A multiperiod autocarrier transportation problem with probabilistic future demands
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Abstract
In this paper we study the problem of delivering finished vehicles from a logistics yard to dealer locations at which they are sold. The requests for cars arrive dynamically and are not announced in advance to the logistics provider who is granted a certain timespan until which a delivery has to be fulfilled. In a realworld setting, the underlying network is relatively stable in time, since it is usually a rare event that a new dealership opens or an existing one terminates its service. Therefore, probabilities for incoming requests can be derived from historical data. The study explores the potential of using such probabilities to improve the daytoday decision of sending out or postponing cars that are ready for delivery. Apart from the order selection, we elaborate a heuristic to optimize delivery routes for the selected vehicles, whereby special loading constraints are considered to meet the particular constraints of car transportation via road. In a case study, we illustrate the value of introducing probabilistic information to the planning process and compare the quality of different configurations of our approach.
Keywords
Automotive industry Vehicle routing problem Case studyJEL Classification
C61 L62 L91 R411 Introduction
1.1 Motivation
The automotive industry is one of the key industrial sectors in the world, see Çoban et al. (2014). There are traditional markets such as Europe, Japan, and the US that keep high production levels as well as emerging markets like China, India, Brazil, and Mexico that push the worldwide dynamics in the industry. In 2016, almost 95 million vehicles were produced all over the world, while it had been about 69 million in 2006, see OICA (2017). The market environment is highly competitive in the traditional as well as in the emerging markets, which imposes a permanent challenge to improve on all segments of the supply chain.
This study focuses the last step of distribution logistics, i.e. the lastmile delivery from an automotive terminal to the dealership from where the end customer finally obtains its vehicle. We follow the process framework implemented in the transport management software solution SyncroTess TMS provided by INFORM GmbH, a German software company that specializes in logistics optimization. Based on years of experience with multiple customers, the software offers a generic modeling framework that is typically customized for the specific requirements of individual customer projects. We consider the common case that the inventory and order releases are controlled by the original equipment manufacturer (OEM), whereas the deliveries are conducted by a single or multiple thirdparty logistics providers (3PL(s)).
The overall planning process is as follows. On a daytoday basis, the OEM determines which cars qualify to be sent out. A transport order is created for each of them and added to the set of orders that already exist. Various conditions are imaginable that must be fulfilled before a transport order is created. The basic ones are the fact that a car has arrived on the terminal, has completed all repairs and required checks, and has been ordered by a dealer.
Even if it cannot perfectly be foreseen when a transport order will emerge in the future, past experience with a particular transport relation can be a promising basis to forecast upcoming order volumes. When cars are build to stock, the dealers are typically free to order any car in stock without previously announcing their request. Nonetheless, the volume of future transport orders towards a dealership may well be estimated on basis of its recent sales volumes. In a maketoorder market, there is typically even more information available, but also here, there are various sources of uncertainty, e.g. caused by arrival time fluctuations, scarce terminal handling capacities, damages to the vehicles or modification requests. Moreover, in many practical situations, precise information would be available somewhere, but it is not properly transmitted in advance to the terminal’s information system. Also here, it is often possible to create a reliable forecast of future transport volumes.
Once a transport order is created, it is assigned with a due date that may again depend on various factors. Most commonly, a general terminal policy is applied, but there can also be specific dealer contracts. In most cases, it will be possible to also set the order due date manually. Hence, it is a typical situation for a dispatcher to have a mixed portfolio of orders. Some of them may be urgent so that they definitely have to be sent out. Others may still be postponed to the next day if no suitable trip can be found for them.
The transport resources are not controlled by the OEM in the setting that we consider. Therefore, it is difficult to determine their availability in advance, particularly if more than one logistics partner is employed. In the transport management system, the planning is therefore done without capacity restrictions in the first place when identifying beneficial loads. These loads are then tried to match to a carrier in a second step.
