dynamic pricing with risk analysis and target revenues baichun xiao long island university
TRANSCRIPT
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Dynamic Pricing with Risk Analysis and Target Revenues
Baichun Xiao Long Island University
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Outline
Risk-neutral- a basic assumption of most RM models; its inability to deal with short-term behavior;
Literature review; What affects short-term risk? A RM model with a target revenue and
penalty function; Concluding remarks.
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Risk Neutral Assumption of RM Models
Decision makers are risk neutral; i.e., all models attempt to maximize the expected revenues at the end of the disposal period;
Optimal in the long run (the law of large numbers): no single realization has the potential for severe revenue impacts on the company;
Not necessarily the best option in the short run.
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Risk-Neutral May Not Apply to Short-Term
Compelling reasons for concerning short-
term revenues: Financial constraints; Uncompromised revenue goals or
minimum probability of achieving these goals;
Shareholders’ requirements; All of the above are escalated by the
perishability of products.
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Risk with Short-term Revenues
Short-term revenues can swing drastically from their long-term estimates because:• Uncertainty of demand;• Forecast errors;• Speculations;• Unexpected capacity changes.
A single poor performance can be very damaging and may not be compromised by the long-term average.
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Example One
20, =3, =20, =5.
(0, ) 295.75
Target revenue = 280
( 280) 53.4%
p M T
V M
P X
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Practitioners’ Solutions
Sacrifice expected revenues in return for higher
probability of achieving a revenue goal. Liquidations; Clearances; Negotiated discounts; Favorable price for large-volume
demand.
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Example Two
15, =5
306.6
, =20, =5.
(0, )
Target revenue = 280
(
6
9 %280) 0
M T
V M
P X
p
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Theoretical Framework?
Current RM models are unable to explain why a dynamic control
policy leads to a steep dive of prices during liquidation periods;
unable to explain discount policies for large-volume demand.
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Related Research
McGill and van Ryzin (1999): addressed importance and challenge of pricing group demand.
Bitran and Caldentey (2002): “essentially all the models that we have discussed assume that the seller is risk neutral.”
Feng and Xiao (1999): a risk-sensitive pricing model to maximize sales revenue of perishable products.
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Related Research
Kleywegt and Papastavrou (1997), Slyke and Young (2000), Brumell and Walczak (2003): pricing group demand, but not from the perspective of risk.
Lim and Shanthikumar (2004): (i) a model built upon erroneous forecast parameters may perform badly and present a risk; (ii) robust dynamic pricing; (iii) equivalent to single product dynamic RM with exponential utility function without parameter uncertainty;
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Related Research
Levin, McGill, and Nediak (2005): (i) motivated by inventory clearance of high-value items (automobiles, electronic equipment, appliances, etc.); (ii) permit control of the probability that total revenues fall below a minimum acceptable level; (iii) augment the expected revenue objective with a penalty term for the probability that revenues drop below a desirable level.
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Related Research
Feng and Xiao (2005): (i) Maximize the risk-averse utility function instead of risk-neutral revenue; (ii) The risk-averse utility model retains monotone properties of the optimal policy; (iii) Risk-neutral models are special cases of risk-averse models; (iv) The risk-averse model explains behaviors that cannot be rationalized by the risk-neutral assumption; e.g., group discount policy.
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How is Risk Measured?
Risk is normally measured by variance and standard deviation;
Variance and standard deviation are policy dependent;
Variance and standard deviation are affected by the remaining inventory and time-to-go;
Penalty function using the standard deviation is not a proper choice when the time-to-go is diminishing.
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Risk is Affected by Policy
20 15
3 5
20 20
5 5
295.75 306.66
69.3 21.8
p
M
T
(0, )V M
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Variance vs. Remaining Inventory
5, 15, 5, ,
5, 15, 5,
10 0.65
20 21.,
5,
76
4015, 74.775, ,
T p M
T p M
T p M
Variance increases with the remaining Inventory.
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Variance vs. Remaining Inventory
A single-policy model ( ):
2 2
1 1
2
1
( ) ( )
! !
( )
: variance with items;
: expected revenue.
k T k Tn
nk n
n
nk
n
n
k
T e T eVAR k n
k k
P
VAR n
V
V k
1p
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Variance vs. Remaining Inventory
11
2 21
11 1
1
1
11 1
( )(2 1)
!
