Transcript Yield management: Properties

Analysis of a Yield Management Model
for On Demand IT Services
Parijat Dube
IBM Watson Research Center
with Laura Wynter and Yezekael Hayel
On Demand computing services
On Demand means offering IT resources to firms
when they need it, in the quantity that is required
On Demand is a business model – it can be viewed as
an alternative to the buy-and-service and lease models
for IT hardware.
It is also an alternative to purchasing software licenses
for use on proprietary hardware.
It means paying for use only, of IT hardware,
software and networking resources.
On Demand computing services
On Demand takes advantage of network speed and
sophisticated “middleware”, which allows seamless
operation of IT resources, remotely.
On Demand is a win-win proposition, for the provider
of the service and for the customer:
• The provider can experience considerable scale
economies through resource sharing;
• The customer saves on outlay expenses, converts
purchases to operating costs, and reaps the savings
of the scale economies passed on by the provider.
Features of On Demand
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Temporary (very short term) increases and
decreases in resource needs can be satisfied
instantaneously,
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Neither space nor human resources need be
consumed, or reassigned when no longer needed,
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There is opportunity to pool resources.
Why Yield Mgmt. for On Demand

Marginal cost of providing On Demand services is
very low,
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Market for On Demand services is segmentable,
with different job requirements and urgencies,
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While mainly large players (IBM, HP,Sun) are
touting On Demand now, field will grow to a large
number of mid-size providers -> synchronization of
pricing is inevitable.
Yield management vs. no Y.M.
Consider charging a single price.
Given a demand curve, one can find the profitmaximizing single price.
Revenue = 25;
Market share = 50%
Yield management vs. no Y.M.
Now consider charging a different price for each
segment.
Based on the same demand curve,
Yield management vs. no Y.M.
Determine the optimal quantities to offer at each
price segment.
Revenue = 40;
Market share = 80%.
Yield management vs. no Y.M.
Revenue increases with the number of
segments used, under some conditions.
Yield management: Properties
We have the following properties of the Yield
Management segmented prices:
Theorem 1: Let demand be any monotone, decreasing, and nonnegative
function, d(p) of price p. Suppose that as the number of price
segments increases from i segments to i+1 segments, i=1..N, all price
levels are maintained, and a new price level is added as the i+1 st
segment. Then, the revenue increases as the number of price
segments increases.
Sketch of proof:
Show that as N increases, R increases.
R(N) = Si=1..N pi ( d(pi) – d(pi+1) ) =
Si=1..N (pi – pi-1 ) d(pi) .
R(N+1) = Si=1..N+1 pi (d(pi) – d(pi+1)) = R(N) + (pN+! – pN ) d(pN+!) ,
price difference is positive by assumption as is d(.).
Yield management: Properties
Theorem 2: Let demand be any monotone,
decreasing, and nonnegative function, d(p) of
price p. Then if the price levels are set
optimally, the revenue increases as the number
of price segments increases, irrespective of
whether price levels are maintained or not.
Sketch of proof:
Let R*(N) be the revenue with N optimally set
prices, and R(N) the revenue with N, possibly
suboptimal, prices. Then we have that
R*(N+1)  R(N+1) R*(N),
where the first inequality follows from the
optimality of the prices, and the second from
Thrm.1.
Yield management: Properties
Corollary 1: For the case where d(p) = ap+b, with
a < 0 and b > 0, i.e., the demand is affine and
decreasing in price, the optimal price levels for i
price segments is:
Pji* 
 jb
, j  1,2,i.
i  1a
Theorem 3: Let demand be any monotone,
decreasing, and nonnegative function, d(p) of
price p. Then if the price levels are set
optimally, the maximum revenue is obtained
when the number of segments goes to infinity,
and is given by the integral under the demand
curve over the region in which d(p)>0.
Yield management: Properties
Theorem 4: Let demand be given by an affine, decreasing
function of price, d(p) = ap+b, with a<0 and b>0. Then, when
price levels are optimally set, the larger the market size, the
greater the benefit of an increasing number of price segments.
That is, let d  d be two different market sizes.
1
2
Then,
*
*
*
*
R1 N  1  R1 N   R2 N  1  R2 N 
where R1* N  1 is the revenue with N+1 optimally-set price
segments and a market size of d1 and similarly
for R * N  1 and d 2
2
Yield management: Opt. Model
The model to determine optimal yield mgmt. quantities
on the IT utility takes as input:
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User (random) discrete choice preference function
describing the probability of a user with workload
type accepting a YM offering
Probability that an arriving job is of that type
Random workload, storage req. of jobs
Parallelizability of jobs
Characteristics of the resources (node speeds,
storage available, memory and CPU available)
Yield management:
Optimization Model
max
    T (W , n
c 1.. C i 1.. t k 1.. N q 1..Q
i
ikq
, c)( rikq nikq  pik sik ) P(Wi , n, s)c
T : sojourn time of a job in the system;
r and p : unit prices/segments for compute power
and storage space;
P: multinomial choice probability function;
c: probability of arrival of a customer of type c
c=customer type, i=time, k=fee, q=machine type
Yield management: Model Features
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nonconcave, nonlinear
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Degree of nonconcavity related primarily to the
choice of sojourn time function for each job,
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Some linear cases do exist,
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When sojourn times are exogenous, nonconvexity of
model is minimal, can in practice be solved to global
optimization by NLP code
A Simplified Model: an example
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Logit Probability function:
Pkc n  
e U k
c
K
e
U cj
j 1
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A simple two class, single period model
Utility of each class
U kc (nk )   1Tc rk nk   2Tc
P1c 

