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Newsvendor with multiple options of expediting
European Journal of Operational Research 226 (2013) 94–99
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Innovative Applications of O.R.
Newsvendor with multiple options of expediting Ke Fu a, Jiayan Xu a, Zhaowei Miao b,⇑ a b
Lingnan College, Sun Yat-sen University, Guangzhou 510275, China School of Management, Xiamen University, Xiamen 361005, China
a r t i c l e
i n f o
Article history: Received 24 February 2012 Accepted 26 October 2012 Available online 12 November 2012 Keywords: Inventory Newsvendor Procurement lead times Expediting Lead-time-dependent pricing
a b s t r a c t We consider a manufacturer facing single period inventory planning problem with uncertain demand and multiple options of expediting. The demand comes at a certain time in the future. The manufacturer may order the product in advance with a relatively low cost. She can order additional amount by expediting after the demand is realized. There are a number of expediting options, each of which corresponds to a certain delivery lead time and a unit procurement price. The unit procurement price is decreasing over delivery lead time. The selling price is also decreasing over time. In this paper, we assume that the manufacturer must deliver all products to the customer in a single shipment. The problem can be formulated as a profit maximization problem. We develop structural properties and show how the optimal solution can be identified efficiently. In addition, we compare our model with the classical newsvendor model and obtain a number of managerial insights. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In the traditional newsvendor model, a manufacturer or retailer procures a single product from her upstream supplier and resells the product to her downstream customers who have one-time uncertain demand. In general, due to the long procurement lead time and customer restriction on delivery lead time, the order of the manufacturer should be placed long ahead of the actual demand being revealed. If the actual demand exceeds the pre-stocked quantity of the manufacturer, unmet demand is lost. In reality, however, if a shortage occurs, some companies have options to place makeup orders by expediting. For example, it is not uncommon that the manufacturer may have other emergency sourcing options or even in-house rush production. The expediting options are quick and responsive, but in general they are much more expensive compared to early procurement. Therefore, the manufacturer should evaluate carefully the trade-off between cost and responsiveness when making inventory and procurement decisions. In this paper, we study the newsvendor problem with multiple options of expediting under the above environment. A manufacturer (newsvendor) procures a product using normal procedure before the demand is realized. After demand realization, if the demand exceeds the quantity of pre-stock products, the manufacturer can place a make-up order using expediting procedures to fully satisfy the demand. There may be a number of expediting options, each of which corresponds to a certain lead time and a unit price. For example, different expediting options correspond ⇑ Corresponding author. Tel.: +86 0592 2187067. E-mail address: [email protected] (Z. Miao). 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.10.039
to different transportation modes and a higher price is charged for a shorter procurement lead time. In addition, the price for selling the product depends on the delivery lead time, and a longer delivery lead time leads to a lower unit price, reflecting the fact that late delivery is penalized. The newsvendor problem is one of the classical problems in the literature on inventory management (Arrow et al., 1951; Silver et al., 1998). There is a rich literature on the newsvendor problem with a wide variety of research issues. We refer readers to Khouja (1999), Qin et al. (2011), Gallego and Moon (1993) and Petruzzi and Dada (1999) for thorough and excellent reviews. In the following we discuss some closely related papers. A number of research papers have addressed the expediting or multiple sourcing issues, such as Hong and Hayya (1992), Ramasesh et al. (1991) and Tomlin and Wang (2005). For multiple sourcing, a stream of researches focus on supplier selection, risk control and auctions (Ghodsypour and O’Brien, 2001; Treleven and Schweikhart, 1988; Tunca and Wu, 2009), other than inventory management. In the field of inventory management, most researches consider only one expediting option (dual sourcing), in either single-period, multi-period or continuous-time models. Closely related to our research, Khouja (1996) considers a newsboy problem with one emergency supply option where a proportion of customers satisfy their demand from an emergency supply source when there is a shortage. The author solves the problem under the objective of maximizing the expected profit or maximizing the probability of achieving a target profit. Fu et al. (2009) consider expediting in a single-period assemble-to-order (ATO) system with one expediting option. DeYong and Cattani (2012) study a special newsvendor problem where the newsvendor can expedite an
K. Fu et al. / European Journal of Operational Research 226 (2013) 94–99
additional order or cancel the previous order based on the updated demand forecast. Examples of researches on multi-period inventory control include Lawson and Porteus (2000), Durán et al. (2004), Huggins and Olsen (2010), Zhou and Chao (2010), Zhu (2012), to name a few. These researches analyze that certain inventory policies, such as the base-stock policy, are optimal or close-to-optimal. Chiang (2010) proposes a single-item continuous-review order expediting inventory policy, which can be considered as an extension of the ordinary (s,Q) policy. The buyer can expedite part of an outstanding order via a fast transportation mode at extra costs when his inventory falls below a certain level. Plambeck and Ward (2007) consider expediting option in a continuous-time ATO system. In addition, a number of recent papers study supply chain models with second production or procurement opportunities, including Cachon and Swinney (2009), Li et al. (2009), Jones et al. (2003) and Jones et al. (2001). This paper considers the lead-time-dependent procurement cost and product price. Previous researches based on this setting are relatively limited. Fang et al. (2008) consider the case of lead-time-dependent final product price and explore how the manufacturer can use a ‘‘Vendor Managed Consignment Inventory’’ scheme to manage the underlying risk and coordinate independent suppliers’ decisions on the production quantities of their components under demand uncertainty. Hsu et al. (2006) consider an ATO system with lead-time-dependent component and final product price without expediting. Chandra and Grabis (2008) study a single-stage variable lead-time inventory system with lead-time dependent procurement cost. They develop a cost-minimization model without expediting and do not consider the varying product prices. Hsu et al. (2007) address inventory decisions in an assemble-to-order system with lead-time-dependent final product pricing under the full shipment case and propose efficient solution procedures. They assume constant component purchasing prices, and thus do not consider the expediting issue, which is the main focus of our research. Our model differs from previous work in the following two aspects. First, our paper considers multiple options of expediting, while most of previous researches address only one option. Second, we consider decreasing procurement cost and product price. As will be seen later, the newsvendor problem with these two features reveals some new characteristics and insights. To the best of our knowledge, this has not been addressed in the literature. The rest of the paper is organized as follows. In Section 2, we discuss our problem and formulate it as a profit maximization model. In Section 3, we develop some structural properties and solve the optimization model. In Section 4, we compare our model with the traditional newsvendor problem and discuss a number of managerial insights. Section 5 concludes the paper and offers some future research directions.
2. The model We consider the problem under the following assumptions. A manufacturer produces or processes a single product from a supplier and resells the product to customers. Demand comes at a certain time in the future and the manufacturer may order some amount of product using normal procedure in advance. These products will arrive before the demand is revealed. When demand is realized, in case of shortage, the manufacturer will place makeup orders using expediting procedure to satisfy the entire demand. There are a number of expediting options, each of which corresponds to a lead time and a unit price. The unit price is lower with a longer lead time. Hence, the manufacturer wants to determine the quantity of the pre-stocked product to maximize her total expected profit.
95
Let D L li
c(L) P(L)
c0 b Q
stochastic demand, which is assumed to be realized at time 0; delivery lead time; ith lead time of expediting, i = 1, 2, . . ., n, we index the lead times such that l1 6 l2 6 6 ln, in addition, define l0 0 and ln+1 + 1; unit purchasing price under expediting if delivery lead time is L, we assume that c(L) is decreasing in L; unit price of final product if delivery lead time is L, we assume that P(L) is decreasing in L. In order to avoid trivial cases, we further assume P(L) P c(L) for all L; unit purchasing price of preseason ordering before time 0; unit salvage value, we further assume b < c0; pre-stocked quantity.
