MODELS FOR PERFORMANCE ANALYSIS OF A CROSS-DOCK

Transcription

1 The Pennsylvania State University The Graduate School Harold and Inge Marcus Department of Industrial and Manufacturing Engineering MODELS FOR PERFORMANCE ANALYSIS OF A CROSS-DOCK A Thesis in Industrial Engineering by Nikita Ankem 2017 Nikita Ankem Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 2017

2 The thesis of Nikita Ankem was reviewed and approved* by the following: Vittaldas V. Prabhu Professor of Industrial and Manufacturing Engineering Thesis Adviser Terry Harrison Professor of Supply Chain and Information Systems M. Jeya Chandra Professor of Industrial and Manufacturing Engineering Acting Department Head of Industrial and Manufacturing Engineering *Signatures are on file in the Graduate School. ii

3 ABSTRACT Increasing demand for high-speed delivery without errors necessitates the need of automation in cross-docking which is inspired by higher automation in distribution centers. Speedy processing and redirection of freight to its destination is the integral part of cross-docking operations and the tremendous labor involvement in the process results in variation and errors. For a completely automated cross-dock with robotic arms for loading and unloading and Automated Guided Vehicles (AGVs) to carry the freight across the cross-dock, the thesis proposes computational models based on shape, size and AGV specifications to determine feasibility and performance parameters under given conditions. The Max-Flow model uses the Max-flow Min-cut theorem and determines best shape of the cross-dock based on maximum possible throughput under given conditions. Max-Flow is the maximum possible throughput that a given system can have and gives an upper-bound of throughput for the system. The shapes are also compared based on the area and effect of the type of door assignments for inbound and outbound trucks. A more detailed probabilistic model is proposed which uses Mean Value Analysis (MVA) to calculate the throughput for given data of inbound and outbound freight. The MVA model considers traffic and congestion caused due to AGVs carrying the freight and calculates the throughput for given number of AGVs as well as the wait times and queue lengths at each intersection along the AGV path. Keywords: AGV, Cross-dock, Computational Model, Max-Flow, Throughput, Mean Value Analysis iii

4 TABLE OF CONTENTS List of Figures...viii List of Tables...xi Acknowledgements xiii CHAPTER 1 INTRODUCTION Cross-docking Operations Literature Summary Overview Organization... 5 CHAPTER 2 LITERATURE REVIEW Characteristics of cross-docking Layout Design for a Cross-dock Dock Door Assignment Scheduling Temporary Storage iv

5 2.6 Performance Analysis Network Diagrams Mean Value Analysis CHAPTER 3 MODEL AND ASSUMPTIONS Cross-Dock Design Cross-dock Operations Freight Size and Service Rate Specifications CHAPTER 4 CROSSDOCK SHAPE AND AREA Variety in shapes Shape Nomenclature and Door Placement Available free space Area of a cross-dock Comments about the H shaped cross-dock Computational modelling of the graphs and utilization CHAPTER 5 MAX-FLOW OF A CROSS-DOCK Model Assumptions Cross-dock Settings under review v

6 5.3 Best shape for various cross-dock sizes Best shape for various AGV speeds Best shape for varying number of AGVs available Best shape for varying AGV capacity Computational Model CHAPTER 6 MEAN VALUE ANALYSIS Model Assumptions Division of pathways into segments Computational Model CHAPTER 7 COMPARISON AND APPLICATION OF MODELS Similarities and differences in the two models Model Results for different cases CHAPTER 8 COST ANALYSIS Model Assumptions Cost-Considerations Details of different cases considered Cost Comparison for different regions vi

7 CHAPTER 9 CONCLUSIONS AND FUTURE WORK Conclusions Scope for Future work REFERENCES APPENDIX_ DEMAND DATASETS USED TO TEST MVA MODEL vii

8 LIST OF FIGURES Figure 1. 1 Process Diagram... 1 Figure 1. 2 Layout Diagram of automated cross-dockock... 2 Figure 1. 3 Thesis Organization... 5 Figure 2. 1 Example of a single source, single sink flow network..15 Figure 2. 2 Computational Model: Variables in yellow with result values and capacity constraints of the arcs Figure 2. 3 Computational Model: Flow constraints at the nodes with result values Figure 3. 1 Cross-dock Layout with Dimensions 20 Figure 3. 2 Path Division into Segments Figure 3. 3 Process Flow with Service Rates Figure 4. 1 Layout Diagram with Operations...26 Figure 4. 2 Cross-dock shapes considered Figure 4. 3 I shape with and without doors on width Figure 4. 4 Doors lost and space available in a cross-dock Figure 4. 5 Area vs No. of Doors for I and I' viii

9 Figure 4. 6 Area vs No. of doors for increasing width of I shape Figure 4. 7 Area vs No. of doors for different shapes Figure 4. 8 Table headings in computational model for an I cross-dock with doors on width35 Figure 4. 9 Table headings in computational model for a X shaped cross-dock Figure 5. 1 Network Flow Diagram of Cross-dock Operations 37 Figure 5. 2 Cases Considered (I) Figure 5. 3 Cases Considered (II) Figure 5. 4 Max Flow vs Cross-dock Size Figure 5. 5 Max-Flow vs AGV speed for 100 doors in a cross-dock Figure 5. 6 Max-Flow vs AGV speed for 200 doors in a cross-dock Figure 5. 7 Max-Flow vs AGV speed for 300 doors in a cross-dock Figure 5. 8 Max-Flow vs Available AGVs Figure 5. 9 Max-Flow vs AGV's carrying capacity Figure Network Flow Diagram of Cross-dock Operations Figure Formulae used in the max-flow computational model Figure 6. 1 MVA Throughput for varying number of AGVs in system..54 Figure 7. 1 Comparison of Throughput from Max-Flow and MVA models 63 ix

10 Figure 7. 2 Max-Flow and MVA throughput for grocery dataset with high demand data values and high deviation Figure 7. 3 Comparison of Max-flow and MVA values for demand data with mean 1 and deviation Figure 7. 4 Comparison of Max-flow and MVA values for demand data with mean 3 and deviation Figure 7. 5 Comparison of MVA throughput for datasets with same mean but different deviation Figure 7. 6 Comparison of MVA throughput for datasets with and Figure 8. 1 Cost Comparison for USA.75 Figure 8. 2 Cost Comparison for Germany Figure 8. 3 Cost Comparison for China x

