Consignment supply chain cooperation for complementary products under online to offline business mode

Abstract

The purpose of this paper is to explore the cooperation issues in a consignment supply chain with complementary products under Online to Offline Business Mode (O2O mode). The centralized, decentralized and cooperative game-theoretic decision models are developed, analyzed and compared for a generic consignment supply chain with complementary products under O2O mode. Similar models are also developed for that under pure online- and offline-channel modes for comparison purpose. The corresponding numerical and sensitivity analyses are conducted to validate the analytical findings and derive managerial insights. It is found that the cooperative strategy always creates many more profits for the supply chain by setting a much lower retail price and generating much more demands than those of the decentralized strategies. The best supply chain strategy for a consignment supply chain with complementary products sold through both online and offline channels is identified as the cooperative strategy. An extensive sensitivity analysis of seven key parameters on an O2O cooperative decision model has derived several insightful findings for the O2O consignment supply chain practitioners with complementary products. These managerial insights will facilitate better decision-making for the O2O retailers and their complementary suppliers. Since the cooperation of a consignment supply chain under O2O mode with complementary products is a fresh research domain, there are more research issues to be explored.

Appendices

Appendix A: Proofs for Sect. 4.1

1.1 Centralized decision model

When the consignment supply chain with complementary products under O2O mode takes a centralized strategy, the optimal profit function of the supply chain can be formulated as follows:

$$\mathop {\hbox{max} }\limits_{p,z} \;\varPi_{SC}^{o} \left( {p,z} \right) = y_{o} \left( p \right)\left\{ {pE\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - c_{o} z} \right\}$$

When the distribution of random variable x satisfies the IGFR condition, the first order conditions $\left( {p_{c}^{o} ,z_{c}^{o} } \right)$ determine a unique solution to the above optimization problem. Solving the first-order condition of the optimization profit function with respect to the retail price $p$ and the stock factor $z$:

$$\frac{{\partial \varPi_{SC} \left( {p,z} \right)}}{\partial z} = y\left( p \right)\left[ {\left( {p - c_{o} } \right) - pF\left( z \right)} \right] = 0$$

$$\frac{{\partial \varPi_{SC} \left( {p,z} \right)}}{\partial p} = \frac{y\left( p \right)}{p}\left\{ {\left( {b - \theta } \right)c_{o} z - \left( {b - \theta - 1} \right)p\left[ {z - {\rm M}\left( z \right)} \right]} \right\} = 0$$

where ${\rm M}\left( z \right) = \int_{A}^{z} {\left( {z - x} \right)f\left( x \right)dx}$

Then, we can get the optimal total retail price and the distribution function of the centralized optimal stock factor as follows:

$$p_{c}^{o} = \frac{b - \theta }{b - \theta - 1}\frac{{c_{o} z_{c}^{o} }}{{z_{c}^{o} - {\rm M}\left( {z_{c}^{o} } \right)}}$$

$$F\left( {z_{c}^{o} } \right) = \frac{{p_{c}^{o} - c_{o} }}{{p_{c}^{o} }} = \frac{1}{b - \theta } + \frac{{\left( {b - \theta - 1} \right){\rm M}\left( {z_{c}^{o} } \right)}}{{\left( {b - \theta } \right)z_{c}^{o} }}$$

where ${\rm M}\left( {z_{c}^{o} } \right) = \int_{A}^{{z_{c}^{o} }} {\left( {z_{c}^{o} - x} \right)f\left( x \right)dx}$

By observing the structure of the optimal total retail price, it shows that there is a positive relationship between the optimal total retail price $p_{c}^{o}$ and the total channel cost $c_{o}$. Thus, the optimal retail price of the ith product should have the same structure and the relationship between the optimal retail price of the ith product $p_{i}^{oc}$ and the corresponding proportion of the total channel cost $\frac{{\alpha_{i} }}{{\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}c_{o}$. Thus, we can get the optimal retail price of the ith product as follows:

$$p_{i}^{oc} = \frac{{\alpha_{i} }}{{\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}p_{c}^{o}$$

Then, we can have the centralized optimal order quantity as follows:

$$q_{c}^{o} = y_{o} \left( {p_{c}^{o} } \right)z_{c}^{o}$$

Substituting the optimal retail price $p_{c}^{o}$ and the optimal stock factor $z_{c}^{o}$ into the profit function of the centralized consignment supply chain under O2O mode, we can obtain the optimal profit of the consignment supply chain as follows:

$$\varPi_{SC}^{oc} = \frac{{c_{o} q_{c}^{o} }}{b - \theta - 1}$$

1.2 Decentralized decision model

• (1) Suppliers’ Simultaneous Decision (SI)

The profit functions of the supplier i and the O2O retailer in the consignment supply chain under the revenue sharing contract can be formulated as follows:

$$\varPi_{{S_{i} }}^{o} \left( {p_{i} ,z} \right) = y_{o} \left( p \right)\left\{ {\left( {1 - \phi } \right)p_{i} E\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - \alpha_{i} c_{o} z} \right\}$$

$$\varPi_{R}^{o} \left( \phi \right) = y_{o} \left( p \right)\left\{ {\phi pE\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)c_{o} z} \right\}$$

Solving the first-order condition of the supplier i’s optimization problem with respect to (w.r.t.) the retail price $p_{i}$:

$$\frac{{\partial \varPi_{{S_{i} }} \left( {p_{i} ,z} \right)}}{{\partial p_{i} }} = \frac{y\left( p \right)}{p}\left\{ \begin{aligned} - \left( {b - \theta - 1} \right)\left( {1 - \phi } \right)p_{i} \left[ {z - {\rm M}\left( z \right)} \right] + \left( {b - \theta } \right)\alpha_{i} c_{o} z \hfill \\ + \left( {1 - \phi } \right)p_{ - i} \left[ {z - {\rm M}\left( z \right)} \right] \hfill \\ \end{aligned} \right\} = 0$$

where ${\rm M}\left( z \right) = \int_{A}^{z} {\left( {z - x} \right)f\left( x \right)dx}$

We can obtain the reaction function of $p_{i}$ and $p_{ - i}$ w.r.t. $z$ as follows:

$$p_{i}^{od} \left( z \right) = \frac{{\left( {b - \theta - n} \right)\alpha_{i} + \sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}{b - \theta - n}\frac{{c_{o} z}}{{\left( {1 - \phi } \right)\left[ {z - {\rm M}\left( z \right)} \right]}}$$

$$p_{ - i}^{od} \left( z \right) = \frac{{\left( {b - \theta - 1} \right)\sum\nolimits_{i = 1}^{n} {\alpha_{i} } - \left( {b - \theta - n} \right)\alpha_{i} }}{b - \theta - n}\frac{{c_{o} z}}{{\left( {1 - \phi } \right)\left[ {z - {\rm M}\left( z \right)} \right]}}$$

Plugging $p_{i}^{od} \left( z \right)$ and $p_{ - i}^{od} \left( z \right)$ into supplier i’s expect profit function, we can get the supplier i’s optimal problem as follows:

$$\mathop {\hbox{max} }\limits_{z} \;\varPi_{{S_{i} }}^{o} \left( {p_{i}^{od} \left( z \right),p_{ - i}^{od} \left( z \right),z} \right) = y_{o} \left( {p_{i}^{od} \left( z \right),p_{ - i}^{od} \left( z \right)} \right)\left\{ {\left( {1 - \phi } \right)p_{i}^{od} \left( z \right) \cdot E\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - \alpha_{i} c_{o} z} \right\}$$

