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Cost-effective ordering policies for inventory systems with emergency order
Computers & Industrial Engineering 57 (2009) 1336–1341
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Cost-effective ordering policies for inventory systems with emergency order B.C. Giri a,*, T. Dohi b a b
Department of Mathematics, Jadavpur University, Kolkata, India Department of Information Engineering, Hiroshima University, Hiroshima, Japan
a r t i c l e
i n f o
Article history: Received 10 September 2007 Received in revised form 7 April 2009 Accepted 2 July 2009 Available online 5 July 2009 Keywords: Inventory Emergency order Cost effectiveness Ordering policies Poisson demand
a b s t r a c t Most of the studies on inventory control reported in earlier contributions deal with the optimization problems minimizing an expected cost criterion such as the long-run average cost. However, when the plant engineers design the inventory systems in practical situations, they must take account of reliability into the inventory system as well as economical aspect, where reliability can be defined as the probability that the stock is not depleted until a pre-specified time. In this paper, we discuss the inventory control policies that provide a balance between economical and reliability requirements. By applying the cost effectiveness criterion, which simultaneously includes the effects of system availability and expected cost, as optimality one, we derive the optimal inventory replenishment policies of two kinds of inventory models. Finally, with a set of numerical examples, we show that the optimal inventory policies of the models under consideration make the stationary availability increase. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The importance of inventory control in business has increased dramatically with the development of sophisticated CIM (Computer Integrated Manufacturing) concept. It gives the logic and basis on controlling the manufacturing system as well as plays the role of releasing the surplus operating capital tied up in excessive inventories. Most of the studies on inventory control reported in the literature (see Hadley & Whitin, 1963; Silver, Pyke, & Peterson, 1998) deal with the optimization problems minimizing an expected cost criterion such as the long-run average cost. However, when the plant engineers design the inventory systems in practical situations, they are often required to take account of reliability into the inventory system as well as usual economical aspect, where reliability can be defined as the probability that the stock is not depleted until a pre-specified time. In inventory systems, the shortages are sometimes caused by a delivery-lag. In the early contributions, Allen and D’Esopo (1968a, 1968b), Barankin (1961), Neuts (1964) and Rosenshine and Obee (1976) address some inventory models to avoid the shortage by an additional order form. Lau and Zhao (1993) and Moinzadeh and Nahmias (1988) analyze the optimal ordering policies for the inventory system with two supply modes, and extended the seminal results by Whittmore and Saunders (1977). In fact, the problem of shortage may be overcome to some extent by splitting orders among a number of suppliers; but such an ordering form is rather costly and is not always realized in practice, even if the stock control system with multiple suppliers is available. * Corresponding author. E-mail address: [email protected] (B.C. Giri). 0360-8352/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2009.07.001
The above works consider the expected cost as optimality criterion. Some researchers deal stochastic inventory control problems with other optimality criterion such as service level criterion which introduces a service level constraint instead of shortage cost (see Chen & Krass, 2001; Cohen, Kleindorfer, & Lee, 1988; Nahmias, 1993). They define service level as the availability of stock in a probabilistic or expected sense. One major disadvantage of this approach is that it is perceived to be less tractable mathematically. The purpose of this paper is to implement the cost effectiveness criterion (Hunter, 1963) which provides a balance between economical and reliability requirements to the standard inventory models, and to propose the cost-effective inventory policy from reliability as well as cost minimization view points. We consider two kinds of inventory models referred to as Model 1 and Model 2 from onwards. Model 1 is a continuous review inventory model in which the stock level is continuously reviewed until a specified time and the items are replenished by the regular order at that time if no shortage has occurred. If the stock level becomes 0 before the regular ordering time, then the items are replenished by the expedited order. On the other hand, Model 2 is a periodic review inventory model in which the stock level is reviewed only at a specified time and one of the regular and expedited orders is placed at that time. Dohi, Kaio, and Osaki (1995) analyzed Model 1 under long-run average cost and expected total discounted cost criteria. Our objective in this study is to derive optimal inventory policies for Models 1 and 2 maximizing the cost effectiveness criterion. The cost effectiveness criterion can be regarded as an optimality criterion which assesses the effective system operating time in one cycle relative to the mean operating costs in that cycle. It is to be mentioned here that the inventory models under consideration have the same mathematical structure as
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the order-replacement models in the context of reliability theory (see Kaio & Osaki, 1977; Osaki, 1992; Osaki, Kaio, & Yamada, 1981; Thomas & Osaki, 1978a, 1978b). The paper is organized as follows: Section 2 describes the nomenclature, notation and assumptions. In Sections 2 and 3, the continuous and periodic review inventory models are explained, cost effectiveness is defined and the optimal ordering times are derived for a fixed order quantity. Assuming a Poisson process demand, we obtain the explicit formulas of cost effectiveness for the two inventory models in Section 5. We also investigate numerically the performance of cost-effective design for the inventory models. Finally, Section 6 concludes the paper.
3. Continuous review model (Model 1)
2. Nomenclature, notation and assumptions
3.1. Model description
2.1. Nomenclature
If the stock is depleted up to a pre-specified time t0 2 ð0; 1Þ, the emergency order is placed immediately at that time and after a lead time L1 ð> 0Þ; Q units are replenished. We call t 0 as ordering time, in this paper. On the other hand, if the stock is not depleted before the time t0 , the regular order is made at the time t0 , and Q units are replenished after a lead time L2 ð> 0Þ. We suppose that the time when the inventory level becomes 0 for the first time obeys a distribution FðtÞ ¼ Fðt; Q Þ with a density function f ðtÞ ¼ f ðt; Q Þ. Of course, specifying the inventory level process XðtÞ leads to characterize FðtÞ and f ðtÞ. We define the first passage time as follows:
IHR Inventory level process
Inventory policy
increasing hazard rate a continuous time stochastic process representing the on-hand inventory level which is decreased by the satisfaction of a demand the policy which consists of both the regular ordering time and the order quantity
(A-2) Without loss of generality, the inventory level is initially set to Q (>0). (A-3) Shortages, if occurred, are not backlogged. (A-4) The time period from time 0 to the time when the inventory level becomes Q next is one cycle and the same cycle repeats itself continually. (A-5) The sum of the ordering and the shortage costs by the expedited order is greater than the sum of those by the regular order, i:e:; c1 Q þ kL1 > c2 Q þ kL2 .
s infft P 0; XðtÞ ¼ 0jXð0Þ ¼ Qg: 2.2. Notation fXðtÞ; t P 0g fNðtÞ; t P 0g Q t0 FðtÞ; f ðtÞ FðÞ 1 FðÞ rðtÞ f ðtÞ=FðtÞ RðtÞ fFðt þ L2 Þ FðtÞg=FðtÞ i
L1 ; L2 c1 ; c2 h k V i ðt 0 ; Q Þ T i ðt 0 ; Q Þ MðQ Þ Ai ðt0 ; Q Þ Z i ðt 0 ; Q Þ ¼ MðQ Þ=V i ðt 0 ; Q Þ gamfðÞ
inventory level process cumulative demand process order quantity ordering time for the regular order cdf and pdf for the time to first shortage survivor function hazard rate for FðtÞ conditional hazard function during interval ðt; t þ L2 model indicator, that is, i ¼ 1; 2 correspond to Model 1 and Model 2, respectively constant lead times for the expedited and regular orders, respectively cost per unit amount for the expedited and regular orders, respectively inventory holding cost per unit amount per unit time penalty cost per unit time suffered for the shortage period expected cost for one cycle mean time for one cycle mean effective time for one cycle stationary availability cost effectiveness upper incomplete gamma function
ð1Þ
We define the indicator function IfAg for the event fAg. Fig. 1 shows the schematic illustration of the inventory model under consideration where the events fA1g; fA2g and fA3g are such that
XðtÞIfA1g fXðtÞ; min XðtÞ ¼ 0g;
ð2Þ
06t6t 0
XðtÞIfA2g fXðtÞ; min XðtÞ > 0 and 06t6t 0
XðtÞIfA3g fXðtÞ;
min
06t6ft 0 þL2 g
min
06t6ft 0 þL2 g
XðtÞ ¼ 0g;
XðtÞ > 0g;
ð4Þ
respectively. As the inventory process has a regenerative point where the same state repeats periodically, we use renewal reward theory (Ross, 1970) to find the expected cost per unit time which is given by
C 1 ðt0 ; Q Þ ¼ V 1 ðt 0 ; QÞ=T 1 ðt 0 ; Q Þ; where V 1 ðt 0 ; Q Þ, the expected cost in one cycle is
2.3. Assumptions
(A-1) The inventory management begins operating at time 0, and the planning horizon is infinite.
