*5.2. Experiment Results*

A total of 60 benchmark datasets with homogeneous demands were used in our experiment. Each instance is replicated for 30 runs for both the sAIS algorithm and the GA method. The solution from the sweep method, which generates proper clusters for shipment grouping of vehicles, is then picked up as the initial solution for both the sAIS and the GA methods. The comparison of performance between the two methods is computed based on the average improvement rate (AIR), as shown in Equation (14). The experimentation results are given in Table 1.

$$\text{Average Improvement Rate} \left( \text{AR} \right) = \frac{\left| \text{AIS}\_{\text{Average Cost}} - \text{GA}\_{\text{Average Cost}} \right|}{\text{GA}\_{\text{Average Cost}}} \times 100\%. \tag{14}$$


**Table 1.** Performance comparisons between sAIS and GA.


As listed, the 60 benchmark problems were sorted into 20 subgroups with three instances in each, based on the node coordinates assigned. Instances within one group are given identical node coordinates, except for the position where the cross-docking facility is located. Each set of the 60 instances contains 100 nodes, comprising both pickup and delivery customers. Each node is associated with a known demand of 10 units, while each vehicle dispatched is constrained by a capacity limit of 100 units. Examples of optimized routing sequences are shown in the Appendix A.

The computational results showed that the solution quality of the sAIS method is superior to the GA, outperforming each with an average improvement rate and lowest improvement rate of 11.98% and 9.47%, respectively, on the basis of average solution quality. In addition, the sAIS optimization, which has the lowest cost, was able to discover new, better solutions than the GA did in all 60 benchmarks, even though the computation time of the sAIS algorithm method takes longer; however, it is still a reasonable amount of time. Moreover, the maximum rate of improvement on average solution quality is 30.03%, whereas the minimum rate is acquired at 1.28%, indicating that the performance of the sAIS method is robust and competitive with the GA method. Nevertheless, all problems' average improvement rate was better able to discover new solutions than the GA did.

Finally, a one-sided, two-sample *t*-test was conducted to verify the performance of the two methods. The hypothesis test is:

$$\mathbf{H}\_{0} \colon \mu\_{\rm GA} - \mu\_{\rm sAlS} = 0$$

$$\text{HA: } \mathfrak{h}\_{\text{GA}} - \mathfrak{h}\_{\text{sAS}} > 0$$

the results showed that the *t*-value = 2.09 and the *p*-value = 0.019. Because the *p*-value is less than the 0.05 significant level, we reject the null hypothesis and conclude that μGA is significantly greater than μsAIS. That is, the total cost, including transportation costs and operational costs, generated by the sAIS approach is smaller than the total cost generated by the GA in our experiments.

#### **6. Conclusions and Future Research**

In this research, a novel sAIS algorithm is proposed to approach the combinatorial optimal solution of the VRPCD. The primary objectives of this work include the integration of the operation of cross-docking and the optimal vehicle routing schedule into the design of supply chain optimization. A significant development lies in the synchronization between upstream suppliers and downstream retailers, where both sides of the supply chain are simultaneously considered to collaborate on the physical flow of goods in each inbound and outbound process. Manufacturers can reduce logistics costs by building IoT monitoring technology into smart logistics IT systems to track its location in real-time, reduce inventory levels in warehouses, monitor environmental conditions, and optimize the routing sequence for trucks for smart city infrastructure.

The computational results show that the sAIS model is effective for solving the VRPCD. The effectiveness of the method comes from the two-phase mechanism. In the initial route generation phase, the initial solution was generated by the sweep method before being input into the route's optimization phase with the AIS algorithm. The combination of the two-phase approach ensures that the sAIS method yields quality solutions.

A total of 60 benchmark problems were deployed to investigate the applicability of the proposed sAIS method. The experimental results showed that the sAIS method was able to produce significant improvements over the GA, surpassing each testing problem. Additionally, the sAIS method was able to discover better solutions than the GA method for all 60 benchmark problems, and the sAIS's search time is faster than the GA. It found sAIS to be a useful methodology. As artificial intelligence research rapidly increases, more simulations from human body systems could be used to improve the solution quality of the AIS optimization, and the AIS optimization could be applied to other research problems in the operations research field, such as multiple-criteria decision-making problems or supplier selection problems, etc.

**Author Contributions:** Conceptualization, S.-C.L. and Y.-L.C.; methodology, S.-C.L. and Y.-L.C.; validation, S.-C.L. and Y.-L.C.; formal analysis, S.-C.L. and Y.-L.C.; experimental, S.-C.L. and Y.-L.C.; writing—original draft preparation, S.-C.L. and Y.-L.C.; writing—review and editing, S.-C.L. and Y.-L.C.; visualization, S.-C.L. and Y.-L.C.; supervision, S.-C.L. and Y.-L.C.; funding acquisition, S.-C.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

Examples of optimized routing sequences are shown in the appendix.

**Figure A1.** Optimized pickup routes from sAIS for instance 53P1.

**Figure A2.** Optimized delivery routes from sAIS for instance 53P1.

**Figure A3.** Optimized pickup and delivery routes from sAIS for instance 53P1.
