In this section, the proposed waste collection approaches are evaluated to investigate how well the proposed solutions align with the optimization model results. The first subsection compares the two heuristics and the optimization model results for some small-scale scenarios. However, due to the inefficiency of the ILP solver under large maps, the second subsection shows more complex scenarios using larger maps and the comparison of the proposed heuristic approaches.
6.2.2. Feasibility Study of Heuristics
In order to compare the effectiveness and efficiency of the heuristic methods, several scenarios were applied to the simulation environment. In the simulation environment, the arrival process of the waste was formulated based on the Poisson distribution. Under multiple runs, distinct instances of the bins were obtained. For each scenario, the results were calculated as the average of five different runs.
For the comparisons, two maps, which included 20 and 40 bins, were used. The map containing 40 bins was two-times larger than the 20-bin map. The map with 40 bins is presented in Figure 7
. On the map, the central truck station is where the trucks start their routes, and the disposal area is the point where trucks visit upon reaching their capacity upper limit. Each vertical segment of the grid is 200 m long, and each horizontal segment is 300 m long. Bins are marked with red rectangles on the map. In each scenario, two trucks were employed. Three different arrival rates, which were 1 kg per 5 min, 3 kg per 5 min and 5 kg per 5 min, were used. For these settings, increasing the bin capacity from 20 kg to 30 kg and increasing the truck capacity from 400 kg to 600 kg were observed separately. Each scenario was solved by the CVF, CUT and CULT methods. In order to compare these three methods, cost, delay per route and the number of trips values were used. Total cost was obtained for 12 h. Delay per route was obtained by dividing the total time that trucks needed to visit the bins by the route count for 12 h. Instead of total delay, the delay per route metric was employed.
In the first two figures, Figure 8
and Figure 9
, cost values are observed under the arrival rate of 3 kg per 5 min. In the first figure, Figure 8
, the change is observed for bin capacity being increased from 20 kg to 30 kg. In Figure 9
, the impact of truck capacity being increased from 400 kg to 600 kg is depicted. The increase in the bin count improved CUT when compared to CULT and CVF. For the 20-bin case, both methods similarly increased their performance in comparison to CVF in terms of cost. Under the 40-bin case, CUT and CULT performed better than CVF and provided a more efficient solution in terms of cost. However, increasing bin capacity did not introduce further improvements to the performance CUT and CULT in comparison to CVF. CUT provided a 14% better solution than CVF, whereas CULT provided a 0.07% better solution than CVF. When the bin capacity increased, the difference between the threshold value that triggered the alarm and the bin capacity increased, as well. This resulted in trucks having a longer period of time to visit that bin before the waste started to spread and caused the penalty. Collecting the waste from the bins that have not yet triggered alarms caused longer delay values for the ones that had alarmed to be collected. This increased the penalty for CUT and CULT. When there was not enough time between the alarm threshold being reached and bin capacity being met, the decision mechanism can affect the solution by decreasing the total cost, and vice versa.
When increasing truck capacity under the 20-bin case in Figure 9
, all three methods provided reasonable solutions. CVF performed the best, and for CUT and CULT, the difference from CVF increased in a negative way. Under the 40-bin case, CUT performed the best, and increasing truck capacity made CULT perform worse than CVF. Under the lower truck capacity, both CUT and CULT performed better than CVF. Moreover, the improvement of CVF by CUT decreased when the truck capacity was increased. Increasing truck capacity was expected to decrease the total cost in general depending on the trip count decrease and observed as expected for the map with 20 bins. However, when there were 40 bins on the map, increasing truck capacity caused higher cost values. This was due to the more non-optimal routes. The distance between bins could be higher in the wider area, and when the truck could not reach its capacity, it needed to wait for new triggers to be generated. This led to longer distances for the trucks to visit. Moreover, visiting further bins caused other bins on the route to spread waste around and cause penalties.
