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A (r,r,Q) inventory model for spare parts involving equipment criticality
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Int. J. Production Economics 97 (2005) 66–74 www.elsevier.com/locate/dsw
A ðr; r; QÞ inventory model for spare parts involving equipment criticality Pao-Long Changa,, Ying-Chyi Choub, Ming-Guang Huangc a
Department of Business Administration, Feng Chia University, 100 Wenhwa Road, Seatwen, Taichung, Taiwan b Department of Business Administration, Tunghai University, Taichung, Taiwan c Department of Finance, Shih Chien University, Taiwan Received 18 October 2000; accepted 5 June 2004 Available online 2 September 2004
Abstract Demand for spare parts can sometimes be classified into critical and non-critical demand, depending on the criticality of the equipment for production. To effectively handle this situation in spare parts inventory control, we propose a ðr; r; QÞ inventory model for spare parts where some of the stock is reserved for critical demand. A solution procedure to determine the reorder point r, which is equal to the critical level, and the reorder quantity Q is presented. Finally, some numerical examples of the spare parts of semiconductor equipment are presented to illustrate the results of using the proposed policy. r 2004 Elsevier B.V. All rights reserved. Keywords: Spare parts; Inventory policy; Equipment criticality; Demand classes
1. Introduction This paper is motivated by a case study on the inventory control of spare parts at a semiconductor equipment manufacturer in Taiwan. The company manufactures many different types of integrated circuits (ICs) utilizing production equipment involving a spare parts inventory of more than 2000 different types of items with a Corresponding author. Tel.: +886-4-22846615; fax: +886-
4-22846660. E-mail address: [email protected] (P.-L. Chang).
value of over US$15 million dollars. Since some equipment in an IC fabrication facility may be quite important, the spare parts installed in this equipment should be treated differently. Therefore, it is required to develop an inventory control model having equipment criticality considered explicitly. Dekker et al. (1998) define equipment criticality as the importance of equipment for sustaining production in a safe and efficient way. In a case at a petrochemical complex one distinguishes between vital, essential and auxiliary equipment. If spare parts belong to equipment of only one
0925-5273/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2004.06.048
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criticality level, then one could always give each criticality class its own service level or stockout penalty. However, if parts are installed in equipment of various criticality, one would encounter problems especially if only few are installed in high critical equipment. A way out of this problem is to reserve spare parts within the stock for demands originating from those critical equipment. In Dekker et al. (1998) a method was provided in case the parts were classified with only two criticality levels: critical and non-critical demand, where critical demand originates from failure of parts installed in vital equipment, and non-critical demand from parts installed in essential and auxiliary equipment. To handle the problem about inventory rationing, Veinott (1965) proposed a critical-level policy for a periodic review model with several demand classes. A similar model was analyzed by Topkis (1968), where the optimal ordering and rationing policies were characterized under modest assumptions. Lee and Hong (2003) studied a ðs; SÞcontrolled stochastic production system with several demand classes and lost sales, and compared the average operating cost under the proposed stock rationing policy with the one without stock rationing. Other contributions incorporating different demand classes are made by Nahmias and Demmy (1981), Cohen et al. (1998), Dekker et al. (1998, 2002), Evans (1968), Kaplan (1969) and Melchiors et al. (2000). They both assume that demands are divided into two classes. Nahmias and Demmy (1981) analyzed an inventory model with two demand classes, backordering, a fixed lead time and a critical level policy, under the assumption that there is at most one outstanding order. Kaplan (1969) suggested letting the critical level depend on the time until the next replenishment. Without considering the cost factors, Dekker et al. (1998) derived the service levels for both classes of demand by using an ðs; SÞ inventory policy. Cohen et al. (1998) proposed an ðs; SÞ inventory system based on a Markov-chain model in which there are two priority classes of customers. However, it is complicated to solve, and the authors used a heuristic algorithm to find the ðs; SÞ. Melchiors
67
et al. (2000), using the same inventory policy as Nahmias and Demmy (1981), investigated the benefit of the rationing policy for various cases and showed that significant cost reductions can be obtained. Recently, Dekker et al. (2002) analyzed a critical level, lot-for-lot policy with multiple demand classes, where demand not satisfied from stock on hand is lost. Efficient solution procedures were developed to obtain a set of optimal policy parameters for models with or without service level constraints. The aim of this article is to develop an inventory model with two demand classes using a ðr; r; QÞ policy, which is a special case of the ðc; s; QÞ policy suggested by Melchiors et al. (2000). In our model, only two parameters are included where Q is the order quantity, r is the critical level and is chosen to be equal to the reorder level. Hence the model is much simpler than that of Melchiors et al. (2000) and is easy to implement in the SAS language. It is also easier to implement in inventory control systems in practice, as no separate data-field needs to be added; a manual procedure satisfies. In the following sections, first, the assumptions and formulation of the model are described. Second, we derive some mathematical properties of the model; in particular, we show that the total relevant cost function is convex. Third, a solution procedure to determine the optimal reorder point r and optimal reorder quantity Q is presented. Finally, numerical examples are given to illustrate the solution procedure.
2. The model In order to effectively deal with the problem, the following policy is proposed. Let Q and r be non-negative numbers. Spare parts are taken out from the inventory one at a time such that the inventory level can be observed when it drops to exactly r units. Consequently, a fixed order of size Q is always placed when the inventory level reaches r. However, when the stock on hand drops to the level r, the so-called reorder level, all remaining stock is reserved for critical demand and thus all non-critical demand will be
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backordered until the stock level again exceeds the level r. The policy proposed in this paper is a critical level policy with equal reorder point and critical level r, and is denoted as ðr; r; QÞ policy. Considering the demand uncertainty and two kinds of shortage cost, it is required to determine the values of Q and r that insure minimum inventory costs. In order to describe the model mathematically, the following assumptions and notation partly listed in Silver et al. (1998) are introduced. Notations D Q L r XC
XN
mC mN sC sN f C ðxÞ f N ðxÞ DC F C ðxÞ A h C WC WN SN
S C ðrÞ TCðQ; rÞ
ðQ ; r Þ
average demand; units per year order quantity; in units purchasing lead time; in days reorder level; in units total number of critical units demanded over the lead time L; a random variable total number of non-critical units demanded over the lead time L; a random variable mean of X C mean of X N standard deviation of X C standard deviation of X N probability density function for X C probability density function for X N average number of critical units demanded over the lead time L cumulative distribution function for X C ordering cost holding cost per unit per year average purchasing price per unit shortage cost per critical unit shortage cost per non-critical unit expected shortage number of noncritical units demanded over the lead time L expected shortage number of critical units demanded over the lead time L approximate expected total annual relevant cost which is a function of both decision variables Q and r optimal solution of TCðQ; rÞ
Assumptions 1. Demand can be classified into two classes, i.e. critical and non-critical demand. 2. The proposed method is based on the stochastic continuously reviewed constant reorder quantity policy ðr; r; QÞ. 3. All outstanding backorders can be satisfied in every replenishment and the stock level after a replenishment order arrives is larger than r. 4. The lead time of replenishment is constant. 5. Critical or non-critical demand over the lead time is high enough so that the normal distribution can be used to describe the demand patterns of spare parts. 6. Inventory demand shortages are back-ordered. Formulation Similar to other stochastic inventory models, e.g. Elsayed and Boucher(1985) and Al-Bahi (1993), the total cost is decomposed into the purchasing cost, the ordering cost, the expected holding cost and the expected shortage cost. The sum of these four terms, forms an approximation of the total cost function: TCðQ; rÞ ¼ CD þ AD=Q þ h½Q=2 þ r DC þ D=Q½W N SN þ W C SC ðrÞ 8rX0; Q40;
(1)
where Z
1
xf N ðxÞ dx;
SN ¼ 0
Z
SC ðrÞ ¼ Z DC ¼
1
ðx rÞf C ðxÞ dx; r 1
xf C ðxÞ dx:
0
This cost function can be written in the form: TCðQ; rÞ ¼ aðrÞ=Q þ bQ þ dðrÞ
ð2Þ
with aðrÞ ¼ AD þ W N S N D þ W C SC ðrÞD;
ð3Þ
h b¼ ; 2
ð4Þ
dðrÞ ¼ h½r DC þ CD:
ð5Þ
Note that aðrÞ40; 8rX0; b40.
