A Supply Chain Optimization Model: Nairi Nazarian A thesis ... · DC cost includes: Receive – Put away – Pick ... DC – Single DC central-stock strategy Table 4 Number of ZipLoc

A Supply Chain Optimization Model:

Minimizing Transportation and

Inventory Cost

Nairi Nazarian

A thesis submitted in partial fulfillment

of the requirements for the degree of

BCHELOR OF APPLIED SCIENCE

Supervisor: R.H. Kwon

Department of Mechanical and Industrial Engineering

University of Toronto

March, 2007

A Supply Chain Optimization Model: Minimizing Transportation and Inventory Costs

1

Abstract

In this thesis I have studied the three in-stock strategies – flow-through,

regional and single DC central stock - currently used by Hudson’s Bay Company

(Hbc) and thus developed a simple transportation-inventory model in order to

compare their total costs. I have also described a distribution model proposed by

(Berman et al [2006]) in which the model is formulated as a non-linear integer

optimization problem. Due to the non-linearity of the inventory cost in the

objective function, two heuristics and an exact algorithm is proposed in order to

solve the problem.

The results obtained from the transportation-inventory models show that the

single DC and regional central stock strategies are more cost-efficient

respectively compared to the flow-through approach. It is recommended to take

the single DC and the regional central stock strategies for slow moving and

demanding products respectively.


2

Acknowledgements

I would like to thank my supervisor, Professor Roy Kwon, for providing me with help

and support especially for his guidance in determining my thesis topic.

I would also like to thank D.A. Power at Hbc’s Supply Chain Optimization for helping

and supporting me throughout the course of the thesis project.


3

TABLE OF CONTENTS

Abstract ………………………………………………………………….……………….i Acknowledgements……………………………………………………………………………….….ii Table of Contents………………………………………………………………………………….….iii List of Symbols…………………………………………………………………….……iv

List of Figures……………………………………………………………………….…..x

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

1. Motivation…………………………………………………………………....….1

2. Introduction………………………………………………………………..…….2

3. Literature Review…………………………………………………………..……3

Supply Chain Network Design…………………………………………..…..3

Hudson’s Bay Company (Hbc) Inventory Structure and Product Flow….…4

4. Models………….…………………………………………………………..……7

Preliminary Distribution Model…………………..…………………….…...7

Cross-dock and Direct Shipment Models (Berman et al[2006])……….…...12

Distribution Strategies………………………………………….……13

Model Assumptions…………………………………………………15

Formulation………………………………………………………….16

Hbc Transportation and Inventory Model……………………..….………...18

Transportation Model Formulation………….…………………..…..20

Inventory and Transportation Model Formulation…………….…….20

Programming Using OPL……………………………………………21

Data Acquisition……………………………………………………..21

5. Comparison of Models – Results……………………..…………………….……22

6. Future Research..………………………………………………………….……..28

7. Conclusion……………………………………………………………………….28

8. References………………………………………………………………….……29

9. Collections of Figures and Tables………………………………………………..30

Appendix A…………………………………………………………………………..37

Appendix B………………………………………………..…………………………39

Appendix C…………………………………………………………………………..44

Appendix D……………………………………………………………………….….46

Appendix E…………………………………………………………………….…….48

Appendix F…………………………………………………………………………...50

Appendix F-1………………….…………………………..………………….….50

Appendix F-2…………………………………………………………………….53


4

List of Symbols

Preliminary model

m = Number of stores or demand points

n = Number of potential DCs

l = Number of suppliers

t = Number of potential cross-dock locations

=jD Weekly/Annual demand from store j

=iK Potential capacity of DC at site i

=hS Supply capacity at supplier h; truckload (TL) and less-than-truckload (LTL)

=eW Potential cross-dock capacity; Dummy Variable

=eF Fixed cost of processing shipment from supplier at cross-dock site e

=eV Variable cost of processing shipment from supplier at cross-dock site e

=iF Fixed cost of investment at DC i

=iV Variable cost of processing shipment from supplier at DC i

=hic Cost of shipping one unit from supplier h to DC i

=hjc Cost of shipping one unit from supplier h to store j

=hec Cost of shipping one unit from supplier h to cross-dock site e

=ejc Cost of shipping one unit from cross-dock site e to store j

=ijc Cost of shipping one unit from DC i to store j

=ic Cost of holding one unit at DC i


5

=jc Cost of holding one unit at store j

=iy 1 if DC is located at site i, 0 otherwise

=ey 1 if cross-dock is done at site e, 0 otherwise

=ejx Quantity shipped from cross-dock e to store j

=ijx Quantity shipped from DC at site i to store j

=hix Quantity shipped from supplier h to DC at site i

=hex Quantity shipped from supplier h to cross-dock at site e

=hjx Quantity shipped from supplier h to store j


6

Cross-Dock and Direct Shipment Models Suggested by (Berman et al

[2006])

I = Set of Suppliers

J = Set of Plants

P = Set of products

K = Set of cross-docks

C= Truck capacity

=d

ijc Direct transportation cost of shipping one truckload of products from

supplier i to plant j

=i

ikc Inbound transportation cost of shipping one truckload of products from

supplier I to cross-dock k

=o

kjc Outbound transportation cost of shipping of one truckload of products from

cross-dock k to plant j

=d

ijt Direct transportation time (periods) from supplier i to plant j

=k

ikt Inbound transportation time (periods) from supplier i to cross-dock k

=o

kjt Outbound transportation time (periods) from cross-dock k to plant j

=kT Time spent transferring from inbound to outbound at cross-dock k

=pb Truck capacity occupied by one unit of product p

=ph Inventory-carrying cost of one unit of product p per period

F = },0,,,:),,{( >∈∈∈ ijpdPpJjIipji set of flows

},0,,:),,{( >∈∈= ijpdPpIipjiF set of flows to plant j


7

},0,:),,{( >∈= ijpij dPppjiF set of flows from supplier i to plant j

}),,(:{ ijij FpjipP ∈=

}),,(:),{( jj FpjipiIP ∈=

=ijpkx {1 if flow (i,j,p) is shipped through cross-dock k; 0 otherwise


8

Transportation Model


n = Number of potential DCs (Distribution Centres)

l = Number of suppliers

t = Number of products

=pjD Demand for product p at store j


=phiS Supply capacity at supplier h to DC i for product p

=pijT Cost of transporting product p from supplier at DC site i to store j

=iC DC Cost of processing shipment from supplier at DC site i;

