- Email: [email protected]
The effects of lumpy demand and shipment size constraint: A response to “Revisit the note on supply chain integration in vendor-managed inventory”
Decision Support Systems 48 (2010) 421–425
Contents lists available at ScienceDirect
Decision Support Systems j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / d s s
The effects of lumpy demand and shipment size constraint: A response to “Revisit the note on supply chain integration in vendor-managed inventory” Boray Huang ⁎, Zhisheng Ye Department of Industrial and Systems Engineering, National University of Singapore, 117576, Singapore, Singapore
a r t i c l e
i n f o
Available online 1 September 2009 Keywords: Vendor management inventory Supply chain integration Lumpy demand
a b s t r a c t This paper responds to a comment by Wang et al. [3] regarding the disagreement between Yao et al. [4] and van der Vlist et al. [2] on the impact of vender-managed-inventory (VMI). We explore the factors which affect the shipment size from the vendor to the buyer and identify the conditions where the shipment size will increase/decrease under VMI. A numerical example also shows when and how the inventory shifts between the supplier and the buyer. © 2009 Elsevier B.V. All rights reserved.
1. Introduction
2. The cost functions and the optimal solutions
Wang et al. [3] try to resolve a disagreement between Yao et al. [4] and van der Vlist et al. [2] regarding the impact of vendor-managedinventory (VMI) on a supply chain. Yao et al. [4] use a two-tier EOQ model to investigate the costs and order behaviours of the supply chain members in both VMI and non-VMI scenarios. They find some interesting analytical results and identify the distribution of the VMI benefit between the supplier and the buyer. van der Vlist et al. [2] argue that Yao et al. overstate the inventory needed at the supplier, and extend Yao et al.'s model to incorporate shipping costs. The findings in Yao et al. [4] and van der Vlist et al. [2] are different in many ways. For example, Yao et al. assert that the buyer's order size shall decrease under VMI, but van der Vlist et al. have an opposite result. In their response [5], Yao et al. argue that van der Vlist et al. [2] make problematic assumptions on the shipping costs. Wang et al. [3] also support Yao et al.'s original results by studying a special case. It is interesting to note that, while the arguments of Yao et al. [5] and Wang et al. [3] focus on the validity of van der Vlist et al.'s cost assumptions, some of the results in van der Vlist et al. do not rely on these assumptions. To clarify the causes of the disagreement, we relax all the extra assumptions in van der Vlist et al. [2], let the delivery cost be zero and revisit the original paper of Yao et al. [4]. In this note we discuss two important factors on the benefit of VMI: The lumpy demand at the suppliers and the shipment size constraint. Both are neglected in all these papers. Our study clearly shows how these two factors affect the optimal order quantities, and provides intuitive insights for the implementation of VMI.
We use the same models and notations in Yao et al. [4]. The supply chain consists of one buyer and one supplier. The buyer's order size is q and the supplier's order size is Q. From Crowston et al. [1] and Zipkin ([6], Theorem 5.3.2), it is obvious that the supplier's order pattern in van der Vlist et al.'s synchronized case is the optimal stationary pattern for both the supplier and the entire supply chain. In other words, the supplier's ordering pattern presumed in Yao et al. [4], which always keeps excessive q units on hand, is an inferior one. With the order pattern in the synchronized case of van der Vlist et al., the total cost function for the entire system in the non-VMI (NV) case is
⁎ Corresponding author. E-mail address: [email protected] (B. Huang). 0167-9236/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.dss.2009.08.003
S
R
TCNV ðQ ; qÞ = TCNV ðQ ; qÞ + TCNV ðqÞ
ð1Þ
where S
TCNV ðQ ; qÞ =
=
Cr Q q +H Q 2
q Cr Q +H H Q 2 2 R
TCNV ðqÞ =
q cr +h q 2
ð2Þ
ð3Þ ð4Þ
Q ≥ q > 0: TCSNV and TCRNV are respectively the cost functions of the supplier and of the buyer. Our cost function TCNV is different from the ones in [2] and [4] for the non-VMI case (e.g., Eq. (1) of [2] when T = 0 and Eq. (3) of [4].) The difference is due to the fact that both [2] and [4] overlook the effect of lumpy demand (i.e., the last term in (3)) faced
422
B. Huang, Z. Ye / Decision Support Systems 48 (2010) 421–425
by the supplier. Interestingly, this effect is included in the supplier's cost functions of their VMI models. 1 Note that the lumpy demand from the buyer does not affect the supplier's optimal decision of the order size Q unless the buyer's order size is sufficiently large (that is, when the condition Q ≥ q is concerned.) However, it does affect the supplier's cost performance. Thus the effect of lumpy demand at the supplier becomes an important factor for the decision of the shipment size in the VMI case. Go back to the non-VMI (NV) case, the optimal order quantities can be determined directly from the classic EOQ solution to (3) and (4). That is, rffiffiffiffiffiffiffi 2cr q⁎ = h
ð5Þ
rffiffiffiffiffiffiffiffi 2Cr : Q⁎ = H
ð6Þ
When q⁎ > Q⁎, the supplier's order size will be raised to q⁎ because, as shown in the proof of Proposition 1, it is never optimal for the supplier to have an order size Q < q. As a result, we have the optimal order quantities and the optimal cost functions for the nonVMI case: (All the proofs in this note are shown in Appendix.)