Carrier contracts may consider various payment schemes. However, the less payments and efforts are related, the more may an optimization approach improve the OEM’s situation at the costs of the 3PL(s). A sustainable contract will therefore closely relate the payments to the key cost factors so that both the OEM and the carrier can benefit from planning advances. Apart from a fixed organizational overhead and costs per traveled kilometer, the number of dealer stops significantly impacts the total efforts induced by a trip. Every dealer approach requires a certain extra time to visit the site and arrange the handover. Moreover, several visits on the same day are also highly disliked from the dealers’ perspective due to the additional effort of handling multiple truck visits.
To summarize, the daytoday challenge to a load planner is to keep the given delivery promises while arriving at reasonable low operative costs at the same time. Considering that future transport orders are uncertain to at least some extent, there is also a speculative component involved.
Therefore, the scope of this study is to explore the costreduction potential of using probabilistic information within the daytoday planning process described above. In the first place, we refer to the situation that is characteristic for maketostock markets: only the volumes that can be shipped today are known, new requests emerge without previous announcement, but we assume that the recent sales volumes of each dealer are known and can be considered representative for future transport order requests.
1.2 Problem description
We will focus on the daily selection of vehicles, their feasible allocation to transporters and the creation of routes for the transporters. The precise loading of the vehicles on the transporter is not part of this study. The aim is to identify a selection strategy and routing method that minimizes the relevant logistics costs throughout a planning horizon of multiple periods while keeping all delivery promises. Logistics costs are incurred per trip created, per stop at a customer, and per kilometer travelled by the autocarriers in use. Since every consignment causes effort and disturbance at the dealership, we assume that split deliveries on the same day have to be avoided if possible.
In the autocarrier version of the vehicle routing problem, it is not easily determined whether or not it is possible to load all vehicles allocated to a transporter. Feasibility of a loading depends on various properties of the according vehicles such as weight, length and height. In our study, we follow the approach introduced by Hu et al. (2015) and also used by Wensing (2018) to model an autocarrier’s capacity via loading patterns based on classes of vehicles.
While the common strategy is to focus on the existing orders to come to the daily decisions, an important aspect of our research is the incorporation of knowledge about the ordering behavior of the dealers into the order selection routine. We investigate, how statistic information about ordering frequencies can improve the longterm operative efficiency of the daytoday decision making.
1.3 Literature review
Two streams of literature relate to our study. First, there are several studies directly dedicated to the socalled autocarrier transport problem (ACTP), i.e. dealing with the special aspects of car transportation via road. Second, there is a number of papers that address aspects of dynamic multiperiod vehicle routing from a general perspective.
The first study on the loading of autocarriers is due to Agbegha et al. (1998). They suppose that the selection and allocation of vehicles as well as the routing has already been done and thus focus on the loading aspect. Their objective is minimizing the number of reloading processes during execution of the routes. An integer program is formulated and a branch and bound approach is proposed. Chen (2016a) presents modifications of the model and an improved solution approach to the one of Agbegha et al. (1998). In a subsequent paper, Chen (2016b) relaxes the constraint that reloaded cars must be positioned at the same slot of the autocarrier again. An iterative heuristic improves the results of the more restrictive version of the problem.
The first paper that addresses a routing problem combined with the specific loading constraints of autocarriers is by Tadei et al. (2002). They consider a static, singleperiod setting and present a mixed integer program (MIP) for the assignment of vehicles to autocarriers that deliver only one cluster at the same time without explicit routing. For checking the feasibility of a loading they compute equivalent lengths for vehicles and autocarriers. In their model they demand that the sum of equivalent lengths of the allocated vehicles is not larger than the equivalent length of the transporter.
Hu et al. (2015) present a new idea for the feasibility check. They introduce loading patterns, where each pattern is defined by the number of cars of different types that can be loaded on a transporter. A MIP is formulated in which time windows for customers are considered and the objective is to minimize the travel time. They solve the problem, which they call finishedvehicle transporter routing problem, by an evolutionary algorithm and evaluate it by a comparison to the solution obtained by Gurobi. Minimizing the routing costs is the objective of Dell’Amico et al. (2014), too, who consider a realworld problem of a logistic company in Italy. For the allocating and routing part of the ACTP they present an iterative local search approach that includes a greedy heuristic, eight local search procedures, and a perturbation method. Whenever an improved solution is proposed by this algorithm, the feasibility of the new routes is checked. To be feasible, a loading must comply to a weight constraint as well as to a two dimensional loading constraint. Furthermore, a vehicle to be unloaded at a dealer must not be blocked by another vehicle on the autocarrier. These constraints are modeled by a MIP that is solved by a branchandbound procedure.