DC cost includes: Receive – Put away – Pick - Ship

=pijX Number of product p shipped from DC i to store j


9

Inventory and Transportation Model


n = Number of potential DCs (Distribution Centres)

l = Number of periods

k = Number of products

=pjtD Demand for product p at store j in period t


=phiS Supply capacity at supplier h to DC i for product p

=ijT Cost of transporting of one unit of product from supplier at DC site i to

store j

=iC DC Cost of processing shipment from supplier at DC site ij;

DC cost includes: Receive – Put away – Pick - Ship

pI = Inventory cost for product p

pjII =Initial inventory product p at store j

=pijtX Number of product p shipped from DC i to store j at period t

pjtY = Number of product p stored at store j in period t


10

List of Figures

Figure 1 Schematic representation of the process that takes

products to get onto stores’ shelves

Figure 2 Distribution strategies (Shapiro [2005])

Figure 3 Distribution strategies (Berman et al [2006])

Figure 4 Product delivery routes


11

List of Tables

Table 1 Distribution Stratégies and Inventory Costs for a full

truck (Berman et al [2006])

Table 2 Products’ shipment through Toronto cross-dock

Table 3 Number of Shower Cleaner Kits shipped from Toronto

DC – Single DC central-stock strategy

Table 4 Number of ZipLoc Sandwich Bags shipped from Toronto

DC – Single DC central-stock strategy

Table 5 Number of Shower Cleaner Kits stored in the stores

(Inventory)

Table 6 Number of ZipLoc Sandwich Bags stored in the stores

(Inventory)

Table 7 Cost Comparison


12

1. Motivation

In summer 2006 I had an opportunity to work as a replenishment intern

at the Hudson’s Bay Co. (Hbc) and gain valuable hands-on experience in the

domain of Supply Chain (SC). Prior to my internship at the Hbc I had also had

summer research experience in the SC area with Professor Kwon. My research

topic was to basically develop a Mathematical Optimization model for the Safety

Stock in a Multi-Echelon Inventory system subject to uncertain demand with a

quoted Guaranteed Service Time at each echelon. My combination of the

research and practical SC experience and interest towards this area as well as

the recommendation of Professor Kwon made me to prepare my thesis on the

Optimization of the Hbc’s Supply Chain network. The project that I am assigned

to is part of the Hbc Strategic Inventory Optimization and Logistics Network

Design project that determines the inventory deployment decisions regarding the

selection of locations and facilities where products get stored or cross-docked.

The project was initially introduced to me by D. A. Power at the Hbc’s

Optimization division.


13

2. Introduction

Achievement of significant cost savings and improvements in profitability

requires a typical retail company to make long-term decisions regarding the

structure of its supply chain network and bringing its facilities, suppliers and

customers closer together under the strategic supply chain planning (Shapiro

[2005]). SC network optimization models allow us to model multiple inventory

deployment decisions under multiple scenarios and finally analyze the results of

each scenario. That being said my purpose here is to model the Hbc’s logistics

network, and to study and analyze the trade-off between taking flow-through

and central stock approaches- which will be explained in “Supply Chain Network

Design” section- in terms of total costs as well as in-stock position. In the next

step the model will be coded in OPL (Optimization Programming Language). In

this thesis I have studied two major distribution strategies of Hbc’s supply chain

network namely cross-dock and central stock approaches for one of its suppliers-

SC Johnson. As part of my study I have developed a transportation-inventory

model for a single source to multiple destinations scenario. I have also studied

the distribution model proposed by (Berman et al [2006]) that can be used for

Hbc’s distribution system.


14

3. Literature Review

3.1 Supply Chain Network Design

Before discussing the importance and effect of the inventory deployment

decisions in the supply chain network optimization, it is beneficial to go over the

major reasons for going through the process of the supply chain redesign and

optimization. According to Shapiro et al [2004] the triggering events for the

design of the supply chain can be mergers, acquisitions, inconsistent service

levels, rapid growth, poor asset utilization, rising costs, etc. These are all key

drivers that can force the companies to reconsider their supply chain in order to

better understand and meet their customers’ needs by efficiently utilizing their

manufacturing and distribution resources and potentials. According to (Shapiro et

al [2004]), in order to study the primary objectives of supply chain design

projects, SLIM Technologies-a leading provider of supply chain management

solutions- has chosen a number of companies that had conducted the project.

Almost all companies had agreed that cost reduction, maximization of asset

utilization and service improvement were the top three motives for the redesign

of the supply chain.

According to (Chopra [2003]) there are four decisions to be made

regarding the design of the supply chain network. The first decision is related to

the facility role-decisions that define the role and processes that are performed


15

at each facility. The second decision is regarding the facility location. Since

facility location decisions affect subsequent SC decisions; therefore, all aspects of

these decisions have to be carefully studied. Two major considerations would be

the cost associated with locating the facility and its proximity to the market.

Capacity allocation decisions also determine the amount of capacity to be

allocated to each facility and finally decisions regarding market and supply

allocations decide what markets and which supply sources should each facility

respectfully serve and be fed from.

Fisher [1997] in his paper “What is the right Supply Chain for your

product?” emphasizes the consideration of the nature of the demand for one’s

products before devising a supply chain. In other words what he identified as the

main cause of problems in supply chain was a mismatch between the type of

product and the type of supply chain. According to Fisher [1997] on the basis of

products demand patterns, they either fall into functional or innovative

categories of products.

