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mathorcup数学建模挑战赛获奖论文-第四届C题_10317 english.pdf
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The judges scoring, note
Team number:
10317
The judges scoring, note
The judges scoring, note
Problem:
C
The judges scoring, note
Title:The Design of Family Summer Travelling Plan
Abstract
A cozy family trip is always the pursuit of our humans. We just want to get the most
value and enjoyment under the same time and space conditions, especially in
designing the travel packages.
Our report selects Chengdu as our travel city, at the meantime, selects 11 popular
places of interest. According to different needs of family, we build3independent
models,
Model 1, visiting all the attractions with least expense: under condition of the 11 sites
identified, the problem is abstracted to a typical travelling salesman problem (TSP).
We build a modified algorithm to suit our problem.
Model 2, visiting most attractions with given budget: In this model, we use reverse
thinking by using method in the first model. We finally give the best plan by changing
the numbers of visiting places.
Model 3, visiting most attractions with least expense within given time: It is the typical
Multi-objective planning model. We give the time, travelling cost, the number of
places and 0-1 variables constraints, resulting in planning the best family journey
during different time period.
These three models almost meet all families’ needs.
When thinking about the improvement of our models, we set up two modified
model------ the model which considering the attraction’s popularity and the
comprehensive model which takes weather risk into consideration. In this way, we
refine our design for the family travel.
Finally, we make a family travel packages consisting of 9 routes.
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Key words : refined circle, TCP, reverse thought, A multi-objective linear
programming, traveling package
The Design of Family Summer Travelling Plan
1. Introduction
Summer holiday is always the best time to have a family journey. Parents usually
bring their children to go on a trip in order to broaden their horizons and develop
relationship between families.
Each family, however, has different needs for a journey, such as the number of family,
the affordable expense, the time to travel, etc. choose a city as the destination, build a
model considering the way of travel, the cost, the time and any other important
factors.
2. The analysis of the problem
This problem is a discrete optimization problems under certain constraints. Our
primary task is to determine different factors in the process of family travel. Then, we
will emphasis on each of these factors, by building mathematic models.
Through the observation and survey on the network in everyday life, we found that
when choosing travel plans, most families focus on three factors----vacation time,
performance-to-price ratio, and the number of places to visit. Due to the actual
situation of different family, we respectively use travel time and travel cost, as well as
the number of tourist attractions, three variables which can be quantified, to elaborate
the three aspects of family concerns, and respectively build mathematical models.
During the process of problem solving, we choose Chengdu and its surrounding
attractions as our research object, namely is our family travel planning design of tour.
We will establish three models to suit families of different needs.
Three models respectively view the number of tourist attractions, travel costs, travel
time as a key consideration of family travel plan.
3. Assumptions
1. we take each family member equally, which means they will not have differences
in travel costs and choices making.
2. we use cab as the only vehicle in our models. Data is referred to local travel
agents.
3. the average speed of our cab is 50km per hour, while the average cost is 0.3 yuan
per kilometer.
4. when we go from place A to place B, we have no visit to places during the trip.
5. during a time period, family members start from Chengdu, and end in Chengdu.
6. in a day, 12 hours for traveling and 12 hours for rest.
7. there is no accident in our travel.
8. we choose the following national 5A and 4A attractions as our potential destination
after thinking about the surrounding tourist attractions: Chengdu, jiuzhaigou,
huanglong, leshan, emeishan, siguniangshan, danba, dujiangyan, qingchengshan,
hailuogou, kangding.
4. Introduction of the parameters
i
,
j
——number site, number j site, ,
i i
j
=1,2,……,11;
They represent Chengdu, JiuZhaiGou, HuangLong, Leshan, EMeishan, SiGuNiangshan, Danba,
DuJiaYan,QingChengShan,HaiLuoGou,KangDing,TianTaiShan,DaoChengYaDing;
c
——the average of total cost of a family member ;
i
t
——every family member spends in number site;
i
t i
i
c
——each family member in the total consumption spots;;
ij
t
——from the spot to the attractions journey time required;
ij
c
——transport costs from the scenic spots to the attractions;
1
0
ij
f
rom i to j
r
else
5. Establishment of the model and models’ solving
5.1 Model Ⅰvisiting all the attractions with least expense
5.1.1 Brief introduction to the model
In this model, we will discuss family in bound for all 11 scenic spots under the
condition of the formation of the least expensive. Under the condition of the 11
sites identified, 11 scenic area can be as 11 points, the problem is abstracted to a
typical traveling salesman problem (TSP).
5.1.2 Establishment of the model
We can abstract the family trip problem to the following questions:
A businessman wants to n city to sell the goods, starting from a particular city,
along the way after each city a return to the departure city, to determine a
walking route, shortest route. Is the traveling salesman problem (TSP).In terms of
graph theory, is an empowerment in the diagram, completely find out the right to
a minimum Hamilton circle. Say this is the optimal times. Contrary to the short
circuit and attachment problems, although there is currently no effective
algorithm to solve traveling salesman problem. But there is a feasible way is to
ask a Hamilton ring, then the appropriate modification to get another Hamilton
circle with a smaller right. Modified method called improved algorithm.
Suppose
12 1n
Cvv vv 。
For we create a new Hamilton circle
11ij ,n
12 1 2 1 1 2 1
,
ij ijjj ijj n
Cvvvvvv vvv vv
It is derived from deleting
1ii
vv
and
1jj
vv
and adding and . If
, we substitute with ,which is called
the refined circle.
ij
vv
ij
C
11ij
vv
11 1
()( ) ( )
ij i j ii
wvv wv v wvv wv
1
(
jj
v
)
C
However
, results using the improved circle algorithm are almost certainly not
optim
al. In order to get higher accuracy, on the premise of not given the starting
position, can choose different initial circle, repeated times algorithm, to obtain
accurate res
ults.
n
Thus, we use the mathematical expression of the abstract:
Suppose there are n cities, is the distance between
i and
ij
d
j
,
where,
01
ij
tor
0 means the way has been taken, 1 means the way has not been selected.
Thus, we have
1
1
1
,
min
. . 1, 1, 2,...,
1, 1, 2,...,
1, 2 1, 1, 2,...,
n
ij ij
i
n
ij
i
n
ij
j
n
ij
ijs
dx
st x i n
xx n
x
ssns
n
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