Variable neighborhood search and tabu search for auction-based waste collection synchronization

Transportation Research Part B 133 (2020) 1–20

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Variable neighborhood search and tabu search for auction-based waste collection synchronization Saijun Shao a, Su Xiu Xu b,∗, George Q. Huang c a

Department of Transportation Economics and Logistics Management, College of Economics, Shenzhen University, Shenzhen, PR China Institute of Physical Internet, School of Intelligent Systems Science and Engineering, Jinan University (Zhuhai Campus), Zhuhai, China c HKU-ZIRI Lab for Physical Internet, Department of Industrial and Manufacturing Systems Engineering, The University of Hong Kong, Hong Kong b

a r t i c l e

i n f o

Article history: Received 3 April 2019 Revised 17 December 2019 Accepted 18 December 2019

Keywords: Commercial waste collection Synchronization requirements Less-than-truckload O-VCG auction Variable neighborhood tabu search

a b s t r a c t We study a less-than-truckload (LTL) transportation service procurement (TSP) in the waste collection industry, where shippers make small-volume requests and a single carrier optimizes the collection routes to realize maximum profitability. In our proposed auctionbased waste collection synchronization (A-WCS) mechanism, the carrier plays a role as the auctioneer to decide on who wins the bids and the corresponding payments while the shippers act as the bidders. This is attributed to the carrier-centric structure where a carrier has to consolidate multiple LTL shipping requests so as to provide cost-effective services. Furthermore, a single shipper (e.g., a construction site) may generate multiple types of waste and highly desires for a synchronous service, requiring multiple vehicles dedicated for different waste types to arrive almost simultaneously. The synchronization is beneficial to prevent repeated handling and reduce disruption caused to the minimum. The synchronization requirements are innovatively factored by incorporating time windows into each customer bid. The A-WCS mechanism can encourage truthful bidding from shippers and obtain fair payments for both sides, therefore can benefit the market in the long run. A-WCS problems are extremely difficult to solve due to the integration of two combinatorial problems. We address the A-WCS with a two-layered algorithm, where the variable neighborhood search (VNS) is purposed to search auction allocations in the upper layer and tabu search (TS) is conducted to generate efficient vehicle routes in the lower layer. The capability of our algorithm is demonstrated by an extensive computational study, comparing with a number of state-of-the-art exact and heuristic approaches. Finally, a number of managerial implications for the practitioners are obtained, with insights into the impacts caused by changes in market size, task complexity, customer distribution pattern, and auction frequency. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The annual total solid waste output reached approximately 17 billion tons globally and this figure was expected to reach 27 billion by 2050, making the waste collection and recycling a serious issue (Glushkov et al., 2019; Karak et al., 2012;

∗

Corresponding author. E-mail address: [email protected] (S.X. Xu).

https://doi.org/10.1016/j.trb.2019.12.004 0191-2615/© 2019 Elsevier Ltd. All rights reserved.

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Rabbani et al., 2018). However, transport-related costs account for as much as 80% of all the operational costs incurred from waste collection activities (Tavares et al., 2009). Therefore, there is a pressing need for effective approaches to improve efficiency in waste collection logistics. As a motivation to carry out this study, the logistics of commercial waste collection in Hong Kong demonstrates such characteristics as carrier-centric, less-than-truckload (LTL), and multi-commodity. (1) Carrier-centric: Relative to municipal waste, commercial waste shows more value for recycling or must be collected for the concerns over environment and safety. Furthermore, highly specialized vehicles and disposal facilities are required for such waste. Consequently, the number of qualified transporters remains at single-digit figures (Environmental Protection Department, 2017), whereas the size of concurrent construction sites alone increased to 1500 in Hong Kong by 2018 (Turner and Townsend, 2018). Distinct from many other transport service markets, the commercial waste collection logistics is frequently managed in a carrier-centric manner. (2) Less-than-truckload (LTL): Owing to the shortage of land supply in metropolitan cities like Hong Kong, there is a strong need for waste to be removed on a frequent basis. The extremely high density of population puts more strain on the prompt collection of hazards and pollutants. Therefore, the demands show such characteristics as high frequency and low volume. A truck normally visits a number of sites with LTL demands to make a full use of the capacity. (3) Multi-commodity: A commercial customer may generate multiple types of waste at the same time (e.g. inert materials, metal, glass and demolition waste could all be generated at a single construction site). Customers expect a synchronous service, thus requiring multiple vehicles (for different waste types) to arrive nearly simultaneously for waste collection, which is beneficial to prevent repeated handlings. Meanwhile, the disruption caused to common activities could also be reduced to the minimum. At present, it is a commonplace across various industries to contract logistics services out to third party carriers. Auction has been practically accepted as one of the most-used approaches in transport service procurement (TSP) (Lafkihi et al., 2019). Despite plenty of efforts made on auction-based TSP over the past decades, a majority of them concentrate on trading of full truckload (TL) services, where shipping requests are frequently known as lanes or O-D pairs (see Jothi Basu et al. (2015) for a review of auction-based TSP for TL services). However, in order to enhance the response speed to markets, there is a surging need for faster deliveries of smaller shipments. Both shippers and carriers are desired for an approach to decide a reasonable fee rate for LTL shipments. The auction-based waste collection synchronization (A-WCS) problem that we encounter is distinct from all the current studies in at least two respects. On the one hand, because of the carrier-centric structure, we propose a one-sided auction mechanism for LTL demands. The carrier decides on winning bids from a number of shippers, which is exactly the opposite way to the majority of one-sided auctions wherein the shipper acts as the auctioneer. In conventional TL auctions, a shipper requires a multitude of carriers with respective advantages on different lanes to satisfy the demands. In A-WCS, however, it is the carrier who integrates LTL requests from multiple shippers to make a full use of the vehicle capacity. In consideration of the carrier-centric pattern in waste collection logistics industry, a one-sided auction with carrier as the auctioneer and shippers as bidders is suggested. From the perspective of market mechanism design, it is fairer and more reasonable to decide the payment price based on the “marginal contribution” of each customer. On the other hand, synchronization requirements are factored into the auction-based LTL TSP for the first time, where all vehicles have to arrive within the same time window. Additionally, as the payment willingness is closely associated with the service level, a customer is allowed to make a number of bids, each with a different synchronization level (defined as the duration of time window) and a bidding value. Once a bid is selected by the auctioneer, all arrivals shall be made within the time window as declared in the winning bid. Therefore, the auctioneer (i.e., the carrier) needs to determine an allocation that could achieve a maximum profitability while ensuring synchronization for all winning bids. The neglection of auction-based LTL TSP in the literature may be due to a few reasons. One potential cause lies in the risk of LTL transport, for example, cargo damage, theft and other forms of loss. Nevertheless, the above risks are significantly mitigated in the transport of waste, particularly when one vehicle is dedicated for a single type of waste. Another possible reason is that, traditional requests for proposals mainly deal with long-term transportation demands in a period of several years. However, as demands are changing more rapidly, it is obviously important to consider online auctions for LTL demands. Recent years have also seen enterprises running such auction platforms for instant LTL demands (https://www.huolala.cn/). With the development of supply chain activities and supporting technologies, the auction-based TSP for LTL shipments exhibits both practical necessity and feasibility. The difficulty in solving the A-WCS stems from the combination of two combinatorial optimization problems: auction allocation in the upper layer and vehicle routing in the lower layer. To address the realistic sized instances, a variable neighborhood tabu search (VNTS) algorithm is proposed, where the variable neighborhood search (VNS) is conducted to search among different allocations while tabu search (TS) is embedded for construction of cost-effective routes under the specified allocations. We innovatively construct the neighborhood structures for VNS by making use of the auction allocation and varies the searching depth by simply making changes to different number of elements in the previous allocation. Besides, to further improve the performance of VNTS, internal connections are set up between the TS and VNS, with an adaptive threshold mechanism put in place to determine which initial solutions are sufficiently promising to be further improved by TS. Finally, the termination condition of TS incorporates the current searching depth of VNS in the upper layer. Numerical studies are extensively performed to assess the quality of the suggested VNTS. The test instances are generated based on the well-known Solomon’s instances, with varying problem sizes and different geographical distribution patterns. The effectiveness of VNTS is firstly validated by comparisons with CPLEX on several sets of small sized instances, for which CPLEX is able to provide (near) optimal solutions. Then, in respect of medium and larger instances, VNTS is compared

