per unit item is ch ðp  dÞT n cs þ nv0 þ Aðn; T 1 ; T 2 ; . . . ; T n Þ ¼ cm þ pT n " 2d Z T i T j n1 X i1 X 1 þ Pj RðT i  T j  sj Þ dF ðsj Þ pT n i¼1 j¼0 T i1 T j (Z )# n1 T n T j X þ Pj RðT n  T j  sj Þ dF ðsj Þ þ v1 F ðT n  T j Þ :

ð7Þ

T n1 T j

j¼0

The fraction of non-conforming items produced in a production lot pT is given by "Z # n X i1 T i T j 1 X Pj fph1 ðsj  T i1j Þ þ ph2 ðT i  T j  sj Þg dF ðsj Þ þ ph1 ðT i  T i1 ÞF ðT i  T j Þ : pT i¼1 j¼0 T i1 T j

ð8Þ

We assume that the repair times are small compare to MTBF (mean time between failures) so that they can be negligible. As a result, the failures over the warranty period w occur according to a non-homogeneous Poisson process with intensity function equal to the failure rate. Therefore, the expected warrantyR servicing cost for a w conforming item is simply cr  E½number of repairs within the warranty period wR¼ cr 0 r1 ðtÞ dt. w Similarly, the expected warranty servicing cost for a non-conforming item is cr 0 r2 ðtÞ dt. The expected total post-sale cost per unit item is given by BðT 1 ; T 2 ; . . . ; T n Þ " (Z )# n X i1 T i T j   1 X ¼ cr 1  Pj ph1 ðsj  T i1j Þ þ ph2 ðT i  T j  sj Þ dF ðsj Þ þ ph1 ðT i  T i1 ÞF ðT i  T j Þ pT n i¼1 j¼0 T i1 T j (Z Z w n X i1 T i T j   cr X  r1 ðtÞdt þ Pj ph1 ðsj  T i1j Þ þ ph2 ðT i  T j  sj Þ dF ðsj Þ pT n i¼1 j¼0 0 T i1 T j ) Z w þ ph1 ðT i  T i1 ÞF ðT i  T j Þ  r2 ðtÞdt: ð9Þ 0

Hence, the expected total cost which is the sum of pre-sale and post-sale costs per unit item is C 1 ðn;T 1 ;T 2 ;. ..;T n Þ ¼ Aðn;T 1 ;T 2 ;. ..;T n Þ þ BðT 1 ; T 2 ;. ..;T n Þ " Z T i T j n1 X i1 ch ðp  dÞT n cs þ nv0 1 X þ þ Pj RðT i  T j  sj ÞdF ðsj Þ ¼ cm þ pT n i¼1 j¼0 2d pT n T i1 T j (Z )# n1 T n T j X Pj RðT n  T j  sj ÞdF ðsj Þ þ v1 F ðT n  T j Þ þ T n1 T j

j¼0

þ cr 

"Z

(Z

w 0

1 r1 ðtÞdt þ pT n

T i T j



Z 0

w

r2 ðtÞdt 

Z

w

r1 ðtÞdt

X n X i1

0

Pj

i¼1 j¼0



)#

ph1 ðsj  T i1j Þ þ ph2 ðT i  T j  sj Þ dF ðsj Þ þ ph1 ðT i  T i1 ÞF ðT i  T j Þ

:

T i1 T j

ð10Þ

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Our objective is to find the optimal number of inspections n and the inspection sequence {T1, T2, . . . , Tn} which minimize C1(n, T1, T2, . . . , Tn). Since n is a discrete variable and T1, T2, . . . , Tn are continuous variables, it is quite hard to determine the optimal values of n, T1, T2, . . . , Tn simultaneously. Let us suppose that the number of inspections n is given. Then the necessary conditions for a minimum of C1(n, T1,T2, . . . , Tn) are i1   oC 1 X  P j a0 f ðT i  T j Þ þ a1 F ðT i  T j Þ  a1 F ðT i1  T j Þ oT i j¼0 Z i X    P j a0 f ðT i  T j Þ þ a1 ðT iþ1  T i Þf ðT i  T j Þ þ cr



