# A Traffic Equilibrium Model for Multi-Modal Networks with Uncertain Demands
## 4 Effective Travel Cost (ETC) and Traffic Equilibrium Model  
### Effective Travel Cost  
Due to the demand for interval travels, the travel cost of a route becomes an interval variable. If only the expected value of the interval travel cost in travel mode and travel route decision is considered similarly to the traditional equilibrium model, it is difficult to illustrate the travellers' real choices comprehensively. Therefore, a new choice criterion needs to be defined and a corresponding equilibrium model needs to be formulated to capture travellers' choices on travel modes and routes under the interval travel cost. By analyzing the characteristic of interval travel cost, the concept of ETC, denoted as \(\tilde{c}_{w}^{p}\), is proposed, which explicitly considers both reliability and unreliability aspects of the interval travel cost in the travel mode and route choosing process. First, a predicted travel cost value is considered from the origin to the destination, which is defined as the TCB. **Definition 1**. : _The TCB is defined as an interval variable \(\widetilde{c}_{w}=[\widetilde{c}_{w}^{-},\widetilde{c}_{w}^{+}]\), of which the mean value is calculated as follows ([PERSON] (1996, 1979) [20; 21]):_  
\[\frac{\widetilde{c}_{w}^{-}+\widetilde{c}_{w}^{+}}{2}=\sum_{m}c_{w}^{m} \tag{24}\]  
_where \(\varsigma_{w}^{m}\) is the expected free-flow travel cost of routes in travel mode \(m\) for OD pair \(\omega\); the deviation value of TCB is calculated as follows:_  
\[\frac{\widetilde{c}_{w}^{+}-\widetilde{c}_{w}^{-}}{2}=\frac{\widetilde{c}_{w} ^{-}+\widetilde{c}_{w}^{+}}{2}\times\omega \tag{25}\]  
_where \(\omega\) is defined as an acceptable deviation value, which reflects the traveller's tolerance for variability of the interval travel cost. The value is assumed to be 0.1._  
Comparing the upper and lower bounds of the budgeted travel cost with the actual travel cost, the actual travel cost can be divided into multiple subintervals. If the value in the sub-interval is likely to be less than the budgeted travel cost, the sub-interval is called the reliability part of the actual travel cost. If the value in the subinterval is likely to be greater than the budgeted trip cost, the subinterval is said to be the unreliable part of the actual trip cost, The definition of reliability and unreliability aspects of the actual travel costs is shown in Figure 3.The effective trip cost is defined as the product of the mean of the reliable subinterval and the probability that the subinterval is less than the budgeted trip cost + the product of the mean of the unreliable subinterval and the probability that the subinterval is greater than the budgeted trip cost. The first group reflects the reliability aspect, under which travellers travel from their origin to their destination with an acceptable level deviation of \(\omega\) from their TCB. It is defined as the subintervals of the actual interval travel cost, which has a certain probability of being less than the TCB. The second group reflects the unreliability aspect, under which travellers travel from their origin to their destination but exceed their TCB by a value greater than \(\omega\). It is defined as subintervals of the actual interval travel costs, which have a certain probability of exceeding the TCB. **Definition 2**. : _The ETC \(\hat{c}_{w}^{p}\) for each route is defined as a weighted sum of the mean values of the reliability and unreliability aspects of the actual travel costs. The respective weights are equal to the corresponding probabilities. Therefore, the ETC equation is presented for the following six cases:_  
**Case 1:** When the actual interval travel cost and the TCB satisfy \(c_{w}^{p^{-}}\leq c_{w}^{p+}\leq\widetilde{c}_{w}^{-}\leq\widetilde{c}_{w}^{+}\), the reliability aspect is given by the interval \(\left[c_{w}^{p^{-}},c_{w}^{p^{+}}\right]\), and the probability of not exceeding the TCB is equal to 1.