Interfaces Joseph Woods, Jing Zhou,€¦ · Vol. 40, No. 1, January–February 2010, pp. 17–32...

This article was downloaded by: [149.164.77.95] On: 05 April 2016, At: 10:02Publisher: Institute for Operations Research and the Management Sciences (INFORMS)INFORMS is located in Maryland, USA

Interfaces

Publication details, including instructions for authors and subscription information:http://pubsonline.informs.org

HP Transforms Product Portfolio Management withOperations ResearchJulie Ward, Bin Zhang, Shailendra Jain, Chris Fry, Thomas Olavson, Holger Mishal, JasonAmaral, Dirk Beyer, Ann Brecht, Brian Cargille, Russ Chadinha, Kathy Chou, Gavin DeNyse,Qi Feng, Cookie Padovani, Sesh Raj, Kurt Sunderbruch, Robert Tarjan, Krishna Venkatraman,Joseph Woods, Jing Zhou,

To cite this article:Julie Ward, Bin Zhang, Shailendra Jain, Chris Fry, Thomas Olavson, Holger Mishal, Jason Amaral, Dirk Beyer, Ann Brecht,Brian Cargille, Russ Chadinha, Kathy Chou, Gavin DeNyse, Qi Feng, Cookie Padovani, Sesh Raj, Kurt Sunderbruch, RobertTarjan, Krishna Venkatraman, Joseph Woods, Jing Zhou, (2010) HP Transforms Product Portfolio Management with OperationsResearch. Interfaces 40(1):17-32. http://dx.doi.org/10.1287/inte.1090.0476

Full terms and conditions of use: http://pubsonline.informs.org/page/terms-and-conditions

This article may be used only for the purposes of research, teaching, and/or private study. Commercial useor systematic downloading (by robots or other automatic processes) is prohibited without explicit Publisherapproval, unless otherwise noted. For more information, contact [email protected].

The Publisher does not warrant or guarantee the article’s accuracy, completeness, merchantability, fitnessfor a particular purpose, or non-infringement. Descriptions of, or references to, products or publications, orinclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, orsupport of claims made of that product, publication, or service.

Copyright © 2010, INFORMS

Please scroll down for article—it is on subsequent pages

INFORMS is the largest professional society in the world for professionals in the fields of operations research, managementscience, and analytics.For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org

http://pubsonline.informs.org

http://dx.doi.org/10.1287/inte.1090.0476

http://pubsonline.informs.org/page/terms-and-conditions

http://www.informs.org

Vol. 40, No. 1, January–February 2010, pp. 17–32issn 0092-2102 �eissn 1526-551X �10 �4001 �0017

informs ®

doi 10.1287/inte.1090.0476© 2010 INFORMS

THE FRANZ EDELMAN AWARDAchievement in Operations Research

HP Transforms Product Portfolio Management withOperations Research

Julie Ward,a Bin Zhang,a Shailendra Jain,a Chris Fry,b Thomas Olavson,c Holger Mishal,d Jason Amaral,e

Dirk Beyer,f Ann Brecht,g Brian Cargille,h Russ Chadinha,i Kathy Chou,j Gavin DeNyse,k

Qi Feng,l Cookie Padovani,m Sesh Raj,n Kurt Sunderbruch,o Robert Tarjan,p Krishna Venkatraman,q

Joseph Woods,r Jing Zhous

aHewlett-Packard Labs, Palo Alto, California 94304 {[email protected], [email protected], [email protected]} bStrategic Management Solutions,Redwood Shores, California 94065, [email protected] cHewlett-Packard SPaM, Palo Alto, California 94304, [email protected]

dHewlett-Packard Personal Systems Group, Cupertino, California 95014, [email protected] eEmeraldwise LLC, Woodside, California 94062,[email protected] fM-Factor, Inc., San Mateo, California 94404, [email protected] gHewlett-Packard Technology Solutions Group, Fort Collins,

Colorado 80528, [email protected] hHewlett-Packard SPaM, Palo Alto, California 94304, [email protected] iHewlett-Packard Personal SystemsGroup, Houston, Texas 77070, [email protected] jHewlett-Packard Personal Systems Group, Cupertino, California 95014, [email protected]

kHewlett-Packard SPaM, Portland, Oregon 97221, [email protected] lMcCombs School of Business, University of Texas at Austin, Austin, Texas 78712,[email protected] mHewlett-Packard Technology Solutions Group, Roseville, California 95747, [email protected] nDSApps, Inc.,

Sunnyvale, California 94087, [email protected] oHewlett-Packard Technology Solutions Group, Roseville, California 95747, [email protected] Labs, Palo Alto, California 94304, and Department of Computer Science, Princeton University, Princeton, New Jersey 08540,

[email protected] qIntuit, Mountain View, California 94043, [email protected] rHewlett-Packard Technology Solutions Group, Roseville,California 95747, [email protected] sBelk College of Business, University of North Carolina at Charlotte, Charlotte, North Carolina 28223, [email protected]

Hewlett-Packard (HP) offers many innovative products to meet diverse customer needs. The breadth of itsproduct offering has helped the company achieve unparalleled market reach; however, it has come with signif-icant costs and challenges. By offering multiple similar products, a manufacturer increases its overall demandvolatility, reduces forecast accuracy, and can adversely affect revenue and costs across the entire product lifecycle. At HP, these impacts included increases in inventory-driven costs and order-cycle time; liabilities to chan-nel partners; and costs of operations, research and development, marketing, and administration. Furthermore,complexity in HP’s product lines confused customers, sales representatives, and channel partners, sometimesdriving business to competitors. HP developed two powerful operations research-based solutions for manag-ing product variety. The first, a framework for screening new products, uses custom-built return-on-investment(ROI) calculators to evaluate each proposed new product before introduction; those that do not meet a thresholdROI level are targeted for exclusion from the proposed lineup. The second, HP’s Revenue Coverage Optimiza-tion (RCO) tool, which is based on a fast, new maximum-flow algorithm, is used to manage product varietyafter introduction. By identifying a core portfolio of products that are important to order coverage, RCO enablesHP businesses to increase operational focus on their most critical products. These tools have enabled HP toincrease its profits across business units by more than $500 million since 2005. Moreover, HP has streamlinedits product offerings, improved execution, achieved faster delivery, lowered overhead, and increased customersatisfaction and market share.

Key words : cost analysis; stochastic inventory analysis; flow algorithms; product portfolio management;inventory management; regression; statistics; binary programming; Lagrangian relaxation; parametricmaximum flow.

Hewlett-Packard (HP) serves more than one billioncustomers (consumers, small-to-medium busi-

nesses, and large-enterprise customers in virtuallyevery industry) in more than 170 countries on six con-tinents. Its four business units offer products span-

ning enterprise storage and servers, personal systems,imaging and printing, software, services, and corpo-rate investments. With 2008 revenues of $118 billion,HP is the global market share leader in PCs, print-ers, and servers. It ships 48 million PCs annually

17

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.

Ward et al.: HP Transforms Product Portfolio Management with Operations Research18 Interfaces 40(1), pp. 17–32, © 2010 INFORMS

and over one million printers weekly. One of everythree servers shipped worldwide is an HP product. Itsproduct lineup includes significant variety in nearlyevery offering, boasting more than 2,000 laser printerstock-keeping units (SKUs), more than 15,000 serverand storage SKUs, and over eight million possibleconfigure-to-order combinations in its notebook anddesktop product lines.

