# Comparison of the Sub-Tour Elimination Methods for the Asymmetric Traveling Salesman Problem Applying the SECA Method

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## Abstract

**:**

## 1. Introduction

## 2. Description of Three Sub-Tour Elimination Constraints (SECs)

#### 2.1. The Danzig–Fulkerson–Johnson (DFJ) Formulation

_{ij}is a binary variable and is equal to 1 when the nodes of i,j are visited. Q is a set of vertices whose cardinalities are between 3 and n − 1 because two nodes cannot take a tour, and the minimum number for making a tour is 3.

#### 2.2. The Miller–Tucker–Zemlin (MTZ) Formulation

_{ij}is a binary variable and is equal to 1 when the nodes of i,j are visited. Constraint acts on the basis of node labeling. This means each node receives a number label, and these numbers should be sequential. Figure 2 shows that each number is greater than the previous one except in the last one (1 is not greater than 4). This simple rule helps prevent the TSP from making arc i,l and eliminate taking a tour.

#### 2.3. The Gavish–Graves (GG) Formulation

_{ij}describe a single commodity’s flow vertex 1 from every other vertex [12]. The GG [15] formulation for a single commodity problem that has sub-tour elimination constraints in it is

_{ij}is a binary variable and is equal to 1 when the nodes of i,j are visited. Constraint (7) ensures that the flow variable (Z

_{ij}) exists between nodes with one unit following. Constraint (8) assures that a flow is possible when the nodes are connected (y

_{ij}= 1).

## 3. Research Gap

## 4. Methodology

- (1)
- Specifying the relevant criteria and alternatives
- (2)
- Assigning numerical measures to the criteria under the impact of alternatives
- (3)
- Ranking each alternative

_{j}is the weight of each unknown criterion. Moreover,${\sigma}_{j}$, each vector elements’ standard deviation ${\pi}_{j}$, shows the degree of conflict between j-th criterion and other criteria. In addition,$\epsilon $, a small positive parameter, is equal to 10

^{−3}as a lower bound for criteria weights. The coefficient β is used for minimizing deviation from reference points. In the source paper, it is mentioned that when the values of β are greater than 3 ($\beta \ge 3$), the performance of alternatives is more stable. Therefore, β is taken to be 3 in this study. S

_{i}shows the overall performance score of each alternative.

## 5. Selection of Related Criteria by Reviewing Articles

- (1)
- DFJ is located in the C group, and the combination of the base model with this sub-tour elimination has ${2}^{n}+2n-2$ constraints and n(n − 1) 0–1 variables, because the exponential number of constraints cannot solve this practically.
- (2)
- MTZ is located in the S group, and the combination of the base model with this sub-tour elimination has ${n}^{2}-n+2$ constraints, n(n − 1) 0–1 variables, and (n − 1) continuous variables.
- (3)
- GG is located in the F1 group, and the combination of the base model with this sub-tour elimination has n(n + 2) constraints, n(n − 1) 0–1 variables, and n(n − 1) continues variables.
- (4)
- To demonstrate the relative strengths of LP relaxations of these SECs, they provide the following results (Table 2) for 10 cities’ TSP.

- (1)
- Number of constraints: when this criterion increases, the time to solve increases, too.
- (2)
- Number of variables: these criteria affect the solution time.
- (3)
- Relaxation value: whatever is closer to an optimum value is better.
- (4)
- Number of nodes: it affects solution time and SEC performance.
- (5)
- Type of variable: integer variables increase the complexity of a problem.
- (6)
- Time in second: the time to find a solution is a significant issue in selecting a method.

## 6. Computational Results

_{ij}≤ 1).

## 7. Using the SECA Method for Ranking Computational Results

_{ij}denotes the performance value of alternatives (i) on each criterion column (j). Use linear normalization for normalizing. In this, the criteria are divided into two sets, beneficial and non-beneficial. Linear normalization formulation is

_{jl}is the correlation between j and l vectors.

