Optimal design and benefits of a short turning strategy for a bus corridor

Abstract

We develop a short turning model using demand information from station to station within a single bus line-single period setting, aimed at increasing the service frequency on the more loaded sections to deal with spatial concentration of demand considering both operators’ and users’ costs. We find analytical expressions for optimal values of the design variables, namely frequencies (inside and outside the short cycle), capacity of vehicles and the position of the short turn limit stations. These expressions are used to analyze the influence of different parameters in the final solution. The design variables and the corresponding cost components for operators and users (waiting and in-vehicle times) are compared against an optimized normal operation scheme (single frequency). Applications on actual transit corridors exhibiting different demand profiles are conducted, calculating the optimal values for the design variables and the resulting benefits for each case. Results show the typical demand configurations that are better served using a short turn strategy.

Appendices

Appendix A. Definition of g _i functions

• g₀ is the total running time:

$$ g_{0} = 2\sum\limits_{k = 1}^{N - 1} {R_{k} } $$

• g₁ and g₂ are in the waiting and cycle times, g₁ is the demand benefited by the strategy (origin and destination inside the short turn), while g₂ are the passenger whose origin or destination are outside the short turn.

$$ g_{1} (s_{0} ,s_{1} ) = \sum\limits_{{k = s_{0} }}^{{s_{1} - 1}} {\lambda_{k}^{ + } (k + 1,s_{1} )} + \sum\limits_{{k = s_{0} + 1}}^{{s_{1} }} {\lambda_{k}^{ + } (s_{0} ,k - 1)} $$

$$ g_{2} (s_{0} ,s_{1} ) = \sum\limits_{k = 1}^{{s_{0} - 1}} {\lambda_{k}^{1 + } } + \sum\limits_{{k = s_{0} }}^{{s_{1} - 1}} {\lambda_{k}^{ + } (s_{1} + 1,N)} + \sum\limits_{{k = s_{1} }}^{N} {\lambda_{k}^{1 + } } + \sum\limits_{{k = s_{1} + 1}}^{N} {\lambda_{k}^{2 + } } + \sum\limits_{{k = s_{0} + 1}}^{{s_{1} }} {\lambda_{k}^{ + } (1,s_{0} - 1)} + \sum\limits_{k = 1}^{{s_{0} }} {\lambda_{k}^{2 + } } $$

• g₃ is the running time for vehicles inside the short turn

$$ g_{3} (s_{0} ,s_{1} ) = 2\sum\limits_{{k = s_{0} }}^{{s_{1} - 1}} {R_{k} } $$

• g₄ is the in-vehicle time experienced by passengers

$$ g_{4} = \sum\limits_{k = 1}^{N} {\sum\limits_{l = 1}^{N} {\lambda_{kl} \sum\limits_{i = k}^{l - 1} {R_{i} } } } $$

• g₅ and g₆ are the factors to calculate the total dwell time of passengers benefited by the strategy and the others, respectively.

$$ g_{5} (s_{0} ,s_{1} ) = g_{5}^{1} (s_{0} ,s_{1} ) + g_{5}^{2} (s_{0} ,s_{1} ) $$

$$ g_{6} (s_{0} ,s_{1} ) = g_{6}^{1} (s_{0} ,s_{1} ) + g_{6}^{2} (s_{0} ,s_{1} ) $$

where

$$ \begin{aligned} g_{5}^{1} (s_{0} ,s_{1} ) = & \sum\limits_{k = 1}^{{s_{0} - 1}} {\sum\limits_{{l = s_{0} + 1}}^{{s_{1} }} {\lambda_{kl} \sum\limits_{{i = s_{0} }}^{l - 1} {\lambda_{i}^{ + } (i + 1,s_{1} )} } } + \sum\limits_{k = 1}^{{s_{0} - 1}} {\sum\limits_{{l = s_{1} + 1}}^{N} {\lambda_{kl} \sum\limits_{{i = s_{0} }}^{{s_{1} - 1}} {\lambda_{i}^{ + } (i + 1,s_{1} )} } } \\ + \sum\limits_{{k = s_{0} }}^{{s_{1} - 1}} {\sum\limits_{l = k + 1}^{{s_{1} }} {\lambda_{kl} \sum\limits_{i = k}^{l - 1} {\lambda_{i}^{1 + } } } } + \sum\limits_{{k = s_{0} }}^{{s_{1} - 1}} {\sum\limits_{{l = s_{1} + 1}}^{N} {\lambda_{kl} \sum\limits_{i = k}^{{s_{1} - 1}} {\lambda_{i}^{ + } (i + 1,s_{1} )} } } \\ \end{aligned} $$

