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Integrated vendor–buyer cooperative inventory models with controllable lead time and ordering cost reduction
European Journal of Operational Research 170 (2006) 481–495 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Integrated vendor–buyer cooperative inventory models with controllable lead time and ordering cost reduction Hung-Chi Chang a, Liang-Yuh Ouyang a
b,* ,
Kun-Shan Wu c, Chia-Huei Ho
d
Department of Logistics Engineering and Management, National Taichung Institute of Technology, Taichung, Taiwan b Department of Management Sciences and Decision Making, Tamkang University, Tamsui, Taipei, 25137 Taiwan c Department of Business Administration, Tamkang University, Tamsui, Taipei, Taiwan d Graduate Institute of Management Sciences, Tamkang University, Tamsui, Taipei, Taiwan Received 28 May 2003; accepted 16 June 2004 Available online 11 September 2004
Abstract This study deals with the lead time and ordering cost reduction problem in the single-vendor single-buyer integrated inventory model. We consider that buyer lead time can be shortened at an extra crashing cost which depends on the lead time length to be reduced and the ordering lot size. Additionally, buyer ordering cost can be reduced through further investment. Two models are presented in this study. The first model assumes that the ordering cost reduction has no relation to lead time crashing. The second model assumes that the lead time and ordering cost reduction are interacted. An iterative procedure is developed to find the optimal solution and numerical examples are presented to illustrate the results of the proposed models. 2004 Elsevier B.V. All rights reserved. Keywords: Inventory; Integrated model; Lead time reduction; Ordering cost reduction
1. Introduction Most inventory models considered to date assume just one facility (e.g., a buyer or a vendor) managing its inventory policy to minimize its own cost or maximize its own profit. This one-sided-optimal-strategy is not suitable for global markets. The issue of just-in-time (JIT) has recently received great attention. Most JIT research has focused on the integration between vendor and buyer. Once a long-term relationship
*
Corresponding author. Tel.: +886 2 86313221; fax: +886 2 86313214. E-mail address: [email protected] (L.-Y. Ouyang).
0377-2217/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.06.029
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between both facilities has been developed, both parties can cooperate and share information to achieve improved benefits. The joint optimization concept for buyer and vendor was initiated by Goyal (1976). Subsequently, numerous scholars developed integrated inventory models under various assumptions. For example, Banerjee (1986) assumed that the vendor was manufacturing at a finite rate and considered a joint economic-lotsize model in which a vendor produces to order for a buyer on a lot-for-lot basis. Goyal (1988) relaxed the lot-for-lot policy and suggested that vendor economic production quantity should be an integer multiple of buyer purchase quantity. Goyal (1995) later proposed a different shipment policy that suggested that the ith shipment size equals (first shipment size) · (production rate/demand rate)(i1). Hill (1997) further extended this concept and proposed a general shipment policy which suggested that the ith shipment size equals (first shipment size) · y(i1), where 16y6 (production rate/demand rate). In the same year, Ha and Kim (1997) developed an integrated JIT lot-splitting model for facilitating multiple shipments in small lots, that is, the vendor produces the contract quantity in one setup and ships to the buyer in small quantities of equal size in several lots. Later, Hill (1999) established a more general batching and shipping policy which suggested that the successive shipment size of the first m shipments increases by a fixed factor, and the remaining shipments would be equal sized. In a recent study, Pan and Yang (2002) developed an integrated inventory model with controllable lead time. Most previous researches on the integrated vendor–buyer inventory problem only considered the production shipment schedule in terms of the number and size of batches transferred between both parties. In modern production management, controllable lead time and ordering cost reduction are keys to business success and have attracted considerable research attention. Ordering quantity, service level and business competitiveness can be shown to possibly be influenced directly or indirectly via lead-time and/or ordering cost control. The previously mentioned integrated inventory models treat the ordering cost and/or lead time as constants. However, in some practical situations, lead time and ordering cost can be controlled and reduced in various ways. Ordering cost reduction can be attained through worker training, procedural changes, and specialized equipment acquisition. In the literature, Porteus (1985) first introduced the concept and developed a framework for investing in reducing EOQ model set-up cost. This development encouraged many researchers to examine setup/ordering cost reduction (e.g. Keller and Noori, 1988; Nasri et al., 1990; Kim et al., 1992; Paknejad et al., 1995). As stated in Tersine (1994), lead time usually comprises several components, such as setup time, process time, wait time, move time and queue time. In many practical situations, lead time can be reduced using an added crashing cost. In other words, lead time is controllable. The Japanese experience of using JIT production showed that the benefits associated with lead time control are clear. Therefore, reducing lead time is both necessary and beneficial. Inventory models incorporating lead time as a decision variable were developed by several researchers. Liao and Shyu (1991) first devised a probability inventory model in which lead time was the unique decision variable. Later, several researchers (e.g., Ben-Daya and Raouf (1994), Ouyang et al. (1996), Moon and Choi (1998), Hariga and Ben-Daya (1999)) developed various analytical inventory models to explore the lead time reduction problem. The underlying assumption in the above studies was that lead time could be decomposed into n mutually independent components, each with a different but fixed crash cost independent of the ordered lot size. However, this view may not be realistic. In a real environment, to reduce lead time, managers may ask workers to work overtime, employ part-time workers, use special delivery, and so on. Intuitively, extra cost should be paid for these services, and these costs may depend on the ordered lot size. Generally, the larger the ordered lot size, the higher the cost needed to reduce the lead time. Therefore, it seems reasonable to consider that lead time crash cost depends not only on the amount of lead time to be shortened, but also on the ordered lot size. Pan et al. (2002) extended this idea further by considering lead time crash cost as a function of both the order quantity and reduced lead time.
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Recently, Ouyang et al. (1999) investigated the influence of ordering cost reduction on modified continuous review inventory systems involving variable lead time with partial backorders. Subsequently, Ouyang and Chang (2002) proposed a modified lot-size reorder-point inventory model with imperfect production processes to study the effects of reducing lead time and set-up cost. The optimal policies derived in these two articles are buyer focused, and the lead time and ordering/set-up cost reduction were assumed to act independently. However, an independent relationship between lead time and ordering/set-up cost is just one possibility. In some practices, lead time and ordering/set-up cost reduction might be closely related. A lead time reduction could accompany a reduction in the ordering/set-up cost, and vice versa. For example, electronic data interchange (EDI) technology could simultaneously reduce both the lead time and the ordering/set-up cost. To date, little research has been done on establishing the relationship between lead time and ordering cost reduction. To provide insight and analytical tractability, as in Chiu (1998) and Chen et al. (2001), this study employed a linear function to formulate the above relationship. Expanding the focus beyond that solely of the buyer, this paper attempts to incorporate lead time and ordering cost reduction into the integrated vendor–buyer inventory model. Two integrated cooperative inventory models are proposed for the continuous review inventory system. The first model considers the case in which the lead time and ordering cost reduction are performed independently, while the second model considers the lead time and ordering cost reduction to interact linearly. The objective is to minimize the joint total expected cost by simultaneously optimizing the order quantity, reorder point, ordering cost, lead time and number of shipments between the two facilities. For both models, an effective iterative procedure is developed to determine the optimal policy, and numerical examples are used to illustrate the results and benefits of integration. 2. Notation and assumptions To develop the proposed models, we adopt the following notation and assumptions which are similar to those used in Pan et al. (2002). Notations D P A0 hb hv p p0 S A L Q r I(A) h d m X
average demand per unit time of the buyer production rate of the vendor buyers original ordering cost buyers holding cost per unit per unit time vendors holding cost per unit per unit time buyers fixed penalty cost per unit short buyers gross marginal profit per unit vendors set-up cost per set-up buyers ordering cost per order (decision variable) length of lead time (decision variable) buyers order quantity (decision variable) buyers reorder point (decision variable) buyers capital investment required to achieve ordering cost A, 0 < A 6 A0 fractional opportunity cost of capital investment per unit time percentage decrease in ordering cost A per dollar increase in investment I(A) the number of lots in which the product is delivered from the vendor to the buyer in one production cycle, a positive integer (decision variable) the leadptime ffiffiffi demand which follows a normal distribution with finite mean DL and standard deviation r L, where r denotes the standard deviation of the demand per unit time
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Assumptions 1. There is single-vendor and single-buyer for a single product in this article. 2. The buyer orders a lot of size Q and the vendor manufactures mQ with a finite production rate P(P > D) at one set-up but ship in quantity Q to the buyer over m times. The vendor incurs a set-up cost S for each production run and the buyer incurs an ordering cost A for each order of quantity Q. 3. Inventory is continuously reviewed. The buyer places the order when the on hand inventory reaches the reorder point r. 4. The reorder point r = the expected demand during lead time pffiffiffi + safety stock (SS), and SS = k · (standard deviation of lead time demand), that is, r ¼ DL þ kr L where k is a safety factor. 5. Shortages are allowed and partially backordered. b denotes the fraction of the demand during the stock-out period that will be backordered. 6. The lead time L consists of n mutually independent components. The ith component has a normal duration Ti and minimum duration ti, i = 1, 2, . . . , n. 7. For the ith component of lead time, the crashing cost per unit time, ci, depends on the ordering lot size Q and is described by ci = ai + biQ, where ai > 0 is the fixed cost, and bi > 0 is the unit variable cost, for i = 1, 2, . . . , n. 8. For any two crash cost lines ci = ai + biQ and cj = aj + bjQ, where ai > aj, bi
3. Basic model For the model that does not consider ordering cost and reorder point as decision variables, the buyer model closely follows Pan et al. (2002), who established the total expected cost per unit time as follows:
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TCb ðQ; LÞ ¼ ordering cost þ inventory holding cost þ stock-out cost þ lead time crashing cost pffiffiffi pffiffiffi AD Q D D þ hb þ r DL þ ð1 bÞr LWðkÞ þ ½p þ p0 ð1 bÞ r LWðkÞ þ CðLÞ; ¼ Q 2 Q Q
485
ð1Þ
where W(k) = /(k) k[1 U(k)], / and U are the standard normal probability density function (p.d.f.) and cumulative distribution function (c.d.f.), respectively. Note that it can be shown that W(k) is a decreasing function of k, and lim k !1W(k) = 0. In contrast to Pan et al. (2002), this study considers that the ordering cost A could vary through capital investment, and adds the reorder point into the buyer decision variable to further improve buyer total cost. Hence, the buyers objective is to minimize the new total expected cost per unit time, namely, the sum of the capital investment cost for reducing A and the inventory relevantpffiffiffi cost as expressed in (1), by optimizing over Q, A, r, and L constrained on 0 < A6A0. Given r ¼ DL þ kr L, the safety factor k can be considered as a decision variable instead of r. Thus, the total expected cost per unit time for the buyer becomes TECb ðQ; A; k; LÞ ¼ hIðAÞ þ TCb ðQ; LÞ pffiffiffi pffiffiffi h A0 AD Q þ hb þ kr L þ ð1 bÞr LWðkÞ ¼ ln þ d Q 2 A pffiffiffi D D þ ½p þ p0 ð1 bÞ r LWðkÞ þ CðLÞ; Q Q
ð2Þ
constrained on 0 < A6A0. On the other hand, for the vendor, since S is the vendor set-up cost per set-up and the production quantity for a vendor in a lot is mQ, the expected set-up cost per unit time is given by SD/(mQ). During the production period, once the first Q units are produced, the vendor delivers them to the buyer, and then continues making the delivery on average every Q/D units of time until the inventory level falls to zero (see Fig. 1). Hence, the average inventory per unit time for the vendor can be calculated as follows: 2 Q Q m2 Q2 Q mQ mQ þ ðm 1Þ ½1 þ 2 þ þ ðm 1Þ P D D 2P D Q D 2D m 1 ¼ 1þ : 2 P P
quantity cumulated inventory level
Q PQ D
time mQ P mQ D
Fig. 1. Inventory level of vendor.
