Velocity-based Storage and Stowage Decisions in a Semi

156 Pages·2016·3.04 MB·English

Velocity-based Storage and Stowage Decisions in a Semi-
automated Fulfillment System
by
Rong Yuan
Submitted to the Sloan School of Management on July 19, 2016 in
partial fulfillment of the requirements for the degree of Doctor of
Philosophy in Operations Research

ABSTRACT

The supply chain management for an online retailing business is centered around the operations
of its fulfillment centers. A fulfillment center receives and holds inventory from vendors, and
then uses this inventory to fill customer orders. Our research focuses on a new operating
architecture of an order fulfillment system, enabled by new technology. We refer to it as the
Semi-automated Fulfillment System. Different from the person-to-goods model in traditional
warehouses, the semi-automated fulfillment system adopts a goods-to-person model for
stowing and picking items from a storage field. In a semi-automated fulfillment system the
inventory is stored on mobile storage pods; those mobile pods are then carried by robotic drives
to static stations at which the operators conduct pick or stow operations.

In the first chapter, we describe and identify three key operational decisions in the semi-
automated fulfillment system, namely from which pods to pick the inventory needed (picking
decision), where to return the pod to the storage field upon the completion of a pick or stow
operation (storage decision), and to which pods to replenish the received inventory (stowage
decision). We present a high-level capacity planning model for determining the number of
robotic drives needed to achieve a given throughput level. This model highlights how the
operational efficiency in this system depends on two key parameters, namely the travel time
for an entire drive trip and the number of unit picks or stows per pod trip.

In the second chapter, we focus on the storage decisions. The storage decision is to decide to
which storage location to return a pod upon the completion of a pick or stow operation. We
extend the academic results on the benefits of adopting velocity-based and class-based storage
policies to the context of the semi-automated fulfillment system. We associate with each
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storage pod a velocity measure that represents an expectation of the number of picks from that
pod in the near future. We then show that by assigning the high velocity pods to the most
desirable storage locations, we can significantly reduce the drive travel time, compared to the
random storage policy that returns the pod to a randomly-chosen storage location. We show
that class-based storage policies with two or three classes, can achieve most of the benefits
from the idealized velocity-based policy. Furthermore, we characterize how the performance
of the velocity-based and class-based storage policies depend on the velocity variability across
the storage pods; in particular we model how the benefits from velocity-based storage policies
increase with increased variation in the pod velocities.

In the third chapter, we build a discrete-time simulator to validate the theoretical models in the
second chapter with real industry data. We observe a 6% to 11% reduction in the travel
distance with 2-class or 3-class system, depending on the parameter settings. From a
sensitivity analysis we establish the robustness of the class-based storage policies as they
continue to perform well under a broad range of warehouse settings including different zoning
strategies, resource utilization levels and space utilization levels.

In the fourth chapter, we examine two stowage decisions, one at the zone level and the other at
the pod level. The zone-level decision is to decide how to allocate the received inventory to
multiple storage zones. The objective is to assure that the resulting picking workload for each
zone is within its capacity. We show by simulation that a chaining-based allocation can be
effective to balance the picking workload across different storage zones. The pod-level
stowage decision is to decide on which pods to stow the inventory. We formulate a mixed-
integer program (MIP) to find the optimal stowage profile that maximizes the number of unit
picks per pod trip. We solve the MIP for a set of test cases to gain insight into the structure
of optimal stowage policy. Motivated by these insights, we further propose a class-based
stowage process that induces variability across the pod velocities.

Thesis Supervisor: Stephen C. Graves
Title: Abraham J. Siegel Professor of Management

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Acknowledgements
I have worked closely with Professor Stephen Graves for six years in my master and doctoral
studies at MIT. Words alone cannot express how fortunate I feel to have him as my advisor. I
thank him for providing me with such valuable opportunities to work on innovative real-world
problems; for offering his wisdom, patience, and encouragement for developing initial ideas
into solid research; for helping me establish the clarity and accuracy of expressions through
commenting on my wittings as well as giving me detailed feedback from our conversations
during meetings and discussions. I am fully indebted to Professor Graves as a student.

I would also like to extend my sincere appreciation to my committee members Professor
Patrick Jaillet, Professor David Simchi-Levi, and Tolga Cezik for their interests, efforts, and
helpful comments. I want to thank Tolga Cezik in particular for his invaluable inputs and
discussions over the past few years, which have greatly improved the relevance of my research
in practice.

I was also privileged to have friends at the Operations Research Center at MIT. In particular I
want to express my heartfelt thanks to Jason Avmovic, Annie Chen, Miles Lubin, Hai Wang,
He Wang, Yehua Wei, and Chiwei Yan for helpful discussions and continuous support for this
research.