We assume here that the manufacturer must deliver products in a single shipment to satisfy the entire customer demand. The formulation consists of two stages. At stage 1, the manufacturer determines the pre-stocked quantity of the product before the actual demand is revealed. At stage 2 when the demand is realized, given the demand D and the pre-stocked quantity Q, the manufacturer will use the pre-stocked quantity to fully satisfy the demand exactly at time 0 if D 6 Q, i.e., the demand is not more than the prestocked quantity. If D > Q, however, the manufacturer must place make-up orders to procure additional products. When these additional products arrive, the manufacturer will then deliver all products in a single shipment to customer. First, we have the following property. Proposition 1. For the stage-two problem, if Q < D, i.e., the prestocked inventory is less than the demand, then there is an optimal procurement and shipment decision such that the manufacturer places a single make-up order at time 0 and makes a single full shipment, where the order lead time and delivery lead time are L⁄ 2 {l1, l2, . . . , ln}. Proof. If there are orders placed at any time t > 0, then we can modify this solution by moving all the procurement times to time 0 and keeping other decisions unchanged. It is easy to verify that the new solution will not increase the costs or decrease the revenue. Hence in the following we assume this property is satisfied, i.e., all orders are placed at time 0. If there is a single order that will arrive at a time li0 2 fl1 ; l2 ; . . . ; ln g, then it is easy to verify that there is an optimal shipment decision that satisfies L ¼ li0 . If there are multiple orders that will arrive at times li0 ; li1 ; . . . ; lim , where 1 6 i0 < i1 < < im 6 n, we can modify the solution by merging all orders into a single one with the longest lead time lim . It is clear that this modification will reduce the total procurement cost without affecting the revenue received from the customer. h Hereafter we shall consider the procurement and delivery decisions satisfying Proposition 1. Namely, the manufacturer will order the additional products at time 0 and deliver the final products as soon as the ordered products arrive. Therefore, the make-up delivery lead time can be restricted to {l1, l2, . . . , ln}. For convenience, define
Pi Pðli Þ;
0 6 i 6 n:
Clearly,
P0 P P1 P P Pn : In addition, the unit procurement cost under expediting is also decreasing in expediting lead time and we assume that the the unit
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cost for pre-stocked products is no more than the lowest unit cost under expediting, namely,
c1 P c2 P P cn P c0 : Note that cn P c0. This makes sense, as usually the manufacturer has enough time to prepare the pre-stocked products. For example, she may use cheap transportation modes and get price discount for advance order information (Sheopuri et al., 2010). Based on Proposition 1, we can express the second stage problem as follows. Our objective is to maximize the net revenue, denoted as R(Q,D). þ
0
þ
ðF Þ RðQ ; DÞ ¼ max fPi D ci ðD QÞ þ bðQ DÞ g: i2f0;1;...;ng
ð1Þ
Proof. To solve problem (F0 ), we need to compare the total profits associated with the two delivery lead times. Given the two lead times li and lj, the profit for each can be expressed respectively as (we ignore the initial ordering cost c0Q as this term is the same for both cases).
Pi D ci ðD Q Þþ and Pj D cj ðD Q Þþ :
Clearly, if D 6 Q, then L⁄ = 0. Suppose D > Q. Then we shall choose li instead of lj if li leads to a higher profit, namely,
Pi D ci ðD Q Þ P Pj D cj ðD QÞ;
ci cj Q ¼ r ij Q; ðci cj Þ ðPi Pj Þ
D6
ðFÞ max ED fc0 Q þ RðQ ; DÞg:
which completes the proof. h
ð2Þ
3. Structural properties In this section, we discuss a number of structural properties of the proposed problem that allow us to simplify problem (F). Proposition 2. If i 6 j, where i,j 2 {1, 2, . . . , n}, and Pi ci P Pj cj, then regardless of the pre-stocked quantity Q, there is an optimal solution that does not use lj as delivery time. Proof. Suppose there is a solution that uses lj as delivery time. Now we change lj to li and examine the profit changes. Let Cj be the total profit when choosing lj as delivery time. Let Ci be the total profit when choosing li as delivery time. Then for arbitrary Q and D, we have þ
þ
C j C i ¼ ½P j D cj ðD Q Þ c0 Q þ bðQ DÞ ½Pi D ci ðD Q Þ
þ
c0 Q þ bðQ DÞþ
ð6Þ
or
At stage one, our objective is to maximize the total expected profit under uncertain demand D. Q P0
ð5Þ
ð7Þ
Example 1. Suppose P(l1) = 4, P(l2) = 3, c(l1) = 3, c(l2) = 1, Q = 10, D = 11, then if we choose l1, total net profit is P(l1)D (D Q)+ c(l1) = 4(11) 1(3) = 41. If we choose l2, then the total net profit is P(l2)D (D Q)+c(l2) = 3(11) 1(1) = 32. Therefore, we should choose l1, which is also consistent with D 6 r1,2Q. But if Q = 0, then we should choose l2 since D > r1,2Q. The following proposition follows from the above proposition. Proposition 4. Given li, lj, lk, 0 < i < j < k 6 n, if rij P rjk then there is an optimal solution that does not use lj as delivery time. Proof. Consider two cases: (1) D 6 rijQ and (2) D > rijQ. (1) D 6 rijQ. From Proposition 3, li is better than lj. Thus, it is optimal not to use lj as delivery time. (2) D > rijQ. It is clear that D > rjkQ since rij P rjk, then lk is better than lj. Again it is optimal not to use lj as delivery time.
6 ½Pj D cj D ½Pi D ci D 6 0; where the first inequality holds because ci P cj, and the second inequality holds by the assumption Pi ci P Pj cj. h By Proposition 2, we can remove lj for any pair (li,lj) such that Pi ci P Pj cj. After removing all lead times that satisfy the above inequality, we will obtain a smaller set of lead times {li} such that Pi ci is increasing in i. Namely, we have
P 1 c1 < P 2 c2 < < P n cn :
ð3Þ
Hereafter, we shall assume the lead times already satisfy this condition. Without loss of generality, we still denote the set of lead times as {l0, l1, . . . , ln} where l0 0. It is intuitively true that for the second stage problem (F0 ), the choice of delivery lead time L depends on the pre-stocked quantity Q as well as the realized demand D. The following property states this relationship precisely and is a fundamental insight for us to simplify the problem significantly. Before describing the proposition, we define the following notation:
r i;j ¼
c i cj ; ðci cj Þ ðP i Pj Þ
1 6 i < j 6 n:
Therefore, for all D P 0, we will not choose lj as delivery time. h By Proposition 4, we can remove any lead time lj such that rij P rjk (0 < i < j < k 6 n). This property, along with the earlier discussions in this section, allows us to modify the problem to satisfy the following conditions without affecting its optimal solution. (a) Pi ci is strictly increasing over i, 1 6 i 6 Proposition 2):
n (by
P 1 c1 < P 2 c 2 < < P n c n : (b) ri,i+1 is strictly increasing over i, 1 < i 6 n 1 (by Proposition 4):
r 1;2 < r 2;3 < < rn1;n : For convenience, we define ri ri,i+1, i = 1, . . . , n 1. Based on the above discussions, we can summarize the solution for the second stage problem (F0 ). For convenience, we define r0 = 1 and rn +1.
ð4Þ
It is easy to see that ri,j P 1 because ci cj > Pi Pj due to inequalities (3). Clearly, given any two lead times L1 < L2, the choice between the two depends on both Q and D. Proposition 3. For the second stage problem (F0 ), for any two lead times li and lj, where li < lj, 1 6 i < j 6 n, if D 6 ri,jQ, then there is an optimal solution that does not choose lj as delivery lead time.
Proposition 5. For any pre-stocked quantity Q and demand realization D, the optimal solution for the second stage problem (F0 ) can be expressed as shown in Table 1: Based on Proposition 5, we can now express the expected profit function for the first stage problem.