ð3Þ
Fig. 1. State diagram for one cycle in Model 1.
ð5Þ
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Z s Z V 1 ðt0 ; Q Þ ¼ h E XðtÞdt þ Q þk
Z
0 t0
L1 dFðtÞ þ
t 0 þL2
Z
0
½t ðt 0 þ L2 ÞdFðtÞ
1
Z s V 1 ð1; QÞ ¼ hE XðtÞdt þ c1 Q þ kL1 :
t 0 þL2
ðt 0 þ L2 tÞdFðtÞ t0
0 Þ; þ c1 QFðt 0 Þ þ c2 Q Fðt
ð6Þ
Z 0
þ
t0
ðt þ L1 ÞdFðtÞ þ Z
Z
t 0 þL2
ðt 0 þ L2 ÞdFðtÞ t0
1
tdFðtÞ;
ð7Þ
t 0 þL2
E being the mathematical expectation operator. It is clear that the processing time of this inventory system represents an ineffective usage time and decreases the useful/effective system operating time or the system effective availability, if the shortage period is relatively large. The stationary availability is the probability that the stock is not depleted, i.e., the inventory system is effective in a cycle. Let us define the stationary availability as follows:
A1 ðt0 ; Q Þ ¼ MðQ Þ=T 1 ðt 0 ; Q Þ;
ð8Þ
where MðQ Þ is the mean effective system operating time for one cycle, which is the time interval that the stock is not depleted for one cycle. Note that MðQ Þ is independent of t 0 and is defined as
MðQ Þ
Z
1
tdFðtÞ:
ð9Þ
0
Thus, the cost effectiveness, which balances between C 1 ðt 0 ; Q Þ and A1 ðt 0 ; Q Þ, can be formulated as
Z 1 ðt 0 ; QÞ A1 ðt 0 ; QÞ=C 1 ðt 0 ; Q Þ
ð10Þ
¼ MðQ Þ=V 1 ðt0 ; Q Þ:
3.2. Optimal ordering time We define the numerator divided by Fðt0 Þ of the partial derivative with respect to t 0 of the right-hand side of Eq. (11) as
qc ðt0 Þ fðhQ þ kÞRðt0 Þ þ ½ðc1 c2 ÞQ þ kðL1 L2 Þrðt0 Þ hQgMðQÞ;
Z 1 ð0; QÞ ¼ MðQ Þ=V 1 ð0; QÞ;
ð13Þ
where
Z s Z V 1 ð0; Q Þ ¼ hE XðtÞdt þ hQE½s þ hQ Z
L2
FðtÞdt
0 L2
FðtÞdt þ c2 Q hQL2 ;
ð14Þ
0
and
Z 1 ð1; QÞ ¼ MðQ Þ=V 1 ð1; Q Þ; where
Proof. Differentiating Z 1 ðt0 ; Q Þ with respect to t 0 and setting it equal to zero implies the equation qc ðt 0 Þ ¼ 0. Further, differentiating qc ðt 0 Þ with respect to t0 , we have
q0c ðt 0 Þ ¼ fðhQ þ kÞR0 ðt 0 Þ þ ðc1 c2 ÞQ þ kðL1 L2 Þr 0 ðt 0 ÞgMðQ Þ;
ð17Þ
0
where is the symbol of the differentiation with respect to t 0 . Since the hazard rate is strictly increasing, from the assumption, we have q0c ðt 0 Þ < 0, i.e., qc ðt 0 Þ is strictly decreasing. If qc ð0Þ > 0 and qc ð1Þ < 0, then there exists a finite and unique optimal ordering time t 0 ¼ tc0 ð0 < t c0 < 1Þ satisfying qc ðt 0 Þ ¼ 0, since qc ðt 0 Þ is strictly increasing and continuous. If qc ð0Þ 6 0, Z 1 ðt 0 ; Q Þ is strictly decreasing and the optimal ordering time is t 0 ¼ 0. If qc ð1Þ P 0; Z 1 ðt 0 ; Q Þ is strictly increasing and t 0 ! 1. Thus, the proof is completed. h Remark 1. t 0 ¼ 0 implies that the regular order is made at the same time instant as beginning of the operation. On the other hand, t 0 ! 1 means that the regular order is not made and the expedited one is done only at the same time instant as stock-out. Remark 2. Theorem 1 indicates the necessary and sufficient conditions for the existence of the optimal ordering time under Assumption (A-5). Note that, in addition to the Assumption (A-5), a restrictive assumption such as C 1 ðt 0 ; Q Þ < k for all t 0 is needed for the existence of the optimal ordering time under the long-run average cost criterion (see Dohi et al., 1995). That is, there probably exists the optimal ordering time maximizing the cost effectiveness in every cost circumstance.
ð12Þ
where rðtÞ and RðtÞ are assumed to be differentiable and have the same monotone properties (e.g. see Osaki, 1992). Two special cases in the cost effectiveness, t0 ¼ 0 and t 0 ! 1, are the following:
0
(ii) If qc ð0Þ 6 0 then t0 ¼ 0 and the optimal value of Z 1 ðt0 ; Q Þ is given in Eq. (13). (iii) If qc ð1Þ P 0 then t0 ! 1 and the optimal value of Z 1 ðt0 ; Q Þ is given in Eq. (15).
ð11Þ
Our objective is to find the optimal inventory policy ðt 0 ; Q Þ which maximize the cost effectiveness (11) for an arbitrary inventory level process fXðtÞ; t P 0g. Let us consider the problem of characterizing the optimal ordering time t 0 by maximizing Z 1 ðt 0 ; Q Þ for a fixed order quantity.
þk
We define t 0 ¼ t c0 satisfying qc ðt 0 Þ ¼ 0. Then, the theorem on the existence of the optimal ordering time is presented for a fixed order quantity as follows: Theorem 1. Suppose that FðtÞ is strictly IHR. (i) If qc ð0Þ > 0 and qc ð1Þ < 0 then there exists a finite and unique optimal ordering time t0 ¼ tc0 ð0 < t c0 < 1Þ satisfying qc ðt 0 Þ ¼ 0 and Z 1 ðt 0 ; Q Þ is given by Z 1 ðt c0 ; Q Þ
and T 1 ðt 0 ; Q Þ, the mean time of one cycle is given by
T 1 ðt 0 ; QÞ ¼
ð16Þ
0
ð15Þ
4. Periodic review model (Model 2) 4.1. Model description In this section, we consider the inventory model in which the stock level is inspected periodically rather than continuously. If the stock is depleted before a pre-specified time t0 2 ð0; 1Þ, the expedited order is placed at time t 0 and after a lead time L1 ð> 0Þ, Q units are replenished. On the other hand, if the stock is not depleted before the time t0 , the regular order is made at the time t0 , and Q units are received after a lead time L2 ð> 0Þ. The other modeling assumptions are the same as those of the continuous review model. Fig. 2 illustrates the inventory profile for the periodic review model. For the periodic review model, the expected cost per cycle is given by
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and
Z 2 ð1; Q Þ ¼ MðQ Þ=V 2 ð1; Q Þ;
ð26Þ
where
Z s V 2 ð1; QÞ ¼ hE XðtÞdt þ c1 Q :
ð27Þ
0
We define t 0 ¼ t p0 which satisfies qp ðt0 Þ ¼ 0, and state the following result for Model 2. Theorem 2. Suppose that FðtÞ is strictly IHR. (i) If qp ð0Þ > 0 and qp ð1Þ < 0 then there exists a finite and unique optimal ordering time t0 ¼ t p0 ð0 < t p0 < 1Þ satisfying qp ðt0 Þ ¼ 0 and Z 2 ðt 0 ; Q Þ is given by Z 2 ðt p0 ; Q Þ. (ii) If qp ð0Þ 6 0 then t0 ¼ 0 and the optimal value of Z 2 ðt 0 ; Q Þ is given in Eq. (24). (iii) If qp ð1Þ P 0 then t 0 ! 1 and the optimal value of Z 2 ðt 0 ; Q Þ is given in Eq. (26). Fig. 2. State diagram for one cycle in Model 2.
