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Optimal lot sizing for products sold under free-repair warranty
European Journal of Operational Research 149 (2003) 131–141 www.elsevier.com/locate/dsw
Production, Manufacturing and Logistics
Optimal lot sizing for products sold under free-repair warranty Chih-Hsiung Wang a
a,*
, Shey-Heui Sheu
b
Department of Commerce Automation and Management, National Pingtung Institute of Commerce, 51 Min-Sheng E. Road, Pingtung 900, Taiwan, ROC b Department of Industrial Management, National Taiwan University of Science and Technology, Taipei 200, Taiwan, ROC Received 28 March 2000; accepted 10 April 2002
Abstract Repairable items sold under a free-repair warranty (FRW) policy are examined. In the literature, the lot-size problem is treated as a quality-control problem and not as an inventory problem for a continuous-type deteriorating production system, in which there is a shift from an in-control state to an out-of-control state and it is assumed that the system has a geometric survival distribution. Consequently, smaller lots result in a reduction in warranty cost per item, but such a reduction is achieved at the expense of an increased manufacturing cost per item. In order to control such a production system economically under an FRW, tradeoffs between manufacturing cost and warranty cost must be analyzed. In this paper, analytical results are extended to a discrete general shift distribution, to provide an optimal lot size so that the long run total cost of the setup, inventory holding, and warranty is minimized. Different conditions for optimality, properties and bounds on the optimal lot size are provided. A numerical example is given to see the adequacy of using the geometric distribution when the actual distribution is discrete Weibull. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Production; Lot size; Warranty
1. Introduction This study investigates the relationship between production and inventory for a deteriorating production system under the assumption that products are repairable and are sold with a free-repair warranty (FRW). Deterioration of production is an inherent process in most manufacturing industries, because a production system usually deteriorates continuously due to usage or age factors, such as corrosion, fatigue and cumulative wear. In the simplest characterization, the process state is classified as either ‘‘in-control’’ or ‘‘outof-control’’. Suppose that a production process starts to produce in an in-control state; however, after a period of time, the process may evolve from this ‘‘in-control’’ state to an ‘‘out-of-control’’ state, in which more non-conforming items are produced than in the case of an in-control state. Here we define a non-conforming item as one which does not satisfy specifications but is still usable. Its performance *
Corresponding author. Tel.: +886-8-723-8700x6117; fax: +886-8-723-8720/7941. E-mail address: [email protected] (C.-H. Wang).
0377-2217/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0377-2217(02)00429-0
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characteristics (e.g., mean time to failure), however, are considerably inferior to those of a conforming item. Once the process is out of control, without considering any inspection or maintenance actions, it remains in that state until the remainder of the lot has been produced. Obviously, a non-conforming item may incur a greater post-sale servicing cost than would a conforming item when sold under FRW. Therefore, controlling the lot size in order to restore a system to and keep it in an in-control state will reduce the warranty cost arising because non-conforming items are sold under FRW. Usually, a small lot size can reduce the number of non-conforming items released for sale, thereby resulting in a smaller warranty cost and inventory holding cost per unit item. However, such a reduction increases the manufacturing cost per unit due to the increased number of set-ups needed. In order to control such a production–inventory system economically when repairable products are sold under FRW, tradeoffs among the manufacturing cost, inventory holding cost and warranty cost must be analyzed. In order to control the process reliability so that the non-conforming items can be reduced in a production lot, many studies have focused on finding an optimal maintenance-inspection policy [1–5] in deteriorating production systems. However, in many production processes, it is either impossible or expensive to interrupt the production process during a production run, or it is not possible to detect the deterioration in the process [6]. Thus, Porteus [7] assumed that the process goes out-of-control with a given probability each time it produces an item. Once the process shifts to an out-of-control state, the process continues to produce defective items until the entire lot is produced. Porteus [7] found that the optimal lot size is smaller than the classical economic manufacturing quantity. Djamaludin et al. [8] further studied the effects of product warranty policy on the optimal lot size for a production system, in which the process goes out-ofcontrol with a given probability each time it produces an item and in which the inventory holding cost is assumed to be zero. In this paper, the work of Djamaludin et al. [8] is extended to treat the lot-size problem as a qualitycontrol and production–inventory problem for a process with discrete general shift distribution. Different conditions for optimality, properties and bounds on the optimal lot size are provided. The objective of this study is to find the optimal lot size in a production run so that the long run total cost per item, including the manufacturing cost, inventory holding cost and warranty cost, is minimized. The remainder of this paper is organized as follows: In Section 2, the model formulation, in which the assumptions and notations of the model are given in Section 2.1, is presented in detail. In Section 2.2, the mathematical model is established, and the formula for the long run total cost per item is obtained. In Section 3, the condition of the uniqueness property for the optimal lot size is explored. Properties for the optimal lot size are also investigated. In Section 4, the models considered by Porteus [7] and Djamaludin et al. [8] are shown to be special cases of our model. Moreover, the properties of optimal solutions for the geometric survival distribution case which was considered by Djamaludin et al. [8] are investigated. Finally, an illustrative numerical example is given.
2. Model formulation 2.1. Basic assumptions and notations As in Djamaludin et al. [8], the following assumptions are made: (1) The process state is classified as either in-control or out-of-control. Due to manufacturing variability, a produced item may be either conforming or non-conforming, depending on whether its performance meets the specifications of the product. It is assumed that h1 ðh2 Þ is the probability that a produced item is conforming while the process is in-control (out-of-control) at the end of the production period for the item, where h1 > h2 . That is, a produced item is more (less) likely to be conforming if the state is in-con-
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trol (out-of-control). Here, a non-conforming item is operational when put into use, and the non-conformance can be detected only through time-testing for a significant time period. (2) It is assumed that the produced items are repairable and that they are sold under a FRW policy, where in all failures occurring within the warranty period W P 0 are rectified through minimal repair by the manufacturer at no cost to the buyer. Furthermore, suppose that the process starts to produce items in an in-control state. On the basis of the concepts developed by Wang and Sheu [9], it is assumed that while producing the jth unit item, the production process shifts from an in-control state to an out-of-control state, with probabilities pj ¼ P fU ¼ jg, where U is the total number of produced items needed for the productionPprocess to shift from an incontrol state to an out-of-control state since the last setup. It is noted that 1 j¼1 pj ¼ 1. Once the process shifts to the out-of-control state, it remains there until the end of the production run. It is assumed that at the start of production of each lot, process inspections and preventive maintenance action are performed with a fixed cost Cs . If the system is found to be out-of-control, then it is restored to an in-control state with an additional cost g. Here, the cost incurred with each lot, in checking the state and bringing it back to incontrol if it is out-of-control, is called the setup cost [8]. Furthermore, let P j ¼ P fU > jg. That is, P j is the probability that the number of products produced in the ‘‘in-control’’ state is larger than j. It is assumed throughout this work that the domain of j is f1; 2; 3; . . .g. The notation {P j } is used as an abbreviation for a sequence of probabilities. The failure rate function of U is defined as rðjÞ ¼ P fU ¼ jjU P jg ¼ pj =P j1 , j ¼ 1; 2; 3; . . . provided P j > 0 (see Shaked and Rocha-Martinez [10]). U has a non-decreasing failure rate (NDFR) if rðj þ 1Þ P rðjÞ, j ¼ 1; 2; . . . In this paper we shall, as far as possible, use the same notations as in Djamaludin et al. [8]. The notations are summarized as follows: R1 ðR2 Þ the average number of warranty repairs per item for conforming (non-conforming) items within the warranty period W, where R1 < R2 D deterministic, constant demand rate (units/time) M deterministic, constant manufacturing rate (units/time), where M > D Ch inventory holding cost ($/unit/time) Cm material and labor cost required to produce a single item ($/item) CR a deterministic, constant repair cost incurred each time a sold item fails within the warranty period W ($) L number of items producing a lot Lm upper limit for lot size; i.e., physical capacity limitation of production system wi random variable, where wi ¼ 1 if set-up for lot i involves changing the process from out-of-control to in-control, and where wi ¼ 0 if no such change is involved, for i P 2, and w1 ¼ 0 Ni number of conforming items in lot i Ki total number of failures under warranty for items belonging to lot i The objective of this study is to determine the optimal lot size L that minimizes the long run total cost per item subject to the constraint 0 < L 6 Lm , where the total cost per item consists of the manufacturing cost, warranty cost and inventory holding cost. This cost is chosen as a criterion for obtaining the optimal lot size because it is a non-random quantity which depends on the lot size. 2.2. Total cost per item First, to identify costs related to the long run manufacturing cost, the expected restoration cost gE½wi in lot i, i P 2 must be derived. Note that w1 ¼ 0.
