For our computational study, we use a real-world data set of over 12,000 orders from a Dutch transportation platform company. This company matches any submitted orders to the available load capacity of empty or partly empty trips of subscribed carriers. The data set contains locations and time windows for both pickup and delivery of each order, as well as order release times and load quantities.
To investigate possible cooperation gains (Sect. 5.1) and the impact of bundling (Sect. 5.3), we define 6 instances of 2000 orders each, and impose different assignments of orders to various numbers of carriers. To be able to compare central and local combinatorial auctions (Sect. 5.2), we define smaller instances consisting of 50–200 orders. Additionally, we use the data set provided by Gansterer and Hartl (2016) as a benchmark. In the study on strategic bidding (Sect. 5.4), we consider unassigned orders to remove any bias from unprofitable initial contracts.
Cooperation gains
For determining possible cooperation gains in large-scale problems, we generated 6 instances with the following properties. Each instance consists of 2000 orders with pickup and delivery locations (in and close to the Netherlands) and load quantities approximately as in the original data set. Original time windows have been kept, except for shifts of whole days, such that all orders fall within a time span of 10 days. Release times have been set to the start of the time span to avoid problems with initial assignments. Per instance, 1000 identical vehicles of capacity 13.6 (loading meters) are available during the complete time span, distributed over 50 randomly chosen depots (such that each depot accommodates 20 vehicles). All vehicles are assumed to have a constant speed of 72 km/h, and Euclidean distances between all locations are used. The open problem variant is used, i.e., vehicles do not have to return to their depots after the last service. The reservation price for an order equals 1.5 times the travel costs between the pickup and delivery location.
Per instance, 5 carrier configurations (10, 50, 100, 500, or 1000 carriers) and 2 assignment configurations (close assignment or random assignment) are considered. With 10 carriers, each carrier owns 100 vehicles, i.e., precisely 5 depots. With 50 carriers, each carrier has exactly 1 depot with 20 vehicles. With 100, 500, and 1000 carriers, each carrier owns 10 vehicles, 2 vehicles, or 1 vehicle, respectively, i.e., each depot contains the vehicles of 2, 10, or 20 carriers. Each order is assigned to a depot—the depot closest to its pickup location for the close assignment configuration and a randomly selected depot for random assignment—and then randomly to a carrier having vehicles in that depot. Hence, the theoretical optimum is dependent on the carrier and assignment configurations if cooperation is not considered, but not if cooperation is considered.
To obtain the cooperative solutions, we apply the MAS with and without bundling three times on all instance configurations. In the runs without bundling, a maximum of 30 reauctions per order is allowed. In the runs with bundling, single orders are reauctioned a maximum of 10 times. In addition, we select the three most promising bundles of size 2 for the order and the most promising bundle of size 3 for the order (see Sect. 4.2), and auction them a maximum of 5 times each. These parameters are selected in such a way that the total number of reauctions for each order is equal with and without bundling. Note, however, that some orders might be offered more than 30 times if they appear in bundles of other orders as well. In both cases, each carrier applies a small LNS improvement phase (100 iterations, at most 5 orders per iteration) only after an auction causes an insertion or deletion in one of its routes.
To obtain the solutions of the non-cooperative scenario, we use the following procedure for each carrier. Initially, the insertion heuristic is used to include all the tasks of the carrier into the routes of its vehicles, and afterwards an LNS improvement phase of 2500 iterations with a maximum of 100 orders per iteration is applied to improve this solution. Since we have to compute this non-cooperative solution only once for each carrier, we could use a much larger LNS improvement phase than the small LNS improvement phases that are iteratively performed after each auction in the cooperative scenario.
We show the average decrease in total travel costs for the cooperative scenarios compared to the non-cooperative scenario in Fig. 4. As expected, cooperation gains increase with the number of participating carriers, but remarkably can be as large as 77% for 1000 carriers with random assignment. Although the non-cooperative solutions with close assignment are expected to be much better than their random assignment equivalents, cooperation can also drastically reduce the travel costs for the larger instances with close assignment: we observe savings of 68% for 1000 carriers. Note that the cooperative scenarios with bundling result in higher gains than the cooperative scenarios without bundling. We will explore this in depth in Sect. 5.3. Furthermore, note that all of the 2000 orders have been accepted in all cases, except for the non-cooperative scenarios with 1000 or 500 carriers (for 1000 carriers, 2 orders on average have been rejected with random assignment and 10 orders on average with close assignment; for 500 carriers, only 2 orders on average have been rejected with close assignment). Hence, cooperation may even improve the service level.
