Coordinating production and shipment decisions in a two-stage supply chain with time-sensitive demand

Mathematical and Computer Modelling 51 (2010) 632–648

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Coordinating production and shipment decisions in a two-stage supply chain with time-sensitive demand Emmett J. Lodree Jr. a,∗ , Christopher D. Geiger b , Kandace N. Ballard c a

Department of Information Systems, Statistics, and Management Science, Culverhouse College of Commerce and Business Administration, The University of Alabama, Tuscaloosa, AL 25487-0226, United States b Department of Industrial Engineering and Management Systems, University of Central Florida, Orlando, FL 32816-2993, United States c

Department of Industrial and Systems Engineering, Auburn University, Auburn, AL 36849-5346, United States

article

info

Article history: Received 30 October 2009 Accepted 30 October 2009 Keywords: Inventory control Direct shipping Supply chain responsiveness

abstract This paper investigates a supply chain system consisting of one manufacturer who receives an order from a single retailer and then coordinates a production and shipment schedule to fulfill the retailer’s order as quickly and cost effectively as possible. It is assumed that the neither the manufacturer nor the retailer has inventory on hand at the time the retailer’s order is received by the manufacturer, and that the demand rate at the retailer is constant. It is also assumed that shortages at the retailer result in lost sales penalties. In this setting, lost sales penalties are incurred during the time in which the retailer has no inventory on hand, which illustrates the time-sensitive nature of the demand. We derive the manufacturer’s optimal production and shipping policy for the case in which the retailer controls the supply chain relationship, and also for the case in which the manufacturer controls the relationship. © 2009 Elsevier Ltd. All rights reserved.

1. Supply chain responsiveness The ultimate measure of supply chain performance is profit maximization (e.g., [1]). However, further examination of the research literature and best practices among leading organizations suggest that there is an intimate connection between supply chain profitability and other performance metrics such as flexibility, coordination, customer satisfaction, and responsiveness (e.g., [2,1]). For example, a responsive and flexible supply chain is more likely to capitalize from spot market opportunities than a supply chain with long lead times and limited flexibility. Furthermore, customer responsiveness is a well recognized corporate strategy that organizations often leverage in order to gain advantage over their competitors, and thereby maintain or increase market share and preserve long-term profitability (e.g., [3]). More recently, events such as the catastrophic hurricanes that devastated the Gulf Coast region of the southeastern United States and the terrorist attacks of September 11, 2001 have also revealed the important role of responsiveness in supply chain management and the logistics of disaster planning. This paper investigates a two-stage supply chain system consisting of one manufacturer and one retailer whose performance is directly influenced by the supply chain’s ability to quickly respond to a sudden realization of (or surge in) demand for a seasonal item, such as high-end fashion apparel or lawn care equipment. Fig. 1 describes, in general, the relationship between the manufacturer and the retailer as related to order fulfillment for a seasonal product. However, the time interval [T2 , T5 ] represents the focus of our research.

∗

Corresponding author. Tel.: +1 205 348 9851; fax: +1 205 348 0560. E-mail address: [email protected] (E.J. Lodree Jr.).

0895-7177/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2009.10.044

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633

Fig. 1. Sequence of events in a responsive production/distribution system.

The overall process in Fig. 1 is described as follows. At the onset (time T0 ), the retailer (or buyer) places an order for a seasonal product from the manufacturer (or supplier) well in advance of the selling season. When the manufacturer receives the retailer’s order, the manufacturer then coordinates production and shipping schedules during the time interval [T0 , T1 ], which represents the off-season. If the manufacturer is the dominant player in the supply chain relationship, then the manufacturer’s production and shipment policies will be constructed to optimize his own efficiencies. However if the retailer is the dominant player, then it is likely that the manufacturer’s shipment schedule will be dictated by the retailer. During the off-season, the retailer’s demand is uncertain. Therefore, when demand realization occurs some time during the selling season, the retailer incurs overstocking costs if her order from the manufacturer exceeds actual demand, or incurs shortage costs if her order from the manufacturer is less than actual demand. This scenario, which spans [T0 , T2 ] in Fig. 1, is closely related to the classic newsvendor problem. The newsvendor problem and many of its variations are discussed in [4,5]. In one variation, the retailer is allowed to place a second order from the manufacturer, which is often referred to as an emergency order or mid-period replenishment. This newsvendor variation spans [T0 , T3 ] in Fig. 1 and seeks to determine order quantities that minimize the buyer’s total expected cost due to ordering, overstocking, and understocking. The focus of this paper is the emergency replenishment associated with the retailer’s second order if a shortage occurs, which corresponds to [T2 , T5 ] in Fig. 1. Since actual demand is observed at time T2 , the emergency replenishment process is a deterministic problem in which the manufacturer and retailer attempt to effectively respond to known time-sensitive demand. Our emphasis is the coordination of production and shipping policies during emergency replenishment. More specifically, we derive the manufacturer’s optimal production and shipping policy for the case in which the retailer dictates the supply chain relationship, and we also derive the optimal policy for the case in which the manufacturer dictates the relationship. 2. Literature review Coordination of production and distribution activities within the firm and across the supply chain has been an active area of research for many years. A variety of systems have been considered including one-to-one (i.e., single-supplier, single-buyer) supply chains (e.g., [6–8]), one-to-many supply chains (e.g., [9–11]), many-to-one supply chains (e.g., [12,13]), and many-to-many supply chains (e.g., [14,15]). These systems have been studied under a variety of conditions such as deterministic demand (e.g., [9,16]), stochastic demand (e.g., [17,7,11]), single product (e.g., [9,14,7]), multiple products (e.g., [18,19]), single period (e.g., [7,16,20]), and multiple periods (e.g., [10,21,11]). Distribution strategies such as direct shipping (e.g., [6,7]) and vehicle routing (e.g., [22–24]) have been considered, while various production decisions have been examined such as batch size (e.g., [25,15,24]) and production run length (e.g., [25,26]). Additionally, a variety of modeling and solution approaches have been applied including deterministic modeling and analysis (e.g., [9,10]), stochastic modeling and analysis (e.g., [27,28,8]), simulation modeling and analysis (e.g., [22,7]), and heuristics (e.g., [9,17,19]). For a more comprehensive presentation of the production–distribution research literature, the reader is referred to the comprehensive surveys presented in [29–31]. In order to address the production–distribution coordination problem associated with emergency replenishment as described in Section 1, this paper introduces models based on the framework of [6], who were the first to examine shipment size as a decision parameter that could be used to minimize logistics costs in direct shipping environments. In [6], the authors developed a model for a one-to-one supply chain whose solution resulted in an EOQ (Economic Order Quantity) type formula for shipment size that optimizes the trade-off between transportation and inventory costs. They also presented models for one-to-many systems with a single supplier and multiple buyers across different locations. In particular, they compared direct shipping strategies to peddling strategies1 and determined the conditions in which one strategy would