Based on Dell’Amico et al. (2014), a multiperiod version of the ACTP with dynamically arriving demands that must be fulfilled in a given time window is investigated by Cordeau et al. (2015). The objective is the minimization of costs including the length of the tours, fixed costs for used autocarriers, service costs at dealers and penalty costs for late deliveries. They present a daytoday planning framework that initially selects a subset of vehicles to be delivered within the next days, depending on their time window and the geographical position of their dealers. Afterwards, routes are determined for these days by an iterated local search. The routes of the current day are implemented, whereas the routes of the following days may change in the next iteration. The feasibility of a loading is checked by the procedure proposed by Dell’Amico et al. (2014).
From a more general perspective, the problem we study belongs to the class of dynamic multiperiod vehicle routing problems, see Pillac et al. (2013) and Ritzinger et al. (2016) for recent reviews of this field of literature. A multiperiod vehicle routing problem with complete information about orders is investigated by Archetti et al. (2015). They compare three different MIPformulations where due dates and limited capacity must be regarded. The objective is the minimization of transportation costs, inventory holding costs and costs for unsatisfied orders. They present valid inequalities and solve the MIPs with CPLEX. Angelelli et al. (2007b) consider a dynamic problem in which orders may be fulfilled at the first 2 days after the request arrives. They analyse the competitive ratio of straightforward algorithms for the special case with two periods and customers that are placed in the Euclidean space or the realline. The objective is the minimization of travel distances in the case of vehicles without capacity limits. The same authors generalize some of their results in Angelelli et al. (2007a) for an enlarged planning horizon.
Wen et al. (2010) examine a problem with dynamically arriving orders that must be fulfilled within a given time window. Furthermore, capacity limits as well as duration limits for the tours must be satisfied. Besides minimizing travel time, they aim to accomplish service requests as early as possible and balance the workload over the planning horizon. Their solution approach is a threephase heuristic that first determines a service day for each customer. In a second step the tours of a predefined number of consecutive days are planned for the selected customers. After that, the tours of the current day are optimized with respect to the travel time. This problem is extended in AlbaredaSambola et al. (2014) by introducing probabilities for the occurrence of future requests of a set of given customers, where the demand quantity is known in advance. They formulate a prize collecting vehicle routing problem for every day that is solved in their computational study by CPLEX and by an adaptive variable neighborhood search. The calculation of customer profits regards the remaining time for the service request as well as the probability for future requests in the neighborhood of a customer.
1.4 Contribution and outline
The problem we address is most closely related to the one studied by Cordeau et al. (2015), where we use the more generic capacity model introduced by Hu et al. (2015). More importantly, we explore the potential of using probabilistic information to improve the overall solution quality of the underlying daytoday planning procedure. As to our experience, this kind of information is well available in the considered field of application and we are not aware of any work on the ACTP that is addressing it in any form.
Moreover, we contribute to the general literature on dynamic stochastic vehicle routing problems by considering demands of unknown volume that may repeatedly occur at the same location, where the closest related work of AlbaredaSambola et al. (2014) assumes that demand volumes are known in advance and only the time window in which they have to be satisfied is uncertain.
The remainder of the paper is organized as follows. In Sect. 2 we define the problem, including our approach for checking the feasibility of a load. In Sect. 3, we present an anticipatory algorithm that along with three different strategies to select or postpone customer orders. The algorithm is evaluated and the three strategies are compared against the backdrop of a case study in Sect. 4. Section 5 concludes our study.
2 Problem definition
The orders arrive independently over the planning horizon, so the underlying process is strictly stationary. As the exact probability distribution is unlikely to be known in practice, we confine ourself to less detailed information. To be precise, we assume probabilities \(p_i\) for at least one incoming order of customer i on every day. Hence, we have information about the frequency of orders, but do not know anything about quantity, type and deadlines of future orders.