brackets of (10) with respect to Q and q separately. We can immediately obtain
Q⁎ =
rffiffiffiffiffiffiffiffi 2Cr H
sffiffiffiffiffiffiffiffiffiffiffiffiffi 2c′ r q⁎ = hH ′
when c ≤
The optimal cost function of the entire system without VMI is
⁎ = TCNV
rffiffiffiffiffiffi 8 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi cr > > > 2crh + 2CrH H > < 2h > > > > :
pffiffiffiffiffiffiffiffiffiffi C +1 2crh 2c
; when c ≤
; when c >
h C H
S
R
=
=
( )
q Cr Q q c′ r +H + +h Q 2 q 2 " # q Cr Q c′ r +H + + ðh HÞ Q 2 q 2
ð7Þ
h C: H
ð8Þ ð9Þ
ð10Þ
where Q ≥ q > 0. We define TCSVMI and TCRVMI as the total costs at the supplier's side and at the buyer's side, respectively. Within the first square bracket of (10) we have a classic EOQ cost which depends only on Q. The second square bracket of (10) also contains a classic EOQ cost but depends only on q. When the decision making is centralized in the VMI case, the optimal order quantities and the optimal cost can be obtained by optimizing the two EOQ costs in the square
1
hH C, H
which is
⁎ = QVMI ⁎ whenever h ≤ H, but they do not solve the problem of the qVMI optimal order quantities in the more general situation of {q⁎ > Q⁎}, nor ⁎ , q⁎ do they provide exact expressions of {QVMI VMI } when h ≤ H. We qffiffiffiffiffiffiffi
2Cr find Q = q = is not an optimal solution to TCVMI whenever c′ >
hH C H
H
(including the case of h ≤ H.) In fact, we have the following
proposition: ⁎ and Proposition 2. In the VMI case, the optimal order quantities QVMI ⁎ qVMI under the constraint Q ≥ q are
⁎ ; qVMI ⁎ QVMI =
8 > > > > > > > > > > <
0rffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi1 2c′ r A @ 2Cr; H hH
; when c ≤
hH C H
; when c′ >
hH C: H
′
> > 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 > > ′ ′ > > > > @ 2ðc + CÞr; 2ðc + CÞr A > > : h h
The optimal cost function under VMIis
When VMI is implemented, the buyer's order cost c is replaced by c′, where c′ ≤ c as assumed in Yao et al. [4]. The total cost function (1) can therefore be rewritten as TCVMI ðQ; qÞ = TCVMI ðQ ; qÞ + TCVMI ðqÞ
′
A tricky case occurs when c >
equivalent to q⁎ > Q⁎. van der Vlist et al. [2] indicates that we can let
rffiffiffiffiffiffiffiffi) 2Cr ⁎ = max qNV ⁎ ; Q NV H
ð12Þ
hH C. H
Proposition 1. In the non-VMI (NV) case, the optimal order quantities ⁎ ⁎ ⁎ Q⁎ NV and qNV under the condition qNV ≤ Q NV are rffiffiffiffiffiffiffi 2cr ⁎ = qNV h (
ð11Þ
Strictly speaking, the buyer's cost function TCRNV should also be modified if the end demand is discrete. Such modification, however, will not affect our main results.
⁎ = TCVMI
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi > > 2c′ rðh HÞ + 2CrH > < > > > :
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðc′ + CÞrh
; when c ′ ≤
hH C H
ð13Þ ;
when c ′
>
h−H C: H
3. The impact of VMI The supplier's cost function TCS provides a quick answer to the impact of lumpy demand. The last term of (3) is independent of Q and represents the impact of the lumpy demand in the non-VMI case. The total cost function of the VMI case has the same structure. It is then easy to obtain the following property which holds in both VMI and non-VMI cases. Corollary 1. The cost at the supplier's side is non-increasing in the buyer's order size. According to Corollary 1, the centralized decision maker in the VMI case may p want toffi increase the buyer's order size from its non-VMI ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi solution ( 2c′ r = h) in order to benefit from the cost reduction at the supplier. In fact, the buyer's order size shall be set to a point where the marginal increase of the buyer's cost equals to the marginal saving of the supplier's cost if the q ≤ Q constraint is not concerned. That is, q should satisfy ′ d cr h H ∂ R S TCVMI = 2 + = = TCVMI dq 2 2 ∂q q
ð14Þ
B. Huang, Z. Ye / Decision Support Systems 48 (2010) 421–425
⁎ and c′ ≤ when q = qVMI ⁎ qVMI
hH C. H
423
As a result, we have
sffiffiffiffiffiffiffiffiffiffiffiffiffi ′ 2c r = hH
which is exactly the same result as (12). To the contrary, the non-optimal ordering pattern in Yao et al. [4] causes the following cost function for the supplier: Yao
TCS ðQ; qÞ =
q Cr Q +H +H Q 2 2
ð15Þ
which owns the opposite property: The total cost at the supplier's side increases with the buyer's order size q. One of the reasons is the excessive inventory the supplier keeps. The larger is q, the higher is the supplier's inventory cost. Thus in the VMI case, it would be better off for the centralized decision maker to lower the buyer's order size from the classic EOQ solution in order to reduce the inventory cost at the supplier. We now have the following proposition which corresponds to Proposition 1 of Yao et al. [4]. hH