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The expected cost of restoring the process at the start of the production of lot i, each lot of size L being given by gE½wi ¼ gð1 P L Þ, i P 2. The long run expected average manufacturing cost per unit item, cðL; fP j gÞ, is given by cðL; fP j gÞ ¼ Cm þ ðCs þ gð1 P L ÞÞ=L:
ð1Þ
Next, the long run warranty cost per item is derived. First, one has the mean number of conforming items in lot i, given by E½Ni ¼
L X
pj fðj 1Þh1 þ ðL j þ 1Þh2 g þ
j¼1
1 X
h1 Lpj ;
i ¼ 1; 2; 3; . . .
ð2Þ
j¼Lþ1
Furthermore, the mean number of failures under warranty for items belonging to lot i is given by E½Ki ¼ E½Ni R1 þ ðL E½Ni ÞR2 ;
i ¼ 1; 2; 3; . . .
ð3Þ
The long run average warranty cost per item, CW ðL; fP j gÞ, is given by CW ðL; fP j gÞ ¼ CR E½Ki =L:
ð4Þ
Substituting (2) into (3) and then (3) into (4) gives " !, ( # ) L X CW ðL; fP j gÞ ¼ CR ðR1 R2 Þ pj ððj 1Þh1 þ ðL j þ 1Þh2 Þ L þ h1 P L þ R 2 :
ð5Þ
j¼1
Given lot size L, the inventory holding cost per item can be easily derived as ðCh L=2Þðð1=DÞ ð1=MÞÞ (e.g., see [11]). Thus, the long run total cost per item, J ðL; fP j gÞ, is given by
Ch L 1 1 J ðL; fP j gÞ ¼ cðL; fP j gÞ þ CW ðL; fP j gÞ þ ð6Þ 2 D M with cðL; fP j gÞ given by (1) and CW ðL; fP j gÞ given by (5). Substituting (1) and (5) into (6), one obtains " (
L X
Ch L 1 1 J ðL; fP j gÞ ¼ Cm þ pj ððj 1Þh1 þ Cs þ gð1 P L Þ L þ CR ðR1 R2 Þ 2 D M j¼1 !, # ) þ ðL j þ 1Þh2 Þ
L þ h1 P L þ R 2 :
ð7Þ
For the long run case, the objective is to find an optimal lot size L that minimizes J ðL; fP j gÞ given in (7) subject to the constraint 0 < L 6 Lm .
3. Optimal lot size In this section, the properties of the lot-size problem, are studied and the condition of uniqueness of the optimal lot size is explored. The effect of model parameters on the optimal lot size is also discussed. Now, the total cost per item function, J ðL; fP j gÞ, given in Eq. (7) is investigated as follows. To find an L
which minimizes J ðL; fP j gÞ, the inequalities J ðL; fP j gÞ 6 J ðL þ 1; fP j gÞ and J ðL; fP j gÞ < J ðL 1; fP j gÞ
are formed;
ð8Þ
implying H ðL; fP j gÞ P Cs þ g
and
H ðL 1; fP j gÞ < Cs þ g;
ð9Þ
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where
8 P Lþ1 > < ðP L þ LpLþ1 Þg þ Ch LðL þ 1Þ D1 M1 2 þ CR ðh1 h2 ÞðR2 R1 Þ j¼1 pj ðj 1Þ H ðL; fP j gÞ ¼ for L ¼ 1; 2; 3; . . . ; > : 0 for L ¼ 0: ð10Þ
The derivation of Eq. (10) is given in Appendix A. Theorem 1 (a) If U is NDFR and rðLm Þ < q ¼
CR ðh1 h2 ÞðR2 R1 Þ ; CR ðh1 h2 ÞðR2 R1 Þ þ g
ð11Þ
then J ðL; fP j gÞ has a unique cost minimizing lot size L ; and this optimal L is equal to the first L such that H ðL; fP j gÞ P Cs þ g. In the event that H ðLm ; fP j gÞ < Cs þ g, then the optimal lot size is Lm . (b) H ðL; fP j gÞ is increasing, and so a unique L exists, if the probability masses are increasing, i.e. if pjþ1 P pj , for j ¼ 1; 2; . . . ; Lm . Theorem 1(a) shows that the optimal lot size is unique and bounded in a finite interval ½1; Lm if the production process has a NDFR and its failure rate at Lm is less than the cost ratio q. q seems to compare the expected cost of ‘‘extra’’ repairs needed for non-conforming units to the cost of correcting an outof-control system; if the relative cost of correcting is low, then it is more likely that a unique L exists. Moreover, conditions on the value of H ðL; fP j gÞ for determining the optimal lot size are provided. Theorem 1(b) provides a more demanding requirement on the distribution, but it would help hold for any choice of costs or other parameters. It is further observed that H ðL; fP j gÞ increases when one or more of the parameters h1 , R2 , g, Ch , CR and M increase and/or when one or more of the parameters h2 , R1 and D decrease. Thus, in the following theorem, the influences of h1 and h2 on the optimal lot size L are shown. Theorem 2. Assume that (11) holds, and that 0 < pj < 1, for j ¼ 1; 2; . . . ; Lm þ 1. L ¼ Lm whenever h1 h2 < d; otherwise, L 2 ½1; Lm . Here, Cs þ gð1 P Lm Lm pLm þ1 Þ Ch Lm ðLm þ 1Þ D1 M1 2 P d¼ : Lm þ1 ðj 1Þp CR ðR2 R1 Þ j j¼1 Theorem 2 shows that it will not be worth stopping the production run early (before Lm ) if the performance of an in-control system is not much better than that of an out-of-control system. By using arguments similar to those used in Theorem 2, one also can find the influences of the parameters R1 , R2 , g, Ch , M, CR , Cs and D used in determining L .
4. Special cases In this section, extensions in the investigations to the geometric failure distributions is given. In particular, the case of the geometric is shown to be an extension of previously reported results [7,8]. Case 1: h1 ¼ 1, h2 ¼ 0, R1 ¼ 0, R2 ¼ 1, P j ¼ qj for j ¼ 1; 2; 3; . . ., Lm ! 1, M ! 1, Cm ¼ 0 and g ¼ 0, which is the case considered by Porteus [7]. In other words, if the state is in-control at the start of
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production of an item, then during the production of the item, it can go out-of-control with probability 1 q or can continue to be in-control with probability q. In this case, using expression (7), one obtains the following result:
Cs Ch L qð1 qL Þ j þ CR 1 J ðL; fq gÞ ¼ þ ð$=itemÞ: ð12Þ 2D Lð1 qÞ L Thus, one has the (long run) total cost per unit time:
Cs D Ch L qð1 qL Þ þ þ DCR 1 JPO ðL; fqj gÞ ¼ L 2 Lð1 qÞ
ð$=timeÞ;
ð13Þ
which was obtained by Porteus [7]. In this case, the condition in Theorem 1(a) holds since rðLm Þ ¼ 1 q < q ¼ 1. By applying Theorem 1(a), one can conclude that there exists a unique and finite optimal lot size L PO that minimizes (12), or equivalently, that L PO minimizes the cost rate function JPO ðL; fqj gÞ given in (13). Case 2: Ch ¼ 0 and P j ¼ qj for j ¼ 1; 2; 3; . . . By using expression (7), one obtains the following result: qð1 qL Þ þ h 2 þ R2 ; J ðL; fqj gÞ ¼ Cm þ ðCs þ gð1 qL ÞÞ=L þ CR ðR1 R2 Þ ðh1 h2 Þ ð14Þ ð1 qÞL which is the formula obtained by Djamaludin et al. [8]. When q 1, Djamaludin et al. [8] used an approximation of 2
qL 1 þ L lnðqÞ þ ðL lnðqÞÞ =2
ð15Þ
to obtain the approximately optimal solution to (14) as 1=2 2Cs L
¼ q : ð1 qÞ½qCR ðh1 h2 ÞðR2 R1 Þ ð1 qÞg
ð16Þ
Obviously, the necessary and sufficient condition for L
to exist is qCR ðh1 h2 ÞðR2 R1 Þ ð1 qÞg > 0:
ð17Þ
It is not difficult to verify that the necessary and sufficient condition for the existence of L given in (17) holds if, and only if, (11), the condition in Theorem 1(a) also holds. In this case, one can apply Theorem 1(a) to obtain a unique L 2 ½1; Lm instead of using an approximate solution given in (16). Furthermore, in the following theorem, it is shown that L
is a lower bound solution of L when q is very close to one, the proof of which is given in Appendix A. Theorem 3. Assume that (17) holds. If q P ql , then the approximate optimal solution L
, obtained by Djamaludin et al. [8] to (14), is a lower bound solution of the real optimal solution, where L
is given in (16) and ql ¼ fððCR ðh1 h2 ÞðR2 R1 ÞÞ=ð2Cs þ gÞÞ þ 1g1 . An illustrative numerical example is presented in the next section.
