Improving the Inbound Supply Chain through Dynamic Pickup Windows
By
Charles R Cummings III
B.S. in Mechanical Engineering
University of Kentucky, 2009
Submitted to the MIT Sloan School of Management and the Mechanical Engineering Department on
May 9, 2014 in Partial Fulfillment of the Requirements for the Degrees of
Master of Business Administration
and
Master of Science in Mechanical Engineering
MASSACHUSETT INSfTIIE
OF TECHNOLOGY
JUN 18 2014
In conjunction with the Leaders for Global Operations Program at the
Massachusetts Institute of Technology
June 2014
LIBRARIES
©2014 Charles R Cummings III. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic
copies of this thesis document in whole or in part in any medium now known or hereafter created.
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a PItai
Magem
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t aplice, Thesis Supervisor
SeIor ecturer, Engineering Systems Division
Exegutive Director, Center for Transportation and Logistics
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essor
hanical Engineering
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Accepted by
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MIT Sloan School of Management
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2
Improving the Inbound Supply Chain through Dynamic Pickup Windows
By
Charles R Cummings III
Submitted to the MIT Sloan School of Management and the Mechanical Engineering Department on
May 9, 2014 in Partial Fulfillment of the Requirements for the Degrees of Master of Business
Administration and Master of Science in Mechanical Engineering
Abstract
Amazon is one of the world's largest retailers with revenues of $74.5 billion in 2013 and 22% growth
over 2012. As Amazon continues to grow and offer greater selection, more products are flowing through
an expanding inbound network. While this growth has added complexity for the inbound transportation
organization, it has also created opportunities to reduce transportation cost and improve performance.
Inbound transportation managed by Amazon currently represents 60% of the company's inbound freight.
For this freight, Amazon uses automated shipment-planning systems to select a carrier for all shipments.
The systems run once per day, selecting carriers for a set of shipment requests where each vendor has
specified a freight ready date of tomorrow. Several inputs are included to achieve a low transportation
cost for the network, but the systems are constrained by the vendor's freight ready date. By introducing
dynamic pickup windows based on when the freight is needed in the fulfillment centers (FCs), Amazon
has the opportunity to reduce transportation cost and mitigate out-of-stock occurrences.
A current state analysis revealed that approximately 70% of Amazon's freight was shipped through
expensive less-than-truckload and small-parcel methods. While truckload shipments are ideal in
transportation, ordering smaller lots more frequently is preferable to maintain high in-stock levels in the
FCs while keeping inventory holding costs low. Therefore, Amazon's shipment-planning systems
minimize transportation cost by building multi-stop routes to pick up smaller shipments from several
vendors before delivering to the FC.
The dynamic pickup window solution changes the planning process by relaxing the constraint of
tendering a shipment today to a high cost transportation mode if that freight does not need to ship today.
If shipment requests are not tendered today and instead sent to tomorrow's pool of requests, two types of
consolidation can occur: (1) a single-vendor consolidation and (2) a multi-vendor consolidation.
A model was developed to simulate shipment planning on the entire network for one week, resulting in a
2% transportation cost reduction and 4% fewer shipments while protecting in-stock levels. Amazon
piloted in late 2013 with success and plans to implement throughout the network in early 2014.
While the dynamic pickup window solution is presented with Amazon as the case study, the solution is
applicable to any business with stochastic demand and lead time, a large vendor base, and control of
managing its inbound transportation.
Thesis Supervisor: Itai Ashlagi
Title: Assistant Professor of Operations Management, Sloan School of Management
Thesis Supervisor: Chris Caplice
Title: Senior Lecturer, Engineering Systems Division
3
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4
Acknowledgments
I want to thank everyone at Amazon for providing the opportunity to research its Inbound Transportation
operations. The Inbound Transportation teams at Amazon were fantastic to work with and played a very
important role in the success of the internship. Specifically, I would like to thank the following
individuals, who had a profound impact on my work for this project: Bob Flannery, Mark Michener, KK
Ananth, Abhi Sulakhe, and Rao Panchalavarapu.
Special thanks also goes to my MIT advisers, Chris Caplice, Itai Ashlagi, and Steve Graves for their
guidance and support throughout the project.
I would also like to acknowledge the Leaders for Global Operations Program with special thanks for
providing such an excellent opportunity.
Most importantly, I would like to thank my wife Rachel for her unwavering support and encouragement
throughout these past two years.
5
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6
Note on Proprietary Information
In the interest of protecting Amazon's competitive and proprietary information, tables and figures
presented throughout this document may have been disguised, are solely for the purpose of illustration,
and may not represent actual Amazon data.
7
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8
Tables of Contents
Abstract .....................................................................................................
3
A cknow ledgm ents...........................................................................................................................
5
N ote on Proprietary Inform ation...............................................................................................
7
List of Tables ................................................................................................................................
11
List of Figures ...............................................................................................................................
12
List of Equations...........................................................................................................................
13
1.
Project O verview ...................................................................................................................
14
1.1
Am azon Background .......................................................................................................................
14
1.2
Inbound Transportation Organization .........................................................................................
14
1.3
Purpose and M otivation of Project ..............................................................................................
14
1.4
Project Goals and Approach........................................................................................................
15
1.5
Thesis Overview ..............................................................................................................................
15
2.
Literature Review ..................................................................................................................
2.1
Inbound Transportation Overview ...............................................................................................
16
16
2.1.1
Full Truckload ..............................................................................................................................
16
2.1.2
Less-Than-Truckload ...................................................................................................................
17
2.1.3
Sm all Parcel..................................................................................................................................
20
2.2
Inventory Policies with Probabilistic Dem and and Lead Tim e...................................................
21
2.3
Freight Consolidation ......................................................................................................................
25
2.3.
Spatial Consolidation ...................................................................................................................
25
2.3.2
Tem poral Consolidation ...............................................................................................................
27
3.
A m azon Inbound Transportation Current State..................................................................
29
3.1
Pickup W indow ...............................................................................................................................
30
3.2
Shipm ent Planning System ..............................................................................................................
30
The D ynam ic Pickup W indow Solution................................................................................
32
4.1
The M ust Ship By Date Concept .................................................................................................
32
4.2
Populating the M ust Ship By Date ..............................................................................................
34
4.2.1
Phase 1 - Latest Vendor Ship Date ..........................................................................................
35
4.2.2
Phase 2 - Projected Stock Out Date..........................................................................................
36
4.
9
4.3
Probability of Consolidation Logic ..............................................................................................
37
4.3.1
Auto-Eligible Transportation Lanes ..........................................................................................
38
4.3.2
M ulti-stop Candidates ..................................................................................................................
40
4.3.3
M inim um Less-than-Truckload Cost ...........................................................................................
42
4.4
5.
M axim um Delay Tim e.....................................................................................................................
42
The Shipm ent Planning Sim ulation M odel ........................................................................
44
M odel Inputs....................................................................................................................................
44
5.1.1
Input Data .....................................................................................................................................
44
5.1.2
User-Controlled Inputs.................................................................................................................46
5.1
5.2
Assigning Initial Transportation Cost..........................................................................................
47
5.2.1
Truckload Cost Equation..............................................................................................................
48
5.2.2
Less-than-Truckload Cost Equation..........................................................................................
50
5.3
M ulti-stop Truckload Optim ization............................................................................................
52
5.3.1
M odel Assum ptions......................................................................................................................
53
5.3.2
M ILP Formulation........................................................................................................................
55
5.3.3
Optim ization M odel Outputs...................................................................................................
57
5.4
Decisions for Delay Eligibility .....................................................................................................
57
5.5
Simulation M odel Outputs...............................................................................................................
58
6.
M odel Results and Analysis ..............................................................................................
60
6.1
Results Overview .............................................................................................................................
60
6.2
Sensitivity Analysis .........................................................................................................................
64
6.2.1
M inim um LTL Cost Constraint....................................................................................................
64
6.2.2
M ulti-stop Truckload Pickup Charge ........................................................................................
65
6.2.3
Offset from M ust Ship By Date ...................................................................................................
66
6.2.4
M ax Days Delayed .......................................................................................................................
67
Effects of the Solution Discussion...............................................................................................
68
6.3
7.
Conclusion.............................................................................................................................
70
8.
Bibliography ..........................................................................................................................
72
A ppendix A : Longest Lead Tim e Derivation ............................................................................
73
Appendix B: Longest Vendor-Controlled Lead Time Derivation .............................................
76
10
List of Tables
Table 1: SKU-Level PSODs .......................................................................................................................
37
Table 2: Percentage of Lanes that Shipped M ultiple Times in a W eek ...................................................
38
Table 3: Test Case Predictor Perform ance on W eek 22..........................................................................
39
Table 4: Auto-Eligible Predictor Effectiveness ......................................................................................
40
Table 5: M ulti-stop Candidate Predictor Effectiveness ...........................................................................
41
Table 6: Exam ple of Data Inputs.................................................................................................................
46
Table 7: Exam ple of User-Controlled Inputs ..........................................................................................
47
Table 8: M STL 3-Digit Zip Code ...............................................................................................................
54
Table 9: M ST L Analysis State ....................................................................................................................
54
Table 10: M STL Analysis Num ber of FCs ..............................................................................................
54
Table 11: M STL Analysis Number of FC Clusters..................................................................................
55
Table 12: Simulation M odel Results - Key Output Param eters ...............................................................
61
Table 13: Delay Effectiveness Comparison ............................................................................................
63
Table 14: M in LTL Cost Sensitivity Comparison....................................................................................
65
Table 15: M STL Pickup Charge Comparison..........................................................................................
66
Table 16: M SBD Shift Comparison............................................................................................................
67
Table 17: Delay Lim it Comparison.............................................................................................................
67
11
List of Figures
Figure 1: LTL Cost W eight Breaks.............................................................................................................
18
Figure 2: LTL Cost W eight Breaks Actual ..............................................................................................
19
Figure 3: Freight Class Density Guidelines ...........................................................................................
20
Figure 4: Lead Tim e Segm entation.............................................................................................................
24
Figure 5: Hub Consolidation Diagram .....................................................................................................
26
Figure 6: M ulti-stop Truckload Consolidation Diagram ........................................................................
26
Figure 7: Temporal Consolidation - Single Vendor to Single FC..........................................................
27
Figure 8: Temporal and Spatial Consolidation to Build M ore M STL ...................................................
27
Figure 9: Am azon Fulfillm ent Center Locations .....................................................................................
29
Figure 10: Ship W indow Diagram ..............................................................................................................
30
Figure 11: Inbound Transportation Process ...........................................................................................
31
Figure 12: Pickup Window Diagram .......................................................................................................
32
Figure 13: Decisions Using the Must Ship By Date ..............................................................................
33
Figure 14: Shipm ent Planning Logic with Dynam ic Pickup W indow ...................................................
33
Figure 15: Distribution of Freight Requests within the Ship W indow ...................................................
35
Figure 16: Phase 2 M ust Ship By Date Diagram .....................................................................................
36
Figure 17: Shipm ent Request to VOFC ......................................................................................................
48
Figure 18: Truckload Cost Regression.....................................................................................................
49
Figure 19: Updated TL Cost Equation .....................................................................................................
49
Figure 20: LTL Cost Structure for Three FAKs .....................................................................................
52
Figure 21: Origin Cluster FC Cluster Sim plification ..............................................................................
53
Figure 22: VOFCC Combination ................................................................................................................
56
Figure 23: Delay Eligibility Evaluation Example ...................................................................................
58
Figure 24: Simulation M odel Results.....................................................................................................
61
Figure 25: Shipments Delayed and Cost Reduced ...................................................................................
62
Figure 26: Percentage of M STL and TL Requests Comparison ............................................................
63
Figure 27: Days of Shipm ent Comparison..............................................................................................
63
Figure 28: M inimum LTL Cost Sensitivity Analysis...............................................................................
64
Figure 29: M STL Pickup Charge Sensitivity Analysis..........................................................................
66
Figure 30: Delay Limit Sensitivity Analysis............................................................................................
67
12
List of Equations
Equation 1: Order Quantity for Periodic Review Inventory Policy with Deterministic Lead Time...........22
Equation 2: Order Quantity for Periodic Review Inventory Policy with Stochastic Lead Time .............
23
Equation 3: Longest Lead Time to Ensure Cycle Service Level .............................................................
23
Equation 4: Longest Vendor-Controlled Portion of Lead Time to Ensure Cycle Service Level............. 24
Equation 5: Phase 2 Must Ship By Date Formula....................................................................................
37
Equation 6: Percentage of Correctly Identified Lanes Shipped Multiple Times ....................................
39
Equation 7: Percentage of Auto-Eligible Lanes Not Shipped Multiple Times .......................................
39
Equation 8: Predictor Effectiveness Form ula...........................................................................................
40
Equation 9: Cost Regression for a Single Truckload Shipment..............................................................
49
E quation 10: F loor Spaces Form ula ............................................................................................................
50
Equation 11: Truck U tilization Form ula .....................................................................................................
50
Equation 12: N um ber of Trucks Form ula ...................................................................................................