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against three other state-of-the-art heuristic algorithms. The results have demonstrated the quality and stability of VNTS in solving A-WCS instances. Finally, from the perspective of practitioners, we conduct research into the impacts of the number of customers and waste types, distribution patter, as well as the auction frequency. Numerous managerial implications are obtained accordingly. The contributions of this study are three-fold. Firstly, this is among the first to introduce the transportation service procurement (TSP) in commercial waste collection industry. Unlike most one-sided auctions considering shipper-centric structures, the commercial waste collection industry exhibits a carrier-centric pattern and thus our suggested mechanism is aimed at solving the winner determination problem from the carrier side. Shippers’ synchronization requirements are innovatively factored by incorporating time windows into their bids. The O-VCG mechanism can encourage truthful bidding from shippers and obtain fair payments for both sides, therefore can benefit the market in the long run. Secondly, an effective two-layered VNTS algorithm is devised for the extremely challenging A-WCS problems. Several refinement techniques including the adaptive threshold mechanism are also introduced. Thirdly, a series of computational studies have been performed to validate the quality of our proposed algorithm. A number of managerial implications for the practitioners are also obtained, with insights into the impacts caused changes in market size, task complexity, customer distribution pattern, and auction frequency. The remaining part of this paper is organized as follows. In Section 2, a literature review of related works is conducted. In Section 3, a description of the A-WCS problem in given, followed by the introduction of the O-VCG mechanism in Section 4. The two-layered VNTS algorithm is elaborated in Section 5. In Section 6, extensive numerical studies are performed, and based on which managerial implications are derived. Finally, a conclusion drawn from this study is presented in Section 7, with several directions worth future efforts indicated. 2. Literature review There are two sorts of VRPs in respect of waste management: node- and arc-routing problems. For node routing problems (see, for example, Markov et al., 2016; Rabbani et al., 2016), vehicles set off from a depot and are expected to visit a set of customers, collect the waste and return to a waste treatment center or the depot. The arc routing problems (Cortinhal et al., 2016; Tirkolaee et al., 2018) are frequently related to the households. The vehicles need to visit plenty of households for collection of the waste along all the arcs (e.g., streets). Apart from the previous two streams, the roll-on roll-off vehicle routing problem (RRVRP) is also addressed in the literature, where the waste is beforehand stored in containers and vehicles directly replace the loaded container with an empty one upon arrivals. To learn more about the relevant studies about waste collection logistics, please refer to the thorough reviews conducted by Beliën et al. (2012); Ghiani et al. (2014); Ma and Hipel (2016); Van Engeland et al. (2020). However, the studies mentioned above use centralized methods to deal with the routing problems in the field of waste management. Little research considers the waste management problems from the perspective of transport service procurement (TSP), especially using decentralized approaches like game theory and auction design. Our study thus fills the gap by proposing the auction-based mechanism for less-than-truckload (LTL) TSP in the commercial waste collection industry. We then review in detail the related literature on auction-based TSP. Allowing for the ever-fierce competing environment, a large proportion of enterprises have decided to contract out non-core businesses, with the logistics activities in particular, to an independent third party. Recently, there has been an increasing attention drawn to the mechanisms of TSP, both from academia and industries. As internet popularizes and communication technologies advance, auction has been accepted as a major approach to TSP (Lafkihi et al., 2019). In an auction, one party (usually the demand side, called shipper) clarifies the demands for transport (often called lanes) while multiple participants from the other party (mostly the supply side, called carriers) make bids accordingly. An online mechanism then determines the winners as well as the payments automatically. From the perspective of bidding structure, researches into auction-based TSP can be categorized into one-sided and twosided auctions. In the former, either the carriers or the shippers make bids, and the other party takes decision on the allocation result upon receipt of the bids. In general, the carriers play the role as bidders and join the competition by submitting bids over combinations of shipping requests. The shipper, as the auctioneer, decides on the winning bids by addressing a winner determination problem (WDP). Remli and Rekik (2013) resolved the WDP in a one-sided combinatorial auction, with uncertain shipment volumes taken into account. In order for robust results, a two-stage robust formulation was suggested and resolved with a constraint generation method. Zhang et al. (2015) conducted investigation into the WDP in a one-sided auction under the context of uncertain demands, which was premised on a more general deterministic model. The authors took a central limit theorem (CLT) approach to construct uncertainty sets of transport demands on the basis of historical data. In recent years, two-sided/double auctions are attracting increasing attention from researchers, which enables bidding from both shippers and carriers at the same time. Garrido (2007) devised a double-auction scheme with flexible demand in the spot market, where the demand of shippers is variable depending on the price of transport service. In order for full use of their excess capacity especially during the return trips, it is likely that carriers offer lower prices to stimulate extra demands from shippers. The suggested double-auction mechanism is effective in improving utilization of excess capacities. Meanwhile, the overall cost borne by shippers is lowered. Xu and Huang (2013) proposed a periodic sealed double auction for TSP under the context of dynamic single-lane demands and supplies, wherein the auctioneer receives bids from both buyers and sellers, and the market is cleared one at a time periodically. It was discovered that a reduced auction length

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generates more profits to the auctioneer (the third-party platform) in a short term. Nevertheless, there was an argument that the auctioneer should perform auctions with relatively longer lengths and clear the market less often, so as to increase the utilities of both carriers and shippers, and benefits can be delivered to the platform with regard to long-term revenues. The studies on auction-based TSP can also be classed based on the bidding item, for example, single-item and multi-item auctions (also known as combinatorial auctions). In the former, a single lane is auctioned among multiple carriers, for which a single carrier would be deployed to the bidding lane. The practice of trading an entire set of lanes simultaneously and eventually allocate them to the same winning carrier can also be viewed as a single-item auction. The benefits created by single-item auctions include the easy-to-implement supplier selection from the perspective of operation, and the robustness of supply by the time of rising demand (Caplice and Sheffi, 2003). In comparison, firstly introduced by Rassenti et al. (1982), the multi-item auctions allow each carrier to make bids for combinations of lanes. Because of the synergies between transport pathways, carriers have strong preferences for bundles instead of individual lanes. Basu et al. (2015) addressed a carrier assignment problem (CAP) for TL services, where combinatorial bids are considered to effectively avoid undesired empty travels. Numerical results had revealed the advantage of applying combinatorial auctions in total cost reduction and carbon footprint. There has also been research tackling the auction-based carrier collaboration problem, which is the horizontal collaboration among carriers within the coalition to collectively optimize their logistics operations through request exchanges. The carrier collaboration in truckload pickup-and-delivery services was studied by Lai et al. (2017), where an iterative auction mechanism was proposed. The authors proved that only small efficiency loss was incurred when carriers keep critical information private, however significant profit improvement can be achieved through the request exchange. Gansterer and Hartl (2018) investigated the bundle generation problem (BuGP) in horizontal collaboration among a number of carriers. As claimed by the authors, the size of bundles largely affected the computational efficiency in combinatorial auctions. As the size of potential bundles increases exponentially with the number of traded lanes, the authors proposed a genetic algorithmbased approach to generate promising bundles so as to reduce the bundle size. More recently, Gansterer et al. (2019) studied an auction-based carrier collaboration problem where transportation requests are bundled before being exchanged among a number of carriers to further increase the overall profitability. The combinatorial auction mechanism can avoid the disclosure of critical information, and it was found that a significant increase in profit can be achieved by letting the auctioneer rather than the carriers to design the bundles, under incomplete information. While TL services have attracted the most attentions, some researchers addressed the carrier collaboration problems with LTL services where typically small-volume pickup and delivery tasks are exchanged among a set of carriers (Chen, 2016; Gansterer and Hartl, 2018; Gansterer et al., 2019; Li and Zhang, 2015). To the best of our knowledge, this article is among the first to address the winner determination problem (WDP) on carrier’s side in LTL TSP. Unlike most one-sided auctions proposed for shipper-centric structures, the commercial waste collection industry exhibits a carrier-centric pattern and thus the proposed one-sided O-VCG mechanism enables a carrier to determine winning bids from a group of shippers. Furthermore, compared with existing auction-based TSP literatures, the synchronization requirements are factored into the auction-based LTL TSP, motivated by the multi-commodity demands in the commercial waste collection industry. Finally, studies into the orienteering problem (OP) also exhibit some relevance to our work. OP, also known as vehicle routing problems with profits (VRPP) or selective vehicle routing problems (SVRP) in literature, allows the transporter to decide which customers to serve for maximum profit. A comprehensive review of OP can be found in the recent work done by Gunawan et al. (2016). A significant distinction is that in our study, the selection of customer is made with the objective of social welfare maximization, and corresponding payment for each winner is decided by an O-VCG mechanism. From the perspective of market mechanism design, it is fairer and more reasonable to decide the payment price based on the “marginal contribution” of each customer. Besides, the A-WCS mechanism proves to encourage truthful bidding from shippers and realize maximum social welfare, which are both beneficial for the market in the long run. 3. Problem description We consider a waste collection network comprised of a single transporter (i.e., the carrier) and a number of customers (i.e., the shippers). Customer demands are featured as less-than-truckload (LTL), therefore a vehicle may visit multiple customers to fully use the capacity. Besides, multiple types of waste may be generated by the same customer, and each has to be transported by a specific type of vehicle. In order to provide synchronous services, vehicles of different types need to arrive within the same time window. The customers announce demands to the transporter by means of sealed bids. Each bid includes information of demands for each type of waste, the required time window, and a bidding price. Each customer is allowed to submit more than one bid, if he/she accepts different synchronization levels at the cost of varying payments. Note that the demands remain the same in all bids from the same customer. The transporter then acts as the auctioneer to decide on winning bids with the objective of maximizing social welfare. Once a customer is selected as the winner, all demands need to be satisfied, and all vehicles have to arrive within the same time window as declared in the winning bid. Fig. 1 presents an illustrative example of the A-WCS problem, with one possible auction result for the auction: Customer 1 places two bids and the transporter chooses the time window [10:0 0, 12:0 0]. Three vehicles are required to arrive within the chosen time window. As customer 5 doesn’t win the auction (i.e., none of his/her bids is selected by the transporter), no vehicle is assigned to the customer.