Although HP’s product offerings drove sales andmarket share, the variety of its product portfoliocaused significant organizational complexity, createdmajor operational and performance challenges, andcaused HP to fall behind its competitors on a num-ber of metrics. Although revenues grew each year,unplanned increases in operating costs eroded its prof-its, partly because of complexities such as increases ininventory-driven costs, product design costs, channelliabilities, and rework. As variety increased, forecast-ing accuracy decreased, resulting in excesses of someproducts and shortages of others. In 2002, HP’s inven-tory turns were lower than many of its competitors,and shortages and excesses were rampant. Despitehigh inventory levels, major deals were lost becauseproducts were not available to meet demand. By 2004,HP’s order-cycle time (OCT) was unpredictable, andits average OCT was nearly twice that of its leadingcompetitor, adversely affecting customer satisfactionand making it difficult to win accounts although prod-uct quality was high. Across all HP businesses, ineffec-tive management of complexity was diminishing thebenefits of HP’s broad, diverse product portfolio.

The challenge was not simply that the variety wasdifficult to manage; more fundamentally, measuringthe true costs and benefits of variety was difficult.Many product line complexity costs are hidden, i.e.,not captured in standard accounting systems and dif-ficult to measure systematically and fairly. To estimatethe impact on overhead costs of adding a single SKUor product feature to the product line seemed almostimpossible. In many cases, new products strainedexisting resources but did not give rise to new directoverhead costs. It was only in aggregate that webegan to see the cost impacts. Estimating the impactof variety on inventory-driven costs required sophis-ticated statistics and stochastic modeling capabilitiesthat the teams (i.e., marketing and product manage-ment) who managed product portfolios did not have.

Because the cost impacts of variety were so difficultto measure, debates over new-product introductionswere often very one-sided. Marketing teams couldargue that new products would generate incrementalrevenue; however, making a counterargument that theintroductions would impact cost, let alone objectivelyweighing these costs and benefits against each other,was difficult. Without clear standards for evaluatingproduct proposals in a balanced way, HP was unableto implement an effective process for systematicallymanaging product proposals.

Once products were launched into the portfolio,it was difficult to measure and manage their impact.Few standards existed for how and when to remove aproduct from the portfolio. Rationalization decisionswere often made based on a product’s individual rev-enue; however, this metric neglects key elements ofthe product’s importance, such as the value of a low-revenue product in fulfilling a high-revenue order.Moreover, the cost structure and impact of variety dif-fered dramatically from business to business withinHP. Some businesses, such as high-end imaging andprinting products and business-critical servers, facedhigh variety-driven costs associated with creating,developing, testing, and launching new SKUs. How-ever, their processes for reviewing and approvingnew SKU introductions did not incorporate a com-prehensive, quantitative cost assessment. SKU intro-duction decisions were often based on a business casefrom marketing that focuses on the benefits ratherthan a balanced and data-driven view of incremen-tal costs and benefits, making it easy for variety toproliferate.

Other businesses, such as HP’s Personal SystemsGroup (PSG), which sells configurable PC products,had comparatively low per-SKU costs but high costsfor simultaneously managing inventory and availabil-ity on many underlying parts. In PSG, most orders donot ship until every product is available; a stockout ofa single product can delay an entire order. Because ofdifficulties in maintaining adequate availability acrossits vast product line, PSG’s average OCT was notcompetitive, and the lack of predictability and longlead times frustrated customers.

Over the past five years, HP has made managingproduct variety a strategic business priority. It hasdeveloped and implemented two operations research

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.

Ward et al.: HP Transforms Product Portfolio Management with Operations ResearchInterfaces 40(1), pp. 17–32, © 2010 INFORMS 19

(OR)-based solutions that have helped HP dramati-cally improve its performance, resulting in bottom-line profit improvements of more than $500 million.In this paper, we present these two methodologies,describe the details of their application in PSG, andpresent the substantial quantitative and qualitativebenefits obtained through the broad use of these toolsacross many HP businesses.

SolutionsThe first solution, a process for screening new-product proposals before introduction, is driven byOR-based supply chain analytics for measuring thetrue cost impact and projected return on investment(ROI) of proposed products. We evaluate the pro-jected complexity-adjusted ROI for each proposed newproduct, prior to its creation, using a complexity ROIcalculator; this calculator is developed for each busi-ness through a one-time analysis of the up-front andongoing cost impacts of introducing and managingproducts. Products that do not meet a threshold ROIlevel are targeted for exclusion from the proposedlineup (Cargille et al. 2005, Olavson and Fry 2006,Cargille and Melia 2007).

The second solution, the Revenue Coverage Opti-mization (RCO) tool, is used to manage product vari-ety after introduction. RCO analyzes order history torank products along the efficient frontier of portfo-lio size and order coverage—defined as the portion ofthe number, revenue, or margin of orders that canbe completely fulfilled by products in the portfolio.By helping to identify a core portfolio of productsthat are important to order coverage, the RCO resultsenable HP businesses to increase operational focuson their most critical products and make data-drivenrationalization decisions.

Because these two tools address different aspectsof managing product variety, their use varies accord-ing to each business’ requirements. Businesses thatincur significant one-time costs for each new-productlaunch might emphasize a screening process that usescomplexity ROI calculators to quantitatively evaluateproposed new variety. Businesses with highly config-urable product lines emphasize RCO to help themidentify the product offering “sweet spot” that coversthe majority of orders; thus, they can achieve oper-ational efficiencies through improved focus on these

Prior to introduction: After introduction:Complexity ROI screening Portfolio management with RCO

• Identify and estimate costimpacts of variety,modeling relationshipsusing activity-basedcosting, stochasticinventory modeling, andother quantitative methods,as needed.

• Codify relationships into acomplexity ROI calculator.

• Screen new SKU or featureproposals for projected ROIusing complexity ROIcalculator.

• Use RCO to segment portfoliointo core high-contributionoffering andlower-contribution extendedoffering.

• Construct differentiatedservice offerings to improveperformance of core offeringand reduce overheadassociated with servingextended offering.

• Target select elements ofextended offering forelimination or rationalization,as appropriate.

Figure 1: HP follows a two-phased approach that involves up-front screen-ing of new-product proposals for ROI and ongoing evaluation and reprior-itization of the product portfolio using RCO. Both phases rely heavily onOR to provide a data-driven framework for decision making.

products. Some businesses, such as PSG, use bothheavily.

Figure 1 shows HP’s overall portfolio managementapproach, highlighting the role of these OR solutionsat each phase.

New-Product ROI ScreeningFrameworkHP screens proposals for new SKUs, features, prod-uct bundles, and platforms prior to investing in them.The screening process begins with a detailed analy-sis of the cost structure and drivers in each businessand product line—cost relationships that are gener-ally obscure and not captured in accounting systems.A team of OR professionals spends one to threemonths developing a model of how business costsrespond to increases in product variety. It capturesthe cost relationships and codifies them into a set ofguidelines and an ROI calculator that the businesscan deploy to evaluate new-product proposals. Ascosts change, the business can update the calculator’sparameters, enabling it to evaluate proposals on anongoing basis.

We identify the major cost drivers that product vari-ety impacts. First, we examine the complete life cycle,from conception through postlife support (i.e., sup-port after the product has been removed from the

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


Nature ofCost type relationship Cost categories

Variable complexitycosts

Volume-driven • Material costs: volumediscounts

• Variability-driven costs:excess costs (financing,storage, depreciation,obsolescence, fire sales)and shortage costs(material pricepremiums, expediting,lost sales because ofshortages)

Fixed complexitycosts

Variety-driven • Resource costs: R&D,testing, productmanagement, etc.

• External cash outlays:tooling, costs to contractmanufacturer

• Indirect impacts ofvariety: manufacturingswitching costs,warranty-programexpenses, qualityimpacts, returns costs

Figure 2: HP systematically assesses the cost impacts of product varietyusing a framework that captures major cost impacts along the full profit &loss statement and throughout the entire life cycle of each product.

portfolio). Second, we examine the entire businesscost structure, including fixed and variable costs. Weuse the cost categories (Figure 2) as a guide to ensurethat we have considered all important cost elements.