Proposition 3. When c′ + C > c ≥ c′ > c, the optimal shipment h ⁎ from the supplier to the buyer increases in the VMI case. That is, size qVMI ⁎ > qNV ⁎ Otherwise, qVMI ⁎ is no greater than qNV ⁎ . qVMI Note that we do not use the replenish frequency to describe the buyer's order behaviour in Proposition 3. Instead, we use the ⁎ and qVMI ⁎ to demonstrate the real frequency of shipment sizes qNV the buyer's orders. When the shipment size is smaller, the buyer places more orders within a fixed duration. From Corollary 1 and Proposition 3, we have a clear picture about the change of the shipment size when VMI is implemented. There are three driving forces which affect the optimal shipment size in the VMI case: First, the lumpy demand at the supplier. When the supplier's order pattern is optimal as in van der Vlist et al.'s synchronized case, the centralized decision maker under VMI tends to increase the shipment size q in order to bring more cost reduction at the supplier's side. Second, the buyer's ordering cost, which is usually lower under VMI and forms the opposite force to drag down the shipment size. hH When the buyer's order cost c′ is not low enough (i.e., c′ > c,) the h ⁎ ⁎ optimal shipment size qVMI is larger than qNV (in the case where the constraint q ≤ Q is not a concern.) Otherwise, the centralized decision maker can take advantage of the low order cost at the buyer and reduce the shipment size. Third, the shipment size limitation q ≤ Q. As shown in the proof, the condition c′ + C > c in Proposition 3 is related to the constraint q ≤ Q. When the supplier's order cost C is so small that C ≤ c − c′, the constraint become a concern which may result in a upper bound for the shipment size q. To demonstrate our results, we use the same numerical example from Figure 3 of Yao et al. [4]. In the figure of Yao et al., the supplier's total holding cost is always higher under VMI while the buyer always enjoys the benefit of reduced inventory. Yao et al. suggest that “VMI shifts inventory from the buyer to the supplier to take advantage of the lower carrying charge at the supplier's site.” However, this reason does not explain why the buyer's inventory holding cost is reduced the most (by shifting more inventory to the supplier?) when the supplier's holding cost rate is the highest (d = H/h = 1) in their figure. Fig. 1 shows the revised version on the effect of the holding cost ⁎ increases with ratio d. In this case we find the buyer's order size qVMI d because it becomes more expensive to hold inventory at the supplier when d is high. When d is small (< 0.5), which yield a higher hH ⁎ is smaller than qNV ⁎ , threshold value of c in Proposition 3, qVMI h and the supplier has a higher inventory under VMI. On the other ⁎ is larger than qNV ⁎ when d is large hand, the shipment size qVMI
Fig. 1. Inventory holding cost savings under VMI (when g = C/c = 1 and g′ = C/c′ = 2).
(> 0.5,) meaning more inventory is shifted from the supplier to the buyer. The increase of the buyer's inventory holding cost, however, has a limit due to the constraint q ≤ Q. In addition, the optimal order ⁎ and qVMI ⁎ are equal when d ≥ 2/3, which implies a crosssizes QVMI docking strategy (with 100% inventory holding cost saving) for the supplier if its inventory holding cost rates is not much lower than the buyer's. One more noteworthy issue is the supplier's order size Q. Yao et al. [4] and van der Vlist et al. [2] claim that the supplier's optimal order size remains the same when VMI is introduced. We find that, as long as the shipment size constraint q ≤ Q is concerned, the optimal order qffiffiffiffiffiffiffi size of the supplier may be greater than its classic EOQ solution 2Cr . H
4. Conclusions In this paper, we have shown that the original models in Yao et al. [4] are problematic in two ways: First, the supplier in their models adopts an order pattern which causes excessive inventory all the time. The amount of excessive inventory is proportional to the buyer's order size q. Second, while the lumpy demand at the supplier is modeled in the VMI case, it is overlooked in the non-VMI case. van der Vlist et al. [2] identify the first drawback, but fail to find the second one. In addition, van der Vlist et al. do not provide clear answers to the situation where the shipment size constraint Q ≥ q is concerned. While Yao et al. [5] and Wang et al. [3] focus their responses on the validity of van der Vlist et al.'s cost assumptions, we find that these assumptions are not required for a different result from Yao et al.'s [4]. We then revisit the original paper of Yao et al. [4] to identify the dynamics of the order behaviour when VMI is implemented. By ⁎ > qNV ⁎ or qVMI ⁎ ≤ qNV ⁎ , we resolve showing the conditions where qVMI the disagreement between Yao et al. [4] and van der Vlist et al. [2] without extra assumptions and special cases. We also provide an intuitive view of the cost functions so that most optimal quantities can be derived from the properties of the EOQ model (e.g., convexity), without going through the first and second order conditions. Appendix A Proof of Proposition 1. From (5) and (6), q⁎ > Q⁎ implies c > (h/H)C, and the proposition follows. Note that it is never optimal for the ⁎ < qNV ⁎ . If Q NV ⁎ < qNV ⁎ , supplier (and the supply chain) to have Q NV the supplier's order frequency is higher than the buyers'. Thus the supplier's inventory will accumulates over time before the arrival of the buyer's next order. Instead, the supplier can increase its order size
