A proactive transshipment model for prototype parts logistics in the automotive industry

The proposed planning model aims to support automotive manufacturers in the short-term planning of shipments of prototype parts from warehouses to customers, where parts may additionally be transshipped between warehouses before customer delivery. This section describes the problem setting. To this end, the characteristics of the logistics network are described. Subsequently, the sequential planning problem of prototype parts logistics is introduced. Finally, the considered objective is justified.

The considered logistics network centers an automotive manufacturer. The manufacturer operates multiple warehouses, which are spatially dispersed and characterized by a limited capacity to store prototype parts. One of the warehouses serves as a central hub. On the inbound side, the incoming shipments must be received at the central hub. Personnel with limited capacity are deployed to handle the parts. Possible actions for incoming parts include storage in the central hub or transshipment for storing in another warehouse. On the outbound side, ordered parts are commissioned by the warehouse personnel and shipped to the customers. A schematic overview of the assumed hub-and-spoke network is shown in Fig. 1.

The modeling approach is based on further assumptions regarding the parts, the shipments and transshipments, the warehouses, and the approximation of future shipping costs.

This section introduces the sets, parameters, and decision variables of the planning model. The set \({N}\) contains locations, where each location \(i, j \in {N}\) is either a warehouse (\(N^{\text {Warehouses}}\subset {N}\)) or a customer (\(N^{\text {Customers}}\subset {N}\), with \(N^{\text {Warehouses}}\cap N^{\text {Customers}}= \varnothing\)). Each warehouse has a specific capacity \(C_i^{\text {Storing}}\) for storing parts (in units) and a specific capacity \(C_i^{\text {Handling}}\) for storing and picking (in units).

The set \(P\) represents the prototype parts. As prototype parts \(x \in P\) are unique, each part can only be stored in one warehouse i. The available inventory of warehouse i is given by binary parameters \(B_{xi}\). For the central hub, \(B_{xi}\) also reflects the incoming shipments from suppliers. The demand is given by binary parameters \(D_{xi}\). Each part can only be ordered once. Vehicles with a given capacity for parts \(C^{\text {Vehicles}}\) (in units per vehicle) can be commissioned for transport to the customers or between warehouses. A fixed cost rate \(r_{ij}^{\text {T}}\) (in monetary units) is incurred for each transport depending on the traveled path (i, j). The approximated cost rate for the future shipment of part x is reflected in parameters \(r_{xi}^{\text {F}}\) depending on the warehouse where the part is stored (in monetary units).

The decision variables \({\varvec{z}}_{xij}\in {\mathbb {B}}\) indicate that prototype parts x are shipped or transshipped between two locations (i, j). The decision variables \({\varvec{w}}_{ij}\in {\mathbb {N}}_0\) indicate the number of vehicles commissioned for transport between locations i and j. Decision variables \({\varvec{R}}^{\text {T}}\in {\mathbb {R}}_0^+\) and \({\varvec{R}}^{\text {F}}\in {\mathbb {R}}_0^+\) reflect the realized and approximated costs of transports, respectively. The notation is summarized in Table 1.

This section describes the planning model for prototype parts logistics. The Objective Function (1) minimizes two optimization criteria. The first term, \({\varvec{R}}^{\text {T}}\), contains the realized costs for transports. The second term, \({\varvec{R}}^{\text {F}}\), is a secondary objective and contains the approximated future shipping costs. The technical parameter \(\varepsilon\) is intended to be chosen sufficiently small so as not to compromise the primary objective. Therefore, the secondary objective anticipates the advantageous transshipment of parts between warehouses utilizing the spare capacity of already commissioned transports.

