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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.18 No.4 pp.884-899
DOI : https://doi.org/10.7232/iems.2019.18.4.884

# Analysis on a Drop-Shipping Method for Supplier-Retailer Coordination

Changkyu Park*
College of Business, University of Ulsan, Republic of Korea
Corresponding Author, E-mail: ckparkuou@ulsan.ac.kr
July 25, 2019 September 27, 2019 October 28, 2019

## ABSTRACT

In a two-stage supply chain, a retailer forecasts a future demand, determines an order quantity, and sends the order to the supplier. Then the supplier fulfills the order accordingly. The supplier may provide a drop-shipping method for order fulfillment to increase profits for the both parties, while the retailer takes inventory-related risks. In order to investigate the effect of a drop-shipping method for supplier-retailer coordination, we build the profit models for the supplier, the retailer, and the total system with/without a drop-shipping method. Then we analyze the independent and vertically integrated models using Stackelberg game and demonstrate equilibrium solutions. Through numerical experiments, we also illustrate the effectiveness of newly developed models and derive some observed implications. Study findings imply that (i) a drop-shipping method for shortages improve the supplier-retailer coordination performance, (ii) in general, decision power give an advantage to the supplier, (iii) the Nash equilibrium is not appealing to anybody, and (iv) the vertically integrated models outperform the independent models.

## 1. INTRODUCTION

Recently, we observe new trends in supply chain management, such as the globalization of market economies, shorter product life cycles, digitalization, and multifaceted customer expectations, along with developments such as resource scarcity, stricter regulatory requirements, and a more long-term focus. These trends drive the incorporation of environmental and social responsibility issues into the management of supply chains. As a result, sustainable supply chain management has become a contemporary issue and an important area of research (Saeed and Kersten, 2019).

Sustainable Supply Chain Foundation (www.sustainablescf. org) defines that “sustainable supply chain management involves integrating environmentally and financially viable practices into the complete supply chain lifecycle, from product design and development, to material selection, (including raw material extraction or agricultural production), manufacturing, packaging, transportation, warehousing, distribution, consumption, return and disposal. Environmentally sustainable supply chain management and practices can assist organizations in not only reducing their total carbon footprint, but also in optimizing their end-toend operations to achieve greater cost savings and profitability.”

The research area of sustainable supply chain management includes the network design (Rabbani et al., 2018;Tsao, 2015), coordination of supplier-retailer (Gharaei et al., 2019a;Giri and Bardhan, 2014;Giri and Masanta, 2018;Sarkar and Giri, 2018;Shah et al. 2018;Yin et al., 2016), inventory control (Gharaei et al., 2019b;2019c, 2019d;Kazemi et al., 2018;Shekarabi et al., 2019), supplier selection (Rabbani et al., 2019), transportation (Awasthi and Omrani, 2019;Sayyadi and Awasthi, 2018a, 2018b), manufacturing (Dubey et al., 2015;Hao et al., 2018), and maintenance (Duan et al., 2018).

We consider a two-stage supply chain that is composed of a supplier and a retailer. The retailer forecasts a future demand, determines an order quantity, and requests the order to the supplier in pursuit of maximizing his own profit. Then the supplier delivers the order quantity to the retailer. In the two-stage supply chain, it is assumed that all inventory risks are placed to the retailer. Due to the huge financial risk, the retailer limits his order quantity, even though the supplier would like the retailer to order as much as possible. Thus the possibility of shortages can increase, which results in lost-sales. If some coordination mechanisms are introduced, the mechanisms motivate the retailer to manage inventory more cost-effectively, which reduces lost-sales and increases profit for both the supplier and the retailer (Simchi-Levi et al., 2009). In this paper, we consider a drop-shipping method for coordination mechanism.

