Chance constrained programming models for uncertain hub covering location problems

Abstract

Hub covering location problem is a typical strategic decision with the purpose of locating hubs and determining the assignments of other nodes to ensure the travel time is not exceeding a specific threshold. Since the parameters such as flows and travel times are difficult to be precisely obtained in advance, a feasible way is to estimate them following the experts’ subjective beliefs. Hence, this paper is devoted to study hub covering location problem by using uncertain measure to characterize the subjective belief and considering the flows and travel times by uncertain variables. The uncertain hub set covering location problem is first discussed under the purpose of covering the flows entirely with the minimum setup cost of hubs. Then the uncertain hub maximal covering problem is studied by maximizing the total flow covered when the number of hubs is confirmed previously. Chance constrained programming models for both problems are constructed, respectively, and their corresponding deterministic forms are derived. A hybrid intelligence algorithm named GA–VNS is proposed by combing the variable neighborhood search with the genetic algorithm. Finally, several numerical experiments are presented to indicate the efficiency of GA–VNS.

Appendix

In this appendix, we give some elementary knowledge about uncertainty theory for the better understanding of this paper.

Definition 1

(Liu 2007) Let ${\mathcal {L}}$ be a $\sigma$-algebra on a nonempty set $\Gamma$. A set function ${\mathcal {M}}$ is called an uncertain measure if it satisfies:

(i):

(Normality Axiom) ${\mathcal {M}} (\Gamma )=1$ for the universal set $\Gamma$;

(ii):

(Duality Axiom) ${\mathcal {M}} (\Lambda ) +{\mathcal {M}} (\Lambda ^c) =1$ for any event $\Lambda$;

(iii):

(Subadditivity Axiom) ${\mathcal {M}}\left\{ {\bigcup \limits _{i = 1}^\infty {{\Lambda _i}} } \right\} \le \bigcup \limits _{i = 1}^\infty {{\mathcal {M}}\left\{ {{\Lambda _i}} \right\} }$ for every countable sequence of events $\Lambda _i$;

(iv):

(Product Axion) ${{\mathcal {M}}}\left\{ {\prod _{i = 1}^\infty {\Lambda _i}} \right\} = \bigwedge _{i = 1}^\infty {{{\mathcal {M}}}_i}\left\{ {{\Lambda _i}} \right\}$ where $\Lambda _i$ are arbitrarily chosen events from ${\mathcal {L}}_i$.

Definition 2

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

Definition 3

(Liu 2007) The uncertainty distribution of uncertain variable $\xi$ is defined as

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

for any real number x.

Additionally, the inverse function ${\Phi ^{ - 1}}(x)$ is called the inverse uncertainty distribution of $\xi$.

Lemma 1

(Liu 2009) Let $\xi _1, \xi _2, ...,\xi _n$ be independent uncertain variables with regular uncertainty distributions $\Phi _1, \Phi _2, ...,\Phi _n$, respectively. If f is a strictly increasing function, then $\xi =f(\xi _1, \xi _2, ...,\xi _n)$ is an uncertain variable with an inverse uncertainty distribution

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