In our analysis, we distinguish between fixed andvariable complexity costs. We define fixed complexitycosts as those that scale with the number of productsoffered; therefore, the key cost driver is the numberof SKUs or variants sold, regardless of each variant’sunit volume. These are the costs, such as research anddevelopment (R&D) or product marketing resources,that are required to bring new products to marketand support them over their life cycles. To estimatevariety-driven cost impacts, we use a mix of activity-based costing, regression analysis, and other tech-niques. For example, as variety increases, customershave more options and are more likely to return orexchange products. To quantify this, we use regres-sion analysis to evaluate the impact of SKU variety onproduct returns cost, allowing us to estimate a roughrelationship between the numbers of products offeredand the cost of returns.

We define variable complexity costs as those wherethe unit variable cost of a SKU or part increasesbecause of insufficient volume to reach an opera-tionally efficient scale; the unit volume for a givenSKU or variant is the key driver. Examples includematerial costs, which are higher for lower-volumeparts because of inadequate negotiating leverage forthe buyer or economies of scale for the supplier, andcosts associated with demand variability, which resultin significantly more excess costs (devaluation, excess,and obsolescence) and shortage costs (freight expe-diting, supplier price premiums, lost sales). Figure 3shows an approach we use to estimate how excessand shortage costs related to demand variability scaleup and down with volume.

To evaluate whether or not to introduce a new-product variant, we balance the complexity costsagainst its projected marketing and sales benefits. Werecommend screening out low-value products, whichare not necessarily the same as low-volume products.Screening products by volume overlooks the signif-icant differences in complexity cost among differentproduct types. Volume thresholds also miss that somehigh-volume SKUs could drive very little incremen-tal revenue and margin if they have a close substitute.Volume thresholds and rules of thumb can be use-ful but only if they adjust for cannibalization effectsand complexity cost differences between differentSKU types.

Our solution is to screen based on complexity-adjusted margin: the incremental margin (with can-nibalization effects subjectively estimated by market-ing) less the incremental complexity costs. To keepthe solution simple, we start with an exhaustive con-sideration of all possible complexity costs and paredown that list to include only the most significantcategories—for example, those that drive roughly 80percent of the complexity costs. We then capture com-plexity cost guidelines for those costs in easy-to-usespreadsheet calculators. Because the key complexitycost drivers vary across businesses, we develop a dif-ferent calculator for each business while leveragingcommon frameworks and techniques for complexitycost modeling.

In some cases, constrained resources (e.g., R&D)might be a significant portion of the complexity costs;

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


1 10 100 1,000 10,000 100,000 1,000,000

Dem

and

vari

abili

ty(s

td.

dev.

/mea

n)

• Inventory holding• Component devaluation• Excess and obsolescence• Price protection

• Lost sales

• Expediting

• Material price premiums

� ( fu(k) + kFu (k))

� ( fu(k) − k [1−Fu(k)])E[shortage] =

E[on hand] =

Notation:� = standard deviation of lead-time demandk = critical fractile based on service-level target

Fu = unit normal distribution functionfu = unit normal density function

Mean demand volume (log scale)

Empirical modeling:volume impact on variability

Inventory modeling:variability impact on cost

Figure 3: HP models the relationship between product volumes and demand variability empirically as an inputto measuring the impact of product variety on costs associated with demand variability. Greater product vari-ety and less-concentrated demand result in higher variability, which in turn results in higher inventory-drivencosts and shortage-related costs. Once the relationship is modeled at this level, calculations of inventory-drivencosts and shortage costs can be embedded into calculators to automate the estimation of complexity cost impactsbecause of different volume assumptions.

thus, we calculate a complexity ROI using our earlierdistinctions between fixed and variable complexitycosts:

Complexity ROI

= �Incremental margin−Variable complexity costs��Fixed complexity costs�

�

Typically, HP businesses set their ROI hurdle fairlyhigh (6:1 or greater) to ensure that the investment inintroducing new variants is justified vis-à-vis otherinvestments that HP could make with the sameresources. Often we limit the costs that we includein the denominator to only those that represent thespecific resource or resources that are constrained,moving some (non-resource-based) fixed costs to thenumerator when we compute the ROI.

Once the cost structure has been identified andmajor costs quantified, a cross-functional team,including supply chain, finance, and marketing rep-resentatives, validates the modeled cost relation-ships. A cross-functional core team and sponsorship

team must go through the complexity cost modelingjourney together, to gain input and commitment fromall functions on the cost guidelines that the ROI cal-culator will use going forward. The sponsorship teamshould include executives from the major functionsimpacted by the costs and benefits of variety: supplychain, R&D, and marketing. To the extent possible,the team conducts the validation by comparing pre-dicted results with actual data. However, because therelationships often do not show up directly in data, itis more important to obtain buy-in from the organiza-tions that the calculator represents a reasonable modelof costs than to prove conclusively that the modeledrelationships are an exact predictor.

Figure 4 shows an example of the main interfaceof a complexity ROI calculator used in HP’s LaserJetbusiness. For ease of use and diffusion across a largenumber of users, we keep the calculator interfaceintuitive and graphical, despite considerable behind-the-scenes sophistication and modeling used to assessand capture the relationships that drive the calcu-lations. The tool should require only inputs (e.g.,

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


LaserJet variety cost-benefit calculator

Model Number/Name of Proposed Model: ABC123 Estimated unaccounted costs of adding model

Platform: XXX Opportunity costs:Lost sales due to stockouts

Description: Total

COGS + Contra-revenue impacts:Product category: Inventory holding, storage, and financing

Excess and obsolescence/fire salesExpeditingPrice protectionSpare parts inventory-driven costsMaterial cost volume discountsMOH (switching and setup)Refurb and returnsWarrantyTotal

Operating-expense impacts:Sales and Mktg OH

Model Statistics Product data mgmtProjected lifetime (months) 15 Commodity mgmtMonthly volume 5,900 Mfg program mgmtSKUs added with Model 31 Planning and ForecastingList price $349 Product completion centerHardware net revenue/unit $329 Sr. mgmt attention for escalationsContrib. margin/unit $79 Test center

Other un accounted costsTotal

Engine Requirements

Percentage ofvolume

Monthlyvolume

Totalvolume

Avg netrev/unit

Grand totalEngines110V NW BL 40 2,360 1,680 $596 ROI:220V NW BL 40 2,360 1,680 $596 Incremental margin110V NW BL MIJ 10 590 420 $596 Adj. incremental margin220V NW BL MIJ 10 590 420 $596 Fixed cost

ROI

Incremental volume Cutoff valuesPercent incremental volume 45 Yellow zone ROI 1:1Total cannibalized units per month 3,245 Green zone ROI 7:1

Minimum percentage incrementalto qualifyForecast Monthly Lifetime Incremental…for green zone

45Volume (units) 5,900 88,500 39,825

…for yellow zone19

Hardware revenue $1,941,100 $29,116,500 $13,102,425Contrib. margin $466,100 $6,991,500 $3,146,175

Cannibalization

Related product number 1 Related product number 2

Name: ABC Name: DEF% of cannib 80 % of cannib 20Cannib’d units 2,596 Cannib’d units 649

Related products Planned Adjusted Planned AdjustedProjected lifetime (months) 17 17.0 14 14.0Monthly volume 5,000 2,404 2,000 1,351List price $369 $369 $299 $299Hardware net revenue/unit $349 $349 $279 $279IFS2 margin/unit $89 $89 $59 $59

Which supply chain?

Is the model being added as part of NPI?

What kind of accessory is being added?

Is new documentation required?

Is a new accessory required?

