Multi-objective ant colony optimization based on decomposition for bi-objective traveling salesman problems

Abstract

This paper proposes a framework named multi-objective ant colony optimization based on decomposition (MoACO/D) to solve bi-objective traveling salesman problems (bTSPs). In the framework, a bTSP is first decomposed into a number of scalar optimization subproblems using Tchebycheff approach. To suit for decomposition, an ant colony is divided into many subcolonies in an overlapped manner, each of which is for one subproblem. Then each subcolony independently optimizes its corresponding subproblem using single-objective ant colony optimization algorithm and all subcolonies simultaneously work. During the iteration, each subproblem maintains an aggregated pheromone trail and an aggregated heuristic matrix. Each subcolony uses the information to solve its corresponding subproblem. After an iteration, a pheromone trail share procedure is evoked to realize the information share of those subproblems solved by common ants. Three MoACO algorithms designed by, respectively, combining MoACO/D with AS, MMAS and ACS are presented. Extensive experiments conducted on ten bTSPs with various complexities manifest that MoACO/D is both efficient and effective for solving bTSPs and the ACS version of MoACO/D outperforms three well-known MoACO algorithms on large bTSPs according to several performance measures and median attainment surfaces.

Appendix: Performance metrics

1.1 S metric

S metric, also named the hypervolume ratio (Zitzler and Thiele 1998), relates to the ratio of the hypervolume of an approximation set ${\mathcal{P}}$ and an optimum Pareto set or pseudo-optimum Pareto set ${\mathcal{P}}_{0},$ depicted in equation

$$ S\mathop{=}\limits^\Updelta = \frac{\hbox{HV}({\mathcal{P}})}{\hbox{HV}({\mathcal{P}}_{0})}, $$

(21)

where $\hbox{HV}({\mathcal{P}})$ and $\hbox{HV}({\mathcal{P}}_{0})$ are the hypervolumes of set ${\mathcal{P}}$ and ${\mathcal{P}}_{0},$ respectively, where the hypervolume relates to the area of coverage of a set and an anti-idea point with respect to the objective for a two-objective MOP, defined as

$$\hbox{HV}({\mathcal{P}}) \mathop{=} \limits^\Updelta = \{ {\mathop{\cup} \limits_{i}} \hbox{vol}_{i}|q_{i} \in {\mathcal{P}} \}, $$

(22)

where q _i is a non-dominated vector in ${\mathcal{P}};\, \hbox{vol}_{i}$ is the volume between the anti-idea solution and vector q _i . For a minimization MOP, it’s assumed that the anti-idea point is the maximum value for each objective.

The S metric can be used as the basis of a dominance compliant comparison and possess the advantage of measuring both diversity and proximity. S value less than one indicates the approximation set ${\mathcal{P}}$ is not as good as ${\mathcal{P}}_{0}$ and if S value equals to 1, then ${\mathcal{P}}={\mathcal{P}}_{0}.$ Therefore, the larger the value S is, the better the ${\mathcal{P}}$ approximation set is.

1.2 R1 _R and R3 _R metrics

The R1 _R metric (Hansen 1998) calculates the probability that an reference set ${\mathcal{P}}_{0}$ is better than an approximation set ${\mathcal{P}}$ over a set of utility functions U, defined as

$$R1_{R}({\mathcal{P}},U,p) = \int\limits_{u \in U} C({\mathcal{P}}_{0}, {\mathcal{P}},u)p(u)\,\hbox{d}u, $$

(23)

where $u \in U$ is a utility function mapping each set to an utility measure; p(u) is the probability density of the utility function u, and

$$ C({\mathcal{P}}_0,{\mathcal{P}},u)= \left\{ \begin{array}{ll} 1 & :\hbox{if } u({\mathcal{P}}_{0}) < u({\mathcal{P}})\\ 1/2 & :\hbox{if } u({\mathcal{P}}_{0}) = u({\mathcal{P}})\\ 0 & :\hbox{if } u({\mathcal{P}}_{0}) > u({\mathcal{P}})\\ \end{array} \right. $$

(24)

with $u({\mathcal {P}}) = \mathop{\min}\limits_{q \in {\mathcal{P}} \{ u(q)\} .}$

The R3 _R metric (Hansen and Jaszkiewicz 1998) measures the probability of superiority of an reference set ${\mathcal{P}}_{0}$ over approximation set ${\mathcal{P}},$ formulated as

$$R3_{R}({\mathcal{P}},U,p) = \int\limits_{u \in U} \frac{u({\mathcal{P}}_{0}) - u({\mathcal{P}})}{u({\mathcal{P}}_{0})}p(u)\hbox{d}u,$$

(25)

where U and p(u) are defined as in R1 _R metric.

