Enhanced Cross-Dock Productivity: Combining Self-Driving Vehicles with Forklifts

Abstract

A cross-dock (CD) in a supply chain avoids storing goods that would be picked for orders soon after. Vehicles inbound to the CD are unloaded and their contents are re-sorted. Appropriate items are then loaded within a short time on outbound vehicles for shipment to customers. The CD material handling operations of unloading, sorting and loading are typically done “manually”, by forklifts with human operators. In this chapter, we consider the replacement of some or many forklifts by “Self-Driving Vehicles” (SDV). Can the resulting semi-automated material handling system attain the same or greater productivity as the fully manual system? At what cost (per unit of output)? We develop simulation models of two CDs, one purely manual and the other containing a mixture of forklifts and SDVs. Several CD performance measures are defined and estimated via simulation. For each CD, response surface methodology is employed to determine a near-optimal set of material handling equipment, when that CD is operated at a specified performance level.

1. Introduction

The increased variety of items demanded poses greater challenges in shipping the right quantities of those goods from the supplier to each customer. Often the demand for a particular product does not meet a full truckload (FTL). Suppliers use cross-dock facilities to improve outbound vehicle utilization.

Cross-docks coordinate shipments between particular suppliers and customers. Goods demanded by several retail centres or customers are sent from various suppliers to a central hub, the CD. Products received are then dispatched to customers within 24–48 hours of arrival at the cross-dock, having been sorted and consolidated using labour and MHE. Requirements of individual retail centres or customers are thus shipped in an almost Just-In-Time manner, without holding stocks at the CDs.

1.1 Material handling in a cross-dock facility

CD performance metrics include throughput rate (pallets processed per unit time), average truck turnaround time, MHE utilization, doorway or dock utilization, and shipping accuracy. Those are all directly influenced by the efficiency of material handling activities. MHE, for example hand pallet trucks and forklifts, are widely used in cross-dock facilities to process inbound items. However, recent advances in robotics for material handling, for example Automated Guided Vehicles (AGV) and Autonomous Mobile Robots or Self-Driving Vehicles (SDV), motivated our research on the scope of SDV in a CD.

Automated Guided Vehicles are programmed to move only on a dedicated path, making them well-suited for handling and material movement in a warehouse or other facility. The designated path is marked by paint or wire. SDVs, however, can move around an aisle or route by sensing any objects in their way. The ability of the SDV’s central control system to choose optimal routes, and the SDV’s obstacle-manoeuvring capabilities, makes Self-Driving Vehicles a potential alternative to forklifts. SDVs can thus be used in a CD to process pallets or goods received as unit loads.

A generic cross-dock facility, if operating manually, would require forklifts to sort and transfer those pallets between inbound and outbound trucks. A “semi-automated CD” [1] would process pallets with a mixture of forklifts and SDVs. Use of SDVs for cross-docking could perhaps reduce the variable operating cost, since less labour would likely be needed for such a semi-automated facility, compared to a manually operated CD. The present research financially compares a cross-dock operating with only forklifts to one operating with a mixture of forklifts and SDVs.

1.2 Solution methodology

To accommodate randomness in a CD material handling operation, we employed the discrete event simulation software ARENA. Two models were developed: Simulation Model 1) Forklift-only cross-dock (FL-only CD); and Simulation Model 2) Forklift-and-SDV cross-dock (FL-and-SDV CD). The models were built to be flexible and scalable. (Modeling assumptions and operating conditions are discussed in Section 3.) Independent simulation experiments were run with each model by varying the MHE quantities in offloading, sorting and loading. Response Surface Methodology enabled CD output performance metrics to be modeled in terms of MHE allocated to each process.

The latter prediction equations then permitted the formulation of an optimization problem. Its objective function is to minimize the operating cost of a CD facility, subject to constraints on the CD performance metrics we employ (throughput rate, throughput rate/MHE and MHE utilization rate). That optimization model was solved for desired throughput rates; the corresponding optimal MHE configurations were thus identified. Our overall solution framework is given in Figure 1.

Figure 1.

Solution methodology.

2. Literature review

There are ample publications that treat an overall cross-docking operation. But only a few papers were found that focused on modeling and analysis of floor-level activities, with different mixtures of MHE. Fewer still optimized those material handling operations, directly or indirectly. In that light, we discuss the relevant papers available on the modeling of material handling in a CD, first with deterministic and then stochastic models. Unless stated otherwise, the MHE are forklifts.

2.1 Deterministic modeling of CD material handling events

Bartholdi and Gue [2] optimized the labour cost of an LTL cross-docking operation, subject to floor-space congestion, that is, the ratio between average pallet flow and the total area available in front of an outbound door. Their objective function minimized the expected costs of moving pallets in rectilinear distance plus waiting due to interference between MHE.

Gelareh et al. [3] developed new formulations for CD door assignment. While all models are deterministic, the transportation times and distances between inbound and outbound doors were carefully calculated. Wolff et al. [4], noted, as we have, that efficient operation of a cross-dock is reflected by how internal resources (labour, MHE) are employed. Via a (deterministic) mixed-integer programming model, those authors sought a feasible truck schedule carried out with minimal resources.