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Therefore, the total expected cost per unit time for the vendor is: SD Q D 2D þ hv mð1 Þ 1 þ TECv ðQ; mÞ ¼ : mQ 2 P P
ð3Þ
Accordingly, the joint total expected cost per unit time for the vendor and buyer requiring minimization is given by JTECðQ; A; k; L; mÞ ¼ TECb ðQ; A; k; LÞ þ TECv ðQ; mÞ pffiffiffi pffiffiffi h A0 AD Q þ hb þ kr L þ ð1 bÞr LWðkÞ ¼ ln þ d Q 2 A pffiffiffi D
SD Q D 2D þ þ hv m 1 p þ p0 ð1 bÞ r LWðkÞ þ 1þ Q mQ 2 P P " # i1 X D þ ðaj þ bj QÞðT j tj Þ ; L 2 ½Li ; Li1 : ðai þ bi QÞðLi1 LÞ þ Q j¼1
ð4Þ
To simplify notation, we let ¼ p þ p0 ð1 bÞ; p U ðLÞ ¼ ai ðLi1 LÞ þ
i1 X
aj ðT j tj Þ;
j¼1
and
D 2D H ðmÞ ¼ hb þ hv m 1 1þ : P P
Now, the problem can be formulated by " # i1 X h A0 Min JTECðQ; A; k; L; mÞ ¼ ln bj ðT j tj Þ þ D bi ðLi1 LÞ þ d A j¼1 pffiffiffi pffiffiffi D S þ A þ þ pr LWðkÞ þ U ðLÞ þ hb r L½k þ ð1 bÞWðkÞ Q m Q þ H ðmÞ 2 subject to 0 < A 6 A0 :
ð5Þ
To solve the above nonlinear programming problem, this study temporarily ignores the constraint 0 < A6A0 and relaxes the integer requirement on m (the number of shipments from the vendor to the buyer during one production cycle). For fixed Q, A, k, and L 2 [Li, Li1], JTEC(Q, A, k, L, m) can be proved to be a convex function of m. Consequently, the search for the optimal shipment number, m , is reduced to find a local minimum. Property 1. For fixed Q, A, k, and L, JTEC(Q, A, k, L, m) is convex in m. Proof. Taking the first and second partial derivatives of JTEC(Q, A, k, L, m) with respect to m, we have oJTECðQ; A; k; L; mÞ SD Q D ¼ 2 þ hv 1 ; ð6Þ om mQ 2 P
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and o2 JTECðQ; A; k; L; mÞ 2SD ¼ 3 > 0: om2 mQ
ð7Þ
Therefore, JTEC(Q, A, k, L, m) is convex in m, for fixed Q, A, k, and L. This completes the proof of Property 1. h Next, the first partial derivatives of JTEC(Q, A, k, L, m) with respect to Q, A, k and L 2 (Li, Li1) are taken for fixed m, respectively. This process yields: pffiffiffi oJTECðQ; A; k; L; mÞ D S H ðmÞ r LWðkÞ þ U ðLÞ þ ¼ 2 Aþ þp ; ð8Þ oQ m 2 Q oJTECðQ; A; k; L; mÞ h D ¼ þ ; oA dA Q
ð9Þ
pffiffiffi oJTECðQ; A; k; L; mÞ D pffiffiffi r L½UðkÞ 1 þ hb r Lf1 þ ð1 bÞ½UðkÞ 1 g; ¼ p ok Q
ð10Þ
oJTECðQ; A; k; L; mÞ D hb r ai 1=2 1=2 rWðkÞL ¼ ½k þ ð1 bÞWðkÞ L þ bi : þ D p oL 2Q 2 Q
ð11Þ
and
Furthermore, for fixed (Q, A, k, m), JTEC(Q, A, k, L, m) is noted to be a concave function in L 2 [Li, Li1], because o2 JTECðQ; A; k; L; mÞ D hb r rWðkÞL3=2 ½k þ ð1 bÞWðkÞ L3=2 < 0: ¼ p 2 4Q 4 oL
ð12Þ
Hence, for fixed (Q, A, k, m), the minimum joint total expected cost per unit time occurs at the end points of the interval [Li, Li1]. Conversely, for fixed m and L 2 [Li, Li1], JTEC(Q, A, k, L, m) can be proved to be a convex function of (Q, A, k). Thus, the minimum value of JTEC(Q, A, k, L, m) occurs at the point (Q , A , k ) which satisfies oJTEC(Q, A, k, L, m)/oQ = 0, oJTEC(Q, A, k, L, m)/oA = 0 and oJTEC(Q, A, k, L, m)/ok = 0, simultaneously. Solving these three equations, yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2D S r LWðk Þ þ U ðLÞ ; A þ þ p Q ¼ H ðmÞ m
A ¼
hQ ; dD
ð13Þ
ð14Þ
and Uðk Þ ¼ 1
hb Q : D þ hb Q ð1 bÞ p
ð15Þ
Pn Clearly, if none of the lead time components crashes, the lead time length is j¼1 T j , and the corresponding crash cost is zero. Meanwhile, if all of the lead time components crash toPtheir minimum durations, the Pn n lead time length becomes j¼1 tj , and the corresponding fixed crash cost is j¼1 aj ðT j tj Þ. Moreover, the
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ordering cost A depends on total capital invested in the ordering cost reduction and the range of A is 0 < A6A0. Based on these scenarios, we let vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 vffiffiffiffiffiffiffiffiffiffiffi 3ffi u uX u u n u 2D 4 S rWðkÞt ð16Þ Qmin ¼ t t j 5; þp H ðmÞ m j¼1 and Qmax
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 vffiffiffiffiffiffiffiffiffiffiffiffiffi u uX u n n X u S u 2D 4 rWðkÞt A0 þ þ p ¼t Tj þ aj ðT j tj Þ5: H ðmÞ m j¼1 j¼1
ð17Þ
Eqs. (13), (16) and (17) yield Qmin < Q 6 Qmax. If Qmin 2 ½QSi1 ; QSi and Qmax 2 ½QSj1 ; QSj , where i 6 j, then order quantity Q is located in ½QSi1 ; QSj . To simplify the search process and optimize the number of shipments m , an intuitively starting value for m is provided using the following Property. Property 2. The optimal number of shipment m must satisfy S hb hv þ hv 2D P pffiffiffi 6 m ðm þ 1Þ: m ðm 1Þ 6 r LWðk Þ þ U ðLÞ 1 DP hv ½A þ p
ð18Þ
Proof. See Appendix A. The optimal solutions for Q, A, and k for given L and m can be obtained by solving Eqs. (13)–(15) iteratively until convergence. Procedure convergence then can be proved by adopting a similar graphical technique used in Hadley and Whitin (1963). An algorithmic procedure was developed as follows to identify the optimal solution for (Q, A, k, L, m) (this study adopted Pan et al.s, 2002 algorithm to determine Qmin, Qmax, e, and f in Step 2–Step 4, and part of Step 5 was adopted from Ouyang et al., 1999). h Algorithm P ^ (i.e., the lower bound of m ) by substituting A = A0, L ¼ ni¼1 T i , U ðLÞ ¼ Step 0. Compute m Pn i¼1 ai ðT i t i Þ, and W(k ) = W(0) = 0.3989 into (18). ^ Step 1. Set m ¼ m. Step 2. Compute the intersection points QS of the crash cost lines ci = ai + biQ and cj = aj + bjQ, for all i and j, where ai > aj, bi < bj, i 5 j and i, j = 1, 2, . . . n. Arrange these intersection points such that S ¼ 1. Q1S < Q2S < < QwS and let Q0S ¼ 0, Qwþ1 Step 3. Computing Qmin and Qmax using Eqs. (16) and (17). S Step 4. Use Qmin and Qmax to determine the values of e and f such that Qe1 6 Qmin 6 QeS , and S S Qf 1 6 Qmax 6 Qf . S Step 5. For each order quantity range ðQj1 ; QjS Þ, j = e,e + 1, . . . , f, rearrange ci such that c16c26 6cn, do (5-1) For each Li, i = 0, 1, . . . , n, perform (5-2) to (5-4). (5-2) Start with ki1 = 0 (implies U(ki1) = 0.5). (i) Using U(ki1) and Eq. (15) to evaluate Qi1. (ii) Using Qi1 to determine Ai1 from Eq. (14). (iii) Using Qi1 and Ai1 to determine W(ki2) from Eq. (13), then finds ki2, and hence U(ki2). (iv) Repeat (i)–(iii) until no change occurs in the values of Qi, Ai and ki.