Drop-shipping method is an order fulfillment method that a retailer reroutes customer demands to a supplier. The supplier progresses the customer demands and delivers products directly to the customers. Even though mail-order firms have traditionally utilized a dropshipping method, in the Internet age the drop-shipping method becomes a well-known online-order fulfillment strategy. Using drop-shipping method, the retailer takes advantage of the fulfillment capabilities of supplier to fulfill orders. In other words, the retailer is able to sell products without dealing with some issues such as product development and manufacture, inventory management, warehouse operations, and shipping/receiving operations (Park, 2017). Drop-shipping method provides some benefits to both the retailer and the supplier. The retailer can reduce an investment in inventory and fulfillment capabilities, and can receive the wider product selection. Meanwhile, the supplier can benefit by the lower costs due to economies of scale and the lower transportation costs. On the other hand, drop-shipping method can bring about some risks such as loss of product margin, loss of control that can negatively affect service quality, and encroachment on customers (Randall et al., 2002).

In this paper, we analyze the effect of drop-shipping method in the following situation. Usually, a retailer satisfies customer demands with his own inventory. However, when customer demands arrive during a shortage, the retailer offers the customers a price discount and promises a quick delivery. Some customers will accept the retailer’s offer, but some others will leave for other retailers. Then the supplier delivers the requested units directly to the customers who accept the retailer’s offer. The above situation was observed in the inventory operations practice of a parts distributor of the marine diesel engine manufactured by the Hyundai Heavy Industries (HHI) (Park and Seo, 2013). When the parts distributor could not satisfy shipping company’s demands immediately because of a stock-out, the HHI delivered the spare parts directly to the harbor that the shipping company’s vessel would arrive. Another example is BlueLight.com, which keeps some quantities of the top 40 CDs to increase sales revenue and utilizes a dropshipping method for shortages (Netessine and Rudi, 2006).

Clearly, the supplier-retailer coordination performance will be improved by adopting a drop-shipping method because the coordination can reduce the number of lost-sales. However, some questions arise as follows. Under what condition, it is desirable for both the supplier and the retailer to employ a drop-shipping method? Under what condition, it is fair for the both? What will happen to the both as the conditions change? In order to better understand the effect of drop-shipping method, we focus on a supplier-retailer supply chain, in which the retailer controls his inventory using a periodic review, order-up-tolevel (T, R) policy and the supplier assumes to have a sufficient capability for the retailer order.

The supplier-retailer supply chain is examined under three scenarios. Scenario 1 does not consider a dropshipping method. Thus, scenario 1 depicts the traditional inventory management in which the retailer (or the integrated retailer-supplier) determines the inventory level and maintains the inventory. The sale is lost when a product is out of stock. Scenario 1 is the baseline for evaluating the effect of drop-shipping method in other scenarios. In Scenarios 2 and 3, the supplier provides the retailer with a drop-shipping method. Scenario 2 is different from scenario 3 in that scenario 2 requires the retailer to keep the order-up-to-level R determined in scenario 1. For each scenario, we analyze an independent model and a vertically integrated model. Further, we consider three submodels for the independent model having a drop-shipping method, because the model solution depends on a decision power that is modeled by Stackelberg game. In the sub-models, either the supplier or the retailer plays a role of Stackelberg leader. Instead, the each can have an equal decision power, resulted in Nash equilibrium.

This paper is organized as follows. The following section discusses the related literature. In Section 3, we outline the notation and modeling assumptions. We also explain profit models for the supplier, the retailer, and the total system without/with a drop-shipping method. For all three scenarios, Section 4 analyzes independent and vertically integrated scenario models. Section 5 describes numerical experiments conducted to examine a new approximation of mean physical stock used in the profit model under the situation that the effect of lost-sales cannot be ignored, to illustrate the scenario models, and to investigate the effect of drop-shipping method. Lastly, Section 6 presents conclusion.

## 2. REVIEW OF RELATED LITERATURE

Internet retailers usually fulfill their orders using a drop-shipping method. Although the drop-shipping method is well known as a primary way of fulfilling on-line orders among Internet retailers, it has received only limited attention. It has been acknowledged recently that the integration of on-line and off-line operations (e.g., Clicks-and-Mortar business model) has furnished retailers with the possibility of increasing their profit and improving customer service (Chen et al., 2011). However, the academic studies on channel coordination mainly concentrate on the marketing issues of Internet retailing. They are largely qualitative studies such as Gulati and Garino (2000), Steinfield et al. (2002), Biyalogorsky and Naik (2003), De Koster (2003), and Randall et al. (2006).

