The prizecollecting vehicle routing problem with single and multiple depots and nonlinear cost
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Abstract
In this paper, we propose a new routing problem to model a highly relevant planning task in small package shipping. We consider the PrizeCollecting Vehicle Routing Problem with NonLinear cost in its single and multidepot version, which integrates the option of outsourcing customers to subcontractors instead of serving them with the private fleet. Thereby, a lower bound on the total customer demand to be served by the private fleet guarantees a high utilization of the fleet capacity. To represent the practical situation, where a discount is given by a subcontractor if larger amounts of packages are outsourced, subcontracting costs follow a nonlinear function. The considered problem is NPhard and we propose an Adaptive Variable Neighborhood Search algorithm to solve instances of realistic size. We propose new benchmark sets for the single and the multidepot problem, which are adapted from test instances of the capacitated VRP and the closely related MultiDepot VRP with Private fleet and Common carrier. In numerical studies, we investigate the performance of our algorithm on the newly generated test instances and on standard benchmark problems of related problems. Moreover, we study the effect of different cost functions and different values of the minimal demand to be served by the private fleet on the routing solutions obtained.
Keywords
Vehicle routing Prizecollecting Heuristic Nonlinear costIntroduction
The market situation for small package shippers (SPS) has drastically changed since the deregulation in the EU and the US. Before, the formerly big players like DHL ^{1} used to operate huge vehicle fleets and perform all lastmile deliveries with their own employees. However, rising competition has forced them to adopt the business model of using subcontractors for lastmile delivery as done by companies like DPD ^{2}. The subcontractors are paid per parcel delivered, saving the SPS the high fixed costs for vehicles and employees. Beside the outsourcing of entire unprofitable delivery areas, subcontractors are often used on the operational level to balance high demand fluctuations, in particular when the capacity of the selfoperated vehicle fleet is not sufficient to serve all customers on a given day. On such days, the operational task is to decide which customers should be served by the private fleet and which customers should be subcontracted. The decision has to balance the tradeoff between the cost of serving a customer [based on the solution of a Vehicle Routing Problem (VRP)] and the costs for subcontracting the customer. Chu (2005) modeled this planning problem, relaxing several practical constraints, as an extension of the Capacitated VRP (CVRP), which was later named VRP with Private fleet and Common carriers (VRPPC) (Bolduc et al. 2008). Extending the VRPPC, Stenger et al. (2012) proposed the MultiDepot VRPPC (MDVRPPC) that considers not only multiple selfowned depots but also different subcontractors with individual subcontracting costs and limited delivery radiuses.
However, both problems disregard important realworld characteristics. First, a lower bound on the customer demand served by the private fleet is generally mandatory in order to maintain the profitability of the vehicle fleet. Second, the costs charged by a subcontractor for serving additional customers follow a nonlinear cost function. Quantity discounts are given if the number of customers to outsource falls into a certain discount range. In this way, the subcontractor tries to improve the capacity utilization of his fleet.
We contribute by incorporating the abovedescribed realworld characteristics in a planning problem called PrizeCollecting MultiDepot Vehicle Routing Problem with NonLinear costs (PCMDVRPNL). It extends the MDVRPPC and the wellknown PrizeCollecting Traveling Salesman Problem (PCTSP), in which a prize is collected when visiting a customer and penalties are incurred for each unvisited customer. The task is to collect at least a given prize while minimizing the sum of distances traveled and penalty costs for unvisited customers. In the PCMDVRPNL, penalty costs are equal to subcontracting costs while at least a given customer demand (prize) has to be served by the private fleet. Besides the problem with multiple selfowned depots and multiple subcontractors, we also consider the singledepot case of the problem, denoted as PrizeCollecting Vehicle Routing Problem with NonLinear cost (PCVRPNL). This problem is modelled as a special case of the PCMDVRPNL, in which only one selfowned depot and one subcontractor with unlimited delivery radius are available.
As we are, to the best of our knowledge, the first dealing with these problems in their given form, we propose new benchmark sets for both problems. The benchmark for singledepot PCVRPNL is adapted from test instances of the classical VRP and the PCMDVRPNL instances are based on test problems of the closely related MDVRPPC. As solution method, we present an Adaptive Variable Neighborhood Search (AVNS) algorithm that is inspired by the AVNS of Stenger et al. (2012) but employs modified neighborhood structures, a random mechanism for ordering these neighborhoods and route and customer selection rules specifically adapted to the nonlinear problem addressed. The performance of the method is evaluated on the newly generated test instances and on available benchmark problems of the related VRPPC and MDVRPPC. Moreover, we study the effect of different cost functions on the route design and the subcontracting decisions as well as the influence of the minimum amount of demand to be served by the private fleet.
The remainder of the paper is structured as follows. First, we review the literature related to our work (see “Literature review”). Subsequently, we formulate the mixed integer program of the problem at hand (see “Mathematical model of the PCMDVRPNL”). In section “Solution method for the PCMDVRPNL”, we present the problemspecific AVNS. The mathematical model and the details of our solution approach are described for PCMDVRPNL, of which the single depot problem can be considered a special case as described above. In section “Computational studies”, we discuss the extensive numerical tests performed with our AVNS algorithm, followed by some concluding remarks in section “Conclusion”.
Literature review
In this section, we provide a brief review of the literature that is of importance to our work. The idea of prizecollecting first arose in the context of the iron and steel industry, where a PCTSP was used to model the operational scheduling of a steel rolling mill. Balas (1989) transferred this idea to the general case of a traveling salesman and studied structural properties. As mentioned above, a traveling salesman collects a prize for each city visited and has to pay a penalty for each city that remains unvisited. The objective is to minimize the total distance traveled and penalty costs incurred for unvisited cities while collecting at least a given amount of prize money.
Several solution methods for the PCTSP have been proposed. For example, Dell’Amico et al. (1998) proposed a heuristic using lagrangian relaxation to produce an initial solution and subsequently apply an extension and collaborate procedure in the improvement phase. Recently, Chaves and Lorena (2008) presented a hybrid metaheuristic for generating initial solutions using a combination of greedy randomized search procedure and VNS. Based on the generated solutions, clusters are formed and promising clusters are identified and further improved by means of a local search procedure. However, no generally used set of PCTSP benchmark problems exists and thus the quality of the proposed solution methods cannot be evaluated in a straightforward fashion. For an extended literature review, the reader is referred to Feillet et al. (2005), who provide an overview of the literature on traveling salesman problems with profits.
The PCTSP was extended to a PrizeCollecting Vehicle Routing Problem (PCVRP) by Tang and Wang (2006) in order to model the hot rolling production scheduling problem. Here, every customer represents an order to be scheduled that has a given length that corresponds to the demand of the customer. Each vehicle route describes a turn and the vehicle capacity corresponds to the maximum length of a turn. The objective is to find the schedule that minimizes the variable production and fixed setup costs and maximizes the profits of scheduled orders.
Chu (2005) presented an extension of the CVRP, in which customers can either be served by the private fleet or be outsourced to a common carrier. The costs for deliveries by the private fleet depend on the traveled distance and fixed vehicle cost. The common carrier is paid a fixed price per outsourced customer. The objective is to minimize the total costs including fixed vehicle costs, variable travel costs of the private fleet and the costs of assigning deliveries to the common carrier. To solve the problem (later named VRPPC by Bolduc et al. (2008), Chu (2005) proposed a simple heuristic based on the wellknown Clarke and Wright algorithm (Clarke and Wright 1964). Another heuristic that outperforms the approach of Chu (2005) was developed by Bolduc et al. (2007). They modeled the VRPPC as heterogeneous VRP and presented a randomized constructionimprovementperturbation heuristic. Furthermore, they generated two large sets of benchmark instances for the VRPPC with up to 480 customers, which are based on classical VRP instances. Recently, two Tabu Search (TS) heuristics have been developed for the VRPPC. Côté and Potvin (2009) presented a heuristic which is mainly based on the unified TS framework proposed by Cordeau et al. (1997). The solutions obtained by this heuristic were further improved by the TS of Potvin and Naud (2011) which enhances the earlier TS by the concept of ejection chains. Numerical studies show that ejection chains help to significantly improve the solution quality, in particular on instances with a heterogeneous vehicle fleet, but also strongly increase computing time.