Pf ðQ Þ ¼ ED fc0 Q þ RðQ ; DÞg Z Q ¼ c0 Q þ ½P0 x þ bðQ xÞdFðxÞ 0
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K. Fu et al. / European Journal of Operational Research 226 (2013) 94–99
þ
n1 Z X
r iþ1 Q
Z
¼ c0 Q þ ( n2 Z X þ Z
In general, the properties of the objective function Pf(Q) depend on the demand distribution function F(x). However, based on the structural properties, it is not difficult to obtain optimal solutions to the problem for a number of commonly used demand distributions including Uniform, Gamma, and Normal. To keep our discussions concise, we refer readers to an online technical report of this paper for a more detailed discussion about this problem. Next, we consider a special case where procurement costs and selling prices are both linear, i.e.,
Q
½P0 x þ bðQ xÞdFðxÞ
0 1
½Piþ1 x ciþ1 ðx Q ÞdFðxÞ
ri Q
i¼0
)
1
½Piþ1 x ciþ1 ðx Q ÞdFðxÞ
r iþ1 Q
þ
Z
þ1
½P n x cn ðx Q ÞdFðxÞ
r n1 Q
Z
cðlÞ ¼ u0 u1 l; PðlÞ ¼ v 0 v 1 l;
Q
¼ c0 Q þ ½P0 x þ bðQ xÞdFðxÞ 0 Z 1 þ ½P 1 x c1 ðx Q ÞdFðxÞ
where u0, u1,
Q
þ
n1 Z 1 X i¼1
½ðPiþ1 Pi Þx ðciþ1 ci Þðx Q ÞdFðxÞ:
0
xdFðxÞ ¼ yFðyÞ
Ry 0
FðxÞdx and
R1 y
xdFðxÞ ¼ E½D yFðyÞ þ
Ry 0
FðxÞdx.
Using the two formulae, we can simplify the above objective function Pf(Q) as follows.
Z Q Pf ðQ Þ ¼ c0 Q þ ðP0 bÞQFðQ Þ ðP0 bÞ FðxÞdx þ bQFðQÞ 0 Z Q þ ðP1 c1 Þ QFðQ Þ þ FðxÞdx þ c1 Q ½1 FðQÞ 0
Z n1 X ½ðP iþ1 P i Þ ðciþ1 ci Þ r i QFðr i Q Þ þ þ
ri Q
FðxÞdx
0
i¼1
þðciþ1 ci ÞQ½1 Fðr i QÞg þ ðPn C n ÞE½D: Differentiating Pf(Q), we obtain P 0f ðQ Þ ¼ c0 þ P0 FðQÞ þ P 0 Qf ðQ Þ ðP0 bÞFðQ Þ þ ðP1 c1 Þ½FðQÞ Qf ðQ Þ þ FðQÞ þ c1 ½1 FðQ Þ c1 Qf ðQÞ þ
n1 X
½ðP iþ1 Pi Þ ðciþ1 ci Þ r i Fðr i Q Þ r 2i Qf ðr i Q Þ þ r i Fðr i Q Þ
i¼1
þðciþ1 ci Þ½1 Fðr i QÞ ðciþ1 ci Þr i Qf ðr i QÞg ¼ c0 þ P0 Qf ðQÞ þ bFðQ Þ ðP 1 c1 ÞQf ðQÞ þ c1 ½1 FðQ Þ c1 Qf ðQÞ þ
n1 X
½ðci ciþ1 Þ ðPi P iþ1 Þr 2i Qf ðr i Q Þ
i¼1
ðci ciþ1 Þ½1 Fðr i QÞ þ ðci ciþ1 Þr i Qf ðr i QÞg:
Note that by definition,
ri ¼
v0, v1 are constants, and u1, v1 > 0. Then
ci ¼ u0 u1 li ;
i ¼ 1; 2; . . . ; n;
P i ¼ v 0 v 1 li ;
i ¼ 0; 1; . . . ; n:
Thus, without loss of generality, we can define analogously to (4)
ri Q
By integration by parts, it is straightforward to show that Ry
n1 X ðci ciþ1 ÞFðr i Q Þ: i¼1
½Piþ1 x ciþ1 ðx Q ÞdFðxÞ
ri Q
i¼0
P 0f ðQ Þ ¼ cn c0 þ ðP 0 P 1 ÞQf ðQ Þ ðc1 bÞFðQ Þ þ
ci ciþ1 : ðci ciþ1 Þ ðPi Piþ1 Þ
Hence the above expression can finally be simplified to the following:
Table 1 The optimal solution for (F0 ). i⁄
R(Q, D)
Range of D
0 1 2 .. . n1 n
P0D + b(Q D) P1D c1(D Q) P2D c2(D Q) .. . Pn1D cn1(D Q) PnD cn(D Q)
D6Q Q < D 6 r1Q r1Q < D 6 r2Q .. . rn2Q < D 6 rn1Q D > rn1Q
ðu0 u1 li Þ ðu0 u1 lj Þ ðu0 u1 li Þ ðu0 u1 lj Þ ½ðv 0 v 1 li Þ ðv 0 v 1 lj Þ u1 ðlj li Þ ¼ u1 ðlj li Þ v 1 ðlj li Þ u1 ¼ : u1 v 1
r i;j ¼
That is, with linear costs, r ij u1u1v 1 will be constants for all 1 6 i < j 6 n. Based on earlier discussions, it is easy to prove the following result. Proposition 6. For the second stage problem, if Q < D, i.e., the prestocked inventory is less than the demand, suppose the procurement cost and the selling price functions are linear as defined above, then (i) if u1 6 v1, there is an optimal solution that always uses l1 as delivery time; (ii) if u1 > v1, we consider three subcases: (a) if D < u1u1v 1 Q , then there is an optimal solution that uses l1 as delivery time if the additional procurement is necessary; (b) if D ¼ u1u1v 1 Q , any of li (i = 1, 2, . . . ,n) can be used as the optimal delivery time; (c) if D > u1u1v 1 Q , then there is an optimal solution that uses ln as the delivery time. Based on Proposition 6 and earlier discussions, the problem can be efficiently solved when the costs and prices are linear. Proposition 6 says that if the procurement cost and selling price functions are both linear, then depending on the relative quantity between the realized demand and the pre-stocked quantity, the manufacturer can choose either the shortest or the longest lead time for expediting. In order to explain how parameters u1 and v1 affect the optimal delivery time, we divide the total demand D (when D > Q) into two parts: the pre-stocked quantity Q and the second order quantity D Q. Then we consider two cases. In case (i), u1 6 v1 implies that Pi ci is decreasing over i. For the pre-stocked quantity Q, it is optimal to be delivered to customers as soon as possible because Pi is decreasing. For the second order quantity D Q, the manufacturer should also deliver this part as soon as possible since Pi ci is decreasing. Therefore, the delivery time l1 is optimal if additional procurement is required. In case (ii), u1 > v1 suggests that Pi ci is increasing over i. Therefore, the pre-stocked quantity Q should be delivered as soon as possible (l1) while the second order quantity D Q should be delivered at time ln. However, all products must be delivered to the customer in a single shipment here. As a result, there is a tradeoff between l1 and ln for the manufacturer. This tradeoff in return depends on the relative quantity of Q and D. If D is relatively large (this implies the second order quantity D Q is relatively large compared to the
K. Fu et al. / European Journal of Operational Research 226 (2013) 94–99
Remark 1. In this paper, we assume that the manufacturer must deliver total products in a single full shipment. For the case of partial shipments, where the manufacturer is allowed to deliver products in multiple shipments, it is quite clear that the model is simply a parameter-adjusted newsvendor problem in which the under-stocking cost is reduced from P0 c0 to (P0 c0) (P⁄ c⁄), where P⁄ c⁄ = max16i6n(Pi ci). Consequently, under this case, the manufacturer will pre-order less than the classical newsvendor case without expediting options. In addition, the optimal profit in the partial shipments case is higher than that in the classical newsvendor case.
4. Comparison and managerial insights In this section, we compare the full shipment case with the classical newsvendor model. Specifically, we compare the following two models.