$$ \begin{aligned} g_{5}^{2} (s_{0} ,s_{1} ) = & \sum\limits_{{k = s_{1} + 1}}^{N} {\sum\limits_{{l = s_{0} }}^{{s_{1} - 1}} {\lambda_{kl} \sum\limits_{i = l + 1}^{{s_{1} }} {\lambda_{i}^{ + } (s_{0} ,i - 1)} } } + \sum\limits_{{k = s_{1} + 1}}^{N} {\sum\limits_{l = 1}^{{s_{0} - 1}} {\lambda_{kl} \sum\limits_{{i = s_{0} + 1}}^{{s_{1} }} {\lambda_{i}^{ + } (s_{0} ,i - 1)} } } \\ + \sum\limits_{{k = s_{0} + 1}}^{{s_{1} }} {\sum\limits_{{l = s_{0} }}^{k - 1} {\lambda_{kl} \sum\limits_{i = l + 1}^{k} {\lambda_{i}^{2 + } } } } + \sum\limits_{{k = s_{0} + 1}}^{{s_{1} }} {\sum\limits_{l = 1}^{{s_{0} - 1}} {\lambda_{kl} \sum\limits_{{i = s_{0} + 1}}^{k} {\lambda_{i}^{ + } (s_{0} ,i - 1)} } } \\ \end{aligned} $$

$$ \begin{aligned} g_{6}^{1} (s_{0} ,s_{1} ) = & \sum\limits_{k = 1}^{{s_{0} - 1}} {\sum\limits_{l = k + 1}^{{s_{0} }} {\lambda_{kl} \sum\limits_{i = k}^{l - 1} {\lambda_{i}^{1 + } } } } + \sum\limits_{k = 1}^{{s_{0} - 1}} {\sum\limits_{{l = s_{0} + 1}}^{{s_{1} }} {\lambda_{kl} \left[ {\sum\limits_{i = k}^{{s_{0} - 1}} {\lambda_{i}^{1 + } + \sum\limits_{{i = s_{0} }}^{l - 1} {\lambda_{i}^{ + } (s_{1} + 1,N)} } } \right]} } \\ + \sum\limits_{k = 1}^{{s_{0} - 1}} {\sum\limits_{{l = s_{1} + 1}}^{N} {\lambda_{kl} \left[ {\sum\limits_{i = k}^{{s_{0} - 1}} {\lambda_{i}^{1 + } + \sum\limits_{{i = s_{0} }}^{{s_{1} - 1}} {\lambda_{i}^{ + } (s_{1} + 1,N)} } + \sum\limits_{{i = s_{1} }}^{l - 1} {\lambda_{i}^{1 + } } } \right]} } \\ + \sum\limits_{{k = s_{0} }}^{{s_{1} - 1}} {\sum\limits_{{l = s_{1} + 1}}^{N} {\lambda_{kl} \left[ {\sum\limits_{i = k}^{{s_{1} - 1}} {\lambda_{i}^{ + } (s_{1} + 1,N) + \sum\limits_{{i = s_{1} }}^{l - 1} {\lambda_{i}^{1 + } } } } \right]} } + \sum\limits_{{k = s_{1} }}^{N} {\sum\limits_{l = k + 1}^{N} {\lambda_{kl} \sum\limits_{i = k}^{l - 1} {\lambda_{i}^{1 + } } } } \\ \end{aligned} $$

