efficient product rationalization within a single product ...1706/fulltext.pdf · has a goal to...

Efficient Product Rationalization within a

Single Product Portfolio

A Thesis Presentedby

Jackson Chou

to the faculty ofMechanical and Industrial Engineering

in Partial Fulfillment of the Requirementsfor the Degree of

Master of Science

in the field ofOperations Research

Northeastern UniversityBoston, Massachusetts

May 15, 2013

c© Copyright 2013 by Jackson ChouAll Rights Reserved

2

Table of Contents

Table of Contents 3

Abstract 6

Acknowledgments 8

1 Introduction 91.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 Thesis Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Literature Review 142.1 Product Rationalization . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Applied Mixed Integer Programming for Product PortfolioManagement . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 Benefits of Product Rationalization . . . . . . . . . . . . . 182.2 Gross Margin Return on Inventory Investment - GMROI . . . . . 18

2.2.1 Common Performance Measures . . . . . . . . . . . . . . . 192.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Mathematical Model 223.1 Fractional Programming . . . . . . . . . . . . . . . . . . . . . . . 223.2 Overall GMROI Model . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Linearized GMROI Model with Auxiliary Variables . . . . 263.2.2 Overall GMROI Model with Dinkelbach’s Algorithm . . . 30

3.3 Individual GMROI Model . . . . . . . . . . . . . . . . . . . . . . 333.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Model Solutions and Analysis 374.1 Scenario Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Overall GMROI Model Solution . . . . . . . . . . . . . . . . . . . 384.3 Individual GMROI Model Solution . . . . . . . . . . . . . . . . . 404.4 Comparison of Model Results and Conclusion . . . . . . . . . . . 42

3

5 Discussion and Conclusion 475.1 Redistribution and Sales Margin Model . . . . . . . . . . . . . . . 485.2 Limitations and Extensions of Future Research . . . . . . . . . . . 50

Bibliography 52

Appendices 55.1 Appendix A: Overall GMROI Model Optimal Solution . . . . . . 56.2 Appendix B: Individual GMROI Model Optimal Solution . . . . . 65.3 Appendix C: Overall GMROI LINGO Code . . . . . . . . . . . . 74.4 Appendix D: Individual GMROI LINGO Code . . . . . . . . . . . 75

4

List of Figures

1.1 Company X’s Product Portfolio . . . . . . . . . . . . . . . . . . . 111.2 Company X’s Inventory Investment from each quintile group . . . 121.3 Product Hierarchy of a Product Portfolio . . . . . . . . . . . . . . 13

4.1 GMROI and Inventory Turnover as inventory investment is re-duced (Overall GMROI) . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 GMROI and Inventory Turnover as inventory investment is re-duced (Individual GMROI) . . . . . . . . . . . . . . . . . . . . . 42

5

Abstract

Product proliferation is a problem seen in many large consumer packaged

goods (CPG) companies. Effectively managing the supply chain when the prod-

uct portfolio is constantly growing is a complex task for management. Without

proper assessment of the product portfolio, companies can easily run into the

danger of adding products that only result in reducing profits. One approach to

effectively manage the product portfolio is stock keeping unit (SKU) rationaliza-

tion or product rationalization. SKU rationalization is an overall performance

assessment on the company’s product portfolio and focuses on reducing the in-

ventory cost by eliminating underperforming products. However, there is no uni-

versal performance measure for wholesalers and retailers to use. The definition

of an underperforming product may vary depending on management’s goals and

performance metric used. Although finding a suitable performance measure is

challenging, successful implementation of weeding out underperforming products

will improve inventory policies and the company’s supply chain operations will

run efficiently at a reduced cost, thus increasing profits. In this study, we explore

the feasibility of using the performance metric gross margin return on inventory

investment (GMROI) as the objective function and develop two product ratio-

nalization models: (1) Overall GMROI and (2) Individual GMROI. The models

maximize GMROI metrics at reduced inventory investment levels and portfolio

size, while ensuring certain inventory turnover level. The nonlinear fractional

6

overall GMROI model is solved by Dinkelbach’s Algorithm. While we are able

to solve the overall GMROI model for the global optimal, its recommended solu-

tions may not be practical. On the other hand, the individual linear (0-1) integer

GMROI model provided good solutions in terms of gross margin and inventory

turnover. The models were tested using the portfolio of 1120 products of a CPG

company.

7

Acknowledgments

I would like to express my sincere gratitude to my thesis advisor, Professor

Emanuel Melachrinoudis, for the advice, guidance, and most importantly his

time and support. His office was always open to me whenever I had questions

and provided me with reference materials for the completion of this thesis.

I would also like to thank the managers of the company that provided me the

data for this research. Without them, this thesis would not have been possible

and I would not have had the opportunity to develop my research abilities. Their

insights and expertise knowledge were also invaluable to the completion of this

thesis.

Lastly, I would like to express my gratitude to my parents for all the support

they have given me throughout my life. They have made great sacrifices in order

for me to receive a higher education, I would not be here without their love and

support.

Jackson Chou

Boston,USA

May 15, 2013

8

Chapter 1

Introduction

Effectively managing the supply chain of any large consumer packaged goods

(CPG) company has always been and will be a daunting task for management. As

these large CPG companies continue to grow and expand, so does their product

portfolio. Product proliferation, defined as “product line depth or “an increased

assortment within a product line [1], is a strategy that allows companies to attract

more consumers, leading to more demand and sales. At the same time, the

increase of product varieties adds to overall costs such as inventory carrying cost

and warehouse expenses [1]. Without proper assessment of the product portfolio,

companies can easily run into the danger of adding products that only result in

reducing profits.

One approach to effectively manage the product portfolio is stock keeping

unit (SKU) rationalization or product rationalization. SKU rationalization is an

overall performance assessment on the company’s product portfolio and focuses

on reducing the inventory cost by eliminating underperforming products. How-

ever, there is no universal performance measure for wholesalers and retailers to

use due to the “multidimensional character of a retailer firm” [2]. The defini-

tion of an underperforming product may vary depending on management’s goals

and performance metric used. Common retail performance metrics are service

9

level, lost sales, product substitute percentage, gross margin, stock-turn, gross

margin return on inventory investment (GMROI) and sell-through percentage

[2], [3]. Although finding a suitable performance measure is challenging, success-

ful implementation of weeding out underperforming products will allow supply

chain operations to run efficiently at a reduced cost, improving the distribution

of products and reducing storage cost thus increasing the profits.

1.1 Problem Statement

The research done in this thesis was motivated from a real world problem fac-

ing a large CPG company. The company desired to remain anonymous, therefore

we will refer to it as Company X in this study. Company X is large CPG com-

pany and as with most CPG companies, maintaining a profitable product mix in

its product portfolio is one of the top objectives for management. The company

has a goal to improve the business performance and profitability growing in its

product portfolio. In addition, Company X is challenged with achieving working

capital targets, while maintaining a high level of customer service. This naturally

leads to a SKU rationalization exercise for Company X.

Figure 1.1 is a Pareto Chart that shows Company X’s current contribution of

total Gross Margin within a product portfolio consisting of 1120 different prod-

ucts. We can see from this chart that 80% of gross margin contribution is coming

from only 22% of the SKUs (246 out of 1120) in the portfolio. This phenomenon

is what we call the 80/20 rule or the Pareto principle, which states that about

80% of effects come from only 20% of the causes [7]. The 80/20 rule is widely

seen in business, especially within the CPG industries. As we venture further

along the Pareto Chart, we can see that the last 20% of products barely have

any effect on the total gross margin. Mapping the current status of the product

10

portfolio allows us to view the condition of tail (the worst performing SKUs or

in this case the last 20% of the product portfolio) and begin consideration on

products to trim and reduce.

