Research on End Distribution Path Problem of Dairy Cold Chain
Zhen-ping Li1, Shan Wang 2
1
2
School of Information, Beijing Wuzi University, Beijing, 101149, China
Department of Postgraduate, Beijing Wuzi University, Beijing, 101149,China
(lizhenping66@163.com, wangshan6969@126.com)
Abstract - The vehicle routing problem of dairy cold
chain end distribution with random demand and time
window is investigated in this paper. Considering the
characteristics of dairy cold chain end distribution, the
chance constrained theory and the penalty function is
introduced to establish a mathematical model of this
problem. A scanning-insert algorithm to solve the model is
proposed. The algorithm can be described as: Firstly,
according to the capacity of the vehicle and time window
restrictions, the customers are divide into several groups by
scan algorithm; Then find a feasible routing line for each
group of customers; Finally, using the idea of recent
insertion method to adjust the vehicle route and find the
final optimal distribution vehicle route.
Keywords - Dairy cold chain, Random demand,
Mathematical model, Scanning-insert algorithm
I. INTRODUCTION
Vehicle routing problem with time windows refers to
the transportation problem in general under the premise of
customer‘s requirements of time window. Solomon and
Desrosiers etc [1][2] consider joined time window
constraint to the general vehicle routing problem in 1987.
Desrochers [3] used to concise summary and summarized
various kinds of method solving vehicle routing problem
with time windows further in 1988. Sexton & Choi [4]
used the Decomposition method proposed by Bender to
solve the single vehicle pick-up and delivery problem
with time window restriction.
Chance constrained mechanism use the default value
constraint error to return to probability in the essence of
vehicles service process, and additional cost caused by
service failure is not within planning [5]. Stewart[6] and
Laporte[7] used respectively chance constrained program
change SVRP into equivalent deterministic VRP under
some assumptions. M.Dror[8] used Clark-Wright algorithm
to solve vehicle routing optimization problem.
This paper’s main consideration is regular route for
distribution mode under the target of minimizing the cost.
It means that the customer or the number of nodes and
their position are fixed in every day visit, but each
customer’s demand is different, and their demands meet
Normal Distribution.
____________________
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).
II. ANALYSIS OF THE COST IN COLD CHAIN
LOGISTICS DISTRIBUTION
A. Fixed costs
Distribution center has to pay for the fixed costs for
the use of each vehicle. These costs include the driver's
wages, insurance, lease rental of the vehicle.
m k
c1   f
k 1
B. The transportation cost
The transportation cost of a vehicle is the relevant
expenses caused by travel, which includes fuel
consumption, maintenance, maintenance fee.
n n k
k
c2    cij dij xij
i 1 j
C. The cost of damage
In the cold chain, the main factors causing fresh
products damaged are storage temperature, food of
microbes in water activity, PH value, oxygen content [9].
Assume the damage rate is  , the unit value of the
products is P, and capacity of vehicle k is Qk.
c3  P Qk
D. The cost of energy consumption
The heat load vehicle refrigeration equipment is
mainly due to difference heat transfer between the vehicle
body inside and outside. Suppose the temperature
difference between inside and outside of the vehicle is
fixed in a certain period, then the cost of energy
consumption can be expressed as:
m k
k
c4  A  (e  s )
k 1
E. Penalty cost
Soft time window can allow the distribution vehicle to
arrive outside the time window, but outside the appoint
time must be punished. Delivery time can be divided into
three categories: service in advance, service by time
window, service delay [10-11], which is shown in Fig.1.
(7) Products in the transportation process can stay in a
fixed transport temperature, and the vehicle’s energy
consumption is only related to their travel time.
Penalty cost
B. Symbols and Mathematical Model
M
f k ：Fixed cost of vehicle k；
cijk : The unit transportation cost of vehicle k in the travel
road from customer i to customer j;
dij : Distance from customer i to customer j;
Time t
a
g
h
b
Fig.1 Time window
(1) Service in advance is that the distribution vehicles
arrive in time window [a, g). Immediate delivery may
cause customers' inconvenience and complaint, but it can
reduce the energy consumption.
(2)Service by time window means that the distribution
vehicle arrives in the time window [g, h]. Immediate
delivery and the energy costs relate to time is a constant.
(3) Service delay means that the distribution vehicle
arrive in time window (h, b]. Immediate delivery and the
energy and relevant penalty costs will increase.
In conclusion, the penalty cost function is:
 M , t k  ta or t k  t
i
i
bi
i

(t k  t k ) , t  t k  t
ai
gi
i
 gi i
k
 (ti )  
k
t g  ti  th
 0,
i
i
 k
k
（ti  th ) , th  ti  tb

i
i
i
III. MATHEMATICAL MODEL
A. Related hypothesis:
A: Unit cost of energy consumption;
ek : The time when vehicle k return to the distribution
center;
s k : The time when vehicle k start off from the
distribution center;
t k : The time when vehicle k arrives at customer i;
Di : Demand of customer i;


k 1, If vehicle k come to customer j from customer i
xij 
0,
ot her wi se
1, If vehicle k service for customer i
k
yi 
0,
ot her wi se
The mathematical model can be formulated as follows:
m k
m n n k k m n
k k
min z   [ f  PQk  A e - s ]     cij dij xij     (tik )
k 1
k 1 i 1 j 1
k 1 i 1
s.t.
m k
m,
i0
(1)
 yi 
1, i  1, 2, , n
k 1



n k
k
y j   xij , i  j ; j  1, 2, , n
i 1
n k
n k
p  1, 2,..., n
 xip   x pj  0
i 0
j 0
(2)
(3)
(1) The model only considers the pure delivery
problem.