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Fig. 1. One auction result for the illustrative A-WCS problem.

Let H be the set of customers and vh (t) be the value of customer h ∈ H for time window t. Though vh (t) is customer h’ s private information, all customers tell the truth in the suggested auction. Each customer is self-serving and strives for maximum net utility, uh (t), where uh (t) = vh (t) − ph (t), and ph (t) refers to the payment made by the customer. Let  be the set of viable allocations and ψ be a viable allocation, ψ ∈  . A viable allocation means that all constraints can be satisfied by the transporter, given the limited number of vehicles for each waste type. Let H(ψ ) be the set of winning  customers for allocation ψ , and V(ψ ) be the sum of winning customers’ values, where V (ψ ) = h∈H (ψ ) vh (th ). To minimize total transportation costs, the transporter is required to determine a set of routes that serve the set of customers H(ψ ) while meeting the various requirements. For example, Fig. 1 presents a viable allocation in which the transporter serves customers 1, 2, 3 and 4; three routes cover the four customers H(ψ ) with synchronization requirements; and the overall value of selected bids V(ψ ) is $90 (=$15 + $25 + $20 + $30). Let C(ψ ) be the true cost function for allocation ψ . Given C(ψ ), the transporter makes an attempt to find an effective  allocation that maximizes social welfare ψ ∈ xψ (V (ψ ) − C (ψ ) ), where xψ ∈ {0, 1} indicates whether allocation ψ is chosen or not. An effective allocation can be obtained by solving the following integer program (IP): IP-1:

max



ψ ∈

s. t.



xψ (V (ψ ) − C (ψ ) )

(1)

xψ ≤ 1

(2)

xψ ∈ {0, 1}

(3)

ψ ∈

The aim of (IP-1) is to achieve maximum social welfare given C(ψ ) for ψ ∈  . Constraints (2) ensure that at maximum one viable allocation can be chosen. A mathematical model of the A-WCS problem can be found in Appendix A. 4. The O-VCG auction for A-WCS The double VCG auction contributes to incentive compatibility and allocative efficiency. Meanwhile, however, it causes the third-party auctioneer to encounter overbudgeting. In line with the generic definition of VCG mechanism (e.g., Clarke, 1971; Groves, 1973; Krishna, 2009; Nisan and Ronen, 2007; Vickrey, 1961), an introduction is made of the one-sided VCG (O-VCG) auction. Let Z(H) be the value of (IP-1), and Z(H\h) be the value of (IP-1) if customer h is discounted in the auction. In case

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that there is more than one equivalently optimal solution, ties are broken arbitrarily and only one optimal solution will be selected. The O-VCG auction for the WCS problem progresses as follows: (1) Every single customer h ∈ H submits a sealed valuation function vh (t), to the transporter (auctioneer); (2) The auctioneer discloses the transport cost function C(ψ ) for any ψ ∈  ; (3) The auctioneer solves the (IP-1) and decides on the set of winners H(ψ ), where ψ indicates an effective allocation achieving Z(H); (4) Every single customer h ∈ H(ψ ) makes a VCG-like payment of ph (th ) = vh (th ) − (Z(H) − Z(H\h)) for time window th , and his/her utility is uh (th ) = Z(H) − Z(H\h) (if bidders’ reports are truthful). The auctioneer’s utility is uauctioneer =  h∈H (ψ ) ph (th ) − C (ψ ). If uauctioneer > 0, the trade succeeds; otherwise, the trade collapses. Theorem 1. The O-VCG auction realizes incentive compatibility on the buy side. The proof simply follows the standard results of VCG-like mechanisms (e.g., see Clarke, 1971; Groves, 1973; Krishna, 2009; Nisan and Ronen, 1999, 2007; Parkes et al., 2001; Vickrey, 1961). The proof is therefore omitted. Theorem 1 indicates that true bidding is an ex post Nash equilibrium for customers in the O-VCG auction. That is to say, it has (weakly) dominance for every single customer to bid truthfully if others do so, regardless of whether the transporter tells the truth or not. Therefore, our O-VCG auction is incentive compatible on the side of the buyers. Theorem 2. If the transporter tells the truth, then the O-VCG auction results in allocative efficiency, individual rationality, and budget balance. Proof. It is worth noting that uh (th ) = Z(H) − Z(H\h) ≥ 0, and the trade will collapse if uauctioneer > 0. Thus, OVCG auction is rational for each customer on an individual basis, and budget-balanced for the auctioneer. According to Theorem 1 and the assumption that the transporter tells the truth, O-VCG auction identifies an efficient allocation that  maximizes h∈H (ψ ) ph (th ) − C (ψ ). Admittedly, even if the auctioneer makes a false disclosure of cost function, the O-VCG auction leads to incentive compatibility on the buy side, individual rationality and (ex post) budget balance. The maximal social welfare can be delivered as long as the transporter tells the truth. Intuitively, high efficiency and a simplified bidding strategy are supposed to draw in more customers to get involved in the auction. This in turn will generate revenues continuously for the transporter and improve the effectiveness of the suggested auction mechanism from a long-term perspective. 5. Solution method Our solution approach is purposed to deliver the maximum social welfare, which is equivalent to the total value of winning bids minus the transportation cost. Three kinds of limitations are taken into account: (1) the synchronization requirements specified as time windows; (2) vehicle capacity limits; and (3) the constraint of fleet size. Solving an A-WCS problem can be viewed as looking for a proper auction allocation (called the upper layer), and then figure out cost-effective vehicle routes with the given allocation (called the lower layer). The sub-problem in the lower layer is NP-hard as it is essentially a capacitated vehicle routing problem with time windows (CVRPTW). Worse still, the number of sub-problems rises dramatically with the number of customers and that of bids offered per customer, because there are as many as (1 + B)|H| possible allocations (recall that B is the number of customers and |H| is the number of bids offered by each customer). Based on the bi-level characteristic of the A-WCS problem, a two-layered VNTS algorithm was developed by embedding TS into a VNS structure, where VNS is adopted to change auction allocations in the upper layer and TS is applied to obtain efficient vehicle routes in the lower layer. The rest of this section is organized as follows: Section 5.1 first introduces the key idea of VNS, and then gives an overall framework for VNTS. Section 5.2 presents the greedy heuristic we designed to construct initial solutions. The neighborhood structures are defined in Section 5.3, among which VNS strategically changes from one to another dynamically during the whole searching process. Section 5.4 illustrates the details of TS, followed by a set of refinements we made to improve the performance of VNTS in Section 5.5. 5.1. Overall framework of VNTS VNS has been used to address various combinational optimization problems since first put forward by Mladenovic´ and Hansen (1997). For this study, VNS was adopted for two major reasons: (1) Compared with most other metaheuristics which abide by the same neighborhood structure, VNS can effectively escape from local optima by systematically changing the searching neighborhoods. The advantage of VNS in addressing complex combinatorial problems was demonstrated by many recent studies (Hof et al., 2017; Sze et al., 2017); (2) The VNS methodology is especially suitable for our A-WCS because a series of neighborhood structures can be very naturally constructed by bringing different magnitude of changes to the auction allocation. The neighborhood structure will be further elaborated in Section 5.3.