The preceding analyses of cross-dock material handling yielded optimal or near-optimal solutions of a deterministic model. However, optimality in real CD facilities must deal with high randomness in material handling operations [5]. Simulation modeling is next discussed.

2.2 Stochastic simulation of CD material handling

Magableh et al. [6] modeled a generic cross-dock in which arriving trucks are assigned available doors based on FIFO. Using MHE, goods are then unloaded, sorted and loaded onto outbound vehicles for dispatch. Although this model represented the randomness in real-world CD material handling activities, the facility parameters (e.g. size, number of workers, MHE composition) were not varied or even specified explicitly. Liu and Takakuwa [7] minimized the labour cost of a retail distribution CD. Their simulation model treated stochastic operator movements, but randomness involved in actual MHE activities was ignored. Yang et al. [8] apply simulation to study the impacts on CD operation of the numbers of forklifts, door layout, and the freight mix for the case of purely manual MHE. Those authors analyzed the mean throughput rate/forklift and mean pallet-handling time in a CD. Forklift movements were modelled as they would occur in a real-world cross-dock, but optimal MHE levels were not identified.

Adewunmi and Aickelin [9] proposed a simulation and optimization framework covering CD order consolidation performed by 5–7 operators using −5 pieces of MHE. Their Total Cost objective was evaluated by simulation within a relatively small solution space.

Torbali and Alpan [10] systematically reviewed models of cross-dock scheduling that take a real-time perspective.

Let us now turn to simulation-optimization problems. These are generally more complicated to solve than deterministic optimization models, due to the difficulty in objective function evaluation. The technique involves independent simulations of each feasible set within the optimization search space.

Rather than simulating all those feasible sets, researchers have built and analysed the meta-models¹ of a simulation model, to estimate the optimal solution of the simulation-optimization problem. Regression Analysis and Neural Networks have been widely used for such purposes [11].

Aickelin and Adewunmi [12] solved the cross-dock door assignment problem to reduce the distance traveled by MHE with pallets. Bottlenecks and delays involved in a real-world CD were modeled, but MHE performance was not studied, hence not optimized. Shi et al. [13] proposed a simulation- optimization framework to design a robust JIT-based CD. Suh [14] employed agent-based techniques and then simulation to model the overall product movement from suppliers to distributors via CD. The simulation model addressed uncertainties in waiting times and logistics inventories but did not include material handling activities.

2.3 Autonomous material handling

Only a few papers on applications of SDV for material handling purposes have been found, but we have identified research on AGV use in the factory, warehouse or CD-like facilities. Peixoto et al. [15] applied simulation modeling techniques to study the impact of a new automated MHS for warehouse order picking, with results compared to the performance of a manually operated facility. He and Prabhu [16] study a CD where AGVs enable outbound vehicles to get the proper packages directly from inbound trucks. They develop a queueing model to establish the CD’s performance measures and determine the number of AGVs. Simulation is then used to check the model’s accuracy.

Rupp et al. [17] treat the Hub-Arrival-Departure problem. A cross-dock is an important example, whereby each vehicle must be assigned two-time intervals: first for arrival and then for departure. A consolidation time is needed at the hub to sort, prepare and rearrange products for outbound dispatch. Because they do not deal with a CD as such, Rupp et al. do not drill down to the details of material handling.

Guo et al. [18] also emphasize the treatment at hubs via parallel-machine scheduling. In the case of material handling via forklifts, the goal is little or no waiting in the transfer of pallets from inbound to outbound vehicles. The two trucks must be given time windows with maximal overlap, in order that the forklift be able to smoothly transfer the goods. Again, it is the details of the MHE utilized that separate our research from others. We concentrate on how the pallets are actually moved and transferred (solely by forklift or by a mixture of forklift-and-SDV?). Articles such as [17] or [18] could furnish important inputs into deciding the door assignment of truck arrivals at the CD. However, our issues concern the particular MHE operations on the facility floor.

Sayed et al. [19] do consider the times required for within-CD handling, that is, for unloading, transfer and loading of products. Those authors stress the importance of recognizing a spatial dependence in estimation of the handling times, in terms of the “route” taken within the CD. Indeed, in our research, the cross-dock layout (Figure 2 or Figure 3) implies such time and distance functionality. For us, the precise mixture of MHE will then determine the makespan, for a given truck arrival and departure pattern, and mixtures of goods involved.

Figure 2.

Facility layout: forklift-only cross-dock facility.

Figure 3.

Facility layout: forklift-and-self-driving vehicle cross-dock facility.

Fragapane et al. [20] distinguish between centralization and decentralization. What we term “SDV”, Self-Directed Vehicles, they call “AMR”, or Autonomous Mobile Robots. The larger the number of AMRs in the system, the more likely that decision-making will be decentralized. Our system, with its adjustable mixture of forklifts and self-directed vehicles, employs centralized decision-making. Our simulation models thus enable smooth coordination between the MHE (SDV and forklifts). Our SDV (see Section 3.2 and Table 1) can do all that a forklift can: fetch, sort and transport. The real benefit of an SDV, of course, is that it requires no human operator. With these attributes of our automated MHE, it is reasonable for us to compare the capabilities of CDs with varying mixtures of forklifts and SDV.