Figure 1.1: Company X’s Product Portfolio

Figure 1.2 is a breakdown of the inventory investment supporting every quin-

tile group in the entire product portfolio. It makes sense to see that the majority

of Company X’s inventory investment is supporting the first quintile group of

products, which contributes $55.14 million dollars of the total gross margin in

the product portfolio. However, it is alarming here that the last three quintile

groups, or in this case, the last 60% of the product portfolio requires $9.15 mil-

lion dollars of inventory investment and only provides $6.47 million dollars of

the total gross margin. This inefficient use of inventory investment capital shows

why product proliferation may not always be profitable and increasing products

may result in reduction of profits. Company X has the opportunity to improve

its product portfolio and efficiently handle the inventory investment associated

11

with these products. The problem now is to determine the best combination of

products to keep in Company X’s portfolio.

Figure 1.2: Company X’s Inventory Investment from each quintile group

1.2 Thesis Motivation

Company X’s main objective is to increase the business performance and im-

prove profitability of its product portfolio. As illustrated in Figure 1.1 and 1.2,

there are opportunities for product rationalization at the tail-end of the portfolio.

Company X seeks to scale back on their inventory investment capital by elim-

inating these underperforming products in their product portfolio. Ultimately,

the reduced working capital will allow for higher profits and higher inventory

turnover in the product portfolio.

Companies in various industries have developed product portfolio optimiza-

tion models to effectively manage the inventory costs and reduce the amount of

products offered in their portfolio [11] [12]. While SKU Rationalization refers to

12

the elimination of products at the SKU level (where the difference of each SKU or

product may also include variants of the same product such as package size, ven-

dor, color, flavor, software, promotions, geographical location, etc.), the research

done in this thesis will be analyzing the elimination of products with only one

minor difference from each other. In other words, the products are exactly the

same but different characteristics such as flavor or the geographical location of the

product sold as illustrated in Figure 1.3. In this thesis, we explore the feasibility

of GMROI as a performance measure and objective function for the optimization

model. We concentrate on eliminating SKUs based on management’s goals and

objectives to maximize the GMROI within a single product portfolio.

Figure 1.3: Product Hierarchy of a Product Portfolio

1.3 Thesis Organization

The following thesis is divided into 5 chapters. Chapter 2 provides the litera-

ture review of SKU Rationalization and the GMROI performance metric. Chapter

3 presents different approaches of mathematical models for Product Rationaliza-

tion to be compared. Chapter 4 presents the comparison of different Product

Rationalization models under a scenario example. Finally, Chapter 5 summa-

rizes the thesis and presents future work and recommendation.

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Chapter 2

Literature Review

In this chapter, we describe the focus and general approach to SKU or Product

Rationalization and take a look at what companies in different retail industries

have done to create an efficient product portfolio. In addition, this chapter pro-

vides both literature and application of mathematical programming models to

solve the product portfolio problems. Finally, we will introduce the performance

measure GMROI and describe the uses of GMROI.

2.1 Product Rationalization

As mentioned in the previous chapter, product proliferation strategies require

careful analysis and assessment, otherwise retail firms run into the danger of

adding unprofitable products into their portfolios. Product proliferation problems

also arises from many other factors: firm’s desire of large product selections to

appeal to every consumer type, product proliferation strategy used as a barrier

to entry, increase in sales through minor product modifications, pressures from

functional areas within firm, improper management of product life cycles, demand

for unique products by resellers, and manager’s reluctance to prune products [1].

In fact, it is not uncommon to see CPG companies to have well over 50,000 SKUs

in their product portfolio [8]. In supply chain, product rationalization is a popular

14

approach to ease the undesirable and unprofitable effects of product proliferation

[1], [4].

One of the biggest challenges in product rationalization involves defining the

complexity costs incurred in the portfolio. There is no perfect way to accurately

account for the true costs associated with management and planning time, mar-

keting, operational costs, etc. Complexity cost is defined differently in each com-

pany, it is difficult to measure and is not always captured in standard accounting

systems [11]. For example, Hewlett-Packard (HP) defined their complexity costs

into two types: Volume-driven and Variety-driven. Volume-driven complexity

costs consist of material costs and variability-driven costs such as excess costs

and shortage costs [11]. Variety-driven costs are defined as resource costs, exter-

nal cash outlays, and indirect impacts of variety such as manufacturing, rework,

warranty-program expenses [11]. Even in the healthcare industry, complexity

cost is defined as the “impact that rationalization of a brand has on cash flow,

labor hours, machine hours, forecast accuracy, inventory, number of SKUs per

healthcare product, and sales per SKU” [9]. Therefore, product rationalization

requires management to strictly define their objectives and specifically eliminate

products that do not meet the threshold.

There are two general approaches in product rationalization. The first ap-

proach is redundant product rationalization, where the strategy is to focus on

the top to medium ranking products (based on the performance metric used) and

seeks to eliminate products that do not “satisfy a unique customer need” [8].

In doing so, the process is believed to redistribute the sales from the eliminated

products among the remaining products in the portfolio.

The second approach is tail-end pruning, which is the common and tradi-

tional method of product rationalization [8]. Tail-end pruning aims to eliminate

15

the bottom ranking products, which is generally the tail from the initial Pareto

analysis of the portfolio. As seen in Figure 1.2, the Pareto analysis of Company

X, you will always expect the 80/20 rule when mapping the product portfolio

based on some revenue metric. We observed that the last 60% of the Company

X’s product portfolio required more inventory investment than margin return.

However, companies may also exhibit negative volume and negative return from

their bottom ranking products because of excess product returns. Pruning the

bottom ranking products will lead to overall lower revenue and margin, however,

the aim is to have the reduced costs outweigh the revenue and margin loss [5].

2.1.1 Applied Mixed Integer Programming for ProductPortfolio Management

The study and practice of applying integer programming models for product ra-

tionalization has long been researched in various industries. Most notably, the

pharmaceutical industry often experiences product proliferation where the SKU

differences are multiple dosage forms of the same medicine such as film-coated

tablets or sugar-coated tablets [8]. In literature, the use of stochastic program-

ming has been proposed as an approach to properly manage pharmaceutical prod-

uct portfolios [10]. Although the stochastic programming approach determines

which project to undertake based on expected net present value, the decision of

eliminating and keeping a product can be seen as such projects. The framework

has two stages, the first stage deals with decision that must be made immediately,

the “here and now” decisions [10]. After the decisions are made, a second stage

deals with uncertainty, the “wait and see” decisions [10].

In the computer hardware and software industry, HP tackled their product

portfolio problem with integer programming to improve business performance

and effectively reduce operational costs. However, due to HP’s sheer size of

16

data set and order history, the formulated integer programming problem was

too large to solve by standard linear programming methods. Instead, HP found

that the problem can be formulated as a parametric maximum-flow problem

in a bipartite network and developed a new, exact algorithm that solves their

parametric maximum-flow problem [11].

John Deere & Company (Deere), the leading manufacturer of agricultural ma-

chinery in the world, believed their product lines included too many configurations

available for customers that resulted in reduced profits. In Deere, configuration

is defined as a “feasible combination of features available on a machine” such as

engine type, transmission, and horsepower etc. [12]. To efficiently build their

product portfolio, Deere developed a mixed integer programming model to select

configurations that maximizes shareholder value added [12].

As we have seen in the previous two industrial applications, mixed integer lin-

ear programming (MILP) can be applied to achieve optimal solutions in product

rationalization. However, in order to even attempt to build a MILP problem, the

data quality must be high and accurate since complexity costs must be incorpo-

rated into the model [8]. The binary decision variables in the MILP model would

indicate whether to keep the product or eliminate from the portfolio. An MILP

model can be formulated with either maximize or minimize objective function.

The maximization objective function will result in an optimal solution that indi-

cates the best products to keep in the portfolio. Alternatively, the minimization

objective function will output the optimal solution that indicates the worst per-

forming products in the portfolio. The flexibility of the objective function and

constraints would be based on management’s aspiring goals.