3.2 Hudson’s Bay Company (Hbc) Inventory Structure and Product

Flow

Before getting into the modeling stage of the Hbc’s inventory across its

supply chain I would like to briefly explain the three approaches to Hbc’s in-stock

strategy namely flow-through, central stock (rapid replenishment and slow


16

moving). Furthermore I will explain what happens in an Hbc DC from the very

first stage where the product is received from the supplier until it reached onto

the stores’ shelves.

As mentioned earlier the Hbc in-stock strategy consists of 3 approaches:

The first approach is flow-through which locates the stock close to suppliers for

the most cost efficient movement of product from suppliers to stores. The

second approach which is regional central stock (rapid replenishment) locates

central stock inventory close to stores for fast allocation when it is needed. And

finally the last approach which locates slow-moving central stock in one facility to

accommodate fluctuations in demand and special buys is called single DC central

stock (slow-moving).

That being said, the industry best practice is to stock fast-moving SKUs

regionally to ensure a reliable in-stock position whereas the Hbc’s current

network is well-designed for flow-through; therefore, Vision 2524 (industry best

practices) is adopted to determine the best stocking strategies for commodities.

The other issue that is worth knowing is the process that a product goes

through in order to reach onto the stores’ shelves. This process contains 4 stages

(Figure 1). The first stage is receipt which is responsible for booking receiving

appointments from the suppliers and also managing the receiving of shipment


17

and printing of labels of products. The above mentioned roles at the first stage

fall under Inbound Management System (IMS) and National Purchase Order

Management System (NPOMS) respectively. The second stage is also responsible

for receiving shipments, allocating pallets to storage slots and printing labels. At

this stage the shipment is either delivered based on flow-through or central stock

approaches. Warehouse Management System (WMS) and NPOMS control this

stage. At the third stage (Splitcase) pickers handle the received batches by store.

This stage falls under pick and pack (P&P) system. As the batches got handled

by store, they move to the next stage where they get sorted and get ready for

shipment. This stage falls under Warehouse Control System (WCS). Lastly the

batches get shipped to the stores by tractors and trailer. The Transportation

Management System (TMS) is responsible for managing the fleet of tractors and

trailers.

Figure 1 - Schematic representation of the process that

takes products to get onto stores’ shelves (Courtesy of Hbc)

CENTRAL

STOCK

FLOW

THROUGH

SPLITCASE SHIP TRANSPO

RT RECEIVE


18

4. Models

4.1 Preliminary Distribution Model

Before getting into the modeling portion of my work, I would like to briefly

explain inventory requirements for each of the product groups and how they

vary according to different distribution channels (Shapiro [2005]). As shown in

Figure 2, products can flow in three different paths. In the first path, product is

shipped through a cross dock to a store, meaning that no inventory is held in

that place. Inventory is only held at a store. Costs associated with this path are

transportation as well as fixed and variable processing costs at cross dock site.

Cost of transportation is also related to the shipment volume (either truckload

(TL) or less-than-truckload (LTL). In the second path, product is directly shipped

by the supplier to stores. The only costs associated with this path are the costs

of transportation and inventory at stores. In the third path inventory is only held

both at DCs and stores. Again transportation, inventory holding and fixed as well

as variable processing costs are considered for this path.


19

Figure 2 – Distribution strategies (Shapiro [2005])

In the modeling of the network, I have considered a supply chain in which

suppliers ship product either directly to stores, or cross dock site or DCs as

explained earlier. Location and capacity allocation decisions have to be made for

distribution centre (DC). Multiple DCs may be used to satisfy demand at a

market.

The goal is to identify distribution locations as well as quantities shipped

between various points that minimize the total fixed and variable costs. Define

the following decision variables:

The preliminary version of the problem is formulated as the following integer

program:

Suppliers

Distribution

Centre (DC)

Cross-Dock

Stores

(1) (2) (3)

vf CC ,

tC

tC

tC

hvf CCC ,,

tC

hC


20

Objective Function

Min

∑ ∑∑∑∑∑= = =

+

=

+

=

++

=

++

l

h

l

h

t

e

xc

n

i

xc

t

e

VFy

n

i

VFy hehehihieeeiii

1 1 1

*

1

*

1

)(*

1

)(*

∑ ∑ ∑∑ ∑∑= = == =

+

=

+

l

h

t

e

m

j

xc

n

i

m

j

xc

m

j

xc ejejijijhjhj

1 1 1

*

1 1

*

1

*

Subject to

1. ∑∑∑= = =

≤++

n

i

t

e

h

m

j

hjhehi Sxxx1 1 1

)( for h=1,…, l

Total amount shipped from a supplier cannot exceed the supplier’s capacity

2. ii

m

j

ij yKx *1

≤∑=

for i=1,…,n

Amount stocked in the DC cannot exceed its capacity

3. ∑ ∑= =

≡−

l

h

m

j

ejhe xx1 1

0 for e=1,…,t

The amount shipped out of a cross-dock site is exactly equal the amount

received from the supplier


21

4. ∑ ∑= =

≥−

l

h

m

j

ijhi xx1 1

0 for i=1,…,n

The amount shipped out of a DC site cannot exceed the amount received from

the supplier

5. ∑∑∑= = =

=+

l

h

n

i

t

e

jijej Dxx1 1 1

for j=1,…,m

The amount shipped to a customer must cover the demand

The objective function minimizes the total fixed and variable costs of the

supply chain network. The constraint in equation 1 specifies that the total

amount shipped from a supplier cannot exceed the supplier’s capacity. The

constraint equation 2 enforces that amount stocked in the DC cannot exceed its

capacity. The constraint in equation 3 states that the amount shipped out of a

cross-dock site is exactly equal the amount received from the supplier. The

constraint in equation 4 specifies that the amount shipped out of a DC site

cannot exceed the amount received from the supplier. The constraint in equation

5 specifies that the amount shipped to a customer must cover the demand.