5. Numerical example If (11) holds, then, a bisection algorithm for computing the optimal lot size L can be obtained based on the monotonicity of H ðL; fP j gÞ, as shown in Theorem 1(a) and the result of Theorem 1(a); i.e., there exists a unique solution L 2 ½1; Lm to H ðL; fP j gÞ P Cs þ g and H ðL 1; fP j gÞ < Cs þ g; L minimizes J ðL; fP j gÞ;
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Table 1 The PCP of using geometric solution when the actual distribution is discrete Weibulla L0 L
PCP
20 21 8 7.2118
30 26 11 4.7961
40 31 14 2.9248
50 35 18 1.3320
60 39 24 0.2037
70 42 63 0.3166
80 45 71 0.6353
90 48 78 0.8860
100 51 84 1.0757
110 54 89 1.2159
Ch L0 L
PCP
1 40 100 0.8248
2 38 19 0.8621
3 36 18 1.1042
4 35 18 1.3320
5 34 17 1.5261
6 33 17 1.6953
7 32 17 1.8273
8 31 17 1.9222
9 30 16 1.9959
10 29 16 2.0337
M (D=M ¼ 11=16) L0 L
PCP
50
55
60
65
70
75
80
85
90
95
32 17 1.6875
33 17 1.6499
33 17 1.5286
34 17 1.5056
34 18 1.4150
35 18 1.4033
35 18 1.3320
35 18 1.2690
35 18 1.2129
36 18 1.2091
CR L0 L
PCP
1 64 100 5.0245
2 52 100 5.0548
3 44 93 3.4408
4 39 74 1.1907
5 35 18 1.3320
6 32 15 3.2530
7 30 14 4.8618
8 28 13 6.0276
9 26 12 6.7407
10 25 11 7.6343
W L0 L
PCP
0.5 60 100 5.3666
1 35 18 1.3320
1.5 23 11 7.8596
2 17 8 8.8100
2.5 14 7 9.0424
3 11 6 6.6259
3.5 10 5 6.8840
4 9 5 6.5563
4.5 8 4 5.6098
5 7 4 4.4877
Lm L0 L
PCP
80 35 18 1.3320
85 35 18 1.3320
90 35 18 1.3320
95 35 18 1.3320
100 35 18 1.3320
105 35 18 1.3320
110 35 18 1.3320
115 35 18 1.3320
120 35 18 1.3320
125 35 18 1.3320
h1 L0 L
PCP
0.5 47 100 3.1179
0.55 45 96 2.8791
0.6 44 92 2.4672
0.65 42 87 2.1441
0.7 41 82 1.6783
0.75 39 76 1.2623
0.8 38 70 0.7534
0.85 37 62 0.2568
0.9 36 19 0.7047
0.95 35 18 1.3320
L0 L
PCP
0 34 17 1.9177
0.05 35 18 1.3320
0.1 36 19 0.7247
0.15 37 62 0.2715
0.2 38 70 0.8194
0.25 39 76 1.4130
0.3 41 82 1.9346
0.35 42 87 2.5465
0.4 44 92 3.0211
0.45 45 96 3.6373
L0 L
PCP
100 34 18 2.7269
110 34 18 2.4149
120 34 18 2.1139
130 35 18 1.9322
140 35 18 1.6270
150 35 18 1.3320
160 35 18 1.0468
170 35 18 0.7708
180 35 71 0.6895
190 36 78 1.2701
L0 L
PCP
0.54 100 100 0
0.59 100 100 0
0.64 100 100 0
0.69 100 100 0
0.74 100 100 0
0.79 100 100 0
0.84 100 100 0
0.89 100 100 0
0.94 72 100 1.3390
0.99 35 18 1.3320
L0 L
PCP
1.5 35 18 1.3320
2 35 100 8.1588
2.5 35 100 10.8006
3 35 100 11.9522
3.5 35 100 12.5727
4 35 100 12.9531
4.5 35 100 13.2073
5 35 100 13.3896
5.5 35 100 13.5160
6 35 100 13.6112
Cm L0 L
PCP
1 35 18 1.7184
2 35 18 1.6022
3 35 18 1.5007
4 35 18 1.4114
5 35 18 1.3320
6 35 18 1.2611
7 35 18 1.1974
8 35 18 1.1398
9 35 18 1.0875
10 35 18 1.0398
Cs
h2
g
q
a
a
L0 is the optimal lot size by using geometric distribution (a ¼ 1).
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otherwise, L ¼ Lm . For the geometric case, in Djamaludin et al. [8], L was obtained by evaluating J ðL; fqj gÞ for L ¼ 1; 2; 3; . . . ; Lm ; thus, the time complexity for obtaining the optimal lot size L can be reduced from OðLm Þ to OðlogðLm ÞÞ by using bisection algorithm when (11) holds. On the other hand, if (11) does not hold, then a sequential search for the optimal production lot size is needed. The geometric distribution is commonly used in the literature for deteriorating production systems, yet it is not always accurate. To study the adequacy of using the geometric distribution, we consider the production process has a discrete Weibull shift distribution, which has an increasing failure rate. That is, a P fU > jg ¼ P j ¼ qj for j ¼ 1; 2; 3; . . ., a > 1, and 0 < q < 1 (see Nakagawa and Osaki [12]). The following nominal values for the parameters are considered as: q ¼ 0:99, a ¼ 1:5, Cm ¼ 5 ($), g ¼ 150 ($), Ch ¼ 4 ($/ item/time), CR ¼ 5 ($), Cs ¼ 50 ($), M ¼ 80 (item/time), D ¼ 55 (item/time), W ¼ 1 (unit time), Lm ¼ 100, h1 ¼ 0:95 and h2 ¼ 0:05. Furthermore, suppose the failure distribution of both conforming and non-conforming items are non-homogeneous failure rate r1 ðtÞ ¼ 0:1t and r2 ðtÞ ¼ t, R 1 Poisson process with increasing R1 respectively. Then, we have R1 ¼ 0 0:1t dt ¼ 0:05 and R2 ¼ 0 t dt ¼ 0:5. In this example, we obtain the optimal production lot size is L ¼ 35 with a total cost per item 17.7899$. When using the geometric distribution, the optimal lot size L0 is 18. The percentage of cost penalty (PCP) for implementing the geometric solution when Weibull is the actual distribution is 1.332%, where a a a PCP ¼ ðJ ðL0 ; qj Þ J ðL ; qj ÞÞ=J ðL ; qj Þ 100. Using this example, we carry out a sensitivity analysis to study the effects of the above parameters on the solution quality. We perform sensitivity analysis of this example by varying one or two parameters at a time. The remaining parameters have the values selected at the beginning of this section. The results are summarized in Table 1. From Table 1, we have: (1) When CR is relatively high (low), the PCP becomes significant. (2) It has been observed that PCP is significant on some warranty periods (e.g., when W ¼ 1:5, the PCP is up to 7.8596%). (3) An increase in a ða > 1Þ results in an increase in the PCP while the PCP is not significant when q is change. In this example, we can conclude that if a process which has an increasing failure rate for its producing products sold under FRW, the geometric solution may not be a good approximation to the discrete WeibullÕs solution. 6. Conclusions In this paper, a lot-size problem has been considered for products which are repairable and which are sold under a FRW policy. A general shift distribution for the production process has been considered. A sufficient condition has been given for the case in which the optimal lot size is unique, thereby extending the condition for the existence of the approximate solution obtained by Djamaludin et al. [8]. Special cases [7,8] of geometrically distributed in-control periods has been studied in detail. A numerical example is performed to see the adequacy of using the geometric distribution when the actual distribution is discrete Weibull. A sensitivity analysis is conducted in order to see the effect of the input parameters on the solution quality of using geometric solution, and some managerial insights are observed. Acknowledgements The authors wish to express their appreciation to the anonymous referees for their very valuable comments and suggestions, which greatly enhanced the clarity of the article. All their suggestions were
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139
incorporated directly into the text. In addition, the authors wish to express appreciation to Dr. Cheryl Rutledge for her editorial assistance.