Cost impact

$12.6K–$49.3K

$3.6K–$7.7K

$1.7K–$3.7K

$0.7K–$1.5K

$1.1K–$2.3K

$12.5K–$18.8K

$55K–$136K

$3.9K–$8.3K

Cost impact

$17.5K–$37.6K

$2K–$4K

$0.0K–$0.0K

$3.0K–$10.2K

$2.47M–$2.39M

ROI Assessment$2.53M–$2.53M

$177K–$320K13.94:1–7.46:1

$0.0K–$0.0K$0.0K– $0.0K$3.4K–$9.6K

$96.5K–$157.1K$5.0K–$10.0K$3.8K–$6.3K

$177K–$320K

$58.6K–$118.4K

$0.0K $0.0K

$10.0K–$18.8K

Cost impact

$234K–$459K

SF mono

NPI

Class I

No

Low-Touch

No

Figure 4: Users enter information on new-model proposals into the calculator. The projected ROI for the modelis shown, and detail on the cost impacts is included. Color coding is used to indicate whether an SKU achieves“green zone,” “yellow zone,” or “red zone” ROI thresholds.

projected volume and product type) that are readilyavailable from marketing when making the case fora new SKU. More complex modeling (Figure 3) goesinto making and calibrating the tool but is not part ofthe user interface.

Our complexity-analysis framework has proven tobe flexible and effective. Wherever possible, we tryto leverage the ROI approach and cost classification(Figures 2 and 3) and reuse components of calculatorspreadsheets. However, because the key cost drivers

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


and the impacts of product variety can vary signif-icantly across business environments, a customizedsolution is typically required. The average cost toconduct a complexity-analysis project (i.e., the timerequired by the OR team and sponsoring organiza-tion) for one HP business is approximately $90,000.

The Revenue CoverageOptimization (RCO) ToolAfter products or features have been launched intoHP’s product portfolio, some costs of variety becomesunk, and the variety management focus shifts fromscreening new products to maximizing value from theactive portfolio. As products begin to sell in the mar-ketplace, transaction-level sales data become avail-able, enabling more sophisticated analysis. RCO wasdesigned to help HP understand the revenue trade-offs in managing product variety when a historyof customer order data is available. Prior to imple-menting RCO, HP’s prevailing method of productportfolio management had been to judge productsby their individual revenue or volume contributionsin recent order history. However, researchers andbusiness managers knew that when determining theimportance of products in businesses with config-urable products, examining each product in isolationwould not suffice. A product that generates rela-tively little revenue on its own, such as a powersupply, might be a critical component in high-revenueorders and essential to order fulfillment. To capturethe interrelationship among products through orders,HP developed a new metric, order coverage, which rep-resents the percentage of a given set of past ordersthat could be completely fulfilled from the portfolio.Similarly, revenue (margin) coverage of a portfolio is therevenue (margin) of its covered orders as a percent-age of the total revenue from the data set. The conceptof coverage provides a meaningful way to measureeach product’s overall impact on a business. RCO is adeterministic optimization tool that finds the smallestportfolio of products that covers any given percentageof historical order revenue. It answers questions suchas, “If I pick only 100 products, which ones shouldI choose to maximize revenue from orders contain-ing only these products?” More generally, given a setof historical orders, RCO computes a nested series of

product portfolios along the efficient frontier of order-revenue coverage and portfolio size.

The black curve in Figure 5 illustrates this effi-cient frontier. In this example, 80 percent of orderrevenue can be covered with less than 27 percent ofthe total product portfolio if we select those prod-ucts according to RCO’s recommendations. One canuse this tool to select the portfolio along the efficientfrontier that offers the best trade-off—relative to busi-ness objectives—between revenue coverage and port-folio size. The strong Pareto effect in the RCO curvepresents an opportunity to improve on-time deliveryperformance. A small investment in improved avail-ability of the top few products will significantly reduceaverage OCT.

The portfolios corresponding to points along theefficient frontier are nested; the portfolio with 95 per-cent revenue coverage contains the one with 90 per-cent coverage. Thus, RCO provides a product rankingthat yields a continuum of portfolio choices that areeasily modified to adjust to changes in desired cover-age level.

The problem of generating a single portfolio onthe efficient frontier is known as a selection prob-lem. Its canonical formulation is an integer program-

0

20

40

60

80

100

0 200 400 600 800 1,000 1,200

Cum

ulat

ive

perc

enta

ge o

f re

venu

e co

vere

d

Number of products

RCO compared to heuristic ranking methods

RCORevenue impactMax order revenueUnits shippedRevenue generated

Figure 5: This chart shows revenue coverage versus portfolio sizeachieved by RCO (black) and four other product-ranking methods, appliedto the same historical data. The four other curves, in decreasingly sat-urated grays, are based on ranking by the following product metrics:revenue impact (the total revenue of orders containing the product),maximum revenue of orders containing the product, number of unitsshipped, and individual product revenue.

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


ming problem that, for HP’s order-history data sets,is too big to solve by standard methods. We foundthat the problem of generating a series of solutionsalong the efficient frontier can be posed as a para-metric maximum-flow problem in a bipartite network(Balinksi 1970, Rhys 1970). The team developed anew, efficient, and exact algorithm to solve the para-metric maximum-flow problem (Tarjan et al. 2006;Zhang et al. 2004; 2005a, b). This algorithm, calledsimultaneous parametric maximum flow (SPMF), isseveral times faster than best-known prior solutiontechniques for the same problem on the large real-world data sets that we faced (Babenko et al. 2007).It is also much easier to implement than previousalgorithms. In the new algorithm, the parametricmaximum-flow problem A is converted to a specialnonparametric maximum-flow problem B. Solving Bgives the chain of nested solutions to problem A atall break points of the parameter. The special non-parametric maximum-flow problem B is solved by anew flow-balancing method, which redistributes theflows over a number of arcs either around a closedloop or among all the arcs incident to a vertex. Thisflow-balancing method differs from two main typesof maximum-flow algorithms in the literature—theaugmenting path method (Ford and Fulkerson 1956)and the preflow-push-relabel method (Goldberg andTarjan 1986, 1988). Tarjan et al. (2006) later generalizedthe flow-balancing method to general nonparametricmaximum-flow problems. Appendix B shows detailsof the portfolio-selection problem, its equivalence toa parametric maximum-flow problem, and the SPMFalgorithm.

RCO compares favorably to other heuristics forranking products (Figure 5). The gray curves showthe cumulative revenue coverage achieved by fourheuristic product rankings, in comparison to thecoverage achieved by RCO. The best alternative toRCO is one that ranks each product according to itsrevenue impact, a metric our team devised to repre-sent the total revenue of orders in which the productappears. The revenue-impact heuristic comes closestto RCO’s coverage curve, because it is best amongthe heuristics at capturing product interdependencies.Still, in our empirical tests, we found that the revenue-impact ranking provides notably less revenue cover-age than RCO’s ranking. Given that RCO runs in less

than two minutes for typical data sets, HP had noreason to settle for inferior coverage.

Although we have emphasized the objective ofmaximizing historical revenue coverage subject to aconstraint on portfolio size, RCO is flexible enoughto allow a much wider range of objectives, such ascoverage of order margin, number of orders, or anyother metric associated with individual orders. It caneasily accommodate up-front strategic constraints onproduct inclusion or exclusion and can be applied atany level of the product hierarchy, from SKUs downto components.

The SPMF algorithm has applications well beyondproduct portfolio management, such as in the selec-tion of parts and tools for repair kits, terminal selec-tion in transportation networks, and database-recordsegmentation. Each problem can be naturally formu-lated as a parametric maximum-flow problem in abipartite network. The team’s extension of SPMF tononparametric max flows in general networks hasan even broader range of applications, e.g., in air-line scheduling, open-pit mining, graph partitioningin social networks, baseball elimination, staff schedul-ing, and homeland security.

In practice, RCO is used to enhance and facilitatehuman judgment in managing product variety. Portfo-lio design depends critically on knowledge of strategicnew-product introductions and planned obsolescence,which historical order data do not reveal.

HP businesses typically use the previous threemonths of orders as input data to RCO, becausethis duration provides a representative set of orders.Significantly longer horizons might place too muchweight on products that are obsolete or nearing endof life. When analysis on longer horizons is desired,RCO allows weighting of orders in the objective, thusplacing more emphasis on covering the most recentorders in a given time window.