r2 ðtÞ dt 

0

j¼0

"

w

i1 X

Z



w

r1 ðtÞ dt

0

P j fph1 ðT i1  T j  T i1j Þf ðT i  T j Þ þ pðh2  h1 ÞF ðT i  T j Þ  ph2 F ðT i1  T j Þ þ ph1 g

j¼0



i X

#

P j fph1 ðT i  T j  T ij Þf ðT i  T j Þ þ ph2 ðT iþ1  T i Þ  f ðT i  T j Þ þ ph1 F ðT iþ1  T j Þg ¼ 0

j¼0

8i ¼ 1; 2; . . . ; n  2;

ð11Þ

i1 X   oC 1  P j a0 f ðT n1  T j Þ þ a1 F ðT n1  T j Þ  a1 F ðT n2  T j Þ oT n1 j¼0 Z w " X Z w n2 r2 ðtÞdt  r1 ðtÞdt P j fph1 ðT n2  T j  T n2j Þ  f ðT n1  T j Þ þ cr 0

0

j¼0

þ pðh2  h1 ÞF ðT n1  T j Þ  ph2 F ðT n2  T j Þ þ ph1 g 

n1 X

#

P j fph1 ðT n1  T j  T n1j Þf ðT n1  T j Þ þ ph2 ðT n  T n1 Þ  f ðT n1  T j Þ þ ph1 F ðT n  T j Þg

j¼0

¼ 0;

ð12Þ

and pT 2n

oC 1 ch ðp  dÞpT 2n  ðcs þ nv0 Þ  oT n 2d " Z T i T j n1 X i1 X  Pj RðT i  T j  tÞ  dF ðtÞ i¼1

þ

n1 X

j¼0

(Z

)#

RðT n  T j  tÞ dF ðtÞ þ v1 F ðT n  T j Þ

Pj T n1 T j

j¼0

þ Tn

T i1 T j T n T j

n1 X

P j fða0  v1 Þf ðT n  T j Þ þ a1 F ðT n  T j Þ  a1 F ðT n1  T j Þg

j¼0

Z w  Z w r2 ðtÞ dt  r1 ðtÞ dt  cr 0 " 0 (Z ) n X i1 T i T j X  Pj ðph1 ðt  T i1j Þ þ ph2 ðT i  T j  tÞÞ dF ðtÞ þ ph1 ðT i  T i1 ÞF ðT i  T j Þ i¼1

 Tn

j¼0

n1 X

T i1 T j

#

P j fph1 ðT n1  T j  T n1j Þ  f ðT n  T j Þ þ pðh2  h1 ÞF ðT n  T j Þ  ph2 F ðT n1  T j Þ þ ph1 g

j¼0

¼ 0:

ð13Þ

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245

Assuming that there exists a local minimum of C1(n, T1, T2, . . . , Tn) for any given n, we propose a sequential approach in the following algorithm to determine the optimal inspection schedule. 4.1. Inspection scheduling algorithm Step 1. Start with n ! 1 and solve Eq. (13) for Tn by using any one dimensional search technique. Calculate the expected cost per unit item. Step 2. Set n ! n + 1 and a trial value of T1. Step 3. Solve Eqs. (11) and (12) recursively to obtain the inspection sequence {T2, T3, . . . , Tn}. Step 4. Substitute the values of T1, T2, . . . , Tn in Eq. (13). If jRj, the absolute value of the LHS of Eq. (13) is less than a pre-assigned positive quantity  then go to Step 5. Otherwise, go to Step 6. Step 5. Calculate the expected cost per unit item. If it is lower than the previous one then go to Step 2. Otherwise, the previous inspection sequence for n  1 inspections is optimal. Stop. Step 6. For any chosen small d (>0), set T1 ! T1  d if R > 0; T1 ! T1 + d if R < 0 and then go to Step 3. The line search on n in the above algorithm may be time consuming and hence may not be appropriate in general. Because of difficulties in computing optimal non-periodic inspection policy, many researchers (Munford and Shahani, 1972; Munford, 1981) have suggested nearly optimal inspection policies in the maintenance literature. In the following, we employ the approximate method proposed by Munford (1981) to determine a sub-optimal inspection policy. 4.2. Sub-optimal inspection policy In our problem, the time to process shift from an ‘in-control’ state to an ‘out-of-control’ state is assumed to follow a general probability distribution with an increasing hazard (failure) rate. Let the inspection time sequence {T1, T2, . . . , Tn} be such that the integrated hazard over each interval is the same for all intervals, i.e., Z T jþ1 Z T1 rðtÞ dt ¼ rðtÞ dt; for j ¼ 0; 1; 2; . . . ; n  1; ð14Þ 0