Order fulfillment is considered as a vulnerable part in Internet business. Agatz et al. (2008) conducted literature review on Internet fulfillment in a multi-channel environment. In this paper, we focus on the literature on quantitative analyses of drop-shipping method, especially in terms of inventory control. Table 1 shows a summary of the literature from the viewpoint of analytic tools. The used analytic tools are Newsvendor model, economic order quantity (EOQ) model, periodic review model, and Stackelberg game. As an order fulfillment, most studies utilized a dual strategy, while some studies used a pure drop-shipping method. The dual strategy implies that a retailer uses in-stock inventory as main order fulfillment and drop-shipping method as secondary for shortages.

There are four studies that used Newsvendor model as an analytic tool. They are Khouja (2001), Lee and Chu (2005), Hovelaque et al. (2007), and Netessine and Rudi (2006). Khouja (2001) considered two channel demands for in-stock inventory and drop-shipping. Assuming that a fraction of demands for in-stock inventory accepts a dropshipping method for shortages, he analyzed optimal solutions that combine in-stock inventory and drop-shipping using Newsvendor model. For several demand distributions, he derived optimal order quantities for in-stock inventory and drop-shipping demands, and illustrated that it could increase significantly an expected profit.

Under the Newsvendor environment, Lee and Chu (2005) studied a supplier-retailer supply chain that is operated by either a traditional way or a drop-shipping way. The traditional way means that the retailer determines an inventory level and maintains the inventory. Lee and Chu (2005) compared the expected payoffs for the two ways. When there are coexistent networks of a store-based and a website-based sales, Hovelaque et al. (2007) compared three different organizational forms: store-picking, dedicated warehouse-picking, and drop-shipping. Using Newsvendor model, they compared pros and cons of the three different organizational forms, and analyzed the effect of inventory policy parameters in the supply chain.

Netessine and Rudi (2006) examined three channels of retailers: traditional, drop-shipping, and dual strategy. The dual strategy was modeled by a competitive game among retailers and a supplier. Netessine and Rudi (2006) illustrated which channel was preferable for certain values of critical parameters. The parameters are demand variability, a number of retailers in the channel, supplier prices, and transportation costs.

In literature, Khouja and Stylianou (2009) and Ayanso et al. (2006) used the EOQ model as an analytic tool. Using the EOQ model, Khouja and Stylianou (2009) examined the effect of drop-shipping method. They developed two EOQ models that a supplier provides a dropshipping method for shortages during lead-time. One model backorders the units short, while the other model loses sales.

Ayanso et al. (2006) considered a situation that Internet retailer utilizes a dual strategy as an order fulfillment. The Internet retailer satisfies the low margin demands by a drop-shipping method if in-stock inventory is not sufficient. In Ayanso et al. (2006), the theme is to decide the threshold level of inventory that determines whether in-stock inventory is used or drop-shipping method is used to fulfill the low margin demands. They conducted a computer simulation with the EOQ model to examine a threshold level inventory rationing policy for stochastic demand and lead-time.

As an analytic tool, Bailey and Rabinovich (2005) and Park (2017) used a periodic review model. Bailey and Rabinovich (2005) formulated multi-item periodic review models to analyze Internet book retailers’ decisions. With imperfect demand information, they considered two strategies that utilize in-stock inventory or drop-shipping method. Bailey and Rabinovich (2005) found that if market shares increase, Internet retailers consider both in-stock inventory and drop-shipping method; meanwhile if products are more popular, Internet retailers depend on instock inventory.

Park (2017) developed a periodic review model to address an environment in which unmet demand orders are partially lost and partially backordered when purchase dependence exists. The partial backorders are fulfilled by a drop-shipping method. Through computational analyses, he demonstrated the effect of drop-shipping method for both a partial backordering and purchase dependence.

Lastly, using a game-theoretic setting, Netessine and Rudi (2004) analyzed a supplier-retailer supply chain within a drop-shipping process. They carried out the analysis under different power structures and proposed a supply chain coordination scheme. The supply chain coordination scheme demonstrates that the system optimum can be reached by the retailer who compensates the supplier for a part of the inventory holding cost and by the supplier who subsidizes the retailer for a part of the marketing expenses.

