Bundle pricing and inventory decisions on complementary products

Abstract

Selling correlated products faces the sellers with the cross-selling, which is a key factor in managing revenue and costs of the sellers. Cross-selling is a phenomenon which happens when the demands of products are correlated so that the demand for one of the correlated products automatically initiates demand of another. In these cases, different selling tactics such as bundling, tying, mixed bundling, etc. are applied to sell the items. In this paper, an integrated pricing-inventory model for two complementary products under three selling strategies is developed. In the first model, it is assumed that the seller sells the products separately while those are packed and sold as bundling in the second model. The third model is extended in presence of a mixed-bundle strategy so that the products are presented as both bundling and single. The aim is to determine the optimal values of selling prices, bundle prices and order quantities of products along with the optimal period length so as the total profit is maximized. Moreover, optimal solutions are derived, solution algorithms are proposed, and at the end, numerical illustrations and also some sensitivity analyses are presented.

Appendices

Appendix 1: The coefficient of the optimal value of selling price $(p_{{M_{1} }}^{*} )$

The coefficients of the optimal value of the selling price $(p_{{M_{1} }}^{*} )$ are as follows:

$$ \psi_{1} = 4bc_{1} \varepsilon^{2} \left( {1 - \theta } \right) $$

(49)

$$ \psi_{2} = 4bc_{1} \theta \left( {b\theta \left( {b + \lambda } \right) - \varepsilon \lambda } \right) $$

(50)

$$ \psi_{3} = 2b\varepsilon^{2} \left( {c_{1} + c_{2} } \right)\left( {1 - \theta } \right) $$

(51)

$$ \psi_{4} = 2\varepsilon \left( {c_{1} + c_{2} } \right)\left( {b^{2} \theta - \varepsilon \lambda } \right) + 2b\lambda \varepsilon \theta \left( {c_{1} + 2c_{2} } \right) + 4b\theta \left( {b + \lambda } \right)\left( {a - bc_{2} } \right) + 2\varepsilon \left( {\lambda \left( {bc_{2} - a} \right) + ab\theta } \right) $$

(52)

$$ \psi_{5} = 2\lambda^{2} \left( {a + bc_{1} + \varepsilon \left( {c_{1} + c_{2} } \right) - b\theta \left( {2c_{1} + c_{2} } \right)} \right) - 2b\lambda \theta \left( {b\left( {c_{1} + c_{2} } \right) - a} \right) - \varepsilon c_{2} $$

(53)

$$ \psi_{6} = 2\lambda^{2} \left( { - a + b\left( {c_{1} \left( {2 - \theta } \right) + c_{2} } \right) - \varepsilon \left( {c_{1} + c_{2} } \right)} \right) + 2b\lambda \left( {2 - \theta } \right)\left( {b\left( {c_{1} + c_{2} } \right) - a} \right) $$

(54)

$$ \begin{aligned} \psi_{7} & = - 4b\left( {b + \lambda } \right)\left( {2bc_{1} + a\left( {2 - \theta } \right)} \right) + 2\varepsilon \left( {c_{1} + c_{2} } \right)\left( {\varepsilon \lambda + b^{2} \left( {\theta - 2} \right)} \right) + 4b^{2} \theta \left( {c_{2} + c_{1} \theta } \right)\left( {b + \lambda } \right) \\ & \quad { + }2b\varepsilon \left( {\theta - 2} \right)\left( {a - c_{1} \lambda } \right) + 2\varepsilon \lambda \left( {a - 3bc_{2} } \right) \\ \end{aligned} $$

(55)

$$ \psi_{8} = b\varepsilon h_{B} \left( {\theta - 2} \right)\left( {b + \lambda } \right) + b\varepsilon^{2} \left( {1 - \theta } \right)\left( {2h_{1} + h_{2} } \right) + \lambda \varepsilon^{2} h_{B} $$

(56)

$$ \psi_{9} = - b\lambda \varepsilon h_{2} \left( {1 + \theta } \right) + \varepsilon \lambda \left( {\lambda h_{B} - 2bh_{1} \theta } \right) + b\theta \left( {b + \lambda } \right)\left( {2b\left( {h_{2} + h_{1} \theta } \right) - \lambda h_{B} } \right) $$