50
Equation 13: Total Truckload Cost Form ula...........................................................................................
50
Equation 14: Cost Regression for Less Than Truckload Shipments.......................................................
51
13
1. Project Overview
1.1 Amazon Background
Created in 1994 by Jeff Bezos, Amazon has grown significantly each year as it strives "to be Earth's most
customer-centric company for four primary customer sets: consumers, sellers, enterprises, and content
creators."' In 2013, Amazon reached total revenues of $74.5B, growing nearly 22% over 2012 revenues 2.
Once an online bookstore, Amazon has continued to expand its selection each year, now offering a variety
of products from electronics to large sporting goods to apparel. This increase in overall volume and
variety of products results in a more complex supply chain that must be managed efficiently to provide
great customer experience and remain profitable.
1.2 Inbound Transportation Organization
The inbound transportation organization is responsible for managing the freight from the time it leaves
the vendor's outbound dock to the time it arrives within an Amazon Fulfillment Center (FC). As Amazon
continues to grow, more products are flowing through the inbound logistic network from an increased
number of vendors located all over the world to an increased number of FCs. While this has generated
many challenges for the organization, it has also created opportunities to reduce inbound transportation
cost and improve supply chain performance.
1.3 Purpose and Motivation of Project
When arriving at Amazon headquarters in February 2013 for a six-month internship, I learned how freight
decisions were made. The organization utilizes an in-house developed shipment-planning program that
runs daily to select the best transportation method - ship mode and carrier - for every shipment request
coming from domestic vendors. The program ensures that all freight in the network ready for pickup
today is transported to the desired fulfillment centers for the lowest total transportation cost.
I (Amazon.com)
2 Source: Amazon 2013 Income
Statement
14
While the planning systems perform well, the inbound transportation organization felt that decisions
could be improved by incorporating transit time of the carriers and when the products in the shipment are
needed in the fulfillment centers to remain in stock. By optimizing for cost and time in transportation
decisions, Amazon would have the opportunity to reduce transportation costs while also improving
inbound supply chain performance.
1.4 Project Goals and Approach
The goal of the project was to develop an innovative solution to incorporate time into the transportation
decisions for domestic shipments and then quantify its effects on the inbound supply chain. While
Amazon also sources products internationally, the project scope was focused on shipments transported
within the United States. I spent the first month of the internship understanding the relationship between
inventory planning and inbound transportation as well as diving deep into Amazon's data. After
interviewing several stakeholders and researching freight consolidation methods, pickup windows, and
total landed cost models, a solution was drafted. From there, I created a model to simulate the shipmentplanning process with the new solution and then analyzed the results to ultimately make recommendations
to Amazon leadership.
1.5 Thesis Overview
This thesis begins with a background on inbound transportation ship modes in terms of cost and transit
time, inventory planning with probabilistic demand and lead time, and freight consolidation methods in
Chapter 2. This is followed by a more in-depth discussion of Amazon's current state transportation
decisions and the constraints on the shipment-planning program. Chapter 4 describes the dynamic pickup
window solution that would allow Amazon to optimize the inbound network for the lowest cost while
ensuring on-time product arrival in the FCs. Chapter 5 steps through the development of the simulation
model while Chapter 6 provides results and sensitivity analysis of the dynamic pickup window solution
on an Amazon data set. The thesis ends with recommendations for Amazon and general conclusions for
other companies in Chapter 7.
15
2. Literature Review
This chapter begins with a review of transportation ship modes in terms of cost and transit time, which
provides the necessary background for understanding the dynamic pickup window solution and the
simulation model described in subsequent chapters. Next, we will review literature on common inventory
planning practices with probabilistic demand and lead time as well as freight consolidation policies
commonly used in industry.
2.1 Inbound Transportation Overview
Inbound Transportation is a critical component of a company's supply chain that must be managed
efficiently to achieve business and operations objectives. For large retail companies like Amazon with a
large quantity of vendors located throughout the country shipping products of all shapes and sizes to
multiple destinations, it is important to understand the advantages and disadvantages of different shipping
options. In this section, three ship modes will be discussed: Full Truckload (TL), Less-Than-Truckload
(LTL), and Small Parcel (SP). Other ship modes that companies use like air, water, and rail are important
for transportation organizations, but are not critical for understanding the solution and model presented in
this thesis.
2.1.1
Full Truckload
By tendering freight to a full truckload carrier, a company is paying for the entire volume of the truck to
move directly from an origin to a destination. The most common size TL trailer is 53 feet long and can
haul up to 40,000 pounds or 26 un-stacked standard sized pallets. The cost of the truckload is primarily
dependent on the distance traveled, independent of the shipment's weight and volume. Specifically, TL
costs consist of a minimum charge, a base cost dependent on the origin-destination pair, and a fuel
surcharge that follows gas prices in the US. Using a truckload is the best ship mode economically if the
company has enough freight to fill the truck.
16
In terms of transit time, TL carriers can move freight around 500 miles per day using a single driver. With
the Il-hour driving limit per day set by The Federal Motor Carrier Safety Administration', the total
distance per day assumes that trucks travel 45 miles per hour on average. TL carriers also have an option
to expedite shipping by assigning a team of drivers at a premium price. A team of two drivers increases
the daily driving limit to 22 hours, allowing freight to move 1000 miles each day.
2.1.2
Less-Than-Truckload
The less-than-truckload ship mode can be best described as the truckload ship mode where a company is
only charged for the space on the trailer that its freight occupies. For instance, if a company needed to
send one pallet, it would be economical to use LTL over TL because the company is charged a small
fraction of the full trailer. LTL carriers collect freight from many companies in the same origin area and
then travel to a nearby terminal. At the tenninal, freight is sorted and consolidated toward subsequent
terminals in the LTL carrier's network until it reaches the destination. The goal of the LTL carrier is to
build fully utilized trailers that can travel as far as possible toward the destination before being sorted
again at the next terminal.
Because the LTL carrier is consolidating many companies' shipments of different shapes and sizes to fill
trailers, it is naturally a difficult challenge to decide what price each shipper pays for its freight. Does it
make sense to charge each company the same amount? What about charging each company based on the
total weight or volume of its shipment? Would it make sense to charge based on how many pallet spaces
each company occupied?
The cost of LTL depends on five primary factors: minimum charge, distance, weight, fuel surcharge, and
freight class.
3 (Federal Motor Carrier Safety Administration, 2013)
17
*
Minimum Charge - This is the lowest cost a shipper must pay to move freight with the LTL
carrier (usually between $50 and $100).
*
Distance - The LTL cost generally increases as distance increases, however this is not always the
case. Each origin-destination pair, commonly referred to as a lane, is priced independently based
not only on distance but also on supply and demand of freight on that lane. This means that two
origin-destination pairs in the US with the same distance will not likely have the same LTL cost.
"
Weight - For any given origin-destination pair, LTL carriers set base rates as cost per hundred
weight (CWT), which simply means cost per one hundred pounds. While the LTL cost is
positively correlated with weight, carriers provide discounts at different weight thresholds.
Standard cost breaks occur as a shipment's weight crosses 500, 1000, 2000, 5000, 10000, and
20000 pounds. Figure 1 illustrates how the LTL base rate decreases once the shipment crosses
the 5001b, I0001b, and 20001b thresholds.
0
500
1000
2000
Weight (Ibs)
4
Figure 1: LTL Cost Weight Breaks
(Ozkaya, Keskinocak, Joseph, & Weight, 2009)
18
Based on Figure I, you can see where the LTL base cost of a 4991b shipment could be higher than
that of a 5001b shipment. In reality, this does not occur. The LTL carriers instead rate the
shipment at the higher weight threshold if that cost is lower. This is illustrated in Figure 2.
0
0
500
1000
2000
Weight (lbs)
Figure 2: LTL Cost Weight Breaks Actual5
"
Fuel Surcharge - The fuel surcharge is directly proportional to the LTL base cost and adjusts for
fuel prices in the US. LTL carriers commonly use the Department of Energy's (DOE) National
Average Diesel Fuel Price report to obtain the standard fuel prices6 and then set fuel surcharges as
a percentage of the base cost or total cost corresponding to that fuel price.
"
Freight Class - To account for the different types of products that are shipped and adjust cost
accordingly, the Commodity Classification Standards Board (CCSB) developed the National
Motor Freight ClassificationTM (NMFC@). Freight is assigned to one of 18 freight classes based
on the products' density, stow-ability, handling, and liability. Simply put, lower density items fall
into higher freight classes resulting in a higher LTL cost. Furthermore, items that are difficult to
stow, difficult to handle, or are a liability are moved to a higher freight class. Figure 3 shows the
density guidelines for all 18 freight classes.
5 (Ozkaya,
6 (U.S.
Keskinocak, Joseph, & Weight, 2009)
Energy Information Administration)
19
COMMODITY CIASSiFICAIION
STANDARDS BOARD
DENSITY GWSDWNEBS
Minimum Average Denuly
C
(n poundk or cubic foo)
50
50
35
30
22.5
55
60
65
15
13.5
12
10.5
70
77.5
85
92.5
100
110
9
8
3
125
150
175
200
250
2
1
300
400
7
6
5
4
Less than 1
500
Figure 3: Freight Class Density Guidelines 7
FAK (Freight All Kinds) - In industry, it is popular for companies when negotiating contracts
with LTL carriers to establish FAKs (PLS Logistics Services, 2013). An FAK is a grouping of
freight classes that will all be charged under a single class. For example, a company may
negotiate that all items with freight classes of 70 through 110 are charged under the 92.5 freight
class. Instead of having 18 different freight classes, a company can simplify the LTL cost
structure by having a few FAKs. The number of FAKs and how the FAKs are set are both
determined by a negotiation between the company and the carrier.
From a transit time perspective, LTL shipments typically take longer than TL shipments because freight
sits at each terminal for some time before moving to the next one.
2.1.3
Small Parcel
Small Parcel carriers like UPS, FedEx, and the USPS are well known to the general public because these
companies handle small packages that are more common in non-commercial shipping. For companies like
Amazon, the Small Parcel ship mode is ideal for shipments under 150 pounds. The network design for
7 Figure Source: (National Motor Freight Traffic Association, 2013)
20
Small Parcel is similar to LTL in that carriers utilize terminals throughout the country to consolidate and
sort freight. One difference is that Small Parcel carriers typically have terminals in more regions of the
country than LTL carriers to move freight more efficiently to less populated areas. Small Parcel terminals
also possess automated equipment optimized to sort small packages quickly so that time spent in these
terminals is minimal.
2.2 Inventory Policies with Probabilistic Demand and Lead Time
For retail companies with fixed capacity, a large variety of products, and probabilistic demand, inventory
planning is critical to profitability and high customer service. On one end, retailers want to ensure that
they have a product in stock whenever a customer wants it. If the product is out of stock, the customer
will likely purchase from another company, resulting in a lost sale for the retailer. On the other end,
retailers need to keep inventory levels to a minimum to maintain competitive prices while remaining in
business.
Inventory policies have become commonplace in industry to detennine how much to order and how often
to meet a desired service level. This desired service level is often determined by balancing the costs of
overstocking and under-stocking an item, or simply set by management as a standard for the company.
There are two common metrics for "service level": cycle service level and itemfill rate. The cycle service
level metric represents the probability that an item is in stock during the cycle, which is the complement
of the probability that the item is out of stock. Item fill rate measures what percentage of demand was
filled from the on-hand inventory, the complement of the percent of items backordered.
The two most commonly used inventory policies are continuous review and periodic review. In the
continuous review policy, an order is placed for a certain quantity (Q) whenever the inventory position
drops below an amount (R). In a periodic review, an order is placed each period to bring the inventory
position up to a certain level (S). The inventory position is defined as the sum of the inventory on hand
and the inventory already on order in both policies.
21
For the remainder of this section, we will focus on the periodic review inventory policy where
management sets the cycle service level. To illustrate how lead time and lead time variation affect the
order-up-to-level (S), let's look at a simple example first with deterministic lead time and then again with
probabilistic lead time.
Example:
Demand for a single stock keeping unit (SKU) is normally distributed with an expected demand of 1000
units per week and a standard deviation of 250 units per week. The total lead time from order placement
to arrival in the desired warehouse is 3 weeks exactly. Currently, the inventory position of this SKU is
3800 units. Management has standardized a desired cycle service level of 95% (k=1.645) and a review
period of I week.
To determine how much to order, we first define the following variables:
D - Expected Demand
- Standard Deviation of Demand
L - Expected Leadtime
UD
k - based on Cycle Service Level, using Standard Normal Distribution
I - Review Period
IP - Inventory Position
XDLI
-Expected
UDLI -
Demand over Leadtime and Review Period
Standard Deviation of Demand over Leadtime and Review Period
S - Order-up-to Level
0 - Order Quantity
The order quantity (0) can be calculated using the following equations:
XDLI
=D*(L+I)
GDLIJ
D
S =xDLI +k*
L+I
(YDLI
o=S -IP
O=D(L+I)+kaL+I-IP
Equation 1: Order Quantity for Periodic Review Inventory Policy with Deterministic Lead Time
22
In our example, the order-up-to-level is 4822 units meaning the order quantity is 1022 units. Let's repeat
the same example but change the lead time from a deterministic value of 3 weeks to a probabilistic value
normally distributed with an expected value of 3 weeks and a standard deviation of 0.5 weeks. To account
for the variability in lead time, we introduce a variable for the standard deviation of lead time and the
Hadley-Whitin equation (Hadley & Whitin, 1963).
