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

ii

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.

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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

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TABLE OF CONTENTS

List of Figures ……………………………………………………………………………...viii

List of Tables………………………………………………………………………………....xi

Acknowledgements…………………………………………………………………………xiii

CHAPTER 1 INTRODUCTION ............................................................................................. 1

1.1 Cross-docking Operations ...................................................................................... 1

1.2 Literature Summary ................................................................................................ 2

1.3 Overview ................................................................................................................ 3

1.4 Organization ........................................................................................................... 5

CHAPTER 2 LITERATURE REVIEW .................................................................................. 6

2.1 Characteristics of cross-docking ............................................................................. 6

2.2 Layout Design for a Cross-dock ............................................................................. 7

2.3 Dock Door Assignment .......................................................................................... 9

2.4 Scheduling ............................................................................................................ 11

2.5 Temporary Storage ............................................................................................... 12

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2.6 Performance Analysis ........................................................................................... 13

2.7 Network Diagrams ................................................................................................ 14

2.8 Mean Value Analysis............................................................................................ 17

CHAPTER 3 MODEL AND ASSUMPTIONS……………………………………………..20

3.1 Cross-Dock Design ............................................................................................... 20

3.2 Cross-dock Operations ......................................................................................... 22

3.3 Freight Size and Service Rate Specifications ....................................................... 23

CHAPTER 4 CROSSDOCK SHAPE AND AREA .............................................................. 25

4.1 Variety in shapes .................................................................................................. 25

4.2 Shape Nomenclature and Door Placement ........................................................... 26

4.3 Available free space .............................................................................................. 29

4.4 Area of a cross-dock ............................................................................................. 30

4.5 Comments about the H shaped cross-dock ........................................................... 34

4.6 Computational modelling of the graphs and utilization ....................................... 35

CHAPTER 5 MAX-FLOW OF A CROSS-DOCK ............................................................... 37

5.1 Model Assumptions .............................................................................................. 37

5.2 Cross-dock Settings under review ........................................................................ 38

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5.3 Best shape for various cross-dock sizes ............................................................... 40

5.4 Best shape for various AGV speeds...................................................................... 42

5.5 Best shape for varying number of AGVs available .............................................. 44

5.6 Best shape for varying AGV capacity .................................................................. 46

5.7 Computational Model ........................................................................................... 48

CHAPTER 6 MEAN VALUE ANALYSIS .......................................................................... 53

6.1 Model Assumptions .............................................................................................. 53

6.2 Division of pathways into segments ..................................................................... 55

6.3 Computational Model- .......................................................................................... 55

CHAPTER 7 COMPARISON AND APPLICATION OF MODELS ................................... 61

7.1 Similarities and differences in the two models ..................................................... 61

7.2 Model Results for different cases ......................................................................... 62

CHAPTER 8 COST ANALYSIS .......................................................................................... 69

8.1 Model Assumptions .............................................................................................. 69

8.2 Cost-Considerations.............................................................................................. 69

8.3 Details of different cases considered .................................................................... 71

8.4 Cost Comparison for different regions ................................................................. 75

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CHAPTER 9 CONCLUSIONS AND FUTURE WORK ...................................................... 79

9.1 Conclusions .......................................................................................................... 79

9.2 Scope for Future work .......................................................................................... 81

REFERENCES ....................................................................................................................... 82

APPENDIX_ DEMAND DATASETS USED TO TEST MVA MODEL ............................. 86

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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.............................................................................................................. 16

Figure 2. 3 Computational Model: Flow constraints at the nodes with result values ............. 17

Figure 3. 1 Cross-dock Layout with Dimensions……………………………………………20

Figure 3. 2 Path Division into Segments ................................................................................ 21

Figure 3. 3 Process Flow with Service Rates .......................................................................... 24

Figure 4. 1 Layout Diagram with Operations …………………………………………….....26

Figure 4. 2 Cross-dock shapes considered .............................................................................. 27

Figure 4. 3 I shape with and without doors on width.............................................................. 29

Figure 4. 4 Doors lost and space available in a cross-dock .................................................... 30

Figure 4. 5 Area vs No. of Doors for I and I' .......................................................................... 31

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Figure 4. 6 Area vs No. of doors for increasing width of I shape ........................................... 32

Figure 4. 7 Area vs No. of doors for different shapes............................................................. 32

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 .................... 36

Figure 5. 1 Network Flow Diagram of Cross-dock Operations………………………………37

Figure 5. 2 Cases Considered (I)............................................................................................. 38

Figure 5. 3 Cases Considered (II) ........................................................................................... 39

Figure 5. 4 Max Flow vs Cross-dock Size .............................................................................. 41

Figure 5. 5 Max-Flow vs AGV speed for 100 doors in a cross-dock ..................................... 42

Figure 5. 6 Max-Flow vs AGV speed for 200 doors in a cross-dock ..................................... 43

Figure 5. 7 Max-Flow vs AGV speed for 300 doors in a cross-dock ..................................... 44

Figure 5. 8 Max-Flow vs Available AGVs ............................................................................. 45

Figure 5. 9 Max-Flow vs AGV's carrying capacity ................................................................ 47

Figure 5. 10 Network Flow Diagram of Cross-dock Operations ............................................ 49

Figure 5. 11 Formulae used in the max-flow computational model ....................................... 51

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

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Figure 7. 2 Max-Flow and MVA throughput for grocery dataset with high demand data values

and high deviation ................................................................................................................... 64

Figure 7. 3 Comparison of Max-flow and MVA values for demand data with mean 1 and

deviation 0 .............................................................................................................................. 65

Figure 7. 4 Comparison of Max-flow and MVA values for demand data with mean 3 and

deviation 0 ............................................................................................................................... 66

Figure 7. 5 Comparison of MVA throughput for datasets with same mean but different

deviation .................................................................................................................................. 67

Figure 7. 6 Comparison of MVA throughput for datasets with and ....................................... 68

Figure 8. 1 Cost Comparison for USA……………………………………………………….75

Figure 8. 2 Cost Comparison for Germany ............................................................................. 76

Figure 8. 3 Cost Comparison for China .................................................................................. 77

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LIST OF TABLES

Table 4. 1 Minimum possible doors for a cross-dock shape................................................... 28

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) ............................................ 40

Table 5. 3 Max-flow values for given shapes and sizes ......................................................... 41

Table 5. 4 Max-flow values for given shapes and AGV speeds ............................................. 42

Table 5. 5 Service rates for the experiment (Max-flow vs AGV speeds) ............................... 43

Table 5. 6 Service rates for the experiment (Max-flow vs no. of AGV/door) ........................ 45

Table 5. 7 Max-flow values for given shapes and no. of AGVs/door .................................... 46

Table 5. 8 Service Rates for Experiment (Max-flow vs AGV capacity) ................................ 47

Table 5. 9 Max-flow values for given shapes and AGV speeds ............................................. 48

Table 5. 10 Symbols and letters for Service rates ................................................................... 49

