OPTIMIZING RESOURCE ALLOCATION DECISIONS

AND REMAPPING INITIATIVES

FOR A MAJOR DISTRIBUTION FACILTY

An Honors Project Manuscript

Presented by

Evan Lynch

Completion Date:

May 2015

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ABSTRACT

Title: Optimizing Resource Allocation Decisions and Remapping Initiatives for a Major Distribution Facility Author: Evan Lynch Thesis/Project Type: Independent Honors Project Approved by: Ahmed Ghoniem, OIM Approved by: Iqbal Agha, OIM This independently contracted honors project seeks to show the value of applied mathematical optimization, as it relates to resource allocation decisions in a large cross-docking distribution facility. The report stresses the value of getting the right data at the right time, and leveraging that data through a mixed-integer goal program. The model constructed makes use of highly variable volume data to make resource allocation decisions that minimize distribution costs. The project also addresses the value of permanent store to door reassignment initiatives in conjunction with optimal resource allocation methods. The distribution facility of interest in this study is Macy’s flagship distribution center in Secaucus, New Jersey.

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1 Introduction

1.1 Distribution and Cross Docking

One of the major reasons why the world is booming with all sorts of products is because of the

effectiveness of distribution centers in aiding the process of logistics and transportation. Distribution

centers exist in order to hold inventories of finished goods before they are shipped out to various store

locations. Cross-docking is a particular type of logistical strategy in which the facility only serves to

quickly transfer goods from inbound vendors to outbound delivery trucks. In a typical distribution center

product coming from vendors would be unloaded, stored in inventory for a time, and shipped out to stores

upon request. Cross-docking facilities have popularized in an effort to minimize inventory costs,

distribution costs, and lead times. Instead of storing inventory in warehouses, a company can simply

reorder from their vendors when needed, and have deliveries made to a central facility. Goods delivered

in bulk from vendors would then go through a massive sorting operation, giving each store location the

correct quantity of goods ordered. The cross-docking facility of interest in this particular project is

Macy’s flagship distribution center in Secaucus, New Jersey.

1.2 Company Overview

Macy’s Department Stores, one of the largest retail companies in the United States, employs over

160,000 people and is comprised of two operating divisions: Macy’s and Bloomingdales. Macy’s operates

850 stores across the U.S., including the largest department store in the world, located in the heart of

Manhattan. Bloomingdales serves its customers through 37 stores and 13 outlets. Bloomingdales focuses

on a more upscale market with an emphasis on distinctive merchandise offerings. Product distribution to

both Macy’s and Bloomingdales is managed by Macy’s Logistics and Operations, which operates 23

distribution centers across the United States. The distribution center in Secaucus, New Jersey serves as

the company’s Logistics and Operations headquarters, and is the main subject of this study.

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1.3 Secaucus Overview

The Secaucus distribution center currently serves 72 stores in the greater New York City area,

which are made up of 59 Macy’s, 11 Bloomingdales, and 2 Bloomingdales Outlets. The distribution

facility is almost entirely cross dock. There are 23 inbound truck bays where deliveries are unloaded to

feed the system. There are 129 outbound truck bays where sorted freight is loaded for delivery to the 72

stores the facility serves. Because there is a surplus in outbound truck bays, larger volume stores can be

divided among a few lines, so as to avoid clogging the system. The sortation system at the Secaucus

facility manages to automate the sortation process of roughly 50,000 cartons per day.

Associates working in the receiving areas of the building unload cartons from trailers that come

from the hundreds of vendors who sell their products at Macy’s and Bloomingdales. From the moment

cartons are unloaded from the trailer, they begin a long journey through the building via a network of high

speed conveyor belts. Cartons ultimately reach their destination at the other end of the building, where

outbound associates reload the cartons onto trailers destined for the stores.

Within the sortation system, there are two inbound sorting machines that read carton labels,

which tell the system where it needs to travel to next. Once the cartons get to the other end of the facility,

they travel through one of the four outbound sorters which divert cartons to their assigned outbound

trailer. These four outbound sorters are called “Matthews”, “South”, “East”, and “West”. Figure 1 below

shows a detailed map the facility. The red lines are where product comes into the system, while the blue

and green lines against the outer walls are where delivery trucks are loaded before being shipped to stores.

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Figure 1

2 Problem Statement

The biggest priority in distribution processes is that they are completed as efficiently as possible.

As expected, the management team at the Secaucus distribution center places a huge emphasis on how

productive the associates are throughout the day. The productivity of an associate is calculated as the

number of cartons an associate processes divided by the number of hours that associate worked.

In every case, there is a goal of how many cartons per hour (CPH) an associate should load into a

trailer. Using historical data, the management team has determined specific productivity goals for each

area to achieve. In fluid loading areas, associates are set to a goal CPH of 266. In zonal loading,

associates performance is benchmarked against a lower goal CPH of 170. Fluid loading is where

associates simply take cartons off of the conveyor belt and load them into a trailer. Zonal loading is a

particular loading process in which cartons are stacked on top of pallets, wrapped in plastic, and driven

into trailers by fork lifts. Some stores that have certain delivery restrictions must receive product in this

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way. Because this process takes longer to complete, the goal CPH is therefore lower for associates

working on doors with stores that have such zonal requirements.