(57)

$$ \psi_{10} = b\lambda \varepsilon h_{2} \left( {1 + \theta } \right) + 2b\varepsilon \lambda h_{1} \left( {2 - \theta } \right) - \lambda^{2} \varepsilon h_{B} + b\left( {b + \lambda } \right)\left( {\lambda h_{B} \left( {2 - \theta } \right) - 4bh_{1} + 2b\theta \left( {h_{1} \theta - h_{2} } \right)} \right) $$

(58)

$$ \psi_{11} = \varepsilon^{2} \left( {bh_{1} - \lambda h_{B} } \right) + b\varepsilon \theta \left( {h_{B} \left( {b + \lambda } \right) - \varepsilon h_{1} } \right) $$

(59)

$$ \psi_{12} = b\lambda^{2} \left( {h_{1} - h_{2} } \right)\left( {1 - \theta } \right) + \varepsilon \left( {h_{1} + h_{2} } \right) $$

(60)

$$ \chi_{1} = b\theta^{2} \left( {b + \lambda } \right) - \varepsilon \lambda \theta + 2\varepsilon $$

(61)

$$ \chi_{2} = 2b\left( {b + \lambda } \right)\left( {\theta^{2} - 2} \right) + 2\varepsilon \lambda \left( {2 - \theta } \right) $$

(62)

Appendix 2: The coefficients of the optimal value of the selling price $(p_{{M_{2} }}^{*} )$

The coefficients of the optimal value of the selling price $(p_{{M_{2} }}^{*} )$ are as follows:

$$ \varphi_{1} = 2b\varepsilon^{2} \left( {c_{1} + c_{2} } \right)\left( {1 - \theta } \right) $$

(63)

$$ \varphi_{2} = 2\varepsilon \left( {c_{1} + c_{2} } \right)\left( {b^{2} \theta - \varepsilon \lambda } \right) + 2b\lambda \varepsilon \theta \left( {c_{2} + 2c_{1} } \right) + 4b\theta \left( {b + \lambda } \right)\left( {a - bc_{1} } \right) + 2\varepsilon \left( {\lambda \left( {bc_{1} - a} \right) + ab\theta } \right) $$

(64)

$$ \varphi_{3} = 4bc_{2} \varepsilon^{2} \left( {1 - \theta } \right) $$

(65)

$$ \varphi_{4} = 4bc_{2} \theta \left( {b\theta \left( {b + \lambda } \right) - \varepsilon \lambda } \right) $$

(66)

$$ \varphi_{5} = 2\lambda^{2} \left( { - a + b\left( {c_{2} \left( {2 - \theta } \right) + c_{1} } \right) - \varepsilon \left( {c_{1} + c_{2} } \right)} \right) + 2b\lambda \left( {2 - \theta } \right)\left( {b\left( {c_{1} + c_{2} } \right) - a} \right) - \varepsilon c_{1} $$

(67)

$$ \varphi_{6} = 2\lambda^{2} \left( {a + bc_{2} + \varepsilon \left( {c_{1} + c_{2} } \right) - b\theta \left( {2c_{2} + c_{1} } \right)} \right) - 2b\lambda \theta \left( {b\left( {c_{1} + c_{2} } \right) - a} \right) $$

(68)

$$ \begin{aligned} \varphi_{7} & = - 4b\left( {b + \lambda } \right)\left( {2bc_{2} + a\left( {2 - \theta } \right)} \right) + 2\varepsilon \left( {c_{1} + c_{2} } \right)\left( {\varepsilon \lambda + b^{2} \left( {\theta - 2} \right)} \right) + 4b^{2} \theta \left( {c_{1} + c_{2} \theta } \right)\left( {b + \lambda } \right) \\ & \quad + 2b\varepsilon \left( {\theta - 2} \right)\left( {a - c_{2} \lambda } \right) + 2\varepsilon \lambda \left( {a - 3bc_{1} } \right) \\ \end{aligned} $$

(69)

$$ \varphi_{8} = b\varepsilon h_{B} \left( {\theta - 2} \right)\left( {b + \lambda } \right) + b\varepsilon^{2} \left( {1 - \theta } \right)\left( {h_{1} + 2h_{2} } \right) + \varepsilon^{2} \lambda h_{B} $$

(70)

$$ \varphi_{9} = b\varepsilon \lambda h_{1} \left( {1 + \theta } \right) + 2b\varepsilon \lambda h_{2} \left( {2 - \theta } \right) - \varepsilon \lambda^{2} h_{B} + b\left( {b + \lambda } \right)\left( {\lambda h_{B} \left( {2 - \theta } \right) - 4bh_{2} + 2b\theta \left( {h_{2} \theta - h_{1} } \right)} \right) $$