Table 5. 11 X and Y coordinates for inbound and outbound door nos. 1 through 4 .............. 50

Table 5. 12 Distance Matrix showing distances between inbound and outbound doors ........ 50

Table 5. 13 Service rates specified by user ............................................................................. 52

Table 6.1……………………………………………………………………………………...53

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Table 6. 2 Experiment assumptions ........................................................................................ 54

Table 6. 3 Demand Matrix in the computational model ......................................................... 55

Table 6. 4 Conversion of S-matrix to S column ..................................................................... 57

Table 6. 5 Values of S, Tau and visit ratio in computational model ....................................... 58

Table 6. 6 Changing Throughput values against no. of AGVs in system ............................... 59

Table 6. 7 Sample of MVA model results showing throughput, ............................................ 60

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 .................................................................................... 73

Table 8. 3 Demand Matrix for Case 3..................................................................................... 74

Table 8. 4 Annual Cost Values for USA ................................................................................. 75

Table 8. 5 Annual Cost Values for Germany .......................................................................... 76

Table 8. 6 Annual Cost Values for China ............................................................................... 77

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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.

xiv

DEDICATION

To my parents,

Rajshree and Sudarshan

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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

2

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 cross-

dock 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 cross-

dock. 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

Figure 1. 2 Layout Diagram of automated cross-dock

3

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

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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.

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1.4 Organization

This thesis is organized as shown in Figure 1.3.

Figure 1. 3 Thesis Organization

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CHAPTER 2

LITERATURE REVIEW

2.1 Characteristics of cross-docking

2.1.1 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 cross-

docking 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 cross-

docks 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.

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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. Cross-

docking 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-

8

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 cross-

docks 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

9

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

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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 out-

bound 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.

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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

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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.

13

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

14

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

2.7.1 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.

15

Figure 2. 1 Example of a single source, single sink flow network

2.7.2 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 real-

world applications, for example it is used in airline scheduling, transportation design and

production scheduling in factories.

2.7.3 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

16

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 max-

flow 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.

17

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.

18

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.

Wait time at each segment k = Wk(m) = Lk(m-1) +1

Lk

System throughput for m AGVs = m = m

∑ Wk(𝑚)∗vk𝐾𝑘=1

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

19

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.

20

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

21

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

22

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.

23

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 cross-

dock 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 cross-

dock. 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

24

Unloading466.7 pkgs/hr

Picking466.7 pkgs/hr Travel

Delivery

466.7 pkgs/hr

Loading

466.7 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.

25

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

26

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.

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 cross-

dock. 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.

Figure 4. 1 Layout Diagram with Operations

27

Figure 4. 2 Cross-dock shapes considered

28

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.

29

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, cross-

docks 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 4.4.

Figure 4. 3 I shape with and without doors on width

30

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.

31

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

Figure 4. 5 Area vs No. of Doors for I and I'

32

0

100

200

300

400

500

600

700

800

900

1000

28 34 40 46 52 58 64 70 76 82 88 94 100 106 112 118 124

Sq.F

t. A

rea

( X

10

0)

No. of doors

Area vs No. of doors

Area4

Area5

Area 6

0

200

400

600

800

1000

1200

1400

1600

1800

Do

ors 26

34

42

50

58

66

74

82

90

98

10

6

11

4

12

2

13

0

13

8

14

6

15

4

16

2

17

0

17

8

18

6

19

4

20

2

21

0

21

8

22

6

Are

a X

10

0

No. of doors

Area vs No. of doors

Scaled Area I

Scaled Area I'

Scaled Area L

Scaled Area T

Scaled Area X

Scaled Area H

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

33

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 cross-

dock.

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 + 6912

For T: Area (in sq.ft) = 576*x + 13824

For X: Area (in sq.ft) = 576*x + 27648

For H: Area (in sq.ft) = 576*x + 27648

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-

34

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.

35

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.

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 cross-

dock 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 4.9.

Figure 4. 8 Table headings in Computational Model for an I cross-dock with doors on width

36

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

37

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

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

Figure 5. 1 Network Flow Diagram of Cross-dock Operations

38

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 Max-

flow 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 max-

flow 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)

39

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)

40

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

50 2720 I-II

60 2886 I-II

70 3018 I-II

80 3125 I-II

90 3213 I-II

100 3288 I-II

110 3351 I-II

120 3405 I-II

130 3453 I-II

140 3498 L-IV

150 3541 L-IV

160 3580 L-IV

170 3624 X-I

Table 5. 2 Service rates for the experiment (Max-flow vs Size)

AGV capacity 1.0 package

Unloading rate 466.7 packages/hr

Delivery rate 466.7 packages/hr

Picking rate 466.7 packages/hr

Loading rate 466.7 packages/hr

No. of Doors

Best Maxflow (Packages/hr)

Best shape

180 3718 X-I

190 3808 X-I

200 3892 X-I

210 3972 X-I

220 4047 X-I

230 4118 X-I

240 4186 X-I

250 4250 X-I

260 4311 X-I

270 4369 X-I

280 4425 X-I

290 4478 X-I

300 4529 X-I

Figure 5.9 Max Flow vs Cross-dock SizeTable

5.2: Best shape based on Max-flow against no.

of doors

No. of Doors

Best maxflow (Packages/hr)

Best shape

180 3718 X-I

190 3808 X-I

200 3892 X-I

210 3972 X-I

220 4047 X-I

41

No. of Doors TC XA XB XC HA HB HC HD

50 16.87812 14.45846 12.43121 14.17373 12.31046 12.26914 10.26441 11.89718

100 23.18173 22.88219 19.60362 22.65337 20.19607 20.09577 18.36604 19.76683

150 27.51871 28.45864 24.33651 28.26637 25.65371 25.51474 23.80911 25.31746

200 30.43329 32.43642 27.7086 32.27424 29.67749 29.5153 27.8661 29.39938

250 32.52864 35.41664 30.23255 35.27517 32.76396 32.58878 31.07086 32.49932

300 34.12629 37.73944 32.20155 37.61364 35.21263 35.03108 33.62298 34.95984

Figure 5. 4 Max Flow vs Cross-dock Size

Table 5. 3 Max-flow values for given shapes and sizes

42

5.4 Best shape for various AGV speeds

AGV speed TB TC XA XB XC HA HB HC HD

2 9.060941 10.53715 10.401 8.910738 10.29699 9.18003 9.13444 8.3482 8.984921

3 13.59141 15.80572 15.60149 13.36611 15.44548 13.77004 13.70166 12.5223 13.47738

4 18.12188 21.0743 20.80199 17.82148 20.59397 18.36006 18.26888 16.6964 17.96984

5 22.65235 26.34287 26.00249 22.27685 25.74247 22.95007 22.8361 20.8705 22.4623

6 27.18282 31.61144 31.20299 26.73221 30.89096 27.54009 27.40332 25.0446 26.95476

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

2 3 4 5 6

Pac

kage

s/h

r

AGV speed (ft/s)