The primary responsibility of the area managers in the facility is to make decisions on how many

associates they need to do the work that is estimated to arrive at their area over the next hour, and which

doors an associate should be assigned in order to maximize the total area productivity. While there are

129 doors, surprisingly few associates are staffed on the outbound docks. One associate might be assigned

five or six doors to load, with cartons periodically diverting down to trailers from the conveyor system. In

a perfect world, nothing would vary, and the same number of associates would work the exact same doors

every single day. If this was the case, the decision making process would be simple. However, the reality

is that the almost constant variation in carton volume being delivered to stores across the area and

throughout the day causes the decision making process to be a much more complex problem. Consider

what is going on at the store level outside of the distribution facility. Each store is located in a different

local economy, and thus sells different volumes of product at varying rates. For example, a store in the

middle of New York City is likely going to sell a lot of product very quickly, because it operates in one of

the largest cities in the world. On the other hand, a Macy’s location in a small town in Central New Jersey

will likely sell less volume and at a slower rate. This economic reality reveals itself in the variation

observed at the distribution facility. For example, in one hour a door could receive 80 cartons, and the

next hour nothing. Moreover, in the course of an hour, the volume seen across an entire area of the

building will vary widely from store to store. Below are two examples of this. Figure 2 shows the

variation of carton volume going to a store in Philadelphia from one day to the next. Figure 3 is an

example of how volume varies across stores assigned to the South Dock over the span of three hours.

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Figure 2

Figure 3

Because the management team does not know exactly what will be arriving at each door in the

hour, decisions as to how many associates are needed and what doors they should be assigned have to be

eyeballed, and readjustments may need to be made over the course of the hour in order to react to

unexpected changes in volume. The current state of the decision making process for managers is

reactionary and inadequate for truly minimizing distribution costs. If managers want to maximize

productivity and thus minimize distribution costs, then they need to be able to get their hands on specific

volume data before the decision making process occurs. With the proper optimization tool, this data can

be leveraged prescriptively to make resource allocation decisions that truly maximize the potential

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Variation of Store Volume

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productivity of their associates. The word “potential” is used because managers do not control exactly

how many cartons an associate loads into a trailer. Rather, managers are responsible for giving an

associate enough work so as to promote a productivity that meets or exceeds the goal.

Recently there has also been much discussion around the value of the remapping of outbound

docks as it relates to the promotion of higher potential productivity. Each store, or subset of a store (i.e. a

department), is assigned a permanent door in the facility where cartons being delivered to that store will

always divert from the sortation system. Remapping is defined as the reassignment of a store to a different

door within the facility, through a reconfiguration of the sortation system. With 129 outbound doors, the

possible maps that could be generated are staggering. Before eliminating some obvious maps that would

not work, there are 129! possible maps, a number that goes far beyond the trillions of trillions. Without

leveraging a massive amount of data finding an optimal map for the facility is simply impossible.

Nevertheless, over the past several months, remaps of the building have yielded positive gains in

productivity. With the current decision making process in place, it has held true that a reconfiguration of

the store assignments in the building does in fact promote higher productivity. However, if there existed a

model that could dynamically allocate resources so as to maximize productivity across the outbound

associates, would the mapping actually matter? And if it did matter, by how much would a better map

improve maximum potential productivity? In other words, when optimal allocation decisions are not

being made, the map of the building can play a big role in the improvement of productivity. However, if

there existed a way to precisely assign the optimal number of doors to the right number of associates, so

as to maximize potential productivity, what would the impact of a better map be in the facilitation of these

management decisions?

As we move into the solution approach, it is important to stress just how much variation is seen in

the system, and why that variation creates such an interesting and frustrating problem. The reality is,

while it is possible to make very good allocation decisions, managers are not able to make optimal

decisions because of the constant shifting of volume in the system. Truly, no hour is like the next, and no

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store receives the same volume as the one adjacent to it. Carton volume varies widely across stores and

hours, and this makes the resource allocation process incredibly difficult to get right. Getting a firm hold

on this variation is like trying to ride a bull, but the benefits that can be seen in taking full control of the

operation far outweigh the costs of exploring solutions to such a challenge.

3 Approach

3.1 Models

In approaching this resource allocation problem, it was not going to be enough to find a solution

for maximizing just one associate’s potential productivity in one hour of the day. A tool needed to be built

that, if fed the proper data, could make optimal allocation decisions that maximize each associate’s

potential productivity for any area of the facility and for any hour of the day. In order to solve such a

problem, a linear programming model was constructed and implemented using AMPL, an optimization

software that has the ability to handle large and complex decision models. The overall objective of the

model is to minimize the underachievement of productivity for each associate, in each area of the

building, and in each hour of the day, as it relates to the management’s goal. Moving forward, this model

will be known as the dynamic allocation model. A slight variation of this model was also created for

analysis purposes, and is hereby known as the static allocation model. This variation shows the worst

possible performance of associates if decisions were made in a very rigid way. The results section

compares the static results, actual productivity figures pulled from the manager’s daily reports, and

dynamic allocation results. A separate linear program was also constructed in search of a better map

based on the average volume going to each store over a given time period. After running this remap

model, the dynamic allocation model was run again with modified data to compare the optimal

performance of an alternative new map versus the current configuration.