1
1 e
 (U1c U 2c )
P2c 
1
1 e
 (U1c U 2c )
Optimization Problem
r1 n1
r2 n2


max  c Tc 

1Tc  r1n1  r2 n2 
1Tc  r2 n2  r1n1  
1 e
1 e

c 1
C
n1 , n2
n1  n2  N
Model Analysis
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Assume total workload is same for different
customer classes
Need to solve in one variable: T  f (n1 )  g (n1 )
Solution of fixed point equation:
(n1 )  0
(n1 )  y (n1 )  M (n1 )
y(n1 )  e
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Tn1 ( r1  r2 ) 1Tr2 N
M (n1 ) 
1
r2
H (n1 ) 2
H (n1 )
 r1 
4r2
2r2
Observe that M (0)  y(0) and M ( N )  y( N )
Further M (n ) is decreasing and y(n ) is
increasing in n1
1
1
Analytical Solution
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Assume
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We can approximate y(n1 )  eTn ( r r ) Tr N
by its Taylor’s expansion
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
 1TN max( r1 , r2 )
1
1
2
1
Need to find the solution of
1   1Tn1 (r1  r2 )   1Tr2 N 
quadratic in n1
1
r2
H (n1 ) 2
H (n1 )
 r1 
4r2
2r2
2
Solution Efficiency
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An Example:   0.05,   1,   2, r  2, r  3, N  1, T  2
We get:
n1*  0.1973 revenue 2.6007
Approximation: n1*  0.2131 revenue 2.6004
1
2
1
Error: n1*  8%
Revnue error is practically zero.
2
Yield management: Model Properties
Model objective with exogenous sojourn times,
multinomial logit choice function.
Induced Demand Curve
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The expected quantity that would subscribe to the
IT service based on multi-variate logit model at a
given price and quality, all other data being fixed.
Optimal Yield Management Solution
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Increase in revenue as the number of price
segments increases
Tradeoff in increasing complexity due to a high
number of price segments is balanced by a little
increase in revenue.
Yield Management for Transactions at
a Service Center
Total demand over time; Revenue with a single (high,
med, low) price vs. 5 price segments
Optimal Number of Price Segments
Vs. Demand
Optimal Number of Price Segments
Vs. Demand (contd.)
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Optimal number of price segments is not monotone in
demand
Yield management system should be re-run as new and
better demand data become available
Summary and conclusions
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Revenue theoretically increases in this type of market
with an increasing number of price segments.
In the optimization model, with discrete choice
preference functions (instead of a single demand
curve, d(p), behavior is more complex:
Ideal number of segments varies with demand;
Program must be rerun periodically to optimize
revenue.
Additional work needed to smooth end-user price
over usage horizon; various financial instruments
(options, futures) may be of value.