SPMF was implemented in C++. A graphicaluser interface (GUI) in a Web browser and RCOoutput visualization in Excel were integrated withSPMF and the corporate financial database for RCOdeployment in HP businesses. RCO’s approximately$1.1 million development cost includes researchers’time to develop and implement the algorithm andcontractor time to build the GUI.

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


As we deployed RCO, we validated the work onmultiple fronts to ensure that all stakeholders wouldfeel confident in the results. We verified the algo-rithm’s correctness in three ways. First, we provedits convergence to correct results theoretically and bymatching its output with those of a well-establishedcommercial optimization solver (CPLEX) applied toan equivalent problem formulation. Second, with helpfrom domain experts in product marketing, we care-fully reviewed the tool’s input and cross-validatedthe results with these experts’ intuition and throughcomparison to other metrics. Third, we validated themodel. Although the model’s objective of computingthe efficient frontier of coverage and portfolio sizehad been defined jointly by the business stakehold-ers and OR professionals on the team, the best evi-dence of its validity is that the business results matchthe model’s predictions: improved focus on the top-ranked products yields significant overall operationalbenefits. The next section highlights these results.

Case Example: PortfolioManagement in PSGPSG, a $42 billion business, includes HP’s commercialand consumer desktop and notebook PC businesses,as well as its workstations, handheld computing, anddigital entertainment product lines.

PSG experienced many of the challenges discussedabove as its variety grew over the past decade. Toincrease market reach and maximize competitivenessin the markets in which it competes, it offered manyproducts and options, allowing customers to selectfrom many technology platforms and product formfactors. Within each platform, customers could selectfrom an array of processors, drives, memory con-figurations, and accessories; using HP’s configure-to-order process, these components could be combined togenerate millions of desktop and notebook PC ordercombinations. Adding to the complexity, each config-uration option could include different subassembliesand components, which might be sourced from multi-ple suppliers.

Managing each component required testing, fore-casting, supplier management, inventory, end-of-lifeand warranty support, production planning, and a

host of other processes and outlays. If one compo-nent was unavailable, production and order ship-ments might be delayed and customers dissatisfied.Some products were sold often and in high volumes;others were involved in only a handful of orders andwere of little strategic importance to the business.However, differentiating between these products in asystematic and data-driven way was difficult; thus,some components might be overstocked and someunderstocked. These availability issues led to unpre-dictable and uncompetitive OCTs and frustrated cus-tomers. Last, the inconsistencies between offerings inthe Americas, Europe, and Asia, despite product sim-ilarities, gave rise to even greater variety and costsfor PSG.

HP’s approach, which it began in its notebook divi-sion and extended later to desktops, started as anengagement with its Strategic Planning and Modeling(SPaM) team to review the divisions’ cost structuresand develop an appropriate complexity ROI calcula-tor; an engagement with HP Labs to integrate the RCOalgorithm into its business planning systems followed.

Initial complexity cost analysis for notebook PCsindicated relatively low up-front variety-driven fixedcosts and relatively high volume-driven variablecosts. Therefore, PSG’s emphasis was on managingthe portfolio to steer demand from low-volume fea-tures to medium- and high-volume features, ratherthan on up-front screening of new entrants. PSGdeployed complexity ROI calculators for notebooksand desktop PCs to allow it to quantitatively evalu-ate feature decisions, and receive guidance on min-imum incremental margins necessary for features tobe viable. It then used RCO to prioritize features inits offerings. Today, these tools continue to providecritical input into three major PSG programs:

• Worldwide Recommended Offering (RO) pro-gram for notebooks and desktops. This programuses RCO to identify the most critical features ineach region; it designates them as the “core offering,”which includes about 20 percent of the feature port-folio and covers 80 percent of all orders. It classi-fies all other features as the “extended offering.” PSGprovides different service levels for the two classesof features. It stocks core features in higher inven-tory levels; thus, they have short lead times. It stocksextended features at lower levels or not at all; thus,

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


they have longer lead times. By reallocating its inven-tory investment, PSG has reduced its average OCTby four days on core notebook features and two dayson core desktop features since 2006. These operationalefficiency gains are self-reinforcing. Demand shifts tothe core because customers who choose core featuresare rewarded with rapid delivery. As demand concen-trates on fewer features, demand variability declines,leading to additional cycle-time improvements. Inthe program’s first six months, the number of fea-tures required to cover 80 percent of orders droppedby one-fifth; the average revenue contribution ofcore features doubled and that of extended featuresstayed flat. RO-based demand steering has improvedforecast accuracy and availability, lowered inventoryexpenses, and improved consistency and predictabil-ity for both customers and suppliers. To date, PSG hasimplemented the notebook RO program in the UnitedStates and the Asia/Pacific region, and the desktopRO program in the United States and Europe/MiddleEast region. PSG estimates that each day of OCTimprovement saves it at least $38 million annuallyin inventory-driven costs, operational expenses, andfinancing and saves a $12 million one-time gross mar-gin increase through cash-flow benefits. PSG manage-ment estimates that it realized savings of $130 millionin Europe, Middle East, and Africa (EMEA) and theUnited States in the first year of RO implementationand $100 million annually thereafter. Rollouts to otherPSG product lines are likely to generate comparablebenefits.

• Global Product Offering program. This initia-tive is a set of products made available worldwideto HP’s largest global customers. Each customer hasa preferred set of standard products and wants thoseproducts offered worldwide with consistent price,components, and life cycle. Since early 2008, RCO hasbeen used to design this offering, replacing the previ-ously used manual “best guess” process. The globalproduct offering adoption rate has increased from18 to 85 percent. Moreover, revenue from PSG’s globalcustomers grew 23 percent in 2008, partly because ofa better-designed global product offering. PSG con-servatively estimates a $130 million annual revenueincrease because of using RCO for this initiative;global customer escalations, because of inconsistentworldwide product offerings, have also decreasedsignificantly.

• Feature Screening program. Using the complex-ity cost assessment and the complexity ROI calculatortools developed for notebooks and desktops, PSG canperform data-driven evaluation of product propos-als and eliminate low-ROI products or features beforelaunching them into the portfolio, thus avoiding arange of unrecoverable costs. To date, these programshave generated over $100 million in margin improve-ments and continue to generate over $40 million peryear for PSG.

The initiatives have generated hundreds of mil-lions of dollars in impact in PSG and dramaticallychanged how its management and operations teamswork. Marketing and supply chain teams can nowhave informed, fact-based discussions around trade-offs in the portfolio and, for the first time, jointly dis-cuss the concept of the sweet spot.

Many organizational and informational hurdles hin-dered implementation of the initiatives. The primarychallenge, in PSG and across HP, was to shift the mind-set from revenue-focused management to margin-focused management. The decisions that drove explo-sions in product variety were made to chase revenueopportunities without a clear understanding of theircost and margin impacts. Margin-focused manage-ment required cross-functional teams to bridge theorganizational divide between supply chain, market-ing, and R&D to bring together a complete picture ofthe costs and benefits of variety. To exacerbate matters,many incentives both in the sales organization and forexecutive management were based on revenue resultsrather than profitability results. The efforts supportedby HP’s OR teams provided data-driven and unbiasedinsights and tools to bridge the organizational divide.We also helped put in place explicit complexity met-rics, which eventually became part of the scorecardsused to evaluate management performance.

A second challenge was around disseminating andgaining agreement on the many process changes thatcame out of the programs in PSG, such as new rulesfor customer communication, inventory management,and supplier management.

Last, the data-driven processes that we put in placealso required new information links and improvedmanagement of product and part information withinPSG. Building and supporting the tools driving thesolution implementation required support from HP’sinformation technology organization.