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From a standard result in probability theory (see Chopra and Meindl, 2001, p.220), it follows that r mC r mC SðrÞ ¼ ðr mC Þ 1F þ sC f ; sC sC where f ð Þ; F ð Þ denote the pdf, cdf of standard normal distributions, respectively. This result directly implies the following formulas: m2C sC ð6Þ DC ¼ pffiffiffiffiffiffi exp 2 þ mC ½1 F C ð0Þ; 2sC 2p sN m2N ð7Þ S N ¼ pffiffiffiffiffiffi exp 2 þ mN ½1 F N ð0Þ; 2sN 2p sC ðr mC Þ2 S C ðrÞ ¼ pffiffiffiffiffiffi exp 2s2C 2p þ ðmC rÞ½1 F C ðrÞ:
(8)
and q2 TCðQ; rÞ qQqr
W C D ðr mC Þ ðr mC Þ2 pffiffiffiffiffiffi ¼ exp 2s2 Q2 2psC C þF C ðrÞ 1 þ ðmC rÞðf C ðrÞÞ ¼
W CD ½1 F C ðrÞ40 Q2
2 2 q2 TCðQ; rÞ q2 TCðQ; rÞ q TCðQ; rÞ qr2 qQqr qQ2 1 ¼ 4 f2aðrÞW C Df C ðrÞ Q
HðQ; rÞ ¼
W 2C D2 ½1 F C ðrÞ2 g;
Theorem 1. If mC 412, then TCðQ; rÞ is convex on r40; Q40.
lim HðQ; rÞ ¼ 0:
ð15Þ
Also
qTCðQ; rÞ aðrÞ ¼ 2 þ b; qQ Q
ð9Þ
2
q TCðQ; rÞ 2aðrÞ ¼ 3 40 qQ2 Q
for all Q40;
ð10Þ
qTCðQ; rÞ 1 qaðrÞ qdðrÞ ¼ þ qr Q qr qr W C D ðr mC Þ ðr mC Þ2 pffiffiffiffiffiffi ¼ exp Q 2s2C 2psC þF C ðrÞ 1 þ ðmC rÞðf C ðrÞÞ þ h ¼
(14)
Q4 is always positive, and limr!1 f C ðrÞ ¼ 0; limr!1 F C ðrÞ ¼ 1. With respect to r, we notice that r!1
Proof. Eq. (2) yields
(13)
for all Q40. Let HðQ; rÞ denote the determinant of the Hessian of TCðQ; rÞ. Then
3. The convexity In this section, the following results will be shown.
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W CD ½F C ðrÞ 1 þ h; Q
q2 TCðQ; rÞ W C D f C ðrÞ40 for all Q40 ¼ qr2 Q
(11)
ð12Þ
qHðQ; rÞ 1 ¼ 2 4 ½2W C DaðrÞf C ðrÞðr mC Þ qr sC Q for all rX0;
(16)
then qHðQ; rÞ qr
40; p0;
0promC ; rXmC :
ð17Þ
Eq. (17) reveals that HðQ; rÞ is increasing on ½0; mC Þ and decreasing on ½mC ; 1Þ. From Eqs. (15) and (17), with increasing r ðrXmC Þ, the determinant of Hessian tends to decrease monotonously to zero. Hence, HðQ; rÞ40
for all rXmC :
ð18Þ
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And HðQ; mC Þ 1 D2 W C ð2A þ W N Þ W C W N D2 pffiffiffiffiffiffi þ ¼ 4 p Q 2psC 2 mN 1 1 2 2 þ W C D 40: exp 2 p 4 2sN
(19)
In the appendix, we show that HðQ; 0Þ HðQ; mC Þ40;
Combining Eqs. (23) and (24), we obtain EOQ CD þ h ðr DC Þ þ 2 AD þ D½W N S N þ W C S C ðr Þ þ EOQ Q oCD þ h ðr DC Þ þ 2 AD þ D½W N S N þ W C S C ðr Þ þ : Q Eq. (25) yields that
Q40 for mC 412:
TCðEOQ; r ÞoTCðQ ; r Þ:
Hence we get
ð26Þ
HðQ; 0Þ40:
(25)
ð20Þ
Eq. (26) is a contradiction since ðQ ; r Þ is the optimal solution of TCðQ; rÞ. Consequently Q XEOQ:
From Eqs. (17) and 20, with r belonging to the interval ½0; mC Þ, the determinant of Hessian tends to increase from zero. Hence,
This completes the proof of Theorem 2.
HðQ; rÞX0
Theorem 3. If hðEOQÞ=W C DX1 F C ð0Þ; rX0; then
for all 0promC :
ð21Þ
Combining Eqs. (18) and (21), we conclude that TCðQ; rÞ is convex on rX0 and Q40. This completes the proof of Theorem 1. & Theorem p2. Let ffi Economic order quantity ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðEOQÞ ¼ 2AD=h, then we have Q XEOQ.
ð27Þ
r ¼ 0
and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2AD þ 2W N S N D þ 2W C S C ð0ÞDÞ : Q ¼ h
Proof. Eq. (11) can be rewritten as
Proof. If 0oQ oEOQ, Eq. (5.4) in Silver et al. (1998, p. 154) indicates that
qTCðQ ; rÞ W C D ¼ ½F C ðrÞ 1 þ h: qr Q
AD EOQ AD Q þ ho þ h: EOQ 2 Q 2
Because F C ðrÞXF C ð0Þ, then
ð22Þ
qTCðQ ; rÞ W C D X ½F C ð0Þ 1 þ h: qr Q
Eq. (22) implies that AD EOQ þ h þ CD þ hðr DC Þ EOQ 2 AD Q o þ h þ CD þ hðr DC Þ: Q 2
and
(23)
W CD ½F C ð0Þ 1 þ h Q W CD ½F C ð0Þ 1 þ h X EOQ
ð29Þ
(30)
because F C ð0Þ 1o0.