As described above, this paper considers a multidepot problem that extends the classical MultiDepot Vehicle Routing Problem (MDVRP) first described by Wren and Holliday (1972). Compared to the singledepot VRP, only few heuristic solution methods were presented for the MDVRP. Chao et al. (1993) proposed a multiphase extension of the recordtorecord travel method of Dueck (1993) and were able to solve benchmark instances with up to 360 customers and 9 depots. Renaud et al. (1996) presented a tabusearch heuristic which first assigns customers to their closest depot and next determines an initial routing solution using the improved petal heuristic of Renaud et al. (1996). Subsequently, a fast improvement phase, an intensification and a diversification phase are applied, all based on a set of local search operators which exchange customers of up to three routes. Cordeau et al. (1997) proposed another tabu search heuristic for MDVRP that is also able to solve Periodic VRP (PVRP) and periodic TSP problems. To diversify the search, attributes that are added to solutions are analyzed and penalties are modified accordingly in order to efficiently guide the search. Pisinger and Ropke (2007) used an Adaptive Large Neighborhood Search (ALNS), which was designed for a wide range of routing problems, to solve the MDVRP. The solutions obtained by ALNS were of higher quality than those found by Cordeau et al. (1997) but required larger computing time. Recently, Vidal et al. (2012) presented a Hybrid Genetic Search with Adaptive Diversity Control (HGSADC) to solve MDVRP, PVRP and periodic MDVRP. The computational studies performed show considerable improvements for all problem classes. Stenger et al. (2012) proposed the MDVRPPC and developed an AVNS algorithm that biases the random shaking step. The approach obtains high quality solutions in short computing time for MDVRPPC and closely related problems, such as the MDVRP and the singledepot VRPPC.
Mathematical model of the PCMDVRPNL
In this section, we provide the mathematical formulation of the PCMDVRPNL, which includes the PCVRPNL as a special case.
We model the problem as an extension of the MDVRPPC proposed in Stenger et al. (2012). To define PCMDVRPNL, let G = (V, E) be an undirected, complete graph. The set of nodes V = J ∪ H is composed of the set J of customers and the set H of depots. Set H itself is partitioned into a subset I of selfowned depots and a subset L of subcontractor depots. The total capacity of a depot i ∈ H, i.e., the maximum total demand it can serve, is limited to w _{ i } units. A customer can either be served by a vehicle k of the set of private vehicles K or by a subcontractor. At most k _{max} identical vehicles of the private fleet K with a restricted capacity Q are available at each selfowned depot, where K ≤ k _{max}·I holds. For each employed vehicle a fixed cost of F is charged as well as the variable costs that are identical to the travel times c _{ ij } for traversing arc (i, j) ∈ E. With each customer j ∈ J a service time of t _{ j } units is associated. The maximum duration of a route is limited to t _{max}. Furthermore, r _{max} denotes the maximum delivery radius of subcontractor depots, i.e., customers outside this range are not served by the respective subcontractor.
The cost of subcontracting customer j ∈ J to depot l ∈ L is p _{ lj }. This cost is discounted with factor (1 − e _{ l }(g _{ l } ^{sub} )), where e _{ l }(g _{ l } ^{sub} ) denotes the discount given by subcontractor l for subcontracted demand g _{ l } ^{sub} . Moreover, at least T demand units have to be delivered by the private fleet.
The objective (1) is to minimize the sum of fixed costs of employed vehicles, travel and subcontracting costs. Constraints (2) ensure that each customer is either visited exactly once by a vehicle of the private fleet or it is subcontracted. Constraints (3) and (4) describe the capacity restrictions of vehicles and depots. Constraints (5) and (6) define that each node entered by a vehicle has to be left and that each route has to end at the depot from which it originated. Constraint (7) determines the minimum customer demand T to be served by the private fleet. Subtour elimination is ensured by Constraints (8). Constraints (9) guarantee that a customer is only assigned to a certain depot if there exist a route starting from this depot that visits the customer. Constraints (10) guarantee that each customer is assigned to a depot. The number of employed vehicles is restricted by (11). Compliance with route duration constraints is ensured in (12). Due to Constraints (13), a customer can only be subcontracted to depot l if it is not farther from l than r _{max} units. Equation (14) defines the subcontracted demand for each subcontractor. Binary variables are defined in Constraints (15) and (16) (Fig. 1).
The PCVRPNL is defined as special case of PCMDVRPNL, with I = 1, L = 1, r _{max} = ∞.
Solution method for the PCMDVRPNL
The PCMDVRPNL extends the already NPhard CVRP by realworld characteristics of SPS. Consequently, only small problems can be solved by means of an exact approach. In order to handle problem instances of realistic size, we propose an Adaptive VNS (AVNS) metaheuristic approach to solve both the single and multidepot version of the problem. AVNS, proposed by Stenger et al. (2012), is a metaheuristic that successfully combines ideas of Adaptive Large Neighborhood Search (ALNS) (Pisinger and Ropke 2007) and VNS. For a detailed description of the wellknown metaheuristic concepts VNS and ALNS, we refer the reader to Mladenović and Hansen (1997) and Pisinger and Ropke (2007).
AVNS follows the solution process of a VNS and, starting from an initial solution, performs local search on systematically changing neighborhoods. In detail, a new neighboring solution is determined in each shaking step by exchanging customers among routes according to a set of predefined neighborhood structures. In contrast to standard VNS, the routes and customers involved in this shaking are not selected randomly in AVNS. Instead, a set of methods that bias the route and customer selection are applied to explore the solution space more efficiently. AVNS incorporates these selection methods in a fashion similar to the treatment of the removal and insertion heuristics in ALNS. In each iteration, one route and one customer selection method are chosen, while the probability of each method depends on its success in former iterations. In this way, the heuristic adapts to the characteristics and requirements of the problem instance to be solved and to the current state of the solution process. In the next step, AVNS performs a greedy local search on the generated neighboring solution and the resulting solution is finally accepted according to a simulated annealing criterion. If the solution is accepted, it replaces the current starting solution and a new shaking step using the first neighborhood structure is performed. Otherwise, the solution found is rejected and the shaking procedure is repeated with a more distant neighborhood and the former starting solution.
In order to handle the outsourcing of customers to subcontracting depots, our heuristic uses the concept of virtual vehicles as proposed by Prosser and Shaw (1996) (see also Tang and Wang 2006; Bolduc et al. 2007). Here, a virtual vehicle with a capacity equal to the depot capacity w _{ i } is assigned to each subcontractor depot i ∈ L and all customers served by i are inserted into a corresponding virtual route. The length of the virtual subcontracting route is not considered in cost calculations as the subcontracting cost is based on outsourced quantities and not on the routing decision of the subcontractor.
In the following, we provide the details of our solution method: the “Initialization” phase , the “Adaptive shaking” phase including the route and customer selection methods, the “Local search” and the applied “Acceptance criterion”.
Initialization
 1.
Determine for each customer j ∈ J and each subcontractor depot l ∈ L the quotient \(\xi_{jl} ={\frac{p_{jl}}{q_j}}\) and insert ξ_{ jl } into the list P sorted in increasing order.
 2.
Choose the first element in P denoted by ξ_{ j_1l_1} and check whether (1) the distance between j _{1} and l _{1} is smaller than the maximal service radius r _{max} of the subcontractor, (2) the remaining capacity of depot l _{1} is greater than the demand q _{ j_1} and (3) the demand of subcontracted customers including the investigated customer is smaller or equal to the maximum demand to be outsourced, i.e., ∑_{ i ∈ O ∪ {j1}} q _{ i } ≤ ∑_{ j ∈ J } q _{ j } − T, where T denotes the minimum customer demand to be served by the private fleet (see section “Mathematical model of the PCMDVRPNL”) and O the list of already subcontracted customers. If all conditions are met, customer j _{1} is subcontracted, added to the list O and all ξ_{ j1l }, l ∈ L are removed from P. Otherwise, only entry ξ_{ j1l1} is removed from P.
 3.
Repeat step 2 until P is empty.
 1.
Select the route with the lowest demand at a randomly chosen depot with violations.
 2.
Remove all customers from this route and iteratively insert the customers at the cost optimal position concerning all routes. To be more precise, the insertion position is chosen such that the increase of total cost is minimized.
 3.
Repeat until the number of vehicles is lower or equal k _{max} at all depots.
Note that the obtained solution respects fleet restrictions but capacity constraints of the generated routes may be violated. In a final step, the solution obtained is improved by means the greedy local search described in section “Local search”.
Adaptive shaking
In the adaptive shaking phase, we perturb the initial solution based on a given set of neighborhood structures, which are applied following a random ordering. To avoid the evaluation of unprofitable neighboring solutions, we bias the shaking by means of problemspecific selection methods that determine the routes and customers involved in the selected neighborhood moves. In each iteration, the route and customer selection method to be applied is chosen by a roulette wheel selection mechanism. In the process, the probability of each method adapts according to the success of the selection method in former iterations, which is evaluated based on a scoring system.
In the following, we detail the utilized “Neighborhood structures”, the “Route and customer selection methods” and the “Adaptive mechanism”.
Neighborhood structures
The neighborhood structures used in our AVNSRN algorithm are either defined by a moveexchange or a cyclicexchange neighborhood operator (Thompson and Psaraftis 1993). For both operators, we define two sets of neighborhoods, one that only considers routes of the private fleet originating from identical depots and the other one allowing exchanges between all routes of the private fleet and the subcontractors. For all neighborhood operators and sets, the sequence length ω to be exchanged by neighborhood κ is randomly selected from interval \([0,\min(((\kappa1) \mod 6)+1, N)], \) where N denotes the number of customers in the route. This means that for each neighborhood operator, we consider exchanges of up to six customers, while the number is obviously restricted by the number of customers in the route.

