Modeling and Algorithm for Logistics Distribution Center Location
Kuang Ying Qiang1 Kuang Qiao2 Guo Yu3
1 School of Business Administration, South China University of Technology,
Tel: 86-020-87113597 Email: [email protected] 2 School of Computer Science and Engineering,
South China University of Technology, Tel: 86-13512713940 Email: [email protected] 3 School of Computer Science and Engineering,
South China University of Technology, Tel: 86-13533054347 Email: [email protected]
ABSTRACT
The factors affecting the system model are the current network of transportation, the volume and direction of logistics, means of delivery, time limit and the cost. On the base of introducing these factors in the thesis, we target at minimizing the total daily expenses when construct the model and develop the algorithm. The inputs of this model are:
number of cities within the post logistics system, number of distribution centers that are going to be constructed, the distance matrix and volume matrix of each pair of cities, construction cost of each distribution center, depreciable life, the freightage per ton kilometer and the radius of time limit. And the outputs are: the minimized total daily expenses and the specific locations of distribution centers, maximum service radius of each distribution center, distribution range, shortest distribution routing and its distance for each distribution center. At last, an example is introduced to cast some light on the problem.
Key Words: Logistics Distribution Center; Site Selection; Algorithm
1 INTRODUCTION
In the course of logistics distribution, location of distribution centers and distribution routes are the main considerations for logistics enterprises to build up an effective and economical logistics distribution networks. Such an optimized physical network, would undoubtedly contribute to the competitiveness of the logistics enterprise.
2 MODEL
Distribution center location and site selection indicate the course to decide number of distribution centers, specific sites of each distribution center and the route between every two of them within an economic region. The factors affecting the model of distribution center location are the current network of transportation, the volume and direction of logistics, means of delivery, time limit and the cost.
Among all the factors mentioned above, the current network of transportation is the most basic one.
The development of the transportation network will directly determine whether the logistics service is punctual, effective, top-quality, whether the distribution assignment can be finished at a low cost, and whether the customers are satisfied with the service. As the transportation network begins to build up in
a place, the freight circulation will also start at that place. As the transportation network develops, the freight circulation will be aggrandized in range and scale. As the transportation network’s density increases, carrying capacity enhances, and facilities advances, the freight circulation scale will also becomes larger and larger.
The volume and direction of logistics flow also put significant influence on the layout of the distribution network. The volume and direction of logistics will confine not only the distribution center’s construction scale, functional genre, site selection, inventory control, and utilization rate of the facilities, but also the formation of the distribution network, adopted distribution pattern, mode of shipping, and so on. With detail analyzing on the features of the volume and direction of logistics, mastering the activities pattern, we can work out a rational site selection strategy, optimize the logistics distribution network, coordinate and utilize the shipping capacity.
Motor vehicle transportation, as the main means of delivery, is better than other means of delivery in that it can provide a door to door service. By using motor vehicles, where there is a road, there might be a transportation route. Time limit is also another consideration, since under different time limits, customers will choose different means of transportation and logistics enterprises will choose different layout and means of delivery. Costs also carry some weight in the course of distribution location, such as the construction fee of a distribution center, freightage, circulation and processing cost, shifting charge, storage charge and administration cost. Yet, due to the complexity it might be for including all the above costs, we only select the most important two factors into our model, that is, construction cost of a distribution center and freightage per ton kilometer. Minimizing these two charges counted by days will be the ultimate goal of the distribution center location model.
Consequently, we should take everything into account in the process of planning physical network of logistics distribution, including the layout of the transportation network, site selection for every distribution center, means of delivery and transportation route optimization. With the total freightage and the construction fee of distribution centers as the major cost during the course of logistics distribution, the distribution center location model is constructed as follows
Goal: minimizing the daily freightage and distribution center construction cost
Hypothesis:
⑴ Number of producing areas: N
Output per producing area: Ai, i=1,2,…,N Number of consuming areas: M
Demand per consuming area: BBj, j=1,2,…,M
Supply from producing area i to consuming area j: Aij, j
n
1 i
ij i
m
1 j
ij
A ; A B
A = ∑ =
∑
= =(If the demand is supplied locally, which means do not need to transport, so we just set all Aii as 0.)
⑵ Number of candidate sites: N+M
The construction cost of the point j: Zj, j=1,2,…,N+M Depreciate life:
y
0 years⑶ Freightage is in direct ratio with distance and volume of transport.
⑷ Distribution center commutes with the producing area and consuming areas with radiation network, and commutes with other distribution centers using mesh network.
Visualized site selection model is depicted as follows in Diagram 1-1: freight was produced by producing area, then through the distribution center, finally arrived at the consuming area.
In some special situations, producing area can also be the consuming area and vice versa. The general situation is depicted as follows in Diagram 1-2:
Diagram 1-1: Distribution center model (1) Diagram 1-2: Distribution center model (2)
The model is based on n (n=N+M) sites, which can be producing area, consuming area, or even both.
3 CORRESPONDING ALGORITHM (1) Start.
(2) Input: n, m. (n=number of sites, m=number of logistics distribution centers).
(3) Enter the initial distance matrix and introduce Floyd’s all pairs shortest path algorithm to evaluate the shortest distance matrix which is denoted by
( ) dij n×n, volume matrix denoted by ( ) fij n×n,
construction cost denoted by Zj, j=1,2…,n, the freightage per ton kilometer denoted by P (yuan), depreciable life denoted by
y
0(year), the radius of time limit denoted byT
0(kilometer).(4) Divide the n sites into m sets according to the shortest distance between that site and logistics distribution center.