4.2 Cross-Dock and Direct Shipment Models (Berman et al [2006])

Before getting into the details of the final model that I developed and

used for 1 supplier, 1 cross-dock and 2 distributions centers case as well as

assumptions that I made in order to maintain the linearity of the objective

function I would like to briefly describe an optimization model suggested by


22

(Berman et al [2006]) that is the solution to a distribution strategy selection

problem where cost functions of both direct delivery and shipment through a

cross-dock are modeled and compared.

According to (Berman et al [2006]) there are two important issues in the

supply chain area that contribute to the total cost of the supply chain network

namely transportation and inventory costs. That being said retail companies can

achieve significant savings by considering these two costs at the same time

rather than trying to minimize each separately. As mentioned above in this paper

the two distribution strategies mainly direct delivery and shipment through cross-

dock are considered where a group of products are shipped from a set of

suppliers to a set of plants. The cost function consists of the total transportation,

pipeline inventory, and plant inventory costs. The presence of the plant inventory

cost has made the model to be formulated as a non-linear integer programming

problem. According to (Berman et al [2006]) the objective function is highly non-

linear and neither convex nor concave; therefore, a greedy heuristic is suggested

to find an initial solution and an upper bound. And then a branch-and-bound

algorithm is developed based on the Lagrangian relaxation of the non-linear

program.

Before getting into the formulation portion of the model, I am going to

provide a brief background of the two distribution strategies discussed in the

paper and then briefly state the assumptions made by (Berman et al [2006]) in

order to have a solvable problem. According to Jonathan Patrick a post-doctorial


23

fellow at Center for Health Care Management, Sauder School of Business, UBC

“As far as your model accurately reflects reality, it is not solvable; and as far as it

is solvable, it will not reflect reality”.

4.2.1 Distribution Strategies

For many retail companies products are shipped by suppliers through one

of the following shipment strategies. The first one is direct shipment where

products get shipped directly from the supplier to the DC/plant without stop. The

second method of shipment is milk-run (peddling) where trucks pick up products

from different suppliers on their ways and finally drop them at one or several

DCs. The last but not least is cross-dock where products get shipped to DCs

through cross-dock by suppliers. Below is a graphical representation of the three

distribution strategies.


24

Figure 3 – Distribution stratégies (Berman et al [2006])

Each of these three distribution strategies has its own advantages and

disadvantages over the other strategies based on the delivery time and the

transportation cost criteria. Products delivered directly from the supplier to the

plant cost less due to its short distance between the supplier and the plant and

also take less time to get to the plant. As for cross-dock the transportation cost

is high due to the long distance between suppliers and plants and therefore

takes longer to deliver the products. An advantage of the cross-dock strategy

over the direct delivery is that products from several suppliers can be shipped

from a single cross-dock site to the plant which means high delivery frequency

and low plant inventory whereas in the case of direct delivery only products from

the same supplier are shipped to the same plant which means low delivery

frequency and high plant inventory. The Table 1 represents the relationship

between each distribution strategy and its associated cost.

Diretc

Milk-run

Cross-dock

Supplier

Plant

Cross-dock


25

Distribution

Strategy

Delivery Time Delivery

Frequency

Inventory

Cost: Pipeline

Inventory

Cost: Plant

Direct Short Low Low High

Milk-run Medium Medium Medium Medium

Cross-dock Long High High Low

Table 1 – Distribution Stratégies and Inventory Costs for a full truck (Berman et al [2006])

4.2.2 Model Assumptions

As mentioned earlier to have a solvable problem, a couple of assumptions

have been made in this paper.

1. It is assumed that the product quantities are infinitely splittable, in other

words a product can be shipped in any quantity within a vehicle shipment.

2. Delivery frequency can be any positive number and is not limited to a set

of potential nembers.

3. Products are always available for shipping at suppliers, no matter which

distribution strategy is chosen.

4. Inbound-outbound coordination at the cross-dock is ignored.


26

5. All units of the same flow (a flow is a combination of supplier, plant and,

product) are assigned to the same transportation option, i.e., direct or

through the same cross-dock.

6. Each truck is fully loaded. Only the volume of products is concerned when

calculating truck capacity usage. The transportation costs are only

determined by the source and destination, regardless of the weight.

4.2.3 Formulation

Below is the nonlinear integer mathematical formulation of (Berman et al

[2006])’s model.

Objective Function

Min ∑∈

0

)(Kk

k Xg

Subject to

(1) ∑∈

=0

1kk

ijpkx ,),,( Fpji ∈∀

(2) }1,0{∈ijpkx 0,),,( KkFpji ∈∈∀

In this model if k is 0-in other words, if 10 =ijpx means that flow (i,j,p) is shipped

directly. X in the objective function represents the vector of all decision variables

ijpkx and )(Xg k is the total of transportation, pipeline inventory, and plant


27

inventory costs to ship flows through cross-dock k. Constraint 1 ensures that

every flow is delivered and constraint 2 makes sure that the same flow follows

the same route (Berman et al [2006]).

The following two equations are the expanded forms of )(Xg k first for direct

shipment and second for shipment of flows that travel through cross-dock. The

detailed objective function can be found in (Berman et al [2006]).

Total cost of direct delivery-transportation costs, pipeline inventory costs and

inventory costs:

∑∑∑∈∈ ∈

++=

ijPpd

ij

ijpijpp

ijpijpp

d

ij

d

ij

Ii Jj

d

ijf

xdhxdhtcfXg )]

*2

*****(*[)(

0

00

In the above formula the frequency of shipment between each supplier-plant

pair (i,j) is formulated as follows:

Cxdbf

ijPp

ijpijpp

d

ij /** 0∑∈

=

The total cost of shipping flows that travel through cross-dock k consists of two

parts: the inbound transportation cost and outbound transportation cost.

According to (Berman et al [2006]) the shipping time through cross-dock k is

formulated as )( 0

kjk

i

ik tTt ++ that is used in the calculation of pipeline

inventory.