1 When vehicles visit more than one customer per dispatch, this is referred to as a peddling strategy. This is in contrast to the direct shipping strategy in which vehicles visit exactly one customer per dispatch.

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be more beneficial than the other. Other extensions of the basic one-to-one model introduced in [6] include the tradeoff between inventory, transportation, and production costs as a function of shipment size [22,25,15,23,32,24]; stochastic demand [7]; and stochastic demand with waiting costs [33,26,34,20,35]. Blumenfeld et al. [15] determined the optimal shipment size and production run length in many-to-many systems with multiple products under direct shipping policies, shipping using a consolidation terminal, and shipping policies that employ a combination of both strategies. Blumenfeld et al. [25] also determined the optimal shipment size and production run length in a two-tier supply chain system. Their objective was to compare the performance of a coordinated production/transportation strategy with synchronized production and shipping to that of a system with decoupled production and shipping schedules. Blumenfeld et al. [33] investigated the impact of response time into the decision process. Their results indicate that reductions in response time lead to reductions in retailer inventory levels. Other models that optimize over shipment size in one-to-many systems have considered vehicle routing. Examples include [23,32,24,22], who models stochastic demand with backlogging within a vehicle routing problem. The direct shipment model with variable shipment size has also been extended to include variable production rate and stochastic demand. Chien [7] extended the single period newsvendor model to include both production rate and shipment size as decision variables. The author developed an iterative procedure that leads to the optimal production rate and shipment size when demand is uniformly or exponentially distributed. Other stochastic demand models in which shipment size is treated as a decision variable have also included waiting costs into the cost structure. For instance, Blumenfeld et al. [26] developed an analytical model to determine shipment size and production cycle length. Their model included an expediting cost for faster-than-normal delivery service and was used to evaluate the trade-off between transportation and inventory costs under uncertainty and direct shipping. Jang [34] incorporated time and quantity dependent waiting costs into a direct shipping model with two customer classes and uncertain demand. Their model specified the quantity of items to be produced by a manufacturer and an allocation policy for distributing items among the two customer types in the event of shortage. Jang and Kim [20] extended [34] to account for n customer classes. Lodree et al. [35] incorporated time and quantity dependent waiting costs into the newsvendor problem with backlogged shortages that are fulfilled through an emergency replenishment process. They introduced a non-linear waiting cost function that allows multiple shipments during the emergency procurement process. Cetinkaya and Lee [27] also investigated shipping strategies that account for waiting costs and stochastic demand. Specifically, they derived optimal shipping policies and replenishment quantities that minimize expected costs due to inventory replenishment, delivery, inventory carrying, and customer waiting for a vendormanaged inventory system. However, they considered a time based shipping policy as opposed to a batch size based shipping policy as described in the above-mentioned papers. This paper makes contributions to the research literature in the following respects. First, note that this paper explicitly accounts for time sensitivity in demand when both shipment size and production rate are treated as decision variables. Based on our review of the literature, it seems that all other models that address time-sensitive demand and optimize over shipment size also consider demand uncertainty. However, none of the stochastic demand models with waiting costs treat production rate and shipment size as simultaneous decision variables. Note that although [7] optimizes over both production rate and shipment size under stochastic demand, waiting costs are not considered. Also note that [35] optimize over production rate and production batch size under stochastic demand with waiting costs, but optimal shipment sizes are determined exogenously (i.e., shipment size is not an explicit decision variable). In general, the above-mentioned stochastic demand models are based on the time period [T0 , T2 ] in Fig. 1. The models introduced in this paper represent continuations of the stochastic demand models in that our focus is the time period after demand realization, which is represented by the time period [T2 , T5 ] in Fig. 1. This paper’s contribution can also be distinguished from existing supply chain models with variable shipment size and multiple suppliers and/or buyers. One distinguishing characteristic is that production rate is either a fixed value in these more complex supply chain models (e.g., [25,15]), or production rate is ignored completely (e.g., [34,20]). As mentioned above, this paper treats both shipment size and production rate as decision variables. On a final note, many of the existing multiple customer models are based on heuristic procedures, bounds for heuristic procedures, and sensitivity analysis (e.g., [9,26,32]), but do not admit closed form solutions as in this paper. 3. Optimal policy with dominant retailer In this section, we assume that the retailer dominates the supply chain relationship and forces the manufacturer into a production and shipping policy that optimizes the retailer’s performance. Based on this assumption, we derive the manufacturer’s optimal production and shipping policy that is most beneficial to the retailer. Relevant notations are listed in Table 1, and the models introduced in this section are based on the following assumptions. Assumptions 1. During the emergency replenishment process, the retailer’s demand rate is a constant R items per unit time. 2. All demand that occurs during a retailer stockout is considered lost sales. 3. Although some demand will be lost as a result of stockout, all items ordered by the retailer will be sold during the emergency replenishment process. That is, Q = D − N, where D is the actual demand realized by the retailer and N is the number of lost sales as a consequence of the manufacturer’s production and shipping policy.

