A Heuristic for the Doubly Constrained Entropy Distribution/Assignment Problem

Abstract

The doubly constrained entropy distribution/assignment (DEDA) problem that combines a gravity-based trip distribution (TD) problem and a traffic assignment (TA) problem has long been formulated as an optimization model and solved by two solution algorithms, i.e., partial- and full-linearization solution algorithms. As an alternative, this research first treats the DEDA problem by the augmented Lagrangian dual (ALD) method as the singly constrained entropy distribution/assignment (SEDA) problem, which in turn is addressed, via a tactical supernetwork representation, as an “extended” 1-origin-to-1-destination TA problem. A quick-precision TA solution algorithm, − called TAPAS (Traffic Assignment by Paired Alternative Segments), − is then adopted for solutions. The proposed approach is demonstrated with a numerical example for the correctness of the result, using Lingo 11 solver and the partial linearization solution (PLS) algorithm. Moreover, through the use of TAPAS in the innermost loop, the proposed approach also has the merit of generating unique path flow solution, which is very useful in route guidance under the intelligent transportation systems environment, among other academic applications. In addition, the proposed approach can be easily applied, or with minor modification at most, to various combined models in travel demand forecasting.

Appendix I: Traffic Distribution and Traffic Assignment Model

In model (1), the equilibrium link flows {f ^*_• a } defined in (1b) can be obtained from the following DEDA model (Sheffi 1985; Bar-Gera 2010).

$$ \underset{\left\{{f}_{\bullet a}\left(\mathbf{h}\right),{d}_{pq}\right\}\in {\varOmega}_4}{ \min}\kern0.75em {z}_4={\displaystyle {\sum}_{a\in A}{\displaystyle {\int}_0^{f_{\bullet a}\left(\mathbf{h}\right)}{c}_a\left(\omega \right)}d\omega }+{\displaystyle {\sum}_{p\in {N}_o}{\displaystyle {\sum}_{q\in {N}_d}{\displaystyle {\int}_0^{d_{pq}} \ln \left(\omega \right)}d\omega }} $$

(A1a)

where the feasible region Ω ₄ is delineated by the following constraints.

Flow conservation constraints.

$$ {\displaystyle {\sum}_{q\in {N}_d}{d}_{pq}}={\overline{O}}_p\kern1.25em \forall p\kern1.25em \left({\mu}_p\right) $$

(A1b)

$$ {\displaystyle {\sum}_{p\in {N}_o}{d}_{pq}}={\overline{D}}_q\kern1.25em \forall q\kern1.25em \left({\lambda}_q\right) $$

(A1c)

Non-negative constraints.

$$ {h}_r\ge 0\kern1.25em \forall r\in {R}_{pq},p,q $$

(A1d)

Definitional constraints.

$$ {\displaystyle {\sum}_{r\in {R}_{pq}}{h}_r={d}_{pq}}\kern1.25em \forall p,q $$

(A1e)

$$ {f}_{\bullet a}={\displaystyle {\sum}_{r\in {R}_{pq}:\ r\supseteq a}{h}_r}\kern1.25em \forall a\in A $$

(A1f)

In the objective function (A1a), the first term is the sum of the integrals of the link cost functions {c _a } over the interval between 0 and link flow f _• a on that link, where h denotes a feasible route solution, the second term is equal to ${\displaystyle {\sum}_{p\in {N}_o}{\displaystyle {\sum}_{q\in {N}_d}{d}_{pq} \ln {d}_{pq}-{d}_{pq}}}$ which is regarded as the “virtual” cost incurred due to the added trip distribution dimension. Eq. (1b) conserves flows Ō _p generated at each origin p, and μ _p is the associated dual variable. Eq. (1c) conserves flows ${\overline{D}}_q$ attracted at each destination q, and λ _q is the associated dual variable. Eq. (A1d) shows non-negative constraints. Eq. (1e) defines O-D demand d _pq as the sum of the associated route flows. These constraints state that the flow on all paths connecting each O-D pair has to be equal to the O-D trip rate d _pq . (A1f) are definitional constraints which enter the network structure.

After forming the associated Lagrangian function and then taking the first derivatives in terms of route flows, h _r , and using the definitional constraints, the optimality conditions can be obtained as follows:

$$ {c}_r+ \ln {d}_{pq}-{\mu}_p-{\lambda}_q\ge 0\kern1.25em \forall r\in {R}_{pq},p,q $$

(A2a)

Alternatively, letting γ _pq  = μ _p  + λ _q , Eq. (A2a) can be expressed as equilibrium conditions that conform to Wardrop’s first principle as follows:

$$ {c}_r+ \ln {d}_{pq}\left\{\begin{array}{cc}\hfill ={\gamma}_{pq}\hfill & \hfill if\ {h}_r>0\hfill \\ {}\hfill \ge {\gamma}_{pq}\hfill & \hfill if\ {h}_r=0\hfill \end{array}\right.\kern1.25em \forall r\in {R}_{pq},p,q $$

(A2b)

Furthermore, by moving ln d _pq to the right hand side, Eq. (A2b) can be rewritten as

$$ {c}_r\left\{\begin{array}{cc}\hfill ={\gamma}_{pq}- \ln {d}_{pq}\hfill & \hfill if\ {h}_r>0\hfill \\ {}\hfill \ge {\gamma}_{pq}- \ln {d}_{pq}\hfill & \hfill if\ {h}_r=0\hfill \end{array}\right.\kern1.25em \forall r\in {R}_{pq},p,q $$

(A2c)

Equation (A2c) is essentially reminiscent of Wardrop’s first principle., i.e., for each O-D pair (p,q), the travel costs c _r on all routes actually used, i.e., h _r  > 0, are less than or equal to those which would be experienced by a single vehicle on any unused route, where γ _pq and γ _pq  − ln d _pq , respectively, denote the minimum perceived route cost and minimum real route cost between O-D pair (p,q).

Note also that by simple arithmetic manipulation, the upper part of Eq. (A2b) can be rewritten as:

$$ {d}_{pq}= \exp \left({\gamma}_{pq}-{c}_{r\in {R}_{pq}}\right)\kern1.25em \forall p,q $$

(A2d)

After further derivation, it obtains

$$ {d}_{pq}={\overline{O}}_p\cdot \frac{ \exp \left({\gamma}_{pq}-{c}_{r\in {R}_{pq}}\right)}{{\displaystyle {\sum}_{q^{\prime }} \exp \left({\gamma}_{p{q}^{\prime }}-{c}_{r^{\prime}\in {R}_{p{q}^{\prime }}}\right)}}\kern1.25em \forall p,q $$

(A2e)

$$ {d}_{pq}={\overline{D}}_q\cdot \frac{ \exp \left({\gamma}_{pq}-{c}_{r\in {R}_{pq}}\right)}{{\displaystyle {\sum}_{p^{\prime }} \exp \left({\gamma}_{p^{\prime }q}-{c}_{r^{\prime}\in {R}_{p^{\prime }q}}\right)}}\kern1.25em \forall p,q $$

(A2f)

Equation (A2e) or (A2f) simply state that trip demand for each O-D pair (p,q) is determined by a logit model.