U1 - Standard Deviation of Lead Time
X DL
=D *(L + I)
2*D
GrDLI
where
orL+2 = UL2
D'
2
when I is kept constant
S =xDI,+ k *UDLI
o = S - IP
This set of equations gives us a single equation for the order quantity.
O=D(L+I)+kj(L+1)Y2
+D2
-L+2_IP
Equation 2: Order Quantity for Periodic Review Inventory Policy with Stochastic Lead Time
In this example the order-up-to-level is now 5163 units and the order quantity is 1363 units, an increase of
341 units caused by the lead time variability. There are two key takeaways from this example.
(1)
Shorter lead times and less lead time variation result in smaller order quantities and less inventory
holding costs.
(2) Based on the quantity that was ordered, we can calculate a date that the material must arrive to
the destination to maintain our cycle service level.
Now that 1363 units have been ordered, we will calculate the longest lead time that can occur to maintain
the 95% service level. To do so, we start with Equation 2 setting the order quantity equal to 1363 and then
solve for lead time. Appendix A includes a detailed derivation of Equation 3.
L L=S +-kaDk
D2
D
2D
D
-4DS+(kJD))-I
Equation 3: Longest Lead Time to Ensure Cycle Service Level
23
Equation 3 tells us that if the product arrives to the destination in 3.31 weeks (23 days), we can still
maintain our 95% service level. Knowing this date is valuable for an inbound transportation organization
because it provides a key target to ensure that the freight arrives. However, it may be more important for
inbound transportation to know when a product should ship to maintain service levels.
In our previous example, we defined lead time from the placement of the order until arrival in the
warehouse. Let's update that definition to make a clear distinction between the time at the vendor and the
time in transit.
LV
LT
Ship
Order
Arrive
Figure 4: Lead Time Segmentation
Figure 4 shows that the overall lead time is simply the sum of the two lead time components. Assuming
the components are independent, the lead time standard deviation is:
L
Lt,+O(TLT
Going back to our example, let's assume that the expected lead time at the vendor is 2 weeks and the
expected transit time is I week. Furthermore, the standard deviation of lead time at the vendor is 0.4
weeks and the standard deviation of transit time is 0.3 weeks, which gives us the same overall lead time
standard deviation of 0.5 weeks.
Since the expected value and standard deviation of the overall lead time are the same, we still order 1363
units and we still need the shipment to arrive in the warehouse no later than 3.31 weeks to maintain the
service level. The latest date for the freight to ship and still maintain the service level is calculated using
Equation 4. A detailed derivation of this formula is provided in Appendix B.
L =
D
+ k 2 kuD2
D2(4DS+(kuD
2)
+4D
-2
_I-L
dTmEsCe
dP
D4:
g
2
Equation 4: Longest Vendor-Con trolled Portion of Lead Time to Ensure Cycle Service Level
24
Applying this equation tells us that we need to ship the products no later than 2.19 weeks (15 days) from
the time we order to ensure the 95% service level.
In this section, we have walked through how lead time and lead time variability affect inventory planning
in a periodic review policy with a management defined cycle service level. Beyond that, we have seen
that once the order has been placed, inbound transportation can look at the latest possible ship date and
latest possible arrival date to still achieve the desired service level. These concepts are the basic building
blocks of the dynamic pickup window solution explained in Chapter 4.
2.3 Freight Consolidation
In this section, we will briefly discuss two types of consolidation in the inbound network, spatial and
temporal (Ford Jr., 2006).
2.3.1
Spatial Consolidation
Spatial consolidation is explained by the use of company-owned or third party hubs where freight from
several vendors is shipped a short distance with Small Parcel or LTL carriers to the hub and then
consolidated into TL shipments as illustrated in Figure 5. The benefits are largely explained through
economies of scale in shipped weight as we saw in Section 2.1 with the LTL cost breaks and eventually
moving into the TL ship mode. The first challenge for companies using this consolidation method is
whether to own and manage its hubs or use a third party. If the company chooses to operate the hubs, the
second challenge is where to locate the hubs in the network to minimize cost and maintain high
performance. Olufemi Oti describes solutions for these challenges in Hub and Spoke Network Designfor
the Inbound Supply Chain (Oti, 2013).
25
Reduce distance
traveled by higher
cost ship-modes
>
>
High Cost Ship-modes (LTL, SP)
Low Cost Ship-modes (TL)
Figure 5: Hub Consolidation Diagram8
Another option in spatial consolidation is using Multi-stop Truckloads (MSTL) to pick up freight from
several vendors before traveling to the destination. Figure 6 illustrates that instead of sending costly LTL
shipments from three vendors into the same fulfillment center, MSTL allows the company to move the
freight with more affordable TL rates.
TL
LT
Figure 6: Multi-stop Truckload Consolidation Diagram
From a cost and time perspective, MSTL is essentially the same as a TL shipment with two key
differences. TL carriers typically have a fixed cost for each pickup and drop off on the route, and transit
time is slightly longer due to the time spent at each vendor's dock and the travel time between vendors.
While using MSTL has large benefits for an inbound supply chain, it requires precise coordination with
vendors and sophisticated route planning to find opportunities in the network. Commercial software for
transportation route planning like TMW Systems Appian and IBM Sterling Transportation Management
System (TMS) utilize combinatorial optimization algorithms, most notably the Vehicle Routing Problem
and its variations (Society for Industrial and Applied Mathematics, 2002) to build optimal routes.
8 Figure
Source: (Oti, 2013)
26
2.3.2
Temporal Consolidation
Temporal consolidation occurs when freight is intentionally not shipped today in order to consolidate
with freight ready to ship tomorrow. By waiting an additional day, the inbound transportation
organization can again take advantage of the LTL cost breaks or obtain enough weight to warrant a TL,
and have fewer shipments arriving to the destination.
Figure 7 illustrates temporal consolidation from a single vendor to a single fulfillment center. On
Monday, the vendor had 1000lbs of freight to ship. If the vendor had 1500lbs to ship on Tuesday, the
freight could be consolidated into a single 25001b shipment at a reduced cost.
Monday - 1000lbs
LTL
LTL
V
F
Tuesday - 2500lbs
Tuesday - 1500lbs
Figure 7: Temporal Consolidation - Single Vendor to Single FC
We can apply both temporal and spatial consolidation methods to create more opportunities for building
MSTLs. Instead of a single vendor, Figure 8 shows three vendors that are shipping to the same
destination. However, each vendor ships on a different day of the week and therefore a MSTL route
cannot be built. By delaying pickup on the first two vendors to Wednesday, a MSTL is built.
Monday
All
I
I Wednesday
LTL
Tuesday
Wednesday
Figure 8: Temporal and Spatial Consolidation to Build More MSTL
27
TL
While both consolidation examples look great in theory, it is very challenging to know when to delay
picking up freight. On one hand, a consolidation may occur resulting in lower transportation cost. On the
other, a consolidation may not occur; or worse, delaying pickup may cause a product to go out-of-stock.
In reality, the retailer may not know that a vendor is ready to ship on a Tuesday and that other vendors
nearby are ready to ship on Wednesday.
In this chapter, we have laid the foundation for the dynamic pickup window solution and the case study of
Amazon North American inbound transportation. We have discussed three primary ship modes
commonly used in inbound transportation in terms of cost and transit time. In Section 2.2, the connection
between inventory planning and inbound transportation was conveyed through simple examples with an
emphasis on how inbound transportation can determine a latest possible arrival date and latest possible
ship date to still ensure a desired service level. Lastly, we discussed spatial and temporal consolidation
using examples of single-vendor consolidation and multi-vendor consolidation via Multi-stop Truckloads.
The questions that need to be addressed to take advantage of consolidation opportunities are:
(1)
How do you ensure that delaying a decision does not cause a product to go out of stock?
(2) How do you know if delaying a decision will result in a consolidation or not?
28
3. Amazon Inbound Transportation Current State
This chapter familiarizes the reader with the network geography of Amazon's North American inbound
transportation operation and provides a picture of current state practices as I began the internship.
Amazon currently operates around 40 FCs in the United States, with FCs in the 15 states shown in Figure
9.9 Just as traditional retailers place stores close to customer demand, FCs are generally located in areas
where they can reach major markets quickly and obtain a low outbound cost structure. Due to Amazon's
large variety of products, the company manages several vendors in the inbound supply chain dispersed
throughout the country. These two sets of nodes create a complex and challenging transportation network.
Without knowing how many vendors supply Amazon, you can see that it would only take 250 vendor
locations to have 10,000 shipping lanes in the network.
4'H
Figure 9: Amazon Fulfillment Center Locations"
The Amazon inbound transportation organization distinguishes the inbound freight into two
classifications: freight that is managed by Amazon and freight that is managed by the vendor. For vendormanaged freight, the vendor is responsible for working with the carriers to ensure its freight arrives to
FCs on time. The rest of this thesis will only discuss freight that is managed by Amazon, which is
approximately 60% of the network. Furthermore, we will only consider freight from vendors located in
the US, and will ignore ocean and air ship modes.
9 (Dunn, 2014)
0 Figure Source: (Amazon.com)
29
3.1 Pickup Window
To investigate the inbound process, let's start when Amazon issues a purchase order. Every purchase
order sent to a vendor includes a ship window, specifying the range of dates where it is acceptable to ship.
The ship window is bounded by two dates - the earliest vendor ship date and the latest vendor ship date as depicted in Figure 10.
SHIP WINDOW
Earliest Vendor
Ship Date
Latest Vendor
Ship Date
Figure 10: Ship Window Diagram
Tying this back to Section 2.2, Amazon can determine the latest vendor ship date using an equation
similar to Equation 4, depending on how it plans inventory in a probabilistic demand and lead time
environment. While the vendor sees these dates, they are not currently used to guide transportation
decisions.
3.2 Shipment Planning System
As a vendor is close to having the products ready for shipment, it sends an electronic request to Amazon
for transportation, specifying the date when the freight is ready for pickup, the destination FC, and all
freight-specific information needed to deternine transportation cost. Every day, a shipment-planning
program considers all requests where the freight is ready tomorrow and assigns contracted carriers to each
lane to minimize the network transportation cost.
The shipment-planning program evaluates whether it is best to ship using a Small Parcel carrier, a Lessthan-Truckload carrier, a Truckload carrier, a spatial consolidation terminal, or a Multi-stop Truckload
route. Once the decisions are made for all requests, Amazon sends electronic notifications to both the
vendor and the carrier providing them the information needed for the scheduled pickup. This process is
illustrated in Figure 11.
30
Qiaaon
VENDOR
Vendor prepares
products for
Shipment-Planning System
shipment
Pool of shipment requests
from many vendors
Each day, system chooses
carriers to minimize
transportation cost
Figure 11: Inbound Transportation Process
The current state limitation of the shipment-planning program is that each shipment request is always
evaluated and assigned the day before the vendor's freight ready date. In other words, temporal
consolidation never occurs because the shipment-planning program does not allow it. This is a
conservative practice because the shipment-planning program lacks visibility to when shipments are
needed in the FCs and the transit time performance for each carrier on each lane in the network.
Therefore, one would not want to delay a shipment blindly and risk causing an out-of-stock occurrence.
While the shipment-planning program lacks the visibility, Amazon has all of this data within the company
to utilize if a solution was developed to allow temporal consolidation while minimizing the risk of a
stock-out-occurrence and ensuring high probability that if a shipment is delayed, it is consolidated to
benefit the network. As we discussed in 2.3.2, temporal consolidation for Amazon would allow the
shipment-planning program to achieve same-vendor consolidation and build more multi-stop truckload
routes.
31
4. The Dynamic Pickup Window Solution
This chapter defines and describes the solution provided to Amazon for optimizing its inbound
transportation decisions for cost and time in a way that leveraged its existing systems and architecture so
it could be implemented swiftly. We will begin describing the solution conceptually through a discussion
of the Must Ship By Date in Section 4.1. Section 4.2 presents the details of how Amazon could populate
this date, including a short-term and long-term solution. Section 4.3 and 4.4 end the chapter through an
explanation of constraints on the solution and logic created to maximize the probability that delaying a
decision results in a consolidation.
4.1 The Must Ship By Date Concept
The dynamic pickup window solution fundamentally changes the way that the Amazon shipmentplanning program moves freight in the network by giving it the option to delay pickup on a shipment
request. As we have discussed, delaying the decision allows for temporal consolidation opportunities,
such as same-vendor consolidations and multi-stop truckload routes. The pickup window, not to be
confused with the shipping window from Section 3.1, starts at the vendor's specified Freight Ready Date
(FRD) and ends at the Must Ship By Date (MSBD) as illustrated in Figure 12.
