1. INTRODUCTION
Ecommerce has become widespread and made rapid progress because of the commercialization of the Internet, the expansion of sales network. Under this situation, direct online sales where manufacturers sell products directly to customers through online channels have been increasing rapidly. As the sales method of products, a dual channel (DC) comprising a retail channel (RC) run by a retailer and a direct online channel (DOC) run by a manufacturer has become popular (Cai et al., 2009;Khouja et al., 2010;Xu et al., 2012;Huang et al., 2012). It is necessary for agents in DC to consider the customer purchasing preference for products sales to operate DC profitably (Khouja et al., 2010).
Regarding this problem, Khouja et al. (2010) assumed two types of customers: customers who preferred purchasing products through RC and customers who preferred purchasing them through DOC. However, as for the customer purchasing preference, Khouja et al. (2010) did not discuss the existence ratio of indecisive customers who purchase the products in either RC or DOC in their analysis modeling. They also didn’t incorporate bias for the customer purchasing preference into analysis modeling regarding DC. Dan et al. (2014) analyzed competition in a dualchannel supply chain composed of a manufacturer, a traditional retailer and an electronic retailer. The traditional retailer retains a strong position. The influence of retailers’ power on channel members was studied and the competitive strategies of traditional retailers was explored. Zhang and He (2016) examined the optimal decisions for such dualchannel supply chain under three kinds of power structures: balance power between the retailer and manufacturer, retailerleader and manufacturer leader, and obtain the equilibrium solutions of prices. They used the Nash and Stackelberg game (Cachon and Netessine, 2004). They found that the impact of the acceptance of the online channel on the channel price strategy was greater than other parameters and there exists a unique and same threshold under the three kinds of power structures. Zhang et al. (2017) proposed retailer’s channel structure choice and pricing decisions in a supply chain with a manufacturer and a retailer who was the leader of the decisionmaking. They investigated how three possible channel structures, a pure offline channel, a pure online channel, and dual channels, affected pricing decisions and what the optimal channel structure was for the retailer.
Differently from previous studies mentioned above, this paper incorporates (i) the existence ratio of indecisive customers and (ii) cooperation and competition between agents in DC into the optimal sales strategy. For product sales under DC comprising RC and DOC, it is necessary to consider customers’ purchasing behaviors and preferences. This paper proposes the optimal sales strategy for a DC under the situation where the customer purchasing preference is unknown. This paper discusses three types of customers: (i) customers who prefer purchasing a single type of products in RC, (ii) customers who prefer purchasing them in DOC, (iii) indecisive customers who purchase in either RC or DOC. A retailer runs a RC and determines the optimal retail price. A manufacturer runs a DOC and determines the optimal direct online price. This paper assumes that each channel faces pricedependent demand. Two sales strategies are compared: the cooperated sales strategy (COSS) and the competed sales strategy (CMSS). Under COSS, a retailer and a manufacturer determine their prices cooperatively. Under CMSS, they determine their prices competitively. Using numerical examples, how (i) the uncertainty in the customer purchasing preference, (ii) the existence ratio of indecisive customers, (iii) the sensitivity of demand by the difference between the retail price and the direct online price and (iv) the decrease ratio of the product demand for the increase in price, affect the optimal decisions under COSS and CMSS.
The rest of this paper is organized as follows. Section 2 provides model descriptions regarding operational flows of a dual channel (DC) with a retail channel (RC) operated by a retailer and a direct online channel (DOC) operated by a manufacturer. Section 2 discusses model formulations in the DC including (i) demands of a single type of product in a RC and a DOC and (ii) the expected profits of the retailer and the manufacturer in the DC. In Section 4, the optimal sales strategies in the DC under COSS and CMSS, respectively. Section 5 shows results of numerical analyses and describes managerial insights. Finally, Section 6 summarizes conclusions and future researches for this paper.
The contribution of this paper provides managerial insights regarding the optimal sales strategies in DC considering (i) the existence ratio of indecisive customers and (ii) cooperation and competition between agents in DC by theoretical analysis.
2. MODEL DESCRIPTIONS
2.1. Operational Flows of a Dual Channel (DC)

(1) A manufacturer produces a single type of products with production cost c per product and sells them to a retailer with wholesale price w per product.