Move customer sequence between vehicle routes of the private fleet originating from same depot Neighborhood structures κ = 1, …, 6 move a sequence of length ω customers from one route to another.

Move customer sequence between all routes of the private fleet and subcontractor Structures κ = 7, …, 12 are similar to the first set, however, customer sequences can be inserted into or removed from both routes that originate at a different depot and virtual routes of a subcontractor.

Cyclic exchange of customer sequence between routes of the private fleet originating from same depot This set of neighborhood structures exchanges customer sequences between up to four different routes in a cyclic fashion. For example, a neighborhood considering three routes r _{1}, r _{2}, r _{3} removes a customer sequence from r _{1}, inserts it into r _{2}, from where a sequence of customers is removed and transferred to route r _{3}. The sequence removed from r _{3} is moved to r _{1}, thus closing the cycle. Neighborhood structures κ = 13, …, 18 apply exchanges between two routes, κ = 19, …, 24 between up to three and κ = 25, …, 30 between up to four routes.

Cyclic exchange of customer sequence between all routes of the private fleet and subcontractor This set of 18 neighborhood structures is similar to the third set but customer sequences can be inserted into or removed from both routes that originate at a different depot and virtual routes of a subcontractor.
Route and customer selection methods
 1.
Use roulette wheel selection to determine route selection method.
 2.
Determine the first route k _{1} to be involved according to chosen route selection method.
 3.
Determine the remaining routes to be involved randomly, considering closeness measures.
 4.
Use roulette wheel selection to choose customer selection method.
 5.
For all involved routes, select randomly the number of customers ω to be exchanged and select the customers to be exchanged according to chosen selection method.
 6.
Perform the shaking move.

Random: The probability of all routes to be selected is equal. This is the standard procedure in VNS.

Longest route: The selection probability of a route is defined as being proportional to its total travel distance.

Unit longest route: The probability of a route to be selected is proportional to the ratio between total travel distance and total demand of the customers in the route. Routes with long distance and low demand are generally considered as inefficient and are therefore more likely to be chosen for modification.

Demand: The probability of a route is proportional to the average customer demand in this route. Biasing towards routes of the private fleet with high demand increases the chance of reaching the next discount step of a subcontractor.

Distance to Discount: The probability of a subcontracted route to be chosen is inversely proportional to the gap between the total demand in this route and the demand required for reaching the next discount step.
The first three methods correspond to the route selection methods successfully applied in Stenger et al. (2012) and the last two methods constitute additional methods that consider the problemspecific characteristics of the PCMDVRPNL. Note that methods longest route, unit longest route and demand are only applicable for routes of the private vehicle fleet and method distance to discount can only be applied to routes of a subcontractor. As described above, the selection method is exclusively used to determine the first route involved in the shaking step, the other routes are chosen randomly. However, to avoid the evaluation of unpromising solutions, we restrict the potential routes using the concept of an embedding rectangle as proposed by Stenger et al. (2012). In this way, we consider merely exchanges of customers among routes that are close to each other.
In the next step, a customer selection method is chosen and subsequently applied to each of the routes involved in the shaking step in order to determine the customers to be exchanged. Our AVNSRN applies the general selection methods described in Stenger et al. (2012) plus two problemspecific methods that aim at reaching higher discount levels. In case of routes of the private fleet, we exchange customer sequences of length ω (determined by selecting the first customer), while for subcontracted routes ω individual customers are selected by iteratively applying the selection method. We present the customer selection methods structured according to their applicability to the different types of routes.
Applicable to all routes:

Random: The selection probability is equal for all customers.

Distance to next route: The probability of a customer or a customer sequence being selected is inversely proportional to its distance to the center of gravity of the target route, i.e., the route in which the customers will be inserted. In this way, we bias towards customers that are close to the target route and thus more likely to be part of distanceefficient routes.
Only applicable to subcontracting routes:

Subcontracting cost: The selection probability of a customer is proportional to its subcontracting cost. Customers with high subcontracting cost are more likely to be served efficiently by the private fleet.

Unit subcontracting cost: The selection probability of a customer is proportional to the ratio between subcontracting cost and demand. This method targets at biasing customers with high subcontracting cost and/or low demand towards routes of the private fleet as they are likely to be efficiently served by them.

Inverse demand: The selection probability of a customer is inversely proportional to its demand. Since customers with low demand can be inserted more easily into existing routes of the private fleet, we favor the removal of such customers from subcontracted routes.
Only applicable to routes of selfowned vehicles:

Distance to neighboring customers: The probability of selecting a customer is inversely proportional to the sum of the distance between the first customer of the sequence and its predecessor and the last customer and its successor in the route. Customer sequences that are distant to other customers in a route are most likely to be exchanged in a profitable way.