Parameter	Value
Model naming; CD working/simulation run time	(IB-FL). (SDV). (OB-FL); 7.5 hr/day
Facility dimension
Length; Width	750 ft.; 97 ft.
Trucks
IB or OB truck inter-arrival time; Pallets per IB or OB truck	0; UNIF (20, 26)
Docks
Number of inbound doorways or docks	50
Number of outbound doorways or docks	50
OB Staging capacity	13 Pallets in 15 × 13 ft^2
Number of pallet holders	1
MHE speed
Forklifts with pallet; Forklifts without pallet; SDV with or without pallets	316 ft/min or 1.61 m/s; 548 ft/min or 2.78 m/s; 316 ft/min or 1.61 m/s
Time delays
IB or OB truck changeover time; MHE travel time; MHE time to manoeuver and pick up pallets; MHE time to manoeuver and drop-off pallets	$\frac{\mathit{\text{TRIA}}\left(2,3,5\right)\mathit{\text{mins}}.}{\mathit{\text{manhattan}}\mathit{\text{Distance}}/\mathit{\text{Speed}}}$; UNIF (8, 12)secs; UNIF (8, 12)sec attempts

Table 1.

Design parameters: forklift-and-SDV cross-dock facility.

We emphasize that SDVs permit stable manoeuvring around obstacles and smart decision-making to choose an optimal path. This makes their movement equivalent to the movement of manual forklifts in a free path. Therefore, simulation modeling of SDV movements requires techniques similar to those for simulating manual forklifts, not AGVs.

3. Simulation modelling

3.1 Cross-dock design

The volume of goods which can be processed in a CD is determined by its cross-dock geometry. How many inbound and outbound doors does it have? How large are the staging area, MHE aisle, MHE parking area, MHE charging stations and the measuring stations? We will focus on the CD’s dedicated material handling region. That is, the number of doors and door size, facility shape, staging area and MHE aisle.

There are two types of doorways in a cross-dock. Goods from arriving vehicles are unloaded at the inbound doors. Trucks will leave the CD at outbound doors; their consolidated loads of products (received from inbound vehicles) are sorted by destination. For this research, a cross-dock is assumed to have a total of 100 doors, each 15 ft wide. The various CD shapes include I (long narrow rectangular), L, H, U, T, H, X and E. An I-shaped narrow rectangular cross-dock with 100 doors is assumed, based on the computational experiments on CD shapes of Bartholdi and Gue [21]. That is, 50 doors each on opposite sides of a facility, for a total length of 750 ft.

In an ideal cross-dock, materials received can be directly loaded onto outbound trucks from those inbound using MHE, when OB trucks are available, that is, parked at outbound doors. In cases where the destination vehicles are unavailable or goods require sorting and consolidation, the products are put in temporary storage, called a Staging Area, for further action. A CD facility with inadequate staging area suffers serious congestion and time delays; an excessive staging area would result in un-utilized floor space. The following are our assumptions on staging areas and aisles.

Forklift-only cross-dock facility (Figure 2):

Staging Area: Floor space of 58 × 15 ft² is dedicated for each pair of doors from opposite sides.

Forklift Aisle: Floor space of 30 × 15 ft² is dedicated in front of each door, for the forklifts to manoeuver, move between doors, sort goods and for truck loading or unloading

Forklift-and-self-driving vehicle cross-dock facility (Figure 3):

Staging Area: Floor space of 13 × 15 ft², allocated to each pair of doors on opposite sides. Forklifts Aisle: Floor space of 30 × 15 ft² is dedicated in front of each door for the forklifts to manoeuver, unload and load the trucks.

Self-Driving Vehicle Aisle: Floor space of 12 × 150 ft² is dedicated for the SDV to move back and forth across the CD. For each door, a floor space of 6 × 15 ft² is dedicated on either side of the SDV aisle for the SDV to park.

3.2 Material handling assumptions

Various assumptions were required to model and simulate the cross-dock material handling activities of the two MHE configurations. We first discuss premises common to both models: the General Assumptions (GA), followed by those unique to each of the particular CD configurations.

• (GA 1) Trucks arrive at a CD with inter-arrival time 0, ready to offload or load (more correctly, the cross-dock never waits for a vehicle). This enables us to concentrate on the material handling tasks of the CD.

• (GA 2) The number of pallets carried on each IB truck is uniformly distributed between 20 and 26, that is, full capacity of a 53 ft trailer. Those pallets are ready to offload.

• (GA 3) Pallets received from IB trucks are sorted, then loaded to respective outbound trucks as a unit load using MHE (forklifts or SDV).

• (GA 4) All outbound trucks (OB) require UNIF (20, 26) pallets to be loaded onto them, before leaving the CD.

• (GA 5) MHE are available throughout the simulation runtime, with no downtime.

• (GA 6) MHE are allocated based on their proximity to the “job call”. By this, we mean the nearness of free MHE to the pallet in question.

• (GA 7) Each IB or OB truck has a changeover time² of TRIA (2, 3, 5) mins. That is the time taken for a processed truck to depart, or for an unprocessed truck to dock at a doorway.

Based on the offloading or loading activity in which it engages, a manual forklift is categorized as either an inbound or outbound forklift, respectively. An IB-FL, dedicated to offloading and staging pallets from an inbound truck to the respective staging area, can move across the inbound doors or between an inbound door and the staging area. An OB-FL sorts and loads pallets from the staging area to destination trucks at outbound doors, and can thus move between the staging area and those doors.