$$ \begin{gathered} g_{6}^{2} (s_{0} ,s_{1} ) = \sum\limits_{{k = s_{1} + 1}}^{N} {\sum\limits_{{l = s_{1} }}^{k - 1} {\lambda_{kl} \sum\limits_{i = l + 1}^{k} {\lambda_{i}^{2 + } } } } + \sum\limits_{{k = s_{1} + 1}}^{N} {\sum\limits_{{l = s_{0} }}^{{s_{1} - 1}} {\lambda_{kl} \left[ {\sum\limits_{{i = s_{1} + 1}}^{k} {\lambda_{i}^{2 + } + \sum\limits_{i = l + 1}^{{s_{1} }} {\lambda_{i}^{ + } (1,s_{0} - 1)} } } \right]} } \hfill \\ + \sum\limits_{{k = s_{1} + 1}}^{N} {\sum\limits_{l = 1}^{{s_{0} - 1}} {\lambda_{kl} \left[ {\sum\limits_{{i = s_{1} + 1}}^{k} {\lambda_{i}^{2 + } + \sum\limits_{{i = s_{0} + 1}}^{{s_{1} }} {\lambda_{i}^{ + } (1,s_{0} - 1)} } + \sum\limits_{i = l + 1}^{{s_{0} }} {\lambda_{i}^{2 + } } } \right]} } \hfill \\ + \sum\limits_{{k = s_{0} + 1}}^{{s_{1} }} {\sum\limits_{l = 1}^{{s_{0} - 1}} {\lambda_{kl} \left[ {\sum\limits_{{i = s_{0} + 1}}^{k} {\lambda_{i}^{ + } (1,s_{0} - 1) + \sum\limits_{i = l + 1}^{{s_{0} }} {\lambda_{i}^{1 + } } } } \right]} } + \sum\limits_{k = 1}^{{s_{0} }} {\sum\limits_{l = 1}^{k - 1} {\lambda_{kl} \sum\limits_{i = l + 1}^{k} {\lambda_{i}^{2 + } } } } \hfill \\ \end{gathered} $$

Appendix B. Load of vehicles between stations

• 1. Load of vehicles serving the entire corridor (fleet A):

Direction 1

$$ \pi_{k}^{1} = \left\{ {\begin{array}{*{20}c} {\pi_{k - 1}^{1} + {\frac{{\lambda_{k}^{1 + } }}{{f_{\text{A}} }}} - {\frac{{\lambda_{k}^{1 - } }}{{f_{\text{A}} }}}} \hfill & {{\text{if}}\;1 \le k \le s_{0} - 1} \hfill \\ {\pi_{k - 1}^{1} + {\frac{{\lambda_{k}^{ + } (k,s_{1} )}}{{f_{\text{A}} + f_{\text{B}} }}} + {\frac{{\lambda_{k}^{ + } (s_{1} + 1,N)}}{{f_{\text{A}} }}} - {\frac{{\lambda_{k}^{ - } (1,s_{0} - 1)}}{{f_{\text{A}} }}} - {\frac{{\lambda_{k}^{ - } (s_{0} ,k - 1)}}{{f_{\text{A}} + f_{\text{B}} }}}} \hfill & {{\text{if}}\;s_{0} \le k \le s_{1} - 1} \hfill \\ {\pi_{k - 1}^{1} + {\frac{{\lambda_{k}^{1 + } }}{{f_{\text{A}} }}} - {\frac{{\lambda_{k}^{1 - } }}{{f_{\text{A}} }}}} \hfill & {{\text{if}}\;s_{1} \le k \le N} \hfill \\ \end{array} } \right. $$

Direction 2

$$ \pi_{k}^{2} = \left\{ {\begin{array}{*{20}c} {\pi_{k + 1}^{2} + {\frac{{\lambda_{k}^{2 + } }}{{f_{\text{A}} }}} - {\frac{{\lambda_{k}^{2 - } }}{{f_{\text{A}} }}}} \hfill & {{\text{if}}\;s_{1} + 1 \le k \le N} \hfill \\ {\pi_{k + 1}^{2} + {\frac{{\lambda_{k}^{ + } (s_{0} ,k - 1)}}{{f_{\text{A}} + f_{\text{B}} }}} + {\frac{{\lambda_{k}^{ + } (1,s_{0} - 1)}}{{f_{\text{A}} }}} - {\frac{{\lambda_{k}^{ - } (s_{1} + 1,N)}}{{f_{\text{A}} }}} - {\frac{{\lambda_{k}^{ - } (k + 1,s_{1} )}}{{f_{\text{A}} + f_{\text{B}} }}}} \hfill & {{\text{if}}\;s_{0} + 1 \le k \le s_{1} } \hfill \\ {\pi_{k + 1}^{2} + {\frac{{\lambda_{k}^{2 + } }}{{f_{\text{A}} }}} - {\frac{{\lambda_{k}^{2 - } }}{{f_{\text{A}} }}}} \hfill & {{\text{if}}\;1 \le k \le s_{0} } \hfill \\ \end{array} } \right. $$