Demand: The probability of a customer sequence of being selected in a route of a selfowned vehicle is proportional to its demand. This favors the subcontracting of customers with high demand, which potentially leads to higher discount levels.
It is worth mentioning that a shaking move is capable of removing all customers of a route. In case such a move is accepted by the algorithm, the number of routes is reduced. In our computational studies, we found that this happens quite frequently, in particular due to the possibility of profitably moving customers into subcontracted routes. Without further measures allowing the reopening of routes, the search often gets stuck in low quality regions of the search space. To counteract, we keep empty routes in the set of potential routes involved in the shaking. To be more precise, for every depot that has at least one vehicle left, we add an empty route. This allows the algorithm to open an additional route by inserting customers in an empty route during shaking.
Adaptive mechanism
Our AVNSRN involves a quite extensive set of route and customer selection methods that clearly differ in their objective, i.e., they bias the shaking step in different directions. Depending on the problem instance addressed and the current state of the solution process, the success of individual methods will vary. Therefore, AVNSRN follows the approach of ALNS and chooses the selection method to be used in each iteration based on associated probabilities that depend on the success of the respective method in former iterations. To be more precise, each selection method i is assigned a weight ζ_{ i } that represents the success of the method. Given a set of s different methods, roulette wheel selection is applied in each shaking step to determine the method to be used, where the probability of selecting method i is equal to \(\frac{\zeta_i}{\sum\nolimits_{j=1}^{s}{\zeta_j}}. \) Note that route selection and customer selection are treated separately in this process.
At the beginning of the search, all methods are assigned an equal weight ζ_{0}. During ϕ iterations, the success of a method is evaluated by a scoring mechanism. In case the shaking step using method i leads to a new overall best solution, the score of the method is increased by the amount δ_{1}. If the obtained solution improves on the incumbent, a somewhat lower score of δ_{2} is assigned. Finally, if a solution is worse than the incumbent but nevertheless accepted by the algorithm, the score is increased by δ_{3}. After the evaluation period, the weights of all methods are updated as follows: Let \(\varsigma_i\) denote the number of times method i was used during the evaluation period and let μ_{ i } describe the obtained score of method i, then the new weight ζ_{ i } is equal to \(\zeta_i^{\rm old}(1\varrho)+\varrho \frac{\mu}{\varsigma_i}. \) Factor ϱ ∈ [0,1] is used to balance between the recent and the past success of a method when updating the weight (Pisinger and Ropke 2007).
Local search
After the shaking, greedy local search is used to determine a local optimum with respect to the applied neighborhood operators. Note that the local search is only performed on the routes involved in shaking. For intraroute moves, we use the wellknown edgeexchange operator 2opt, that exchanges two existing edges with two new ones (Lin 1965). For interroute moves, a relocate and a swap operator are applied. Relocate removes a single customer from a route and inserts it into the other route at the costoptimal position. The swap operator simply exchanges the position of two customers from different routes. The interroute moves are applied to all routes of the private fleet and the virtual routes of the subcontractors. A more detailed description of the local search process is provided in pseudocode in Fig. 3.
Acceptance criterion
Contrary to standard VNS approaches, where only improving solutions are accepted, we use a simulated annealingbased acceptance criterion, whose improvement potential has been shown in recent works (see, e.g., Pirkwieser and Raidl 2008; see, e.g., Hemmelmayr et al. 2009; see, e.g., Stenger et al. 2012). The solution x′′ obtained in the local search is compared to the currently best solution x and accepted if it improves the latter. Additionally, we accept deteriorating moves according to the Metropolis probability \(\hbox{e}^{\frac{(f(x^{\prime\prime})f(x))}{\theta}},\) where \(f(\cdot)\) denotes the objective function value and θ the current temperature, which is used to control the degree of diversification. Starting from an initial value θ_{init} > 0, the temperature is decreased by the factor η_{dec} after each AVNS iteration. In this way, the probability of accepting deteriorating solutions is reduced during the search ending in an intensification phase rejecting all nonimproving solutions.
Computational studies
We perform extensive numerical tests to assess the performance of the proposed AVNSRN algorithm and to study the influence of several problem parameters. We design new benchmark sets for PCMDVRPNL and its singledepot version as we are, to the best of our knowledge, the first dealing with these problems in their given form (see section “Generation of benchmark instances”). To find the best parameter values for our algorithm, we performed numerous tests on new benchmark instances (see section “Parameter setting”). Our first experiments assess the performance of our AVNSRN on the generated benchmark instances and investigate the influence of the nonlinear cost function on the routing solutions obtained (see section “Performance of AVNSRN on PC(MD)VRPNL and the influence of nonlinear subcontracting cost”). This is achieved by a comparison with the results realized with a linear cost function. In Section “Varying the minimum demand to be delivered by the private fleet”, we study the impact of different values of the mandatory demand to be served by the private fleet T and show that this parameter significantly influences the subcontracting decisions and hence the overall solutions. Finally, to substantiate the competitive performance of our AVNSRN, we present results obtained on benchmark instances proposed for the MDVRPPC by Stenger et al. (2012), which is closely related to the PCMDVRPNL (see section “Evaluating the algorithmic performance on benchmark instances of related problems”) and on standard test instances of the VRPPC, which is closely related to the singledepot PCVRPNL.
All numerical tests were conducted on a desktop computer with an Intel i5 Processor clocked at 2.67 GHz and 4GB RAM. The algorithm is implemented in Java.
Generation of benchmark instances
To generate test problems for the PCMDVRPNL, we use the MDVRPPC instances proposed by Stenger et al. (2012) as a base. The subcontracting price p _{ lj } of these instances depends mainly on the customer demand and can thus be used for our problem. The minimum demand T to be served by the private fleet is set to \(0.7 \cdot \tilde{q}. \) In addition, we give an upper bound which represents a high quality solution without subcontracting, i.e., a high quality MDVRP solution (Cordeau et al. 1997). The upper bound provides a simple comparison value to evaluate the solution quality of our algorithm on the newly generated instances.
As PCVRPNL is an extension of the CVRP, we use the VRP benchmark instances proposed by Christofides and Eilon (1969) and Golden et al. (1998) as basis for designing a new benchmark set for our single depot problem. The benchmark design is inspired by the procedure for generating VRPPC instances described in Bolduc et al. (2008). However, utilization of the VRPPC instances presented there is not appropriate since their subcontracting costs mainly depend on the customers’ distance to the depot, whereas in realworld small package shipping, prices charged by a subcontractor are based on customer demand.
Of the original CVRP instances, we utilize the depot and customer coordinates, the customer demand values and the vehicle capacities. The fixed vehicle cost F and the standard subcontracting price p _{ i } for each customer are computed as follows. Let C(x ^{*}) be the objective function value and k ^{*} the number of vehicles of a highquality solution to the respective CVRP base instance.
Parameter setting
Results obtained on a subset of PCMDVRPNL instances with different parameter settings
Simulated annealing  
θ_{init}  20  0  50 
\(\Updelta_{\rm avg}\)  0.0 %  0.87 %  −0.17 % 
η_{dec}  0.9995  0.999  0.9998 
\(\Updelta_{\rm avg}\)  0.0 %  0.03 %  −0.34 % 
Penalties  
Pen _{min}  1  10  100 
\(\Updelta_{\rm avg}\)  0.0 %  0.11 %  0.45 % 
Pen _{max}  1,000  500  2,500 
\(\Updelta_{\rm avg}\)  0.0 %  −0.05 %  −0.10 % 
Adaptive mechanism  
ϱ  0.3  0.5  0.7 
\(\Updelta_{\rm avg}\)  0.0 %  −0.16 %  −0.08 % 
ϕ  30  15  45 
\(\Updelta_{\rm avg}\)  0.0 %  −0.55 %  −0.02 % 
δ_{1}/δ_{2}/δ_{3}  9/2/1  9/5/1  9/2/0 
\(\Updelta_{\rm avg}\)  0.0 %  −0.13 %  −0.14 % 
Performance of AVNSRN on PC(MD)VRPNL and the influence of nonlinear subcontracting cost
In the following, we present the results obtained with our AVNSRN on the newly designed benchmark sets of PCMDVRPNL and PCVRPNL in their original form and with a linearized cost function. For all problem instances, we present the upper bound computed as explained above (Cost_{UB} and the number of vehicles k _{UB}), the best solution found in ten runs (Cost_{best} and k _{best}), the deviation of this solution from the upper bound in percentage (\(\Updelta_{\rm best}\)), the average solution cost over the ten runs plus the deviation from the upper bound (Cost_{avg} and \(\Updelta_{\rm avg}\)) and finally the average percentage of subcontracted customers (SC _{avg}). For the original instances, we additionally report the average computing time in seconds (CPU(s)), computing times for the linearized instances are not reported due to their similarity.
Results obtained on the original benchmark set for the PCMDVRPNL and the set with a linearized cost function
Instance  Cost_{UB}  k _{UB}  PCMDVRPNLoriginal  PCMDVRPNLlinearized  