Compared to literature, the quantitative models developed in this paper have different features: (1) we utilize a game-theoretic setting like Netessine and Rudi (2004), but we implement a dual strategy rather than a pure drop-shipping method; and (2) we use a dropshipping method as a backup for order fulfillment like studies in Table 1, but the retailer in this paper offers a price discount to keep customers from leaving in case of a shortage, thus influencing on a backordering rate.

## 3. PROFIT MODEL FORMULATION

In order to investigate the effect of drop-shipping method, we develop profit models for the supplier and the retailer, where the total system is derived by summing the supplier and the retailer. The drop-shipping method is provided by the supplier to increase profit for both the supplier and the retailer. We derive the profit model of retailer by developing a periodic review, order-up-to-level (T, R) model accommodating a drop-shipping method. On the other hand, the profit model of supplier is derived by calculating the revenue generated by the retailer policy. In formulating the profit models, we make the following assumptions:

• (a) Only a single product is considered.

• (b) Customer demand is a random variable with a known distribution.

• (c) Review period is given and greater than a constant replenishment lead-time.

• (d) Unit selling price is exogenously given.

• (e) Backordering rate β is linearly influenced by the price discount rate (1 - α) offered by the retailer, that is, β = 1 - α (see Lin, 2009;Giri and Masanta, 2018).

• (f) The supplier has a sufficient capability for the retailer order.

• (g) All parameters are same to the supplier and the retailer and known to the both (i.e., no information asymmetry).

The notation used in this paper is summarized in Table 2. Superscripts S, R, and T will denote supplier, retailer, and total system, respectively. Similarly, subscripts 1, 2, and 3 are for scenarios, I and V for independent and vertically integrated models, and S, R, and N for supplier, retailer, none as a Stackelberg leader.

### 3.1 Profit Models without a Drop-Shipping Method

This section depicts the traditional operations in which the retailer examines his inventory level at the review period interval T and orders a sufficient quantity to bring the inventory up to the order-up-to-level R. If a product is out of stock, the sale is lost. The average annual profit of the retailer equals to the sales revenue minus the sum of the purchasing cost, the ordering cost, the cost of carrying inventory, and the cost of lost-sales. Then,

$P R = ( s − C p ) D − C o T − C h R 2 2 D T − C l B ( R ) T$
(1)

where $C l = s − C p + C g$. In Eq. (1), we calculate the mean physical stock by an approximation under the situation that the effect of lost-sales cannot be ignored. We examine this approximation in detail through a computer simulation in Section 5.1.

The supplier’s average annual profit is determined by the retailer policy as follows:

$P S = ( D − B ( R ) T ) ( C p − C m )$
(2)

Notice that the only sales revenue is considered in Eq. (2) because we intend to investigate the effect of drop-shipping method that is provided by the supplier to increase profits.

Lastly, since the total system is a summation of the supplier and the retailer, the average annual profit of the total system is

$P T = ( s − C m ) D − C o T − C h R 2 2 D T − C l B ( R ) T$
(3)

where $C l = s − C m + C g .$

### 3.2 Profit Models with a Drop-Shipping Method

In case of shortages, the retailer offers a price discount to customers and a part of customers accepts the offer. Then the supplier delivers the product directly to the customer according to the request of retailer. Compared to the profit models in Section 3.1, the profit models in this section should include additional sales revenue. The additional profit for the retailer is

$β ( s − C p + C g + α s − C d ) B ( R ) T$
(4)

If we add Eqs. (1) and (4), then the average annual profit of the retailer is represented by the same form of Eq. (1), but

$C l = ( s − C p + C g ) − β ( s − C p + C g + α s − C d )$
(5)

The additional profit for the supplier is

$β ( C d − C m − C t ) B ( R ) T$
(6)

If we add Eqs. (2) and (6), then the supplier’s average annual profit is represented as follows:

$P S = ( D − B ( R ) T ) ( C p − C m ) + β ( C d − C m − C t ) B ( R ) T$
(7)