∑ ∑ ∑∑∈ ∈ ∈ ∈

+++++=

Ii Jj Jj IPpi

kjijpkijppijpkijppkjk

i

ikkjkj

i

ik

i

ikk

j

fxdhxdhtTtcfcfXg),(

0000 )]2/(****).[(**)(


28

Where

∑ ∑∈ ∈

=

Jj Pp

ijpkjp

i

ik Cxdbf /** and

∑∑∈ ∈

=

Ii Pp

ijpkijppkj Cxdbf /**0

For each Ii∈ and Jj∈ are the frequency of inbound and outbound shipment.

4.3 Hbc Transportation and Inventory Model

As discussed earlier a distribution model that integrates both

transportation cost and inventory cost in its objective function can be modeled as

a non-linear integer optimization program and solved by branch-and-bound

algorithms and other heuristic methods due to the inherent difficulty of such

problems.

That being said for this project in order to have a linear objective function

and be able to get answers using optimization programming languages (OPL) as

well as due to the time limitation, I have prepared my models for single source

and single destination cases and at the end compared their total costs. My

model consists of one supplier in Ontario, two DCs-one in Toronto and the other

one in Calgary and finally nine retail stores in Calgary. The DC in Toronto also

can act as a cross-dock site. Figure 4 is a graphical representation of the

possible shipment routes.


29

Figure 4 - Product delivery routes

Three scenarios are considered:

Scenario 1: Products are delivered from the supplier through cross-dock at

Toronto to stores in Calgary- Flow-through approach

Scenario 2: Products are shipped from the supplier to the Toronto DC-products

are stored in Toronto DC - and shipped to the stores in Calgary – Single DC

Central Stock (slow moving products) approach

Scenario 3: Products are shipped from the supplier to the Calgary DC – products

are stored in Calgary DC – and shipped to the stores in Calgary – Regional

Central Stock (Rapid Replenishment) approach

Due to the simplified nature of the model, I have formulated and coded two

simple transportation and inventory models. Below are the mathematical

formulations of the models.

Supplier- SC Johnson

Calgary DC Toronto

DC/Cross-

dock

Calgary Stores (9)

1 2 9


30

4.3.1 Transportation Model Formulation

Objective Function

Min

∑ ∑ ∑= = =

+∗

t

p

n

i

m

j

ipijpij CTX1 1 1

)(

Subject to

6. ∑=

=

m

j

phipij SX1

for h=1,…, l ; p=1,…,t; i=1,…,n

7. ∑=

=

n

i

pjpij DX1

for p=1,…,t; j=1,…,m

8. i

t

p

pij KX ≤∑= 1 for i=1,…,n; j=1,…,m

4.3.2 Inventory and Transportation Model Formulation

Objective Function

Min

∑ ∑ ∑ ∑= = = =

++

k

p

n

i

m

j

l

t

pjtppijtijr YIXTC1 1 1 1

**)(


31

Subject to

1. ∑∑= =

≤

k

p

m

j

ipijt KX1 1

for i=1,…,n; t=1,…,l

2. ∑=

−+=+

n

i

pjtpjtpijttpj YDXY1

)1( for p=1,…,k; j=1,…,m;t=1,…,l

3. pjpjt IIY = for p=1,…,k; j=1,…,m

4.3.3 Programming Using OPL

At the next stage of my research I coded the mathematical formulations

using ILOG’s OPL (Optimization Programming Language) software. In OPL

the model and the data are saved in two separate files and once it is run

successfully solutions to the objective function as well as the decision

variables defined in the model appear in a separate window at the bottom of

the coding window. Details of the codes can be found in Appendices A and D.

4.3.4 Data Acquisition

The data required to populate the models with were obtained through BIS

system at Hbc. Through BIS it is possible to run weekly, monthly, seasonal

and annual sales reports for any set of SKU/LOC (i.e. product-location)

combination as well as for any set of vendors. As mentioned earlier the data


32

are saved in a separate file. Details of the data can be found in Appendices B,

B and E.

5. Comparison of Models – Results

After populating the models with data and running them, the following results

were obtained from each scenario. Table 2 provides the quantity shipped

from the supplier through Toronto cross-dock to Calgary stores (scenario 1).

Since scenario 1 is a single source to multiple destination case and there is no

inventory held at the cross-dock site; therefore, the amount shipped to each

store is equal to its demand. The total delivery cost-transportation and DC

costs (receive, put away, pick and shipment) - for one period is $ 513.34


33

Shower Cleaner Kit ZipLoc Sandwich Bag

Calgary Store 108 1 11


Medicine Store 243 1 4

Lethbridge Store 286 1 13






Table 2-Products’ shipment through Toronto cross-dock

In scenarios 2 and 3 products are delivered from the supplier to the DCs

in Toronto and Calgary respectively. Products are stored and then shipped to

the stores in Calgary based on the demand at each store. The model for

these two scenarios incorporates a new decision variable namely the amount

of inventory and a cost parameter associated with it into the transportation

model from the first scenario. Tables 3 and 4 contain the number of shower

cleaner kits and ZipLoc sandwich bags that are shipped from Toronto DC to

all 9 Calgary stores in each of the 5 periods respectively. Tables 5 and 6 show

the amount of inventory held at each store after satisfying the demand in

that period. As we see in table 5 there are one or two units of shower cleaner


34

kit left at certain periods whereas for ZipLoc sandwich bag (table 6) the

inventory is zero. That being said the total delivery cost – transportation, DC

costs (receive, put away, pick and shipment) and inventory cost – for

scenarios 2 and 3 over all 5 periods are $ 1785.9 and $ 2201.9 respectively in

other words $ 357.18 and $ 440.38 for a single period.

The cost comparison of the three scenarios (table 7) shows that the

scenario 2 – shipment from the supplier to Toronto DC (central stock) – is

less costly compared to the scenarios 1 and 3 ($ 513.34 and $ 440.38

respectively).

Details of the solutions can be found in Appendices C and F-1 and F-2.