Tj

where r(t) is the hazard function defined by rðtÞ ¼ f ðtÞ=F ðtÞ. In particular, we suppose that the time to shift follows a Weibull distribution whose probability density function is given by b1 ðktÞb

f ðtÞ ¼ kbðktÞ

e

;

t > 0; b P 1; k > 0:

For this failure probability distribution, the hazard rate function r(t) is rðtÞ ¼ kbðktÞ

b1

:

Substituting r(t) in Eq. (14) and simplifying, we get T j ¼ ðT bj1 þ T b1 Þ

1=b

;

j ¼ 1; 2; . . . ; n:

ð15Þ

Eq. (15) transforms the cost function C1(n, T1, T2, . . . , Tn) into a function of only two independent variables n and T1 only which can be handled easily. Then, using the recurrence relation (15), a suboptimal inspection schedule fT 01 ; T 02 ; . . . ; T 0n g can be obtained. Proposition 1. The suboptimal inspection sequence fT 01 ; T 02 ; . . . ; T 0n g is such that (i) t01 P t02 P P t03 P    P t0n , where t0j ¼ T 0j  T 0j1 ; 8j ¼ 1; 2; . . . ; n (ii) limn!1 nj¼1 t0j ¼ 1, (iii) t0j ¼ t01 ; 8j ¼ 2; 3; . . . ; n when b = 1 (exponential failure case), i.e., the periodic inspection policy is optimal. Proof. The proof is straightforward and hence it is omitted for brevity.

h

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5. Model analysis under inspection policy II Under inspection policy II, either restoration or preventive maintenance is performed at each inspection. So, the process is in the original ‘in-control’ state immediately after each inspection. In this case, the expected cost for restoration and preventive maintenance in a production cycle can be obtained as Z 1 n Z t i X Rðti  tÞ dF ðtÞ þ v1 dF ðtÞ : i¼1

0

ti

Therefore, the expected total pre-sale cost per unit item is given by Z 1 n Z ti ch ðp  dÞT cs þ nv0 1 X þ þ cm þ Rðti  tÞ dF ðtÞ þ v1 dF ðtÞ : 2d pT i¼1 0 pT ti The expected total number of non-conforming items produced in a production lot is n Z X i¼1

ti

fh1 pt þ h2 pðti  tÞg dF ðtÞ þ

0

Z

1

h1 pti dF ðtÞ :

ti

Proceeding in a fashion similar to the previous section, the expected post-sale cost for a production lot pT is given by " # Z w Z 1 n Z t i 1 X cr 1  ðh1 pt þ h2 pðti  tÞÞ dF ðtÞ þ h1 pti dF ðtÞ r1 ðtÞ dt pT i¼1 0 ti 0 Z w Z 1 n Z t i cr X þ ðh1 pt þ h2 pðti  tÞÞ dF ðtÞ þ h1 pti dF ðtÞ r2 ðtÞ dt: pT i¼1 0 ti 0 Therefore, the expected total cost which includes the pre-sale and post-sale costs per unit item is Z 1 n Z t i ch ðp  dÞT cs þ nv0 1 X þ þ Rðti  tÞ dF ðtÞ þ v1 dF ðtÞ C 2 ðn; t1 ; t2 ; . . . ; tn Þ ¼ cm þ 2d pT i¼1 0 pT ti "Z Z w  Z w w 1 þ cr r1 ðtÞ dt þ r2 ðtÞ dt  r1 ðtÞ dt pT 0 0 0 # Z 1 n Z t i X  ðh1 pt þ h2 pðti  tÞÞ dF ðtÞ þ h1 pti dF ðtÞ ; 0