D40;
thus
ð28Þ
Theorem 2 implies that Q 4EOQ.
On the other hand, we have W N S N þ W C S C ðr Þ40
&
D½W N SN þ W C S C ðr Þ D½W N S N þ W C SC ðr Þ o : EOQ Q ð24Þ
If
hðEOQÞ X1 F C ð0Þ, Eqs. (29) and (30) imply W CD
qTCðQ ; rÞ X0 qr
for all rX0:
ð31Þ
ARTICLE IN PRESS P.-L. Chang et al. / Int. J. Production Economics 97 (2005) 66–74
Therefore, Eq. (31) reveals that qTCðQ ; rÞ=qr is non-decreasing with respect to rX0 and TCðQ ; rÞ has a minimum at r ¼ 0. So r ¼ 0. On the other hand, Eq. (10) shows that TCðQ; 0Þ is convex with respect to Q40. Q can be determined by solving the following equation: qTCðQ; 0Þ ¼ 0: qQ
ð32Þ
We obtain sffiffiffiffiffiffiffiffiffi að0Þ Q ¼ b rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2AD þ 2W N SN D þ 2W C S C ð0ÞDÞ ¼ : h
(33)
The above argument completes the proof of Theorem 3. & Theorem 3 states that if the holding cost of at least the EOQ is larger than the stock out penalty times the whole demand, then it does not pay off to have a positive reorder level.
4. Optimization procedure Since TCðQ; rÞ is convex, the first-order conditions for a minimum are given by qTCðQ; rÞ aðrÞ ¼ 2 þ b ¼ 0; qQ Q
ð34Þ
qTCðQ; rÞ W C D ¼ ½F C ðrÞ 1 þ h ¼ 0: qr Q
ð35Þ
Solve to get: sffiffiffiffiffiffiffiffi aðrÞ ; Q ¼ QðrÞ ¼ b F C ðrÞ ¼ 1
hQ : W CD
ð36Þ
ð37Þ
The iterative psolution ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi procedure is to initially set Q ¼ EOQ ¼ 2AD=h, then solve for a corresponding r value in Eq. (37), and then use this value in Eq. (36) to find a new Q value, and so
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forth. The details about the solution procedure can be described as follows: qffiffiffiffiffiffiffi Step 1: Let 40; Qopt ¼ EOQ ¼ 2AD h . Step 2: If hQopt =W C DX1 F C ð0Þ, set ropt ¼ 0. Otherwise, solve for a corresponding r value in Eq. (37) and set ropt ¼ the corresponding r value. Step 3: Use the ropt value to find a new Q value from equation (36). Set Qnew ¼ Qðropt Þ. Step 4: If hQnew =W C DX1 F C ð0Þ, set rnew ¼ 0. Otherwise, solve for a corresponding r value in Eq. (37) and set rnew ¼ the corresponding r value. Step 5: If jQnew Qopt jo and jrnew ropt jo, go to Step 6. Otherwise, set Qopt ¼ Qnew ; ropt ¼ rnew and go to Step 2. Step 6: The optimal order quantity Q ¼ Qopt , the optimal reorder point r ¼ ropt . 5. Numerical examples The two examples we present are based on a real-life situation at a semiconductor equipment manufacturer in Taiwan. The data of spare parts in Cases 1 and 2 are given in Table 1. The ordering cost, expected holding cost and expected shortage cost are calculated by ordering cost ¼ 5% C, holding cost ¼ 30% C, shortage cost for non-critical demand ¼ 10% C, shortage cost for critical demand ¼ 100% C, where C is the unit cost. Table 1 Data of spare parts in Case 1 and Case 2 (lead time = 1 week)
C (NT dollars per unit) h (NT dollars per unit) A (NT dollars per unit) W N (NT dollars per unit) W C (NT dollars per unit) D (units per year) mN (units over lead time) sN mC (units over lead time) sC SN (units over lead time) DC (units over lead time)
Case 1
Case 2
8410 2523 420.5 841 8410 571 10.19 19.85 1.71 7.47 14.03 3.91
32500 9750 1625 3250 32500 529 9.75 16.72 1.27 5.54 12.65 2.90
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The parts in Case 1 are gas filters and in Case 2 are mass flow controllers. It is assumed that the demand rate of the part is independent of the operating time of the equipment. In Case 1, the parts belong to CVD (chemical vapor deposition) machines are critical, and the parts used in metrology machines are non-critical, and in Case 2, the parts belong to PVD (physical vapor deposition) machines are critical, and the parts used in metrology machines are non-critical. Because a computerized ordering system is
Table 2 Results of Cases 1 and 2
EOQ Iteration number 1 2 3 4
ðQ; rÞ ðQ; rÞ ðQ; rÞ ðQ; rÞ ðQ ; r Þ
Total cost of optimal solution
Case 1
Case 2
13.80
13.28
(13.80, (74.83, (77.35, (77.46, (77.47,
19.97) 14.84) 14.73) 14.72) 14.72)
5024833
(13.28, (68.45, (70.30, (70.37, (70.37,
14.74) 11.05) 10.98) 10.98) 10.98)
17957319
adopted, spare parts suppliers in United States will know the demand immediately, and all spare parts will be transported to Taiwan by airlift. Thus, under the normal condition, the lead time is approximately a week. The computing algorithm is implemented in SAS language. Using the algorithm, we can obtain the optimal solution ðQ ; r Þ ¼ ð77:465; 14:724Þ for Case 1 and ðQ ; r Þ ¼ ð70:368; 10:978Þ for Case 2, respectively. Results are presented in Table 2. We also conducted the sensitivity analysis in critical level r to demonstrate the optimality of total cost function at r . The results are shown in Table 3. Since ðr; r; QÞ policy can be viewed as a special case of ðc; s; QÞ policy suggested by Melchiors et al. (2000), we have used simulation to compare the performance of the ðr; r; QÞ and ðc; s; QÞ policies. Case 1 in this paper and Case 1 in Melchiors et al. (2000) were considered in the simulation. The simulation algorithm was implemented in Clanguage. The results in Tables 4 and 5 were obtained by calculating an average over 100 simulation runs. In Tables 4 and 5, the simulation results for these two cases reveal that the holding cost and the shortage cost obtained using the ðr; r; QÞ policy are
Table 3 Sensitivity analysis in critical level of Case 1 and Case 2 Case 1
Case 2
ðQ ; r Þ ¼ ð77:46; 14:72Þ, let Q ¼ 77
ðQ ; r Þ ¼ ð70:37; 10:98Þ, let Q ¼ 70
r
Total cost (% of minimum cost)
r
Total cost (% of minimum cost)
0 1 10 11 12 13 14 15 16 17 18 19 20 50
5 223 854 5 191 290 5 036 573 5 031 627 5 028 219 5 026 102 5 025 050 5 024 859 5 025 355 5 026 387 5 027 830 5 029 584 5 031 567 5 106 165
0 1 6 7 8 9 10 11 12 13 14 15 16 50
18 541 549 18 414 944 18 035 624 18 002 887 17 980 512 17 966 606 17 959 400 17 957 328 17 959 062 17 963 530 17 969 899 17 977 546 17 986 026 18 315 871
(103.96%) (103.31%) (100.23%) (100.07%) (100.07%) (100.03%) (100.01%) (100.00%) (100.01%) (100.03%) (100.06%) (100.09%) (100.13%) (101.62%)
(103.25%) (102.55%) (100.44%) (100.25%) (100.13%) (100.05%) (100.01%) (100.00%) (100.03%) (100.03%) (100.07%) (100.11%) (100.16%) (102.00%)
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Table 4 Simulation results of Case 1 in this paper Policy
ðr; r; QÞ Policy
Solution
Optimal
Simulation
Simulation
(15,15,77)
(15,15,79)
(2,13,74)
120301.41 99322.38 3099.55 222723.34
122701.33 97151.45 3051.27 222904.05
119981.27 97040.51 3125.14 220146.92
Holding cost per year Shortage cost per year Ordering cost per year Total cost per year
ðc; s; QÞ Policy
Table 5 Simulation results of Case 1 in Melchiors et al. (2000) Policy
ðr; r; QÞ Policy
ðc; s; QÞ Policy
Solution
Simulation
Optimal
Simulation
(6,6,53)
(2,14,48)
(2,14,48)
2498.75 201.19 1672.69 4372.63
2311.94 173.38 1868.96 4354.28
2356.04 169.18 1802.78 4328.00
Holding cost per year Shortage cost per year Ordering cost per year Total cost per year
higher than those obtained using the ðc; s; QÞ policy, but the ordering cost is lower in the former policy. Summing these values shows that the ðc; s; QÞ policy out-performs the ðr; r; QÞ policy only by a less than 2% in terms of total cost of inventory.