Solving the first-order condition of the supplier i’s optimization problem w.r.t. the stock factor $z$, we can obtain:

$$\begin{aligned} \frac{{d\varPi_{{S_{i} }}^{o} \left( {p_{i}^{od} \left( z \right)|p_{ - i}^{od} \left( z \right),z} \right)}}{dz} \hfill \\ = \frac{{\partial \varPi_{{S_{i} }}^{o} \left( {p_{i}^{od} \left( z \right)|p_{ - i}^{od} \left( z \right),z} \right)}}{\partial z} + \frac{{\partial \varPi_{{S_{i} }}^{o} \left( {p_{i}^{od} \left( z \right)|p_{ - i}^{od} \left( z \right),z} \right)}}{{\partial p_{ - i} }}\frac{{dp_{ - i}^{od} \left( z \right)}}{dz} + \frac{{\partial \varPi_{{S_{i} }}^{o} \left( {p_{i}^{od} \left( z \right)|p_{ - i}^{od} \left( z \right),z} \right)}}{{\partial p_{i} }}\frac{{dp_{i}^{od} \left( z \right)}}{dz} \hfill \\ = \frac{{y\left( p \right)c_{s} }}{{\left( {b - \theta - n} \right)\left[ {z - {\rm M}\left( z \right)} \right]}}\left[ {z + \left( {b - \theta - 1} \right){\rm M}\left( z \right) - \left( {b - \theta } \right)zF\left( z \right)} \right] = 0 \hfill \\ \end{aligned}$$

Let $L\left( z \right) = z + \left( {b - \theta - 1} \right){\rm M}\left( z \right) - \left( {b - \theta } \right)zF\left( z \right)$

For any $z \in \left[ {A,B} \right)$, solving the first derivative of $L\left( z \right)$ w.r.t. $z$:

$$L^{'} \left( z \right) = \left[ {1 - F\left( z \right)} \right]\left[ {1 - \left( {b - \theta } \right)g\left( z \right)} \right]$$

According to IGFR assumption, $g^{'} \left( z \right) = h\left( z \right) + zh^{'} \left( z \right) > 0$, i.e., $g\left( z \right)$ is increasing in $z$, and thus $L\left( z \right)$ is decreasing in $z$. Then, $L^{'} \left( z \right)$ crosses 0 at most once (from positive to negative). Since $L\left( A \right) = A > 0$, $L\left( B \right) = - \left( {b - \theta - 1} \right)\mu < 0$, the first derivative $L^{'} \left( z \right)$ either is always negative or changes from positive to negative. Thus, $L\left( z \right)$ crosses 0 exactly once, from positive to negative. Therefore, the equation $L\left( z \right) = 0$ has a unique solution.

Therefore, the distribution function of the equilibrium stock factor as follows:

$$F\left( {z_{d}^{o} } \right) = \frac{1}{b - \theta } + \frac{{\left( {b - \theta - 1} \right){\rm M}\left( {z_{d}^{o} } \right)}}{{\left( {b - \theta } \right)z_{d}^{o} }} = F\left( {z_{c}^{o} } \right)$$

On this basis, we can get the supplier’s reaction function of equilibrium retail price of product i w.r.t. the revenue keeping rate, the reaction function of total equilibrium retail price of product set w.r.t. the revenue keeping rate, and the distribution function of the equilibrium stock factor as follows:

$$p_{i}^{od} \left( \phi \right) = \frac{{\left( {b - \theta - n} \right)\alpha_{i} + \sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}{b - \theta - n}\frac{{c_{o} z_{c}^{o} }}{{\left( {1 - \phi } \right)\left[ {z_{c}^{o} - {\rm M}\left( {z_{c}^{o} } \right)} \right]}} = \frac{{\left( {b - \theta - 1} \right)\left[ {\left( {b - \theta - n} \right)\alpha_{i} + \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right]}}{{\left( {b - \theta } \right)\left( {b - \theta - n} \right)\left( {1 - \phi } \right)}}p_{c}^{o}$$

$$p_{d}^{o} \left( \phi \right) = \frac{{\left( {b - \theta } \right)\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}{b - \theta - n}\frac{{c_{o} z_{c}^{o} }}{{\left( {1 - \phi } \right)\left[ {z_{c}^{o} - {\rm M}\left( {z_{c}^{o} } \right)} \right]}} = \frac{{\left( {b - \theta - 1} \right)\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}{{\left( {b - \theta - n} \right)\left( {1 - \phi } \right)}}p_{c}^{o}$$

$$F\left( {z_{d}^{o} } \right) = F\left( {z_{c}^{o} } \right)$$

Then, we have the reaction function of the decentralized equilibrium order quantity of product set w.r.t. the revenue keeping rate as follows:

$$q_{d}^{o} \left( \phi \right) = a\left( {p_{d}^{o} } \right)^{{ - \left( {b - \theta } \right)}} z_{d}^{o} = \left[ {\frac{{\left( {b - \theta - n} \right)\left( {1 - \phi } \right)}}{{\left( {b - \theta - 1} \right)\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}} \right]^{b - \theta } a\left( {p_{c}^{o} } \right)^{{ - \left( {b - \theta } \right)}} z_{c}^{o} = \frac{{\left( {b - \theta - n} \right)^{b - \theta } \left( {1 - \phi } \right)^{b - \theta } }}{{\left( {b - \theta - 1} \right)^{b - \theta } \left( {\sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)^{b - \theta } }}q_{c}^{o}$$

Substituting $p_{d}^{o} \left( \phi \right)$ and $z_{d}^{o}$ into the O2O retailer’s expected profit function, we can obtain the O2O retailer’s optimal problem as follows:

$$\begin{aligned} \mathop {\hbox{max} }\limits_{\phi } \;\varPi_{R}^{o} \left( \phi \right) = y_{o} \left( {p_{d}^{o} \left( \phi \right)} \right)\left\{ {\phi p_{d}^{o} \left( \phi \right)\left[ {z_{d}^{o} - {\rm M}\left( {z_{d}^{o} } \right)} \right] - \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)c_{o} z_{d}^{o} } \right\} \hfill \\ = c_{o} q_{c}^{o} \left[ {\frac{b - \theta - n}{{\left( {b - \theta - 1} \right)\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}} \right]^{b} \left( {1 - \phi } \right)^{b} \left[ {\frac{\phi }{1 - \phi }\frac{{\left( {b - \theta } \right)\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}{b - \theta - n} - \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)} \right] \hfill \\ \end{aligned}$$

Solving the first-order condition of the O2O retailer’s profit function w.r.t. the revenue keeping rate $\phi$:

$$\frac{{d\varPi_{R} \left( \phi \right)}}{d\phi } = \frac{{\left( {b - \theta } \right)c_{o} q_{c}^{o} \left( {1 - \phi } \right)^{b - \theta - 2} \left( {b - \theta - n} \right)^{b - \theta - 1} }}{{\left( {b - \theta - 1} \right)^{b - \theta } \left( {\sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)^{b - \theta } }}\left\{ \begin{aligned} - \phi \left[ {b - \theta - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)} \right] \hfill \\ + \left[ {1 + \left( {b - \theta - n - 1} \right)\left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)} \right] \hfill \\ \end{aligned} \right\} = 0$$