Z s Z V 2 ðt0 ; Q Þ ¼ h E XðtÞdt þ Q þk
Z
0
1
½t ðt 0 þ L2 ÞdFðtÞ
t 0 þL2
t0
ðt0 t þ L1 ÞdFðtÞ þ
0
Z
The proof is omitted as it is similar to that of Theorem 1.
t 0 þL2
ðt 0 þ L2 tÞdFðtÞ
t0
þ c1 QFðt 0 Þ þ c2 Q Fðt 0 Þ Z s Z ¼ hE XðtÞdt þ hQ 0
1
tdFðtÞ hQ ðt 0 þ L2 ÞFðt0 þ L2 Þ
t 0 þL2
Z þ k ðt0 þ L1 ÞFðt 0 Þ Fðt0 Þ
Z
t0
tdFðtÞ þ ðt 0 þ L2 Þ½Fðt 0 þ L2 Þ 0
t 0 þL2
tdFðtÞ þ c1 QFðt 0 Þ þ c2 Q Fðt0 Þ;
ð18Þ
t0
Remark 3. Dohi et al. (1995) obtained the optimal ordering time minimizing C 1 ðt0 ; Q Þ. It is, however, hard to characterize the optimal ordering time minimizing C 2 ðt0 ; Q Þ because of the complexity in T 2 ðt0 ; Q Þ. Moreover, the unique optimal ordering times maximizing A1 ðt0 ; Q Þ and A2 ðt 0 ; Q Þ do not always exist under the assumptions made in this paper. Remark 4. When h ¼ 0, the inventory models (Model 1, Model 2) considered in this paper are essentially reduced to the orderreplacement models (Kaio & Osaki, 1977; Osaki et al., 1981; Thomas & Osaki, 1978a, 1978b). In other words, the inventory models under consideration are generalizations to the replacement systems with delay, since the corresponding cost effectiveness includes the state variable, i.e., XðtÞ.
and the mean time of one cycle is given by
T 2 ðt 0 ; Q Þ ¼
Z 0
þ
t0
ðt0 þ L1 ÞdFðtÞ þ Z
Z
5. Numerical illustrations
t0 þL2
ðt0 þ L2 ÞdFðtÞ
t0 1
tdFðtÞ:
ð19Þ
t 0 þL2
The mean effective time for one cycle is given in Eq. (9). The cost effectiveness for Model 2 is defined as
Z 2 ðt 0 ; Q Þ ¼ A2 ðt 0 ; QÞ=C 2 ðt 0 ; Q Þ ¼ MðQÞ=V 2 ðt 0 ; QÞ;
ð20Þ
where
C 2 ðt 0 ; Q Þ ¼ V 2 ðt0 ; Q Þ=T 2 ðt 0 ; QÞ;
ð21Þ
A2 ðt 0 ; Q Þ ¼ MðQ Þ=T 2 ðt 0 ; Q Þ:
ð22Þ
4.2. Optimal ordering time In a fashion similar to Model 1, we define the numerator divided by Fðt0 Þ of the partial derivative with respect to t 0 of the right-hand side of Eq. (20) as
qp ðt0 Þ qc ðt 0 Þ kfFðt0 Þ=Fðt0 ÞgMðQ Þ:
ð23Þ
Two special cases (t 0 ¼ 0 and t 0 ! 1) of the model are as follows:
Z 2 ð0; Q Þ ¼ MðQÞ=V 2 ð0; QÞ;
ð24Þ
where
Z s Z V 2 ð0; Q Þ ¼ hE XðtÞdt þ hQ 0
1
L2
ðt L2 ÞdFðtÞ þ c2 Q ;
ð25Þ
In this section, we discuss the procedure to obtain the optimal inventory policies maximizing the cost effectiveness. For this, we must specify the inventory level process fXðtÞ; t P 0g. Let fNðtÞ; t P 0g be the cumulative demand process. Then it is obvious that
XðtÞ ¼ Q NðtÞ:
ð28Þ
Throughout this paper, we assume a Poisson process as cumulative demand. In fact, we would expect that most real systems could be accurately described by Poisson demand pattern, which is a common assumption in the inventory theory. The cumulative demand process fNðtÞ; t P 0g is a Poisson process k P ltÞ . with rate lð> 0Þ, where PrfNðtÞ 6 xjNð0Þ ¼ 0g ¼ xk¼0 ðltÞ expð k! Furthermore, since s is the time required for Q demands to occur, it follows that the distribution of stock-out time is the Erlang distribution with parameters l and Q. That is,
FðtÞ ¼
Z 0
t
lQ sQ1 expðlsÞ CðQ Þ
ds:
ð29Þ
It is well known that the Erlang distribution is strictly IHR for Q > 1. Without loss of generality, we assume that Q > 1. For the Poisson demand, the mean effective time is given by MðQ Þ ¼ Q =l. Thus we need to obtain the explicit expressions of V i ðt 0 ; Q Þ for i ¼ 1; 2. Consider the continuous review cyclic inventory model, i.e. Model 1. For the Poisson demand, the cost effectiveness is given by Eq. (11) as a function of the ordering time and the order quan-
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tity, and the corresponding expected cost per cycle is (see Appendix A)
V 1 ðt0 ; Q Þ ¼ hQ ðQ þ 1Þ=ð2lÞ þ c1 Q þ kL1 ðhQ þ kÞðt0 þ L2 Þ
Table 1 Optimal ordering time for fixed order quantity (c1 ¼ 9; c2 ¼ 4; L1 ¼ 5; L2 ¼ 9; h ¼ 3 and k ¼ 2). Q
f1 gamfðlðt0 þ L2 Þ; Q Þg kCðQ þ 1Þ f1 gamfðlt 0 ; Q þ 1Þg=ðlCðQ ÞÞ þ fðc2 c1 ÞQ
0.139 0.120 0.096 0.077 0.065 0.055
3.985 0.000 0.000 0.000 0.000 0.000
0.137 0.120 0.096 0.077 0.064 0.055
4
0.2 0.4 0.6 0.8 1.0 1.2
5.142 0.982 0.000 0.000 0.000 0.000
0.090 0.085 0.085 0.079 0.071 0.064
4.754 0.977 0.000 0.000 0.000 0.000
0.090 0.085 0.085 0.079 0.071 0.064
8
0.2 0.4 0.6 0.8 1.0 1.2
128.844 20.410 9.093 4.989 2.830 1.234
0.064 0.058 0.056 0.054 0.052 0.050
49.941 18.248 8.847 4.942 2.821 1.233
0.061 0.057 0.055 0.054 0.052 0.050
12
0.2 0.4 0.6 0.8 1.0 1.2
208.462 36.180 17.552 10.699 7.188 5.043
0.047 0.043 0.041 0.040 0.040 0.039
78.678 31.897 17.011 10.573 7.149 5.029
0.045 0.042 0.041 0.040 0.040 0.039
V 2 ðt0 ; Q Þ ¼ hQ ð3Q þ 1Þ=ð2lÞ þ ðhQ þ kÞðt0 þ L2 Þgamfðlðt 0 þ kðL1 L2 Þggamfðlt 0 ; QÞ ðhQ þ kÞCðQ þ 1Þgamfðlðt 0 þ L2 Þ; Q þ 1Þ=ðlCðQ ÞÞ ðt 0 þ L2 ÞhQ þ c2 Q :
ð31Þ
Note that it is difficult to obtain analytically the optimal order quantity Q for a fixed t 0 from the above expressions. We, therefore, numerically examine the behavior of V i ðt0 ; Q Þ for the order quantity. Fig. 3 shows the uni-modal property of the cost effectiveness for known order quantity. We first consider the optimal ordering time for a fixed order quantity. Table 1 shows the dependence of the mean parameter l on the optimal ordering times and their associated cost effectivenesses. This example shows that the cost effectiveness for each model decreases and the optimal ordering time decreases or remains the same as l increases. To this end, this means that one should execute the regular order as soon as possible when the demand increases in the sense of expectation. Then the cost effectiveness ought to decrease since the probability that the shortage occurs becomes high, as shown in Table 1. Comparing Model 1 with Model 2, Z 2 ðt 0 ; Q Þ is obviously less than Z 1 ðt0 ; Q Þ for same t 0 and Q since the shortage period for Model 2 is longer than Model 1. For this result, it seems to be interesting that the optimal ordering time minimizing Z 2 ðt0 ; Q Þ is rather less than that minimizing Z 1 ðt 0 ; Q Þ. The result indicates that the optimal policy functions to reduce the shortage cost. Let us now consider the performance for the cost-effective inventory policies. We define the inventory policies: ðtc0i ; Q ci Þ minimizing C i ðt0 ; Q Þ, and ðte0i ; Q ei Þ and ðta0i ; Q ai Þ maximizing Z i ðt 0 ; Q Þ and Ai ðt0 ; Q Þ, respectively.