6. Conclusions In spare parts inventory systems the situation may occur that identical parts can be installed in different equipment, with different importance for the production process. Hence, some of the demand may have very high penalty costs in case of a stock-out. In this paper we discussed a model that enabled us to handle different classes of demand. We analyzed a stocking policy where some of the stock is reserved for spare parts demand originating from failures of critical equipment.
First, the article shows that the model relevant cost function is convex. Second, it is shown that the classical economic order quantity (EOQ) is not the optimal solution in the model. Third, the paper reveals that if hðEOQÞ=W C Do1 F C ð0Þ, then we can get the optimal solution Q ; r from the pffiffiffiffiffiffiffiffiffiffiffiffiffi equations QðrÞ ¼ aðrÞ=b and F C ðrÞ ¼ 1 hQ= W C D. Otherwise, ðQ ; r Þ ¼ ðQð0Þ; 0Þ; Finally, numerical examples are given to illustrate the solution procedure. From the results of numerical examples, we can see that the optimization procedure yields in most practical cases the optimum, although convergence could not be proven.
Acknowledgements This study was supported by The National Science Council, R.O.C., under the grant NSC 892213-E-009-042. The authors are also grateful to Applied Materials Taiwan for supplying data.
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sC m2 þ f C ð0ÞW C D pffiffiffiffiffiffi exp C2 2sC 2p þ W C D½1 F C ð0Þ 1 1 mC þ F C ð0Þ 40; 2 2
Appendix HðQ; 0Þ HðQ; mC Þ40; for Q40; mC 412 Proof. HðQ; 0ÞHðQ; mC Þ Q8 W 2C D2 ½1
2
¼ f2að0ÞW C Df C ð0Þ F C ð0Þ g 1 1 2 2 2aðmC ÞW C D pffiffiffiffiffiffi W C D 2psC 4 1 ¼ 4að0ÞaðmC ÞW 2C D2 f C ð0Þ pffiffiffiffiffiffi 2psC 1 4 4 2 þ W C D ½1 F C ð0Þ 4 1 að0Þf C ð0ÞW 3C D3 2 1 2aðmC ÞW 3C D3 ½1 F C ð0Þ2 pffiffiffiffiffiffi 2psC 1 1 2 2 ¼ W C D 4aðmC Þ pffiffiffiffiffiffi W C D að0Þf C ð0Þ 2psC 2 1 2 W C Dð1 F C ð0ÞÞ ; (A.1) 2 where að0Þ ¼ AD þ W N SN D þ W C S C ð0ÞD ¼ AD þ W N SN D þ W C D m2 sC pffiffiffiffiffiffi exp C2 2sC 2p
þ mC ð1 F C ð0ÞÞ ;
sC aðmC Þ ¼ AD þ W N S N D þ W C D pffiffiffiffiffiffi : 2p
(A.2)
ðA:3Þ
Note that 1 1 4aðmC Þ pffiffiffiffiffiffi W C D 2psC 2 ¼ 4ðAD þ W N SN DÞ þ
2 1 W C D40; p 2
1 W C D½1 F C ð0Þ2 2 ¼ f C ð0ÞAD þ f C ð0ÞW N S N D
að0Þf C ð0Þ
(A.4)
(A.5)
since 0o1 F C ð0Þo1 and mC 412, implying mC 12 þ 12 F C ð0Þ40. Combining Eqs. (A.1), (A.4) and (A.5), the proof is completed. & References Al-Bahi, A.M., 1993. Spare provisioning policy based on maximization of availability per cost ratio. Computers and Industrial Engineering 24 (1), 81–90. Chopra, S., Meindl, P., 2001. Supply Chain Management. Prentice-Hall, Englewood Cliffs, NJ. Cohen, M.A., Kleindorfer, P.R., Lee, H.L., 1998. Service constrained ðs; SÞ inventory systems with priority demand classes and lost sales. Management Science 34 (4), 482–499. Dekker, R., Kleijn, M.J, de Rooij, P.J., 1998. A spare parts stocking policy based on equipment criticality. International Journal of Production Economics 56–57, 69–77. Dekker, R., Hill, R.M., Kleijn, M.J., Teunter, R.H., 2002. On the ðs 1; sÞ lost sales inventory model with priority demand classes. Naval Research Logistics 49, 593–610. Elsayed, A.A., Boucher, T.O., 1985. Analysis and Control of Production Systems. Prentice-Hall, Englewood Cliffs, NJ. Evans, R.V., 1968. Sales and restocking policies in a single item inventory system. Management Science 14 (7), 463–472. Kaplan, A., 1969. Stock rationing. Management Science 15 (5), 260–267. Lee, J.E., Hong, Y., 2003. A stock rationing policy in a ðs; SÞcontrolled stochastic production system with 2-phase Coxian processing times and lost sales. International Journal of Production Economics 83, 299–307. Melchiors, P.M., Dekker, R., Kleijn, M.J., 2000. Inventory rationing in an ðs; QÞ inventory model with lost sales and two demand classes. Journal of the Operational Research Society 51 (1), 111–122. Nahmias, S., Demmy, S., 1981. Operating characteristics of an inventory system with rationing. Management Science 27, 1236–1245. Silver, E.A., Pyke, D.F., Peterson, R., 1998. Inventory Management and Production Planning and Scheduling. John Wiley & Sons Inc, New York. Topkis, D.M., 1968. Optimal ordering and rationing policies in a nonstationary dynamic inventory model with n demand classes. Management Science 15 (3), 160–176. Veinott Jr., A.F., 1965. Optimal policy in a dynamic, single product, nonstationary inventory model with several demand classes. Operations Research 13 (3), 761–778.