(2) There are enough delivery vehicles in the
distribution center, and each vehicle’s capacity is limited.
(3) The stock in the distribution center is enough for all
the customers and all customers’ time windows are
known.
(4) All vehicles start off from the distribution center,
and return to the distribution center again after
completion.
(5) The position of each customer is given, but
quantities demand Di of each customer i is random, it
d
k k ij
k
t j  ti   (1  xij ) M , j  1, 2, , n; k  1, 2, , m
(4)
v
k k d
k
(5)
ti  s  oi  (1  xoi )M , i  1, 2, , n; k  1, 2, , m
v
d jo
k k
k
e tj 
 (1  x jo ) M , j  1, 2, , n; k  1, 2, , m (6)
v
k
ta  ti  tb , i  1, 2, , n; k  1, 2, , m
(7)
i
i
n
n 2 k
k
1
k
 i yi   (  )   i yi  Q
(8)
i 1
i 1
satisfies a normal distribution Di N ( i ,  i2 ) , and they
are mutual independent.
(6) The route of each vehicle is determined and will not
change in the deliver road.
k
yi  0,1 i  1, 2, , n; k  1, 2, , m;
k
xij  0,1 i, j  1, 2, , n; i  j; k  1, 2, , m;
The objective function minimum the total cost.
Constraint (1) means that each customer will be
serviced by one vehicle, and each vehicle’s route start
from and ended at the distribution center.
Constraint (2) means that if vehicle k arrives at
customer j, then it must service for customer j.
Constraint (3) means that if vehicle k arrives at
customer p, then it must leave from customer p after
finishing service.
Constraints (4-5) are the conditions that the arriving
time for vehicle k come to customers i and j must satisfy.
Constraints (6-7) are the time window restrictions.
Constraint (8) means that the probability of each
vehicle’s capacity is no less than the total demands of all
the customers it serviced is great than  .
Ⅳ. ALGORITHM
The algorithm can be described as follows:
A. Set up the polar coordinates system.
B. Partition the customers into several groups.
1) Starting from zero Angle and rotating along
counterclockwise direction, pick up the customer into a
group one by one until the total demands of all customers
in this group exceed a vehicle’s capacity limit.
2) For each group of customers, ordered them into
sequence according to the demand time window and form
initial solution route. Then determine whether the solution
route satisfies the time window constraints.
3) If the solution route satisfies the time window
constraints, then a new group is created. Go to 1).
Continue to rotate along counterclockwise direction, and
the rest of the customers will be added one by one into a
new group. Otherwise, if the solution route does not
satisfy the time window constraints, we can adjust the
order of customers and find another feasible solution
route sequence. If no feasible solution route satisfying
time window constraints exists, delete a customer who
does not satisfy time window constraints from the group,
and add it into the next new group as a necessary
customer. Go to 1).
C. Repeat step B until all customers are partitioned
into groups.
D. Optimize the vehicle’s route by the recent
insertion method in each group.
1) Select the earliest time requirements customer to
form a sub-route with distribution center 0.
2) Insert customer point vk as the next demand point
according to time window series. Find an arc (vi , v j ) in
the sub-route, insert customer node vk between customer
nodes vi and v j to form a new sub- route such that the
new sub-route satisfies time window and the cost
increment is minimum.
3) Repeat step 2), until all customer nodes are added
into a route.
E. Repeat step D, until all groups are optimized.
Ⅴ. SIMULATION RESULTS
There are 30 customers needed to be serviced.
Suppose all vehicles are of the same type. The capacity of
each vehicle is 48; fixed costs is 100; the vehicle speed is
30 km/h; Unit of energy consumption cost $0.5 per
minute; unit distance transportation cost is $5 per
kilometer; punishment coefficient  is 0.4 and  is 0.5; 
is 95%,  is 0.01, P is 100. The experimental data is
random generation through the computer under
experimental hypothesis.
A. Set up the polar coordinates system.
B. Partition the customers into several groups
1) Starting from zero Angle and rotating along
counterclockwise direction, we can find the first group
customers are 2, 3, 5, 6, 7, and 9. The detail information is
listed in TABLEⅠ.