Max-flow vs AGV speed for 100 doors

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

AGV speed IA IB LA LB LC LD TA

2 10.50432 12.45306466 10.29135 11.38415 9.310622 12.28992 8.935621

3 15.75648 18.67959699 15.43703 17.07623 13.96593 18.43488 13.40343

4 21.00865 24.90612931 20.5827 22.76831 18.62124 24.57984 17.87124

5 26.26081 31.13266164 25.72838 28.46038 23.27655 30.7248 22.33905

6 31.51297 37.35919397 30.87405 34.15246 27.93186 36.86976 26.80686

Figure 5. 5 Max-Flow vs AGV speed for 100 doors in a cross-dock

Table 5. 4 Max-flow values for given shapes and AGV speeds

43

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 466.7 packages/hr

Picking rate 466.7 packages/hr

Delivery rate 466.7 packages/hr

Loading rate 466.7 packages/hr

No. of AGVs 1

Figure 5. 6 Max-Flow vs AGV speed for 200 doors in a Cross-dock

0

1000

2000

3000

4000

5000

6000

2 3 4 5 6

Pac

kage

s/h

r

AGV speed (ft/s)

Max-flow vs AGV speed for 200 doors

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

44

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

5.6

0

1000

2000

3000

4000

5000

6000

7000

2 3 4 5 6

Pac

kage

s/h

r

AGV speed (ft/s)

Max-flow vs AGV speed for 300 doors

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

45

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 466.7 packages/hr

Picking rate 466.7 packages/hr

Delivery rate 466.7 packages/hr

Loading rate 466.7 packages/hr

AGV capacity 100 packages

0

2000

4000

6000

8000

10000

12000

14000

16000

1 2 3 4 5 6 7 8 9 10

Pac

kage

s/h

r

No. of AGVs/door

Max-Flow vs No. of AGVs

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

46

5.6 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 5.8

No of AGVs I I-II L L-II L-III L-IV T

1 2264 2501 2294 2409 1842 2520 2281

2 4527 5002 4588 4817 3684 5039 4563

3 6791 7503 6881 7226 5526 7559 6844

4 9055 10004 9175 9635 7368 10078 9125

5 11318 12150 11469 11753 9185 12229 11379

6 12789 13207 13264 13035 10633 13255 13068

7 13532 13757 13869 13658 11743 13780 13783

8 13898 13981 14000 13929 12596 13982 13990

9 14000 14000 14000 13999 13223 14000 14000

10 14000 14000 14000 14000 13649 14000 14000

No of AGVs T-II T-III X X-II X-III H H-II H-III H-IV

1 2329 2490 2654 2267 2641 2428 2415 2280 2405

2 4657 4980 5308 4534 5281 4856 4830 4560 4811

3 6986 7470 7962 6801 7922 7284 7245 6840 7216

4 9314 9960 10616 9068 10562 9713 9660 9120 9622

5 11440 12120 12693 11212 12657 11982 12004 11275 11929

6 12790 13311 13702 12707 13684 13315 13399 12857 13318

7 13572 13869 14000 13601 14000 13910 13952 13641 13913

8 13942 14000 14000 13977 14000 14000 14000 13941 14000

9 14000 14000 14000 14000 14000 14000 14000 14000 14000

10 14000 14000 14000 14000 14000 14000 14000 14000 14000

Table 5. 7 Max-flow values for given shapes and no. of AGVs/door

47

Figure 5. 9 Max-Flow vs AGV's carrying capacity

Table 5. 8 Service Rates for Experiment (Max-flow vs AGV capacity)

0

50

100

150

200

250

300

350

400

450

100 200 300 400 500 600 700 800 900 1000 2000 3000

po

un

ds/

s

AGV capacity (in pounds)

MaxFlow vs AGV capacity

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

No of Operating Doors

200

AGV speed 3 ft/s

Unloading rate 466.7 packages/hr

Picking rate 466.7 packages/hr

Delivery rate 466.7 packages/hr

Loading rate 466.7 packages/hr

No. of AGVs 1

48

Table 5. 9 Max-flow values for given shapes and AGV speeds

5.7 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

AGV capacity I I-II L L-II L-III L-IV T

100 62.88019 69.47142964 63.71618 66.9091 51.16473 69.98938 63.37065

200 125.7604 138.9428593 127.4324 133.8182 102.3295 139.9788 126.7413

300 188.6406 208.4142889 191.1485 200.7273 153.4942 209.9681 190.1119

400 251.5207 277.8857186 254.8647 267.6364 204.6589 279.9575 253.4826

500 314.4009 337.5053498 318.5809 326.4741 255.1513 339.6958 316.08

600 355.2564 366.856613 368.4338 362.0924 295.3714 368.2057 363.0037

700 375.8766 382.142616 385.2606 379.3825 326.2044 382.7721 382.8558

800 386.045 388.3644409 388.8889 386.9239 349.8998 388.3901 388.6085

900 388.8889 388.8888889 388.8889 388.8616 367.2957 388.8889 388.8889

1000 388.8889 388.8888889 388.8889 388.8889 379.1475 388.8889 388.8889

2000 388.8889 388.8888889 388.8889 388.8889 388.8889 388.8889 388.8889

3000 388.8889 388.8888889 388.8889 388.8889 388.8889 388.8889 388.8889

AGV capacity T-II T-III X X-II X-III H H-II H-III H-IV

100 64.68239 69.16656192 73.71913 62.97408 73.35054 67.44884 67.08022 63.33204 66.81676

200 129.3648 138.3331238 147.4383 125.9482 146.7011 134.8977 134.1604 126.6641 133.6335

300 194.0472 207.4996858 221.1574 188.9223 220.0516 202.3465 201.2407 189.9961 200.4503

400 258.7295 276.6662477 294.8765 251.8963 293.4021 269.7954 268.3209 253.3282 267.2671

500 317.7769 336.6742384 352.5971 311.4306 351.5709 332.8204 333.4364 313.2015 331.3732

600 355.2783 369.7552683 380.6133 352.9615 380.1102 369.8497 372.191 357.1421 369.9343

700 376.9911 385.2632822 388.8889 377.8191 388.8889 386.3775 387.5579 378.925 386.4667

800 387.2694 388.8888889 388.8889 388.2567 388.8889 388.8889 388.8889 387.2405 388.8889

900 388.8889 388.8888889 388.8889 388.8889 388.8889 388.8889 388.8889 388.8889 388.8889

1000 388.8889 388.8888889 388.8889 388.8889 388.8889 388.8889 388.8889 388.8889 388.8889

2000 388.8889 388.8888889 388.8889 388.8889 388.8889 388.8889 388.8889 388.8889 388.8889

3000 388.8889 388.8888889 388.8889 388.8889 388.8889 388.8889 388.8889 388.8889 388.8889

49

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.

Table 5. 10 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 466.7 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.

Figure 5. 10 Network Flow Diagram of Cross-dock Operations

50

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.

Using data tables, the distances between every two doors was calculated. This is shown in

figure below.