Int. J. Production Economics 97 (2005) 66–74 www.elsevier.com/locate/dsw
A ðr; r; QÞ inventory model for spare parts involving equipment criticality Pao-Long Changa,, Ying-Chyi Choub, Ming-Guang Huangc a
Department of Business Administration, Feng Chia University, 100 Wenhwa Road, Seatwen, Taichung, Taiwan b Department of Business Administration, Tunghai University, Taichung, Taiwan c Department of Finance, Shih Chien University, Taiwan Received 18 October 2000; accepted 5 June 2004 Available online 2 September 2004
Abstract Demand for spare parts can sometimes be classified into critical and non-critical demand, depending on the criticality of the equipment for production. To effectively handle this situation in spare parts inventory control, we propose a ðr; r; QÞ inventory model for spare parts where some of the stock is reserved for critical demand. A solution procedure to determine the reorder point r, which is equal to the critical level, and the reorder quantity Q is presented. Finally, some numerical examples of the spare parts of semiconductor equipment are presented to illustrate the results of using the proposed policy. r 2004 Elsevier B.V. All rights reserved. Keywords: Spare parts; Inventory policy; Equipment criticality; Demand classes
1. Introduction This paper is motivated by a case study on the inventory control of spare parts at a semiconductor equipment manufacturer in Taiwan. The company manufactures many different types of integrated circuits (ICs) utilizing production equipment involving a spare parts inventory of more than 2000 different types of items with a Corresponding author. Tel.: +886-4-22846615; fax: +886-
4-22846660. E-mail address: [email protected] (P.-L. Chang).
value of over US$15 million dollars. Since some equipment in an IC fabrication facility may be quite important, the spare parts installed in this equipment should be treated differently. Therefore, it is required to develop an inventory control model having equipment criticality considered explicitly. Dekker et al. (1998) define equipment criticality as the importance of equipment for sustaining production in a safe and efficient way. In a case at a petrochemical complex one distinguishes between vital, essential and auxiliary equipment. If spare parts belong to equipment of only one
0925-5273/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2004.06.048
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criticality level, then one could always give each criticality class its own service level or stockout penalty. However, if parts are installed in equipment of various criticality, one would encounter problems especially if only few are installed in high critical equipment. A way out of this problem is to reserve spare parts within the stock for demands originating from those critical equipment. In Dekker et al. (1998) a method was provided in case the parts were classified with only two criticality levels: critical and non-critical demand, where critical demand originates from failure of parts installed in vital equipment, and non-critical demand from parts installed in essential and auxiliary equipment. To handle the problem about inventory rationing, Veinott (1965) proposed a critical-level policy for a periodic review model with several demand classes. A similar model was analyzed by Topkis (1968), where the optimal ordering and rationing policies were characterized under modest assumptions. Lee and Hong (2003) studied a ðs; SÞcontrolled stochastic production system with several demand classes and lost sales, and compared the average operating cost under the proposed stock rationing policy with the one without stock rationing. Other contributions incorporating different demand classes are made by Nahmias and Demmy (1981), Cohen et al. (1998), Dekker et al. (1998, 2002), Evans (1968), Kaplan (1969) and Melchiors et al. (2000). They both assume that demands are divided into two classes. Nahmias and Demmy (1981) analyzed an inventory model with two demand classes, backordering, a fixed lead time and a critical level policy, under the assumption that there is at most one outstanding order. Kaplan (1969) suggested letting the critical level depend on the time until the next replenishment. Without considering the cost factors, Dekker et al. (1998) derived the service levels for both classes of demand by using an ðs; SÞ inventory policy. Cohen et al. (1998) proposed an ðs; SÞ inventory system based on a Markov-chain model in which there are two priority classes of customers. However, it is complicated to solve, and the authors used a heuristic algorithm to find the ðs; SÞ. Melchiors
67
et al. (2000), using the same inventory policy as Nahmias and Demmy (1981), investigated the benefit of the rationing policy for various cases and showed that significant cost reductions can be obtained. Recently, Dekker et al. (2002) analyzed a critical level, lot-for-lot policy with multiple demand classes, where demand not satisfied from stock on hand is lost. Efficient solution procedures were developed to obtain a set of optimal policy parameters for models with or without service level constraints. The aim of this article is to develop an inventory model with two demand classes using a ðr; r; QÞ policy, which is a special case of the ðc; s; QÞ policy suggested by Melchiors et al. (2000). In our model, only two parameters are included where Q is the order quantity, r is the critical level and is chosen to be equal to the reorder level. Hence the model is much simpler than that of Melchiors et al. (2000) and is easy to implement in the SAS language. It is also easier to implement in inventory control systems in practice, as no separate data-field needs to be added; a manual procedure satisfies. In the following sections, first, the assumptions and formulation of the model are described. Second, we derive some mathematical properties of the model; in particular, we show that the total relevant cost function is convex. Third, a solution procedure to determine the optimal reorder point r and optimal reorder quantity Q is presented. Finally, numerical examples are given to illustrate the solution procedure.
2. The model In order to effectively deal with the problem, the following policy is proposed. Let Q and r be non-negative numbers. Spare parts are taken out from the inventory one at a time such that the inventory level can be observed when it drops to exactly r units. Consequently, a fixed order of size Q is always placed when the inventory level reaches r. However, when the stock on hand drops to the level r, the so-called reorder level, all remaining stock is reserved for critical demand and thus all non-critical demand will be
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backordered until the stock level again exceeds the level r. The policy proposed in this paper is a critical level policy with equal reorder point and critical level r, and is denoted as ðr; r; QÞ policy. Considering the demand uncertainty and two kinds of shortage cost, it is required to determine the values of Q and r that insure minimum inventory costs. In order to describe the model mathematically, the following assumptions and notation partly listed in Silver et al. (1998) are introduced. Notations D Q L r XC
XN
mC mN sC sN f C ðxÞ f N ðxÞ DC F C ðxÞ A h C WC WN SN
S C ðrÞ TCðQ; rÞ
ðQ ; r Þ
average demand; units per year order quantity; in units purchasing lead time; in days reorder level; in units total number of critical units demanded over the lead time L; a random variable total number of non-critical units demanded over the lead time L; a random variable mean of X C mean of X N standard deviation of X C standard deviation of X N probability density function for X C probability density function for X N average number of critical units demanded over the lead time L cumulative distribution function for X C ordering cost holding cost per unit per year average purchasing price per unit shortage cost per critical unit shortage cost per non-critical unit expected shortage number of noncritical units demanded over the lead time L expected shortage number of critical units demanded over the lead time L approximate expected total annual relevant cost which is a function of both decision variables Q and r optimal solution of TCðQ; rÞ
Assumptions 1. Demand can be classified into two classes, i.e. critical and non-critical demand. 2. The proposed method is based on the stochastic continuously reviewed constant reorder quantity policy ðr; r; QÞ. 3. All outstanding backorders can be satisfied in every replenishment and the stock level after a replenishment order arrives is larger than r. 4. The lead time of replenishment is constant. 5. Critical or non-critical demand over the lead time is high enough so that the normal distribution can be used to describe the demand patterns of spare parts. 6. Inventory demand shortages are back-ordered. Formulation Similar to other stochastic inventory models, e.g. Elsayed and Boucher(1985) and Al-Bahi (1993), the total cost is decomposed into the purchasing cost, the ordering cost, the expected holding cost and the expected shortage cost. The sum of these four terms, forms an approximation of the total cost function: TCðQ; rÞ ¼ CD þ AD=Q þ h½Q=2 þ r DC þ D=Q½W N SN þ W C SC ðrÞ 8rX0; Q40;
(1)
where Z
1
xf N ðxÞ dx;
SN ¼ 0
Z
SC ðrÞ ¼ Z DC ¼
1
ðx rÞf C ðxÞ dx; r 1
xf C ðxÞ dx:
0
This cost function can be written in the form: TCðQ; rÞ ¼ aðrÞ=Q þ bQ þ dðrÞ
ð2Þ
with aðrÞ ¼ AD þ W N S N D þ W C SC ðrÞD;
ð3Þ
h b¼ ; 2
ð4Þ
dðrÞ ¼ h½r DC þ CD:
ð5Þ
Note that aðrÞ40; 8rX0; b40.