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The VNTS algorithm is comprised of two layers: the upper layer works to change the auction allocation based on the VNS scheme, while the lower one makes attempts to identify cost-effective routes with a TS-based heuristic. The result obtained by the second layer is then reverted to the first layer to assist with decision-making upon how to alter the current allocation for the next iteration. Algorithm 1 gives the overall framework of VNTS. To begin with, a greedy heuristic is used to generate the initial solution, which will be kept as the best-so-far solution since the feasibility of the initial solution is guaranteed (see details in Section 5.2). During each iteration, the Shake operator generates a random neighbor solution of s in the kth neighborhood. The value of k starts from 1 and will be added by 1 every time VNTS fails to improve the best-so-far record, which means a new neighbor solution will be generated by Shake in the next neighborhood (generally a larger one). Once a new global best solution is identified by TS, k will be reset to 1. Our VNTS is slightly different from the classical version in the sense that an adaptive threshold mechanism is incorporated (see Line 5). In particular, only those solutions no worse than λ times of the best-so-far record will be passed over to the local search operator (i.e., TS). Due to the extremely large number of possible allocations, VNTS has to skip those unpromising ones to enhance the computational efficiency. The threshold parameter λ is dynamically adapted, which will be elaborated in Section 5.5. Algorithm 1 Overall framework of VNTS.

0 1 2 3 4

s ← Construct the initial solution with a greedy heuristic s∗ ← s , k ← 1 REPEAT s ← Shake(s, k) IF z(s ) > λ · z(s∗ ) s ← TS(s ) IF z(s) > z(s∗ ) s∗ ← s k←1 ELSE k←k+1 ELSE k←k+1 UNTIL k > kmax or time limit is reached

5.2. Greedy heuristic for initial solution We have developed a greedy heuristic to generate the initial solution while satisfying various constraints. During each iteration, the bid with highest value from the remaining customer pool will be chosen, and his/her demands will be inserted into existing routes. No capacity violation is allowed during the insertion, namely, the total load of the route will not exceed the vehicle capacity after insertion. Also, time window constraints for all customers (including the newly inserted one) on the route must be respected. If more than one insertion position is available, the one with least incremental cost will be selected. In case that no such insertion point can be found in existing routes, a new route will be created to accommodate the customer. Since different types of waste is transported separately, the insertion will be performed repeatedly for each waste type of the customer. Note that a single vehicle can perform multiple routes during the planning horizon, and all existing routes need to be packed into tours to check whether the fleet size constraint is satisfied. In this paper, “route” and “tour” are two different concepts, following the definitions given by Brandao and Mercer (1997): a route starts from the depot, visits one or multiple customers and then returns to the depot, while a tour refers to a set of routes performed by the same vehicle in a given sequence during the planning horizon. In order to do the route bin-packing more effectively, we minimize the duration of each route with a double-shifting operator. Fig. 2 illustrates how the double-shifting operator shrinks the total duration of the route, which starts from depot and visits customers C1, C2 and C3 before finally returns to the depot. After pushing arrival times at all nodes to the earliest possible instant, the duration is reduced by t1 , relative to that of the original route. The duration is further reduced by t2 after postponing the departure time from each node to the latest possible time instant. The route duration after double shifting is reduced by t2 + t3 as compared to the original one. 5.3. Neighborhood structures



A sequence of neighborhoods is defined and denoted as N1 , N2 , . . . Nkmax , where kmax = |H |B. Nk indicates the neighborhood obtained by making a number of k changes to a given allocation. We denote an auction allocation with an array of integers. For instance, in a situation with 3 bidders, the array [3, 1, 0] indicates that the 3rd bid of bidder 1 and the 1st bid of bidder 2 are chosen, while bidder 3 is eliminated from the allocation. Note that, 0 is used to label a bidder if none of his/her bids is selected. Fig. 3 illustrates the neighborhood structures defined for A-WCS. The value of k is constant-changing throughout the process of search, which represents the core idea of VNS. Noticeably, a small k guides the search in nearer neighborhoods, while a large k is beneficial to avoid a local valley and guide the search to a remote region of the solution space.

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S. Shao, S.X. Xu and G.Q. Huang / Transportation Research Part B 133 (2020) 1–20

Fig. 2. An illustrative example of double-shifting operator.

5.4. Tabu search TS is purposed to make improvements to the promising solutions obtained from the upper layer. The implementation of TS in this paper is based on Glover (1989, 1990). TS has been identified as an effective metaheuristic for VRPs with regard to both solution quality and computational speed (Nguyen et al., 2013). As described in Algorithm 2, TS begins with the initial solution s passed over from the upper layer or derived by the greedy constructive heuristic. Then TabuList, AspirationList and Penalties are initialized based on s. To be specific, TabuList maintains a record of the solutions visited recently by the algorithm to prevent cycling, while AspirationList is employed to override the Tabu status if the implementation of the corresponding move contributes to a better incumbent.

Algorithm 2 TS.

0 1 2 3 4 5 6 7 8

s ← Initial solution from upper layer or greedy constructive heuristic Initialize TabuList, AspirationList and Penalties with s s∗ ← s, NiIter ← 0, TotalIter ← 0 REPEAT TotalIter + +, NiIter + + Set of neighbor solutions of s: M(s) ← ∅ FOR each neighbor s of s IF s is Non-Tabu or activates the aspiration criterion M ( s ) ← M ( s )∪s s ← neighbor solution with largest objective from M(s) IF z(s ) > z(s) s ← s Update TabuList and Penalties IF s is feasible AND z(s) > z(s∗ ) s∗ ← s Update AspirationList NiIter ← 0  

UNTIL N iIter > 100 k|H |B OR TotalIter > 10 0 0 k|H |B

S. Shao, S.X. Xu and G.Q. Huang / Transportation Research Part B 133 (2020) 1–20

Fig. 3. Illustration of the defined neighborhood structures.

9

10

S. Shao, S.X. Xu and G.Q. Huang / Transportation Research Part B 133 (2020) 1–20

For each iteration, a set of four operators are adopted to obtain multiple neighbor solutions: intra-route exchange, intraroute 2-opt, inter-route relocation, and inter-route CROSS. These improvement operators are widely applied in the VRP literature, for their excellent performance in searching the local neighborhood of a given solution. In case that a neighbor solution is non-Tabu or activates the aspiration criteria, it will be added into M(s). The one with largest objective value in M(s) will be chosen for the next iteration. The objective value for solution s is expressed as z(s) = V(s) − C(s) − α · f(s) − β · g(s) − γ · t(s). Here V(s) indicates the total value of selected bids, and C(s) denotes the total transport cost. Besides, f(s), g(s) and t(s) refer to the violations of constraints of fleet size, vehicle capacity and time window, respectively. Penalty coefficients α , β and γ are all set to 1 initially and adjusted dynamically based on the feasibility of accepted solutions. In this study, a specific penalty coefficient will be multiplied by 1.1 if the corresponding constraint is violated or divided by 1.1 otherwise. Finally, TS is terminated after a specified number of iterations have elapsed since commencement or last improvement. Note that we have incorporated the k value into the termination conditions (Line 18). The intuition is that a larger k generally means the VNTS has already failed to find a better solution for a number of k consecutive iterations, therefore longer searching time should be allowed to help the algorithm escape from the local valley. 5.5. Further refinements of VNTS In this section, we introduce how we further improve the performance of VNTS by tuning kmax and λ, which are two crucial parameters for the algorithm. As demonstrated by our preliminary results, a majority of the new best solutions are discovered in the nearer neighborhoods (i.e., when the value of k is relatively small). Setting kmax excessively large results in quitelong computational time but almost no superior solutions. Our computational experiments show that by setting kmax to |H |B, one can achieve a