Each of the linear programs has the following three files associated with it: a model file, data file,

and run file. The model file is where all of the generalized mathematical formulations occur, which will

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be explained momentarily. Each model defines the parameters and sets used, variables, objective

function, and constraints imposed on the optimization of the stated objective. In every case, the objective

function is to minimize a variable, getting the variable as close to zero as possible. The role the

constraints play is to try to prevent the objective function from hitting zero. They are sort of the “reality

check” for the objective. The data file is where the values of parameters are held. Instead of creating a

massive model file that would be impossible to modify if any numbers happened to change, the data file

serves to be a holding place for all of the specific information needed to solve the problem. The beauty of

the data file is that if any of the numbers change, say because a door closes or managers are trying to

optimize productivity for the month of May instead of January, edits can be made in a matter of minutes.

The run file contains all of the commands necessary to execute the problem solving process. In some

instances, parameters that are dependent on specific data for a specific area and hour are calculated

through the run file before the problem is solved. This file also contains the commands necessary to

display the results in a way that is intuitive to the user.

3.2 Data Structuring

When approaching the problem of making optimal resource allocation decisions, it is important to

understand what data can be leveraged and when it can become available. The sortation system in the

Secaucus Distribution Center is equipped to record exactly how many cartons diverted to which doors

after it happens. Thus, in the hour after decisions are made and cartons are processed, the management

team is able to see exactly what went where and how productive the associates were. The issue is, the data

becomes available one hour too late, and currently cannot help managers in the decision making process.

Recognizing this, it is important to point out that the solution shows what could have been, or rather what

decisions, had they been made, could bring potential productivity up to or past management’s goal CPH.

The results section of this report will reveal how valuable having this data sooner in the process really is.

However, for the time being the dynamic allocation model is useful for benchmarking how optimal

manager’s decisions actually were when looking back on them historically.

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Volume data for the month of January was given in the form of manager’s daily reports.

Information on the number of cartons diverted to each store in every hour worked over the month was

pulled from the spreadsheets and placed in a single and clean file. In order to structure the data in a way

that a mathematical model could understand, areas, doors, and hours needed to be coded properly. For

example, door 141 on the South dock had to become door 1 in area 1. When looking at area productivity,

management separates the productivity results into “South Fluid”, “123 Fluid”, and “123 Zonal”. In order

to create a feasible problem, the building also had to be separated into sizeable sections where doors were

adjacent to each other and the productivity goal was the same. After each area was optimized, the model

automatically brought areas back together in order to effectively show the results in a way that is familiar.

Figure 4 shows an example of doors that were coded into a useable structure.

Figure 4

Data was being compiled from each hour worked on each day in the month of January. In order to

build an effective model, saying x store received y number of cartons on January 26th at 1:00pm wasn’t

going to work. Instead the hours in the month also needed to be numerically ordered, with hour 1 starting

at 7:00am on January 5th, the first hour worked in the month. These hours go up to hour 112, which is

2:00pm on January 30th, the last hour worked in the month.

Ultimately a 129 x 112 matrix of store volume by hour was generated from the daily results

spreadsheets. A small subset of this matrix is shown in figure 5 below.

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Figure 5

3. 3 Dynamic Allocation Model Approach

The ultimate objective of the management team at the Secaucus distribution center is to minimize

the cost of distributing product from vendors to various Macy’s locations in the Tri-State area, while at

the same time maintaining a high level of service. The biggest way to minimize these costs is by

maximizing the productivity of each associate that is paid to work in the facility. As stated, managers

have a goal CPH in mind when measuring how productive an associate is. To give an example, if an

associate loads 100 cartons in an hour, this is far below their manager expects them to achieve, while a

CPH of 400 cartons would far exceed expectations. While managers cannot force associates to be

productive, they can enable their potential to be productive by assigning them to a number of doors where

the expected sum of the volume going to those doors meets or exceeds the goal. It is important that when

managers make allocation decisions for one associate, they also keep in mind the work that other

associates need to be assigned as well. For example, one associate may be assigned 5 doors, where the

sum of the expected store volumes going to them is 350 cartons. If the goal CPH is 266, the associate has

more than enough work to do. In fact, they have the potential to even exceed the goal by 84 cartons.

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Meanwhile, the person working adjacent to them may be assigned the next 3 doors, where the sum of the

expected volumes is only 182 cartons. Because this is all they would receive in the hour, it would be

impossible for this associate to meet the goal of 266 cartons, making the goal CPH unrealistic to achieve.

The second associate would expect to miss their target by a total of 84 cartons. Now, what if that 5th store

on the far end of the first associate’s assigned doors has an expected volume of 84 cartons? Couldn’t the

manager reassign that 5th door to the second associate, so that the first associate gets the first four doors

and the second gets the next four? If this allocation decision was made, then the potential productivity of

both associates is 266 cartons. In this way, the model does not only attempt to maximize the potential

productivity for the first associate, but maximizes it for all associates across the area, only giving an

associate more than enough work when it does not take work away from the associate working in the next

group of adjacent doors.