Modeling assumptions: forklift-only cross-dock facility (Figure 2)

1. An IB-FL assigned to a particular truck offloads pallets, one by one, from that inbound vehicle to the staging area. The time required depends on speed and travel distance.

2. The staging area (58 × 15 ft²) in front of an inbound door is reserved for pallets received from only that particular door.

3. An IB forklift assigned to offload an IB truck is relieved only after the offloading is complete.

4. When the staging area is full, an OB-FL needs to move those pallets causing congestion. The assigned IB-FL will wait for that, before completing the offloading process.

Now, consider the forklift-and-self-driving vehicle cross-dock (Figure 3). The CD sortation and consolidation activities are the most cumbersome, resulting in huge floor and traffic congestion, and requiring immense labour and supervision to ensure transshipment accuracy. SDV are therefore particularly useful in combining several palletized loads. After the unloading of an inbound pallet, an SDV is programmed (with RFIDs or instructions from the master computers used for job allocation) to learn the destination of that pallet. The SDV will thereby transfer pallets to appropriate outbound doors, significantly reducing the labour requirement.

MHE in a forklift-and-SDV CD is categorized by the activities of offloading, sorting or loading, respectively performed by inbound forklifts, SDV and outbound forklifts. Pallets are offloaded from inbound trucks by IB-FLs, then are transferred to an SDV if available; otherwise, they go to the pallet holder. IB-FLs travel around or between inbound doors to offload the trucks. SDVs that are dedicated to sort and deliver pallets, from IB to OB doors, traverse the assigned 12 ft SDV aisle via a Manhattan-distance based path. A pallet that is on a pallet holder or on an IB-FL can be grasped by an SDV, but it can offload that pallet only to an OB-FL. As for OB-FL, they load pallets to outbound vehicles from SDV or from the staging area. An SDV sorts pallets and moves them to OB doors; OB-FL load pallets directly from the SDV to an outbound truck if available, or else places them in the staging area.

Our modeling assumptions for the forklift-and-self-driving vehicle cross-dock facility are as follows:

1. Pallets picked up by IB-FLs from inbound trucks are loaded to an SDV, if available at the designated SDV waiting zone, otherwise on the free pallet holder. When both are unobtainable, the inbound forklift awaits an SDV at the designated drop-off point.

2. After delivering the pallets, free SDV will return to the shortest active IB door (having the least number of SDV) within a 10-door distance.

3. Unavailable outbound vehicles cause pallets to be placed at the staging area to release the SDV. Sometimes there is no outbound truck and the staging area is full. There is then a delay in offloading, as the SDV anticipates arrival of an inbound vehicle.

4. Pallets staged are loaded by OB-FL to an OB truck, once latter is available at the door.

Simulation of the forklift-and-SDV CD proceeds via the general and modeling assumptions. Table 1 presents particular design parameters and time delays.

3.3 Output performance measures

Only a few metrics provide insight on overall cross-dock performance; most of the others emphasize particular aspects of CD operation: MHE, doors or trucks. For example, output measures such as Average Throughput rate, Average Pallet Flow Time or CD Utilization rate offer intuition on overall effectiveness and general capability of a cross-docking operation. But metrics such as Outbound Truck Tardiness and Average Loading or Unloading time indicates the performance of only certain facets of that operation (e.g. truck scheduling procedure) in a CD. Recognition of suitable output measure(s) is thus important for proper evaluation of SDV in a cross-dock.

After discussion with CD subject matter experts, we chose Average Throughput rate, Average Throughput rate/MHE and Average MHE Utilization rate. Together, these measures will provide good intuition on CD material handling effectiveness and will enable assessment of SDV capability.

δ, the mean throughput rate, furnishes an understanding of the facility’s overall material handling capability for a given MHE configuration.

Let P_j = Number of pallets processed on day j; N = Total number of replications or days.

$$\text{Then}\delta =\frac{{\sum }_j^N=i{p}_j}{N}\text{in units of}\mathit{\text{pallets}}/\mathit{day}\mathit{\text{or}}\mathit{ppd}$$E1

The cross-dock objective is to maximize δ, the mean throughput rate. However, efficient MHE utilization cannot be assured by studying only δ. Consider the ratio between total MHE uptime and the total time available in a CD, that is the Overall MHE Utilization rate, _. This offers intuition on aggregate utilization of the CD’s equipment, rather than attempting to analyse any machine individually. (Naturally, MHE idle time can incur cost in terms of non-ideal utilization of labour and MHE.) The category-wise MHE utilization rates are also computed for IB-FL (UI), SDV (US) and OB-FL (UOB). The CD objectives are to maximize the overall utilization, and each of those category rates.

Let UT_ij = Uptime of i^th MHE on the j^th replication; T = Simulation run time/replication; c = Total number of MHE in a cross-dock; X_I =Total number of inbound forklifts in a CD; X_S= Total number of self-driving vehicles in a CD;

X_o = Total number of outbound forklifts in a CD.