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


ImpactAcross all its businesses, HP has achieved, and con-tinues to enjoy, substantial benefits from the imple-mentation of the portfolio management techniquesdescribed in this paper. These include direct finan-cial benefits, operational improvements, and a rangeof important “soft” benefits. For example, using acomplexity ROI calculator custom-built with execu-tives and program managers from their business, thecompany’s trademark LaserJet business reduced SKUcounts by 40 percent between 2006 and 2009, dramat-ically streamlining the offering and generating annualnet profits of approximately $20 million.

HP’s enterprise server business, Business CriticalSystems (BCS), runs RCO quarterly to evaluate itsexisting product portfolio and make product ratio-nalization decisions. RCO enabled the eliminationof more than 3,300 products from BCS’ portfolio ofover 10,000 products from late 2004 through 2008.BCS supply chain managers estimate that this reduc-tion has resulted in at least $11 million in admin-istrative cost savings, excluding any inventory costor other operational expense. They have also usedRCO to help them prioritize the order in which prod-ucts should be brought into compliance with newEuropean environmental standards. Moreover, BCSemploys RCO for improved, data-driven design ofoptions for new-product platforms based on the orderhistory of previous-generation platforms.

The use of the ROI screening framework and RCOin HP’s portfolio management programs has yieldedcompany-wide profit improvements, conservativelyestimated, of over $500 million between 2005 and 2008,and continues to generate benefits of about $180 mil-lion per year, with the potential for greater benefitsthrough expanded deployment of these methods.

The application of these tools has also yieldedimportant qualitative benefits for HP.

• Improved customer satisfaction: PSG customersappreciate the overall cycle-time reduction and themore predictable product availability that has resultedfrom the RO program. These changes have improvedcustomer loyalty, market share, and competitive posi-tioning for PSG. Moreover, the global product offeringallows PSG’s large global customers to satisfy theirproduct requirements in multiple countries, leading toincreased demand, higher customer satisfaction, and

markedly fewer global customer escalations to HP’stop management.

• Product line complexity reduction: Eliminationof thousands of low-value products from HP’s line-ups, both through prelaunch screening and post-launch rationalization, reduced complexity and drovecost reductions across many areas, such as productdevelopment, qualification, testing, forecasting, plan-ning, order management, manufacturing, data man-agement, marketing, and supplier management.

• Reduced confusion among customers and salesrepresentatives: Significant SKU reductions through-out the company have lessened the confusion thatexcess product variety caused among customers andHP sales professionals.

• Partner and supplier benefits: HP’s suppliers andchannel partners benefit from improved forecastingaccuracy, as well as reduced up-front tooling andqualification costs achieved through these portfoliomanagement techniques.

• Increased organizational effectiveness: The useof the OR-based tools to manage product varietyhas heightened awareness of the cost of complexitythroughout HP businesses, brought about better orga-nizational discipline in SKU introduction, improvedcollaboration between product marketing and supplychain teams, and resulted in a significant reduction incostly manual errors that arose from overloading theforecasting and planning teams.

• Improved visibility of OR within HP: The suc-cess of RCO and the complexity ROI calculators hasimproved organizational understanding, up to thesenior executive level, of how OR can provide opera-tional efficiencies to increase revenue and profit. Thehigh visibility of RCO and the complexity ROI frame-work benefits has led to several new deployments ofthese tools.

• Portability: HP’s approaches to managing prod-uct variety are applicable to many businesses insideand outside of HP. In configurable product busi-nesses, where fulfillment of an order depends onavailability of several products, RCO can help achieveoperational efficiencies by identifying the sweet spotin the offering. The complexity ROI framework hasbeen and continues to be applied successfully outsideof HP. The consulting authors have worked with com-panies in a variety of industries to implement similarapproaches in their businesses.

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


OR at HPHP has been a leader in applying OR to its impor-tant business problems for decades. In addition toits many OR professionals throughout the company,HP has two major centers of excellence in OR: SPaMand HP Labs. SPaM is an analytics team that worksthrough an internal consulting model to support oper-ational innovation and operations strategy at HPwhile delivering immediate impact through businessprojects. It combines the talents of OR PhDs withtop-tier management-consulting experience (Olavsonand Cargille 2008). SPaM sometimes partners withOR academics or analytical consultants at special-ized firms such as Strategic Management Solutions,an important contributor to the development of thevariety-management ROI screening framework thatSPaM uses. HP Labs is HP’s central R&D organizationchartered to conduct high-impact scientific researchto address the most important opportunities thatHP and its customers will face in the next decade.Within HP Labs, researchers with PhDs in OR, math-ematics, statistics, economics, and computer sciencedeliver sophisticated analytical tools to HP’s busi-nesses and advance the state of the art in businessprocess research (Jain 2008). SPaM and HP Labs, incollaboration with the authors of this paper, devel-oped the approaches to product portfolio manage-ment described herein.

ConclusionsHP’s approach to managing its product portfolio isa real-world example of applying OR to dramaticallyimprove business performance. Although groundedin science, our solutions are highly tailored to thetrue nature of the business problems facing HP. Theyrepresent a powerful, comprehensive, and flexibleapproach to managing product variety. It has provensuccessful at HP and is extendable to businesses inother industries.

Appendix A. A Method for CalculatingVariability-Driven CostsWe define variability-driven costs as costs resultingfrom physical mismatches of demand and availablesupply, including both costs of shortage (expedit-ing, lost sales, and material price premiums) and

inventory holding and excess (excess and obsoles-cence, component devaluation, product discounting,or price protection). We also present an extension ofthe method to allow variability-driven costs to beapproximated as a simple function of part or productvolume, making it easy to operationalize the methodin decision support tools such as the ROI calculators.

The advantage of calculating variability-drivencosts directly is that this method can be applied to ana-lyze the impact of process improvements on knowncost pools or high-risk events (e.g., new-product intro-duction or product end of life). Most importantly,the method provides an alternative for quantifyingthe financial benefits of any process change thatpools demand risk to reduce demand variability, suchas reducing part variety or consolidating inventorystocking locations. The typical rough-cut approachestimates how much an inventory buffer can bereduced and applies an average inventory-driven costrate against this reduction to calculate a cost sav-ings. Although useful in some contexts, this rough-cut method also has serious shortcomings. First, itaddresses only the costs of excess and not the costsof shortage. Second, the calculated risk-pooling ben-efits approach zero as the planned inventory bufferapproaches zero; however, some of the highest sourcesof variability-driven cost concern events in which thebuffer is relatively small, e.g., end-of-life planning,buffer planning in advance of announced componentprice drops, or large demand spikes because of bigaccount orders. By contrast, our method does not havethese shortcomings.

The foundation for calculating variability-drivencosts is a closed-form calculation of expectedstockouts and expected on-hand units. We use the fol-lowing definitions:

X: random demand over replenishmentlead time (lead-time demand).

�= E�X�: expected lead-time demand.� = stdev(X): standard deviation of lead-time

demand.S = E�X�+ b: order-up-to point of inventory in a

periodic-review system.b: target safety stock buffer quantity in

units, where b= S−E�X�.

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


k: the fractile on the unit normal asso-ciated with the buffer quantity: k =�S−��/� = b/� .

Fu�k�, fu�k�: distribution and density function forthe unit normal distribution.