i¼1

ð16Þ

ti

where T = t1 + t2 +    + tn. Remark 1. When n = 1, i.e., the process is inspected only once at the end of each production run, the model is similar to that of Wang (2004) and we have from Eqs. (10) and (16), C1(1, T) = C2(1, T). Proposition 2. For any given n and T, there exists an optimal inspection schedule fT 1 ; T 2 ; . . . ; T n g such that t1 ¼ t2 ¼    ¼ tn ¼ T =n, provided that the following condition holds: ða0  v1 Þf 0 ðT =nÞ þ AðwÞf ðT =nÞ > 0; where AðwÞ ¼ a1 þ cr ðh2  h1 Þp

Z 0

w

r2 ðtÞ dt 

ð17Þ Z



w

r1 ðtÞ dt

0

and prime denotes differentiation with respect to t.

>0

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247

Proof. For any given n and T, the optimal inspection schedule can be obtained by solving the following nonlinear programming (NP) problem: min

t1 ;t2 ;...;tn

subject to

C 2 ðn; t1 ; t2 ; . . . ; tn Þ n X

ti ¼ T ; ti > 0

ð18Þ

8i ¼ 1; 2; . . . ; n:

i¼1

The Lagrangian function for the NP problem (18) is given by ! n X Lðt1 ; t2 ; . . . ; tn Þ ¼ C 2 ðn; t1 ; t2 ; . . . ; tn Þ þ m ti  T ; i¼1

where m is the Lagrange multiplier. From the first-order necessary conditions of optimality, we have oL ¼ 0; oti oL ¼ 0; om

for i ¼ 1; 2; . . . ; n; ð19Þ

which yield ða0  v1 Þf ðti Þ þ a1 F ðti Þ þ cr

Z

w

r2 ðtÞ dt  0

Z

w

 r1 ðtÞ dt fðh2  h1 ÞpF ðti Þ þ h1 pg þ mpT ¼ 0; i ¼ 1; 2; . . . ; n

0

ð20Þ and n X

ti ¼ T :

ð21Þ

i¼1

It is clear from Eqs. (20) and (21) that the optimal inspection intervals are of equal length and they are each equal to T/n, i.e., t1 ¼ t2 ¼    ¼ tn ¼ T =n. This periodic inspection schedule is a local minimizer of C2(t1, t2, . . . , tn) provided that ½o2 C 2 =ot2i ti ¼T =n > 0, i.e., the condition (17) is satisfied. h Remark 2. If the process shift distribution is uniform, i.e., the process is equally likely to shift from an ‘in-control’ state to an ‘out-of-control’ state at any time during a production run then the condition (17) is always satisfied. Proposition 3. Suppose that the sequence {tn} is non-increasing. Then for any given n, a periodic inspection schedule satisfying oC2/oT = 0 is a local minimizer of C2(n, t1, t2, . . . , tn) provided that a0 > v1 and the production runlength T satisfies the relation (17). Proof. Replacing ti by Ti  Ti1 and T by Tn, we have, from Eq. (16), ch ðp  dÞT n cs þ nv0 þ 2d pT n  Z Z 1 n T i T i1 X 1 þ RðT i  T i1  tÞ dF ðtÞ þ v1 dF ðtÞ pT n i¼1 0 T i T i1 "Z Z w  Z w w 1 þ cr r1 ðtÞ dt þ r2 ðtÞ dt  r1 ðtÞ dt pT 0 0 0 # Z 1 n Z T i T i1 X  ðh1 pt þ h2 pðT i  T i1  tÞÞ dF ðtÞ þ h1 pðT i  T i1 Þ dF ðtÞ :

