Integer Linear Programming Model and Greedy Algorithm
for Camping Along the Big Long River Problem
Zhen-ping Li1, Xiao-dong Huang 2
1
School of Information, Beijing Wuzi University, Beijing, China
Department of Postgraduate, Beijing Wuzi University, Beijing, China
([email protected], [email protected])
2
Abstract - In this paper, we investigate the problem of
camping along the Big Long River: How to schedule the X
trips in a rafting season of the Big Long River so that the
total meets of any boats are minimal? By introducing the
proper variables, the problem is formulated into an integer
linear programming model. For small size problem, this
integer linear programming can be solved by Lingo software;
for large size problem, we design a greedy algorithm to
arrange the schedule of the given X boats. Finally, we do
some simulations of the above model and algorithm and
obtain the optimal solution.
Keywords – Camping along the river, integer linear
programming model, greedy algorithm, simulation, the
optimal solution
I. INTRODUCTION
Visitors to the Big Long River (225 miles) can enjoy
scenic views and exciting white water rapids. The river is
inaccessible to hikers, so the only way to enjoy it is to
take a river trip that requires several days of camping.
River trips all start at First Launch and exit the river at
Final Exit, 225 miles downstream. Passengers take either
oar- powered rubber rafts, which travel on average 4 mph
or motorized boats, which travel on average 8 mph. The
trips range from 6 to 18 nights of camping along the river.
Currently, X trips travel down the Big Long River each
year during a six month period (the rest of the year is too
cold for river trips). There are Y camp sites on the Big
Long River, distributed fairly uniformly throughout the
river corridor. In order to make sure the passengers enjoy
a wilderness experience and also for the sake of their
safety [1], we should try to avoid the meet of two groups of
boats on the river. Besides, due to the capacity
constraints, no two sets of campers can occupy the same
site at the same time.
Every year, before the rafting season, the park
managers must arrange the schedule of these X trips that
will raft along the Big Long River in the rafting season.
The key problem is how to schedule these X trips so that
the total meets among all boats in the river is minimum.
In this paper, we will solve this problem. This paper is
organized as follows: In section II, we make some
assumptions [2], [3] and introduce several variables [4], and
then we formulate the problem into an integer linear
____________________
This work is supported by National Natural Science Foundation of
China under Grant No.11131009 and the Funding Project for Academic
Human Resources Development in Institutions of Higher Learning under
the Jurisdiction of Beijing Municipality (No.PHR201006217).
model; Then we will design a greedy algorithm in section
III; Section IV is the simulation results; The conclusion is
given in section V.
II. THE INTEGER LINEAR PROGRAMMING MODEL
A. Assumptions and variables
a) Assumptions
 Once people choose one of the propulsion (oarpowered rubber rafts or motorized boats) at first,
they can not change again on the way;
 The duration of each trip ranges from 6 to 18 nights
on the river;
 There are X trips, each trip has a given duration;
 There are Y camps distributed fairly uniformly
throughout the river corridor;
 There is so enough fuel and power for each boat that
no breakdown might occur on the whole river trip;
 Each boat, controlled by the specialized staff, will
run exactly on schedule;
 There are 180 days in the Big Long River’s rafting
season;
 There are 8 hours open for the river trip at daytime;
 Each rafting boat must stay in one camping site at
night.
b) Variables
X: the total number of available trip boats;
Y: the total number of camping sites;
rik: the time when boat i arrives at camping site k;
dik: the time when boat i leaves from camping site k;
1,boat i occupys camping site k
xik = 
0, otherwise
1,boat i and j meet at river between camping site k and k +1
cijk = 
0,otherwise

