Optimization of Logistic Solutions for Incoming and Outgoing Trucks System Using a Simulated Cross-Docking Centre Environment

2.1 Strategy and implementation of cross-docking

The cross-docking strategy also includes positioning cross-docks. It is realized based on the forbidden search algorithms to determine the cross-dock location from a set of multiple possible locations [5, 18, 19]. One such algorithm is the taboo search algorithm based on a metaheuristic algorithm and used to prevent local optimization. It consists of several rounds. At each iteration, a potential solution with the associated solution neighborhood is found. The algorithm moves from solution to solution erasing the previous neighborhood and keeping the new one at each iteration until the best solution is obtained [10, 18]. The idea of this study includes optimization of time windows of entrance and exit doors to optimize the order assembling operations and minimize time costs in the cross-docking technology. The study was conducted at the Raben Ukraine facility on June 24, 2021.

To determine the best cross-dock shape, the number of doors of the object must be calculated. Cross-dock has two types of gateways: receiving or entry doors and shipping or outgoing (exit) doors. To determine the number of doors for shipment is easier regardless of the volume of orders. In certain cases, there is no more than one gateway (exit) per a destination point [17].

Determination of the number of entry doors is a much more difficult task due to the fact that it requires the information on the exact number of trucks that can arrive at the same time to the docks as well as on the place where the goods are directly collected. For ordinary cargo transfer, the Little's law (1) enables to calculate the number of receiving doors by multiplying the capacity of trucks by the average time needed to unload a truck [4].

$L=\lambda W$     (1)

The long-term average number L of requirements in a stationary system is equal to the long-term average intensity λ of the input flow multiplied by the average time W of the application in the system [19].

The most common is the I-shape of cross-docking (Figure 1(a)). It is used at the facilities with 100 and more doors, which is typical for cross-docking. This shape makes a direct path from the entry to the exit door thus minimizing material touching, decreases the costs of goods transportation and reduces the needed space inside the facility. If the buildings have more than 150 doors, T-shape (Figure 1(b)) is more appropriate. X-shape (Figure 1(c)) should be used in the buildings with more than 200 doors. In some cases, L-shaped cross-dock is considered to be used for the building with 100 doors and more (Figure 1(d)) [5, 9, 10, 18].

It is convenient to determine the most used doors. For this purpose, the distances from one gateway to all other ones in the cross-dock should be calculated. We consider here a rectangular-shaped cross-docking-center shown in Figure 2.

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Figure 1. Cross-docking-center shapes

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Figure 2. Simplified diagram of a cross-docking-center

2.2 Mathematical model for determination of time windows for trucks

The integer program using heuristics is more accurate and stable, which helps to calculate the operation time of time windows [9].

min

$\begin{array}{cc}\alpha \sum_{i \in I} \sum_{k \in K_i} & P_{i k}^I w_{i k}^I+\beta \sum_{o \in O} \sum_{k \in K_o} P_{o k}^O w_{o k}^O+ \\ & \gamma \sum_{h \in H, i \in I, d \in D} S_{h i d}^I \\ \text { s.t. } & \sum_{i \in I} \sum_{k \in K_i} W_{i k h}^I w_{i k}^I \leq N^I \\ & \forall h \in H\end{array}$      (2)

$\begin{gathered}\sum_{o \in O} \sum_{k \in K_O} W_{o k h}^O w_{o k}^O \leq N^O \\ \forall h \in H\end{gathered}$    (3)

$\begin{gathered}x_{h i o}+S_{h i d}^I \leq M \sum_{k \in K_i} W_{i k h}^I w_{i k}^I \\ \forall h \in H, i \in I, o \in O\end{gathered}$      (4)

$x_{h i o}+S_{h o}^O \leq M \sum_{k \in K_O} W_{o k h}^I w_{o k}^O$

$\forall h \in H, i \in I, o \in O$     (5)

$\begin{gathered}\sum_{h \in H, o \in O} Z_{d o} x_{h i o}+\sum_{h \in H} S_{h i d}^I=Q_{i d} \\ \forall h \in I, d \in D\end{gathered}$   (6)