And we can obtain the equilibrium revenue keeping rate as follows:

$$\phi_{d}^{o} = \frac{{1 + \left( {b - \theta - n - 1} \right)\left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)}}{{b - \theta - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)}}$$

Plugging $\phi_{d}^{o}$ into $p_{i}^{od} \left( \phi \right)$, $p_{d}^{o} \left( \phi \right)$ and $q_{d}^{o} \left( \phi \right)$, then we can get the supplier i’s equilibrium retail price of product i, the equilibrium total retail price of product set and the equilibrium order quantity of product set as follows:

$$p_{i}^{od} = \frac{{\left[ {\left( {b - \theta - n} \right)\alpha_{i} + \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right]\left[ {b - \theta - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)} \right]}}{{\left( {b - \theta } \right)\left( {b - \theta - n} \right)\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}p_{c}^{o}$$

$$p_{d}^{o} = \frac{{b - \theta - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)}}{b - \theta - n}p_{c}^{o}$$

$$q_{d}^{o} = \frac{{\left( {b - \theta - n} \right)^{b - \theta } }}{{\left[ {b - \theta - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)} \right]^{b - \theta } }}q_{c}^{o}$$

Plugging the supplier i’s equilibrium retail price of product i, the equilibrium total retail price of product set, the equilibrium stock factor, and the equilibrium order quantity of product set into the profit functions of the suppliers, the O2O retailer and the consignment supply chain, we can obtain the equilibrium profits of the O2O retailer, the ith supplier and the consignment supply chain as follows:

$$\varPi_{R}^{od} = \frac{{\left( {b - \theta - n} \right)^{b - \theta - 1} }}{{\left[ {b - \theta - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)} \right]^{b - \theta - 1} }}\varPi_{SC}^{oc}$$

$$\varPi_{{S_{i} }}^{od} = \frac{{\left( {b - \theta - n} \right)^{b - \theta - 1} \left( {b - \theta - 1} \right)\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}{{\left[ {b - \theta - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)} \right]^{b - \theta } }}\varPi_{SC}^{oc}$$

$$\varPi_{SC}^{od} = \frac{{\left( {b - \theta - n} \right)^{b - \theta } + \left( {b - \theta - n} \right)^{b - \theta - 1} \left( {b - \theta } \right)n\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}{{\left[ {b - \theta - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)} \right]^{b - \theta } }}\varPi_{SC}^{oc}$$

• (2) Suppliers’ Sequential Decision (SE)

The profit functions of the supplier i and the O2O retailer in the consignment supply chain under the revenue sharing contract can be formulated as follows:

$$\varPi_{{S_{i} }}^{o} \left( {\left. {p_{i} } \right|p_{1} , \ldots ,p_{i - 1} ,z} \right) = y_{o} \left( p \right)\left\{ {\left( {1 - \phi } \right)p_{i} E\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - \alpha_{i} c_{o} z} \right\}$$

$$\varPi_{R}^{o} \left( \phi \right) = y_{o} \left( p \right)\left\{ {\phi pE\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)c_{o} z} \right\}$$

We will use induction proof method to derive the corresponding equilibrium solutions.

First, we consider the case of $i = n$. Given stock factor $z$, the supplier n’s optimal problem is as follows:

$$\mathop {\hbox{max} }\limits_{{p_{n} }} \;\varPi_{{S_{n} }}^{o} \left( {\left. {p_{n} } \right|p_{1} , \ldots ,p_{n - 1} ,z} \right) = y_{o} \left( p \right)\left\{ {\left( {1 - \phi } \right)p_{n} E\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - \alpha_{n} c_{o} z} \right\}$$

Solving the first-order condition of the supplier n’s optimization problem with respect to (w.r.t.) and the retail price $p_{n}$:

$$\frac{{\partial \varPi_{{S_{n} }}^{o} \left( {p_{n} |p_{1} , \ldots ,p_{n - 1} ,z} \right)}}{{\partial p_{n} }} = \frac{{y_{o} \left( p \right)}}{p}\left\{ \begin{aligned} & - \left( {b - \theta - 1} \right)p_{n} \left( {1 - \phi } \right)\left[ {z - {\rm M}\left( z \right)} \right] \hfill \\ & + \left( {b - \theta } \right)\alpha_{n} c_{o} z + p_{ - n} \left( {1 - \phi } \right)\left[ {z - {\rm M}\left( z \right)} \right] \hfill \\ \end{aligned} \right\} = 0$$

Then we can get the reaction function of $p_{n}^{od'}$ w.r.t. $\left\{ {p_{1} , \ldots ,p_{n - 1} } \right\}$ as follows:

$$p_{n}^{od'} \left( {p_{1} , \ldots ,p_{n - 1} } \right) = \frac{{\left( {b - \theta } \right)\alpha_{n} c_{o} z}}{{\left( {b - \theta - 1} \right)\left( {1 - \phi } \right)\left[ {z - {\rm M}\left( z \right)} \right]}} + \frac{{\sum\nolimits_{i = 1}^{n - 1} {p_{i} } }}{b - \theta - 1}$$

Substituting $p_{n}^{od'} \left( {p_{1} , \ldots ,p_{n - 1} } \right)$ into $\varPi_{{S_{n - 1} }} \left( {p_{n - 1} |p_{1} , \ldots ,p_{n - 2} ,z} \right)$, we can get the supplier n-1’s optimal problem is as follows:

$$\begin{aligned} \mathop {\hbox{max} }\limits_{{p_{n - 1} }} \;\varPi_{{S_{n - 1} }}^{o} \left( {p_{n - 1} |p_{1} , \ldots ,p_{n - 2} ,z} \right) \hfill \\ = y_{o} \left( {p_{1} , \ldots ,p_{n - 1} ,p_{n}^{od'} \left( {p_{1} , \ldots ,p_{n - 1} } \right)} \right)\left\{ {\left( {1 - \phi } \right)p_{n - 1} E\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - \alpha_{n - 1} c_{o} z} \right\} \hfill \\ \end{aligned}$$

Solving the first-order condition of the supplier n-1’s optimization problem with respect to (w.r.t.) and the retail price $p_{n - 1}$, we can get:

$$\frac{{\partial \varPi_{{S_{n - 1} }}^{o} \left( {p_{n - 1} |p_{1} , \ldots ,p_{n - 2} ,z} \right)}}{{\partial p_{n - 1} }} = \frac{{y_{o} \left( p \right)}}{p}\left\{ \begin{aligned} \frac{{\left( {b - \theta } \right)\alpha_{n} + \left( {b - \theta } \right)^{2} \alpha_{n - 1} }}{b - \theta - 1}c_{o} z \hfill \\ - \left( {b - \theta } \right)\left( {1 - \phi } \right)p_{n - 1} \left[ {z - {\rm M}\left( z \right)} \right] \hfill \\ + \frac{b - \theta }{b - \theta - 1}\left( {1 - \phi } \right)\sum\nolimits_{i = 1}^{n - 2} {p_{i} } \left[ {z - {\rm M}\left( z \right)} \right] \hfill \\ \end{aligned} \right\} = 0$$