(2) The manufacturer sells the same products to customers with direct online price P_{d} per product and then incurs total operational cost Z_{d} in the direct online sale.

(3) A retailer sells the products to customers with retail price P_{r} per product and incurs total operational cost Z_{r} in the retail sale.

(4) After finishing sales of the products in DC, some products are returned to the retailer from RC at ratio r_{r} and to the manufacturer from DOC at ratio r_{d}.

(5) The manufacturer buys back the returned products in RC from the retailer with buyback price b per product. The manufacturer sells all the returned products with disposal price s per product in a second market.
2.2. Model Assumptions

(1) The customer purchasing preference x follows a probability distribution. The probability density function of x is f(x) . The closer to 0 x is, the higher the customer purchasing preference through the DOC is. Also, the closer to 1 x is, the higher the customer purchasing preference through the RC is.

(2) Using index t of the existence ratio of indecisive customers and the standard deviation σ of x, the indecisiveness of customers is expressed as tσ. Therefore, customers with purchasing preferences through DOC are distributed in 0≤x≤0.5−tσ and indecisive customers, who purchase in either RC or DOC, are distributed in 0.5−tσ ≤ x ≤0.5+tσ, and customers with purchasing preferences through RC are distributed in 0.5+tσ≤x≤1. In this case, the expected ratio of customers E_{d} with purchasing preferences through the DOC, the expected ratio of indecisive customers Y, and the expected ratio of customers E_{r} with purchasing preferences through the RC are calculated as
From Eqs. (1)(3), demands in both RC and DOC are affected by the probability distribution regarding the customer purchasing preference.

(3) The demand of indecisive customers is influenced by sensitivity ℓ of demand by the price difference between retail price P_{r} and direct online price P_{d}.