424
B. Huang, Z. Ye / Decision Support Systems 48 (2010) 421–425
⁎ so that the total ordering cost is lower (due to a larger order,) to qNV and the total holding cost is zero (since no inventory is accumulated.) As a result, we have
⁎ = Q NV
8 > > > > > <
rffiffiffiffiffiffiffiffi 2Cr H
; when c ≤
rffiffiffiffiffiffiffi > > > 2cr > > ⁎ = : qNV h
h C H
6. where Q⁎ and qe⁎ represent respectively the optimal solution of TC1 and the optimal order quantity under VMI if Q = q. They can be obtained from (11) and (18). In addition, we have
′
c≥
hH C⇐⇒ H
ð16Þ
′
ðc + CÞH≥Ch
⇐⇒ c′ h≥ðc′ + CÞh ðc′ + CÞH
h ; when c > C: H
′
c hH
⇐⇒ ⇐⇒
≥
′
c +C h
q⁎ ≥ qe⁎
The optimal cost function without VMI can then be obtained from ⁎ and Q = Q NV ⁎ . (1) by letting q = qNV Proof of Proposition 2. We first find the optimal solution of TCVMI under the condition Q = q. When Q = q, TCVMI(Q, q) in (9) becomes q q Cr c′ r ðC + c′ Þr + +h = +h q q 2 q 2
TCVMI ðq; qÞ =
ð17Þ
which is a classic EOQ cost with a ordering cost (C + c′) per order and a holding cost rate h. Let qe⁎ denote the optimal solution of (17). In other words, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðC + c′ Þr qe⁎ = : h
7. where q⁎ is the optimal solution of TC2 and can be obtained from hH C⇐⇒q⁎≥qe⁎≥Q ⁎ when h > H. (12). As a result, we have c′ ≥ H ⁎ ⁎ The inequalities among q , qe and Q⁎ together with the convexity of TC1 and TC2 imply that, for any pair of order quantities (Q, q) and Q ≥ q, we have: • If Q ≥ q⁎, TCVMI ðQ ; qÞ = TC1 ðQ Þ + TC2 ðqÞ
ð19Þ
≥TC 1 ðq⁎Þ + TC 2 ðq⁎Þ = TC VMI ðq⁎; q⁎Þ
ð18Þ ≥TCVMI ðqe ⁎; qe ⁎Þ
The proof of Proposition 1 has shown that any (Q , q) with Q < q is never the optimal solution to TCVMI because we can always set Q to q and get a lower total cost. Therefore, with the condition Q ≥ q for TCVMI, the following proof includes three parts: hH
1. c′ < C. The condition implies that h > H because c′ cannot be H negative. In this case, we have Q⁎ ≥ q⁎ from (11) and (12). The constraint Q ≥ q is not a concern for TC VMI (Q ⁎ , q ⁎ ). Thus ( ) qffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi 2Cr 2c′ r ⁎ ; qVMI ⁎ g = fQ ⁎; q⁎g = fQVMI ; H
hH
• The inequality (19) comes from the fact that Q ≥q⁎ ≥Q⁎ and TC1 is convex in Q. In addition, q⁎ is the optimal solution for TC2. The inequality (20) results from the optimality of (qe⁎, qe⁎) for TCVMI (Q, q) when Q =q. • If Q < q⁎, TCVMI ðQ ; qÞ = TC1 ðQÞ + TC2 ðqÞ ≥TC1 ðQ Þ + TC2 ðQÞ
hH
3. c′ ≥ C and h > H. Note that we can re-write the cost function H TCVMI as
ð21Þ
= TCVMI ðQ ; Q Þ ≥TCVMI ðqe ⁎; qe ⁎Þ
hH
C and h ≤ H. Because h − H is negative, it is easy to see 2. c′ ≥ H from (10) that TCVMI is decreasing in q. That is, for any q ≤ Q, we have TCVMI(Q, q) ≥ TCVMI(Q,Q) ≥ TCVMI(qe⁎, qe⁎). As a result, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ⁎ and qVMI ⁎ . qe = 2ðc′ + CÞr = h is the optimal solution of QVMI
ð20Þ
• The inequality (21) comes from the fact that q⁎ ≥ Q ≥ q and TC2 is convex in q. After all, we have TCVMI(Q, q) ≥ TCVMI(qe⁎, qe⁎) for any Q ≥ q. Thus hH (qe⁎, qe⁎) is the optimal solution for TCVMI(Q, q) when c′ ≥ C and H
h > H. ⁎ can be obtained by letting Q = QVMI ⁎ and The optimal cost TCVMI ⁎ in (10). This concludes the proof. q = qVMI
TCVMI ðQ ; qÞ = TC1 ðQ Þ + TC2 ðqÞ 4. where TC 1 ðQ Þ =
Cr Q
TC 2 ðqÞ =
cr q
′
+H
Proof of Proposition 3. From Proposition 1 and Proposition 2, it is hH easy to check that c′ + C > c ≥ c′ > c is a necessary condition for h ⁎ > qNV ⁎ . Now we need to prove its sufficiency. We can compare qVMI
Q 2
q + ðh HÞ
⁎ and qVMI ⁎ in Proposition 1 and 2. There are only two cases in qNV ⁎ > qNV ⁎ : which qVMI
2
5. We now show the following equivalence when h > H: ′
c≥
hH C⇐⇒ H ⇐⇒ ⇐⇒
′
ðc + CÞH≥Ch ′
c +C h
C H
≥
qe⁎ ≥ Q ⁎
hH C H
• c′ ≤
and c′ >
hH c. h
Note that the first condition implies
h > H (because c′ cannot be negative) and then have C > ′
c +C>
H c h
and
hH H c+ c=c h h
hH H
C>
hH c. h
We
B. Huang, Z. Ye / Decision Support Systems 48 (2010) 421–425
• c′ >
hH C H
and c′ + C > c. In this case, we also have
hH ′ ðc c Þ H hH ′ hH ′ c > c ⇒c + H H h ′ hH c ⇒ c > H H hH c ⇒c′ > h ′
c >
Combining these two cases, we have the necessary and sufficient hH ⁎ > qNV ⁎ . c for qVMI
condition c′ + C > c ≥ c′ >
h
425
References [1] W.B. Crowston, M. Wagner, J.F. Williams, Economic lot size determination in multistage assembly systems, Management Science 19 (1973) 517–527. [2] P. van der Vlist, R. Kuik, B. Verheijen, Note on supply chain integration in vendormanaged inventory, Decision Support Systems 44 (2007) 360–365. [3] W. Wang, H. Wee, J. Tsao, Revisiting the note on supply chain integration in vendormanaged inventory, Decision Support Systems 48 (2010) 419–420. [4] Y. Yao, P.T. Evers, M.E. Dresner, Supply chain integration in vendor-managed inventory, Decision Support Systems 43 (2007) 663–674. [5] Y. Yao, P.T. Evers, M.E. Dresner, Response to “Note on supply chain integration in vendor-managed inventory”, Decision Support Systems 44 (2007) 366–367. [6] P.H. Zipkin, Foundations of Inventory Management, McGraw Hill, 2000.