Period 1 Period 2 Period 3 Period 4 Period 5

Calgary_108 2 0 0 5 0

Calgary_109 1 1 1 0 3

Medicine_243 1 0 0 1 3

Lettbridge_286 1 0 0 1 3

Calgary_433 2 0 0 0 1

Calgary_435 1 2 1 0 1

Calgary_462 2 1 1 0 0

Calgary_480 1 0 0 0 0

Calgary_498 3 1 2 0 3

Table 3 - Number of Shower Cleaner Kits shipped from Toronto DC – Single DC central-stock strategy


35


Calgary_108 10 14 10 1 5

Calgary_109 6 13 7 16 5

Medicine_243 4 1 7 4 6

Lettbridge_286 14 13 4 7 7

Calgary_433 6 16 8 12 16

Calgary_435 7 2 4 0 6

Calgary_462 7 4 5 3 4

Calgary_480 12 7 5 3 4

Calgary_498 17 6 7 11 8

Table 4 - Number of ZipLoc Sandwich Bags shipped from Toronto DC – Single DC central-stock strategy


36


Calgary_108 0 0 0 2 2

Calgary_109 0 0 0 0 1

Medicine_243 0 0 0 0 0


Calgary_433 0 0 0 0 0

Calgary_435 0 0 1 0 0

Calgary_462 0 0 1 1 0

Calgary_480 0 0 0 0 0

Calgary_498 0 0 1 0 0

Table 5 - Number of Shower Cleaner Kits stored in the stores (Inventory)


37


Calgary_108 0 0 0 0 0

Calgary_109 0 0 0 0 0

Medicine_243 0 0 0 0 0


Calgary_433 0 0 0 0 0

Calgary_435 0 0 0 0 0

Calgary_462 0 0 0 0 0

Calgary_480 0 0 0 0 0

Calgary_498 0 0 0 0 0

Table 6-Number of ZipLoc Sandwich Bags stored in the stores

Scenario 1 Scenario 2 Scenario 3

Total cost for one

period(CAD$)

513.34

357.18

440.38

Table 7-Cost Comparison


38

6. Future Research

Although it has been concluded that delivery through cross-dock has the

longest distance, and therefore the highest transportation cost and the longest

delivery time but on the other hand cross-dock can combine products from

different suppliers which means high delivery frequency and low plant inventory.

Such trade-offs between transportation and inventory costs should be studied

more in order to obtain a near-optimal or local-optimal solution for a typical

distribution problem. Given that non-linear integer optimization models are hard

to solve due to the non-linearity of their objective functions; therefore,

developing heuristic methods and branch-and-bound algorithms so far have

proven to be very successful (Berman [2006]) for that reason the search for new

heuristic methods and algorithms should be continued.

7. Conclusion

In this thesis, I have considered the problem of selecting the most cost

efficient in-stock strategy – flow-through, regional and single DC central stock

approaches– for delivering two products from a single supplier to the Hbc’s

Toronto and Calgary DCs respectively and then shipping them to the stores in

Calgary so that the total transportation, distribution center and inventory costs

are minimized. A distribution model, proposed by (Berman [2006]) for the flow-


39

through (cross-dock) strategy is recommended and a simple transportation-

inventory model is developed to compare the cost associated with each strategy.

The results have demonstrated that taking the Single DC central stock strategy

with the DC in Toronto is the most cost-efficient. This strategy is suitable for

slow-moving and less demanding items. The second cost efficient strategy after

the Single DC central stock is the regional central stock. Even though this

approach is costlier than the first one but it is best suited for rapid moving items

for fast allocation when it is needed.

It is recommended to adopt Single DC and Regional central stock strategies for

slow and fast moving products respectively.


40

8. References Shapiro, J. F., Wagner S. [2005], “Strategic Inventory Optimization” SLIM

Technologies, October, 2005

Shapiro, J. F., Wagner S., Jacobs K. [2004], “A Practical Guide to Supply Chain

Network Design” SLIM Technologies, September, 2004

Fisher M. L. [1997], “What is the Right Supply Chain for Your Product?” Harvard

Business Review, 1997

Chopra S., Meindl P. [2003], Supply Chain Management: Strategy, Planning, and

Operations, 2nd ed. Prentice Hall

Berman O., Whang Q. [2006], “Inbound Logistic Planning: Minimizing

Transportation and Inventory Cost” transportation Science, 2006


41

9. Collection of Figures & Tables

Figure 1 (Courtesy of the Hbc)

Figure 2 (Shapiro [2005])

Suppliers

Distribution

Centre (DC)

Cross-Dock

Stores

(1) (2) (3)

vf CC ,

tC

tC

tC

hvf CCC ,,

tC

hC

CENTRAL

STOCK

FLOW

THROUGH

SPLITCASE SHIP TRANSPO

RT RECEIVE


42

Figure 3 (Berman et al [2006])

Distribution

Strategy

Delivery Time Delivery

Frequency

Inventory

Cost: Pipeline

Inventory

Cost: Plant

Direct Short Low Low High

Milk-run Medium Medium Medium Medium

Cross-dock Long High High Low

Table 1 (Berman et al [2006])

Diretc

Milk-run

Cross-dock

Supplier

Plant

Cross-dock


43

Figure 4 (Product delivery routes)

Shower Cleaner Kit ZipLoc Sandwich Bag



Medicine Store 243 1 4

Lethbridge Store 286 1 13






Table 2-Products’ shipment through Toronto cross-dock

Supplier- SC Johnson

Calgary DC Toronto

DC/Cross-

dock

Calgary Stores (9)

1 2 9


44


Calgary_108 2 0 0 5 0

Calgary_109 1 1 1 0 3

Medicine_243 1 0 0 1 3


Calgary_433 2 0 0 0 1

Calgary_435 1 2 1 0 1

Calgary_462 2 1 1 0 0

Calgary_480 1 0 0 0 0

Calgary_498 3 1 2 0 3

Table 3-Number of Shower Cleaner Kits shipped from Toronto DC using central-stock strategy