The average annual profit of the total system is obtained by summing those of the supplier and the retailer, resulted in the same form of Eq. (3), but

$C l = ( s − C m + C g ) − β ( s − C p + α s − C m − C t + C g )$
(8)

## 4. SCENARIO MODEL ANALYSIS

In this paper, we mainly focus on the supplierretailer coordination by utilizing a drop-shipping method as an order fulfillment in case of a shortage. Sometimes the supplier and the retailer cooperate with each other, and at other times, they compete with each other for their own profits. In such a situation, game theory is regarded as a relevant method to find equilibrium solutions (Yao et al., 2008). Table 3 shows the taxonomy of scenario models that are considered in this paper. The models for the scenario 1 (not using a drop-shipping method) provide the baselines for evaluating the effects of drop-shipping method in the scenarios 2 and 3.

### 4.1 Scenario 1

Model 1I represents the traditional inventory management in which the retailer determines the inventory level and maintains the inventory. The sale is lost when a product is out of stock. The retailer is a single decision maker. The average annual profits of the retailer and the supplier are calculated by Eqs. (1) and (2), respectively.

Proposition 1.For a given T, the value of R which maximizes Eq. (1) must satisfy that

$H ^ ( R ) = C h R C l D$
(9)

where $H ^ ( R ) = ∫ R ∞ h ( x ) d x$.

Proof is presented in Appendix A.

In the model 1V, the supplier and the retailer are vertically integrated. The integrated supplier-retailer is a single decision maker who determines the inventory level The model 1V seeks to maximize the objective function formulated as Eq. (3). For a given T, we can determine the value of R which maximizes Eq. (3) using Proposition 1.

### 4.2 Scenario 2

Scenario 2 describes the situation that the supplier provides the retailer with a drop-shipping method in pursuit of additional revenue. The supplier requires the retailer to keep the order-up-to-level R determined in the scenario 1. In the independent model, the supplier controls the unit selling price of supplier for a drop-shipping method, Cd and the retailer also controls the price discount rate, (1 - α) influencing the average backorder rate, in order to maximize their own profits. Since the supplier and the retailer strategically make their decisions, a gametheoretic situation is occurred.

We first visualize the problem by presenting the game graphically for an easy understanding. Figure 1 is taken from the example in Section 5. The best response functions of the retailer and the suppler are denoted by BRFR(Cd) and BRFS(α), respectively. The point (CdN, αN) is a Nash equilibrium that is a cross-point of the best response lines. A Stackelberg equilibrium with the supplier as a leader (CdS, αS) is placed on the retailer’s best response line, while a Stackelberg equilibrium with the retailer as a leader (CdR, αR) is placed on the supplier’s best response line.

The model 2S depicts the problem that the supplier is a Stackelberg leader and offers the retailer a dropshipping method with the price of Cd. For a given Cd, the retailer maximizes his profit by choosing an optimal value of α.

Proposition 2.For the model 2S, a Stackelberg equilibrium solution is a point (CdS, αS) that is characterized as follows:

$C d S = 2 s − C p + C g + C m + C t 2$
(10)

$α S = 2 s + C p − C g + C m + C t 4 s$
(11)

Proof is presented in Appendix B.

Before turning to the model 2R, we verify the range of Cd for the scenario 2 that allows both the supplier and the retailer to be profitable.

Proposition 3.To be profitable for both the supplier and the retailer, the value of Cd for the Scenario 2 should satisfy that

$C m + C t ≤ C d ≤ s − C p + C g + α s$
(12)

Proof is presented in Appendix C.

Corollary 1.The value of α for the Scenario 2 should satisfy that

$α ≥ C m + C t − s + C p − C g s$
(13)

Now, assume that the retailer is a Stackelberg leader. In the model 2R, the retailer offers a price discount to customers who take a drop-shipping method. For a given α, the supplier maximizes his profit by choosing an optimal value of Cd.

