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Judges a score, signature
and notes
Team No:
10476
Three judges score,
signature and notes
Two judges scoring,
signature and notes
Topics:
C
Four judges score,
signature and notes
Topic:
The article is about how to choose the best tourist track,which belong to travelling
salesman problem. For seeking shortest and most low-cost track,we simulated three
models containing economical model,time-saving model and easy model.In order to
solve different problems,we used ant colony optimization, modified circle algorithm
and multi-objective programming.Then, we combined the fact to the three models and
analysed residual.What ‘s more,we analysed the complexity of above algorithm.
For the first problem,referring to Global mapper,making sure 10 scenic spots’
longitude and latitude, calculating a shortest track.In another word,starting with the
hotel,visiting every spots only once and ending up in the hotel.According to the
longitude and latitude,we ‘ll make full use of geometry to calculate the distance from
two cities.we can describle the problem as setting up 10 dots representing 10 cities.The
line between each two dots means the tourist track.The route attributes such as the
length of the said for the edge weights, you can travel the city network abstraction as a
weighted directed graph. Set a weighted directed graph G for binary group G = (V,
{E}), which contains V is the set of n nodes, E is a collection of h side , (I, j) is from
edge nodes I and j E, edge (I, j) is nonnegative weights. Set S, T V respectively in the
starting node and destination node, then the optimal path problem is refers to in the
weighted directed graph G, then the optimal path problem is refers to in the weighted
directed graph G, find from the specified initial node to the destination node of a path
with the minimum weight sum The shortest tour line length is 92.2 kilometers.
In view of the problem, the purpose of this problem is to design the most economical
travel plan, namely the minimum distance cost we use improved algorithm to solve
traveling salesman problem, in any distance between two points of minimum cost
matrices as the weights, using undirected graph adjacency matrix structure, according
to the question I do not know the starting location, so using the Matlab software
repeated 10 times improved circle algorithm in each city as a starting point, namely
from 10 Hamilton got the optimal times, namely.
Between any two points on question three, give priority to in order to enjoy here, in a
model based on the results of the model, we set principles: priority is convenient,
when both the cost than taking a taxi on a luxury bus of the high cost of used within a
certain range, is taking a taxi Here by dynamic programming to implement the plan,
on the basis of the economy of the shortest route, through a change in a way, to make
the final cost deviates from the value of the minimum cost of in our allowed range,
thus to save money The purpose of saving time and convenient Ultimately satisfied
tourists themselves need to travel plan, the total cost 1643 yuan (do not include
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accommodation Under the condition of cost).
Then we combined with the actual situation of the three model for scientific error
analysis, and analyzes the complexity of the algorithm is used, at the same time for us
to solve traveling salesman algorithm is evaluated, which made us to the traveling
salesman problem have the further understanding.
Keys:travelling salesman problem, ant colony optimization, modified circle algorithm,
dynamic planning, error analysis.
The best travel route choice model
1 problem repeat
Summer vacation is coming, many parents will choose this time to take children to travel to a
certain city, but different family has different requirements (number, cost constraints, time
constraints, etc.), please choose a tourist cities (such as your city), considering travel routes,
cost, time, and other factors do you think of the more important, for families with different
demand design a best travel packages.
According to the different needs, we can put the problem into consideration from three
different aspects:
![](https://csdnimg.cn/release/download_crawler_static/88965827/bg3.jpg)
The short circuit is designed according to the geographical location (longitude and latitude)
travel plan.
Hypothesis between any two cities have luxury bus and taxi, 2 times the price of taking a taxi
is the distance between two points (unit: yuan), is the price of luxury bus segment, is a
distance of 2.5 times within 30 kilometers, more than 30 km and within 70 kilometers of is
1.7 times that of the distance is the distance of more than 70 km of 1.4 times, if a family can
choose taxi luxury bus, design the most economical travel plan
(3) in the comprehensive practical cases, consider to save money To save time and convenient,
set up corresponding evaluation criteria and indicators, establish corresponding mathematical
model, the improved and under the premise that the revision of the above two options
Assuming that luxury bus and taxi can increases with the increasing to walk, taxi speed is 80
km/h, luxury bus speed is 50 km/h
2 symbol
2.1 Condition assumptions
(1) Calculating the attractions of the distance between ignore terrain such as hilly basin
natural factors influence on the calculation results;
(2) Hypothesis speed must be in the journey, and does not consider emergencies interference
taxis or luxury bus trip;
(3) Hypothesis between any two spots have luxury bus and taxi, 2 times the price of taking a
taxi is the distance between two points (unit: yuan), is the price of luxury bus segment, is a
distance of 2.5 times within 30 kilometers, more than 30 km and is within 70 kilometers of
the distance of 1.7 times, more than 70 km distance of 1.4 times;
(4) Assumes that the residence time in each scenic spot for a day.