With c = X_I + X₀ + X_S (and X_S = 0 for FL—only CD);

$$\text{Then}{U}_0=\frac{{\sum }_i^{c=1}{\sum }_j^N{\mathit{UT}}_{\mathit{ij}}}{T\times c\times N}\times 100\%$$E2

Also, all for the j^th replication:

Let UTI_ij = Uptime of i^th IB-FL; UTS_ij = Uptime of i^th SDV; and UTO_ij = Uptime of i^th OB-FL.

$$\text{Then}{U}_I=\frac{{\sum }_i^{x=1}{\sum }_j^N{\mathit{UT}}_{\mathit{ij}}}{T\times X_I\times N}\times 100\%$$E3

$$U_S=\frac{{\sum }_{i=1}^{x_S}{\sum }_{j=1}^N{\mathit{UTS}}_i}{T\times X_S\times N}\times 100\%$$E4

$$U_{\mathit{OB}}=\frac{{\sum }_{i=1}^{x_0}{\sum }_{j=1}^N{\mathit{UTO}}_{\mathit{ij}}}{T\times X_O\times N}\times 100\%$$E5

While the MHE utilization rate specifies an awareness of equipment usage and idleness, it may not indicate efficient employment of MHE. Poor MHE allocation in a CD (between IB-FL and OB-FL, or between IB-FL, SDV and OB-FL) may still result in achieving a reasonable MHE utilization rate. That is, might a lesser amount of material handling equipment process a greater number of pallets? The following criterion, Average Throughput Rate/MHE,δ_M, seeks to resolve this.

δ_M attempts to compromise between the preceding two performance metrics. It is a ratio between Average Throughput rate (δ) and the total amount of MHE in the cross-dock. δ_M helps us understand how efficiently pallets are processed using the MHE in that CD. The cross-dock objective is to maximize

$${\delta }_M=\frac{{\sum }_{i=1}^N{P}_i}{N\times c}$$E6

δ_M is measured in pallets/day/MHE, and c is the number of MHE.

3.4 ARENA simulation model overview

ARENA 15.0 was employed for CD modeling. For each of the forklift-only and forklift-and-SDV cross-docks, separate simulation models were developed. Those models followed the material handling assumptions and working parameters of Section 3.2. A Visual Basic API in ARENA was used to model a scalable facility, but (Tables 1 and 2) we fixed the size as 50 IB doors ×50 OB doors. Each respective model would terminate when the simulation clock time reached 7.5 hrs, assuming one shift or 8 hrs of operation with 30mins break and no simulation warm-up period.

Parameter	Value
Model naming; CD working or simulation run time	(IB-FL). (OB-FL); 7.5 hr/day.
Facility dimension
Length; Width	750 ft.; 118 ft.
Trucks
Inbound or outbound truck inter-arrival time; Pallets per inbound or outbound truck	0.; UNIF (20, 26).
Docks
No. inbound doors or docks; No. Outbound doors or docks extent of staging area	50.; 50.; 30 Pallets in 15 × 26 ft^2
Forklift average speed
IB-FL or OB-FL with pallet; IB-FL or OB-FL without pallet	316 ft/min or 1.61 m/s.; 548 ft/min or 2.78 m/s.
Time delays
IB or OB truck changeover time; IB or OB Forklift travel time; Forklift time to manoeuver and pickup pallets; Forklift time to manoeuver and drop pallets	$\frac{\mathit{\text{TRIA}}\left(2,3,5\right)\mathit{\text{mins}}.}{\mathit{\text{manhattan}}\mathit{\text{Distance}}/\mathit{\text{Speed}}}$; UNIF (8, 12) secs.; UNIF (8, 12) secs.

Table 2.

Design parameters: forklift-only cross-dock facility.

Validation of a simulation model is critical to ensure that the model represents the actual system [22]. Our models for the FL-only and FL-and-SDV cross-docks were validated by the group of members involving cross-dock SMEs and a simulation analyst from the SDV manufacturer.

Besides the input parameters in Tables 1 and 2, execution of the cross-dock simulation model requires the MHE configuration. For an FL-only CD, this includes X_I and X₀ (Table 3); and those values plus X_S (Table 4) for an FL-and-SDV CD. These configurations determine the cross-dock’s material handling performance, and (most of) the facility’s variable operating cost.

Factors	Notation	Number of levels	Factor levels
Number of IB-FL; Number of OB-FL	XI; Xo	5; 10	25, 30, 35, 40, 45; 50, 60, 70, …, 130, 140

Table 3.

RSM Factor Definition: forklift-only cross-dock. 5 × 10 = 50 treatment combinations in full-factorial design.

Factors	Notation	Number of levels	Factor levels
No. IB-FL No. SDV; No. OB-FL	XI; XS; X0	5; 11; 7	25, 30, 35, 40, 45; 50, 60, 70, …, 140, 150; 25, 30, 35, 40, 45, 50, 55

Table 4.

RSM Factor Definition: Forklift-and-SDV cross-dock. 5 × 11 × 7 = 385 treatment combinations.

Optimization of a simulation model, or statistical comparison of two models, requires a minimum of 30 replications each. The mean runtime for each replication of our ARENA CD simulation model took 1 minute, in a computer system with Intel(R) Core(TM)i7-3537U CPU @ 2.0 GHz processor, 6GB RAM. However, there are clearly very many feasible MHE configurations for the forklift-only CD and for the forklift-and-SDV cross-dock. To avoid the enormous computer run time, we propose a simulation-optimization technique in Sections 4 and 5 to overcome those computational difficulties. We suggest a meta-modeling approach to find optimal MHE configuration in a fast, time-efficient manner.

