5.4.1. Problem Description

The location of logistics distribution center is an important link in the logistics system. The location of distribution center determines the efficiency of the entire logistics network system and the utilization of resources. The location selection model of logistics distribution center is a nonlinear model with more complex constraints and non-smooth characteristics, which can be attributed to HP-hard problem. The problem of logistics distribution center location can be described as: *m* cargo distribution centers are searched in *n* demand points, so that the distance between *m* searched distribution centers and other *n* cargo demand points is the shortest. At the same time, the following constraint conditions must be met: the supply of goods in the distribution center can meet the requirements of the cargo demand points; the goods required for a cargo demand point can only be provided by one distribution center; and the cost of transporting the goods to the distribution center is not considered. According to the above assumptions, the mathematical model of the problem for logistics distribution center location can be described as:

$$\min(\cos t) = \sum\_{i=1}^{m} \sum\_{j=1}^{n} (n \text{ced}\_{j} \cdot \text{dist}\_{i,j} \cdot \mu\_{i,j}) \tag{18}$$

$$\text{s.t. } \sum\_{i=1}^{m} \mu\_{i,j} = 1, i \in M, j \in \mathcal{N} \tag{19}$$

$$h\_{i,j} \le h\_{j,\*} \, i \in \mathcal{M}\_\* \, j \in \mathcal{N} \tag{20}$$

$$\sum\_{t=1}^{m} h\_i \le p, i \in M \tag{21}$$

$$\mathcal{M}\_{\rangle} \in \{0, 1\}, i \in M \tag{22}$$

$$\{\mu\_{i,j} \in \{0, 1\}, i \in M, j \in N\}\tag{23}$$

$$M = \{j | j = 1, 2, \dots, m\} \text{ N} = \{j | j = 1, 2, \dots, n\} \tag{24}$$

where Equation (18) is the objective function, cost represents the transportation cost, *m* is the number of logistics distribution center, *n* determines the number of goods demand point, *nestj* is the demand quantity of demand point *j*, and *disti*,*<sup>j</sup>* indicates the distance between distribution center *i* and goods demand point *j*. When *ui*,*j* is equal to 1, the goods of demand point *j* are distributed by distribution point *i*. Equations (19)–(24) are the constraints. Equation (19) defines that a demand point of goods can only be distributed by a distribution center. Equation (20) indicates that each demand point of goods must have a distribution center to distribute goods. Equation (21) represents the number of goods demand points for a distribution center. Equations (22)–(24) are the relevant definitions.

#### 5.4.2. Analysis of Experimental Results

To verify the performance of the DMQL-CS algorithm in solving the problem of logistics distribution center location, 40 demand points were adopted. The geographical position coordinates and demands are shown in Table 14. To make a fair comparison, all experiments were carried out on a P4 Dual-core platform with a 1.75 GHz processor and 4 GB memory, running under the Windows 7.0 operating system. The algorithms were written by MATLAB R2017a. The maximum number of iterations, population size, and the times of running were set to 30,000, 15, and 30, respectively. The probability of foreign eggs being found was = 0.25.

**Table 14.** The geographical position coordinates and demands.


To further verify the efficiency of the DMQL-CS algorithm, the effectiveness of the proposed method was verified by comparing the standard cuckoo search algorithm (CS) [69], the improved cuckoo search algorithm (ICS) [101], a modified chaos-enhanced cuckoo search algorithm (CCS) [68], and the immune genetic algorithm (IGA) [64]. Figure 5 shows the average convergence curve and

optimal convergence curve of DMQL-CS algorithm for running 20, 30, and 50 times, respectively, in 40 cities and six distribution centers. The six optimal distribution center points and optimal routes found by DMQL-CS algorithm are also shown in Figure 6. Figure 7 shows the average convergence curve and optimal convergence curve of DMQL-CS algorithm running 20, 30, and 50 times, respectively, in 40 cities and 10 distribution centers. Table 15 shows distribution ranges for six distribution centers in 40 cities, and Table 16 shows distribution ranges for 10 distribution centers in 40 cities.

