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Optimal Price Design for Variable Capacity

Outsourcing Contracts

Chris Kenyon & Giuseppe Paleologo

IBM Research (ZRL, WRL)

{chk@zurich|gappy@us}.ibm.com

© IBM 2004

Outline

• Background & Motivation

• Problem Setup

• Solution Method & Example

• Summary

N.B. in this field “utility” gets used to mean Utility Theory and utility-like services (acting like gas, electricity, etc.)

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Background – Commercial• Outsourcing deals are big business, typical order pipeline for a major

provider (e.g. Accenture, EDS, HP, IBM) ~ tens of billions USD.• Contract duration typically 2 to > 10 years• Contracts usually cover many services:

– Hardware• Mainframes / Servers / Desktops / Laptops / Devices • Hardware maintenance & support (on-site / call center / etc.)

– Software• OS / Infrastructure / Dedicated / Third party• Applications maintenance & support, application development

– Business processes• HR / Procurement / Customer Relationship Management / Finance & Accounting• Industry/company specific (e.g. insurance claim handling)

• Current expansion up value chain from infrastructure into business processes

• Increasing emphasis on variable volume, variable price, contracts– “adaptive enterprise”, “real-time enterprise”, “on demand”, etc.

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Background – Academic

• Outsourcing and coordination– Subcontracting decisions [Bertrand&01], [Mieghem98, 99],

[Atamtürk&01], [Wang&03]– Production & delivery in VMI [Fry&01]– Competition & outsourcing [Cachon&02], [Elmaghraby00]

• Variable volume and risk sharing– Empirical, automotive [Okamuro00]– Labor [Milner&01]– Risk sharing [Pratt00], [Tomlin03], [Gaur&04]– Flexibility [Tsay&99a,b], [Li&99], [Mooinzadeh&00]

• This is just a few recent references from a very extensive literature, and for Utility Theory the literature is even larger.– General [Camerer95]

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MotivationClient Objectives

- Minimize base price

- Minimize expected price

- Price certainty – i.e. a known price-per-volume at different volumes

Challenge

• In general these objectives are not compatible for variable volume contracts because costs are history dependent

• Fixed costs leave traces, as do variable costs that are not perfectly variable, e.g. lags on delivery, setup costs, etc.

Business Transformation

On DemandOperating

Environment

Business P

rocessesFlexible Financial and Delivery Options

Provider Objectives

- High certainty of minimum margin

- Maximize expected margin

- Client satisfaction

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Problem Setup

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Assumptions

1. Single customer• Multiple entities within client but not considered here. • No resource sharing across multiple clients.

2. Demand does not depend on price• Strong assumption – what about incentive of Client to add new services

to scope of outsourcing contract?• We do not consider Client-internal charging policies.

3. Single service• Multiple components within service.• No resource sharing across multiple services.

4. Myopic service fulfillment policy• Clearly suboptimal but this is not the focus of the current work.

5. Single pricing decision at start of contract• No changes during contract.

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Objective

Maximize UTotal = UC + UP

• UC client utility• UP provider utility

• By choosing an optimal pricing scheme– scheme can be cost based (e.g. cost + margin)– or based on a defined price-volume curve

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Client Utility, UC

i=0, … , n measurement/billing time points of contractp(vi) price for volume vi given volume history up to iω ΩC demand event space (Client’s point of view)FC demand distribution function as estimated by the Clientδ relative weight of lack of “cleanness” of price-volume curvea Client target price-volume curve (unit price target)vi(ω) demand volume history up to time point i in scenario ωvi(ω) demand volume at time point i in scenario ω||.|| evaluation metric, typically ()+, i.e. positive part

v

Cii

Ci

ni

ii

ni

iiCCC

dFvpvp

dFvavpvpaFU

)(|)()(|

)(||)())((||))((),,,(

'

00

UtilityC = - (cost + deviation + smoothness)

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Features of UC

• δ = 0, γ=0 only expected price of contract important

• δ high (a 0) certainty of unit price important

• || ()2 and a = expected unit price mean-variance framework for contract assessment– typically use ()+, i.e. positive part (first semi-deviation)

• γ high smoothness of price per unit important for different unit level– Some overlap between γ and ()2 for specification

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Provider Utility, UP

vi(ω) demand volume history up to time point i in scenario ωω ΩP demand event space (in Provider’s view)FP distribution function as estimated by the Providerp(vi) price given demand volume history up to point ic(vi) cost given demand volume history up to point Iwm relative weight of not meeting minimum margin requirement

(generally negative and large)Imin margin indicator function for contract NOT margin meeting minimum margin

marginmin

1

1 )())((

))((),( IwdF

vc

vpFU mPni

ii

ni

ii

PPP

UtilityP = expected margin provided VaR of margin distribution OK

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Features of UP

• Price and Cost both depend on volume history and both can be arbitrarily history dependent.

• Contract utility is zero if contract does not meet given thresholds– Important in practice to meet VaR type restrictions not only expectation-

type.

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Cost and Demand• Costs

– Can be arbitrarily history dependent in general and stochastic, e.g. software license costs.

Examples covered:

1. Linear (ideal case, no history dependence), e.g. software licensed on per-transaction basis

2. Constant, e.g. processor activation in existing box on site (i.e. no change)

3. Granular, e.g. processors from offsite computational utility

4. Change-costs, e.g. boxes at site – cost to install/remove (can be different)

5. Lagged, e.g. people (think training, notice periods)

• Assume demand ~ stationary (S)ARMA process– Useful for analytic tractability

– Not a limitation for the method in general

• Model relevant for low-pass filtered true/potential demand where bandwidth limit defined by service characteristics.