Lastly, I am forever grateful to my parents who are always there for me. Their love and support
allow me to have the opportunity to be the best I can be. None of this would have been possible
without them.

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Table of Contents
Acknowledgements .................................................................................................................. 3
Table of Contents ..................................................................................................................... 3
Table of Figures ....................................................................................................................... 7
Tables of Tables ....................................................................................................................... 9
Chapter 1 – Introduction of Online Retailing and Semi-automated Fulfillment System 12
1.1 Overview of the Online Retailing Service ................................................................. 12
1.2 Introduction of the Semi-automated Fulfillment System........................................... 14
1.2.1 Automated Storage and Retrieval System (AS/RS) ....................................... 15
1.2.2 Semi-automated Storage System ................................................................... 16
1.3 Key Processes and Operational Decisions in the Semi-automated Storage System .. 18
1.3.1 Picking Decision ............................................................................................ 20
1.3.2 Storage Decision ............................................................................................ 22
1.3.3 Stowage Decision........................................................................................... 23
1.4 High Level Capacity Planning of the Semi-automated Storage System.................... 24
1.4.1 Estimation of the Number of Robotic Drives ................................................ 24
1.4.2 Estimation of the Queue Level at the Stations ............................................... 33
1.5 Key Contributions of the Thesis ................................................................................ 36
Chapter 2 – Velocity-based Storage Decisions .................................................................... 38
2.1 Literature Review....................................................................................................... 39
2.2 Model Assumptions ................................................................................................... 43
2.3 Model Preliminaries ................................................................................................... 48
2.4 Evaluation of Storage Policies under Random Stowage............................................ 49
2.5 Evaluate Storage Policies under Velocity-based Stowage ......................................... 57
2.5.1 Full-velocity Storage Policy Under Velocity-based Stowage ....................... 59
2.5.2 Class-based Storage Policy under Velocity-based Stowage .......................... 65
2.6 Numerical Example and Sensitivity Analysis............................................................ 72
2.7 Discussion and Future Research ................................................................................ 77
Chapter 3 - Simulation of the Class-based Storage Policy ................................................. 80
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3.1 Introduction of the Simulation Study on Storage Decisions ...................................... 80
3.2 Description of Input Data ........................................................................................... 82
3.3 Simulation Framework............................................................................................... 84
3.3.1 Assumptions ................................................................................................... 84
3.3.2 Pod Velocity Measures .................................................................................. 86
3.3.3 Storage Policies .............................................................................................. 88
3.3.4 Picking Algorithms ........................................................................................ 91
3.3.5 Parameter Settings ......................................................................................... 92
3.3.6 Simulation Logic ............................................................................................ 95
3.4 Simulation Results ..................................................................................................... 96
3.4.1 The Actual Picking Algorithm ....................................................................... 98
3.4.2 The Pod Set Cover Algorithm...................................................................... 104
3.5 Sensitivity Analysis ................................................................................................. 105
3.5.1 Zoning Strategy ............................................................................................ 105
3.5.2 Station Utilization ........................................................................................ 107
3.5.3 Space Utilization .......................................................................................... 108
3.5.4 Velocity Threshold Changes ........................................................................ 109
3.6 Discussion and Future Research .............................................................................. 109
Chapter 4 – Stowage Decisions ........................................................................................... 112
4.1 Introduction of the Stowage Decisions .................................................................... 113
4.2 Literature Review..................................................................................................... 115
4.3 Zone-balancing Stowage Policy .............................................................................. 117
4.3.1 Problem Description .................................................................................... 117
4.3.2 Assumptions for the Numerical Experiment ................................................ 119
4.3.3 Stowage Policies .......................................................................................... 121
4.3.4 Comparison of Different Stowage Policies.................................................. 123
4.3.5 Sensitivity Analysis ..................................................................................... 125
4.3.6 Conclusion ................................................................................................... 127
4.4 Optimal Stowage Profile .......................................................................................... 128
4.4.1 Introduction .................................................................................................. 128
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4.4.2 Model ........................................................................................................... 128
4.4.3 Optimality Conditions and Piece-wise Linear Formulation ........................ 131
4.4.4 Numerical Examples .................................................................................... 136
4.4.5 Heuristic for the Stowage Profile Problem .................................................. 141
4.4.6 Conclusions and Remarks ............................................................................ 143
4.5 Class-based Stowage Policy .................................................................................... 144
4.5.1 Stowage Processes in a Semi-automated Fulfillment System ..................... 144
4.5.2 Class-based Stowage Policy ........................................................................ 146
4.5.3 Optimal Stow Batch Size ............................................................................. 149
4.6 Conclusion ............................................................................................................... 152
Reference .............................................................................................................................. 153

