Subtour Elimination In The Case Of Multiple Visits

Introduction

In the context of Mixed Integer Programming (MIP), subtour elimination is a crucial technique used to prevent the formation of subtours in the solution of a problem. A subtour is a subsequence of nodes that are visited in a specific order, but do not form a Hamiltonian path. In this article, we will discuss the subtour elimination technique in the case of multiple visits, where we have two types of nodes: $S$ for multiple visits and $C$ for a single visit.

Problem Formulation

Let's consider a problem where we have a set of nodes $N = \{1, 2, ..., n\}$, where each node $i$ has a type $t_i \in \{S, C\}$. The goal is to find a Hamiltonian path that visits each node exactly once, while avoiding subtours. We can formulate this problem as a MIP model using the following variables:

• $x_{ij}$: a binary variable indicating whether node $i$ is visited before node $j$
• $y_i$: a binary variable indicating whether node $i$ is visited
• $z$: a binary variable indicating whether the solution is a Hamiltonian path

The objective function is to minimize the total cost of the solution, subject to the following constraints:

• Subtour Elimination Constraint: $(A)$: $\sum_{i \in N} x_{ij} \leq |N| - 1$ for all $j \in N$
• Hamiltonian Path Constraint: $\sum_{i \in N} y_i = |N|$
• Single Visit Constraint: $y_i = 1$ for all $i \in N$ with $t_i = C$
• Multiple Visit Constraint: $\sum_{i \in N} x_{ij} \leq |N| - 1$ for all $j \in N$ with $t_j = S$
• Non-Negativity Constraints: $x_{ij} \geq 0$ for all $i, j \in N$
• Binary Constraints: $y_i \in \{0, 1\}$ for all $i \in N$

Subtour Elimination in the Case of Multiple Visits

In the case of multiple visits, we need to modify the subtour elimination constraint $(A)$ to account for the fact that some nodes can be visited multiple times. We can do this by introducing a new variable $w_i$ for each node $i$ with $t_i = S$, which indicates whether node $i$ is visited more than once. The modified subtour elimination constraint is:

• Modified Subtour Elimination Constraint: $(B)$: $\sum_{i \in N} x_{ij} + w_i \leq |N| - 1$ for all $j \in N$ with $t_j = S$

We also need to add the following constraints to ensure that the variable $w_i$ is correctly updated:

• Multiple Visit Constraint: $w_i \geq x_{ij}$ for all $i, j \in N$ with $t_i = S$ and $t_j = S$
• Non-Negativity Constraints: $w_i \geq 0$ for all $i \in N$ with $t_i =$

Example

Let's consider an example with 5 nodes: $N = \{1, 2, 3, 4, 5\}$, where nodes 1 and 2 have type $S$, and nodes 3, 4, and 5 have type $C$. The cost matrix is:

	1	2	3	4	5
1	0	10	15	20	25
2	10	0	15	20	25
3	15	15	0	5	10
4	20	20	5	0	5
5	25	25	10	5	0

The solution to this problem is a Hamiltonian path that visits each node exactly once, while avoiding subtours. The solution is:

• $x_{12} = 1$, $x_{23} = 1$, $x_{34} = 1$, $x_{45} = 1$
• $y_1 = 1$, $y_2 = 1$, $y_3 = 1$, $y_4 = 1$, $y_5 = 1$
• $z = 1$

The total cost of this solution is 60.

Conclusion

In this article, we discussed the subtour elimination technique in the case of multiple visits, where we have two types of nodes: $S$ for multiple visits and $C$ for a single visit. We introduced a new variable $w_i$ for each node $i$ with $t_i = S$, which indicates whether node $i$ is visited more than once. We also modified the subtour elimination constraint $(A)$ to account for the fact that some nodes can be visited multiple times. The modified subtour elimination constraint is:

• Modified Subtour Elimination Constraint: $(B)$: $\sum_{i \in N} x_{ij} + w_i \leq |N| - 1$ for all $j \in N$ with $t_j = S$

We also added the following constraints to ensure that the variable $w_i$ is correctly updated:

• Multiple Visit Constraint: $w_i \geq x_{ij}$ for all $i, j \in N$ with $t_i = S$ and $t_j = S$
• Non-Negativity Constraints: $w_i \geq 0$ for all $i \in N$ with $t_i = S$

We also provided an example with 5 nodes, where nodes 1 and 2 have type $S$, and nodes 3, 4, and 5 have type $C$. The solution to this problem is a Hamiltonian path that visits each node exactly once, while avoiding subtours. The solution is:

• $x_{12} = 1$, $x_{23} = 1$, $x_{34} = 1$, $x_{45} = 1$
• $y_1 = 1$, $y_2 = 1$, $y_3 = 1$, $y_4 = 1$, $y_5 = 1$
• $z = 1$

The total cost of this solution is 60.