$\begin{gathered}\sum_{i \in I, h \in H} x_{h i o}+\sum_{h \in H} S_{h o}^O=C \\ \forall o \in O\end{gathered}$    (7)

$\sum_{k \in K_i} w_{i k}=1 \quad \forall i \in I$     (8)

$\sum_{k \in K_0} w_{o k}=1 \quad \forall o \in O$       (9)

$\begin{gathered}S_{h d}=S_{(h-1) d}+\sum_{i \in I} S_{h i d}^I-\sum_{o \in O} Z_{d o} S_{h o}^O \\ \forall d \in D, h \in H \backslash\{0\}\end{gathered}$      (10)

$S_{o d}=\sum_{i \in I} S_{0 i d}^I-\sum_{o \in O} Z_{d o} S_{0 o}^O \quad \forall d \in D$        (11)

$\begin{gathered}x_{h i o}, S_{h i d}^I, S_{h o}^O, S_{h d} \in N^{+} \\ \forall h \in H, i \in I, o \in O, d \\ \omega_{i k}^I, \omega_{o k}^O \in\{0,1\} \\ \forall i \in I, o \in O, k \in K\end{gathered}$          (IP*)

where, H - a set of time periods (for example, half an hour) within the defined planned limits; I - a set of incoming trucks; O - a set of outgoing trucks; D - a set of directions; Q_id - the number of pallets for certain directions d $\in$ D in the truck i $\in$ I; Z_do - (=1), if the truck o $\in$ O for the direction of d $\in$ D, 0 in other cases; N^I - the number of entrance doors; N^O - number of exit doors; M - the maximum number of machines that can be moved during one time period from one machine to another; C - volume of outbound truck deliveries; x_hio - is the number of units that go from truck i to truck o in a period of time h; w^I_ik =1, if slot k $\in$ K_i is selected for truck i, 0 otherwise; w^O_ok =1, if slot k $\in$ K_о is selected for truck o, 0 otherwise; s^I_hid - is the number of goods with destination d that move from truck i to storage location for a period of time h; s^O_ho - is the number of products that go from the storage location to the truck o in time period h.

K_I (respectively K_O) as a set of possible intervals of truck presence i $\in$ I (respectively o $\in$ O). These possible intervals are described by the matrices W_I and W_O.

= 1, if the hour h $\in$ H in the slot k $\in$ K_i for the incoming truck i $\in$ I;

= 1, if the hour h $\in$ H in the slot k $\in$ K_o for the outgoing truck o $\in$ O.

Penalties for both P^I і P^O are determined as follows:

- the penalty is paid for the use of the time interval k $\in$ K_i for a truck i $\in$ I, if k is the desired time window set by the carrier;

- the penalty is paid for the use of the time interval k $\in$ K_o for a truck o $\in$ O.

The first heuristic is responsible for fixing the schedule of inbound trucks, while the schedule of departures is fixed in the other heuristic.

Use of the Z matrix enables to specify the destination of each outbound truck. One can easily calculate X^O by a binary matrix defined as follows: X^O_dh = 1 if the truck sent to a destination d is present for a period h, otherwise X^O_dh = 0.

The integer program (IP1) uses w^I_ik as a solution variable in the same way as two new variables that measure the difference between the arrival and departure plans.

δ⁺_dh - for the time when h $\in$ H, the positive difference between the number of pallets for a destination d $\in$ D available for unloading and the number of pallets that can be loaded into trucks for d present near the exit doors, while δ^-_dh - for the time when h $\in$ H, the negative difference [9, 20].

min

$\begin{array}{ll} & \sum_{d \in D, h \in H}\left(\delta_{d h}^{+}+\delta_{d h}^{-}\right) \\ \text {s.t. } & \sum_{i \in I} \sum_{k \in K_i} Q_{i d} W_{i k h}^I w_{i k}^I \\ = & M X_{d h}^O+\delta_{d h}^{+}-\delta_{d h}^{-} \\ & \forall d \in D, h \in H\end{array}$      (12)