45


Calgary_108 10 14 10 1 5

Calgary_109 6 13 7 16 5

Medicine_243 4 1 7 4 6

Lettbridge_286 14 13 4 7 7

Calgary_433 6 16 8 12 16

Calgary_435 7 2 4 0 6

Calgary_462 7 4 5 3 4

Calgary_480 12 7 5 3 4

Calgary_498 17 6 7 11 8

Table 4-Number of ZipLoc Sandwich Bags shipped from Toronto DC using central-stock strategy


46


Calgary_108 0 0 0 2 2

Calgary_109 0 0 0 0 1

Medicine_243 0 0 0 0 0


Calgary_433 0 0 0 0 0

Calgary_435 0 0 1 0 0

Calgary_462 0 0 1 1 0

Calgary_480 0 0 0 0 0

Calgary_498 0 0 1 0 0

Table 5-Number of Shower Cleaner Kits stored in the stores


47


Calgary_108 0 0 0 0 0

Calgary_109 0 0 0 0 0

Medicine_243 0 0 0 0 0


Calgary_433 0 0 0 0 0

Calgary_435 0 0 0 0 0

Calgary_462 0 0 0 0 0

Calgary_480 0 0 0 0 0

Calgary_498 0 0 0 0 0

Table 6 - Number of ZipLoc Sandwich Bags stored in the stores (Inventory)

Scenario 1 Scenario 2 Scenario 3

Total cost for one

period(CAD$)

513.34

357.18

440.38

Table 7 - Cost Comparison


48

Appendix A


49

Transportation Model in OPL Code

Model:

{string} DCs =...; {string} Stores =...; {string} Products = ...; float Capacity[DCs] = ...; float Supply[Products][DCs] = ...; float Demand[Products][Stores] = ...; assert forall(p in Products) sum(o in DCs) Supply[p][o] == sum(d in Stores) Demand[p][d]; float Trans_Cost[Products][DCs][Stores] = ...; float DC_Cost[Products][DCs][Stores] = ...; dvar float+ Trans[Products][DCs][Stores]; minimize sum( p in Products , o in DCs, d in Stores ) Trans[p][o][d] * (Trans_Cost[p][o][d] + Trans_Cost[p][o][d]); subject to { forall( p in Products , o in DCs ) ctSupply: sum( d in Stores ) Trans[p][o][d] == Supply[p][o]; forall( p in Products , d in Stores ) ctDemand: sum( o in DCs ) Trans[p][o][d] == Demand[p][d]; forall( o in DCs, d in Stores ) ctCapacity: sum( p in Products ) Trans[p][o][d] <= Capacity[o]; } execute DISPLAY { writeln("trans = ",Trans); }


50

Appendix B


51

Transportation Model’s Data

Data: DCs = { TORONTO CALGARY }; Stores = {CALGARY_108 CALGARY_109 MEDICINE LETHBRIDGE CALGARY_433 CALGARY_435 CALGARY_462 CALGARY_480 CALGARY_498}; Products = { SHOWER_KIT SANDWICH_BAG }; Capacity = [100 100]; Supply = #[ SHOWER_KIT: #[ TORONTO: 8 CALGARY: 0 ]# SANDWICH_BAG: #[ TORONTO: 97 CALGARY: 0 ]# ]#; Demand = #[ SHOWER_KIT: #[ CALGARY_108: 1 CALGARY_109: 1 MEDICINE: 1 LETHBRIDGE: 1 CALGARY_433: 1 CALGARY_435: 0 CALGARY_462: 1 CALGARY_480: 1 CALGARY_498: 1 ]# SANDWICH_BAG: #[ CALGARY_108: 11 CALGARY_109: 11 MEDICINE: 4 LETHBRIDGE: 13 CALGARY_433: 17 CALGARY_435: 9


52

CALGARY_462: 8 CALGARY_480: 9 CALGARY_498: 15 ]# ]#; Trans_Cost = #[ SHOWER_KIT: #[ TORONTO: #[ CALGARY_108: 2.3002 CALGARY_109: 2.3002 MEDICINE: 2.3214 LETHBRIDGE: 2.332 CALGARY_433: 2.332 CALGARY_435: 2.4592 CALGARY_462: 2.5122 CALGARY_480: 2.6394 CALGARY_498: 2.756 ]# CALGARY: #[ CALGARY_108: 0.636 CALGARY_109: 0.636 MEDICINE: 1.1342 LETHBRIDGE: 0.954 CALGARY_433: 0.6572 CALGARY_435: 0.6678 CALGARY_462: 0.6678 CALGARY_480: 0.689 CALGARY_498: 0.6996 ]# ]# SANDWICH_BAG: #[ TORONTO: #[ CALGARY_108: 2.3002 CALGARY_109: 2.3002 MEDICINE: 2.3214 LETHBRIDGE: 2.332 CALGARY_433: 2.332 CALGARY_435: 2.4592


53

CALGARY_462: 2.5122 CALGARY_480: 2.6394 CALGARY_498: 2.756 ]# CALGARY: #[ CALGARY_108: 0.636 CALGARY_109: 0.636 MEDICINE: 1.1342 LETHBRIDGE: 0.954 CALGARY_433: 0.6572 CALGARY_435: 0.6678 CALGARY_462: 0.6678 CALGARY_480: 0.689 CALGARY_498: 0.6996 ]# ]# ]#; DC_Cost = #[ SHOWER_KIT: #[ TORONTO: #[ CALGARY_108: 2.26807141 CALGARY_109: 2.26807141 MEDICINE: 2.26807141 LETHBRIDGE: 2.26807141 CALGARY_433: 2.26807141 CALGARY_435: 2.26807141 CALGARY_462: 2.26807141 CALGARY_480: 2.26807141 CALGARY_498: 2.26807141 ]# CALGARY: #[ CALGARY_108: 2.26807141 CALGARY_109: 2.26807141 MEDICINE: 2.26807141 LETHBRIDGE: 2.26807141 CALGARY_433: 2.26807141 CALGARY_435: 2.26807141 CALGARY_462: 2.26807141


