Uncertain multi-objective Chinese postman problem

Abstract

Chinese postman problem is one of the significant combinatorial optimization problems with a wide range of real-world applications. Modelling such real-world applications quite often needs to consider some uncertain factors for which the belief degrees of the experts are essential. Liu (Uncertainty Theory, 2nd edn. Springer, Berlin, 2007) proposed uncertainty theory to model such human beliefs. This paper presents a multi-objective Chinese postman problem under the framework of uncertainty theory. The objectives of the problem are to maximize the total profit earned and to minimize the total travel time of the tour of a postman. Here, we have proposed an expected value model (EVM) for the uncertain multi-objective Chinese postman problem (UMCPP). The deterministic transformation of the corresponding EVM is done by computing the expected value of the uncertain variable using 999-method for which we have proposed an algorithm, 999-expected value model-uncertain multi-objective Chinese postman problem. Subsequently, the model is solved by two classical multi-objective solution techniques, namely global criterion method and fuzzy programming method. Two multi-objective genetic algorithms (MOGAs): nondominated sorting genetic algorithm II and multi-objective cross-generational elitist selection, heterogeneous recombination and cataclysmic mutation are also used to solve the model. A numerical example is presented to illustrate the proposed model. Finally, the performance of MOGAs is compared on six randomly generated instances of UMCPP.

Appendix A

In this section, we present some corollaries related to Theorem 4.1 (cf. Sect. 4).

Corollary A1

If$\xi_{{a_{ij} }}$, $\xi_{{c_{ij} }}$and$\xi_{{t_{ij} }}$are the independent linear uncertain variables, which are, respectively, represented as${\mathcal{L}}\left( {p_{{a_{ij} }} ,q_{{a_{ij} }} } \right)$, ${\mathcal{L}}\left( {p_{{c_{ij} }} ,q_{{c_{ij} }} } \right)$and${\mathcal{L}}\left( {p_{{t_{ij} }} ,q_{{t_{ij} }} } \right).$Then, model (16) presented in Theorem4.1can be written as model (A1).

$$ \left\{ {\begin{array}{*{20}l} {{\text{Max}}\; Z_{1} = \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = 1,j \ne i}^{n} \left\{ {\left( {\frac{{p_{{a_{ij} }} + q_{{a_{ij} }} }}{2} - \frac{{p_{{c_{ij} }} + q_{{c_{ij} }} }}{2}} \right)\left( {x_{ij} + x_{ji} } \right)} \right\}} \hfill \\ {{\text{Min }}\;Z_{2} = \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = 1,j \ne i}^{n} \left( {\frac{{p_{{t_{ij} }} + q_{{t_{ij} }} }}{2}} \right)\left( {x_{ij} + x_{ji} } \right)} \hfill \\ {{\text{subject}}\; {\text{to}}} \hfill \\ {\mathop \sum \limits_{j = 1}^{n} x_{ij} - \mathop \sum \limits_{k = 1}^{n} x_{ki} = 0, \quad i = 1,2, \ldots ,n} \hfill \\ {x_{ij} + x_{ji} \ge 1 ,\quad e_{ij} \in E_{G} } \hfill \\ {x_{ij} \in \left\{ {0,1} \right\} , \quad e_{ij} \in E_{G} }. \hfill \\ {} \hfill \\ \end{array} } \right. $$

(A1)

Corollary A2

If$\xi_{{a_{ij} }}$, $\xi_{{c_{ij} }}$and$\xi_{{t_{ij} }}$are the independent zigzag uncertain variables, which are expressed as${\mathcal{Z}}\left( {p_{{a_{ij} }} ,q_{{a_{ij} }} ,r_{{a_{ij} }} } \right)$, ${\mathcal{Z}}\left( {p_{{c_{ij} }} ,q_{{c_{ij} }} ,r_{{c_{ij} }} } \right)$ and ${\mathcal{Z}}\left( {p_{{t_{ij} }} ,q_{{t_{ij} }} ,r_{{t_{ij} }} } \right)$, respectively. Then, model (16) reported in Theorem4.1can be represented as model (A2).

$$ \left\{ {\begin{array}{*{20}l} {{\text{Max}}\; Z_{1} = \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = 1,j \ne i}^{n} \left\{ {\left( {\frac{{p_{{a_{ij} }} + 2q_{{a_{ij} }} + r_{{a_{ij} }} }}{4} - \frac{{p_{{c_{ij} }} + 2q_{{c_{ij} }} + r_{{c_{ij} }} }}{4}} \right)\left( {x_{ij} + x_{ji} } \right)} \right\}} \hfill \\ {{\text{Min }}\;Z_{2} = \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = 1,j \ne i}^{n} \left( {\frac{{p_{{t_{ij} }} + 2q_{{t_{ij} }} + r_{{t_{ij} }} }}{2}} \right)\left( {x_{ij} + x_{ji} } \right)} \hfill \\ {{\text{subject}}\;{\text{to}}} \hfill \\ {\mathop \sum \limits_{j = 1}^{n} x_{ij} - \mathop \sum \limits_{k = 1}^{n} x_{ki} = 0, \quad i = 1,2, \ldots ,n} \hfill \\ {x_{ij} + x_{ji} \ge 1 ,\quad e_{ij} \in E_{G} } \hfill \\ {x_{ij} \in \left\{ {0,1} \right\} , \quad e_{ij} \in E_{G} }. \hfill \\ {} \hfill \\ \end{array} } \right. $$

(A2)

Corollary A3

If$\xi_{{a_{ij} }}$, $\xi_{{c_{ij} }}$and$\xi_{{t_{ij} }}$are the independent normal uncertain variables, which are of the form${\mathcal{N}}\left( {m_{{a_{ij} }} ,\sigma_{{a_{ij} }} } \right)$, ${ \mathcal{N}}\left( {m_{{c_{ij} }} ,\sigma_{{c_{ij} }} } \right)$and${\mathcal{N}}\left( {m_{{t_{ij} }} ,\sigma_{{t_{ij} }} } \right)$, respectively. Then, model (16) shown in Theorem4.1can be written as follows:

$$ \left\{ {\begin{array}{*{20}l} {{\text{Max}}\; Z_{1} = \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = 1,j \ne i}^{n} \left\{ {\left( {m_{{a_{ij} }} - m_{{c_{ij} }} } \right)\left( {x_{ij} + x_{ji} } \right)} \right\}} \hfill \\ {{\text{Min }}\;Z_{2} = \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = 1,j \ne i}^{n} m_{{t_{ij} }} \left( {x_{ij} + x_{ji} } \right)} \hfill \\ {{\text{subject}}\;{\text{to}}} \hfill \\ {\mathop \sum \limits_{j = 1}^{n} x_{ij} - \mathop \sum \limits_{k = 1}^{n} x_{ki} = 0, \quad i = 1,2, \ldots ,n} \hfill \\ {x_{ij} + x_{ji} \ge 1 ,\quad e_{ij} \in E_{G} } \hfill \\ {x_{ij} \in \left\{ {0,1} \right\} , \quad e_{ij} \in E_{G} }. \hfill \\ {} \hfill \\ \end{array} } \right. $$

(A3)