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Table 1 Notational conventions. V p pmin pmax R

δ

pδ=R pδ>R pδR VδR TCδ
Shipment size; decision variable Production rate (items per unit time, say weeks); decision variable Minimum possible production rate Maximum possible production rate Demand rate at the retailer’s location (items per week) Delivery rate to the retailer location (items per week) Optimal production rate if δ = R Optimal production rate if δ > R Optimal production rate if δ < R Optimal production rate among the above three cases Optimal shipment size if δ = R Optimal shipment size if δ > R Optimal shipment size if δ < R Optimal shipment size among the above three cases Retailer’s order quantity (the manufacturer’s demand) Travel time between the manufacturer and retailer Fixed cost incurred for each direct shipment from the manufacturer to the retailer The number of direct shipments from the manufacturer to the retailer Time that the kth shipment is received by the retailer Inventory level at time t Total cost if the manufacturer’s delivery rate equals the retailer’s demand rate Total cost if the manufacturer’s delivery rate exceeds the retailer’s demand rate Total cost if the manufacturer’s delivery rate is less than the retailer’s demand rate Retailer total cost associated with (p∗ , V ∗ ) Retailer’s waiting cost Retailer’s ordering cost Retailer’s inventory holding cost Unit revenue earned for each item the retailer sells Unit ordering cost incurred by the retailer Unit holding cost per unit time incurred by the retailer

4. Waiting cost is calculated as the opportunity cost of lost sales incurred during stockout while the retailer is waiting on a shipment from the manufacturer. 5. The retailer’s inventory level is zero at the time the plant receives the retailer’s emergency order because the previous period is treated as a newsboy problem where demand is realized and then an emergency order is placed if necessary. 6. The emergency order fulfillment process is purely a make-to-order system. That is, the manufacturer’s initial stock level is zero upon receiving the retailer’s emergency order. 7. Once the manufacturer’s inventory level reaches the desired shipment size V as a result of make-to-order production, the shipment is immediately transported to the retailer where it arrives T time units later. 8. The plant’s variable production rate is bounded by a minimum rate pmin and maximum rate pmax . Given the above-mentioned assumptions, Fig. 2 demonstrates the retailer’s inventory level as a function of time for the case δ = R, where δ is the rate at which deliveries arrive at the retailer location. The retailer’s inventory levels for δ > R and δ < R are shown in Figs. 3 and 4 respectively. It is evident from Figs. 2–4 that each of the three cases result in different cost functions. Thus the remainder of this section analyzes each case separately. The manufacturer’s coordinated production and shipping policy that minimizes the retailer’s costs can then be determined by first computing the optimal policy associated with each of these three cases, and then selecting among these three policies the one that yields the minimum total cost. This approach is explained more formally at the end of this section. 3.1. Delivery rate = Consumption rate When δ = R, the dynamics of the retailer’s inventory level is almost identical to that of the classic Economic Order Quantity (EOQ) model, as shown in Fig. 2. This model differs from EOQ in that (i) waiting costs are incurred up until the first shipment is received by the retailer and (ii) the decision variables are the shipment size V and production rate p. The retailer’s total cost based on the manufacturer’s production/shipping policy (p, V ) is then TCδ=R (p, V ) = Cwait + Corder + Cinvy . Since s =

Q V

, ordering costs are simply given by Corder = scO =

observe that the plant ships lot sizes of V to the retailer every k = 1, . . . , s, we have tk =

(3.1)

kV p

+ T.

V p

cO Q V

. Now in order to derive expressions for Cwait and Cinvy ,

weeks by Assumptions 6 and 7. Hence, it follows that for

(3.2)

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Fig. 2. Retailer’s inventory dynamics when δ = R.

Fig. 3. Retailer’s inventory dynamics when δ > R.

Thus for k = 1, . . . , s, the interarrival times of shipments at the retailer are given by tk − tk−1 =

kV p

+T −

(k − 1)V p

−T =

From (3.3), the retailer receives V items every Waiting costs are then

 Cwait = rRt1 = rR

V p

 +T

 = rR

V R

V p V p

. weeks so that δ =

 +T

(3.3)

= r (V + RT )

V V /p

= p. Since δ = R for this case, we have δ = p = R.

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637

Fig. 4. Retailer’s inventory dynamics when δ < R.

and the inventory holding costs are Cinvy = h

s Z X

tk+1

I (t )dt = hs

tk

k=1

tk+1

Z

I (t )dt =

tk

hQ V

·

1 2

·

V2 p

=

hQV 2R

.

Therefore, Eq. (3.1) can be expressed as a function of one variable V as follows. TCδ=R (V ) =



hQ

r+

2R

 V+

cO Q V

+ rRT .

(3.4)

The first and second order derivatives associated with Eq. (3.4) are dTCδ=R

=r+

dV d2 TCδ=R dV 2

=

hQ 2R

2cO Q V3

−

cO Q V2

.

(3.5) (3.6)

It is obvious from Eq. (3.6) that Eq. (3.4) is a convex function for all V > 0. Therefore (3.5) and the first order condition can be used to characterize the manufacturer’ s optimal production/shipping policy (pδ=R , Vδ=R ) for the case δ = R if the retailer dominates the relationship. More formally, the result is stated as follows. Theorem 1. The manufacturer’s production and shipping policy that is most beneficial to the retailer for the case δ = R is pδ=R = R

s Vδ=R =

(3.7) 2RQcO 2Rr + hQ

.

(3.8)

The proof of Theorem 1 is straightforward and follows immediately from the first order condition. Therefore, no formal proof is presented here. Also, note that if R 6∈ [pmin , pmax ], then the case δ = p = R is not applicable. 3.2. Delivery rate > Consumption rate Similar to the previous case, we have TCδ>R (p, V ) = Cwait + Corder + Cinvy .