Z 2 ðt0 ; Q Þ
7.900 0.000 0.000 0.000 0.000 0.000
ð30Þ
þ L2 Þ; Q Þ þ fðc1 c2 ÞQ
t0
0.2 0.4 0.6 0.8 1.0 1.2
þ kðt0 þ L2 L1 Þgf1 gamfðlt0 ; Q Þg þ ðhQ
Again, consider the periodic review cyclic inventory model, i.e., Model 2. For the Poisson demand, the cost effectiveness is given by Eq. (20) as a function of the ordering time and the order quantity, and the corresponding expected cost per cycle is
Z 1 ðt0 ; Q Þ
2
þ kÞCðQ þ 1Þf1 gamfðlðt 0 þ L2 Þ; Q þ 1Þg=ðlCðQ ÞÞ:
t0
l
Example 1. Let the values of the parameters be as follows: c1 ¼ 9; c2 ¼ 4; L1 ¼ 5; L2 ¼ 7; h ¼ 3; k ¼ 2 and l ¼ 0:6. The optimal policies for the respective criteria are
ðte01 ; Q e1 Þ ¼ ð0:0; 2:0Þ;
Z 1 ðt e01 ; Q e1 Þ ¼ 1:056 101 ;
ðte02 ; Q e2 Þ ¼ ð0:0; 2:0Þ;
Z 2 ðt e02 ; Q e2 Þ ¼ 1:056 101 ;
ðtc01 ; Q c1 Þ ðtc02 ; Q c2 Þ ðta01 ; Q a1 Þ ðta02 ; Q a2 Þ
¼ ð1; 1:0Þ;
C 1 ðtc01 ; Q c1 Þ ¼ 2:001;
¼ ð1; 1:0Þ;
C 2 ðtc02 ; Q c2 Þ ¼ 2:817;
¼ ð0:0; 1Þ;
A1 ðt a01 ; Q a1 Þ ¼ 1:000;
¼ ð0:0; 1Þ;
A2 ðt a02 ; Q a2 Þ ¼ 1:000:
From above, we have
C 1 ðt e01 ; Q e1 Þ ¼ 4:423;
C 2 ðt e02 ; Q e2 Þ ¼ 4:423;
A1 ðt e01 ; Q e1 Þ A1 ðt c01 ; Q c1 Þ
¼ 0:466;
A2 ðt e02 ; Q e2 Þ ¼ 0:466;
¼ 0:000;
A2 ðt c02 ; Q c2 Þ ¼ 0:237:
The stationary availabilities for ðtc0i ; Q ci Þ take extremely small values. On the other hand, the optimal inventory policies maximizing the cost effectiveness increase the stationary availability more than those minimizing the long-run average cost, though the values of C i ðt e0i ; Q ei Þ are larger than the minimum long-run average costs. For a fixed order quantity, remember that MðQ Þ is independent of t 0 . Therefore, the optimal ordering time maximizing the cost effectiveness is equivalent to one minimizing the expected cost for one cycle V i ðt 0 ; Q Þ. Thus, the cost effectiveness criterion has some attractive properties and seems to be a useful/practical measure for the inventory management.
6. Conclusion
Fig. 3. Behavior of cost effectiveness for order quantity ðc1 ¼ 9; c2 ¼ 4; L1 ¼ 5; L2 ¼ 9; h ¼ 3; k ¼ 2; l ¼ 0:6; t0 ¼ 50Þ.
In this paper, we have considered two kinds of inventory models (continuous review and periodic review) with the emergency order and analytically derived the optimal ordering time maximizing the cost effectiveness for a fixed order quantity. Further, for each inventory model, we have numerically obtained the optimal
B.C. Giri, T. Dohi / Computers & Industrial Engineering 57 (2009) 1336–1341
inventory policy which consists of the ordering time and the order quantity. It is seen from the numerical examples that the cost effectiveness criterion has some attractive properties for the inventory management. In general, the opportunity loss results from the shortage and often gives the production/inventory management a severe blow. The cost-effective inventory policies suggested in this paper propose a measure for the decision-maker without an acceptance designed reliability level.
1341
Moreover, we have
E½expðbsÞ ¼
l
Q
lþb
:
ð38Þ
Finally, by taking b ! 0 and applying the l’Hospital’s theorem, we have
Z s Q ðQ 1Þ E NðtÞdt ¼ : 2l 0
ð39Þ
Appendix A References In this Appendix, we derive the long-run average cost for the Poisson process demand. Given the distribution functions of the stock-out time in Eq. (29), it is easy to calculate the other part of Rs the expected costs except for E½ 0 XðtÞdt. Therefore, we get here the analytical expression of it. To deal with this problem we apply the first Dynkin formula (e.g. Karlin & Taylor, 1981). From Eq. (28), we have
E
Z s 0
Z s expðbtÞXðtÞdt ¼ QE expðbtÞdt 0 Z s expðbtÞNðtÞdt : E
ð32Þ
0
For a bounded and well-defined function gðÞ, we have
E
Z s
expðbtÞgðNðtÞÞdt ¼ Uð1Þ UðQ ÞE½expðbsÞ;
ð33Þ
0
for a counting process fNðtÞ; t P 0g, where
UðxÞ ¼ ðRb gÞðxÞ ¼
Z
Q
Gb ðx; yÞgðyÞdy:
ð34Þ
1
Rb and Gb ðx; yÞ are called the resolvent operator and its kernel of NðtÞ, respectively, given Nð0Þ ¼ x. Note that the kernel Gb ðx; yÞ is equivalent to the Laplace–Stieltjes transform of the distribution function of NðtÞ. When the demand process follows the Poisson process, we have
UðxÞ ¼
Q X gðyÞ l yx : lþb lþb y¼x
ð35Þ
Thus, from Eq. (33) and putting gðyÞ ¼ y, we have
E
Z s
expðbtÞNðtÞdt ¼ U p ð1Þ U p ðQÞE½expðbsÞ;
ð36Þ
0
where
( ) 1 l l Qxþ1 l l Q x : U p ðxÞ ¼ xþ Q b b b lþb lþb
ð37Þ
Allen, S. G., & D’Esopo, D. A. (1968a). An ordering policy for repairable stock items. Operations Research, 16, 669–674. Allen, S. G., & D’Esopo, D. A. (1968b). An ordering policy for stock items when delivery can be expedited. Operations Research, 16, 880–883. Barankin, E. W. (1961). A delivery-lag inventory model with an emergency provision. Naval Research Logistics Quarterly, 8, 285–311. Chen, F. Y., & Krass, D. (2001). Inventory models with minimal service level constraints. European Journal of Operational Research, 134, 120–140. Cohen, M. A., Kleindorfer, P. R., & Lee, H. L. (1988). Service constrained (s, S) systems with priority demand classes and lost sales. Management Science, 34, 482–499. Dohi, T., Kaio, N., & Osaki, S. (1995). Continuous review cyclic inventory models with emergency order. Journal of the Operations Research Society of Japan, 38, 212–229. Hadley, G., & Whitin, T. M. (1963). Analysis of inventory systems. Englewood Cliffs: Prentice-Hall. Hunter, L. C. (1963). Optimum checking procedure. In M. Zelen (Ed.), Stochastic theory of reliability. Madison: The University of Wisconsin Press. Kaio, N., & Osaki, S. (1977). Ordering policies with two types of lead times. Microelectronics and Reliability, 16, 225–229. Karlin, S., & Taylor, H. M. (1981). A second course in stochastic processes. New York: Academic Press. Lau, H., & Zhao, L. (1993). Optimal ordering policies with two suppliers when lead time and demand are all stochastic. European Journal of Operational Research, 68, 120–133. Moinzadeh, K., & Nahmias, S. (1988). A continuous review model for an inventory system with two supply modes. Management Science, 34, 761–773. Nahmias, S. (1993). Production and operations management. Homewood, IL: Irwing. Neuts, M. F. (1964). An inventory model with optional time lag. SIAM Journal of Applied Mathematics, 12, 179–185. Osaki, S. (1992). Applied stochastic system modeling. Berlin/Heidelberg: SpringerVerlag. Osaki, S., Kaio, N., & Yamada, S. (1981). A summary of optimal ordering policies. IEEE Transactions on Reliability, R-30, 272–277. Rosenshine, M., & Obee, D. (1976). Analysis of a standing order inventory system with emergency orders. Operations Research, 24, 1143–1155. Ross, S. M. (1970). Applied probability models with optimization applications. San Francisco: Holden-Day. Silver, E. A., Pyke, D. F., & Peterson, R. (1998). Inventory management and production planning and scheduling (3rd ed.). John Wiley & sons. Thomas, L. C., & Osaki, S. (1978a). A note on ordering policy. IEEE Transactions on Reliability, R-27, 380–381. Thomas, L. C., & Osaki, S. (1978b). An optimal ordering policy for a spare unit with lead time. European Journal of Operational Research, 2, 409–419. Whittmore, A. S., & Saunders, S. (1977). Optimal inventory under stochastic demand with two supply options. SIAM Journal of Applied Mathematics, 32, 293–305.
