Optimizing stop plan and tickets allocation for high-speed railway based on uncertainty theory

Abstract

Aiming to provide a generic modeling framework for finding stop plan and tickets allocation of high-speed railway, we first propose a stop plan and tickets allocation collaborative optimization model in this paper, which is established to maximize the passenger satisfaction degree and the average seated occupancy rate. Due to the randomness and uncertainty of passenger demand, uncertain variables are set and the primal model is an uncertain model. And then, the model is transformed into equivalent deterministic model based on uncertainty theory. Because of the computational complexity of the model, especially for the large-scale real-world instances, we develop a Lagrangian relaxation (LR-based) heuristic algorithm that decomposes the primal problem into two sub-problems and thus is able to find good solutions in short time. Finally, a numerical experiment based on the operation data of high-speed railway from Beijing south Station to Shanghai Hongqiao Station is implemented to verify the effectiveness and feasibility of the proposed approaches.

Appendix A: Uncertainty theory

Uncertainty theory provides an axiomatic system to deal with the imprecise information in experts’ empirical data. Similar to probability theory, an event is denoted by a set with a designated nonnegative value, which indicates the occurrence possibility of the event. Different from that in probability theory, the occurrence possibility is obtained by the judgement of experts other than historical or experimental data. There are some fundamental concepts and properties in Uncertainty theory.

Let $\varGamma$ be a nonempty set and L is a $\sigma$-algebra over $\varGamma$. Each element $\varLambda$$\in$L is designated a number $M\{\varLambda \}$. In order to ensure the set function $M\{\varLambda \}$ has certain mathematical properties, the following four axioms were presented (Liu 2007, 2009):

Axiom 1

(Normality) $M\{\varLambda \}=1$.

Axiom 2

(Self-duality) $M\{\varLambda \} + M\{\varLambda ^c\}=1$ for any event $\varLambda$.

Axiom 3

(Countable subadditivity) For every countable sequence of events $\{\varLambda _i\}$, we have

$$\begin{aligned} M\left\{ \bigcup \limits _{i=1}^\infty \varLambda _i\right\} \le \sum \limits _{i=1}^\infty M\{\varLambda _i\}. \end{aligned}$$

Axiom 4

(Product Axiom) Let ($\varGamma _i, L_i, M_i$) be uncertainty spaces for $i=1, 2, \ldots , n$. Then, the product uncertain measure M is an uncertain measure on the product $\sigma$-algebra $L_1\times L_2 \times \cdots \times L_n$ satisfying

$$\begin{aligned} M\left\{ \prod \limits _{i=1}^n \varLambda _i\right\} =\min \limits _{1 \le i \le n}M_i\{\varLambda _i\}. \end{aligned}$$

Fig. 4

Zigzag uncertainty distribution

The set function $M\{\varLambda \}$ is called an uncertain measure if it satisfies the first three axioms. Simply speaking, uncertain measure $M\{\varLambda \}$ is just the occurrence possibility of event $\varLambda$, which is from experts’ evaluation. Based on the axioms, the concept of uncertain variable was proposed, whose mathematical properties can also be deduced.

Definition 1

(Liu 2007) An uncertain variable is a measurable function $\xi$ from an uncertainty space ($\varGamma , L, M$) to the set of real numbers, i.e., for any Borel set B of real numbers, the set $\{\xi \in B \}=\{\gamma \in \varGamma \vert \xi (\gamma )\in B\}$ is an event.

Similar to the random variable, the distribution of an uncertain variable $\xi$ is defined by $\varPhi (x)=M\{\xi \le x\}$ for any real number x. For example, the zigzag uncertain variable $\xi \sim Z(a, b, c)$ has an uncertainty distribution $\varPhi (x)$, which is shown in Fig. 4.

Definition 2

(Liu 2010) An uncertain variable $\xi$ with distribution $\varPhi$ is said to be regular, if its inverse function $\varPhi ^{-1}(\alpha )$ exists and is unique for each $\alpha \in (0, 1)$.

$$\begin{aligned} \varPhi (x)=\left\{ \begin{array}{lll} 0, &{}\quad &{} {\mathrm{if}~x \le a}\\ \displaystyle \frac{x - a}{2(b - a)}, &{} &{}\quad {\mathrm{if}~a< x \le b}\\ \displaystyle \frac{x + c - 2b}{2(c - b)}, &{} &{}\quad {\mathrm{if}~b < x \le c}\\ 1, &{} &{}\quad {\mathrm{if}~x > c.} \end{array} \right. \end{aligned}$$

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According to Definition 2, zigzag uncertain variable Z(a, b, c) is a regular variable, whose inverse distribution function is $\varPhi ^{-1}(\alpha )$, i.e.,

$$\begin{aligned} \varPhi ^{-1}(\alpha )=\left\{ \begin{array}{lll} (1 - 2\alpha )a + 2\alpha b, &{} &{}\quad {\mathrm{if}~0< \alpha< 0.5}\\ (2 - 2\alpha )b + (2\alpha - 1)c, &{} &{}\quad {\mathrm{if}~0.5 \le \alpha < 1.} \end{array} \right. \end{aligned}$$

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Obviously, if $\xi$ is regular, the distribution function $\varPhi (x)$ is continuous and strictly increasing at each point x with $0<\varPhi (x)<1$. Meanwhile, $\varPhi ^{-1}(\alpha )$ is continuous and strictly increasing for $\alpha \in (0, 1)$. We usually assume that all uncertain variables in practical applications are regular. Otherwise, a small perturbation can be imposed to obtain a regular one. Next, we introduce an operational law for regular uncertain variables.

A real function $f(x_1, x_2, \ldots , x_n)$ is said to be strictly increasing if f satisfies the following conditions:

• (1) $f(x_1, x_2, \ldots , x_n)$$\ge$$f(y_1, y_2, \ldots , y_n)$ when $x_i \ge y_i$ for $i = 1, 2, \ldots , n$.

• (2) $f(x_1, x_2, \ldots , x_n)$$> f(y_1, y_2, \ldots , y_n)$ when $x_i > y_i$ for $i = 1, 2, \ldots , n$.

Given a strictly increasing function, Liu (2010) and Ding (2013) independently proved the following operational law for regular variables, which makes uncertainty theory efficient in monotonic system.

Theorem 1

Let $\xi _1, \xi _2, \ldots , \xi _n$ be independent uncertain variables with regular uncertainty distributions $\varPhi _1, \varPhi _2, \ldots , \varPhi _n$, respectively, and let f be a continuous and strictly increasing function. Then, the uncertain variable $\xi =f(\xi _1, \xi _2, \ldots , \xi _n)$ has an inverse uncertainty distribution, i.e.,

$$\begin{aligned} \varPhi ^{-1}(\alpha )=f(\varPhi _1^{-1}(\alpha ), \varPhi _2^{-1}(\alpha ), \ldots , \varPhi _n^{-1}(\alpha )). \end{aligned}$$

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