Appendix A. Derivation of H(L;{Pj }) From (7), the inequality J ðL þ 1; fP j gÞ P J ðL; fP j gÞ if, and only if, ! ( " Lþ1 X 1 1 2 þ CR ðR1 R2 Þ h1 Cs þ g 6 ðP L þ LpLþ1 Þg þ Ch LðL þ 1Þ ðj 1Þpj þ ðL þ 1ÞP Lþ1 L D M j¼1 " #) ! # L Lþ1 L X X X pj ðj 1Þ þ LP L ðL þ 1Þ þ h2 Lpj ðL j þ 2Þ pj ðL j þ 1ÞðL þ 1Þ
j¼1
j¼1
j¼1
(
1 1 2 þ CR ðR1 R2 Þ h1 ¼ ðP L þ LpLþ1 Þg þ Ch LðL þ 1Þ D M !) Lþ1 L X X Lpj pj ðL j þ 1Þ ¼ H ðL; fP j gÞ: þ h2 j¼1
L X LpLþ1 pj ðj 1Þ
!
j¼1
j¼1
Similarly, for J ðL; fP j gÞ < J ðL 1; fP j gÞ, one can derive H ðL 1; fP j gÞ < Cs þ g, where H ðL; fP j gÞ is given as (10). The derivation is omitted here. Thus, the proof is completed. Proof of Theorem 1(a). From (10), one has H ðL þ 1; fP j gÞ H ðL; fP j gÞ ¼ ðL þ 1Þ CR ðh1 h2 ÞðR2 R1 ÞðP Lþ1 P Lþ2 Þ gðpLþ1 pLþ2 Þ
1 1 þ Ch ðL þ 1Þ ; for L ¼ 1; 2; 3; . . . D M
ðA:1Þ
If U is NDFR and rðLm Þ < q ¼ ðCR ðh1 h2 ÞðR2 R1 ÞÞ=ðCR ðh1 h2 ÞðR2 R1 Þ þ gÞ, then we have CR ðh1 h2 ÞðR2 R1 ÞP L gpL is a decreasing function of L, since CR ðh1 h2 ÞðR2 R1 ÞP L gpL ¼ ðCR ðh1 h2 ÞðR2 R1 Þ þ gÞP L1 ðq rðLÞÞ:
ðA:2Þ
In this case, the R.H.S. of (A.1) is positive. That is, H ðL; fP j gÞ is an increasing function of L for L 2 ½1; Lm . Furthermore, if H ðL; fP j gÞ P Cs þ g, then a solution to (9) exists, and from the monotonicity of H ðL; fP j gÞ, it is unique and less or equal to Lm . Also, the inequalities of (9) imply (8); therefore, there exists a finite and unique L that satisfies (8). In addition, it is obvious that H ðL 1; fP j gÞ < Cs þ g and H ðL ; fP j gÞ P Cs þ g. Moreover, from the monotony of H ðL; fP j gÞ, it is easy to see that J ðL þ 1; fP j gÞ J ðL; fP j gÞ ¼ H ðL; fP j gÞ ðCs þ gÞ < 0, for L ¼ 1; 2; . . . ; L 1 and J ðL þ 1; fP j gÞ J ðL; fP j gÞ ¼ H ðL; fP j gÞ ðCs þ gÞ P 0, for L ¼ L ; L þ 1; . . . ; Lm . Therefore, J ðL ; fP j gÞ 6 J ðL; fP j gÞ, for L ¼ 1; 2; . . . ; Lm and 0 < L 6 Lm ; i.e., L minimizes J ðL; fP j gÞ given in (7). On the other hand, if H ðLm ; fP j gÞ < Cs þ g, then H ðL; fP j gÞ < Cs þ g for L ¼ 1; 2; . . . ; Lm 1 since H ðL; fP j gÞ is monotonically increasing in L. Moreover, from the monotony of H ðL; fP j gÞ, it is easy to see that J ðL þ 1; fP j gÞ J ðL; fP j gÞ ¼ H ðL; fP j gÞ ðCs þ gÞ < 0, for L ¼ 1; 2; . . . ; Lm 1. Therefore, J ðL þ 1; fP j gÞ < J ðL; fP j gÞ, for L ¼ 1; 2; . . . ; Lm 1; i.e., the optimal lot size L ¼ Lm . Thus, the Proof of Theorem 1(a) is completed. Proof of Theorem 1(b). It is clear form the Proof of Theorem 1(a).
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Proof of Theorem 2. It is easy to see that H ðLm ; fP j gÞ < Cs þ g for 0 6 h1 h2 < d and H ðLm ; fP j gÞ P Cs þ g for d 6 h1 h2 < 1. From Theorem 1(a), one has L ¼ Lm for 0 6 h1 h2 < d and 0 < L 6 Lm for d 6 h1 h2 < 1. Thus, the Proof of Theorem 2 is completed. Proof of Theorem 3. Djamaludin et al. [8] used (15) and dJ ðL
; fqj gÞ=dL ¼ 0 to obtain L
, or equivalently,
2
qL ¼ 1 þ L
lnðqÞ þ ðL
lnðqÞÞ =2
ðA:3Þ
and g
q 2 CR ðh1 h2 ÞðR2 R1 Þ ¼ 2Cs =ðL
lnðqÞÞ : 1q
ðA:4Þ
Setting Ch ¼ 0 and P j ¼ qj , for j ¼ 1; 2; 3; . . . into (10) and substituting (A.3) into (10) and then rearranging (10) gives
q 2
j CR ðh1 h2 ÞðR2 R1 Þ ðL
lnðqÞ þ ðL
lnðqÞÞ =2Þ H ðL ; fq gÞ ¼ g þ g 1q 2
þ L
ðð1 qÞg qCR ðh1 h2 ÞðR2 R1 ÞÞð1 þ L
lnðqÞ þ ðL
lnðqÞÞ =2Þ
q CR ðh1 h2 ÞðR2 R1 Þ ¼ g þ L
g 1q 2
2
flnðqÞ þ L
ðlnðqÞÞ =2 þ ð1 qÞ½1 þ L
lnðqÞ þ ðL
lnðqÞÞ =2 g:
ðA:5Þ
By substituting (A.4) into (A.5) and replacing lnðqÞ þ L
ðlnðqÞÞ2 =2 þ ð1 qÞ½1 þ L
lnðqÞ þ ðL
lnðqÞÞ2 =2 2
with /ðL
; qÞ L
ðlnðqÞÞ =2, (A.5) can be rewritten as H ðL
; fqj gÞ ¼ g þ L
ð2Cs =ðL
lnðqÞÞ2 Þð/ðL
; qÞ L
ðlnðqÞÞ2 =2Þ; 2
2
ðA:6Þ
2
where /ðL
; qÞ ¼ ð1 qÞðlnðqÞÞ ðL
Þ =2 þ ðð1 qÞ lnðqÞ þ ðlnðqÞÞ ÞL
þ ð1 q þ lnðqÞÞ, or equivalently,
2 /ðL
; qÞ ¼ ð1 qÞqL þ L
ðlnðqÞÞ þ lnðqÞ.
It is obvious that /ðL ; qÞ P 0 for L
6 1= lnðqÞ. Furthermore, L
6 1= lnðqÞ ()
Cs
6 1=2 2Cs =ðL
lnðqÞÞ2 Cs 6 1=2 () q C ðh h R 1 2 ÞðR2 R1 Þ g 1q ()
ðBy ð21ÞÞ
ðA:7Þ
q 2Cs þ g P : 1q CR ðh1 h2 ÞðR2 R1 Þ
Let fðqÞ ¼ q=ð1 qÞ, 0 6 q 6 1. It is not difficult to see that f0 ðqÞ ¼ 1=ð1 qÞ2 > 0, fð0Þ ¼ 0 and limq!1 fðqÞ ! 1, which collectively imply that there exists ql , 0 6 ql 6 1, such that fðqÞ P ð2Cs þ gÞ= ðCR ðh1 h2 ÞðR2 R1 ÞÞ, where ql 6 q 6 1, and where fðql Þ ¼ ð2Cs þ gÞ=ðCR ðh1 h2 ÞðR2 R1 ÞÞ, or equiva1 lently, ql ¼ fðCR ðh1 h2 ÞðR2 R1 ÞÞ=ð2Cs þ gÞ þ 1g , which implies that if ql 6 q 6 1, then /ðL
; qÞ P 0. 2 2
Thus, if ql 6 q 6 1, then from (A.6) one has H ðL ; fqj gÞ 6 g þ L
ð2Cs =ðL
lnðqÞÞ ÞððL
lnðqÞÞ =2Þ ¼
g þ Cs . Therefore, from Theorem 1(a), one concludes that L 6 L . Thus, the Proof of Theorem 3 is completed.