satisfactory balance between solution quality and computational time. We thus let kmax equal to |H |B for all numerical experiments in this study. Concerning λ, this parameter determines to what degree we trust a solution is promising and worth further improvement with TS. It is apparent that a smaller λ leads to more solutions passed over to TS, while a larger λ ignores solutions with unsatisfactory objective values at the moment. A range of experiments were conducted to verify a set of different values of λ, with the results revealing that setting an overly large λ (e.g., λ ≥ 1.2) usually ends up in poor results. This is because local search is less frequently invoked and the VNTS is too myopic. On the contrary, setting an excessively small λ (e.g., λ ≤ 0.4) is also deemed inappropriate as the algorithm loses its effective guidance. Therefore, λ is chosen at random from [0.6, 0.8] for all the experiments in this study. 6. Computational studies This section carries out a set of computational studies to assess the performance of our suggested VNTS algorithm and provide managerial implications from the perspective of practitioners. Particularly, Section 6.1 introduces how the instances were generated. In Section 6.2, we validated the efficiency of VNTS by comparing it with CPLEX 12.8 on small instances, for which CPLEX can obtain optimal or near-optimal results. Then in Section 6.3, an extensive comparison was conducted among VNTS and three other start-of-the-art heuristics. Finally, in Sections 6.4, 6.5 and 6.6, a series of sensitivity analyses were performed to investigate the impacts caused by changes in market size, task complexity, customer distribution pattern, and auction frequency. A number of managerial implications were obtained accordingly from the perspective of practitioners. All computational experiments were conducted on a 64-bit Windows machine, with an Intel Core processor i7-4790, 3.6 GHz, and 12GB of RAM. 6.1. Test instances To facilitate a thorough comparison, a set of instances were created on the basis of the well-known Solomon’s instances for VRP (Solomon, 1987). Especially, 9 different Solomon instances were used, including R101, R102 and R103 where customers are located on a random basis; C101, C102 and C103 where customers are subject to clustering; and RC101, RC102 and RC103 which exhibit a mix of randomly distributed and clustered customers. Fig. 4 shows three different distribution patterns of customers in Solomon’s instances, with the red square representing the depot and black dots representing the customers. The Solomon’s instances have stated information including positions of customers and the depot, the demand and time window of each customer, vehicle capacities as well as the fleet size. We then incorporated the following characteristics to extend Solomon’s instances for our A-WCS as follows: •

•

multiple waste types: Solomon’s instances consider a single commodity scenario, and hence we expanded the number of waste types to 2 and 3, respectively. A customer’s demand volumes for each type of waste are assumed to be the same and equivalent to the value given in Solomon’s instances; synchronization time windows: besides the original time windows given in Solomon’s instances, we also created wider and narrower time windows accordingly, by either doubling or halving the duration of the original time window, respectively. Note that different window widths indicate different levels of expected synchronization services;

S. Shao, S.X. Xu and G.Q. Huang / Transportation Research Part B 133 (2020) 1–20

11

Fig. 4. Different distribution patterns of customers in Solomon’s instances.

•

•

•

customer bids: every customer is permitted to provide a maximum of 3 different bids. The bidding price is a randomized value between 100 and 500, and the corresponding time window is chosen on a random basis from the original, wider and narrower time windows. The demands remain the same in a customer’s multiple bids; vehicles: each vehicle is dedicated to a specific waste type, and the number of vehicles applied to transfer each waste is 20 at maximum. All vehicles have the same capacity of 200 units and travel speed of 30 km/h. The service duration at each customer is set to 1.5 h, regardless of the amount of demand; number of customers: the number of customers in Solomon’s instances is 100. Hence, we firstly generated instances involving 100 customers. Then, instances were obtained by means of random selection of 10, 15, 25 and 50 customers from the initial instances, respectively.

In this study, the generated instances are named in the format of “the base Solomon’s instance” - “number of customers” - “number of waste types”. For example, “C101-25-2” signifies that the instance was created by a random selection of 25 customers from Solomon’s C101 instance and extended to the situation with 2 different waste types. Moreover, every single experiment was conducted repeatedly for 10 times, and the best result among them was reported. The computation time limit was set to 240, 300, 600, 900 and 1200 s for instances with 10, 15, 25, 50 and 100 customers, respectively, whichever heuristic algorithm was applied. 6.2. Comparison of VNTS and CPLEX Given that CPLEX 12.8 can only obtain (near) optimal solutions for small sized instances within reasonable computation time, the comparison between VNTS and CPLEX was conducted based on instances with 10 and 15 customers. The maximum computation time for CPLEX was set to 3600 and 5400 s, for instances with 10 and 15 customers, respectively. The results are presented in Table 1 and Table 2 as follows, where the underlined numbers refer to the optimal solutions obtained by CPLEX, and “Gap vs. CPLEX” is calculated by (VNTS-CPLEX)/CPLEX. One can see from Table 1 and Table 2 that CPLEX only solved 13 out of 36 instances to optimality, all of which were instances with 10 customers (none of the 15-customer instances was solved optimally by CPLEX), revealing the extreme challenge of tackling A-WCS problems. The stability of VNTS can thus be validated by the fact that it obtained solutions within −0.5% gap from optimal solutions for all instances optimally solved by CPLEX. Specifically, when two waste types were involved, VNTS outperformed CPLEX by 0.4% in average for the 10-customer instances, while an average improvement as much as 12.6% was made in instances with 15 customers. Similarly, when three types of waste were considered, the average gap between VNTS and CPLEX was 0.2% in cases with 10 customers. For larger instances with 15 customers, the gap between VNTS and CPLEX increased dramatically to an average value of 13.3%. The effectiveness of VNTS was validated based on instances for which CPLEX can work out (near) optimal solutions. VNTS has proved to perform stably and be able to provide solutions of high quality. 6.3. Comparison of GSA, GTS, VNSA and VNTS This section is set for two reasons: Firstly, it is purposed to examine the advantage of two-layered algorithms relative to single-layered algorithms in solving A-WCS. Secondly, it is aimed at verifying the effectiveness of integrating VNS and TS, compared with the combination of VNS and SA. To achieve the above purposes, three other benchmarking algorithms have been devised as well, including the greedy simulated annealing (GSA), the greedy tabu search (GTS) and the variable neighborhood simulated annealing (VNSA). As for the former two algorithms, the greedy constructive heuristic described in Section 5.2 is used to find a decently good allocation, with which vehicle routes are then generated with SA and TS, respectively. Obviously, since GSA and GTS abide by

12

S. Shao, S.X. Xu and G.Q. Huang / Transportation Research Part B 133 (2020) 1–20 Table 1 Comparison of VNTS and CPLEX when |W| = 2. Instances

|H|

CPLEX

VNTS

Gap vs. CPLEX

C101–10–2 C102–10–2 C103–10–2 R101–10–2 R102–10–2 R103–10–2 RC101–10–2 RC102–10–2 RC103–10–2

10

2969.6 3749.8 3133.5 1408.6 2296.9 1760.0 1101.4 1496.6 1648.3

2969.6 3802.0 3130.6 1404.2 2295.6 1754.2 1096.1 1543.3 1648.3

0.0% 1.4% −0.1% −0.3% −0.1% −0.3% −0.5% 3.1% 0.0%

2173.9

2182.7

0.4%

2180.0 4629.6 4683.0 2180.0 2550.7 3000.6 1220.4 2991.3 2541.8

2292.0 4672.6 4901.5 3185.4 2966.1 3376.7 1551.3 3027.6 2533.6

5.1% 0.9% 4.7% 46.1% 16.3% 12.5% 27.1% 1.2% −0.3%

Sub average C101–15–2 C102–15–2 C103–15–2 R101–15–2 R102–15–2 R103–15–2 RC101–15–2 RC102–15–2 RC103–15–2

