michel bierlaire - university of california, los...
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Integrated demand and supply optimization
Michel Bierlaire
Transport and Mobility LaboratorySchool of Architecture, Civil and Environmental Engineering
Ecole Polytechnique Federale de Lausanne
November 16, 2015
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Introduction
Outline
1 Introduction
2 Demand
3 Supply
4 Integrated framework
5 A simple example
A linear formulationExample: one theaterExample: two theaters
6 Summary7 Appendix: dealing with capacities
Example: two theaters
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Introduction
Transportation systems
Two dimensions
Supply = infrastructure
Demand = behavior, choices
Congestion = mismatch
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Introduction
Transportation systems
Objectives
Minimize costsMaximize satisfaction
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Introduction
Transportation systems
Maximize revenues
Revenues = Benefits - Costs
Costs: examples
Building infrastructure
Operating the system
Environmental externalities
Benefits: examples
Income from ticket sales
Social welfare
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Introduction
Demand-supply interactions
Operations Research
Given the demand...
configure the system
Behavioral models
Given the configuration ofthe system...
predict the demand
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Introduction
Research objectives
Framework for demand-supply interactions
General: not designed for a specific application or context.
Flexible: wide variety of demand and supply models.
Scalable: the level of complexity can be adjusted.
Integrated: not sequential.
Operational: can be solved efficiently.
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Demand
Outline
1 Introduction
2 Demand
3 Supply
4 Integrated framework
5 A simple example
A linear formulationExample: one theaterExample: two theaters
6 Summary7 Appendix: dealing with capacities
Example: two theaters
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Demand
Aggregate demand
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Demand
Aggregate demand
Homogeneous population
Identical behavior
Price (P) and quantity (Q)
Demand function: Q = f (P)
Demand curve: P = f −1(Q)
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Demand
Disaggregate demand
Heterogeneous population
Different behaviors
Many variables:
Attributes: price, travel time,reliability, frequency, etc.Characteristics: age, income,education, etc.
Complex demand/inversedemand functions.
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Demand
Disaggregate demand
Behavioral models
Demand = combination ofindividual choices.
Modeling demand = modelingchoice.
Behavioral models: choicemodels.
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Demand
Choice models
Daniel McFadden
UC Berkeley 1963, MIT 1977,UC Berkeley 1991
Laureate of The Bank of Sweden
Prize in Economic Sciences in
Memory of Alfred Nobel 2000
Owns a farm and vineyard inNapa Valley
“Farm work clears the mind,and the vineyard is a great placeto prove theorems”
2000
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Demand
Decision rules
Neoclassical economic theory
Preference-indifference operator &
1 reflexivitya & a ∀a ∈ Cn
2 transitivity
a & b and b & c ⇒ a & c ∀a, b, c ∈ Cn
3 comparabilitya & b or b & a ∀a, b ∈ Cn
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Demand
Decision rules
Utility
∃ Un : Cn −→ R : a Un(a) such that
a & b ⇔ Un(a) ≥ Un(b) ∀a, b ∈ Cn
Remarks
Utility is a latent concept
It cannot be directly observed
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Demand
Decision rules
Choice
Individual n
Choice set Cn = {1, . . . , Jn}
Utilities Uin, ∀i ∈ Cn
i is chosen iff Uin = maxj∈Cn Ujn
Underlying assumption: no tie.
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Demand
Example
Two transportation modes
U1 = −βt1 − γc1U2 = −βt2 − γc2
with β, γ > 0
Mode 1 is chosen if
U1 ≥ U2 iff − βt1 − γc1 ≥ −βt2 − γc2
that is
−β
γt1 − c1 ≥ −
β
γt2 − c2
or
c1 − c2 ≤ −β
γ(t1 − t2)
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Demand
Example
Trade-off
c1 − c2 ≤ −β
γ(t1 − t2)
c1 − c2 in currency unity (CHF)
t1 − t2 in time units (hours)
β/γ: CHF/hours
Value of time
Willingness to pay to save travel time.
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Demand
Example
-4
-2
0
2
4
-4 -2 0 2 4
1 is chosen
2 is chosen
c1-c
2
t1-t2
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Demand
Example
-4
-2
0
2
4
-4 -2 0 2 4
1 is chosen
2 is chosen
c1-c
2
t1-t2
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Demand
Assumptions
Decision-maker
perfect discriminating capability
full rationality
permanent consistency
Analyst
knowledge of all attributes
perfect knowledge of & (orUn(·))
no measurement error
Must deal with uncertainty
Random utility models
For each individual n and alternative i
Uin = Vin + εin
andP(i |Cn) = P[Uin = max
j∈CnUjn] = P(Uin ≥ Ujn ∀j ∈ Cn)
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Demand
Logit model
Utility
Uin = Vin + εin
Choice probability: logit model
Pn(i |Cn) =yine
Vin
∑
j∈C yjneVjn
.
