**4. Experimental Results**

To assess the performance of the proposed QDGWO algorithm, two groups of datasets are used for solving the KP01.

All experiments were conducted with Matlab 2016b, running on an Intel Core i7-4790 CPU @ 3.60 GHz, and Windows 7 Ultimate Edition.

In the first experiment described in [37], there were 50, 250, 500, 1000, 1500, 2000, 2500, and 3000 dimension sets of data by Equation (26) to test the performance of the QDGWO in high-dimension situations.

Given a set of *m* items, *W* = (*<sup>x</sup>*1, *x*2, *x*3,... , *xm*).

$$\begin{array}{l}w\_{i} = \text{rand\\_i}[1, \, 10] \\ p\_{i} = w\_{i} + 5 \\ \text{C} = \frac{1}{2} \sum\_{i=1}^{m} w\_{i} \end{array} \tag{26}$$

where *wi* is the weight of the *i*th item *xi*; *p*i is the value of *xi*; *C* is the weight capacity of the knapsack; and *m* is the number of items.

In the first experiment, *m* ranged from 50 to 3000, and the maximum number of iterations in all cases was set to 1000.

To verify the effectiveness and efficiency of the QDGWO, the results of the proposed algorithm were compared with three algorithms: QEA [37], AQDE [45], and QSE [48]. The parameters of algorithms used in the experiments are presented in Table 2, where the population size is 20. The best profits, the average profits, the worst profits, and the standard deviations of 30 independent runs are shown in Table 3 and Figures 4–7. The Wilcoxon signed-rank test [59] is performed for the results of the competing algorithms in Table 3 with a significance level *α* = 0.05, where +, −, and = indicate that this algorithm is superior, inferior, or equal to the QDGWO, respectively.

**Table 3.** Experimental results for 0-1 knapsack problems (Experiment 1).


**Figure 4.** Best profits for the 0-1 knapsack problems (250 items in Experiment 1).

**Figure 5.** Best profits for the 0-1 knapsack problems (500 items in Experiment 1).

**Figure 7.** Best profits for the 0-1 knapsack problems (3000 items in Experiment 1).

To illustrate the importance of the role of the crossover operation of the DE in exploring the global optimum, comparative tests between the QDGWO with and without the crossover operation were performed. Moreover, we compared the binomial crossover operator of the DE with the exponential crossover operator of the DE. The best profits, the average profits, the worst profits, and the standard deviations of 30 independent runs are presented in Table 4. The Wilcoxon signed-rank test [59] is performed for the results in Table 4 with a significance level *α* = 0.05, where +, −, and = indicate that this strategy is superior, inferior, or equal to the QDGWO with a binomial crossover of the DE, respectively.


**Table 4.** Experimental results of QDGWO algorithm without crossover of DE, with binomial crossover of DE, and with exponential crossover of DE.

In the second experiment described in [49], there were 50, 200, 500, 1000, 1500, and 2000 dimension sets of data by Equation (27) to test the performance of the QDGWO in high-dimension situations.

Given a set of *m* items, *W* = (*<sup>x</sup>*1, *x*2, *x*3,... , *xm*).

$$\begin{aligned} w\_i &= \text{rand\\_i}[1, 10] \\ p\_i &= w\_i + 5 \\ \text{C} &= \frac{3}{4} \sum\_{i=1}^{m} w\_i \end{aligned} \tag{27}$$

where *wi* is the weight of the *i*th item *xi*; *p*i is the value of *xi*; *C* is the weight capacity of the knapsack; and *m* is the number of items.

In the second experiment, *m* ranged from 50 to 2000, and the maximum number of iterations in all cases was set to 1000.

To present the performance of the proposed algorithm in the global optimization, we compared the QDGWO algorithm with the QIHSA [49] for knapsack problems. The optimization results of the success rate (*SR*%) and the best profit are shown in Table 5. The Wilcoxon signed-rank test [59] is performed for the results of the QIHSA in Table 5 with a significance level *α* = 0.05, where +, −, and = indicate that this algorithm is superior, inferior, or equal to the QDGWO, respectively.

The obtained results demonstrate the competitive performance of the proposed QDGWO algorithm. According to the results, the proposed algorithm is more efficient for the high-dimensional 0-1 knapsack problems, as shown in Table 4 and Figures 3–5. Compared with the QEA [25], AQDE [33], QSE [36], and QIHSA [37], the QDGWO was the most effective and efficient algorithm in the experiments. The advantages of the QDGWO became more obvious when the number of items was large, especially in high dimensional cases of the knapsack problems.


**Table 5.** Experimental results of QDGWO and QIHSA for 0-1 knapsack problems (Experiment 2).

The proposed algorithm obtains both rapid exploration and high exploitation in searching solutions. The QDGWO converges quickly to the global optimal solution. For example, the algorithm approaches the global optimum at about the 500th iteration in the case of 500 items (see Figure 4). However, the algorithm continues searching near the global optimal solution, i.e., the exploitation. To illustrate this, in the case of 500 items (see Figure 4), the QDGWO continues seeking further optimization after approaching the optimal solution and obtains better solutions in the exploitation until the end of the iterations.

Based on the results shown in Table 3, it can be concluded that the crossover operation plays a significant role in searching the solution space efficiently. However, the performance of the QDGWO is not very sensitive to which kind of crossover operator is used in the algorithm. From the experiment results, the binomial crossover operator of the DE yields slightly better optimal solutions than the exponential crossover operator of the DE in all cases. The results can be interpreted to show that quantum updating with the quantum rotation gate remains the most decisive and crucial operation in exploring the search space even if the crossover operation is required to improve the solutions.

Finally, compared with the other four methods, the experimental results show the advantages of the collaborative optimization with operations of adaptive mutation, crossover, and quantum rotation gate with the adaptive GWO in investigating the search space.