**Figure 6.** Convergence curves and optimal distribution centers scheme for the DMQL-CS algorithm in 6 distribution centers.

**Figure 7.** Convergence curves and optimal distribution centers scheme for the DMQL-CS algorithm in 10 distribution centers.

**Table 15.** The distribution scheme for six distribution centers in 40 cities.



**Table 16.** The distribution scheme for 10 distribution centers in 40 cities.

For the first set of experiments, the DMQL-CS algorithm was run 20, 30, and 50 times independently in 40 cities for six distribution centers. As shown in Figure 6, the average convergence curve can converge at 30 iterations. It indicates that the fitness value decreases rapidly for the logistics distribution center location method based on DMQL-CS algorithm at early stage of the algorithm. The optimal distribution cost and average distribution cost obtained by the DMQL-CS algorithm are 4.5013 × 10<sup>4</sup> and 4.8060 × 104, respectively, which indicates that DMQL-CS has high solution accuracy for six distribution centers and reduces the cost of logistics distribution. The optimal distribution center points found in Figure 3 are: 10, 21, 20, 22, 1, and 15.

For the second set of experiments, the DMQL-CS algorithm was run 20, 30, and 50 times independently for 40 cities and 10 distribution centers. As shown in Figure 7, the average convergence curve can converge at 20 iterations. The optimal distribution cost and average distribution cost obtained by the DMQL-CS algorithm are 2.9811 × 10<sup>4</sup> and 3.0157 × 104, respectively, which indicates that DMQL-CS has high solution accuracy not only for six distribution centers but also for 10 distribution centers. The 10 optimal distribution centers and distribution addressing schemes are shown in Table 16 and Figure 7. The optimal distribution center points are: 30, 23, 14, 18, 11, 28, 21, 1, 20, and 15.

Due to limited space, only three comparison algorithms (CS [69], CCS [68], and IGA [64]) are listed in this paper. IGA algorithm introduced crossover and variation strategy into immune algorithm, which improves performance of the immune algorithm. In this experiment, the convergence curves and optimal distribution scheme diagrams of 6 and 10 distribution centers in 40 cities are shown, respectively. Figures 8 and 9 show the convergence curves and optimal distribution centers scheme for the IGA algorithm for 6 and 10 distribution centers. Figures 10 and 11 show the convergence curves and optimal distribution centers scheme for the CS algorithm for 6 and 10 distribution centers. Figures 12 and 13 show the convergence curves and optimal distribution centers scheme for CCS algorithm for 6 and 10 distribution centers. The six optimal distribution centers and distribution addressing schemes for these algorithms are shown in Table 17, while the 10 optimal distribution centers and distribution addressing schemes are shown in Table 18.

**Figure 8.** Convergence curves and optimal distribution centers scheme for the IGA algorithm for six distribution centers.

**Figure 9.** Convergence curves and optimal distribution centers scheme for the IGA algorithm for 10 distribution centers.

**Figure 10.** Convergence curves and optimal distribution centers scheme for the CS algorithm for six distribution centers.

**Figure 11.** Convergence curves and optimal distribution centers scheme for the CS algorithm for 10 distribution centers.

**Figure 12.** Convergence curves and optimal distribution centers scheme for the CCS algorithm for six distribution centers.

**Figure 13.** Convergence curves and optimal distribution centers scheme for the CCS algorithm for 10 distribution centers.