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From a standard result in probability theory (see Chopra and Meindl, 2001, p.220), it follows that r mC r mC SðrÞ ¼ ðr mC Þ 1F þ sC f ; sC sC where f ð Þ; F ð Þ denote the pdf, cdf of standard normal distributions, respectively. This result directly implies the following formulas: m2C sC ð6Þ DC ¼ pffiffiffiffiffiffi exp 2 þ mC ½1 F C ð0Þ; 2sC 2p sN m2N ð7Þ S N ¼ pffiffiffiffiffiffi exp 2 þ mN ½1 F N ð0Þ; 2sN 2p sC ðr mC Þ2 S C ðrÞ ¼ pffiffiffiffiffiffi exp 2s2C 2p þ ðmC rÞ½1 F C ðrÞ:
(8)
and q2 TCðQ; rÞ qQqr
W C D ðr mC Þ ðr mC Þ2 pffiffiffiffiffiffi ¼ exp 2s2 Q2 2psC C þF C ðrÞ 1 þ ðmC rÞðf C ðrÞÞ ¼
W CD ½1 F C ðrÞ40 Q2
2 2 q2 TCðQ; rÞ q2 TCðQ; rÞ q TCðQ; rÞ qr2 qQqr qQ2 1 ¼ 4 f2aðrÞW C Df C ðrÞ Q
HðQ; rÞ ¼
W 2C D2 ½1 F C ðrÞ2 g;
Theorem 1. If mC 412, then TCðQ; rÞ is convex on r40; Q40.
lim HðQ; rÞ ¼ 0:
ð15Þ
Also
qTCðQ; rÞ aðrÞ ¼ 2 þ b; qQ Q
ð9Þ
2
q TCðQ; rÞ 2aðrÞ ¼ 3 40 qQ2 Q
for all Q40;
ð10Þ
qTCðQ; rÞ 1 qaðrÞ qdðrÞ ¼ þ qr Q qr qr W C D ðr mC Þ ðr mC Þ2 pffiffiffiffiffiffi ¼ exp Q 2s2C 2psC þF C ðrÞ 1 þ ðmC rÞðf C ðrÞÞ þ h ¼
(14)
Q4 is always positive, and limr!1 f C ðrÞ ¼ 0; limr!1 F C ðrÞ ¼ 1. With respect to r, we notice that r!1
Proof. Eq. (2) yields
(13)
for all Q40. Let HðQ; rÞ denote the determinant of the Hessian of TCðQ; rÞ. Then
3. The convexity In this section, the following results will be shown.
69
W CD ½F C ðrÞ 1 þ h; Q
q2 TCðQ; rÞ W C D f C ðrÞ40 for all Q40 ¼ qr2 Q
(11)
ð12Þ
qHðQ; rÞ 1 ¼ 2 4 ½2W C DaðrÞf C ðrÞðr mC Þ qr sC Q for all rX0;
(16)
then qHðQ; rÞ qr
40; p0;
0promC ; rXmC :
ð17Þ
Eq. (17) reveals that HðQ; rÞ is increasing on ½0; mC Þ and decreasing on ½mC ; 1Þ. From Eqs. (15) and (17), with increasing r ðrXmC Þ, the determinant of Hessian tends to decrease monotonously to zero. Hence, HðQ; rÞ40
for all rXmC :
ð18Þ
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And HðQ; mC Þ 1 D2 W C ð2A þ W N Þ W C W N D2 pffiffiffiffiffiffi þ ¼ 4 p Q 2psC 2 mN 1 1 2 2 þ W C D 40: exp 2 p 4 2sN
(19)
In the appendix, we show that HðQ; 0Þ HðQ; mC Þ40;
Combining Eqs. (23) and (24), we obtain EOQ CD þ h ðr DC Þ þ 2 AD þ D½W N S N þ W C S C ðr Þ þ EOQ Q oCD þ h ðr DC Þ þ 2 AD þ D½W N S N þ W C S C ðr Þ þ : Q Eq. (25) yields that
Q40 for mC 412:
TCðEOQ; r ÞoTCðQ ; r Þ:
Hence we get
ð26Þ
HðQ; 0Þ40:
(25)
ð20Þ
Eq. (26) is a contradiction since ðQ ; r Þ is the optimal solution of TCðQ; rÞ. Consequently Q XEOQ:
From Eqs. (17) and 20, with r belonging to the interval ½0; mC Þ, the determinant of Hessian tends to increase from zero. Hence,
This completes the proof of Theorem 2.
HðQ; rÞX0
Theorem 3. If hðEOQÞ=W C DX1 F C ð0Þ; rX0; then
for all 0promC :
ð21Þ
Combining Eqs. (18) and (21), we conclude that TCðQ; rÞ is convex on rX0 and Q40. This completes the proof of Theorem 1. & Theorem p2. Let ffi Economic order quantity ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðEOQÞ ¼ 2AD=h, then we have Q XEOQ.
ð27Þ
r ¼ 0
and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2AD þ 2W N S N D þ 2W C S C ð0ÞDÞ : Q ¼ h
Proof. Eq. (11) can be rewritten as
Proof. If 0oQ oEOQ, Eq. (5.4) in Silver et al. (1998, p. 154) indicates that
qTCðQ ; rÞ W C D ¼ ½F C ðrÞ 1 þ h: qr Q
AD EOQ AD Q þ ho þ h: EOQ 2 Q 2
Because F C ðrÞXF C ð0Þ, then
ð22Þ
qTCðQ ; rÞ W C D X ½F C ð0Þ 1 þ h: qr Q
Eq. (22) implies that AD EOQ þ h þ CD þ hðr DC Þ EOQ 2 AD Q o þ h þ CD þ hðr DC Þ: Q 2
and
(23)
W CD ½F C ð0Þ 1 þ h Q W CD ½F C ð0Þ 1 þ h X EOQ
ð29Þ
(30)
because F C ð0Þ 1o0.