2.2 Notational Conventions
n
:Represents the number of cities;
ij
d
:he distance between the two cities,
1 1 10ij
1,
0
ij
ij
x
表示走过城市 到城市 的路
, 表示没有选择走这条路
C
: Initial ring;
ij
CC: 的改良圈
,
1 1 10ij
![](https://csdnimg.cn/release/download_crawler_static/88965827/bg4.jpg)
1
1
1 1,2, ,10
1 1,2, ,10
n
ij
j
n
ij
i
xi
xj
: 每个点只有一条边出去, ;
: 每个点只有一条边出去, ;
10 10
( , )F i j
:Minimum cost between any two points constitute the matrix tak
3 .Analysis of the Problem
3.1 Analysis of Problem Ⅰ
The problem is a combination of a travel path optimization problem, and its formal
description is: represented by a node 10 attractions, connecting edge between two nodes
indicates itinerary and length of the route, said the right to property and the value for the edge ,
then you can put Attractions Travel network Abstraction is a weighted directed graph.Given a
weighted directed graph G is a tuple G = (V, {E}), where V is a collection containing n nodes,
E is contained h edges (arcs) collection, (i, j ) E is the edge from node i to j, and the edge (i, j)
non-negative weights.Let S, T V, respectively the starting node and the destination node, then
the optimal path problem refers to the weighted directed graph G, then the optimal path
problem refers to the weighted directed graph G, looking from specify the starting node to a
target node paths with minimum weight sum of the values.
Under the influence of the different needs of conditions, attractions travel route network
not only has the physical attributes, such as road routes, but also has a variety of other logical
attributes route length, travel time, ticket prices, etc., thus changing the optimal route of
travel.
One problem, we have the latitude and longitude location based on actual use of the
knowledge of geometry to calculate the distance between the two cities each, and the distance
to the right, and then establish a model for solving. Explore 10 attractions set out the optimal
travel routes.
3.2 Analysis of Problem Ⅱ
According to 10 attractions in latitude and longitude, the most economical design an
itinerary, from a resident, after each attraction just once and spend the least.It can be seen that
this issue is part of the traveling salesman problem, we can consider the use of improved
algorithm for solving this problem circle. According to various attractions attractions order
number, depending on the distance to attractions problem between one obtained, we can
consider in order to drive between any two points at least spend as weights constructed using
undirected graph, taking into account the starting point is not given, If certain attractions as a
starting point, using the optimal loop algorithm was modified ring may not be the optimal
solution, so we will use the Matlab software programming improved lap times repeated
algorithm will get the best lap, thus ensuring optimal solution, that is the most economical
travel routes.End constituted by a closed loop back to the beginning of the most economical
route. This will design one of the most economical travel routes.
3.3 Analysis of Problem Ⅲ
For Question 3 on the basis of a model, the results of the two models, we can consider
setting principles: convenience priority, when the cost of two taxi used luxury bus ride than
![](https://csdnimg.cn/release/download_crawler_static/88965827/bg5.jpg)
the high cost of not using a range, take a taxi.Consider the adoption of dynamic programming
to implement this program, based on the most economical on the shortest route through the
change to take way, if the final cost deviate from the minimum allowable spending within our
range, to accept this solution, to save money provincial convenient time and purpose. Finally
get to meet their own needs travel program.
4 Modeling and Solution
4.1Modeling and Solution to problem Ⅰ
According problem analysis, we first create Attractions Travel Network Diagram, given a
weighted graph, which represents the set of vertices (points of interest) is expressed
as:
{1,2, ,10}
G
V
.
G
E
is empower the side (the distance between two spots) collection that
the distance between the two places(km).Expressed as
RwVjiwjiE
ijGijG
,,,,,
.
Hamilton path map can be expressed as:
121
vvvvS
n
;among
Gi
Vv
,
ji
vvji ,341
,and
341 i
,
Gvvnii
Ewvv
nii
),,(
1)mod(
1)mod(
.
)(GH
is a collection of G note of all Hamilton circuits,definitions
11
11
)(
ni
vvvv
nii
wwSw
;The purpose is to find a shortest Hamilton circuit
*
S
,
Make
)(min)(
)(
*
SwSw
GHS
.
By Ant Colony Algorithm to calculate the resulting data to search.
4.1.1Estimate the distance between cities
Find attractions by Global mapper 10 specific latitude and longitude values (Table 1) as
follows:
Figure 1 10 attractions map position display
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