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Table of Figures
Figure 1.1: Sketches of the AS/RS and Carousal System (Courtesy of The Material
Handling Industry of America) ................................................................................. 16
Figure 1.2: Sketch of a Grid Storage Field ....................................................................... 18
Figure 1.3: Key Operational Processes in the Fulfillment Center .................................... 19
Figure 1.4: Layout of the Storage Field; Red Frame Indicates A Drive District .............. 26
Figure 1.5: Sensitivity Analysis for Parameter 𝑝 and 𝑧 ................................................. 32
Figure 1.6: Sensitivity Analysis for Parameter 𝛼 ............................................................. 33
Figure 2.1: The Ratio of Standard Deviation to Mean of the Daily Demand of Over 120,000
Items in a Week .......................................................................................................... 44
Figure 2.2: A Typical Layout of a Grid Storage Field; Length of Storage Field = 160, Width
of Storage Field = 100, Number of Stations = 22 ...................................................... 46
Figure 2.3: Travel Distances from Storage Locations to the Closest Stations; Length of
Storage Field = 160, Width of Storage Field = 100, Number of Stations = 22 ......... 47
Figure 2.4: Graph Illustration of the Expected Picks of the Items.................................... 49
Figure 2.5: Improvement Ratios of Applying the Full-velocity Storage Policy under
Random Stowage ....................................................................................................... 54
Figure 2.6: Improvement Ratios of Applying the 2-class Storage Policy under Random
Stowage ...................................................................................................................... 54
Figure 2.7: Improvement Ratios of Applying the 3-class Storage Policy under Random
Stowage ...................................................................................................................... 55
Figure 2.8: Improvement Ratios for the 2-class Storage Policies under Different Choice of
the Break Points; 𝛼 = 0.05,𝐶 = 0.001, and 𝜈 = 1.0 ............................................... 55
Figure 2.9: Improvement Ratios for the 3-class Storage Policies under Different Choice of
the Break Points; 𝛼 = 0.05,𝐶 = 0.001, and 𝜈 = 1.0 ............................................... 56
Figure 2.10: Convergence of the Improvement Ratios to the Limits (𝛼 = 0.1)............... 73
Figure 2.11: Convergence of the Improvement Ratios to the Limits (𝛼 = 0.01) ............ 73
Figure 2.12: Convergence of the Improvement Ratios to the Limits (𝛼 = 0.2)............... 74
Figure 2.13: Improvement Ratios for the Full-velocity and the Optimal 2-class and 3-class
Storage Policies Under Different 𝛼 When 𝐽 → ∞ .................................................. 75
Figure 3.1: Histogram of the Number of Pods Associated with an Item for a Typical
Inventory Snapshot .................................................................................................... 83
Figure 3.2: Percentage of Order Arrival by Due Time Hours for a Typical Day .............. 84
Figure 3.3: Illustrations for the 2-class and 3-class Velocity Sub-zones of a Two-station
System ........................................................................................................................ 89
Figure 3.4: Storage and Station Locations by Velocity Sub-zones for the 2-class System
(40% Zone 1; 60% Zone 2) ........................................................................................ 97
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Figure 3.5: Storage and Station Locations by Velocity Sub-zones for the 3-class System
(20% Zone 1; 30% Zone 2; 50% Zone 3) .................................................................. 98
Figure 3.6: Velocity Threshold by Time for the 2-class System ....................................... 99
Figure 3.7: Velocity Thresholds by Time for the 3-class System ................................... 100
Figure 3.8: Distribution of the Empty Storage Space by Time for the 2-class System .. 101
Figure 3.9: Distribution of the Empty Storage Space by Time for the 3-class System .. 101
Figure 3.10: Distribution of the Empty Storage Space by Time for the Closest-open-
location Storage Policy (assume 2-class system zoning code) ................................ 102
Figure 3.11: Distribution of Velocity Class 1 Pods at the End of the 24th Hour of the
Simulation for the 2-class System............................................................................ 103
Figure 3.12: Distribution of Velocity Class 2 Pods at the End of the 24th Hour of the
Simulation for the 2-class System............................................................................ 103
Figure 3.13: Locations of the Base Case and Additional Pick Stations in the Simulation
.................................................................................................................................. 107
Figure 4.1: Material Flow from a Receive Station to the Multiple Storage Zones ......... 113
Figure 4.2: Demand Patterns for Zone Balancing Stowage Policy ................................ 120
Figure 4.3: Piece-wise Linear Approximation to the Exponential Function ex ............ 137
Figure 4.4: An Illustration of the Current Stowage Process ........................................... 146
Figure 4.5: Class-based Stowage Algorithm ................................................................... 148
Figure 4.6: An Illustration of the 2-Class-based Stowage Process ................................. 149
Figure 4.7: Expected Buffer Cost and Mismatched Cost for a 3-class System Example
.................................................................................................................................. 151