Freight
Must Ship
Ready Date
By Date
Figure 12: Pickup Window Diagram
The idea is that the Must Ship By Date is the latest possible date that Amazon can pick up freight from a
vendor and still ensure that products arrive to the fulfillment center before an out-of-stock occurrence.
Since the situation for each shipment will be unique and dependent on many factors, the pickup window
will be dynamic. This contrasts the solution with static pickup windows offered with commonly used
transportation routing software, where a user can manually select how long shipments are allowed to pool
before they must be tendered.
32
By planning shipments around the MSBD, Amazon can know when to delay a transportation decision and
consolidate freight, when freight should move immediately with the lowest cost carrier, and when freight
may need to be expedited. Figure 13 conveys that Amazon has the opportunity to delay a shipment
request if the MSBD is in the future. Likewise, if the MSBD has passed, Amazon may need to choose a
faster mode of transportation at an additional cost.
FRD
Jan 31
MSBD
Feb 3
FRD & MSBD
Jan 31
MSBD
Jan 28
FRD
Jan 31
Figure 13: Decisions Using the Must Ship By Date
In the future, using the dynamic pickup window solution, all shipment requests with a freight ready date
of tomorrow would first be evaluated just as they are today. However, instead of being tendered today for
pickup tomorrow, all requests would flow through the logical structure in Figure 14.
YES
NO
Must Ship By
Date Logic
NO
Probability of
Consolidation wo
NO
Logkc
F r4hYES
YES
Figure 14: Shipment Planning Logic with Dynamic Pickup Window
33
If the freight is assigned to a multi-stop truckload shipment, the freight is tendered today because this is
an efficient ship mode. Remember that anything tendered today will ship tomorrow. If the freight is
assigned to a truckload shipment and the truck is over a certain percent utilization, this freight is also
tendered today. Lastly, any freight assigned to a consolidation terminal should be shipped today because
the efficiency of these terminals depends on the volume coming in so highly utilized truckload shipments
can be built for the second leg.
If the freight is assigned to a less-than-truckload or small parcel carrier, then the shipment request's Must
Ship By Date is now evaluated. If the MSBD has already occurred, the freight is tendered today and may
require expediting. After the MSDB logic, the freight goes through the three decision blocks titled
"Probability of Consolidation Logic". This logic was created so that shipment requests are only delayed if
that freight has a high probability of consolidating. If the freight has a low probability of consolidating,
Amazon does not have an incentive to delay and therefore it should be tendered today. A detailed
discussion of this logic is described in Section 4.3.
Using the Must Ship By Date as a target ensures that if a consolidation opportunity does not present itself,
Amazon can still use the lowest cost carrier today and ensure that the shipment will arrive on time.
However, if an opportunity does present itself, the shipment will likely arrive earlier to the FC than
current state because TL and MSTL ship modes have shorter transit times than LTL.
4.2 Populating the Must Ship By Date
To populate the Must Ship By Date, a two-phased solution was presented to Amazon. Phase 1 had the
potential to be implemented more quickly because it used data that was already established and easily
accessible within the company. Phase 2 was an improvement over Phase 1, but required additional system
requirements before implementation would be feasible.
34
4.2.1
Phase 1 - Latest Vendor Ship Date
In Phase 1, the Must Ship By Date of the dynamic pickup window is set equal to the Latest Vendor Ship
Date (LVSD) of the shipping window described in Section 3.1. The idea is that the freight will arrive to
the FC in time to stay in stock at the desired cycle service level as long as the vendor ships on or before
the Latest Vendor Ship Date. This is because the date was assigned to each purchase order based on how
Amazon plans inventory, accounting for demand and lead time variability as discussed in Section 2.2. For
each shipment request, the MSBD is set equal to the minimum LVSD of its contained purchase orders to
ensure that all purchase orders arrive on time. This is a conservative approach to protect all products from
a stock out occurrence.
An analysis of ship window compliance was conducted to understand what percentage of shipment
requests were shipped before the LVSD. Figure 15 shows the distribution of this study and highlights that
approximately 80% of the requests were shipped before the LVSD. This means that 80% of the volume
has the opportunity to wait at least one day before it must ship.
20.0%
17.0%
18.0%
18.0%
-15.2%
16.0%
14.0%
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
EARLY
Earliest 2nd Day 3rd Day
Vendor
Ship Date
4th Day
5th Day
6th Day
7th Day
Latest
Vendor
Ship Date
Figure 15: Distribution of Freight Requests within the Ship Window
35
LATE
4.2.2
Phase 2 - Projected Stock Out Date
While Phase 1 is an improvement over current state for optimizing cost and time, the LVSD is set at the
time of planning inventory and placing the purchase order. For long lead time SKUs in particular, this
means that a lot of demand variability can occur from the time the purchase order is sent to when the
freight is ready for pickup. While Phase 1 addresses the lead time variability, it assumes that the demand
distribution is the same as when inventory was planned. Phase 2 is an improvement because it addresses
both lead time and demand variability by looking at the latest forecast of demand, inventory in transit, and
inventory levels within the FC to detennine a ProjectedStock Out Date (PSOD) for each SKU in each FC
for a specified cycle service level. This PSOD would be updated frequently throughout the day to keep up
with changes in the Amazon network. The Phase 2 recommendation starts with using the PSODs of all
SKUs on a shipment request and then working backwards to calculate the Must Ship By Date.
Transit time of the lowest cost carrier at the
X ' percentile
PSOD
MSBD
Figure 16: Phase 2 Must Ship By Date Diagram
Figure 16 shows that the MSBD is simply the transit time of the lowest cost carrier at the Xh percentile
subtracted from the PSOD. Transit time for every shipment already exists within Amazon's inbound
transportation management systems by origin-destination pair and by carrier. Phase 2 requires pulling the
mean and standard deviation" of transit time distributions into the shipment-planning system as each
shipment request is evaluated. It is important that the transit time of the lowest cost carrier is used so that
if the decision is delayed but a consolidation opportunity does not occur, the freight can be shipped with
the lowest cost carrier tomorrow and still arrive at the FC before the PSOD. To calculate the MSBD
accounting for the transit time variability, management should set the Xth percentile shown in Figure 12
equal to its desired cycle service level (e.g. 95%). Since each SKU is assigned a PSOD within each FC,
I Analysis shows that transit time can be described using the normal distribution
36
Amazon systems need to aggregate all SKUs' PSODs into a shipment request PSOD. Similar to Phase 1,
the shipment request PSOD should be set to the earliest date of all SKU's included. To clarify how the
MSBD is calculated in Phase 2, let's go through a quick example.
Example:
Amazon's shipment planning system is currently evaluating a shipment request that includes five SKUs
for a fulfillment center in Kentucky. Based on FC inventory levels, up-to-date demand forecasts, and
inventory in transit, PSODs are calculated for each SKU.
Table 1: SKU-Level PSODs
SKU-LEVEL PSOD
SKU
2/10/14
1
2
3
2/13/14
2/24/14
4
5
2/19/14
2/11/14
Table 1 shows that SKU 1 has the minimum PSOD of February
1 0 1h
2014. Therefore, we assign this
PSOD to the entire shipment request and use that date to calculate the MSBD. For this example, let's
assume the transit time of the lowest cost carrier has a mean and standard deviation of 5 days and I day
respectively. At a 95% service level, the MSBD is calculated to be February 3P, 2014 by the following
equation:
MSBD = PSOD - (LT + kUL)
MSBD = 2 /10 /14 - (5+1.645 *1)
MSBD = 2 /10 /14 -6.645
days
MSBD = 2 / 3 /14
Equation 5: Phase 2 Must Ship By Date Formula
The remaining sections in Chapter 4 apply to both Phase 1 and Phase 2.
4.3 Probability of Consolidation Logic
As mentioned in Section 4.1, all shipment requests that have the opportunity for delay per the Must Ship
By Date logic are evaluated for a high probability of consolidation. The idea is that Amazon should not
increase the lead time of a shipment unless there is a good chance it will consolidate with other freight if
37
delayed. The dynamic pickup window solution uses three decision blocks to determine a high probability
of consolidation qualitatively.
4.3.1
Auto-Eligible Transportation Lanes
The primary purpose of the auto-eligible decision block is to capture the same-vendor consolidation
opportunities depicted in Figure 7 of Section 2.3.2. If the vendor is sending freight to the same FC
multiple times in a given week, any freight request with that Vendor-FC transportation lane has a high
likelihood of consolidating with another shipment request on the same lane. Therefore, Amazon should
automatically not tender this freight today and instead evaluate it with tomorrow's pool of requests.
To determine the best method for deciding which Vendor-FC lanes would become auto-eligible, an
analysis was conducted looking at all Amazon shipments, excluding Small Parcel, for Week 14 through
Week 21. The recommended method would best predict the Vendor-FC lanes where freight was shipped
multiple times in Week 22.
Four cases were tested based on whether the Vendor-FC lane was shipped multiple times in a number of
weeks during the eight-week period. Table 2 shows the four test cases and the percentage of Vendor-FC
lanes that were classified auto-eligible. As conveyed, 38% of Vendor-FC lanes shipped multiple times per
week in at least one week out of the eight-week period. Similarly, 16% of Vendor-FC lanes shipped
multiple times per week in at least two weeks out of the eight-week period. If some of these shipments
were delayed, same-vendor temporal consolidation may have occurred.
Table 2: Percenta e of Lanes that Shi
At
At
At
At
least
least
least
least
I
2
3
4
out of 8 weeks
out of 8 weeks
out of 8 weeks
out of 8 weeks
ed Multi le Times in a Week
38%
16%
8
5%
In Week 22, 12% of Vendor-FC lanes were shipped multiple times. Table 3 shows how each test case
performed on predicting the correct Vendor-FC lanes that shipped multiple times. In the table, two values
are presented and ultimately used to determine the overall effectiveness of the predictor. The first value
38
represents the predictor's ability to correctly identify those Vendor-FC lanes shipped multiple times in
Week 22. The second value penalizes the predictor for declaring too many lanes auto-eligible that do not
ship multiple times in Week 22. In other words, a Vendor-FC lane is predicted to have shipments on
multiple days in Week 22 but in fact only ships one day or not at all. The perfect predictor would
maximize the first value and minimize the second. In Table 3, the two values are calculated using the
following notation and formulas:
M22-
Vendor-FC lanes shipped multiple times in Week 22
AE - Vendor-FC lanes declared auto-eligible from Week 14 through 21 for test case i
Correcti - Auto-eligible Vendor-FC lanes that shipped multiple times in Week 22
FalseAEj - Auto-eligible Vendor FC lanes that did not ship multiple times in Week 22
Correct,= MA2 c AE
%Correct~=
# of Correct.
#'f
#of M22
Equation 6: Percentage of Correctly Identified Lanes Shipped Multiple Times
%FalseAE
=
#of AE, - #of Correct.
#of AE,
Equation 7: Percentage of Auto-Eligible Lanes Not Shipped Multiple Times
Table 3: Test Case Predictor Performance on Week 22
%of
Idetfe
Corcl
1/
of
I
Auoigi
At least 2 out of 8 weeks
50%
69%
At least 3 out of 8 weeks
34%
62%
At least 4 out of 8 weeks
25%
55%
ae
Table 3 shows that given the Vendor-FC lanes from Week 14 through Week 21, Test Case "At least 1 out
of 8 weeks" was able to correctly predict 72% of the Vendor-FC lanes that were shipped multiple times in
Week 22. However, 77% of lanes declared auto-eligible in this test case were in fact not shipped multiple
times. To select the best method, the PredictorEffectiveness metric was created where the highest
percentage conveys the best predictor. The predictor effectiveness is defined as:
39
.
%Correct.
'
Predictor Effectiveness. =
'%FalseAEi
Equation 8: Predictor Effectiveness Formula
Table 4 shows this metric for all test cases, showing that "At least I out of 8 weeks" is the best predictor.
Even though this predictor will allow the most Vendor-FC lanes to become auto-eligible, this larger data
set results in the most correct predictions, while it's false predictions are not much worse than the other
test cases.
Table 4: Auto-Eligible Predictor Effectiveness
Test
C'
'd
Effectivsn9s%
At least 1 out of 8 weeks
92%
At least 2 out of 8 weeks
72%
At least 3 out of 8 weeks
55%
At least 4 out of 8 weeks
45%
Each week, the set of auto-eligible Vendor-FC lanes should be updated based on the previous eight weeks
of data. This way, lanes will be added and removed from the auto-eligible set to keep up with the current
environment of the inbound network.
4.3.2
Multi-stop Candidates
If the shipment request is not deemed "auto-eligible", this means it has a low likelihood of resulting in a
same-vendor consolidation if delayed. Therefore, this next decision block tries to accurately predict
shipment requests that have a high likelihood of becoming a multi-stop truckload shipment if delayed.
The question "Is the vendor located in a highly populated vendor area?" determines the shipment requests
that are "Multi-stop Candidates".
Similar to the Vendor-FC auto-eligible analysis, an analysis was conducted to determine the best
predictor of whether or not a shipment request would become a MSTL shipment. All Amazon shipments
from Week 17 through Week 21 were studied, excluding Small Parcel, and then used to predict MSTL
shipments in Week 22.