3.4 Model Formulation

Dynamic Allocation Model

Sets

A = Set of associates. Pre-calculated as the volume expected in an area and hour divided by the goal CPH

in that area.

D = Set of outbound doors.

I ⊆ D:𝑉𝑉𝑖𝑖 > 0 = Subset of outbound doors D, where volume exists.

j = Index for an associate in the set of associates A.

i = Index for a door in the subset of doors I.

Parameters

β = The maximum number of doors an associate is allowed to be assigned.

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λ = The maximum volume of cartons an associate can be assigned.

𝑉𝑉𝑖𝑖 = Volume of cartons that is will go to store i.

δ = The goal CPH of an area.

Decision Variables

𝑋𝑋𝑖𝑖𝑖𝑖 = Binary assignment variable of associate j to door i.

𝑍𝑍𝑖𝑖 = Binary variable for the decision to use associate j or not.

𝛼𝛼𝑖𝑖 = The underachievement of the goal CPH of associate j.

Model Formulation

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 � 𝛼𝛼𝑖𝑖𝑖𝑖∈𝐴𝐴

(1)

Subject to:

� 𝑉𝑉𝑖𝑖𝑋𝑋𝑖𝑖𝑖𝑖 ≥ 𝑍𝑍𝑖𝑖𝛿𝛿 − 𝛼𝛼𝑖𝑖 ∀𝑗𝑗 ∈ 𝐴𝐴 (2) 𝑖𝑖∈𝐼𝐼

� 𝑋𝑋𝑖𝑖𝑖𝑖 = 1𝑖𝑖∈𝐴𝐴

∀𝑀𝑀 ∈ 𝐼𝐼 (3)

𝑋𝑋𝑖𝑖2𝑖𝑖 ≥ 𝑋𝑋𝑖𝑖1𝑖𝑖 + 𝑋𝑋𝑖𝑖3𝑖𝑖 − 1 ∀𝑗𝑗 ∈ 𝐴𝐴, 𝑀𝑀1 ∈ 𝐼𝐼, 𝑀𝑀2 ∈ 𝐼𝐼, 𝑀𝑀3 ∈ 𝐼𝐼: 𝑀𝑀1 < 𝑀𝑀2 < 𝑀𝑀3 (4)

𝑋𝑋𝑖𝑖𝑖𝑖 ≤ 𝑍𝑍𝑖𝑖 ∀𝑗𝑗 ∈ 𝐴𝐴, 𝑀𝑀 ∈ 𝐼𝐼 (5)

� 𝑋𝑋𝑖𝑖𝑖𝑖 ≥ 𝑍𝑍𝑖𝑖 ∀𝑗𝑗𝑖𝑖∈𝐼𝐼

∈ 𝐴𝐴 (6)

� 𝑍𝑍𝑖𝑖 ≥ [𝐴𝐴] − 1𝑖𝑖∈𝐴𝐴

(7)

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� 𝑉𝑉𝑖𝑖𝑋𝑋𝑖𝑖𝑖𝑖 ≤ 𝜆𝜆 ∀𝑗𝑗𝑖𝑖∈𝐼𝐼

∈ 𝐴𝐴 (8)

� 𝑋𝑋𝑖𝑖𝑖𝑖 ≤ 𝛽𝛽 ∀𝑗𝑗 ∈ 𝐴𝐴𝑖𝑖∈𝐼𝐼

(9)

The objective function (1) minimizes the underachievement of the goal CPH for all associates j in

an area and hour. Constraints (2) define what underachievement means. For any associate in the set of

associates that are calculated as needed in that area and hour, the sum of volume going to the doors that

associate is assigned must be greater than or equal to the goal CPH, minus the underachievement. Keep in

mind, the objective attempts to bring 𝛼𝛼𝑖𝑖 to zero, so that expected CPH is at least at the goal CPH or

above. The expected CPH will only exceed the goal CPH if it does not take away from the minimization

of another associate’s alpha. Constraints (3) ensure that a door is assigned to one associate and only one.

Constraints (4) ensure that associates are only assigned to doors that are adjacent to each other. If a door

has a volume of zero, it will skip the consideration of assigning an associate to that door, and thus the

next adjacent door can be assigned. Constraints (5) are binary switching constraints. If a door is assigned

an associate, that associate must be considered as used. Constraints (6) state that if an associate is to be

used, they must be assigned to at least one door. Constraint (7) states that the model has to use the number

of associates that are calculated as needed in the area and hour, minus one associate if it can find a way to

feasibly do so. Ultimately the model would like to only assign one associate to do all of the work in the

facility. That would keep that associate very busy! While this decision may maximize that one associate’s

productivity, it is clearly not a reasonable decision to make, and thus the model must not be allowed to do

this. Thus, this constraint allows the number of associates used to be one less than is required by volume,

but associates used cannot go any lower. Constraints (8) ensure that the volume assigned to an associate

across doors to not exceed a certain number of cartons, which is defined as λ. Constraints (9) ensure that

the number of doors an associate is assigned never exceeds 𝛽𝛽.