Fundamentally, we are interested in evaluating twoquantities: E[stockout�S�], the expected stockout if theorder-up-to point is S, and E[on-hand inventory(S�],the expected on-hand inventory units if the order-up-to point is S. Special properties of the normaland Gamma distributions allow closed-form solutionsto these expressions. We illustrate for the normalcase below. First, observe that stockout(S�− on-handinventory�S�=X− S, from which we have

E[on-hand inventory �S�� = S−E�X�+E[stockout�S��

= b+E[stockout�S�� (1)

In the case of normally distributed demand, we cancalculate the expected stockout by scaling the solutionderived in Silver et al. (1998, Equation B.7) for the unitnormal by the factor � :

E[stockout�S��= ��fu�k�− k�1− Fu�k�� (2)

Stockouts are typically measured in percentageterms as a fill rate, calculated as

fill rate�S�= 1−E[stockout�S��/��

Combining Equations (1) and (2) yields an expres-sion for expected on-hand inventory:

E[on-hand inventory�S��= ��fu�k�+ kFu�k�� (3)

From Equations (2) and (3), we can calculate howthe expected excess and shortage rates (units as a per-cent of mean demand) vary with changes in lead-timedemand variability. If we assume that historical totalvariability-driven cost pools (e.g., total excess andobsolescence costs) scale in proportion to the shortageand excess rates, then we have a model for how eachcost pool can be impacted based on the lead time andbuffer stock relevant to that cost pool. For example,for end-of-life discounting costs, the target buffer iszero, and the lead time for critical components couldbe longer than usual as suppliers demand end-of-lifebuys at longer lead times.

An extension of the method allows the variability-driven cost per unit to be modeled as a function ofaverage part volume. This allows us to quantify howthe variability-driven cost varies with volume. Forexample, consider the guideline “an LCD panel sell-ing less than 10 K units per month has a $10/unithigher variability-driven cost than a panel selling morethan 50 K units per month.” This is easy to under-stand and easy to factor into the decision process ofwhether to offer an additional panel in the productline. Figure A.1 summarizes the analysis process.

To derive the cost per unit as a function of volumerelationship, it is also necessary to model a statisticalrelationship between part volume and variability—low-volume parts have a higher coefficient of vari-ation (CoV, or the ratio of standard deviation toaverage demand). We conducted a statistical analy-sis and concluded that a statistical estimator for apart’s demand variability derived from such an aggre-gate volume-variability relationship (using data frommany different parts) carries as much predictive valuefor the part’s true variability going forward as did anestimator derived from a limited history of actual dataon that specific part. Next, we collect data on the dis-tribution of part volumes across the portfolio so thatthe following equality holds

n∑

i=1

vic�vi�=Cportfolio�

where we sum across all the n parts, each with aver-age volume vi and variability-driven cost per unitc�vi�, to arrive at the total portfolio variability-driven

3. Variability-drivencost/unit = f (volume)

1. CoV = f (avg. volume)Historical data on

weekly part volume

Tool orguidelines

2. Relative shortage oron hand = f (volume)

Distribution of partvolumes across the

portfolio

Top-down data ontotal variability-driven costs/year

Figure A.1: The flowchart shows an overview of an approach for estimat-ing variability-driven cost per unit.

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


cost Cportfolio. Although we know the part volumes andthe total cost, we do not know c�v�. However, wecan calculate a relative cost curve, r�v�, using the on-hand inventory and shortage-cost modeling, and wecan link the two through a constant d for which wewill solve

c�vi�= dr�vi�→n∑

i=1

vidr�vi�=Cportfolio → d= Cportfolio∑vir�vi�

�

We now have a simple link between part volumeand cost; we can use it in the complexity cost calcu-lator to compare the costs of multiple small-volumeSKUs versus pooling the volume in a single SKU.

Appendix B. RCOProblem Formulationand SPMFAlgorithmIn this appendix, we describe the technical details ofthe RCO problem formulation and its equivalence toa parametric maximum-flow problem in a bipartitenetwork, and we provide an overview of a new algo-rithm for parametric maximum flow.

The problem of finding a product portfolio of sizeof at most n that maximizes the revenue of orders cov-ered can be formulated as the integer program IP(n�:

Maximize∑

o

royo subject to

yo ≤ xp for each product-order combination �o� p�� (4)

∑

p

xp ≤ n� (5)

xp ∈ �0�1�� yo ∈ �0�1�� (6)

where ro is the revenue of order o, and binary deci-sion variables xp and yo represent whether product pis included in the portfolio and whether order o iscovered by the portfolio, respectively.

Because typical data sets have hundreds of thou-sands of product-order combinations, IP(n� can takemany hours to solve or might even exceed memorylimitations. However, it does have a nice structuralproperty—the constraints (4) are totally unimodular.We exploit this property by creating the followingLagrangian relaxation LR(��:

Maximize∑

o

royo −�∑

p

xp subject to

yo ≤ xp for each product-order combination �o� p�,xp ∈ �0�1�� yo ∈ �0�1��Because LR(�� is totally unimodular, it has an inte-

ger optimal solution. Moreover, if a set of coveredorders and selected products, �O�P�, is optimal forLR(�), then it is optimal for IP(�P ��. Thus, solvingLR(�� for a series of values of � generates a series ofsolutions to IP(n� for several values of n. Solutionsgenerated by this method are nested—the optimal setof covered orders for a given �0 is a subset of theoptimal covered orders for smaller �≤ �0. Moreover,these solutions lie along the efficient frontier of rev-enue coverage versus portfolio size. This series doesnot provide an integer solution for every possiblevalue of n; solutions below the concave envelope ofthe efficient frontier are skipped. However, in prac-tice the number of distinct solutions is typically about85 percent of the total product count. A wise selectionof values of � produces quite a dense curve of solu-tions. To obtain a complete product ranking, we useproduct revenue impact as a heuristic to break tiesamong products, because this metric proved to pro-vide the best coverage among the heuristics we tried.

Our original implementation of RCO used a com-mercial LP solver (CPLEX) to solve the series ofproblems, LR(�). However, for typical data sets withmillions of order line items, each such problem tookseveral minutes to solve. To solve it for many val-ues of � to create a dense efficient frontier took manyhours. We needed a more efficient approach.

The key to a more efficient approach is the equiv-alence between the problem, LR(�), and the problemof finding a minimum cut in a particular bipartite net-work (Balinksi 1970). To see how LR(�) can be viewedas a minimum-cut problem, consider the network inFigure B.1, with a source node s at the far left and asink node t at the far right. Adjacent to the source nodeis a set of nodes, each corresponding to one product.Adjacent to the sink node is a set of nodes, each corre-sponding to one order. The capacity of the links adja-cent to s is �. The capacity of the link from the nodefor order i to t is the revenue of order i. The capac-ity of links between product nodes and order nodes isinfinite.

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


.

.

.

.

.

.

Products Orders

t

∞∞

∞∞

�

�

� r1

rn

s

Selectedproducts

Coveredorders

Figure B.1: The diagram illustrates a bipartite minimum-cut/maximum-flow problem that corresponds to the Lagrangian relaxation LR(�).

For this network, the set T in a minimum cut cor-responds to the products selected and orders cov-ered by an optimal solution to LR(�). To see why,first observe that because the links from productnodes to order nodes have infinite capacity, theywill not be included in a finite capacity cut. There-fore, for any order node in the T set of a finitecapacity cut, each product in the order must alsohave its node in T . Therefore, a finite capacity cutcorresponds to a feasible solution to LR(�). More-over, the value of an s − t cut is

∑o ro�1 − yo� +

�∑p xp. Minimizing this quantity is equivalent to

maximizing∑o royo − �

∑p xp; therefore, a minimum

cut is an optimal solution to LR(�).It is a well-known result of Ford and Fulkerson

(1956) that the value of a maximal flow equals thevalue of a minimum cut. Moreover, the minimum cutcan be obtained by finding a maximal flow.

Because we seek a solution to LR(�) for multiplevalues of �, it is a parametric maximum-flow prob-lem because the arc capacities depend on a parameter.Several known algorithms for parametric maximumflow exist, including those in Gallo et al. (1989) forgeneral networks and Ahuja et al. (1994) for bipar-tite networks. In most prior algorithms for parametricmaximum flow, a series of maximum-flow problemsis solved, and each problem’s solution is used tospeed up the solution to the next one. By comparison,the algorithm presented here simultaneously finds themaximum flow in the network for all break points ofthe parameter value. The value of the maximum flowfrom s to t is a piecewise-linear function of �. A breakpoint of the parameter value is where the derivativeof the piecewise-linear function changes.