$\begin{gathered}\sum_{i \in I} \sum_{k \in K_i} W_{i k h}^I w_{i k}^I \leq N^I \\ \forall h \in H\end{gathered}$      (13)

$\begin{gathered}\sum_{k \in K_i} w_{i k}=1 \\ \quad \forall i \in I\end{gathered}$    (14)

$\begin{gathered}\delta_{d h}^{+}, \delta_{d h}^{-} \in N^{+} \\ \forall d \in D, h \in H \\ w_{i k}^I \in\{0,1\} \\ \forall i \in I, k \in K\end{gathered}$   (IP1)

The first stage of the heuristic (IP2) forms an acceptable outbound truck schedule that minimizes early and late outbound trucks regardless of the input data. Given a fixed output data, (IP*) is used to create a schedule for arriving trucks [9].

min

$\begin{array}{cc} & \sum_{o \in O} \sum_{k \in K_O} P_{o k}^O P_{o k}^O \\ \text { s.t. } & \sum_{o \in O} \sum_{k \in K_O} W_{o k h}^O w_{o k}^O \leq N^O \\ & \forall h \in H\end{array}$   (15)

$\begin{gathered}\sum_{k \in K_O} w_{o k}=1 \\ \forall O \in O\end{gathered}$   (16)

$\begin{gathered}w_{o k}^O \in\{0,1\} \\ \forall o \in O, k \in K\end{gathered}$    (IP2)

The integer program (IP2) uses w^O_ok as a solution variable to minimize resentment of transporters with shipments. The constraint (15) ensures that the number of trucks in operation during any period of time does not exceed the number of exit doors, while the constraint (16) guarantees that only one time window is assigned to each outbound truck [12].

In the next stage, the output from (IP2) w^O_ok is used as the IP*.

2.3 Software with initial parameters for the cross-docking center model

To model the processes of the cross-docking platform, AnyLogic software was used, which provides ready-made templates for simulating processes with the setting of output parameters. Figure 3 shows the main processes that make up the cross-dock center, which allows you to display all the main key parameters in the form of an interface during the simulation. For example, the Dock element contains the following parameters for customization: Presentation (scale, level1), Parameters (dockNode, loadPoint: null, numDock:1, storage: null; truckPoint; turn point); Variables (isOccupied, processingForklift); Functions (releaseForklift, reserve, spaceAvailable, unreserve); Collections (orders, reserved).

Other elements are configured in the same way, Forklift, LoadingTruck, Main, Order, Pallet, ResourceHandler, Truck, Type, UnloadingTruck, Simulation etc. Figure 4 shows the main process and elements of the layout that contains the cross-docking center.

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Figure 3. Main project processes and their settings

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Figure 4. Setting up processes and elements of the model interface

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Figure 5. Run the simulation and set the parameters

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Figure 6. Additional process customization during simulation

After building the layout, you need to model all the processes and elements for the simulation. Simulation: Main displays the initial parameters for running the simulation, as shown in Figure 5, which shows the settings of such parameters as Initial utilization of the storage, number of pallet types, number of forklifts, number of unloading docks, number of loading docks.

After the simulation is launched, the model suggests changes during the execution of processes in the software environment. The process parameters that can be changed include the following (Figure 6): Forklifts - per unloading dock and per order assembling; Unloading trucks, per hour; Loading trucks, per hour; Order, per hour; Truck capacity; Minimum order size; Maximum order size; Create orders, etc.

This model allows us to demonstrate the optimization of the cross-docking center's processes for unloading and loading docks and allows us to establish the relationship where the number and volume of transported products affects the time windows for trucks.

2.4 Modeling basic cross-docking processes

In this article, modeling was carried out to obtain combinations of agent and discrete events. This enabled to mimic the behavior of identifiers directly involved in cross-docking operations. For discrete events, simulating operations related to packet aggregation became possible. Modeling makes possible to obtain the necessary information about the operations related to transshipment, which in turn enables to choose more appropriate dimensions and shape of the distribution center for cross-docking [20, 21]. AnyLogic simulation environment was used as a software. It is a good imitating modeling tool with support of modeling methodology using a graphical language in Java code.

