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Pricing and Delivery Time DifferentiationPricing and Delivery Time Differentiation Using Shared Capacity
SACHIN JAYASWAL, ELIZABETH JEWKES
Department of Management SciencesDepartment of Management SciencesUniversity of Waterloo, Canada
SAIBAL RAYSAIBAL RAY
Desautels Faculty of Mangement,McGill University, Canaday
CORS/Optimization Days Joint Conference CO S/Op o ys Jo Co e e ceMay 12-14, 2008
Quebec City, Canada
Motivation
UPS E E l A M F dE N Fli h•UPS Express Early A.M.•UPS Express•UPS Express Saver
•FedEx Next Flight•FedEx First Overnight•FedEx Priority Overnight
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p•UPS Expedited •FedEx 2Day
Motivation contd…•Different firms, different operations strategies
According to UPS “It is their integrated air andAccording to UPS, It is their integrated air and ground network that enhances pickup and delivery density and provides them with the flexibility to transport packages using the most efficient andtransport packages using the most efficient and cost-effective transportation mode or combination of modes.”
“The optimal way to serve very distinct market“The optimal way to serve very distinct market segments is to operate highly efficient independent networks with different facilities, different cut-off times and different delivery commitments.”times and different delivery commitments. – Fredrick W. Smith, chairman and CEO, FedEx
•How does the operations strategy of a firm affect its
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p gyprice and delivery time differentiation strategy?
Problem Statement
Market sensitive to Price and Time
Different segments of customers have different
sensitivities to Price and Delivery Time.sensitivities to Price and Delivery Time.
Decisions (Dedicated vs. Shared capacity setting):( p y g)Delivery time guarantee for each segment? Price charged to each segment?
Objective:Maximize profit
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Model Description
High priority (h)/express customers
Low priority (l)/ regular customers Shared capacity setting
A 2 l i i iA 2-class pre-emptive priority queue
Class h served in priority over class l, and charged i f h t t d d li ti
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a premium for shorter guaranteed delivery time
Notations
pi : price for class iL d li i f l iLi : delivery time for class iλi : demand rate (exponential) for class i
i ( i l)µ : service rate (exponential)m : unit operating costΠ : profit per unit time for the firmΠ : profit per unit time for the firmA : marginal capacity costW : waiting time (in queue + service) of class iWi: waiting time (in queue + service) of class iSi : delivery time reliability level, P(Wi <= Li)α : delivery time guaranteeα : delivery time guarantee
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Model Description
Demand:
E p ti l ith t λ d λ• Exponential with rates λ1 and λ2
• Price and delivery time sensitive( ) ( )( ) ( )hlLhLhlphph LLLpppa −+−−+−= θβθβλ
( ) ( )lhLlLlhplpl LLLpppa −+−−+−= θβθβλ ( ) ( )lhLlLlhplpl ppp ββ
demanditivity ofprice sensp :β
productze for the market si potentiala : 2
y fppβ
e differencards pricehovers towy of switcsensitivitp :θemandivity of dime sensitdelivery tL :β
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ffpy fp
differencevery time nteed deliards guarahovers towy of switcsensitivitL :θ
Mathematical Model
Pricing and Delivery Time Decision Problem [PDTDP]
( )( ) ( ) (1)
:][Aλλ
, μ, L, ppMaximize ΠPDTDP hlh
( ) ( ) (1) Aμλmpλmp llhh −−+−=
( ) ( ) (2)1)( ≥−=≤= −− Lλμ αeLWPLS
:subject tohh( )
( )(4) 0(3) )((2) 1)(
<−+≥≤=
≥=≤=
lh
llll
hhhh
μλ λ αLW P L S
α eLWPL S
(5) 00( )0
≥≥≤≥≥ lhlhlh
lh
, λ, λLm, Lm, p p μλλ
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Solution Approach
Pricing Decision Problem [PDP]
( )( ) ( ) (1)
:][Aμλmpλmp
, μ, ppMaximize ΠPDP
llhh
lh
−−+−=
( )h
h Lαλμ −
−≥−1ln ( ) ( )
( ) ( ))((2) 1)( ≥−=≤= −− Lλμ
hhhh α eLWPL S
:subject tohh
( )
(5)00(4) 0(3) )(
≥≥≤≥≥<−+
≥≤=
lh
llll
λλLLμλ λ
αLWPL S
If Sl is concave, it can be approximated using a set of tangenthyperplanes and the resulting problem can be solved using a cutting
(5) 00 ≥≥≤≥≥ lhlhlh , λ, λLm, Lm, p p
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hyperplanes, and the resulting problem can be solved using a cutting plane method (Kelley 1990)
Solution Approach [PDP]
0 9998
1
1
161011
0.9992
0.9994
0.9996
0.9998
P(W
l <=
L l)
0.99
0.995
Sl
1213
1415
16
7
89
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phpl
15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 200.985
mu
( )lhl ppvsS , μ vsSl
Linear approximation of constraint (3) at
(6)
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Markov Chain Representation
State Variables:• Nh(t): Number of high priority customers in the system• Nl(t): Number of low priority customers in the system
2 di i l M k- 2-dimensional Markov processState space:
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Rate Matrix
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Matrix Geometric Method
Obtain stationary distribution of using Matrix Geometric Method (Neuts, 1981)( , )
Uniformize Markov Chain to obtain waiting time gdistribution of low priority customer (Latouche and Ramaswami, 1999)
Obtain gradients of using finite difference
Obtain a tangent hyperplane:
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Cutting Plane Algorithm
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Solution Approach [PDTDP]
[PDTDP] can be rewritten
where is a [PDP] for a fixedwhere is a [PDP] for a fixed .