Future Work

In the future, we plan to extend this to more complex problems, such as the Vehicle Routing Problem (VRP) and the Capacitated Vehicle Routing Problem (CVRP). We also plan to investigate the use of other techniques, such as the Miller-Tucker-Zemlin (MTZ) constraint, to improve the efficiency of the subtour elimination technique.

References

• [1] Miller, C. E., Tucker, A. W., & Zemlin, R. A. (1960). Integer programming formulation of traveling salesman problems. Journal of the Association for Computing Machinery, 7(2), 203-211.
• [2] Dantzig, G. B. (1963). Linear programming and extensions. Princeton University Press.
• [3] Nemhauser, G. L., & Wolsey, L. A. (1988). Integer and combinatorial optimization. Wiley.

Introduction

In our previous article, we discussed the subtour elimination technique in the case of multiple visits, where we have two types of nodes: $S$ for multiple visits and $C$ for a single visit. We introduced a new variable $w_i$ for each node $i$ with $t_i = S$, which indicates whether node $i$ is visited more than once. We also modified the subtour elimination constraint $(A)$ to account for the fact that some nodes can be visited multiple times. In this article, we will answer some frequently asked questions about subtour elimination in the case of multiple visits.

Q: What is the subtour elimination technique?

A: The subtour elimination technique is a method used in Mixed Integer Programming (MIP) to prevent the formation of subtours in the solution of a problem. A subtour is a subsequence of nodes that are visited in a specific order, but do not form a Hamiltonian path.

Q: Why is subtour elimination important?

A: Subtour elimination is important because it ensures that the solution to a problem is a Hamiltonian path, which is a path that visits each node exactly once. This is particularly important in problems where the goal is to visit a set of nodes in a specific order, such as the Traveling Salesman Problem (TSP).

Q: What is the difference between a subtour and a Hamiltonian path?

A: A subtour is a subsequence of nodes that are visited in a specific order, but do not form a Hamiltonian path. A Hamiltonian path, on the other hand, is a path that visits each node exactly once.

Q: How does the subtour elimination technique work?

A: The subtour elimination technique works by introducing a new variable $w_i$ for each node $i$ with $t_i = S$, which indicates whether node $i$ is visited more than once. We also modify the subtour elimination constraint $(A)$ to account for the fact that some nodes can be visited multiple times.

Q: What is the modified subtour elimination constraint?

A: The modified subtour elimination constraint is:

• Modified Subtour Elimination Constraint: $(B)$: $\sum_{i \in N} x_{ij} + w_i \leq |N| - 1$ for all $j \in N$ with $t_j = S$

Q: What are the additional constraints required for subtour elimination?

A: The additional constraints required for subtour elimination are:

• Multiple Visit Constraint: $w_i \geq x_{ij}$ for all $i, j \in N$ with $t_i = S$ and $t_j = S$
• Non-Negativity Constraints: $w_i \geq 0$ for all $i \in N$ with $t_i = S$

Q: Can the subtour elimination technique be used in other problems?

A: Yes, the subtour elimination technique can be used in other problems, such as the Vehicle Routing Problem (VRP) and the Capacitated Vehicle Routing Problem (CVRP).

Q What are some common challenges associated with subtour elimination?

A: Some common challenges associated with subtour elimination are:

• Computational complexity: The subtour elimination technique can be computationally expensive, particularly for large problems.
• Integer programming: The subtour elimination technique requires the use of integer programming, which can be challenging to implement.

Q: How can the subtour elimination technique be improved?

A: The subtour elimination technique can be improved by:

• Using more efficient algorithms: Using more efficient algorithms, such as the Miller-Tucker-Zemlin (MTZ) constraint, can improve the efficiency of the subtour elimination technique.
• Using more advanced techniques: Using more advanced techniques, such as branch and bound, can improve the efficiency of the subtour elimination technique.

Conclusion

In this article, we answered some frequently asked questions about subtour elimination in the case of multiple visits. We discussed the subtour elimination technique, its importance, and how it works. We also provided some additional constraints required for subtour elimination and some common challenges associated with it. Finally, we discussed some ways to improve the subtour elimination technique.