An emergency logistics distribution routing model for unexpected events

Abstract

Though there are many specialities of logistics distribution in unexpected events, the greatest uniqueness of the problem should be the feature that no historical data about some parameters are available. The paper defines the emergency logistics as the one of which the parameters have no historical data due to the occurrence of the unexpected events, and discusses an emergency logistics distribution routing problem in which demands of the affected areas and road travel times lack historical data and are given by experts’ estimations. Uncertain variables are used to describe the experts’ estimates of the parameters and the use of them is justified. An emergency logistics distribution routing model is developed based on uncertainty theory. To solve the problem, the equivalent model is provided and a cellular genetic algorithm is designed. In addition, an example is presented to illustrate the application of the proposed model and the effectiveness of the proposed algorithm.

Appendix: Fundamentals of uncertainty theory

Uncertainty theory is a branch of mathematics based on the following four axioms.

Definition 1

Let $\mathcal{L}$ be a $\sigma$-algebra over a nonempty set $\Gamma$. Each element $\Lambda \in \mathcal{L}$ is called an event. A set function $\mathcal{M} \{\Lambda \}$ is called an uncertain measure if it satisfies the following four axioms (Liu 2007, 2009a):

• (i) $\mathcal{M} \{\Gamma \}=1.$

• (ii) $\mathcal{M} \{\Lambda \}+\mathcal{M} \{\Lambda ^c\}=1.$

• (iii) For every countable sequence of events$\{\Lambda _i\}$, we have

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

The triplet $(\Gamma , \mathcal{L}, \mathcal{M})$ is called an uncertainty space.

• (iv) Let $(\Gamma _k, \mathcal{L}_k, \mathcal{M}_k)$ be uncertainty spaces for $k=1, 2, \ldots$ The product uncertain measure is

  $$\begin{aligned} \displaystyle \mathcal{M}\left\{ \prod _{k=1}^{\infty }\Lambda _k\right\} =\bigwedge _{k=1}^{\infty } \mathcal{M}_k\{\Lambda _k\} \end{aligned}$$

where $\Lambda _k$ are arbitrarily chosen events from $\mathcal{L}_k$ for $k=1,2,\ldots ,$ respectively.

Definition 2

(Liu 2007) An uncertain variable is a measurable function $\xi$ from an uncertainty space $(\Gamma , \mathcal{L}, \mathcal{M})$ to the set of real numbers.

In application, an uncertain variable is characterized by an uncertainty distribution function.

Definition 3

(Liu 2007) The uncertainty distribution $\Phi : \mathfrak {R}\rightarrow [0,1]$ of an uncertain variable $\xi$ is defined by

$$\begin{aligned} \Phi (t)=\mathcal{M}\{\xi \le t\}. \end{aligned}$$

For example, a normal uncertain variable has the following uncertainty distribution

$$\begin{aligned} \Phi (t)=\displaystyle \left( 1+\exp \left( \frac{\pi (\mu -t)}{\sqrt{3}\sigma } \right) \right) ^{-1},\quad t\in \mathfrak {R}, \end{aligned}$$

where $\mu$ and $\sigma$ are real numbers and $\sigma >0.$ In the paper, we denote it by $\xi \sim N(\mu , \sigma )$.

Definition 4

(Liu 2010) An uncertainty distribution $\Phi (t)$ is said to be regular if it is a continuous and strictly increasing function with respect to t at which $0<\Phi (t)<1,$ and

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }\Phi (t)=0, \, \lim \limits _{t\rightarrow +\infty }\Phi (t)=1. \end{aligned}$$

It is seen that the distribution of a normal uncertain variable is regular. In application, we often suppose that all uncertainty distributions are regular. Otherwise, we can give some perturbations to get a regular one.

Definition 5

(Liu 2010) Let $\xi$ be an uncertain variable with regular uncertainty distribution $\Phi (t).$ Then the inverse function $\Phi ^{-1}(\alpha )$ is called the inverse uncertainty distribution of $\xi .$

The operational law of the uncertain variables is given by Liu (2010) as follows:

Theorem 2

(Liu 2010) Let $\xi _1, \xi _2, \ldots , \xi _n$ be independent uncertain variables with regular uncertainty distributions $\Phi _1, \Phi _2,\ldots , \Phi _n,$ respectively. If $f(t_1,t_2,\ldots ,t_n)$ is strictly increasing with respect to $t_1,t_2,\ldots ,t_n,$ then

$$\begin{aligned} \xi =f(\xi _1,\xi _2,\ldots ,\xi _n) \end{aligned}$$

is an uncertain variable that has the following inverse uncertainty distribution function

$$\begin{aligned} \Psi ^{-1}(\alpha )=f(\Phi ^{-1}_1(\alpha ), \Phi ^{-1}_2(\alpha ), \ldots , \Phi ^{-1}_n(\alpha )), \quad 0< \alpha <1. \end{aligned}$$

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