_{ij}= 0 in the SECA model. For this reason, the cells that are zero are converted to 1. One may ask why it is allowed to substitute 0 with 1 in the decision matrix. As mentioned above, the minimum amount of x in SECA method is 1 and cannot use a smaller number. The significant number of nodes (100 and 500) in which there are no criteria values less than 1 are calculated to confirm this research result. In the table given below, gap is the difference between the optimal and the relaxed value, and the less the value, the better and beneficial it is. An example of the decision matrix is used to show the case of Node = 15 in Table 4.

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**CPLEX code:**

DFJ: |

//define number of nodes |

int nbnode=10; // define parameter |

range nodes=1..nbnode; // define index |

//c is the distance between nodes |

float c[i in nodes][j in nodes]=rand(10)+rand(200); |

//in this section define set and subset |

range ss = 1..ftoi(round(2^nbnode)); |

{int} sub [s in ss] = {i | i in 1..nbnode: (s div ftoi(2^(i-1))) mod 2 == 1}; |

//model |

dvar boolean y[nodes][nodes]; // y is decision variable |

which |

is binary. |

minimize sum(i,j in nodes)c[i][j]*y[i][j]; //objective function |

subject |

to{ //constraints |

forall (j in nodes) sum(i in nodes)y[i][j]==1; |

forall (i in nodes) sum(j in nodes)y[i][j]==1; |

forall (s in ss: 2<card(sub[s])<nbnode) sum(i, j in sub[s]) y[i][j] <= card(sub[s])-1; |

} |

MTZ: |

//define number of nodes |

int nbnode=10; // define parameter |

range nodes=1..nbnode; // define index |

//c is the distance between nodes |

float c[i in nodes][j in nodes]=rand(10)+rand(200); |

//model |

dvar boolean y[nodes][nodes]; //y is binary variable |

dvar int+ u[nodes]; // u is integer variable |

minimize sum(i,j in nodes)c[i][j]*x[i][j]; //objective function |

subject |

to{ //constraints |

forall (j in nodes) sum(i in nodes)y[i][j]==1; |

forall (i in nodes) sum(j in nodes)y[i][j]==1; |

forall (i,j in nodes: j!=1) u[i]-u[j]+nbnode*y[i][j]<=nbnode-1; |

} |

GG: |

//define number of nodes |

int nbnode=10; // define parameter |

range nodes=1..nbnode; // define index |

//c is the distance between nodes |

float c[i in nodes][j in nodes]=rand(10)+rand(200); |

//model |

dvar boolean y[nodes][nodes]; //y is binary variable |

dvar float+ z[nodes][nodes]; //z is positive variable |

minimize sum(i,j in nodes)c[i][j]*y[i][j]; //objective function |

subject |

to{ //constraints |

forall (j in nodes) sum(i in nodes)y[i][j]==1; |

forall (i in nodes) sum(j in nodes)y[i][j]==1; |

forall (i in nodes: i>=2 ) sum(j in nodes)z[i][j]-sum(j in nodes: j!=1)z[j][i]==1; |

forall (i,j in nodes: i!=1) z[i][j]<=(nbnode-1)*y[i][j]; |

} |

## Appendix B

Normalize | Constraint | Variable | Type | Time | Gap |
---|---|---|---|---|---|

DFJ | 0.007375 | 1 | 1 | 0.166667 | 1 |

MTZ | 1 | 0.9375 | 0.0625 | 1 | 0.061463 |

GG | 0.948819 | 0.517241 | 1 | 1 | 0.699301 |

STD | 0.456343 | 0.214367 | 0.441942 | 0.392837 | 0.39131 |
---|---|---|---|---|---|

STD-N | 0.240598 | 0.113021 | 0.233006 | 0.207116 | 0.206311 |

r_{ij} | Constraint | Variable | Type | Time | Gap |
---|---|---|---|---|---|

Constraint | 1 | −0.5622 | −0.53913 | 0.998951 | −0.77613 |

Variable | −0.56225 | 1 | −0.39336 | −0.59953 | −0.08508 |

Type | −0.53913 | −0.3933 | 1 | −0.5 | 0.949517 |

Time | 0.998951 | −0.5995 | −0.5 | 1 | −0.74644 |

Gap | −0.77613 | −0.0850 | 0.949517 | −0.74644 | 1 |

1 − r_{ij} | Constraint | Variable | Type | Time | Gap | Sum Each Row | Πn |
---|---|---|---|---|---|---|---|