D40;
thus
ð28Þ
Theorem 2 implies that Q 4EOQ.
On the other hand, we have W N S N þ W C S C ðr Þ40
&
D½W N SN þ W C S C ðr Þ D½W N S N þ W C SC ðr Þ o : EOQ Q ð24Þ
If
hðEOQÞ X1 F C ð0Þ, Eqs. (29) and (30) imply W CD
qTCðQ ; rÞ X0 qr
for all rX0:
ð31Þ
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Therefore, Eq. (31) reveals that qTCðQ ; rÞ=qr is non-decreasing with respect to rX0 and TCðQ ; rÞ has a minimum at r ¼ 0. So r ¼ 0. On the other hand, Eq. (10) shows that TCðQ; 0Þ is convex with respect to Q40. Q can be determined by solving the following equation: qTCðQ; 0Þ ¼ 0: qQ
ð32Þ
We obtain sffiffiffiffiffiffiffiffiffi að0Þ Q ¼ b rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2AD þ 2W N SN D þ 2W C S C ð0ÞDÞ ¼ : h
(33)
The above argument completes the proof of Theorem 3. & Theorem 3 states that if the holding cost of at least the EOQ is larger than the stock out penalty times the whole demand, then it does not pay off to have a positive reorder level.
4. Optimization procedure Since TCðQ; rÞ is convex, the first-order conditions for a minimum are given by qTCðQ; rÞ aðrÞ ¼ 2 þ b ¼ 0; qQ Q
ð34Þ
qTCðQ; rÞ W C D ¼ ½F C ðrÞ 1 þ h ¼ 0: qr Q
ð35Þ
Solve to get: sffiffiffiffiffiffiffiffi aðrÞ ; Q ¼ QðrÞ ¼ b F C ðrÞ ¼ 1
hQ : W CD
ð36Þ
ð37Þ
The iterative psolution ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi procedure is to initially set Q ¼ EOQ ¼ 2AD=h, then solve for a corresponding r value in Eq. (37), and then use this value in Eq. (36) to find a new Q value, and so
71
forth. The details about the solution procedure can be described as follows: qffiffiffiffiffiffiffi Step 1: Let 40; Qopt ¼ EOQ ¼ 2AD h . Step 2: If hQopt =W C DX1 F C ð0Þ, set ropt ¼ 0. Otherwise, solve for a corresponding r value in Eq. (37) and set ropt ¼ the corresponding r value. Step 3: Use the ropt value to find a new Q value from equation (36). Set Qnew ¼ Qðropt Þ. Step 4: If hQnew =W C DX1 F C ð0Þ, set rnew ¼ 0. Otherwise, solve for a corresponding r value in Eq. (37) and set rnew ¼ the corresponding r value. Step 5: If jQnew Qopt jo and jrnew ropt jo, go to Step 6. Otherwise, set Qopt ¼ Qnew ; ropt ¼ rnew and go to Step 2. Step 6: The optimal order quantity Q ¼ Qopt , the optimal reorder point r ¼ ropt . 5. Numerical examples The two examples we present are based on a real-life situation at a semiconductor equipment manufacturer in Taiwan. The data of spare parts in Cases 1 and 2 are given in Table 1. The ordering cost, expected holding cost and expected shortage cost are calculated by ordering cost ¼ 5% C, holding cost ¼ 30% C, shortage cost for non-critical demand ¼ 10% C, shortage cost for critical demand ¼ 100% C, where C is the unit cost. Table 1 Data of spare parts in Case 1 and Case 2 (lead time = 1 week)
C (NT dollars per unit) h (NT dollars per unit) A (NT dollars per unit) W N (NT dollars per unit) W C (NT dollars per unit) D (units per year) mN (units over lead time) sN mC (units over lead time) sC SN (units over lead time) DC (units over lead time)
Case 1
Case 2
8410 2523 420.5 841 8410 571 10.19 19.85 1.71 7.47 14.03 3.91
32500 9750 1625 3250 32500 529 9.75 16.72 1.27 5.54 12.65 2.90
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The parts in Case 1 are gas filters and in Case 2 are mass flow controllers. It is assumed that the demand rate of the part is independent of the operating time of the equipment. In Case 1, the parts belong to CVD (chemical vapor deposition) machines are critical, and the parts used in metrology machines are non-critical, and in Case 2, the parts belong to PVD (physical vapor deposition) machines are critical, and the parts used in metrology machines are non-critical. Because a computerized ordering system is
Table 2 Results of Cases 1 and 2
EOQ Iteration number 1 2 3 4
ðQ; rÞ ðQ; rÞ ðQ; rÞ ðQ; rÞ ðQ ; r Þ
Total cost of optimal solution
Case 1
Case 2
13.80
13.28
(13.80, (74.83, (77.35, (77.46, (77.47,
19.97) 14.84) 14.73) 14.72) 14.72)
5024833
(13.28, (68.45, (70.30, (70.37, (70.37,
14.74) 11.05) 10.98) 10.98) 10.98)
17957319
adopted, spare parts suppliers in United States will know the demand immediately, and all spare parts will be transported to Taiwan by airlift. Thus, under the normal condition, the lead time is approximately a week. The computing algorithm is implemented in SAS language. Using the algorithm, we can obtain the optimal solution ðQ ; r Þ ¼ ð77:465; 14:724Þ for Case 1 and ðQ ; r Þ ¼ ð70:368; 10:978Þ for Case 2, respectively. Results are presented in Table 2. We also conducted the sensitivity analysis in critical level r to demonstrate the optimality of total cost function at r . The results are shown in Table 3. Since ðr; r; QÞ policy can be viewed as a special case of ðc; s; QÞ policy suggested by Melchiors et al. (2000), we have used simulation to compare the performance of the ðr; r; QÞ and ðc; s; QÞ policies. Case 1 in this paper and Case 1 in Melchiors et al. (2000) were considered in the simulation. The simulation algorithm was implemented in Clanguage. The results in Tables 4 and 5 were obtained by calculating an average over 100 simulation runs. In Tables 4 and 5, the simulation results for these two cases reveal that the holding cost and the shortage cost obtained using the ðr; r; QÞ policy are
Table 3 Sensitivity analysis in critical level of Case 1 and Case 2 Case 1
Case 2
ðQ ; r Þ ¼ ð77:46; 14:72Þ, let Q ¼ 77
ðQ ; r Þ ¼ ð70:37; 10:98Þ, let Q ¼ 70
r
Total cost (% of minimum cost)
r
Total cost (% of minimum cost)
0 1 10 11 12 13 14 15 16 17 18 19 20 50
5 223 854 5 191 290 5 036 573 5 031 627 5 028 219 5 026 102 5 025 050 5 024 859 5 025 355 5 026 387 5 027 830 5 029 584 5 031 567 5 106 165
0 1 6 7 8 9 10 11 12 13 14 15 16 50
18 541 549 18 414 944 18 035 624 18 002 887 17 980 512 17 966 606 17 959 400 17 957 328 17 959 062 17 963 530 17 969 899 17 977 546 17 986 026 18 315 871
(103.96%) (103.31%) (100.23%) (100.07%) (100.07%) (100.03%) (100.01%) (100.00%) (100.01%) (100.03%) (100.06%) (100.09%) (100.13%) (101.62%)
(103.25%) (102.55%) (100.44%) (100.25%) (100.13%) (100.05%) (100.01%) (100.00%) (100.03%) (100.03%) (100.07%) (100.11%) (100.16%) (102.00%)
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Table 4 Simulation results of Case 1 in this paper Policy
ðr; r; QÞ Policy
Solution
Optimal
Simulation
Simulation
(15,15,77)
(15,15,79)
(2,13,74)
120301.41 99322.38 3099.55 222723.34
122701.33 97151.45 3051.27 222904.05
119981.27 97040.51 3125.14 220146.92
Holding cost per year Shortage cost per year Ordering cost per year Total cost per year
ðc; s; QÞ Policy
Table 5 Simulation results of Case 1 in Melchiors et al. (2000) Policy
ðr; r; QÞ Policy
ðc; s; QÞ Policy
Solution
Simulation
Optimal
Simulation
(6,6,53)
(2,14,48)
(2,14,48)
2498.75 201.19 1672.69 4372.63
2311.94 173.38 1868.96 4354.28
2356.04 169.18 1802.78 4328.00
Holding cost per year Shortage cost per year Ordering cost per year Total cost per year
higher than those obtained using the ðc; s; QÞ policy, but the ordering cost is lower in the former policy. Summing these values shows that the ðc; s; QÞ policy out-performs the ðr; r; QÞ policy only by a less than 2% in terms of total cost of inventory.