In a simulated environment, using the agents with active components, their behavior is determined with formation of interlink. The simulation is a sequence of discrete events that perform basic actions (unloading and loading a truck are discrete events being a sequential algorithm, which ultimately results in emptying or filling the cargo compartment of the truck).

The built layout includes logistic elements of the operations with the corresponding zones: unloading dock, loading dock, assembly dock with pallets (8 types), forklifts (total number of 29 units) and backup storage. The unloading dock consists of five doors to receive trucks with goods. The loading dock consists of seven doors. Here, the assembled goods are loaded into a truck for shipment to the customer.

Figures 7 and 8 present 2D and 3D models of a cross-docking-center. The main technological processes of unloading, assembling and loading goods with performing technological operations in three stages are shown.

Unloading stage. A truck delivers pallets to a free unloading dock. The pallets are unloaded from the truck by a forklift and placed in the receiving dock area. After that, other forklifts move the pallets to the main racks.

Assembly stage. Orders are collected from pallets, which may be of the same or different types. An order can only be collected if the required pallets can be placed in the assembly area near the docks (or, if the space is not enough, in the additional storage area), provided that the required types are available in the main warehouse. The order is assembled by forklifts.

Loading stage. Orders are fully assembled and loaded into a truck for delivery to the customer. A forklift loads the orders into trucks from the assembly area. The total number of orders for one truck must be at least half of its capacity.

Perishables such as vegetables and fruit supplied from different regions or countries and assembled for delivery to customers, for example, in supermarkets and stores, are considered as goods.

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Figure 7. 2D model of a cross-docking-center showing the processes

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Figure 8. 3D model of a cross-docking-center showing the processes

The process of assembling goods is the most important, since just after unloading a problem caused by delay in the assembly of orders arises. This problem significantly affects the rate of assembly and timely delivery of goods directly to the client. Therefore, the more goods are in the queue and waiting for assembly, the longer time it takes to complete the order in time.

The technology of cross-docking in the tasks of planning and managing logistic operations requires continuous improvement, which is explained by the optimization of the tasks of planning multiple-door transshipment for inbound and outbound trucks. To carry out analysis in planning and control problems, we consider some models. In modeling, metaheuristic algorithms are introduced. Golshahi-Roudbaneh et al. [20] analyzed the numerical results using the criteria of relative percentage deviation and time. Compared to our proposed model, metaheuristic algorithms are reasonable to use for the lower boundary and strict heuristics in the truck planning problems.

To minimize penalties caused by delays in customer service, Cota et al. [12] analyzed an integrated problem, in which scheduling trucks in a cross-docking-center is combined with an open problem of vehicle routing [1]. The authors propose a linear programming model with mixed integers for the optimal solving control problems. The model containing two heuristics contributes to obtaining the complex solution of planning and control problems. However, use of cross-docking vehicle routing heuristics enables to focus on solving only one problem in planning operations, which is an inefficient solution for managing multiple processes.

In this regard, optimization of cross-docking using imitating models is studied practically. The structure of the imitating model is created using a hybrid model based on discrete events and agents. This enables to more accurately and individually describe the operations and processes inside the cross-docking-center. Suh [22] studied the structure of the imitating model was used to assess the possibility of implementing the operations of planning problems for a supply chain of goods as well as to optimize the cross-docking performance according to the set restrictions. The authors evaluate the impact of controlled input parameters on key cross-docking characteristics and optimize these parameters to achieve the best results from cross-docking operations.

However, some models are aimed at studying the logistic strategy of cross-docking, which enables to understand the influence on the sending order process in order to successfully implement the logistic strategy. Little and Graves [23] considered two different situations in which use of this strategy can be beneficial. The first situation is based on the analysis of what the order processing system should be in order to support cross-docking and modeling. This facilitates presentation of cross-dock internal processes based on the data provided. The second situation is based on modeling to provide cross-docking a global perspective. The process of the delivery of an order to a customer from the moment the product leaves the distribution center to its arrival at the final destination point is considered using real data.