Table 5. 12 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

IC door X Y OG door X' Y'

1 2 6 1 94 6

2 2 18 2 94 18

3 2 30 3 94 30

4 2 42 4 94 42

Table 5. 11 X and Y coordinates for inbound and outbound

door nos. 1 through 4

51

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 3600.

Figure 5. 11 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

52

it to calculate max-flow for a required user input based on the cross-dock size needed,

specifications of the AGV being used and different service rates.

Table 5. 13 Service rates specified by user

No of Operating Doors 150

AGV speed 4.4 ft/s

AGV capacity 30 lbs

Unloading rate 3.89 lbs/s

Picking rate 3.89 lbs/s

Delivery rate 3.89 lbs/s

Loading rate 3.89 lbs/s

53

CHAPTER 6

MEAN VALUE ANALYSIS

6.1 Model Assumptions

Mean Value Analysis (Referred to as MVA henceforth) is a recursive technique based on

queuing theory that is used to calculate the throughput, wait times and queue lengths in a cross-

dock’s travel operation. The max-flow is a deterministic model that successfully calculates the

best-case throughput from a cross-dock under stated assumptions. The MVA model can be

seen as a practical worst case for the cross-dock operation. We model the travel operation based

on MVA. The MVA model, unlike max-flow cannot be calculated without the demand matrix.

The matrix shown in Table 6.1 is the outline of the demand matrix specifying number of

packages moving from each inbound door to each outbound door. The Table shows a matrix

for a 4X4 cross-dock. For the following model, each of the cells in this matrix is 1 and the

matrix is for a 50X50 cross-dock. The number of inbound and outbound doors and the AGV

speeds are as specified in the Table 6.2

Table 6. 1 Demand matrix outlining package quantities going

from inbound to outbound doors

Oubound Door

Inbound

Door

I-O 1 2 3 4

1 1 1 1 1

2 1 1 1 1

3 1 1 1 1

4 1 1 1 1

54

Table 6. 2 Experiment assumptions

Figure 6. 1 MVA Throughput for varying number of AGVs in system

The Figure 6.1 shows the trend of the throughput values across varying number of AGVs. The

model can be extended to different sizes of cross-docks.

A point to be noted here is that the Mean Value Analysis theorem is only applicable to closed

queueing systems. So, we apply the MVA to the system for a fixed time period in which a

fixed number of trucks with a defined number of packages exists in the systems and the number

of AGVs in the system is also fixed for that time frame.

0

500

1000

1500

2000

2500

3000

3500

4000

4500

1

13

25

37

49

61

73

85

97

10

9

12

1

13

3

14

5

15

7

16

9

18

1

19

3

20

5

21

7

22

9

24

1

25

3

26

5

27

7

28

9

Pac

kage

s/h

r

No. of AGVs in system

TH vs No. of AGVs

AGV speed 4.4 ft/s

IC doors 50 OG doors 50

55

6.2 Division of pathways into segments

The paths of AGVs from an inbound door to an outbound door, are divided by intersection

points into separate segments. As explained earlier, each door has 8 segments corresponding

to it out of which two are called spurs which are semicircular and 9.42 ft long each. There are

four segments which are called the arteries which extend parallel to that side of a cross-dock

which houses all the doors. The arteries are all 6 ft each. Two of the segments are branches,

are 86 ft long and they are parallel to the width of the cross-dock and connect the arteries on

the inbound and outbound side. The MVA model demonstrates this segment division for an I

shaped cross-dock only, but can be extended to any other shape.

6.3 Computational Model-

The demand matrix is a matrix which specifies how many packages head from an inbound

truck at door i to an outbound truck at door j in the specified time frame. Based on this matrix,

the usage of each segment is calculated.

Table 6. 3 Demand Matrix in the computational model

Outbound door

Inbound door

I-O 1 2 3 4

1 1 1 1 1

2 1 1 1 1

3 1 1 1 1

4 1 1 1 1

56

The matrix which gives the usage of each segment is called the S matrix. The formulae to

calculate the usage is as specified below. Sik here, denotes the usage of segment k

corresponding to door i where it is assumed that there are 8 segments corresponding to each

inbound door and hence k can range from 1 to 8. Another practical assumption made is that

the segment usage includes the return travel of the AGVs.

The model for the S matrix is as follows.

Si1 = ∑ 𝐷(𝑖𝑦)𝑛𝑦=1

Si2 = ∑ 𝐷(𝑥𝑖)𝑛𝑥=1

Si3 = ∑ 𝐷(𝑖𝑦)𝑛𝑦=𝑖+1 + ∑ ∑ 𝐷(𝑥𝑦)𝑛

𝑦=𝑖+1𝑖−1𝑥=1 + ∑ ∑ 𝐷(𝑥𝑦)𝑖

𝑦=1𝑛𝑥=𝑖+1

Si4 = ∑ 𝐷(𝑖𝑦)𝑛𝑦=𝑖+1 + ∑ ∑ 𝐷(𝑥𝑦)𝑛

𝑦=𝑖+1𝑖−1𝑥=1 + ∑ ∑ 𝐷(𝑥𝑦)𝑖

𝑦=1𝑛𝑥=𝑖+1

Si5 = ∑ ∑ 𝐷(𝑥𝑦)𝑛𝑦=𝑖

𝑖−1𝑥=1 + ∑ ∑ 𝐷(𝑥𝑦)𝑖

𝑦=1𝑛𝑥=𝑖+1

Si6 = ∑ ∑ 𝐷(𝑥𝑦)𝑛𝑦=𝑖+1

𝑖𝑥=1 + ∑ ∑ 𝐷(𝑥𝑦)𝑖−1

𝑦=1𝑛𝑥=𝑖

Si7 = ∑ 𝐷(𝑥𝑖)𝑖𝑥=1

Si8 = ∑ 𝐷(𝑥𝑖)𝑛𝑥=𝑖

Where x and y are variables which can assume values as specified in the formula, i denotes the

corresponding door number for which usage is calculated, D(xy) denotes the corresponding

value from the demand matrix for inbound door x and outbound door denoted by y.

57

Once the S matrix is derived, we convert the matrix into a S column with S11 as S1, S18 as S8,

S21 as S9, S22 as S10 and so on.

j

S matrix 1 2 3 4 5 6 7 8

i

1 4 4 6 6 3 3 1 4

2 4 4 8 8 7 7 2 3

3 4 4 6 6 7 7 3 2

4 4 4 0 0 3 3 4 1

The visit ratio denoted by v is calculated by dividing the S matrix by sum of all packages

moving from inbound to outbound trucks. Then we also calculate k which denotes the time

spent by an AGV travelling in a particular segment k. As the AGV speed is constant and

segment lengths repeat after every 8 segments, ik is identical for all I, hence denoted by k .