Then we can get the reaction function of $p_{n - 1}^{od'}$ w.r.t. $\left\{ {p_{1} , \ldots ,p_{n - 2} } \right\}$ as follows:

$$p_{n - 1}^{od'} \left( {p_{1} , \ldots ,p_{n - 2} } \right) = \frac{{\left[ {\alpha_{n} + \left( {b - \theta } \right)\alpha_{n - 1} } \right]c_{o} z}}{{\left( {b - \theta - 1} \right)\left( {1 - \phi } \right)\left[ {z - {\rm M}\left( z \right)} \right]}} + \frac{{\sum\nolimits_{i = 1}^{n - 2} {p_{i} } }}{b - \theta - 1}$$

Repeating the same derivation process, finally, we can get:

$$p_{1}^{od'} = \frac{{\left[ {\left( {b - \theta - 1} \right)\alpha_{1} + \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right]c_{o} z}}{{\left( {b - \theta - 1} \right)\left( {1 - \phi } \right)\left[ {z - {\rm M}\left( z \right)} \right]}}$$

Substituting $p_{1}^{od'}$ into the reaction function $p_{2}^{od'} \left( {p_{1} } \right)$, we can get $p_{2}^{od'}$; substituting $p_{1}^{od'}$ and $p_{2}^{od'}$ into the reaction function $p_{3}^{od'} \left( {p_{1} ,p_{2} } \right)$,…., repeating these substituting process, we can get the reaction function of $p_{i}^{od'}$ and $p_{d'}^{o}$ w.r.t. $z$ as follows:

$$p_{i}^{od'} \left( z \right) = \frac{{\left[ {\left( {b - \theta - 1} \right)^{i} \alpha_{i} + \left( {b - \theta } \right)^{i - 1} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right]c_{o} z}}{{\left( {b - \theta - 1} \right)^{i} \left( {1 - \phi } \right)\left[ {z - {\rm M}\left( z \right)} \right]}}$$

$$p_{d'}^{o} \left( z \right) = \left( {\frac{b - \theta }{b - \theta - 1}} \right)^{n} \frac{{\alpha c_{o} z}}{{\left( {1 - \phi } \right)\left[ {z - {\rm M}\left( z \right)} \right]}}$$

Substituting $p_{i}^{od'} \left( z \right)$ and $p_{d'}^{o} \left( z \right)$ into the supplier i’s expected profit function, we can obtain the supplier i’s optimal problem as follows:

$$\mathop {\hbox{max} }\limits_{z} \;\varPi_{{S_{i} }}^{o} \left( {p_{i}^{od'} \left( z \right),p_{d'}^{o} \left( z \right),z} \right) = y_{o} \left( {p_{d'}^{o} \left( z \right)} \right)\left\{ {\left( {1 - \phi } \right)p_{i}^{od'} \left( z \right) \cdot E\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - \alpha_{i} c_{o} z} \right\}$$

Likewise, solving the first-order condition of the supplier i’s optimization problem w.r.t. the stock factor $z$:

$$\begin{aligned} \frac{{d\varPi_{{S_{i} }}^{o} \left( {p_{i}^{od'} \left( z \right),p_{d'}^{o} \left( z \right),z} \right)}}{dz} \hfill \\ = a\frac{{\left( {b - \theta - 1} \right)^{{\left( {b - \theta } \right)n - i}} }}{{\left( {b - \theta } \right)^{{\left( {b - \theta } \right)n - i + 1}} }}\frac{{\left( {1 - \phi } \right)^{b - \theta } c_{si} }}{{\left( {c_{s} } \right)^{b - \theta } }}\frac{{\left[ {z - {\rm M}\left( z \right)} \right]^{b - \theta - 1} }}{{z^{b - \theta } }}\left\{ {z + \left( {b - \theta - 1} \right){\rm M}\left( z \right) - \left( {b - \theta } \right)zF\left( z \right)} \right\} \hfill \\ = 0 \hfill \\ \end{aligned}$$

Then we can get the equilibrium stock factor as follows:

$$F\left( {z_{d'}^{o} } \right) = \frac{1}{b - \theta } + \frac{{\left( {b - \theta - 1} \right){\rm M}\left( {z_{d'}^{o} } \right)}}{{\left( {b - \theta } \right)z_{d'}^{o} }} = F\left( {z_{c}^{o} } \right)$$

On this basis, we can get the supplier’s reaction function of equilibrium retail price of product i w.r.t. the revenue keeping rate, the reaction function of total equilibrium retail price of product set w.r.t. the revenue keeping rate, and the distribution function of the equilibrium stock factor as follows:

$$p_{i}^{od'} \left( \phi \right) = \frac{{\left( {b - \theta - 1} \right)^{i} \alpha_{i} + \left( {b - \theta } \right)^{i - 1} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}{{\left( {b - \theta } \right)\left( {b - \theta - 1} \right)^{i - 1} \left( {1 - \phi } \right)}}p_{c}^{o}$$

$$p_{d'}^{o} \left( \phi \right) = \frac{{\left( {b - \theta } \right)^{n - 1} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}{{\left( {b - \theta - 1} \right)^{n - 1} \left( {1 - \phi } \right)}}p_{c}^{o}$$

$$F\left( {z_{d'}^{o} } \right) = \frac{1}{b - \theta } + \frac{{\left( {b - \theta - 1} \right){\rm M}\left( {z_{d'}^{o} } \right)}}{{\left( {b - \theta } \right)z_{d'}^{o} }} = F\left( {z_{c}^{o} } \right)$$

Then, we have the reaction function of the decentralized equilibrium order quantity w.r.t. the revenue keeping rate as follows:

$$q_{d'}^{o} \left( \phi \right) = \frac{{\left( {b - \theta - 1} \right)^{{\left( {n - 1} \right)\left( {b - \theta } \right)}} \left( {1 - \phi } \right)^{b - \theta } }}{{\left( {b - \theta } \right)^{{\left( {n - 1} \right)\left( {b - \theta } \right)}} \left( {\sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)^{b - \theta } }}q_{c}^{o}$$

Substituting $p_{d'}^{o} \left( \phi \right)$ and $z_{d'}^{o}$ into the O2O retailer’s profit function, we can obtain the O2O retailer’s optimal problem as follows:

$$\mathop {\hbox{max} }\limits_{\phi } \;\varPi_{R}^{o} \left( \phi \right) = y_{o} \left( {p_{d'}^{o} \left( \phi \right)} \right)\left\{ {\phi p_{d'}^{o} \left( \phi \right)\left[ {z_{d'}^{o} - {\rm M}\left( {z_{d'}^{o} } \right)} \right] - \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)c_{o} z_{d'}^{o} } \right\}$$

Solving the first-order condition of the O2O retailer’s profit function w.r.t. the revenue keeping rate $\phi$, we can obtain the equilibrium revenue keeping rate as follows:

$$\frac{{d\varPi_{R} \left( \phi \right)}}{d\phi } = \left( {1 - \phi } \right)^{b - \theta - 2} ac_{o} z_{d'}^{o} \left[ {\frac{{z_{d'}^{o} - {\rm M}\left( {z_{d'}^{o} } \right)}}{{c_{s} z_{d'}^{o} }}} \right]^{b - \theta } \frac{{\left( {b - \theta - 1} \right)^{{\left( {b - \theta - 1} \right)n}} }}{{\left( {b - \theta } \right)^{{\left( {b - \theta } \right)n - 1}} }}\left\{ \begin{aligned} - \phi \left[ {\left( {b - \theta } \right)^{n} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } + \left( {b - \theta - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)} \right] \hfill \\ + \left[ {\left( {b - \theta - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right) + \left( {b - \theta } \right)^{n - 1} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right] \hfill \\ \end{aligned} \right\} = 0$$