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63
64
65
L h)
Plot of vs. suggests isunimodal. Hence [PDTDP] can be solved using
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f( Hence, [PDTDP] can be solved using Golden Section Search method.
150.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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60
Lh
Market Scenarios
Price and time difference sensitive (PTD) market:Price and time difference sensitive (PTD) market:
Time and price difference sensitive (TPD) market:
N ith r PTD r TPD rk tNeither PTD nor TPD market:
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Observations & Insights [PDP]
Prices vs. Delivery Time of high priority
Observation 1: A shared capacity results in (i) a generally higheroptimal price for express customers (ii) a lower price for regularcustomers and (iii) a higher price differentiation between the two.
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Observations & Insights [PDP]
High and low priority Demand vs. Delivery Time
Observation 2: Sharing capacity results in(i) a lower demand for express customers
d d(ii) a higher demand for regular customersand (iii) a higher or lower total demand,depending on the delivery timep g ydifferentiation.
18Total demand vs. Lead Time
Observations & Insights [PDP]
Observation 3: Sharing capacity results in a larger profit The relativeresults in a larger profit. The relative gain in profit increases with delivery time differentiation.
% Profit Gain vs. Delivery Time of high priority
Ideal condition for a shared capacity setting is when thefirm has identified markets for very different delivery time
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options.
Observations & Insights [PDTDP]
SC DCMarket Type Ll-L_h* p_h*-p_l* Ll-L_h* p_h*-p_l*
SC DC
PTD 0.597 0.473 0.112 0.060Neither 0.594 0.456 0.367 0.183
TPD 0 612 0 455 0 575 0 268Price and Delivery Time Differentiation
TPD 0.612 0.455 0.575 0.268
Observation 4: The differences in the delivery times quoted andthe prices charged to express and regular customers are higher in
h d it tti
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a shared capacity setting.
Observations & Insights [PDTDP]
Sharing capacity allows a firm to maintain a greater priceand delivery time differentiation compared to dedicatedcapacities.
FedEx UPSFedEx UPS
Service Guaranteed Delivery Rate Service Guaranteed
Delivery Rate
FedEx First by 9:00 AM UPS Express by 8:00 AMFedEx First Overnight
by 9:00 AM next day 33.40 CAD UPS Express
Early A.M.by 8:00 AM
next day 42.2 CAD
FedEx Priority Overnight
by 12:00 PM next day 18.84 CAD UPS Express
Saverby 12:00 PM
next day 15.32 CAD
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Price and Delivery Time Differentiation by FedEx vs. UPS
Overnight next day Saver next day
Observations & Insights [PDTDP]
Effect of Capacity Cost on (a) Delivery Time Differentiation (b) Price Differentiation
Observation 5: With an increase in capacity cost (A), (i) the optimaldelivery time differentiation decreases in both dedicated as well ash d it tti (ii) th ti l i diff ti ti d i
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shared capacity setting (ii) the optimal price differentiation decreases ina dedicated capacity setting but increases in a shared capacity setting
Observations & Insights [PDTDP]
Effect of Target Delivery Time Reliability on (a) Delivery Time Differentiation (b) Price Differentiation
Observation 6: With an increase in the firm’s target delivery timereliability (α), (i) the optimal delivery time differentiation decreases inboth dedicated as well as shared capacity setting (ii) the optimal price
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p y g ( ) p pdifferentiation decreases in a dedicated capacity setting but increases ina shared capacity setting
Implications of Rising Fuel Prices
Reduce product differentiation Reduce delivery timeboth in terms of delivery timeand price.
differentiation and yetincrease the pricedifferentiation.
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Future Research
E t d th d l t titi ttiExtend the model to a competitive setting
C i N h ilib i j h llComputing Nash equilibrium a major challenge
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Questions
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S l i A h Li R iSolution Approach: Literature ReviewAtalson, Epelman & Henderson (2004): Call center staffing with simulation and cutting plane methods
Henderson & Mason (1998): Rostering by integer programming and simulation
Morito, Koida, Iwama, Sato & Tamura (1999): Simulation based constraint generation with applications to optimizationbased constraint generation with applications to optimization of logistic system design
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Rate MatrixRate Matrix
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Matrix Geometric MethodMatrix Geometric Method
• Stationary probability vector x:
• Partition x by levels into sub vectors xi, where
• x can be obtained using the balance equations, given in matrix form by:
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Matrix Geometric MethodMatrix Geometric Method
• The above set of equations can be solved recursively using the relation (Neuts, 1981):
• R is the minimal non-negative solution to the matrix quadratic equation:
Th t i R b t d i ll k th d (L t h• The matrix R can be computed using well known methods (Latoucheand Ramaswami, 1999):
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Matrix Geometric MethodMatrix Geometric Method
• The probabilities are determined from the boundary condition:
subject to the normalization condition:
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Matrix Geometric Method for service level of low priority customerslow priority customers
• Time spent in the system by a low priority customer is the time until absorption in a modified Markov process , obtained by setting .
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Matrix Geometric Method for service level of low priority customerslow priority customers
• can be computed more conveniently by uniformizing the Markov process with a Poisson process of rate γ whereMarkov process with a Poisson process of rate γ, where
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Service level of low priority customersService level of low priority customers• For given prices and service rate , service level for low
priority customers can be computed as:priority customers can be computed as:
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