140
optimal pre−order quantities
pre-stocked quantity Q), the manufacturer should choose ln as the delivery time because the second order quantity has a larger contribution to the total profit. If D is relatively small, then the manufacturer should choose l1 as the delivery time because the prestocked quantity has a larger contribution to the total profit. At the critical point when D ¼ u1u1v 1 Q , the pre-stocked and the second order quantities are equally important and any delivery time will result in the same profit.
130
classical newsvendor
120 110 100
full shipment 90 80 10
20
30
40
50
60
70
standard deviation σ Fig. 1. The impact of the demand standard deviation on the optimal order quantities.
950 900
classical newsvendor
850
profits
98
800
full shipment
750 700 650 10
20
30
40
50
60
70
standard deviation σ
Classical newsvendor. In this case, there is no second order opportunity and hence the demand may not be fully satisfied. We assume that there is no shortage cost for the unfulfilled demand. Newsvendor with multiple expediting options and a full shipment. In this case, there is a second order opportunity to fully satisfy the customer demand and a single full shipment is required by the customer. It can be easily verified that the optimal profit in the partial shipments case is higher than those in the full shipment case and classical newsvendor case. However, whether the full shipment is better than the classical newsvendor is not clear. Although the manufacturer in the full shipment case has a second order opportunity, she has to wait for the make-up procurement, if her prestocked product is not enough, to arrive and satisfy the entire demand in a single shipment. A delay in the final delivery means a loss in the revenue from the customer since the final product price is decreasing in the delivery time. Therefore, there is a probability that the profit from full shipment may not be as large as that from classical newsvendor. However, the relationship between the two cases depends on the specific distribution of the demand. The profit from classical newsvendor may be more sensitive to the demand variation because the newsvendor cannot procure for a second time if the demand exceeds his pre-stocked quantity. We will illustrate this relationship in the following numerical analysis. In this analysis, we are interested in the characteristics of optimal order quantities and profits in the two models. The basic parameters are set as follows. Let P0 = 20, c0 = 10, b = 5. There are three options for expediting: P1 = 19, c1 = 15; P2 = 18, c2 = 12; P3 = 15, c3 = 11. Assume that the demand follows Normal distribution N(l, r2), where l = 100, r = 30. Then we can get the optimal order quantities Qn = 112.9, Qf = 113.6 and the optimal profits pn = 836.4, pf = 800.8. Next, we examine how the coefficient of variation (COV) affects the optimal decisions and profits. We fix the demand mean and vary the standard deviation. As shown in Fig. 1, Qn increases in a 0 linear fashion. This is intuitive because PP00c > 0:5. However, Qf first b
Fig. 2. The impact of the demand standard deviation on the profits.
increases and then decreases in the standard deviation r. This phenomenon occurs due to the requirement of a full shipment and can be interpreted as follows. Consider an order decision Q and a demand realization D. If there is an overage of the pre-stocked products (D < Q), then as is the case with classical newsvendor, the manufacturer will satisfy all the demand and salvage the leftover products. If, however, there is a shortage of the pre-stocked products (D > Q), then the pre-stocked products may have less value compared to the classical newsvendor, as the manufacturer will have to wait for the make-up order to arrive and then deliver in a single full shipment. Thus, the demand variation may affect the pre-stocked quantity in opposite ways. When the variation is relatively small, the manufacturer may wish to pre-stock more to prevent shortages when the variation increases. When the variation is relatively large, the manufacturer may wish to pre-stock less when the variation increases because the value of the pre-stocked products diminishes with large variation. By similar arguments, we can explain the relationship between the pre-stocked quantities in the classical newsvendor and full shipment cases. Fig. 2 shows that the profits of the two cases all decrease in standard deviation, which is consistent with our common understanding that demand variation decreases the expected profit. Furthermore, the profit in the newsvendor case is more sensitive to standard deviation. This is due to the fact that it has only one order opportunity and cannot satisfy the exceeding demand with expediting options. If COV is small, the optimal profit in the newsvendor case is higher than that in the full shipment case. If COV is large, however, the newsvendor case is worse than the full shipment case in terms of profits. 5. Conclusions In this paper, we consider the newsvendor problem with multiple options of expediting. A manufacturer procures a product from a supplier and resells the product to customers who have uncertain
K. Fu et al. / European Journal of Operational Research 226 (2013) 94–99
demand. The manufacturer can order additional products using expediting procedures if necessary. The procurement cost and the product price are lead-time dependent. We analyze the optimal properties and solution procedures for this problem. We compare our model with the classical newsvendor and obtain a number of managerial insights. We now offer a few research directions. First, in our model, although we assume that no shortage is allowed, it would be interesting to study the case where both expediting and shortage are allowed. In addition, we may extend our current models to consider various issues such as capacity constraints in the newsvendor literature. Another direction is to study other types of inventory problems with multiple options of expediting. Our initial attempts show that some of these directions may require very different approaches. Acknowledgements The authors thank the Editor and an anonymous referee for their valuable comments and suggestions that have greatly improved this article. This research was supported in part by National Natural Science Foundation of China (71072090, 70701039, 71222105, 70802052), and Humanities and Social Science Fund of Ministry of Education of China (09YJC630237). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.ejor.2012.10.039. References Arrow, K., Harris, T., Marshak, J., 1951. Optimal inventory policy. Econometrica 19, 250–272. Cachon, G.P., Swinney, R., 2009. Purchasing, pricing, and quick response in the presence of strategic consumers. Management Science 55, 497–511. Chandra, C., Grabis, J., 2008. Inventory management with variable lead-time dependent procurement cost. Omega 36, 877–887. Chiang, C., 2010. An order expediting policy for continuous review systems with manufacturing lead-time. European Journal of Operational Research 203, 526– 531. DeYong, G.D., Cattani, K.D., 2012. Well adjusted: using expediting and cancelation to manage store replenishment inventory for a seasonal good. European Journal of Operational Research 220, 93–105. Durán, A., Gutiérrez, G., Zequeira, R.I., 2004. A continuous review inventory model with order expediting. International Journal of Production Economics 87, 157– 169. Fang, X., So, K.C., Wang, Y., 2008. Component procurement strategies in decentralized assemble-to-order systems with time-dependent pricing. Management Science 54, 1997–2011.
