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Journal of Business Economics

, Volume 88, Issue 7–8, pp 1009–1028 | Cite as

A multiperiod auto-carrier transportation problem with probabilistic future demands

  • Christian Billing
  • Florian Jaehn
  • Thomas Wensing
Original Paper
  • 75 Downloads

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 time-span until which a delivery has to be fulfilled. In a real-world 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 day-to-day 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 study 

JEL Classification

C61 L62 L91 R41 

1 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 last-mile 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 third-party logistics providers (3PL(s)).

The overall planning process is as follows. On a day-to-day 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 make-to-order 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 day-to-day 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 cost-reduction potential of using probabilistic information within the day-to-day planning process described above. In the first place, we refer to the situation that is characteristic for make-to-stock 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

The transport and fulfillment problem can be subdivided into four tasks, as illustrated in Fig. 1. First, the set of requests to fulfill is selected from the pool of available transport orders. Second, the according vehicles are allocated to transporters, which, third, are arranged to a precise delivery route. Finally, the allocated vehicles are arranged on the transporter as to comply with technical and physical restrictions and to facilitate loading and unloading as much as possible.
Fig. 1

Daily decisions in the multi-period auto-carrier transportation problem based on Agbegha et al. (1998)

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 auto-carriers 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 auto-carrier 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 auto-carrier’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 long-term operative efficiency of the day-to-day decision making.

1.3 Literature review

Two streams of literature relate to our study. First, there are several studies directly dedicated to the so-called auto-carrier 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 multi-period vehicle routing from a general perspective.

The first study on the loading of auto-carriers 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 auto-carrier 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 auto-carriers is by Tadei et al. (2002). They consider a static, single-period setting and present a mixed integer program (MIP) for the assignment of vehicles to auto-carriers 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 auto-carriers. 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 finished-vehicle 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 real-world 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 auto-carrier. These constraints are modeled by a MIP that is solved by a branch-and-bound procedure.

Based on Dell’Amico et al. (2014), a multi-period 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 auto-carriers, service costs at dealers and penalty costs for late deliveries. They present a day-to-day 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 multi-period vehicle routing problems, see Pillac et al. (2013) and Ritzinger et al. (2016) for recent reviews of this field of literature. A multi-period vehicle routing problem with complete information about orders is investigated by Archetti et al. (2015). They compare three different MIP-formulations 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 real-line. 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 three-phase 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 Albareda-Sambola 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 day-to-day 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 Albareda-Sambola 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

We are given a complete and undirected graph \(G=(V,E)\) with \(V=\{0,1,\ldots ,n\}\), where node 0 denotes the depot and the other nodes represent the dealers. The travel distances between all pairs of vertices \(i,j \in V\) are given by \(c_{ij}\) and they satisfy the triangle inequality. We assume an arbitrarily large homogeneous fleet of auto-carriers located at the depot that must deliver cars of different types \(t \in T\). We consider a finite time horizon with time periods \(d = 1,2,\ldots ,D\) that correspond to D days. The orders for vehicles arrive dynamically over the planning horizon. As no adjustments of the routes are possible during the day, we can assume that all orders are placed at the beginning of a day and can already be fulfilled at the same day. Every order consists of exactly one car, so an order \(o \in \mathcal {O}\)1 is specified by the type \(t_o\), the deadline \(\bar{d}_o\) until the car must be delivered, and the customer \(i_o\) who places the order. The deadlines, that are set exogenously, are within the planning horizon, i.e. \(\bar{d}_o \le D\ \forall o \in \mathcal {O}\), and must not be expired. The set of orders placed at day d is denoted by \(\mathcal {O}_d\), so the set of all orders is given by \(\mathcal {O} = \mathcal {O}_1 \cup \ldots \cup \mathcal {O}_D\). Note that the orders \(\mathcal {O}_d, d \in \{1,2,...,D\},\) are not known before day d. An order is called active from its placement until its delivery. At every day d we denote the set of all active orders by
$$\begin{aligned} O_d := \mathcal {O}_1 \cup \ldots \cup \mathcal {O}_d{\setminus}(X_1 \cup \ldots \cup X_{d-1}), \end{aligned}$$
where \(X_d\) is the set of orders fulfilled on day d. Furthermore, we define the set of urgent orders by
$$\begin{aligned} \overline{O}_d := \{o \in O_d|\bar{d}_o = d\}. \end{aligned}$$
Additionally, we define the set of customers with at least one active order by
$$\begin{aligned} V_d := \{i \in V | \exists o \in O_d: i_o = i\}. \end{aligned}$$
The deadline of customer i is defined as the earliest deadline of any active order of customer \(i \in V_d\) and denoted by
$$\begin{aligned} \bar{d}_i := \min \{\bar{d}_o | o \in O_d \wedge i_o = i\}. \end{aligned}$$
If customer i has no active order we set \(\bar{d}_i := -\infty\). We label i an urgent customer on day d if \(\bar{d}_i = d\) holds.

