This paper deals with a reallife warehouse location problem, which is an automobile spare part warehouse location problem. Since the automobile spare part warehouse location problem is a very complex problem, particle swarm optimization is used and some improved strategies are proposed to improve the performance of this algorithm. At last, the computational results of the benchmark problems about warehouse location problems are used to examine the effectiveness of particle swarm optimization. Then the results of the reallife automobile spare part warehouse location problem also indicate that the improved particle swarm optimization is a feasible method to solve the warehouse location problem.
The profitability of automobile spare part warehouse location for automobile spare part factories is important for several reasons. The last decade has witnessed greatly growth in the demand of automobile and commensurate with this growth is the large demand of automobile spare parts. The cost of the management and stock has been an excess burden for automobile spare part factories. Furthermore, the fixed cost of the vehicles involved in automobile spare part delivery is high enough. Automobile spare part warehouses are a part of an overall effort to gain place and time utility (a decision aid in warehouse site selection). It has some obvious advantages. For example, a warehouse can hold stocks to match the imbalance between supply and demand. And the travelling cost can be decreased by collecting from multiple sources into a single vehicle to the final destination. Therefore, these increasing pressures and the advantages of warehouse have caused these factories to look for an automobile spare part warehouse to minimize their management cost.
There have been many literatures on the location of warehouse. Michel and van Hentenryck [
From the literatures, it can be attained that warehouse location problem has been recognized by academics and practitioners. Most of them paid attentions to the fixed cost of warehouse or transportation cost from warehouse to customers. However, in the reallife automobile spare part warehouse problem, the transportation cost from the factories to warehouse should also be considered. However, only few literature works attempted to incorporate routing from the suppliers to the warehouse and the warehouse to customers in location analysis. This paper attempted to solve a reallife automobile spare part warehouse location problem, in which there are some automobile spare part factories and their customers.
Therefore, the objective of the warehouse location problem is to minimize the sum of the transportation cost from the factories to the warehouse and from the warehouse to the customers and the fixed cost of warehouse. Based on the objective, it can be attained that the automobile spare part warehouse location problem is a generalization of wellknown and difficult location problems. It is therefore a large and complex problem. Many literature works suggested that heuristic algorithm was often a first choice to solve this kind of complicated problems [
The remainder of the paper is organized as follows. Section
The automobile spare part warehouse location problem can be described as follows. There are a set of
An example of the automobile spare part warehouse location problem.
From Figure
PSO is a populationbased search method, proposed by Kennedy and Eberhart [
In PSO, the position of each particle is a solution of the problem. The fitness of each particle is based on the objective function of the problem. The position (
According to (
From the formula (
The functions of
Crossover operation is a reproduction operation in GA, which is used to exchange genetic information between two parent chromosomes at a predefined probability [
The positions and speed vectors of the children particles are shown as follows:
Crossover operation can help the PSO algorithm to search in a larger space; however, in the late iteration, frequent crossover operation will make the algorithm run with more computing time and hard to be converged. Therefore, an adaptive method to adjust the crossover probability is used in this paper. The adaptive method can be found as follow:
This paper attempts to use an improved PSO algorithm to solve the automobile spare part warehouse location problem. To examine the feasibility of the improved PSO algorithm (IPSO), some benchmarks for uncapacitated warehouse location from the standard OR library are selected in this paper. Then, the IPSO is used to solve a reallife automobile spare part warehouse location problem. The following will describe the two examples, respectively.
In order to examine the performance of the proposed IPSO in this paper, the instances from the standard OR library [
The information of the test problems.
Bench  Size  Bench  Size  Bench  Size 

Cap71  16 × 50  MO1  100 × 100  MR1  500 × 500 
Cap72  16 × 50  MO2  100 × 100  MR2  500 × 500 
Cap73  16 × 50  MO3  100 × 100  MR3  500 × 500 
Cap74  16 × 50  MO4  100 × 100  MR4  500 × 500 
Cap101  25 × 50  MO5  100 × 100  MR5  500 × 500 
Cap102  25 × 50  MP1  200 × 200  MS1  1000 × 1000 
Cap103  25 × 50  MP2  200 × 200  MS2  1000 × 1000 
Cap104  25 × 50  MP3  200 × 200  MS3  1000 × 1000 
Cap131  50 × 50  MP4  200 × 200  MS4  1000 × 1000 
Cap132  50 × 50  MP5  200 × 200  MS5  1000 × 1000 
Cap133  50 × 50  MQ1  300 × 300  MT1  2000 × 2000 
Cap134  50 × 50  MQ2  300 × 300  MT2  2000 × 2000 
Cap a  100 × 1000  MQ3  300 × 300  MT3  2000 × 2000 
Cap b  100 × 1000  MQ4  300 × 300  MT4  2000 × 2000 
Cap c  100 × 1000  MQ5  300 × 300  MT5  2000 × 2000 
Computational results using IPSO algorithm and tabu search.
Bench  Tabu search  IPSO  Bench  Tabu search  IPSO 