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References [1] M.J. Rosenblatt, H.L. Lee, A comparative study of continuous and periodic inspection policies in deteriorating production systems, IIE Transactions 18 (1986) 2–9. [2] H.L. Lee, M.J. Rosenblatt, Simultaneous determination of production cycle and inspection schedules in a production system, Management Science 33 (9) (1987) 1125–1136. [3] H.L. Lee, M.J. Rosenblatt, A production and maintenance planning model with restoration cost dependent on detection delay, IIE Transactions 21 (4) (1989) 368–375. [4] V. Makis, Optimal lot sizing and inspection policy for an EMQ model with imperfect inspections, Naval Research Logistics 45 (1998) 165–186. [5] V. Makis, J. Fung, An EMQ model with inspections and random machine failures, Journal of the Operational Research Society 49 (1998) 66–76. [6] M.J. Rosenblatt, H.L. Lee, Economic production cycles with imperfect production process, IIE Transactions 18 (1986) 48–55. [7] E.L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Operations Research 34 (1986) 137–144. [8] I. Djamaludin, D.N.P. Murthy, R.J. Wilson, Quality control through lot sizing for items sold with warranty, International Journal of Production Economics 33 (1994) 97–107. [9] C.H. Wang, S.H. Sheu, Simultaneous determination of the optimal production-inventory and product inspection policies for a deteriorating production system, Computers and Operations Research 28 (2001) 1093–1110. [10] M. Shaked, J.M. Rocha-Martinez, A discrete-time model of failure and repair, Applied Stochastic Models and Data Analysis 11 (1995) 167–180. [11] E.A. Silver, D.F. Pyke, R. Peterson, Inventory Management and Production Planning and Scheduling, Wiley, New York, 1998. [12] T. Nakagawa, S. Osaki, The discrete Weibull distribution, IEEE Transactions on Reliability R-24 (5) (1975) 300–301.
Production, Manufacturing and Logistics
Optimal lot sizing for products sold under free-repair warranty Chih-Hsiung Wang a
a,*
, Shey-Heui Sheu
b
Department of Commerce Automation and Management, National Pingtung Institute of Commerce, 51 Min-Sheng E. Road, Pingtung 900, Taiwan, ROC b Department of Industrial Management, National Taiwan University of Science and Technology, Taipei 200, Taiwan, ROC Received 28 March 2000; accepted 10 April 2002
Abstract Repairable items sold under a free-repair warranty (FRW) policy are examined. In the literature, the lot-size problem is treated as a quality-control problem and not as an inventory problem for a continuous-type deteriorating production system, in which there is a shift from an in-control state to an out-of-control state and it is assumed that the system has a geometric survival distribution. Consequently, smaller lots result in a reduction in warranty cost per item, but such a reduction is achieved at the expense of an increased manufacturing cost per item. In order to control such a production system economically under an FRW, tradeoffs between manufacturing cost and warranty cost must be analyzed. In this paper, analytical results are extended to a discrete general shift distribution, to provide an optimal lot size so that the long run total cost of the setup, inventory holding, and warranty is minimized. Different conditions for optimality, properties and bounds on the optimal lot size are provided. A numerical example is given to see the adequacy of using the geometric distribution when the actual distribution is discrete Weibull. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Production; Lot size; Warranty
1. Introduction This study investigates the relationship between production and inventory for a deteriorating production system under the assumption that products are repairable and are sold with a free-repair warranty (FRW). Deterioration of production is an inherent process in most manufacturing industries, because a production system usually deteriorates continuously due to usage or age factors, such as corrosion, fatigue and cumulative wear. In the simplest characterization, the process state is classified as either ‘‘in-control’’ or ‘‘outof-control’’. Suppose that a production process starts to produce in an in-control state; however, after a period of time, the process may evolve from this ‘‘in-control’’ state to an ‘‘out-of-control’’ state, in which more non-conforming items are produced than in the case of an in-control state. Here we define a non-conforming item as one which does not satisfy specifications but is still usable. Its performance *
Corresponding author. Tel.: +886-8-723-8700x6117; fax: +886-8-723-8720/7941. E-mail address: [email protected] (C.-H. Wang).
0377-2217/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0377-2217(02)00429-0
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characteristics (e.g., mean time to failure), however, are considerably inferior to those of a conforming item. Once the process is out of control, without considering any inspection or maintenance actions, it remains in that state until the remainder of the lot has been produced. Obviously, a non-conforming item may incur a greater post-sale servicing cost than would a conforming item when sold under FRW. Therefore, controlling the lot size in order to restore a system to and keep it in an in-control state will reduce the warranty cost arising because non-conforming items are sold under FRW. Usually, a small lot size can reduce the number of non-conforming items released for sale, thereby resulting in a smaller warranty cost and inventory holding cost per unit item. However, such a reduction increases the manufacturing cost per unit due to the increased number of set-ups needed. In order to control such a production–inventory system economically when repairable products are sold under FRW, tradeoffs among the manufacturing cost, inventory holding cost and warranty cost must be analyzed. In order to control the process reliability so that the non-conforming items can be reduced in a production lot, many studies have focused on finding an optimal maintenance-inspection policy [1–5] in deteriorating production systems. However, in many production processes, it is either impossible or expensive to interrupt the production process during a production run, or it is not possible to detect the deterioration in the process [6]. Thus, Porteus [7] assumed that the process goes out-of-control with a given probability each time it produces an item. Once the process shifts to an out-of-control state, the process continues to produce defective items until the entire lot is produced. Porteus [7] found that the optimal lot size is smaller than the classical economic manufacturing quantity. Djamaludin et al. [8] further studied the effects of product warranty policy on the optimal lot size for a production system, in which the process goes out-ofcontrol with a given probability each time it produces an item and in which the inventory holding cost is assumed to be zero. In this paper, the work of Djamaludin et al. [8] is extended to treat the lot-size problem as a qualitycontrol and production–inventory problem for a process with discrete general shift distribution. Different conditions for optimality, properties and bounds on the optimal lot size are provided. The objective of this study is to find the optimal lot size in a production run so that the long run total cost per item, including the manufacturing cost, inventory holding cost and warranty cost, is minimized. The remainder of this paper is organized as follows: In Section 2, the model formulation, in which the assumptions and notations of the model are given in Section 2.1, is presented in detail. In Section 2.2, the mathematical model is established, and the formula for the long run total cost per item is obtained. In Section 3, the condition of the uniqueness property for the optimal lot size is explored. Properties for the optimal lot size are also investigated. In Section 4, the models considered by Porteus [7] and Djamaludin et al. [8] are shown to be special cases of our model. Moreover, the properties of optimal solutions for the geometric survival distribution case which was considered by Djamaludin et al. [8] are investigated. Finally, an illustrative numerical example is given.
2. Model formulation 2.1. Basic assumptions and notations As in Djamaludin et al. [8], the following assumptions are made: (1) The process state is classified as either in-control or out-of-control. Due to manufacturing variability, a produced item may be either conforming or non-conforming, depending on whether its performance meets the specifications of the product. It is assumed that h1 ðh2 Þ is the probability that a produced item is conforming while the process is in-control (out-of-control) at the end of the production period for the item, where h1 > h2 . That is, a produced item is more (less) likely to be conforming if the state is in-con-
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133
trol (out-of-control). Here, a non-conforming item is operational when put into use, and the non-conformance can be detected only through time-testing for a significant time period. (2) It is assumed that the produced items are repairable and that they are sold under a FRW policy, where in all failures occurring within the warranty period W P 0 are rectified through minimal repair by the manufacturer at no cost to the buyer. Furthermore, suppose that the process starts to produce items in an in-control state. On the basis of the concepts developed by Wang and Sheu [9], it is assumed that while producing the jth unit item, the production process shifts from an in-control state to an out-of-control state, with probabilities pj ¼ P fU ¼ jg, where U is the total number of produced items needed for the productionPprocess to shift from an incontrol state to an out-of-control state since the last setup. It is noted that 1 j¼1 pj ¼ 1. Once the process shifts to the out-of-control state, it remains there until the end of the production run. It is assumed that at the start of production of each lot, process inspections and preventive maintenance action are performed with a fixed cost Cs . If the system is found to be out-of-control, then it is restored to an in-control state with an additional cost g. Here, the cost incurred with each lot, in checking the state and bringing it back to incontrol if it is out-of-control, is called the setup cost [8]. Furthermore, let P j ¼ P fU > jg. That is, P j is the probability that the number of products produced in the ‘‘in-control’’ state is larger than j. It is assumed throughout this work that the domain of j is f1; 2; 3; . . .g. The notation {P j } is used as an abbreviation for a sequence of probabilities. The failure rate function of U is defined as rðjÞ ¼ P fU ¼ jjU P jg ¼ pj =P j1 , j ¼ 1; 2; 3; . . . provided P j > 0 (see Shaked and Rocha-Martinez [10]). U has a non-decreasing failure rate (NDFR) if rðj þ 1Þ P rðjÞ, j ¼ 1; 2; . . . In this paper we shall, as far as possible, use the same notations as in Djamaludin et al. [8]. The notations are summarized as follows: R1 ðR2 Þ the average number of warranty repairs per item for conforming (non-conforming) items within the warranty period W, where R1 < R2 D deterministic, constant demand rate (units/time) M deterministic, constant manufacturing rate (units/time), where M > D Ch inventory holding cost ($/unit/time) Cm material and labor cost required to produce a single item ($/item) CR a deterministic, constant repair cost incurred each time a sold item fails within the warranty period W ($) L number of items producing a lot Lm upper limit for lot size; i.e., physical capacity limitation of production system wi random variable, where wi ¼ 1 if set-up for lot i involves changing the process from out-of-control to in-control, and where wi ¼ 0 if no such change is involved, for i P 2, and w1 ¼ 0 Ni number of conforming items in lot i Ki total number of failures under warranty for items belonging to lot i The objective of this study is to determine the optimal lot size L that minimizes the long run total cost per item subject to the constraint 0 < L 6 Lm , where the total cost per item consists of the manufacturing cost, warranty cost and inventory holding cost. This cost is chosen as a criterion for obtaining the optimal lot size because it is a non-random quantity which depends on the lot size. 2.2. Total cost per item First, to identify costs related to the long run manufacturing cost, the expected restoration cost gE½wi in lot i, i P 2 must be derived. Note that w1 ¼ 0.