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Tables of Tables
Table 1.1: Parameter Settings for the Numerical Example for M/G/∞ System .............. 31
Table 1.2: Travel Time of Each Component in a Drive Trip ............................................ 31
Table 1.3: Key Results of the Numerical Example ........................................................... 31
Table 1.4: Input Parameters for the Numerical Example for the M/M/n Queueing Model
.................................................................................................................................... 36
Table 1.5: Performance of the Numerical Example for the M/M/n Queueing Model ...... 36
Table 2.1: Improvement Ratios for the Full-velocity and Class-based Storage Policies;
𝛼 = 0.05,𝐶 = 0.001, and 𝜈 = 1.0 ............................................................................ 53
Table 2.2: Improvement Ratios and Optimal Break Points for the Full-velocity, the optimal
2-class and 3-class Storage Policies Under Different 𝛼 When 𝐽 → ∞ ................... 76
Table 2.3: Comparison of the Improvement Ratios for an Example where α = 0.05,C =
0.001,and ν = 1.0 ..................................................................................................... 76
Table 3.1: Performance Comparison for the Closest-open-location, the 2-class, and the 3-
class Storage Policy under the Base Case Scenario (10 Stations, 1% Empty Space, 𝛼 =
0.5) for AP Algorithm ................................................................................................ 98
Table 3.2: Storage Accuracy for Each Velocity Class for the 2-class System ................ 102
Table 3.3: Storage Accuracy for Each Velocity Class for the 3-class System ................ 102
Table 3.4: Performance Comparison for the Closest-open-location, the 2-class, and the 3-
class Storage Policy under the Base Case Scenario (10 Stations, 1% Empty Space, 𝛼 =
0.5) for PSC Algorithm ............................................................................................ 104
Table 3.5: Performance Comparison for the AP and the PSC Picking Algorithm .......... 105
Table 3.6: Comparison of the Improvement on Pod Travel Distance for Different Zoning
Settings for 2-class System ...................................................................................... 106
Table 3.7: Comparison of the Improvement on Pod Travel Distance for Different Zoning
Settings for 3-class System ...................................................................................... 106
Table 3.8: Comparison of the Improvement on Pod Travel Distance for Different Station
Utilization ................................................................................................................ 108
Table 3.9: Comparison of the Improvement on Pod Travel Distance for Different Space
Utilization ................................................................................................................ 109
Table 3.9: Comparison of the Improvement on Pod Travel Distance for Different 𝛼 ... 109
Table 4.1: Parameter Settings for Demand Patterns ....................................................... 120
Table 4.2: p and p Represented as Percentage of the Expected Total Demand .... 124
Table 4.3: p for Different Resource Utilization Levels ............................................. 126
Table 4.4: p for Different Demand Skewness ............................................................ 126
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Table 4.5: p for Inventory Level Uniformly Distributed in the Range of 2-period to 6-
period Over the Demand Mean ................................................................................ 127
Table 4.6: The Probabilities That the (Ranked) Units Are Not Required,q Pr(d k), for
ik i
Example 1 ................................................................................................................ 138
Table 4.7: The Probabilities That the (Ranked) Units Are Not Required,q Pr(d k), for
ik i
Example 2 ................................................................................................................ 138
Table 4.8: Optimal Stowage Profile for Example 1 ........................................................ 139
Table 4.9: Optimal Stowage Profile for Example 2 ........................................................ 140
Table 4.10: Optimal Stowage Profile for Example 1 with q values, Pod 1(Yellow), 2
ik
(Green), 3(Blue), and 4 (Orange) ............................................................................. 140
Table 4.11: Optimal Stowage Profile for Example 2 with q values, Pod 1(Yellow), 2
ik
(Green), 3(Blue), and 4 (Orange) ............................................................................. 141
Table 4.12: Optimality Gap of the Greedy Heuristics for the Examples ........................ 142
Table 4.13: Stowage Profile by the Greedy Algorithm for Example 1 with q values, Pod
ik
1(Yellow), 2 (Green), 3(Blue), and 4 (Orange) ........................................................ 142
Table 4.14: Numerical Testing for the Optimality Gap of the Greedy Heuristics .......... 143

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