In Fig. 5, we give an indication of profits for the platform and for the carrier collective as a fraction of the sum of all reservation prices (i.e., the total price the shippers have paid). Analogously to the gains in travel costs, the profits for both carriers and the platform increase if cooperation is applied, and slightly more with bundling than without bundling. Furthermore, the profit increases are larger when more carriers participate. Note that the exact values highly depend on the WGS and CGS parameters for larger numbers of carriers, as well as on the prices that shippers pay for transportation. Under the current settings, shippers have paid 1.5 times the travel costs from pickup to delivery locations of the orders. With random assignment among 1000 carriers, this does not compensate the high travel costs if cooperation is not allowed. With low gain shares for the carriers and shippers (\(\mathrm{WGS}=\mathrm{CGS}=0.1\); Fig. 5a), carriers even make no profit after exchange of tasks (although the platform does). With higher WGS and CGS values, carriers do make profit when collaborating (Fig. 5b, c).
Comparing central and local combinatorial auctions
Now we have seen that large cooperation gains could be obtained if we apply the MAS on large-scale instances, we naturally come to the question what the quality of the MAS itself is. Since there are no optimality guarantees, both the results for the non-cooperative and for the cooperative scenarios might differ from the optimal solutions, leaving some space for worse or even better possible cooperation gains. To get more grip on the quality of the MAS, we compare it with established methods, both on our own instances, and on a benchmark data set.
Company-based instances (50–200 orders)
First, we compare the MAS with the central combinatorial auction as proposed by Gansterer et al. (2020a, 2020b) (see Sect. 4.4) on instances of size varying from 50 to 200 orders. Larger problem sizes turn out to take too much time for the central combinatorial auction, unless the number of bundles would be reduced drastically. We consider 5 or 10 carriers per instance, each having their own depot. The number of vehicles equals 10% of the number of orders, and time windows are omitted, but all orders need to be done within 24 h. Other settings are equal to the settings of Sect. 5.1.
In Table 2, we show the increases in total profit by cooperation, both for the central combinatorial auction (CCA) and for the MAS with local combinatorial auctions, compared to a non-cooperative LNS solution. As expected, the CCA performs better on the smallest instances. The MAS, however, performs increasingly better when instance size increases. For the largest instances, the number of submitted orders and the total number of bundles generated wihtin the CCA needed already to be lowered to be able to solve the winner determination problem to optimality.
There is a notable difference between instances with random assignment and instances with close assignment. While the CCA finds comparable improvements for both assignments on the instances of size 100 and 200, the improvements for the MAS are much better on the instances with random assignment. An analysis of the profit values discloses that the cooperative solutions for random and close assignment instances are similar for the MAS, but different for the CCA. Hence, the CCA is much more dependent on the initial assignment than the MAS. Of course, this effect is dependent on the parameters used for the CCA, and in particular on the number of submitted orders per carrier. For the instance with 50 orders, where 5 carriers submit each at most 10 orders, the auctioneer has an almost complete view on the total set of orders, resulting in a larger improvement with random assignment.
In one case (200 orders, 5 carriers, close assignment), the MAS obtains a negative improvement. Although this appears counterintuitive, it is explainable since we did not use the non-cooperative LNS solution referred to in the table as starting point for the MAS; instead, we used the same fast LNS approximations as are used by carriers after an auction causes any change for them. These generally arrive at about 7% lower profits than the non-cooperative LNS solutions referred to in Table 2. Although the MAS compensates this in all other cases, it did not even obtain the non-cooperative solution under these specific settings. Hence, it largely depends on the parameters whether the MAS is competitive with the CCA, but in general, the MAS seems to be a reasonable alternative when the CCA suffers from scalability issues.
Table 2 Solution improvement due to cooperation (in terms of profit increase, relative to a non-cooperative LNS solution) for both the central combinatorial auction and the MAS with local combinatorial auctions on instances with 50–200 orders and 5–10 carriers Each row comprises the average over 10 instances Benchmark data set (30–45 orders)
Next, we apply our method on the static data set proposed by Gansterer and Hartl (2016). We benchmark against the best known solutions (BKSs) that have been found for those instances by any method, as described by Gansterer et al. (2020a, 2020b). All instances consist of 3 carriers with depots located at 200 distance units from the others. Each carrier initially has 10 (set Ox_10) or 15 (set Ox_15) orders for which the pickup and delivery locations are in a radius of 150 (set O1_xx), 200 (set O2_xx) or 300 (set O3_xx) distance units around its depot. Thus, the area of overlap is smallest for sets O1_10 and O1_15 and largest for sets O3_10 and O3_15.