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.

At every day d we have to decide about \(X_d\), the set of orders to satisfy. Additionally, all orders \(o \in X_d\) must be allocated to auto-carriers and the routes of the carriers must be constructed. We have to ensure that every order is fulfilled before its deadline, i.e. \(\overline{O}_d \subseteq X_d\), and a feasible loading for the vehicles allocated to a transporter must exist. Our objective is to determine routes of minimum costs over the whole planning horizon. For that, we regard
  • costs per travelled kilometer (\(\alpha\)),

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

  • costs per tour (\(\gamma\)).

The notation is summarized in the Table 1.
Table 1

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 auto-carriers, 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 auto-carrier for fulfillment. Apart from being feasible, this solution provides us with the maximum of auto-carriers 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

Once we have decided about the set of orders \(X_d\) to fulfill at day d, a routing problem has to be solved. Depending on the choice \(X_d\), the set of customers to serve at day d is
$$\begin{aligned} V^d = \{i \in V| \exists o \in X_d: i_o = i\}. \end{aligned}$$
(1)
Within the daily routing subproblem, we do not allow to split deliveries for the same customer, except it is not possible to fulfill all selected orders of a customer with only one auto-carrier. In that case, extra routes for such customers are constructed in advance. Thus, for the subsequent consideration of the routing problem, we assume that all selected orders of a customer can be loaded on a single auto-carrier.

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.

Thus, we have to solve a variant of the VRP, where the feasibility of a loading does not depend on a one-dimensional capacity constraint, but on the available loading patterns of the auto-carriers. To model this subproblem as a MILP we define \(q_{j,t}\) as the number of cars of type t that must be delivered to customer j. The binary variable \(x_{i,j,e}\) is set to one iff edge (ij) is traversed by an auto-carrier loaded with pattern e. The variable \(u_{j,t}\) gives the number of vehicles of type t delivered on a tour after visiting customer j.
$$\begin{aligned} \text{Min } \alpha \cdot \sum _{i,j \in V^d \cup \{0\}} \sum _{e \in E}&c_{i,j} \cdot x_{i,j,e} + \gamma \cdot \sum _{i \in V^d} \sum _{e \in E} x_{0,i,e}&\end{aligned}$$
(2)
$$\begin{aligned} \text{subject to }&\nonumber \\ \sum _{e \in E} \sum _{i \in V^d \cup \{0\}} x_{i,j,e}&= 1&\forall j \in V^d \end{aligned}$$
(3)
$$\begin{aligned} \sum _{e \in E} \sum _{j \in V^d \cup \{0\}} x_{i,j,e}&= 1&\forall i \in V^d \end{aligned}$$
(4)
$$\begin{aligned} \sum _{i \in V^d \cup \{0\}} x_{i,j,e} - x_{j,i,e}&= 0&\forall j \in V^d \cup \{0\}, e \in E \end{aligned}$$
(5)
$$\begin{aligned} u_{j,t}&\ge q_{j,t}&\forall j \in V^d, t \in T \end{aligned}$$
(6)
$$\begin{aligned} u_{j,t} \cdot \sum _{i \in V^d \cup \{0\}} x_{j,i,e}&\le Q_{e,t}&\forall j \in V^d, t \in T, e \in E \end{aligned}$$
(7)
$$\begin{aligned} (Q_{e,t} + u_{i,t}) \cdot x_{i,j,e} - u_{j,t}&\le Q_{e,t} - q_{j,t}&\forall i,j \in V^d, t \in T, e \in E \end{aligned}$$
(8)
$$\begin{aligned} x_{i,j,e}&\in \{0,1\}&\forall i,j \in V^d \cup \{0\}, e \in E. \end{aligned}$$
(9)
As in basic flow models for vehicle routing problems, constraints (3) and (4) guarantee that every customer in \(V^d \subseteq V\) is visited exactly once. Constraints (5) are necessary to ensure that the used loading pattern does not change during a tour. Constraints (6) and (7) guarantee the fulfillment of demands and the observation of capacity of loading pattern e for vehicle type t. An adaption of the Miller–Tucker–Zemlin subtour elimination constraints, introduced in Miller et al. (1960) and later corrected in Kara et al. (2004), leads to (8). Note that constraints (7) and (8) can be linearized by the big-M method. The objective function (2) regards the length and the number of routes. As no splitting of customers is allowed, the number of stops is predefined by the selected orders \(X_d\). Hence, no consideration in the objective function of the routing subproblem is necessary.