(4) Demands in both RC and DOC decrease at the decrease rate m as the individual sales price increases.
3. MODEL FORMULATIONS IN DC
3.1. Demands of the Products in RC and DOC
Formulations of the product demands in RC and DOC are discussed. Denote A as market volume (potential demand) of the products, X_{r} as E_{r} + 0.5Y and X_{d} as E_{d} + 0.5Y. From subsection 2.2, the product demand D_{r} in RC in terms of the retail price P_{r} and the direct online price P_{d} is formulated as
Here, D_{r} in Eq. (4) is obtained as the sum of the product demand in RC not influenced by P_{r} and P_{d} (first term), the product demand of indecisive customers fluctuating as to the price difference between P_{r} and P_{d} (second term), and the product demand in RC decreasing as to P_{r} (third term). Similarly, the product demand D_{d} in DOC in terms of P_{r} and P_{d} is formulated as(5)
3.2. The Expected Profits of a Retailer and a Manufacturer
First, the expected profit of a retailer is discussed. From subsection 3.1, the expected profit of the retailer ${\Pi}_{R}\left({P}_{r},\hspace{0.17em}{P}_{d}\right)$ for P_{r} and P_{d} is calculated as
Next, the expected profit of a manufacturer is discussed. From subsection 3.1, the expected profit of the manufacturer ${\Pi}_{M}\left({P}_{r},{P}_{d}\right)$ for Pr and Pd is calculated as
4. OPTIMAL SALES STRATEGY IN DC
4.1. Optimal Price Decisions under COSS
Under COSS, a retailer and a manufacturer cooperatively determine the optimal retail price and optimal direct online price by the following decision procedures proposed in this paper.
[Step 1] The firstsecond order partial differential equations of the retailer’s expected profit in Eq. (6) in terms of the retail price P_{r} under the direct online price P_{d} are derived as
$\left(\because \left(A\ell Y+m\right)>0,\hspace{0.17em}\hspace{0.17em}\left(1{r}_{r}\right)>0\right)$. The elicitation process of Eq. (8) is derived in Appendix A.
Similarly, those of the manufacturer’s expected profit in Eq. (7) in terms of P_{d} under P_{r} are derived as
$\left(\because \left(A\ell Y+m\right)>0,\hspace{0.17em}\hspace{0.17em}\left(1{r}_{d}\right)>0\right)$. The elicitation process of Eq. (10) is derived in Appendix B.
Here, it is verified that Eqs. (9) and (11) are negative from the following conditions: $\left(A\ell Y+m\right)>0,\hspace{0.17em}\hspace{0.17em}\left(1{r}_{r}\right)>0$ and (1  r_{d}) > 0. Therefore, Eq. (6) is a concave function for P_{r} under P_{d} , and Eq. (7) is that for P_{d} under P_{r} .
[Step 2] The tentative retail price P_{r}(P_{d}) under P_{d} is obtained as P_{r}, satisfying $\partial {\Pi}_{R}\left({P}_{r},{P}_{d}\right)\hspace{0.17em}/\hspace{0.17em}\partial {P}_{r}=0$ as
The elicitation process of Eq. (12) is derived in Appendix C. Similarly, the tentative direct online price P_{d}(P_{r}) under Pr is obtained as Pd, satisfying $\partial {\Pi}_{M}\left({P}_{r},{P}_{d}\right)\hspace{0.17em}/\hspace{0.17em}\partial {P}_{d}=0$ as
The elicitation process of Eq. (13) is derived in Appendix D.