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This general model can be used to maximize the potential productivity for associates in a

particular outbound area of the Secaucus Distribution Center in a particular hour of the day. When

running the model over the month of January, it was not reasonable to optimize for each individual area in

each individual hour of the day. Running models for 7 areas x 112 hours, or 784 separate optimization

problems would have taken hours or even days to complete manually. Instead a looping strategy was

used, which took chunks of the 129x112 volume matrix by area and hour, optimized and moved on to the

next area and hour. Using this strategy, the time it took to maximize potential productivity for the entire

month of January was cut down to under one hour. In an applied setting, where the next hour’s allocation

decisions need to be made, the model would take under 30 seconds to decide which associates are needed

and where they should be assigned.

3.5 Example Solution

Figure 6 shows an optimal allocation decision from AMPL for areas 1 and 2 (these two areas

make up the South) for the 10:00 hour on January 5th 2015. Across the top of the matrix is the number of

associates that were elected to be used in this particular hour. Along the left side of the matrix are the

door numbers that will have positive volume at some point in the hour. If a door is not expecting any

volume, it will be temporarily skipped over. A value of 1 indicates the door will be assigned to the

corresponding associate. Notice that an associate is only assigned doors that are adjacent to each other.

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Figure 6

In this case 10 associates should be allocated to the South dock at 10 am, and these are the

particular doors each associate should be assigned. It is true that one associate might be assigned two

doors while another is assigned six, but the actual number of cartons they will be loading will be as

evenly spread as possible. Actual CPH for the South in this particular hour was 239, missing the goal by a

little more than 10%. If ten associates were used instead of 13 and these allocation decisions were made,

productivity could have been as high as 311 cartons, nearly 17% above the goal.

3.6 Static Allocation Model

The static allocation model is a modified version of dynamic allocation, and can allow the

management team to see the worst that potential productivity would be, if rigid allocation decisions were

made across the building. In the truly optimal dynamic allocation model, decisions are made on how

many associates were needed and precisely where they should be assigned in order to maximize their

potential productivity. If a door did not have volume coming to it within the next hour, that door was not

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considered in the assignment process. The first associate on the dock could be assigned three heavy doors,

while the next associate is assigned six doors that are expecting light volume within the hour. In the static

model, however, the number of associates needed in an hour by volume will all be used, and they will be

distributed evenly across the doors on the dock. For example, if 14 associates are needed in the hour on

the South dock, the number of doors each associate will be assigned will be 56 doors divided by the 14

workers, or 4 doors each. Each associate will be assigned 4 consecutive doors, even if one of those doors

has zero cartons coming to it. Ultimately this is the lower bound, or the worst that productivity would be.

The model is essentially the same, with the removal of the Z variable, and with the addition of the

following parameters and constraints.

d = The number of doors in an area.

a = The number of associates in an area and hour.

� 𝑋𝑋𝑖𝑖𝑖𝑖 ≤ �𝑑𝑑𝑎𝑎�

𝑖𝑖∈𝐼𝐼

∀𝑗𝑗 ∈ 𝐴𝐴 (10)

� 𝑋𝑋𝑖𝑖𝑖𝑖 ≥ �𝑑𝑑𝑎𝑎�

𝑖𝑖∈𝐼𝐼

∀𝑗𝑗 ∈ 𝐴𝐴 (11)

The Z variable, or the variable that decides whether or not to actually use an associate, is removed

because the model is forced to use all associates that the volume to goal ratio suggests are needed.

Constraints (10) force the maximum number of doors that can be assigned to be the ceiling of the number

of doors in the area divided by the number of associates needed in that area and hour. For example, if the

South dock was to require 16 workers in a particular hour, then the number of doors each of the 16

associates would be assigned would be no more than 4. Constraints (11) ensure that the minimum number

of doors each associate can be assigned is the floor of the same calculation. In our example, this would be

mean that each associate would have to be assigned at least 3 doors. Thus, an associate working on the

South dock in this particular hour would be assigned either 3 or 4 doors. These two constraints, work

together to smooth out the number of doors assigned as much as possible, so that the workload (from a

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door perspective) appears to be fair. As it is known from the problem statement, the level of variation in

the facility makes it clear that such an allocation policy would do much damage to the potential

productivity of associates, and in reality would not be fair at all. In implementing this policy, an associate

can be assigned five very heavy volume doors, while the next associate might just happen to be assigned

five light volume doors. Exploring this static policy was meant to show how poorly the facility might

perform, when making rigid decisions that do not give the variation the attention it demands. The results

of employing such a policy are discussed in the results section of this report.

3.7 Remap Model

A major curiosity of the management team involved the effectiveness of remapping the

permanent store to door assignments as it relates to the improvement of potential productivity. Results of

a remap executed in the summer of 2014 had already resulted in considerable jumps in area productivity,

but these results had only been improvements on the ad hoc allocation policy currently employed by

managers. The question is, with a dynamic allocation policy, where assignment decisions are optimized to

promote maximum potential for the associates, does the actual permanent assignment of doors matter

anymore? Is one map actually going to make the total potential increase, or will the associates be able to

achieve the same productivity with slightly different assignments? This is a very large question, and the

model constructed only touches the tip of the iceberg. Because there are almost countless possible

permutations within the reassignment of 129 stores, finding the best possible map overall requires a

massive optimization effort. Even the reassignment of a store to the next door over would result in a new

map. For a moment, just imagine the possibility. There is essentially an ocean of possible maps, with only

one map being the best of them all. To give some insight into what would actually need to be done in

order to find this one best map, the model would not only need to run the previous 784 separate

optimization models for the current map, as the dynamic allocation model did for the month of January.