6. Conclusions In spare parts inventory systems the situation may occur that identical parts can be installed in different equipment, with different importance for the production process. Hence, some of the demand may have very high penalty costs in case of a stock-out. In this paper we discussed a model that enabled us to handle different classes of demand. We analyzed a stocking policy where some of the stock is reserved for spare parts demand originating from failures of critical equipment.
First, the article shows that the model relevant cost function is convex. Second, it is shown that the classical economic order quantity (EOQ) is not the optimal solution in the model. Third, the paper reveals that if hðEOQÞ=W C Do1 F C ð0Þ, then we can get the optimal solution Q ; r from the pffiffiffiffiffiffiffiffiffiffiffiffiffi equations QðrÞ ¼ aðrÞ=b and F C ðrÞ ¼ 1 hQ= W C D. Otherwise, ðQ ; r Þ ¼ ðQð0Þ; 0Þ; Finally, numerical examples are given to illustrate the solution procedure. From the results of numerical examples, we can see that the optimization procedure yields in most practical cases the optimum, although convergence could not be proven.
Acknowledgements This study was supported by The National Science Council, R.O.C., under the grant NSC 892213-E-009-042. The authors are also grateful to Applied Materials Taiwan for supplying data.
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sC m2 þ f C ð0ÞW C D pffiffiffiffiffiffi exp C2 2sC 2p þ W C D½1 F C ð0Þ 1 1 mC þ F C ð0Þ 40; 2 2
Appendix HðQ; 0Þ HðQ; mC Þ40; for Q40; mC 412 Proof. HðQ; 0ÞHðQ; mC Þ Q8 W 2C D2 ½1
2
¼ f2að0ÞW C Df C ð0Þ F C ð0Þ g 1 1 2 2 2aðmC ÞW C D pffiffiffiffiffiffi W C D 2psC 4 1 ¼ 4að0ÞaðmC ÞW 2C D2 f C ð0Þ pffiffiffiffiffiffi 2psC 1 4 4 2 þ W C D ½1 F C ð0Þ 4 1 að0Þf C ð0ÞW 3C D3 2 1 2aðmC ÞW 3C D3 ½1 F C ð0Þ2 pffiffiffiffiffiffi 2psC 1 1 2 2 ¼ W C D 4aðmC Þ pffiffiffiffiffiffi W C D að0Þf C ð0Þ 2psC 2 1 2 W C Dð1 F C ð0ÞÞ ; (A.1) 2 where að0Þ ¼ AD þ W N SN D þ W C S C ð0ÞD ¼ AD þ W N SN D þ W C D m2 sC pffiffiffiffiffiffi exp C2 2sC 2p
þ mC ð1 F C ð0ÞÞ ;
sC aðmC Þ ¼ AD þ W N S N D þ W C D pffiffiffiffiffiffi : 2p
(A.2)
ðA:3Þ
Note that 1 1 4aðmC Þ pffiffiffiffiffiffi W C D 2psC 2 ¼ 4ðAD þ W N SN DÞ þ
2 1 W C D40; p 2
1 W C D½1 F C ð0Þ2 2 ¼ f C ð0ÞAD þ f C ð0ÞW N S N D
að0Þf C ð0Þ
(A.4)
(A.5)
since 0o1 F C ð0Þo1 and mC 412, implying mC 12 þ 12 F C ð0Þ40. Combining Eqs. (A.1), (A.4) and (A.5), the proof is completed. & References Al-Bahi, A.M., 1993. Spare provisioning policy based on maximization of availability per cost ratio. Computers and Industrial Engineering 24 (1), 81–90. Chopra, S., Meindl, P., 2001. Supply Chain Management. Prentice-Hall, Englewood Cliffs, NJ. Cohen, M.A., Kleindorfer, P.R., Lee, H.L., 1998. Service constrained ðs; SÞ inventory systems with priority demand classes and lost sales. Management Science 34 (4), 482–499. Dekker, R., Kleijn, M.J, de Rooij, P.J., 1998. A spare parts stocking policy based on equipment criticality. International Journal of Production Economics 56–57, 69–77. Dekker, R., Hill, R.M., Kleijn, M.J., Teunter, R.H., 2002. On the ðs 1; sÞ lost sales inventory model with priority demand classes. Naval Research Logistics 49, 593–610. Elsayed, A.A., Boucher, T.O., 1985. Analysis and Control of Production Systems. Prentice-Hall, Englewood Cliffs, NJ. Evans, R.V., 1968. Sales and restocking policies in a single item inventory system. Management Science 14 (7), 463–472. Kaplan, A., 1969. Stock rationing. Management Science 15 (5), 260–267. Lee, J.E., Hong, Y., 2003. A stock rationing policy in a ðs; SÞcontrolled stochastic production system with 2-phase Coxian processing times and lost sales. International Journal of Production Economics 83, 299–307. Melchiors, P.M., Dekker, R., Kleijn, M.J., 2000. Inventory rationing in an ðs; QÞ inventory model with lost sales and two demand classes. Journal of the Operational Research Society 51 (1), 111–122. Nahmias, S., Demmy, S., 1981. Operating characteristics of an inventory system with rationing. Management Science 27, 1236–1245. Silver, E.A., Pyke, D.F., Peterson, R., 1998. Inventory Management and Production Planning and Scheduling. John Wiley & Sons Inc, New York. Topkis, D.M., 1968. Optimal ordering and rationing policies in a nonstationary dynamic inventory model with n demand classes. Management Science 15 (3), 160–176. Veinott Jr., A.F., 1965. Optimal policy in a dynamic, single product, nonstationary inventory model with several demand classes. Operations Research 13 (3), 761–778.