11 LIST OF TABLES Table 4. 1 Minimum possible doors for a cross-dock shape Table 5. 1 Best shape based on Max-flow against no. of doors..40 Table 5. 2 Service rates for the experiment (Max-flow vs Size) Table 5. 3 Max-flow values for given shapes and sizes Table 5. 4 Max-flow values for given shapes and AGV speeds Table 5. 5 Service rates for the experiment (Max-flow vs AGV speeds) Table 5. 6 Service rates for the experiment (Max-flow vs no. of AGV/door) Table 5. 7 Max-flow values for given shapes and no. of AGVs/door Table 5. 8 Service Rates for Experiment (Max-flow vs AGV capacity) Table 5. 9 Max-flow values for given shapes and AGV speeds Table Symbols and letters for Service rates Table X and Y coordinates for inbound and outbound door nos. 1 through Table Distance Matrix showing distances between inbound and outbound doors Table Service rates specified by user Table xi

12 Table 6. 2 Experiment assumptions Table 6. 3 Demand Matrix in the computational model Table 6. 4 Conversion of S-matrix to S column Table 6. 5 Values of S, Tau and visit ratio in computational model Table 6. 6 Changing Throughput values against no. of AGVs in system Table 6. 7 Sample of MVA model results showing throughput, Table 7. 1 Throughput values from MVA and Max-flow models 63 Table 8. 1 Demand Matrix for Case I 72 Table 8. 2 Demand Matrix for Case II Table 8. 3 Demand Matrix for Case Table 8. 4 Annual Cost Values for USA Table 8. 5 Annual Cost Values for Germany Table 8. 6 Annual Cost Values for China xii

13 ACKNOWLEDGEMENTS I am extremely grateful to my adviser, Prof. Vittaldas V. Prabhu, who expressed immense trust in my abilities and motivated me to work smarter and harder and guided me throughout the research. I am thankful to Prof. Terry Harrison for his contributions in improving the quality of the thesis work. I am thankful to my boyfriend, Akshay Dongarwar for his whole-hearted support and friends Sandeep Shastry and Athul Gopala Krishna for their kind assistance during the research. xiii

14 DEDICATION To my parents, Rajshree and Sudarshan xiv

15 CHAPTER 1 INTRODUCTION 1.1 Cross-docking Operations Cross-docking is a logistics practice of unloading a shipment from an incoming trailer and loading the packages in that shipment to outbound trucks headed for respective destinations with little or no storage of the packages at the cross-docking facility. Cross-docking is intended to either consolidate the shipments from disparate sources and achieve economies of scale in outbound transportation or eliminate the inventory holding function of a warehouse while still serving its purpose of receiving, redirecting and shipping of packages. This thesis focuses on an automated cross-dock with trailers coming in at the inbound doors, packages unloaded by a robotic arm and placed on the inbound dock, AGVs picking a package up from the inbound dock and carrying it to the respective outbound dock, a robotic arm loading the packages into the outbound truck at the door. The design efficiency and performance of a cross-dock can be assessed by several factors like its area, throughput, cycle-time and congestion. Figure 1.1 shows the different processes involved and Figure 1.2 shows the layout and different entities of the automated cross-docking facility where the processes occur. Figure 1. 1 Process Diagram 1

16 Figure 1. 2 Layout Diagram of automated cross-dock 1.2 Literature Summary Several authors have published experiments and proposed computational models to assess the performance of a distribution center as well as investigated effects of different factors on the performance of a cross-dock. Bartholdi and Gue [1] studied how different shapes of a crossdock affect cross-dock performance and give tremendous insight into how size and shape affect labor costs. For the use of that performance analysis, they use the freight flows based on the one-pass interchange heuristic from their paper on reducing labor costs for a LTL cross-dock in February 2000 [21]. Some authors talk about optimizing the temporary storage at the crossdock. Vis and Roodbergen [3] propose a layout design in which goods will be stored in parallel rows between which workers can move and they also propose an algorithm for optimal number and length of rows. Gue and Kang [2] call these rows staging queues and compare single stage storage in cross-docking versus a double stage one for size and number of staging queues as well as throughput. Vis and Roodbergen [3] form a problem to find the optimal location for storage such that the travel distances of goods are minimized. Pandit and Palekar [14], establish a metric called the response time to determine optimal layout design of a DC and establish 2

17 formulae to calculate the estimated travel times for different types of layouts and for different number of racks in a row or column for a distribution center. In his Master s thesis, Athul Gopala Krishna [16] has proposed a queuing model for clearly defined outbound processes in a distribution center and has given results for performance analysis of such a DC for different sizes and workforce utilizations. Mean Value Analysis has been used by some authors to test performance of manufacturing systems. M. Jain, Sandhya Maheshwari and K.P.S. Baghel [22] develop a queueing model which contains multiple material handling devices they call MHDs and suggest models to analyze the interference of the MHDs in a flexible manufacturing system. R.D. van der Mei, E.M.M. Winands [23] propose a MVA based approach for a multi-queue single-server system in which the server visits the queues and processes the requests. J. R. Artalejo and J.A.C. Resing [24] apply the MVA approximation to carry out performance analysis of M/G/1 type retrial queues. They show that the mean value analysis technique provides a reliable alternative to obtain the expected queue lengths and wait times by avoiding the use of heavy algebra. 1.3 Overview This thesis compares cross-docks for parameters of area, % area utilization, max-flow from a deterministic model, throughput from a probabilistic model and congestion across cross-docks with different shapes, sizes and AGV speeds. The objective is to study performance and area utilization across different types of cross-docks and recommend the best fit for a given set of user requirements. A user-friendly computational tool is also made to compare the different shapes based on a given set of user requirements. The area is assessed to analyze the space 3

18 requirement and utilization for the same size across different shapes. The max-flow gives the upper-bound of the throughput from a cross-dock and can be used to determine the best shape when the throughput is an integral performance parameter. Mean Value Analysis is a probabilistic model to calculate the throughput and it considers the congestion and its effect on the AGV travel time. In this thesis, a fully automated cross-dock is analyzed. The unloading speed from the inbound truck to receiving dock is called UR (Unloading Rate), picking rate is called PR (Picking Rate), rate of delivery of the dock is called DR (Delivery Rate) and rate of loading the packages to the outbound truck is LR (Loading Rate). The rate of transfer of a package from the inbound to outbound dock is called travel rate. All the rates are measured in packages/hour. The travel paths have been divided into segments for analyzing the congestion at the junctions. The Mean Value Analysis model calculates the waiting time at each segment of the travel track and the queue length at each junction. The areas, max-flow and throughput from MVA model are all plotted in respective graphs across number of doors to see how cross-dock size affects these parameters for given values of AGV speed, shape, LR, PR, DR and LR. 4

