Abstract
This paper addresses a network pricing problem where users are assigned to the paths of a transportation network according to a mixed logit model, i.e., price sensitivity varies across the user population. For its solution, we propose algorithms based on combinatorial approximations, and show that the smoothing effect induced by both the discrete choice and price sensitivity features of the model help in determining near-global solutions. This stands in contrast with simpler formulations where the main difficulty is due to the combinatorial nature of the problem. From an economic point of view, we provide an estimate of the proportion of revenue raised from the various population segments, an information that can be used for policy purposes.
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Notes
Throughout the paper, a tilde denotes a random quantity.
This can be achieved through an arbitrary small perturbation of the fixed costs, as has been shown in [10].
Abbreviations
- \(\mathcal{A} \) :
-
Set of arcs
- \(\mathcal{A}_{\mathrm{toll}}\) :
-
Subset of toll arcs
- θ :
-
Logit scale parameter
- t a :
-
Toll on arc \(a\in\mathcal{A}_{\mathrm{toll}}\)
- c a :
-
Fixed cost on arc \(a\in\mathcal{A}\)
- \(\mathcal{Q} \) :
-
Set of OD pairs
- \(\mathcal{R} \) :
-
Set of paths
- \(\mathcal{R}^{q} \) :
-
Set of paths with respect to OD pair q
- d q :
-
Demand on OD pair q
- \(\mathcal{A}( r) \) :
-
Set of arcs incident to path r
- \(\mathcal{A}_{\mathrm{toll}}( r)\) :
-
Set of toll arcs incident to path r
- \(\mathcal{N} \) :
-
Finite set of commuter classes
- \(\tilde{\alpha} \) :
-
Price sensitivity (random variable)
- f :
-
Price sensitivity density
- α n :
-
partition of the support of f
- \(\tilde{\mu} \) :
-
Value of time (random variable)
- u qr (t,α):
-
Expected disutility of path \(r \in \mathcal{R}^{q}\), under toll policy t, for a commuter having price sensitivity \(\tilde{\alpha}=\alpha\)
- τ q (t,α):
-
Shortest path expected disutility, for OD pair \(q\in\mathcal{Q}\), under toll policy t, for a commuter having price sensitivity \(\tilde{\alpha}=\alpha\), according to a logit discrete choice model
- \(\pi^{n}_{q} \) :
-
Shortest path expected disutility, for OD pair \(q\in \mathcal{Q}\), under toll policy t, for a commuter of class \(n\in \mathcal{N}\), under a deterministic assignment
- logit(t,α):
-
Proportion of commuters on OD pair q having price sensitivity \(\tilde{\alpha}=\alpha\) assigned to path \(r\in\mathcal{R}^{q}\), under toll policy t, according to the logit choice model
- y qa (t,α):
-
Proportion of commuters on OD pair q having price sensitivity \(\tilde{\alpha}=\alpha\) assigned to a path incident to toll arc a, under toll policy t, according to the logit choice model
- x qr (μ):
-
Proportion of commuters on OD pair q having price sensitivity \(\tilde{\alpha}=\alpha\) assigned to path \(r\in\mathcal {R}^{q}\), under a deterministic assignment
- \(x^{n}_{qr} \) :
-
Proportion of class n commuters on OD pair q assigned to path \(r\in\mathcal{R}^{q}\), under a deterministic assignment
- F CD :
-
Continuous-multi-class deterministic revenue (objective of Model 2.)
- F ML :
-
Mixed-logit revenue (objective of Model 4.)
- F DS :
-
Discrete-stochastic approximation revenue (objective of Approximation 1)
- F DD :
-
Discrete-deterministic approximation revenue (objective of Approximation 2)
- F US :
-
Uniform-stochastic approximation revenue (objective of Approximation 3)
- F UD :
-
Uniform-deterministic approximation revenue (objective of Approximation 4)
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Gilbert, F., Marcotte, P. & Savard, G. Mixed-logit network pricing. Comput Optim Appl 57, 105–127 (2014). https://doi.org/10.1007/s10589-013-9585-0
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DOI: https://doi.org/10.1007/s10589-013-9585-0