costs per travelled kilometer (\(\alpha\)),

costs per customer stop (\(\beta\)) and

costs per tour (\(\gamma\)).
List of symbols
Symbol  Explanation 

V  Set of customers including the depot 0 
T  Set of vehicle types 
\(\mathcal {O}\)  (Multi)set of orders 
\(\mathcal {O}_d\)  Set of orders placed at day d 
\(O_d\)  Set of active orders at day d 
\(\overline{O}_d\)  Set of urgent orders at day d 
\(V_d\)  Set of customers with at least one active order at day d 
\(X_d\)  Set of orders fulfilled at day d 
D  Number of days 
\(c_{i,j}\)  Distance between customers i and j 
\(t_o\)  Type of car of order o 
\(\bar{d}_o\)  Deadline of order o 
\(i_o\)  Customer of order o 
\(\bar{d}_i\)  Deadline of customer i 
\(p_i\)  Probability of an incoming order of customer i 
\(\alpha\)  Costs per travelled kilometer 
\(\beta\)  Costs per customer stop 
\(\gamma\)  Costs per tour 
As we assume an arbitrarily large fleet of autocarriers, a feasible solution can always be found by a simple construction routine: select all urgent orders, i.e. \(X_d = \overline{O}_d\), and assign each of it to an own autocarrier for fulfillment. Apart from being feasible, this solution provides us with the maximum of autocarriers to employ in an optimum solution, due to Property 1.
Proposition 1
There is an optimal solution of the problem with at least one urgent customer on each route.
The costs of a route do not depend on the day it is performed. So, a route without any urgent customer can be shifted to a subsequent period without increasing the overall costs. Postponing routes until the first deadline of one of its customers is reached implies Property 1.
2.1 Routing subproblem
Of course, a feasible loading of all vehicles delivered in a common tour must exist. To model the loading constraints we use a set of loading patterns E, as introduced by Hu et al. (2015). A loading pattern \(e \in E\) is defined by the number of vehicles of type t, \(Q_{e,t}\), that can be allocated to a transporter using pattern e.
3 Solution approach
3.1 Phase 1: selection of orders
In this phase we aim to determine a subset of orders to fulfill in the current period. The underlying principle is to initially identify promising customers that are to be delivered. Of these customers all active orders are marked as candidates for fulfillment on the particular day.

\(\displaystyle f_{i,d}^1 := \sum _{j \in N(i)} \pi _{j,d}\)

\(\displaystyle f_{i,d}^2 := \sum _{j \in N(i)} \pi _{j,d} \cdot \left( 1  \widetilde{c}_{ij}\right)\)