C 2 ðn; T 1 ; T 2 ; . . . ; T n Þ ¼ cm þ

i¼1

0

T i T i1

ð22Þ

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For any given n, the necessary conditions for a minimum of C2(n, T1, T2, . . . , Tn) give ða0  v1 Þff ðT i  T i1 Þ  f ðT iþ1  T i Þg þ AðwÞfF ðT i  T i1 Þ  F ðT iþ1  T i Þg ¼ 0;

i ¼ 1; 2; . . . ; n  1 ð23Þ

and n X ch ðp  dÞpT 2n  ðcs þ nv0 Þ  2d i¼1

Z

T i T i1

RðT i  T i1  tÞ dF ðtÞ þ v1

Z



1

dF ðtÞ

T i T i1

0

Z w  Z w þ T n fða0  v1 Þf ðT n  T n1 Þ þ a1 F ðT n  T n1 Þg þ cr r2 ðtÞ dt  r1 ðtÞ dt 0 0 " n Z T i T i1 X  T n fh1 p þ ðh2  h1 ÞpF ðT n  T n1 Þg  ðh1 pt þ h2 pðT i  T i1  tÞÞ dF ðtÞ þ

Z

1

h1 pðT i  T i1 Þ  dF ðtÞ

0

i¼1

#

ð24Þ

:

T i T i1

If the sequence {tn} is non-increasing then we have 0 < f(Ti  Ti1)  f(Ti+1  Ti) 6 F(Ti  Ti1)  F(Ti+1  Ti),"Ti’s. This implies that if the minimum restoration cost exceeds the preventive maintenance cost i.e., a0 > v1 then for a chosen value of T1, Eq. (23) provides the unique solution {T2, T3, . . . , Tn} such that ti = ti+1"i = 1, 2, . . . , n  1. To find a local minimizer of C2, T1 should be chosen in such a way that it, together with T2, T3, . . . , Tn obtained from Eq. (23), satisfy Eq. (24). Moreover, T = Tn must satisfy the relation (17). h Remark 3. The optimal and suboptimal inspection policies of the developed model can be obtained following the algorithm (exact method) and approximate method, respectively, as described in the previous section. Remark 4. In case of exponential process shift, the approximate method is consistent with the exact method. 6. Numerical example To illustrate the proposed model, we consider the Weibull shift distribution: b

F ðtÞ ¼ 1  eðktÞ ;

t > 0; k > 0; b P 1;

Expecected cost per unit item

Approximate value Exact value 11.28 11.26 11.24 11.22 11.2 1

2

3

4

5

6

7

8

9

10

Number of inspections Fig. 2. Number of inspections vs. expected cost per unit item under inspection policy I (b = 2, k = 0.2).

B.C. Giri, T. Dohi / European Journal of Operational Research 183 (2007) 238–252

249 2

and the lifetime distribution of conforming and non-conforming items as F 1 ðtÞ ¼ 1  eðt=6Þ and F 2 ðtÞ ¼ 2 1  eðt=2Þ , respectively (see Wang, 2004). Suppose that the values of the parameters involved in the model are as follows: d = 900 (units/month), p = 1500 (units/month), cs = 250 ($), ch = 1 ($/unit/month), cm = 5 ($/unit), cr = 1 ($), v0 = 10 ($), v1 = 50 ($), a0 = 200 ($), a1 = 20 ($), h1 = 0.05, h2 = 0.95, and w = 12 (months). Using the above data in the numerical computational software MATHEMATICA (Wolfram, 1996) we find that the expected cost function C1(n, T1, T2, . . . , Tn) is convex with respect to n. The behaviour of the cost function, for instance, is shown in Fig. 2 when b = 2 and k = 0.2. From the figure it can be observed that the approximate method provides a slightly higher expected cost per unit item in comparison to that obtained by the exact solution algorithm, except for n = 1. Table 1 presents the suboptimal inspection policy I for different values of k and b whereas Table 2 exhibits the corresponding results for the optimal inspection policy I. When b increases, for fixed k, the number of inspections and the expected cost per unit item decrease. There is a significant increase in the inspection intervals. When k increases, for fixed b, the number of inspections and expected cost per unit item increase, and the inspection intervals decrease. It is also observed that the first inspection interval in each inspection sequence is the longest one. This is because the process is in ‘in-control’ state at the beginning; the subsequent inspection interval decreases as the cumulative hazards increase with time. Fig. 3 depicts the decreasing sequence of the inspection intervals for the optimal inspection policy I when k = 0.2 and b = 2. Fig. 4 verifies that the convex property with respect to n holds for the expected cost function C2(n, t1, t2, . . . , tn). The effects of scale parameter k and shape parameter b on the suboptimal inspection policy