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The expected cost of restoring the process at the start of the production of lot i, each lot of size L being given by gE½wi ¼ gð1 P L Þ, i P 2. The long run expected average manufacturing cost per unit item, cðL; fP j gÞ, is given by cðL; fP j gÞ ¼ Cm þ ðCs þ gð1 P L ÞÞ=L:
ð1Þ
Next, the long run warranty cost per item is derived. First, one has the mean number of conforming items in lot i, given by E½Ni ¼
L X
pj fðj 1Þh1 þ ðL j þ 1Þh2 g þ
j¼1
1 X
h1 Lpj ;
i ¼ 1; 2; 3; . . .
ð2Þ
j¼Lþ1
Furthermore, the mean number of failures under warranty for items belonging to lot i is given by E½Ki ¼ E½Ni R1 þ ðL E½Ni ÞR2 ;
i ¼ 1; 2; 3; . . .
ð3Þ
The long run average warranty cost per item, CW ðL; fP j gÞ, is given by CW ðL; fP j gÞ ¼ CR E½Ki =L:
ð4Þ
Substituting (2) into (3) and then (3) into (4) gives " !, ( # ) L X CW ðL; fP j gÞ ¼ CR ðR1 R2 Þ pj ððj 1Þh1 þ ðL j þ 1Þh2 Þ L þ h1 P L þ R 2 :
ð5Þ
j¼1
Given lot size L, the inventory holding cost per item can be easily derived as ðCh L=2Þðð1=DÞ ð1=MÞÞ (e.g., see [11]). Thus, the long run total cost per item, J ðL; fP j gÞ, is given by
Ch L 1 1 J ðL; fP j gÞ ¼ cðL; fP j gÞ þ CW ðL; fP j gÞ þ ð6Þ 2 D M with cðL; fP j gÞ given by (1) and CW ðL; fP j gÞ given by (5). Substituting (1) and (5) into (6), one obtains " (
L X
Ch L 1 1 J ðL; fP j gÞ ¼ Cm þ pj ððj 1Þh1 þ Cs þ gð1 P L Þ L þ CR ðR1 R2 Þ 2 D M j¼1 !, # ) þ ðL j þ 1Þh2 Þ
L þ h1 P L þ R 2 :
ð7Þ
For the long run case, the objective is to find an optimal lot size L that minimizes J ðL; fP j gÞ given in (7) subject to the constraint 0 < L 6 Lm .
3. Optimal lot size In this section, the properties of the lot-size problem, are studied and the condition of uniqueness of the optimal lot size is explored. The effect of model parameters on the optimal lot size is also discussed. Now, the total cost per item function, J ðL; fP j gÞ, given in Eq. (7) is investigated as follows. To find an L
which minimizes J ðL; fP j gÞ, the inequalities J ðL; fP j gÞ 6 J ðL þ 1; fP j gÞ and J ðL; fP j gÞ < J ðL 1; fP j gÞ
are formed;
ð8Þ
implying H ðL; fP j gÞ P Cs þ g
and
H ðL 1; fP j gÞ < Cs þ g;
ð9Þ
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135
where
8 P Lþ1 > < ðP L þ LpLþ1 Þg þ Ch LðL þ 1Þ D1 M1 2 þ CR ðh1 h2 ÞðR2 R1 Þ j¼1 pj ðj 1Þ H ðL; fP j gÞ ¼ for L ¼ 1; 2; 3; . . . ; > : 0 for L ¼ 0: ð10Þ
The derivation of Eq. (10) is given in Appendix A. Theorem 1 (a) If U is NDFR and rðLm Þ < q ¼
CR ðh1 h2 ÞðR2 R1 Þ ; CR ðh1 h2 ÞðR2 R1 Þ þ g
ð11Þ
then J ðL; fP j gÞ has a unique cost minimizing lot size L ; and this optimal L is equal to the first L such that H ðL; fP j gÞ P Cs þ g. In the event that H ðLm ; fP j gÞ < Cs þ g, then the optimal lot size is Lm . (b) H ðL; fP j gÞ is increasing, and so a unique L exists, if the probability masses are increasing, i.e. if pjþ1 P pj , for j ¼ 1; 2; . . . ; Lm . Theorem 1(a) shows that the optimal lot size is unique and bounded in a finite interval ½1; Lm if the production process has a NDFR and its failure rate at Lm is less than the cost ratio q. q seems to compare the expected cost of ‘‘extra’’ repairs needed for non-conforming units to the cost of correcting an outof-control system; if the relative cost of correcting is low, then it is more likely that a unique L exists. Moreover, conditions on the value of H ðL; fP j gÞ for determining the optimal lot size are provided. Theorem 1(b) provides a more demanding requirement on the distribution, but it would help hold for any choice of costs or other parameters. It is further observed that H ðL; fP j gÞ increases when one or more of the parameters h1 , R2 , g, Ch , CR and M increase and/or when one or more of the parameters h2 , R1 and D decrease. Thus, in the following theorem, the influences of h1 and h2 on the optimal lot size L are shown. Theorem 2. Assume that (11) holds, and that 0 < pj < 1, for j ¼ 1; 2; . . . ; Lm þ 1. L ¼ Lm whenever h1 h2 < d; otherwise, L 2 ½1; Lm . Here, Cs þ gð1 P Lm Lm pLm þ1 Þ Ch Lm ðLm þ 1Þ D1 M1 2 P d¼ : Lm þ1 ðj 1Þp CR ðR2 R1 Þ j j¼1 Theorem 2 shows that it will not be worth stopping the production run early (before Lm ) if the performance of an in-control system is not much better than that of an out-of-control system. By using arguments similar to those used in Theorem 2, one also can find the influences of the parameters R1 , R2 , g, Ch , M, CR , Cs and D used in determining L .
4. Special cases In this section, extensions in the investigations to the geometric failure distributions is given. In particular, the case of the geometric is shown to be an extension of previously reported results [7,8]. Case 1: h1 ¼ 1, h2 ¼ 0, R1 ¼ 0, R2 ¼ 1, P j ¼ qj for j ¼ 1; 2; 3; . . ., Lm ! 1, M ! 1, Cm ¼ 0 and g ¼ 0, which is the case considered by Porteus [7]. In other words, if the state is in-control at the start of
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production of an item, then during the production of the item, it can go out-of-control with probability 1 q or can continue to be in-control with probability q. In this case, using expression (7), one obtains the following result:
Cs Ch L qð1 qL Þ j þ CR 1 J ðL; fq gÞ ¼ þ ð$=itemÞ: ð12Þ 2D Lð1 qÞ L Thus, one has the (long run) total cost per unit time:
Cs D Ch L qð1 qL Þ þ þ DCR 1 JPO ðL; fqj gÞ ¼ L 2 Lð1 qÞ
ð$=timeÞ;
ð13Þ
which was obtained by Porteus [7]. In this case, the condition in Theorem 1(a) holds since rðLm Þ ¼ 1 q < q ¼ 1. By applying Theorem 1(a), one can conclude that there exists a unique and finite optimal lot size L PO that minimizes (12), or equivalently, that L PO minimizes the cost rate function JPO ðL; fqj gÞ given in (13). Case 2: Ch ¼ 0 and P j ¼ qj for j ¼ 1; 2; 3; . . . By using expression (7), one obtains the following result: qð1 qL Þ þ h 2 þ R2 ; J ðL; fqj gÞ ¼ Cm þ ðCs þ gð1 qL ÞÞ=L þ CR ðR1 R2 Þ ðh1 h2 Þ ð14Þ ð1 qÞL which is the formula obtained by Djamaludin et al. [8]. When q 1, Djamaludin et al. [8] used an approximation of 2
qL 1 þ L lnðqÞ þ ðL lnðqÞÞ =2
ð15Þ
to obtain the approximately optimal solution to (14) as 1=2 2Cs L
¼ q : ð1 qÞ½qCR ðh1 h2 ÞðR2 R1 Þ ð1 qÞg
ð16Þ
Obviously, the necessary and sufficient condition for L
to exist is qCR ðh1 h2 ÞðR2 R1 Þ ð1 qÞg > 0:
ð17Þ
It is not difficult to verify that the necessary and sufficient condition for the existence of L given in (17) holds if, and only if, (11), the condition in Theorem 1(a) also holds. In this case, one can apply Theorem 1(a) to obtain a unique L 2 ½1; Lm instead of using an approximate solution given in (16). Furthermore, in the following theorem, it is shown that L
is a lower bound solution of L when q is very close to one, the proof of which is given in Appendix A. Theorem 3. Assume that (17) holds. If q P ql , then the approximate optimal solution L
, obtained by Djamaludin et al. [8] to (14), is a lower bound solution of the real optimal solution, where L
is given in (16) and ql ¼ fððCR ðh1 h2 ÞðR2 R1 ÞÞ=ð2Cs þ gÞÞ þ 1g1 . An illustrative numerical example is presented in the next section.