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Fu, K., Hsu, V.N., Lee, C.Y., 2009. Optimal component acquisition for a singleproduct, single-demand assemble-to-order problem with expediting. Manufacturing & Service Operations Management 11, 229–236. Gallego, G., Moon, I., 1993. The distribution free newsboy problem: review and extensions. Journal of the Operational Research Society 44, 825–834. Ghodsypour, S.H., O’Brien, C., 2001. The total cost of logistics in supplier selection, under conditions of multiple sourcing, multiple criteria and capacity constraint. International Journal of Production Economics 73, 15–27. Hong, J.D., Hayya, J.C., 1992. Just-in-time purchasing: single or multiple sourcing? International Journal of Production Economics 27, 175–181. Hsu, V.N., Lee, C.Y., So, K.C., 2006. Optimal component stocking policy for assembleto-order systems with lead-time-dependent component and product pricing. Management Science 52, 337–351. Hsu, V.N., Lee, C.Y., So, K.C., 2007. Managing components for assemble-to-order products with lead-time-dependent pricing: the full-shipment model. Naval Research Logistics 54, 510–523. Huggins, E.L., Olsen, T.L., 2010. Inventory control with generalized expediting. Operations Research 5, 1414–1426. Jones, P.C., Lowe, T.J., Traub, R.D., Kegler, G., 2001. Matching supply and demand: the value of a second chance in producing hybrid seed corn. Manufacturing & Service Operations Management 3, 122–137. Jones, P.C., Kegler, G., Lowe, T.J., Traub, R.D., 2003. Managing the seed-corn supply chain at syngenta. Interfaces 33, 80–90. Khouja, M., 1996. A note on the newsboy problem with an emergency supply option. Journal of the Operational Research Society 47, 1530–1534. Khouja, M., 1999. The single-period (news-vendor) problem: literature review and suggestions for future research. Omega 27, 537–553. Lawson, D.G., Porteus, E.L., 2000. Multistage inventory management with expediting. Operations Research 48, 878–893. Li, J., Chand, S., Dada, M., Mehta, S., 2009. Managing inventory over a short season: models with two procurement opportunities. Manufacturing & Service Operations Management 11, 174–184. Petruzzi, N.C., Dada, M., 1999. Pricing and the newsvendor problem: a review with extensions. Operations Research 47, 183–194. Plambeck, E.L., Ward, A.R., 2007. Note: a separation principle for a class of assembleto-order systems with expediting. Operations Research 55, 603–609. Qin, Y., Wang, R., Vakharia, AJ., Chen, Y., Seref, MM.H., 2011. The newsvendor problem: review and directions for future research. European Journal of Operational Research 213, 361–374. Ramasesh, R.V., Ord, J.K., Hayya, J.C., Pan, A., 1991. Sole versus dual sourcing in stochastic lead-time (s, Q) inventory models. Management Science 37, 428–443. Sheopuri, A., Janakiraman, G., Seshadri, S., 2010. New policies for the stochastic inventory control problem with two supply sources. Operations Research 58, 734–745. Silver, E.A., Pyke, D.F., Peterson, R., 1998. Inventory Management and Production Planning and Scheduling. Wiley, New York. Tomlin, B., Wang, Y., 2005. On the value of mix flexibility and dual sourcing in unreliable newsvendor networks. Manufacturing & Service Operations Management 7, 37–57. Treleven, M., Schweikhart, S.B., 1988. A risk/benefit analysis of sourcing strategies: single vs. multiple sourcing. Journal of Operations Management 7, 93–114. Tunca, T.I., Wu, Q., 2009. Multiple sourcing and procurement process selection with bidding events. Management Science 55, 763–780. Zhou, S.X., Chao, X., 2010. Newsvendor bounds and heuristics for serial supply chains with regular and expedited shipping. Naval Research Logistics 57, 71–87. Zhu, S.X., 2012. Joint pricing and inventory replenishment decisions with returns and expediting. European Journal of Operational Research 216, 105–112.
Contents lists available at SciVerse ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Innovative Applications of O.R.
Newsvendor with multiple options of expediting Ke Fu a, Jiayan Xu a, Zhaowei Miao b,⇑ a b
Lingnan College, Sun Yat-sen University, Guangzhou 510275, China School of Management, Xiamen University, Xiamen 361005, China
a r t i c l e
i n f o
Article history: Received 24 February 2012 Accepted 26 October 2012 Available online 12 November 2012 Keywords: Inventory Newsvendor Procurement lead times Expediting Lead-time-dependent pricing
a b s t r a c t We consider a manufacturer facing single period inventory planning problem with uncertain demand and multiple options of expediting. The demand comes at a certain time in the future. The manufacturer may order the product in advance with a relatively low cost. She can order additional amount by expediting after the demand is realized. There are a number of expediting options, each of which corresponds to a certain delivery lead time and a unit procurement price. The unit procurement price is decreasing over delivery lead time. The selling price is also decreasing over time. In this paper, we assume that the manufacturer must deliver all products to the customer in a single shipment. The problem can be formulated as a profit maximization problem. We develop structural properties and show how the optimal solution can be identified efficiently. In addition, we compare our model with the classical newsvendor model and obtain a number of managerial insights. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In the traditional newsvendor model, a manufacturer or retailer procures a single product from her upstream supplier and resells the product to her downstream customers who have one-time uncertain demand. In general, due to the long procurement lead time and customer restriction on delivery lead time, the order of the manufacturer should be placed long ahead of the actual demand being revealed. If the actual demand exceeds the pre-stocked quantity of the manufacturer, unmet demand is lost. In reality, however, if a shortage occurs, some companies have options to place makeup orders by expediting. For example, it is not uncommon that the manufacturer may have other emergency sourcing options or even in-house rush production. The expediting options are quick and responsive, but in general they are much more expensive compared to early procurement. Therefore, the manufacturer should evaluate carefully the trade-off between cost and responsiveness when making inventory and procurement decisions. In this paper, we study the newsvendor problem with multiple options of expediting under the above environment. A manufacturer (newsvendor) procures a product using normal procedure before the demand is realized. After demand realization, if the demand exceeds the quantity of pre-stock products, the manufacturer can place a make-up order using expediting procedures to fully satisfy the demand. There may be a number of expediting options, each of which corresponds to a certain lead time and a unit price. For example, different expediting options correspond ⇑ Corresponding author. Tel.: +86 0592 2187067. E-mail address: [email protected] (Z. Miao). 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.10.039
to different transportation modes and a higher price is charged for a shorter procurement lead time. In addition, the price for selling the product depends on the delivery lead time, and a longer delivery lead time leads to a lower unit price, reflecting the fact that late delivery is penalized. The newsvendor problem is one of the classical problems in the literature on inventory management (Arrow et al., 1951; Silver et al., 1998). There is a rich literature on the newsvendor problem with a wide variety of research issues. We refer readers to Khouja (1999), Qin et al. (2011), Gallego and Moon (1993) and Petruzzi and Dada (1999) for thorough and excellent reviews. In the following we discuss some closely related papers. A number of research papers have addressed the expediting or multiple sourcing issues, such as Hong and Hayya (1992), Ramasesh et al. (1991) and Tomlin and Wang (2005). For multiple sourcing, a stream of researches focus on supplier selection, risk control and auctions (Ghodsypour and O’Brien, 2001; Treleven and Schweikhart, 1988; Tunca and Wu, 2009), other than inventory management. In the field of inventory management, most researches consider only one expediting option (dual sourcing), in either single-period, multi-period or continuous-time models. Closely related to our research, Khouja (1996) considers a newsboy problem with one emergency supply option where a proportion of customers satisfy their demand from an emergency supply source when there is a shortage. The author solves the problem under the objective of maximizing the expected profit or maximizing the probability of achieving a target profit. Fu et al. (2009) consider expediting in a single-period assemble-to-order (ATO) system with one expediting option. DeYong and Cattani (2012) study a special newsvendor problem where the newsvendor can expedite an
K. Fu et al. / European Journal of Operational Research 226 (2013) 94–99
additional order or cancel the previous order based on the updated demand forecast. Examples of researches on multi-period inventory control include Lawson and Porteus (2000), Durán et al. (2004), Huggins and Olsen (2010), Zhou and Chao (2010), Zhu (2012), to name a few. These researches analyze that certain inventory policies, such as the base-stock policy, are optimal or close-to-optimal. Chiang (2010) proposes a single-item continuous-review order expediting inventory policy, which can be considered as an extension of the ordinary (s,Q) policy. The buyer can expedite part of an outstanding order via a fast transportation mode at extra costs when his inventory falls below a certain level. Plambeck and Ward (2007) consider expediting option in a continuous-time ATO system. In addition, a number of recent papers study supply chain models with second production or procurement opportunities, including Cachon and Swinney (2009), Li et al. (2009), Jones et al. (2003) and Jones et al. (2001). This paper considers the lead-time-dependent procurement cost and product price. Previous researches based on this setting are relatively limited. Fang et al. (2008) consider the case of lead-time-dependent final product price and explore how the manufacturer can use a ‘‘Vendor Managed Consignment Inventory’’ scheme to manage the underlying risk and coordinate independent suppliers’ decisions on the production quantities of their components under demand uncertainty. Hsu et al. (2006) consider an ATO system with lead-time-dependent component and final product price without expediting. Chandra and Grabis (2008) study a single-stage variable lead-time inventory system with lead-time dependent procurement cost. They develop a cost-minimization model without expediting and do not consider the varying product prices. Hsu et al. (2007) address inventory decisions in an assemble-to-order system with lead-time-dependent final product pricing under the full shipment case and propose efficient solution procedures. They assume constant component purchasing prices, and thus do not consider the expediting issue, which is the main focus of our research. Our model differs from previous work in the following two aspects. First, our paper considers multiple options of expediting, while most of previous researches address only one option. Second, we consider decreasing procurement cost and product price. As will be seen later, the newsvendor problem with these two features reveals some new characteristics and insights. To the best of our knowledge, this has not been addressed in the literature. The rest of the paper is organized as follows. In Section 2, we discuss our problem and formulate it as a profit maximization model. In Section 3, we develop some structural properties and solve the optimization model. In Section 4, we compare our model with the traditional newsvendor problem and discuss a number of managerial insights. Section 5 concludes the paper and offers some future research directions.