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S (5-3) Check whether Ai < A0 and Qi 2 ½Qj1 ; QjS . S (i) If Ai < A0 and Qi 2 ½Qj1 ; QjS , then the solution found in (5-2) is optimal for given Li. Go to (5-4). (ii) If AiPA0, then set Ai = A0, and the corresponding (Qi, ki) can be obtained by solving (13) and (15) iteratively until convergence (the solution procedure is similar to the previous steps). S S (iii) If Qi 6 Qj1 , let Qi ¼ Qj1 ; if QjS 6 Qi , let Qi ¼ QjS . Using Qi to compute the corresponding Ai and ki in Eqs. (14) and (15). If AiPA0, then set Ai = A0. Go to (5-4). (5-4) Compute the corresponding JTEC(Qi, Ai, ki, Li, m). (5-5) Find min i=0,1,. . . , n JTEC(Qi, Ai, ki, Li, m). Set JTECðQðjÞ ; AðjÞ ; k ðjÞ ; LðjÞ ; mÞ ¼ mini¼0; 1;...;n JTECðQi ; Ai ; k i ; Li ; mÞ, then ðQðjÞ ; AðjÞ ; k ðjÞ ; LðjÞ ; mÞ is the optimal solution for the order quanS tity range Qj1 ; QjS .
Step 6. Find minj¼e;eþ1;;f JTECðQðjÞ ; AðjÞ ; k ðjÞ ; LðjÞ ; mÞ. Set JTECðQm ; Am ; k m ; Lm ; mÞ ¼ minj¼e;eþ1;;f JTECðQðjÞ ; AðjÞ ; k ðjÞ ; LðjÞ ; mÞ, then ðQm ; Am ; k m ; Lm Þ is the optimal solution for fixed m. Step 7. Set m = m + 1, repeat Steps 2–6 to get JTECðQm ; Am ; k m ; Lm ; mÞ. Step 8. If JTECðQm ; Am ; k m ; Lm ; mÞ 6 JTECðQm1 ; k m1 ; Am1 ; Lm1 ; m 1Þ, then go to Step 7, otherwise go to Step 9. Step 9. Set ðQ ; A ; k ; L ; m Þ ¼ ðQm1 ; Am1 ; k m1 ; Lm1 ; m 1Þ, then (Q , A , k , L , m ) is the optimal solution and JTEC(Q , A , k , L , m ) is the minimum joint total expected cost. pffiffiffiffiffi Once the optimal solution (Q , A , k , L , m ) is obtained, the optimal reorder point r ¼ DL þ k r L follows. 4. Linear function case This section considers that the lead time and ordering cost reduction act dependently. In actual situations, the relationship between the lead time and ordering cost reduction could vary case by case, and this study adopts a linear relationship to provide insight and analytical tractability. The following simple linear relationship used in Chiu (1998) and Chen et al. (2001) is employed to address this issue: L0 L A0 A ¼a ; ð19Þ L0 A0 where a(>0) is a constant scaling parameter to describe the linear relationship between percentage reductions in lead time and ordering cost. By considering this relationship (19), the ordering cost A can be expressed as a linear function of L, that is, AðLÞ ¼ u þ vL; ð20Þ A0 1 where u ¼ 1 a A0 and v ¼ aL0 . Now, assuming that the ordering cost is reduced due to the lead time reduction rather than investment, using (20) in (5), the problem then becomes minimizing the following joint total expected cost: " # i1 X pffiffiffi D S r LWðkÞ þ U ðLÞ u þ vL þ þ p JTECH ðQ; k; L; mÞ ¼ D bi ðLi1 LÞ þ bj ðT j tj Þ þ Q m j¼1 pffiffiffi Q þ hb r L½k þ ð1 bÞWðkÞ þ H ðmÞ; 2 where the subscript ‘‘H ’’ denotes the linear relationship between ordering cost and lead time.
ð21Þ
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To solve this problem, for fixed m, the first partial derivatives of (21) with respect to Q, k and L 2 (Li, Li1), respectively, are taken. This process obtains: pffiffiffi oJTECH ðQ; k; L; mÞ D S H ðmÞ r LWðkÞ þ U ðLÞ þ ¼ 2 u þ vL þ þ p ; ð22Þ oQ m 2 Q pffiffiffi oJTECH ðQ; k; L; mÞ D pffiffiffi r L½UðkÞ 1 þ hb r Lf1 þ ð1 bÞ½UðkÞ 1 g; ¼ p ok Q
ð23Þ
oJTECH ðQ; k; L; mÞ Dv D hb r ai rWðkÞL1=2 þ ¼ þ p ½k þ ð1 bÞWðkÞ L1=2 D þ bi : oL Q 2Q 2 Q
ð24Þ
and
Furthermore, for fixed (Q, k, m), JTECH(Q, k, L, m) can be shown to be a concave function in L 2 [Li, Li1]. Therefore, for fixed (Q, k, m), the minimum total expected cost per unit time will occur at the end points of the interval [Li, Li1]. Additionally, for fixed m and L 2 [Li, Li1], JTECH(Q, k, L, m) can be shown to be convex in (Q, k). Thus, for a given value of L 2 [Li, Li1] and fixed m, by setting Eqs. (22) and (23) equal to zero, the optimal values for Q and k (denoted by Q and k ) can be obtained using the following equations: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi 2D S r LWðk Þ þ U ðLÞ ; u þ vL þ þ p Q ¼ ð25Þ H ðmÞ m and Uðk Þ ¼ 1
hb Q : D þ hb Q ð1 bÞ p
ð26Þ
Moreover, using the same approach as in the previous section, Qmin and Qmax can be obtained as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 vffiffiffiffiffiffiffiffiffiffiffi 3 u uX u n X u n S u 2D 4 rWðkÞt uþv ð27Þ Qmin ¼ t tj þ þ p tj 5; H ðmÞ m j¼1 j¼1 and Qmax
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 vffiffiffiffiffiffiffiffiffiffiffiffiffi u uX u n n n X X u S u 2D 4 rWðkÞt uþv ¼t Tj þ þ p Tj þ aj ðT j tj Þ5: H ðmÞ m j¼1 j¼1 j¼1
ð28Þ
A similar algorithm procedure as proposed in Section 3 then can be performed to obtain the optimal solution for (Q, k, L, m).
5. Numerical examples Example 1. In order to illustrate the above solution procedure, let us consider an inventory system with the data used in Pan et al. (2002): D = 600 units/year, A0 = $200/order, hb = $ 20/unit/year, p = $50/unit, p0 = $150/unit, r = 7 units/week, and the lead time has three components with data shown in Table 1.
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Table 1 Lead time data Lead time component i
1
2
3
Normal duration Ti (days) Minimum duration ti (days) Unit fixed crash cost ai ($/day) Unit variable crash cost bi ($/unit/day)
20 6 0.5 0.012
20 6 1.3 0.004
16 9 5.1 0.0012
Besides, for integrated vendor–buyer cooperative inventory system, we take P = 2000 units/year, S = $1000/set-up, hv = $15/unit/year, 1/d = 2800, and h = 0.2/dollar/year. The above data are first used to evaluate the intersection points, order quantity range interval and component crash priorities in each interval. The analytical results are listed in Table 2, and Fig. 2 shows the crash sequence corresponding to each order quantity range. Then, for different b = 0, 0.5, 0.8 and 1, the previously developed algorithm is applied to obtain the results of the solution procedure as shown in Table 3, and a summary is listed in Table 4. To see the ordering cost reduction effect, we list the fixed ordering cost model results (i.e., A = 200) in Table 4. Notably, the service level can be measured using 1 P(X > r), where P(X > r) denotes the probability of shortage during a cycle. In this sense, the higher reorder point r will result in lower P(X > r), and thus higher service level. Comparing the results shown in Table 4 reveals that, for any given backorder rate b, the ordering cost reduction model has higher reorder point and lower joint total expected cost, meaning that higher service level along with joint total expected cost savings are achieved by ordering cost reduction. Example 2. The case involving a linear relationship between the lead time and ordering cost is also considered. The data is the same as that in Example 1, and a = 0.75, 1.00, 1.25, 2.50 and 5.00. Applying a similar algorithmic procedure as that proposed in Section 3 produces the results listed in Table 5. To clarify
Table 2 The values of QS, order quantity ranges and crash sequence Intersection points (QS)
Order quantity range
Crash sequence of components
100 426 1357 –
(0, 100] (100, 426] (426, 1357] (1357, 1)
1, 2, 2, 3,
2, 1, 3, 2,
3 3 1 1
crash cost per unit time
25.00 20.00
c1
15.00 10.00 5.00 0.00 0
c3 c2 250 500 750 1000 1250 1500 1750 2000
Fig. 2. Crash cost per unit time vs. order quantity.