Cost_{best}  \(\Updelta_{\rm best}\)(%)  k _{best}  Cost_{avg}  \(\Updelta_{\rm avg}\)(%)  SC _{avg}(%)  CPU(s)  Cost_{best}  \(\Updelta_{\rm best}\)(%)  k _{best}  Cost_{avg}  \(\Updelta_{\rm avg}\)(%)  SC _{avg}(%)  CPU(s)  
Sp01  1,236.87  11  1,100.48  −11.03  7  1,103.52  −10.78  29.20  17.4  1,109.77  −10.28  7  1,113.31  −9.99  31.80  34.6 
Sp02  973.53  5  937.21  −3.73  4  939.89  −3.46  21.80  23.5  924.80  −5.01  4  925.39  −4.94  23.00  77.2 
Sp03  1,301.19  11  1,153.46  −1,1.35  7  1,162.33  −10.67  32.40  32.6  1,169.48  −10.12  7  1,175.79  −9.64  34.00  141.8 
Sp04  1,901.04  15  1,767.80  −7.01  11  1,772.70  −6.75  26.20  62.7  1,765.13  −7.15  11  1,771.40  −6.82  25.70  127.6 
Sp05  1,550.03  8  1,371.05  −11.55  6  1,371.38  −11.53  24.00  125.5  1,369.13  −11.67  6  1,369.32  −11.66  24.90  245.3 
Sp06  1,836.50  16  1,626.84  −11.42  11  1,632.94  −11.08  30.50  77.7  1,633.25  −11.07  11  1,639.85  −10.71  31.50  136.7 
Sp07  1,841.97  16  1,647.37  −10.56  11  1,655.12  −10.14  27.70  48.0  1,634.94  −11.24  11  1,647.24  −10.57  28.10  69.0 
Sp08  8,872.78  25  8,309.24  −6.35  18  8,391.77  −5.42  24.78  786.9  8,341.30  −5.99  18  8,401.82  −5.31  24.38  1,169.9 
Sp09  7,498.66  26  7,177.88  −4.28  18  7,213.60  −3.80  30.64  621.9  7,061.09  −5.84  18  7,157.66  −4.55  30.52  1,007.7 
Sp10  7,271.22  26  6,506.43  −10.52  18  6,553.59  −9.87  32.73  522.2  6,488.34  −10.77  18  6,534.17  −10.14  33.69  879.3 
Sp11  7,186.06  26  6,313.71  −12.14  18  6,362.00  −11.47  36.51  461.9  6,292.37  −12.44  18  6,322.78  −12.01  36.22  692.5 
Sp12  2,598.95  8  2,247.61  −13.52  6  2,252.07  −13.35  35.00  61.6  2,255.24  −13.22  6  2,256.68  −13.17  33.13  111.8 
Sp13  2,598.95  8  2,219.79  −14.59  6  2,229.15  −14.23  40.63  73.3  2,201.04  −15.31  6  2,209.11  −15.00  41.38  92.0 
Sp14  2,800.12  8  2,363.92  −15.58  6  2,363.92  −15.58  40.00  68.1  2,354.37  −15.92  6  2,393.80  −14.51  41.25  110.8 
Sp15  5,065.42  16  4,363.31  −13.86  11  4,374.82  −13.63  38.63  130.9  4,350.50  −14.11  11  4,352.80  −14.07  41.06  223.2 
Sp16  5,132.23  16  4,354.42  −15.16  11  4,358.93  −15.07  40.88  137.6  4,340.03  −15.44  11  4,360.61  −15.03  41.56  205.4 
Sp17  5,269.09  16  4,372.57  −17.01  11  4,377.25  −16.93  46.94  237.8  4,358.64  −17.28  11  4,373.26  −17.00  46.63  257.4 
Sp18  7,382.85  23  6,204.83  −15.96  16  6,218.06  −15.78  47.79  264.9  6,181.17  −16.28  16  6,213.61  −15.84  48.04  345.6 
Sp19  7,667.06  24  6,126.71  −20.09  16  6,149.59  −19.79  54.58  323.6  6,108.85  −20.32  16  6,137.21  −19.95  54.63  495.8 
Sp20  7,898.07  24  6,194.82  −21.57  16  6204.68  −21.44  55.17  265.5  6,170.75  −21.87  16  6,177.82  −21.78  54.67  439.0 
Sp21  1,0914.84  34  8,955.33  −17.95  24  8,972.64  −17.79  56.44  399.9  8,939.03  −18.10  24  8,960.02  −17.91  56.08  1056.3 
Sp22  1,1462.16  36  9,238.15  −19.40  24  9,259.96  −19.21  55.14  328.6  9,228.07  −19.49  24  9,258.82  −19.22  54.97  895.7 
Sp23  11,838.75  36  9,458.06  −20.11  24  9,500.82  −19.75  53.42  502.3  9,436.26  −20.29  24  9,477.05  −19.95  53.53  958.7 
Spr01  1,741.32  4  1,606.23  −7.76  3  1,606.23  −7.76  14.58  44.7  1,595.28  −8.39  3  1,596.66  −8.31  14.38  79.6 
Spr02  2,587.34  8  2,305.25  −10.90  6  2,315.72  −10.50  13.13  101.0  2,301.32  −11.05  6  2,305.21  −10.90  12.50  201.6 
Spr03  3,723.80  12  3,239.03  −13.02  8  3,286.14  −11.75  10.21  173.8  3,240.77  −12.97  8  3,251.62  −12.68  17.71  297.9 
Spr04  3,978.31  16  3,595.55  −9.62  13  3,629.45  −8.77  8.65  218.9  3,575.24  −10.13  12  3,595.98  −9.61  10.26  414.2 
Spr05  4,731.20  20  4,548.49  −3.86  18  4,572.15  −3.36  8.13  287.0  4,522.94  −4.40  18  4,554.05  −3.74  9.00  407.5 
Spr06  5,556.30  24  5,204.22  −6.34  20  5,268.48  −5.18  6.04  364.9  5,185.81  −6.67  20  5,216.69  −6.11  11.35  617.3 
Spr07  2,169.56  6  1,969.81  −9.21  5  1,969.81  −9.21  8.33  33.5  1,968.15  −9.28  5  1,968.15  −9.28  8.33  63.8 
Spr08  3,344.85  12  2,987.89  −10.67  9  3,031.42  −9.37  14.79  126.7  2,955.41  −11.64  8  2,967.18  −11.29  24.51  212.1 
Spr09  4,293.20  18  3,940.58  −8.21  14  3,963.50  −7.68  15.56  209.7  3,899.89  −9.16  13  3,927.11  −8.53  20.09  435.1 
S−pr10  5,748.26  24  5,383.28  −6.35  19  5,435.59  −5.44  20.35  365.5  5,344.27  −7.03  18  5,390.83  −6.22  22.99  870.3 
Average  17.5  −11.84  12.6  −11.41  30.33  227.3  −12.12  12.5  −11.71  31.57  405.2 
Results obtained on the original benchmark set for the PCVRPNL and the set with a linearized cost function
Instance  Cost_{UB}  k _{UB}  PCVRPNLoriginal  PCVRPNLlinearized  