In the following two figures, Figure 10
and Figure 11
, the change in bin capacity and truck capacity is studied under the arrival rate of 5 kg per 5 min. When the arrival rate increased, the results became closer to each other. The bins that did not trigger, but on the way of a truck that was assigned a route were highly probable to have generated a trigger at the next arrival of waste. Therefore, adding these kinds of bins independently of
to the route increased the efficiency in terms of cost and delay. Moreover, selecting closer
resulted in similar cost and delay results. In the scenarios applied,
was set to
was set to
= 20, and
= 5 kg/5 min,
= 5; when a bin received an amount of waste, it reached the
directly. This removed the difference between CUT and CULT methods in this scenario. For the 5 kg/5 min arrival rate scenario, increasing bin capacity decreased the total cost for all methods, but did not have a significant impact on the comparison of the three algorithms.
However, increasing truck count caused a more significant impact on the cost values, which can be seen from Figure 11
. Increasing truck capacity increased the total cost in general for both maps containing 20 and 40 bins. However, in the 40-bin case, the improvement of the CUT and CULT according to CVF increased. They both performed better in each situation, but they increased the efficiency percentages that they provided when compared to CVF.
The change in the settings provided less difference in delay when compared to cost values. Delay per route values was effected by how well the route was optimized and how close the visiting bins were. When the delay was decreased, it could be said that the route was more optimized in those simulations with the provided configuration settings. The following four figures are presented to show the comparison in terms of delay per route.
In Figure 12
, the increasing bin capacity affect can be seen for the 20-bin and 40-bin maps with the arrival rate of 3 kg per 5 min. The impact was very small; however, in general, the CVF provided the shortest and CUT provided the longest delay per the route values. It is seen in Figure 8
that CUT was the most effective among the three algorithms on the 40-bin map. Therefore, CUT can be chosen as a solution when the time is not important, but the cost efficiency is more important. When time is the most important metric, CVF can be chosen. This was the result of CUT providing more optimal routes for each truck. Since trucks visited any other bin on the way to the alarmed bins, trucks did not have to go further first and come back again for the bins that they passed by before. Furthermore, when the bin count was increased, the delay per route values decreased, and this meant that the algorithms could work better and manage more optimal routes when the area was wider and there were more triggers coming from the bins in a time window. When there were less bins on the map, the possibility of getting many triggers at the same time decreased, and getting less triggers would cause the bins to be added on the route in an unoptimized manner.
In Figure 13
, the increase in truck capacity can be seen at the arrival rate of 3 kg per 5 min. It is observed that when the truck capacity was increased, the delay per route increased. This is precisely the expected result to be observed, since there was more capacity that one truck could collect, and it visited more bins in a route. Moreover, increasing truck capacity resulted in more effective performance of CVF and less effective performance of CULT, which was the opposite condition of the lower truck capacity scenario.
In the following figures, Figure 14
and Figure 15
, the scenarios where the arrival rate was 5 kg/5min are presented. The delay per route values was smaller in general when compared to Figure 12
and Figure 13
. Since the arrival rate was increased, the collected bins had higher amount of wastes stored in them. This resulted in a shorter time for the trucks reaching their capacity. In Figure 14
, the affect of bin capacity increase is shown. Increasing the truck capacity effected smaller map scenarios more than the wider area map. In the first section on the left, the biggest impact can be observed by the CULT, and while CULT and CVF were performing better, CUT increased the delay per route value for the smaller area case. For the wider area, all the methods performed better and provided smaller delay values when the bin capacity was increased.
In Figure 15
, the effect of truck capacity increase is shown. The increase in delay per route values in all cases where the truck capacity was increased independently of other configuration settings was the natural result. For this figure, where the arrival rate was 5 kg/5min and the bin capacity was 30 kg, increasing the truck capacity from 200 kg to 400 kg did not change the order of the effectiveness of the three algorithms. CVF provided the smallest delay per route values, and CUT provided the longest ones.
As the last comparison metric, the number of routes in each scenario are shown in the following four figures.