0,if dik >d jk
cijk1 = 

1,otherwise
and rik+1 >rjk+1

0,if dik <d jk and rik+1 <rjk+1
cijk2 = 

1,otherwise
Ti: the total trip duration of boat i (measured in nights on
the river);
Pi: from which day boat i start off (Pi is an integer) at the
First Launch;
vi, min: the minimal speed of boat i;
vi, max: the maximum speed of boat i.
B.
The Integer Linear Programming Model
The problem of camping along the Big Long River
can be formulated into an Integer Linear Programming
Model[5-7]], in which the schedule time table of all given X
boats (the total number of available trip boats) can be
obtained with the total meets among all boats in the river
be minimal.
(7)
(8)
III. GREEDY ALGORITHM
(9)
As a matter of fact, we could use Lingo software to
solve this problem; however, the problem scale is so large
that the time consuming will be too long. So it’s not wise
enough to use Lingo software in this situation. Here we
design a greedy algorithm [8-12] to solve this problem.
Firstly, we define the open time for river trip.
According to the given information, we could define that
the river trip is only allowed at daytime from 08:00 to
16:00 clock; for the other time, passengers have to stay at
the camping site.
b) The integer linear programming model
As analyzed above, the objective function is
constructed to minimize the total meets between any pair
of boats on the river. The constraints including: (1) the
river trip can only be allowed at daytime from 08:00 to
16:00; (2) the river trip duration of each boat is an integer,
ranging from 6 to18 nights; (3) no two sets of campers
can occupy the same site at the same time; (4) the season
for river trip only lasts 6 months (180 days). Based on the
descriptions above, we formulate the integer linear
programming mode as follows:
X
X
Y
i
j
k
(1)
s.t
0  rik  dik , i  1, 2, , X ; k  1, 2,
w
w
225
 rik  di ( k  1) 
, w=
vi max
vi min
Y 1
24 Pi  8  dio  24 Pi  16
k
s 1
s 1
k
k 1
s 1
s 1
24( Pi   xis )  8  rik  24( Pi   xis )  8
Y
s 1
is
 Ti
xik  dik  rik  Mxik
(2)
(3)
(4)
k
24( Pi   xis )  8  dik  24( Pi   xis )  16
x
,Y
(10)
 rik  rjk    M   xik  x jk 
(11)
 rik  0

 xik  0,1
c k  0,1
 ij
 Pi  0 (integer)
T  0
 i
i, j  1, 2,..., X

 k  1, 2,..., Y
(12)
The objective function (1) is to minimum the total
meets between any pair of boats on the river;
Constraint (2) means the time when boat i leaves site
k is later than that when boat i arrives at site k;
Constraint (3) guarantees the time for boat i to travel
from site k-1 to site k is between the lower and upper
bound;
Constraint (4) means that boat i will begin its trip in
the Pith day at the open time, where Pi =1,2,…174;
Constraint (5) guarantees that boat i leaves camping
site k at the open time;
Constraint (6) guarantees that boat i arrives at
camping site k at the open time;
Constraint (7) means the duration of boat i is Ti,
where Ti is an integer ranging from 6 to 18;
Constraint (8) describes the condition whether boat i
occupies the camping site k;
Constraint (9) guarantees all the boats will finish
their river trips in six months;
Constraint (10) describes the condition whether boat
i and boat j meets on the river;
Constraint (11) guarantees no two sets of campers
can occupy the same camping site at the same time;
Constraint (12) describes the value range of
variables.
a) One essential definition
min z =  Cijk
 dik  djk  Mcijk 1