S1 4

S2 4

S3 6

S4 6

S5 3

S6 3

S7 1

S8 4

S9 4

S10 4

S11 8

S12 8

S13 7

S14 7

S15 2

S16 3

S column

And so on till S32

And so on till S32

And so on till S32

And so on till S32

And so on till S32

Table 6. 4 Conversion of S-matrix to S column

58

Table 6. 5 Values of S, Tau and visit ratio in computational model

Segment k S column V column Tau column

S1 4 0.25 2.14

S2 4 0.25 2.14

S3 6 0.38 1.36

S4 6 0.38 1.36

S5 3 0.19 1.36

S6 3 0.19 1.36

S7 1 0.06 19.55

S8 4 0.25 19.55

S9 4 0.25 2.14

S10 4 0.25 2.14

S11 8 0.50 1.36

S12 8 0.50 1.36

S13 7 0.44 1.36

S14 7 0.44 1.36

S15 2 0.13 19.55

S16 3 0.19 19.55

And so on till segment 32.

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.

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.

Mean Value Analysis formulae-

59

Wait time at each segment k = Wk(m) = Lk(m-1) +1

Lk

System throughput for m AGVs = m = m

∑ Wk(𝑚)∗vk𝐾𝑘=1

Average Queue Lengths at each segment k = vk*m*Wk(m)

Using these formulae, we get the values of wait times and queue lengths at each segment and

the system throughput for different number of AGVs in the system. We plot the throughput

against the number of AGVs to see the trend of changing throughput.

Table 6. 6 Changing Throughput values against no. of AGVs in system

m TH

1 0.028143789

2 0.052329636

3 0.073044967

4 0.090738014

5 0.105817048

6 0.118649665

7 0.12956279

8 0.138843664

9 0.14674184

10 0.153471986

11 0.159217227

12 0.16413273

13 0.1683493

14 0.171976796

15 0.175107262

16 0.177817709

60

Table 6. 7 Sample of MVA model results showing throughput,

queue length and wait times for different segments

m V-> 1 2 3 4 5 6 7 8 9

1 Wk 2.14 2.14 1.36 1.36 1.36 1.36 19.55 19.55

TH1 0.03

Lk 0.02 0.02 0.01 0.01 0.01 0.01 0.03 0.14

2 Wk 2.17 2.17 1.38 1.38 1.37 1.37 20.22 22.23

TH2 0.05

Lk 0.03 0.03 0.03 0.03 0.01 0.01 0.07 0.29

The data from the iterations sheet can be used to see the trends in queue length, wait times and

throughput across varying values of k and m.

61

CHAPTER 7

COMPARISON AND APPLICATION OF MODELS

7.1 Similarities and differences in the two models

In the thesis, both the Max-flow and Mean Value Analysis models are used to calculate the

performance parameters of a cross-dock. The max-flow model measures throughput of the

cross-dock based on a deterministic approach. It takes into account the flow-network of the

entire set of cross-docking operations. The Mean Value Analysis model only takes into account

the travel operation.

The Max-Flow model is equivalent to the best-case performance of the system. It considers a

system wherein the process times are absolutely regular and the bottle-neck rate directly affects

the throughput of the system. Also, it considers that the system is congestion free and there is

no upper limit to the number of AGVs in the system. The MVA model on the other hand is a

probabilistic approach to finding the throughput. The prime difference in the two models is

that MVA considers congestion in the system and calculates the wait times and queue lengths.

The throughput measured by the MVA model lies between the best case and the worst case

(the worst case being the throughput of 1 package per cycle time in case of extreme wait times

at any one operation). It can be said that the MVA throughput is equivalent to the practical

worst-case performance of a system.

The Max-Flow model also has some shortcomings. It only considers a model with a certain

number of AGVs available to each door. In the MVA model, the total number of AGVs in the

62

system is considered. In a real world, AGVs will be used as and when required and at the door

they are required, hence the MVA model is more practical. In the Max-Flow model, the travel

distance is calculated using a rectilinear path and the change in speed at the turns is not

accounted for. In the MVA model, the paths are divided into segments and each segment travel

is treated as a separate operation and wait times are considered at each of the operations. While

Max-Flow is a macro-view of the system, MVA is a micro-view of the same.

The Max-Flow model makes it simpler to consider different shapes and door assignments as it

calculates the travel rate based on the geometry of the cross-dock alone. The model can be

easily extended to include the demand data and a simple door assignment strategy. In contrast,

the MVA model is based on the demand matrix, lengths of each segment of a path and its

usage, which makes it difficult to isolate the performance of a cross-dock based on its shape

alone. But nevertheless, shapes can be compared for different demand matrices and segment

usage.

7.2 Model Results for different cases

We will compare the two models below. We consider the travel distance of a round trip as

AGVs are assumed to return to their starting point in both the models. We will compare the

throughput in packages/s for different values of AGVs in the system. The plot shows a

comparison of these throughputs from different models. In figure 7.1, the MVA plot shows the

resulting throughput for a 4X4 cross-dock for a demand matrix with each call value 1 i.e. one

package going from each inbound dock to each outbound dock. The demand matrices for each

of these cases are shown in the Appendix Section.

63

Table 7. 1 Throughput values from MVA and Max-flow models

m MVA-TH MF-TH

1 0.028 0.021

2 0.052 0.042

3 0.073 0.063

4 0.091 0.084

5 0.106 0.105

6 0.119 0.126

7 0.130 0.148

8 0.139 0.169

9 0.147 0.190

10 0.153 0.211

11 0.159 0.232

12 0.164 0.253

13 0.168 0.274

14 0.172 0.295

15 0.175 0.316

16 0.178 0.337

m MVA-TH MF-TH

17 0.180 0.358

18 0.182 0.379

19 0.184 0.400

20 0.186 0.422

21 0.187 0.443

22 0.188 0.464

23 0.189 0.485

24 0.190 0.506

25 0.191 0.527

26 0.192 0.548

27 0.193 0.569

28 0.193 0.590

29 0.194 0.611

30 0.194 0.632

31 0.195 0.653

32 0.195 0.674

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

TH (

Pac

kage

s/s)

No. of AGVs

TH vs m (4*4 dock)

TH MF

Figure 7. 1 Comparison of Throughput from Max-Flow and MVA models

64

The Max-flow and MVA models were tested to obtain results of throughput for a 50X50 cross-

dock with a demand dataset of a grocery store’s cross-docking facility. The results for the data

are shown in Figure 7.2. The MVA values are much lesser than the Max-flow values and

become constant at 37 Packages/hr for around 97 AGVs in the system. The demand matrix is

highly dispersed and has very high cell values meaning that the number of packages going

from an inbound door to another outbound door are very high causing traffic and congestion

in paths. The data was tested for a total of 8 periods with similar results, one of which is shown

in the figure.

Figure 7. 2 Max-Flow and MVA Throughput for grocery dataset with high demand data

values and high deviation

We apply the model for demand matrices with different quantities and demand deviation to

see the effect of each factor on the MVA throughput. As a reminder, the max-flow values do

not change with change in demand matrix solely because the max-flow model gives only the

65

upper-bound maximum throughput that can go through a particular system and hence not

affected by the demand data. dispersed and has very high cell values meaning that the number

of packages going from an inbound door to another outbound door are very high causing traffic

and congestion in paths.