Cost_{best}  \(\Updelta_{\rm best}\)(%)  k _{best}  Cost_{avg}  \(\Updelta_{\rm avg}\)(%)  SC _{avg}(%)  CPU(s)  Cost_{best}  \(\Updelta_{\rm best}\)(%)  k _{best}  Cost_{avg}  \(\Updelta_{\rm avg}\)(%)  SC _{avg}(%)  CPU(s)  
CEP01  1,049.22  5  946.16  −9.42  4  946.1629  −9.42  30.00  136.5  990.37  −5.61  4  990.37  −5.61  30.00  211.5 
CEP02  1,670.52  11  1,482.8  −10.69  7  1,483.93  −10.62  36.27  136.3  1,486.50  −11.02  7  1,490.03  −10.80  36.67  294.8 
CEP03  1,652.28  8  1,455  −11.83  6  1,459.75  −11.54  35.50  321.8  1,480.13  −10.42  6  1,485.41  −10.10  31.80  431.3 
CEP04  2,056.84  12  1,770.7  −13.56  8  1,798.071  −12.22  34.53  582.5  1,783.30  −13.30  8  1,786.04  −13.17  37.53  1,135.6 
CEP05  2,590.82  17  2,324.6  −10.02  12  2,333.105  −9.69  31.61  439.2  2,367.08  −8.64  12  2,374.45  −8.35  35.43  1025.1 
CEP06  1,110.86  6  908.71  −17.94  4  908.7138  −17.94  30.00  138.2  955.58  −13.98  4  955.58  −13.98  30.00  202.8 
CEP07  1819.36  11  1514.4  −16.41  7  1,515.452  −16.35  35.73  134.9  1,519.03  −16.51  7  1,522.01  −16.34  36.67  267.5 
CEP08  1,731.88  9  1,432.2  −17.21  6  1,436.033  −16.99  35.90  285.9  1,460.87  −15.65  6  1,461.23  −15.63  29.40  491.2 
CEP09  2,325.10  14  1,826.9  −21.41  8  1,869.036  −19.60  34.13  497.8  1,841.23  −20.81  8  1,843.85  −20.70  37.13  1,046.6 
CEP10  2,805.62  18  2,335.2  −16.05  12  2,344.136  −15.73  31.86  509.9  2,388.57  −14.86  12  2,396.00  −14.60  34.22  995.5 
CEP11  2,084.22  7  1918.3  −7.69  5  1,922.233  −7.50  31.08  601.6  1,946.83  −6.59  5  1,947.56  −6.56  31.83  812.8 
CEP12  1,639.12  10  1,471.6  −9.69  7  1,474.877  −9.49  25.30  247.8  1,512.88  −7.70  7  1,516.38  −7.49  29.90  310.6 
CEP13  3,099.88  11  2126.8  −30.97  5  2,130.561  −30.85  30.83  794.1  2,125.99  −31.42  5  2,166.10  −30.12  31.25  1,188.1 
CEP14  1,732.74  11  1,470.4  −14.73  7  1,473.93  −14.52  24.70  266.2  1,517.90  −12.40  7  1,521.28  −12.20  29.20  288.9 
Average CEP  10.7  −14.83  7.0  −14.46  31.96  363.7  −13.49  7.0  −13.26  32.93  621.6  
GP01  11,255.08  9  9,901.5  −12.03  7  9,971.30  −11.41  28.08  1,459.0  1,0322.18  −8.29  7  10,368.34  −7.88  32.08  2161.1 
GP02  16,895.84  10  14,157  −16.21  7  14,237.13  −15.74  36.31  2,503.3  14,623.34  −13.45  7  14,712.13  −12.92  42.25  2,502.3 
GP03  22,072.44  10  18,246  −17.34  7  18,449.12  −16.42  35.80  25,13.9  19,081.50  −13.55  7  19,307.47  −12.53  39.00  2,504.7 
GP04  27,249.04  10  23,416  −14.07  7  23,531.87  −13.64  41.58  2,504.9  24,008.95  −11.89  7  24,218.14  −11.12  43.04  2,505.8 
GP05  12,921.96  5  11,224  −13.14  4  11,375.99  −11.96  33.90  2,081.5  11,778.76  −8.85  4  11,804.36  −8.65  34.25  2,449.7 
GP06  16,825.76  7  13878  −17.52  5  13,920.43  −17.27  37.43  2,463.0  14,603.88  −13.21  5  14,677.29  −12.77  44.39  2,529.3 
GP07  20,391.12  9  16,825  −17.49  6  16,923.12  −17.01  40.00  2,503.9  17,299.10  −15.16  6  17,412.11  −14.61  45.17  2,502.8 
GP08  23,327.10  8  20,687  −11.32  8  20,761.16  −11.00  28.50  2,502.2  20,499.41  −12.12  7  21,677.00  −7.07  32.39  2,503.7 
GP09  1,166.78  14  1,023.8  −12.25  10  1,027.76  −11.91  45.22  2,083.9  1,027.17  −11.97  10  1,031.03  −11.63  47.80  2,524.5 
GP10  1,483.12  16  1,288.7  −13.11  11  1,299.12  −12.41  45.17  2,524.3  1,289.98  −13.02  11  1,297.25  −12.53  48.08  2,564.4 
GP11  1,836.90  18  1,611.7  −12.26  12  1,642.25  −10.60  45.29  2,502.0  1,643.03  −10.55  13  1,650.10  −10.17  47.44  2,502.6 
GP12  2,214.38  18  2,076.3  −6.24  14  2,092.78  −5.49  39.54  2,503.5  2,032.75  −8.20  14  2,054.12  −7.24  47.95  2,507.9 
GP13  1,718.22  26  1,551.7  −9.69  18  1,563.60  −9.00  40.79  1,024.3  1,546.71  −9.98  18  1,565.77  −8.87  45.24  1,374.7 
GP14  2,162.62  30  1,942.8  −10.16  21  1,960.02  −9.37  41.72  1,197.3  1,932.97  −10.62  21  1,944.59  −10.08  45.66  2,093.1 
GP15  2,690.46  33  2,414.3  −10.26  24  2,424.33  −9.89  41.87  1,908.8  2,420.44  −10.04  24  2,429.85  −9.69  46.24  2,509.4 
GP16  3,245.38  37  2,905  −10.49  27  2,919.19  −10.05  42.17  2,452.8  2,883.69  −11.14  26  2,914.68  −10.19  46.65  2,526.2 
GP17  1,415.58  22  1,292.5  −8.69  16  1,298.40  −8.28  27.17  497.6  1,306.92  −7.68  16  1,310.22  −7.44  29.92  836.7 
GP18  1,997.46  27  1,776.3  −11.07  19  1,811.41  −9.31  28.90  1,001.0  1,779.08  −10.93  19  1,815.24  −9.12  30.53  1543.1 
GP19  2,733.72  33  2,420.9  −11.44  23  2,451.35  −10.33  27.33  966.4  2,423.38  −11.35  23  2,440.80  −10.71  30.28  1,937.8 
GP20  3,640.18  38  3,191.9  −12.31  27  3,200.68  −12.07  28.69  1,767.6  3,202.29  −12.03  27  3,216.71  −11.63  30.40  2,410.8 
Average GP  19.0  −12.36  13.7  7,643.05  −11.66  36.77  1,948.1  7,785.2764  −11.20  13.6  7,892.36  −10.34  40.44  2,249.5  
Total average  15.9  −13.51  10.9  −12.95  34.79  1295.7  −12.15  10.9  −11.54  37.35  1,579.2 
For both discount functions, the solutions improve the VRPbased upper bound by more than 12 % while requiring moderate computing times. In addition, the number of vehicles required is reduced by more than 30 %. This shows again that our algorithm is able to identify customers that can be profitably subcontracted and to construct costefficient vehicle routes. Comparing the solutions obtained with the two different cost functions, the deviation from the upper bound Cost_{UB} as well as the number of vehicles required are almost equal for both cost functions. However, almost 7 % less customers are subcontracted when the stepwise cost function is considered. This can be explained by the fact that the stepwise function reaches e _{max} earlier, i.e., with less subcontracted demand. In case of the linear function, increasing the subcontracted demand can always be profitable up to \(\tilde{q}T\) units as the discount factor continuously increases.
Compared to the multi depot case, the average improvement of the upper bound is higher. In the multidepot problem different subcontractors with a limited delivery radius and capacity exist. Thus, the demand of outsourced customers is distributed among several subcontractors and maximal possible discounts are not reached. Our algorithm tries to maximize the discount by filling the vehicles of a subcontractor but the restricted delivery radiuses often counteracts this objective.
Varying the minimum demand to be delivered by the private fleet
With increasing value of \(\frac{T}{\tilde{q}}, \) the flexibility to outsource customers decreases and the solution quality clearly suffers. Similarly, the number of subcontracted customers decreases when the minimum demand to be served by the private fleet is increased. However, reducing T below \(0.7 \tilde{q}\) has only a very slight influence on the gap to the upper bound while still significantly increasing the number of subcontracted customers. This indicates that after the most unprofitable customers have been outsourced, a high number of solutions with similar solution quality exist. Their quality only slightly depends on further subcontracting customers.
Although the rough tendency of the outcome of this study appears expectable, the results show the strong influence of the important realworld constraint defining a lower bound on the demand served by the private fleet. In addition, the results prove again the suitability of our algorithm to handle the subcontracting decision while paying attention to the prizecollecting constraint.
Evaluating the algorithmic performance on benchmark instances of related problems
Results obtained on MDVRPPC benchmark instances compared to AVNS algorithm of Stenger et al. (2012)
Instance  BKS  SVES  AVNSRN  

