To stop or not to stop: a time-constrained trip covering location problem on a tree network

Abstract

Location of new stations/stops in public transportation networks has attracted much interest from both the point of views of theory and applications. In this paper we consider a set of pairs of points in the plane demanding traveling between the elements of each pair, and a tree network embedded in the plane representing the transportation system. An alternative mode of transportation competes with the combined plane-network mode so that the modal choice is made by distance (time) comparisons. The aim of the problem dealt with in this paper is to locate a new station/stop so that the traffic through the network would be maximized. Since stops at new stations increases the time of passengers that already used the combined mode, and may persuade them to change the mode, a constraint on the increase of the overall time is imposed. A quadratic in the number of pairs time algorithm is proposed.

Appendix: Proofs of lemmas stated in the main text

1.1 1 Proof of Lemma 6

Proof

Let us consider $(i , j) \in {\overline{C}}$ holding (5), and let x be a given point of edge ${\varvec{e}}$. From Property 2, we have $h_{ij}^{min}(v_k, v_r) > \tau _{ij}$, $\forall v_k, v_r \in V_s$. It only remains to prove $h_{ij}^{min}(x, w) > \tau _{ij}$, for any $w \in V_s$. From definition of $h_{ij}^{min}(x, w)$ (see (1)), it is sufficient to prove that $h_{ij}(x, w) > \tau _{ij}$. Hence, we have

$$\begin{aligned} h_{ij}(x, w)= & {} ||i-x|| + \displaystyle \frac{d(x, w)}{\kappa } + ||w-j|| \\&+\displaystyle \sum _{\begin{array}{c} v_s \in (V \cup \{x\}) \cap P(w, x) \\ v_s \not =w, x \end{array} } \delta _s \ge ||i-x|| + ||w-j|| \\\ge & {} \text{ dist }(i, {\varvec{e}}) + \displaystyle \min _{w \in V_s} ||w-j|| > \tau _{ij}. \end{aligned}$$

$\square$

1.2 2 Proof of Lemma 7

Proof

Let x be a given point of edge ${\varvec{e}}=[u,v]$. Condition (6) can be rewritten as

$$\begin{aligned} \min \{ \alpha _{ij}+\text{ dist }(i, {\varvec{e}}), \alpha _{ij}+\text{ dist }(j, {\varvec{e}}) \} > \tau _{ij}. \end{aligned}$$

Let $(i, j) \in {\overline{C}}$ be a pair holding this inequality. From Property 4, for concluding that $(i, j) \in {\overline{C}}(x)$, we need to prove $P_{ij}(x) = \emptyset$, which is equivalent to prove $h_{ij}^{min}(v_k, v_r) > \tau _{ij}$, for all $v_k, v_r \in V_s \cup \{x\}$.

First, $(i, j) \in {\overline{C}}$ implies $h_{ij}^{min}(v_k, v_r) > \tau _{ij}, \; \forall v_k, v_r \in V_s$. Therefore it remains to prove that $h_{ij}^{min}(x, w) = \min \{ h_{ij}(x, w), h_{ji}(x, w)\}> \tau _{ij}$, for all $w \in V_s$. Since the same reasoning applies to both $h_{ij}(x, w)$ and $h_{ji}(x, w)$ by interchanging i and j, the proof is completed by showing that $h_{ij}(x, w) > \tau _{ij}$.

From definition of the $\alpha _j$-values, both $w_j$ and the end vertex (u or v) at which $\alpha _j$ is reached belong to the same subtree, and it occurs the same with $w_i$ and $\alpha _i$. Hence, we have

$$\begin{aligned} h_{ij}(x, w)= & {} ||i-x|| + \displaystyle \frac{d(x, w)}{\kappa } + ||w-j||+ \displaystyle \sum _{\begin{array}{c} v_s \in (V \cup \{x\}) \cap P(x, w) \\ v_s \not = x, w \end{array}} \delta _s\\\ge & {} \text{ dist }(i, {\varvec{e}}) + \displaystyle \frac{d(x, w)}{\kappa } + ||w-j||+ \displaystyle \sum _{\begin{array}{c} v_s \in (V \cup \{x\})\cap P(x, w) \\ v_s \not = x, w \end{array}} \delta _s\\\ge & {} \text{ dist }(i, {\varvec{e}}) + \min \left\{ \displaystyle \frac{d(x,u)}{\kappa }+ \alpha _j(u), \displaystyle \frac{d(x,v)}{\kappa }+ \alpha _j(v) \right\} \\\ge & {} \text{ dist }(i, {\varvec{e}}) + \alpha _{ij} > \tau _{ij}. \end{aligned}$$

$\square$

1.3 3 Proof of Lemma 22

Proof

The first statement of the lemma is the trivial case, and for the second one we note that Corollary 11 implies $C^{+}(x_k)=C_{cand}({{\varvec{e}}}; \uparrow ) \cap C(x_k)$, and $C^{-}(x_k)=C_{cand}({{\varvec{e}}}; \downarrow ) \cap {\overline{C}}(x_k)$. Therefore the given characterization of $C^{+}(x_k)$ follows straightforward from Corollary 18 and definition of sets $L(x_k)$, (here we note that $L(x_k)$, $x_k \in Q({\varvec{e}})$, are the sets introduced in the proof of Theorem 19).

We next discuss in more detail $C^{-}(x_k)$. Let $(i, j) \in C_{cand}({{\varvec{e}}}; \downarrow )$ be a given pair. Thus, we analyze the conditions so that $(i, j) \in {\overline{C}}(x_k)$. First, $(i, j) \in {\overline{C}}(x_k)$ requires $x_k \in P(v_k, v_r)$, for all $P(v_k, v_r) \in P_{ij}$, otherwise from Lemma 8, $(i, j) \in C(x_k)$. And, from Corollary 18, $(i, j) \in {\overline{C}}(x_k)$ also requires $(i, j) \notin L(x_k)$ and $h_{ij}^{min}+\delta _{x_k} > \tau _{ij}$. This concludes the proof. $\square$

1.4 4 Proof of Lemma 23

Proof

For $k = 1, \ldots , \ell -1$, each set $L_{ini}(x_k)$ contains the pairs such that some associated sublevel set has an endpoint at $x_k$. Thus, in passing through $k = 1, \ldots , \ell -1$, the sets $L(x_k)$ are updated by incorporating the pairs $(i, j) \in L(x_{k-1})$ with some sublevel curve defined at $x_k$, which means $x_k \in S_{ij}$. As consequence, from $L(x_k)$ both $C^{+}(x_k)$ and $C^{-}(x_k)$ can also be computed according with Lemma 22. Finally, for the case $C^{-}(x_k)$, condition $x_k \in P_{ij}(v_k, v_r)$ is equivalent to ${\varvec{e}} \subseteq P_{ij}(v_k, v_r)$ when $k = 1, \ldots , \ell -1$. $\square$