The OPTNET Procedure

Traveling Salesman Problem

The traveling salesman problem (TSP) finds a minimum-cost tour in an undirected graph $G$ with node set $N$ and links set $A$. A tour is a connected subgraph for which each node has degree two. The goal is then to find a tour of minimum total cost, where the total cost is the sum of the costs of the links in the tour. With each link $(i,j) \in A$, a binary variable $x_{ij}$, which indicates whether link $x_{ij}$ is part of the tour, and a cost $c_{ij}$ are associated. Let $\delta (S) = \{ (i,j) \in A \; |\;  i \in S, \;  j \notin S\} $. Then an integer linear programming formulation of the TSP is as follows:

\[  \begin{array}{llllll} \text {minimize} &  \displaystyle \sum _{(i,j) \in A} c_{ij} x_{ij} \\ \text {subject to} &  \displaystyle \sum _{(i,j) \in \delta ({i})} x_{i,j} &  = &  2 &  i \in N &  (\mr {two\_ match}) \\ &  \displaystyle \sum _{(i,j) \in \delta (S)}x_{ij} &  \geq &  2 &  S \subset N, \;  2 \leq |S| \leq |N| - 1 &  (\mr {subtour\_ elim})\\ &  x_{ij} \in \{ 0,1\}  & & &  (i,j) \in A \end{array}  \]

The equations (two_match) are the matching constraints, which ensure that each node has degree two in the subgraph, and the inequalities (subtour_elim) are the subtour elimination constraints (SECs), which enforce connectivity.

In practical terms, you can think of the TSP in the context of a routing problem in which each node is a city and the links are roads that connect cities. Given the pairwise distances between each city, the goal is to find the shortest possible route that visits each city exactly once. The TSP has applications in planning, logistics, manufacturing, genomics, and many other areas.

In PROC OPTNET, the traveling salesman problem solver can be invoked by using the TSP statement. The options for this statement are described in the section TSP Statement.

The traveling salesman problem solver reports status information in a macro variable called _OROPTNET_TSP_. See the section Macro Variable _OROPTNET_TSP_ for more information about this macro variable.

The algorithm used in PROC OPTNET for solving TSP is based on a variant of the branch-and-cut process described in (Applegate et al., 2006).

The resulting tour is represented in two ways: In the data set that is specified in the OUT_NODES= option in the PROC OPTNET statement, the tour is given as a sequence of nodes. In the data set that is specified in the OUT= option of the TSP statement, the tour is given as a list of links in the optimal tour.

Traveling Salesman Problem of a Simple Undirected Graph

As a simple example, consider the weighted undirected graph in Figure 2.59.

Figure 2.59: A Simple Undirected Graph

A Simple Undirected Graph

The links data set can be represented as follows:

data LinkSetIn;
   input from $ to $ weight @@;
A B 1.0 A C 1.0 A D 1.5 B C 2.0 B D 4.0
B E 3.0 C D 3.0 C F 3.0 C H 4.0 D E 1.5
D F 3.0 D G 4.0 E F 1.0 E G 1.0 F G 2.0
F H 4.0 H I 3.0 I J 1.0 C J 5.0 F J 3.0
F I 1.0 H J 1.0

The following statements calculate an optimal traveling salesman tour and output the results in the data sets TSPTour and NodeSetOut:

proc optnet
   loglevel   = moderate
   data_links = LinkSetIn
   out_nodes  = NodeSetOut;
     out      = TSPTour;
%put &_OROPTNET_;

The progress of the procedure is shown in Figure 2.60.

Figure 2.60: PROC OPTNET Log: Optimal Traveling Salesman Tour of a Simple Undirected Graph

NOTE: ------------------------------------------------------------------------- 
NOTE: ------------------------------------------------------------------------- 
NOTE: Running OPTNET.                                                           
NOTE: ------------------------------------------------------------------------- 
NOTE: ------------------------------------------------------------------------- 
NOTE: Reading the links data set.                                               
NOTE: There were 22 observations read from the data set WORK.LINKSETIN.         
NOTE: Data input used 0.00 (cpu: 0.00) seconds.                                 
NOTE: Building the input graph storage used 0.00 (cpu: 0.00) seconds.           
NOTE: The input graph storage is using 0.0 MBs of memory.                       
NOTE: The number of nodes in the input graph is 10.                             
NOTE: The number of links in the input graph is 22.                             
NOTE: ------------------------------------------------------------------------- 
NOTE: ------------------------------------------------------------------------- 
NOTE: Processing TSP statement.                                                 
NOTE: The initial TSP heuristics found a tour with cost 16 using 0.00 (cpu:     
      0.00) seconds.                                                            
NOTE: The MILP presolver value NONE is applied.                                 
NOTE: The MILP solver is called.                                                
          Node  Active    Sols    BestInteger      BestBound      Gap    Time   
             0       1       2     16.0000000     16.0000000    0.00%       0   
             0       0       2     16.0000000     16.0000000    0.00%       0   
NOTE: Optimal.                                                                  
NOTE: Objective = 16.                                                           
NOTE: Processing the traveling salesman problem used 0.13 (cpu: 0.00) seconds.  
NOTE: ------------------------------------------------------------------------- 
NOTE: ------------------------------------------------------------------------- 
NOTE: Creating nodes data set output.                                           
NOTE: Creating traveling salesman data set output.                              
NOTE: Data output used 0.02 (cpu: 0.00) seconds.                                
NOTE: ------------------------------------------------------------------------- 
NOTE: ------------------------------------------------------------------------- 
NOTE: The data set WORK.NODESETOUT has 10 observations and 2 variables.         
NOTE: The data set WORK.TSPTOUR has 10 observations and 3 variables.            
STATUS=OK  TSP=OPTIMAL                                                          
BEST_BOUND=16  NODES=1  ITERATIONS=15  CPU_TIME=0.00  REAL_TIME=0.13            

The data set NodeSetOut now contains a sequence of nodes in the optimal tour and is shown in Figure 2.61.

Figure 2.61: Nodes in the Optimal Traveling Salesman Tour

Traveling Salesman Problem

node tsp_order
A 1
B 2
C 3
H 4
J 5
I 6
F 7
G 8
E 9
D 10

The data set TSPTour now contains the links in the optimal tour and is shown in Figure 2.62.

Figure 2.62: Links in the Optimal Traveling Salesman Tour

Traveling Salesman Problem

from to weight
A B 1.0
B C 2.0
C H 4.0
H J 1.0
J I 1.0
I F 1.0
F G 2.0
G E 1.0
E D 1.5
D A 1.5

The minimum-cost links are shown in green in Figure 2.63.

Figure 2.63: Optimal Traveling Salesman Tour

Optimal Traveling Salesman Tour