19 1.4 Organization This thesis is organized as shown in Figure 1.3. Figure 1. 3 Thesis Organization 5

20 CHAPTER 2 LITERATURE REVIEW 2.1 Characteristics of cross-docking Definitions Cross-docking is most widely defined as a logistics practice of unloading materials from an incoming semi-trailer truck or rail car and loading these materials directly into outbound trucks, trailers, or rail cars, with little or no storage in between. The main idea behind crossdocking is to transfer incoming shipments to outgoing freight directly, thus reducing storage in a supply chain. Cross-docking was defined by Kinnear E. [4] as Receiving product from a supplier or manufacturer for several end destinations and consolidating this product with other suppliers product for common final delivery destinations. This definition focuses on the consolidation of shipments to achieve economies of scale and have cost benefits because of the same. Cross-docking was also defined as The process of moving merchandise from the receiving dock to shipping [dock] for shipping without placing it first into storage locations by the Material Handling Industry of America (MHIA). This definition does not take into account any storage at all between the incoming and outgoing freight and looks at an ideal scenario than a practical one as zero storage at a cross-dock means perfect synchronization between inbound and outbound vehicles which is difficult to establish. Hence, most crossdocks have storage docks which store goods for a little time until they are picked up for delivery. This short storage time is difficult to define exactly, but is defined as less than 24 hours by a few authors. 6

21 2.1.2 Benefits and Drawbacks of Cross-docking Cross-docking serves different goals. One can be achieving savings in transportation costs by consolidating shipments. Another can be a shorter delivery lead time as there is no time taken for storage and retrieval of materials from racks like in a traditional distribution center. The inventory expenditure also goes down with reduction in warehousing costs, handling and labor costs as well as faster inventory turnover, reduced risks of over stock, loss and damage of items. The space requirement for storage greatly reduces. This works best for industries where the unit costs of materials or products are very high. The cross-docking benefits align with the concept of a lean supply chain. These are some benefits of a cross-docking strategy. A cross-dock however cannot replace a warehouse in industries where the demand is extremely irregular as it requires buffers to be maintained in the supply chain. Cross-docking is unsuitable where there exist imbalances between the incoming load and outgoing load. Also, a very advanced and most often a computerized logistics system is integral for the operations of a cross-dock which inhibits its use by small and medium suppliers and businesses. Crossdocking is specifically unsuitable where unit stock-out cost is high as the level of inventory carried is reduces and stock-outs are more probable. Also, cross-docking is not cost efficient where the distances from suppliers and to customers are short. 2.2 Layout Design for a Cross-dock A factor affecting the efficiency and operational success of a cross-dock is its layout design, specifically its shape and size. Bartholdi and Gue [1] studied how different shapes of a cross- 7

22 dock affect cross-dock performance. They estimated the total travelling distance and labor cost for different shapes of the cross-dock. They concluded that an I shape is best for most crossdocks with less than 150 doors and an X shape is good for a bigger sized cross-dock. They base these results on an assumption that corners of a cross-dock are less functional than other areas and the efficiency decreases as the corners increase. They state however that the best shape for a cross-dock will change with different material flow patterns which they have not considered. Most other papers attempt to determine the best shape of a cross-dock considering that they temporarily stage goods on the floor or on racks. Vis and Roodbergen [3] propose a layout design in which goods will be stored in parallel rows between which workers can move and they also propose an algorithm for optimal number and length of rows. Gue and Kang [2] call these rows staging queues and compare single stage storage in cross-docking versus a double stage one for size and number of staging queues as well as throughput. As the thesis focuses on analyzing best shape of a cross-dock based on the area, max-flow and throughput, let us dive deeper into some of the established concepts in this area of research. Bartholdi and Gue [1] establish that the research is necessary because the people who design the cross-dock are often architects and do not pay attention to performance measures like travel times, throughput etc. They focus on travel times and associated labor costs to determine the best shape without considering other factors like truck turning radius, parking requirements and office space needs. They prove that a narrower dock reduces labor costs, state that the minimum width of a dock depends on the staging requirements. Considering a fixed offset between two consecutive doors, they compare the distance of a centrally located door from all other doors for different shapes, compare the centrality and number of outside corners to see 8

23 the tradeoff between the number of doors lost and change in diameter on increasing number of doors. Their experimental design compares the labor costs based on the intensity of freight flow and total flow for different cross-dock shapes. They analyze this for two cases- that of a uniform freight flow (similar to worst case analysis) and also an exponential freight flow while varying the fraction of receiving doors. They establish the freight flows based on the one-pass interchange heuristic from their paper on reducing labor costs for a LTL cross-dock in February 2000 [1]. According to their analysis, the most labor-efficient shapes with increasing size are I, T and X respectively. 2.3 Dock Door Assignment Peck developed an assignment model using simulation which derived the heuristic solution to the problem of minimizing travel times. Tsui and Chang [6] formulated a bilinear programming problem with a heuristic solution to an assignment problem where they assign the origin and delivery locations to the dock doors instead of assigning trucks and base the model on minimizing the number of trips and corresponding distances that forklifts have to make to carry the specified freight load from incoming to outgoing docks. Bermudez and Cole [8] propose a genetic algorithm for a similar problem, with the difference being that the doors are not assigned specifically as for inbound or outbound trucks. With automation on the rise and use of bar codes or RFID systems in warehouse management, a floating dock assignment is common in technologically advanced companies. A floating dock 9