( ) ( )
( )(2 1)
!
( ) ( ) ( 1)
0.
k T
n nk n
n n
n nk k
k T
k n
n n
n n nk k
T eVAR VAR n
k
P V k P V k
T en
k
P V k P V k P V n
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Variance vs. Remaining Inventory
( )2
2( )
2
( , ) ( ) ( , 1)
( ) ( , 1)
s
it
i
s
it
i
dT
i iti
dT
i iti
VAR t n s V s n p e ds
s V s n p e ds
General cases (inventory control)
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Variance and Time Remaining
Standard Deviation of Revenue (p=1,lambda=3, n=5)
0
0.5
1
1.5
2
1 5 9 13 17 21 25 29 33 37 41
Time Remaining
ST
D
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Coefficient of Variation
Coefficient of Variation
0
2
4
6
8
10
12
1 4 7 10 13 16 19 22 25 28 31 34 37 40
Time Remaining
ST
D/E
(X)
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Risk of Selling Perishable Products
For a given policy, risk increases if the remaining inventory increases;
For a given policy, risk increases if the time-to-go diminishes;
Risk can be controlled by policy.
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Alternatives for Reducing Risk
Expected revenue with a penalty function when the target revenue is not met;
Expected revenue with a penalty function when the probability of not achieving the target revenue is above a threshold;
Risk-averse expected utility function;
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Distribution of Revenues
1
3
10
5
p
M
T
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Distribution of Revenues
1
3
10
1
p
M
T
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Distribution of Revenues
1
3
20
5
p
M
T
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A Continuous-Time Pricing Model with Target Revenue
Assumptions: Management has a target revenue in the
short run; If the target is not met, a penalty is
incurred; The penalty is proportional to the deficit
of revenue; Management makes price decisions to
maximize the expected revenue with a penalty of not meeting the target.
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Notations
1
0
= { , }, price set, for
( ) = demand intensity at given
= target revenue
( ) = remaining inventory at
( ) = revenue collected up to
= penalty coeffici
m i j
i i
p p p p i j
t t p
r
n t t
r t t
c
ent
= initial inventory
= end of sales period
M
T
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Objective Function
optimal expected revenue at T given the remaining inventory and realized revenue at t be n(t) and r(t), respectively;
The revenue function has three parameters t, n, and r.
( , ( ), ( )) :V t n t r t
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Boundary Conditions when t = T
0
0
0 0
( , , ) ( )
(1 )
V T n r r c r r
r r r
c r cr r r
Note: (i) Penalty is incurred if r < r0; (ii) Thepenalty function is piecewise linear.
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Boundary Conditions when t = T
0
0 0
( , 1, ) ( , , )
(1 ) , ;
( ) ,
0.
i
i i
i i
V T n r p V T n r
c p r p r
p c r r r p r
Implication: sell remaining inventory in theneighborhood of T for any price.
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Boundary Conditions when n = 0
0
0 0
, ;( ,0, )
(1 ) , .
r r rV t r
c r cr r r
Note: V(t, 0, r) is independent of t.
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Boundary Conditions when n = 0
1 2
1 2 1 2 0
1 2 0 2 1 0 2
1 2 0 1 2
( ,0, ) ( ,0, )
, ;
( ) ( ), ;
(1 )( ), .
V t r V t r
r r r r r
r r c r r r r r
c r r r r r
For r1 > r2,
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Optimality Condition
Only one price is accepted at any given time;
Optimal price is chosen from the price set P (may not be the highest price).
( , , )max ( )[ ( , 1, ) ( , , )] 0.i i i
V t n rt V t n r p V t n r
t
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Optimality Condition
If pi is the optimal solution at t with realized revenue r, then
( ) ( , 1, ) ( ) ( , 1, )( , , ), ,
( ) ( )
( ) ( , 1, ) ( ) ( , 1, )( , , ), .
