In this section, we evaluate the performance of proposed solution approaches by conducting a case study based on the Shanghai Jinshan industrial park network. Both the SAA method and the hybrid GA are coded in MATLAB_2014b, andthe SAA method is combined with CPLEX 12.6. All computational experiments are conducted on a personal computer with a 3.60 GHz processor and 8.00 GB RAM under the Windows 7 operating system. The total computational times of both proposed solution approaches are limited to 3600 s.
5.1. Data Generation
We conduct a case study based on the drone delivery information of Ele.me to evaluate the applicability of the proposed solution methods. Ele.me has developed a fleet of drones to delivery food on fixed routes in Shanghai Jinshan industrial park. Based on the transportation and the geographical location information of Shanghai Jinshan industrial park, a network is proposed. The network has a total of 20 customers, as shown in Figure 7
. There are 11 service routes, three types of drones, and 4 categories of parcels, as shown in Table 3
, Table 4
and Table 5
, respectively. Note that the total travel time on each route depends on the drone speed and the total route distance. The planning time period is set to be one week, i.e., 168 h (10,080 min). There are 4 service modules for drones with departure time intervals, i.e., 5, 10, 15 and 20 min with corresponding service frequencies calculated as
. The labor cost for a courier delivering a parcel on each leg of each route depends on the distance of the leg. Moreover, we assume the total number of scenarios to be 50. Under each scenario, the number of parcels of each category on each leg of each route per minute is uniformly generated from 1 to 3.
5.2. Computational Results
The computational experiments are conducted and reported in this part. For the two solution methods, after the drone fleet deployment is determined, the objective value is calculated as the sum of the drone leasing and operating costs and the expected labor cost (under all 50 scenarios). Note that the scenario set comprises 50 scenarios and that we evaluate the objective value based on the entire .
An illustrative example based on a small network is first studied to compare the proposed two solution method with the deterministic situation. Note that the deterministic situation is considered where the demand
is fixed and equal to its mean value. In the example, there are 6 customers and 3 drone routes,
. The computational results are reported in Table 6
denotes the number of drones deployed in total. Note that drones selected by the three methods are of type 3. We can obtain that the number obtained under the deterministic situation is 5003, which is 103 and 16 larger than those obtained by the SAA and the HGA. That is, considering the uncertain demand is realistic and cost-saving.
The performance of the proposed solution methods under different number of scenarios is tested, and the results are reported in Table 7
. Scenarios for the solution methods are randomly selected from the given 50 scenarios. The number of scenarios for the solution methods range from 3 to 50. From Table 7
, we observe that the SAA obtains a similar objective with the deterministic situation. Moreover, it can be obtained that the computational time of SAA increases dramatically with the number of scenarios and that SAA loses its power to solve the problem within 3600 s if there are 25 scenarios. That is because, when the number of scenarios increases, the number of decision variables and constraints increase rapidly. The average computational time of SAA is 1424.2 s, which is about 9 times larger than that of the hybrid GA. Besides, as the scale of the tested instance is not large, the drone deployment decisions obtained by SAA and the hybrid GA are very similar. The average objective value of the hybrid GA is
, which is about 0.98% larger than that of SAA. In summary, (i) for each test, the quality of solutions obtained by SAA and the hybrid GA is very similar, (ii) SAA is relatively time-consuming and the computational time of SAA increases rapidly with the number of scenarios, and (iii) the average computational time of the hybrid GA is quite smaller than that of SAA.
The impact of the total flight time of drones on each route is examined and reported in Table 8
. Since the flight time of a drone on each route depends on the distance and the average speed, the increase of average drone speed is set from 0, 5, 10, …, 30 km/h. Based on the analysis on the tradeoff between the solution quality and the computational time, the number of scenarios for SAA is set to be 10 and that for the hybrid GA is set to be 50. For SAA, 10 distinct scenarios are randomly generated from
. After the first-stage decisions, i.e., the drone fleet deployment and drone service module, are determined, the expected labor cost is calculated as the sample average of the labor costs under 50 scenarios. From Table 8
, it can be obtained that the number of drones deployed and the objective values decrease when the average speed increases. That may be because, when the average drone speed increases, the total flight time of a drone on each route decreases and the number of drones deployed on each route may decrease in ensuring the constraint of Equation (9). Therefore, the drone operating cost decreases accordingly. Besides, the average computational times of SAA and the hybrid GA are 467.7 and 312.7 seconds, respectively.
The impact of the volume and weight capacities of drones are tested and presented in Table 9
. We increase the volume and weight capacities by four combinations, i.e., (0.5, 5), (0.5, 10), (1, 5), and (1, 10). It can be observed that the number of drones deployed decreases when the volume and weight capacities increase. That may be because, when the capacities increases, the number of parcels can be delivered by a drone increases and thus the time interval may increase. Therefore, the number of drones deployed may decrease in ensuring the constraint of Equation (9) and the objective value may decrease accordingly, as shown in Table 9
The sensitivity of the value of time intervals
of each service module is then tested. We reduce the time interval of each service module by 1, 2, 3, and 4, respectively. It can be observed from Table 10
that, when the time interval values decrease, the number of drones deployed increases. That may be because, when the time intervals decrease, the number of drones deployed increases to guarantee the constraint of Equation (9), the number of parcels delivered by drones decreases, and thus the number of parcels handled by couriers decreases. With the decrease of the time intervals, the objective value first decreases and then increases. That may be because there is a tradeoff between the number of drones deployed and the labor cost; when the number of drones on each route is too large, the drone operating cost increases rapidly.
with the increase of the number of scenarios, the computational time of SAA increases dramatically;
given the same number of scenarios, the computational time of the hybrid GA is smaller than SAA with high solution quality;
with the increase of average drone speed and the total flight time, the number of drones deployed and the total cost decrease;
when the volume and weight capacities of drones increase, the number of drones deployed and the total cost decrease;
when the time intervals decrease, the number of drones deployed increases; and
the developed hybrid GA outperforms the SAA in terms of the computational time with high solution quality.
Therefore, for the stochastic drone fleet deployment and planning problem in multiple-type parcel delivery service, we recommend applying SAA when the number of scenarios is small and applying the hybrid GA method when the number of scenarios is large.