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Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Cost-effective ordering policies for inventory systems with emergency order B.C. Giri a,*, T. Dohi b a b
Department of Mathematics, Jadavpur University, Kolkata, India Department of Information Engineering, Hiroshima University, Hiroshima, Japan
a r t i c l e
i n f o
Article history: Received 10 September 2007 Received in revised form 7 April 2009 Accepted 2 July 2009 Available online 5 July 2009 Keywords: Inventory Emergency order Cost effectiveness Ordering policies Poisson demand
a b s t r a c t Most of the studies on inventory control reported in earlier contributions deal with the optimization problems minimizing an expected cost criterion such as the long-run average cost. However, when the plant engineers design the inventory systems in practical situations, they must take account of reliability into the inventory system as well as economical aspect, where reliability can be defined as the probability that the stock is not depleted until a pre-specified time. In this paper, we discuss the inventory control policies that provide a balance between economical and reliability requirements. By applying the cost effectiveness criterion, which simultaneously includes the effects of system availability and expected cost, as optimality one, we derive the optimal inventory replenishment policies of two kinds of inventory models. Finally, with a set of numerical examples, we show that the optimal inventory policies of the models under consideration make the stationary availability increase. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The importance of inventory control in business has increased dramatically with the development of sophisticated CIM (Computer Integrated Manufacturing) concept. It gives the logic and basis on controlling the manufacturing system as well as plays the role of releasing the surplus operating capital tied up in excessive inventories. Most of the studies on inventory control reported in the literature (see Hadley & Whitin, 1963; Silver, Pyke, & Peterson, 1998) deal with the optimization problems minimizing an expected cost criterion such as the long-run average cost. However, when the plant engineers design the inventory systems in practical situations, they are often required to take account of reliability into the inventory system as well as usual economical aspect, where reliability can be defined as the probability that the stock is not depleted until a pre-specified time. In inventory systems, the shortages are sometimes caused by a delivery-lag. In the early contributions, Allen and D’Esopo (1968a, 1968b), Barankin (1961), Neuts (1964) and Rosenshine and Obee (1976) address some inventory models to avoid the shortage by an additional order form. Lau and Zhao (1993) and Moinzadeh and Nahmias (1988) analyze the optimal ordering policies for the inventory system with two supply modes, and extended the seminal results by Whittmore and Saunders (1977). In fact, the problem of shortage may be overcome to some extent by splitting orders among a number of suppliers; but such an ordering form is rather costly and is not always realized in practice, even if the stock control system with multiple suppliers is available. * Corresponding author. E-mail address: [email protected] (B.C. Giri). 0360-8352/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2009.07.001
The above works consider the expected cost as optimality criterion. Some researchers deal stochastic inventory control problems with other optimality criterion such as service level criterion which introduces a service level constraint instead of shortage cost (see Chen & Krass, 2001; Cohen, Kleindorfer, & Lee, 1988; Nahmias, 1993). They define service level as the availability of stock in a probabilistic or expected sense. One major disadvantage of this approach is that it is perceived to be less tractable mathematically. The purpose of this paper is to implement the cost effectiveness criterion (Hunter, 1963) which provides a balance between economical and reliability requirements to the standard inventory models, and to propose the cost-effective inventory policy from reliability as well as cost minimization view points. We consider two kinds of inventory models referred to as Model 1 and Model 2 from onwards. Model 1 is a continuous review inventory model in which the stock level is continuously reviewed until a specified time and the items are replenished by the regular order at that time if no shortage has occurred. If the stock level becomes 0 before the regular ordering time, then the items are replenished by the expedited order. On the other hand, Model 2 is a periodic review inventory model in which the stock level is reviewed only at a specified time and one of the regular and expedited orders is placed at that time. Dohi, Kaio, and Osaki (1995) analyzed Model 1 under long-run average cost and expected total discounted cost criteria. Our objective in this study is to derive optimal inventory policies for Models 1 and 2 maximizing the cost effectiveness criterion. The cost effectiveness criterion can be regarded as an optimality criterion which assesses the effective system operating time in one cycle relative to the mean operating costs in that cycle. It is to be mentioned here that the inventory models under consideration have the same mathematical structure as
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the order-replacement models in the context of reliability theory (see Kaio & Osaki, 1977; Osaki, 1992; Osaki, Kaio, & Yamada, 1981; Thomas & Osaki, 1978a, 1978b). The paper is organized as follows: Section 2 describes the nomenclature, notation and assumptions. In Sections 2 and 3, the continuous and periodic review inventory models are explained, cost effectiveness is defined and the optimal ordering times are derived for a fixed order quantity. Assuming a Poisson process demand, we obtain the explicit formulas of cost effectiveness for the two inventory models in Section 5. We also investigate numerically the performance of cost-effective design for the inventory models. Finally, Section 6 concludes the paper.
3. Continuous review model (Model 1)
2. Nomenclature, notation and assumptions
3.1. Model description
2.1. Nomenclature
If the stock is depleted up to a pre-specified time t0 2 ð0; 1Þ, the emergency order is placed immediately at that time and after a lead time L1 ð> 0Þ; Q units are replenished. We call t 0 as ordering time, in this paper. On the other hand, if the stock is not depleted before the time t0 , the regular order is made at the time t0 , and Q units are replenished after a lead time L2 ð> 0Þ. We suppose that the time when the inventory level becomes 0 for the first time obeys a distribution FðtÞ ¼ Fðt; Q Þ with a density function f ðtÞ ¼ f ðt; Q Þ. Of course, specifying the inventory level process XðtÞ leads to characterize FðtÞ and f ðtÞ. We define the first passage time as follows:
IHR Inventory level process
Inventory policy
increasing hazard rate a continuous time stochastic process representing the on-hand inventory level which is decreased by the satisfaction of a demand the policy which consists of both the regular ordering time and the order quantity
(A-2) Without loss of generality, the inventory level is initially set to Q (>0). (A-3) Shortages, if occurred, are not backlogged. (A-4) The time period from time 0 to the time when the inventory level becomes Q next is one cycle and the same cycle repeats itself continually. (A-5) The sum of the ordering and the shortage costs by the expedited order is greater than the sum of those by the regular order, i:e:; c1 Q þ kL1 > c2 Q þ kL2 .
