An entropy based solid transportation problem in uncertain environment

Abstract

The uncertain solid transportation problem considers material dispatching with uncertain elements like demands. Now, it plays an increasingly important role in logistics managements. Traditionally, the transportation cost is used as the optimization objective, while the dispersals of trips between origins and destinations are usually neglected. In order to minimize the transportation penalties and ensure uniform distribution of goods between origins and destinations, this paper employs entropy function of dispersals of trips between origins and destinations as a second objective function. Within the framework of uncertainty theory, the uncertain entropy based solid transportation model is transformed into its deterministic equivalent, which can be solved by general optimization methods. Finally, a numerical example is given for illustrating purpose.

Appendix: Uncertainty theory

Uncertain theory, founded by Liu (2007) and refined by Liu (2009a), is a branch of axiomatic mathematics for modeling human uncertainty. Let $\varGamma$ be a nonempty set, $\mathcal{ L }$ a $\sigma$-algebra over $\varGamma$, and each element $\varLambda$ in $\mathcal{ L }$ is called an event. Uncertain measure is defined as a function from $\mathcal{ L }$ to [0,1].

Definition A.1

(Liu 2007) The set function $\mathcal{ M }$ is called an uncertain measure if it satisfies:

• Axiom 1. $\mathcal{ M }\{\varGamma \}=1$ for the universal set $\varGamma$;

• Axiom 2. $\mathcal{ M }\{\varLambda \}+\mathcal{ M }\{\varLambda ^c\}=1$ for any event $\varLambda$;

• Axiom 3. For any countable sequence of events $\varLambda _1,\varLambda _2,\ldots ,$ we have

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

  (A.1)

Besides, in order to provide the operational law, Liu (2009a) defined the product uncertain measure on the product $\sigma$-algebra $\mathcal{ L }$ as follows.

• Axiom 4. Let $(\varGamma _k,\mathcal{ L }_k,\mathcal{ M }_k)$ be uncertainty spaces for k = 1, 2,$\ldots$ The product uncertain measure $\mathcal{ M }$ is an uncertain measure satisfying:

  $$\begin{aligned} \mathcal{ M }\left\{ \mathop {\prod }\limits _{k=1}^{\infty } \varLambda _k \right\} =\mathop {\bigwedge }\limits _{k=1}^{\infty } \mathcal{ M }_k \left\{ \varLambda _k \right\} . \end{aligned}$$

  (A.2)

  where $\varLambda _k$ are arbitrarily chosen events from $\mathcal{ L }_k$ for k = 1, 2$\ldots$, respectively. Based on the concept of uncertain measure, we can define an uncertain variable.

Definition A.2

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

In order to describe uncertain variable in practice, uncertainty distribution $\Phi : {\mathfrak {R}}\rightarrow [0,1]$ of an uncertain variable is defined as

$$\begin{aligned} \Phi (x)=\mathcal{ M }\{ \xi \le x \}, \end{aligned}$$

(A.3)

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

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

(A.4)

If $\xi$ is an uncertain variable with regular uncertainty distribution $\Phi (x)$, we call the inverse function $\Phi ^{-1}(\alpha )$ is the inverse uncertainty distribution of $\xi$.

An uncertain variable $\xi$ is called normal if it has a normal uncertainty distribution

$$\begin{aligned} \Phi (x)=\left( 1+\text {exp}\left( \frac{\pi (e-x)}{\sqrt{3}\sigma }\right) \right) ^{-1},x \in {\mathfrak {R}}\end{aligned}$$

(A.5)

denoted by $\text{ N }(e,\sigma )$ where e and $\sigma$ are real numbers with $\sigma>0$. The inverse uncertainty distribution of normal uncertain variable $\text{ N }(e,\sigma )$ is

$$\begin{aligned} \Phi ^{-1}(\alpha ) = e + \frac{\sigma \sqrt{3}}{\pi } \text {ln} \frac{\alpha }{1-\alpha }. \end{aligned}$$

(A.6)

Definition A.3

(Liu 2010) The uncertain variables $\xi _1, \xi _2, \ldots , \xi _n$ are said to be independent if

$$\begin{aligned} \mathcal{ M }\left\{ \bigcap \limits _{i=1}^{n} \left\{ \xi _{i} \in B_{i} \right\} \right\} =\bigwedge \limits _{i=1}^{n} \mathcal{ M }\left\{ \xi _{i} \in B_{i} \right\} \end{aligned}$$

(A.7)

for any Borel sets $B_1, B_2, \ldots , B_m$ of real numbers.

Definition A.4

(Liu 2007) Let $\xi$ be an uncertain variable. The expected value of $\xi$ is defined by

$$\begin{aligned} \text {E}[\xi ]=\displaystyle {\int }_0^{+ \infty } \mathcal{ M }\{\xi \ge r \}\text {d}x-\displaystyle {\int }_{-\infty }^{0}\mathcal{ M }\{\xi \le r \}\text {d}x, \end{aligned}$$

(A.8)

provided that at least one of the above two integrals is finite.

Theorem A.1

(Liu 2010) Let$\xi$ be an uncertain variable with regular uncertainty distribution$\Phi$. If the expected value exists, then

$$\begin{aligned} \mathrm{E}[\xi ]=\displaystyle {\int }_0^1 \Phi ^{-1}(\alpha ) d \alpha . \end{aligned}$$

(A.9)

Theorem A.2

(Liu 2010) Let$\xi$ and $\eta$ be independent uncertain variables with finite expected values. Then for any real numbers a and b, we have

$$\begin{aligned} \mathrm{E}[a\xi +b\eta ]=a\mathrm{E}[\xi ]+b\mathrm{E}[\eta ]. \end{aligned}$$

(A.10)

Theorem A.3

(Liu 2009b) Assume the constraint function$g({\varvec{x}}, \xi _1, \xi _2, \ldots , \xi _n)$ is strictly increasing with respect to $\xi _1, \xi _2, \ldots , \xi _k$ and strictly decreasing with respect to$\xi _{k+1}$, $\xi _{k+2}$, $\ldots$, $\xi _{n}$. If $\xi _1$, $\xi _2$, $\ldots$, $\xi _n$ are independent uncertain variables with uncertain distributions$\Phi _1$, $\Phi _2$, $\ldots$, $\Phi _n$, respectively, then the chance constraint

$$\begin{aligned} \mathcal{ M }\left\{ g({\varvec{x}}, \xi _1, \xi _2, \ldots , \xi _n) \le 0 \right\} \ge \alpha \end{aligned}$$

(A.11)

holds if and only if

$$\begin{aligned} \begin{array}{l} g\left( {\varvec{x}}, \Phi _1^{-1}(\alpha ), \ldots , \Phi _k^{-1}(\alpha ),\right. \\ \qquad \left. \Phi _{k+1}^{-1}(1- \alpha ), \ldots , \Phi _{n}^{-1}(1- \alpha ) \right) \le 0. \end{array} \end{aligned}$$

(A.12)