40
The following five predicting attributes were analyzed to determine the best predictor:
*
5-digit Zip Code
*
3-digit Zip Code
*
*
*
Origin City
Origin City - FC Cluster Pair
Origin City - FC Cluster Pair that shipped over 12,000 pounds
The first three predicting attributes are based on the vendor's location alone, independent of the
destination FC on the shipment request. They simply state the vendor locations where MSTL shipments
are most likely to occur. The last two attributes, alternatively, attempt to identify the specific lanes for a
MSTL shipment. An FC Cluster, as seen in the last two predicting attributes, is defined as a combination
of Amazon fulfillment centers that are located near each other.
For each of the five attributes, two cases were tested based on whether the predicting attribute had a
MSTL shipment in at least one of the four weeks or at least two of the four weeks. Table 5 shows the
percentage of correct MSTL shipments captured, the percentage of false predictions where a MSTL was
not shipped, and the predictor effectiveness for each test case. All three metrics were calculated using the
same equations as the auto-eligible candidates in Section 4.3.1.
Table 5: Multi-stop Candidate Predictor Effectiveness
5-digit Zip Code -1 of 4
80o
9%sePedcinsPedco
84%
5-digit Zip Code -2 of 4
3-digit Zip Code - 1 of 4
3-digit Zip Code - 2 of 4
Origin City - 1 of 4
69%
91%
85%
94%
96%
96%
74%
94%
89%
82%
71%
96%
95%
86%
75%
Origin City/FC Cluster -1 of 4
81%
96%
84%
Origin Cluster/FC Cluster -2 of 4
72%
96%
75%
Origin Cluster/FC Cluster -
74%
96%
77%
Oriin City - 2 of 4
over 12,000lbs
Table 5 shows that the 3-digit zip code where a MSTL shipment occurred in at least one week of the fourweek period is the best predictor with a predictor effectiveness of 94%. Furthermore, this test case
predicted 91% of the MSTL shipments in Week 22. Therefore, the vendors that are located within these
3-digit zip codes are labeled "Multi-stop Candidates".
41
The idea is that a vendor located in a remote area of the country will have a significantly lower
probability of consolidating freight with other vendors than a vendor located in a highly populated vendor
area. This decision block ensures that vendors located in 3-digit zip codes that have not sent a MSTL
shipment anytime in the previous four weeks are tendered today and therefore not delayed. As
recommended for the auto-eligible dataset, this dataset is to be updated each week to reflect the current
environment of Amazon's inbound network.
4.3.3
Minimum Less-than-Truckload Cost
If a shipment request is not auto-eligible but is a multi-stop candidate, then the request will flow to the
third probability of consolidation decision block. Even if the vendor is located in a highly populated
vendor area, the specific shipment request may have a low likelihood or even an impossibility of
consolidating into a MSTL shipment.
In Section 2.3.1, it was stated that a multi-stop truckload shipment is essentially the same as a truckload
shipment from a cost perspective, except that the truckload carrier will charge the shipper a fixed cost for
each pickup and drop off point along the route. If the fixed cost for pickup from a vendor is greater than
the cost to ship with an LTL or SP carrier, the freight will never consolidate into a MSTL shipment.
Therefore, it makes sense to set the "Minimum Less-than-TruckloadCost" at least equal to the pickup
charge for a MSTL shipment. However, we may want to increase this cost threshold even more so that the
shipment-planning program does not delay a shipment if the opportunity for a MSTL consolidation would
only reduce the transportation cost by a small amount. We will analyze the effects of this threshold in
Chapter 6.
4.4 Maximum Delay Time
The dynamic pickup window solution includes one final constraint for each shipment request. Each
request can only be delayed at most two days in the shipment-planning system before it must be tendered
for pickup tomorrow. This constraint holds even if the Must Ship By Date is farther in the future. This
42
criterion was set to protect vendors from freight sitting on their outbound docks for an extended period of
time. The 2-day limit was recommended after research of practices in industry and discussions with
several vendors.
In this chapter, the dynamic pickup solution was presented with in-depth discussions of the Must Ship By
Date logic and the probability of consolidation logic. A Phase-I solution using the Latest Vendor Ship
Date of the ship window and a Phase-2 solution incorporating a Projected Stock Out Date was
recommended for the short-term and long-term respectively. Integrating the dynamic pickup window
solution into Amazon's current shipment-planning program will increase the opportunities for temporal
consolidation while ensuring that products arrive as needed in the FCs. The remainder of this thesis will
describe the model that was developed to simulate the solution with Amazon data, analyze the results
with comparisons to current state, and conclude with final thoughts and recommendations.
43
5. The Shipment Planning Simulation Model
To understand the effects of the dynamic pickup window solution on Amazon's inbound supply chain, a
shipment-planning simulation model was developed that closely resembles the company's actual
shipment-planning program. The model uses Visual Basic and a mixed integer linear program (MILP) to
simulate the process for an actual Amazon data set.
The objective of the model is to assign a ship mode to each shipment request that minimizes the total
transportation cost in the network. The model first calculates the transportation cost of the inbound
network as Amazon's shipment-planning program would have in that it tenders all freight with a freight
ready date of tomorrow. Then, the model runs the dynamic pickup window solution to delay freight as
described in Chapter 4 so that the effects on the network can be analyzed. In the simulation model, we
assume that freight requests can only be assigned to LTL, TL, or MSTL ship modes.
In Chapter 5, we will walk through the inputs to the model and the cost equations used for each ship
mode. Next, the multi-stop truckload optimization will be described including why it was modeled using
a MILP. The chapter will conclude with the modeling of the dynamic pickup window logic and a
description of the model outputs.
5.1 Model Inputs
Two types of inputs are important to running the simulation model: Input Data and User-Controlled
Inputs.
5.1.1
Input Data
The input data to the simulation model included all shipment requests for Week 23 from all US vendors
shipping to the US fulfillment centers. All shipment requests shipped through small parcel carriers or
consolidation terminals were omitted from the dataset. The primary source of the data was Amazon's
shipment-planning program as it keeps all data required to assign a ship mode to each shipment request.
44
However, data was also collected from other Amazon systems to set the Must Ship By Date for each
request based on the purchase orders contained within. Each input is listed and briefly described below.
"
Shipment Request Number - A unique identifier of the shipment request, allowing the model to
track how each request was routed.
*
Vendor Name - The name of the vendor.
*
Origin Zip Code - The 5-digit zip code describing where the freight originates. Since many
vendors have multiple facilities within the United States, the combination of the vendor name and
origin zip code is used to identify a single pickup location.
"
Fulfillment Center - The destination of the shipment.
*
Distance - The travel distance (in miles) was obtained using the PCMILER transportation
mapping and routing tool based on the origin zip code from the vendor and the destination zip
code that is linked to the specified FC.
*
Freight Ready Date - The earliest date the freight is ready to ship as specified by the vendor in
the Amazon systems.
*
MSBD - The latest possible date that Amazon can pick up freight from a vendor and still ensure
that products arrive to the fulfillment center before an out-of-stock occurrence. This date was
captured from another Amazon database, as it is not currently used in the shipment-planning
program. The Latest Vendor Ship Date in the ship window was used to set the Must Ship By
Date, meaning that Phase I is used for the model. Specifically, the MSBD was set equal to the
minimum LVSD of all purchase orders included in the shipment request.
*
Weight - The weight of the shipment in pounds, as input by the vendor.
"
Un-stacked Pallets - The number of un-stacked pallets used to prepare the products for shipping.
*
Stacked Pallets - The number of stacked pallets used to prepare the products for shipping.
.
Freight Class - A single freight class used to describe the shipment as described in Section 2.1.2.
45
An example of the input data, which we will use throughout this chapter, is shown in Table 612.
Table 6: Exam le of Data In
uts
6/4/2013
s~~ reusRanb
~ ~~hipment
ea
e DlayLimt -Thede
l
5
sCtoday
U
imi2 s-
yeiht
Tisutrll
dasCTi ue-ontrolled
ed.ste iNumSetor
ady
.,tercmmne
Deeatue thtspffew
h
mxmm
aiu
ea
iilta
n
e inta
inInpuwtss
aayeth
oe otusbycagn
the delay limit between zero days (Amazon's current state), one day, and two days.
*
MSBD Shift - This controlled input is also an integer value, which will shift the MSBD earlier or
later in day increments. The recommended solution from Chapter 4 would correspond to a MSBD
Shift of zero.
*
MSTL Pickup Cost - The cost of an additional pickup or drop off on a MSTL shipment.
*
Minimum LTL Cost - The minimum LTL cost that the model will allow a delay if the shipment
has not reached the MSBD, is not auto-eligible, but is a multi-stop candidate as explained in
Section 4.3.
12
Example data is purely for illustrative purposes and does not reflect Amazon's actual data.
46
Table 7 shows an example of the user-controlled inputs for the model.
Table 7: Example of User-Controlled Inputs
$200
5.2 Assigning Initial Transportation Cost
Once the user-controlled inputs are entered, the simulation model begins the process of assigning a
transportation cost to only the pool of shipment requests with a freight ready date of tomorrow. These
requests are then prepared for the cost equations by combining all requests with the same vendor name,
origin zip code, and FC. For this discussion, let's define a variable VOFC13 that equals this combination.
In our example, the VOFC is VENDOR91745PHX7. We define the VOFC, which represents a shipping
lane, to combine all requests so that the products are shipped together. The model will see multiple
shipment requests with the same VOFC on the day it is evaluating for two reasons:
(1)
The vendor sent multiple requests with the same origin, FC, and freight ready date. This may
occur if the vendor inputs shipment requests as products arrive to the outbound dock within its
facility.
(2) The dynamic pickup window solution delayed freight previously evaluated to today's pool of
shipments. Combining these shipment requests into a single VOFC creates the same-vendor
consolidation discussed in Section 2.3.2.
Since the model will calculate cost at the VOFC level, the last preparation step is to combine the shipment
attributes of the requests as follows:
*
VOFC Distance - Since the distance is the same for all requests within a VOFC, this is simply
the distance.
1
*
VOFC Weight - The sum of the weight from all requests.
*
VOFC Un-stacked Pallets - The sum of the un-stacked pallets from all requests
VOFC stands for Vendor, Origin, FC.
47
"
VOFC Stacked Pallets - The sum of the stacked pallets from all requests
"
VOFC Freight Class - The maximum freight class from all requests.
Figure 17 continues our example by introducing another shipment request and then combining the two so
that the model can calculate the cost of the shipment on this VOFC lane.
845
*1993
9174535
91745
1
359
3599
*
VNDOR9I74SPHX7
VENDOR
*VENDOR
6/4/20134
*6/4/2013
18351
6/6/2013
7050
92.5
50
*6/5/2013
9.
Figure 17: Shipment Request to VOFC
With all shipment requests for the day prepared as VOFCs, the model uses the shipment attributes to
calculate the transportation using both the TL and LTL ship modes.
5.2.1
Truckload Cost Equation
In Section 2. 1.1,5 we discussed that the transportation cost structure of a truckload shipment consists of a
minimum charge, a base cost dependent on the origin-destination pair, and a fuel surcharge. The model
simplifies this cost structure by using a piece-wise linear equation, which includes a linear regression and
a minimum charge based on Amazon data.
To create the truckload cost equation, a dataset was used that included 1042 truckload shipments over
several weeks in 2013. Figure 18 shows the actual cost data and a linear equation, which has an R2 of
0.93.
48
0
500
1500
1000
2000
2500
3000
Distance (miles)
Figure 18: Truckload Cost Regression
After analyzing the TL shipments that travel distances less than 250 miles, there were patterns of a clear
minimum charge that the linear regression was not capturing. Therefore, the TL cost equation was
updated to include this minimum charge as follows:
COD+CI
TLCosts = MAX
TLin
Equation 9: Cost Regression for a Single Truckload Shipment
C1 - constant cost per distance [-]
mi
D - distance [mi]
Co - constant for position of line
[$]
TLmin - minimum charge [$]
For these 1042 shipments, the updated equation has a mean absolute percent error (MAPE) of 14.9% and
a mean deviation (MD) of -$1. 1. Figure 19 shows this cost equation with the actual cost from the data set.
Actual TL Cost
0
500
1000
'TL
1500
Cost Equation
2000
Distance (miles)
Figure 19: Updated TL Cost Equation
49
2501)
3000
Beyond the cost equation, the simulation model accounts for truckload trailer capacity constraints to
determine whether the VOFC-level freight requires only a single truckload trailer or multiple. As a
reminder, the capacity of a truckload trailer is 40,000lbs and 26 pallet floor spaces. If either of these
constraints is surpassed, the freight requires an additional truck. The number of trucks required is
determined by the following equations.
FloorSpaces= VOFC Unstacked Pallets+ VOFC Stacked Pcilets
2
Equation 10: Floor Spaces Formula
Utilization = MAX
Floor Spaces /26
VOFC Weight / 40,000lbs
Equation 11: Truck Utilization Formula
Number of Trucks = the next integer greater than Utilization
Equation 12: Number of Trucks Formula
Combining the cost model for a single TL trailer and the number of trucks, the final TL cost equation
used in the simulation model is presented.
