A Dantzig-Wolfe Decomposition Based Heuristic Scheme for Bi-level Dynamic Network Design Problem

Abstract

We present a heuristic to solve the NP-hard bi-level network design problem (NDP). The heuristic is developed based on the Dantzig-Wolfe decomposition principle such that it iteratively solves a master problem and a pricing problem. The master problem is the budget allocation linear program solved by CPLEX to determine the budget allocation and construct a modified cell transmission network for the pricing problem. The pricing problem is the user-optimal dynamic traffic assignment (UODTA) solved by an existing combinatorial algorithm. To facilitate the decomposition principle, we propose a backward connectivity algorithm and complementary slackness procedures to efficiently approximate the required dual variables from the UODTA solution. The dual variables are then employed to augment a new column in the master program in each iteration. The iterative process repeats until a stopping criterion is met. Numerical experiments are conducted on two test networks. Encouraging results demonstrate the applicability of the heuristic scheme on solving large-scale NDP. Though a single destination problem is considered in this paper, the proposed scheme can be extended to solve multi-destination problems as well.

Appendix: Dantzig-Wolfe decomposition principle

Consider a standard form of an LP:

$$Max\,c_1^T x_1 + c_2^T x_2 $$

(6.1)

subject to

$$A_1 x_1 \leqslant b_1 $$

(6.2)

$$\bar A_1 x_1 + \bar A_2 x_2 \leqslant b_2 $$

(6.3)

$$\begin{array}{*{20}l} {{} \hfill} & {{x_{1} \geqslant 0} \hfill} \\ \end{array} $$

(6.4)

$$\begin{array}{*{20}l} {{} \hfill} & {{x_{2} \geqslant 0} \hfill} \\ \end{array} $$

(6.5)

where the vectors of decision variables are $x_1 \in R^{n_1 }$, $x_2 \in R^{n_2 }$; the parameters are $c_1 \in R^{n_1 } ,c_2 \in R^{n_2 } ,b_1 \in R^{m_1 } ,b_2 \in R^{m_2 } ,A_1 \in R^{m_1 \times n_1 } ,\bar A_1 \in R^{m_2 \times n_1 } ,\bar A_2 \in R^{m_2 \times n_2 }$.

Applying the DWD principle (Dantzig 1963) to this standard LP (with an assumed bounded and convex feasibility set for Eq. (6.2)), the resulting master and pricing problem are:

Restricted Master Program

$$Max\;\sum\limits_{i = 1}^k {\left( {c_1^T X_1^i } \right)\lambda _i + \left( {c_2^T X_2 } \right)\beta } $$

(7.1)

subject to

$${\sum\limits_{i = 1}^k {{\left( {\ifmmode\expandafter\bar\else\expandafter\=\fi{A}_{1} X^{i}_{1} } \right)}\lambda _{i} + {\left( {\ifmmode\expandafter\bar\else\expandafter\=\fi{A}_{2} X_{2} } \right)}\beta \leqslant b_{2} } }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,:{\text{ $ \eta $ }}$$

(7.2)

$$\begin{array}{*{20}l} {{{\sum\limits_{i = 1}^k {\lambda _{i} = 1} }} \hfill} & {{:\psi } \hfill} \\ \end{array} $$

(7.3)

$$\lambda _i \geqslant 0;\;i = 1,2, \ldots ,k$$

(7.4)

$$\beta \geqslant 0$$

(7.5)

where the decision variables are β, λ ₁ , λ ₂ ,…, λ _k $\in R$; $X_1^1 ,X_1^2$,…, $X_1^k \in R^{n_1 } \;\forall k \in K$ are subset of extreme points of Eq. (6.2); $X_2 \in R^{n_2 }$ is the extreme ray of Eq. (6.5). $\eta \in R^{m_2 }$ and $\psi \in R$ are dual variables of Eqs. (7.2) and (7.3). We can consider X ₂ as the vector of decision variables, and remove β and Eq. (7.5) from the restricted master program without affecting the optimal solution. The reduced cost associated with λ _i is $c_1^T X_1^i - \eta ^T \left( {\bar A_1 X_1^i } \right) - \psi$, which is used to construct the objective function of the pricing problem.

Pricing Problem

$$z = Max\;c_1^T x_1 - \eta ^T \bar A_1 x_1 - \psi $$

(8.1)

subject to

$$A_1 x_1 \leqslant b_1 $$

(8.2)

$$x_1 \geqslant 0$$

(8.3)

The BLPNDP in this work is decomposed according the above principle.