Such a model would have to optimize the 784 models again for each possible new map, always looking

into the future to see if there is a slightly better configuration of store assignments that could have

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improved potential even more. Even making the effort to eliminate the several million possible maps that

do not work because of various constraints, still leaves the ocean of maps almost unchanged.

While exploring such a massive optimization model might prove to save the company a very

considerable amount of money in distribution costs every year, it first must be shown that the map of the

facility actually matters after a dynamic allocation policy is implemented. In order to aid in this effort,

another model was created to reassign the stores to different doors based on the average volume going to

each store over the month of January. This specific model takes into account the high level constraints

that management is concerned about, such as maximum volume being sent to certain areas in the building

and certain groups of stores that need to remain together. The model is similar to the dynamic allocation

presented earlier, with a few key modifications. Most essentially, constraints (4), which forced associates

to only be assigned doors that were adjacent to each other, were removed. The removal of these

constraints effectively frees up associates to be assigned any store that maximized their potential

productivity. For example, an associate could be assigned a store at the end of the South dock and another

in the middle of the Matthews. Clearly an assignment like this would not be realistic under the dynamic

allocation policy, which is why the adjacency constraint exists in the first place. However, in this model

the solution should be interpreted as stores almost being lifted from their assigned doors and being

grouped together in such a way that, when placed back down in their respective groups, would maximize

productivity for the associates assigned to these groups.

To create such an optimization problem, the following modifications were made to the model:

Sets

S = Associates assigned to the South Dock.

M = Associates assigned to the Matthews area where stores can be remapped.

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Parameters

𝐶𝐶𝑠𝑠 = The volume capacity that can be handled on the South sorter.

𝐶𝐶𝑚𝑚 = The volume capacity that can be handled on the Matthews sorter.

𝐷𝐷𝑠𝑠= The number of doors located on the South.

𝐷𝐷𝑚𝑚= The number of doors located on the portion of the Matthews being considered for remapping.

Constraints

� � 𝑉𝑉𝑖𝑖𝑋𝑋𝑖𝑖𝑖𝑖 ≤ 𝐶𝐶𝑠𝑠

𝑖𝑖∈𝐼𝐼𝑖𝑖∈𝑆𝑆

(12)

� � 𝑉𝑉𝑖𝑖𝑋𝑋𝑖𝑖𝑖𝑖 ≤ 𝐶𝐶𝑚𝑚

𝑖𝑖∈𝐼𝐼𝑖𝑖∈𝑀𝑀

(13)

� � 𝑋𝑋𝑖𝑖𝑖𝑖 ≤ 𝐷𝐷𝑠𝑠

𝑖𝑖∈𝐼𝐼𝑖𝑖∈𝑆𝑆

(14)

� � 𝑋𝑋𝑖𝑖𝑖𝑖 ≤ 𝐷𝐷𝑚𝑚

𝑖𝑖∈𝐼𝐼𝑖𝑖∈𝑀𝑀

(15)

Constraint (12) ensures that the total volume of stores assigned to South associates does not

exceed the number of cartons that the south can handle in an hour. Constraint (13) does the same, but with

Matthew’s associates and for the Matthews sorter. Constraint (14) ensures that more stores are not

assigned to associates on the South dock than there are doors on the dock itself. Constraint (15) does the

same for the Matthews.

Admittedly, the construction of this model was only meant to find a quick mapping alternative

that was based off some hard average data. After the remap model proposed an “optimal” map, the data

file for dynamic allocation was adjusted in order to account for the changes in what stores were assigned

to what doors. With a revised data file, the dynamic allocation model was run again as a test of the

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monthly performance with a better map. The comparison of the current map vs. a new one are found in

the results section of this report.

4 Results

This next portion of this report dives into the comparison of expected productivity results using a

dynamic allocation policy versus static. There is also an analysis of expected productivity if an

optimization tool such as the one created was implemented, as it compares to the real productivity figures

seen over the month of January. Last is the comparison between the dynamic allocation results under the

current map and the results after a remap occurred, giving a little insight into whether or not remapping

when using a dynamic allocation optimization tool actually matters.

4.1 Dynamic vs. Static Allocation Policies

Based on the decision making method employed in the dynamic allocation model, it should be

reasonable to expect that CPH in the dynamic model should be significantly better than a more rigid,

static model. In fact, dynamic resource allocation should always perform better than static resource

allocation.

Over the month of January, a static resource allocation policy would never allow associates to

exceed the goal CPH. In figure 7 it is made clear that the problem with using a strict policy that evenly

distributes door assignments across associates, while it seems fair from the door perspective, squashes the

facilities chances of maximizing productivity and thus minimizing cost. Under this policy, associates

would come within 5% of the goal CPH less than a quarter of the time, and would never actually reach it.