Cost_{best}  \(\Updelta_{\rm best}\)(%)  Cost_{avg}  \(\Updelta_{\rm avg}\)(%)  CPU(s)  Cost_{best}  \(\Updelta_{\rm best}\)(%)  Cost_{avg}  \(\Updelta_{\rm avg}\)(%)  CPU(s)  
sp01  1,145.01  1,145.01  0.00  1,148.78  0.33  13.3  1,145.01  0.00  1,148.30  0.29  10.7 
sp02  937.21  937.21  0.00  940.52  0.35  20.5  937.21  0.00  938.32  0.12  16.0 
sp03  1,192.51  1,192.51  0.00  1,199.86  0.62  30.6  1,192.51  0.00  1,199.49  0.59  20.7 
sp04  1,827.27  1,828.74  0.08  1,834.95  0.42  34.7  1,831.94  0.26  1,838.41  0.61  27.7 
sp05  1,406.68  1406.68  0.00  1,411.72  0.36  88.7  1,406.68  0.00  1,413.04  0.45  82.1 
sp06  1,686.32  1,686.59  0.02  1,689.93  0.21  36.5  1,689.37  0.18  1,692.14  0.35  33.0 
sp07  1,666.03  1668.53  0.15  1,676.36  0.62  32.4  1,672.18  0.37  1,683.12  1.03  27.6 
sp08  8,625.75  8,661.27  0.41  8,702.42  0.89  233.2  8,673.50  0.55  8,706.75  0.94  156.7 
sp09  7,251.74  7,258.54  0.09  7,334.25  1.14  268.4  7,275.68  0.33  7,324.22  1.00  321.2 
sp10  6,606.12  6,655.90  0.75  6,679.96  1.12  179.0  6,639.91  0.51  6,688.54  1.25  249.9 
sp11  6,353.59  6,353.59  0.00  6,386.66  0.52  203.5  6,355.02  0.02  6,409.87  0.89  318.9 
sp12  2,303.31  2,303.31  0.00  2,307.33  0.17  53.2  2,303.31  0.00  2,304.99  0.07  32.6 
sp12  2,278.14  2,278.14  0.00  2,278.14  0.00  38.8  2,278.14  0.00  2,278.14  0.00  40.9 
sp14  2,404.72  2,404.72  0.00  2,407.28  0.11  41.0  2,404.72  0.00  2,404.72  0.00  35.8 
sp15  4,456.02  4,459.41  0.08  4,471.65  0.35  73.8  4,459.41  0.08  4,473.90  0.40  58.3 
sp16  4,449.41  4,449.41  0.00  4,465.02  0.35  67.3  4,449.41  0.00  4,461.24  0.27  67.8 
sp17  4,442.24  4,448.10  0.13  4,480.52  0.86  102.9  4,446.10  0.09  4,485.21  0.97  76.3 
sp18  6,269.70  6,272.70  0.05  6,313.71  0.70  122.3  6,270.70  0.02  6,284.63  0.24  94.4 
sp19  6,164.91  6,166.91  0.03  6,185.15  0.33  121.7  6,166.91  0.03  6,178.20  0.22  148.3 
sp20  6,240.66  6,245.90  0.08  6259.78  0.31  102.0  6,240.66  0.00  6,257.98  0.28  114.9 
sp21  8,758.25  8,759.25  0.01  8,790.24  0.37  178.9  8,767.11  0.10  8,818.00  0.68  187.0 
sp22  9,275.24  9,275.24  0.00  9,299.56  0.26  118.1  9,278.24  0.03  9,295.21  0.22  203.2 
sp23  9,546.29  9,581.64  0.37  9,609.59  0.66  162.1  9,557.54  0.12  9,598.85  0.55  188.4 
spr01  1,606.23  1,606.23  0.00  1,627.51  1.32  34.7  1,606.23  0.00  1,613.14  0.43  21.8 
spr02  2,303.99  2,306.76  0.12  2,330.14  1.14  74.8  2,304.99  0.04  2,323.08  0.83  62.5 
spr03  3,278.81  3,283.52  0.14  3,319.86  1.25  129.7  3,282.01  0.10  3,303.94  0.77  113.9 
spr04  3,573.21  3,598.16  0.70  3,666.68  2.62  213.5  3,603.69  0.85  3,652.40  2.22  158.6 
spr05  4,519.67  4,519.67  0.00  4,569.21  1.10  170.4  4,525.87  0.14  4,563.84  0.98  168.7 
spr06  5,202.88  5,232.78  0.57  5,271.81  1.32  329.2  5,239.48  0.70  5,268.96  1.27  197.1 
spr07  1,969.81  1,969.81  0.00  1,969.81  0.00  29.1  1,969.81  0.00  1,969.81  0.00  22.5 
spr08  2,979.94  2,994.37  0.48  3,038.74  1.97  99.8  2,996.38  0.55  3,043.59  2.14  84.3 
spr09  3,913.70  3,914.22  0.01  3,967.92  1.39  166.9  3,913.70  0.00  3,957.01  1.11  150.9 
spr10  5,399.10  5,457.46  1.08  5,499.17  1.85  263.8  5,417.12  0.33  5,497.26  1.82  222.9 
Average  0.16  0.76  116.2  0.16  0.69  112.6 
Although our AVNSRN is specifically adapted to the PCMDVRPNL and the parameter tuning is carried out on PCMDVRPNL instances, it obtains competitive results that even slightly improve on the average solution quality, requiring basically identical runtime. Furthermore, we found new best solutions for two benchmark instances during these 10 runs (BKS marked bold) and a total of 16 new best solutions during our overall testing (BKS marked in italics). These results further confirm the competitiveness of the proposed method.
Results of AVNSRN on VRPPC benchmark instances
Instance  BKS  TS+  TS  AVNS  AVNSRN  