s infft P 0; XðtÞ ¼ 0jXð0Þ ¼ Qg: 2.2. Notation fXðtÞ; t P 0g fNðtÞ; t P 0g Q t0 FðtÞ; f ðtÞ FðÞ 1 FðÞ rðtÞ f ðtÞ=FðtÞ RðtÞ fFðt þ L2 Þ FðtÞg=FðtÞ i
L1 ; L2 c1 ; c2 h k V i ðt 0 ; Q Þ T i ðt 0 ; Q Þ MðQ Þ Ai ðt0 ; Q Þ Z i ðt 0 ; Q Þ ¼ MðQ Þ=V i ðt 0 ; Q Þ gamfðÞ
inventory level process cumulative demand process order quantity ordering time for the regular order cdf and pdf for the time to first shortage survivor function hazard rate for FðtÞ conditional hazard function during interval ðt; t þ L2 model indicator, that is, i ¼ 1; 2 correspond to Model 1 and Model 2, respectively constant lead times for the expedited and regular orders, respectively cost per unit amount for the expedited and regular orders, respectively inventory holding cost per unit amount per unit time penalty cost per unit time suffered for the shortage period expected cost for one cycle mean time for one cycle mean effective time for one cycle stationary availability cost effectiveness upper incomplete gamma function
ð1Þ
We define the indicator function IfAg for the event fAg. Fig. 1 shows the schematic illustration of the inventory model under consideration where the events fA1g; fA2g and fA3g are such that
XðtÞIfA1g fXðtÞ; min XðtÞ ¼ 0g;
ð2Þ
06t6t 0
XðtÞIfA2g fXðtÞ; min XðtÞ > 0 and 06t6t 0
XðtÞIfA3g fXðtÞ;
min
06t6ft 0 þL2 g
min
06t6ft 0 þL2 g
XðtÞ ¼ 0g;
XðtÞ > 0g;
ð4Þ
respectively. As the inventory process has a regenerative point where the same state repeats periodically, we use renewal reward theory (Ross, 1970) to find the expected cost per unit time which is given by
C 1 ðt0 ; Q Þ ¼ V 1 ðt 0 ; QÞ=T 1 ðt 0 ; Q Þ; where V 1 ðt 0 ; Q Þ, the expected cost in one cycle is
2.3. Assumptions
(A-1) The inventory management begins operating at time 0, and the planning horizon is infinite.
ð3Þ
Fig. 1. State diagram for one cycle in Model 1.
ð5Þ
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Z s Z V 1 ðt0 ; Q Þ ¼ h E XðtÞdt þ Q þk
Z
0 t0
L1 dFðtÞ þ
t 0 þL2
Z
0
½t ðt 0 þ L2 ÞdFðtÞ
1
Z s V 1 ð1; QÞ ¼ hE XðtÞdt þ c1 Q þ kL1 :
t 0 þL2
ðt 0 þ L2 tÞdFðtÞ t0
0 Þ; þ c1 QFðt 0 Þ þ c2 Q Fðt
ð6Þ
Z 0
þ
t0
ðt þ L1 ÞdFðtÞ þ Z
Z
t 0 þL2
ðt 0 þ L2 ÞdFðtÞ t0
1
tdFðtÞ;
ð7Þ
t 0 þL2
E being the mathematical expectation operator. It is clear that the processing time of this inventory system represents an ineffective usage time and decreases the useful/effective system operating time or the system effective availability, if the shortage period is relatively large. The stationary availability is the probability that the stock is not depleted, i.e., the inventory system is effective in a cycle. Let us define the stationary availability as follows:
A1 ðt0 ; Q Þ ¼ MðQ Þ=T 1 ðt 0 ; Q Þ;
ð8Þ
where MðQ Þ is the mean effective system operating time for one cycle, which is the time interval that the stock is not depleted for one cycle. Note that MðQ Þ is independent of t 0 and is defined as
MðQ Þ
Z
1
tdFðtÞ:
ð9Þ
0
Thus, the cost effectiveness, which balances between C 1 ðt 0 ; Q Þ and A1 ðt 0 ; Q Þ, can be formulated as
Z 1 ðt 0 ; QÞ A1 ðt 0 ; QÞ=C 1 ðt 0 ; Q Þ
ð10Þ
¼ MðQ Þ=V 1 ðt0 ; Q Þ:
3.2. Optimal ordering time We define the numerator divided by Fðt0 Þ of the partial derivative with respect to t 0 of the right-hand side of Eq. (11) as
qc ðt0 Þ fðhQ þ kÞRðt0 Þ þ ½ðc1 c2 ÞQ þ kðL1 L2 Þrðt0 Þ hQgMðQÞ;
Z 1 ð0; QÞ ¼ MðQ Þ=V 1 ð0; QÞ;
ð13Þ
where
Z s Z V 1 ð0; Q Þ ¼ hE XðtÞdt þ hQE½s þ hQ Z
L2
FðtÞdt
0 L2
FðtÞdt þ c2 Q hQL2 ;
ð14Þ
0
and
Z 1 ð1; QÞ ¼ MðQ Þ=V 1 ð1; Q Þ; where
Proof. Differentiating Z 1 ðt0 ; Q Þ with respect to t 0 and setting it equal to zero implies the equation qc ðt 0 Þ ¼ 0. Further, differentiating qc ðt 0 Þ with respect to t0 , we have
q0c ðt 0 Þ ¼ fðhQ þ kÞR0 ðt 0 Þ þ ðc1 c2 ÞQ þ kðL1 L2 Þr 0 ðt 0 ÞgMðQ Þ;
ð17Þ
0
where is the symbol of the differentiation with respect to t 0 . Since the hazard rate is strictly increasing, from the assumption, we have q0c ðt 0 Þ < 0, i.e., qc ðt 0 Þ is strictly decreasing. If qc ð0Þ > 0 and qc ð1Þ < 0, then there exists a finite and unique optimal ordering time t 0 ¼ tc0 ð0 < t c0 < 1Þ satisfying qc ðt 0 Þ ¼ 0, since qc ðt 0 Þ is strictly increasing and continuous. If qc ð0Þ 6 0, Z 1 ðt 0 ; Q Þ is strictly decreasing and the optimal ordering time is t 0 ¼ 0. If qc ð1Þ P 0; Z 1 ðt 0 ; Q Þ is strictly increasing and t 0 ! 1. Thus, the proof is completed. h Remark 1. t 0 ¼ 0 implies that the regular order is made at the same time instant as beginning of the operation. On the other hand, t 0 ! 1 means that the regular order is not made and the expedited one is done only at the same time instant as stock-out. Remark 2. Theorem 1 indicates the necessary and sufficient conditions for the existence of the optimal ordering time under Assumption (A-5). Note that, in addition to the Assumption (A-5), a restrictive assumption such as C 1 ðt 0 ; Q Þ < k for all t 0 is needed for the existence of the optimal ordering time under the long-run average cost criterion (see Dohi et al., 1995). That is, there probably exists the optimal ordering time maximizing the cost effectiveness in every cost circumstance.
ð12Þ
where rðtÞ and RðtÞ are assumed to be differentiable and have the same monotone properties (e.g. see Osaki, 1992). Two special cases in the cost effectiveness, t0 ¼ 0 and t 0 ! 1, are the following:
0
(ii) If qc ð0Þ 6 0 then t0 ¼ 0 and the optimal value of Z 1 ðt0 ; Q Þ is given in Eq. (13). (iii) If qc ð1Þ P 0 then t0 ! 1 and the optimal value of Z 1 ðt0 ; Q Þ is given in Eq. (15).
ð11Þ
Our objective is to find the optimal inventory policy ðt 0 ; Q Þ which maximize the cost effectiveness (11) for an arbitrary inventory level process fXðtÞ; t P 0g. Let us consider the problem of characterizing the optimal ordering time t 0 by maximizing Z 1 ðt 0 ; Q Þ for a fixed order quantity.