15

Sub average

2886.4

3167.4

12.6%

Total average

2530.1

2675.0

6.5%

Table 2 Comparison of VNTS and CPLEX when |W| = 3. Instances

|H|

CPLEX

VNTS

Gap vs. CPLEX

C101–10–3 C102–10–3 C103–10–3 R101–10–3 R102–10–3 R103–10–3 RC101–10–3 RC102–10–3 RC103–10–3

10

3797.1 3699.8 3564.7 654.1 1021.8 1438.3 371.7 691.1 1055.9

3788.2 3699.8 3553.4 654.1 1040.1 1453.3 371.7 687.8 1054.4

−0.2% 0.0% −0.3% 0.0% 1.8% 1.0% 0.0% −0.5% −0.1%

1810.5

1811.4

0.2%

1334.7 4695.9 5023.8 1334.7 2235.8 1596.5 537.9 1169.1 1693.8

1375.1 5260.4 5009.7 2183.1 2417.0 1714.0 646.9 1181.2 1774.3

3.0% 12.0% −0.3% 63.6% 8.1% 7.4% 20.3% 1.0% 4.8%

Sub average C101–15–3 C102–15–3 C103–15–3 R101–15–3 R102–15–3 R103–15–3 RC101–15–3 RC102–15–3 RC103–15–3

15

Sub average

2180.2

2395.7

13.3%

Total average

1995.4

2103.6

6.8%

the identical allocation during the whole searching process, they are single-layered algorithms without the ability to search among different allocations. Concerning VNSA, it adopts the identical two-layered structure as VNTS. The only difference is that simulated annealing (SA) is adopted to generate vehicle routes. Please refer to Appendix B for more details about the SA algorithm. For each instance in Tables 3 and 4, the results obtained by four algorithms were reported. “Gap vs. GTS” is calculated by (VNTS-GTS)/GTS, and “Gap vs. VNSA” is obtained by (VNTS-VNSA)/VNSA. The first indicator shows that, with adopting the same heuristic (namely TS) to search for cost-effective vehicle routes, to which extent the incorporation of VNS as an upper layer search scheme can contribute to the final result. The second indicator shows that, by adopting the identical two-layered structure, to which degree the integration of VNS and TS can outperform the combination of VNS and SA. The results of instances with 2 and 3 different waste types were indicated in Tables 3 and 4, respectively. From Table 3, it can be discovered that the two-layered algorithms (including both the VNTS and the VNSA) clearly outperform the single-layered algorithms (i.e., GSA and GTS). Especially, the average gap between VNTS and GTS is widened from 34.7%

S. Shao, S.X. Xu and G.Q. Huang / Transportation Research Part B 133 (2020) 1–20

13

Table 3 Comparison of four algorithms when |W| = 2. Instances

|H|

GSA

GTS

VNSA

VNTS

Gap vs. GTS

Gap vs. VNSA

C101–25–2 C102–25–2 C103–25–2 R101–25–2 R102–25–2 R103–25–2 RC101–25–2 RC102–25–2 RC103–25–2

25

7047.6 1746.8 5914.6 7018.4 6562.8 4194.5 4494.4 3145.8 3912.8

7047.6 1832.0 6019.2 7018.4 6562.8 4194.5 4494.4 3145.8 4134.0

7209.2 6468.3 7192.7 7018.4 6562.8 4336.6 4527.2 3431.1 3823.1

7433.2 6348.8 7739.2 7074.6 6562.8 4449.7 4742.9 3561.3 4400.8

5.5% 246.6% 28.6% 0.8% 0.0% 6.1% 5.5% 13.2% 6.5%

3.1% −1.8% 7.6% 0.8% 0.0% 2.6% 4.8% 3.8% 15.1%

sub average

4893.1

4938.7

5618.8

5812.6

34.7%

4.0%

10,625.6 14,850.6 5297.2 9546.9 4637.3 9714.4 4786.1 1585.1 3994.9

10,625.6 14,850.6 5666.0 9546.9 4637.3 9915.1 4786.1 1585.1 4561.1

12,363.3 16,302.6 9260.0 9870.2 6214.3 11,085.5 5401.9 4243.7 5475.1

13,442.6 17,689.1 9860.6 10,939.2 6505.6 11,262.4 5863.2 4344.9 5596.1

26.5% 19.1% 74.0% 14.6% 40.3% 13.6% 22.5% 174.1% 22.7%

8.7% 8.5% 6.5% 10.8% 4.7% 1.6% 8.5% 2.4% 2.2%

7226.5

7352.6

8913.0

9500.4

45.3%

6.0%

8623.7 21,005.8 7572.8 11,686.5 8498.1 3334.8 1745.4 6910.8 701.0

8623.7 21,200.0 7572.8 11,989.2 8498.1 3334.8 1745.4 6910.8 1066.5

11,579.1 28,666.1 9598.9 16,148.7 10,504.9 6249.6 5581.3 8881.3 4646.4

12,674.8 28,709.4 11,684.3 16,415.3 11,020.8 6523.9 6952.9 9268.6 5288.5

47.0% 35.4% 54.3% 36.9% 29.7% 95.6% 298.4% 34.1% 395.9%

9.5% 0.2% 21.7% 1.7% 4.9% 4.4% 24.6% 4.4% 13.8%

Sub average

7786.6

7882.4

11,317.4

12,059.8

114.1%

9.4%

Total average

6635.4

6724.6

8616.4

9124.3

64.7%

6.5%

C101–50–2 C102–50–2 C103–50–2 R101–50–2 R102–50–2 R103–50–2 RC101–50–2 RC102–50–2 RC103–50–2

50

sub average C101–100–2 C102–100–2 C103–100–2 R101–100–2 R102–100–2 R103–100–2 RC101–100–2 RC102–100–2 RC103–100–2

100

to 114.1% when the number of customers increases from 25 to 100. This could be also seen in Table 4 where the gap between VNTS and GTS rises from 77.1% to 116.4% as the customer size rises from 25 to 100. This is what is expected as a single-layered algorithm only sticks to same allocation selected by the greedy constructive heuristic, which is rather myopic. However, a two-layered algorithm receives huge benefits from the capability to conduct search among various allocations. A second discovery is that, despite both being a two-layered algorithm, VNTS outperformed VNSA in solving A-WCS problems in most cases, demonstrating the advantage of integrating VNS and TS, relative to the combination of VNS and SA. Particularly, VNTS produced more desirable results than VNSA in 26 out of 27 instances with the average gap of 6.5% when the number of waste types equals two. Likewise, VNTS exceeded VNSA by 8.3% on average and led to better results in 24 out of 27 instances when three waste types were involved. Besides the refinements made as described in Section 5.5, this is probably due to TS performing slightly better than SA when constructing vehicle routes given the identical allocation. The superiority could also be verified from the fact that GTS achieved better solutions in the majority of instances relative to GSA, and at least found the identical solution otherwise. Also, the incorporation of VNS features into the TS layer also leads to more effective searching (for example, taking the k value into the termination condition of TS). Finally, it is noteworthy that VNTS improves its competitiveness as the number of customers is on the rise, or when more types of waste are involved, demonstrating the capability of VNTS when practical instances are dealt with. Moreover, the advantage of VNTS over other approaches appears to be stable, irrespective of how customers are geographically distributed. 6.4. The impact of number of customers and waste types We obtain average social welfare values from Table 3 and Table 4 by groups of customer sizes (i.e., 25, 50 and 100) and waste type sizes (i.e., 2 and 3), respectively. The results are presented in Fig. 5, from which two managerial implications can be obtained: (1) The social welfare increases when more customers are involved, and this is because the auctioneer can derive better allocations and vehicle routes within a larger customer pool. Put another way, it is always beneficial in terms of social welfare maximization to attract more shippers into the same auction, which can be viewed as the economy of scale effect.