Then we can obtain the equilibrium revenue keeping rate as follows:

$$\phi_{d'}^{o} = \frac{{\left( {b - \theta - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right) + \left( {b - \theta } \right)^{n - 1} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}{{\left( {b - \theta - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right) + \left( {b - \theta } \right)^{n} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}$$

Plugging $\phi_{d'}^{o}$ into $p_{i}^{od'} \left( \phi \right)$, $p_{d'}^{o} \left( \phi \right)$ and $q_{d'}^{o} \left( \phi \right)$, then we can get the supplier i’s equilibrium retail price of product i, the equilibrium total retail price of product set and the equilibrium order quantity of product set as follows:

$$p_{i}^{od'} = \frac{{\left[ {\left( {b - \theta - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right) + \left( {b - \theta } \right)^{n} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right]\left[ {\left( {b - \theta - 1} \right)^{i} \alpha_{i} + \left( {b - \theta } \right)^{i - 1} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right]}}{{\left( {b - \theta - 1} \right)^{i} \left( {b - \theta } \right)^{n} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}p_{c}^{o}$$

$$p_{d'}^{o} = \frac{{\left( {b - \theta - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right) + \left( {b - \theta } \right)^{n} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}{{\left( {b - \theta - 1} \right)^{n} }}p_{c}^{o}$$

$$q_{d'}^{o} = \frac{{\left( {b - \theta - 1} \right)^{{\left( {b - \theta } \right)n}} }}{{\left( {b - \theta - 1} \right)^{{\left( {b - \theta } \right)n}} \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)^{b - \theta } + \left( {b - \theta } \right)^{{\left( {b - \theta } \right)n}} \left( {\sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)^{b - \theta } }}q_{c}^{o}$$

Plugging the supplier i’s equilibrium retail price of the ith product, the equilibrium total retail price of product set, the equilibrium stock factor of product set, and the equilibrium order quantity of product set into the profit functions of the suppliers, the O2O retailer and the consignment supply chain, we can obtain the equilibrium profits of the O2O retailer, the ith supplier and the consignment supply chain as follows:

$$\varPi_{R}^{od'} = \frac{{\left( {b - \theta - 1} \right)^{{\left( {b - \theta - 1} \right)n}} }}{{\left[ {\left( {b - \theta - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right) + \left( {b - \theta } \right)^{n} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right]^{b - \theta - 1} }}\varPi_{SC}^{oc}$$

$$\varPi_{{S_{i} }}^{od'} = \frac{{\left( {b - \theta - 1} \right)^{{\left( {b - \theta } \right)n}} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}{{\left[ {\left( {b - \theta - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right) + \left( {b - \theta } \right)^{n} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right]^{b - \theta } }}\left( {\frac{b - \theta }{b - \theta - 1}} \right)^{i - 1} \varPi_{SC}^{oc}$$

$$\varPi_{SC}^{od'} = \frac{{\left( {b - \theta - 1} \right)^{{\left( {b - \theta - 1} \right)n}} \left\{ {\left( {b - \theta - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right) + \left[ {\left( {b - \theta } \right)^{n + 1} - \left( {b - \theta - 1} \right)^{n + 1} } \right]\sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right\}}}{{\left[ {\left( {b - \theta - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right) + \left( {b - \theta } \right)^{n} \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right]^{b - \theta } }}\varPi_{SC}^{oc}$$

1.3 Cooperative decision model

The O2O retailer’s and the ith supplier’s profit reaction functions w.r.t. the revenue keeping rate $\phi$ are as follows:

$$\varPi_{R}^{oc} \left( \phi \right) = \left[ {\left( {b - \theta } \right)\phi - \left( {b - \theta - 1} \right)\left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)} \right]\varPi_{SC}^{oc}$$

$$\varPi_{{S_{i} }}^{oc} \left( \phi \right) = \left[ {\left( {b - \theta } \right)\left( {1 - \phi } \right)\frac{{\alpha_{i} }}{{\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }} - \left( {b - \theta - 1} \right)\alpha_{i} } \right]\varPi_{SC}^{oc}$$

According to the Nash bargaining theory (Nash, 1950; Kalai and Smorodinsky 1975; Binmore et al. 1986; Muthoo 1999), the asymmetric Nash bargaining problem for bargaining over the revenue keeping rate $\phi$ can be expressed as follows:

$$\begin{aligned} & \mathop {\hbox{max} }\limits_{\phi } \;\;\varOmega \left( \phi \right) = \prod\nolimits_{i = 1}^{n} {\left[ {\varPi_{{S_{i} }}^{oc} \left( \phi \right) - T_{{S_{i} }}^{od} } \right]^{{\tau_{i} }} } \cdot \left[ {\varPi_{R}^{oc} \left( \phi \right) - T_{R}^{od} } \right]^{{1 - \sum\nolimits_{i = 1}^{n} {\tau_{i} } }} \\ & s.t.\;\;\;\left\{ \begin{aligned} & \sum\nolimits_{i = 1}^{n} {\varPi_{{S_{i} }}^{oc} \left( \phi \right)} + \varPi_{R}^{oc} \left( \phi \right) = \varPi_{SC}^{oc} \hfill \\ & \varPi_{{S_{i} }}^{oc} \left( \phi \right) \ge T_{{S_{i} }}^{od} \hfill \\ & \varPi_{R}^{oc} \left( \phi \right) \ge T_{R}^{od} \hfill \\ & 0 \le \phi \le 1 \hfill \\ \end{aligned} \right. \\ \end{aligned}$$

Hereinto, $\tau_{i} \in \left( {0,1} \right)$ is the bargaining power of the ith supplier.$T_{{S_{i} }}^{od} = \hbox{max} \left\{ {\varPi_{{S_{i} }}^{od} ,\varPi_{{S_{i} }}^{od'} } \right\}$ and $T_{R}^{od} = \hbox{max} \left\{ {\varPi_{R}^{od} ,\varPi_{R}^{od'} } \right\}$ are the disagreement points or threat points.