Table 1 Suboptimal inspection policy I for different values of b and ka b

k 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

11 0.06980 11.3256

16 0.04871 11.4198

18 0.04386 11.4964

19 0.03812 11.5632

19 0.03562 11.6285

23 0.03105 11.6779

26 0.02768 11.7315

26 0.02641 11.7826

26 0.02618 11.8296

2

2 0.50897 11.1650

5 0.31464 11.2168

7 0.25288 11.2693

8 0.22051 11.3216

10 0.19184 11.3738

11 0.17407 11.4254

12 0.15983 11.4764

13 0.14821 11.5269

14 0.13859 11.5767

3

1 0.77243 11.1284

1 0.70190 11.1487

3 0.48154 11.1773

4 0.40974 11.2106

4 0.37023 11.2472

5 0.32586 11.2859

6 0.29210 11.3265

6 0.27016 11.3683

7 0.24718 11.4107

a

Each cell in each column shows the computational results of ðn0 ; T 01 ; C 1 ðT 01 ; T 02 ; . . . ; T 0n ÞÞ.

Table 2 Optimal inspection policy I for different values of b and ka b

k 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

11 0.07202 11.3253

16 0.05188 11.4195

20 0.04372 11.4951

23 0.03982 11.5611

26 0.03728 11.6211

28 0.03620 11.6769

31 0.03514 11.7296

33 0.03500 11.7802

36 0.03486 11.8288

2

2 0.47750 11.1647

5 0.26438 11.2155

7 0.20288 11.2669

8 0.17357 11.3180

10 0.14608 11.3688

11 0.13062 11.4191

12 0.11827 11.4689

13 0.10824 11.5181

14 0.09994 11.5667

3

1 0.77243 11.1284

1 0.70190 11.1487

3 0.43330 11.1765

4 0.35648 11.2089

5 0.30262 11.2445

5 0.27606 11.2822

6 0.24182 11.3214

7 0.21542 11.3620

7 0.20070 11.4031

a

Each cell in each column shows the computational results of (n ; T 1 ; C 1 ðT 1 ; T 2 ; . . . ; T n ÞÞ.

250

B.C. Giri, T. Dohi / European Journal of Operational Research 183 (2007) 238–252

Fig. 3. Decreasing sequence of inspection intervals for the optimal inspection policy I (b = 2, k = 0.2).

Approximate value Exact value Expected cost per unit item

11.5

11.4

11.3

11.2 1

2

3 4 5 6 7 8 Number of inspections

9

10

Fig. 4. Number of inspections vs. expected cost per unit item under inspection policy II (b = 2, k = 0.2).

Table 3 Influence of k and b on the suboptimal inspection IIa b

k 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

4 0.17211 11.5604

6 0.11841 11.7675

7 0.09858 11.9288

8 0.08566 12.0660

9 0.07646 12.1876

10 0.06953 12.2983

11 0.06408 12.4008

12 0.05967 12.4968

13 0.05602 12.5877

2

1 0.66854 11.1755

2 0.48121 11.2581

2 0.41426 11.3312

3 0.35906 11.3979

3 0.32437 11.4632

4 0.29879 11.5247

4 0.27684 11.5839

4 0.25827 11.6430

5 0.24678 11.6982

3

1 0.77243 11.1284

1 0.70190 11.1487

1 0.61097 11.1872

1 0.53221 11.2368

2 0.46736 11.2904

2 0.43200 11.3370

2 0.38621 11.3861

2 0.35551 11.4364

2 0.32961 11.4875

a

Each cell in each column shows the computational results of (n0 ; t01 ; C 2 ðt01 ; t02 ; . . . ; t0n ÞÞ.