\(\displaystyle f_{i,d}^3 := \sum _{j \in N(i)} \frac{\pi _{j,d}}{\widetilde{c}_{ij}}.\)
Example 1
While in the first way, \(f^1\), the relative distance between two customers does not matter (apart from the neighborhood), this distance affects the day factors in the other two cases. As \(\widetilde{c}_{ij} \le \rho\) holds for all neighbors, the day factors \(f^2\) are just slightly adjusted compared to \(f^1\). However, the relative distance has stronger impact using the third variant, since \(\widetilde{c}_{ij}\) and \(f^3\) are indirect proportional to each other.
The idea for this selection process is similar to the one presented by Wen et al. (2010). However, the calculation of the day factors is quite different compared to the “compatibility index” of Wen et al. (2010). Once the selection of customers is completed, in phase two routes are built to fulfill all active orders of the selected customers at day \(d_{\text{cur}}\).
3.2 Phase 2: construction of routes
In the second phase we construct routes to fulfill the selected orders at minimum costs. We adapt the idea of the savings algorithm by Clarke and Wright (1964), which is still one of the most popular algorithms for vehicle routing problems as stated by Toth and Vigo (2002, p. 125). In contrast to conventional vehicle routing problems, we can be faced with the problem that the total demand of a customer exceeds the capacity of the autocarrier, i.e. no feasible pattern exists to load all selected orders of a customer on one autocarrier. In that case, we construct an individual route in advance using a pattern that allows to load as many vehicles as possible. This is repeated until the demand does not exceed the capacity anymore. The only difference to the standard version of the savings algorithm is due to the special loading constraints of the ACTP. So, two routes with the largest saving are merged when a feasible loading for the vehicles of both routes exists. As explained in Sect. 2, this is checked by looking for an appropriate pattern.
After that, the routes are improved by local search procedures. The first one, customer_switch, looks for an improvement by inserting a customer to the best position of a different route. Additionally, we make use of the bcyclic ktransfer introduced by Thompson and Psaraftis (1993). This procedure improves the solution by choosing b routes \(r_1,...,r_b\) and transferring k customers of \(r_i\) to \(r_{i+1}\) for \(i = 1,...,b1\) and k customers of \(r_b\) to \(r_1\). The customers are always inserted in the best possible way into the new routes. As the routes have only few stops in our problem due to the very limited capacity of autocarriers, we just implemented bcyclic 1transfers for \(b \in \{2,3\}\). These procedures are repeated iteratively until no improvement is obtained.
3.3 Phase 3: finalization
In this phase we try to further improve the solution with regard to the costs over all periods. For this, we implement two strategies. First, in view of Property 1, we delete all routes constructed in phase two that do not include an urgent customer. Such a route could also be accomplished in a future period with the same costs. By postponing these customers we keep up the opportunity to combine them with other customers in the future and potentially save costs.
Additionally, in a last step we try to avoid idle capacity that may exist on some routes after phase 2. For that purpose, we look for customers not chosen in phase 1 whose orders can be inserted into an existing route with idle capacity. If an insertion is feasible with respect to loading patterns, we determine the best position of the new customer visit. The insertion is executed if the additional length is not more than \(tol \in \mathcal {Q}\) times the distance of the inserted customer to the depot.
Algorithm 1 summarizes the entire procedure that is repeated day by day in the overall planning process.
4 Case study
We want to investigate the potential of our anticipatory approach in a case study. Therefore, we compare our heuristic with some basic solution strategies, which are to reflect decisions similar to those performed in practice. Furthermore, we examine the influence of the three cost factors on the solution value, depending on the solution strategy. By a sensitivity experiment, we also investigate the influence of the dealer’s demand volume on the improvement potential of our approach. In order to determine suitable values for parameters \(\rho\) and tol, tests on diverging artificial instances are run in advance.
4.1 Generation of instances
The case study refers to a demo case from the context of the transport planning system mentioned in the introduction. It has also been considered in Wensing (2018). We choose to consider a generated case instead of a real data sample in order to exclude irregular fluctuations from our study that are not well predictable on basis of historic sales figures. While there is no doubt that such effects are critical for the practical applicability of our approach, we think that adding unexplained randomness would lead to numerical results that are less significant in terms of our original study question.
The case is built upon all cities in the German state of North RhineWestphalia with a population greater than 30,000. Each of these exhibits at least one dealership and additionally one more per 200,000 inhabitants. Altogether, the case comprises to a total number of 159 customer locations. The travel distance between two customers i and j is calculated by \(c_{ij} = 1.2 \times d_{ij} + 5\), with \(d_{ij}\) being the distance as the crow flies between i and j. The probabilities \(p_i\) are proportional to the population of the according city divided by the number of dealers of the city and scaled such that 50 dealers place an order on average per day. The number of vehicles ordered by a selected dealer is uniformly distributed in \(\{1,2,3\}\). Once it is placed, the deadline of an order is set to 0, 1, 2 or 3 days in the future, also with equal probability. We consider three different types of cars (small, medium, and large) all ordered with the same probability. Thus, for each car, a random number from set \(\{1,2,3\}\) is drawn with equal distribution, which defines the type of the car. We have used 41 loading patterns that allow at most 9 small, 8 medium or 7 large sized cars on one autocarrier. Our planning horizon comprises 30 periods. The cost factors are set to \(\alpha = 2, \beta = 20\), and \(\gamma = 50\).
4.2 Parameter setting
Before testing our solution approach against basic strategies, we have to set appropriate values for the parameters \(\rho\), which influences the size of the neighborhood, and tol, which is used as acceptance criteria in the finalization phase. In order to obtain fair and meaningful results we generated artificial instances that differ from the instances of the case study in some aspects. We consider 50 customers, located unclustered in a \([0,500]\times [0,500]\) square grid. The travel distance \(c_{i,j}\) is equal to the Euclidean distance between i and j. The probabilities \(p_i\) are uniformly distributed in the interval [0, 0.5]. The remaining aspects stay the same as in our case study. Again, the number of ordered vehicles is uniformly distributed in \(\{1,2,3\}\) and the deadline is at most 3 days in the future. The same types of cars and loading patterns are used. Also, the planning horizon covers 30 day and the cost factors remain unchanged.
Average costs of 50 instances for different values of \(\rho\)
\(\rho = 0.05\)  \(\rho = 0.1\)  \(\rho = 0.15\)  \(\rho = 0.2\)  