**Table 17.** Comparisons between DMQL-CS and other algorithms for six distribution centers.

**Table 18.** Comparisons between DMQL-CS and other algorithms for 10 distribution centers.


For the third set of experiments, the CS algorithm was run 20, 30 and 50 times independently for the 40 city and six distribution center example. As shown in Figure 10, both the average convergence curve and the optimal convergence curve can converge at 80 iterations. The optimal distribution cost and average distribution cost obtained by the CS algorithm are 4.9629 × 10<sup>4</sup> and 6.1392 × 104, respectively. As shown in Figure 11, average convergence curve can converge at 100 iterations for 10 distribution centers. The optimal distribution cost and average distribution cost obtained by the CS algorithm are 3.2435 × 10<sup>4</sup> and 3.9502 × 104, which indicates that logistics distribution location strategy of CS algorithm is the worst in both the optimal convergence curve and the average convergence curve. Convergence curve of CCS algorithm can converge at 20 iterations in both the optimal convergence curve and the average convergence curve. CCS algorithm is much inferior to DMQL-CS algorithm in solving accuracy for 6 and 10 distribution centers. Although the IGA algorithm can converge, it has a lot of noise for the average convergence curve. The convergence e ffect of IGA is worse compared with CCS algorithm. The results of standard deviation indicate that DMQL-CS has a better robustness than the other algorithms. The optimal distribution centers and distribution addressing schemes are shown in Tables 17 and 18. According to Tables 17 and 18, the optimal distribution center points found by CS algorithm for 6 and 10 distribution centers are (3, 11, 22, 1, 15, 20) and (6, 8, 18, 11, 21, 28, 16, 1, 20, 15). The optimal distribution center points found by IGA algorithm for 6 and 10 distribution centers are (10, 22, 21, 2, 20, 17) and (30, 23, 14, 1, 2, 11, 25, 24, 15, 4). The optimal distribution center points found by CCS algorithm for 6 and 10 distribution centers are (23, 22, 21, 16, 15, 20) and (6, 10, 23, 14, 22, 25, 7, 16, 15, 20).

To further analyze the e ffectiveness of DMQL-CS algorithm, DMQL-CS was compared with four algorithms: CS [69], CCS [68], ICS [101], and IGA [64]. The comparison results with average fitness value (Mean), the best fitness value (Best), the worst fitness value (Worst), standard deviation (Std), and running time (Time) are shown in Table 19. It can be seen that the average distribution cost of DMQL-CS at six distribution centers is 4.8060 × 10<sup>4</sup> which is 13,332 lower than CS, and the average distribution cost in 10 distribution centers is 3.0157 × 104, which is 9345 lower than CS. Therefore, DMQL-CS is obviously superior to CS algorithm. Although ICS can provide far better final results for most of the cases, it takes more execution time because of the use of more expensive exploration operation during the initial phases. For 6 and 10 distribution centers, there is not much di fference

between ICS and DMQL-CS for average distribution cost, but the optimal distribution cost of DMQL-CS is significantly higher than that of ICS algorithm. Meanwhile, from the standard deviation and running time data, it can be known that DMQL-CS has better robustness. The IGA algorithm achieved the worst performance compared with other comparison algorithms, except for the CS algorithm. For the six distribution centers, the average value obtained by IGA algorithm is 5.3008 × 104, which is 4948 more than DMQL-CS. The average value obtained by IGA algorithm is 3.6460 × 104, which is 6330 more than DMQL-CS for 10 distribution centers. CCS algorithm obtains the third best performance for 6 and 10 distribution center, respectively. In summary, the results of DMQL-CS are better than the comparison algorithms in terms of optimal value, worst value, average value, or running time. The reason may be that the *Q*-learning step size strategy improves the precision of the algorithm. The crossover and mutation operator accelerate the convergence speed of the algorithm. Overall, the selection method of logistics distribution center based on cuckoo search algorithm with *Q*-Learning and genetic operation has better optimal value compared with the five other algorithms for both 6 and 10 distribution centers, which indicates that the selection strategy based on DMQL-CS has higher solution accuracy and wider range of optimization. Meanwhile, it can be seen in Table 19 that the running time of DMQL-CS is significantly lower than the four other algorithms, and the number of iterations is significantly reduced. In general, DMQL-CS algorithm can select the address of logistics distribution center more quickly and accurately compared with the comparison algorithm. Finally, we can say that our proposed algorithm interestingly outperforms other competitive algorithms in terms of convergence rate and robustness.


**Table 19.** Comparisons between DMQL-CS and other algorithms for 6 and 10 distribution centers in 40 cities.