Constraint | 0 | 1.562251 | 1.539128 | 0.001049 | 1.77613 | 4.878559 | 0.199069 |

Variable | 1.56225 | 0 | 1.393365 | 1.599526 | 1.085082 | 5.640223 | 0.230148 |

Type | 1.53913 | 1.39336 | 0 | 1.5 | 0.050483 | 4.482973 | 0.182927 |

Time | 0.001049 | 1.59953 | 1.5 | 0 | 1.746444 | 4.847023 | 0.197782 |

Gap | 1.77613 | 1.08508 | 0.050483 | 1.74644 | 0 | 4.658133 | 0.190074 |

SCORE | Ranking | |
---|---|---|

DFJ | 0.6233 | 2 |

MTZ | 0.6233 | 2 |

GG | 0.8356 | 1 |

## Appendix C

Code of SECA in LINGO 11 for node=15: |

MODEL: |

SETS: |

AL/1..3/:S; #AL is alternatives and s is score |

CR/1..5/: W,zig,p; # CR is criteria |

LINK (AL, CR): X; # X denoted the performance value of |

alternatives on each criterion column |

ENDSETS |

#read data from excel file |

DATA: |

B=3; #B is beta in the SECA method |

#zig is the normalization of standard deviation |

#p is the normalization of sum (1- correlation) of each row |

X, zig, p=@OLE(‘C:\MATRIX.XLSX’,‘DECISION’,‘SIG’,‘PI’); |

ENDDATA |

@FOR (AL (I): |

s(I)=@SUM (CR(J):W(J)*X(I,J)); |

LA <=S (I); |

); |

@FOR(CR (J): |

W (J) <=1; |

W (J) >=0.001; # W shouldn’t be zero |

); |

@sum(CR(J):W(J))=1; |

LB=@SUM (CR (J):((W (J)- zig (J))^2 )); |

LC=@SUM(CR(J):((W (J)- p (J))^2 )); |

Z=LA-(B*(LB+LC)); #x2003; #objective function |

@FREE(Z); #x2003; # Z is free variable |

MAX=Z; |

END |

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Category | Formulations | Variables | Constraints |
---|---|---|---|

Exponential size | DFJ | O(n^{2}) | O(2^{n}) |

Miller–Tucker–Zemlin-based | MTZ | O(n^{2}) | O(n^{2}) |

Single commodity flow | GG | O(n^{2}) | O(n^{2}) |

Model | Size | LP.obj | Iterations | Time (s) | IP.obj | Nodes | Time (s) |
---|---|---|---|---|---|---|---|

C | 502 × 90 | 766 | 37 | 1 | 766 | 0 | 1 |

S | 92 × 99 | 773.6 | 77 | 3 | 881 | 665 | 16 |

F1 | 120 × 180 | 794.22 | 148 | 1 | 881 | 449 | 13 |

Instances | SEC | Constraints | Variables | Type-V | Time (s) | Vopt | R |
---|---|---|---|---|---|---|---|