2. The model We consider the problem under the following assumptions. A manufacturer produces or processes a single product from a supplier and resells the product to customers. Demand comes at a certain time in the future and the manufacturer may order some amount of product using normal procedure in advance. These products will arrive before the demand is revealed. When demand is realized, in case of shortage, the manufacturer will place makeup orders using expediting procedure to satisfy the entire demand. There are a number of expediting options, each of which corresponds to a lead time and a unit price. The unit price is lower with a longer lead time. Hence, the manufacturer wants to determine the quantity of the pre-stocked product to maximize her total expected profit.
95
Let D L li
c(L) P(L)
c0 b Q
stochastic demand, which is assumed to be realized at time 0; delivery lead time; ith lead time of expediting, i = 1, 2, . . ., n, we index the lead times such that l1 6 l2 6 6 ln, in addition, define l0 0 and ln+1 + 1; unit purchasing price under expediting if delivery lead time is L, we assume that c(L) is decreasing in L; unit price of final product if delivery lead time is L, we assume that P(L) is decreasing in L. In order to avoid trivial cases, we further assume P(L) P c(L) for all L; unit purchasing price of preseason ordering before time 0; unit salvage value, we further assume b < c0; pre-stocked quantity.
We assume here that the manufacturer must deliver products in a single shipment to satisfy the entire customer demand. The formulation consists of two stages. At stage 1, the manufacturer determines the pre-stocked quantity of the product before the actual demand is revealed. At stage 2 when the demand is realized, given the demand D and the pre-stocked quantity Q, the manufacturer will use the pre-stocked quantity to fully satisfy the demand exactly at time 0 if D 6 Q, i.e., the demand is not more than the prestocked quantity. If D > Q, however, the manufacturer must place make-up orders to procure additional products. When these additional products arrive, the manufacturer will then deliver all products in a single shipment to customer. First, we have the following property. Proposition 1. For the stage-two problem, if Q < D, i.e., the prestocked inventory is less than the demand, then there is an optimal procurement and shipment decision such that the manufacturer places a single make-up order at time 0 and makes a single full shipment, where the order lead time and delivery lead time are L⁄ 2 {l1, l2, . . . , ln}. Proof. If there are orders placed at any time t > 0, then we can modify this solution by moving all the procurement times to time 0 and keeping other decisions unchanged. It is easy to verify that the new solution will not increase the costs or decrease the revenue. Hence in the following we assume this property is satisfied, i.e., all orders are placed at time 0. If there is a single order that will arrive at a time li0 2 fl1 ; l2 ; . . . ; ln g, then it is easy to verify that there is an optimal shipment decision that satisfies L ¼ li0 . If there are multiple orders that will arrive at times li0 ; li1 ; . . . ; lim , where 1 6 i0 < i1 < < im 6 n, we can modify the solution by merging all orders into a single one with the longest lead time lim . It is clear that this modification will reduce the total procurement cost without affecting the revenue received from the customer. h Hereafter we shall consider the procurement and delivery decisions satisfying Proposition 1. Namely, the manufacturer will order the additional products at time 0 and deliver the final products as soon as the ordered products arrive. Therefore, the make-up delivery lead time can be restricted to {l1, l2, . . . , ln}. For convenience, define
Pi Pðli Þ;
0 6 i 6 n:
Clearly,
P0 P P1 P P Pn : In addition, the unit procurement cost under expediting is also decreasing in expediting lead time and we assume that the the unit
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K. Fu et al. / European Journal of Operational Research 226 (2013) 94–99
cost for pre-stocked products is no more than the lowest unit cost under expediting, namely,
c1 P c2 P P cn P c0 : Note that cn P c0. This makes sense, as usually the manufacturer has enough time to prepare the pre-stocked products. For example, she may use cheap transportation modes and get price discount for advance order information (Sheopuri et al., 2010). Based on Proposition 1, we can express the second stage problem as follows. Our objective is to maximize the net revenue, denoted as R(Q,D). þ
0
þ
ðF Þ RðQ ; DÞ ¼ max fPi D ci ðD QÞ þ bðQ DÞ g: i2f0;1;...;ng
ð1Þ
Proof. To solve problem (F0 ), we need to compare the total profits associated with the two delivery lead times. Given the two lead times li and lj, the profit for each can be expressed respectively as (we ignore the initial ordering cost c0Q as this term is the same for both cases).
Pi D ci ðD Q Þþ and Pj D cj ðD Q Þþ :
Clearly, if D 6 Q, then L⁄ = 0. Suppose D > Q. Then we shall choose li instead of lj if li leads to a higher profit, namely,
Pi D ci ðD Q Þ P Pj D cj ðD QÞ;
ci cj Q ¼ r ij Q; ðci cj Þ ðPi Pj Þ
D6
ðFÞ max ED fc0 Q þ RðQ ; DÞg:
which completes the proof. h
ð2Þ
3. Structural properties In this section, we discuss a number of structural properties of the proposed problem that allow us to simplify problem (F). Proposition 2. If i 6 j, where i,j 2 {1, 2, . . . , n}, and Pi ci P Pj cj, then regardless of the pre-stocked quantity Q, there is an optimal solution that does not use lj as delivery time. Proof. Suppose there is a solution that uses lj as delivery time. Now we change lj to li and examine the profit changes. Let Cj be the total profit when choosing lj as delivery time. Let Ci be the total profit when choosing li as delivery time. Then for arbitrary Q and D, we have þ
þ
C j C i ¼ ½P j D cj ðD Q Þ c0 Q þ bðQ DÞ ½Pi D ci ðD Q Þ
þ
c0 Q þ bðQ DÞþ
ð6Þ
or
At stage one, our objective is to maximize the total expected profit under uncertain demand D. Q P0
ð5Þ
ð7Þ
Example 1. Suppose P(l1) = 4, P(l2) = 3, c(l1) = 3, c(l2) = 1, Q = 10, D = 11, then if we choose l1, total net profit is P(l1)D (D Q)+ c(l1) = 4(11) 1(3) = 41. If we choose l2, then the total net profit is P(l2)D (D Q)+c(l2) = 3(11) 1(1) = 32. Therefore, we should choose l1, which is also consistent with D 6 r1,2Q. But if Q = 0, then we should choose l2 since D > r1,2Q. The following proposition follows from the above proposition. Proposition 4. Given li, lj, lk, 0 < i < j < k 6 n, if rij P rjk then there is an optimal solution that does not use lj as delivery time. Proof. Consider two cases: (1) D 6 rijQ and (2) D > rijQ. (1) D 6 rijQ. From Proposition 3, li is better than lj. Thus, it is optimal not to use lj as delivery time. (2) D > rijQ. It is clear that D > rjkQ since rij P rjk, then lk is better than lj. Again it is optimal not to use lj as delivery time.