TLCost = Number of Trucks * TLCosts
Equation 13: Total Truckload Cost Formula
For model simplicity, we assume that VOFC combination of shipment requests cannot be separated and
therefore must ship with the same mode.
5.2.2
Less-than-Truckload Cost Equation
As explained in Section 2.1.2, the less-than-truckload cost structure is more complex than the TL
structure because it depends on the origin-destination pair, weight, volume, and the freight class. To
create an equation in the simulation model, 126,266 LTL shipments from Amazon were analyzed over
several weeks in 2013.
50
From the data, two important observations were used to create the LTL cost equation:
(1)
The percentage discount for moving into a higher weight threshold was relatively constant
independent of the FAK and distance.
(2) The percentage increase in cost between two FAKs was relatively constant independent of the
weight and distance.
Observation
1 follows from Figure 2 in Section 2.1.2 and ensures that economies of scale discounts are
applied as shipments increase in weight. A simple linear regression based on weight would not provide
this discount that exists in reality.
For the shipment-planning simulation model, the LTL cost follows the following structure:
LTL Cost = MAX(W * CFAK * x
,
LTL nII)
Equation 14: Cost Regression for Less Than Truckload Shipments
MIN (W400 x VOFC Weight, W 000 x 500ibs) for VOFC Weight < 500lbs
MIN(WIeO( x VOFC Weight,W,142000 x 1000lbs) for 5001bs
W =
<MIN(W
VOFC Weight < 1000lbs
< 2000lbs
VOFC Weight, W5(( x 2000lbs) for 10001bs VOFC Weight
xVOFC Weight,Wiseco x 5000lbs) for 20001bs-! VOFC Weight < 5000lbs
20 0 0 x
MIN(W
MIN (W,0000 x VOFC Weight,W2,(10
W42000 x VOFC Weight for 100001bs
CFA-
x 10000lbs) for 50001bs
VOFC Weight < 10000bs
VOFC Weight < 20000lbs
CFAK,
for FAK,
CFAK2
for FAKI
CFAK, for FAK,,
x
f (VOFC
Distance)
The first part of the LTL cost equation, W, structures the economies of scale discounts that occur for
larger weight shipments. As discussed in Section 2.1.2 and depicted in Figure 2, LTL carriers use cost per
pound constants for each weight threshold in their base rates. The set of equations for W follow this
51
structure, and thus W500 is a constant cost per pound used when the shipment is less than 500lbs.
Observation 1 states that the ratio of any two of these constants (e.g. W5oo / W 1ooo) is also constant
independent of the distance and FAK. The second factor, CFAK, is a constant dependent on the FAK of the
shipment. For a single distance, the LTL cost structure is shown in Figure 20.
-FAK
I
= 50
= 100
"FAK
FAK = 300
00000"
0
2000
4000
6000
8000
12000
10000
Weight (Ibs)
14000
16000
18000
20000
Figure 20: LTL Cost Structure for Three FAKs
The final LTL cost equation was determined by setting x = k * vrVOFC Distance and then solving for
the k and LTLmin values that minimized the mean absolute percent error (MAPE). The final equation
resulted in a MAPE of 30.4% and a mean deviation (MD) of $3.04. Testing this equation on a new data
set gave a similar MAPE of 34%.
After a TL cost and an LTL cost are calculated, the ship mode with the minimum cost is assigned to the
VOFC.
5.3 Multi-stop Truckload Optimization
After every VOFC is assigned to a TL or LTL ship mode, the model is ready to build multi-stop truckload
routes. Instead of solving an NP-hard capacitated vehicle routing problem optimization, a few
assumptions were made to simplify the model based on historical Amazon data.
52
5.3.1
Model Assumptions
Outside of the TL capacity constraints, the model assumes that a MSTL shipment can only pick up freight
from a maximum of four vendors and can only drop off freight to a maximum of two FCs. These
constraints fall in line with Amazon's actual shipment-planning program. The model assumes two
additional constraints that simplify an NP-hard optimization down to a mixed integer linear program
(MILP).
(1)
All vendors on any one MSTL shipment must originate in the same "Origin Cluster".
(2) Both FCs on any one MSTL shipment must be located in the same "FC Cluster".
An origin cluster is a group of 3-digit zip codes that are adjacent to each other covering a small area
approximately the size of a city. Origin clusters were selected based on areas that included many Amazon
vendors. An FC cluster, similarly, is a group of two FCs that are located very close to each other.
Applying these two constraints in the model changes the optimization from building a transportation
sequence that reduces cost to simply deciding which freight should move through the Origin Cluster - FC
Cluster lane. This simplification is illustrated in Figure 21.
nal Forest
--
-----
na
-Phoni
smazo
Origin Cluster
FC Cluster
Figure 21: Origin Cluster FC Cluster Simplification
Now that the assumptions have been explained, let's explore their validity for the simulation model.
Using the vendors' origin zip codes, 1479 Amazon MSTL shipments were visualized on a map of the
53
United States over several weeks showing origin location. Looking at the data this way, it became clear
that the majority of the MSTL shipments originated from a relative few areas of the country. Each MSTL
shipment was studied to understand how many of its vendors originated from the same 3-digit zip code.
Table 8 shows the results of this analysis.
Table 8: MSTL 3-Digit Zip Code
# of 3ZIP origins on MSTL
# of MSTLs
Percentage
1
2
3
506
775
190
34.2%
52.4%
12.9%
4
8
0.5%
Table 8 shows that 34.2% of all MSTL shipments included vendors that were located within the same 3digit zip code. Moreover, 86.6% of all MSTL shipments included vendors that were located in either one
or two 3-digit zip code areas. Table 9 shows a similar analysis by origin state.
Table 9: MSTL Analysis State
# of Origin States on MSTL
# of MSTLs
1
2
3
Percentage
1133
327
19
76.61%
22.11%
1.28%
In Table 9, we see that 76.6% of MSTL shipments included vendors located in the same state, and 98.7%
included vendors in two states. Many times, the shipments with two states included vendors that were
located in cities close to state border lines.
Therefore, the origin clusters were created to group several 3-digit zip codes within the same vendor area.
If the 3-digit zip code was not within a larger area with many vendors, the origin cluster was set equal to
the 3-digit zip code itself.
An analysis was also conducted to understand the percentage of FCs for the MSTLs.
Table 10: MSTL Analysis Number of FCs
FCs on MSTL
1
2
# of MSTLs
639
840
54
Percentage
43.2%
56.8%
Table 10 shows that 43% of MSTLs ship to a single FC while 57% ship to two FCs. After this analysis,
the FC clusters were created with FC pairs that were located in the same area. Table 11 shows that 68% of
MSTLs were shipped to FCs within the same FC Cluster.
Table 11: MSTL Analysis Number of FC Clusters
FC Clusters on MSTL
# of MSTLs
Percentage
1
1009
68.22%
470
2
31.78%
From this analysis, we can conclude that a strong majority of the MSTL shipments included vendors
within the same Origin Cluster and shipped to FCs within the same FC Cluster. This validates our
assumption. When making this assumption, we are assuming that the distance traveled out of the way
between vendors and between FCs is negligible. The model, instead, captures the maximum distance of
the shipments on the MSTL and uses that in the truckload cost equation.
Even though the model misses some multi-stop shipments through these assumptions, the benefit of using
a MILP for the simulation is significant. The purpose of the model is simply to analyze the impact of
using the dynamic pickup window solution.
5.3.2
MILP Formulation
As a refresher, the idea behind the dynamic pickup window solution is that delaying freight that does not
yet need to ship can create additional opportunities for same-vendor and multi-vendor consolidations. The
MILP is seeking to find good candidates that reduce the transportation cost by shipping via MSTL instead
of shipping directly via TL or LTL. The TL and LTL costs for every VOFC has already been calculated
by this point, so the simulation program will only choose a combination of shipment requests if the
MSTL cost is less than the total cost of shipping directly.
55
Based on our assumptions in 5.3.1, let's define a variable VOFCC14 that combines all VOFCs that are
traveling to either FC within the same FC Cluster. Figure 22 shows an example of two VOFCs combining
into a VOFCC. The freight attributes are combined in the same fashion as shipment requests were
combined into a VOFC.
VENDOR91745PHX7
359
25401
14
4
92.5
Some TL Cost
VENDOR91745PHX5
359
4230
0
2
100
Some LTL Cost
VENDOR91745PHX5/7
359
29631
14
6
SLCos1
Figure 22: VOFCC Combination
In Figure 22, we can observe that the previous cost equations have assigned a TL cost to the first VOFC
and a LTL cost to the second VOFC. When combined into a VOFCC, those costs are added together and
given the variable SLCost, which stands for single-lane cost. This is an important variable used in the
objective function of the MILP.
The last variable we will define is OCFC 5 , which is a combination of several VOFCCs where the
vendors are located in the same origin cluster as explained in Section 5.1.2. Essentially, an Origin Cluster
is a combination of several 5-digit zip codes that make up a larger vendor area. While most Origin
Clusters are simply the 3-digit zip code area of the vendor location, some consist of several 3-digit zip
codes (e.g. Los Angeles). A map of Amazon's current vendor network was used to assign all Origin
Clusters combining multiple 3-digit zip codes. In this way, note that each VOFCC can belong to only one
OCFC. The MILP evaluates each OCFC to determine which combination of VOFCCs will provide the
maximum savings over their current transportation cost while still satisfying all of the constraints.
14 VOFCC stands for Vendor - Origin - FC Cluster
15 OCFC stands for Origin Cluster - FC Cluster
56
Objective Function:
maximize
2
*
SLCos,
-
TLCostk +
* MSTLPICKUPCOST
for all k
i - VOFCC (Vendor - Origin - FC Cluster)
k - OCFC (Origin Cluster - FC Cluster)
SLCosti - Single-Lane Cost for VOFCC
MSTLPICKUPCOST - User-defined input as stated in Section 5.1.2.
Decision Variable:
Z e {0,1} which VOFCCs are on the MSTL
Constraints:
z, < 4 number of vendors constraint
e {0,l} binary variable constraint
,
,
5.3.3
* Floorspacei 26 floor space constraint
* Weight < 40000 weight constraint
Optimization Model Outputs
The optimization outputs which VOFCCs are shipped on the MSTL shipment and which ones are not.
Since the model knows the VOFCCs on the shipment, it also knows all VOFCs and all shipment requests
on that shipment as well. Since the cost of the MSTL is fixed, the simulation model allocates the cost to
each VOFCC, VOFC, and shipment request proportional to its shipment weight.
5.4 Decisions for Delay Eligibility
Once the MSTL optimization has run, all ship modes and transportation costs have been assigned to each
shipment request for the pickup date evaluated (e.g. Monday). The last routine for the model is to run the
dynamic pickup window logic as described in Figure 14 of Chapter 4. The model follows this decision
tree. The main assumption in this routine is that all shipment requests within the same VOFC must stay
together. Therefore, when evaluating whether shipment requests must ship today or can be moved to
tomorrow's pool of shipment requests, the model looks at:
57
(1)
The maximum delay of the VOFC
(2) The minimum MSBD of the VOFC
(3) Total transportation cost of the VOFC
Figure 23 shows an example of how two shipment requests with the same VOFC are evaluated when
flowing through the dynamic pickup window decision tree.
276
123
SUPPLIER90601 PHX7 SUPPLIER90601 PHX7
6/6/2013
6/5/2013
0
$199
$158
276
123
SUPPLIER90601PHX7 SUPPLIER90601PHX7
6/5/2013
6/5/2013
$357
$357
Figure 23: Delay Eligibility Evaluation Example
5.5 Simulation Model Outputs
After running the dynamic pickup window logic, the simulation model has completed evaluating all
shipment requests for that day. It then proceeds to the next day and repeats the process for every
additional day of the week. Each day's pool includes all shipment requests with that freight ready date
and all requests that were delayed from the previous day's evaluation.
Once a request has been tendered to ship, a transit time is assigned based on the origin-destination pair
and the ship mode. The transit times used are the average transit times from Amazon historical data for
each of these lanes and ship modes.
After the entire week's pool of shipments has been evaluated, the following variables are assigned to a
shipment request:
*
Shipment ID - A shipment ID is assigned to every truck that moves through the inbound
network. Any shipment request that moves together in the network is assigned the same shipment
ID.
58
"
Ship Mode - The assigned ship mode (TL, LTL, or MSTL)
*
Transportation Cost - The cost assigned depending on the ship mode. If multiple shipment
requests were sent together, each request is assigned a fraction of the cost proportional to its
shipment weight.
*
Pickup Date - The date the request was tendered for pickup. This will not be equal to the freight
ready date if the request was delayed.
"
Delay - The number of days the request was delayed.
"
Reason for Delay Decision - If a shipment request was not delayed, the model gives a reason.