54

CALGARY_480: 2.26807141 CALGARY_498: 2.26807141 ]# ]# SANDWICH_BAG: #[ TORONTO: #[ CALGARY_108: 2.26807141 CALGARY_109: 2.26807141 MEDICINE: 2.26807141 LETHBRIDGE: 2.26807141 CALGARY_433: 2.26807141 CALGARY_435: 2.26807141 CALGARY_462: 2.26807141 CALGARY_480: 2.26807141 CALGARY_498: 2.26807141 ]# CALGARY: #[ CALGARY_108: 2.26807141 CALGARY_109: 2.26807141 MEDICINE: 2.26807141 LETHBRIDGE: 2.26807141 CALGARY_433: 2.26807141 CALGARY_435: 2.26807141 CALGARY_462: 2.26807141 CALGARY_480: 2.26807141 CALGARY_498: 2.26807141 ]# ]# ]#;


55

Appendix C


56

Transportation Model’s Solution Report

Cost calculation of products cross-docked at Toronto DC: Final solution with objective = 513.34: Trans = [[[1 1 1 1 1 0 1 1 1] [0 0 0 0 0 0 0 0 0]] [[11 11 4 13 17 9 8 9 15] [0 0 0 0 0 0 0 0 0]]];


57

Appendix D


58

Transportation & Inventory Model in OPL Code

Model:

{string} Products = ...; {string} DCs = ...; int NbPeriods = ...; range Periods = 1..NbPeriods; {string} Stores = ...; float Capacity[DCs] = ...; float Demand[Products][Stores][Periods] = ...; float TransCost[DCs][Stores] = ...; float Inventory[Products][Stores] = ...; float InvCost[Products] = ...; float DC_Cost[DCs] = ...; dvar float+ Trans[Products][DCs][Stores][Periods]; dvar float+ Inv[Products][Stores][0..NbPeriods]; minimize sum( p in Products, r in DCs, s in Stores, t in Periods ) ((DC_Cost[r] + TransCost[r][s])*Trans[p][r][s][t] + InvCost[p]*Inv[p][s][t]); subject to { forall( r in DCs, t in Periods ) ctCapacity: sum( p in Products, s in Stores ) Trans[p][r][s][t] <= Capacity[r]; forall( p in Products , s in Stores, t in Periods ) ctDemand: Inv[p][s][t-1] + (sum( r in DCs) Trans[p][r][s][t]) == Demand[p][s][t] + Inv[p][s][t]; forall( p in Products, s in Stores ) ctInventory: Inv[p][s][0] == Inventory[p][s]; };


59

Appendix E


60

Transportation & Inventory Model’s Data

Data:

Products = { SHOWER_KIT SANDWICH_BAG }; DCs = { TORONTO CALGARY }; Stores = {CALGARY_108 CALGARY_109 MEDICINE LETHBRIDGE CALGARY_433 CALGARY_435 CALGARY_462 CALGARY_480 CALGARY_498}; NbPeriods = 5; Capacity = [ 400, 400 ]; Demand = [ [[2 0 0 3 0] [1 1 1 0 2] [1 0 0 1 3] [1 0 0 1 3] [2 0 0 0 1] [1 2 0 1 1] [2 1 0 0 1] [1 0 0 0 0] [3 1 1 1 3]] [[10 14 10 1 5 ] [6 13 7 16 5 ] [4 1 7 4 6] [14 13 4 7 7] [6 16 8 12 16 ] [7 2 4 0 6 ] [7 4 5 3 4] [12 7 5 3 4] [17 6 7 11 8]] ]; Inventory = [ [0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 ] ]; InvCost = [ 0.2 0.2]; TransCost = [ [2.3002 2.3002 2.3214 2.332 2.332 2.4592 2.5122 2.6394 2.756] [0.636 0.636 1.1342 0.954 0.6572 0.6678 0.6678 0.689 0.6996] ]; DC_Cost = [2.26807141 5.0648]; Supply = [ [ [14 5 5 7 14] [0 0 0 0 0] ] [ [83 76 57 57 61] [0 0 0 0 0] ] ];


61

Appendix F-1


62

Transportation & Inventory Solution Reports

Solution Report 1: Final solution with objective = 1785.9: Trans = [[[[2 0 0 5 0] [1 1 1 0 3] [1 0 0 1 3] [1 0 0 1 3] [2 0 0 0 1] [1 2 1 0 1] [2 1 1 0 0] [1 0 0 0 0] [3 1 2 0 3]] [[0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0]]] [[[10 14 10 1 5] [6 13 7 16 5] [4 1 7 4 6] [14 13 4 7 7] [6 16 8 12 16] [7 2 4 0 6] [7 4 5 3 4] [12 7 5 3 4] [17 6 7 11 8]] [[0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0]]]]; Inv = [[[0 0 0 0 2 2] [0 0 0 0 0 1] [0 0 0 0 0 0]


63

[0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 1 0 0] [0 0 0 1 1 0] [0 0 0 0 0 0] [0 0 0 1 0 0]] [[0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0]]];


64

Appendix F-2


65

Transportation & Inventory Solution Reports

Solution Report 2:

Final solution with objective = 2201.9: Trans = [[[[0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0]] [[2 0 0 5 0] [1 1 2 0 2] [1 0 1 0 3] [1 0 0 1 3] [2 0 0 0 1] [1 2 1 0 1] [2 1 0 0 1] [1 0 0 0 0] [3 1 1 1 3]]] [[[0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0]] [[10 14 10 1 5] [6 13 7 16 5] [4 1 7 4 6] [14 13 4 7 7] [6 16 8 12 16] [7 2 4 0 6] [7 4 5 3 4] [12 7 5 3 4] [17 6 7 11 8]]]]; Inv = [[[0 0 0 0 2 2] [0 0 0 1 1 1] [0 0 0 1 0 0]


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[0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 1 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0]] [[0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0]]];

Documents

A Supply Chain Optimization Model: Nairi Nazarian A thesis ... · DC cost includes: Receive – Put away – Pick ... DC – Single DC central-stock strategy Table 4 Number of ZipLoc