In Figure 16
and Figure 17
, the arrival rate is 3 kg/5 min, and the simulation is observed for 20 bins and 40 bins maps. First, in Figure 16
, the increase in bin capacity is presented. The highest route number was always in CVF, and the lowest was in CUT. However, the difference between the methods is more visible and observable when the wider map was used. Route count being small meant that each trip took a longer time for that method, and a trip took longer when the bins either had a small amount of waste inside in each collection or the routes that the trucks were assigned to were not that optimal. The increase in the bin capacity increased the number of routes in both cases when there were 20 or 40 bins. The reason behind this behaviour was that the thresholds were set according to the bin capacities (
’s), which is
/2. Once an alarm has been triggered, the bin had a greater amount of waste to be collected. This consumed the truck capacity more quickly and resulted in higher values of route counts.
In Figure 17
, the increase in the truck capacity is presented. In this figure, like the previous one, CUT had the smallest number of routes, and CVF had the most number of routes. Again, the difference among them is more visible when the map has 40 bins. For both maps, when the truck capacity increased, the route number decreased. With the increased truck capacity, one truck could hold more waste in a single trip, which precisely resulted in the decrease in the number of routes.
For the following two figures, Figure 18
and Figure 19
, the arrival rate is set to 5 kg/5 min, and the maps with 20 and 40 bins are observed under increasing bin capacity or truck capacity. First, in Figure 18
, the increase in the bin capacity is observed.
In Figure 19
, the increase in the truck capacity is tested under the 5 kg/5 min arrival rate setting. The increase in the truck capacity decreased the count of routes as explained before. For these scenarios, again, the highest number of routes was observed by applying CVF, whereas the lowest number of routes were observed under CUT and CULT.
In the following Table 3
, total cost, delay per route and number of route counts are presented when
= 1 kg/5 min. The applied scenarios are the same as the previous cases. The increase in
were tested on two maps containing 20 and 40 bins.
In Table 3
, the first part presents the results on the map with 20 bins. When
were lower, the CVF results were more efficient in terms of cost. However, increasing these parameters separately resulted in more efficient solutions when compared to CULT. Since
was small, there was enough time for the trucks to visit bins that had alarmed and collect the ones on their way according to the decision mechanism in CULT. Delay per route values were similar for CVF and CULT in most cases. As the truck capacity increased, the difference between them started to become wider. In addition, CUT yielded longer durations for a single trip in each scenario on the smaller map. The number of routes was quite similar on the smaller map. Comparing the methods for this particular performance metric on the wider map was more worthwhile.
When the simulation scenarios were tested on the map with 40 bins, CVF demonstrated the most competent performance, whereas CUT resulted in the worst solutions in terms of cost. Increasing and also increased the gap between CVF and its counterparts. Delay per route under CVF and that under CULT on the wider map were quite similar. CUT exhibited a longer delay time per route, which meant adding new bins on-the-fly ended up increasing the delay by collecting waste from bins that did not have a significant amount of waste accumulated. Delay per route may not significantly change when the bin capacity was increased. Thus, trip durations mostly depended on the truck capacity. Similar to the delay performance, the number of routes that trucks had in a single simulation run was smaller under CUT. Since trucks travelled longer in a single trip and the total delay did not change significantly, the number of routes was smaller under CUT. Moreover, the route numbers almost never changed under CVF and CULT. Besides, increasing the truck capacity had a direct impact on the route count, but not the bin capacity.