5. Numerical example If (11) holds, then, a bisection algorithm for computing the optimal lot size L can be obtained based on the monotonicity of H ðL; fP j gÞ, as shown in Theorem 1(a) and the result of Theorem 1(a); i.e., there exists a unique solution L 2 ½1; Lm to H ðL; fP j gÞ P Cs þ g and H ðL 1; fP j gÞ < Cs þ g; L minimizes J ðL; fP j gÞ;
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137
Table 1 The PCP of using geometric solution when the actual distribution is discrete Weibulla L0 L
PCP
20 21 8 7.2118
30 26 11 4.7961
40 31 14 2.9248
50 35 18 1.3320
60 39 24 0.2037
70 42 63 0.3166
80 45 71 0.6353
90 48 78 0.8860
100 51 84 1.0757
110 54 89 1.2159
Ch L0 L
PCP
1 40 100 0.8248
2 38 19 0.8621
3 36 18 1.1042
4 35 18 1.3320
5 34 17 1.5261
6 33 17 1.6953
7 32 17 1.8273
8 31 17 1.9222
9 30 16 1.9959
10 29 16 2.0337
M (D=M ¼ 11=16) L0 L
PCP
50
55
60
65
70
75
80
85
90
95
32 17 1.6875
33 17 1.6499
33 17 1.5286
34 17 1.5056
34 18 1.4150
35 18 1.4033
35 18 1.3320
35 18 1.2690
35 18 1.2129
36 18 1.2091
CR L0 L
PCP
1 64 100 5.0245
2 52 100 5.0548
3 44 93 3.4408
4 39 74 1.1907
5 35 18 1.3320
6 32 15 3.2530
7 30 14 4.8618
8 28 13 6.0276
9 26 12 6.7407
10 25 11 7.6343
W L0 L
PCP
0.5 60 100 5.3666
1 35 18 1.3320
1.5 23 11 7.8596
2 17 8 8.8100
2.5 14 7 9.0424
3 11 6 6.6259
3.5 10 5 6.8840
4 9 5 6.5563
4.5 8 4 5.6098
5 7 4 4.4877
Lm L0 L
PCP
80 35 18 1.3320
85 35 18 1.3320
90 35 18 1.3320
95 35 18 1.3320
100 35 18 1.3320
105 35 18 1.3320
110 35 18 1.3320
115 35 18 1.3320
120 35 18 1.3320
125 35 18 1.3320
h1 L0 L
PCP
0.5 47 100 3.1179
0.55 45 96 2.8791
0.6 44 92 2.4672
0.65 42 87 2.1441
0.7 41 82 1.6783
0.75 39 76 1.2623
0.8 38 70 0.7534
0.85 37 62 0.2568
0.9 36 19 0.7047
0.95 35 18 1.3320
L0 L
PCP
0 34 17 1.9177
0.05 35 18 1.3320
0.1 36 19 0.7247
0.15 37 62 0.2715
0.2 38 70 0.8194
0.25 39 76 1.4130
0.3 41 82 1.9346
0.35 42 87 2.5465
0.4 44 92 3.0211
0.45 45 96 3.6373
L0 L
PCP
100 34 18 2.7269
110 34 18 2.4149
120 34 18 2.1139
130 35 18 1.9322
140 35 18 1.6270
150 35 18 1.3320
160 35 18 1.0468
170 35 18 0.7708
180 35 71 0.6895
190 36 78 1.2701
L0 L
PCP
0.54 100 100 0
0.59 100 100 0
0.64 100 100 0
0.69 100 100 0
0.74 100 100 0
0.79 100 100 0
0.84 100 100 0
0.89 100 100 0
0.94 72 100 1.3390
0.99 35 18 1.3320
L0 L
PCP
1.5 35 18 1.3320
2 35 100 8.1588
2.5 35 100 10.8006
3 35 100 11.9522
3.5 35 100 12.5727
4 35 100 12.9531
4.5 35 100 13.2073
5 35 100 13.3896
5.5 35 100 13.5160
6 35 100 13.6112
Cm L0 L
PCP
1 35 18 1.7184
2 35 18 1.6022
3 35 18 1.5007
4 35 18 1.4114
5 35 18 1.3320
6 35 18 1.2611
7 35 18 1.1974
8 35 18 1.1398
9 35 18 1.0875
10 35 18 1.0398
Cs
h2
g
q
a
a
L0 is the optimal lot size by using geometric distribution (a ¼ 1).
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otherwise, L ¼ Lm . For the geometric case, in Djamaludin et al. [8], L was obtained by evaluating J ðL; fqj gÞ for L ¼ 1; 2; 3; . . . ; Lm ; thus, the time complexity for obtaining the optimal lot size L can be reduced from OðLm Þ to OðlogðLm ÞÞ by using bisection algorithm when (11) holds. On the other hand, if (11) does not hold, then a sequential search for the optimal production lot size is needed. The geometric distribution is commonly used in the literature for deteriorating production systems, yet it is not always accurate. To study the adequacy of using the geometric distribution, we consider the production process has a discrete Weibull shift distribution, which has an increasing failure rate. That is, a P fU > jg ¼ P j ¼ qj for j ¼ 1; 2; 3; . . ., a > 1, and 0 < q < 1 (see Nakagawa and Osaki [12]). The following nominal values for the parameters are considered as: q ¼ 0:99, a ¼ 1:5, Cm ¼ 5 ($), g ¼ 150 ($), Ch ¼ 4 ($/ item/time), CR ¼ 5 ($), Cs ¼ 50 ($), M ¼ 80 (item/time), D ¼ 55 (item/time), W ¼ 1 (unit time), Lm ¼ 100, h1 ¼ 0:95 and h2 ¼ 0:05. Furthermore, suppose the failure distribution of both conforming and non-conforming items are non-homogeneous failure rate r1 ðtÞ ¼ 0:1t and r2 ðtÞ ¼ t, R 1 Poisson process with increasing R1 respectively. Then, we have R1 ¼ 0 0:1t dt ¼ 0:05 and R2 ¼ 0 t dt ¼ 0:5. In this example, we obtain the optimal production lot size is L ¼ 35 with a total cost per item 17.7899$. When using the geometric distribution, the optimal lot size L0 is 18. The percentage of cost penalty (PCP) for implementing the geometric solution when Weibull is the actual distribution is 1.332%, where a a a PCP ¼ ðJ ðL0 ; qj Þ J ðL ; qj ÞÞ=J ðL ; qj Þ 100. Using this example, we carry out a sensitivity analysis to study the effects of the above parameters on the solution quality. We perform sensitivity analysis of this example by varying one or two parameters at a time. The remaining parameters have the values selected at the beginning of this section. The results are summarized in Table 1. From Table 1, we have: (1) When CR is relatively high (low), the PCP becomes significant. (2) It has been observed that PCP is significant on some warranty periods (e.g., when W ¼ 1:5, the PCP is up to 7.8596%). (3) An increase in a ða > 1Þ results in an increase in the PCP while the PCP is not significant when q is change. In this example, we can conclude that if a process which has an increasing failure rate for its producing products sold under FRW, the geometric solution may not be a good approximation to the discrete WeibullÕs solution. 6. Conclusions In this paper, a lot-size problem has been considered for products which are repairable and which are sold under a FRW policy. A general shift distribution for the production process has been considered. A sufficient condition has been given for the case in which the optimal lot size is unique, thereby extending the condition for the existence of the approximate solution obtained by Djamaludin et al. [8]. Special cases [7,8] of geometrically distributed in-control periods has been studied in detail. A numerical example is performed to see the adequacy of using the geometric distribution when the actual distribution is discrete Weibull. A sensitivity analysis is conducted in order to see the effect of the input parameters on the solution quality of using geometric solution, and some managerial insights are observed. Acknowledgements The authors wish to express their appreciation to the anonymous referees for their very valuable comments and suggestions, which greatly enhanced the clarity of the article. All their suggestions were
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139
incorporated directly into the text. In addition, the authors wish to express appreciation to Dr. Cheryl Rutledge for her editorial assistance.