[Step 3] The optimal retail price ${P}_{r}{{}^{COSS}}^{*}$ and the optimal direct online price ${P}_{d}{{}^{COSS}}^{*}$ are determined as solutions of simultaneous equations in Eqs. (12) and (13) as
The elicitation processes of Eqs. (14) and (15) are derived in Appendix E.
4.2. Optimal Price Decisions under CMSS
Under CMSS, a retailer and a manufacturer competitively determine the optimal retail price and the optimal direct online price. This paper discusses two types of CMSSs: CMSS 1: a retailer is the leader and a manufacturer is the follower and CMSS 2: a manufacturer is the leader and a retailer is the follower. In this paper, the leader of the decisionmaking determines the own price optimally so as to maximize the own expected profit. The follower of the decisionmaking determines the own price optimally so as to maximize the own expected profit under the optimal price determined by the leader.
4.2.1. Optimal Price Decisions under CMSS 1
The decision procedures under CMSS 1 proposed in this paper is shown below.
[Step 1] A retailer, who is the leader of the decisionmaking, determines the tentative retail price P_{r}(P_{d}) which maximizes the own expected profit under the direct online price P_{d}. P_{r}(P_{d}) is obtained as the following solution:
satisfying $\partial {\Pi}_{R}\left({P}_{r},\hspace{0.17em}\hspace{0.17em}{P}_{d}\right)\hspace{0.17em}/\hspace{0.17em}\partial {P}_{r}=0$ under P_{d}.
[Step 2] Substituting the tentative retail price P_{r}(P_{d}) in Eq. (16) into Eq. (7), the manufacturer’s expected profit is rewritten as
Here, k_{d}_{1}, n_{d}_{1}, k_{r}_{1}, n_{r}_{1}, a_{d}, b_{d} and c_{d} in Eq. (17) are defined as
$\begin{array}{l}{k}_{d1}=\left(A\ell Y+m\right)+\frac{{\left(A\ell Y\right)}^{2}}{2\left(A\ell Y+m\right)}\\ {n}_{d1}=A{X}_{d}+\frac{A\ell Y}{2\left(A\ell Y+m\right)}A{X}_{r}+\frac{\left(wb{r}_{r}\right)A\ell Y}{2\left(1{r}_{r}\right)}\\ {k}_{r1}=\frac{1}{2}A\ell Y\\ {n}_{r1}=\frac{1}{2}A{X}_{r}\frac{\left(wb{r}_{r}\right)\left(A\ell Y+m\right)}{2\left(1{r}_{r}\right)}\\ {a}_{d}=\left(1{r}_{d}\right){k}_{d1}\\ {b}_{d}=\left(1{r}_{d}\right){n}_{d1}+\left(s{r}_{d}c\right){k}_{d1}+\left(wb{r}_{r}+s{r}_{r}s{r}_{d}\right){k}_{r1}\\ {c}_{d}=\left(s{r}_{d}c\right)\hspace{0.17em}{n}_{d1}+\left(wb{r}_{r}+s{r}_{r}s{r}_{d}\right){n}_{r1}{Z}_{d}\end{array}$
The elicitation process of Eq. (17) is derived in Appendix F.
[Step 3] The manufacturer determines the optimal direct online price in CMSS 1 ${P}_{d}{{}^{CMSS1}}^{*}$ under the tentative retail price ( ) r P_{d} P so as to maximize the expected profit in DOC. Eq. (17) is a quadratic function in terms of P_{d} . From the characteristic, the optimal direct online price under CMSS 1 CMSS1* P_{d} is determined as
[Step 4] Substituting ${P}_{d}{}^{CMSS1*}$ in Eq. (18) into Eq.(16), the optimal retail price under CMSS 1 ${P}_{r}{}^{CMSS1*}$ which maximizes the expected profit in RC is determined as(19)
4.2.2. Optimal Price Decisions under CMSS 2
The decision procedures under CMSS 2 proposed in this paper is shown below.
[Step 1] A manufacturer, who is the leader of the decisionmaking, determines the tentative retail price P_{d}(P_{r}) which maximizes the own expected profit under the retail price P_{r}. P_{d}(P_{r}) is obtained as the following solution:
satisfying $\partial {\Pi}_{M}\left({P}_{r},\hspace{0.17em}{P}_{d}\right)\hspace{0.17em}/\hspace{0.17em}\partial {P}_{d}=0$ under P.
[Step 2] Substituting the tentative retail price P_{d}( P_{r}) in Eq. (20) into Eq. (6), the retailer’s expected profit is rewritten as
Here, k_{r}_{2}, n_{r}_{2}, a_{r}, b_{r} and c_{r} in Eq. (21) are defined as
$\begin{array}{c}{k}_{r2}=\left(A\ell Y+m\right)+\frac{{\left(A\ell Y\right)}^{2}}{2\left(A\ell Y+m\right)}\\ {n}_{r2}=A{X}_{r}+\frac{A\ell Y}{2\left(A\ell Y+m\right)}A{X}_{d}\frac{\left(s{r}_{d}c\right)A\ell Y}{2\left(1{r}_{d}\right)}\\ +\frac{\left(wb{r}_{r}+s{r}_{r}s{r}_{d}\right){\left(A\ell Y\right)}^{2}}{2\left(A\ell Y+m\right)\left(1{r}_{d}\right)}\\ {a}_{r}=\left(1{r}_{r}\right){k}_{r2}\\ {b}_{r}=\left(1{r}_{r}\right){n}_{r2}\left(wb{r}_{r}\right){k}_{r2}\\ {c}_{r}=\left(wb{r}_{r}\right){n}_{r2}{Z}_{r}.\end{array}$
The elicitation process of Eq. (21) is derived in Appendix G.
[Step 3] The retailer determines the optimal retail price in CMSS 2 ${P}_{r}{{}^{CMSS2}}^{*}$ under the tentative direct online price P_{d}( P_{r}) so as to maximize the expected profit in RC. Eq. (21) is a quadratic function in terms of P_{r} . From the characteristic, the optimal retail price under CMSS 2 CMSS ${P}_{r}{{}^{CMSS2}}^{*}$ is determined as
[Step 4] Substituting ${P}_{r}^{CMSS2}{}^{*}\hspace{0.17em}={b}_{r}/\left(2{a}_{r}\right)$ in Eq. (22) into Eq. (20), the optimal direct online price in CMSS 2 ${P}_{d}{}^{CMSS2*}$ which maximizes the expected profit in DOC is determined as(23)
5. NUMERICAL SIMULATIONS
The numerical analysis illustrates the results of the optimal decisions under the strategies, COSS and CMSS. The expected profits of a retailer and a manufacturer under COSS are compared with those under CMSS. In addition, the analysis clarifies numerically how (i) the uncertainty in the customer purchasing preference, (ii) the existence ratio of indecisive customers, (iii) the sensitivity of demand by the difference between the retail price and the direct online price and (iv) the decrease ratio of the product demand for the increase in price, affect the optimal decisions under COSS and CMSS.
The following system parameters are used as the numerical examples:
$A=100000,\hspace{0.17em}\hspace{0.17em}w=300,\hspace{0.17em}\hspace{0.17em}b=50,\hspace{0.17em}\hspace{0.17em}s=20,\hspace{0.17em}\hspace{0.17em}c=40,\hspace{0.17em}\hspace{0.17em}{r}_{r}=0.01,$ ${r}_{d}=0.10,\hspace{0.17em}\hspace{0.17em}{Z}_{r}=300000,\hspace{0.17em}\hspace{0.17em}{Z}_{d}=800000,\hspace{0.17em}t=1.5,\hspace{0.17em}\hspace{0.17em}\ell =0.01,$ $m=65.$
Customer’s purchasing preference x is modeled by using the beta distribution with shape parameter m and scale parameter n. The probability density function $f\left(xm,\hspace{0.17em}n\right)$ of $x\left(0\le x\le 1\right)$ is given as(24)
Where Γ(□) denotes the gamma function.
In numerical examples, the following six cases of the combination of (m, n) regarding x are considered:

Case1: $B\left(x\hspace{0.17em}1.5,\hspace{0.17em}\hspace{0.17em}4.5\right)$ assuming very higher purchasing preference through DOC

Case2: $B\left(x\hspace{0.17em}3.5,\hspace{0.17em}4.5\right)$ assuming relatively higher purchasing preference through DOC

Case3: $B\left(x\hspace{0.17em}1,\hspace{0.17em}1\right)$ assuming that x is uniformly distributed within the range where 0 ≤ x ≤ 1

Case4: $B\left(x\hspace{0.17em}5,\hspace{0.17em}5\right)$ assuming that x is distributed like normal distribution, which has no biases for purchasing preference, within the range where 0 ≤ x ≤ 1

Case5: $B\left(x\hspace{0.17em}4.5,\hspace{0.17em}3.5\right)$ assuming relatively higher purchasing preference through RC

Case6: $B\left(x\hspace{0.17em}4.5,\hspace{0.17em}\hspace{0.17em}1.5\right)$ assuming very higher purchasing preference through RC.
Figure 1 shows the distribution of customer’s purchasing preference x in Cases 16. Figure 1 indicates that the closer to 0 x is, the higher the customer purchasing preference through the DOC is, meanwhile, the closer to 1 x is, the higher the customer purchasing preference through the RC is.
By substituting numerical examples into Eqs. (1), (2),
and (3), the expected ratios of customers with purchasing preference E_{d}, Y, and E_{r} are calculated as
$\begin{array}{l}\text{Case1}:\hspace{0.17em}{E}_{d}=0.5732,Y=0.4219,{E}_{r}=0.0049\\ \text{Case2:}\hspace{0.17em}{E}_{d}=0.1413,Y=0.8256,{E}_{r}=0.0331\\ \text{Case3:}\hspace{0.17em}{E}_{d}=0.0670,Y=0.8660,{E}_{r}=0.0670\\ \text{Case4:}\hspace{0.17em}{E}_{d}=0.0699,Y=0.8601,{E}_{r}=0.0699\\ \text{Case5:}\hspace{0.17em}{E}_{d}=0.0331,Y=0.8256,{E}_{r}=0.1413\\ \text{Case6:}\hspace{0.17em}{E}_{d}=0.0049,Y=0.4219,{E}_{r}=0.5732\end{array}$
It can be seen that values of E_{d}, Y and E_{r} in Case 1 are symmetrical about those in Case 6. The combinations of both (Case 2, Case 5) and (Case 3, Case 4) are almost the same relations. Therefore, numerical analysis in aftermentioned 5.1 and 5.2 are conducted by using the expected ratios of customers of Cases 1, 2 and 4. Regarding Tables 14 below, the loss of demand in the whole system means the sum of the loss of demand in RC and that in DOC.
5.1. Effect of Uncertainty in Customer Purchasing Preference on the Optimal Price Decisions under Each Sales Strategy and the Expected Profits
Table 1 shows the effect of the customer purchasing preference on the optimal price decisions under COSS, CMSS 1 and CMSS 2 and the expected profits in DC.
From Table 1, the following results can be seen:
The optimal direct online price under COSS, CMSS 1 and CMSS 2 are the highest in Case 1 which has very higher the customer purchasing preference through DOC.
This leads to results that the expected profits in DOC under COSS, CMSS 1 and CMSS 2 are the highest in Case 1. In addition, the difference between the optimal retail price and the optimal direct online price in COSS becomes smaller, as the distribution of the customer purchasing preference changes from Case 1 through Case 2 to Case 4. This is because the smaller the bias of the distribution of the customer purchasing preference is, not only the more the expected ratio of indecisive customers are, but also the more intensified price competition between a retailer and a manufacturer is.
5.2. Comparison of Benefits of Optimal Price Decisions under Each Sales Strategy
Benefits of the optimal price decisions under COSS, CMSS 1 and CMSS 2 are compared.
From Table 1, the following results can be seen: Regardless of the distribution of the customer purchasing preference, the expected profits of a retailer and a manufacturer have the following magnitude relations between COSS, CMSS 1 and CMSS 2. For a retailer, (CMSS 1) > (COSS) > (CMSS 2), meanwhile, for a manufacturer, (CMSS 2) > (COSS) > (CMSS 1). These results lead to the following consequence regarding benefits of the optimal price decisions under COSS and CMSS. (CMSS where the own agent is the leader of the decisionmaking) > (COSS) > (CMSS where the own agent is the follower of the decisionmaking). Therefore, CMSS is the most beneficial sales strategy for a retailer and a manufacturer in the situation where they can determine which agent is the leader of the decisionmaking. In contrast, COSS is the most profitable sales strategy for a retailer and a manufacturer in the situation where they cannot determine which agent is the leader of the decisionmaking.
5.3. Effect of Existence Ratio of Indecisive Customers on the Optimal Price Decision under Each Sales Strategy and the Expected Profits
Table 2 shows the effect of the index t of the existence ratio of indecisive customers on the optimal price decisions and the expected profits in DC under each sales strategy. Here, Case 4 of the distribution of the customer purchasing preference is adopted. From Table 2, the following results can be seen:

The higher t is, the lower the optimal retail price and the optimal direct online price are. This is because the higher t is, the higher the expected ratio of indecisive customers, Y, is and then the increase in Y leads to the situation where price competition between the optimal retail price determined by a retailer and the optimal direct online price determined by a manufacturer is more intensified. From the results on the optimal price decisions, the higher t is, the lower the loss of demand in the whole system is.

Regardless of t, the magnitude relation between the expected profits of the retailer and the manufacturer under each sales strategy is the same as that in 5.2.
5.4. Effect of Sensitivity ℓ in Price Difference on the Optimal Price Decisions under Each Sales Strategy and the Expected Profits
Table 3 shows the effect of the sensitivity ℓ in price difference on the optimal price decisions and the expected profits in DC under each sales strategy. Here, Case 4 of the distribution of the customer purchasing preference is adopted. From Table 3, the following results can be seen:

The higher ℓ is, the lower the optimal retail price and the optimal direct online price are. This is because the increase in ℓ leads to the situation where price competition between the optimal retail price determined by a retailer and the optimal direct online price determined by a manufacturer is more intensified. From the results on the optimal price decisions, the higher ℓ is, the lower the loss of demand in the whole system is.