Q
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Table 3 The results of solution procedure m
Qmin
Qmax
e, f
Qm
Am
rm
Li
JTEC(Æ)
b=0 ^ ¼1 m
1 2 3 4
221 131 94 73
373 283 239 212
2,2 2,2 1,2 1,2
249 154 113 87
200 144 105 81
71 74 75 136
4 4 4 8
$6740.58 $6275.10 $6213.13 $6231.39
b = 0.5 m ^ ¼1
1 2 3 4
221 131 94 73
332 245 204 179
2,2 2,2 1,2 1,2
250 155 111 87
200 144 103 81
67 71 130 132
4 4 8 8
$6684.48 $6223.12 $6157.80 $6163.48
b = 0.8 ^ ¼1 m
1 2 3 4
221 131 94 73
304 219 180 156
2,2 2,2 1,2 1,2
251 153 111 87
200 143 104 81
91 123 126 128
6 8 8 8
$6618.29 $6164.88 $6086.53 $6094.76
b = 1.0 ^ ¼2 m
2 3 4
131 94 73
199 162 139
2,2 1,2 1,2
154 111 87
144 104 82
117 121 123
8 8 8
$6078.04 $6005.14 $6016.82
Table 4 The optimal solutions b
Ordering cost reduction model
0 0.5 0.8 1.0
Fixed ordering cost model (A = 200)
Q
A
r
L
m
JTEC(Æ)
QA
rA
LA
mA
JTECA(Æ)
113 111 111 111
$105 $103 $104 $104
75 130 126 121
4 8 8 8
3 3 3 3
$6213.13 $6157.80 $6086.53 $6005.14
161 161 161 160
73 70 95 117
4 4 6 8
2 2 2 2
$6304.38 $6252.10 $6193.38 $6107.74
Notes: (1) The unit of L and LA are weeks. (2) Subscript A denotes the case of fixed ordering cost.
the ordering cost reduction effect, the fixed ordering cost model (i.e., a = 1) results are listed at the bottom of Table 5. Table 5 reveals that, for any given b, the order quantity, ordering cost, joint total expected cost increase and the number of shipments decrease as the a value increases. Furthermore, for any given a, the joint total expected cost decreases with increasing value of b. Additionally, it is observed that for any given b, the fixed ordering cost model (a = 1) has a larger joint total expected cost compared with the ordering cost reduction model. This result shows the influence of ordering cost reduction.
6. Conclusions This study investigated how lead time and ordering cost reduction affect the integrated inventory model. Lead time crash cost was assumed to depend on ordered lot size and the amount of lead time to be shortened. Moreover, ordering cost was included among the decision variables. Two models were proposed in this study. The first model employed a logarithmic investment cost function for ordering cost reduction, in which the ordering cost and lead time reductions were performed independently. The second model as-
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493
Table 5 The optimal solutions of Example 2 a
b
Q
A
0.75
0 0.5 0.8 1.0
87 87 88 109
$33 $33 $33 $33
1.00
0 0.5 0.8 1.0
113 113 113 114
1.25
0 0.5 0.8 1.0
2.50
r
L
m
JTECH(Æ)
61 59 56 52
3 3 3 3
4 4 4 3
$5565.76 $5524.13 $5482.01 $5433.83
$75 $75 $75 $75
60 57 55 60
3 3 3 3
3 3 3 3
$5795.71 $5752.64 $5708.84 $5658.80
116 115 115 115
$100 $120 $120 $120
60 72 70 66
3 4 4 4
3 3 3 3
$5926.57 $5881.66 $5830.97 $5773.05
0 0.5 0.8 1.0
119 119 120 120
$160 $160 $160 $160
75 72 69 66
4 4 4 4
3 3 3 3
$6136.83 $6086.75 $6035.75 $5977.39
5.00
0 0.5 0.8 1.0
158 159 159 159
$180 $180 $180 $180
73 70 67 64
4 4 4 4
2 2 2 2
$6229.18 $6177.01 $6123.50 $6061.64
1
0 0.5 0.8 1.0
161 161 161 160
$200 $200 $200 $200
73 70 95 117
4 4 6 8
2 2 2 2
$6304.38 $6252.10 $6193.38 $6107.74
Note: The unit of L is weeks.
sumed that the ordering cost and lead time reductions interacted with each other linearly. An iterative algorithm was devised to determine the optimal solution for lot size, reorder point, ordering cost, lead time, and number of shipments between the vendor and buyer. The numerical example results succinctly explained the importance of lead time and ordering cost reduction. The first model showed that both higher service level and lower joint total expected cost could be obtained by ordering cost reduction. Meanwhile, the second model showed that for any given backorder ratio, b, the order quantity, ordering cost and joint total expected cost increase and the number of shipments decrease with increasing scaling parameter, a. For any given scaling parameter, a, the joint total expected cost decreases with increasing backorder ratio, b. Finally, this study does not address quality-related issues. A possible extension of this work may consider the imperfect quality items. On the other hand, this study assumes lead times to be deterministic. This limitation can be overcome by considering procurement lead time as a random variable and discussing the effects in reducing lead-time variability. Furthermore, this study assumed that the relationship between lead time and ordering cost reduction is linear, while other functional forms may exist in practical situations. Acknowledgement The authors would like to thank the anonymous referees for their very valuable and helpful suggestions on an earlier version of the paper.
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Appendix A Proof of Property 2. For a particular value of L, substituting Eqs. (13)–(15) into JTEC(Q, A, k, L, m) and then ignore the terms that are independent of m. This process yields: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi S r LWðk Þ þ U ðLÞ : JTEC1 ðmÞ JTECðQ ; A ; k ; L; mÞ ¼ 2DH ðmÞ A þ þ p m Taking square of JTEC1(m), yields
pffiffiffi S r LWðk Þ þ U ðLÞ : JTEC2 ðmÞ ½JTEC1 ðmÞ ¼ 2DH ðmÞ A þ þ p m 2
ðA:1Þ
Since m is a positive integer, the optimal value m is obtained when JTEC2 ðm Þ 6 JTEC2 ðm þ 1Þ and
JTEC2 ðm Þ 6 JTEC2 ðm 1Þ:
ðA:2Þ
From (A.1) and (A.2), m is noted to satisfy the following condition: S hb hv þ hv 2D P pffiffiffi 6 m ðm þ 1Þ: m ðm 1Þ 6 r LWðk Þ þ U ðLÞ 1 DP hv A þ p This completes the proof.
ðA:3Þ
h
References Banerjee, A., 1986. A joint economic-lot-size model for purchaser and vendor. Decision Sciences 17, 292–311. Ben-Daya, M., Raouf, A., 1994. Inventory models involving lead time as a decision variable. Journal of the Operational Research Society 45 (5), 579–582. Chen, C.K., Chang, H.C., Ouyang, L.Y., 2001. A continuous review inventory model with ordering cost dependent on lead time. International Journal of Information and Management Sciences 12 (3), 1–13. Chiu, P.P., 1998. Economic production quantity models involving lead time as a decision variable. Master Thesis: National Taiwan University of Science and Technology. Goyal, S.K., 1976. An integrated inventory model for a single supplier-single customer problem. International Journal of Production Research 15 (1), 107–111. Goyal, S.K., 1988. A joint economic-lot-size model for purchaser and vendor: A comment. Decision Sciences 19, 236–241. Goyal, S.K., 1995. A one-vendor multi-buyer integrated inventory model: A comment. European Journal of Operational Research 82, 209–210. Ha, D., Kim, S.L., 1997. Implementation of JIT purchasing: An integrated approach. Production Planning and Control 8 (2), 152–157. Hadley, G., Whitin, T., 1963. Analysis of Inventory Systems. Prentice-Hall, New Jersey. Hall, R.W., 1983. Zero Inventories. Dow Jones-Irwin, Homewood, Illinois. Hariga, M., Ben-Daya, M., 1999. Some stochastic inventory models with deterministic variable lead time. European Journal of Operational Research 113, 42–51. Hill, R.M., 1997. The single-vendor single-buyer integrated production–inventory model with a generalized policy. European Journal of Operational Research 97, 493–499. Hill, R.M., 1999. The optimal production and shipment policy for the single-vendor single-buyer integrated production–inventory problem. International Journal of Production Research 37 (11), 2463–2475. Keller, G., Noori, H., 1988. Justifying new technology acquisition through its impact on the cost of running an inventory policy. IIE Transitions 20, 284–291. Kim, K.L., Hayya, J.C., Hong, J.D., 1992. Setup reduction in economic production quantity model. Decision Sciences 23, 500–508. Liao, C.J., Shyu, C.H., 1991. An analytical determination of lead time with normal demand. International Journal of Operations and Production Management 11, 72–78. Moon, I., Choi, S., 1998. A note on lead time and distributional assumptions in continuous review inventory models. Computers and Operations Research 25, 1007–1012.
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Nasri, F., Affisco, J.F., Paknejad, M.J., 1990. Setup cost reduction in an inventory model with finite-range stochastic lead times. International Journal of Production Research 28, 199–212. Ouyang, L.Y., Chang, H.C., 2002. Lot size reorder point inventory model with controllable lead time and set-up cost. International Journal of Systems Science 33 (8), 635–642. Ouyang, L.Y., Chen, C.K., Chang, H.C., 1999. Lead time and ordering cost reductions in continuous review inventory systems with partial backorders. Journal of the Operational Research Society 50, 1272–1279. Ouyang, L.Y., Yeh, N.C., Wu, K.S., 1996. Mixture inventory model with backorders and lost sales for variable lead time. Journal of the Operational Research Society 47, 829–832. Paknejad, M.J., Nasri, F., Affisco, J.F., 1995. Defective units in a continuous review (s,Q) system. International Journal of Production Research 33 (10), 2767–2777. Pan, J.C., Hsiao, Y.C., Lee, C.J., 2002. Inventory models with fixed and variable lead time crash costs considerations. Journal of the Operational Research Society 53 (9), 1048–1053. Pan, J.C., Yang, J.S., 2002. A study of an integrated inventory with controllable lead time. International Journal of Production Research 40 (5), 1263–1273. Porteus, E.L., 1985. Investing in reduced setups in the EOQ model. Management Science 31, 998–1010. Tersine, R.J., 1994. Principles of Inventory and Materials Management. Prentice Hall, Englewood Cliffs, New Jersey.