4. Response surface methodology

Fu [11] presents various simulation-optimization techniques, of which Response Surface Methodology (RSM) meets our requirement. The goal of our meta-modeling approach is to develop RSM prediction models for CD performance metrics using the MHE configuration as exploratory variables. We will then employ those regression prediction models to formulate constraints of an optimization problem.

Solution of the latter will yield the best MHE configuration, that is, one whose total variable operating cost is minimum, and whose cross-dock performance metrics are as desired.

A full-factorial experiment is designed for both CD simulation models with differing levels of MHE configuration. Performance of the FL-only CD is studied by varying the numbers of IB-FL and OB-FL in the CD. Those two factors have respectively 5 and 10 levels, as shown in Table 3. However, three factors (Table 4) are needed to evaluate the performance of the FL-and-SDV cross-dock: numbers of IB-FL, SDV and OB-FL in the CD. For both simulation models, each treatment combination is replicated³ 15 times, recording the performance metrics (δ, δ_M, U₀, U_I, U_0B, U_S) in each case.

Four⁴ regression models were estimated for each performance metric for both cross-dock facilities. Statistical assessment of those models showed that the regression coefficients of each model to be statistically significant. In light of R squared and the predicted regression sum of squares, we chose quadratic regression models. In view of the preceding statistical significance, further analysis enabled identification of the best MHE configurations.

Suppose R is a CD performance metric; its estimate is defined as $\widehat{R^{\prime }}$ for the FL-only CD, and as ${\widehat{R}}^{″}$ for the FL-and-SDV CD. The following remarks refer to both of those estimates.

As quadratic regression models, the fitted equations included terms linear in each of the variables, plus all possible products (every term of total exponent 2) of each variable multiplied by itself or any of the others. (Detailed regression equations are omitted for lack of space.)

5. Optimization of cross-dock MHE configuration

The fitted regression models for each facility’s performance metrics were used to formulate respective optimization models. Two independent minimization models were formulated in terms of MHE configuration, for the FL-only CD and for the FL-and-SDV CD. Each cost objective was subject to constraints by its predicted performance metrics R^̂ that yielded the desired level of performance R^̅. Each optimization model thus furnished an optimal or near-optimal⁵ MHE configuration and the corresponding total variable cost for the CD.

5.1 MHE optimization models

CD variable costs include labour and MHE operating costs. Each forklift truck naturally requires an operator; we assume that no additional labour is needed for material handling in a cross-dock. The total variable cost (TVC) of operating an FL-only CD is estimated to be $522.63/day. (Figure includes operator wages, and forklift power consumption and repair and maintenance.) Decision VariablesX_I - Number of IB-FLs; X₀ - Number of OB-FLs. The resulting optimization model for an FL-only CD is then formulated as:

$$\mathbf{Min}\mathrm{TVC}=522.63\left({\mathit{\text{X}}}_{\mathit{\text{I}}}+{\mathit{\text{X}}}_{\mathit{\text{0}}}\right)$$E7

Subject To$\widehat{{\mathit{\delta }}^{\prime }}\ge \overline{\mathit{\delta }};\widehat{{\mathit{\delta }}^{\prime }}\ge 200\mathit{M};\widehat{{\mathit{U}}^{\prime }}\ge 80\%\mathit{O};\widehat{{\mathit{U}}^{\prime }}\ge 60\%I;\widehat{{\mathit{U}}^{\prime }}\ge 60\%\mathit{OB};$.

And 25 ≤ X_I≤ 45; 50 ≤ X₀ ≤ 140; X_I, X₀ ∈ Z⁺ (positive integers).

Manual forklifts in a CD incur the costs of labour and of forklift operation; just operating costs are incurred by SDV. Provided that, over time, the manual forklifts and SDV have about the same operating costs, the following optimization model will minimize the total variable cost of a forklift-and-SDV cross-dock.

Decision Variables: X_I - Number of IB-FL; X_S - Number of SDVs; X₀ - Number of OB-FLs.

$$\mathbf{Min}\mathrm{TVC}=522.63\left({\mathit{\text{X}}}_{\text{I}}{\mathit{\text{+X}}}_{\mathit{\text{O}}}\right)+271.20{\mathit{\text{X}}}_{\mathit{\text{S}}}$$E8

Subject To:$\widehat{{\mathit{\delta }}^{\prime }}\ge \overline{\mathit{\delta }};\widehat{{\mathit{\delta }}^{″}}\ge 120\mathit{M};{\widehat{\mathit{U}}}^{″}\ge 80\%\mathit{O};{\widehat{\mathit{U}}}^{″}\ge 60\%I;{\widehat{\mathit{U}}}^{″}\ge 60\%\mathit{OB};{\widehat{\mathit{U}}}^{″}\ge 60\%S$.

And 25 ≤ X_I ≤ 45; 50 ≤ X_s ≤ 155; 25 ≤ X₀ ≤ 55; X_I, X_S, X₀ ∈ Z⁺.

(The coefficient 271.20 of X_S is the estimated average cost/day to operate SDV including electricity, repair and maintenance).