In order to present our approaches’ suitability to real-life scenarios, the test scenarios were modified to show results with larger maps that contained more bins. Three additional larger maps were used: (1) the map given in Figure 7
for the 40-bin scenario was modified, and 40 more bins (total of 80 bins) were deployed to the region; (2) the map given in Figure 7
was duplicated in terms of the area and the number of bins; moreover, a third map of 160 bins was constructed. All three approaches (i.e., CVF, CUT and CULT) were tested using these three maps with the following parameter settings: bin capacity (
= 30 kg), truck capacity (C
= 600 kg), waste arrival rate (
= 3 kg/5min) and the number of trucks (N
= 2). These scenarios were compared based on their total cost, delay per route and the number of routes. The results are presented in Figure 20
, Figure 21
and Figure 22
shows the total cost obtained under these three scenarios. CUT outperformed its counterparts, CULT and CVF. As mentioned earlier in this section, under the moderate arrival rate, as the map became larger, CUT tended to provide more cost-efficient solutions. As expected, increasing map size and the number of bins increased the total cost for all methods. Definitely, the higher the number of bins, the higher the number of triggers during a certain period of time. On the other hand, by considering more triggers when making decisions, the routes could be constructed more efficiently (leading to consecutively visited bins becoming closer). Hence, better cost values were achieved, and as we increased the area, we did not observe a remarkable increase in the total cost values obtained.
and Figure 22
present the delay per route and the number of routes, respectively, calculated for the same three scenarios. In terms of total delay, CUT provided the most efficient results for all three scenarios with two trucks. However, the number of routes was the lowest for CUT and highest for CVF. Moreover, total delay did not change significantly as observed for the previous cases. Hence, CVF resulted in the smallest delay per route. Previously, it was observed that the delay per route decreased as the area became larger due to the increase in the route counts. The results of 80-bin and 160-bin scenarios also supported this. As the number of routes increased, shorter routes were established since collected bins became closer to each other and the trucks reached their capacity limits quickly.
As for the complexities of these three methods, the Big O notation is applied. All three methods use Dijkstra’s algorithm to generate the shortest path from the given weighted adjacency matrix. The weights are the distances between the nodes, which represent the bins. Dijkstra’s algorithm calculates the shortest distance, but does not calculate the path information. It is modified for the simulation environment to show the shortest path between any given two nodes for the methods CUT and CULT. At the beginning of the simulation, Dijkstra’s algorithm runs and constructs a matrix representing the distances between any two points. For CUT and CULT, an additional matrix representing the path information between any two nodes is constructed. Dijkstra’s algorithm is not covering to negative weighted edges, and Bellman–Ford algorithm can be used in that case and improve the constructed route for a truck in the simulation environment. The time complexity of Dijkstra’s algorithm is , where V denotes the number of vertices in a connected graph.
If binary heap representation were used, it might have improved the complexity and have reduced it to , E being the edge count and V being the number of vertices. This complexity is for constructing the matrices at the beginning. Time complexity for adding a bin to the route is , M representing the bin count. Time complexity for checking the bins on the shortest path in CUT and CULT is . In summary, all three methods have the time complexity of , where V denotes the number of vertices in a connected graph.
According to the results, when the bin capacity and bin threshold’s (, ) had a high difference compared to each other, in the small sized area with smaller arrival rates (), CVF gave a more efficient solution compared to CUT and CULT. This means there was enough time for the trucks to reach the bins that raised alarms before waste was spread on the ground. The penalty values for spread waste affected the total cost in terms of the spread waste amount. In this situation, collecting the waste from the bins that did not raise an alarm, even if they were on the way of the truck, increased the total cost and delay values since dumping a single bin had its own cost and time consumption. CUT gave the least efficient solution among them. CULT provided a better solution and became closer to the the solution of CVF if the truck capacity also increased. This was the result of a decrease in the number of trips to the disposal area. When truck capacity was higher, the trucks could pick up more bins in one trip. The additional bins that did not raise an alarm, but were picked up on the way did not occupy a significant amount of space in the truck when was higher. It is worth noting that the longest path was the ones between the disposal area and the central station (h), and also, the most time-consuming activity was emptying the waste at the disposal area and making the truck ready for the next trip (), so increasing the trip count increased the cost and delay significantly.
In larger maps, with higher and proportional , the proposed heuristic CULT provided more efficient solutions regarding cost and delay. However, when the constants became closer and the map became smaller, CVF and CUT tended to ensure better solutions. The increase in the had more impact on the results when the map and the were larger.