(3.9)

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From  Figs.2 and 3, it is evident that Cwait and Corder are the same as the δ = R case (with the exception that Cwait = rR

V p

V R

+ T 6= rR Cinvy cO

=

s Z X

sRV 2 sRV

sRV

cO

V

+ Q V

2

 1−

2

 1−

p

R

 +

p R

 1−

p

X s

p R

2V 2

sV

k+

s(s + 1)

p

2

+

 + ··· +

p

 1−

2p

k=1



2

R

sV

2

R

p

 1−

R p

 +

sV 2 2p

 1−

R



p



p



2p

sV 2

1−

R



p

.

and simplifying, we obtain

 1−

p

1−

p

2

By substituting s = Q

V

+

2p2



p

2

2p2

=

I (t )dt

V2

+

2p2

=

=

tk+1 tk

k=1

=

Cinvy


+ T ). Therefore we only need to derive an expression for Cinvy as follows.



R

V+

2p

Q2



2p

R

1−



p

.

After algebraic manipulation, Eq. (3.9) can then be written as TCδ>R (p, V ) =



hQ p

 1−

R



2p

+

rR

 V+

p

cO Q V

+

hQ 2



2p

1−



R 2p

+ rRT .

(3.10)

In order to determine (pδ>R , Vδ>R ) that minimize Eq. (3.10), we first assume that the shipment size V is fixed and calculate the optimal production rate pδ>R as a function of V . We then use pδ>R to determine the optimal shipment size, Vδ>R , for the case δ > R. To determine pδ>R , it is convenient to express Eq. (3.10) as follows.



TCδ>R (p, V ) = (hQ + rR)V +

hQ 2 2

 ·

1 p

−

hQR(V + Q ) 2

·

1 p2

+

cO Q V

+ rRT .

(3.11)

The corresponding partial derivatives are

  ∂ TCδ>R hQ 2 1 1 = − (hQ + rR)V + · 2 + hQR(V + Q ) · 3 ∂p 2 p p   ∂ 2 TCδ>R hQ 2 2 3 = ( hQ + rR ) V + · 3 − hQR(V + Q ) · 4 . 2 ∂p 2 p p

(3.12)

(3.13)

In this case, the first order condition does not guarantee optimality since Eq. (3.13) is not necessarily non-negative for p > 0. However, it is still possible to obtain pδ>R . To do so, we define α, β , and γ as follows:

α = (hQ + rR)V + β= γ =

hQR(V + Q )

hQ 2 2

2 cO Q V

+ rRT .

Then Eq. (3.11) becomes TCδ>R (p) =

α p

−

β p2

+ γ.

(3.14)

The following properties of the total cost function TCδ>R (p) given by Eqs. (3.11) and (3.14) will enable us to identify the optimal value pδ>R (proofs for these properties are shown in the Appendix). Lemma 1. Let TCδ>R (p) be defined as in Eq. (3.14). Then 1. lim TCδ>R (p) = −∞. p→0+

2. lim TCδ>R (p) = γ . p→∞

E.J. Lodree Jr. et al. / Mathematical and Computer Modelling 51 (2010) 632–648



2β



3. TCδ>R (p) is strictly increasing for p ∈ 0, α and strictly decreasing for p ∈ 2β 4. If V is fixed, then TCδ>R (p) achieves its maximum on (0, ∞) at p = α .



2β

α

639



,∞ .

Now define pˆ as

pˆ =

  pmin ,        pmax ,

2β

if pmax ≤

α

2β

if pmin ≥

   pmin ,       pmax ,

α

≤ pmax and TCδ>R (pmin ) ≤ TCδ>R (pmax ) α 2β ≤ ≤ pmax and TCδ>R (pmin ) ≥ TCδ>R (pmax ). α

if pmin ≤ if pmin

(3.15)

2β

The above properties and pˆ can now be used to describe the manufacturer’s production rate pδ>R that is most beneficial to the retailer for the case δ > R when the shipment size V is fixed. Proposition 1. Let R be the retailer’s demand rate and pˆ be defined as in Eq. (3.15). 1. If R ∈ (0, pmin ), then pδ>R = pˆ . 2. If R ∈ [pmin , pmax ] and pˆ = pmax , then pδ>R = pmax . 3. If R ∈ [pmin , pmax ] and pˆ = pmin , then

pδ>R =

  R,        pmax ,

2β

∈ (pmax , ∞) α 2β 2β if ∈ [pmin , pmax ] and ∈ (0, R) α α 2β 2β if ∈ [pmin , pmax ], ∈ (R, ∞), and TCδ>R (R) ≤ TCδ>R (pmax ) α α 2β 2β if ∈ [pmin , pmax ], ∈ (R, ∞), and TCδ>R (R) ≥ TCδ>R (pmax ). α α if

   R,      

pmax ,

The proof of Proposition 1 is shown in the Appendix. Now suppose p is fixed and we are to determine the optimal shipment size V . From Eq. (3.10), the first and second order partial derivatives are

∂ TCδ>R hQ = ∂V p

 1−



R 2p

+

rR p

−

cO Q

(3.16)

V2

∂ 2 TCδ>R 2cO Q = . 2 ∂V V3

(3.17)

Since Eq. (3.17) is non-negative for all V > 0, it is evident that TCδ>Q (V ) given by Eq. (3.10) is convex in V when p is fixed. Therefore, Eq. (3.16) and the first order condition can be used to determine the manufacturer’s shipment size that is most beneficial to the retailer when p is fixed. The following proposition states this result more formally (note that the proof is omitted because it follows immediately from the first order condition). Proposition 2. If p is fixed, then the optimal shipment size is

v u u V (p) = t

cO Qp

∗



hQ 1 −

R 2p



.

(3.18)

+ rR

Proposition 1 suggests that pδ>R ∈ {pmin , pmax , R}. Thus the optimal production and shipping policy for the case δ > R can be obtained by substituting (p, V (p)) into TC (p, V ) given by Eq. (3.10), where p ∈ {pmin , pmax , R} satisfies Proposition 1 and V (p) satisfies Eq. (3.18). Therefore, we have that

(pδ>R , Vδ>R ) =

argmin

p∈{pmin ,pmax ,R}

TC p, V ∗ (p) ,




where TC (p, V ) is Eq. (3.10) and V ∗ (p) is Eq. (3.18). More formally, the optimal policy for the case δ > R can be stated as follows. Theorem 2. The manufacturer’s production and shipping policy that is most beneficial to the retailer if δ > R is (pδ>R , Vδ>R ), where pδ>R satisfies Proposition 1, Vδ>R = V ∗ (pδ>R ), and V ∗ (p) is Eq. (3.18).