Contents lists available at ScienceDirect
Decision Support Systems j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / d s s
The effects of lumpy demand and shipment size constraint: A response to “Revisit the note on supply chain integration in vendor-managed inventory” Boray Huang ⁎, Zhisheng Ye Department of Industrial and Systems Engineering, National University of Singapore, 117576, Singapore, Singapore
a r t i c l e
i n f o
Available online 1 September 2009 Keywords: Vendor management inventory Supply chain integration Lumpy demand
a b s t r a c t This paper responds to a comment by Wang et al. [3] regarding the disagreement between Yao et al. [4] and van der Vlist et al. [2] on the impact of vender-managed-inventory (VMI). We explore the factors which affect the shipment size from the vendor to the buyer and identify the conditions where the shipment size will increase/decrease under VMI. A numerical example also shows when and how the inventory shifts between the supplier and the buyer. © 2009 Elsevier B.V. All rights reserved.
1. Introduction
2. The cost functions and the optimal solutions
Wang et al. [3] try to resolve a disagreement between Yao et al. [4] and van der Vlist et al. [2] regarding the impact of vendor-managedinventory (VMI) on a supply chain. Yao et al. [4] use a two-tier EOQ model to investigate the costs and order behaviours of the supply chain members in both VMI and non-VMI scenarios. They find some interesting analytical results and identify the distribution of the VMI benefit between the supplier and the buyer. van der Vlist et al. [2] argue that Yao et al. overstate the inventory needed at the supplier, and extend Yao et al.'s model to incorporate shipping costs. The findings in Yao et al. [4] and van der Vlist et al. [2] are different in many ways. For example, Yao et al. assert that the buyer's order size shall decrease under VMI, but van der Vlist et al. have an opposite result. In their response [5], Yao et al. argue that van der Vlist et al. [2] make problematic assumptions on the shipping costs. Wang et al. [3] also support Yao et al.'s original results by studying a special case. It is interesting to note that, while the arguments of Yao et al. [5] and Wang et al. [3] focus on the validity of van der Vlist et al.'s cost assumptions, some of the results in van der Vlist et al. do not rely on these assumptions. To clarify the causes of the disagreement, we relax all the extra assumptions in van der Vlist et al. [2], let the delivery cost be zero and revisit the original paper of Yao et al. [4]. In this note we discuss two important factors on the benefit of VMI: The lumpy demand at the suppliers and the shipment size constraint. Both are neglected in all these papers. Our study clearly shows how these two factors affect the optimal order quantities, and provides intuitive insights for the implementation of VMI.
We use the same models and notations in Yao et al. [4]. The supply chain consists of one buyer and one supplier. The buyer's order size is q and the supplier's order size is Q. From Crowston et al. [1] and Zipkin ([6], Theorem 5.3.2), it is obvious that the supplier's order pattern in van der Vlist et al.'s synchronized case is the optimal stationary pattern for both the supplier and the entire supply chain. In other words, the supplier's ordering pattern presumed in Yao et al. [4], which always keeps excessive q units on hand, is an inferior one. With the order pattern in the synchronized case of van der Vlist et al., the total cost function for the entire system in the non-VMI (NV) case is
⁎ Corresponding author. E-mail address: [email protected] (B. Huang). 0167-9236/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.dss.2009.08.003
S
R
TCNV ðQ ; qÞ = TCNV ðQ ; qÞ + TCNV ðqÞ
ð1Þ
where S
TCNV ðQ ; qÞ =
=
Cr Q q +H Q 2
q Cr Q +H H Q 2 2 R
TCNV ðqÞ =
q cr +h q 2
ð2Þ
ð3Þ ð4Þ
Q ≥ q > 0: TCSNV and TCRNV are respectively the cost functions of the supplier and of the buyer. Our cost function TCNV is different from the ones in [2] and [4] for the non-VMI case (e.g., Eq. (1) of [2] when T = 0 and Eq. (3) of [4].) The difference is due to the fact that both [2] and [4] overlook the effect of lumpy demand (i.e., the last term in (3)) faced
422
B. Huang, Z. Ye / Decision Support Systems 48 (2010) 421–425
by the supplier. Interestingly, this effect is included in the supplier's cost functions of their VMI models. 1 Note that the lumpy demand from the buyer does not affect the supplier's optimal decision of the order size Q unless the buyer's order size is sufficiently large (that is, when the condition Q ≥ q is concerned.) However, it does affect the supplier's cost performance. Thus the effect of lumpy demand at the supplier becomes an important factor for the decision of the shipment size in the VMI case. Go back to the non-VMI (NV) case, the optimal order quantities can be determined directly from the classic EOQ solution to (3) and (4). That is, rffiffiffiffiffiffiffi 2cr q⁎ = h
ð5Þ
rffiffiffiffiffiffiffiffi 2Cr : Q⁎ = H
ð6Þ
When q⁎ > Q⁎, the supplier's order size will be raised to q⁎ because, as shown in the proof of Proposition 1, it is never optimal for the supplier to have an order size Q < q. As a result, we have the optimal order quantities and the optimal cost functions for the nonVMI case: (All the proofs in this note are shown in Appendix.)