6 ½Pj D cj D ½Pi D ci D 6 0; where the first inequality holds because ci P cj, and the second inequality holds by the assumption Pi ci P Pj cj. h By Proposition 2, we can remove lj for any pair (li,lj) such that Pi ci P Pj cj. After removing all lead times that satisfy the above inequality, we will obtain a smaller set of lead times {li} such that Pi ci is increasing in i. Namely, we have
P 1 c1 < P 2 c2 < < P n cn :
ð3Þ
Hereafter, we shall assume the lead times already satisfy this condition. Without loss of generality, we still denote the set of lead times as {l0, l1, . . . , ln} where l0 0. It is intuitively true that for the second stage problem (F0 ), the choice of delivery lead time L depends on the pre-stocked quantity Q as well as the realized demand D. The following property states this relationship precisely and is a fundamental insight for us to simplify the problem significantly. Before describing the proposition, we define the following notation:
r i;j ¼
c i cj ; ðci cj Þ ðP i Pj Þ
1 6 i < j 6 n:
Therefore, for all D P 0, we will not choose lj as delivery time. h By Proposition 4, we can remove any lead time lj such that rij P rjk (0 < i < j < k 6 n). This property, along with the earlier discussions in this section, allows us to modify the problem to satisfy the following conditions without affecting its optimal solution. (a) Pi ci is strictly increasing over i, 1 6 i 6 Proposition 2):
n (by
P 1 c1 < P 2 c 2 < < P n c n : (b) ri,i+1 is strictly increasing over i, 1 < i 6 n 1 (by Proposition 4):
r 1;2 < r 2;3 < < rn1;n : For convenience, we define ri ri,i+1, i = 1, . . . , n 1. Based on the above discussions, we can summarize the solution for the second stage problem (F0 ). For convenience, we define r0 = 1 and rn +1.
ð4Þ
It is easy to see that ri,j P 1 because ci cj > Pi Pj due to inequalities (3). Clearly, given any two lead times L1 < L2, the choice between the two depends on both Q and D. Proposition 3. For the second stage problem (F0 ), for any two lead times li and lj, where li < lj, 1 6 i < j 6 n, if D 6 ri,jQ, then there is an optimal solution that does not choose lj as delivery lead time.
Proposition 5. For any pre-stocked quantity Q and demand realization D, the optimal solution for the second stage problem (F0 ) can be expressed as shown in Table 1: Based on Proposition 5, we can now express the expected profit function for the first stage problem.
Pf ðQ Þ ¼ ED fc0 Q þ RðQ ; DÞg Z Q ¼ c0 Q þ ½P0 x þ bðQ xÞdFðxÞ 0
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þ
n1 Z X
r iþ1 Q
Z
¼ c0 Q þ ( n2 Z X þ Z
In general, the properties of the objective function Pf(Q) depend on the demand distribution function F(x). However, based on the structural properties, it is not difficult to obtain optimal solutions to the problem for a number of commonly used demand distributions including Uniform, Gamma, and Normal. To keep our discussions concise, we refer readers to an online technical report of this paper for a more detailed discussion about this problem. Next, we consider a special case where procurement costs and selling prices are both linear, i.e.,
Q
½P0 x þ bðQ xÞdFðxÞ
0 1
½Piþ1 x ciþ1 ðx Q ÞdFðxÞ
ri Q
i¼0
)
1
½Piþ1 x ciþ1 ðx Q ÞdFðxÞ
r iþ1 Q
þ
Z
þ1
½P n x cn ðx Q ÞdFðxÞ
r n1 Q
Z
cðlÞ ¼ u0 u1 l; PðlÞ ¼ v 0 v 1 l;
Q
¼ c0 Q þ ½P0 x þ bðQ xÞdFðxÞ 0 Z 1 þ ½P 1 x c1 ðx Q ÞdFðxÞ
where u0, u1,
Q
þ
n1 Z 1 X i¼1
½ðPiþ1 Pi Þx ðciþ1 ci Þðx Q ÞdFðxÞ:
0
xdFðxÞ ¼ yFðyÞ
Ry 0
FðxÞdx and
R1 y
xdFðxÞ ¼ E½D yFðyÞ þ
Ry 0
FðxÞdx.
Using the two formulae, we can simplify the above objective function Pf(Q) as follows.
Z Q Pf ðQ Þ ¼ c0 Q þ ðP0 bÞQFðQ Þ ðP0 bÞ FðxÞdx þ bQFðQÞ 0 Z Q þ ðP1 c1 Þ QFðQ Þ þ FðxÞdx þ c1 Q ½1 FðQÞ 0
Z n1 X ½ðP iþ1 P i Þ ðciþ1 ci Þ r i QFðr i Q Þ þ þ
ri Q
FðxÞdx
0
i¼1
þðciþ1 ci ÞQ½1 Fðr i QÞg þ ðPn C n ÞE½D: Differentiating Pf(Q), we obtain P 0f ðQ Þ ¼ c0 þ P0 FðQÞ þ P 0 Qf ðQ Þ ðP0 bÞFðQ Þ þ ðP1 c1 Þ½FðQÞ Qf ðQ Þ þ FðQÞ þ c1 ½1 FðQ Þ c1 Qf ðQÞ þ
n1 X
½ðP iþ1 Pi Þ ðciþ1 ci Þ r i Fðr i Q Þ r 2i Qf ðr i Q Þ þ r i Fðr i Q Þ
i¼1
þðciþ1 ci Þ½1 Fðr i QÞ ðciþ1 ci Þr i Qf ðr i QÞg ¼ c0 þ P0 Qf ðQÞ þ bFðQ Þ ðP 1 c1 ÞQf ðQÞ þ c1 ½1 FðQ Þ c1 Qf ðQÞ þ
n1 X
½ðci ciþ1 Þ ðPi P iþ1 Þr 2i Qf ðr i Q Þ
i¼1
ðci ciþ1 Þ½1 Fðr i QÞ þ ðci ciþ1 Þr i Qf ðr i QÞg:
Note that by definition,
ri ¼
v0, v1 are constants, and u1, v1 > 0. Then
ci ¼ u0 u1 li ;
i ¼ 1; 2; . . . ; n;
P i ¼ v 0 v 1 li ;
i ¼ 0; 1; . . . ; n:
Thus, without loss of generality, we can define analogously to (4)
ri Q
By integration by parts, it is straightforward to show that Ry
n1 X ðci ciþ1 ÞFðr i Q Þ: i¼1
½Piþ1 x ciþ1 ðx Q ÞdFðxÞ
ri Q
i¼0
P 0f ðQ Þ ¼ cn c0 þ ðP 0 P 1 ÞQf ðQ Þ ðc1 bÞFðQ Þ þ
ci ciþ1 : ðci ciþ1 Þ ðPi Piþ1 Þ
Hence the above expression can finally be simplified to the following:
Table 1 The optimal solution for (F0 ). i⁄
R(Q, D)
Range of D
0 1 2 .. . n1 n
P0D + b(Q D) P1D c1(D Q) P2D c2(D Q) .. . Pn1D cn1(D Q) PnD cn(D Q)
D6Q Q < D 6 r1Q r1Q < D 6 r2Q .. . rn2Q < D 6 rn1Q D > rn1Q
ðu0 u1 li Þ ðu0 u1 lj Þ ðu0 u1 li Þ ðu0 u1 lj Þ ½ðv 0 v 1 li Þ ðv 0 v 1 lj Þ u1 ðlj li Þ ¼ u1 ðlj li Þ v 1 ðlj li Þ u1 ¼ : u1 v 1
r i;j ¼
That is, with linear costs, r ij u1u1v 1 will be constants for all 1 6 i < j 6 n. Based on earlier discussions, it is easy to prove the following result. Proposition 6. For the second stage problem, if Q < D, i.e., the prestocked inventory is less than the demand, suppose the procurement cost and the selling price functions are linear as defined above, then (i) if u1 6 v1, there is an optimal solution that always uses l1 as delivery time; (ii) if u1 > v1, we consider three subcases: (a) if D < u1u1v 1 Q , then there is an optimal solution that uses l1 as delivery time if the additional procurement is necessary; (b) if D ¼ u1u1v 1 Q , any of li (i = 1, 2, . . . ,n) can be used as the optimal delivery time; (c) if D > u1u1v 1 Q , then there is an optimal solution that uses ln as the delivery time. Based on Proposition 6 and earlier discussions, the problem can be efficiently solved when the costs and prices are linear. Proposition 6 says that if the procurement cost and selling price functions are both linear, then depending on the relative quantity between the realized demand and the pre-stocked quantity, the manufacturer can choose either the shortest or the longest lead time for expediting. In order to explain how parameters u1 and v1 affect the optimal delivery time, we divide the total demand D (when D > Q) into two parts: the pre-stocked quantity Q and the second order quantity D Q. Then we consider two cases. In case (i), u1 6 v1 implies that Pi ci is decreasing over i. For the pre-stocked quantity Q, it is optimal to be delivered to customers as soon as possible because Pi is decreasing. For the second order quantity D Q, the manufacturer should also deliver this part as soon as possible since Pi ci is decreasing. Therefore, the delivery time l1 is optimal if additional procurement is required. In case (ii), u1 > v1 suggests that Pi ci is increasing over i. Therefore, the pre-stocked quantity Q should be delivered as soon as possible (l1) while the second order quantity D Q should be delivered at time ln. However, all products must be delivered to the customer in a single shipment here. As a result, there is a tradeoff between l1 and ln for the manufacturer. This tradeoff in return depends on the relative quantity of Q and D. If D is relatively large (this implies the second order quantity D Q is relatively large compared to the
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Remark 1. In this paper, we assume that the manufacturer must deliver total products in a single full shipment. For the case of partial shipments, where the manufacturer is allowed to deliver products in multiple shipments, it is quite clear that the model is simply a parameter-adjusted newsvendor problem in which the under-stocking cost is reduced from P0 c0 to (P0 c0) (P⁄ c⁄), where P⁄ c⁄ = max16i6n(Pi ci). Consequently, under this case, the manufacturer will pre-order less than the classical newsvendor case without expediting options. In addition, the optimal profit in the partial shipments case is higher than that in the classical newsvendor case.