Constraints (2) ensure the inventory balance. Accordingly, part x can only be transported from warehouse i if it is stored there or transshipped to it from another warehouse j. Constraints (3) enforce the fulfillment of customer orders. Therefore, each part x ordered by customer i has to be delivered from either of the warehouses j. Constraints (4)–(6) ensure compliance with different types of capacity. To this end, Constraints (4) address the capacity of warehouses for storing parts. These constraints consider the parts already stored in the warehouse (first term), transshipped to it (second term), and transshipped from it to another location (third term). Constraints (5) refer to the warehouses’ capacity of handling operations, i.e., to transfer parts from the goods receipt area into the warehouse or pick parts from the warehouse for subsequent transport. Accordingly, these constraints consider the parts transshipped to warehouse i from another warehouse (first term) and the parts shipped or transshipped from warehouse i to another location. Constraints (6) determine the number of required vehicles \({\varvec{w}}_{ij}\) traveling paths (i, j) considering the vehicles’ capacity and the number of transported parts.subject toConstraints (7)–(8) compute the values of the optimization criteria. To this end, Constraint (7) calculates the realized costs for transports \({\varvec{R}}^{\text {T}}\) considering the number of traveling vehicles and the associated cost rate per vehicle \(r_{ij}^{\text {T}}\) for the paths (i, j). Constraint (8) approximates the potential future shipping costs \({\varvec{R}}^{\text {F}}\) considering the approximated cost rate for future customer delivery of part x when stored in warehouse i (i.e., \(r_{xi}^{\text {F}}\)) and the warehouse the part is stored in. Therefore, this term incentivizes the transshipment of parts to warehouses with lower approximated future shipping costs.Constraints (9)–(12) define the range of the decision variables. The formulation can be classified as a mixed-integer linear programming model. The model is static and thus parameterized and applied periodically. This accommodates the short-term nature of the planning situation and the availability of demand information. The process of sequential planning is schematically illustrated in Fig. 2.

This section introduces the design of the numerical study and the used data. To this end, the general assumptions on the considered logistics network and the scenarios are presented.

Warehouses and customers The study considers a hub-and-spoke network with three warehouses storing 1,000 parts. Warehouse 1 serves as a central hub for incoming shipments. Initially, 200, 500, and 300 parts are randomly allocated to Warehouse 1, 2, and 3, respectively (i.e., \(\sum _{x \in P}B_{xi}\) for each warehouse i). The handling capacity of the warehouses for storing and picking operations is assumed to be similar to the initial inventory. The capacity of the warehouses for storing parts \(C_i^{\text {Storing}}\) is computed based on the initial stock and an exogenously parameterized warehouse utilization. The warehouse utilization is varied among {60%, 80%, 95%} to create different scenarios that evaluate the impact of flexibility in the operation of the logistics network. An exemplary parameterization of the network for a warehouse utilization of 80% is given in Table 2.

10 different customers are considered. The warehouses and customers are spatially dispersed. A visualization of their locations, the distances between them, and the resulting cost rates \(r_{ij}^{\text {T}}\) for one vehicle traveling (i, j) are provided in Fig. 7, Tables 7, and 8 in Appendix A. The capacity of vehicles \(C^{\text {Vehicles}}\) is limited to 50 parts per vehicle.

Furthermore, in the numerical experiments, the inventory turnover varies among different scenarios. To this end, {1%, 5%, 10%} of the initial 1000 parts are ordered every period. The ordered parts are randomly chosen based on a uniform distribution considering all parts in inventory. To keep the average utilization of the warehouses constant during the planning horizon, the same number of parts are replenished into the network as are ordered by customers.

Order generation and approximation of future shipping costs For generating customer orders, it must be considered that the customer for each part is unknown before the order arrives. To achieve this, probabilities are generated for each part-customer combination, representing the likelihood of that customer ordering that specific part. A beta distribution is used to assign probabilities to customers. The probabilities resulting from the beta distribution are randomly assigned to the ten customers for each part. The used parameters and the resulting probabilities are described and summarized in Table 9 of Appendix A. The beta distribution is chosen based on observations from practice, indicating that planners can distinguish between more probable and less probable customers for a part.

In the proactive planning model, an approximation of future shipping costs of parts is considered depending on the warehouse a part is stored in, i.e., parameter \(r_{xi}^{\text {F}}\). The same probabilities previously used for generating customer orders are utilized for this approximation. Thus, it is assumed that the planner can perfectly predict the probabilities of customer orders of parts. Furthermore, the approximation is based on the known cost rates of transports between two locations \(r_{ij}^{\text {T}}\). An exemplary visualization of the approximation of future shipping costs is given in Fig. 3. The approximation serves to identify potentially beneficial warehouses for each part as a basis for decisions on a proactive transshipment of parts between warehouses.

Summary of scenarios and instance generation The study considers nine scenarios varying the inventory turnover and warehouse utilization. For each of these scenarios, 20 instances are generated. Each instance randomly assigns the probabilities of customers to order parts. In the sequential application of the model, the ordered parts are revealed successively for each period and serve as deterministic input data. The considered scenarios and their identifiers are summarized in Table 3.