Proposition 4.For the model 2R, a Stackelberg equilibrium solution is a point (CdR, αR) that is characterized as follows:

$C d R = s − C p + C g + α R s$
(14)

where

$C m + C t ≤ C d R ≤ 2 s − C p + C g$
(15)

$C m + C t − s + C p − C g s ≤ α R ≤ 1$
(16)

Proof is presented in Appendix D.

The model 2N illustrates that the supplier and the retailer have an equal decision power. There is a unique Nash equilibrium.

Proposition 5.For the model 2N, a Nash equilibrium is a point (CdN, αN) that is characterized as follows:

$C d N = 2 s − C p + C g$
(17)

$α N = 1$
(18)

Proof is presented in Appendix E.

The model 2V, the vertically integrated model of the supplier and the retailer, seeks to maximize the objective function formulated as Eqs. (3) and (8).

Proposition 6.The value of α that maximizes the profit of model 2V is

$α = C p − C g + C m + C t 2 s$
(19)

Proof is similar to the way that the retailer behaves to maximize his profit in Proposition 2.

### 4.3 Scenario 3

This scenario extends the scenario 2 by removing the supplier’s requirement that the retailer should keep the order-up-to-level R determined in the scenario 1. Thus, in the scenario 3, the retailer is free to adjust the order-up-tolevel R in order to maximize his profit. As the scenario 3 allows the order-up-to-level R to be the retailer’s decision variable, it is hardly possible to derive analytically the equilibrium solutions. However, we can derive some observations for equilibrium solutions through inferences from the scenario 2 and numerical examples.

Observation 1.For the model 3S, a Stackelberg equilibrium solution is a point (CdS, RS, αS) that is characterized as follows:

$H ^ ( R S ) = C h R S C l D$
(20)

$α S = C p − C g + C d S 2 s$
(21)

$max C d S P S$
(22)

where Cl is defined by Eq. (5) and PS is defined by Eq. (7).

For a given CdS, Eq. (20) can be derived from Proposition 1 and Eq. (21) can be derived from Proposition 2. Since the supplier’s average annual profit, Eq. (7) shows empirically a concave form in terms of CdS, we can find numerically the value of CdS, which maximizes the supplier’s average annual profit.

Observation 2.For the model 3R, a Stackelberg equilibrium solution is a pointCdR, RR, αRthat is characterized as follows:

$C d R = ( s − C p + C g + α R s ) − 1 ( 1 − α R ) B ( R R ) ( C h ( R R 2 − R 1 I 2 ) 2 D − ( s − C p + C g ) ( B ( R R ) − B ( R 1 I ) ) ) δ ( α R )$
(23)

$max R R , α R P R$
(24)

where R1I is determined by the model 1I and PR is defined by Eqs. (1) and (5). δ(αR) = 0 if αR = 1, or 1 otherwise.

For given RR and αR, Eq. (23) is the maximum of CdR that can be allowed by the retailer. In order to find RR and αR, we process the following steps:

Step 1: Choose any value of αR that satisfies the range of CdR. The maximum of CdR is given by Eq. (23) and the minimum is determined by the supplier’s average annual profit as follows:

$C m + C t + ( C p − C m ) ( B ( R R ) − B ( R 1 I ) ) ( 1 − α R ) B ( R R )$
(25)

Step 2: For αR chosen in the step 1, find the maximum integer value of RR that satisfies the range of CdR. The retailer’s average annual profit function shows an increasing form with respect to RR.

Step 3: With RR found in the step 2, find the range of αR that satisfies the range of CdR.

Observation 3.For the model 3N, a Nash equilibrium is a point (CdN, RN, αN) that is characterized as follows:

$C d N = 2 s − C p + C g$
(26)

$H ^ ( R N ) = C h R N C l D$
(27)

$α N = 1$
(28)

where Cl is defined by Eq. (5).

We observe that Nash equilibrium is a cross-point of the best response functions of the supplier and the retailer. The numerical solution for the simultaneous equations of Eqs. (20), (21), and (23) is observed to deduce the Observation 3.

Proposition 7.The values of α and R that maximize the profit of model 3V are

$α = C p − C g + C m + C t 2 s$
(29)

$H ^ ( R ) = C h R C l D$
(30)

where Cl is defined by Eq. (8).

