The document discusses solving a traveling salesman problem (TSP) using two methods: a Hungarian heuristic approach and AMPL programming. The Hungarian method finds an optimal route of 770 miles for a salesman visiting 5 cities. Using AMPL, the optimal route is found to be 725 miles, showing AMPL finds a better solution that is 45 miles shorter. The conclusion is that AMPL programming provides a more optimal approach for solving TSP problems compared to the Hungarian heuristic method.

1. “SOLVING A TSP WITH HEURISTIC MODEL
APPROACH AND COMPARING WITH AMPL
SOLUTION”
Presented by
Dr. Govind Shay Sharma
Department of Mathematics
Pornima College of Engineering, Jaipur

2. ABSTRACT
 The paper will discuss on the two methods to solve the
TSP problem of a book seller. The TSP problem solution
finds the optimum route which optimizes the route and
cost. The paper shows the comparison result of
Hungarian method hand approach and AMPL program.
The user-defined function is involved with AMPL to solve
a more complicated problem. This shows the better
result between the both. The purpose of the paper is to
find the calculation by AMPL programming and
approach for optimum route. The AMPL programming is
used for the solution of Linear and Non linear equations.
In this paper we are discussing the problem of book
seller who has to visit the five cities to fulfill the demand.

3. INTRODUCTION
 The TSP problem deals with finding of the shortest tour
in n- cities, where each city is visited exactly once
before returning to the starting point. The associated
TSP model is defined by two pieces of data.
 The number of cities n. The distances dij between cities
I and j ( dij = ∞ if cities I and j are not linked).
 The maximum number of tours in an n-city situations is .
[1] presented assignment based integer linear
formulation. This concluded the assignment modal that
can lead to infeasible solution. Infeasibility can be
removed by introducing additional constraints. [1]
studied about a salesman of five cities which is consider
designated visit of ith to jth city by decision variable Xij ,
where I = 1,2,3,4,5 and j = 1,2,3,4,5 but I to j. To solve
the ILPP, software LINGO version 11 was used.
Asymmetric Travelling salesman problem is studied with
TIME WINDOW ( ATSP-TW) by [2].

4. INTRODUCTION
 The real word application aim at is the control of a
stacker crane in a warehouse.TSP is the
optimization problem but with increasing the
number of cities it is converted in the complex
travelling problem. [3]. [4] optimize delivering of
packages at twelve randomly chosen pizza centers
in the city of Jaipur. [5] Presented a mapping of
chicken supply chains and explores inputs and
outputs of the systems which have important
implications for the sustainability.

5. INTRODUCTION
 [6] described Genetic algorithms (GA) are basically
based on the survival of the fittest chromosome
among the species which are generated by random
changes in their gene-structure in the evolutionary
biology. [7] reviewed on various algorithms like Ant
Colony Optimization Algorithms (ACO), Particle
Swarm Optimization (PSO) and Genetic Algorithms
(GA) available with respective attributes to find the
nearest optimal solution for the traveling salesman
problem

6. INTRODUCTION
 According to [8] the TSP is the best possible way to
visiting all cities by salesman and returning to the
starting point with optimum cost.
 [9] study is to find the role of Operations Research
techniques for made a job assignment model which
help to increase the labor productivity and cost
reduction with respect to time. [10] Studied the role
of Operations Research techniques for designing a
Material Handling model which will help to increase
the productivity and reduce cost. [11] Presented
experiences with mixed integer linear programming
based short-term hydro scheduling.

7. Define
=
if city j is reached from city I, then value assign 1 and otherwise 0.
Minimize
for all I = j

8. The Hungarian method here we apply to solve the problem of a book
salesman. Here we approach to solve the problem a sales man who lives
in Basin must to visit the four cities. Once a month he has to tour among
four cities Wald, Bon, Mena and Kiln. The distances from Basin to Wald
120 Milles, Basin to Wald 220 Miles, Basin to Mena 150 Miles and Basin
to Kiln 210 Miles

9. The aim of this study is to find the result of problem by hands approach solution
by Hungarian method and AMPL model. In Hungarian approach first we change
the all diagonal zero’s to dashes or infinity symbols. Now The Distance matrix
can be written as

10. Now by the assigning process, allot the Zero from row one and cross the entire zero in that
row and column. And find the resultant matrix

11. So we consider the path for salesman is 1-4-5-2-3 -1. Here the distance find 1-4 = 150, 4-5 =
190, 5-2= 130, 2-3 = 80, 3-1 = 220. The total distance cover by the salesman is 770 miles.
Now we calculate the path of this problem with AMPL programming. And
compare the solution with the hand approach optimal solution by Hungarian method.
And compare the solution with the hand approach optimal solution by Hungarian
method. AMPL is a modeling programming approach for more complex mathematical
problems. It is a declarative and imperative programming style. AMPL is most popular
format representation of complex mathematical problems.

12. AMPL PROGRAMMING :-
FILE NAME:- tsp.mod
minimize z:
sum{i in city, j in city} DIST[i,j]*x[i,j];
subject to
# exactly one outgoing
c1{k in city}: sum{i in city} x[i,k] = 1;
# exactly one incoming
c2{k in city}: sum{j in city} x[k,j] = 1;
# no subtours
c3{k in city, j in city: j > 1 and k > 1}:
U[j] - U[k] + N*x[j,k] <= N-1;
data;
param: city: names :=
1 "DELHI"
2 "GAJIYABAD"
3 "MEERUT"
4 "NOIDA"
5 "LUDHIYANA"
;

13. param DIST: 1 2 3 4 5 :=
1 50000 120 220 150 210
2 120 50000 80 110 130
3 220 80 50000 160 185
4 150 110 160 50000 190
5 210 130 185 190 50000
;
Result:-
ampl: include welcome.run
CPLEX 12.6.3.0: optimal integer solution; objective 725
24 MIP simplex iterations
0 branch-and-bound nodes
z = 725
x [*,*]
: 1 2 3 4 5 :=
1 0 0 0 1 0
2 1 0 0 0 0

14. 3 0 1 0 0 0
4 0 0 0 0 1
5 0 0 1 0 0
;
U [*] :=
1 0
2 4
3 2
4 0
5 1
;
1 (DELHI) -> 4 (NOIDA) [1.00]
2 (GAJIYABAD) -> 1 (DELHI) [1.00]
3 (MEERUT) -> 2 (GAJIYABAD) [1.00]
4 (NOIDA) -> 5 (LUDHIYANA) [1.00]
5 (LUDHIYANA) -> 3 (MEERUT) [1.00]
CPLEX 12.6.3.0: optimal integer solution; objective 725
22 MIP simplex iterations
0 branch-and-bound nodes
z = 725
;

15. Result & Conclusion: - Travelling Salesmen problem is the major concept of Operations research.
In this study we solve the TSP with the traditional method and AMPL programming. With both
the way and we find the result of TSP and compare the result which is more optimistic. The total
distance cover by the salesman is 770 miles with the Hungarian heuristic approach. And to solve
the problem with the AMPL programming we found the distance cover by salesmen 725 miles.
The AMPL programming provide the minimum distance cover by salesmen which is 45 miles
lower than the result by heuristic approach. The result shows that AMPL programming is the
better tool to solve the TSP.