The proactive model is compared to a reactive model. To this end, the model formulation introduced previously is adapted. The reactive model consists of the Objective Function (13) and Constraints (2)–(7) and (9)–(11). In contrast to the proactive model, only two situations allow the reactive model to transship parts between warehouses. First, this holds for parts ordered by a customer in the current period to facilitate consolidated shipments to customers. Second, parts may be transshipped to an alternative warehouse when the capacity of a warehouse is otherwise exceeded. Accordingly, the reactive model does not use the approximation of future shipping costs.Both models are implemented in Python 3.10.9 and solved using Gurobi 10.0.1 on machines with 64 GB RAM and eight Intel Xenon Platinum 8180 CPU threads at 2.5 GHz.

This section presents the results of the numerical experiments comparing the proactive and reactive planning models. To this end, the results refer to the costs and transports. All instances of all scenarios were solved optimally.

The results on the costs of proactive and reactive planning models are summarized in Table 4 reporting the cumulative costs over the considered planning horizon of 20 periods. Generally, costs for both models increase with higher warehouse utilization (ceteris paribus). This can be explained by the increasing need to transfer parts to another warehouse due to insufficient capacity at the central hub. Furthermore, costs increase with an increase in inventory turnover (ceteris paribus) for both models. The explanation for this is intuitive, as the increased number of ordered parts very likely results in an increased number of required transports and, consequently, costs. When comparing the results of the planning models for the individual scenarios, the proactive model yields lower costs for each instance. The achievable cost reduction ranges from 5.1\(-\)19.0%. Therefore, the proactive planning model is advantageous in terms of costs.

Figure 4 compares the average costs per period for reactive and proactive approaches considering the 20 instances of Scenario \(C_{5\%,80\%}\). The costs per period vary only slightly in the reactive approach. In the proactive approach, the costs are identical to those in the reactive approach in the first two periods. Subsequently, they decrease until they stabilize at a lower level. Therefore, the benefits of the proactive approach emerge over time. In the early periods, the proactive and reactive approaches consider the same initial allocation of parts to warehouses and are obliged to deliver the ordered parts to the customers. Consequently, the proactive approach must initially carry out the same transports as the reactive approach and cannot reduce costs in the early periods. However, the transports in the proactive approach include not only the parts explicitly ordered by customers or those necessarily transferred due to insufficient capacity at the central hub but also additional parts anticipating advantageous warehouse locations. In the subsequent periods, the benefits of proactively transshipping parts between warehouses can be realized, resulting in continued cost reductions when using the proactive approach.

These findings are supported when evaluating the number of transshipped parts. The transshipment of parts can, generally, occur for three reasons. Firstly, a non-ordered part can be proactively transshipped if storing it in a different warehouse is considered advantageous (proactive approach only). Secondly, currently ordered parts may be transshipped between warehouses to consolidate orders and allow for an advantageous shipment to the customer (proactive and reactive approaches). Thirdly, transshipping parts may become necessary if the capacity of a warehouse is otherwise exceeded (proactive and reactive approaches). Figure 5 visualizes the average number of ordered parts being transshipped per period for reactive and proactive approaches considering the 20 instances of Scenario \(C_{5\%,80\%}\). The number of ordered parts being transshipped using the reactive approach remains relatively stable. The transshipment is caused by a consolidation of parts before their customer shipment. In the proactive approach, the number of ordered parts being transshipped decreases over time. This is due to the option of proactively transshipping non-ordered parts using the proactive approach. Figure 6 visualizes the number of non-ordered transshipped parts. Accordingly, the proactive approach transships a significantly increased number of non-ordered parts in the earlier periods. This is done by utilizing the remaining capacities of necessary transports, thus incurring no additional costs. After this initial phase, both approaches transship approximately the same number of non-ordered parts per period. However, the proactive approach enables the shipment of customer orders at lower costs. Therefore, the proactive approach remains advantageous beyond the initial phase, as it continuously selects advantageous parts for transshipment to the associated beneficial warehouses, considering the approximation of future shipping costs.

The results on the number of transshipped parts, the number of transports, and their utilization are summarized in Table 5 for all considered scenarios. The proactive approach consistently requires significantly fewer transports than the reactive approach, effectively avoiding their associated costs.

In the previous experiments, networks with 1,000 parts were evaluated. To assess the suitability of the proposed approach for larger settings, additional experiments are conducted for instances with 10,000 and 50,000 parts. Therefore, ten instances of each size are compared regarding required computational times for Scenario \(C_{5\%,80\%}\).

Table 6 summarizes the computational results. Generally, the computational time increases significantly with the considered number of parts. However, for instances of a practical size, the computational time remains reasonable.