Proof is similar to the way that the retailer behaves to maximize his profit in Proposition 2 and Proposition 1.

## 5. NUMERICAL EXPERIMENTS

To illustrate the models and investigate the effect of drop-shipping method, we consider a numerical example. Table 4 shows the parameter setting for a numerical example. First of all, we examine the approximation of mean physical stock used in Eq. (1) under the situation that the effect of lost-sales cannot be ignored.

### 5.1 Comparison Experiments for Inventory Models

It is known that inventory models including lostsales are difficult to analyze and to solve (Bijvank and Vis, 2011). Even though some studies propose the exact analytic approaches for the periodic review inventory model that includes lost-sales, it is too difficult to utilize those exact analytic approaches for analyzing a supplier-retailer supply chain. Thus approximate approaches arise as an alternative. Those approximate approaches show differences in calculating the mean physical stock. The wellknown approaches are the model proposed by Hadley and Whitin (1963) and the linear approximation (e.g., used in Moses and Seshadri, 2000). Those approximations are based on the assumption that the lost-sales are incurred only in very small quantities.

In this paper, however, we consider a supplier-retailer supply chain in which the effect of lost-sales cannot be ignored and thus the supplier offers a drop-shipping method for shortages. In order to analyze the supplier-retailer supply chain, we developed the inventory model as Eq. (1) instead of using the well-known approximations. This section conducts computation simulations to verify whether the newly developed model is suitable to use.

In the comparison experiment, we considered only the average annual cost (AC) that is consisted of the cost of carrying inventory and the cost of lost-sales, because other conditions are identical. The AC of H&W (Hadley and Whitin) model is

$C h ( R − μ − D T 2 + B ( R ) ) − C l B ( R ) T$
(31)

The AC of linear approximation is

$C h ( R − μ + ∫ 0 R ( R − x ) h ( x ) d x ) − C l B ( R ) T$
(32)

The AC of newly developed model is

$C h ( R 2 2 D T ) − C l B ( R ) T$
(33)

We conducted computer simulations to calculate the actual average annual cost for the values of R obtained from each model. The computer simulation was run for 10,000 periods, in which we considered the beginning period of 1,000 as a warming-up period for stabilizing the inventory system and collected the data for the lasting period of 9,000. For each R, the value of AC was calculated by averaging 10 simulation results. It was assumed that the average annual demand follows Normal distribution with a mean of 600 and a standard deviation of 40.

Tables 5 and 6 show the comparisons of average annual cost when L = 0.05 and 0.10, respectively. In this comparison, we used the service level to show how much lost-sales occur. The service level is the ratio of demands delivered directly from stock on hand. The service level is impacted by the value of R, and then the value of R is impacted by the cost of carrying inventory and lost-sales. We controlled the service level as increasing the cost of lostsales with fixing the cost of carrying inventory. From the results in Tables 5 and 6, we can observe that the H&W model and the linear approximation show better performance for the higher service level. However, as the service level is a little lower, the newly developed model shows the better performance. It is interested to notice that there will be significant errors in the inventory models ignoring lostsales, even for the high service level of 95%.

### 5.2 Numerical Analyses for the Models

With the parameters given in Table 4, we solved the numerical example. Table 7 shows the solutions for the numerical example. Note here again that the scenario 1 presents the baselines for evaluating the effect of dropshipping method in the scenarios 2 and 3. From the results in Table 7, we can observe several implications. First, as we expected, the drop-shipping method improves the supplier-retailer coordination performance. That is, the profits of PR, PS, and PT in the scenarios 2 and 3 are greater than or equal to those in the scenario 1.

Next, in terms of the retailer’s profit (i.e., PR in the independent model), the retailer is better off in the model 2S for the scenario 2, but in the model 3R for the scenario 3. Figures 2 and 3 show the profit curves in the scenarios 2 and 3, respectively. In order to present the profit curves of the model 3R in a two-dimension, the profit curves are drawn when R = 176 and α = 0.60, respectively. In the models 2R and 3R that the retailer is a Stackelberg leader, the retailer’s profit shows a constant line along the axis of α. This means that the supplier can cannibalize all the benefits of drop-shipping method for any value of α offered by the retailer. However, the retailer can optimize his profit by maximizing the value of R in the scenario 3.