(71)

$$ \varphi_{10} = - b\varepsilon \lambda h_{1} \left( {1 + \theta } \right) + \varepsilon \lambda \left( {\lambda h_{B} - 2bh_{2} \theta } \right) + b\theta \left( {b + \lambda } \right)\left( {2b\left( {h_{1} + h_{2} \theta } \right) - \lambda h_{B} } \right) $$

(72)

$$ \varphi_{11} = \varepsilon^{2} \left( {bh_{2} - \lambda h_{B} } \right) + b\varepsilon \theta \left( {h_{B} \left( {b + \lambda } \right) - \varepsilon h_{2} } \right) $$

(73)

$$ \varphi_{12} = b\lambda^{2} \left( {h_{2} - h_{1} } \right)\left( {1 - \theta } \right) + \varepsilon \left( {h_{1} + h_{2} } \right) $$

(74)

Appendix 3: The coefficients of the optimal value of the selling price $(p_{MB}^{*} )$

The coefficients of the optimal value of the selling price $(p_{MB}^{*} )$ are as follows:

$$ \xi_{1} = 2\varepsilon \left( {1 - \theta } \right)\left( { - 2a + bc_{2} \theta } \right) - 2bc_{1} \varepsilon \left( {\theta \left( {1 + \theta } \right) - 2} \right) $$

(75)

$$ \xi_{2} = 2\theta \left( {c_{1} + c_{2} } \right)\left( {b^{2} \theta - \lambda \varepsilon } \right) + 2a\theta \left( {b\theta + \lambda } \right) + 2b\lambda \theta \left( {c_{1} \left( {2\theta - 1} \right) + c_{2} \theta } \right) $$

(76)

$$ \xi_{3} = 2\varepsilon \left( {1 - \theta } \right)\left( { - 2a + bc_{1} \theta } \right) - 2bc_{2} \varepsilon \left( {\theta \left( {1 + \theta } \right) - 2} \right) $$

(77)

$$ \xi_{4} = 2b^{2} \theta^{2} \left( {c_{1} + c_{2} } \right) - 2c_{2} \lambda \theta \left( {b + \varepsilon } \right) + 2c_{1} \lambda \theta \left( {b\theta - \varepsilon } \right) + 2a\theta \left( {b\theta + \lambda } \right) + 4bc_{2} \lambda \theta^{2} $$

(78)

$$ \xi_{5} = 4\lambda^{2} \left( {c_{1} + c_{2} } \right)\left( {1 - \theta } \right) $$

(79)

$$ \xi_{6} = - 8a\left( {b + \lambda } \right) + 2\left( {c_{1} + c_{2} } \right)\left( {b\theta^{2} \left( {2b + 3\lambda } \right) + \lambda \theta \left( {b - 2\varepsilon } \right) + 4\left( {\lambda \varepsilon - b^{2} } \right) - 6b\lambda } \right) + 4a\theta \left( {b\theta + \lambda } \right) $$

(80)

$$ \xi_{7} = 2bh_{B} \left( {b + \lambda } \right)\left( {\theta^{2} - 2} \right) - b\varepsilon \left( {h_{1} + h_{2} } \right)\left( {\theta \left( {1 + \theta } \right) - 2} \right) + 2\varepsilon \lambda h_{B} \left( {2 - \theta } \right) $$

(81)

$$ \xi_{8} = 2\lambda^{2} h_{B} \left( {1 - \theta } \right) + b\lambda \theta \left( {\theta \left( {h_{1} + h_{2} } \right) + \left( {h_{2} - h_{1} } \right)} \right) - 2b\lambda h_{2} $$

(82)

$$ \xi_{9} = \lambda \left( {1 - \theta } \right)\left( {2\lambda h_{B} - bh_{2} \theta } \right) + b\lambda h_{1} \left( {\theta \left( {\theta + 1} \right) - 2} \right) $$

(83)

$$ \xi_{10} = bh_{B} \theta^{2} \left( {b + \lambda } \right) + \varepsilon \theta \left( {bh_{2} \left( {1 - \theta } \right) - \lambda h_{B} } \right) $$

(84)

$$ \xi_{11} = bh_{B} \theta^{2} \left( {b + \lambda } \right) + \varepsilon \theta \left( {bh_{1} \left( {1 - \theta } \right) - \lambda h_{B} } \right) $$

(85)