k1
 ri ( k  1)  rj ( k  1)  Mcij

k2
 dik  djk  Mcij

k2
 ri ( k  1)  rj ( k  1)  Mcij
 k
k1
k2
cij  cij  cij  1
(5)
(6)
Y
6  Pi   xis  180
s 1
A. Several Essential Supposes
According to our model, we will write a procedural
by using the MATLAB software, but before this we give
some essential supposes again:
 To avoid the meet between any pair of boats on the
river, the boats have to start off by some gap time,
and we define the gap time as follows: Gap time
=225/(Y+1)/v, where v is the speed of boat;
 The boat will run at the same speed during the whole
trip;
 The actual time of river trip for each boat every day is
not more than 5 hours;
 Passengers who want to have a river trip need to
make a booking in advance, and then we will divide
the six months’ rafting season into several cycles
according to the passengers’ booking. During one
cycle time, we will arrange the boats by their
duration like this: boat of 6 nights first, and then it
will be 7, 8,… in turn, boats of 18 nights will be the
last to be arranged;
 The number of boats (represented with Q) arranged
every day will depend on its duration and could be
calculated by Q=[Y/Tdur]. For example, for boats
whose duration is 6, the maximum number of this
type boats we can arrange every day is [Y/6]; for
boats whose duration is 7, the maximum number we
can arrange every day is [Y/7].Why? We can explain
this by the following graph (see Fig. 1).
Suppose Y=24，then for boats whose duration is 6,
the maximum number of boats we can arrange every day
is 4:
…
Fig.1 Trip process of boats arranged in the first and the second days
In Figure 1, the bold horizontal line denotes the
riverbank while the thin vertical line represents the
camping site and the arrow symbolizes the boat. At the
first day we arrange 4 boats. Then we can see these four
boats as a whole and its whole trip process could be
described vividly in the above graph. By this method we
can guarantee to utilize the camping sites in the best
possible way. As to other boats of different duration, we
can draw the similar graph like the above.
B.
The Greedy Algorithm
The greedy algorithm can be described as follows.
BEGIN
Classes all X boats into 13 groups according to their
duration. Denote the number of boats whose duration is i
by X(i), i=6,7,…,18.
Provided that the maximum river trip time every day
is no more than 5 hours, calculate the maximum number
of boats whose duration is i we can arrange every day,
denoted it by m(i), where m(i)=[Y/i].
for i = 6:18
Arrange the schedule of all boats with duration i.
For all X(i) boats with duration i, arrange them to
start off from the First Launch in X (i) / m(i) consecutive
days, with the time gap between two successive boats is
225/(Y+1)/v daytime.
After all X(i) boats with duration i arranged, we can
arrange the following X(i+1) boats with duration i+1 in
the following X (i  1) / m(i  1) days…., till all X boats
are arranged.
end
END
IV. SIMULATION RESULTS
In this section, we will do some simulations of the
model and algorithm described above.
Supposing Xi denotes the number of boats whose
duration is i (i=6, 7…18), where
X6=5, X7=6, X8=11, X9=12, X10=10,X11=8, X12=6,
X13=15, X14=7, X15=12,X16=12, X17=11, X18=9.
Given Y=53.
We run our procedural coded by MATLAB software,
the simulation result are as follows:
D1: 5 boats whose duration is 6 nights and 3 boats
whose duration is 7 nights will start off in the first day;
D2: 3 boats whose duration is 7 nights and 4 boats
whose duration is 8 nights will start off in the second day;
D3: 6 boats whose duration is 8 nights will start off in
the third day;
D4: 1 boat whose duration is 8 nights and 5 boats
whose duration is 9 nights will start off in the fourth day;
D5: 5 boats whose duration is 9 nights will start off in
the fifth day;
…
D31: 2 boats whose duration is 18 nights will start off
in the 31st day;
D32: 2 boats whose duration is 18 nights will start off
in the 32nd day;
D33: 1 boat whose duration is 18 nights will start off
in the 33rd day.
The detail rafting schedule of all boats can be
described in Figure 2.
According to the simulation results, we find that we
can arrange the 124 boats in about 50 days. This inspired
us that we can divided the rafting season (180 days) into
several (for example 3) periods. Arrange X/3 boats in
camping
site
day
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Fig.2 The detail schedule of all boats obtained by greedy algorithm
every period respectively according to the greedy
algorithm. This can avoid all boats with same duration be
arranged in several concentrate days.
Remarks: By using the greedy algorithm, we can
give a solution to the problem; however this solution
might not be the optimal one. But based on this solution,
we can take some measures to improve. By continuous
adjusting, we can finally find a satisfied solution.
V. CONCLUSION
The problem of Camping along the Big Long River
is very complex and the solution should be of great
openness. In this paper, we formulate this problem into an
integer linear programming model and design a greedy
algorithm to arrange the schedule of boats. Then by doing
some simulations with this algorithm, we give a solution
to the problem. The results show that this method can
obtain the optimal solution by continuous improving.
Furthermore; we can estimate the capacity of the river by
this greedy algorithm.
Although river trip is quite interesting and exciting,
it is also very risky and need some spirit of adventure.
Any accident may happen during this process, such as bad
weather, passenger’s injuries and so on. These potential
factors might have a great impact on the supervisor’s
decision and management. We don’t take these factors
into account. In the future, we will consider these factors
in the model and algorithm.
ACKNOWLEDGMENT
The author gratefully acknowledges the valuable
discussions and comments that Huang qiuai, Chen Qing,
Li Tingting and Wang Shan provide for this paper.
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