brackets of (10) with respect to Q and q separately. We can immediately obtain
Q⁎ =
rffiffiffiffiffiffiffiffi 2Cr H
sffiffiffiffiffiffiffiffiffiffiffiffiffi 2c′ r q⁎ = hH ′
when c ≤
The optimal cost function of the entire system without VMI is
⁎ = TCNV
rffiffiffiffiffiffi 8 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi cr > > > 2crh + 2CrH H > < 2h > > > > :
pffiffiffiffiffiffiffiffiffiffi C +1 2crh 2c
; when c ≤
; when c >
h C H
S
R
=
=
( )
q Cr Q q c′ r +H + +h Q 2 q 2 " # q Cr Q c′ r +H + + ðh HÞ Q 2 q 2
ð7Þ
h C: H
ð8Þ ð9Þ
ð10Þ
where Q ≥ q > 0. We define TCSVMI and TCRVMI as the total costs at the supplier's side and at the buyer's side, respectively. Within the first square bracket of (10) we have a classic EOQ cost which depends only on Q. The second square bracket of (10) also contains a classic EOQ cost but depends only on q. When the decision making is centralized in the VMI case, the optimal order quantities and the optimal cost can be obtained by optimizing the two EOQ costs in the square
1
hH C, H
which is
⁎ = QVMI ⁎ whenever h ≤ H, but they do not solve the problem of the qVMI optimal order quantities in the more general situation of {q⁎ > Q⁎}, nor ⁎ , q⁎ do they provide exact expressions of {QVMI VMI } when h ≤ H. We qffiffiffiffiffiffiffi
2Cr find Q = q = is not an optimal solution to TCVMI whenever c′ >
hH C H
H
(including the case of h ≤ H.) In fact, we have the following
proposition: ⁎ and Proposition 2. In the VMI case, the optimal order quantities QVMI ⁎ qVMI under the constraint Q ≥ q are
⁎ ; qVMI ⁎ QVMI =
8 > > > > > > > > > > <
0rffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi1 2c′ r A @ 2Cr; H hH
; when c ≤
hH C H
; when c′ >
hH C: H
′
> > 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 > > ′ ′ > > > > @ 2ðc + CÞr; 2ðc + CÞr A > > : h h
The optimal cost function under VMIis
When VMI is implemented, the buyer's order cost c is replaced by c′, where c′ ≤ c as assumed in Yao et al. [4]. The total cost function (1) can therefore be rewritten as TCVMI ðQ; qÞ = TCVMI ðQ ; qÞ + TCVMI ðqÞ
′
A tricky case occurs when c >
equivalent to q⁎ > Q⁎. van der Vlist et al. [2] indicates that we can let
rffiffiffiffiffiffiffiffi) 2Cr ⁎ = max qNV ⁎ ; Q NV H
ð12Þ
hH C. H
Proposition 1. In the non-VMI (NV) case, the optimal order quantities ⁎ ⁎ ⁎ Q⁎ NV and qNV under the condition qNV ≤ Q NV are rffiffiffiffiffiffiffi 2cr ⁎ = qNV h (
ð11Þ
Strictly speaking, the buyer's cost function TCRNV should also be modified if the end demand is discrete. Such modification, however, will not affect our main results.
⁎ = TCVMI
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi > > 2c′ rðh HÞ + 2CrH > < > > > :
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðc′ + CÞrh
; when c ′ ≤
hH C H
ð13Þ ;
when c ′
>
h−H C: H
3. The impact of VMI The supplier's cost function TCS provides a quick answer to the impact of lumpy demand. The last term of (3) is independent of Q and represents the impact of the lumpy demand in the non-VMI case. The total cost function of the VMI case has the same structure. It is then easy to obtain the following property which holds in both VMI and non-VMI cases. Corollary 1. The cost at the supplier's side is non-increasing in the buyer's order size. According to Corollary 1, the centralized decision maker in the VMI case may p want toffi increase the buyer's order size from its non-VMI ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi solution ( 2c′ r = h) in order to benefit from the cost reduction at the supplier. In fact, the buyer's order size shall be set to a point where the marginal increase of the buyer's cost equals to the marginal saving of the supplier's cost if the q ≤ Q constraint is not concerned. That is, q should satisfy ′ d cr h H ∂ R S TCVMI = 2 + = = TCVMI dq 2 2 ∂q q
ð14Þ
B. Huang, Z. Ye / Decision Support Systems 48 (2010) 421–425
⁎ and c′ ≤ when q = qVMI ⁎ qVMI
hH C. H
423
As a result, we have
sffiffiffiffiffiffiffiffiffiffiffiffiffi ′ 2c r = hH
which is exactly the same result as (12). To the contrary, the non-optimal ordering pattern in Yao et al. [4] causes the following cost function for the supplier: Yao
TCS ðQ; qÞ =
q Cr Q +H +H Q 2 2
ð15Þ
which owns the opposite property: The total cost at the supplier's side increases with the buyer's order size q. One of the reasons is the excessive inventory the supplier keeps. The larger is q, the higher is the supplier's inventory cost. Thus in the VMI case, it would be better off for the centralized decision maker to lower the buyer's order size from the classic EOQ solution in order to reduce the inventory cost at the supplier. We now have the following proposition which corresponds to Proposition 1 of Yao et al. [4]. hH