For a demand data matrix with all cell values small and equal, i.e. number of packages from

each inbound to outbound door equal and small, in this case 1, the results are shown in Figure

7.3. To see the change of increase in values, we test this on a similar matrix with cell values

‘3’. The result is shown in Figure 7.4. With increase in demand values, the MVA TH decreases.

Figure 7. 3 Comparison of Max-flow and MVA values for

demand data with mean 1 and deviation 0

66

Figure 7. 4 Comparison of Max-flow and MVA values for

demand data with mean 3 and deviation 0

To see the effect of deviation, we tested the model on a dataset which has a little deviation of

demand values instead of equal with deviation of 2 and average of 3. The result is almost

exactly equal to the result as the model with no deviation and cell values of ‘3’. We also

compare other data sets with same average value but different deviation and find that the

amount of deviation actually has no effect on the MVA value as long as the mean value is the

same. The comparison of two cases, both with mean value of cells 5.5, one with deviation 4.5

and other with deviation 2.5 is shown in Figure 7.5. As seen, it is almost exactly the same.

67

Figure 7. 5 Comparison of MVA throughput for datasets with same mean but different

deviation

Lastly, we see the result for a data which is clustered, which means in a 50X50 cross-dock,

only some inbound doors and outbound doors exchange freight, for example, packages are sent

from one of the inbound doors between 1 and 5 to one of the inbound doors 2 and 6. The

packages from one of these inbound doors are not directed to all other outbound doors. This

group forms one cluster. Such clusters exist in the 50X50 cross-dock. The MVA throughput

for such a dataset compared to another dataset with same mean and same total packages in

system but not clustered is compared. The results are shown in Figure 7.6. As clear from the

figure, the throughput reduces considerably. This indicates that a door assignment with clusters

results in a higher throughput which seems to be intuitive as clustering isolates the traffic and

lessens congestion as well as the usage of arteries.

68

Figure 7. 6 Comparison of MVA throughput for datasets with and

without clustered door assignments

69

CHAPTER 8

COST ANALYSIS

5.4 Model Assumptions

To determine the benefits and drawbacks of the automated cross-dock, we compare the total

annual costs of three cross-docks with different freight carriers- one with automated guided

vehicles doing the task, the other with workers either pushing the freight using low cost carts

or carrying them depending on the weight and the one with human operated forklifts. The three

types of cross-docks are compared for annual cost for a 20X20 cross-dock with three different

scenarios for different package weights and demand densities. The total weight of freight

travelling across the cross-dock is constant and hence the number of trip of an AGV from an

inbound door to outbound are different. The assumptions in each case are mentioned in the

corresponding table. In any compared case, the production per hour is assumed to be constant.

5.4 Cost-Considerations

The cost and energy considerations taken in account based on industrial average cost values

are as follows-

70

• An AGV costs $30,000/unit and is assumed to have a life cycle of 5 years. Hence, he

depreciation cost per year is 3000 and the service/maintenance cost is assumed to be

on the higher end as 10% of its yearly cost. So, the total equipment cost for an AGV is

assumed to be 3300 per unit per year.

• An AGV uses around 700 kWh of electricity per month and average electricity cost for

industrial purposes is $0.1 per kWh. So, an AGV consumes $70 worth of electricity per

month and $840 per year.

• A forklift costs around $10,000/unit and are assumed to have a life cycle of about 4

years. One human is assumed to operate the forklift and the corresponding labor

charges are accounted for in the model.

• A forklift uses 1.5 gallons of fuel (propane) per hour at $2/gallon and is assumed to

work 80% of the time in the day. Though both are Material Handling Devices (MHDs),

the prime differences between the forklift and AGV is that the forklift is manually

driven, runs on a fossil fuel and can have speeds up to 10fps (The average speed for

modelling purposes is assumed to be 7fps) while AGVs are fully automated, electrically

charged and have an average speed of 3 fps.

• The labor costs are assumed to be $15/hr for USA, $18/hr for Germany and $0.7/hr for

China.

71

• The human labor works in three shifts of 8 hrs each and gets a total of 40 min break.

The total productive time is 22 hrs/day for a labor-intensive cross-dock. It is the same

for a cross-dock where human operated forklifts are used. The cross-dock with AGVs

can run continuously for 24 hrs straight assuming that 5% of the AGVs are out for

charging at any given time and hence the productivity is affected by that. For the same

per-hour throughput, the daily and hence yearly productivity for the manual and fork-

lift run warehouses is lower by around 8%.

• The industrial average cost of electricity spent on utilities like lighting, cooling,

refrigeration, computers, ventilation etc. is known to be $0.7 per sq. ft. area. For our

analysis, we have assumed an I shaped 20X20 cross-dock which has an area of 1920

sq. ft.

• The AGVs being completely automated vehicles that can operate under dark and do

not need any temperature or ventilation control save up to 40% of these costs. This is

accounted for in the cost model.

5.4 Details of different cases considered

The three scenarios considered are called Case I, Case II and Case III and have varying levels

of productivity measured in packages per hour. The details for each are explained below.

72

Case I

For a sparse demand scenario as shown in Table 8.1 where the package weight is assumed to

be 100 lbs. The productivity is 1119 packages per hour with hundred AGVs or 95 humans or

40 forklifts. For an automated system which runs 24 hours a day, the productivity will be

9802440 packages per year while for a forklift based or fully manual cross-dock, the annual

productivity will be 8985570 i.e. an 8% loss.

Table 8. 1 Demand Matrix for Case I

Door Nos. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 50 50 50 25 25 25 25 100 0 0 0 0 0 0 0 0 0 0 0 0

2 0 50 50 50 25 25 25 25 100 0 0 0 0 0 0 0 0 0 0 0

3 0 0 50 50 50 25 25 25 25 100 0 0 0 0 0 0 0 0 0 0

4 0 0 0 50 50 50 25 25 25 25 100 0 0 0 0 0 0 0 0 0

5 0 0 0 0 50 50 50 25 25 25 25 100 0 0 0 0 0 0 0 0

6 100 0 0 0 0 50 50 50 25 0 0 0 100 0 0 0 0 0 0 0

7 25 100 0 0 0 0 0 0 0 0 0 0 50 50 50 25 25 25 25 0

8 0 0 100 0 0 0 0 0 0 0 0 0 0 50 50 50 25 25 25 25

9 25 25 25 100 0 0 0 0 0 0 0 0 0 0 50 50 50 25 25 0

10 25 25 25 25 0 0 0 0 50 0 0 0 0 0 0 50 50 50 25 25

11 25 0 0 0 0 0 0 50 50 0 50 0 0 0 0 0 50 50 50 25

12 0 25 0 0 0 0 0 50 50 75 125 0 0 0 0 0 0 0 0 50

13 0 0 0 0 0 0 0 0 0 0 50 25 25 50 100 75 25 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 100 100 100 50 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 75 75 100 50 50 25 0 0 0

16 50 25 25 50 100 75 25 0 0 0 0 0 0 0 0 0 0 0 0 0

17 0 0 0 0 0 0 0 0 0 0 0 50 0 0 0 50 100 75 100 0

18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 100 100 150

19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 50 50 25 0 50 50 125 0 0 0 0 0 0 0 0 0 0 0 0 0

73

Case II

For a dense demand scenario but less total packages due to high package weight as shown in

Table 8.2 where the package weight is assumed to be 500 lbs. The productivity is 5900

packages per hour with 100 AGVs or 95 humans or 43 forklifts. For an automated system

which runs 24 hours a day, the productivity can be 11317920 packages per year while for a

forklift based or fully manual cross-dock, the annual productivity will be 10374760 i.e. an 8%

loss.