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3.3. Delivery rate < Consumption rate As in the previous cases, the total cost function consists of waiting, ordering, and holding costs so that TCδ
(3.19)

In this case, the ordering cost, Corder , is the same as the previous two cases. However, the previous equations for holding cost, Cinvy , and waiting cost, Cwait , are different and must be derived for the δ < R case. Since δ < R, the retailer will experience shortages even after the manufacturer begins the production and shipping process. Define tˆk as the time stockout begins after the retailer receives the kth delivery from the manufacturer (see Fig. 4). Then tˆk = kV + VR + T and tˆk − tk = VR for k ∈ {2, 3, . . . , s + 1}. Therefore, holding costs are computed as p Cinvy = h

s Z X k=1

tˆk

I (t )dt =

tk

h 2

·

Q V

(tˆ1 − t1 )V =

h 2

·

Q

 

V

R

V

V =

hQV 2R

.

(3.20)

Now waiting costs are incurred during the time interval (0, t1 ) for all three cases. These are the only waiting costs associated with the δ = R and δ > R cases. However, additional waiting costs are incurred for the  δ < R case. In particular, there are s =

Cwait

Q V

stockouts of duration tk − tˆk−1 , where k ∈ {2, 3, . . . , s + 1}. Since tk − tˆk−1 =

  
1 1 V ˆ +T +Q − . = rRt1 + srR tk − tk−1 = r p

p

R

V p

−

V R

, we have

(3.21)

Using Eqs. (3.20), (3.21), and our usual expression for ordering costs, we obtain TCδ


hQ

+

2R

rR p

 V+

   1 1 + rR T + Q − .

cO Q V

p

R

(3.22)

To determine (pδ
rR(V + Q ) p

+

hQV 2R

+

cO Q V

  Q + rR T − . R

(3.23)

The first and second order partial derivatives are

∂ TCδ
(3.24)

∂ 2 TCδ
(3.25)

The following properties of TCδ
3. lim TCδ


V 2R

+

cO Q V

 −1 .

Based on Lemma 2, it is shown in the Appendix that the optimal production rate is as follows. Proposition 3. The manufacturer’s production rate that is most beneficial to the retailer for the case δ < R and fixed shipment size V is

 pδ
pmax , R,

if R ∈ [pmax , ∞) if R ∈ [pmin , pmax ] .

Note that the case δ = p < R is not applicable if R < pmin .

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Now suppose that p is fixed and we wish to determine the optimal shipment size Vδ
∂ TCδ
(3.26)

∂ 2 TCδ
(3.27)

From (3.27), it is evident that TCδ
s V (p) =

2cO QRp

∗

hQp + 2rR2

.

(3.28)

From Proposition 3, we see that no matter what value V assumes, the optimal production rate is either R or pmax . Therefore, the optimal policy for the case δ < R is as follows. Theorem 3. The manufacturer’s production and shipping policy that is most beneficial to the retailer if δ < R is (pδ
C = {(pδ=R , Vδ=R ) , (pδ>R , Vδ>R ) , (pδ
Let TC be the cost function associated with the optimal policy and C ∗ be the cost associated with the optimal policy. Then TC ∗ ∈ {TCδ=R , TCδ>R , TCδR (pδ>R , Vδ>R ) , TCδ
(3.29)

The manufacturer’s optimal policy for the dominant retailer case can now be stated. Theorem 4. The manufacturer’s production and shipping policy that is most beneficial to the retailer is p∗ , V ∗ = argmin TC ∗ (p, V )




(p,V )∈C

where TC satisfies TC ∗ (p∗ , V ∗ ) = C ∗ and C ∗ is given by (3.29). ∗

3.5. Numerical example In order to illustrate the resulting optimal policy, consider the following numerical values for the problem parameters: pmin = 15, pmax = 45, R = 20, Q = 70, T = 1, r = 50, cO = 20, and h = 1. Substituting the above parameter values into Eqs. (3.7) and (3.8) for (pδ=R , Vδ=R ), Proposition 1 and Eq. (3.18) for (pδ>R , Vδ>R ), and Proposition 3 and Eq. (3.28) for (pδ
C = {(20, 5.2), (45, 7.7), (20, 5.2)}. Observe that the policies for (pδ=R , Vδ=R ) and (pδR , Vδ>R ) = (45, 7.7), which corresponds to a production rate that is greater than the demand rate. 4. Optimal policy with dominant manufacturer In this section, we derive the manufacturer’s production and shipping policy for the case in which the manufacturer dominates the supply chain relationship. In this case, the optimal coordinated production and shipping policy minimize the

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Table 2 Additional nomenclature. F cmax cmin Ctran Cprod m

Fixed cost incurred for each direct shipment from the manufacturer to the retailer The (maximum) production cost associated with the maximum production rate pmax The (minimum) production cost associated with the minimum production rate pmin Transportation cost Manufacturer’s production cost Slope of the manufacturer’s production cost function Cprod

manufacturer’s costs. Note that the assumptions presented at the beginning of Section 3 are still applicable, and additional notations introduced in this section are shown in Table 2. Also note that in this section, the notations introduced in Table 1 apply to the manufacturer as opposed to the retailer. For example, TC represents the manufacturer’s total cost as opposed to the retailer’s total cost, r represents the manufacturer’s revenue per item sold and h is the unit holding cost incurred by the manufacturer. The manufacturer’s costs include the cost of holding inventory at the facility and in transit to the retailer, transportation costs, production costs, and waiting costs. Thus, the total cost function can be expressed as TC (p, V ) = Cinvy + Ctrans + Cprod + Cwait .