Proposition 3. When c′ + C > c ≥ c′ > c, the optimal shipment h ⁎ from the supplier to the buyer increases in the VMI case. That is, size qVMI ⁎ > qNV ⁎ Otherwise, qVMI ⁎ is no greater than qNV ⁎ . qVMI Note that we do not use the replenish frequency to describe the buyer's order behaviour in Proposition 3. Instead, we use the ⁎ and qVMI ⁎ to demonstrate the real frequency of shipment sizes qNV the buyer's orders. When the shipment size is smaller, the buyer places more orders within a fixed duration. From Corollary 1 and Proposition 3, we have a clear picture about the change of the shipment size when VMI is implemented. There are three driving forces which affect the optimal shipment size in the VMI case: First, the lumpy demand at the supplier. When the supplier's order pattern is optimal as in van der Vlist et al.'s synchronized case, the centralized decision maker under VMI tends to increase the shipment size q in order to bring more cost reduction at the supplier's side. Second, the buyer's ordering cost, which is usually lower under VMI and forms the opposite force to drag down the shipment size. hH When the buyer's order cost c′ is not low enough (i.e., c′ > c,) the h ⁎ ⁎ optimal shipment size qVMI is larger than qNV (in the case where the constraint q ≤ Q is not a concern.) Otherwise, the centralized decision maker can take advantage of the low order cost at the buyer and reduce the shipment size. Third, the shipment size limitation q ≤ Q. As shown in the proof, the condition c′ + C > c in Proposition 3 is related to the constraint q ≤ Q. When the supplier's order cost C is so small that C ≤ c − c′, the constraint become a concern which may result in a upper bound for the shipment size q. To demonstrate our results, we use the same numerical example from Figure 3 of Yao et al. [4]. In the figure of Yao et al., the supplier's total holding cost is always higher under VMI while the buyer always enjoys the benefit of reduced inventory. Yao et al. suggest that “VMI shifts inventory from the buyer to the supplier to take advantage of the lower carrying charge at the supplier's site.” However, this reason does not explain why the buyer's inventory holding cost is reduced the most (by shifting more inventory to the supplier?) when the supplier's holding cost rate is the highest (d = H/h = 1) in their figure. Fig. 1 shows the revised version on the effect of the holding cost ⁎ increases with ratio d. In this case we find the buyer's order size qVMI d because it becomes more expensive to hold inventory at the supplier when d is high. When d is small (< 0.5), which yield a higher hH ⁎ is smaller than qNV ⁎ , threshold value of c in Proposition 3, qVMI h and the supplier has a higher inventory under VMI. On the other ⁎ is larger than qNV ⁎ when d is large hand, the shipment size qVMI
Fig. 1. Inventory holding cost savings under VMI (when g = C/c = 1 and g′ = C/c′ = 2).
(> 0.5,) meaning more inventory is shifted from the supplier to the buyer. The increase of the buyer's inventory holding cost, however, has a limit due to the constraint q ≤ Q. In addition, the optimal order ⁎ and qVMI ⁎ are equal when d ≥ 2/3, which implies a crosssizes QVMI docking strategy (with 100% inventory holding cost saving) for the supplier if its inventory holding cost rates is not much lower than the buyer's. One more noteworthy issue is the supplier's order size Q. Yao et al. [4] and van der Vlist et al. [2] claim that the supplier's optimal order size remains the same when VMI is introduced. We find that, as long as the shipment size constraint q ≤ Q is concerned, the optimal order qffiffiffiffiffiffiffi size of the supplier may be greater than its classic EOQ solution 2Cr . H
4. Conclusions In this paper, we have shown that the original models in Yao et al. [4] are problematic in two ways: First, the supplier in their models adopts an order pattern which causes excessive inventory all the time. The amount of excessive inventory is proportional to the buyer's order size q. Second, while the lumpy demand at the supplier is modeled in the VMI case, it is overlooked in the non-VMI case. van der Vlist et al. [2] identify the first drawback, but fail to find the second one. In addition, van der Vlist et al. do not provide clear answers to the situation where the shipment size constraint Q ≥ q is concerned. While Yao et al. [5] and Wang et al. [3] focus their responses on the validity of van der Vlist et al.'s cost assumptions, we find that these assumptions are not required for a different result from Yao et al.'s [4]. We then revisit the original paper of Yao et al. [4] to identify the dynamics of the order behaviour when VMI is implemented. By ⁎ > qNV ⁎ or qVMI ⁎ ≤ qNV ⁎ , we resolve showing the conditions where qVMI the disagreement between Yao et al. [4] and van der Vlist et al. [2] without extra assumptions and special cases. We also provide an intuitive view of the cost functions so that most optimal quantities can be derived from the properties of the EOQ model (e.g., convexity), without going through the first and second order conditions. Appendix A Proof of Proposition 1. From (5) and (6), q⁎ > Q⁎ implies c > (h/H)C, and the proposition follows. Note that it is never optimal for the ⁎ < qNV ⁎ . If Q NV ⁎ < qNV ⁎ , supplier (and the supply chain) to have Q NV the supplier's order frequency is higher than the buyers'. Thus the supplier's inventory will accumulates over time before the arrival of the buyer's next order. Instead, the supplier can increase its order size
