Total cost minimizing transit route structures considering trips towards CBD and periphery

Abstract

The total cost minimizing approach to design transit systems is extended here beyond the usual dimensions of fleet (frequency) and vehicle size in order to examine the most appropriate spatial setting of transit lines as well. Motivated by the case of large cities in Latin America, characterized by high volumes of relatively long urban trips, we analyze the best ways to provide public transport services in a simplified urban setting represented by an extended cross-shaped network, where short trips (periphery–center) and long trips (periphery–periphery) coexist, generating economies of density. Three families of strategic lines structures are compared: mostly direct, feeder–trunk and hub and spoke. For each structure fleet and vehicle sizes are optimized, considering total (users’ and operators’) costs. The best structure is found parametrically in total passenger volume, the proportion of long trips and the value of the transfer penalty. The advantages of each dominating structure are explained in terms of factors like idle capacity, waiting or in-vehicle times and number of transfers.

Appendix: cost functions and optimal frequencies for the remaining structures

$$\begin{aligned} VRC_{FT - D}^{*} & = 2(2 + \alpha )C_{0} tY + \sqrt {2T_{A} YC_{0} \left( {4C^{{\prime }} tY + 8\varepsilon P_{w} + P_{v} tY} \right)} + \sqrt {2nT_{A} YC_{0} \left[ {4C^{{\prime }} tY + \frac{{8\left( {1 + \alpha } \right)}}{2}\varepsilon P_{w} + P_{v} tY} \right]} \\ & \quad + \sqrt {2\alpha T_{A} YC_{0} \left( {4\alpha C^{{\prime }} tY + 8\varepsilon P_{w} + \alpha P_{v} tY} \right)} + YT_{A} \left[ {C^{{\prime }} \left( {1 + n + \alpha } \right) + \frac{{P_{v} }}{4}\left( {1 + \alpha } \right)\left( {2 + n} \right)} \right] + P_{t} \eta Y \\ \end{aligned}$$

(24)

$$f_{F}^{*} = \sqrt {\frac{Y}{{32T_{A} C_{0} }}\left( {4C^{{\prime }} tY + 8\varepsilon P_{w} + P_{v} tY} \right)}$$

(25)

$$f_{T}^{*} = \sqrt {\frac{Y}{{32nT_{A} C_{0} }}\left[ {4C^{{\prime }} tY + 4\left( {1 + \alpha } \right)\varepsilon P_{w} + P_{v} tY} \right]}$$

(26)

$$f_{D}^{*} = \sqrt {\frac{\alpha Y}{{32T_{A} C_{0} }}\left( {4\alpha C^{{\prime }} tY + 8\varepsilon P_{w} + \alpha P_{v} tY} \right)}$$

(27)

$$\begin{aligned} VRC_{HS}^{*} & = 2(1 + \alpha )C_{0} tY + \sqrt {\left( {2 + n} \right)T_{A} YC_{0} \left( {4C^{{\prime }} tY + 8P_{w} \varepsilon + P_{v} tY} \right)} \\ & \quad + \sqrt {\alpha \left( {2 + n} \right)T_{A} YC_{0} \left( {4\alpha C^{{\prime }} tY + 8P_{w} \varepsilon + \alpha P_{v} tY} \right)} \,\, \\ & \quad + \left( {2 + n} \right)T_{A} Y\frac{{\left( {1 + \alpha } \right)}}{2}\left( {C^{{\prime }} + \frac{{P_{v} }}{2}} \right) + P_{t} \eta Y \\ \end{aligned}$$

(28)

$$f_{in}^{*} = \sqrt {\frac{Y}{{16\left( {2 + n} \right)T_{A} C_{0} }}\left( {4C^{{\prime }} tY + 8\varepsilon P_{w} + P_{v} tY} \right)}$$

(29)

$$f_{out}^{*} = \sqrt {\frac{\alpha Y}{{16\left( {2 + n} \right)T_{A} C_{0} }}\left( {4\alpha C^{{\prime }} tY + 8\varepsilon P_{w} + \alpha P_{v} tY} \right)}$$

(30)

$$\begin{aligned} VRC_{DIR - 4}^{*} & = \frac{{\left( {4 + 3\alpha } \right)}}{2}tYC_{0} + \left( {2 + n} \right)T_{A} Y\left[ {\frac{{P_{v} \left( {1 + \alpha } \right)}}{4} + C^{{\prime }} } \right] \\ &\quad + 2\sqrt {\left( {2 + n} \right)T_{A} YC_{0} \left[ {\left( {4 + 3\alpha } \right)\left( {\frac{{C^{{\prime }} tY}}{2} + P_{w} \varepsilon } \right) + \frac{{\left( {16 + 12\alpha + 9\alpha^{2} } \right)}}{32}P_{v} tY} \right]} + P_{t} \eta Y \\ \end{aligned}$$

(31)

$$f_{DIR - 4}^{*} = \sqrt {\frac{Y}{{16\left( {2 + n} \right)T_{A} C_{0} }}\left[ {\left( {4 + 3\alpha } \right)\left( {\frac{{C^{{\prime }} tY}}{2} + \varepsilon P_{w} } \right) + \frac{{\left( {16 + 21\alpha^{2} } \right)}}{32}P_{v} tY} \right]}$$

(32)

$$\begin{aligned} VRC_{DIR - 8}^{*} & = \left( {2 + \alpha } \right)C_{0} tY + \left( {2 + n} \right)T_{A} Y\left[ {C^{{\prime }} + P_{v} \frac{{\left( {1 + \alpha } \right)}}{4}} \right] \\ & + \sqrt {4\left( {2 + n} \right)T_{A} YC_{0} \left[ {C^{{\prime }} tY\left( {2 + \alpha } \right) + 2\left( {2 + 3\alpha } \right)P_{w} \varepsilon + \frac{{\left( {4 + 5\alpha^{2} } \right)P_{v} tY}}{8}} \right]} + P_{t} \eta Y \\ \end{aligned}$$

(33)

$$f_{DIR - 8}^{*} = \sqrt {\frac{Y}{{64\left( {2 + n} \right)T_{A} C_{0} }}\left[ {\left( {2 + \alpha } \right)C^{{\prime }} tY + 2\left( {2 + 3\alpha } \right)\varepsilon P_{w} + \frac{{\left( {4 + 5\alpha^{2} } \right)}}{8}P_{v} tY} \right]}$$

(34)

$$\begin{aligned} VRC_{DIR - 16}^{*} & = 2C_{0} tY + \sqrt {16\left( {2 + n} \right)T_{A} YC_{0} \left[ {\frac{{C^{{\prime }} tY}}{2} + \left( {1 + 3\alpha } \right)P_{w} \varepsilon + \frac{{P_{v} tY}}{8}} \right]} \\ &\quad + \left( {2 + n} \right)T_{A} Y\left[ {C^{{\prime }} + P_{v} \frac{{\left( {1 + \alpha } \right)}}{4}} \right] + P_{t} \eta Y \\ \end{aligned}$$

(35)

$$f_{DIR - 16}^{*} = \sqrt {\frac{Y}{{512\left( {2 + n} \right)T_{A} C_{0} }}\left[ {4C^{{\prime }} tY + 8\left( {1 + 3\alpha } \right)\varepsilon P_{w} + P_{v} tY} \right]}$$

(36)