Table 3 Average computation times for the MAS in seconds Table 4 Average maximum improvements in profit using the MAS with respect to the BKSs. The average results per instance set are calculated using the maximum profit value out of 25 runs of the MAS per instance Table 5 Average improvements in profit using the MAS with respect to the BKSs. The average results per instance set are calculated using the average profit value out of 25 runs of the MAS per instance We ran the MAS under standard settings (see Sect. 5.1) on those instances (except for the fact that no maximum number of orders is specified for an LNS iteration). Furthermore, we calculated the solutions where the number of allowed auctions was increased by a factor 10. For all settings, we conducted 25 runs of the algorithm. The average computation times (on an Intel Core i7-8665U CPU at 1.90 GHz; 8 cores) are given in Table 3.
For each instance, the best result out of 25 runs was used to compute the improvement I with respect to the BKS, given by
$$\begin{aligned} I = (P_{\mathrm{MAS}} - P_{\mathrm{BKS}}) / P_{\mathrm{BKS}} \times 100\%, \end{aligned}$$
(7)
where \(P_{\mathrm{MAS}}\) and \(P_{\mathrm{BKS}}\) denote the profit obtained by the MAS and the profit of the BKS, respectively. The average improvement per instance set is given in Table 4. For instance sets O2_10, O2_15, O3_10, and O3_15, our best solutions outperform the BKSs, with up to 6% on average for set O3_10. It should be noted, however, that the number of order exchanges might have been limited in the approaches to find the BKSs, while this was not the case with our MAS. For instance sets O1_10 and O1_15, our best solutions are slightly lower than the BKSs. We observe that allowing bundling generally results in better solutions, while allowing more auctions has a much lower impact. The detailed results provided in Tables 8, 9, and 10 in “Appendix” show that our MAS finds improvements of up to 15% on individual instances. Although we have used the best results out of 25 runs of the algorithm here, the average profits among the 25 runs are not much lower than the profits of the BKSs, as can be observed from Table 5. Thus, the MAS is competitive with the other approaches used to solve the benchmark data set, especially for the instances with large areas of customer overlap.
Bundling benefits
We expected that applying small bundling within the MAS could improve solutions, which is further supported by the results from Fig. 4. In the following, we consider both problems in which all tasks are initially assigned, as before, and problems in which part of the orders is initially unassigned, i.e., shippers connect to the platform to find a carrier.
First, we consider the same instances as in Sect. 5.1, but now we take the scenario without bundling as base case. In Fig. 6, we show how much of the travel costs can be avoided by offering bundles. We observe that gains again increase with increasing numbers of carriers, up to 7% for 1000 carriers with close assignment and even to 13% for 1000 carriers with random assignment.
Second, we consider a more dynamic problem set in which part of the orders is not initially assigned to carriers. Again, we create 6 instances of 2000 orders each, of which only 1000 are initially assigned to carriers. We use 3 carrier configurations, namely 125, 250, or 500 carriers per instance. Each carrier has a single depot, in which it has 1–3 vehicles available. Each of the initially assigned orders is associated with a random carrier from the 10% closest carriers with respect to the pickup location. One third of the carriers have limited availability time windows, the other two third are available during the complete time span. Original order release times have been kept, except for initially assigned orders. For these, the release times equal the corresponding carrier’s release time.
We run the MAS on these instances with different numbers of carriers and various reservation price factors, both with and without bundling. The results are summarized in Table 6. The decrease in travel costs using bundling is generally between 0 and 1%, and there is a small positive influence from bundling on the service level. There is, however, no consistent pattern for increasing numbers of carriers or increasing price factors.
While bundling clearly outperforms no bundling on the instances with assigned orders, it does not on the instances where part of the orders is unassigned. To explain the difference, we again consider an instance of Sect. 5.1, but remove all initial assignments. We run the MAS both with and without bundling, and define a non-cooperative scenario as well. The latter one uses in this case only 1 auction per order to get an initial assignment, followed by an LNS improvement phase by the winning carrier. In Table 7, we compare the results of these experiments to the results of the instance with initial assignment. The travel costs of the non-cooperative solution for the instance without initial assignment are generally much lower than the travel costs of the non-cooperative solution for the instances with random or close assignment. Furthermore, the number of vehicles used in the solutions for the instance without initial assignment is much lower—it is actually quite close to the final number of vehicles used in the cooperative scenarios. Hence, the average route length is larger (see Fig. 7).
This might explain the relative small difference between bundling and no bundling for the instances without initial assignment: first, the possible improvements are already lower than for instances with close or random assignment, and second, bundles of orders might be less easily accepted in longer routes, since these generally are more constrained. Note, however, that bundling still has a slight advantage on instances without initial assignment, not only in travel costs, but also in service level.
