A freight transport price optimization model with multi bounded-rational customers

Abstract

To analyze the effect of customer enterprises’ behavior on freight transport markets with various products, a bi-level programming model for freight pricing is developed. The transport companies make decisions in the upper level, a large number of customer enterprises make decisions at the lower level. Cumulative prospect theory is adopted to depict customers’ bounded rationality. The complex network model is introduced to construct the interactions among customer enterprises where the changes in their risk attitudes are depicted by dynamic evolution of their heterogeneous reference points. The existence of the generalized Nash equilibrium is proved and verified. A comparison of the optimal solutions of the new model and the traditional logit based choice model is conducted. The results show that (1) the new model fully embodies the characteristics of cumulative prospect theory and can also improve customers’ utility; (2) the equilibrium freight price and the total profits of transport companies increase with increases in the customers’ interaction intensity.

Appendix

Theorem 1

There exists Generalized Nash Equilibrium in the bi-level programming of (19) and (20).

Proof

According to formula (19) and (20), for company $i$, the freight optimization problem can be written as:

$$ {\rm{Min}}\;Z_{i} = - \mathop \sum \limits_{j = 1}^{M} \left[ {q_{ij}^{t - 1} + \frac{{\partial q_{ij} }}{{\partial p_{ij} }}\left( {p_{ij}^{t} - p_{ij}^{t - 1} } \right) + \mathop \sum \limits_{k \ne i} \frac{{\partial q_{ij} }}{{\partial p_{kj} }}\left( {p_{kj}^{t} - p_{kj}^{t - 1} } \right)} \right]\left( {p_{ij}^{t} - c_{ij} } \right) $$

(29)

The variables of the optimization problem (29) are $p_{i1}^{t} ,$ $p_{i2}^{t} , \ldots ,p_{iM}^{t}$. Let $p_{i1}^{t} = x_{1}$, $p_{i2}^{t} = x_{2}$, $\ldots ,p_{iM}^{t} = x_{M}$; let $q_{i1}^{t - 1} = a_{1}$, $q_{i2}^{t - 1} = a_{2}$, $\ldots ,q_{1M}^{t - 1} = a_{M}$; $\frac{{\partial q_{i1} }}{{\partial p_{i1} }} = b_{1}$, $\frac{{\partial q_{12} }}{{\partial p_{12} }} = b_{2}$, $\ldots$, $\frac{{\partial q_{1M} }}{{\partial p_{1M} }} = b_{M}$, $p_{i1}^{t - 1} = d_{1}$,$p_{12}^{t - 1} = d_{2}$, $\ldots$, $p_{1M}^{t - 1} = d_{M}$; $\mathop \sum \limits_{k \ne i} \frac{{\partial q_{i1} }}{{\partial p_{k1} }}\left( {p_{k1}^{t} - p_{k1}^{t - 1} } \right) = e_{1}$, $\mathop \sum \limits_{k \ne i} \frac{{\partial q_{i2} }}{{\partial p_{k2} }}\left( {p_{k2}^{t} - p_{k2}^{t - 1} } \right) = e_{2}$,$\ldots$, $\mathop \sum \limits_{k \ne i} \frac{{\partial q_{iM} }}{{\partial p_{kM} }}\left( {p_{kM}^{t} - p_{kM}^{t - 1} } \right) = e_{M}$; $c_{i1} = c_{1}$, $c_{i2} = c_{2} , \ldots , c_{iM} = c_{M}$.

Therefore, optimization problem (29) can be shown as:

$$ {\rm{Min}}\;Z_{i} = - \mathop \sum \limits_{j = 1}^{M} \left[ {a_{j} + b_{j} \left( {x_{j} - d_{j} } \right) + e_{j} } \right]\left( {x_{j} - c_{j} } \right) $$

(30)

The Hessian matrix of quadratic programming (30) is $\left( {\begin{array}{*{20}c} { - 2b_{1} } & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & { - 2b_{M} } \\ \end{array} } \right)$, for each transport company, rising freight price will lead to the decline of transportation demand, which means that the derivative of the transportation volume with respect to the freight price is less than 0 ($b_{1} < 0$,$\ldots$,$b_{M} < 0$). The Hessian matrix of quadratic programming (30) is positive definite. Therefore, the quadratic programming (30) has the unique optimal solution and the Generalized Nash Equilibrium does exist.