Reasons include:
"
o
MSBD Does Not Exist
o
Reached MSBD
o
Utilized TL
o
MSTL
o
Reached Delay Limit
o
Auto-Eligible
o
Low Probability MSTL Area
o
MSTL Candidate Below Min LTL Cost
o
MSTL Candidate Above Min LTL Cost
Transit Time - The mean transit time based on Amazon historical data by origin-destination pair
and ship mode.
*
Lead Time of Transportation - The sum of the transit time and the delay.
In Chapter 5, the simulation model used to compare Amazon's current-state shipment-planning logic and
the dynamic pickup window solution was presented. Specifically, we discussed how a shipment request is
assigned a ship mode, a transportation cost, and a transit time. In Chapter 6, we will see the results of the
simulations including sensitivity analysis around the user-controlled inputs.
59
6. Model Results and Analysis
As discussed in Chapter 5, the simulation model uses a dataset of shipment requests from all US vendors
to all US FCs in Week 23, excluding any requests that were shipped with Small Parcel carriers. While this
dataset is a great representation of Amazon's US network, we are assuming that freight cannot be shipped
through rail or consolidation terminals. The dataset also includes one week of data, and thus we assume
that this week is representative of the entire year. In this chapter, we will review the results of the
simulation model, test the sensitivity of the results to the user-controlled inputs, and finally discuss the
managerial impacts of the recommended solution.
6.1 Results Overview
The first case run in the simulation model represented Amazon's current state by setting the usercontrolled inputs to the following values.
Delay Limit = 0 days
MSBD Shift = 0 days
MSTL Pickup Cost = Amazon's current contracted cost16
Minimum LTL Cost - N/A
In this current state, we captured the weekly transportation cost, average transit time, the number of trucks
arriving into the FCs, and the percentage breakdown of requests shipped with each ship mode. These
output parameters, along with a few others discussed in this section, were captured for each subsequent
test case. Table 12 and Figure 24 show the transportation cost and time results for three important cases:
the current state, the solution without the probability of consolidation logic, and finally the recommended
dynamic pickup window solution.
16
This value is private to Amazon.
60
Table 12: Simulation Model Results - Key
Output Parameters
Trans Cost Transportation Time Average TT No. of Shipments
Current State
--
Solution without
Probability of
Consolidation Logic
-2.2%
13.8%
-0.9%
-5.0%
Dynamic Pickup
Window Solution
-1.8%
5.9%
-1.1%
-3.8%
Trans Cost
-
Transportation Time
115%
100.0%
110%
In99.5%
99.0%
1-
P
-
105%
98.5%
C
98.0%
97.5%
a97.0%
100%
C
4
95%
4--
S96.5%
900/0
Current State
Solution without Probability Dynamic Pickup Window
Solution
of Consolidation Logic
Figure 24: Simulation Model Results
In Table 12 and Figure 24, we can see that the dynamic pickup window solution reduces the
transportation cost in the network by 1.8% while only increasing the average transportation time by 5.9%.
Assuming a current state transportation time of 4 days, a 5.9% increase in transportation time is only an
increase of 0.24 days. As a reminder, transportation time for each request is the sum of the transit time
and the solution-driven delay of pickup. Table 12 also shows that the dynamic pickup window solution
reduces the number of trucks arriving to FCs and the transit time by 3.8% and 1.1% respectively.
To express the impact of the probability of consolidation logic used in the solution, a case that maintained
the MSBD logic but excluded the probability of consolidation logic was simulated. In Table 12 and
Figure 24, we can see that this case reduced transportation cost by 2.2% and the number of shipments by
5%, but increased the average transportation time by 13.8%. While the transportation cost is 0.4% lower
than the dynamic pickup window solution, the transportation time, and thus lead time, increases by 7.9%.
61
Both effects are caused because more shipments are delayed when the probability of consolidation logic
does not exist. Figure 25 helps quantify the effect of this logic.
0 %of Shipment Requests that Experienced Cost Reduction 0% of Shipment Requests Delayed
45%
40%
-
30%
--
25%
-
20%
-
---
--
-
-
-
-
---
-----
-
35%
---
-_-
- -- -
--
15%
10%
5%
-
0%
--
Current State
Solution without Probability of Dynamic Pickup Window
Consolidation Logic
Solution
Figure 25: Shipments Delayed and Cost Reduced
This figure shows us the difference between the run with and without the probability of consolidation
logic in terms of the percentage of shipment requests that were delayed and that experienced a cost
reduction. The run without the logic delayed the pickup date for 42.4% of shipment requests in the data
set, while only 20.2% of requests experienced a reduction in transportation cost. When using the logic in
the full dynamic pickup window solution, only 20.9% of shipment requests were delayed and 15.9%
experienced a reduction in cost.
This confirms that even though the dynamic pickup window solution misses some opportunities for
temporal consolidation, it captures a high percentage of the opportunities while delaying far less requests.
This proves that the three decision blocks of the probability of consolidation logic described in Section
4.3 are beneficial to the recommended solution. Table 13 conveys the Delay Effectiveness, which is
defined as:
Delay Effectiveness = Requests that Experienced a Cost Reduction
Requests Delayed
62
Table 13: Delay Effectiveness Comparison
Delay Effectiveness
Solution without Probability of Consolidation Logic
48%
76%
Dynamic Pickup Window Solution
Beyond the cost and time parameters, the dynamic pickup solution creates more MSTL and TL
shipments. Figure 26 shows that the recommended solution expands the MSTL ship mode from 7.7% to
9.6% of all requests, an increase of 25%. Furthermore, requests shipped with the TL ship mode increase
from 1.2% to 1.4%, an increase of 17%.
E Current State
8 Dynamic Pickup Window Solution
12.0%
-
7%
10.0%
8.0%
6.0% 4.0%
2.0%
-
1.2%
2.0%
1.4%
0.0%
MSTL Requests
TL Requests
Figure 26: Percentage of MSTL and TL Requests Comparison
Figure 27 shows a comparison between the current state and the recommended solution of the day of the
week that each shipment request was shipped. While both cases show that the products are relatively
smooth over the week, we see that the recommended solution shows an increase in requests shipped on
Monday and Friday.
N Current State
v Recommended Solution
30%
25%
_
20%
is%
-
10%
S5%
0%
Monday
Tuesday
Thursday
Wednesday
Day of Shipment
Figure 27: Days of Shipment Comparison
63
Friday
6.2 Sensitivity Analysis
The benefit of the simulation model is the ability to adjust the user-controlled inputs for sensitivity
analysis. In this chapter, we will adjust each of the four input parameters individually while holding the
others constant to understand the impact of each on the transportation cost, the transportation time, and
the number of shipments arriving to FCs.
6.2.1
Minimum LTL Cost Constraint
As described in Section 4.3.3, the minimum LTL cost constraint's purpose is to ship all LTL freight today
that costs less than a user-set value because we know that higher LTL cost shipments are more likely to
be selected for a MSTL and thus generate savings. At a minimum, we discussed that this value should be
set to the MSTL pickup charge since any LTL cost less will never be selected for a MSTL.
For sensitivity around this input, two test cases were run setting the other three inputs constant as follows.
Delay Limit
=
2 days
MSBD Shift = 0 days
MSTL Pickup Cost = Amazon's current contracted cost
Figure 28 shows the results of this analysis.
Trans Time Delta
Trans Cost Delta
Shipments Delta
8.0%
7.0%
0.0%
6.0%
-1.0%
5.0%
-1.5% .
-0.5%
-2.0% A
-3
-2.5% §
4.0%
~
3.0%-3.0%
35%Ur
92.0%
C
1.0%
0.0%
$50
$100
$150
Min LT Cost Threshold
$200
Figure 28: Minimum LTL Cost Sensitivity Analysis
64
-4.0% It
-4.5%
$250
From Figure 28, we see that lower minimum LTL cost values result in larger transportation time
increases, less transportation cost, and fewer shipments for the US network. The minimum LTL cost of
$100 represents the recommended solution presented in Section 6.1. Table 14 highlights the change in the
model effects between the recommended solution and a minimum LTL cost of $200.
Table 14: Min LTL Cost Sensitivity Comparison
Min LTL Cost Trans Cost Reduction Trans Time Increase Shipments Reduction
3.8%
1.8%
5.9%
$100
3.2%
4.3%
1.5%
$200
-0.6%
-1.6%
-0.3%
Change
-16%
-27%
-17%
Percent Change
Increasing the minimum LTL cost to $200 would increase the transportation cost by 17% and the number
of shipments by 16% over the recommended solution. This is simply because a minimum LTL cost of
$200 would reduce the number of shipment requests that are delayed. To determine the optimal value, a
company like Amazon would need to understand the tradeoff of the transportation cost reduction and the
lead time increase. In Section 6.3, we will discuss why $100 was chosen for the recommended solution.
6.2.2
Multi-stop Truckload Pickup Charge
The fixed cost that carriers charge for Multi-stop pickups is a contracted rate negotiated between shippers
and carriers. This charge for Amazon was set at the time of this project and therefore was also set in the
model. However, performing sensitivity analysis around this charge is valuable to understand the effects
on the network if the charge were renegotiated in the future. Figure 29 shows the effects on cost, time,
and shipments as the MSTL pickup charge is increased for the following input values.
Delay Limit 2 days
MSBD Shift 0 days
Minimum LTL Cost = $100
65
O"Trans Time Delta
9.0%
Trans Cost Delta
in
Shipments Delta
-
--- 2.0%
8.0%
1.0%
7.0%
0.0%
E 6.0%
P'U 5.0%
-,go
-1.0%
4.0%
-2.0%
3.0%
-...
2.0%
-3.0%
. .......... ......
.
-4.0%
1.0%
0.0%
100%
'U
C
-5.0%
150%
200%
250%
300%
MSTL Pickup Cost
Figure 29: MSTL Pickup Charge Sensitivity Analysis
As the pickup cost increases, we see an increase in the transportation time because fewer shipment
requests are converted to MSTL shipments while a similar quantity are still delayed. We also see that, for
the same reason, the transportation cost and number of shipments increase as well. Table 15 shows the
exact values for three pickup charges.
Table 15: MSTL Pickup Charge Comparison
Pickup Cost Trans Cost Delta Trans Time Delta Shipments Delta
100%
-1.8%
5.9%
-3.8%
200%
300%
6.2.3
-0.1%
1.2%
7.3%
8.3%
-2.0%
-0.6%
Offset from Must Ship By Date
While allowing shipments to delay up to the MSBD ensures products will arrive to the FCs before any
unplanned out-of-stock occurrences, a test case was run to shift the MSBD to one day earlier. This is a
more conservative deadline for when requests must ship, and hence will result in a lower transportation
savings and number of shipments.
66
Table 16 shows the output values for the MSBD shift given the following input values:
Delay Limit = 2 days
MSTL Pickup Cost = Amazon's current contracted cost
Minimum LTL Cost = $200
Table 16: MSBD Shift Comparison
MSBD Shift Trans Cost Delta Trans Time Delta Shipments Delta
-3.2%
4.3%
-1.5%
0
1
-1.2%
3.6%
-2.6%
6.2.4
Max Days Delayed
The recommended solution allows a shipment request to be delayed up to two days before it must be
shipped independent of the MSBD. As stated in Section 4.4, this limit was recommended after
discussions with Amazon vendors and research of best practices in industry. Sensitivity analysis was
conducted to understand the effects if the limit was set to one day. Table 17 and Figure 30 show the
values and trends of the model outputs.
Delay Limit
0
1
2
Table 17: Delay Limit
Trans Cost Delta Trans
0%
-1.1%
-1.5%
inTrans Cost Delta
in
Comparison
Time Delta Shipments Delta
0%
0%
2.9%
-2.0%
-3.2%
4.3%
Trans Time Delta
imnoShipments Delta
5
4
%
3
.9 2%
1%
C
NW
0%
-2
%
-34
-A
Delay LImit (days)
Figure 30: Delay Limit Sensitivity Analysis
67
We learn from this data that increasing the delay limit increases the transportation cost savings, reduces
the number of trucks in the network, and increases the average transportation time. However, Figure 30
highlights that all model outputs have diminishing returns. A large reason for this trend is that as the delay
limit is increased, more shipment requests are constrained by the MSBD and probability of consolidation
logic instead. For example, if the delay limit was increased to a very large number, we can reason that the
model constraints would force shipment requests to ship before getting close to the delay limit.
6.3 Effects of the Solution Discussion
Using the dataset from Week 23 and assuming that it is representative of the off-peak portion of the year,
we can see that the dynamic pickup window solution can provide significant benefits to Amazon's
inbound supply chain. While the model outputs show these improvements, there are four reasons why we
can expect that the model outputs are conservative estimates of the true benefit to Amazon.
(1)
Amazon's volume significantly increases during peak with FC receipts close to 70% higher than
the off-peak weeks. This large increase in products flowing through the network will present
more opportunities for same-vendor and multi-vendor consolidation.
(2) As discussed in Section 5.3.1, the multi-stop truckload optimization included a simplification that
misses around 25% of the multi-stop shipments that actually occur at Amazon. Therefore, we can
expect that the dynamic pickup window solution will build more MSTLs than the model.