17

2.1.2 Benefits of Product Rationalization

The process of product rationalization may be challenging but the benefits are

quite rewarding. In June 2010, Novartis, a pharmaceutical company, had around

14,000 different SKUs in its product portfolio. By applying both redundant prod-

uct rationalization and tail-end pruning, Novartis was able to reduce about 1100

SKUs in their active product portfolio that would result in a reduced inventory

capital of $20 million USD [8]. HP’s product portfolios consist of 2,000 laser

printer SKUs, 15,000 server and storage SKUs, and over eight million customiz-

able combinations of notebook and desktop products [11]. Through the process of

product portfolio management, HP was able to reduce their LaserJet SKU count

by 40%, resulting in annual profits of $20 million [11]. Deere’s product portfolio

consisted of multiple product lines. Each product line carried thousands to mil-

lions of different configurations for its products. By developing a mixed integer

programming model, Deere was able to reduce its configuration set that allowed

for reduced costs, which ultimately lead to increased profits of millions of dollars

[12].

Each of the companies mentioned above had invested time in analyzing and

assessing their current product portfolio to effectively prune out their worst per-

forming products or simply inefficient products. By doing so, the companies

felt the positive impact of product rationalization and improved their business

performance immensely due to the reduced inventory capital.

2.2 Gross Margin Return on Inventory Invest-

ment - GMROI

As with all types of models, whether it be mathematical programming, simu-

lation modeling, or forecasting – if the data input is inaccurate, then the output

18

from the model will be useless. For supply chain, an important issue is defining

a performance measure for analyzing the decision variables. Since there is no

universal performance measure, supply chain practitioners have used several dif-

ferent measures in analyzing retail performance. The following section describes

some of the common performance metrics and how it relates to GMROI.

2.2.1 Common Performance Measures

Gross margin is a common performance metric used across all businesses.

However, gross margin only takes into account the sales price of the products sold

and does not incorporate any inventory management. Gross Margin is defined

as:

Gross Margin = Revenue− Cost

Inventory Turnover is another common metric in business that measures the

efficiency of inventory management [3]. Higher turnover represents efficient use of

inventory investment and results in lower inventory carrying costs due to inven-

tory ”turning” at a high pace. Inventory turnover has variations of calculations,

however, research done in this thesis is calculated as:

Inventory Turnover =Total Units Sold

Average Inventory in Stock

Gross Margin Return on Inventory Investment (GMROI) is a performance

measure that takes into account both the revenue and inventory unit impact. GM-

ROI was first introduced to give management the flexibility to increase profit lev-

els by making decisions based on gross margin percentage or inventory turnover.

GMROI is defined as:

GMROI =Total Gross Margin

Average Inventory Cost

19

where the total gross margin and average inventory cost are calculated within the

same period of time (for example, within the last 52 weeks). The Turn and Earn

ratio is another variation of GMROI:

Turn and Earn Ratio = Gross Margin %× Inventory Turnover

Ultimately, GMROI can be interpreted as the amount of money earned per

dollar invested in inventory [3]. For example, if the product portfolio has a

GMROI of 5, this means for every dollar invested in inventory, you are earning five

dollars. GMROI has been used as a performance measure for analyzing sourcing

decisions [3], [13]. While GMROI has been used to generate profit insights, to our

best knowledge, the performance metric has not been applied to mathematical

programming for product rationalization decisions.

Although the idea behind GMROI is novel, the performance metric is consid-

ered bias. GMROI can make a product with high margin dollars look worse, and

a low margin product look great. The reasoning behind this is due to the average

inventory cost in the denominator of the GMROI ratio. If two products have

the same inventory investment, the higher gross margin would make the product

the better one. However, this does not take into the account of the inventory

productivity. Even though the higher gross margin product has a better GMROI

metric, the inventory turnover for that product may be lower than the product

with the lower gross margin. GMROI does not truly differentiate between high

margin/low turnover and low margin/high turnover products.

2.3 Conclusion

In this chapter, the general process and approach to product rationalization

20

are discussed. Product rationalization is one approach to solving the product pro-

liferation problems seen in every large retail firm. One of the largest challenge in

product rationalization is determining the true costs associated with each prod-

uct. There are two approaches to product rationalization. First is redundant

product rationalization that seeks to eliminate redundant products within the

top to medium ranking products. The second approach, known as tail-end prun-

ing, tackles the bottom ranking products of the portfolio.

The use of mixed integer programming to solve product rationalization prob-

lems was also discussed, showing several examples of what companies in different

industries have done to efficiently manage and build their product portfolios.

Given the management’s goals and objectives, mathematical programming can

be used to determine the optimal combination of products to keep in the portfo-

lio. Finally, we introduced several performance metrics typically used in supply

chain as well as GMROI, the primary performance measure to be used as the

objective function for the optimization model formulated in the next chapter.

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Chapter 3

Mathematical Model

In this chapter, we introduce several mathematical models for product ra-

tionalization with the GMROI performance metric. As we have defined in the

previous chapter, product rationalization is an overall assessment of the product

portfolio with a goal to reduce inventory cost by pruning underperforming prod-

ucts. Therefore, this problem can be formulated as a type of resource allocation

problem through the use of binary decision variables from Integer Programming

(IP). To be more precise, because the performance metric GMROI is in the objec-

tive function, our model becomes a Fractional Programming (FP) problem with

binary decision variables. In the following sections of this chapter, we show sev-

eral different formulations for this Fractional Programming problem as described

below.

3.1 Fractional Programming

When a mathematical programming problem is to maximize or minimize a ratio

of two linear functions, subject to linear constraints, we define this as a Frac-

tional Programming problem. Fractional programming is a special structure and

class of Non-linear Programming (NLP), and when the problem consists of both

22

continuous and discrete variables, the problem becomes a mixed integer non-

linear programming problem (MINLP). Therefore, a mixed integer linear frac-

tional programming problem (MILFP) is a special structure of MINLP problems.

Fractional Programming problems typically have the form:

FP : Max:α +

∑i nixi

β +∑

i dixi

s.t. Ax = b

xi ≥ 0

where A is an m × n matrix, x is the decision vector, and α and β are scalars.

In linear fractional programming problems, every local minimum (maximum) is

a global minimum (maximum). The following lemma from Chapter 11.4, “Lin-

ear Fracational Programming [16] gives some important properties for obtaining

global optimal solutions.

Lemma 3.1.1. Let F (x) = (α +∑

i nixi)/(β +∑

i dixi), and let S be a convex

set such that β +∑

i dixi 6= 0 over S. Then, F (x) is both pseudoconvex and

pseudoconcave over S.

Due to the above lemma, there are multiple implications for the fractional

programming problem. The following lemmas are based on the theorems in “Non-

linear Programming: Theory and Algorithms in Bazaraa et al. [16].

Lemma 3.1.2. Since F(x) is pseudoconvex and pseudoconcave over S, then it is

also quasiconvex, quasiconcave, strictly quasiconvex, and strictly quasiconcave.

Proof. Based on Theorem 3.5.11, “Generalizations of a Convex Function” [16].

Lemma 3.1.3. Every local optimal solution of F(x) is a global optimal solution

over the feasible region S.

23

Proof. By Theorem 3.5.6, in “Generalizations of a Convex Function” [16].

Therefore, based on lemma 3.1.3, we are able to obtain a global optimal

solution for the FP problem with 0-1 binary variables by using MINLP solvers

able to handle pseudoconvex and pseudoconcave objective function such as the

branch and bound algorithm [17]. We now formulate the overall GMROI product

portfolio problem in the MILFP form.