(4.1)

For this model, inventory holding and transportation costs are represented as in previous models from the literature [e.g., [6,15,7]]. The transportation cost is simply FQ

, (4.2) V which includes a fixed cost F for each delivery to the retailer. Inventory holding costs are incurred both at the manufacturer’s facility and while in transit to the retailer. The holding cost function is Ctran =

Cinvy =

hQV 2p

+ hQT .

(4.3)

We now introduce a production cost function Cprod (p). Capacitated production is assumed, and for illustrative purposes, we assume that Cprod (p) is a linear function with slope m. The minimum production rate and corresponding production cost are pmin and cmin respectively, and the maximum production rate and corresponding production cost are pmax and cmax c −c respectively. Therefore, we have m = pmax −pmin and max

min

Cprod (p) = m(p − pmin ) + cmin .

(4.4)

In order to determine an appropriate expression for the manufacturer’s waiting costs, we assume that the manufacturer incurs waiting costs during any period in which the retailer has no inventory on hand. In this case, Figs. 2–4 suggest that waiting costs are the same whenever δ = R and δ > R, but the case δ < R leads to a different calculation of waiting costs. Therefore, we follow the approach of Section 3 by examining the cases separately, and then using the optimal policy for each case to determine the manufacturer’s overall optimal policy. 4.1. Delivery rate ≥ Consumption rate When δ ≥ R, it was shown in Section 3 that Cwait = rR



V p

 + T . Using this expression for waiting costs along with

Eqs. (4.1)–(4.4), we have TCδ≥R (p, V ) = hQ



V 2p

 +T

+

FQ V

+ m(p − pmin ) + cmin + rR



V p

 +T .

(4.5)

Similar to Section 3, we first assume that p is fixed and determine the optimal shipment size as a function of p. Then the optimal shipment size will be used to derive the optimal production rate. When taking partial derivatives with respect to V , it is convenient to express (4.5) as TCδ≥R (p, V ) =



hQ 2p

+

rR p

 V+

FQ V

+ (hQ + rR)T + m(p − pmin ) + cmin .

(4.6)

The partial derivatives are δ≥R ∂ TCMAN 1 = ∂V p



hQ 2

δ≥R ∂ 2 TCMAN 2FQ = 3 . 2 ∂V V

 FQ + rR − 2 V

Since Eq. (4.8) is positive for all V > 0, the first order condition yields the optimal shipment size when p is fixed.

(4.7)

(4.8)

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643

Proposition 5. If p is a fixed value, then the optimal shipment size is

s V (p) =

2pFQ

∗

hQ + 2rR

.

(4.9)

Substituting Eq. (4.9) into Eq. (4.5) and rearranging terms yield TCδ≥Q (p, V ) = ∗



s

hQ

+ rR

2

s

2FQ p(hQ + 2rR)

+ mp +

FQ (hQ + 2rR) 2p

+ (hQ + rR)T − mpmin + cmin .

(4.10)

It can be shown that the derivatives of Eq. (4.10) in terms of p are

s

dTCδ≥R (p, V ∗ )

=m−

dp

d2 TCδ≥R (p, V ∗ ) dp2

=

·

2

(4.11)

2p3

s

3

FQ (hQ + 2rR)

FQ (hQ + 2rR) 2p5

.

(4.12)

Eq. (4.12) suggests that the first order condition with respect to p > 0 leads to the optimal production rate pδ≥R . Eq. (4.11) can then be used to calculate the corresponding optimal production rate pδ≥R as follows.

r pˆ =

3

FQ (hQ + 2rR)

(4.13)

2m2

The convexity of (4.10) enables us to obtain the following optimal policy: Theorem 5. The manufacturer’s optimal production and shipping policy when δ ≥ R is pδ≥R and Vδ≥R = V ∗ pδ≥R , where V ∗ (p) is Eq. (4.9), pˆ is Eq. (4.13), and




pδ>R

 pmin ,    R,  = pˆ ,    R,   pmax ,

if pˆ ∈ (0, pmin ) and

R ∈ (0, pmin )

if pˆ ∈ (0, pmin ) and

R ∈ [pmin , pmax ]

if pˆ ∈ [pmin , pmax ]

and R ∈ [pmin , pˆ ]

if pˆ ∈ [pmin , pmax ]

and R ∈ [ˆp, pmax ]

if pˆ ∈ (pmax , ∞).

The proof of Theorem 5 is shown in the Appendix. 4.2. Delivery rate < Consumption rate In this case, the total cost function and corresponding partial derivatives are TCδ


hQ 2p

+

rR



p

V+

FQ V

+ (hQ + rR)T + m(p − pmin ) + cmin + rQR



1 p

−

1 R

 (4.14)

∂ TCδ
(4.15)

δ
(4.16)

Since (4.16) is positive for all V > 0, we see that the cost function TCδ 0. The first order condition on V and (4.15) then leads to the following result. Proposition 6. If p is a fixed value, then the optimal shipment size for the case δ < R is

s V (p) = ∗

2pFQ hQ + 2rR

.

(4.17)

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After substituting (4.17) into (4.14) and simplifying, we obtain δ≥R

TCδ
 =

s

hQ

+ rR

2

s +



R p

 −1 

2FQ

+ mp + rQ

p(hQ + 2rR)

FQ (hQ + 2rR) 2p

R p

 −1

+ (hQ + rR)T − mpmin + cmin .

(4.18)

It can be shown that the derivatives of (4.18) in terms of p are dTCδ
s =m−

d2 TCδ
=

3 2

FQ (hQ + 2rR) 2p3

s ·

FQ (hQ + 2rR) 2p5

−

+

rQR

(4.19)

p2 2rQR p3

.

(4.20)

Since (4.20) is positive for all p > 0, the first order condition and Eq. (4.19) can be used to obtain the optimal production rate pδ
(4.21)

Now define pˆ as the positive real solution of Eq. (4.21) that yields the smallest value of TCδ 0



and

2m2 p4 − FQ (hQ + 2rR)p − 2r 2 Q 2 R2 = 0 .