(5) Given the first solution h=1 of m logistics distribution center.
(6) If
h > C
mn go to (10), else go to (7).(7) If the radius of time limit
≤ T
0 go to (8); elseS
h= 1000000000 0
,h=h+1, go to (6).(8) Evaluate the freight volume: for a site, say
V
K, there are:①
∑
.=
=
n1 j
kj
K
f
W
②
( ) ∑ [ ( ) ( ) ] .
=
⋅ +
+
⋅
=
nj
kj j
j k K
k
hk
d k h W d h h d h j f
Y
1
, ,
,
(9) Calculate the total cost for the h logistics distribution center
0,
1 1
365
/ y
Z P Y S
m
k h n
k hk
h =
∑
+∑
k ×=
=
1 h
h= + , and then go to (6).
(10) Calculate the lowest cost of m logistics distribution centers
( )
hC h
m 1
min S
S
m≤ n
=
≤(11) Output
① Output the positions chosen for the m logistics distribution centers and the lowest cost (Yuan).
S
m② Output the radius of time limit of each of the m logistics distribution centers (kilometers).
③ Output the distribution range of each of the m logistics distribution centers.
④ Output the shortest route and distance of logistics distribution for each of the m logistics distribution centers.
(12) End of algorithm.
Explanation:
(1) Let the freightage per ton kilometer be P=1, and the construction cost be Zj=0, j=1, 2… n, then in Output (1), the lowest cost would be the lowest freight volume (ton kilometer).
= means that there are no such solution which satisfy the time limit.
S
mS
m1000000000 0
(2) The initial matrix of n point is symmetric in general. However, even the asymmetric one was inputted, the program will still run on it without being affected.
(3) If certain points are not wanted to become the logistic distribution center, a very large construction cost for them should be inputted, say 10 billion, in that case, they never appear to be chosen for any logistic distribution center.
4 AN EXAMPLE
Consuming Area Producing
Area
1 2 3 4 5 6 7 8 9 10 11
1 0 65 65 54 45 35 35 32 12 35 74
2 35 0 90 36 75 65 54 35 35 69 65
3 54 65 0 35 65 98 55 65 62 87 23
4 55 45 35 0 25 40 68 98 42 55 65
5 68 58 26 68 0 53 90 51 15 4 84
6 90 63 0 56 35 0 43 32 35 56 15
7 100 51 65 32 62 51 0 63 62 15 36
8 35 32 0 35 42 32 63 0 15 25 65
9 69 63 36 65 35 63 45 61 0 35 25
10 87 85 81 98 95 85 13 45 65 0 64
11 55 61 91 40 60 21 15 31 68 12 0
Table 1 average volume matrix per day Unit: ton The distances (kilometers) between the above 11 areas are depicted in Diagram 1-2 as the underlined numbers. Average volume matrix per day is shown above in table 1 (ton). Assume that the freightage per ton kilometer is 0.35 yuan, construction cost for each logistics distribution center to be 300 million, depreciation life is 20 years, and radius of time limit is 150 kilometers. The best distribution network is wanted.
Solution: After inputting the data using Excel, we can get the result that building a logistics distribution center in area 6 will make the average freightage and construction cost per day the smallest, which is 365499 yuan (notice that it will become 402187 yuan constructing two distribution center, and 430643 yuan for three, and so on). The output solution for constructing two logistics distribution centers is as follows:
The lowest cost for two logistics distribution centers: 402197 yuan Areas selected are: area 5 and 6
Distribution Range for the distribution center in area 5: area 1, 2, 5, and 8 Its radius of time limit is 76 kilometers.
Distribution Range for the distribution center in area 6: area 3, 4, 6, 7, 9, 10, 11 Its radius of time limit is 132 kilometers.
The distributing routes and distances (kilometers) for each logistics distribution center are as follows:
Area 5 —— Area 1 The distance is: 58 Area 5 —— Area 2 The distance is: 76 Area 5 —— Area 8 The distance is: 48 Area 6 —— Area 3 The distance is: 80 Area 6 —— Area 7 —— Area 4 The distance is: 132 Area 6 —— Area 7 The distance is: 76 Area 6 —— Area 9 The distance is: 78 Area 6 —— Area 10 The distance is: 55 Area 6 —— Area 10 —— Area11 The distance is: 125
5 CONCLUSION
In order to employ the model to minimize the daily total freightage and distribution center construction cost of the physical network of logistics distribution centers, one only need to get the distances between each pair of cities, investigate the volume and direction of logistics flow and estimate the construction cost, then, the computer will work the solution out at a fast speed (about 2 seconds needed in the above example). The outputted site selection strategy satisfies the qualitative conditions, and the outputted route can be applied in the course of distribution. In reality, when circuit network is employed rather than radiation network, the real cost will be even lower than the output of the model.
6 REFERENCES
[1] WANG Yan, JIANG Xiaomei, 2004, Global Planning for Distribution Center Location, China Machine Press, Beijing
[2] LI Jun, GUO Huihuang, 2001, The theory and methodology of optimization and scheduling for vehicles in logistics distribution, China Logistics Publishing House, Beijing