Regardless of ℓ , the magnitude relation between the expected profits of the retailer and the manufacturer under each sales strategy is same as that in 5.2.
5.5. Effect of Decrease Ratio m on the Optimal Price Decisions under Each Sales Strategy and the Expected Profits
Table 4 shows the effect of decrease ratio m of demand on the optimal price decisions and the expected profits in DC under each sales strategy. Here, Case 4 of the distribution of the customer purchasing preference is adopted. From Table 4, the following results can be seen:
The higher m is, the lower the optimal retail price and the optimal direct online price are. This is because the increase in m means the increase in the loss of demand in the whole system. To avoid it, a retailer and a manufacturer determine their prices lower.

The higher m is, the higher the loss of the demand in the whole system is. This is because the increase in the loss of demand by increasing m is higher than the increase in the product demand by determining the optimal retail price and the optimal direct online price lower.

Regarding benefits of optimal price decisions under each sales strategy, the following different results are obtained as to the range of m . Two results about the expected profits of the retailer and the manufacturer.

(1) When m is low, the expected profits of the retailer and the manufacturer are the highest under CMSS 1 where the retailer is the leader of the decisionmaking.

(2) When m is high, the magnitude relation between the expected profits of the retailer and the manufacturer under each sales strategy is the same as that in 5.2.

6. CONCLUSIONS
This paper discussed the optimal sales strategies for a dual channel under cooperation and competition in a retail channel (RC) and a direct online channel (DOC), considering the customer purchasing preference between both channels. This paper proposed two sales strategies: the cooperated sales strategy (COSS) between two agents: a retailer and a manufacturer and the competed sales strategy (CMSS) between the two agents. Under COSS, the optimal decisions for retail price and direct online price were made so as to maximize the expected profits of the retailer in RC and the manufacturer in DOC. Under CMSS, Under CMSS, a retailer and a manufacturer competitively determined the optimal retail price and the optimal direct online price. Concretely, this paper discussed two types of CMSSs: CMSS 1: a retailer is the leader of the decisionmaking and a manufacturer is the follower of the decisionmaking and CMSS 2: a manufacturer is the leader of the decisionmaking and a retailer is the follower of the decisionmaking. The analysis clarified numerically how (i) the uncertainty in the customer purchasing preference, (ii) the existence ratio of indecisive customers, (iii) the sensitivity in demand by the difference between retail price and direct online price and (iv) the decrease ratio of the products demand for increase of price, affected the optimal decisions under COSS and CMSS. Results of theoretical analysis and numerical analysis in this paper verified the following managerial insights:

When the bias of the customer purchasing preference is large towards either RC or DOC, the agent who has the high bias determines the own optimal price highly.

The smaller the bias of the customer purchasing preference is, the lower the deference between the optimal retail price and the optimal direct online price is.

COSS is the most profitable sales strategy for two agents: a retailer and a manufacturer in the situation where they cannot determine which agent is the leader of the decisionmaking.

CMSS is the most beneficial sales strategy for two agents: a retailer and a manufacturer in the situation where they can determine which agent is the leader of the decisionmaking.
As future researches, it will be necessary to incorporate the following topics into the DC in this paper:

Effect of difference in lead time between a retail channel and a direct online channel

Effect of advertising in a DC

Customer purchasing preference considering a brand strength as well as a price difference between agents

