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mathorcup数学建模挑战赛获奖论文-第四届C题_10463e.pdf
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The judges scoring, note Team number:
10463
The judges scoring, note
The judges scoring, note Problem
:
C
The judges scoring, note
Title
:
Optimal Travel Package Design for Family Summer Vacation
Abstract
Nowadays, more and more parents choose to take their kids to a new city for
traveling with the development of improvement of living standard. But different
families have different requirements which includes expense of the travel package,
time, number of family members, the weather and so forth. So, we choose Jinan as a
city for traveling and comprehensively consider the different requirements of different
families. Then we design an optimal travel package for families with certain specific
requirements.
We have selected the typical combinations of family requirements to analyze
using control variate method and referring to the knowledge of psychology. And we
choose expense, number of sight spots and time as the three basic requirements, the
relationships of which is considered from the first model to the third. Model four
mainly considers the number of family members while the fifth is about weather.
In model one, two, three we focus on different family requirements listed as
follows: 1.cover all the sight spots with limitless expense and a minimum amount of
time. 2.cover a maximum amount of sight spots with limited expense and time. 3.use
minimum money to travel with limited time or enough spots to go to. Considering all
these requirements, we choose the shortest route and use the fastest means of
transportation. Then we use Lingo to process the data. Next, we establish a triune
requirement model combining the three factors which includes expense, time, number
of sight spots using greedy method and graph theoretic approach to satisfy the needs of
families.
Model four and five focus on the condition that several families will go together
and the effects of rainy days based on the aforesaid models. The models are trying to
save more money to improve the travelers’ satisfaction. We come up with
several-family-travel-together strategy and define the loss index of rainy days. Finally,
we get the optimal result with a minimum loss.
Key words
:
travel route design, greedy method, graph theoretic approach, requirement
model, degree of satisfaction
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- 1 -
Optimal Travel Package Design for Family Summer Vacation
1. Restatement of the problem
The summer vacation is drawing near. Lots of parents seize the chance to take
their kids to a new city for a wonderful holiday. Apparently, however, different
families have different requirements, such as the expense of the travel package, time,
number of family members and the weather. Choose a city and comprehensively
consider the different requirements of different families. Then design a travel package
for families with certain specific requirements.
2. Analysis
According to the our understanding of the problem, we need to design a optimal
travel package by considering different family requirements in order to minimize the
expense, to shorten the time, to maximize the number of sight spots or satisfy other
requirements. Taking a variety of factors, we divide the requirements into five
categories listed as follows: expense of travel package, numbers of sight spots, time,
number of family members, the weather.
We have selected the typical combinations of family requirements to analyze
using control variate method and referring to the knowledge of psychology. And we
choose expense, number of sight spots and time as the three basic requirements, the
relationships of which is considered from the first model to the third. Model four
mainly considers the number of family members while the fifth is about weather.
Taking various factors into account, we establish an efficient mathematical
model choosing Jinan as the travel city.
Model one focuses on requirements listed as follows: limitless expense, covering
all the sight spots and a minimum amount of time. Having satisfied all these
requirements, we can choose the route with the shortest length and faster means of
transportation. After analysis, we design an optimal route to satisfy the requirements
using a software named Lingo.
Model two focuses on requirements listed as follows: limited expense, limited
time and more sight spots. Because of the two limited factors, we analyze one factor
at a time and then consider both. The first one is about the control of time.
Specifically speaking, we need to find a travel package which covers the most spots
with limited expense and limitless time. Then we build the objective function and
condition constraint formulas. The second one is about the control of expense. To be
specific, we need to find the optimal strategy which covers the most spots with limited
time and limitless expense. Then comes the function and formulas. Finally we
consider the both aforesaid factors and obtain the final objective function.
- 2 -
Consequently we are able to design an optimal route by greedy method and graph
theoretic approach.