Solving the first-order condition and the second-order derivative of the optimal problem w.r.t. the revenue keeping rate $\phi$ respectively,

$$\frac{d\varOmega \left( \phi \right)}{d\phi } = \left( {b - \theta } \right)\varPi_{SC}^{oc} \varOmega \left( \phi \right) \cdot \left\{ {\frac{{1 - \sum\nolimits_{i = 1}^{n} {\tau_{i} } }}{{\varPi_{R}^{oc} \left( \phi \right) - T_{R}^{od} }} - \sum\nolimits_{i = 1}^{n} {\left[ {\frac{{\alpha_{i} }}{{\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}\frac{{\tau_{i} }}{{\varPi_{{S_{i} }}^{oc} \left( \phi \right) - T_{{S_{i} }}^{od} }}} \right]} } \right\} = 0$$

$$\frac{{d^{2} \varOmega \left( \phi \right)}}{{d\phi^{2} }} = \left[ {\left( {b - \theta } \right)\varPi_{SC}^{oc} } \right]^{2} \varOmega \left( \phi \right) \cdot \left\{ \begin{aligned} \left\{ {\frac{{1 - \sum\nolimits_{i = 1}^{n} {\tau_{i} } }}{{\varPi_{R}^{oc} \left( \phi \right) - T_{R}^{od} }} - \sum\nolimits_{i = 1}^{n} {\left[ {\frac{{\alpha_{i} }}{{\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}\frac{{\tau_{i} }}{{\varPi_{{S_{i} }}^{oc} \left( \phi \right) - T_{{S_{i} }}^{od} }}} \right]} } \right\}^{2} \hfill \\ - \left\{ {\frac{{1 - \sum\nolimits_{i = 1}^{n} {\tau_{i} } }}{{\left[ {\varPi_{R}^{oc} \left( \phi \right) - T_{R}^{od} } \right]^{2} }} + \sum\nolimits_{i = 1}^{n} {\left[ {\left( {\frac{{\alpha_{i} }}{{\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}} \right)^{2} \frac{{\tau_{i} }}{{\left[ {\varPi_{{S_{i} }}^{oc} \left( \phi \right) - T_{{S_{i} }}^{od} } \right]^{2} }}} \right]} } \right\} \hfill \\ \end{aligned} \right\}$$

When the following condition holds:

$$\left\{ {\frac{{1 - \sum\nolimits_{i = 1}^{n} {\tau_{i} } }}{{\varPi_{R}^{oc} \left( \phi \right) - T_{R}^{od} }} - \sum\nolimits_{i = 1}^{n} {\left[ {\frac{{\alpha_{i} }}{{\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}\frac{{\tau_{i} }}{{\varPi_{{S_{i} }}^{oc} \left( \phi \right) - T_{{S_{i} }}^{od} }}} \right]} } \right\}^{2} < \frac{{1 - \sum\nolimits_{i = 1}^{n} {\tau_{i} } }}{{\left[ {\varPi_{R}^{oc} \left( \phi \right) - T_{R}^{od} } \right]^{2} }} + \sum\nolimits_{i = 1}^{n} {\left[ {\left( {\frac{{\alpha_{i} }}{{\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}} \right)^{2} \frac{{\tau_{i} }}{{\left[ {\varPi_{{S_{i} }}^{oc} \left( \phi \right) - T_{{S_{i} }}^{od} } \right]^{2} }}} \right]}$$

We can obtain the bargaining revenue keeping rate $\phi_{c}^{o}$ by solving the follow equation as follows:

$$\sum\nolimits_{i = 1}^{n} {\left[ {\frac{{\alpha_{i} }}{{\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }}\frac{{\tau_{i} }}{{\varPi_{{S_{i} }}^{oc} \left( {\phi_{c}^{o} } \right) - T_{{S_{i} }}^{od} }}} \right]} = \frac{{1 - \sum\nolimits_{i = 1}^{n} {\tau_{i} } }}{{\varPi_{R}^{oc} \left( {\phi_{c}^{o} } \right) - T_{R}^{od} }}$$

Hence, we can get the bargaining profits of the O2O retailer and the ith supplier under O2O mode as follows:

$$\varPi_{R}^{oc} = \left[ {\left( {b - \theta } \right)\phi_{c}^{o} - \left( {b - \theta - 1} \right)\left( {1 - \sum\nolimits_{i = 1}^{n} {\alpha_{i} } } \right)} \right]\varPi_{SC}^{oc}$$

$$\varPi_{{S_{i} }}^{oc} = \left[ {\left( {b - \theta } \right)\left( {1 - \phi_{c}^{o} } \right)\frac{{\alpha_{i} }}{{\sum\nolimits_{i = 1}^{n} {\alpha_{i} } }} - \left( {b - \theta - 1} \right)\alpha_{i} } \right]\varPi_{SC}^{oc}$$

Appendix B: Proofs for Sect. 4.2

Under the pure online/offline mode, the online channel demand share of the market $\lambda = 1$ or $\lambda = 0$, and the mutual promotional coefficient between channels $\theta = 0$. When $\lambda = 1$, i.e., under the pure online channel mode, the retailer’s channel cost is $c_{r} = c_{on}$; when $\lambda = 0$, i.e., under the pure offline channel mode, $c_{r} = c_{off}$. The total cost of multiple complementary suppliers’ products is $c_{s} = \sum\nolimits_{i = 1}^{n} {c_{si} }$. Thus, the total channel cost under the pure online/offline mode is $c_{p} = c_{r} + c_{s}$. For the convenience of modeling, we define the supplier i’s cost ratio $\beta_{i} = \frac{{c_{si} }}{{c_{p} }}$, then, the retailer’s channel cost ratio is $\frac{{c_{r} }}{{c_{p} }} = 1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} }$, and the ratio of the ith product’s cost to the total channel cost is $\frac{{\beta_{i} }}{{\sum\nolimits_{i = 1}^{n} {\beta_{i} } }}$. Due to the mutual promotional coefficient between channels $\theta = 0$, then $y_{p} \left( p \right) = ap^{ - b}$.

2.1 Centralized decision model

When the consignment supply chain under the pure online/offline mode takes a centralized strategy, the optimal profit function of the supply chain can be formulated as follows:

$$\mathop {\hbox{max} }\limits_{p,z} \;\varPi_{SC}^{p} \left( {p,z} \right) = y_{p} \left( p \right)\left\{ {pE\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - c_{p} z} \right\}$$

Solving the first-order condition of the optimization profit function with respect to the retail price $p$ and the stock factor $z$, we can get the optimal total retail price and the distribution function of the centralized optimal stock factor as follows:

$$p_{c}^{p} = \frac{b}{b - 1}\frac{{c_{p} z_{c}^{p} }}{{z_{c}^{p} - {\rm M}\left( {z_{c}^{p} } \right)}}$$

$$F\left( {z_{c}^{p} } \right) = \frac{1}{b} + \frac{{\left( {b - 1} \right){\rm M}\left( {z_{c}^{p} } \right)}}{{bz_{c}^{p} }}$$

Where ${\rm M}\left( {z_{c}^{p} } \right) = \int_{A}^{{z_{c}^{p} }} {\left( {z_{c}^{p} - x} \right)f\left( x \right)dx}$

Likewise, we can get the optimal retail price of the ith product as follows:

$$p_{i}^{pc} = \frac{{\beta_{i} }}{{\sum\nolimits_{i = 1}^{n} {\beta_{i} } }}p_{c}^{p}$$

Then, we can have the centralized optimal order quantity as follows:

$$q_{c}^{p} = y_{p} \left( {p_{c}^{p} } \right)z_{c}^{p}$$

Substituting the optimal retail price $p_{c}^{p}$ and the optimal stock factor $z_{c}^{p}$ into the profit function of the centralized consignment supply chain under the pure online/offline mode, we can obtain the optimal profit of the consignment supply chain as follows:

$$\varPi_{SC}^{pc} = \frac{{c_{p} q_{c}^{p} }}{b - 1}$$

2.2 Decentralized decision model

• 1. Suppliers’ Simultaneous Decision (SI)

The profit functions of the supplier i and the retailer in the consignment supply chain under the revenue sharing contract can be formulated as follows:

$$\varPi_{{S_{i} }}^{p} \left( {p_{i} ,z} \right) = y_{p} \left( p \right)\left\{ {\left( {1 - \phi } \right)p_{i} E\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - \beta_{i} c_{p} z} \right\}$$

$$\varPi_{R}^{p} \left( \phi \right) = y_{p} \left( p \right)\left\{ {\phi pE\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - \left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)c_{p} z} \right\}$$