\(f^1\)  180695  180427  181326  181696 
\(f^2\)  180746  180521  181072  181594 
\(f^3\)  180781  180485  180846  180881 
Two main conclusions can be drawn from these results. Firstly, the quality of the solutions does slightly depend on the way of computing the day factors. Secondly, independent of the choice of day factors the best results are obtained by setting \(\rho = 0.1\). The choice of \(\rho\) has the least impact on the solution quality when \(f^3\) is selected to compute day factors. This might be due to the fact that \(f^3\) weights the nearest neighbors much stronger, so the size of the neighborhood has only little impact.
Average costs of 50 instances for different values of tol
\(tol = 0.1\)  \(tol = 0.2\)  \(tol = 0.3\)  \(tol = 0.4\)  \(tol = 0.5\)  

\(f^1\)  178422  177872  177699  177606  177896 
\(f^2\)  178509  177973  177808  177738  177990 
\(f^3\)  178452  177853  177902  177773  178001 
The best results are obtained for a tolerance value of \(tol = 0.4\). Independent of the variant for the day factors the costs could be reduced by approximately 1.5% during the finalization phase. As the results for \(f^1\), \(f^2\), and \(f^3\) are very similar, we investigate all versions in the following case study. Note that the parameter selection is most probably not the best choice for all possible datasets, where it is a difficult problem of its own to determine in advance which parameters will lead to the best performance on a given data set or test case structure. To allow for a fair comparison of our algorithm independent of the underlying dataset, these tests are performed to determine good values in general. Following the results we have obtained, we choose \(\rho = 0.1\) and \(tol = 0.4\) for the purpose of our case study.
4.3 Results of the case study
As we investigate a problem occurring in practice, we compare our approach to basic strategies that are obvious for solving practical instances. The first_period strategy fulfills all orders as soon as they are placed. The opposite last_period strategy postpones all orders until the deadline is reached, so that only urgent customers are visited. When a customer is visited, we fulfill all its active orders. These strategies differ in the selection process but not in the way to construct the routes.
Results of the case study
Instance  First_period  Last_period  \(f^1\)  \(f^2\)  \(f^3\) 

159NRW01  118821  117024  116744  116589  116365 
159NRW02  118491  116201  114670  114724  114727 
159NRW03  121873  119613  118780  118614  118915 
159NRW04  121718  118757  118186  118186  118376 
159NRW05  115695  111768  112286  112285  112411 
159NRW06  118982  115541  114750  114750  114811 
159NRW07  124082  119953  118340  118337  118698 
159NRW08  120935  120064  118089  118172  118501 
159NRW09  118432  115562  114632  114660  115333 
159NRW10  120708  116353  114336  114084  114857 
159NRW11  123365  118694  118398  118683  118484 
159NRW12  118647  113416  113847  113920  113496 
159NRW13  127442  122240  120799  120913  120857 
159NRW14  120954  117785  117827  117758  117127 
159NRW15  127920  123185  122304  122304  122388 
159NRW16  118760  116662  116050  115721  115239 
159NRW17  120676  117816  116817  116733  117558 
159NRW18  121890  117293  117501  117501  117647 
159NRW19  122063  117982  116682  116800  117010 
159NRW20  115118  111633  112685  112678  112535 
\(\varnothing\) costs  120828  117377  116686  116671  116767 
While the first_period strategy is obviously dominated by the other solution approaches, the different ways of computing the day factor \(f^x\) lead to very similar results. In four of the 20 instances variants \(f^1\) and \(f^2\) even result in the same solutions. Comparing \(f^2\), being the slightly best variant of our solution approach for these instances, with the first_period and the last_period strategy yields improvements of about 3.4 and \(0.6\%\), respectively. To verify the significance of the cost savings acquired by our approach with \(f^2\) compared to the last_period strategy, we ran a paired tTest. The resulting pvalue of \(8.278\cdot 10^{4}\) clearly implies to reject the null hypothesis that the last_period strategy is at least as good as our solution approach without the finalization phase.
Composition of costs of the different strategies
First_period  Last_period  \(f^1\)  \(f^2\)  \(f^3\)  