14

S. Shao, S.X. Xu and G.Q. Huang / Transportation Research Part B 133 (2020) 1–20 Table 4 Comparison of four algorithms when |W| = 3. Instances

|H|

GSA

GTS

VNSA

VNTS

Gap vs. GTS

Gap vs. VNSA

C101–25–3 C102–25–3 C103–25–3 R101–25–3 R102–25–3 R103–25–3 RC101–25–3 RC102–25–3 RC103–25–3

25

6209.4 5935.1 1130.2 2315.0 1829.8 1038.8 1289.4 380.5 527.1

6435.2 5935.1 1130.2 2315.0 1829.8 1038.8 1289.4 380.5 527.1

7343.4 7421.3 2516.1 2616.7 2626.7 2130.8 1747.7 511.5 1610.6

7491.4 7834.7 2444.2 2718.6 2714.3 2370.6 1712.5 676.5 1710.0

16.4% 32.0% 116.3% 17.4% 48.3% 128.2% 32.8% 77.8% 224.4%

2.0% 5.6% −2.9% 3.9% 3.3% 11.3% −2.0% 32.3% 6.2%

2295.0

2320.1

3169.4

3297.0

77.1%

6.6%

50

9564.8 3152.4 817.8 5796.6 4376.3 2238.2 760.4 2240.9 3528.9

9564.8 3180.1 839.2 5796.6 4376.3 2238.2 760.4 1985.2 3528.9

10,067.6 5196.0 5075.9 6377.7 5255.2 2941.7 1580.5 2612.5 3728.9

10,556.8 5875.4 5697.7 6374.4 5441.3 3148.9 1595.6 2820.7 4587.9

10.4% 84.8% 579.0% 10.0% 24.3% 40.7% 109.8% 42.1% 30.0%

4.9% 13.1% 12.2% −0.1% 3.5% 7.0% 1.0% 8.0% 23.0%

3608.5

3585.5

4759.5

5122.1

103.4%

8.1%

100

14,096.2 3912.3 3964.3 6033.2 826.9 976.7 5500.5 1775.7 1241.0

13,762.6 3912.3 3964.3 6033.2 826.9 976.7 5209.0 1775.7 945.2

14,667.1 5140.1 7956.1 7137.2 3002.7 2652.5 5681.0 3039.2 2786.5

16,140.9 6157.5 8447.3 7447.8 3177.3 3201.7 6051.7 3375.8 3008.1

17.3% 57.4% 113.1% 23.4% 284.2% 227.8% 16.2% 90.1% 218.3%

10.0% 19.8% 6.2% 4.4% 5.8% 20.7% 6.5% 11.1% 8.0%

Sub average

4258.5

4156.2

5784.7

6334.2

116.4%

10.3%

Total average

3387.3

3353.9

4571.2

4917.8

99.0%

8.3%

sub average C101–50–3 C102–50–3 C103–50–3 R101–50–3 R102–50–3 R103–50–3 RC101–50–3 RC102–50–3 RC103–50–3 sub average C101–100–3 C102–100–3 C103–100–3 R101–100–3 R102–100–3 R103–100–3 RC101–100–3 RC102–100–3 RC103–100–3

(2) The social welfare decreases when more types of waste are involved, which is not surprising because the bidding values are randomly generated from the same interval (from 100 to 500) regardless of the number of waste types. However, the transportation cost obviously increases when an additional fleet of vehicles are operated. Besides it is more challenging to synchronize more types of vehicles. From the perspective of practitioners, it is thus suggested that a higher bidding value is required when waste collection tasks become more complicated (i.e., involving more types of waste), so as to cover the increased transportation cost.

Fig. 5. The impact of number of customers and waste types.

S. Shao, S.X. Xu and G.Q. Huang / Transportation Research Part B 133 (2020) 1–20

15

Fig. 6. The impact of distribution of customer locations.

6.5. The impact of distribution of customer locations Fig 6 presents the average social welfare given different customer sizes (i.e., 25, 50 and 100) and geographical distribution patterns (i.e., random, clustered and mix), by grouping results from Tables 3 and 4. The following managerial implications can be obtained: (1) The higher degree that customers are clustered, the higher social welfare can be achieved. This is because a vehicle can usually be fully occupied by demands from a cluster, which largely reduces the average route length. However, when customers are randomly dispersed, demands have to be consolidated from a wider range, which often results in longer route lengths. (2) The economy of scale effect is the most significant when customers are clustered, compared with other distribution patterns. Namely, the rise in customer size can bring about largest increase in social welfare when customers are clustered. It is therefore particularly suggested to attract more customers into the same auction, by means of fee discount, for instance, when potential customers could contribute to forming a cluster. 6.6. The impact of auction frequency At the present time, the need for timely service has increased as significant as synchronous service in waste collection industry. When auctions are less frequently performed, a customer has to wait for longer time before the announcement of auction results. A higher auction frequency generally saves waiting time for customers and therefore appeals to them. Nevertheless, overly frequent auctions could lead to less efficient vehicle routes. Two managerial questions are thus raised: (1) how often the auction is supposed to be performed for the maximum social welfare? and, (2) how different auction frequencies have effect on the benefits received by participants (i.e. shippers and the carrier) in the auction? In this study, the auction frequency is defined as the number of auctions performed during the planning horizon (e.g., a day). This section investigates the impact of auction frequency by dividing customers into several periods and performing an auction for each period independently. Note that all experiments conducted in Section 6.3 can all be viewed as single-period auctions, since all customers participate in the same auction. Or equivalently, we say the auction frequency is 1. In this section, the customers will be classified into 2 and 3 periods with equal length, respectively, according to their time windows. For example, if two auctions are to be held during a single day, customers with time windows before noon will be classified into the first period, while the others with time windows after noon will attend the auction in the second period. Note that we have slightly modified the instances used in Section 5.3, so that all time windows of the same customer are located in the same period. In this way, a customer will only participate in exactly one auction. The results are presented in Fig. 7 and Fig. 8, where in Fig. 7 customers are clustered, while in Fig. 8 the customers are randomly distributed. Note that all experiments in this section involve three types of waste, and the percentages in the figures stand for the proportion of carrier’s utility over the social welfare. Two managerial implications can be obtained from the numerical results: (1) A higher auction frequency leads to a reduction in both social welfare and carrier’s utility at all times. Also, as the auction frequency increases, the proportion of carrier’s utility over the total social welfare also decreases. This is largely attributed to that, when a fixed number of bids are classified into multiple independent auctions, the solution space is reduced significantly. From practitioner’s point of view, the auctioneer (i.e., the carrier) needs to identify

16

S. Shao, S.X. Xu and G.Q. Huang / Transportation Research Part B 133 (2020) 1–20

Fig 7. The impact of auction frequency when customers are clustered.

Fig 8. The impact of auction frequency when customers are randomly distributed.

a proper balance among the customer satisfaction, self-interest (i.e., carrier’s own utility), and platform’s long-term development (i.e., the overall social welfare). It is interesting to make a comparison with Xu and Huang (2013), where the authors found that overly frequent auctions harm the benefits of both sides (shippers and carriers) as well as the social welfare, while only the platform (third-party auctioneer) enjoys benefits. The authors also claimed that less frequent auctions should be performed by the auctioneer for the sake of long-term benefits. (2) The impact of higher auction frequencies is mitigated when customers are clustered. This is because when customers are clustered, one has a higher likelihood to be a replacement for another, and as a result the social welfare is less likely to be affected when customers are classified into smaller groups. From the perspective of O-VCG auction, if a participant can be substituted by others to a larger extent, his/her “exclusive contribution” to the auction is less significant, and his/her utility is generally lower. Put another way, the carrier can take a larger share from the social welfare under such circumstances. Therefore, it can be observed that the decrease in social welfare is less remarkable with higher auction frequencies when customers are clustered. In addition, the proportions of carrier’s utility in the clustered instances are slightly larger than the ones in randomly distributed instances.

