scholarly journals An Algorithm for Mapping the Asymmetric Multiple Traveling Salesman Problem onto Colored Petri Nets

Algorithms ◽  
2018 ◽  
Vol 11 (10) ◽  
pp. 143 ◽  
Author(s):  
Furqan Essani ◽  
Sajjad Haider
Keyword(s):  
Petri Nets ◽  
Traveling Salesman Problem ◽  
Feasible Solution ◽  
Optimal Solution ◽  
Traveling Salesman ◽  
Colored Petri Nets ◽  
Cost Matrix ◽  
Multiple Traveling Salesman Problem ◽  
Hard Problems ◽  
The Cost

The Multiple Traveling Salesman Problem is an extension of the famous Traveling Salesman Problem. Finding an optimal solution to the Multiple Traveling Salesman Problem (mTSP) is a difficult task as it belongs to the class of NP-hard problems. The problem becomes more complicated when the cost matrix is not symmetric. In such cases, finding even a feasible solution to the problem becomes a challenging task. In this paper, an algorithm is presented that uses Colored Petri Nets (CPN)—a mathematical modeling language—to represent the Multiple Traveling Salesman Problem. The proposed algorithm maps any given mTSP onto a CPN. The transformed model in CPN guarantees a feasible solution to the mTSP with asymmetric cost matrix. The model is simulated in CPNTools to measure two optimization objectives: the maximum time a salesman takes in a feasible solution and the collective time taken by all salesmen. The transformed model is also formally verified through reachability analysis to ensure that it is correct and is terminating.

scholarly journals A Data-Guided Lexisearch Algorithm for the Asymmetric Traveling Salesman Problem

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Zakir Hussain Ahmed
Keyword(s):  
Traveling Salesman Problem ◽  
Optimal Solution ◽  
Traveling Salesman ◽  
Computational Time ◽  
Cost Matrix ◽  
Asymmetric Traveling Salesman Problem ◽  
Path Representation ◽  
Random Instances ◽  
Representation Method ◽  
The Cost

A simple lexisearch algorithm that uses path representation method for the asymmetric traveling salesman problem (ATSP) is proposed, along with an illustrative example, to obtain exact optimal solution to the problem. Then a data-guided lexisearch algorithm is presented. First, the cost matrix of the problem is transposed depending on the variance of rows and columns, and then the simple lexisearch algorithm is applied. It is shown that this minor preprocessing of the data before the simple lexisearch algorithm is applied improves the computational time substantially. The efficiency of our algorithms to the problem against two existing algorithms has been examined for some TSPLIB and random instances of various sizes. The results show remarkably better performance of our algorithms, especially our data-guided algorithm.

A Compressed Annealing Approach with Pre-Process for the Asymmetric Traveling Salesman Problem with Time Windows

2011 ◽  
Vol 5 (5) ◽  
pp. 669-678
Author(s):  
Tadanobu Mizogaki ◽  
◽  
Masao Sugi ◽  
Masashi Yamamoto ◽  
Hidetoshi Nagai ◽  
...  
Keyword(s):  
Traveling Salesman Problem ◽  
Feasible Solution ◽  
Time Window ◽  
Time Windows ◽  
Computing Time ◽  
Traveling Salesman ◽  
Asymmetric Traveling Salesman Problem ◽  
Insertion Method ◽  
Time Window Constraints ◽  
The Cost

This paper proposes a method of rapidly finding a feasible solution to the asymmetric traveling salesman problem with time windows (ATSP-TW). ATSP-TW is a problem that involves determining the route with the minimum travel cost for visiting n cities one time each with time window constraints (the period of time in which the city must be visited is constrained). “Asymmetrical” denotes a difference between the cost of outbound and return trips. For such a combinatorial optimization problem with constraints, we propose a method that combines a pre-process based on the insertion method with metaheuristics called “the compressed annealing approach.” In an experiment using a 3-GHz computer, our method derives a feasible solution that satisfies the time window constraints for all of up to about 300 cities at an average of about 1/7 the computing time of existing methods, an average computing time of 0.57 seconds, and a maximum computing time of 9.40 seconds.

scholarly journals Sequentially Updated Weighted Cost Opportunity Based Algorithm in Transportation Problem