Q1=3+4+9+5+7+11+1.65 1  2  3  3  4  5 =45.4<48
TABLE I
BASIC INFORMATION TABLE
Customer
X
Y
Requirements T
Accept Time
Demand
quantity
2
22
10
5：00-5：30
4：00-6：00
Q~N(3,1)
3
12
20
5：50-6：30
5：00-7：00
Q~N(4,2)
7
25
30
6：10-6：40
5：10-7：20
Q~N(9,3)
5
10
2
6：30-7：20
6：00-7：50
Q~N(5,3)
6
8
15
7：10-7：40
6：20-8：00
Q~N(7,4)
9
13
35
7：20-7：50
6：20-8：40
Q~N(11,5)
2) Find an initial solution sequence 0-2-3-7-5-6-9-0,
The initial route is shown in Fig.2.
40
35
9
30
7
25
20
3
15
6
10
2
5
5
0
0
5
10
15
20
25
30
Fig.2 Initial route
3) Continue to rotate counterclockwise to build new
group. Repeat the process until all customers are picked
into a group.
C. Optimize the initial route of each group by the
recent Insertion method.
1) Select customer 2 whose requirement time
window is the earliest to form a sub-route with
distribution center 0. Insert customer 3 as the next
customer point according to its requirement time window.
Then customer 3 will be inserted between distribution
center 0 and customer 2, forming a new sub-route 0-2-3-0
satisfying time window and with minimal cost increment.
See TABLE II.
TABLE II
THE TIME TABLE OF SUB-ROUTE
dij
24
customer
14
—
0
4:30
△t
5:18
5:46
6:32
0
-4
0
23
—
2
time
—
3
0
TABLE III
TIME TABLE OF THE OPTIMAL ROUTE
dij
24
20
—
customer
0
time
4:30
13
—
2
—
7
15
9
3
—
13
6
—
10
5
5:18
5:58
6:24
6:54
7:06
7：32
0
-12
-56
24
0
12
△t
9
30
—
0
7：52
with the highest level of service and minimum cost. In
addition, this paper did not consider the asymmetry of
road network and the time handling factors. In the future,
we will investigate the problem with these factors.
40
35
—
6
7
25
20
3
15
REFERENCES
6
10
2
5
5
0
0
5
10
15
20
25
30
Fig.3 The optimal route
2) Insert customers 7,5,6,9 into the sub-route one by
one. We can find the optimal route 0-2-7-9-3-6-5-0. The
objective function value is 796.6. The optimal route is
shown in Fig.3 and TABLE III.
Similarly, we can use the same method to find the
optimal routes in the other groups. The results are shown
in Fig.4.
11
9
14
4
7
12
8
3
6
13
10
2
1
15
5
17
28
26
18
20
22
25
29
16
30
27
23
19
21
24
Fig.4 The optimal routes of all groups
Ⅵ. CONCLUSION
The vehicle routing problem of dairy cold chain end
distribution with random demand and time window is
investigated in this paper. A mathematical model is
constructed, and an algorithm is proposed.
Vehicle routing problem with time windows is a real
problem the enterprises facing with at the end of city
distribution. It is obvious that to pursuit minimum cost
may cause to drop the quality of service and eventually
lead to the loss of customers. To establish a suitable mode
of long-term sustainable development, the enterprise
should find a balance between service quality and cost. As
a result, the enterprises could meet customer requirements
[1] Solomon, M.M. Algorithm for the vehicle routing and
scheduling problems with time-windows constraints,
Operations Research, vol:35, Iss:2, 1987, P:254-265.
[2] Solomon, M.M., and J.Desrosiers, Time Windows
Constrained
Routing
and
Scheduling
Problems,
Transportation Science, Vo1:22, Issa, 1988, P:1-13.
[3] M.Desorchers, J. Lenstra, M. Savelsbergh, and Esoumis.
Vehicle routing with time windows optimization and
apporximation, Vehicle Routing: Methods and Studies [J].
North-Holland Amsterdam, 1988:64-84.
[4] Sexton,T.R.,Choi,Y.M .Pickup and delivery of partial loads
with“Soft” time windows. American Journal of
Mathematical and Management Sciences,1986,6(4): 369398.
[5] Baowen Chen, Ant colony optimization algorithm for the
vehicle routing problem in the research on the application
(in Chinese) [D]. Harbin industrial university, 2009
[6] Stewart W R, Golden B L. Stochastic vehicle routing: a
comprehensive approach. European Journal of Operational
Research, 1983, (14): 371-385
[7] Laporte G, Louveaux F, Mercure H. Models and exact
solutions for a class of stochastic location-routing problems.
European Journal of Operational Research, 1989, (39): 7178.
[8] Dror M, Trudeau P. Stochastic vehicle routing with modified
savings algorithm. European Journal of Operational
Research , 1986,(23): 228-235
[9] Yangjun Wang, Cold-chain logistics distribution center
mode study (in Chinese)[D].Changsha university of science
and technology, 2008.
[10] Shufang Zhan, Parallel algorithm with the soft time
Windows in the vehicle routing problem of the application
(in Chinese)[D].Wuhan university of science and
technology,2006
[11] Thangiah S, Nygard K, Juell PG. Agenetic algorithms
system for vehicle routing with time windows[C]//Miami
Proceedings of the Seventh Conference on Artificial
Intelligence Applications, Florida, 1991, 322-325