24 assignment means that when information about incoming and outgoing items is known in much advance, the WMS systems use this knowledge to optimally assign a door to a truck in a way that the trucks which exchange maximum freight are assigned to closest possible doors to minimize process time. However, advanced IT systems are a necessity to handle the changing assignment of these doors and making sure correct goods are delivered to the correct door every time. Assuming that there will always be more doors than trucks, a dock door assignment problem assigns trucks to the doors in a way that optimizes the productivity of the cross-dock. The assignment which is most common and easier to manage is a fixed type of assignment where all trucks coming from the same origin location are assigned to one door and so are all the trucks going to the same delivery location. In a mid-term horizon, one dock door serves one same location for around six months until the assignments are revised according to changes in the shipping patterns. As one door serves the same location, chances of confusion among pickers and sorters and other errors are minimized but this type of assignment is not optimal like the floating dock and is also less flexible. Bartholdi and Gue [1] proposed a model with a mix of both, a mid-term assignment for outbound trucks with a short-term assignment done by a dock supervisor just when the trucks arrive. They show that the actual time of travel depends on type of freight, material handling system and congestion and propose a non-linear model which accounts for all the three factors and aims at minimizing labor cost, travel cost and congestion cost. They also propose guidelines for efficient cross-docks based on their model results. 10

25 2.4 Scheduling Scheduling comes into picture when there are less doors than incoming trucks and trucks have to be assigned to a door sequentially. Such a problem is typically called scheduling instead of dock door assignment. Several authors have studied scheduling by assuming a single door each for the incoming and outgoing trucks. The problem here is modeled as a two-machine flow shop problem with a constraint that the outbound truck task cannot be processed before all the preceding tasks are completed. Chen and Lee [11] establish that the problem is NP hard and propose a heuristic solution for it. Chen and Song [10] extend this problem enabling multiple trucks to be loaded or unloaded simultaneously o parallelly working machines at the doors. They propose a mixed integer programming model and several heuristics to solve it. Boysen and Fliedner [12] talked about a similar problem, but on a more aggregate level. He divided the time horizon into discrete time slots and assumed that the trucks will be fully loaded or unloaded within one such time slot. Then there are other authors which consider a more realistic case for scheduling with multiple doors each for inbound and outbound trucks. Most of them however assume that the outbound trucks are assigned on a mid-term basis which they mostly are to facilitate easier management. Hence, these papers deal with scheduling of trucks only for the inbound doors. Boysen and Fliedner [12] present an optimization model for such a case. The objective of their model is to schedule the inbound trucks in order to minimize the number of delayed shipments based on the travel time between the assigned inbound and outbound doors and go on to show that the model is NP hard. Most authors have proposed models with heuristic solutions but Lim et al. used the CPLEX solver to find solutions to the NP hard integer programming model for scheduling. He proposes the model with scheduling both for the inbound and outbound trucks 11

26 and shows that the CPLEX solver performs slightly better than DRS heuristic but is also slower and infeasible for larger problems. 2.5 Temporary Storage The inbound and outbound trucks do not arrive in perfect synchronization and the sequence in which the goods arrive are usually not in the sequence they must be loaded. That is why, though cross-docking aims to directly transfer incoming freight to outgoing trucks without any storage at all, more often than not, a temporary storage in the form of racks or staging docks is needed. The purpose of this temporary storage is to store freight when the outbound truck hasn t arrived yet or assigned yet and also to sort and direct the correct items to the correct outbound truck. Vis and Roodbergen [3] form a problem to find the optimal location for storage such that the travel distances of goods are minimized. The authors propose a polynomial time algorithm and prove using experiments that the proposed algorithm decreases the travel distances by 40%. Werners and Wulfing [13] take a different approach. They propose that the facility be divided into four sections and the temporary storage be near the doors for outbound trucks and called as end-points. The objective is to minimize the time distances between the endpoints and the dock-doors. They modeled the problem as a linear assignment problem determining at which endpoint would goods be stored and at which door should a corresponding truck be assigned. 12

27 2.6 Performance Analysis Pandit and Palekar [14], establish a metric called the response time to determine optimal layout design of a DC. They establish formulae to calculate the estimated travel times for different types of layouts and for different number of racks in a row or column. Using simulation, they also studied the effect of congestion due to blocking on the travel times. The simulation model showed that increasing the number of vehicles does not affect response time as it significantly reduces the waiting time while increasing the congestion and mean service time. Also, if the pick and place time is large, increasing the number of vehicles leads to higher congestion. In this paper, they also propose an idea of dividing a floor into parts called districts which contain some specified forklifts and service only specified doors and racks. They show with examples that such districting can reduce the response time. L.A. Medina [15], in his Master s thesis used simulation to find optimal layout arrangements and storage allocation strategies in a warehouse. He developed a simulation approach called SimPC to characterize the operations of a large scale and stochastic non-automated distribution center. The tool also aids in control of daily DC operations and with detailed characterization of interactions between resources and decisions on the DC floor. In his Master s thesis, Athul Gopala Krishna [16] has proposed a queuing model for clearly defined outbound processes in a distribution center and has given results for performance analysis of such a DC for different sizes and workforce utilizations. He developed a generic computational model to calculate travel times for forklifts involved in picking of goods from the racks. The model computes the workforce capacity at different stages of operations to meet specified performance levels using metrics such as truck Processing Time and Labor Hours 13

28 Per Truck. He also calculates the workforce capacity using Square Root Staffing rule used by call center staffing and finds that the results approximately match. He also uses simulation to validate the results from the mathematical model. 2.7 Network Diagrams Definitions and Representation Flow network is a directed graph, used in graph theory to represent a system with a directed flow of material, people or information with vertices called nodes and edges called arcs. Most networks have a single source and a single sink (destination) or at least can be modelled to have a single dummy source and sink. The arcs represent operations in the system or travel in a transportation network. The nodes are locations from where items are sent or received. The arcs have a defined and finite capacity. For the purpose of a cross-docking system, the various docks for staging, sorting, delivery etc. represent the nodes and the operations like unloading, picking, travel etc. represent the arcs. For our system, capacities of the arcs are the service rates for the operations they represent. A flow network should satisfy the condition that the quantity of flow into a node equals the quantity of flow out of it, except for the source which has only outgoing flow or a sink which has only incoming flow. An example of a flow network is in the Figure 2.1. The source and sink are as indicated. The circles denoted by letters are the nodes and the arrows are the arcs with the numbers on them denoting their capacities. 14

29 Figure 2. 1 Example of a single source, single sink flow network Maximum-Flow in a network The max-flow in optimization theory, is the maximum feasible flow through a network with a single source and single sink. The maximum flow problem attempts to maximize the flow from the source to sink without exceeding the capacity of any arc. The max-flow has many realworld applications, for example it is used in airline scheduling, transportation design and production scheduling in factories Maximum-Flow Minimum-Cut theorem One approach to find the max-flow is through the max-flow min-cut theorem. The theorem states that in a flow network, the maximum quantity of flow passing from the source to the sink is equal to the total capacity of the edges in the minimum cut. The minimum cut here, is the cut through those arcs which have the smallest total capacity and which if removed would 15