Solving the first-order condition of the supplier’s optimization problem with respect to (w.r.t.) and the retail price $p_{i}$ and the stock factor $z$, we can get the supplier’s reaction function of equilibrium retail price with respect to the revenue keeping rate, the reaction function of equilibrium total retail price with respect to the revenue keeping rate, and the distribution function of the equilibrium stock factor as follows:

$$p_{i}^{pd} \left( \phi \right) = \frac{{\left( {b - 1} \right)\left[ {\left( {b - n} \right)\beta_{i} + \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right]}}{{b\left( {b - n} \right)\left( {1 - \phi } \right)}}p_{c}^{p}$$

$$p_{d}^{p} \left( \phi \right) = \frac{{\left( {b - 1} \right)\sum\nolimits_{i = 1}^{n} {\beta_{i} } }}{{\left( {b - n} \right)\left( {1 - \phi } \right)}}p_{c}^{p}$$

$$F\left( {z_{d}^{p} } \right) = F\left( {z_{c}^{p} } \right)$$

Then, we have the reaction function of the decentralized equilibrium order quantity w.r.t. the revenue keeping rate as follows:

$$q_{d}^{p} \left( \phi \right) = \frac{{\left( {b - n} \right)^{b} \left( {1 - \phi } \right)^{b} }}{{\left( {b - 1} \right)^{b} \left( {\sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)^{b} }}q_{c}^{p}$$

Substituting $p_{d}^{p} \left( \phi \right)$ and $z_{c}^{p}$ into the retailer’s profit function, we can obtain the retailer’s optimal problem for as follows:

$$\mathop {\hbox{max} }\limits_{\phi } \;\varPi_{R}^{p} \left( \phi \right) = y_{p} \left( {p_{d}^{p} \left( \phi \right)} \right)\left\{ {\phi p_{d}^{p} \left( \phi \right)\left[ {z_{d}^{p} - \varLambda \left( {z_{d}^{p} } \right)} \right] - \left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)c_{p} z_{d}^{p} } \right\}$$

Solving the first-order condition of the retailer’s profit function w.r.t. the revenue keeping rate $\phi$, we can obtain the equilibrium revenue keeping rate as follows:

$$\phi_{d}^{p} = \frac{{1 + \left( {b - n - 1} \right)\left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)}}{{b - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)}}$$

Plugging $\phi_{d}^{p}$ into $p_{i}^{pd} \left( \phi \right)$, $p_{d}^{p} \left( \phi \right)$ and $q_{d}^{p} \left( \phi \right)$, then we can get the supplier i’s equilibrium retail price, the equilibrium total retail price and the equilibrium order quantity as follows:

$$p_{i}^{pd} = \frac{{\left[ {\left( {b - n} \right)\beta_{i} + \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right]\left[ {b - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)} \right]}}{{b\left( {b - n} \right)\sum\nolimits_{i = 1}^{n} {\beta_{i} } }}p_{c}^{p}$$

$$p_{d}^{p} = \frac{{b - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)}}{b - n}p_{c}^{p}$$

$$q_{d}^{p} = \frac{{\left( {b - n} \right)^{b} }}{{\left[ {b - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)} \right]^{b} }}q_{c}^{p}$$

Plugging the supplier i’s equilibrium retail price, the equilibrium total retail price, the equilibrium stock factor, and the equilibrium order quantity into the profit functions of the suppliers, the retailer and the consignment supply chain, we can obtain the equilibrium profits of the retailer, the ith supplier and the consignment supply chain as follows:

$$\varPi_{R}^{pd} = \frac{{\left( {b - n} \right)^{b - 1} }}{{\left[ {b - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)} \right]^{b - 1} }}\varPi_{SC}^{pc}$$

$$\varPi_{{S_{i} }}^{pd} = \frac{{\left( {b - n} \right)^{b - 1} \left( {b - 1} \right)\sum\nolimits_{i = 1}^{n} {\beta_{i} } }}{{\left[ {b - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)} \right]^{b} }}\varPi_{SC}^{pc}$$

$$\varPi_{SC}^{pd} = \frac{{\left( {b - n} \right)^{b} + \left( {b - n} \right)^{b - 1} bn\sum\nolimits_{i = 1}^{n} {\beta_{i} } }}{{\left[ {b - n\left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)} \right]^{b} }}\varPi_{SC}^{pc}$$

• (2) Suppliers’ Sequential Decision (SE)

The profit functions of the supplier i and the retailer in the consignment supply chain under the revenue sharing contract can be formulated as follows:

$$\varPi_{{S_{i} }}^{p} \left( {\left. {p_{i} } \right|p_{1} , \ldots ,p_{i - 1} ,z} \right) = y_{p} \left( p \right)\left\{ {\left( {1 - \phi } \right)p_{i} E\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - \beta_{i} c_{p} z} \right\}$$

$$\varPi_{R}^{p} \left( \phi \right) = y_{p} \left( p \right)\left\{ {\phi pE\left[ {\hbox{min} \left\{ {z,x} \right\}} \right] - \left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)c_{p} z} \right\}$$

Solving the first-order condition of the supplier i’s optimization problem with respect to (w.r.t.) and the retail price $p_{i}$ and the stock factor $z$ sequentially, we can get the supplier i’s reaction function of equilibrium retail price of the ith product with respect to the revenue keeping rate, the reaction function of equilibrium total retail price of product set with respect to the revenue keeping rate, and the distribution function of the equilibrium stock factor of product set as follows:

$$p_{i}^{pd'} \left( \phi \right) = \frac{{\left( {b - 1} \right)^{i} \beta_{i} + b^{i - 1} \sum\nolimits_{i = 1}^{n} {\beta_{i} } }}{{b\left( {b - 1} \right)^{i - 1} \left( {1 - \phi } \right)}}p_{c}^{p}$$

$$p_{d'}^{p} \left( \phi \right) = \frac{{b^{n - 1} \sum\nolimits_{i = 1}^{n} {\beta_{i} } }}{{\left( {b - 1} \right)^{n - 1} \left( {1 - \phi } \right)}}p_{c}^{p}$$

$$F\left( {z_{d'}^{p} } \right) = F\left( {z_{c}^{p} } \right)$$

Then, we have the reaction function of the decentralized equilibrium order quantity w.r.t. the revenue keeping rate as follows:

$$q_{d'}^{p} \left( \phi \right) = \frac{{\left( {b - 1} \right)^{{\left( {n - 1} \right)b}} \left( {1 - \phi } \right)^{b} }}{{b^{{\left( {n - 1} \right)b}} \left( {\sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)^{b} }}q_{c}^{p}$$

Substituting $p_{d'}^{p} \left( \phi \right)$ and $z_{d'}^{p}$ into the retailer’s profit function, we can obtain the retailer’s expected profit function for as follows:

$$\varPi_{R}^{p} \left( \phi \right) = y_{p} \left( {p_{d'}^{p} \left( \phi \right)} \right)\left\{ {\phi p_{d'}^{p} \left( \phi \right)\left[ {z_{d'}^{p} - {\rm M}\left( {z_{d'}^{p} } \right)} \right] - \left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)c_{p} z_{d'}^{p} } \right\}$$