þk
We define t 0 ¼ t c0 satisfying qc ðt 0 Þ ¼ 0. Then, the theorem on the existence of the optimal ordering time is presented for a fixed order quantity as follows: Theorem 1. Suppose that FðtÞ is strictly IHR. (i) If qc ð0Þ > 0 and qc ð1Þ < 0 then there exists a finite and unique optimal ordering time t0 ¼ tc0 ð0 < t c0 < 1Þ satisfying qc ðt 0 Þ ¼ 0 and Z 1 ðt 0 ; Q Þ is given by Z 1 ðt c0 ; Q Þ
and T 1 ðt 0 ; Q Þ, the mean time of one cycle is given by
T 1 ðt 0 ; QÞ ¼
ð16Þ
0
ð15Þ
4. Periodic review model (Model 2) 4.1. Model description In this section, we consider the inventory model in which the stock level is inspected periodically rather than continuously. If the stock is depleted before a pre-specified time t0 2 ð0; 1Þ, the expedited order is placed at time t 0 and after a lead time L1 ð> 0Þ, Q units are replenished. On the other hand, if the stock is not depleted before the time t0 , the regular order is made at the time t0 , and Q units are received after a lead time L2 ð> 0Þ. The other modeling assumptions are the same as those of the continuous review model. Fig. 2 illustrates the inventory profile for the periodic review model. For the periodic review model, the expected cost per cycle is given by
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and
Z 2 ð1; Q Þ ¼ MðQ Þ=V 2 ð1; Q Þ;
ð26Þ
where
Z s V 2 ð1; QÞ ¼ hE XðtÞdt þ c1 Q :
ð27Þ
0
We define t 0 ¼ t p0 which satisfies qp ðt0 Þ ¼ 0, and state the following result for Model 2. Theorem 2. Suppose that FðtÞ is strictly IHR. (i) If qp ð0Þ > 0 and qp ð1Þ < 0 then there exists a finite and unique optimal ordering time t0 ¼ t p0 ð0 < t p0 < 1Þ satisfying qp ðt0 Þ ¼ 0 and Z 2 ðt 0 ; Q Þ is given by Z 2 ðt p0 ; Q Þ. (ii) If qp ð0Þ 6 0 then t0 ¼ 0 and the optimal value of Z 2 ðt 0 ; Q Þ is given in Eq. (24). (iii) If qp ð1Þ P 0 then t 0 ! 1 and the optimal value of Z 2 ðt 0 ; Q Þ is given in Eq. (26). Fig. 2. State diagram for one cycle in Model 2.
Z s Z V 2 ðt0 ; Q Þ ¼ h E XðtÞdt þ Q þk
Z
0
1
½t ðt 0 þ L2 ÞdFðtÞ
t 0 þL2
t0
ðt0 t þ L1 ÞdFðtÞ þ
0
Z
The proof is omitted as it is similar to that of Theorem 1.
t 0 þL2
ðt 0 þ L2 tÞdFðtÞ
t0
þ c1 QFðt 0 Þ þ c2 Q Fðt 0 Þ Z s Z ¼ hE XðtÞdt þ hQ 0
1
tdFðtÞ hQ ðt 0 þ L2 ÞFðt0 þ L2 Þ
t 0 þL2
Z þ k ðt0 þ L1 ÞFðt 0 Þ Fðt0 Þ
Z
t0
tdFðtÞ þ ðt 0 þ L2 Þ½Fðt 0 þ L2 Þ 0
t 0 þL2
tdFðtÞ þ c1 QFðt 0 Þ þ c2 Q Fðt0 Þ;
ð18Þ
t0
Remark 3. Dohi et al. (1995) obtained the optimal ordering time minimizing C 1 ðt0 ; Q Þ. It is, however, hard to characterize the optimal ordering time minimizing C 2 ðt0 ; Q Þ because of the complexity in T 2 ðt0 ; Q Þ. Moreover, the unique optimal ordering times maximizing A1 ðt0 ; Q Þ and A2 ðt 0 ; Q Þ do not always exist under the assumptions made in this paper. Remark 4. When h ¼ 0, the inventory models (Model 1, Model 2) considered in this paper are essentially reduced to the orderreplacement models (Kaio & Osaki, 1977; Osaki et al., 1981; Thomas & Osaki, 1978a, 1978b). In other words, the inventory models under consideration are generalizations to the replacement systems with delay, since the corresponding cost effectiveness includes the state variable, i.e., XðtÞ.
and the mean time of one cycle is given by
T 2 ðt 0 ; Q Þ ¼
Z 0
þ
t0
ðt0 þ L1 ÞdFðtÞ þ Z
Z
5. Numerical illustrations
t0 þL2
ðt0 þ L2 ÞdFðtÞ
t0 1
tdFðtÞ:
ð19Þ
t 0 þL2
The mean effective time for one cycle is given in Eq. (9). The cost effectiveness for Model 2 is defined as
Z 2 ðt 0 ; Q Þ ¼ A2 ðt 0 ; QÞ=C 2 ðt 0 ; Q Þ ¼ MðQÞ=V 2 ðt 0 ; QÞ;
ð20Þ
where
C 2 ðt 0 ; Q Þ ¼ V 2 ðt0 ; Q Þ=T 2 ðt 0 ; QÞ;
ð21Þ
A2 ðt 0 ; Q Þ ¼ MðQ Þ=T 2 ðt 0 ; Q Þ:
ð22Þ
4.2. Optimal ordering time In a fashion similar to Model 1, we define the numerator divided by Fðt0 Þ of the partial derivative with respect to t 0 of the right-hand side of Eq. (20) as
qp ðt0 Þ qc ðt 0 Þ kfFðt0 Þ=Fðt0 ÞgMðQ Þ:
ð23Þ
Two special cases (t 0 ¼ 0 and t 0 ! 1) of the model are as follows:
Z 2 ð0; Q Þ ¼ MðQÞ=V 2 ð0; QÞ;
ð24Þ
where
Z s Z V 2 ð0; Q Þ ¼ hE XðtÞdt þ hQ 0
1
L2
ðt L2 ÞdFðtÞ þ c2 Q ;
ð25Þ
In this section, we discuss the procedure to obtain the optimal inventory policies maximizing the cost effectiveness. For this, we must specify the inventory level process fXðtÞ; t P 0g. Let fNðtÞ; t P 0g be the cumulative demand process. Then it is obvious that
XðtÞ ¼ Q NðtÞ:
ð28Þ
Throughout this paper, we assume a Poisson process as cumulative demand. In fact, we would expect that most real systems could be accurately described by Poisson demand pattern, which is a common assumption in the inventory theory. The cumulative demand process fNðtÞ; t P 0g is a Poisson process k P ltÞ . with rate lð> 0Þ, where PrfNðtÞ 6 xjNð0Þ ¼ 0g ¼ xk¼0 ðltÞ expð k! Furthermore, since s is the time required for Q demands to occur, it follows that the distribution of stock-out time is the Erlang distribution with parameters l and Q. That is,
FðtÞ ¼
Z 0
t
lQ sQ1 expðlsÞ CðQ Þ
ds:
ð29Þ
It is well known that the Erlang distribution is strictly IHR for Q > 1. Without loss of generality, we assume that Q > 1. For the Poisson demand, the mean effective time is given by MðQ Þ ¼ Q =l. Thus we need to obtain the explicit expressions of V i ðt 0 ; Q Þ for i ¼ 1; 2. Consider the continuous review cyclic inventory model, i.e. Model 1. For the Poisson demand, the cost effectiveness is given by Eq. (11) as a function of the ordering time and the order quan-
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tity, and the corresponding expected cost per cycle is (see Appendix A)
V 1 ðt0 ; Q Þ ¼ hQ ðQ þ 1Þ=ð2lÞ þ c1 Q þ kL1 ðhQ þ kÞðt0 þ L2 Þ
Table 1 Optimal ordering time for fixed order quantity (c1 ¼ 9; c2 ¼ 4; L1 ¼ 5; L2 ¼ 9; h ¼ 3 and k ¼ 2). Q
f1 gamfðlðt0 þ L2 Þ; Q Þg kCðQ þ 1Þ f1 gamfðlt 0 ; Q þ 1Þg=ðlCðQ ÞÞ þ fðc2 c1 ÞQ
0.139 0.120 0.096 0.077 0.065 0.055
3.985 0.000 0.000 0.000 0.000 0.000
0.137 0.120 0.096 0.077 0.064 0.055
4
0.2 0.4 0.6 0.8 1.0 1.2
5.142 0.982 0.000 0.000 0.000 0.000
0.090 0.085 0.085 0.079 0.071 0.064
4.754 0.977 0.000 0.000 0.000 0.000
0.090 0.085 0.085 0.079 0.071 0.064
8
0.2 0.4 0.6 0.8 1.0 1.2
128.844 20.410 9.093 4.989 2.830 1.234
0.064 0.058 0.056 0.054 0.052 0.050
49.941 18.248 8.847 4.942 2.821 1.233
0.061 0.057 0.055 0.054 0.052 0.050
12
0.2 0.4 0.6 0.8 1.0 1.2
208.462 36.180 17.552 10.699 7.188 5.043
0.047 0.043 0.041 0.040 0.040 0.039
78.678 31.897 17.011 10.573 7.149 5.029
0.045 0.042 0.041 0.040 0.040 0.039
V 2 ðt0 ; Q Þ ¼ hQ ð3Q þ 1Þ=ð2lÞ þ ðhQ þ kÞðt0 þ L2 Þgamfðlðt 0 þ kðL1 L2 Þggamfðlt 0 ; QÞ ðhQ þ kÞCðQ þ 1Þgamfðlðt 0 þ L2 Þ; Q þ 1Þ=ðlCðQ ÞÞ ðt 0 þ L2 ÞhQ þ c2 Q :
ð31Þ
Note that it is difficult to obtain analytically the optimal order quantity Q for a fixed t 0 from the above expressions. We, therefore, numerically examine the behavior of V i ðt0 ; Q Þ for the order quantity. Fig. 3 shows the uni-modal property of the cost effectiveness for known order quantity. We first consider the optimal ordering time for a fixed order quantity. Table 1 shows the dependence of the mean parameter l on the optimal ordering times and their associated cost effectivenesses. This example shows that the cost effectiveness for each model decreases and the optimal ordering time decreases or remains the same as l increases. To this end, this means that one should execute the regular order as soon as possible when the demand increases in the sense of expectation. Then the cost effectiveness ought to decrease since the probability that the shortage occurs becomes high, as shown in Table 1. Comparing Model 1 with Model 2, Z 2 ðt 0 ; Q Þ is obviously less than Z 1 ðt0 ; Q Þ for same t 0 and Q since the shortage period for Model 2 is longer than Model 1. For this result, it seems to be interesting that the optimal ordering time minimizing Z 2 ðt0 ; Q Þ is rather less than that minimizing Z 1 ðt 0 ; Q Þ. The result indicates that the optimal policy functions to reduce the shortage cost. Let us now consider the performance for the cost-effective inventory policies. We define the inventory policies: ðtc0i ; Q ci Þ minimizing C i ðt0 ; Q Þ, and ðte0i ; Q ei Þ and ðta0i ; Q ai Þ maximizing Z i ðt 0 ; Q Þ and Ai ðt0 ; Q Þ, respectively.