Parametric Bipartite Maximum-Flow AlgorithmThe new parametric bipartite maximum-flow algo-rithm takes advantage of the special structure of thecapacity constraints that Figure B.1 shows.

The logic behind the algorithm is as follows. Firstassume that �=�. Then, the only constraints on flowsresult from the capacity limitations on arcs incidentto t. Finding flow assignments that saturate all capaci-tated links, resulting in a maximum total flow, is easy.

The next step is to find such a maximum-flowassignment that distributes flows as evenly as pos-sible across all arcs leaving s. The property “evenlyas possible” means that it is impossible to rebal-ance flows between any pair of arcs in such away that the absolute difference between these twoflows decreases. Note that even in this most evenmaximum-flow assignment, not all flows will be thesame.

Now, with the most even assignment discussedabove, impose capacity constraints of � < � on thearcs leaving s. If the flow assignment for a given oneof these arcs exceeds �, reduce the flow on this arcto � and propagate the flow reduction appropriatelythrough the rest of the graph. Because the originalflow assignment was most evenly balanced, the totalflow lost to the capacity constraint is minimal andthe total flow remaining is maximal for the givenparameter �.

More formally, the algorithm works as follows:Step 1. For a graph as in Figure B.1 with � = �,

select an initial flow assignment that saturates thearcs incident to t. This is most easily done backwards,starting from t and choosing an arbitrary path for aflow of size ri from t through oi to s.

Step 2. Rebalance the flow assignment iterativelyto obtain a “most evenly balanced” flow assign-ment. Let f �a → b� denote the flow along the linkfrom node a to node b. The rule for redistribut-ing the flows is as follows. Pick i and j for whichthere exists an order node ok and arcs pi → okand pj → ok such that f �s → pi� < f �s → pj� andf �pj → ok� > 0. Then, reduce f �s→ pj� and f �pj → ok�by min��f �s→ pj�− f �s→ pi��/2, f �pj → ok�� andincrease f �s→ pi� and f �pi → ok� by the same amount.Repeat Step 2 until no such rebalancing can be found.

The procedure in Step 2 converges, as Zhang et al.(2004, 2005a) prove. The limit is a flow assignmentthat is most evenly balanced. In addition, because

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


total flow is never reduced, the resulting flow assign-ment is a maximum flow for the graph with �=�.

Step 3. To find a maximum-flow assignment for agiven value of �, replace flows exceeding � on arcsleaving the source s by � and reduce subsequentflows appropriately to reconcile flow conservation.The resulting flow assignment is a maximum flowfor �.

Zhang et al. (2005a) provide more details and a rig-orous mathematical treatment of the problem. Zhanget al. (2004) show that the algorithm generalizes tothe case in which arc capacities are a more generalfunction of a single parameter.

Because our application requires only knowledge ofthe minimum cut, one only needs to identify thosearcs that exceed the capacity limit of � after Step 2.Those arcs will be part of the minimum cut; the onesleaving s with flows less than � will not. To find theremaining arcs that are part of the minimum cut, onehas only to identify which order nodes connect to sthrough one of the arcs with flows less than �, andcut through those nodes’ arcs to t.

We can show that the t-partition of the minimumcut with respect to � contains products whose flowsfrom the source equal � and the orders containingonly those products. These products constitute theoptimal portfolio for parameter �. Note that Steps 1and 2 are independent of �. The result of Step 2 allowsus to immediately determine the optimal portfolio forany value of �.

Because the flows are balanced between two arcs,s→ pi and s→ pj , in the algorithm described above,we call it an arc-balancing method. Arc-balancingSPMF reduced the time for finding the entire efficientfrontier from hours to a few minutes.

We developed a second version of the SPMF algo-rithm based on the idea of redistributing the flowsgoing into a node o in a single step; for all pairspi → o and pj → o, flows f �s → pj� and f �pj → o�

are most evenly balanced. This method of redistribut-ing flows around a vertex o is the vertex-balancingmethod (Zhang et al. 2005b). Vertex-balancing SPMFfurther reduces the time for finding the entire efficientfrontier to seconds for typical problems.

ReferencesAhuja, R. K., J. B. Orlin, C. Stein, R. E. Tarjan. 1994. Improved

algorithms for bipartite network flow. SIAM J. Comput. 23(5)906–933.

Babenko, M., J. Derryberry, A. Goldberg, R. Tarjan, Y. Zhou.2007. Experimental evaluation of parametric max-flow algo-rithms. C. Demetrescu, ed. Proc. WEA 2007, Lecture Notes inComputer Science, Vol. 4525. Springer, Heidelberg, Germany,256–269.

Balinksi, M. L. 1970. On a selection problem. Management Sci. 17(3)230–231.

Cargille, B., J. Melia. 2007. Less is more, later is better: Keepingsales high and costs low at Hewlett-Packard. APICS (October 7)40–43.

Cargille, B., C. Fry, A. Raphel. 2005. Managing product linecomplexity. OR/MS Today 32(3). http://www.lionhrtpub.com/orms/orms-6-05/frcomplexity.html.

Ford, L. R., D. R. Fulkerson. 1956. Maximum flow through a net-work. Canadian J. Math. 8 339–404.

Gallo, G., M. D. Grigoriadis, R. E. Tarjan. 1989. A fast parametricmaximum flow algorithm and applications. SIAM J. Comput.18(1) 30–55.

Goldberg, A. V., R. E. Tarjan. 1986. A new approach to the max-imum flow problem. Proc. 18th Annual ACM Sympos. TheoryComput., Berkeley, CA, ACM, New York, 136–146.

Goldberg, A. V., R. E. Tarjan. 1988. A new approach to the maxi-mum flow problem. J. ACM 35(4) 921–940.

Jain, S. 2008. Decision sciences: A story of excellence at Hewlett-Packard. OR/MS Today 35(2). http://www.lionhrtpub.com/orms/orms-4-08/frroundtable.html.

Olavson, T., B. Cargille. 2008. OR inside. OR/MS Today 35(5). http://www.lionhrtpub.com/orms/orms-10-08/franalytics.html.

Olavson, T., C. Fry. 2006. Understanding the dynamics of valuedriven variety management. MIT Sloan Management Rev. 48(1)63–69.

Rhys, J. M. W. 1970. A selection problem of shared fixed costs andnetwork flows. Management Sci. 17(3) 200–207.

Silver, E. A., D. F. Pyke, R. Peterson. 1998. Inventory Managementand Production Planning and Scheduling. John Wiley & Sons,New York.

Tarjan, R., J. Ward, B. Zhang, Y. Zhou, J. Mao. 2006. Balancingapplied to maximum network flow problems. Proc. 14th Conf.Ann. Eur. Sympos. ESA, Lecture Notes in Computer Science, Vol.4168. Springer, Heidelberg, Germany, 612–623.

Zhang, B., J. Ward, Q. Feng. 2004. A simultaneous parametric max-imum flow algorithm for finding the complete chain of solu-tions. HP Technical Report HPL-2004-189, Palo Alto, CA.

Zhang B., J. Ward, Q. Feng. 2005a. Simultaneous parametric max-imum flow algorithm for the selection model. HP TechnicalReport HPL-2005-91, Palo Alto, CA.

Zhang B., J. Ward, Q. Feng. 2005b. Simultaneous parametric max-imum flow algorithm with vertex balancing. HP TechnicalReport HPL-2005-121, Palo Alto, CA.

Dow

nloa

ded

from

info

rms.

org

by [

149.

164.

77.9

5] o

n 05

Apr

il 20

16, a

t 10:

02 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.

Interfaces Joseph Woods, Jing Zhou,€¦ · Vol. 40, No. 1, January–February 2010, pp. 17–32...

Documents

Transcript of Interfaces Joseph Woods, Jing Zhou,€¦ · Vol. 40, No. 1, January–February 2010, pp. 17–32...