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7. Conclusions The present commercial waste collection industry is often featured as carrier-centric, less-than-truckload (LTL) and multicommodity. A single carrier has to effectively consolidate a number of shipments of small volumes from multiple shippers to make a better use of vehicle capacity. Also, a customer may generate multiple types of waste, hence requiring several vehicles to arrive at almost the same time. To this end, an auction-based waste collection synchronization (A-WCS) approach is suggested, with carrier being the auctioneer and shippers being the bidders. A customer declares his/her demands, synchronization time window and price in each bid, and each customer is allowed to make multiple bids. The auctioneer (i.e., the carrier) then decides on winning bids with the objective to maximize the social welfare. The O-VCG auction achieves incentive compatibility (on the buy side), allocative efficiency, budget balance, and individual rationality. More reasonable payments are decided in the auction manner, and the social welfare maximization also contributes to long-term development of the industry. A two-layered VNTS algorithm is proposed by embedding TS into the VNS structure. The upper layer of VNS functions to provide guidance on the searching process with systemic changes made to the auction allocation, while the lower layer of TS conducts local search of vehicle routes. Based on test cases extended from Solomon’s instances, an extensive computation study is performed. As seen from the numerical results, the suggested VNTS proves to provide stable and quality solutions for the A-WCS problems. A range of sensitivity analyses have been conducted to quantify the impacts caused by changes in the number of customers and waste types, the distribution pattern of customers as well as the auction frequency. A number of managerial implications are then obtained. As for future work, acceleration techniques can be considered to further enhance the computational efficiency. For example, by analyzing the dominance among bids, a certain number of bids could be eliminated from consideration. Also, paralleled computing can also be applied to accelerate the searching process, by dividing the original problem into smaller ones from the allocation level. Another interesting extension is to consider the uncertain demands in commercial waste collection. It is practically difficult to accurately assess the quantity of demands before actual handling. Compensation and penalty mechanisms can be developed for situations of over- or under-estimations. Besides, from the perspective of the transporter, it is of importance to develop robust A-WCS models so as to realize profit maximization under the uncertain environment. CRediT authorship contribution statement Saijun Shao: Conceptualization, Methodology, Software, Writing - original draft, Writing - review & editing. Su Xiu Xu: Conceptualization, Methodology, Writing - original draft, Writing - review & editing. George Q. Huang: Conceptualization, Supervision. Acknowledgement Authors would like to acknowledge the support from National Science Foundation of China (No. 71701079; No. 71901146), and the HKSAR RGC GRF Project (No. 17203518; No. 17203117; No. 17212016). Appendix A: The model of the A-WCS problem The mathematical model of the A-WCS problem is given in this section. It is noteworthy that the notations below are only effective within the scope of this appendix and not applicable to the remainder of this paper. Indices: A i, j, h w n v Sets: C A W Ni Vw

Element or REW boundary customers and the depot waste type bid vehicle set of all customers, {1, 2, …, |C|} set of all nodes, including customers and the depot, {0, 1, 2, …, |C|, |C|+1}, where 0 indicates the starting depot and |C|+1 denotes the returning depot set of all waste types, {1, 2, …, |W|} set of bids of customer i, {1, 2, …, |Ni |}, ∀i ∈ C set of vehicles for waste w, {1, 2, …, |Vw |}, ∀w ∈ W

Parameters: diw demand of waste type w from customer i, ∀i ∈ C, w ∈ W Qw

identical capacity of vehicles transporting waste type w, ∀w ∈ W

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tij

τ

travel time between i and j, ∀i, j ∈ A , i = j identical operation time at each customer

[Ein , Lni ]

time window for customer i stated in its nth bid, ∀i ∈ C, n ∈ Ni

Pin Uw

the bidding price of nth bid of customer i, ∀i ∈ C, n ∈ Ni transportation cost per unit travel time of vehicles for waste type w, ∀w ∈ W

Decision variables: v xw equals 1 if vehicle v transports waste w directly from i to j, and equals 0 otherwise, ∀i, j ∈ A , w ∈ W, v ∈ Vw ij equals 1 if the nth bid of customer i is selected, and equals 0 otherwise, ∀i ∈ C, n ∈ Ni

yni

the time vehicle v of type w starts to provide service to customer i, ∀i ∈ A, w ∈ W, v ∈ Vw

v sw i

The mixed integer programming (MIP) model is given as follows:

max



Pin · yni −

i∈C n∈Ni

   i∈A j∈A w∈W v

v U w · ti j · xw ij

(4)

∈V w

s.t.



∀i ∈ C

yni ≤ 1,

n∈Ni



v

∈V w



v xw ij =



(5)

∀i ∈ C, w ∈ W

yni ,

(6)

n∈Ni

j∈A

∀w ∈ W, v ∈ V w

v xw 0 j = 1,

(7)

j∈A

 i∈A



v xw ih =

i∈A



∀w ∈ W, v ∈ V w

v xw i,|C |+1 = 1,



(8)

∀h ∈ C, w ∈ W, v ∈ V w

v xw hj ,

(9)

j∈A

diw ·



i∈C

∀w ∈ W, v ∈ V w

v w xw ij ≤ Q ,

(10)

j∈A

∀w ∈ W, v ∈ V w

v sw 0 = 0,



(11)



v wv sw ≤ swj v , i + ti j + τ − M · 1 − xi j





v wv sw ≤ swj v , 0 + ti j − M · 1 − xi j





∀i, j ∈ C, i = j, w ∈ W, v ∈ V w ∀ j ∈ A, i = j, w ∈ W, v ∈ V w





v ≤ Ln + M · 1 − yn , Ein − M · 1 − yni ≤ sw i i i v xw i j ∈ {0, 1},

∀i, j ∈ A, w ∈ W, v ∈ V w

∀i ∈ C, n ∈ Ni , w ∈ W, v ∈ V w

(12) (13) (14) (15)

The objective in (4) is to achieve the maximum social welfare, which is equivalent to the difference of total bidding values of selected bids and the total transportation cost to complete the selected bids. Constraints (5) make sure that a maximum of one bid will be chosen for one customer, as mentioned in Section 3. Constraints (6) ensure that if any bid is chosen for a customer, his/her demands for different types of waste must all be satisfied. These constraints also guarantee that for every single type of waste, a customer will be visited exactly once. Constraints (7) and (8) are applied to make each single vehicle start and return to the depot in the planning horizon. Constraints (9) make sure that if a vehicle visits a customer, it has to depart from the same customer for another destination. Vehicle capacity constraints cannot be violated at any time during the route, as stated in inequations (10). Eqs. (11) require that each vehicle starts its service at the depot at the beginning of the planning horizon. The relationship between the departure time of a customer and the arrival of its immediate successor is described in (12) and (13), with M indicating an arbitrarily large positive figure. Inequations (14) have shown the major difference between A-WCS and traditional VRP models. In particular, in the AWCS problem, a customer is allowed to submit multiple bids, with one time window in each single bid. The time window attached to the selected bid must be satisfied by the transporter, namely all required vehicles need to arrive within the same corresponding time window (again, M is an arbitrarily large positive figure). Finally, the domain of decision variables v is expressed in (15). xw ij

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Appendix B: Simulated annealing (SA) for A-WCS Algorithm 3 outlines the steps of SA applied to solve our A-WCS problems. The initial solution s is obtained from upper layer of VNS in VNSA, or from the greedy constructive heuristic in GSA. An iteration of the inner loop begins on line 5 by randomly selecting an improvement operator from intra-route exchange, intra-route 2-opt, inter-route relocation and interroute CROSS, which are also used by TS and mentioned in Section 5.4. A random neighbor solution s is then generated by the selected operator based on the current solution s. Lines 9–12 describe the criteria to accept the newly generated neighbor solution. If s improves s, it will be accepted with probability 1. While in case that s deteriorates s, it will be accepted with probability exp([z(s ) − z(s)]/T), where the positive figure T is a control parameter called temperature. Note that in this paper we are dealing with a maximization problem, an inferior solution with reduced objective value will lead to a probability less than 1. Lines 13–14 update the best-found solution and record the corresponding temperature in Tb . After a number of Inner _limit iterations, the temperature is decreased by multiplying the cooling rate α to enforce the convergence of the search (line 17). Lines 17–18 show that if the temperature reaches the pre-set minimum value Tmin , we double the value of Tb , and increase the temperature to min(Tb ,Tmax ) to help the search escape from locally optimal solutions. The maximum temperature Tmax is used to avoid the search from restarting from a randomly scratched solution. Finally, the search terminates when it reaches the time limit, as introduced in Section 6.1. The initial temperature T0 is set to 25, so that nearly 50% of inferior solutions will be accepted, as many studies advise. Tmax and Tmin are set to 100 and 0.01, respectively. Inner _limit is set to (|H|B)2 , where |H| is the number of customers, and B is the number of bids per bidder. Finally, the cooling rate α is set to 0.9.

Algorithm 3 SA. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

s ← Initial solution from upper layer or greedy constructive heuristic s∗ ← s, T ← T0 , Tb ← T0 REPEAT Inner _count ← 0 REPEAT Inner _count + + Select an improvement operator randomly Generate a feasible solution s with the selected operator IF z(s ) > z(s) s ← s ELSE Set s ← s with probability p = exp([z(s ) − z(s)]/T) IF z(s) > z(s∗ ) s∗ ← s , T b = T UNTIL Inner _count > Inner _limit T=α ·T IF T ≤ Tmin Tb = 2Tb ,T = min(Tb ,Tmax ) UNTIL time limit is exceeded

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