Z 2 ðt0 ; Q Þ
7.900 0.000 0.000 0.000 0.000 0.000
ð30Þ
þ L2 Þ; Q Þ þ fðc1 c2 ÞQ
t0
0.2 0.4 0.6 0.8 1.0 1.2
þ kðt0 þ L2 L1 Þgf1 gamfðlt0 ; Q Þg þ ðhQ
Again, consider the periodic review cyclic inventory model, i.e., Model 2. For the Poisson demand, the cost effectiveness is given by Eq. (20) as a function of the ordering time and the order quantity, and the corresponding expected cost per cycle is
Z 1 ðt0 ; Q Þ
2
þ kÞCðQ þ 1Þf1 gamfðlðt 0 þ L2 Þ; Q þ 1Þg=ðlCðQ ÞÞ:
t0
l
Example 1. Let the values of the parameters be as follows: c1 ¼ 9; c2 ¼ 4; L1 ¼ 5; L2 ¼ 7; h ¼ 3; k ¼ 2 and l ¼ 0:6. The optimal policies for the respective criteria are
ðte01 ; Q e1 Þ ¼ ð0:0; 2:0Þ;
Z 1 ðt e01 ; Q e1 Þ ¼ 1:056 101 ;
ðte02 ; Q e2 Þ ¼ ð0:0; 2:0Þ;
Z 2 ðt e02 ; Q e2 Þ ¼ 1:056 101 ;
ðtc01 ; Q c1 Þ ðtc02 ; Q c2 Þ ðta01 ; Q a1 Þ ðta02 ; Q a2 Þ
¼ ð1; 1:0Þ;
C 1 ðtc01 ; Q c1 Þ ¼ 2:001;
¼ ð1; 1:0Þ;
C 2 ðtc02 ; Q c2 Þ ¼ 2:817;
¼ ð0:0; 1Þ;
A1 ðt a01 ; Q a1 Þ ¼ 1:000;
¼ ð0:0; 1Þ;
A2 ðt a02 ; Q a2 Þ ¼ 1:000:
From above, we have
C 1 ðt e01 ; Q e1 Þ ¼ 4:423;
C 2 ðt e02 ; Q e2 Þ ¼ 4:423;
A1 ðt e01 ; Q e1 Þ A1 ðt c01 ; Q c1 Þ
¼ 0:466;
A2 ðt e02 ; Q e2 Þ ¼ 0:466;
¼ 0:000;
A2 ðt c02 ; Q c2 Þ ¼ 0:237:
The stationary availabilities for ðtc0i ; Q ci Þ take extremely small values. On the other hand, the optimal inventory policies maximizing the cost effectiveness increase the stationary availability more than those minimizing the long-run average cost, though the values of C i ðt e0i ; Q ei Þ are larger than the minimum long-run average costs. For a fixed order quantity, remember that MðQ Þ is independent of t 0 . Therefore, the optimal ordering time maximizing the cost effectiveness is equivalent to one minimizing the expected cost for one cycle V i ðt 0 ; Q Þ. Thus, the cost effectiveness criterion has some attractive properties and seems to be a useful/practical measure for the inventory management.
6. Conclusion
Fig. 3. Behavior of cost effectiveness for order quantity ðc1 ¼ 9; c2 ¼ 4; L1 ¼ 5; L2 ¼ 9; h ¼ 3; k ¼ 2; l ¼ 0:6; t0 ¼ 50Þ.
In this paper, we have considered two kinds of inventory models (continuous review and periodic review) with the emergency order and analytically derived the optimal ordering time maximizing the cost effectiveness for a fixed order quantity. Further, for each inventory model, we have numerically obtained the optimal
B.C. Giri, T. Dohi / Computers & Industrial Engineering 57 (2009) 1336–1341
inventory policy which consists of the ordering time and the order quantity. It is seen from the numerical examples that the cost effectiveness criterion has some attractive properties for the inventory management. In general, the opportunity loss results from the shortage and often gives the production/inventory management a severe blow. The cost-effective inventory policies suggested in this paper propose a measure for the decision-maker without an acceptance designed reliability level.
1341
Moreover, we have
E½expðbsÞ ¼
l
Q
lþb
:
ð38Þ
Finally, by taking b ! 0 and applying the l’Hospital’s theorem, we have
Z s Q ðQ 1Þ E NðtÞdt ¼ : 2l 0
ð39Þ
Appendix A References In this Appendix, we derive the long-run average cost for the Poisson process demand. Given the distribution functions of the stock-out time in Eq. (29), it is easy to calculate the other part of Rs the expected costs except for E½ 0 XðtÞdt. Therefore, we get here the analytical expression of it. To deal with this problem we apply the first Dynkin formula (e.g. Karlin & Taylor, 1981). From Eq. (28), we have
E
Z s 0
Z s expðbtÞXðtÞdt ¼ QE expðbtÞdt 0 Z s expðbtÞNðtÞdt : E
ð32Þ
0
For a bounded and well-defined function gðÞ, we have
E
Z s
expðbtÞgðNðtÞÞdt ¼ Uð1Þ UðQ ÞE½expðbsÞ;
ð33Þ
0
for a counting process fNðtÞ; t P 0g, where
UðxÞ ¼ ðRb gÞðxÞ ¼
Z
Q
Gb ðx; yÞgðyÞdy:
ð34Þ
1
Rb and Gb ðx; yÞ are called the resolvent operator and its kernel of NðtÞ, respectively, given Nð0Þ ¼ x. Note that the kernel Gb ðx; yÞ is equivalent to the Laplace–Stieltjes transform of the distribution function of NðtÞ. When the demand process follows the Poisson process, we have
UðxÞ ¼
Q X gðyÞ l yx : lþb lþb y¼x
ð35Þ
Thus, from Eq. (33) and putting gðyÞ ¼ y, we have
E
Z s
expðbtÞNðtÞdt ¼ U p ð1Þ U p ðQÞE½expðbsÞ;
ð36Þ
0
where
( ) 1 l l Qxþ1 l l Q x : U p ðxÞ ¼ xþ Q b b b lþb lþb
ð37Þ
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