5.2 Optimal MHE configurations

The optimization models for the respective cross-docks are mixed-integer nonlinear programmes (MINLP). Each objective function is linear, with nonlinear constraints and integer decision variables. Each MINLP model is solved using the Lingo 17.0 solver for an increasing level of desired throughput rate $\left(\overline{\delta }\right)$.${\overline{\delta }}_M$ is set as 200 ppd/MHE for the FL-only CD, and as 120 ppd/MHE for FL-and-SDV CD.

The latter cross-dock naturally requires a greater number of MHE, to achieve a δ similar to the former. Optimal MHE configurations (solutions of MINLP models) are in Table 5.

$\overline{\mathit{\delta }}$/10^3	FL-only		FL-and-SDV			TVC ($/day)		Optimal model
	XI	X0	XI	XS	X0	FL-only	FL-and-SDV
20	32	60	33	81	31	48,082	55,416	FL-only
22	33	67	35	90	34	52,263	60,469	FL-only
24	34	74	36	98	39	56,444	65,775	FL-only
26	36	81	38	110	41	61,148	71,120	FL-only
28	38	89	39	115	52	66,374	78,747	FL-only
30	42	97	41	127	55	72,646	84,615	FL-only

Table 5.

Optimal MHE configuration, if FL operating cost = $522.63/day, SDV operating cost M = $271.20/day.

The operating costs for Forklifts and SDVs, at the time of this research, were assumed at extreme limits to account for maximum cost. Adjusting to industrial standards, the average cost to operate a forklift drops to $348.70/day and $97.28/day for SDVs. The revised objective functions for the cross-docks are presented in Eqs. (9) and (10). The corresponding optimal MHE configurations, total variable costs and payback period (PP)⁶ are found in Table 6. (Optimal configurations for the FL-only CD, as shown above in Table 5, still hold; the optimal MHE arrangement changes only for the FL-and-SDV CD.)

δ^̅/10^3	FL-only CD			FL-and-SDV				Optimal model	Savings/year ($)	Fixed cost (in $M)	PP1 (in years)
	XI	X0	TVC ($)	XI	XS	X0	TVC ($)
20	32	60	32,080	32	96	26	29,563	FL-and-SDV	604,061	17.1	14.15
22	33	67	34,870	33	102	31	32,239	FL-and-SDV	631,354	17.93	14.2
24	34	74	37,660	36	112	33	34,956	FL-and-SDV	648,946	19.8	15.26
26	36	81	40,798	38	117	38	37,883	FL-and-SDV	699,586	20.4	14.58
28	38	89	44,285	40	130	41	40,891	FL-and-SDV	814,512	22.73	13.95
30	42	97	48,469	41	127	55	45,830	FL-and-SDV	633,490	21.23	16.75

Table 6.

Optimal MHE configuration (FL operating cost = $348.70/day; SDV operating cost = $ 97.28/ day). Payback period is time required to recoup extra fixed cost¹ of SDV in FL-and-SDV CD, relative to FL-only CD, for that δ.

¹

No of days in a year = 5 days × 4 weeks × 12 months; one 8 hr. shift per day. Cost of a Forklift = $ 75,000. (Source: http://www.costowl.com/b2b/forklift-electric-cost.html).

$$\text{For}\mathrm{FL}-\text{only}\mathrm{CD}:\mathbf{Min}348.70{\mathit{\text{(X}}}_{\mathit{\text{I}}}+{\mathit{\text{X}}}_{\mathit{\text{O}}})$$E9

$$\text{For}\mathrm{FL}-\text{and}-\mathrm{SDV}\mathrm{CD}:\mathbf{Min}348.70{\mathit{\text{(X}}}_{\mathit{\text{i}}}+{\mathit{\text{X}}}_{\mathit{\text{O}}})+97.28X_S$$E10

5.3 Validation of optimal MHE configuration

Suppose the regression model predictions were accurate (i.e. δ^̂ is not statistically different from δ). MHE configurations suggested by solving the MINLP model for FL-only and FL-and-SDV CDs would then be optimal, and would furnish performance measures equivalent to the predicted performance. Alternatively, prediction errors may yield only “near-optimal” MHE configurations. As mentioned earlier, there is a non-random source of variation, left unexplained by the quadratic regression models for the CD performance metrics $\widehat{R^{\prime }}$ or $\widehat{R^{″}}$. Consequently, those fitted models do not predict CD performance fully accurately.

The solution of the MINLP models furnished “the best” MHE configurations, which were input into the two respective cross-dock simulation models. Could those CDs, that is the FL-only and FL-and-SDV cross-docks, operating under the configurations determined, furnish throughput rates that are similar? Statistical comparisons of mean throughput rates for all pairs of ‘proposed optimal MHE configurations’ are given in Table 7. That table shows dissimilar respective performance metrics (i.e. statistically significant differences between the mean throughput rates of the FL-only and FL-and-SDV CD).