4. Comparison and managerial insights In this section, we compare the full shipment case with the classical newsvendor model. Specifically, we compare the following two models.
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optimal pre−order quantities
pre-stocked quantity Q), the manufacturer should choose ln as the delivery time because the second order quantity has a larger contribution to the total profit. If D is relatively small, then the manufacturer should choose l1 as the delivery time because the prestocked quantity has a larger contribution to the total profit. At the critical point when D ¼ u1u1v 1 Q , the pre-stocked and the second order quantities are equally important and any delivery time will result in the same profit.
130
classical newsvendor
120 110 100
full shipment 90 80 10
20
30
40
50
60
70
standard deviation σ Fig. 1. The impact of the demand standard deviation on the optimal order quantities.
950 900
classical newsvendor
850
profits
98
800
full shipment
750 700 650 10
20
30
40
50
60
70
standard deviation σ
Classical newsvendor. In this case, there is no second order opportunity and hence the demand may not be fully satisfied. We assume that there is no shortage cost for the unfulfilled demand. Newsvendor with multiple expediting options and a full shipment. In this case, there is a second order opportunity to fully satisfy the customer demand and a single full shipment is required by the customer. It can be easily verified that the optimal profit in the partial shipments case is higher than those in the full shipment case and classical newsvendor case. However, whether the full shipment is better than the classical newsvendor is not clear. Although the manufacturer in the full shipment case has a second order opportunity, she has to wait for the make-up procurement, if her prestocked product is not enough, to arrive and satisfy the entire demand in a single shipment. A delay in the final delivery means a loss in the revenue from the customer since the final product price is decreasing in the delivery time. Therefore, there is a probability that the profit from full shipment may not be as large as that from classical newsvendor. However, the relationship between the two cases depends on the specific distribution of the demand. The profit from classical newsvendor may be more sensitive to the demand variation because the newsvendor cannot procure for a second time if the demand exceeds his pre-stocked quantity. We will illustrate this relationship in the following numerical analysis. In this analysis, we are interested in the characteristics of optimal order quantities and profits in the two models. The basic parameters are set as follows. Let P0 = 20, c0 = 10, b = 5. There are three options for expediting: P1 = 19, c1 = 15; P2 = 18, c2 = 12; P3 = 15, c3 = 11. Assume that the demand follows Normal distribution N(l, r2), where l = 100, r = 30. Then we can get the optimal order quantities Qn = 112.9, Qf = 113.6 and the optimal profits pn = 836.4, pf = 800.8. Next, we examine how the coefficient of variation (COV) affects the optimal decisions and profits. We fix the demand mean and vary the standard deviation. As shown in Fig. 1, Qn increases in a 0 linear fashion. This is intuitive because PP00c > 0:5. However, Qf first b
Fig. 2. The impact of the demand standard deviation on the profits.
increases and then decreases in the standard deviation r. This phenomenon occurs due to the requirement of a full shipment and can be interpreted as follows. Consider an order decision Q and a demand realization D. If there is an overage of the pre-stocked products (D < Q), then as is the case with classical newsvendor, the manufacturer will satisfy all the demand and salvage the leftover products. If, however, there is a shortage of the pre-stocked products (D > Q), then the pre-stocked products may have less value compared to the classical newsvendor, as the manufacturer will have to wait for the make-up order to arrive and then deliver in a single full shipment. Thus, the demand variation may affect the pre-stocked quantity in opposite ways. When the variation is relatively small, the manufacturer may wish to pre-stock more to prevent shortages when the variation increases. When the variation is relatively large, the manufacturer may wish to pre-stock less when the variation increases because the value of the pre-stocked products diminishes with large variation. By similar arguments, we can explain the relationship between the pre-stocked quantities in the classical newsvendor and full shipment cases. Fig. 2 shows that the profits of the two cases all decrease in standard deviation, which is consistent with our common understanding that demand variation decreases the expected profit. Furthermore, the profit in the newsvendor case is more sensitive to standard deviation. This is due to the fact that it has only one order opportunity and cannot satisfy the exceeding demand with expediting options. If COV is small, the optimal profit in the newsvendor case is higher than that in the full shipment case. If COV is large, however, the newsvendor case is worse than the full shipment case in terms of profits. 5. Conclusions In this paper, we consider the newsvendor problem with multiple options of expediting. A manufacturer procures a product from a supplier and resells the product to customers who have uncertain
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demand. The manufacturer can order additional products using expediting procedures if necessary. The procurement cost and the product price are lead-time dependent. We analyze the optimal properties and solution procedures for this problem. We compare our model with the classical newsvendor and obtain a number of managerial insights. We now offer a few research directions. First, in our model, although we assume that no shortage is allowed, it would be interesting to study the case where both expediting and shortage are allowed. In addition, we may extend our current models to consider various issues such as capacity constraints in the newsvendor literature. Another direction is to study other types of inventory problems with multiple options of expediting. Our initial attempts show that some of these directions may require very different approaches. Acknowledgements The authors thank the Editor and an anonymous referee for their valuable comments and suggestions that have greatly improved this article. This research was supported in part by National Natural Science Foundation of China (71072090, 70701039, 71222105, 70802052), and Humanities and Social Science Fund of Ministry of Education of China (09YJC630237). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.ejor.2012.10.039. References Arrow, K., Harris, T., Marshak, J., 1951. Optimal inventory policy. Econometrica 19, 250–272. Cachon, G.P., Swinney, R., 2009. Purchasing, pricing, and quick response in the presence of strategic consumers. Management Science 55, 497–511. Chandra, C., Grabis, J., 2008. Inventory management with variable lead-time dependent procurement cost. Omega 36, 877–887. Chiang, C., 2010. An order expediting policy for continuous review systems with manufacturing lead-time. European Journal of Operational Research 203, 526– 531. DeYong, G.D., Cattani, K.D., 2012. Well adjusted: using expediting and cancelation to manage store replenishment inventory for a seasonal good. European Journal of Operational Research 220, 93–105. Durán, A., Gutiérrez, G., Zequeira, R.I., 2004. A continuous review inventory model with order expediting. International Journal of Production Economics 87, 157– 169. Fang, X., So, K.C., Wang, Y., 2008. Component procurement strategies in decentralized assemble-to-order systems with time-dependent pricing. Management Science 54, 1997–2011.
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