30 disconnect the source from the sink. There are several approaches to finding the minimum cut of a network flow and we will not discuss them in detail for the purpose of the thesis. For a simple flow network, the max-flow minimum cut theorem is used to calculate the maxflow using a computational model. The figures 2.2 and 2.3 shows the results of the computational model using Excel Solver and gives the max-flow result of the network shown in Figure 2.1. Using solver, we find that the max-flow of this network is 18 units. Figure 2. 2 Computational Model: Variables in yellow with result values and capacity constraints of the arcs. 16

31 Figure 2. 3 Computational Model: Flow constraints at the nodes with result values 2.8 Mean Value Analysis In queueing theory, a part of the theory of probability, mean value analysis (MVA) is established as a recursive technique used to calculate the queue lengths and waiting time at queueing nodes and throughput for a closed system of queues in equilibrium. It considers a closed queueing network of K number of M/M/1 queues and M total customers in the system. V is the visit ratio at a node, L(k) is the queue length, W(k) is the wait time at node k and the TH(m) denotes the throughput for a system with m AGVs. We then use the iterative algorithm to find the waiting time and queue lengths for each segment and throughput for a system with m AGVs. Starting from m=1, the iterative algorithm calculates the throughput for a system with increasing number of AGVs. The model also calculates the wait times and queue lengths for all different segments. The formulae are as follows. Note that Lk(m) denotes the queue length of segment k in a system with m AGVs. 17

32 Then we set Lk(0)=0 and initialize the iteration for k=1,2,3 k. We then repeat the process for m=1,2,3 M. Lk(m-1) +1 Wait time at each segment k = Wk(m) = L k System throughput for m AGVs = m = K k=1 m W k (m) v k Average Queue Lengths at each segment k = vk* m*wk(m) M. Jain, Sandhya Maheshwari and K.P.S. Baghel [22] develop a queueing model which contains multiple material handling devices they call MHDs. They suggest two queueing models to analyze the interference of the MHDs in a flexible manufacturing system. One of the models considers long service times while neglecting queueing at the MHDs and the other considers queueing at the MHDs. They determine the performance of a flexible manufacturing system using an iterative algorithm. They validate the results using a neuro-fuzzy controller system. R.D. van der Mei, E.M.M. Winands [23] propose a MVA based approach for a multi-queue single-server system in which the server visits the queues and processes the requests. Closed form equations are proposed in the paper for heavy traffic and compared to examples of numerical algorithms to verify that the approximations are accurate. J. R. Artalejo and J.A.C. Resing [24] apply the MVA approximation to carry out performance analysis of M/G/1 type 18

33 retrial queues. They show that the mean value analysis technique provides a reliable alternative to obtain the expected queue lengths and wait times by avoiding the use of heavy algebra. 19

34 CHAPTER 3 MODEL AND ASSUMPTIONS 3.1 Cross-Dock Design Cross-docks most commonly range from 60 doors to 300 doors. They may have varied dimensions, synchronous or asynchronous flows, with little or large area for storage. For the purpose of this thesis, a synchronized flow with no storage in between is considered. The length varies based on the number of doors while the width of the dock is assumed to be the minimum required width. In a given period, a certain number of trailers are received at the inbound end and the goods received are all shipped out in outbound trucks in that period. All trailers are reasonably assumed to be 48 ft in length and 9 ft in width. The door openings are 9ft*9ft and each door s side is 12 ft away from the next door s same side. In other words, two doors have an offset of 12 ft. Figure 3.1 illustrates the floor plan of a cross dock with four Figure 3. 1 Cross-dock Layout with Dimensions 20

35 inbound doors and four outbound doors, which in this work will be referred to with the notation of 4x4, which is in-bound doors x out-bound doors. Depending on the geometry of the floor-plan, some of area may not be usable for docking trucks, which is discussed in detail in the next chapter. Based on our assumptions of truck dimensions, the cross-dock loses 48 ft i.e. 4 doors on each side for an inner corner. For an outer corner, a cross-dock loses the staging space of 2 doors due to overlap with the adjacent doors. Hence, we assume that 2 doors are lost for each outer corner so that the staging area for the lost doors can compensate that for two of the next doors on the side. Therefore, we have a loss of 8 doors for each inner corner and loss of 4 doors for each outer corner. For the purpose of Mean Value Analysis, the travel paths are divided into segments. Each segment is separated by a point of intersection of AGVs. Paths and segments are all unidirectional. But each intersection can have vehicles coming from or going in different directions. The figure 3.2 illustrates this division of paths into segments. There are three types of segments- the spurs, the arteries and the branches. In the figure, segment 1 and 2 are the spurs. Segments 3,4,5 and 6 are the arteries. Segments 7 and 8 are the branches. The path in front of the first incoming door is named in this way for an I shaped dock. For the next Figure 3. 2 Path Division into Segments 21

36 door, it is the second multiple of that number i.e. from 9 to 16. All the 8n-2 segments in a nxn cross-dock are named as such. The use of such segmentation is explained in detail in the Mean Value Analysis chapter. 3.2 Cross-dock Operations There are 5 stages of cross-dock operations, the first is unloading a trailer at an inbound door and placing the packages in the staging area we name as inbound dock, it is considered that there is one inbound dock for each inbound door. The operation has service rate UR (Unloading Rate). In all calculations, this dock is assumed to be at 3 feet along the width from the inbound door. The second operation is picking wherein an AGV picks one package from the inbound dock, scans it and designates the outbound dock based on its destination. It is assumed that one AGV carries only one package. This operation has service rate PR (Picking Rate). The third stage is the travel from inbound to outbound dock. The TR (Travel Rate) is directly proportional to the vehicle speed and inversely to its path length. The fourth operation involves offloading the package from AGV onto the outbound dock and is given by DR (Delivery Rate) which signifies the rate of the delivery operation at the outbound dock. The last operation is Loading which again involves loading of the packages from the outbound dock in the outbound truck at respective door. Packing and unpacking processes are adjusted in the unloading and loading operations respectively and not recognized as separate operations. 22