3.2 Overall GMROI Model

The following formulation is a MILFP problem that maximizes the overall

GMROI of the product portfolio. As described in the previous chapter, GMROI

is a ratio of gross margin and average inventory investment. Therefore, we will be

maximizing the product portfolio that will give Company X the best income for

every inventory dollar spent. Since product rationalization requires a decision to

eliminate or to keep a product, we use binary decision variables for each product

where a “0” indicates the decision to eliminate the product and “1” indicates

to keep the product in the portfolio. In addition, each product is attributed

with several parameters for constraint and performance purposes. The following

formulation is a 0-1 fractional programming problem:

Index:Notation Meaning

i index for products

24

Parameters:Notation Meaning

mi Gross Margin in dollars for product ici Average Inventory Investment in dollars for product isi Total Units Sold (Last 52 weeks) for product igi Average Inventory Units (Last 52 weeks) for product iIC Available Investment Capitalb Inventory TurnoverN Desired product count in new product portfolio

Variables:Notation Meaning

xi Decision to invest in product i

Overall GMROI Model:

Max :

∑i(mi × xi)∑i(ci × xi)

(3.2.1)

s.t.∑i

ci × xi ≥ IC (3.2.2)

∑i(si × xi)∑i(gi × xi)

≥ b (3.2.3)∑i

xi ≤ N (3.2.4)

xi ∈ {0, 1} (3.2.5)

In the parameters table above, each product will have an attribute of the

gross margin, average inventory investment, units sold, and average inventory

units. The product’s attributes will allow the model to determine the optimal

mix of products to keep depending on management’s goals and objective. When

making product rationalization decisions, we want to make sure we are analyzing

products that are considered active and not waste time on products that are (1)

already in the process of being discontinued, and (2) new products that were

25

recently introduced. Therefore, the products included in the decision variables

are sold within the last 52 weeks.

The objective function (3.2.1) is maximizing the overall GMROI of the product

portfolio. In other words, we are looking for the best combination of products

that will provide the highest ratio by dividing the sum of mi (total gross margin)

by the sum of ci (total average inventory cost). The constraints of the model

are based on management’s goal in operating under a reduced investment capital

(3.2.2), achieving a higher or same level of inventory turnover (3.2.3), and the

desired number of products in the new product portfolio (3.2.4). Equation 3.2.5

is the constraint that restricts our decision variables (xi) to binary.

One thing to note in this formulation is the investment capital constraint

(3.2.2). Due to the structure of maximizing a ratio, if we let the investment

capital constraint to be a less than or equal, this causes an issue where the

model will simply maximize the overall GMROI metric by choosing products

with extremely high margin product and combining this selection with low margin

products towards the tail-end of the portfolio. By setting the capital constraint

to be greater than or equal, this allows us to analyze the portfolio’s GMROI as

we reduce investment capital at certain levels. The optimal solution will never

equal to this investment capital reduction, instead, the optimal solution will arrive

closely above the investment capital parameter, IC. We show this behavior in

the next chapter when comparing model solutions.

3.2.1 Linearized GMROI Model with Auxiliary Variables

Solving the GMROI model in its MILFP form becomes inefficient and does not

scale well depending on the size of the product portfolio. Since our model consists

of 1,120 different products, we can reformulate the overall GMROI model with

26

a linear objective function to solve the model efficiently. In order to linearize

the objective function, we use a global 0-1 fractional programming approach

developed by Wu [14] :

Theorem 3.2.1. A polynomial mixed 0-1 term z = xy, where x is a 0-1 variable,

and y is a continuous variable taking any positive value, can be represented by the

following linear inequalities:

(1) y - z ≤ K - Kx

(2) z ≤ y

(3) z ≤ Kx

(4) z ≥ 0

Where K is a large number greater than y.

Proof. (a) Suppose z = xy, K is a large number greater than y, then:

1. y − z = y − xy = y(1− x) ≤ K(1− x) = K −Kx,

2. z = xy ≤ y(∴ x = 0 or 1),

3. z = xy ≤ Kx,

4. z = xy ≥ 0

Conditions (1) to (4) are satisfied for x = 0 or 1.

(b) Suppose conditions (1) to (4) are true. If x = 0, then from conditions (3) and

(4), we have z = 0. If x = 1, then from conditions (1) and (2), we have z = y. It

is thus concluded that z = xy. If value of y is less than 1, then K = dye = 1.

Following Wu’s theorem, we can transform the fractional programming model

to a linearized formulation by creating the same auxiliary variables:

y =1∑

i(xi × ci)

27

y becomes the denominator of the GMROI ratio (or in this case, the Average

Inventory Cost), then the objective function becomes:

GMROI = mi × xi × y

Now we introduce the second auxiliary variable:

zi = xi × y

∴ GMROI = mi × zi.

We now have a linear objective function that is equivalent to optimizing the

overall GMROI of the product portfolio. The next step is to incorporate the

same conditions from Wu’s proof into constraints in the linearized optimization

model.




mi Gross Margin in dollars for product ici Average Inventory Investment in dollars for product isi Total Units Sold (Last 52 weeks) for product igi Average Inventory Units (Last 52 weeks) for product iIC Available Investment Capitalb Inventory TurnoverK Arbitrary large number (1, in our case)N Desired product count in new portfolio

28


xi Decision to invest in product iy Dummy decision variable treated as 1

Average Investment Capital

zi Auxiliary variable

Linearized GMROI Model:

Max:∑i

mizi (3.2.6)

s.t.∑i

xici ≥ IC (3.2.7)

∑i

xisi − b∑i

xigi ≥ 0 (3.2.8)

∑i

xi ≤ N (3.2.9)

∑i

ciy = 1 (3.2.10)

zi ≤ Kxi (3.2.11)

Kxi + y − zi ≤ K (3.2.12)

zi ≤ y (3.2.13)

xi ∈ 0, 1 (3.2.14)

zi ≥ 0 (3.2.15)

y ≥ 0 (3.2.16)

The linearized problem has the same functional constraints as the previous

overall GMROI model (Section 3.1) where we want to operate under a new re-

duced investment capital while also reaching a desired inventory turnover. The

objective function (3.2.6) is equivalent to equation (3.2.1) due to the conditions

from Wu’s theorem which is represented in the dummy constraints of the model

29

(Equations 3.2.1–3.2.16). The addition of a new set of decision variables (zi)

does not add to any computing time. However, while this method allows for the

use of regular linear programming (LP) and mixed integer programming (MIP)

solvers, the binary variables in the model can become computationally complex

when dealing with thousands of variables.

3.2.2 Overall GMROI Model with Dinkelbach’s Algorithm

Dinkelbach developed an algorithm to solve fractional programming problems

with optimal solutions by solving multiple subproblems in MILP form [15]. The

algorithm has been shown to solve at most O(log(nM)) subproblems in the worst

case [6]. In addition, Dinkelbach’s algorithm has been proven that it has super-

linear convergence rate [17]. In this subsection, we show the steps of Dinkelbach’s

Algorithm and how to formulate the MINLP GMROI model to solve using this

algorithm.

Algorithm 3.2.1. Dinkelbach’s Algorithm

• Step 1: Set initial x ∈ S, such that λ = λ1 = N(x)/D(x) (can also set

λ1 = 0).

• Step 2: Solve the MILP F(λ) = max{N(x)−λD(x)} to obtain the optimal

x solution.

• Step 3: If objective value of problem F(λ) ≤ δ (optimality tolerance), stop,

we have obtained the optimal solution in x. Else, set λ = N(x)D(x)

and go to

Step 2.

Recall from section 3.1 that the Fractional Programming objective function

is formulated as:

α +∑

i nixiβ +

∑i dixi

30

To implement Dinkelbach’s algorithm, we need to reformulate the problem into a

linear objective function, thus allowing the algorithm to solve a sequence of MILP

subproblems as described in the algorithm. First, let us consider the numerator

linear function and the denominator linear function:

N(x) = α +∑i

nixi

D(x) = β +∑i

dixi

Reformulating FP’s linear objective function along with linear constraints:

Max: F (λ) = N(x)− λD(x)

s.t. D(x) > 0

x ∈ {0, 1}

We denote z(λ) as the optimum objective value of F (λ). Let x∗ be an optimal

solution of FP and let λ∗ = N(x∗)/D(x∗). Then the following is known [6]:

z(λ) > 0 if and only if λ < λ∗ ,

z(λ) = 0 if and only if λ = λ∗ ,

z(λ) < 0 if and only if λ > λ∗

We can see that by solving the z(λ) = 0, we will obtain the optimal solution

equivalent to solving the original FP problem in the ratio form of N(x)/D(x).