• 2. Load of vehicles performing short turning (fleet B):

Direction 1

$$ \tilde{\pi }_{k}^{1} = \left\{ {\begin{array}{*{20}c} 0 \hfill & {{\text{if}}\;1 \le k \le s_{0} - 1} \hfill \\ {\tilde{\pi }_{k - 1}^{1} + {\frac{{\lambda_{k}^{ + } (k,s_{1} )}}{{f_{\text{A}} + f_{\text{B}} }}} - {\frac{{\lambda_{k}^{ - } (s_{0} ,k - 1)}}{{f_{\text{A}} + f_{\text{B}} }}}} \hfill & {{\text{if}}\;s_{0} \le k \le s_{1} - 1} \hfill \\ 0 \hfill & {{\text{if}}\;s_{1} \le k \le N} \hfill \\ \end{array} } \right. $$

Direction 2

$$ \tilde{\pi }_{k}^{2} = \left\{ {\begin{array}{*{20}c} 0 \hfill & {{\text{if}}\;s_{1} + 1 \le k \le N} \hfill \\ {\pi_{k + 1}^{2} + {\frac{{\lambda_{k}^{ + } (s_{0} ,k - 1)}}{{f_{\text{A}} + f_{\text{B}} }}} - {\frac{{\lambda_{k}^{ - } (k + 1,s_{1} )}}{{f_{\text{A}} + f_{\text{B}} }}}} \hfill & {{\text{if}}\;s_{0} + 1 \le k \le s_{1} } \hfill \\ 0 \hfill & {{\text{if}}\;1 \le k \le s_{0} } \hfill \\ \end{array} } \right. $$

After applying the short-turning strategy, the maximum load will correspond to some segment, which can belong to either direction 1 or 2. However, by observing the recursive form of the load equations above, we can establish the following:

• If the maximum load occurs along direction 1, then we can say that

$$ \eta K = \pi_{\max }^{1} = {\frac{{\varphi_{0}^{1} (s_{0} ,s_{1} )}}{{f_{\text{A}} }}} + {\frac{{\varphi_{1}^{1} (s_{0} ,s_{1} )}}{{f_{\text{A}} + f_{\text{B}} }}} $$

• Otherwise, if the maximum load occur along direction 2, then we can say that

$$ \eta K = \pi_{\max }^{2} = {\frac{{\varphi_{0}^{2} (s_{0} ,s_{1} )}}{{f_{\text{A}} }}} + {\frac{{\varphi_{1}^{2} (s_{0} ,s_{1} )}}{{f_{\text{A}} + f_{\text{B}} }}} $$

Thus, a generic relation between the design capacity of buses and the frequencies f _A and f _B can be established in the way summarized in Eq. 14:

$$ K = {\frac{1}{\eta }}\left( {{\frac{{\vartheta_{0} (s_{0} ,s_{1} )}}{{f_{\text{A}} }}} + {\frac{{\vartheta_{1} (s_{0} ,s_{1} )}}{{f_{\text{A}} + f_{\text{B}} }}}} \right) $$

where, if the maximum load happens on a segment that belongs to direction 1, then ϑ ₀(s ₀, s ₁) = φ ¹₀ (s ₀, s ₁) and ϑ ₁(s ₀, s ₁) = φ ¹₁ (s ₀, s ₁). Otherwise, if the maximum load occurs on a segment that belongs to direction 2, then ϑ ₀(s ₀, s ₁) = φ ²₀ (s ₀, s ₁) and ϑ ₁(s ₀, s ₁) = φ ²₁ (s ₀, s ₁).