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B. Huang, Z. Ye / Decision Support Systems 48 (2010) 421–425
⁎ so that the total ordering cost is lower (due to a larger order,) to qNV and the total holding cost is zero (since no inventory is accumulated.) As a result, we have
⁎ = Q NV
8 > > > > > <
rffiffiffiffiffiffiffiffi 2Cr H
; when c ≤
rffiffiffiffiffiffiffi > > > 2cr > > ⁎ = : qNV h
h C H
6. where Q⁎ and qe⁎ represent respectively the optimal solution of TC1 and the optimal order quantity under VMI if Q = q. They can be obtained from (11) and (18). In addition, we have
′
c≥
hH C⇐⇒ H
ð16Þ
′
ðc + CÞH≥Ch
⇐⇒ c′ h≥ðc′ + CÞh ðc′ + CÞH
h ; when c > C: H
′
c hH
⇐⇒ ⇐⇒
≥
′
c +C h
q⁎ ≥ qe⁎
The optimal cost function without VMI can then be obtained from ⁎ and Q = Q NV ⁎ . (1) by letting q = qNV Proof of Proposition 2. We first find the optimal solution of TCVMI under the condition Q = q. When Q = q, TCVMI(Q, q) in (9) becomes q q Cr c′ r ðC + c′ Þr + +h = +h q q 2 q 2
TCVMI ðq; qÞ =
ð17Þ
which is a classic EOQ cost with a ordering cost (C + c′) per order and a holding cost rate h. Let qe⁎ denote the optimal solution of (17). In other words, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðC + c′ Þr qe⁎ = : h
7. where q⁎ is the optimal solution of TC2 and can be obtained from hH C⇐⇒q⁎≥qe⁎≥Q ⁎ when h > H. (12). As a result, we have c′ ≥ H ⁎ ⁎ The inequalities among q , qe and Q⁎ together with the convexity of TC1 and TC2 imply that, for any pair of order quantities (Q, q) and Q ≥ q, we have: • If Q ≥ q⁎, TCVMI ðQ ; qÞ = TC1 ðQ Þ + TC2 ðqÞ
ð19Þ
≥TC 1 ðq⁎Þ + TC 2 ðq⁎Þ = TC VMI ðq⁎; q⁎Þ
ð18Þ ≥TCVMI ðqe ⁎; qe ⁎Þ
The proof of Proposition 1 has shown that any (Q , q) with Q < q is never the optimal solution to TCVMI because we can always set Q to q and get a lower total cost. Therefore, with the condition Q ≥ q for TCVMI, the following proof includes three parts: hH
1. c′ < C. The condition implies that h > H because c′ cannot be H negative. In this case, we have Q⁎ ≥ q⁎ from (11) and (12). The constraint Q ≥ q is not a concern for TC VMI (Q ⁎ , q ⁎ ). Thus ( ) qffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi 2Cr 2c′ r ⁎ ; qVMI ⁎ g = fQ ⁎; q⁎g = fQVMI ; H
hH
• The inequality (19) comes from the fact that Q ≥q⁎ ≥Q⁎ and TC1 is convex in Q. In addition, q⁎ is the optimal solution for TC2. The inequality (20) results from the optimality of (qe⁎, qe⁎) for TCVMI (Q, q) when Q =q. • If Q < q⁎, TCVMI ðQ ; qÞ = TC1 ðQÞ + TC2 ðqÞ ≥TC1 ðQ Þ + TC2 ðQÞ
hH
3. c′ ≥ C and h > H. Note that we can re-write the cost function H TCVMI as
ð21Þ
= TCVMI ðQ ; Q Þ ≥TCVMI ðqe ⁎; qe ⁎Þ
hH
C and h ≤ H. Because h − H is negative, it is easy to see 2. c′ ≥ H from (10) that TCVMI is decreasing in q. That is, for any q ≤ Q, we have TCVMI(Q, q) ≥ TCVMI(Q,Q) ≥ TCVMI(qe⁎, qe⁎). As a result, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ⁎ and qVMI ⁎ . qe = 2ðc′ + CÞr = h is the optimal solution of QVMI
ð20Þ
• The inequality (21) comes from the fact that q⁎ ≥ Q ≥ q and TC2 is convex in q. After all, we have TCVMI(Q, q) ≥ TCVMI(qe⁎, qe⁎) for any Q ≥ q. Thus hH (qe⁎, qe⁎) is the optimal solution for TCVMI(Q, q) when c′ ≥ C and H
h > H. ⁎ can be obtained by letting Q = QVMI ⁎ and The optimal cost TCVMI ⁎ in (10). This concludes the proof. q = qVMI
TCVMI ðQ ; qÞ = TC1 ðQ Þ + TC2 ðqÞ 4. where TC 1 ðQ Þ =
Cr Q
TC 2 ðqÞ =
cr q
′
+H
Proof of Proposition 3. From Proposition 1 and Proposition 2, it is hH easy to check that c′ + C > c ≥ c′ > c is a necessary condition for h ⁎ > qNV ⁎ . Now we need to prove its sufficiency. We can compare qVMI
Q 2
q + ðh HÞ
⁎ and qVMI ⁎ in Proposition 1 and 2. There are only two cases in qNV ⁎ > qNV ⁎ : which qVMI
2
5. We now show the following equivalence when h > H: ′
c≥
hH C⇐⇒ H ⇐⇒ ⇐⇒
′
ðc + CÞH≥Ch ′
c +C h
C H
≥
qe⁎ ≥ Q ⁎
hH C H
• c′ ≤
and c′ >
hH c. h
Note that the first condition implies
h > H (because c′ cannot be negative) and then have C > ′
c +C>
H c h
and
hH H c+ c=c h h
hH H
C>
hH c. h
We
B. Huang, Z. Ye / Decision Support Systems 48 (2010) 421–425
• c′ >
hH C H
and c′ + C > c. In this case, we also have
hH ′ ðc c Þ H hH ′ hH ′ c > c ⇒c + H H h ′ hH c ⇒ c > H H hH c ⇒c′ > h ′
c >
Combining these two cases, we have the necessary and sufficient hH ⁎ > qNV ⁎ . c for qVMI
condition c′ + C > c ≥ c′ >
h
425
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