Model three focuses on the requirements of the minimum expense, under the
condition of which we analyze the combination of numbers of spots and time. Firstly,
we consider a family who wants to travel the most spots with limited time and a
minimum amount of expense. To get a best package, we decide the number of spots at
the beginning and the calculate the minimum amount of expense under corresponding
condition constraint. We can get several resultant strategies so that families are able to
choose one according to their own interests. Secondly, we consider the requirements
listed as follows: a minimum amount of expense, covering all spots and limitless time.
Actually it is based on the analysis the first one for the only difference between them
is the time. So we can use the model above.
Model four focuses on the number of the family members. When two families
choose to go together but their time does not fit we need to find a strategy that costs
least. As we know from the assumptions that more travelers at a spot means less total
expense, we need to take it into account in order to cut the expense.
Model five focuses on the requirements of time. It is obvious that bad weather
will lead to a less satisfying vacation, so we try our best to minimize the effect of the
weather. We define the loss index first and then add the two objectives including total
expense and total loss by their weights. Finally, we get the optimal strategy with a
minimum loss.
1---Baotu Spring park+Quancheng Square
2---Qianfo Mount
- 3 -
3---Daming Lake
4---Quancheng Park
5---Yellow River Forest Park
6---Jinan Zoo
7---Zhujia Yu
8---Jinxiangshan Amusement Park
9---Yingxiongshan Memorial Park
10---Hongjialou Square
3. Assumptions
1.There is no accident during the travel.
2.Assume that the length of the summer vacation is no longer than two months.
3.The ability of each spot to serve travelers is excellent which means that it is possible
to serve travelers from different routes.
4.More travelers means lower total expense.
5.The speed of the bus is 50km per hour, and the cost is 3RMB per kilometer.
6.Going directly from one spot to another means the spots in the middle of the route is
just a station instead of a real spot.
7.Quancheng Square is the beginning and the ending as well. There is no travelers
who leave the family during the travel or those who don’t go inside the spot.
4. Definitions and notations
T
: total time a family spends;
1
p
: time that a family spends on the road;
2
p
: time that a family spends at the sight spots;
3
p
: time that a family spent on possible accommodation;
i
e
: possible accommodation time that a family may stay at i spot sight i;
n
: the number of sight spots;
ij
r
: a 0-1 variable thar demonstrates whether a family goes from sight spot i to sight
spot j;
ij
t
: the time that a family spends from sight spot i to sight spot j;
ij
c
: expense that a family spends on transportation from sight spot i to sight spot j;
- 4 -
m
: total expense that a family spends on the travel;
1
m
: total expense that a family spends on transportation;
2
m
: total expense that a family spends on all the sight spots;
3
m
: total expense that a family probably spends(accommodation, food and so on);
'
i
: the weight of sight spot i for the first family;
''
i
: the weight of sight spot j for the first family;
'
1
m
: total expense that the first family spends on transportation;
''
1
m
: total expense that the second family spends on transportation;
'
2
m
: total expense that the first family spends on all the sight spots;
''
2
m
: total expense that the second family spends on all the sight spots;
3
m
:the amount of saved expense when two families go to the same spot at the same
time;
i
: a 0-1 variable thar demonstrates whether two families go to the same spot at the
same time;
is
P
: possibility that it is overcast and rainy at sight spot i;
i
T
: time that a family arrive at sight spot i;
V
: set of sight spots where a family will go. For example,
V
={1
,
8
,
5
,
7}means a
family has gone to sight spot 1, 8, 5, 7.
5.Modeling and making a solution
Based on requirements of the subject,we comprehensively consider travel
route,time,quantity,weather factors and so on when we design a family travel package
during summer holiday to adapt to different families’ different requirements.
5.1 Model 2 building and making a solution
5.1.1 Object function building
Firstly,we consider requirement as visiting all sight spots and spend the minimum
of money without limitation of time.Based on our understanding of the subject ,we
define the spending time involve three parts,namely those spent on the road,staying at
the sight spots and possible accomdation.Here,we define the following symbols:
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