Production, Manufacturing and Logistics
Integrated vendor–buyer cooperative inventory models with controllable lead time and ordering cost reduction Hung-Chi Chang a, Liang-Yuh Ouyang a
b,* ,
Kun-Shan Wu c, Chia-Huei Ho
d
Department of Logistics Engineering and Management, National Taichung Institute of Technology, Taichung, Taiwan b Department of Management Sciences and Decision Making, Tamkang University, Tamsui, Taipei, 25137 Taiwan c Department of Business Administration, Tamkang University, Tamsui, Taipei, Taiwan d Graduate Institute of Management Sciences, Tamkang University, Tamsui, Taipei, Taiwan Received 28 May 2003; accepted 16 June 2004 Available online 11 September 2004
Abstract This study deals with the lead time and ordering cost reduction problem in the single-vendor single-buyer integrated inventory model. We consider that buyer lead time can be shortened at an extra crashing cost which depends on the lead time length to be reduced and the ordering lot size. Additionally, buyer ordering cost can be reduced through further investment. Two models are presented in this study. The first model assumes that the ordering cost reduction has no relation to lead time crashing. The second model assumes that the lead time and ordering cost reduction are interacted. An iterative procedure is developed to find the optimal solution and numerical examples are presented to illustrate the results of the proposed models. 2004 Elsevier B.V. All rights reserved. Keywords: Inventory; Integrated model; Lead time reduction; Ordering cost reduction
1. Introduction Most inventory models considered to date assume just one facility (e.g., a buyer or a vendor) managing its inventory policy to minimize its own cost or maximize its own profit. This one-sided-optimal-strategy is not suitable for global markets. The issue of just-in-time (JIT) has recently received great attention. Most JIT research has focused on the integration between vendor and buyer. Once a long-term relationship
*
Corresponding author. Tel.: +886 2 86313221; fax: +886 2 86313214. E-mail address: [email protected] (L.-Y. Ouyang).
0377-2217/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.06.029
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between both facilities has been developed, both parties can cooperate and share information to achieve improved benefits. The joint optimization concept for buyer and vendor was initiated by Goyal (1976). Subsequently, numerous scholars developed integrated inventory models under various assumptions. For example, Banerjee (1986) assumed that the vendor was manufacturing at a finite rate and considered a joint economic-lotsize model in which a vendor produces to order for a buyer on a lot-for-lot basis. Goyal (1988) relaxed the lot-for-lot policy and suggested that vendor economic production quantity should be an integer multiple of buyer purchase quantity. Goyal (1995) later proposed a different shipment policy that suggested that the ith shipment size equals (first shipment size) · (production rate/demand rate)(i1). Hill (1997) further extended this concept and proposed a general shipment policy which suggested that the ith shipment size equals (first shipment size) · y(i1), where 16y6 (production rate/demand rate). In the same year, Ha and Kim (1997) developed an integrated JIT lot-splitting model for facilitating multiple shipments in small lots, that is, the vendor produces the contract quantity in one setup and ships to the buyer in small quantities of equal size in several lots. Later, Hill (1999) established a more general batching and shipping policy which suggested that the successive shipment size of the first m shipments increases by a fixed factor, and the remaining shipments would be equal sized. In a recent study, Pan and Yang (2002) developed an integrated inventory model with controllable lead time. Most previous researches on the integrated vendor–buyer inventory problem only considered the production shipment schedule in terms of the number and size of batches transferred between both parties. In modern production management, controllable lead time and ordering cost reduction are keys to business success and have attracted considerable research attention. Ordering quantity, service level and business competitiveness can be shown to possibly be influenced directly or indirectly via lead-time and/or ordering cost control. The previously mentioned integrated inventory models treat the ordering cost and/or lead time as constants. However, in some practical situations, lead time and ordering cost can be controlled and reduced in various ways. Ordering cost reduction can be attained through worker training, procedural changes, and specialized equipment acquisition. In the literature, Porteus (1985) first introduced the concept and developed a framework for investing in reducing EOQ model set-up cost. This development encouraged many researchers to examine setup/ordering cost reduction (e.g. Keller and Noori, 1988; Nasri et al., 1990; Kim et al., 1992; Paknejad et al., 1995). As stated in Tersine (1994), lead time usually comprises several components, such as setup time, process time, wait time, move time and queue time. In many practical situations, lead time can be reduced using an added crashing cost. In other words, lead time is controllable. The Japanese experience of using JIT production showed that the benefits associated with lead time control are clear. Therefore, reducing lead time is both necessary and beneficial. Inventory models incorporating lead time as a decision variable were developed by several researchers. Liao and Shyu (1991) first devised a probability inventory model in which lead time was the unique decision variable. Later, several researchers (e.g., Ben-Daya and Raouf (1994), Ouyang et al. (1996), Moon and Choi (1998), Hariga and Ben-Daya (1999)) developed various analytical inventory models to explore the lead time reduction problem. The underlying assumption in the above studies was that lead time could be decomposed into n mutually independent components, each with a different but fixed crash cost independent of the ordered lot size. However, this view may not be realistic. In a real environment, to reduce lead time, managers may ask workers to work overtime, employ part-time workers, use special delivery, and so on. Intuitively, extra cost should be paid for these services, and these costs may depend on the ordered lot size. Generally, the larger the ordered lot size, the higher the cost needed to reduce the lead time. Therefore, it seems reasonable to consider that lead time crash cost depends not only on the amount of lead time to be shortened, but also on the ordered lot size. Pan et al. (2002) extended this idea further by considering lead time crash cost as a function of both the order quantity and reduced lead time.
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Recently, Ouyang et al. (1999) investigated the influence of ordering cost reduction on modified continuous review inventory systems involving variable lead time with partial backorders. Subsequently, Ouyang and Chang (2002) proposed a modified lot-size reorder-point inventory model with imperfect production processes to study the effects of reducing lead time and set-up cost. The optimal policies derived in these two articles are buyer focused, and the lead time and ordering/set-up cost reduction were assumed to act independently. However, an independent relationship between lead time and ordering/set-up cost is just one possibility. In some practices, lead time and ordering/set-up cost reduction might be closely related. A lead time reduction could accompany a reduction in the ordering/set-up cost, and vice versa. For example, electronic data interchange (EDI) technology could simultaneously reduce both the lead time and the ordering/set-up cost. To date, little research has been done on establishing the relationship between lead time and ordering cost reduction. To provide insight and analytical tractability, as in Chiu (1998) and Chen et al. (2001), this study employed a linear function to formulate the above relationship. Expanding the focus beyond that solely of the buyer, this paper attempts to incorporate lead time and ordering cost reduction into the integrated vendor–buyer inventory model. Two integrated cooperative inventory models are proposed for the continuous review inventory system. The first model considers the case in which the lead time and ordering cost reduction are performed independently, while the second model considers the lead time and ordering cost reduction to interact linearly. The objective is to minimize the joint total expected cost by simultaneously optimizing the order quantity, reorder point, ordering cost, lead time and number of shipments between the two facilities. For both models, an effective iterative procedure is developed to determine the optimal policy, and numerical examples are used to illustrate the results and benefits of integration. 2. Notation and assumptions To develop the proposed models, we adopt the following notation and assumptions which are similar to those used in Pan et al. (2002). Notations D P A0 hb hv p p0 S A L Q r I(A) h d m X
average demand per unit time of the buyer production rate of the vendor buyers original ordering cost buyers holding cost per unit per unit time vendors holding cost per unit per unit time buyers fixed penalty cost per unit short buyers gross marginal profit per unit vendors set-up cost per set-up buyers ordering cost per order (decision variable) length of lead time (decision variable) buyers order quantity (decision variable) buyers reorder point (decision variable) buyers capital investment required to achieve ordering cost A, 0 < A 6 A0 fractional opportunity cost of capital investment per unit time percentage decrease in ordering cost A per dollar increase in investment I(A) the number of lots in which the product is delivered from the vendor to the buyer in one production cycle, a positive integer (decision variable) the leadptime ffiffiffi demand which follows a normal distribution with finite mean DL and standard deviation r L, where r denotes the standard deviation of the demand per unit time
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Assumptions 1. There is single-vendor and single-buyer for a single product in this article. 2. The buyer orders a lot of size Q and the vendor manufactures mQ with a finite production rate P(P > D) at one set-up but ship in quantity Q to the buyer over m times. The vendor incurs a set-up cost S for each production run and the buyer incurs an ordering cost A for each order of quantity Q. 3. Inventory is continuously reviewed. The buyer places the order when the on hand inventory reaches the reorder point r. 4. The reorder point r = the expected demand during lead time pffiffiffi + safety stock (SS), and SS = k · (standard deviation of lead time demand), that is, r ¼ DL þ kr L where k is a safety factor. 5. Shortages are allowed and partially backordered. b denotes the fraction of the demand during the stock-out period that will be backordered. 6. The lead time L consists of n mutually independent components. The ith component has a normal duration Ti and minimum duration ti, i = 1, 2, . . . , n. 7. For the ith component of lead time, the crashing cost per unit time, ci, depends on the ordering lot size Q and is described by ci = ai + biQ, where ai > 0 is the fixed cost, and bi > 0 is the unit variable cost, for i = 1, 2, . . . , n. 8. For any two crash cost lines ci = ai + biQ and cj = aj + bjQ, where ai > aj, bi
3. Basic model For the model that does not consider ordering cost and reorder point as decision variables, the buyer model closely follows Pan et al. (2002), who established the total expected cost per unit time as follows:
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TCb ðQ; LÞ ¼ ordering cost þ inventory holding cost þ stock-out cost þ lead time crashing cost pffiffiffi pffiffiffi AD Q D D þ hb þ r DL þ ð1 bÞr LWðkÞ þ ½p þ p0 ð1 bÞ r LWðkÞ þ CðLÞ; ¼ Q 2 Q Q
485
ð1Þ
where W(k) = /(k) k[1 U(k)], / and U are the standard normal probability density function (p.d.f.) and cumulative distribution function (c.d.f.), respectively. Note that it can be shown that W(k) is a decreasing function of k, and lim k !1W(k) = 0. In contrast to Pan et al. (2002), this study considers that the ordering cost A could vary through capital investment, and adds the reorder point into the buyer decision variable to further improve buyer total cost. Hence, the buyers objective is to minimize the new total expected cost per unit time, namely, the sum of the capital investment cost for reducing A and the inventory relevantpffiffiffi cost as expressed in (1), by optimizing over Q, A, r, and L constrained on 0 < A6A0. Given r ¼ DL þ kr L, the safety factor k can be considered as a decision variable instead of r. Thus, the total expected cost per unit time for the buyer becomes TECb ðQ; A; k; LÞ ¼ hIðAÞ þ TCb ðQ; LÞ pffiffiffi pffiffiffi h A0 AD Q þ hb þ kr L þ ð1 bÞr LWðkÞ ¼ ln þ d Q 2 A pffiffiffi D D þ ½p þ p0 ð1 bÞ r LWðkÞ þ CðLÞ; Q Q
ð2Þ
constrained on 0 < A6A0. On the other hand, for the vendor, since S is the vendor set-up cost per set-up and the production quantity for a vendor in a lot is mQ, the expected set-up cost per unit time is given by SD/(mQ). During the production period, once the first Q units are produced, the vendor delivers them to the buyer, and then continues making the delivery on average every Q/D units of time until the inventory level falls to zero (see Fig. 1). Hence, the average inventory per unit time for the vendor can be calculated as follows: 2 Q Q m2 Q2 Q mQ mQ þ ðm 1Þ ½1 þ 2 þ þ ðm 1Þ P D D 2P D Q D 2D m 1 ¼ 1þ : 2 P P
quantity cumulated inventory level
Q PQ D
time mQ P mQ D
Fig. 1. Inventory level of vendor.