Therefore, we reach an optimal solution when:

F (λ∗) = max{N(x)−λ∗D(x)|x ∈ S} = 0, if and only if λ∗ = N(x∗)D(x∗)

= max{N(x)/D(x)|x ∈

S}, where S is the feasible region of F (λ).

Below is the formulation of the overall GMROI problem (section 3.2) using Dinkel-

bach’s Algorithm:

31




mi Gross Margin in dollars for product ici Average Inventory Investment in dollars for product isi Total Units Sold (Last 52 weeks) of product igi Average Inventory Units (Last 52 weeks) of product iIC Investment Capitalb Inventory TurnoverN Desired product count in new product portfolioλ Intial parameter for arbitrary F(x) solutionδ Optimality tolerance (usually 0.01)



GMROI with Dinkelbach’s Algorithm:

Max: F (λ) =∑i

(mixi)− λ∑i

(cixi) (3.2.17)

s.t.∑i

xici ≥ IC (3.2.18)

∑i

(sixi)− b∑i

(gixi) ≥ 0 (3.2.19)

∑i

xi ≤ N (3.2.20)

xi ∈ {0, 1} (3.2.21)

Here we have two new parameters: λ and δ. λ is the overall GMROI of the

product portfolio since λ =∑

i(mixi)/∑

i(cixi). δ is the optimality tolerance for

the stopping rule when taking the difference between∑

i(mixi) and λ∑

i(cixi).

The optimality tolerance is determined by the user, for the GMROI problem, we

set δ = 0.01.

32

We set our initial λ = 0 to start the algorithm and solve the MILP, thus at

every iteration of Dinkelbach’s algorithm, λ will continue to update at each solved

iteration until λ is at a value that makes the objective value of F (λ) equal to 0.

The optimal solution of x from this λ is equivalent to the optimal solution to

the original FP model. Based on the optimal condition described above, Dinkel-

bach’s algorithm allows us to solve a sequence of MILP subproblems to obtain a

global optimal solution for the overall MINLP GMROI model. Although the al-

gorithm requires scripting, the code is simple (only requires IF and ELSE logical

statements) and using the algorithm allows us to move away from NLP solvers.

Dinkelbach’s algorithm has been shown to use less computing memory compared

to using MINLP solvers [17].

3.3 Individual GMROI Model

In section 3.2, we have shown one approach of maximizing the GMROI for

product rationalization with two different methods of solving the GMROI model.

The second approach is to take account of the individual GMROI ranking of

products. The rationale behind this approach is because maximizing the overall

GMROI of the portfolio also masks the sensitivity of the individual items in

the portfolio. By combining the summation of margin in the numerator (mixi)

and the summation of the average inventory cost in the denominator (cixi), the

relationship between a product’s margin dollars and average inventory cost is

segregated in the overall GMROI model. Therefore, the overall GMROI can easily

be maximized by taking in a few products that are highly profitable, allowing for

the combination of less profitable products.

Furthermore, due to the structure of the GMROI model (MILFP), the 0-1

global fractional programming problem will always require a MINLP solver or

33

an algorithm to reach an optimal solution. The following formulation is less

complex and looks into maximizing the margin dollars of the product portfolio

by the individual GMROI ranking of products.




mi Gross Margin in dollars for product ici Average Inventory Investment in dollars for product isi Total Units Sold (Last 52 weeks) of product igi Average Inventory Units (Last 52 weeks) of product iIC Investment Capitalb Inventory TurnoverN Desired product count in new product portfolio



Individual GMROI model:

Max :∑i

(mi

ci)xi (3.3.1)

s.t.∑i

xici ≥ IC (3.3.2)

∑i

xisi − b∑i

xigi ≥ 0 (3.3.3)

∑i

xi ≤ N (3.3.4)

xi ∈ {0, 1} (3.3.5)

34

While the Individual GMROI model has a linear objective function, this re-

quires management or decision maker to generally know how many products to

eliminate from the portfolio. As indicated in the parameters table above, N is the

remaining product amount in the portfolio after elimination. If management has

no idea how many products to eliminate, the parameter N becomes a decision

variable in Equation 3.3.4.

3.4 Conclusion

The structure and convex properties of a Fractional Programming problem

was described in this chapter. Due to the implications of the lemma 3.1.1, this

allows us to obtain global optimal solutions when solving MILFP using MINLP

solvers. Two approaches of applying mathematical programming into product

rationalization using the GMROI performance measure were presented. The first

approach is maximizing the GMROI of the overall product portfolio, making the

objective function into a sum of ratios. Unfortunately, the combinatorial nature

of this MINLP model does not scale well when decision variables reaches into the

thousands. Thus we introduce two alternative methods of solving the maximum

GMROI product portfolio problem.

The first method reformulates the MINLP GMROI model into a linearized

objective function based on Wu’s theorem and introducing auxiliary variables

into the model. The second method to solving the overall GMROI model uses

Dinkelbach’s algorithm which solves multiple MILP subproblems to reach the

optimal solution. Although the Dinkelbach’s algorithm involves some coding, the

algorithm allows us to obtain a global optimal solution without the use of MINLP

solvers.

The second approach is maximizing the sum of individual GMROI of the

35

product portfolio. The model is a MILP that takes into account of the individual

GMROI and total gross margin dollar of each product in the product portfolio.

The individual GMROI model has a linear objective function and is less complex

compared to the MINLP GMROI problem. In addition, the individual GMROI

model avoids segregating the relationship between a product’s margin dollars and

average inventory cost. The overall GMROI model inherently causes the product

to lose the link between the margin dollars and average inventory cost due to

the fact that the objective function is a summation of margin over summation

of cost as opposed to sum of individual margin over individual cost. Therefore,

the overall GMROI can easily be maximized by taking in a few products that

are highly profitable, allowing for the combination of less profitable products.

We show this in the next chapter when we compare solutions from the overall

GMROI model and the individual GMROI model.

36

Chapter 4

Model Solutions and Analysis

Chapter 4 discusses the solutions obtained from the two primary GMROI

models: (1) Overall GMROI model and (2) Individual GMROI model. In the fol-

lowing sections, we provide a scenario example and compare the optimal solutions

from each model based on that same scenario. We then discuss the differences

between the overall GMROI and individual GMROI and show which model is

better in practical sense.

4.1 Scenario Example

Company X is seeking to improve their business performance by reducing their

inventory investment in one of their product portfolios. In order to do so, the

company hopes to identify and discontinue some of the underperforming products

within a single product portfolio. The product portfolio consists of 1120 different

products, where some products are the same type but have minor differences

such as flavor, color, region, retailer, etc. All products considered for elimination

in the portfolio are in their active status, sold within the last 52 weeks, and do

not include recently introduced products. During the last 52 weeks, 80.1 million

units of the products in the portfolio were sold, generating $72 million dollars

gross margin while requiring a $33 million dollars inventory investment at the

37

average inventory units level of 18 million units. In other words, the current

GMROI of the product portfolio is at 2.16 and the inventory turnover is at 4.51.

As mentioned in the individual GMROI model (Section 3.3), the desired num-

ber of products to eliminate is based on management’s goals. Management may

never know exactly how many products to eliminate or where to cut the tail of

the Pareto chart. If management does know the minimum or exact number of

products to keep in the new product portfolio, we will have a parameter input for

N , otherwise, N becomes a decision variable and the individual GMROI model

will also solve for the optimal number of products to eliminate in the product

portfolio.

For the purpose of comparing the two model solutions, we will assume man-

agement has some idea of how many products to eliminate; the model will cap

the new product count to be N = 672, which is 60% of the original product port-

folio size, 1120. In addition to reducing the portfolio size, management wants the

product portfolio to have a greater than or equal inventory turnover than before,

b = 4.51. We also want to reduce our inventory investment, we start solving

the model with a reduced capital of 10%, IC = .90 × 33 million and continue

reducing investment capital by 10% until we reach a 60% reduced capital.