3 Solution approach

We solve the problem by a heuristic approach, which is repeated day by day. This heuristic consists of three phases as depicted in Fig. 2. In a first phase the algorithm selects a subset of orders that are fulfilled at the current day \(d_{\text{cur}}\). After that, routes are built to serve all selected orders at minimal costs. The last phase tries to improve the solution by cancelling routes with no urgent order assigned and by inserting customers not selected in the first phase into existing routes if this is beneficial.
Fig. 2

Daily steps of the 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.

To enable efficient routes, customers close to each other should be served at the same day. For a good selection, we compute day factors for all customers \(i \in V_{d_{\text{cur}}}\) with at least one active order. These day factors take into consideration active orders as well as probabilities for future orders of neighbors. For the definition of the neighborhood we use the relative distance \(\widetilde{c}_{ij} = \frac{c_{ij}}{c_{0i} + c_{0j}}\) of customers i and j with respect to their depot distances. So, we define the set of neighbors N(i) of a customer i as
$$\begin{aligned} N(i) := \left\{ j \in V{\setminus}\{i\} : \widetilde{c}_{ij} \le \rho \right\} , \end{aligned}$$
where \(\rho\) is a parameter that determines the size of the neighborhood. Albareda-Sambola et al. (2014) define the compatibility index \(I_{ij} = \frac{c_{0i} + c_{0j} - c_{ij}}{2(c_{0j} + c_{0i})}\) and i is neighbor of \(j \ne i\), if \(I_{ij} \ge \rho '\). Their neighborhood is equivalent to ours for \(\rho = 1- 2\rho '\) as
$$\begin{aligned} \frac{c_{0i} + c_{0j} - c_{ij}}{2(c_{0i} + c_{0j})} \ge \rho ' \Leftrightarrow 1 - \frac{c_{ij}}{c_{0i} + c_{0j}} \ge 2\rho ' \Leftrightarrow \frac{c_{ij}}{c_{0i} + c_{0j}} \le 1 - 2\rho '. \end{aligned}$$
(10)
Because of the triangle inequality, a pending tour to a customer is never preferable if the customer could be integrated into another route. Hence, a very small (or even empty) neighborhood should be avoided and we set a minimum of \(N_{\text{min}}\) neighbors. In the case of \(|N(i)| < N_{\text{min}}\) we add the \(N_{\text{min}}\) customers with the smallest relative distances to N(i) even if the relative distance is larger than \(\rho\).
Based on this neighborhood we define day factors that give the preference of customer i to be delivered on day d. The day factors are computed for all days d with \(d_{cur} \le d \le \bar{d}_i\) and for all customers \(i \in V_{d_{\text{cur}}}\). We will examine three ways to compute the day factors, \(f^1\), \(f^2\), and \(f^3\):
  • \(\displaystyle f_{i,d}^1 := \sum _{j \in N(i)} \pi _{j,d}\)

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

  • \(\displaystyle f_{i,d}^3 := \sum _{j \in N(i)} \frac{\pi _{j,d}}{\widetilde{c}_{ij}}.\)

Here, \(\pi _{j,d}\) displays the probability of an active order of customer j on day d that is computed according to the following formula
$$\begin{aligned} \pi _{j,d} = {\left\{ \begin{array}{ll} 1 &{} \text{if } \bar{d}_j \ge d \\ 1 - (1 - p_j)^{d-d_{\text{cur}}} &{} \text{else}. \\ \end{array}\right. } \end{aligned}$$
(11)
Choosing variant \(x \in \{1,2,3\}\), a customer i is selected if
$$\begin{aligned} d_{cur} = \mathop{\text{argmax}} \limits _{d_{cur} \le d \le \bar{d}_i} f_{i,d}^x, \end{aligned}$$
(12)
i.e. the current day is considered the most suitable day for delivering customer i. Note that all customers with an urgent order are selected by this approach as \(\bar{d}_i = d_{\text{cur}}\) holds. However, the day factors are not increasing in d in general, which is why a customer i may also be selected before it is urgent. This can occur when at least one of the neighbors \(j \in N(i)\) has an urgent order which implies \(\pi _{j,d_{\text{cur}}} > \pi _{j,d_{\text{cur}}+1}\). By this, routes to the same region on consecutive days should be avoided if possible. We illustrate the usage of the day factors by the following example.