δ^̅/10^3	FL-only				FL-and-SDV				t-Test p-value
	Model	δ^̂′	δ′	TVC ($)	Model	δ^̂″	δ″	TVC	n = 8	n = 30
Operating cost for FL = $522.63/day and SDV = $271.20/day
20	32.60	20,105	19,512	48,082	33.81.31	20,001	19,629	55,416	0.03	0.00
22	33.67	22,162	21,681	52,263	35.90.34	22,021	21,793	60,469	0.02	0.01
24	34.74	24,049	23,988	56,444	36.98.39	24,000	19,056	65,775	0.00	0.00
26	36.81	25,989	26,244	61,148	38.110.41	26,002	24,928	71,120	0.00	0.00
28	38.89	27,956	28,820	66,374	39.115.52	28,027	27,268	78,747	0.00	0.00
30	42.97	30,039	31,486	72,646	41.127.55	30,033	30,431	84,615	0.00	0.00
Operating cost for FL = $348.70/day and SDV = $97.28/day
20	32.60	20,105	19,512	32,080	32.96.26	20,029	20,524	29,563	0.00	0.00
22	33.67	22,162	21,681	34,870	33.102.31	22,023	21,376	32,239	0.00	0.00
24	34.74	24,049	23,988	37,660	36.112.33	24,013	23,485	34,956	0.00	0.00
26	36.81	25,989	26,244	40,798	38.117.38	26,012	24,808	37,883	0.00	0.00
28	38.89	27,956	28,820	44,285	40.130.41	28,009	27,397	40,891	0.00	0.00
30	42.97	30,039	31,486	48,469	41.127.55	30,033	30,431	45,830	0.00	0.00

Table 7.

Statistical validation of optimal MHE configuration (for model “naming convention,” see rows of Tables 5 and 6 for corresponding $\stackrel{-}{\mathit{\delta }}$).

We conclude, therefore, that by solving the MINLP models, MHE configurations obtained are efficiently allocated. Although not necessarily the optimal ones, those configurations are near-optimal.

6. Discussion and suggestions for further research

Solutions of the formulated MINLP model yield many interesting insights:

• Total numbers of MHE in each category almost always increase together for an increase in desired throughput δ^̅ (Sole exception: last two rows of Table 7, for FL-and-SDV CD)

• Statistical comparisons of manual and semi-automated CDs, under optimal MHE configurations (Tables 5 and 6), show significant differences in actual performances. For reduced MHE operating costs, a mixture of forklifts and SDVs in a CD results in lower total variable operating cost than for a purely-forklift facility.

The proposed solution methodology to find the optimal MHE configuration for FL-only and FL-and-SDV CDs operating at similar performance levels yields an efficient MHE arrangement.

Our solution to the MINLP model may be sub-optimal, but visual comparison (Table 8) of the manual and semi-automated cross-docks yielding similar δ show it could be financially beneficial to choose FL-and-SDV over the FL-only CD. A 5% confidence interval (CI) was constructed for the difference in mean throughput rates of the two CD types. That CI contained the value 0, leading to the conclusion that the two facility types had similar output performance δ. Savings/year and payback period⁷ (PP), given in that table, support those possible benefits. (Note that Table 8 assumes two shifts/day, whereas all other tables assume a single shift).

FL-only CD			FL-and-SDV CD			CI for difference in mean			Savings/year ($)1	PP (in years)1
Model	δ′	TVC ($)	Model	δ”	TVC ($)	Lower bound	Mean	Upper bound
32.72	23,412	36,265	36.112.33	23,485	34,956	−147	−73	1	628,387	22.56
36.81	26,244	40,798	39.125.39	26,260	39,359	−71	−17	38	690,864	19.65
39.94	30,441	46,377	41.127.55	30,431	45,830	−48	9	67	262,723	61.95

Table 8.

MHE configurations of cross-docks with similar δ.

¹

No of working hours = 8 × 2 × 5 × 4 × 12 (two shifts per day); Operating cost, M = $ 97.28/day.

For future research, we propose the following: The predictive accuracy of the regression models might be improved by incorporating additional explanatory variables (e.g. product mix, or labour activities other than for forklift operators); the MHE configuration computed for each CD would then be closer to the true optimum. An expanded MINLP cross-dock model could include additional constraints on costs or space. Finally, the model could be enhanced by considering further randomness in supply and demand, MHE breakdowns, CD traffic congestion and LTL shipment operations.

6.1 Pros and cons of proposed solution methodology

Among the positive features, the methodology of this chapter may in fact be used to solve complex simulation-optimization problems of any system, not just CDs, with a larger search space in a shorter time span. Moreover, multiple systems could be optimized and compared, subject to constraints on various performance metrics, eliminating family-wise error rate due to multiple comparisons.

On the negative side, in-depth knowledge of the system is required to model precisely, and to identify the factors which impact the overall performance of the system. The results could be near-optimal, but may be sub-optimal, if the regression models do not adequately address the sources of variation.

Though this solution methodology is very versatile and could be adapted to optimize and compare any other system, we caution future researchers that the need for in-depth understanding of the actual system is paramount when it comes to simulation modeling. Otherwise, the simulation model risks misrepresenting the system with distorted results. This risk could be mitigated by:

1. Involving subject matter expert(s) from the system and other stakeholders closely in the design, verification and validation of the modeling process.

2. Cross validate the model’s functions and performances against the system.

3. Iteratively addressing the design gaps until the simulation model adequately represents the defined functional scope of the system.

Funding Statement

Research is supported by the Natural Sciences and Engineering Research Council of Canada, Grant # RGPIN-2019-06207.