In general, decision power gives an advantage to the supplier. The supplier can confirm his profit in the models 2S and 3S that the supplier is a Stackelberg leader. In the models 2R and 3R, however, the supplier’s profit depends on the retailer’s decision. Even though there is an opportunity that the supplier can maximize his profit in the model 2R, the opportunity cannot be realized without the retailer’s assistance.

The Nash equilibrium (i.e., the models 2N and 3N) is the same as the case of not using a drop-shipping method. The Nash equilibrium is not appealing to anybody. Figure 4 shows the best response functions of the retailer and the suppler for the scenario 3. (The best response functions for the scenario 2 were already shown in Figure 1.) Figure 4(a) presents the line in a three-dimension into the lines in a two-dimension. The Nash equilibrium is a cross-point of the best response lines, which R = 169, α = 1.0, and Cd = 1700.

As expected, the vertically integrated models outperform the independent models. In the scenarios 2 and 3 that use a drop-shipping method, the vertically integrated model does not include the Cd any longer. This means that there is no obvious way to divide the total profit into each player. If the supplier and the retailer are independent firms, it is required to introduce a new profit-sharing mechanism that allots the total profit fairly to each firm. However, we will leave it for the future study.

As the last observation, it is interested to indicate that the value of Cd is greater than the selling price of retailer, s in the model 3S, whereas the value of Cd is less than the supply expense of supplier (i.e., Cm + Ct) in the model 3R. In the model 3S, the retailer compensates the supplier for a drop-shipping method while reducing the inventory level. On the contrary, in the model 3R, the supplier compensates the retailer for the higher inventory level by the value of Cd less than the supply expense. Even though these models can be partly a coordinating mechanism between the supplier and the retailer, it is required to devise a new coordinating mechanism that induces the total profit as much as the vertically integrated model. We will also leave it for the future study.

## 6. CONCLUSION

In order to investigate the effect of a drop-shipping method for supplier-retailer coordination, we considered a two-stage supply chain that is composed of a supplier and a retailer. The fact that the retailer takes almost all of the inventory risks in a two-stage supply chain forces the retailer to limit an order quantity and subsequently lose potential sales. Thus the supplier provides the retailer with a drop-shipping method for shortages in pursuit of increasing profits for the both. The arising questions are: (i) under what condition, is the employment of a dropshipping method desirable for both the retailer and the supplier?; (ii) under what condition, is it fair for the both?; and (iii) what will happen to the both as the conditions change?

In order to investigate the effect of a drop-shipping method for shortages, we developed the profit models for the supplier, the retailer, and the total system with/without a drop-shipping method. For the three scenarios, we analyzed the scenario models using a Stackelberg game and presented the equilibrium solutions. Through numerical experiments, we first examined a new approximation of mean physical stock used in the profit model under the situation that the effect of lost-sales cannot be ignored. Then we could derive several implications: (1) a dropshipping method for shortages improves the supplierretailer coordination performance; (2) in terms of the retailer’s profit, the retailer is better off in the model 2S for the scenario 2, but in the model 3R for the scenario 3; (3) in general, decision power gives an advantage to the supplier; (4) the Nash equilibrium is not appealing to anybody; (5) the vertically integrated models outperform the independent models; and (6) in the model 3S, the retailer compensates the supplier for a drop-shipping method while reducing the inventory level, meanwhile, in the model 3R, the supplier compensates the retailer for the higher inventory level by the value of Cd less than the supply expense.

## ACKNOWLEDGEMENTS

This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2018S1A5A2A01028500).

## Figure

Stackelberg and nash equilibria in the scenario 2.

Profit curves in the scenario 2.

Profit curves in the scenario 3.

Best response functions in the scenario 3.

## Table

Summary of quantitative studies on drop-shipping method

Summary of notations

Taxonomy of scenario models

Parameter setting

Comparison of average annual cost when L = 0.05

Comparison of average annual cost when L = 0.10

Solutions for the numerical example

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