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Therefore, the total expected cost per unit time for the vendor is: SD Q D 2D þ hv mð1 Þ 1 þ TECv ðQ; mÞ ¼ : mQ 2 P P
ð3Þ
Accordingly, the joint total expected cost per unit time for the vendor and buyer requiring minimization is given by JTECðQ; A; k; L; mÞ ¼ TECb ðQ; A; k; LÞ þ TECv ðQ; mÞ pffiffiffi pffiffiffi h A0 AD Q þ hb þ kr L þ ð1 bÞr LWðkÞ ¼ ln þ d Q 2 A pffiffiffi D
SD Q D 2D þ þ hv m 1 p þ p0 ð1 bÞ r LWðkÞ þ 1þ Q mQ 2 P P " # i1 X D þ ðaj þ bj QÞðT j tj Þ ; L 2 ½Li ; Li1 : ðai þ bi QÞðLi1 LÞ þ Q j¼1
ð4Þ
To simplify notation, we let ¼ p þ p0 ð1 bÞ; p U ðLÞ ¼ ai ðLi1 LÞ þ
i1 X
aj ðT j tj Þ;
j¼1
and
D 2D H ðmÞ ¼ hb þ hv m 1 1þ : P P
Now, the problem can be formulated by " # i1 X h A0 Min JTECðQ; A; k; L; mÞ ¼ ln bj ðT j tj Þ þ D bi ðLi1 LÞ þ d A j¼1 pffiffiffi pffiffiffi D S þ A þ þ pr LWðkÞ þ U ðLÞ þ hb r L½k þ ð1 bÞWðkÞ Q m Q þ H ðmÞ 2 subject to 0 < A 6 A0 :
ð5Þ
To solve the above nonlinear programming problem, this study temporarily ignores the constraint 0 < A6A0 and relaxes the integer requirement on m (the number of shipments from the vendor to the buyer during one production cycle). For fixed Q, A, k, and L 2 [Li, Li1], JTEC(Q, A, k, L, m) can be proved to be a convex function of m. Consequently, the search for the optimal shipment number, m , is reduced to find a local minimum. Property 1. For fixed Q, A, k, and L, JTEC(Q, A, k, L, m) is convex in m. Proof. Taking the first and second partial derivatives of JTEC(Q, A, k, L, m) with respect to m, we have oJTECðQ; A; k; L; mÞ SD Q D ¼ 2 þ hv 1 ; ð6Þ om mQ 2 P
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and o2 JTECðQ; A; k; L; mÞ 2SD ¼ 3 > 0: om2 mQ
ð7Þ
Therefore, JTEC(Q, A, k, L, m) is convex in m, for fixed Q, A, k, and L. This completes the proof of Property 1. h Next, the first partial derivatives of JTEC(Q, A, k, L, m) with respect to Q, A, k and L 2 (Li, Li1) are taken for fixed m, respectively. This process yields: pffiffiffi oJTECðQ; A; k; L; mÞ D S H ðmÞ r LWðkÞ þ U ðLÞ þ ¼ 2 Aþ þp ; ð8Þ oQ m 2 Q oJTECðQ; A; k; L; mÞ h D ¼ þ ; oA dA Q
ð9Þ
pffiffiffi oJTECðQ; A; k; L; mÞ D pffiffiffi r L½UðkÞ 1 þ hb r Lf1 þ ð1 bÞ½UðkÞ 1 g; ¼ p ok Q
ð10Þ
oJTECðQ; A; k; L; mÞ D hb r ai 1=2 1=2 rWðkÞL ¼ ½k þ ð1 bÞWðkÞ L þ bi : þ D p oL 2Q 2 Q
ð11Þ
and
Furthermore, for fixed (Q, A, k, m), JTEC(Q, A, k, L, m) is noted to be a concave function in L 2 [Li, Li1], because o2 JTECðQ; A; k; L; mÞ D hb r rWðkÞL3=2 ½k þ ð1 bÞWðkÞ L3=2 < 0: ¼ p 2 4Q 4 oL
ð12Þ
Hence, for fixed (Q, A, k, m), the minimum joint total expected cost per unit time occurs at the end points of the interval [Li, Li1]. Conversely, for fixed m and L 2 [Li, Li1], JTEC(Q, A, k, L, m) can be proved to be a convex function of (Q, A, k). Thus, the minimum value of JTEC(Q, A, k, L, m) occurs at the point (Q , A , k ) which satisfies oJTEC(Q, A, k, L, m)/oQ = 0, oJTEC(Q, A, k, L, m)/oA = 0 and oJTEC(Q, A, k, L, m)/ok = 0, simultaneously. Solving these three equations, yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2D S r LWðk Þ þ U ðLÞ ; A þ þ p Q ¼ H ðmÞ m
A ¼
hQ ; dD
ð13Þ
ð14Þ
and Uðk Þ ¼ 1
hb Q : D þ hb Q ð1 bÞ p
ð15Þ
Pn Clearly, if none of the lead time components crashes, the lead time length is j¼1 T j , and the corresponding crash cost is zero. Meanwhile, if all of the lead time components crash toPtheir minimum durations, the Pn n lead time length becomes j¼1 tj , and the corresponding fixed crash cost is j¼1 aj ðT j tj Þ. Moreover, the
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ordering cost A depends on total capital invested in the ordering cost reduction and the range of A is 0 < A6A0. Based on these scenarios, we let vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 vffiffiffiffiffiffiffiffiffiffiffi 3ffi u uX u u n u 2D 4 S rWðkÞt ð16Þ Qmin ¼ t t j 5; þp H ðmÞ m j¼1 and Qmax
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 vffiffiffiffiffiffiffiffiffiffiffiffiffi u uX u n n X u S u 2D 4 rWðkÞt A0 þ þ p ¼t Tj þ aj ðT j tj Þ5: H ðmÞ m j¼1 j¼1
ð17Þ
Eqs. (13), (16) and (17) yield Qmin < Q 6 Qmax. If Qmin 2 ½QSi1 ; QSi and Qmax 2 ½QSj1 ; QSj , where i 6 j, then order quantity Q is located in ½QSi1 ; QSj . To simplify the search process and optimize the number of shipments m , an intuitively starting value for m is provided using the following Property. Property 2. The optimal number of shipment m must satisfy S hb hv þ hv 2D P pffiffiffi 6 m ðm þ 1Þ: m ðm 1Þ 6 r LWðk Þ þ U ðLÞ 1 DP hv ½A þ p
ð18Þ
Proof. See Appendix A. The optimal solutions for Q, A, and k for given L and m can be obtained by solving Eqs. (13)–(15) iteratively until convergence. Procedure convergence then can be proved by adopting a similar graphical technique used in Hadley and Whitin (1963). An algorithmic procedure was developed as follows to identify the optimal solution for (Q, A, k, L, m) (this study adopted Pan et al.s, 2002 algorithm to determine Qmin, Qmax, e, and f in Step 2–Step 4, and part of Step 5 was adopted from Ouyang et al., 1999). h Algorithm P ^ (i.e., the lower bound of m ) by substituting A = A0, L ¼ ni¼1 T i , U ðLÞ ¼ Step 0. Compute m Pn i¼1 ai ðT i t i Þ, and W(k ) = W(0) = 0.3989 into (18). ^ Step 1. Set m ¼ m. Step 2. Compute the intersection points QS of the crash cost lines ci = ai + biQ and cj = aj + bjQ, for all i and j, where ai > aj, bi < bj, i 5 j and i, j = 1, 2, . . . n. Arrange these intersection points such that S ¼ 1. Q1S < Q2S < < QwS and let Q0S ¼ 0, Qwþ1 Step 3. Computing Qmin and Qmax using Eqs. (16) and (17). S Step 4. Use Qmin and Qmax to determine the values of e and f such that Qe1 6 Qmin 6 QeS , and S S Qf 1 6 Qmax 6 Qf . S Step 5. For each order quantity range ðQj1 ; QjS Þ, j = e,e + 1, . . . , f, rearrange ci such that c16c26 6cn, do (5-1) For each Li, i = 0, 1, . . . , n, perform (5-2) to (5-4). (5-2) Start with ki1 = 0 (implies U(ki1) = 0.5). (i) Using U(ki1) and Eq. (15) to evaluate Qi1. (ii) Using Qi1 to determine Ai1 from Eq. (14). (iii) Using Qi1 and Ai1 to determine W(ki2) from Eq. (13), then finds ki2, and hence U(ki2). (iv) Repeat (i)–(iii) until no change occurs in the values of Qi, Ai and ki.