37 In the travel operation, the AGVs follow a rectilinear path with unidirectional segments. The unidirectional nature of the segments and a rectilinear geometry along the walls of the crossdock help reduce congestion. For Max-Flow Analysis, it is assumed that an inbound package can belong to a truck at any of the inbound doors and can be directed to any of the outbound trucks with equal probability. A detailed door assignment based on any assumptions is not followed. However, the inbound and outbound doors are located on different sides of the cross-dock in different cases. While modelling, inbound and outbound doors are always assumed to be on separate sides for simplicity. But all the models can be extended to cross-docks where there is little or no separation of inbound and outbound doors. 3.3 Freight Size and Service Rate Specifications The arrival rate is assumed to be the rate of arrival of trailers at the inbound gate of the crossdock. FTL trailers are the entity at the gates while the packages which are entities inside another entity are assumed to be the customers in other operations. One trailer carries on an average 35,000 lbs. We assume a package weight of 30 lbs, meaning that a trailer carries 1000 packages. After entry through the gate, the truck docks at an inbound door wherein the door assignment is random. One AGV carries one package only and heads out. We assume the unloading, picking, delivery and loading rate as similar. We can fairly quote that a freight of 35,000 lbs takes a time of 2.5 hours to unload. The unloading time and hence the unloading rate is derived by dividing the 1167 packages over 2.5 hours. The Figure 3.3 illustrates all these 23

38 Unloading pkgs/hr Picking pkgs/hr Travel Delivery pkgs/hr Loading pkgs/hr Figure 3. 3 Process Flow with Service Rates rates in a service flow model. Also, it is assumed that inbound trucks always have an assigned door available and have zero waiting time at this stage. 24

39 CHAPTER 4 CROSSDOCK SHAPE AND AREA 4.1 Variety in shapes Cross-docks shapes are decided based on various reasons, some being the availability of land, use of a pre-built facility, architect s design, requirement of covered and uncovered areas and government regulations on construction. Different shapes have different advantages and drawbacks. For example, a simple rectangular shaped cross-dock called the I shape is simplistic in organization and placement of the different operational docks and paths. Also, with less corners to the shape, the total number of doors lost is low for the I shape. But the distance between the farthest doors is extensive. In contrast, the X or H shape have more corners and hence, more number of doors are lost but have less distance between their farthest doors for a cross-dock of the same size. Also, many companies transitioned to cross-docking from warehousing later as a strategy to reduce inventory. Also, some companies do not operate cross-docks as completely independent units. Instead, they operate a part of a warehousing facility as the cross-dock. Hence in some cases, the requirements of a warehouse play a role in deciding the best shape. Warehouses are most commonly U shaped, the advantage being the inbound and outbound docks are located next to each other and facilitate use of shared resources such as labor and material handling machinery. I shaped and L shaped layouts have separate receiving and shipping areas providing for higher security and larger sorting spaces near the doors. Figure 4.1 shows how a warehouse 25

40 of I shape has operations distributed in the facility. In this thesis, all cross-docks we consider are without any storage and hence not considered to be functioning partly as a warehouse. These are independently functioning, synchronous cross-docks. Figure 4. 1 Layout Diagram with Operations 4.2 Shape Nomenclature and Door Placement The shapes that we compare are I, L, T, H and X. They are named such because the top views of these cross-docks resemble respective alphabets. Doors are located on each wall of a crossdock. The figure 4.2 shows each of these cross-docks with the door placement. We consider the horizontal axis as X axis and the vertical axis as the Y axis. The doors are denoted by (X, Y) coordinates for all calculations shown later. 26

41 Figure 4. 2 Cross-dock shapes considered 27

42 Table 4. 1 Minimum possible doors for a cross-dock shape Shape Minimum Possible Size I 2 I' 16 L 24 T 22 H 40 X 8 Some of the shapes like the X, T and H have a minimum number of doors to form that shape and the minimum size of every shape is given in the Table 4.1. For our modelling purposes, we do not consider very small cross-docks. The minimum number of doors we consider is 50. The maximum number of doors considered is 300. The models can be easily extended to more number of doors. The I shape is compared for the minimum width and a greater width. Results support the claim by Bartholdi and Gue [1] and we find that the narrower cross-dock is more efficient in terms of area as well as travel distance. Hence, all the other shapes are considered only for their narrowest width. For the narrowest width, we do not consider doors on the shorter edges of the cross-dock. For a 96 ft width, having doors on the shorter edge keeps the total number of doors same as the four doors on this side compensate for the 4 doors lost. However, as it will complicate the path geometry and increase the average travel distance, we assume that there are no doors on the shorter edge and hence, no doors are lost on the outer corners in all these cases. This is illustrated in figure 4.3 for an I shaped cross-dock. The I dock with doors on width is called I for sake of distinction. 28

43 Figure 4. 3 I shape with and without doors on width 4.3 Available free space In each of these shapes, the available free space depends on the location of internal corners as well as the path geometry inside the cross-dock. The available free spaces are important as they are desirable prospective locations for any storage area. Specially in geometries like the X and the H, the internal corners are at the most central locations of the cross-dock and hence apt locations for storage. Even if we neglect storage, these locations are great for housing maintenance equipment or personnel or to locate office staff. In most practical cases, crossdocks are huge and the central location of maintenance or staff will facilitate faster service rate in times of machine breakdowns or other emergencies. Hence, it is important to locate the available free spaces in a cross-dock. These are shown in figure

44 Figure 4. 4 Doors lost and space available in a Cross-dock 4.4 Area of a cross-dock The larger the area, more the setup cost for a cross-dock. Besides, operational costs are also proportional to the area. Hence it is insightful to see how the area changes for a cross-dock with increase in the cross-dock size. For a shape, the variation in the ratio of area against number of doors shows how size affects area for the shape. 30

45 Figure 4. 5 Area vs No. of Doors for I and I' Figure 4.5 shows the comparison between areas of two I shaped cross-docks of same size or number of doors, but one with doors on the shorter side and the other without them. The graph clearly shows that there is no difference in the area for two such cross-docks of same size. This happens because the doors lost on the longer side in a cross-dock are compensated by doors on the shorter side. So, if area is a parameter, both the I shaped cross-docks with and without doors on the shorter side are comparable. Figure 4.6 shows comparison of I shaped cross-docks with different widths. As the width of a dock varies, the number of doors on the width also changes. The graph shows comparison 31