Proposal of the more beneficial cooperated sales strategy

Time limit for sales considering the product life cycle

DC model with multiple retailers and multiple manufacturers
APPENDIX A
· Elicitation process of Eq. (8)
Substituting Eq. (4) into Eq. (6), the retailer’s expected profit for P_{r}, P_{d}, ${\Pi}_{R}\left({P}_{r},\hspace{0.17em}{P}_{d}\right),$ is rewritten as
Eq. (8) can be derived by differentiating partially Eq. (A1) in terms of P_{r}.
APPENDIX B
· Elicitation process of Eq. (10)
Substituting Eq. (5) into Eq. (7), the manufacturer’s expected profit for P_{r}, P_{d}, ${\Pi}_{M}\left({P}_{r},{P}_{d}\right)\hspace{0.17em}$, is rewritten as
Eq. (10) can be derived by differentiating partially Eq. (B1) in terms of P_{d}.
APPENDIX C
· Elicitation process of Eq. (12)
The firstorder condition of Eq. (8) is obtained as
By solving Eq. (C1) in terms of retail price P_{r}, the retailer’s provisional retail price P_{r}(P_{d}) under direct online price d P can be derived as
Eq. (C2) is corresponding to Eq. (12). Therefore, the elicitation process of Eq. (12) is shown.
APPENDIX D
· Elicitation process of Eq. (13)
The firstorder condition of Eq. (10) is obtained as
By solving Eq. (D1) in terms of direct online price P_{d}, the provisional direct online price P_{d}(P_{r}) under retail price P_{r} can be derived as
Eq. (D2) is corresponding to Eq. (13). Therefore, the elicitation process of Eq. (13) is shown.
APPENDIX E
· Elicitation processes of Eq. (14) and Eq. (15)
The solution of the simultaneous equations of Eqs. (12) and (13) are shown as follows:
First, the solution method of the following simultaneous equations regarding variables x and y
is discussed as follows
Multiply both sides of Eq.(E1a) by c
can be obtained.
Similarly, Multiply both sides of Eq. (E1b) by a
can be obtained.
When ad − bc ≠ 0 , subtracting Eq. (E3) from Eq. (E2), variable y can be obtained as follows:(E4)
Similarly, multiply both sides of Eq. (E1a) by d
can be obtained.
Similarly, Multiply both sides of Eq. (E2b) by f
can be obtained.
When ad − bc ≠ 0, subtracting Eq. (E7) from Eq. (E6), variable x can be obtained as follows:(E9)
Therefore, the solution of the simultaneous equations in Eq. (E1) regarding variables x and y can be obtained as Eqs. (E5) and (E8).
Using the solution method mentioned above, by rearranging Eq. (12), the following result can be obtained.(E10)
Using the solution method mentioned above, by rearranging Eq. (13), the following result can be obtained.(E11)
Here, using the following notation
$\begin{array}{l}x={P}_{d}\\ y={P}_{r}\\ a=1\\ b=\frac{A\ell Y}{2\left(A\ell Y+m\right)}\\ c=\frac{A\ell Y}{2\left(A\ell Y+m\right)}\\ d=1\\ e=\frac{A{X}_{d}}{2\left(A\ell Y+m\right)}\frac{s{r}_{d}c}{2\left(1{r}_{d}\right)}\hspace{0.17em}+\frac{\left(wb{r}_{r}+s{r}_{r}s{r}_{d}\right)A\ell Y}{2\left(A\ell Y+m\right)\left(1{r}_{d}\right)}\\ f=\frac{A{X}_{r}}{2\left(A\ell Y+m\right)}+\frac{wb{r}_{r}}{2\left(1{r}_{r}\right)}\end{array}$
${P}_{r}{{}^{COSS}}^{*}$ can be obtained as
Eq. (E12) is corresponding to Eq. (14). Therefore, the elicitation process of Eq. (14) is shown.
${P}_{d}{{}^{CMSS}}^{*}$ can be obtained as
Eq. (E13) is corresponding to Eq. (15). Therefore, the elicitation process of Eq. (15) is shown.
APPENDIX F
· Elicitation process of Eq. (17)
First, substituting Eq. (16) into Eq. (5), the demand of the direct online channel can be obtained as
Next, substituting Eq. (16) into Eq. (4) and rearranging it, the demand of the retail channel can be obtained as
Next, substituting Eqs. (F1) and (F2) into Eq. (7) and rearranging it, the manufacturer’s expected profit can be obtained as
Eq. (F3) is corresponding to Eq. (17). Therefore, the elicitation process of Eq. (17) is shown.
APPENDIX G
· Elicitation process of Eq. (21)
First, substituting Eq. (14) into Eq. (4), the demand of the retail channel can be obtained as
Next, substituting Eqs. (G1) into Eq. (6) and rearranging it, the retailer’s expected profit can be obtained as
Eq. (G2) is corresponding to Eq. (21). Therefore, the elicitation process of Eq. (21) is shown.