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5 10 15 20Time

2.55

7.510

12.515

17.520

Cost & Demand

5 10 15 20Time

2.55

7.510

12.515

17.520

Cost & Demand

5 10 15 20Time

2.55

7.510

12.515

17.520

Cost & Demand

5 10 15 20Time

2.55

7.510

12.515

17.520

Cost & Demand

1. Linear 3. Granular

4. Granular with Change Costs 5. Granular with Change Costs and Lags

N.B. actually discrete billing/costs not continuous.

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Solution Method & Example

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Solution Method

1. Non-Markov service component costs … state expansion

… but separate component costs are (often) independent … so not many dimensions

2. Cost granularity & “well behaved” costs efficient summarization & approximation of state space

3. Utility functional forms & “sensible” restrictions on prices• Define prices at finest level over same state space as costs.• Most “interesting” price schemes much “coarser” than costs solution tractable (linearly constrained) MIP or MIQP problem

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History Dependence State Expansion

• Costs are history dependent … we can express this history dependence as a complex state

– Usual state expansion to move non-Markov processes to Markov processes

• Examples1. Linear, already Markov2. Constant, already Markov3. Granular, already Markov4. Change costs, add 1 state: previous demand level5. Lags, add the same number of states as there are lags

– Note that we can include change costs here as well, just the lag-1 subset.– E.g. say maximum lag is 3 months where there is monthly billing, need 3

added states for a 4-dimensional state space (lag-0, lag-1, lag-2, lag3).

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Demand Characterization

• In general use simulation to populate demand state space.• For specific cases some analytic results are available, e.g.

– Stationary (S)ARMA processes have Multi-Normal distribution for arbitrary lags, and correlation matrix given by autocorrelation

• Assuming series “long enough” so that conditioning on first (contract day 1) value not important.

– Brownian-motion derived processes; in some cases have distributions and transition-density-functions (fairly restrictive).

• BM, GBM• Mean-reverting processes (e.g. Ornstein-Uhlenbeck)

• Important to capture variance of demand as well as expected demand.

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Formulation

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Example

10 20 30 40 50 60Timemonths

4

2

2

4DemandVolume

SARIMAModel0, 0, 12,.7,.7,.1,.1, .9

• Service: granular, costly-to-change (case 4.), e.g. desktop support– Cost depends on current level, and difference from pervious volume level

• Unit granularity, cost = 1 unit per unit, cost to change = 0.5 unit per difference

• Contract: 5-year, monthly billing/measurement

• Demand: Seasonal, stationary– Mean 10 units

5 10 15 20Lag

0.2

0.4

0.6

0.8

1AutoCorrelation

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State Maps for Cost, Demand

6 8 10 12 14V

6

8

10

12

14

Vsuoiverp

6 8 10 12 14V

6

8

10

12

14

Vsuoiverp

Cost Demand Distribution Function• correlation ~~0.75

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Efficient Negotiation Frontier

• Line gives pareto-optimal set of prices to negotiate over• History-independent prices

• Multiple lines are for different cost-to-change ratios• 0.05, 0.25, 0.50, 1.00, 2.00

• Given utility component weights same solution characteristics across range (uniform prices), hence co-incident frontiers

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 1.2 1.4 1.6 1.8 2

Provider Utility

Clie

nt U

tility

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Unit Price vs. Provider Margin

1

1.2

1.4

1.6

1.8

2

2.2

2.4

1 1.2 1.4 1.6 1.8 2

Provider Minimum Margin

Un

it P

rice

• Optimal unit prices to offer as a function of Provider requirement• History-independent (i.e. smooth) prices

• Higher lines are for increasing cost-to-change ratios• 0.05, 0.25, 0.50, 1.00, 2.00

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Price Smoothness versus Expected Price

• As weight of expected contract price increases relative to price smoothness for Client, prices move from uniform to state-dependent– (Irregularities in curves from numerical round-off.)– Cost-to-change 0.5 of unit demand cost.

• Single unit price increases expected overall price by about 1% in this case– i.e. relatively cheap to avoid state-dependent unit prices

1.55

1.6

1.65

1.7

1.75

1.8

-3 -2 -1 0 1 2 3

Distance of Current Load from Previous Load

Uni

t P

rice

&

Cos

t x

Mar

gin

(bla

ck)

1.6

1.605

1.61

1.615

1.62

1.625

1.63

0.1 0.15 0.2Price Smoothness

Exp

ecte

d U

nit

Pric

e

single unit price

state-dependent unit price

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Why state-dependent prices?

1. In the expected margin E[p/c] prices get influenced by costs of different states.

2. Support of demand realizations are different from the support of the demand distribution

• State-dependent prices become more significant when:1. state dependence of costs increases2. less is known about the demand distribution 2. support of demand realizations are small relative to the demand

distribution

• This methodology is also suitable for use when only limits on the demand distribution form are known

– Called non-probabilistic methods in mathematical finance literature.

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Summary• Formulated the optimal IT / BP outsourcing price problem for history

dependent costs and different Provider and Client utility functions• Provided method to solve pricing problem via state expansion

– Key insights: • costs are not very history dependent• usual cost granularity is an advantage

– Can be efficiently solved via MIP, MIQP in a set of cases of interest.• Pseudo-polynomial complexity in history dependence

– Deals with arbitrary demand functions (lowpass filtered)– Deals with general history dependent price policies

• In an example showed construction of efficient negotiation frontier– … and that it was relatively cheap to offer uniform unit prices

• Possible future research directions:– Client internal pricing recommendations for multiple business units– Non-myopic provisioning policies

Contact: Chris Kenyon, chk@zurich.ibm.com