46 Doors Area X100 Sq.Ft. Area ( X100) 1000 Area vs No. of doors Area4 Area5 Area No. of doors Figure 4. 6 Area vs No. of doors for increasing width of I shape Area vs No. of doors Scaled Area I Scaled Area I' Scaled Area L Scaled Area T Scaled Area X Scaled Area H 0 No. of doors Figure 4. 7 Area vs No. of doors for different shapes 32

47 between docks of the equal number of doors in total but varying widths. It is clear that with increasing number of total doors, the difference in areas of the cross-docks rapidly increases. So, larger the cross-dock, narrower the better. Hence, for all the other shapes of cross-docks we will only consider the narrowest ones to compare. The Figure 4.7 illustrates how area changes with increasing size for cross-docks of varying shapes. It is clear from the graph that for the same number of doors, the area of an I shaped cross-dock is the least, followed by an L shaped one, then a T shaped one and the largest area is occupied by a X or H shaped crossdock. The areas can be represented by linear equations, with number of doors as x given by- For I: Area (in sq.ft) = 576*x For I with doors on width: Area (in sq.ft) = 576*x For L: Area (in sq.ft) = 576*x For T: Area (in sq.ft) = 576*x For X: Area (in sq.ft) = 576*x For H: Area (in sq.ft) = 576*x A generalized linear equation will be given by- Area (in sq.ft) = 0.5*door offset * minimum cross-dock width * (no. of doors at the cross-dock + no. of doors lost on corners as per assumptions) From the graph, we can infer three things- 33

48 1. The slope of all the lines is the same, and hence we can say that the change in area of all shapes is same for changing number of doors. 2. Also, the slope is constant and does not change, which signifies that area increases at a constant rate with increase in number of doors across the size of 50 to 300 doors. 3. For a same sized cross-dock, with area as the parameter, an I shape is the most superior and an X or H shape is the most inferior. 4.5 Comments about the H shaped cross-dock The middle portion of an H shaped cross-dock may or may house doors and docks. However, as increasing the length of this portion increases the travel distance between doors on opposite sides of it, it is best to keep this to a minimum required length. The minimum required length here depends on the area required outside the dock to park the trucks at the doors. The length needed to park two trucks is around 96 ft. Also, there are no doors at the inner corners of the H shaped dock, So the 96 ft length is enough for turning the trucks when needed. The turning radius for a 48 ft long trailer is around 90 ft. Hence, the 96 ft length of the middle portion of H meets both these requirements. The width is assumed to be 96 ft as the width of all other arms is the same and it will be easier to design AGV paths with similar geometries. The middle space in H can be used for temporary storage if needed. The central location is ideal for any storage as the travel distances from here are not extreme to any door. However, as we cannot have any doors in the central location, we are losing out on the best doors of the cross-dock from the prospect of minimum travel distances. 34

49 4.6 Computational modelling of the graphs and utilization To calculate the area of a cross-dock of a particular shape and size, the length and breadth of each arm was calculated, then multiplied and added. To calculate the length of an arm, we multiplied the number of doors on the side by the offset 12 ft and added the length of the side lost due to corners. We calculate the breadth of the cross-dock based on our assumption of minimum required width. For the case with doors on width, the number of doors and offset were multiplied and added to the corners lost to calculate the breadth. The Figure 4.8 shows the Table headings in the computational model for an I shaped cross-dock with doors on width. Figure 4. 8 Table headings in Computational Model for an I cross-dock with doors on width For a cross-dock of H or X shape, the minimum area which is the central area for the X shape and the joint between two I s for the H shape, is added to the calculation similar to above. This is because the minimum area is proportional only to the minimum required width of the crossdock in the case of X and the length of the joint portion for H. A sample calculation for a X shaped cross-dock is shown in Figure

50 Figure 4. 9 Table headings in Computational Model for a X shaped cross-dock 36

51 CHAPTER 5 MAX-FLOW OF A CROSS-DOCK 5.1 Model Assumptions The Max-flow model, as explained in the literature review can calculate the maximum output from a cross-dock under the best-case scenario. Figure 5.1 shows the network-flow diagram of cross-dock operations for a 4X4 cross-dock. In the travel operation, we consider the best case as we neglect congestion and queueing, however to distinguish between different shapes we do not consider the travel distance as that between the closest doors. The travel distance for an AGV starting from an incoming door is the average of all distances between that incoming door and each outgoing door Figure 5. 1 Network Flow Diagram of Cross-dock Operations Also, the number of AGVs in the system is equally divided between the number of incoming doors. The flow rate from each incoming door to an outgoing door also depends on the AGVs 37

52 available to each door. In calculating the max-flow rate, the travel distance between any two doors is doubled with the assumption that an AGV that travels from an incoming door to an outgoing door returns to its home position after delivering the package. Then, using the Maxflow Min-cut algorithm, we can get the maximum flowrate out of the cross-dock. 5.2 Cross-dock Settings under review Along with the shapes, the max-flow model can also consider the door assignments in the model. The freight data drives the exact door assignment of incoming and outgoing trucks to specific doors according to some established heuristics and algorithms. As this generic maxflow analysis does not need demand data, we will consider a high-level door assignment which will consider placement of inbound and outbound trucks to doors in general. Figure 5. 2 Cases Considered (I) 38

53 The following cases for cross-docks are considered in the max-flow analysis. The figures 5.2 and 5.3 show the generic inbound and outbound door assignment for the different cases of cross-docks we have compared. The figures without any labels have separate areas for inbound and outbound doors. The same door can be used for inbound freight at times and outbound freight in other times. All these cases are named as mentioned below the respective figures. Figure 5. 3 Cases Considered (II) 39

54 5.3 Best shape for various cross-dock sizes The best shape of a cross-dock based on max-flow as the parameter is illustrated in the graph 1.4 and the Tables 5.1 and 5.3 across a range of different number of doors. The values of AGV speed, capacity, number of AGVs and service rates of all different operations are specified in Table 5.2. Table 5. 1 Best shape based on Max-flow against no. of doors No. of Doors Best Maxflow (Packages/hr) Best shape I-II I-II I-II I-II I-II I-II I-II I-II I-II L-IV L-IV L-IV X-I No. Doors of Best Maxflow (Packages/hr) X-I X-I X-I X-I X-I X-I X-I X-I X-I X-I X-I X-I Best shape X-I Table 5. 2 Service rates for the experiment (Max-flow vs Size) AGV capacity 1.0 package Unloading rate packages/hr Delivery rate packages/hr Picking rate packages/hr Loading rate packages/hr 40