Table 8. 2 Demand Matrix for Case II

Door Nos. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

5 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

6 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

7 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

8 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

9 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

10 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

11 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

12 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

13 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

14 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

15 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

16 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

17 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

18 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

19 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

20 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

74

Case III

For a dense demand scenario and high total packages due to low package weight as shown in

Table 8.3 where the package weight is assumed to be 30 lbs. The productivity is 439 packages

per hour with 200 AGVs or 190 humans or 60 forklifts. For an automated system which runs

24 hours a day, the productivity can be 3845640 packages per year while for a forklift based

or fully manual cross-dock, the annual productivity will be 3525170 i.e. an 8% loss.

Table 8. 3 Demand Matrix for Case 3

Door Nos. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

2 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

3 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

4 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

5 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

6 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

7 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

8 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

9 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

10 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

11 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

12 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

13 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

14 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

15 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

16 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

17 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

18 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

19 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

20 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

75

5.4 Cost Comparison for different regions

The figures 8.1, 8.2 and 8.3 show the cost comparison for all these cases for USA, Germany

and China respectively as shown below. The details are specified in Table 8.4, 8.5 and 8.6

quantitatively.

For USA,

Table 8. 4 Annual Cost Values for USA

USA Case 1 Case 2 Case 3

AGV $744,806 $744,806 $1,488,806

Forklift $6,128,224 $6,587,740 $9,191,664

Labor $12,484,344 $12,484,344 $24,967,344

Figure 8. 1 Cost Comparison for USA

$-

$5,000,000

$10,000,000

$15,000,000

$20,000,000

$25,000,000

$30,000,000

Case 1 Case 2 Case 3

An

nu

al C

ost

Cost Comparison (USA)

AGV Forklift Labor

76

For Germany,

Table 8. 5 Annual Cost Values for Germany

Germany Case 1 Case 2 Case 3

AGV $744,000 $744,806 $1,488,806

Forklift $7,179,424 $7,717,780 $10,768,464

Labor $14,980,944 $14,980,944 $29,960,544

Figure 8. 2 Cost Comparison for Germany

$-

$5,000,000

$10,000,000

$15,000,000

$20,000,000

$25,000,000

$30,000,000

$35,000,000

Case 1 Case 2 Case 3

An

nu

al C

ost

Cost Comparison (Germany)

AGV Forklift Labor

77

For China,

Table 8. 6 Annual Cost Values for China

China Case 1 Case 2 Case 3

AGV $744,000 $744,806 $1,488,806

Forklift $1,117,504 $1,201,216 $1,675,584

Labor $583,884 $583,884 $1,166,424

Figure 8. 3 Cost Comparison for China

On comparing annual costs, it is clear that in developed countries which have higher labor

costs, the annual expenditure of a labor-intensive cross-dock is 15 to 20 times more than that

of an automated cross-dock for the same level of hourly productivity. For the economies with

cheaper labor like China, the labor-intensive cross-dock will be around 22% cheaper. In

markets like Germany and USA, use of forklifts in a cross-dock is more cost-effective than a

fully labor-driven by around 50% to 70% but 6 to 10 times costlier than the automated cross-

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

1800000

Case 1 Case 2 Case 3

An

nu

al C

ost

Cost Comparison (China)

AGV Forklift Labor

78

dock. In China, the forklift driven cross-dock is seen to be the most expensive 15% to 50%

costlier than a fully automated one and 1.5 to 2 times costlier than a labor-intensive one.

Though the costs are definitely a decisive factor in comparing the better cross-dock type, there

are some other considerations too. The first and most important consideration is the package

weight. In cases with where one package is more than 100 lbs., we may not be able to use a

labor-intensive cross-dock due to restrictions on how much and for how long a human can

carry or push weights of certain amount. For example, the United States Department of Labor

restricts the permissible weight to be lifted to 51 lbs. which is adjusted to account for how often

a person lifts the weight, twists their back during lifting, vertical distance the load is lifted, the

distance of the load from your body, etc. The weights that can be pushed on a 4-wheeled hand-

cart is 500 lb if the frequency is 200 units per 8-hour shift and maximum transport distance

100 ft. Depending on the demand density, total number of packages, available workers and

package weights, the decision to adopt a labor-intensive or other cross-dock can be made.

Apart from that, certain countries of the world have subsidies on either certain fuels or

electricity, which may affect energy costs considered also.

Though there will be variables, the model helps to roughly compare the three different types

of cross-docks for various scenarios of demand.

79

CHAPTER 9

CONCLUSIONS AND FUTURE WORK

9.1 Conclusions

The thesis compares different cross-dock shapes based on area, generic door assignments and

max-flow as well as proposes a probabilistic method to calculate the throughput, queue lengths

and wait times at any cross-dock with given parameters and demand data. The following points

reiterate the various results from the thesis.

• 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 cross-dock. The difference is due to the unutilized area at corners.

• The areas that are unutilized are prospective spots for storing freight temporarily. In

most cases, this area is at a central location which makes it an even better option to

store freight as it can be later directed to any door where it is needed conveniently.

• The thesis proposes a max-flow model, which calculates the maximum possible

throughput from a cross-dock depending on given size, shape, type of generic door

assignments that we have considered for purpose of the thesis, number of AGVs, AGV-

speed and AGV-capacity. It considers the cross-docking operations as a flow network

with deterministic flows and capacities.

80

• The max-flow model suggests the best shape and type of door assignment for a given

set of dock and AGV specifications as required by the stakeholder.

• The thesis proposes a probabilistic model using Mean Value Analysis (MVA) to

determine the throughput for a given data matrix of number of packages going from

each inbound to each outbound door based on the number of AGVs available in the

system. It considers congestion and calculates the queueing length and wait times at

each intersection of the travel path of AGVs in the system.

• The max-flow gives an upper-limit of throughput of any cross-dock while the MVA

model gives the value of throughput considering queueing and congestion.

• The MVA results are affected by the total number of packages in the system, mean of

the demand matrix, way inbound to outbound assignment is made i.e. clustered or not.

Increasing value of mean of the demand matrix cell values causes an increase in the

throughput as a result of increased number of total packages moving in the system but

simultaneously causes a steep decrease in the throughput. The deviation in the data

marginally affects the throughput. Clustered assignments however, improve the

throughput for same total packages.