( ) ( )
i i j j
i j
i i j j
i j
t V t n r p t V t n r pV t n r i j
t t
t V t n r p t V t n r pV t n r i j
t t
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Optimality Condition
Result: If then the optimal price in the neighborhood of T is the lowest price ;
The lowest price has the highest revenue rate.
mp
( ) ( ) ,i i j jt p t p i j
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Solve V(t, 1, r)
In the neighborhood of T, ( , , )
( )[ ( , 1, ) ( , , )] 0m m
V t n rt V t n r p V t n r
t
leads to
( )( ,1, ) ( ) ( ,0, ) .
s
mt
T d
m mtV t r s V s r p e ds
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Solve V(t, 1, r)
( )
0
( )
0 0
( ,1, )
( ) ( ) , ;
[(1 )( ) ] ( ) , .
s
mt
s
mt
T d
m m mt
T d
m m mt
V t r
r p s e ds r p r
c r p cr s e ds r p r
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Solve V(t, 1, r)
Let
In the left neighborhood of , the optimal price becomes
Other thresholds are similarly defined.
,1
1 1
1
max{ | 0 } such that
( )( ) ( )( )( ,1, )
( ) ( )
m
m m m m
m m
z t t T
t r p t r pV t r
t t
,1mz
1mp
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Solve V(t, n, r)
Assume has been obtained;
In the neighborhood of T, is the optimal price, V(t, n, r) is given by
( , 1, )V t n r
( )( , , ) ( ) ( , 1, ) .
s
mt
T d
m mtV t n r s V s n r p e ds
mp
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Solve V(t, n, r)
Let
In the left neighborhood of , the optimal price becomes
Other thresholds are similarly defined.
,
1 1
1
max{ | 0 } such that
( ) ( , 1, ) ( )( , 1, )( , , )
( ) ( )
m n
m m m m
m m
z t t T
t V t n r p t t n r pV t n r
t t
,m nz
1mp
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Numerical Experiment
Data:
0
P={10, 9, 8, 7, 6, 5}; 1
{0.05,0.07,0.09,0.11,0.13,0.14}, 1/ 2;
{0.06,0.08,0.09,0.10,0.11,0.12}, 1/ 2;
100, 2, 700.
T
t
t
M c r
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Expected revenue with the target r0
r0 V(0, M, 0)500 738.93550 738.88600 738.24650 734.16700 718.27750 678.39800 609.03850 516.27900 416.81
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Expected Revenue with the Target r0
Expected Revenue
400
450
500
550
600
650
700
750
800
400 500 600 700 800 900 1000r0
V(0
,M,0
)
V(t,n,r) is a decreasing function of r0.
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Expected revenue as Function of c
c V(0, M, 0)1.0 728.531.2 726.471.4 724.411.6 722.361.8 720.312.0 718.272.2 716.232.4 714.192.6 712.15
Note: r0=700
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Expected revenue as Function of c
Expected Revenue
705
710
715
720
725
730
1 1.5 2 2.5 3
c
V(0
,M,0
)
V(t,n,r) is a decreasing and linear function of c.
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Expected revenue as Function of n
-600
-400
-200
0
200
400
600
0 20 40 60 80 100
V(600,n,300)
V(700,n,300)
V(t,n,r) is increasing and concave in n forfixed t and r. (r0=700)
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Expected Revenue as Function of r
-800
-400
0
400
800
1200
0 100 200 300 400 500
V(600,50,r)
V(700,50,r)
V(t, n, r) is an increasing function of r for fixed t and n (r0=700).
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Expected Revenue as Function of t
-600
-400
-200
0
200
400
600
800
1000
0 200 400 600 800 1000
0
1
2
3
4
5
6
7
8
9
10
V(t,50,300)
V(t,49,300)
Delta
V(t,n,r) is a decreasing and concave function of t for given n and r; but V(t,n,r)-V(t,n-1,r) may not decrease in t (r0=700).
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Property of V(t,n,r+s) -V(t,n,r)
V(t,50,300+s)-V(t,50,300)
048
121620242832
0 200 400 600 800 1000t
s=4
s=6
s=8
s=10
( , , ) ( , , )V t n r s V t n r s
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Concluding Remarks
Decision makers are risk-averse in financial market and many other areas, revenue management should not be an exception;
The proposed pricing model handles risk with a target revenue and a penalty function;
Many properties of risk-neutral models seem to hold except the marginal expected revenue;
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Concluding Remarks
More structural properties of the value function need to be uncovered;
Whether pricing policy for group demand can be dealt with by the proposed model need to be explored.