Table 6 Results for bundling on the partly assigned instance set where reservation prices are equal to the distance between pickup and delivery multiplied by a price factor Table 7 Average results (over 3 runs) on instance 1 in terms of travel costs, service level, and used vehicles for the three scenarios. (For instances without initial assignment, the non-cooperative scenario consists of only 1 auction per order, followed by an improvement phase by the winning carrier.) Strategic behaviour
To get insights into the possible cooperation gains for large collaborative vehicle routing problems and the impact of bundling within a MAS, we have assumed that (estimates of the) real marginal costs are always reported. In practice, however, carriers and shippers might bid strategically to improve their individual profits. This is, however, not straightforward, as shown by Gansterer and Hartl (2018a) for central combinatorial auctions. We analyze the possible benefits of strategic behaviour within the proposed MAS, and show with a computational example that strategic bidding might be complicated.
For carriers placing a bid to acquire a bundle B, we can reason as follows, where \(\text{ MC}^t_c(B)\) denotes the carrier’s marginal costs, \(b_0\) denotes the carrier’s bid, and g denotes the profit that a winning carrier makes, i.e., g is a fraction of \(\text{ CC}^t(B)-b_0\), dependent on the used profit distribution function.
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They will not bid a value \(b_0 < \text{ MC}^t_c(B)\) if g is expected to be relatively small, since the compensation \(b_0+g\) will not cover the extra costs \(\text{ MC}^t_c(B)\).
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They might place a bid \(b_0 < \text{ MC}^t_c(B)\) if g is expected to be relatively high. If \(b_0+g > \text{ MC}^t_c(B)\), lowering the bid is a good strategy to outbid another carrier with a bid between \(b_0\) and \(\text{ MC}^t_c(B)\).
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They might speculate on getting a high gain from reselling the bundle later on, or foresee good interaction effects with orders that will appear later on, and hence place a bid \(b_0 < \text{ MC}^t_c(B)\).
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They might bid a value \(b_0 > \text{ MC}^t_c(B)\) to get a higher compensation, but this comes at the risk of not winning the auction anymore.
For carriers or shippers mentioning the marginal costs or reservation prices for outsourcing orders, we make the following observations.
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They do not report a value above their true value, since they need to pay this value.
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They might report a lower value, but this comes with the risk that the lowest bid \(b_0\) is not lower than \(\text{ CC}^t(B)\), hindering the trade. Indeed, they might report lower values and slightly increase them in next auction rounds, but due to the dynamic environment, there is no guarantee on success.
To prevent any bias from unprofitable initial contracts, we use problem instances without initial assignment. Again, we have 2000 orders per instance, and assume 250 carriers with 1–3 vehicles each, of which one third have restricted availability time windows. Further instance characteristics are as described before.
First, we analyze whether carriers can benefit from placing false bids. We run the MAS with different percentages of carriers (10%, 20%, or 30%) placing bids with a value of 0.8, 0.85, 0.9, 0.95, or 1.05 times their true marginal cost estimation. We test three configurations for winner gain share and contracted gain share (\(\mathrm{WGS}=0.1\), \(\mathrm{CGS}=0\); \(\mathrm{WGS}=\mathrm{CGS}=0.1\); and \(\mathrm{WGS}=0\), \(\mathrm{CGS}=0.1\)). In the last configuration, winning carriers do not take any of the profit generated by a succesfull auction (they even lose some profit if their bid is lower than their real costs), but they might obtain a gain if they resell the order later on.
In Fig. 8, we give the average profit per carrier, both for the fairly bidding carriers and for the strategically bidding carriers. We observe that strategic bidding pays off for a bid fraction of 0.9 or 0.95 if \(\mathrm{WGS}=0.1\), but not for other bid fractions. The fairly bidding carriers are worse off if the strategic carriers bid lower than their true prices, even if the strategic carriers themselves also do not gain any extra profit. With \(\mathrm{WGS}=0\) and \(\mathrm{CGS}=0.1\), there is no incentive to bid another value than the true value. Note that the highest profits can be obtained if only low numbers of carriers bid strategically. Similarly, the losses that strategic carriers can obtain will be largest with low numbers of strategic carriers, i.e., if most other carriers just report their true costs. These losses can be already very large with slightly lower bid fractions. Hence, finding a beneficial bid fraction value could be a critical process. Note that with higher values for WGS, lower bid fraction values are expected to be beneficial. If the system assigns large shares of the gains to the winning carriers, cheating might appear too easy.
Next, we analyze how much shippers and carriers can benefit from communicating false (lower) reservation prices or current costs. In Fig. 9, we show average obtained profits per shipper and per carrier when 10–30% of the participants use reservation prices of 75–95% of their true values. Strategic shippers can obtain considerably high profits if they lower their communicated reservation prices. This can be explained by the large difference between reservation price and insertion costs for a carrier that already had planned a route in which the order fits quite well. The shipper then might easily outsource its order at a low price. Likewise, carriers can obtain extra profits by outsourcing orders for a lower price than their actual costs, but the differences are smaller. The drawback of using lower reservation prices, however, is that less orders will be served, as can be observed from Fig. 10.