Solving the first-order condition of the retailer’s profit function w.r.t. the revenue keeping rate $\phi$, we can obtain the equilibrium revenue keeping rate as follows:

$$\phi_{d'}^{p} = \frac{{\left( {b - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right) + b^{n - 1} \sum\nolimits_{i = 1}^{n} {\beta_{i} } }}{{\left( {b - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right) + b^{n} \sum\nolimits_{i = 1}^{n} {\beta_{i} } }}$$

Plugging $\phi_{d'}^{p}$ into $p_{i}^{pd'} \left( \phi \right)$, $p_{d'}^{p} \left( \phi \right)$ and $q_{d'}^{p} \left( \phi \right)$, then we can get the supplier i’s equilibrium retail price of product i, the equilibrium total retail price of product set and the equilibrium order quantity of product set as follows:

$$p_{i}^{pd'} = \frac{{\left[ {\left( {b - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right) + b^{n} \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right]\left[ {\left( {b - 1} \right)^{i} \beta_{i} + b^{i - 1} \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right]}}{{\left( {b - 1} \right)^{i} b^{n} \sum\nolimits_{i = 1}^{n} {\beta_{i} } }}p_{c}^{p}$$

$$p_{d'}^{p} = \frac{{\left( {b - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right) + b^{n} \sum\nolimits_{i = 1}^{n} {\beta_{i} } }}{{\left( {b - 1} \right)^{n} }}p_{c}^{p}$$

$$q_{d'}^{p} = \frac{{\left( {b - 1} \right)^{bn} }}{{\left( {b - 1} \right)^{bn} \left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)^{b} + b^{bn} \left( {\sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)^{b} }}q_{c}^{p}$$

Plugging the supplier i’s equilibrium retail price of the ith product, the equilibrium total retail price of product set, the equilibrium stock factor of product set, and the equilibrium order quantity of product set into the profit functions of the suppliers, the retailer and the consignment supply chain, we can obtain the equilibrium profits of the retailer, the ith supplier and the consignment supply chain as follows:

$$\varPi_{R}^{pd'} = \frac{{\left( {b - 1} \right)^{{\left( {b - 1} \right)n}} }}{{\left[ {\left( {b - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right) + b^{n} \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right]^{b - 1} }}\varPi_{SC}^{pc}$$

$$\varPi_{{S_{i} }}^{pd'} = \frac{{\left( {b - 1} \right)^{bn} \sum\nolimits_{i = 1}^{n} {\beta_{i} } }}{{\left[ {\left( {b - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right) + b^{n} \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right]^{b} }}\left( {\frac{b}{b - 1}} \right)^{i - 1} \varPi_{SC}^{pc}$$

$$\varPi_{SC}^{pd'} = \frac{{\left( {b - 1} \right)^{bn + 1} \left\{ {\left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right) + \left[ {\frac{{b^{n + 1} }}{{\left( {b - 1} \right)^{n + 1} }} - 1} \right]\sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right\}}}{{\left[ {\left( {b - 1} \right)^{n} \left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right) + b^{n} \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right]^{b} }}\varPi_{SC}^{pc}$$

2.3 Cooperative decision model

The retailer’s and the ith supplier’s profit reaction functions w.r.t. the revenue keeping rate $\phi$ are as follows:

$$\varPi_{R}^{pc} \left( \phi \right) = \left[ {b\phi - \left( {b - 1} \right)\left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)} \right]\varPi_{SC}^{pc}$$

$$\varPi_{{S_{i} }}^{pc} \left( \phi \right) = \left[ {b\left( {1 - \phi } \right)\frac{{\beta_{i} }}{{\sum\nolimits_{i = 1}^{n} {\beta_{i} } }} - \left( {b - 1} \right)\beta_{i} } \right]\varPi_{SC}^{pc}$$

According to the Nash bargaining theory (Nash 1950; Kalai and Smorodinsky 1975; Binmore et al. 1986; Muthoo 1999), the asymmetric Nash bargaining problem for bargaining over the revenue keeping rate $\phi$ can be expressed as follows:

$$\begin{aligned} \mathop {\hbox{max} }\limits_{\phi } \;\;\varOmega \left( \phi \right) = \prod\nolimits_{i = 1}^{n} {\left[ {\varPi_{{S_{i} }}^{pc} \left( \phi \right) - T_{{S_{i} }}^{pd} } \right]^{{\tau_{i} }} } \cdot \left[ {\varPi_{R}^{pc} \left( \phi \right) - T_{R}^{pd} } \right]^{{1 - \sum\nolimits_{i = 1}^{n} {\tau_{i} } }} \hfill \\ s.t.\;\;\;\left\{ \begin{aligned} & \sum\nolimits_{i = 1}^{n} {\varPi_{{S_{i} }}^{pc} \left( \phi \right)} + \varPi_{R}^{pc} \left( \phi \right) = \varPi_{SC}^{pc} \hfill \\ & \varPi_{{S_{i} }}^{pc} \left( \phi \right) \ge T_{{S_{i} }}^{pd} \hfill \\ & \varPi_{R}^{pc} \left( \phi \right) \ge T_{R}^{pd} \hfill \\ & 0 \le \phi \le 1 \hfill \\ \end{aligned} \right. \hfill \\ \end{aligned}$$

Hereinto, $\tau_{i} \in \left( {0,1} \right)$ is the bargaining power of the ith supplier.$T_{{S_{i} }}^{pd} = \hbox{max} \left\{ {\varPi_{{S_{i} }}^{pd} ,\varPi_{{S_{i} }}^{pd'} } \right\}$ and $T_{R}^{pd} = \hbox{max} \left\{ {\varPi_{R}^{pd} ,\varPi_{R}^{pd'} } \right\}$ are the disagreement points or threat points.

Solving the first-order condition and the second-order derivative of the optimal problem w.r.t. the revenue keeping rate $\phi$ respectively, we can obtain the bargaining revenue keeping rate $\phi_{c}^{p}$ by solving the follow equation as follows:

$$\sum\nolimits_{i = 1}^{n} {\left[ {\frac{{\beta_{i} }}{{\sum\nolimits_{i = 1}^{n} {\beta_{i} } }}\frac{{\tau_{i} }}{{\varPi_{{S_{i} }}^{pc} \left( {\phi_{c}^{p} } \right) - T_{{S_{i} }}^{pd} }}} \right]} = \frac{{1 - \sum\nolimits_{i = 1}^{n} {\tau_{i} } }}{{\varPi_{R}^{pc} \left( {\phi_{c}^{p} } \right) - T_{R}^{pd} }}$$

Hence, we can get the bargaining profits of the retailer and the ith supplier under the pure online/offline mode as follows:

$$\varPi_{R}^{pc} = \left[ {b\phi_{c}^{p} - \left( {b - 1} \right)\left( {1 - \sum\nolimits_{i = 1}^{n} {\beta_{i} } } \right)} \right]\varPi_{SC}^{pc}$$

$$\varPi_{{S_{i} }}^{pc} = \left[ {b\left( {1 - \phi_{c}^{p} } \right)\frac{{\beta_{i} }}{{\sum\nolimits_{i = 1}^{n} {\beta_{i} } }} - \left( {b - 1} \right)\beta_{i} } \right]\varPi_{SC}^{pc}$$