4.2 Overall GMROI Model Solution

For the scenario example shown in this section, we use Dinkelbach’s algorithm

to obtain the optimal solution. For the initial algorithm parameters, we have

λ = 0, and δ = .01. Table 4.1 is a summary of the 6 different scenarios where

the available inventory capital is changed for each model solution. The optimal

solutions for which products to keep and eliminate can be found in Appendix A.

Increasing the overall product portfolio’s GMROI will require a loss of gross

38

Table 4.1: Overall GMROI Solutions Summary

margin dollars. At the same time, since the product portfolio is working with

a reduced inventory investment, the amount of capital saved can help alleviate

the margin dollar lost. As we eliminate the underperforming products from the

portfolio, both GMROI and inventory turnover increases. However, management

will need to decide at which point to stop reducing the inventory investment,

as shown in Table 4.1 and Figure 4.1. For example, comparing the differences

between scenario 3 and 4, Company X needs to decide whether an increase of

GMROI from 2.84 to 3.11 is worth the $4 million gross margin loss.

Figure 4.1 shows the improvement in GMROI and Inventory Turnover as

we reduce the available inventory investment capital. GMROI increases almost

linearly at each 10% reduction of the original investment capital, whereas the in-

ventory turnover increases also but not at the same rate for each 10% reduction,

showing a sharper increase after the 10% and 30% reduction. As the invest-

ment capital reduction increases further, the marginal improvement of inventory

turnover gradually decreases. The turnover constraint for this model (Equation

3.2.19) only requires the new product portfolio to have a greater than or equal

inventory turnover than the original product portfolio. Although the constraint

may be redundant in our scenario example since we expect inventory turnover to

39

Figure 4.1: GMROI and Inventory Turnover as inventory investment is reduced(Overall GMROI)

always increase as we reduce inventory investment and number of products in the

product portfolio. This is due to the fact that low inventory turnover products

are usually tied to underperforming products (in terms of gross margin dollars).

However, this can be a potential deciding factor for management as they take

inventory turnover into account. For example, management may have inventory

turnover targets of 6 or higher, therefore the inventory turnover constraint will

lead to choosing a combination of products that produce both high turnover and

margin dollar profit.

4.3 Individual GMROI Model Solution

The solutions presented in this section are from the individual GMROI model.

The model is an MILP problem with binary decision variables. However, the

parameter N can be a decision variable when management does not know the

exact amount to reduce and the problem will have an integer decision variable in

40

N along with the binary decision variables of products. Table 4.2 is a summary of

the solutions output from the individual GMROI model. The optimal solutions

for which products to eliminate and keep can be found in Appendix B.

Table 4.2: Individual GMROI Solutions Summary

Recall the objective function for the individual GMROI model (Equation

3.3.1). The model solves for the best combination of products that will maxi-

mize the sum of individual GMROI of the products in the portfolio. Since the

objective value given is the sum of individual GMROI ratios, we recalculate the

overall GMROI of the product portfolio to compare the solutions obtained from

the overall GMROI model. From Figure 4.2, the optimal solutions as we take

away investment capital are more consistent than the solutions seen in Figure

4.1.

As expected, GMROI and inventory turnover increases as inventory invest-

ment capital is reduced. Substantial improvements in both GMROI and inventory

turnover is evident when the initial inventory investment capital is reduced (Sce-

narios 1 through 4). However, the last two scenarios results in identical optimal

solutions. In other words, further reduction in the inventory investment amount

will not yield a better solution in terms of GMROI and inventory turnover unless

the number of products in the portfolio decreases. Management may even include

41

Figure 4.2: GMROI and Inventory Turnover as inventory investment is reduced(Individual GMROI)

the fourth scenario (40% reduction) as the same solution as the last two scenarios

due to the fact that the differences are .03 in GMROI, .02 in inventory turnover,

and .38 gross margin dollars.

4.4 Comparison of Model Results and Conclu-

sion

In terms of GMROI and inventory turnover, the summary of the results shown

in Table 4.2 is similar to the summary from Table 4.1. For example, in the first

three scenarios, both of the model’s optimal solution output in the new portfolio’s

GMROI and Inventory Turnover are extremely similar. The only major difference

is in scenario 2, where inventory turnover for overall GMROI is .09 higher than

individual GMROI’s inventory turnover. However, there are also several differ-

ences in the solution output. While the GMROI and Inventory Turnover metrics

may be similar, the gross margin dollar of the new product portfolio from the

individual GMROI’s model is more consistent than the overall GMROI model.

42

Since the individual GMROI’s objective function takes the GMROI performance

of each product into account, we see a more consistent decrease in gross margin

dollars of the portfolio as available investment capital is reduced.

Table 4.3 is a snippet of the decision variables from the two models for sce-

nario 3 (reduced capital of 30%), both of the model’s optimal solution eliminates

products from the top 10 gross margin products in the portfolio. However, com-

paring the GMROI metric from the two models (Table 4.1 and 4.2), the overall

GMROI model provides a better GMROI and gross margin than the individual

GMROI model from scenarios 1 to 3. Although the overall GMROI eliminates

the two worst performing products (in terms of GMROI) in Table 4.3, it also in-

cludes some of the lower GMROI products (Styl205 Clr004 and Sty173 Clr017).

The logic and validity of the optimal solutions from the overall GMROI model

depends on the scenario and how the constraints are formed in the mathematical

programming model.

Table 4.3: Optimal solutions for the top 10 products in Gross Margin dollars

The Individual GMROI model is a more conservative approach and places

emphasis on the individual GMROI of the products. We exhibit this in Table

4.1, where the last two scenarios (40% and 50% reduced capital) have a vast

difference in the gross margin dollars retained. Although the overall GMROI

43

model may be analytically optimal, some of the rationalization decisions do not

make business sense and result in lower total margin dollars when compared to

the individual GMROI model. Table 4.4 provides an example where the worst

gross margin products are kept in the overall GMROI solution, but are eliminated

in the individual GMROI model.

Table 4.4: Optimal solutions for the bottom 10 products in Gross Margin dollars

When maximizing the overall GMROI, the model does not eliminate the worst

performing products seen in the bottom ranking of the product portfolio. In fact,

the optimal solution is to drop some of the most profitable products (in terms of

margin dollars) from the portfolio. This is due to the fact that when optimizing

the overall GMROI of a product portfolio, the performance metric can easily be

maximized by choosing only a few products that give a high overall GMROI per-

formance. The model can simply include products that have the highest gross

margin and lowest inventory cost, allowing the combination to include the worst

performing products in margin dollars. Therefore, maximizing the overall GM-

ROI of the product portfolio will mask the sensitivity and performance of the

individual products that are considered underperforming.

44

Depending on the situation and management’s objective, the overall GM-

ROI model will require the decision maker to check the optimal solution of each

product to see if it makes business sense (as evident in Table 4.4). The overall

GMROI model is essentially mixing a high margin dollar product with a low cost

dollar product, thus allowing for a “high” GMROI portfolio. For example, in Ta-

ble 4.4, the overall GMROI model cannot differentiate between Sty356 Clr004’s

margin dollars and Sty505 Clr050’s inventory cost. In fact, the overall GMROI

model may possibly be mixing and matching Sty356 Clr004’s margin dollars with

Sty505 Clr050’s inventory cost as a single product, thus we have an extremely

high GMROI of 3.64 (Table 4.1).

As mentioned in the previous chapter, the overall GMROI model inherently

causes the product to lose the link between the margin dollars and average inven-

tory cost due to the fact that the objective function is a summation of margin over

summation of cost as opposed to the sum of individual margin over individual

cost (individual GMROI model). Therefore, in situations where the constraints

allow for very small changes in cost reduction, the overall GMROI model will

provide a better solution (in terms of GMROI, margin dollars, and turnover).