Cap71  932 615.75  932 615.75  MP4  2633.56  2633.56 
Cap72  977 799.40  977 799.40  MP5  2290.16  2290.16 
Cap73  1 010 641.45  1 010 641.45  MQ1  3591.27  3591.27 
Cap74  1 034 976.97  1 034 976.98  MQ2  3543.66  3543.66 
Cap101  796 648.44  796 648.44  MQ3  3476.81  3476.82 
Cap102  854 704.20  854 704.20  MQ4  3742.47  3742.47 
Cap103  893 782.11  893 782.11  MQ5  3751.33  3751.33 
Cap104  928 941.75  928 941.75  MR1  2349.86  2349.86 
Cap131  793 439.56  793 439.56  MR2  2344.76  2344.76 
Cap132  851 495.32  851 495.33  MR3  2183.24  2183.24 
Cap133  893 076.71  893 076.71  MR4  2433.11  2433.11 
Cap134  928 941.75  928 941.75  MR5  2344.35  2344.35 
Cap a  17 156 454.4  17 156 454.48  MS1  4378.63  4378.63 
Cap b  12 979 071.5  12 979 071.58  MS2  4658.35  4658.35 
Cap c  11 505 594.3  11 505 594.33  MS3  4659.16  4659.16 
MO1  1156.91  1156.91  MS4  4536.00  4536.01 
MO2  1227.67  1227.67  MS5  4888.91  4888.91 
MO3  1286.37  1286.37  MT1  9176.51  9176.51 
MO4  1177.88  1177.88  MT2  9618.85  9618.85 
MO5  1147.60  1147.61  MT3  8781.11  8781.11 
MP1  2460.10  2460.10  MT4  9225.49  9225.49 
MP2  2419.32  2419.32  MT5  9540.67  9540.67 
MP3  2498.15  2498.15 
From Table
The PSO algorithm is examined by the warehouse location problems from the standard OR library, which indicates that IPSO algorithm is suitable for solving the warehouse location problem. Then a reallife automobile spare part warehouse location problem needs to be solved by the PSO algorithm. In the reallife warehouse location problem, there is one automobile company that wants to large its business. Thus, the company wants to build some warehouses in a chosen area. Then the automobile spare parts will be sent from the company to the warehouses and from the warehouses to customers (e.g., repair station). There is one warehouse in the chosen area and three alternative points for selection. In the chosen area, there are four potential points for selection and ten customers need to be served. If the horizontal and vertical coordinates of the factory are assumed as (0,0), then, the information of the problem is shown in Tables
The information of warehouse.
Item 





Horizontal coordinate  23  88  72  45 
Vertical coordinate  74  46  90  16 
Land price  1.95  1.8  1.35  2.25 
The transportation cost from warehouses to customers and factories (Yuan/T).
Item 





1  45  37.5  28.5  30 
2  39  32.25  49.5  27 
3  38.25  40.5  30  34.5 
4  29.25  30.75  34.5  32.25 
5  46.5  40.5  38.25  31.5 
6  33.6  35.4  39.6  39 
7  42  24  52  24.5 
8  28  30  30  36 
9  35  16  32  35 
10  36  18  38  30 
 
Factory  33.6  300  39.6  240 
The information of customers.
Customers  Horizontal  Vertical  Demand 

1  25  80  3 
2  30  23  0.8 
3  70  60  0.75 
4  92  51  3 
5  42  64  1.05 
6  87  45  2.1 
7  39  7  1.5 
8  28  71  3 
9  55  20  4.5 
10  14  56  1.5 
Then, the improved PSO continues calculating the automobile spare par warehouse location problem ten times. However, in the reallife seafood product delivery routing problems, the distance between two points is based on the length of the routes. The results are shown in Figure
The optimized results of the automobile spare part warehouse location problem.
Item 





The served customers  (1,5,8,10)  (3,4,6)  ()  (2,7,9) 
The treatment capability  5.2  3.9  0.0  4.6 
Computing results of the improved PSO after running 10 times.
From Table
The automobile spare part warehouse location problem is the warehouse selection in potential depots in order to meet the demand of the customers and keep the automobile spare part factory competitiveness in a chosen area. Thus, the automobile spare part warehouse location problem is attempted to minimize the total cost which includes the transportation cost (from factories to warehouse and from warehouse to customers) and the fixed cost while meeting the constraints. Since the automobile spare part warehouse location problem is difficult to be solved. PSO is selected in this paper and some improved strategies are used to improve the performance of PSO. The computational results of some benchmark instances for uncapacitated warehouse location problem from the standard OR library suggest that the improved PSO is effective to solve the warehouse location problem. The results of the automobile spare part warehouse location problem can also indicate that the improved PSO is an effective method for the automobile spare part warehouse location problem.
The main contribution of this paper is to test the feasibility of PSO for the automobile spare part warehouse location problem. Our future research work will be on the models with stochastic demand, model and solve the model by other convenient methods, and possibly have added a capacity constraint on the multiple central warehouses.
This work was supported by the National Science Foundation for Postdoctoral Scientists of China 2013M530924 and the National Natural Science Foundation of China 51208079 and 11272075.