81

9.2 Scope for Future work

The thesis work paves way for future work to design a completely automated cross-dock with

optimal design and performance. Some of the future work which will add to this work is

described below.

• This thesis work assumes a fully automated cross-dock which has robotic arms and

AGVs. The programming of these automated devices according to door assignments

and freight schedules should be worked upon for smooth functioning.

• The Max-flow model assumes a flow network with picking, delivering, loading and

unloading speeds based on values from practical knowledge. The exact values for an

automated cross-dock will depend on the service rate of all these operations which will

depend on the type and make of the robots or AGVs used.

• The max-flow model assumes a deterministic flow rate and the model calculating does

not integrate the demand data or detailed door assignments in it. For a tighter upper-

bound, both these possibilities can be explored.

• The MVA model and the Max-flow model can be integrated for different shapes. This

will take a deeper understanding and making a blue-print of the path design and door

placements in a cross-dock of all shapes as the MVA model considers the path

segments and their lengths to calculate the throughput, queue lengths and wait times.

82

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86

APPENDIX

DEMAND DATASETS USED TO TEST MVA MODEL

Demand Dataset for Figure 7.1

Sample of a 50X50 Demand Dataset for Figure 7.2

I-O 1 2 3 4

1 1 1 1 1

2 1 1 1 1

3 1 1 1 1

4 1 1 1 1

Outgoing

Incoming

I-O 1 2 3 4 5 6 7 8

1 7 9 6 9 2 79 3 64

2 3 7 2 42 9 3 18 18

3 12 1 19 115 1 3 2 64

4 230 133 1 22 32 10 7 11

5 79 4 1 8 8 17 32 1

6 8 21 2 1 2 10 22 10

7 11 1 1 67 1 11 3 5

8 3 10 12 1 3 4 20 65

9 3 9 6 26 33 1 2 16

10 6 3 18 53 16 10 11 26

11 5 8 12 28 1 20 7 67

12 10 7 2 48 269 6 2 2

13 8 4 2 1 3 2 28 1

Outgoing

Incoming

87

Sample of a 50X50 Demand Dataset for Figure 7.3

Sample of a 50X50 Demand Dataset for Figure 7.4

I-O 1 2 3 4 5 6 7 8

1 1 1 1 1 1 1 1 1

2 1 1 1 1 1 1 1 1

3 1 1 1 1 1 1 1 1

4 1 1 1 1 1 1 1 1

5 1 1 1 1 1 1 1 1

6 1 1 1 1 1 1 1 1

7 1 1 1 1 1 1 1 1

8 1 1 1 1 1 1 1 1

9 1 1 1 1 1 1 1 1

10 1 1 1 1 1 1 1 1

11 1 1 1 1 1 1 1 1

12 1 1 1 1 1 1 1 1

13 1 1 1 1 1 1 1 1

Outgoing

Incoming

I-O 1 2 3 4 5 6 7 8

1 3 3 3 3 3 3 3 3

2 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3

4 3 3 3 3 3 3 3 3

5 3 3 3 3 3 3 3 3

6 3 3 3 3 3 3 3 3

7 3 3 3 3 3 3 3 3

8 3 3 3 3 3 3 3 3

9 3 3 3 3 3 3 3 3

10 3 3 3 3 3 3 3 3

11 3 3 3 3 3 3 3 3

12 3 3 3 3 3 3 3 3

13 3 3 3 3 3 3 3 3

Outgoing

Incoming

88

Samples of 2 50X50 Demand Datasets compared in Figure 7.5

I-O 1 2 3 4 5 6 7 8

1 2 5 8 2 7 5 8 5

2 2 7 2 6 5 3 5 10

3 10 5 8 10 9 5 10 10

4 1 9 9 3 9 3 1 7

5 7 3 6 5 8 5 3 1

6 3 2 10 5 4 1 6 4

7 3 3 1 2 8 7 1 5

8 1 4 7 2 9 8 4 1

9 10 1 7 4 6 4 5 5

10 8 7 2 5 3 4 1 4

11 6 9 2 1 10 10 1 6

12 7 2 4 2 10 4 4 7

13 9 4 10 3 5 9 1 6

Outgoing

Incoming

I-O 1 2 3 4 5 6 7 8

1 5 4 4 5 4 6 3 7

2 6 8 4 5 5 3 4 3

3 7 8 7 8 6 8 5 8

4 4 8 4 7 7 5 5 5

5 7 8 3 6 8 8 8 5

6 5 5 4 6 4 7 6 6

7 7 6 3 5 7 7 8 8

8 4 3 3 6 4 5 3 3

9 5 4 6 8 3 8 5 4

10 8 5 4 5 3 3 3 6

11 3 4 3 7 6 7 7 7

12 3 7 3 8 5 6 6 3

13 7 4 7 7 5 4 3 6

Outgoing

Incoming

89

Samples of 2 50X50 Demand Datasets compared in Figure 7.5

(Clustered and less Clustered)

I-O 1 2 3 4 5 6 7 8 9 10 11 12

1 184 200 169 171 0 0 0 195 156 154 158 0

2 174 167 175 181 0 0 0 171 195 199 174 0

3 199 164 152 171 0 0 0 196 172 163 195 0

4 179 188 196 176 0 0 0 184 183 170 161 0

5 0 0 0 0 0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0 0 0 0 0

8 0 0 0 0 0 0 0 0 0 0 0 0

9 0 0 0 0 158 199 165 196 0 0 0 0

10 0 0 0 0 160 150 151 190 0 0 0 0

11 0 0 0 0 200 171 191 198 0 0 0 0

12 0 0 0 0 178 173 185 153 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 0

16 0 0 0 0 0 0 0 0 0 0 0 0

17 0 0 0 0 0 0 0 157 196 163 188 0

18 0 0 0 0 0 0 0 180 168 179 169 0

19 0 0 0 0 0 0 0 168 199 156 199 0

20 0 0 0 0 0 0 0 199 189 170 197 0

21 0 0 0 0 0 0 0 0 0 0 0 0

Outgoing

Incoming

90

I-O 1 2 3 4 5 6 7 8 9 10 11 12

1 342 399 334 367 0 0 0 352 352 317 346 0

2 334 317 326 371 0 0 0 351 363 378 343 0

3 399 335 343 369 0 0 0 364 371 319 394 0

4 357 361 381 329 0 0 0 383 372 340 358 0

5 0 0 0 0 0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0 0 0 0 0

8 0 0 0 0 0 0 0 0 0 0 0 0

9 0 0 0 0 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 0 0 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 0

16 0 0 0 0 0 0 0 0 0 0 0 0

17 0 0 0 0 0 0 0 0 0 0 0 0

18 0 0 0 0 0 0 0 0 0 0 0 0

19 0 0 0 0 0 0 0 0 0 0 0 0

20 0 0 0 0 0 0 0 0 0 0 0 0

21 0 0 0 0 0 0 0 0 0 0 0 0

Outgoing

Incoming