According to the metrics, the more the value R1 _R is near to $\frac{1}{2}$ or the smaller the value R3 _R is, the better the approximation set ${\mathcal{P}}$ is. Additionally, both R1 _R metric and R3 _R metric require a set of utility functions U which must be defined. In this contribution, the Tchebycheff utility function set (Hansen and Jaszkiewicz 1998) is used, that is,

$$ \left\{u_{\lambda}(q,r) = \mathop{\max}\limits_{j = 1,2,\ldots ,m} \{\lambda_{j}(q_{j}-r_{j})\}|\lambda =(\lambda_{1}, \lambda_2,\ldots,\lambda_{m}) \cap\lambda_{i}\in\left\{\frac{1}{k},\frac{2}{k},\ldots, \frac{k - 1}{k}\right\}\cap\sum\limits_{i = 1}^{m} {\lambda_{i} = 1}\right\}. $$

(26)

Therefore, the original integration in (23) and (25) can be superseded by

$$ R1({\mathcal{P}}_{1},{\mathcal{P}}_2,U,p) = \sum\limits_{u_{\lambda ,r} \in U} C({\mathcal{P}}_{1},{\mathcal{P}}_{2},u_{\lambda ,r})p({u_{\lambda ,r}}) $$

(27)

$$R3({\mathcal{P}}_{1},{\mathcal{P}}_{2},U,p) = \sum\limits_{u_{\lambda ,r \in U}} \frac{u_{\lambda ,r}({\mathcal{P}}_1) - {u_{\lambda ,r}}({{\mathcal P}_{2}})}{u_{\lambda ,r}({\mathcal{P}}_{1})}p(u_{\lambda ,r})$$

(28)

1.3 ${\varvec{\epsilon}}$ indicator

For a minimization problem with m positive objectives, an objective vector ${\user2{a}} = ({a_{1}},{a_{2}},\ldots,{a_{m}})$ is said to $\epsilon$ dominate another objective vector ${\user2{b}} = (b_{1},b_{2},\ldots,b_{m}),$ written as $a\succcurlyeq_\epsilon b,$ if and only if

$$\forall 1 \le i \le m:{a_{i}} \le \epsilon \cdot {b_{i}} $$

(29)

for a given $\epsilon > 0.$ Given two approximation sets, ${\mathcal{P}}_1 \hbox{ and } {\mathcal{P}}_2, \,\epsilon$ indicator measures (Zitzler et al. 2003) the smallest amount $\epsilon$ such that any solution $q_{2} \in {\mathcal {P}}_2$ is $\epsilon$ dominated by at least one solution $q_{1} \in {\mathcal{P}}_1,$ i.e.,

$$ I_\epsilon({\mathcal{P}}_1,{\mathcal{P}}_2) = \min \{ \epsilon |\forall b \in {\mathcal{P}}_{2} \exists a \in {\mathcal{P}}_{1}:a_\epsilon b \}. $$

(30)

When $I_\epsilon({\mathcal {P}}_{1},{\mathcal{P}}_{2}) < 1,$ all solutions in ${\mathcal{P}}_{2}$ are dominated by a solution in ${\mathcal{P}}_1.$ If $I_\epsilon({\mathcal{P}}_1,{\mathcal{P}}_2) = 1 \hbox{ and } I_{\epsilon}({\mathcal {P}}_{2},{\mathcal{P}}_{1}) = 1, {\mathcal{P}}_{1} \hbox{ and } {\mathcal{P}}_{2}$ represent the same approximation set. If $I_{\epsilon}({\mathcal{P}}_1,{\mathcal{P}}_2) > 1$ and $I_{\epsilon}({\mathcal{P}}_2,{\mathcal{P}}_1) > 1, {\mathcal{P}}_1 \hbox{ and } {\mathcal{P}}_{2}$ are incomparable. When the optimum Pareto set or pseudo-optimum Pareto set is considered, i.e., $I_\epsilon({\mathcal {P}}_0, {\mathcal {P}}) \hbox{ and } {I_{\epsilon}}({\mathcal{P}}, {\mathcal {P}}_{0}), I_{\epsilon}({\mathcal{P}}_{0}, {\mathcal{P}}) \le 1 \hbox{ and } I_{\epsilon}({\mathcal{P}},{\mathcal{P}}_{0}) \ge 1,$ and the more the value is near to 1, the better the set ${\mathcal{P}}$ is.

1.4 C metric

C metric (Zitzler and Thiele 1999) aims to evaluate the performance of two multi-objective algorithms by comparing the approximation Pareto sets. Let ${\mathcal{P}}_1, {\mathcal{P}}_2$ be two approximation Pareto sets obtained by two algorithms, the function C defines a mapping from the ordered pair $({\mathcal{P}}_1, {\mathcal{P}}_2)$ to the interval [0, 1], i.e.,

$$ C({\mathcal{P}}_{1},{\mathcal{P}}_{2}) = \frac{| \{ {\mathbf{b}} \in {\mathcal{P}}_{2}|\exists {\mathbf{a}} \in {\mathcal{P}}_{1}: {\mathbf{a}} \succcurlyeq {\mathbf{b}}\} |}{| {\mathcal{P}}_{2}|}. $$

(31)

where ${\mathbf{a}} \succcurlyeq {\mathbf{b}}$ implies that the solution ${\mathbf{a}}$ dominates the solution ${\mathbf{b}}.$ The value $C({\mathcal{P}}_1, {\mathcal{P}}_{2}) = 1$ means that all objective vectors in ${\mathcal{P}}_{2}$ are dominated by at least one objective vector in ${\mathcal{P}}_{1}.$ On the contrary, $C({\mathcal{P}}_{1}, {\mathcal{P}}_{2}) = 0$ represents the case that no point in ${\mathcal{P}}_{2}$ is dominated by any point in ${\mathcal{P}}_1.$ Note that both directions have to be considered, since $C({\mathcal{P}}_{1}, {\mathcal{P}}_{2})$ is not equal to $1- C({\mathcal{P}}_{2} ,{\mathcal{P}}_{1}).$