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S (5-3) Check whether Ai < A0 and Qi 2 ½Qj1 ; QjS . S (i) If Ai < A0 and Qi 2 ½Qj1 ; QjS , then the solution found in (5-2) is optimal for given Li. Go to (5-4). (ii) If AiPA0, then set Ai = A0, and the corresponding (Qi, ki) can be obtained by solving (13) and (15) iteratively until convergence (the solution procedure is similar to the previous steps). S S (iii) If Qi 6 Qj1 , let Qi ¼ Qj1 ; if QjS 6 Qi , let Qi ¼ QjS . Using Qi to compute the corresponding Ai and ki in Eqs. (14) and (15). If AiPA0, then set Ai = A0. Go to (5-4). (5-4) Compute the corresponding JTEC(Qi, Ai, ki, Li, m). (5-5) Find min i=0,1,. . . , n JTEC(Qi, Ai, ki, Li, m). Set JTECðQðjÞ ; AðjÞ ; k ðjÞ ; LðjÞ ; mÞ ¼ mini¼0; 1;...;n JTECðQi ; Ai ; k i ; Li ; mÞ, then ðQðjÞ ; AðjÞ ; k ðjÞ ; LðjÞ ; mÞ is the optimal solution for the order quanS tity range Qj1 ; QjS .
Step 6. Find minj¼e;eþ1;;f JTECðQðjÞ ; AðjÞ ; k ðjÞ ; LðjÞ ; mÞ. Set JTECðQm ; Am ; k m ; Lm ; mÞ ¼ minj¼e;eþ1;;f JTECðQðjÞ ; AðjÞ ; k ðjÞ ; LðjÞ ; mÞ, then ðQm ; Am ; k m ; Lm Þ is the optimal solution for fixed m. Step 7. Set m = m + 1, repeat Steps 2–6 to get JTECðQm ; Am ; k m ; Lm ; mÞ. Step 8. If JTECðQm ; Am ; k m ; Lm ; mÞ 6 JTECðQm1 ; k m1 ; Am1 ; Lm1 ; m 1Þ, then go to Step 7, otherwise go to Step 9. Step 9. Set ðQ ; A ; k ; L ; m Þ ¼ ðQm1 ; Am1 ; k m1 ; Lm1 ; m 1Þ, then (Q , A , k , L , m ) is the optimal solution and JTEC(Q , A , k , L , m ) is the minimum joint total expected cost. pffiffiffiffiffi Once the optimal solution (Q , A , k , L , m ) is obtained, the optimal reorder point r ¼ DL þ k r L follows. 4. Linear function case This section considers that the lead time and ordering cost reduction act dependently. In actual situations, the relationship between the lead time and ordering cost reduction could vary case by case, and this study adopts a linear relationship to provide insight and analytical tractability. The following simple linear relationship used in Chiu (1998) and Chen et al. (2001) is employed to address this issue: L0 L A0 A ¼a ; ð19Þ L0 A0 where a(>0) is a constant scaling parameter to describe the linear relationship between percentage reductions in lead time and ordering cost. By considering this relationship (19), the ordering cost A can be expressed as a linear function of L, that is, AðLÞ ¼ u þ vL; ð20Þ A0 1 where u ¼ 1 a A0 and v ¼ aL0 . Now, assuming that the ordering cost is reduced due to the lead time reduction rather than investment, using (20) in (5), the problem then becomes minimizing the following joint total expected cost: " # i1 X pffiffiffi D S r LWðkÞ þ U ðLÞ u þ vL þ þ p JTECH ðQ; k; L; mÞ ¼ D bi ðLi1 LÞ þ bj ðT j tj Þ þ Q m j¼1 pffiffiffi Q þ hb r L½k þ ð1 bÞWðkÞ þ H ðmÞ; 2 where the subscript ‘‘H ’’ denotes the linear relationship between ordering cost and lead time.
ð21Þ
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To solve this problem, for fixed m, the first partial derivatives of (21) with respect to Q, k and L 2 (Li, Li1), respectively, are taken. This process obtains: pffiffiffi oJTECH ðQ; k; L; mÞ D S H ðmÞ r LWðkÞ þ U ðLÞ þ ¼ 2 u þ vL þ þ p ; ð22Þ oQ m 2 Q pffiffiffi oJTECH ðQ; k; L; mÞ D pffiffiffi r L½UðkÞ 1 þ hb r Lf1 þ ð1 bÞ½UðkÞ 1 g; ¼ p ok Q
ð23Þ
oJTECH ðQ; k; L; mÞ Dv D hb r ai rWðkÞL1=2 þ ¼ þ p ½k þ ð1 bÞWðkÞ L1=2 D þ bi : oL Q 2Q 2 Q
ð24Þ
and
Furthermore, for fixed (Q, k, m), JTECH(Q, k, L, m) can be shown to be a concave function in L 2 [Li, Li1]. Therefore, for fixed (Q, k, m), the minimum total expected cost per unit time will occur at the end points of the interval [Li, Li1]. Additionally, for fixed m and L 2 [Li, Li1], JTECH(Q, k, L, m) can be shown to be convex in (Q, k). Thus, for a given value of L 2 [Li, Li1] and fixed m, by setting Eqs. (22) and (23) equal to zero, the optimal values for Q and k (denoted by Q and k ) can be obtained using the following equations: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi 2D S r LWðk Þ þ U ðLÞ ; u þ vL þ þ p Q ¼ ð25Þ H ðmÞ m and Uðk Þ ¼ 1
hb Q : D þ hb Q ð1 bÞ p
ð26Þ
Moreover, using the same approach as in the previous section, Qmin and Qmax can be obtained as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 vffiffiffiffiffiffiffiffiffiffiffi 3 u uX u n X u n S u 2D 4 rWðkÞt uþv ð27Þ Qmin ¼ t tj þ þ p tj 5; H ðmÞ m j¼1 j¼1 and Qmax
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 vffiffiffiffiffiffiffiffiffiffiffiffiffi u uX u n n n X X u S u 2D 4 rWðkÞt uþv ¼t Tj þ þ p Tj þ aj ðT j tj Þ5: H ðmÞ m j¼1 j¼1 j¼1
ð28Þ
A similar algorithm procedure as proposed in Section 3 then can be performed to obtain the optimal solution for (Q, k, L, m).
5. Numerical examples Example 1. In order to illustrate the above solution procedure, let us consider an inventory system with the data used in Pan et al. (2002): D = 600 units/year, A0 = $200/order, hb = $ 20/unit/year, p = $50/unit, p0 = $150/unit, r = 7 units/week, and the lead time has three components with data shown in Table 1.
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Table 1 Lead time data Lead time component i
1
2
3
Normal duration Ti (days) Minimum duration ti (days) Unit fixed crash cost ai ($/day) Unit variable crash cost bi ($/unit/day)
20 6 0.5 0.012
20 6 1.3 0.004
16 9 5.1 0.0012
Besides, for integrated vendor–buyer cooperative inventory system, we take P = 2000 units/year, S = $1000/set-up, hv = $15/unit/year, 1/d = 2800, and h = 0.2/dollar/year. The above data are first used to evaluate the intersection points, order quantity range interval and component crash priorities in each interval. The analytical results are listed in Table 2, and Fig. 2 shows the crash sequence corresponding to each order quantity range. Then, for different b = 0, 0.5, 0.8 and 1, the previously developed algorithm is applied to obtain the results of the solution procedure as shown in Table 3, and a summary is listed in Table 4. To see the ordering cost reduction effect, we list the fixed ordering cost model results (i.e., A = 200) in Table 4. Notably, the service level can be measured using 1 P(X > r), where P(X > r) denotes the probability of shortage during a cycle. In this sense, the higher reorder point r will result in lower P(X > r), and thus higher service level. Comparing the results shown in Table 4 reveals that, for any given backorder rate b, the ordering cost reduction model has higher reorder point and lower joint total expected cost, meaning that higher service level along with joint total expected cost savings are achieved by ordering cost reduction. Example 2. The case involving a linear relationship between the lead time and ordering cost is also considered. The data is the same as that in Example 1, and a = 0.75, 1.00, 1.25, 2.50 and 5.00. Applying a similar algorithmic procedure as that proposed in Section 3 produces the results listed in Table 5. To clarify
Table 2 The values of QS, order quantity ranges and crash sequence Intersection points (QS)
Order quantity range
Crash sequence of components
100 426 1357 –
(0, 100] (100, 426] (426, 1357] (1357, 1)
1, 2, 2, 3,
2, 1, 3, 2,
3 3 1 1
crash cost per unit time
25.00 20.00
c1
15.00 10.00 5.00 0.00 0
c3 c2 250 500 750 1000 1250 1500 1750 2000
Fig. 2. Crash cost per unit time vs. order quantity.