Instead, under a dynamic allocation policy, associates can almost always be assigned enough work to

reach the goal. When assigned the optimal number of doors, associates would meet or exceed their goal

97% of the time. Associates would exceed the goal of 266 cartons in the South by an average of 11% over

the course of the month. In only three hours would associates in the South not be able to meet the goal.

This happens only when volume in the area is not large enough to realistically meet the goal, or when

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there is no such iteration of door assignments that brings all associate’s underachievement of productivity

to zero. In summary, the average CPH that management should expect to see in the data analyzed under

the dynamic allocation policy is 294 cartons, with a standard deviation of 17 cartons. Under a static

allocation policy the management team should expect to see an average of only 243 cartons, with a

standard deviation of 13. It is true that the static model in this case yielded slightly more consistent

productivity results given the smaller standard deviation, but the results were consistently poor. However,

it is clear that the benefits of much higher average CPH would outweigh the cost of a slightly higher

variation in results.

Figure 7

Building 123, where the Matthews, East and West sorters are located, usually has less total

volume going to it, which is spread across a larger number of doors. Because of this, the dynamic

allocation policy is not able to exceed productivity goals as nicely as it can in the South, effectively

creating more variation in optimal results. Nevertheless, the assignment of doors to associates is always

optimized in order to enable associates to process the most number of cartons possible in the hour. It is

interesting to observe in this area of the facility, that there are instances where productivity under both

policies is the same. This occurs for two reasons. Either first, because the volume and fluctuation is not

able to bring productivity high enough, or second, because the model ends up deciding on the same

050

100150200250300350400

CPH

Hour

South

Static Dynamic Goal

24

assignment of doors under both policies. It is also possible under some circumstances that the model

would decide on a slightly different assignment of doors, which still yielded the same results.

The following results, shown in figures 8 and 9, were seen when comparing the dynamic policy to

the static. For fluid loading, the model found assignment solutions that yielded an average of 255 CPH

with a standard deviation of 59 cartons. While this average is still 11 cartons away from the goal, the

reality of volume distribution in building 123 simply does not allow productivity to be any higher. At the

very least, this is an improvement over the 205 average CPH of associates being rigidly assigned doors

under a static policy. For associates working in zonal loading areas, average expected CPH would be 169

under the dynamic policy and 119 under the static. With a zonal loading goal CPH of 170, the dynamic

allocation policy is clearly more desirable. Standard deviation under the dynamic versus static policies

were 43 and 19 cartons respectively. While the dynamic allocation of resources tends to lead to more

variation on how high productivity can possibly be, at least associates are able to work to their maximum

potential.

Figure 8

0

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100

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200

250

300

350

400

CPH

Hour

123 Fluid

Static Dynamic Goal

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Figure 9

Ultimately it is clear that dynamically assigning associates to doors based on the expected volume

yields significantly better results than simply assigning every associate the same number of doors. At the

end of the day, the model is able to find ways to allocate an optimal number of associates to an area, so

that the same number of cartons can be processed with less labor. This is essentially another way to say

that each worker is being more productive. For the month of January, the impact for the entire building

from using the dynamic resource allocation model versus the static was just above a 20% reduction in

labor hours. In monetary terms, assuming an average hourly wage of $12, the cost savings of employing

such an allocation strategy in the month of January would be more than $5,500.

4.2 Dynamic Allocation vs. Actual Performance on the South Dock

In reality, the management team does not employ a static allocation policy. The static policy is

meant to show the most rigid level of decision making, and how the dynamic model measures against it.

While manager’s decision making processes are not as computational as the optimization model

constructed, they do perform fairly well compared to the goal productivity rates. The effect of decisions

actually made, should always be between the static and dynamic results. The effects of management’s

0

50

100

150

200

250

300

CPH

Hour

123 Zonal

Static Dynamic Goal

26

decisions, within reason, can only ever be as bad as the static model, and can only ever be as optimal as

the dynamic allocation decisions. This section is meant to show the effect of actual allocation decisions

made on the South dock in the month of January, as they compare to the dynamic model. In other words,

this section shows what performance could have been for the South, had managers had and utilized

volume distribution data prior to making staffing and door assignment decisions.

The reason why results in Building 123 are not being considered is because actual productivity

results in the daily sheets are skewed by the presence of a large number of support associates. Support

associates are additional workers assigned to an area of the building who normally close trailers, but also

at times help loading associates take the edge of high volume doors. Essentially some number of support

associates might help the outbound associates load trailers for a certain amount of time, effectively

increasing the number of cartons processed, but their hours would not be added to the calculation. The

help from these associates artificially boosts average CPH in Building 123, and thus makes the data less

useful in analysis. For this reason, the zonal and fluid results cannot be used to show the effectiveness of

implementing a dynamic allocation policy.

Figure 10 shows the actual hourly productivity observed for South associates over the month of

January and the potential productivity of associates across the month using the dynamic allocation model.