(3) We assumed that 2013 volume would be representative of the future. However, Amazon
continues to grow at astonishing rates. Further growth in overall volume will present more
opportunities to consolidate shipments.
(4) The model excluded small parcel shipments, which represents the majority of Amazon's
shipments. While it is very unlikely for shipment requests with less than 150 pounds to
consolidate into a MSTL shipment, it is very likely for same-vendor consolidations to occur with
SP, converting it into a less-than-truckload shipment. A quick analysis of small parcel data
showed an addition 1.3% transportation cost reduction.
68
The recommended minimum LTL cost parameter for Amazon was $100. However, we saw a clear trade
off in transportation cost and transportation time in Table 14. A higher minimum LTL cost results in a
higher transportation cost but a lower time increase. We know from Section 2.2 that an increase in lead
time increases safety stock inventory due to the larger standard deviation of demand over lead time and
review period. After discussions with the inventory-planning department at Amazon, a solution was
created to eliminate this negative lead time increase. Since the increase in transportation time is strictly
due to the dynamic pickup window solution's decision to delay pickup, the time delayed (either one or
two days) will be removed from the overall lead time of any shipment request that was delayed but still
tendered with the same ship mode.
By doing this, Amazon will understand the performance of the actual network and not increase lead times
due to a transportation decision to delay one or two days. All other impacts to lead time outside of the
delays caused by the dynamic pickup window solution are still included. By applying this solution to the
lead time calculation, we prefer a transportation cost reduction to a transportation time increase when
establishing the appropriate minimum LTL cost value.
The larger concern for setting this value too low is that it is a waste to delay shipments that do not result
in a consolidation for Amazon. Therefore, the best value is that which results in the highest ratio of
requests that consolidate to requests that were delayed as shown in Table 13. Therefore, a value lower
than $100 was not chosen.
In Chapter 6, we have reviewed the results of the simulation model running the current state shipmentplanning logic and the dynamic pickup window solution. Sensitivity analysis was also presented for each
of the four user-controlled variables. Lastly, we discussed how the model outputs are a conservative
estimate of the true effects and why $100 was chosen for the minimum LTL cost parameter. In Chapter 7,
we will close the thesis with closing remarks for the Amazon case study and future research opportunities.
69
7. Conclusion
As Amazon continues to grow, it has opportunities to reduce inbound transportation cost and improve
supply chain performance. In this thesis, an innovative solution to incorporate time into the transportation
decisions through the introduction of a dynamic pickup window was presented. The introduction of the
Must Ship By Date will enable Amazon to determine when freight should be delayed for the opportunity
to consolidate, when freight should move today with the lowest cost carrier, and when freight should be
further evaluated for expediting. Furthermore, a simulation model was created to model the effects of this
solution on the inbound supply chain in terms of cost, time, and the number of shipments flowing through
the network. The recommended parameters of the dynamic pickup window solution will reduce
Amazon's transportation cost and the number of shipments by approximately 2% and 4% respectively
while ensuring that products arrive to the FCs before unplanned out of stock occurrences. Due to the
assumptions in the simulation model, it is likely that the true benefits of the solution are greater. As stated
in previous chapters, the results and analysis are based off a dataset from a single week at Amazon. Since
most retailers order products on a weekly basis, we can assume that the results presented in Chapter 6 for
Week 23 will be appropriate for other weeks throughout the year. However, it is recommended to
simulate the dynamic pickup window solution on additional weeks.
The greatest leadership challenge of this solution is changing the behavior of the vendor network to
understand that pickups will now occur within a three-day window instead of exactly on their freight
ready date. To manage this change, a launch plan was created with the technology departments including
recommended areas of the country to start using Phase I of the dynamic pickup window solution. Pilot
launches were initiated in late 2013 with plans for larger rollout in 2014. The greatest next steps for this
project are to begin working on the Phase 2 solution system architecture, quantify the impact of reduced
shipments in the network, and create a total landed cost model to determine if freight should be expedited.
70
As mentioned in Chapter 1, the dynamic pickup window solution described in this thesis can benefit
many retailers and companies from other industries. Amazon was simply a test case to illustrate the
solution and the results that could be achieved. Any business with stochastic demand and lead time, a
large vendor base, and control of managing its inbound transportation can apply this solution to its unique
situation to help answer questions such as:
(1)
If products are delayed for shipment, will this likely cause an out of stock occurrence?
(2) Will delaying the products result in a consolidated shipment with other products?
(3) How does this consolidated shipment affect my inbound supply chain?
(4) Should the products from Vendor X ship right now, or can they be delayed?
A major benefit to this solution is that it does not require any capital investment to the network to achieve
improvement, but instead a less intrusive change within a company's shipment planning and execution
processes. Furthermore, the solution is network independent, and therefore can continue to benefit
companies even when the network structure changes.
Future research opportunities around the dynamic pickup window solution include:
(1)
Providing additional test cases with other companies in various industries
(2) Comparing this solution with an inbound transportation decision to mandate the time that a
vendor's products will ship (removing the ship window idea altogether)
(3) Exploring alternative methods to improve the prediction of whether or not a shipment will be
consolidated if it is delayed
(4) Providing test cases on how a total landed cost model can help a company understand when to
expedite a shipment with particular focus on how to account for the cost of lost sales
71
8. Bibliography
Amazon.com. (2014). Retrieved 02 17, 2014, from Amazon.com: http://phx.corporateir.net/phoenix.zhtml?c=97664&p=irol-irhome
Amazon.com. (2014). Retrieved 02 17, 2014, from Amazon Fulfillment Careers:
http://www.amazonfulfillmentcareers.com/amazon-fulfillment/locations/
Dunn, J. (2014, 03 17). Locations ofAmazon Fulfillment Centers. Retrieved 03 22, 2014, from The
Bottom Line: http://outright.com/blog/locations-of-amazon-fulfiIlment-centers-2/
Federal Motor Carrier Safety Administration. (2013, February). Retrieved February 10, 2014, from
FMCSA: http://www.fmcsa.dot.gov/interstate-truck-drivers-guide-hours-service
Ford Jr., D. J. (2006). Inbound Freight Consolidation: A Simulation Model to Evaluate Consolidation
Rules. Cambridge, MA, USA: Massachusetts Institute of Technology.
Hadley, G., & Whitin, T. (1963). Analysis of Inventory Systems. Englewood Cliffs, NJ, USA: PrenticeHall.
National Motor Freight Traffic Association. (2013, 09 23). Retrieved 02 01, 2014, from
http://www.nmfta.org/documents/CCSB/CCSB%20Guidelines.pdf
Oti, 0. (2013). Hub and Spoke Network Designfor the Inbound Supply Chain. Cambridge, MA, USA:
Massachusetts Institute of Technology.
Ozkaya, E., Keskinocak, P., Joseph, V. R., & Weight, R. (2009, June 09). Estimating and benchmarking
Less-than-Truckload market rates. Elsevier.
PLS Logistics Services. (2013, 05 09). Retrieved 02 01, 2014, from PLS Logistics Services:
http://info.plslogistics.com/Blog/bid/178910/7-Factors-that-Determine-LTL-Pricing
Society for Industrial and Applied Mathematics. (2002). The Vehicle Routing Problem. (P. Toth, & D.
Vigo, Eds.) Philadelphia, PA, USA: Society for Industrial and Applied Mathematics.
U.S. Energy Information Administration. (n.d.). Gasoline and Diesel Fuel Update. Retrieved from U.S.
Energy Information Administration: http://www.eia.gov/petroleum/gasdiesel/
72
Appendix A: Longest Lead Time Derivation
To determine the longest lead time that can occur, we can use the same inventory equation with the orderup-to quantity (S) set to 5163 and then back solve for lead time.
S= D * (L + I)+ k *
(L + I) *
YD
2
+ D2
*
For derivation purposes, let's define a variable x, where:
x =(L+I)
S=Dv+k* X
oD2+D
2*L
The longest lead time allowed to maintain a cycle service level of 95% is a deterministic value and
therefore the
ao2
term is set to zero when solving for x.
S = D* x+ k *
(S
-
Dx)
=
S 2 -2SDx+D 2
Cx*yD
k
2U
=k 2u D2
2
2
D 2X 2 +(-2SD-k
2
D 2 )x+S
2
=0
Now, we can solve for x using the quadratic formula, where:
a= D
b=-2SD-k 2au
C = s2
b 2 -4ac
-b
2a
(2SD + k 2
)D
± I(-2SD 2D
x=
(2SD+k2(
1
2)
73
1)
2
)2 -
4D 2S 2
2
±4D2S2 +4DSk
2D
k2
2
2
+k 4
4
CD
-4D
2S
2
+ k2,2) g 4 DSkD
2D 2
( 2SD
2
+k4
D
(2SD + k2UD2) ± k2OUD 2 (4DS + k 2o(D2
2D 2
(2SD + k2
D2)
±ka
2D
x=
2SD
2D
S
x=D
D
2
kTD( koD
+
(4DS+k23D2)
2
±4DS+k
2
)
2D 2
ku (
kCD
2DY\
+ kD-
4DS+k2 .D2
Now, we plug L-I in for x to solve for lead time.
L+I= S +k
(/k
D 2D2
L=
+
D
D
2D 2
±4DS+k2
±4DS+k2
k
D
e
rD2)
2)J_
se
Let's use this equation to solve for the longest lead time in the example presented in Section 2.2.
L =
5163 (1(1.645)2
50
)2
+
-(1.645)250±V4(1000)(5163)+(1.645)2(250
2(100)
1000
)1) _I
When solving this equation, we obtain two possible solutions (L=3.31 weeks and L=5.19 weeks). To
check if these answers are valid, we plug the solution back into the original inventory equation below to
ensure the order-up-to point (S) equals 5163 units.
S = D*(
+ 1I
&
+ I
74
(*D 2
+D2
I+k*
When checking our solutions, we find that L=3.31 weeks is a solution while L=5.19 weeks is not a
reasonable solution). Therefore, our final equation for the longest lead time allowed ensuring our
originally planned 95% cycle service level is:
S
kaD
L =D+ 22k
D
2D2k
-
4DS+ (kC22
-I
DD)
Knowing that our lead time has a normal distribution with a mean of 3 weeks and a standard deviation of
0.5 weeks, the probability that lead time is less than or equal to 3.31 weeks (23 days) is 73.2%.
75
Appendix B: Longest Vendor-Controlled Lead Time Derivation
To determine the latest date that the freight should be shipped from the vendor, we start with the same
periodic review policy as before with the updated terms to account for the lead time portion at the vendor
and the transit time.
S=D*(L
+LT+I)+k*
(Lv+LT+I)*yD2+D2*(L,2
+UI)
Similar to the last derivation, let's define a variable y as follows:
y=(Lv+ LT +I)
S= D*y+k*
y*UD2 +D 2 *(o2
+
2)
The longest vendor-controlled portion of lead time (Lv) allowed to maintain a cycle service level of 95%
is a deterministic value and therefore the Uv term is set to zero when solving for y. The lead time portion
controlled by the carrier (i.e. transit time) remains stochastic.
S= Dy+k\yan2+D%,L +DQ
S = Dy+k
(S-Dy)
S2 -2SDy+D
D2Y2 + (-2SD
-
2
ycD2 + D
2
= k2D2
2 2
y
=k
k 2 D2 )y+
=
b = -2SD - k 2 uD
c = S2 + k2DL
b 2 -4ac
2a
76
T
+ k2DY21g
D
-b±
2
Dy+k2D2,
As before, we can solve for y using the quadratic formula:
a
L
2
2
)
=
0
2SD
k2
3,
2
2
± J(2SD + k2
2
2)2 -4D
(S2
+ k2D2yL2
2D 2
2SD + k 2
D2 ±
4D2S2 +4DSk 2U)
+k4y)
2SD + k%~, 2+4DSk
±a
2a 2±
2D
2D
2
k
+
yD
D2
D4
+ k 2UD2
V-4'
k
D
22
2
4
2
)
-
4D
4
2
LT
2
\)-4Da
L
4kSk
Now, to obtain an equation for Lv, we plug in y = Lv + LI +
L =S +
-4D
(4DS+(k3D)2
2g k
2D
4D4 k'u
-
4
2
(4D
k
2D
S
2
2
2
2SD+k 2 D2 ±-kI4DSD2 +k+k
2SD + k
S 2 -4D 4 k 2
2
-4D
2
~2D
Y
2
4D+
I.
32)
4D4a
- I - LT
As described in detail for the previous derivation, we can plug in the values from the example in Section
2.2 to calculate a value for Lv. Once the two values are determined, we again check the values in the
original inventory equation to determine if our possible solutions are reasonable.
After this check, the final equation for the longest vendor-controlled portion of lead time to ensure a cycle
service level of 95% is:
L =
D
+
2D
2(ka2
-
2(4DS+(k
))
-4D4L'2
I - LT
This equation tells us that the shipment should ship within 2.19 weeks (15 days) after the order is placed.
Knowing that the vendor-controlled portion of lead time has a mean of 2 weeks and a standard deviation
of 0.4 weeks, the probability that L, is less than 2.19 weeks is 68.3%.
77