Travel costs  94772  92811  91689  91683  91773 
Stop costs  14909  13296  13740  13738  13706 
Tour costs  11147  11270  11257  11250  11287 
Total costs  120829  117377  116686  116671  116767 
Some more conclusions can be drawn from these results. The first_period strategy has by far the highest stop costs as expected because a customer is visited whenever it places an order. Also, the travel costs increase clearly. This might be due to the lower number of cars delivered at a customer at the same time and, hence, serving more dealers per route on average. As stated above, our solution approach has higher stop costs than the last_period strategy. However, the savings in the travel costs (\(1.2\%\) on average for \(f^2\)) overcompensate this. Consequently, our solution approach tends to serve more customers on a route but their locations enable to reduce the traveled distances compared to the last_period strategy.
Composition of costs of the different strategies
\(f^1\)  \(f^2\)  \(f^3\)  

Travel costs  90387  90377  90385 
Stop costs  13949  13943  13919 
Tour costs  10915  10907  10905 
Total costs  115251  115227  115209 
Influence of order volumes on the improvement potential
Last_period  \(f^1\)  \(f^2\)  \(f^3\)  Improvement in %  

159NRW_25  68312  66317  66313  66415  2.93 
159NRW_50  117377  115251  115227  115209  1.85 
159NRW_75  168123  167188  167219  167239  0.56 
The results suggest that the use of factors \(f^x\) generates higher cost savings when the probabilities for incoming orders are low. Seemingly, the selection process becomes less important when more orders are placed, which is probably due to a general bundling effect that makes it easier to find efficient routes on arbitrary subsets of customers.
5 Conclusions
The delivery of finished vehicles is a relevant cost driver in the automotive industry. The reappearing task of planning deliveries to a stable network of dealers is well studied in the literature. However, none of the studies so far explores the potential of using information that can be gained from repeatedly executing the task. To fill this gap, we have included probabilities for future orders into a dynamic multiperiod autocarrier transportation problem with the objective of minimizing delivery costs that consist of costs for travelling, costs for customer stops, and fixed costs for routes. Loading patterns were used to model the capacity constraints of the routing subproblem that was formulated as a MIP. Our solution approach uses the stochastic knowledge about incoming orders in the daily selection phase by calculating day factors that indicate the best day of delivery of a customer.
The benefit of using probabilistic information was verified in a case study. Even if the realized savings compared to basic strategies do not appear overwhelming at first sight, there is a clear indication that potential savings exist, especially when the frequency of orders decreases. Particularly if transports have a limited number of stops, they may exhibit long distances that have to be covered anyway so that the scope for improvement by avoiding detour and extra stops is small. Considering this and also that the transport industry is highly competitive in most markets, even (low) single digit improvements may still be valuable to accomplish. Furthermore, we have seen that the finalization phase, which tries to fill empty space on the autocarriers, leads to relevant cost savings. So, gaining highly utilized autocarriers is also important to achieve efficient solutions.
While we tested our heuristic against basic strategies that reflect solution approaches occurring in practice, it would be interesting to use other benchmarks to evaluate the proposed heuristic. Lower bounds on the solution value as well as exact algorithms to solve small instances under complete knowledge could provide deeper insights. Moreover, the problem can be formulated as a stochastic dynamic program. Techniques that minimize the expected costs could also be compared to our solution approach.
Future research could also investigate other ways of using stochastic information for the decision of serving or postponing a customer. Since our experiments did not reveal that one of the \(f^x\) parameters should be preferred over the others, it may be fruitful to explore the potential of combining the underlying three principles to arrive at a fourth, possibly superior factor. Moreover, aiming at a high utilization of the autocarriers already in the selection of vehicles might yield cost savings. In addition, it could be interesting to examine if more detailed stochastic information, e.g. about the number of orders and the vehicle types, can help to improve the quality of the selection phase. Techniques of machine learning may be a promising way of considering various sources of information to arrive at better decisions.
Furthermore, in seems interesting to also study the value of probabilistic information against the background of a maketoorder market. Where we have considered probabilities for new transport volumes to emerge in the future, we find a somewhat inverse perspective here. Future transport orders are known in advance, but it may still be uncertain if the according vehicle is ready for shipment in time.
Finally, a future study could move on from the generated demo setting and consider a set of real transport orders. A whole new aspect would be the proper handling of multivariate fluctuations, e.g. caused by longterm trends, intraweek effects or seasonality over the year. Creating a reliable forecast function will most probably be a precondition to the successful adaption of our approach to such a setting.
Footnotes
 1.
Actually, \(\mathcal {O}\) is a multiset as a customer may order more than one car of the same type with the same deadline. However, to simplify the notation we use the term set instead of multiset in the following.
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