2019 ◽  
Vol 38 ◽  
pp. 47-55
Author(s):  
ARM Jalal Uddin Jamali ◽  
Pushpa Akhtar
Keyword(s):  
Opportunity Cost ◽  
Feasible Solution ◽  
Supply And Demand ◽  
Optimal Solution ◽  
Basic Feasible Solution ◽  
New Approach ◽  
Cost Matrix ◽  
Transportation Problems ◽  
The Cost ◽  
First Time

Transportation models are of multidisciplinary fields of interest. In classical transportation approaches, the flow of allocation is controlled by the cost entries and/or manipulation of cost entries – so called Distribution Indicator (DI) or Total Opportunity Cost (TOC). But these DI or TOC tables are formulated by the manipulation of cost entries only. None of them considers demand and/or supply entry to formulate the DI/ TOC table. Recently authors have developed weighted opportunity cost (WOC) matrix where this weighted opportunity cost matrix is formulated by the manipulation of supply and demand entries along with cost entries as well. In this WOC matrix, the supply and demand entries act as weight factors. Moreover by incorporating this WOC matrix in Least Cost Matrix, authors have developed a new approach to find out Initial Basic Feasible Solution of Transportation Problems. But in that approach, WOC matrix was invariant in every step of allocation procedures. That is, after the first time formulation of the weighted opportunity cost matrix, the WOC matrix was invariant throughout all allocation procedures. On the other hand in VAM method, the flow of allocation is controlled by the DI table and this table is updated after each allocation step. Motivated by this idea, we have reformed the WOC matrix as Sequentially Updated Weighted Opportunity Cost (SUWOC) matrix. The significance difference of these two matrices is that, WOC matrix is invariant through all over the allocation procedures whereas SUWOC   matrix is updated in each step of allocation procedures. Note that here update (/invariant) means changed (/unchanged) the weighted opportunity cost of the cells. Finally by incorporating this SUWOC matrix in Least Cost Matrix, we have developed a new approach to find out Initial Basic Feasible Solution of Transportation Problems.  Some experiments have been carried out to justify the validity and the effectiveness of the proposed SUWOC-LCM approach. Experimental results reveal that the SUWOC-LCM approach outperforms to find out IBFS. Moreover sometime this approach is able to find out optimal solution too. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 47-55

scholarly journals On Tightly Bounding the Dubins Traveling Salesman's Optimum

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Satyanarayana G. Manyam ◽  
Sivakumar Rathinam
Keyword(s):  
Lower Bound ◽  
Traveling Salesman Problem ◽  
Optimal Solution ◽  
Traveling Salesman ◽  
Motion Constraints ◽  
Euclidean Traveling Salesman Problem ◽  
Problem Instances ◽  
Feasible Solutions ◽  
The Cost ◽  
Dubins Traveling Salesman Problem

The Dubins traveling salesman problem (DTSP) has generated significant interest over the last decade due to its occurrence in several civil and military surveillance applications. This problem requires finding a curvature constrained shortest path for a vehicle visiting a set of target locations. Currently, there is no algorithm that can find an optimal solution to the DTSP. In addition, relaxing the motion constraints and solving the resulting Euclidean traveling salesman problem (ETSP) provide the only lower bound available for the DTSP. However, in many problem instances, the lower bound computed by solving the ETSP is far below the cost of the feasible solutions obtained by some well-known algorithms for the DTSP. This paper addresses this fundamental issue and presents the first systematic procedure for developing tight lower bounds for the DTSP.

scholarly journals A generalization of the convex-hull-and-line traveling salesman problem

1998 ◽  
Vol 2 (2) ◽  
pp. 177-191 ◽  
Author(s):  
Md. Fazle Baki ◽  
S. N. Kabadi
Keyword(s):  
Convex Hull ◽  
Traveling Salesman Problem ◽  
Equivalence Relation ◽  
General Class ◽  
Traveling Salesman ◽  
Cost Matrix ◽  
Solvable Case ◽  
Special Cases ◽  
The Traveling Salesman Problem ◽  
The Cost

Two instances of the traveling salesman problem, on the same node set {1,2,…,n} but with different cost matrices C and C′ , are equivalent iff there exist {ai,bi: i=1,…, n} such that for any 1≤i, j≤n,i≠j,C′(i,j)=C(i,j)+ai+bj [7]. One of the well-solved special cases of the traveling salesman problem (TSP) is the convex-hull-and-line TSP. We extend the solution scheme for this class of TSP given in [9] to a more general class which is closed with respect to the above equivalence relation. The cost matrix in our general class is a certain composition of Kalmanson matrices. This gives a new, non-trivial solvable case of TSP.

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