Inst-10 | DFJ | 987 | 100 | B | 0 | 462 | 462 |

Inst-10 | MTZ | 111 | 100 + 10 | B + I | 0 | 462 | 462 |

Inst-10 | GG | 119 | 100 + 90 | B + P | 0 | 462 | 462 |

Inst-15 | DFJ | 32,676 | 225 | B | 5 | 371 | 371 |

Inst-15 | MTZ | 241 | 225 + 15 | B + I | 0 | 371 | 354.73 |

Inst-15 | GG | 254 | 225 + 210 | B + P | 0 | 371 | 369.57 |

Inst-16 | DFJ | 65,430 | 256 | B | 11 | 425 | 425 |

Inst-16 | MTZ | 273 | 256 + 16 | B + I | 0 | 425 | 425 |

Inst-16 | GG | 287 | 256 + 240 | B + P | 0 | 425 | 425 |

Inst-17 | DFJ | 130,951 | 289 | B | 34 | 451 | 450 |

Inst-17 | MTZ | 307 | 289 + 17 | B + I | 0 | 451 | 416.4 |

Inst-17 | GG | 322 | 289 + 272 | B + P | 0 | 451 | 416.56 |

Inst-18 | DFJ | 262,007 | 324 | B | 72 | 352 | 352 |

Inst-18 | MTZ | 343 | 324 + 18 | B + I | 0 | 352 | 335 |

Inst-18 | GG | 359 | 324 + 306 | B + P | 0 | 352 | 336.76 |

Inst-19 | DFJ | 524,134 | 361 | B | - | - | 421 |

Inst-19 | MTZ | 381 | 361 + 19 | B + I | 0 | 421 | 394.68 |

Inst-19 | GG | 398 | 361 + 342 | B + P | 0 | 421 | 394.78 |

Inst-20 | DFJ | 1,048,194 | 400 | B | - | - | - |

Inst-20 | MTZ | 421 | 400 + 20 | B + I | 0 | 399 | 350.6 |

Inst-20 | GG | 439 | 400 + 380 | B + P | 0 | 399 | 350.68 |

Inst-50 | DFJ | - | - | - | - | - | - |

Inst-50 | MTZ | 2551 | 2500 + 50 | B + I | 0 | 516 | 502.16 |

Inst-50 | GG | 2599 | 2500 + 2450 | B + P | 1 | 516 | 502.25 |

Inst-100 | DFJ | - | - | - | - | - | - |

Inst-100 | MTZ | 10,100 | 10,000 + 100 | B + I | 2 | 665 | 656.08 |

Inst-100 | GG | 10,119 | 10,000 + 9900 | B + P | 8 | 665 | 656.08 |

Inst-500 | DFJ | - | - | - | - | - | - |

Inst-500 | MTZ | 250,501 | 250,000 + 500 | B + I | 44 | 1173 | 1173 |

Inst-500 | GG | 250,999 | 250,000 + 249,500 | B + P | 115 | 1173 | 1173 |

Nodes = 15 | Constraint | Variable | Type | Time | Gap |
---|---|---|---|---|---|

NB | NB | NB | NB | NB | |

DFJ | 32,676 | 225 | 1 | 6 | 1 |

MTZ | 241 | 240 | 16 | 1 | 16.27 |

GG | 254 | 435 | 1 | 1 | 1.43 |

Node | 10 | 15 | 16 | 17 | 18 | 19 | 20 | 50 | 100 | 500 |
---|---|---|---|---|---|---|---|---|---|---|

DFJ | 0.6655 | 0.6233 | 0.37 | 0.605 | 0.6 | - | - | - | - | - |

MTZ | 0.6655 | 0.6233 | 0.81 | 0.605 | 0.6 | 0.62 | 0.67 | 0.64 | 0.59 | 0.57 |

GG | 0.84 | 0.8356 | 0.88 | 0.685 | 0.7 | 0.85 | 0.84 | 0.85 | 0.69 | 0.73 |

Node | 10 | 15 | 16 | 17 | 18 | 19 | 20 | 50 | 100 | 500 |
---|---|---|---|---|---|---|---|---|---|---|

DFJ | 2 | 2 | 3 | 2 | 2 | - | - | - | - | - |

MTZ | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

GG | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

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**MDPI and ACS Style**

Bazrafshan, R.; Hashemkhani Zolfani, S.; Al-e-hashem, S.M.J.M.
Comparison of the Sub-Tour Elimination Methods for the Asymmetric Traveling Salesman Problem Applying the SECA Method. *Axioms* **2021**, *10*, 19.
https://doi.org/10.3390/axioms10010019

**AMA Style**

Bazrafshan R, Hashemkhani Zolfani S, Al-e-hashem SMJM.
Comparison of the Sub-Tour Elimination Methods for the Asymmetric Traveling Salesman Problem Applying the SECA Method. *Axioms*. 2021; 10(1):19.
https://doi.org/10.3390/axioms10010019

**Chicago/Turabian Style**

Bazrafshan, Ramin, Sarfaraz Hashemkhani Zolfani, and S. Mohammad J. Mirzapour Al-e-hashem.
2021. "Comparison of the Sub-Tour Elimination Methods for the Asymmetric Traveling Salesman Problem Applying the SECA Method" *Axioms* 10, no. 1: 19.
https://doi.org/10.3390/axioms10010019