Q
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Table 3 The results of solution procedure m
Qmin
Qmax
e, f
Qm
Am
rm
Li
JTEC(Æ)
b=0 ^ ¼1 m
1 2 3 4
221 131 94 73
373 283 239 212
2,2 2,2 1,2 1,2
249 154 113 87
200 144 105 81
71 74 75 136
4 4 4 8
$6740.58 $6275.10 $6213.13 $6231.39
b = 0.5 m ^ ¼1
1 2 3 4
221 131 94 73
332 245 204 179
2,2 2,2 1,2 1,2
250 155 111 87
200 144 103 81
67 71 130 132
4 4 8 8
$6684.48 $6223.12 $6157.80 $6163.48
b = 0.8 ^ ¼1 m
1 2 3 4
221 131 94 73
304 219 180 156
2,2 2,2 1,2 1,2
251 153 111 87
200 143 104 81
91 123 126 128
6 8 8 8
$6618.29 $6164.88 $6086.53 $6094.76
b = 1.0 ^ ¼2 m
2 3 4
131 94 73
199 162 139
2,2 1,2 1,2
154 111 87
144 104 82
117 121 123
8 8 8
$6078.04 $6005.14 $6016.82
Table 4 The optimal solutions b
Ordering cost reduction model
0 0.5 0.8 1.0
Fixed ordering cost model (A = 200)
Q
A
r
L
m
JTEC(Æ)
QA
rA
LA
mA
JTECA(Æ)
113 111 111 111
$105 $103 $104 $104
75 130 126 121
4 8 8 8
3 3 3 3
$6213.13 $6157.80 $6086.53 $6005.14
161 161 161 160
73 70 95 117
4 4 6 8
2 2 2 2
$6304.38 $6252.10 $6193.38 $6107.74
Notes: (1) The unit of L and LA are weeks. (2) Subscript A denotes the case of fixed ordering cost.
the ordering cost reduction effect, the fixed ordering cost model (i.e., a = 1) results are listed at the bottom of Table 5. Table 5 reveals that, for any given b, the order quantity, ordering cost, joint total expected cost increase and the number of shipments decrease as the a value increases. Furthermore, for any given a, the joint total expected cost decreases with increasing value of b. Additionally, it is observed that for any given b, the fixed ordering cost model (a = 1) has a larger joint total expected cost compared with the ordering cost reduction model. This result shows the influence of ordering cost reduction.
6. Conclusions This study investigated how lead time and ordering cost reduction affect the integrated inventory model. Lead time crash cost was assumed to depend on ordered lot size and the amount of lead time to be shortened. Moreover, ordering cost was included among the decision variables. Two models were proposed in this study. The first model employed a logarithmic investment cost function for ordering cost reduction, in which the ordering cost and lead time reductions were performed independently. The second model as-
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493
Table 5 The optimal solutions of Example 2 a
b
Q
A
0.75
0 0.5 0.8 1.0
87 87 88 109
$33 $33 $33 $33
1.00
0 0.5 0.8 1.0
113 113 113 114
1.25
0 0.5 0.8 1.0
2.50
r
L
m
JTECH(Æ)
61 59 56 52
3 3 3 3
4 4 4 3
$5565.76 $5524.13 $5482.01 $5433.83
$75 $75 $75 $75
60 57 55 60
3 3 3 3
3 3 3 3
$5795.71 $5752.64 $5708.84 $5658.80
116 115 115 115
$100 $120 $120 $120
60 72 70 66
3 4 4 4
3 3 3 3
$5926.57 $5881.66 $5830.97 $5773.05
0 0.5 0.8 1.0
119 119 120 120
$160 $160 $160 $160
75 72 69 66
4 4 4 4
3 3 3 3
$6136.83 $6086.75 $6035.75 $5977.39
5.00
0 0.5 0.8 1.0
158 159 159 159
$180 $180 $180 $180
73 70 67 64
4 4 4 4
2 2 2 2
$6229.18 $6177.01 $6123.50 $6061.64
1
0 0.5 0.8 1.0
161 161 161 160
$200 $200 $200 $200
73 70 95 117
4 4 6 8
2 2 2 2
$6304.38 $6252.10 $6193.38 $6107.74
Note: The unit of L is weeks.
sumed that the ordering cost and lead time reductions interacted with each other linearly. An iterative algorithm was devised to determine the optimal solution for lot size, reorder point, ordering cost, lead time, and number of shipments between the vendor and buyer. The numerical example results succinctly explained the importance of lead time and ordering cost reduction. The first model showed that both higher service level and lower joint total expected cost could be obtained by ordering cost reduction. Meanwhile, the second model showed that for any given backorder ratio, b, the order quantity, ordering cost and joint total expected cost increase and the number of shipments decrease with increasing scaling parameter, a. For any given scaling parameter, a, the joint total expected cost decreases with increasing backorder ratio, b. Finally, this study does not address quality-related issues. A possible extension of this work may consider the imperfect quality items. On the other hand, this study assumes lead times to be deterministic. This limitation can be overcome by considering procurement lead time as a random variable and discussing the effects in reducing lead-time variability. Furthermore, this study assumed that the relationship between lead time and ordering cost reduction is linear, while other functional forms may exist in practical situations. Acknowledgement The authors would like to thank the anonymous referees for their very valuable and helpful suggestions on an earlier version of the paper.
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Appendix A Proof of Property 2. For a particular value of L, substituting Eqs. (13)–(15) into JTEC(Q, A, k, L, m) and then ignore the terms that are independent of m. This process yields: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi S r LWðk Þ þ U ðLÞ : JTEC1 ðmÞ JTECðQ ; A ; k ; L; mÞ ¼ 2DH ðmÞ A þ þ p m Taking square of JTEC1(m), yields
pffiffiffi S r LWðk Þ þ U ðLÞ : JTEC2 ðmÞ ½JTEC1 ðmÞ ¼ 2DH ðmÞ A þ þ p m 2
ðA:1Þ
Since m is a positive integer, the optimal value m is obtained when JTEC2 ðm Þ 6 JTEC2 ðm þ 1Þ and
JTEC2 ðm Þ 6 JTEC2 ðm 1Þ:
ðA:2Þ
From (A.1) and (A.2), m is noted to satisfy the following condition: S hb hv þ hv 2D P pffiffiffi 6 m ðm þ 1Þ: m ðm 1Þ 6 r LWðk Þ þ U ðLÞ 1 DP hv A þ p This completes the proof.
ðA:3Þ
h
References Banerjee, A., 1986. A joint economic-lot-size model for purchaser and vendor. Decision Sciences 17, 292–311. Ben-Daya, M., Raouf, A., 1994. Inventory models involving lead time as a decision variable. Journal of the Operational Research Society 45 (5), 579–582. Chen, C.K., Chang, H.C., Ouyang, L.Y., 2001. A continuous review inventory model with ordering cost dependent on lead time. International Journal of Information and Management Sciences 12 (3), 1–13. Chiu, P.P., 1998. Economic production quantity models involving lead time as a decision variable. Master Thesis: National Taiwan University of Science and Technology. Goyal, S.K., 1976. An integrated inventory model for a single supplier-single customer problem. International Journal of Production Research 15 (1), 107–111. Goyal, S.K., 1988. A joint economic-lot-size model for purchaser and vendor: A comment. Decision Sciences 19, 236–241. Goyal, S.K., 1995. A one-vendor multi-buyer integrated inventory model: A comment. European Journal of Operational Research 82, 209–210. Ha, D., Kim, S.L., 1997. Implementation of JIT purchasing: An integrated approach. Production Planning and Control 8 (2), 152–157. Hadley, G., Whitin, T., 1963. Analysis of Inventory Systems. Prentice-Hall, New Jersey. Hall, R.W., 1983. Zero Inventories. Dow Jones-Irwin, Homewood, Illinois. Hariga, M., Ben-Daya, M., 1999. Some stochastic inventory models with deterministic variable lead time. European Journal of Operational Research 113, 42–51. Hill, R.M., 1997. The single-vendor single-buyer integrated production–inventory model with a generalized policy. European Journal of Operational Research 97, 493–499. Hill, R.M., 1999. The optimal production and shipment policy for the single-vendor single-buyer integrated production–inventory problem. International Journal of Production Research 37 (11), 2463–2475. Keller, G., Noori, H., 1988. Justifying new technology acquisition through its impact on the cost of running an inventory policy. IIE Transitions 20, 284–291. Kim, K.L., Hayya, J.C., Hong, J.D., 1992. Setup reduction in economic production quantity model. Decision Sciences 23, 500–508. Liao, C.J., Shyu, C.H., 1991. An analytical determination of lead time with normal demand. International Journal of Operations and Production Management 11, 72–78. Moon, I., Choi, S., 1998. A note on lead time and distributional assumptions in continuous review inventory models. Computers and Operations Research 25, 1007–1012.
H.-C. Chang et al. / European Journal of Operational Research 170 (2006) 481–495
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Nasri, F., Affisco, J.F., Paknejad, M.J., 1990. Setup cost reduction in an inventory model with finite-range stochastic lead times. International Journal of Production Research 28, 199–212. Ouyang, L.Y., Chang, H.C., 2002. Lot size reorder point inventory model with controllable lead time and set-up cost. International Journal of Systems Science 33 (8), 635–642. Ouyang, L.Y., Chen, C.K., Chang, H.C., 1999. Lead time and ordering cost reductions in continuous review inventory systems with partial backorders. Journal of the Operational Research Society 50, 1272–1279. Ouyang, L.Y., Yeh, N.C., Wu, K.S., 1996. Mixture inventory model with backorders and lost sales for variable lead time. Journal of the Operational Research Society 47, 829–832. Paknejad, M.J., Nasri, F., Affisco, J.F., 1995. Defective units in a continuous review (s,Q) system. International Journal of Production Research 33 (10), 2767–2777. Pan, J.C., Hsiao, Y.C., Lee, C.J., 2002. Inventory models with fixed and variable lead time crash costs considerations. Journal of the Operational Research Society 53 (9), 1048–1053. Pan, J.C., Yang, J.S., 2002. A study of an integrated inventory with controllable lead time. International Journal of Production Research 40 (5), 1263–1273. Porteus, E.L., 1985. Investing in reduced setups in the EOQ model. Management Science 31, 998–1010. Tersine, R.J., 1994. Principles of Inventory and Materials Management. Prentice Hall, Englewood Cliffs, New Jersey.