However, having large inventory investment reductions in the constraint will give

a false sense of high GMROI and impractical product selections in the optimal

solution.

When maximizing the GMROI ratio, we need to be careful of the inventory

investment capital constraint (Equation 3.2.2). As mentioned in Chapter 3, the

behavior of the inventory capital constraint (Equation 3.2.2) will inherently affect

the optimal solution given from the output depending if the constraint is modeled

to be ≤ or ≥. Table 4.5 shows what happens when we solve the overall GMROI

model with a less than or equal investment capital constraint. In scenario 7,

45

GMROI is at 4 with a less than or equal to 10% reduced capital, however, this

is unrealistic since we are only spending 7.6 million dollars and our gross margin

dollar for the portfolio is now less than half of the original baseline. This is due

to the fact that when maximizing the GMROI ratio, the sum of product costs

(cimi) is non-binding and will always be reduced at the lowest level in order to

maximize the GMROI ratio. Therefore, you will see unreasonable products being

selected, resulting in high GMROI metric, but an extremely low margin dollar

and low investment capital product portfolio.

Table 4.5: Overall GMROI model with ≤ constraint

46

Chapter 5

Discussion and Conclusion

In this thesis, we explored the feasibility of using GMROI as a performance metric

in product rationalization decisions. We developed two different approaches to

achieve a better product portfolio: (1) Overall GMROI model and (2) Individual

GMROI model. In addition, we also showed two mathematical methods to solve

the Overall GMROI model: (1) linearized GMROI with auxiliary variables and

(2) Dinkelbach’s Algorithm. We then compared the solutions under different sce-

narios between the two models and found that maximizing the overall GMROI

of the product portfolio has its own limits. For example, under the first 3 scenar-

ios, the overall GMROI’s optimal solutions can be considered logical and made

business sense. However, the flaw of GMROI will always remain. GMROI does

not truly differentiate between high margin/lower turnover and low margin/high

turnover products.

As seen in Table 4.4, when we reduced the available inventory investment

down to 60%, the overall GMROI model maximizes the GMROI metric by taking

in a few high margin products but at the same time keeping the worst gross

margin products in the product portfolio. The overall GMROI model may work

well under certain scenarios and conditions set by the constraints. Ultimately,

the individual GMROI model provides a more consistent optimal solution under

47

different scenarios of reduced inventory investment capital.

5.1 Redistribution and Sales Margin Model

As mentioned earlier, one of the mathematical programming approaches pro-

posed was to formulate a two-stage stochastic model that looks into both the cur-

rent impact and the aftermath of eliminating underperforming products. While

there are multiple ways of optimizing the product rationalization problem, we also

need to be careful about the correlation of product demands. The research done

in this thesis does not take into account the correlation factor between products.

For example, consumers who buy item A also tend to buy item B. If the decision

is to eliminate item A, we must consider the possible sales impact it can have on

item B. If two items go hand in hand (complementary), we need to make sure

that we are eliminating both rather than just one.

Furthermore, if the company chooses to eliminate item A, which shares similar

product characteristics with item C and item D, then perhaps the consumer would

see item C or item D as a substitute and make a purchase. Thus we would have

a redistribution of sales from an eliminated product, back into the remaining

product within the product portfolio. We give a general model in the following

formulation:

Index:Notation Meaningi, j index for products

48


aij Margin factor increase for product j from dropping product i, i 6= jeij Units sold factor increase for product j from dropping product i, i 6= jmj Gross Margin in dollars for product jcj Average Inventory Investment in dollars for product jsj Expected Units Sold (Last 52 weeks) of product jgj Average Inventory (Units) of product jIC Investment Capitalb Inventory TurnoverN Desired product count in new product portfolio


xj Decision to invest in product j

Redistribution of Sales Margin model:

Max :∑j

mjxj +∑ij

aij(1− xi)mj (5.1.1)

s.t.∑j

xjcj ≤ IC (5.1.2)

(∑j

xjsj +∑ij

eij(1− xi)sj

)− b∑j

xjgj ≥ 0 (5.1.3)

∑j

xj ≥ N (5.1.4)

xj, xj ∈ {0, 1} (5.1.5)

The objective function (Equation 5.1.1) includes the current gross margin

(mj) and the margin factor increase (aij) if a correlated product was selected to

be pruned. This formulation is more realistic since it includes both the current

and future impacts of eliminating products from the portfolio. The index sets of i

and j are both the same amount based on the number of products in the portfolio,

49

and the correlation factor aij represents how much margin sales is redistributed

back into product j if product i is dropped and considered a substitute for product

j. A product’s correlation factor to itself will be 0, otherwise this will create an

error where the kept product’s gross margin sale is doubled.

5.2 Limitations and Extensions of Future Re-

search

There are multiple approaches to solving the product rationalization prob-

lem with mathematical programming. Multi-criteria optimization could be one

approach since product rationalization will affect the product portfolio’s gross

margin dollars and inventory turnover. The scenario examples shown in this the-

sis assumes that management knows the exact amount of products to drop from

the portfolio. However, this typically should not be the case and we must allow

for the solver to find the optimal amount of products to prune.

While most of the emphasis was placed on saving inventory investment, we

could easily switch to placing emphasis on the inventory turnover. If management

believes that they can increase their sales margin through other means, then there

will be more incentive to prune slow moving products. In addition, management

may already have a given investment capital allowance and the problem will shift

to which products to keep, or perhaps keep all the products but not service all

sales or a combination of both. The optimization model will incorporate both

binary and continuous decision variables, where binary decision represents which

products to keep and eliminate, and the continuous decision variables represents

how many of the kept products are serviced. Management will then have a fill

rate objective for certain products that are kept. If management does not know

their optimal fill rate, then the inventory fill rate can also be considered as a

50

continuous decision variable. Therefore, the model will have both binary and

continuous decision variables. This all depends on the goals and objectives the

management is trying to achieve.

The redistribution of sales margin model maximizes the margin dollars of

retained products in the portfolio. However, in some cases, the company may

also face customer (retailer) constraints where certain products are correlated

differently. For example, in order to sell product A to a customer, we must also

offer product B. Therefore, the model must also include constraints where if a

product is kept, it must also keep another product due to the customer’s demand.

Otherwise, if the solution is to eliminate product B because of low margin, the

company faces the consequence of also losing margin dollars from product A (high

margin) because of the customer demand.

The largest challenge in applying mathematical programming is the need for

accurate and high quality data for the parameters. The conditions of the prod-

uct portfolio must also be under the optimal inventory policy of the company.

Otherwise, the optimal solutions from the optimization models will not be ac-

curate and eliminating the suggested products from the portfolio will have dire

consequences. Therefore, product rationalization requires management to clearly

define the goals they want to achieve and strictly place a threshold on products

to distinguish between good and bad products.