Appendix A. Derivation of H(L;{Pj }) From (7), the inequality J ðL þ 1; fP j gÞ P J ðL; fP j gÞ if, and only if, ! ( " Lþ1 X 1 1 2 þ CR ðR1 R2 Þ h1 Cs þ g 6 ðP L þ LpLþ1 Þg þ Ch LðL þ 1Þ ðj 1Þpj þ ðL þ 1ÞP Lþ1 L D M j¼1 " #) ! # L Lþ1 L X X X pj ðj 1Þ þ LP L ðL þ 1Þ þ h2 Lpj ðL j þ 2Þ pj ðL j þ 1ÞðL þ 1Þ
j¼1
j¼1
j¼1
(
1 1 2 þ CR ðR1 R2 Þ h1 ¼ ðP L þ LpLþ1 Þg þ Ch LðL þ 1Þ D M !) Lþ1 L X X Lpj pj ðL j þ 1Þ ¼ H ðL; fP j gÞ: þ h2 j¼1
L X LpLþ1 pj ðj 1Þ
!
j¼1
j¼1
Similarly, for J ðL; fP j gÞ < J ðL 1; fP j gÞ, one can derive H ðL 1; fP j gÞ < Cs þ g, where H ðL; fP j gÞ is given as (10). The derivation is omitted here. Thus, the proof is completed. Proof of Theorem 1(a). From (10), one has H ðL þ 1; fP j gÞ H ðL; fP j gÞ ¼ ðL þ 1Þ CR ðh1 h2 ÞðR2 R1 ÞðP Lþ1 P Lþ2 Þ gðpLþ1 pLþ2 Þ
1 1 þ Ch ðL þ 1Þ ; for L ¼ 1; 2; 3; . . . D M
ðA:1Þ
If U is NDFR and rðLm Þ < q ¼ ðCR ðh1 h2 ÞðR2 R1 ÞÞ=ðCR ðh1 h2 ÞðR2 R1 Þ þ gÞ, then we have CR ðh1 h2 ÞðR2 R1 ÞP L gpL is a decreasing function of L, since CR ðh1 h2 ÞðR2 R1 ÞP L gpL ¼ ðCR ðh1 h2 ÞðR2 R1 Þ þ gÞP L1 ðq rðLÞÞ:
ðA:2Þ
In this case, the R.H.S. of (A.1) is positive. That is, H ðL; fP j gÞ is an increasing function of L for L 2 ½1; Lm . Furthermore, if H ðL; fP j gÞ P Cs þ g, then a solution to (9) exists, and from the monotonicity of H ðL; fP j gÞ, it is unique and less or equal to Lm . Also, the inequalities of (9) imply (8); therefore, there exists a finite and unique L that satisfies (8). In addition, it is obvious that H ðL 1; fP j gÞ < Cs þ g and H ðL ; fP j gÞ P Cs þ g. Moreover, from the monotony of H ðL; fP j gÞ, it is easy to see that J ðL þ 1; fP j gÞ J ðL; fP j gÞ ¼ H ðL; fP j gÞ ðCs þ gÞ < 0, for L ¼ 1; 2; . . . ; L 1 and J ðL þ 1; fP j gÞ J ðL; fP j gÞ ¼ H ðL; fP j gÞ ðCs þ gÞ P 0, for L ¼ L ; L þ 1; . . . ; Lm . Therefore, J ðL ; fP j gÞ 6 J ðL; fP j gÞ, for L ¼ 1; 2; . . . ; Lm and 0 < L 6 Lm ; i.e., L minimizes J ðL; fP j gÞ given in (7). On the other hand, if H ðLm ; fP j gÞ < Cs þ g, then H ðL; fP j gÞ < Cs þ g for L ¼ 1; 2; . . . ; Lm 1 since H ðL; fP j gÞ is monotonically increasing in L. Moreover, from the monotony of H ðL; fP j gÞ, it is easy to see that J ðL þ 1; fP j gÞ J ðL; fP j gÞ ¼ H ðL; fP j gÞ ðCs þ gÞ < 0, for L ¼ 1; 2; . . . ; Lm 1. Therefore, J ðL þ 1; fP j gÞ < J ðL; fP j gÞ, for L ¼ 1; 2; . . . ; Lm 1; i.e., the optimal lot size L ¼ Lm . Thus, the Proof of Theorem 1(a) is completed. Proof of Theorem 1(b). It is clear form the Proof of Theorem 1(a).
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Proof of Theorem 2. It is easy to see that H ðLm ; fP j gÞ < Cs þ g for 0 6 h1 h2 < d and H ðLm ; fP j gÞ P Cs þ g for d 6 h1 h2 < 1. From Theorem 1(a), one has L ¼ Lm for 0 6 h1 h2 < d and 0 < L 6 Lm for d 6 h1 h2 < 1. Thus, the Proof of Theorem 2 is completed. Proof of Theorem 3. Djamaludin et al. [8] used (15) and dJ ðL
; fqj gÞ=dL ¼ 0 to obtain L
, or equivalently,
2
qL ¼ 1 þ L
lnðqÞ þ ðL
lnðqÞÞ =2
ðA:3Þ
and g
q 2 CR ðh1 h2 ÞðR2 R1 Þ ¼ 2Cs =ðL
lnðqÞÞ : 1q
ðA:4Þ
Setting Ch ¼ 0 and P j ¼ qj , for j ¼ 1; 2; 3; . . . into (10) and substituting (A.3) into (10) and then rearranging (10) gives
q 2
j CR ðh1 h2 ÞðR2 R1 Þ ðL
lnðqÞ þ ðL
lnðqÞÞ =2Þ H ðL ; fq gÞ ¼ g þ g 1q 2
þ L
ðð1 qÞg qCR ðh1 h2 ÞðR2 R1 ÞÞð1 þ L
lnðqÞ þ ðL
lnðqÞÞ =2Þ
q CR ðh1 h2 ÞðR2 R1 Þ ¼ g þ L
g 1q 2
2
flnðqÞ þ L
ðlnðqÞÞ =2 þ ð1 qÞ½1 þ L
lnðqÞ þ ðL
lnðqÞÞ =2 g:
ðA:5Þ
By substituting (A.4) into (A.5) and replacing lnðqÞ þ L
ðlnðqÞÞ2 =2 þ ð1 qÞ½1 þ L
lnðqÞ þ ðL
lnðqÞÞ2 =2 2
with /ðL
; qÞ L
ðlnðqÞÞ =2, (A.5) can be rewritten as H ðL
; fqj gÞ ¼ g þ L
ð2Cs =ðL
lnðqÞÞ2 Þð/ðL
; qÞ L
ðlnðqÞÞ2 =2Þ; 2
2
ðA:6Þ
2
where /ðL
; qÞ ¼ ð1 qÞðlnðqÞÞ ðL
Þ =2 þ ðð1 qÞ lnðqÞ þ ðlnðqÞÞ ÞL
þ ð1 q þ lnðqÞÞ, or equivalently,
2 /ðL
; qÞ ¼ ð1 qÞqL þ L
ðlnðqÞÞ þ lnðqÞ.
It is obvious that /ðL ; qÞ P 0 for L
6 1= lnðqÞ. Furthermore, L
6 1= lnðqÞ ()
Cs
6 1=2 2Cs =ðL
lnðqÞÞ2 Cs 6 1=2 () q C ðh h R 1 2 ÞðR2 R1 Þ g 1q ()
ðBy ð21ÞÞ
ðA:7Þ
q 2Cs þ g P : 1q CR ðh1 h2 ÞðR2 R1 Þ
Let fðqÞ ¼ q=ð1 qÞ, 0 6 q 6 1. It is not difficult to see that f0 ðqÞ ¼ 1=ð1 qÞ2 > 0, fð0Þ ¼ 0 and limq!1 fðqÞ ! 1, which collectively imply that there exists ql , 0 6 ql 6 1, such that fðqÞ P ð2Cs þ gÞ= ðCR ðh1 h2 ÞðR2 R1 ÞÞ, where ql 6 q 6 1, and where fðql Þ ¼ ð2Cs þ gÞ=ðCR ðh1 h2 ÞðR2 R1 ÞÞ, or equiva1 lently, ql ¼ fðCR ðh1 h2 ÞðR2 R1 ÞÞ=ð2Cs þ gÞ þ 1g , which implies that if ql 6 q 6 1, then /ðL
; qÞ P 0. 2 2
Thus, if ql 6 q 6 1, then from (A.6) one has H ðL ; fqj gÞ 6 g þ L
ð2Cs =ðL
lnðqÞÞ ÞððL
lnðqÞÞ =2Þ ¼
g þ Cs . Therefore, from Theorem 1(a), one concludes that L 6 L . Thus, the Proof of Theorem 3 is completed.
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141
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