(4.22)

The convexity of Eq. (4.18) enables us to determine the optimal production/shipping policy for the case δ < R as follows (note that this result is proved in the Appendix). Theorem 6. The manufacturer’s optimal production and shipping policy for the case δ < R is pδ
pδ
 pmin ,    R , = pˆ ,    R , pmax ,

if if if if if

pˆ pˆ pˆ pˆ pˆ

∈ (0, pmin ) ∈ [pmin , pmax ] and R ∈ (pmin , pˆ ) ∈ [pmin , pmax ] and R ∈ (ˆp, ∞) ∈ (pmax , ∞) and R ∈ (pmin , pmax ) ∈ (pmax , ∞)and R ∈ (pmax , ∞).

4.3. Optimal policy Observe from Propositions 5 and 6 that Vδ≥R = Vδ








.

(4.23)

Then the optimal production rate is pδ≥R if TCδ≥R (pδ≥R , V ) ≤ TCδ
∗

Theorem 7. The manufacturer’s optimal production and shipping policy is p∗ , V ∗ =




argmin

p∈(pδ≥R ,pδ
TC ∗ (p, V ∗ )

where V ∗ = Vδ≥R = Vδ



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645

4.4. Numerical example The results associated with the manufacturer’s optimal policy will be illustrated with a numerical example. The parameter values used in the numerical example for the retailer’s problem will also be used here with the addition of the following parameters that are specific to the manufacturer’s problem: cmin = 30, cmax = 10, and F = 20. Substituting the parameter values into Eqs. (4.9) and (4.13) for (pδ≥R , Vδ≥R ) and Eqs. (4.17) and (4.22) for (pδ
C = {(45, 7.8), (20, 5.2)}, which in turn leads to C ∗ = min {$1, 458.89, $1, 621.66} . Therefore, the optimal policy for the manufacturer is pδ≥R , Vδ>R = (45, 7.8), which corresponds to a production rate that is greater than the demand rate. Note that this policy is nearly identical to the retailer’s optimal policy, which is (45, 7.8).




5. Summary and future research This paper studies a make-to-order supply chain problem between one manufacturer and one retailer in a time-sensitive demand setting. Demand is time-sensitive in that the retailer’s demand rate is constant during the production and shipping process so that waiting costs continuously accumulate as long as the retailer has no inventory on hand. We derive the manufacturer’s optimal production and shipping policy for two different cases. For the first case, the retailer dominates the supply chain relationship and influences the manufacturer’s actions. For the second case, the manufacturer dominates the supply chain relationship such that the manufacturer coordinates production and shipping at his own convenience. If the retailer is dominant, then the optimal coordinated production and shipping policy minimizes the retailer’s costs associated with ordering, inventory holding, and waiting. If the manufacturer is dominant, then the optimal coordinated production and shipping policy minimizes the manufacturer’s costs associated with production, inventory holding (at the facility and in transit), transportation, and waiting. The models introduced in this paper are intended to serve as building blocks to constructing more complex models that can be used in practice. Consequently, there are some opportunities for further research. For instance, our model assumes that the manufacturer’s production and shipping process begin precisely when the retailer’s inventory level is zero. A natural generalization of this is to assume that the retailer places the emergency order some time before her inventory level reaches zero. Our models also assume linear waiting, inventory holding, production, and transportation costs, which presents opportunities to explore more general cost functions. Another characteristic of our models is that we assume either the retailer or the manufacturer dominates the relationship. A promising extension is to explore the case in which neither player is dominant so that game-theoretic analysis can be used to coordinate the supply chain by exploring possible equilibrium solutions. Notice that the results of the numerical examples presented in Sections 3.5 and 4.4 suggest the possibility of a Nash equilibrium, which indeed warrants further investigation. The models and results introduced in this paper could be used directly for this purpose. Appendix. Proofs Lemma 1. Parts 1 and 2 of Lemma 1 are justifiable by inspection of Eq. (3.14). Therefore, we only prove parts3 and4. Eq. (3.7) and the first order condition on p yields − pα2 +

ζ =

2β

α



− ε , where ε ∈ 0,

2β

α



2β p3

= 0, whose solution is p =



2β

α

−ε

2

α

. Let ζ ∈

2β

0, α

. Then

. By Eq. (3.7), we have

2β ∂ TCδ>R (ζ ) α = − 2 +  3 ∂p 2β 2β − ε − ε α α ! 1 2β =  −α . 2 2β 2β −ε α − ε α Since

2β

> 0, we can use Eq. (A.1) to show that

(A.1)

∂ TCδ>R (ζ ) ∂p

  > 0 reduces to 2β > 2β − αε ∀ ζ ∈ 0, 2αβ . The latter

inequality   always holds since each parameter   is a positive value.Therefore,  Eq. (3.7) is strictly increasing in the interval ∂ TC R 2β 2β 2β 2β 0, α since ∂δ> > 0 ∀ ζ ∈ 0 , . Similarly, for ζ ∈ , ∞ , we have ζ = α + ε , where ε ∈ (0, ∞) so p α α   ∂ TCδ>R (ζ ) that < 0 reduces to 2β < 2β + αε . Therefore, (3.7) is strictly decreasing over 2αβ , ∞ . Since TCδ
TCδ
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Proposition 1. Part 1 2β If R ∈ [0, pmin ], then each p ∈ [pmin , pmax ] is feasible. Now suppose pmax < α . Since TCδ>R (p) is increasing in the interval (0, 2αβ ) by Lemma 1 part 3 and (pmin , pmax ) ⊂ (0, 2αβ ), we have inf

p∈[pmin ,pmax ]

TCδ>R (p) = TCδ>R (pmin ). 2β

Therefore, pδ>R = pmin . Similarly, if pmin > α , it follows that pδ>R = pmax because TCδ>R (p) is decreasing in the interval ( 2αβ , ∞) by Lemma 1, part 3 and (pmin , pmax ) ⊂ ( 2αβ , ∞) so that inf

p∈[pmin ,pmax ]