51

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[1] Berman, B. (2011), ”Strategies to reduce product proliferation”, Business

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[2] Mattila, H. (1999), ”Merchandising strategies and retail performance for

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[3] Mattila, H., King, R., and Ojala, N. (2002), ”Retail performance measures

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6 Iss: 4, pp. 340–351

[4] Hamstra, M. (2009), ”Retailers increase focus on SKU Rationalization”. Su-

permarket News, 57(41), p. 28

[5] Gilliland, M. (2011), “SKU Rationalization: Pruning your way to better

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[7] Pareto principle. (n.d.) In Wikipedia., Retrieved March 2 2013 from

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[8] Leither, K.M. (2011), “Assessing and Reducing Product Portfolio Complex-

ity in the Pharmaceutical Industry”, Massachusetts Institute of Technology,

Sloan School of Management; and, (S.M.)–Massachusetts Institute of Tech-

nology, Engineering Systems Division; in conjuction with the Leaders for

Global Operations Program at MIT

[9] Hilliard, D. (2012), “Achieving and sustaining an optimal product portfolio

in the healthcare industry through SKU rationalization, complexity costing,

and dashboards”, Massachusetts Institute of Technology, Sloan School of

Management; and, (S.M.)–Massachusetts Institute of Technology, Engineer-

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ing Efficient Product Portfolios at John Deere and Company”, Operations

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[13] Gokce, M.A., King, R.E., Kay, M.G. (2002), “Supply Chain Multi-Objective

Simulation Optimiation”, Proceedings of the 2002 Winter Simulation Con-

ference, pp. 1306–1314

[14] Wu, T. (1997), “A note on a global approach for general 0–1 fractional

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54

Appendices

55

.1 Appendix A: Overall GMROI Model Opti-

mal Solution

Product 1 2 3 4 5 6 Product 1 2 3 4 5 6

Product 1 2 3 4 5 6 Sty010_Clr036 1 1 1 1 1 1 Sty082_Clr017 0 0 0 0 1 1

Sty003_Clr003 0 1 1 1 1 1 Sty010_Clr037 1 1 1 1 1 1 Sty084_Clr050 0 0 0 0 0 1













































Solution (xi) for each scenario

56

Product 1 2 3 4 5 6 Product 1 2 3 4 5 6 Product 1 2 3 4 5 6















































57
















































58
















































59
















































60
















































61
















































62
















































63

Product 1 2 3 4 5 6

Sty534_Clr050 0 0 0 1 1 1

Sty535_Clr017 0 0 0 0 1 1

Sty535_Clr050 0 0 0 0 1 1

Sty536_Clr017 0 0 0 0 0 0

Sty536_Clr050 0 0 0 0 0 0

Sty537_Clr017 0 0 0 0 1 1

Sty537_Clr050 0 0 0 0 0 1

Sty538_Clr017 0 0 0 0 0 1

Sty538_Clr050 0 0 0 0 1 1

Sty541_Clr017 0 0 0 0 0 1

Sty541_Clr050 0 0 0 0 0 0

Sty544_Clr017 0 0 0 0 0 0

Sty544_Clr050 1 1 0 0 0 0

Sty546_Clr017 0 0 0 1 1 1

Sty546_Clr050 0 0 0 0 1 1

Sty547_Clr017 0 0 0 0 1 1

Sty547_Clr050 0 0 0 0 1 1

64

.2 Appendix B: Individual GMROI Model Op-

timal Solution

Product 1 2 3 4 5 6 Product 1 2 3 4 5 6

Product 1 2 3 4 5 6 Sty010_Clr036 1 1 1 1 1 1 Sty082_Clr017 0 0 0 0 0 0














































Solution (xi) for each scenario

65
















































66
















































67
















































68
















































69
















































70
















































71
















































72

Product 1 2 3 4 5 6

Sty534_Clr050 0 1 1 1 1 1

Sty535_Clr017 0 0 0 0 0 0

Sty535_Clr050 0 0 0 0 0 0

Sty536_Clr017 0 0 0 0 0 0

Sty536_Clr050 0 0 0 0 0 0

Sty537_Clr017 0 0 0 0 0 0

Sty537_Clr050 0 0 0 0 0 0

Sty538_Clr017 0 0 0 0 0 0

Sty538_Clr050 0 0 0 0 0 0

Sty541_Clr017 0 0 0 0 0 0

Sty541_Clr050 0 0 0 0 0 0

Sty544_Clr017 0 0 0 0 0 0

Sty544_Clr050 1 0 0 0 0 0

Sty546_Clr017 0 1 1 1 1 1

Sty546_Clr050 0 0 0 0 0 0

Sty547_Clr017 0 0 0 0 0 0

Sty547_Clr050 0 0 0 0 0 0

73

.3 Appendix C: Overall GMROI LINGO Code

!GMROI Model solved using Dinkelbach Algorithm;

Model:

Sets:

SKU/1..1120/: MARGIN, COST, UNITS_SOLD, AVG_INVENTORY, X; !x(i) = decision

SKUs;

Endsets

Data:

MARGIN, COST, UNITS_SOLD, AVG_INVENTORY =

@OLE('C:\Users\Jackson\Dropbox\Northeastern University\OR Masters

Thesis\Models\Data_3.07.2013.xlsx', 'MARGIN', 'COST', 'UNITS_SOLD',

'AVG_INVENTORY');

q = 0; !Initial parameter for q;

tol = .01; !Optimality Tolreance;

continue = 1;

OldStandardGM = 72433436.29; !Old Standard Gross Margin of Product Portfolio; OldInvestmentCap = 33494976.74; !Old Investment Capital of Product Portfolio; NewSKUCount = 672; !Parameter to choose how many to drop in the product

portfolio;

Turns = 4.51; !Inventory Turnover RHS paramter;

Enddata

Submodel Z:

InvestmentCAP = .90*OldInvestmentCap; !Investment Capital RHS parameter;

Check = @SUM(SKU:X);

[OBJ] MAX=@SUM(SKU:(MARGIN*X)) - q*@SUM(SKU:(COST*X));

@SUM(SKU: X*COST) >= InvestmentCAP; !New Product Portfolio must be greater

than certain amount of InvestmetCAP;

@SUM(SKU: X*UNITS_SOLD) - Turns*@SUM(SKU: X*AVG_INVENTORY) >= 0; !Inventory

Turnover constraint (Linearized);

@SUM(SKU: X) <= NewSKUCount; !New SKU count in product portfolio;

@FOR(SKU: @BIN(X)); !Binary only for all X(i);

ENDSUBMODEL

!Dinkelbach Algorithm;

Calc:

!Create loop;

@while ( continue #EQ# 1 :

!Solve until optimal q reached;

@Solve(Z); !Solve the model Z;

@IFC( OBJ #LT# tol : !If Objective value of Z is less than

optimality tolerance, then stop the model;

continue = 0;

@OLE('C:\Users\Jackson\Dropbox\Northeastern University\OR Masters Thesis\Models\Data_3.07.2013.xlsx', 'SOLUTIONS', 'Q') = X, q;

@ELSE !Otherwise, update parameter q and solve again;

q = @SUM(SKU:(MARGIN*X))/@SUM(SKU:(COST*X));

);

);

Endcalc

End

74

.4 Appendix D: Individual GMROI LINGO Code

!Product portfolio optimization model;

Model:

Sets:

SKU/1..1120/: MARGIN, COST, UNITS_SOLD, AVG_INVENTORY, X; !x(i) = decision

SKUs;

Endsets

OldStandardGM = 72433436.29; !Old Standard Gross Margin of Product Portfolio; OldInvestmentCap = 33494976.74 ; !Old Inventory Investment Capital of Prod Portfolio; NewSKUCount = 672; !Parameter to choose how many to drop in the product

portfolio; !20% reduced SKUs = -224;

Turns = 4.51; !Inventory Turnover RHS paramter;

InvestmentCAP = .90*OldInvestmentCap; !Investment Capital RHS parameter;

Check_X = @SUM(SKU:X);

Used = @SUM(SKU: X*COST);

[OBJ] MAX=@SUM(SKU: ((MARGIN/COST)*X));!Maximize Avg. GMROII, select worst

performing SKUs;

@SUM(SKU: X*COST) >= InvestmentCAP; !New Product Portfolio must be greater

than certain amount of InvestmetCAP;

@SUM(SKU: X*UNITS_SOLD) - Turns*@SUM(SKU: X*AVG_INVENTORY) >= 0; !Inventory

Turnover constraint (Linearized);

@SUM(SKU: X) <= NewSKUCount; !New SKU count in product portfolio;

@FOR(SKU: @BIN(X)); !Binary only for all X(i);

Data:

MARGIN, COST, UNITS_SOLD, AVG_INVENTORY =


Thesis\Models\Data_3.07.2013.xlsx', 'MARGIN', 'COST', 'UNITS_SOLD',

'AVG_INVENTORY');

!Export solutions into Excel;


Thesis\Models\Data_3.07.2013.xlsx', 'AVGSOLUTION')= X;

Enddata

end

75

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