Exact and Heuristic Lexicographic Methods for the Fuzzy Traveling Salesman Problem

Abstract

The traveling salesman problem is notorious not only for its significance in theoretical computer science but also for the vast number of real-world problems it is involved in. In this paper, we propose a branch and bound method and a 2-opt method for solving traveling salesman problems, in situations where pairwise distances, costs, or travel times between cities are not known precisely and are modeled with triangular fuzzy numbers. A lexicographic criterion is used for ranking such fuzzy numbers. This approach is shown to produce intuitively better results than those obtained by using linear ranking functions. Computer codes are provided for reproducibility and practical applications. This research also provides a starting point for the extension to the fuzzy environment of other methods for solving classical (non-fuzzy) traveling salesman problems.

Appendices

A Lexicographic Method for the FLAP

The FLAP can be formulated as the fuzzy integer linear programming problem

$$\begin{aligned} \begin{aligned} \min ~&\sum \limits _{i=1}^n\sum \limits _{j=1}^n\tilde{c}_{ij}\times x_{ij}\\ \text {s.t.}~&\text {constraints}~(2) \;\text {and}\; (3), \end{aligned} \end{aligned}$$

(5)

where the $x_{ij}$ ($i,j=1,2,\ldots ,n$) are binary variables. Definition 1 was used in [13] to transform problem (5) into LLAP (6).

$$\begin{aligned} \begin{aligned} \text {lexmin}~&\sum \limits _{i=1}^n\sum \limits _{j=1}^n c_{ij}x_{ij}\\ \text {s.t.}~&\text {constraints}~(2)\; \text {and}\; (3), \end{aligned} \end{aligned}$$

(6)

where $c_{ij}=\Big (f_1\left( \tilde{c}_{ij}\right) ,f_2\left( \tilde{c}_{ij}\right) ,f_3\left( \tilde{c}_{ij}\right) \Big )$ ($i,j=1,2,\ldots ,n$).

Theorem 2

An optimal solution to LLAP (6) is optimal for FLAP (5).

Proof

See [13].

Burkard et al.’s [5] general algebraic framework can be used to efficiently solve LLAP (6). In their framework, it is assumed that the LAP cost coefficients are elements of a totally ordered commutative semigroup $\big (H,*,\le \big )$, with internal composition $*$ and total order relation $\le$, which fulfills the following properties:

• 1. $\forall x,y,z\in H$, $x\le y\implies x*z\le y*z$ (i.e. $\le$ is compatible with $*$),

• 2. $\forall x,y\in H$ with $x\le y$ $\exists z\in H$ such that $x*z = y$.

The authors developed a method, similar to the Hungarian method, to find optimal assignments by using the concept of an admissible transformation, i.e. a transformation that does not alter the relative order of the assignments. The following theorem describes admissible transformations for algebraic assignment problems (see [5]).

Theorem 3

Let $I,J\subseteq \{1,2,\ldots ,n\}$, $m=|I|+|J|-n\ge 1$, and $c=\min \{c_{ij}:i\in I,j\in J\}$. Then the transformation of matrix C into matrix $\bar{C}$, $T:C\mapsto \bar{C}$, defined by

$$\begin{aligned} \bar{c}_{ij}*c=c_{ij}\quad i\in I,\,j\in J,\\ \bar{c}_{ij}=c_{ij}*c\quad i\notin I,\,j\notin J,\\ \bar{c}_{ij}=c_{ij}\quad \text {otherwise}, \end{aligned}$$

is admissible with $z_T=c*c*\cdots *c$, where the expression in the right-hand side contains m factors.

It can be shown that $\big (\mathbb {R}^{3},+,\le _{\text {lex}}\big )$ is a totally ordered commutative semigroup that meets the properties mentioned above. Consequently, LLAP (6) can be solved with the following method proposed in [5], using $+$ and $\le _{\text {lex}}$ in place of $*$ and $\le$, respectively.

• Step 1. Perform row reductions in matrix C, i.e. perform admissible transformations as in Theorem 3 with $I=\left\{ k\right. \}$, $J=\left\{ 1,2,\ldots ,n\right. \}$. Start with $k=1$ and let $z:=z_T$ be the corresponding index. Continue with $k=2,\ldots ,n$ and update $z:=z*z_T$. Afterward, all elements in the transformed matrix are non-negative with respect to z, i.e. $z\le \bar{c}_{ij}*z$.

• Step 2. Perform column reductions, i.e. perform admissible transformations with $I=\left\{ 1,2,\ldots ,n\right. \}$, $J=\left\{ k\right. \}$, for $k=1,2,\ldots ,n$. Afterward, every row and column in the transformed cost matrix contains at least one 0-element. All other elements remain non-negative with respect to z.

• Step 3. Determine a maximum matching in the following bipartite graph $G = (V; W; E)$, where V contains the row indices of the transformed cost matrix, W the column indices, and $(i,j)\in E$, if and only if $\bar{c}_{ij}*z=z$.

• Step 4. If the maximum matching is perfect, then stop: The optimal solution is given by this matching and z is the optimal value of the objective function. Otherwise, go to Step 5.

• Step 5. Determine a minimum cover of the transformed cost coefficients with a value of 0. This cover produces the new index sets I and J. I contains the indices of the uncovered rows and J contains the indices of the uncovered columns.

• Step 6. Perform an admissible transformation determined by the new index sets I and J as in Theorem 3, update $z:=z*z_T$, and go to Step 3.

B Preliminary Experimental Results

Fig. 2.

Experimental results.