55 Figure 5. 4 Max Flow vs Cross-dock Size Table 5. 3 Max-flow values for given shapes and sizes No. of Doors TC XA XB XC HA HB HC HD

56 Packages/hr Max-flow vs AGV speed for 100 doors AGV speed (ft/s) I I-II L L-II L-III L-IV T T-II T-III X X-II X-III H H-II H-III H-IV Figure 5. 5 Max-Flow vs AGV speed for 100 doors in a cross-dock 5.4 Best shape for various AGV speeds AGV speed TB TC XA XB XC HA HB HC HD AGV speed IA IB LA LB LC LD TA Table Max-flow values for given shapes and AGV speeds

57 Packages/hr To see the effect of AGV speeds, the graphs 5.5, 5.6 and 5.7 illustrate the max-flow values for varying AGV speeds. The values of AGV capacity, number of AGVs and service rates of all different operations are specified in Table 5.5 Table 5. 5 Service rates for the experiment (Max-flow vs AGV speeds) No of Operating Doors 100 AGV capacity 1 package Unloading rate packages/hr Picking rate packages/hr Delivery rate packages/hr Loading rate packages/hr No. of AGVs Max-flow vs AGV speed for 200 doors AGV speed (ft/s) I I-II L L-II L-III L-IV T T-II T-III X X-II X-III H H-II H-III H-IV Figure 5. 6 Max-Flow vs AGV speed for 200 doors in a Cross-dock 43

58 Packages/hr 7000 Max-flow vs AGV speed for 300 doors AGV speed (ft/s) I I-II L L-II L-III L-IV T T-II T-III X X-II X-III H H-II H-III H-IV Figure 5. 7 Max-Flow vs AGV speed for 300 doors in a Cross-dock 5.5 Best shape for varying number of AGVs available To see the effect of number of AGVs per door, the following Figure 5.8 and Table 5.8 illustrates the max-flow values for varying number of AGVs. The values of AGV speed, capacity, number of doors and service rates of all different operations are specified in Table

59 Packages/hr Max-Flow vs No. of AGVs No. of AGVs/door I I-II L L-II L-III L-IV T T-II T-III X X-II X-III H H-II H-III H-IV Figure 5. 8 Max-Flow vs Available AGVs Table 5. 6 Service rates for the experiment (Max-flow vs no. of AGV/door) No of Operating Doors 200 AGV speed 3 ft/s Unloading rate packages/hr Picking rate packages/hr Delivery rate packages/hr Loading rate packages/hr AGV capacity 100 packages 45

60 Table 5. 7 Max-flow values for given shapes and no. of AGVs/door No of AGVs I I-II L L-II L-III L-IV T No of AGVs T-II T-III X X-II X-III H H-II H-III H-IV Best shape for varying AGV capacity To see the effect of number of AGVs per door, the following Figure 5.9 and Table 5.9 illustrate the max-flow values for varying number of AGVs. The values of AGV speed, number of doors and service rates of all different operations are specified in Table

61 pounds/s 450 MaxFlow vs AGV capacity AGV capacity (in pounds) I-II L L-II L-III L-IV T T-II T-III X X-II X-III H H-II H-III H-IV Figure 5. 9 Max-Flow vs AGV's carrying capacity Table 5. 8 Service Rates for Experiment (Max-flow vs AGV capacity) No of Operating Doors 200 AGV speed 3 ft/s Unloading rate packages/hr Picking rate packages/hr Delivery rate packages/hr Loading rate packages/hr No. of AGVs 1 47

62 Table 5. 9 Max-flow values for given shapes and AGV speeds AGV capacity I I-II L L-II L-III L-IV T AGV capacity T-II T-III X X-II X-III H H-II H-III H-IV Computational Model The figure represents the network diagram of the operations of a cross-dock. The receiving and dispatch depends on external factors like frequency of incoming freight and orders for outgoing freight. We focus on the internal activities of a cross-dock which include unloading, picking, travel, delivery and loading. We can find out the maximum output from the cross-dock based 48

63 on the service rates of these operations. As explained in the Max-Flow algorithm in the literature review, we use the max-flow min cut theorem. Figure Network Flow Diagram of Cross-dock Operations Table Symbols and letters for Service rates Unloading rate u packages/hr Picking rate p packages/hr Travel rate tij packages/hr Delivery rate d packages/hr Loading rate l packages/hr No. of AGVs/ inbound door m In the figure 5.10, number of AGVs for each inbound door is 2. Hence, m=2. Values of u, p, d and l are assumed to be fixed at packages/hr. The computational model calculates the travel rate tij for each i and j with i being a specific door on the inbound side and j being the specific door on the outbound side. 49

64 Based on the max-flow min-cut theorem, we can say- max-flow= min-cut= min (u, mp, mt, md, l) First of all, based on the geometries and size of the cross-dock, the X and Y coordinates of each door was fixed. The Table below shows the same for a 4X4 cross-dock. Table X and Y coordinates for inbound and outbound door nos. 1 through 4 IC door X Y OG door X' Y' Using data tables, the distances between every two doors was calculated. This is shown in figure below. Table Distance Matrix showing distances between Inbound and Outbound doors The distances were calculated for each shape and any given size to get the travel distances between any two doors. Then the travel distance from inbound door i was calculated by 50

65 averaging all distances from i to other doors. To calculate travel times for a rectilinear path, travel distances from door i to door j were calculated. To balance flows at incoming and outgoing docks, the no of incoming doors is assumed to be equal to number of outgoing doors. The travel time is determined by dividing the travel distance by AGV speed. The travel time t from door i to an outgoing door was assumed to be average of ti1 to tin where n is the total OG doors. The flowrate is calculated first in pounds/sec by dividing the AGV s weight capacity by the travel time calculated. Then using the max-flow min-cut theorem, the max-flow is calculated. This max-flow is in pounds/sec which is useful when comparing cases with different AGV capacities. It is then converted to packages/hours by dividing it by the AGV weight capacity and multiplying it by Figure Sample from Max-flow Computational Model The Computational model can be used as a tool to calculate the max-flow values for different cases for a given user value of certain input parameters. The user-input sheet of the model contains the following input cells shown colored. This makes the model interactive and enables 51