TCδ>R (p) = TCδ>R (pmax ). 2β

Now suppose α ∈ (pmin , pmax ). Then inf

TCδ>R (p) = TCδ>R (pmin )

inf

TCδ>R (p) = TCδ>R (pmax )

2β p∈[pmin , α ] 2β p∈[ α ,pmax ]

2β

2β

since TCδ>R (p) is increasing in the interval (pmin , α ) and decreasing in the interval ( α , pmax ). Thus pδ>R = argminTCδ>R (pδ>R ), where TCδ>R (pδ>R ) = min{TCδ>R (pmin ), TCδ>R (pmax )}. Part 2 The effect of the condition R ∈ (pmin , pmax ) is that p ∈ (0, pmin ) is no longer feasible; only p ∈ (R, pmax ) is feasible. Thus if pˆ = pmax , then pδ>R = pmax since pmax is still feasible. Part 3 2β 2β Similar to part 2, the feasible values of p lie in (R, pmax ). If pˆ = pmin , part 1 implies that either pmax ≤ α or α ∈ (pmin , pmax ) with TCδ>R (pmin ) ≤ TCδ>R (pmax ). If pmax ≤ 2αβ , it follows that TCδ>R (p) is increasing in the interval (R, pmax ) 2β

since it is increasing in the interval (0, α ). Therefore, inf

p∈[R,pmax ]

TCδ>R (p) = TCδ>R (R),

so that pδ>R = R. 2β 2β Now suppose α ∈ (pmin , pmax ) and TCδ>R (pmin ) ≤ TCδ>R (pmax ). If R ∈ (pmin , α ), then TCδ>R (p) is increasing so that inf

2β p∈[R, α ]

TCδ>R (p) = TCδ>R (R). 2β

However, TCδ>R (p) is decreasing in the interval ( α , pmax ), which implies that inf

2β p∈[ α ,pmax ]

TCδ>R (p) = TCδ>R (pmax ).

Consequently, pδ>R = R if TCδ>R (R) ≤ TCδ>R (pmax ) and pδ>R = pmax if TCδ>R (R) ≥ TCδ>R (pmax ). 2β Finally, if R ∈ ( α , pmax ), it follows that inf

2β p∈[ α ,pmax ]

TCδ>R (p) =

inf

p∈(R,pmax )

TCδ>R (p) = TCδ>R (pmax ) 2β

since TCδ>R (p) is decreasing for p ∈ ( α , ∞) by Lemma 1, part 3. Proposition 2. Since TCδ 0 by Lemma 2, part 1, it follows that TCδ p1 with p1 > 0 and p2 > 0. Thus for all p1 > 0 and p2 > 0, inf

p∈(p1 ,p2 )

TCδ
Note that the condition δ < R implies that p < R. Thus if R ∈ (pmax , ∞), we have from Eq. (A.2) that inf

p∈(pmin ,pmax )

TCδ
since pmax > pmin . Hence, pδ
p∈(pmin ,R)

TCδ
since R > pmin . Thus pδ
(A.2)

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647

Theorem 5. Eq. (4.18) suggests that TCδ≥R (p) is convex for all p > 0 and that pˆ is the unique minimizer of TCδ≥R (p). However, we must ensure that the solution is feasible, which means that p ∈ [pmin , pmax ] and p > R. Of course if pˆ ∈ [pmin , pmax ] and pˆ > R, then pˆ is feasible so that pδ≥R = pˆ . Now suppose pˆ ∈ [pmin , pmax ], but R ∈ (ˆp, pmax ), which implies pˆ < R. In this case, each p ∈ [R, pmax ] is feasible. Since TCδ≥R (p) is a convex function with unique minimizer pˆ , it follows that TCδ≥R (p) is increasing in the interval (ˆp, ∞), which implies that it is increasing over (R, pmax ) since (R, pmax ) ⊂ (ˆp, ∞). Therefore, pδ≥R = R since inf

p∈[R,pmax ]

TCδ≥R (p) = TCδ≥R (R).

Now suppose pˆ > pmax . If R < pmin , then each p ∈ [pmin , pmax ] is feasible. Since TCδ≥R (p) is convex with minimizer pˆ , TCδ≥R (p) is decreasing in the interval (0, pˆ ), and is also decreasing in the interval (pmin , pmax ) since (pmin , pmax ) ⊂ (0, pˆ ). Therefore, pδ≥R = pmax since inf

p∈[pmin ,pmax ]

TCδ≥R (p) = TCδ≥R (pmax ).

If, on the other hand, R ∈ [pmin , pmax ], then each p ∈ [R, pmax ] is feasible. But TCδ≥R (p) is again decreasing since [R, pmin ] ⊂ (0, pˆ ), so that pδ≥R = pmax . Finally, suppose pˆ < pmin . If R < pmin , then each p ∈ [pmin , pmax ] is feasible. As mentioned previously, TCδ≥R (p) is increasing over (ˆp, ∞). Thus TCδ≥R (p) is increasing over [pmin , pmax ] ⊂ (ˆp, ∞), so that inf

p∈[pmin ,pmax ]

TCδ≥R (p) = TCδ≥R (pmin ),

and pδ≥R = pmin . If R ∈ [pmin , pmax ], then each p ∈ [R, pmax ] is feasible. Since TCδ≥R (p) is increasing in this interval, it follows that inf

p∈[R,pmax ]

TCδ≥R (p) = TCδ≥R (R),

so that pδ≥R = R. Theorem 6. From Eq. (4.20), we have that TCδ
p∈[pmin ,pmax ]

TCδ
and pδ
p∈[pmin ,R]

TCδ
and pδ
p∈[pmin ,R]

TCδ
and pδ pmax . If R ∈ [pmin , pmax ], then each p ∈ [pmin , R] is feasible. It follows that TCδ
p∈[pmin ,R]

TCδ
and pδ pmax , then each p ∈ [pmin , pmax ] is feasible. Now (pmin , pmax ) ⊂ (0, pˆ ) so that TCδ
p∈[pmin ,pmax ]

and pδ
TCδ
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