Actual productivity over the month averaged 249 cartons, with a standard deviation of 48 cartons. Both

the visible variation seen in the graph, and the high standard deviation show that results are inconsistent

across the month. While average CPH is not far from the goal, the numbers reveal that the decision

making method currently employed does not consistently deliver the same results. Instead, the dynamic

allocation model not only delivers better results, but performance is significantly more consistent hour

over hour. As stated in the previous section, average CPH using the dynamic allocation model could have

gone as high as 294, with a standard deviation of only 16 cartons. There are some cases on the South

where actual productivity beats optimal productivity. While these moments are only outliers over a series

of very successful results, the reason the model was not able to beat productivity always lied in the need

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to assign one additional associate than was desired. If there was one store with exceptionally high volume

in a given hour, which was high enough to bring the maximum volume constraint into play, but not high

enough to bring the additional associate’s productivity up to the goal, the model would have decided to

assign only one person to that door for that hour. Ultimately the South actually only hit the productivity

target 36% of the time in January, while optimal decision making would have given them the potential to

meet or exceed the target 97% of the time. In reality, associates exceeded the goal of 266 cartons in the

South by an average of 4% over the course of the month of January, while the model exceeded the goal

by an average of 11%.

Figure 10

Through these figures, the value of getting the right data at the right time is clear. Using the

dynamic allocation model and finding a way to obtain the volume distribution data before the staffing and

assignment decisions are made could have resulted in a 18% increase in average outbound productivity

for the South dock in the month of January. Ultimately having a better assignment of associates to doors,

means that number of associates required to do the same amount of work would drop. By making optimal

assignment decisions, the total productive hours required to do the same amount of work in the South

0

50

100

150

200

250

300

350

400

CPH

Hour

South

Actual Model Goal

28

would have gone from 1,314 to 1,109 over the course of a month, a near 16% drop. Zooming into any

random hour in the month, the number of outbound associates needed on the South dock would drop from

an average of 12 to 10. Assuming an average hourly wage of $12, and taking the difference in total hours

saved, this would have resulted in almost $2,500 in cost savings over the month.

4.3 Current Map vs. Remap under Dynamic Allocation Policy

This section provides an analysis of the dynamic allocation performance of the current map as it

compares to the performance of an improved map executed through the remap optimization model. As

stated earlier, one last question that should be answered is whether or not the specific location of stores in

the facility actually matters when employing an optimization tool that maximizes productivity.

Management had already seen significant gains in productivity through remaps done in the past, but these

results were observed under a manual assignment allocation process. In other words, if the dynamic

model just revealed the best possible performance that the building could obtain, would a rearranging of

stores to different doors increase that maximum? The answer is, at least given this one iteration of a new

map, yes.

The remap that was executed does not rearrange every door, but only those doors that the

management team is willing to change at this time. Some stores that belonged to the South dock moved to

the Matthews, while others might have just moved to the next door over. Mapping decisions were based

off of a hard average volume that would be going to the store over the course of the month. Because no

zonal stores were moved, the results of 123 zonal stayed exactly the same. However, something very

interesting happened between the South dock and the Matthews.

Average productivity on the South dock actually dropped slightly after remapping. Before the

remap, the South had an average CPH of 294 cartons, which dropped to 291 after the remap. However,

the Matthews experienced slightly higher average CPH with 260 after the remap as opposed to 255 with

the current map. Overall there was a 0.73% improvement in fluid productivity after the fluid stores were

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remapped. Had this particular remap been executed before the month of January, average potential

productivity would have jumped by 2 cartons. While this number is admittedly very small, its significance

is large. The reason why is because it indicates that the location of stores in the facility actually does

matter when it comes to potential performance. It means that given enough data and computational

bandwidth, there is an optimization model that could find the absolute best mapping of the outbound

docks. While the construction of such a model was outside of the original scope of this project, pursuing

this would be one of the logical next steps to take for the company.

5 A Note on Goal CPH

Through this study, it also became apparent that the productivity goals set by the management

team are not always feasible to obtain. Because the dynamic allocation model’s objective is to bring each

associate’s expected CPH up to or above the goal CPH, when this does not happen it indicates that it is

not actually possible. In these instances it is not reasonable to shoot for what cannot physically be

obtained. Instead this model presents a new way to benchmark performance. Throughout the report, the

word productivity was hardly ever used without the word potential before it. As was mentioned, the

results seen in the model are not the actual results of each associate, but rather the productivity that they

could reach had they been assigned a particular set of doors. Instead of benchmarking off of a set goal

CPH, it would make more sense to benchmark off of the best possible area result for that particular hour.

If the management team does decide to explore the implementation of a linear programming model, this

new approach to defining the goal should also be considered.

6 Conclusion

Through the analysis of these models, it became clear that the way forward for Macy’s is to

employ a linear program that dynamically makes decisions for managers in response to the constant

fluctuations in volume. While the current manual decision making process yields much better results than

a very basic and static process, there would be significant dollars savings in finding a way to capture

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expected volume data before resource allocation decisions are made. The proposed dollar savings

introduced in the results section were only shown for the span of one month. Savings over the course of a

year could be in the tens of thousands. This type of optimization and systematic problem solving is where

the future of minimizing distribution costs lies. With organizations buried in raw data, it is applications

such as this where true value can be found.