Example 1

We consider five customers 1, 2, 3, 4 and 5 that are all neighbors of each other except customers 4 and 5. Customers 1 and 2 have an urgent order at day 1, i.e. \(\bar{d}_1 = \bar{d}_2 = 1\), while orders of customers 4 and 5 can be postponed until the second and the third day, respectively, i.e. \(\bar{d}_4 = 2\) and \(\bar{d}_5 = 3\). Furthermore, there is no active order for customer 3. The probabilities for incoming orders are \(p_1 = p_3 = 0.7\) and \(p_2 = p_4 = p_5 = 0.3\). Of course, customers 1 and 2 are served at day 1, while the decision for customers 4 and 5 depends on the day factors. Using the first variant of the day factors we receive
$$\begin{aligned} f_{4,1}^1 = \sum _{j \in \{1,2,3\}} \pi _{j,1} = 1 + 1 + 0 = 2, \ f_{4,2}^1 = \sum _{j \in \{1,2,3\}} \pi _{j,2} = p_1 + p_2 + p_3 = 1.7 \end{aligned}$$
for customer 4 and
$$\begin{aligned} f_{5,1}^1 = 2, \ f_{5,2}^1 = 1.7, f_{5,3}^1 = 1 - (1-p_1)^2 + 1 - (1-p_2)^2 + 1 - (1-p_3)^2 = 2.33 \end{aligned}$$
for customer 5. Hence, customer 4 is served at day 1 because of \(1 = \mathop{\text{argmax}}\nolimits _{1 \le d \le 2} f_{4,d}^1\), but customer 5 is postponed as \(3 = \mathop{\text{argmax}}\nolimits _{1 \le d \le 3} f_{5,d}^1\).

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 auto-carrier, i.e. no feasible pattern exists to load all selected orders of a customer on one auto-carrier. 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 b-cyclic k-transfer 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 \(i = 1,...,b-1\) 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 auto-carriers, we just implemented b-cyclic 1-transfers 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 Rhine-Westphalia 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 auto-carrier. 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.

We have solved 50 instances that differ in the set of orders with each variant of computing the day factors. Each instance was solved for different values of \(\rho\), namely \(\rho \in \{0.05, 0.1, 0.15, 0.2\}\). The minimum number of neighbors was set to \(N_{\text{min}} = 2\). As we want to focus on the selection process here, the algorithm is executed without the finalization phase. The average costs are shown in Table 2.
Table 2

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.

In a next step, we have solved the same instances including the finalization phase. For this purpose, we fixed \(\rho = 0.1\) and investigated values of \(tol \in \{0.1, 0.2, 0.3, 0.4, 0.5\}\). The average costs are presented in Table 3.
Table 3

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.

Table 4 shows the results of applying the two base strategies as well as the 3 day factors \(f^x\) on 20 generated instances. As our main intention is to investigate the benefit of stochastic knowledge, we run our algorithm without the finalization phase in a first step. The best solutions are highlighted in each row.
Table 4

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

Bold values indicate the best solutions

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 t-Test. The resulting p-value 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.

However, the improvements compared to the last_period strategy are not overwhelming. One reason for that is the fact that this strategy results in the fewest customer stops. Hence, the stop costs are minimized as can be seen in Table 5.
Table 5

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.

Including the finalization phase into our solution approach yields further improvements on the solution value. While the costs for stops at customers are slightly increasing, the total number of stops as well as the travel costs are decreasing. This might be due to a better utilization of the auto-carriers. The total costs can be reduced by 1.2–1.3% on average. Hence, besides an anticipatory selection of customers, a high utilization of auto-carriers seems to be essential to obtain low delivery costs. The results are depicted in Table 6.
Table 6

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

Finally, we want to investigate the influence of the frequency of orders on the quality of our solution approach. In the previous instances the probabilities were set such that every day 50 customers placed an order on average. We scale the probabilities by factor 1.5 (0.5), so 75 (25) customers place an order on average. The average total costs of 20 instances are shown in Table 7. The last column displays the cost savings of the best variant of our approach compared to the last_period strategy.
Table 7

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 auto-carrier 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 auto-carriers, leads to relevant cost savings. So, gaining highly utilized auto-carriers 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 auto-carriers 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 make-to-order 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 long-term trends, intra-week 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. 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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Augsburg, Sustainable Operations and LogisticsAugsburgGermany
  2. 2.Helmut Schmidt University-University of the Federal Armed Forces Hamburg, Management Science and Operations ResearchHamburgGermany
  3. 3.INFORM GmbHAachenGermany

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