II and optimal policy II are summarized in Tables 3 and 4, respectively. Comparing Tables 1 and 2 with Tables 3 and 4, respectively, it is easy to see that the inspection policy I performs better than inspection policy II. This means that preventive maintenance during a production run is not cost-effective. This conclusion strongly depends on the preventive maintenance cost, restoration cost and defective items cost. For relatively higher

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251

Table 4 Influence of k and b on the optimal inspection policy IIa b

k 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

4 0.17211 11.5604

6 0.11841 11.7675

7 0.09858 11.9288

8 0.08566 12.0660

9 0.07646 12.1876

10 0.06953 12.2983

11 0.06408 12.4008

12 0.05967 12.4968

13 0.05602 12.5877

2

1 0.66854 11.1755

2 0.36087 11.2337

3 0.25662 11.2881

3 0.23478 11.3300

4 0.18989 11.3743

4 0.17762 11.4117

4 0.16673 11.4505

5 0.14484 11.4853

5 0.13768 11.5200

3

1 0.77243 11.1284

1 0.70190 11.1487

1 0.61097 11.1872

2 0.37702 11.2041

2 0.35020 11.2258

2 0.32462 11.2517

3 0.25317 11.2728

3 0.23999 11.2920

3 0.22759 11.3130

a

Each cell in each column shows the computational results of ðn ; t1 ; C 2 ðt1 ; t2 ; . . . ; tn ÞÞ.

Table 5 Impact of FRW on the optimal inspection policiesa w

Inspection policy I

0 6 12 18 24 a

Inspection policy II

n*

fT 1 ; T 2 ; . . . ; T n g

C 1

n*

fT 1 ; T 2 ; . . . ; T n g

C 2

1 2 5 7 8

{0.78258} {0.47600, 0.71844} {0.26438, 0.39864, 0.51132, 0.61230, 0.70549} {0.20340, 0.30663, 0.39324, 0.47080, 0.54235, 0.60947, 0.67316} {0.17426, 0.26269, 0.33685, 0.40327, 0.46451, 0.52196, 0.57644, 0.62853}

5.5281 6.9671 11.2155 18.2632 28.1100

1 1 2 3 3

{0.78258} {0.66723} {0.36087, 0.72175} {0.25694, 0.51389, 0.77083} {0.23530, 0.47060, 0.70589}

5.5281 6.9774 11.2337 18.2869 28.1273

C 1 ¼ C 1 ðn ; T 1 ; T 2 ; . . . ; T n Þ, C 2 ¼ C 2 ðn ; t1 ; t2 ; . . . ; tn Þ.

values of the later two, preventive maintenance option when the system is found in an ‘in-control’ state at any inspection may be a better choice. Table 5 explores the impact of FRW on the optimal and sub-optimal inspection policies. As w increases, the optimal production lot size decreases because a longer warranty period implies increased warranty costs for non-conforming items. The optimal number of inspections increases with w to reduce the number of defective items produced in a production lot. Though zero-warranty policy is optimal from manufacturer’s point of view but considering buyers’ concern and company’s goodwill, managers have to decide a FRW policy commensurate with company’s profit level. 7. Concluding remarks In this paper, we have studied the problem of inspection scheduling in an imperfect production process where the process shift time from an ‘in-control’ state to an ‘out-of-control’ state follows a general probability distribution with increasing hazard (failure) rate and the products are sold with a free minimal repair warranty. Inspections are undertaken to monitor the state of the production process during a production run, which in turn reduce the number of non-conforming items in a production lot. The proposed model has been formulated under two different inspection policies (I and II) designating no action or preventive repair action at each inspection during a production run if the process is found in ‘in-control’ state. Since it is hard to derive the optimal inspection policy analytically, we have proposed a computer algorithm to find it numerically. Further, to ease the computational difficulties, we have employed an approximate method which determines a suboptimal inspection policy. The optimal and suboptimal inspection policies have been compared and the impact of FRW on the optimal inspection policy has been investigated in a numerical example. From the numerical study, we have observed that the expected cost per unit item can be reduced by monitoring the production process through inspections during a production run. In this paper, we have assumed that