\(\Updelta_{\rm best}\)(%)  CPU(s)  \(\Updelta_{\rm best}\)(%)  CPU(s)  \(\Updelta_{\rm best}\)(%)  CPU(s)  \(\Updelta_{\rm best}\)(%)  \(\Updelta_{\rm avg}\)(%)  CPU(s)  
CE01  1,119.47  0.00  24.9  0.00  24.3  0.40  92.5  0.00  0.41  81.2 
CE02  1,814.52  0.00  33.9  0.00  33.0  0.00  48.6  0.00  0.13  63.4 
CE03  1,919.05  0.60  81.0  0.11  78.6  0.09  212.1  0.00  0.65  258.1 
CE04  2,507.44  0.71  200.6  0.71  193.2  0.18  279.7  0.07  0.74  179.6 
CE05  3,097.67  0.63  353.2  0.51  309.9  0.07  228.6  0.45  0.76  132.9 
CE06  1,207.47  0.00  25.2  0.00  25.5  0.03  75.9  0.00  0.11  79.8 
CE07  2,004.53  0.10  34.0  0.10  32.7  0.47  50.9  0.00  0.21  61.0 
CE08  2,052.05  0.22  81.6  0.40  85.1  0.00  253.1  0.00  0.50  251.1 
CE09  2,422.74  0.55  188.2  0.65  185.3  0.40  259.0  0.35  0.71  190.8 
CE10  3,383.00  0.56  345.7  0.70  311.1  0.25  201.0  0.18  0.71  162.2 
CE11  2,330.94  0.06  131.0  0.96  126.3  0.05  316.0  0.00  0.06  370.5 
CE12  1,952.86  0.00  59.5  0.00  60.4  0.04  92.9  0.04  0.04  107.5 
CE13  2,858.94  0.07  132.1  0.83  130.0  0.00  278.5  0.00  0.07  351.4 
CE14  2,213.02  0.18  64.2  0.18  65.0  0.11  93.2  0.11  0.11  148.8 
Average CE  0.26  125.4  0.37  118.6  0.15  177.3  0.09  0.37  174.2  
G01  14,123.38  0.47  1,183.8  0.68  638.3  0.24  652.6  0.23  0.39  979.3 
G02  19,155.17  0.28  5,220.6  3.00  1,215.2  0.26  1,558.4  0.17  0.67  1,809.6 
G03  24,411.37  0.74  5,940.4  5.09  2,241.3  0.78  2,356.1  0.51  1.21  2,267.4 
G04  34,275.11  1.54  5,508.7  5.10  3,833.9  0.41  2,500.9  0.76  1.19  2,241.1 
G05  14,229.24  0.23  847.8  3.12  875.0  0.30  1,301.1  0.33  0.63  2263.5 
G06  21,396.38  0.48  1,591.1  4.13  14,45.3  0.21  1,783.5  0.00  0.62  2,049.0 
G07  23,375.60  0.59  5,514.0  3.49  2,052.8  0.00  2,262.8  0.25  0.99  2,107.7 
G08  29,712.97  1.21  5,729.0  3.08  3,059.9  0.28  2,339.7  0.36  0.70  1,907.3 
G09  1,321.73  0.29  819.1  0.48  611.0  1.04  602.0  0.78  1.49  704.1 
G10  1,583.82  0.44  1,762.3  0.44  938.8  1.31  978.4  1.30  1.97  864.9 
G11  2,163.34  0.48  3,284.3  0.41  1,492.7  1.19  1,534.3  1.17  1.95  1,311.0 
G12  2,479.62  0.62  8,587.6  0.53  2,309.7  1.64  2,043.9  1.98  2.52  1,975.8 
G13  2,268.32  0.26  504.5  0.47  360.8  1.04  116.5  0.68  1.15  170.8 
G14  2,692.46  0.40  976.9  0.47  610.4  0.59  183.5  0.94  1.34  215.6 
G15  3,153.55  0.24  1,952.0  0.17  924.8  1.31  357.3  0.76  1.44  221.0 
G16  3,631.47  0.19  4,675.1  0.21  1,313.7  1.10  561.2  1.19  1.61  370.0 
G17  1,632.83  0.03  472.0  0.20  402.6  3.04  110.3  2.40  3.78  182.3 
G18  2,691.61  0.69  892.0  0.53  622.0  1.86  156.4  2.23  2.58  211.5 
G19  3,442.16  1.61  1,418.4  1.61  1,012.5  1.91  194.1  2.17  2.58  234.3 
G20  4,265.11  0.98  2,476.3  1.08  1,368.9  1.58  290.3  1.94  2.20  241.3 
Average G  0.59  2,967.8  1.71  1,366.5  1.00  1,094.2  1.01  1.55  1,116.4  
Total average  0.45  1,797.4  1.16  852.6  0.65  716.6  0.63  1.07  728.4 
The results obtained on the VRPPC instances show again the high efficiency of the AVNSRN algorithm. Compared to the currently best performing algorithm, the TS with ejection chains of Potvin and Naud (2011), the average gap is slightly worse, but we require significantly less computing time. Furthermore, we are able to improve the results of the AVNS presented in Stenger et al. (2012), while the computing time remains on an equal level. It is also worth mentioning that we are able to find a new overall best solution on instance G06. Moreover, the average results obtained with AVNSRN confirm the robustness of the algorithm.
Conclusion
In this paper, we proposed a single and multidepot version of the PCVRPNL to model an important route planning problem arising in small package shipping. To solve the proposed NPhard problems, we presented a powerful AVNS algorithm applying cyclicexchange neighborhoods, which bases on the framework of Stenger et al. (2012). To tackle the requirements of the PCVRPNL, we designed specific route and customer selection methods. In addition, we implemented a random ordering of the neighborhoods used in shaking. Both extensions proved their positive impact on solution quality during testing.
For the computational studies, we designed a set of 34 benchmark instances for the single depot problem and 33 instances for the multidepot problem. Numerical studies are performed on the newly designed benchmark instances to investigate the suitability of the algorithm for the proposed problems. The tests further demonstrate the strong influence of the value chosen for the minimum demand to be served by the private fleet. In addition, we solved benchmark instances of the closely related VRPPC and MDVRPPC. The results clearly prove the high performance of the proposed algorithm.
Footnotes
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