# A Traffic Equilibrium Model for Multi-Modal Networks with Uncertain Demands
## 4 Effective Travel Cost (ETC) and Traffic Equilibrium Model
### Effective Travel Cost
**Case 2:** When the actual interval travel cost and the TCB satisfy \(c_{w}^{p^{-}}\leq\widetilde{c}_{w}^{-}\leq c_{w}^{p+}\leq\widetilde{c}_{w}^{+}\), the reliability aspects are given by the intervals \(\left[c_{w}^{p^{-}},\widetilde{c}_{w}^{-}\right]\)and \(\left[\widetilde{c}_{w}^{-},c_{w}^{p^{+}}\right]\), and their probabilities of not exceeding the TCB are, respectively, given by \(\dfrac{\widetilde{c}_{w}^{-}-c_{w}^{p^{-}}}{c_{w}^{p^{+}}-c_{w}^{p^{-}}}\) and \(\dfrac{\left(c_{w}^{p+}-\widetilde{c}_{w}^{-}\right)^{2}}{2\left(c_{w}^{p+}- c_{w}^{p^{-}}\right)\left(\widetilde{c}_{w}^{+}-\widetilde{c}_{w}^{-}\right)}\) + \(\dfrac{\left(c_{w}^{p+}-\widetilde{c}_{w}^{-}\right)}{\left(c_{w}^{p+}-c_{w}^{ p^{-}}\right)}\) (\(\dfrac{c_{w}^{+}-c_{w}^{p^{+}}}{\left(c_{w}^{p+}-c_{w}^{p^{-}}\right)}\)); the ATC's unreliability aspect is \(\left[\widetilde{c}_{w}^{-},c_{w}^{p^{+}}\right]\), and the probability of exceeding the TCB is \(\dfrac{\left(c_{w}^{p+}-\widetilde{c}_{w}^{-}\right)^{2}}{2\left(c_{w}^{p+}- c_{w}^{p^{-}}\right)\left(\widetilde{c}_{w}^{+}-\widetilde{c}_{w}^{-}\right)}\). **Case 3:** When the actual interval travel cost and the travel cost budget satisfy \(\widetilde{c}_{w}^{-}\leq c_{w}^{p-}\leq c_{w}^{p+}\leq\widetilde{c}_{w}^{+}\), the reliability aspect of ACT is given by the interval \(\left[c_{w}^{p^{-}},c_{w}^{p^{+}}\right]\), and the probability of not exceeding the TCB is \(\dfrac{c_{w}^{p+}-c_{w}^{p^{-}}}{2\left(\widetilde{c}_{w}^{+}-\widetilde{c}_{ w}^{-}\right)}+\dfrac{\widetilde{c}_{w}^{+}-c_{w}^{p^{+}}}{\widetilde{c}_{w}^{ +}-\widetilde{c}_{w}^{-}}\); the ATC's unreliability is \(\left[c_{w}^{p^{-}},c_{w}^{p^{+}}\right]\), and the probability of exceeding the TCB is \(\dfrac{c_{w}^{p^{-}}-\widetilde{c}_{w}^{-}}{\widetilde{c}_{w}^{+}-\widetilde{ c}_{w}^{-}}+\dfrac{c_{w}^{p^{+}}-c_{w}^{p^{-}}}{2\left(\widetilde{c}_{w}^{+}- \widetilde{c}_{w}^{-}\right)}\). **Case 4:** When the actual interval travel cost and the TCB satisfy \(\widetilde{c}_{w}^{-}\leq c_{w}^{p^{-}}\leq\widetilde{c}_{w}^{+}\leq c_{w}^{ p^{+}}\), the ATC's reliability aspect is given by \(\left[c_{w}^{p^{-}},\widetilde{c}_{w}^{+}\right]\), and the probability of not exceeding the TCB is \(\dfrac{\left(c_{w}^{p+}-\widetilde{c}_{w}^{-}\right)^{2}}{2\left(c_{w}^{p+}- c_{w}^{p^{-}}\right)\left(\widetilde{c}_{w}^{+}-\widetilde{c}_{w}^{-}\right)}\); the ATC's unreliability aspects are given by \(\left[c_{w}^{p^{-}},\widetilde{c}_{w}^{+}\right]\) and \(\left[\widetilde{c}_{w}^{+},c_{w}^{p^{+}}\right]\), and the corresponding probabilities of exceeding the TCB are \(\dfrac{\left(\widetilde{c}_{w}^{+}-c_{w}^{p^{+}}\right)}{\left(c_{w}^{p+}-c_{ w}^{p^{-}}\right)}\) (\(\dfrac{c_{w}^{p^{-}}-\widetilde{c}_{w}^{-}}{\widetilde{c}_{w}^{+}-\widetilde{c}_{ w}^{-}}\)) + \(\dfrac{\left(c_{w}^{p+}-\widetilde{c}_{w}^{-}\right)^{2}}{2\left(c_{w}^{p+}-c_{ w}^{p^{-}}\right)\left(\widetilde{c}_{w}^{+}-\widetilde{c}_{w}^{-}\right)}\) and \(\dfrac{c_{w}^{p+}-\widetilde{c}_{w}^{+}}{c_{w}^{p^{+}}-c_{w}^{p^{-}}}\). **Case 5:** When the actual interval travel cost and the TCB satisfy \(c_{w}^{p^{-}}\leq\widetilde{c}_{w}^{-}\leq\widetilde{c}_{w}^{+}\leq c_{w}^{p^{+}}\), the ATC's reliability aspects are given by \(\left[c_{w}^{p^{-}},\widetilde{c}_{w}^{-}\right]\) and \(\left[\widetilde{c}_{w}^{-},\widetilde{c}_{w}^{+}\right]\), and the corresponding probabilities of not exceeding the TCB are \(\dfrac{\widehat{c}_{w}^{-}-c_{w}^{p^{-}}}{c_{w}^{p^{+}}-c_{w}^{p^{-}}}\) and \(\dfrac{\widehat{c}_{w}^{+}-\widetilde{c}_{w}^{-}}{2\left(c_{w}^{p^{+}}-c_{w}^{ p^{-}}\right)}\); the ATC's unreliability aspects are \(\left[\widetilde{c}_{w}^{-},\widetilde{c}_{w}^{+}\right]\) and \(\left[\widetilde{c}_{w}^{+},c_{w}^{p^{+}}\right]\), and the corresponding probabilities of exceeding the TCB are \(\dfrac{\widehat{c}_{w}^{+}-\widetilde{c}_{w}^{-}}{2\left(c_{w}^{p+}-c_{w}^{p^{-} }\right)}\) and \(\dfrac{c_{w}^{p+}-\widetilde{c}_{w}^{+}}{c_{w}^{p^{+}}-c_{w}^{p^{-}}}\).