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all non-conforming items are saleable. However, the model can be extended to a more realistic situation where a fraction of non-conforming items are to be scrapped before sale. Acknowledgements The authors are thankful to the anonymous referees for their valuable comments and suggestions. This work was done when the first author visited the Hiroshima University, Japan, as a JSPS Post-Doctoral Fellow. The authors would like to thank the financial support by a Grant-in-Aid for Scientific Research from the Ministry of Education, Sports, Science and Culture of Japan under Grant No. 02296. References Djamaludin, I., Murthy, D.N.P., Wilson, R.J., 1994. Quality control through lot sizing for items sold with warranty. International Journal of Production Economics 33, 97–107. Groenevelt, H., Pintelon, L., Seidmann, A., 1992. Production lot sizing with machine breakdowns. Management Science 38, 104–123. Lee, J.S., Park, K.S., 1991. Joint determination of production cycle and inspection intervals in a deteriorating production system. Journal of the Operational Research Society 42, 775–783. Lee, H.L., Rosenblatt, M.J., 1987. Simultaneous determination of production cycle and inspection schedules in a production system. Management Science 33, 1125–1136. Makis, V., 1998. Optimal lot sizing and inspection policy for an EMQ model with imperfect inspections. Naval Research Logistics 45, 165– 186. Makis, V., Fung, J., 1998. An EMQ model with inspections and random machine failure. Journal of the Operational Research Society 49, 66–76. Munford, A.G., 1981. Comparison among certain inspection policies. Management Science 27, 261–267. Munford, A.G., Shahani, A.K., 1972. A nearly optimal inspection policy. Operational Research Quarterly 23 (3), 373–379. Porteus, E.L., 1986. Optimal lot sizing process quality improvement and setup cost reduction. Operations Research 34, 137–144. Porteus, E.L., 1990. The impact of inspection delay on process and inspection lot sizing. Management Science 36, 999–1007. Rahim, M.A., 1994. Joint determination of production quantity inspection schedule and control chart design. IIE Transactions 26 (6), 2– 11. Rahim, M.A., Banerjee, P.K., 1993. A generalized model for the economic design of X control chart for production systems with increasing failure rate and early replacement. Naval Research Logistics 40, 787–809. Rosenblatt, M.J., Lee, H.J., 1986. Economic production cycles with imperfect production processes. IIE Transactions 18, 48–55. Tseng, S.H., 1996. Optimal preventive maintenance policy for deteriorating production systems. IIE Transactions 28, 687–694. Wang, C.H., 2004. The impact of free-repair warranty policy on EMQ model for imperfect production systems. Computers and Operations Research 31, 2021–2035. Wang, C.H., 2005. Integrated production and production inspection policy for a deteriorating production system. International Journal of Production Economics 95 (1), 123–134. Wang, C.H., Sheu, S.H., 2003. Optimal lot sizing for products sold under free-repair warranty. European Journal of Operational Research 149 (1), 131–141. Wolfram, S., 1996. The Mathematica Book, third ed. Cambridge University Press, Wolfram Media. Yeh, R.H., Lo, H.C., 1998. Quality control products under free repair warranty. International Journal of Operations and Quantitative Management 4 (3), 265–275. Yeh, R.H., Ho, W.T., Tseng, S.T., 2000. Optimal production run length for products sold with warranty. European Journal of Operational Research 120, 575–582.