Decision-maker n
Alternative i ∈ Cn
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Demand
Variables: xin = (zin, sn)
Attributes of alternative i : zin
Cost / price
Travel time
Waiting time
Level of comfort
Number of transfers
Late/early arrival
etc.
Characteristics of decision-maker n:sn
Income
Age
Sex
Trip purpose
Car ownership
Education
Profession
etc.
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Demand
Demand curve
Disaggregate model
Pn(i |cin, zin, sn)
Total demand
D(i) =∑
n
Pn(i |cin, zin, sn)
Difficulty
Non linear and non convex in cin and zin
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Supply
Outline
1 Introduction
2 Demand
3 Supply
4 Integrated framework
5 A simple example
A linear formulationExample: one theaterExample: two theaters
6 Summary7 Appendix: dealing with capacities
Example: two theaters
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Supply
Optimization problem
Given...
the demand
Find...
the best configuration of the transportation system.
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Supply
Example: airline
Context
An airline considers to propose various destinations i = {1, . . . , J} toits customers.
Each potential destination i is served by an aircraft, with capacity ci .
The price of the ticket for destination i is pi .
The demand is known: Wi passengers want to travel to i .
The fixed cost of operating a flight to destination i is Fi .
The airline cannot invest more than a budget B .
Question
What destinations should the airline serve to maximize its revenues?
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Supply
Example: airline
Decisions variables
yi ∈ {0, 1}: 1 if destination i is served, 0 otherwise.
Maximize revenues
max
J∑
i=1
min(Wi , ci )piyi
Constraints
J∑
i=1
Fiyi ≤ B
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Supply
Example: airline
Integer linear optimization problem
Decision variables are integers.
Objective function and constraints are linear.
Here: knapsack problem.
Solving the problem
Branch and bound
Cutting planes
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Supply
Example: airline
Pricing
What price pi should the airline propose?
max
J∑
i=1
min(Wi , ci )piyi
Issues
Non linear objective
Unbounded problem
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Supply
Example: airline
Unbounded problem
As demand is constant, the airline can make money with very highprices.
We need to take into account the impact of price on demand.
Logit model
Wi =∑
n
Pn(i |pi , zin, sn)
Pn(i |pi , zin, sn) =yie
Vin(pi ,zin,sn)
∑
j∈C yjeVjn(pj ,zjn,sn)
.
The problem becomes highly non linear.
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Integrated framework
Outline
1 Introduction
2 Demand
3 Supply
4 Integrated framework
5 A simple example
A linear formulationExample: one theaterExample: two theaters
6 Summary7 Appendix: dealing with capacities
Example: two theaters
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Integrated framework
The main idea
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Integrated framework
The main idea
Linearization
Hopeless to linearize the logit formula (we tried...)
First principles
Each customer solves an optimization problem
Solution
Use the utility and not the probability
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Integrated framework
A linear formulation
Utility function
Uin = Vin + εin =∑
k
βkxink + f (zin) + εin.
Simulation
Assume a distribution for εin
E.g. logit: i.i.d. extreme value
Draw R realizations ξinr ,r = 1, . . . ,R
The choice problem becomesdeterministic
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Integrated framework
Scenarios
Draws
Draw R realizations ξinr , r = 1, . . . ,R
We obtain R scenarios
Uinr =∑
k
βkxink + f (zin) + ξinr .
For each scenario r , we can identify the largest utility.
It corresponds to the chosen alternative.
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Integrated framework
Comparing utilities
Variables
µijnr =
{
1 if Uinr ≥ Ujnr ,0 if Uinr < Ujnr .
Constraints
(µijnr − 1)Mnr ≤ Uinr − Ujnr ≤ µijnrMnr , ∀i , j , n, r .
where|Uinr − Ujnr | ≤ Mnr , ∀i , j ,
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Integrated framework
Comparing utilities
(µijnr − 1)Mnr ≤ Uinr − Ujnr ≤ µijnrMnr , ∀i , j , n, r .
Constraints: µijnr = 1
0 ≤ Uinr − Ujnr ≤ Mnr , ∀i , j , n, r .
Ujnr ≤ Uinr , ∀i , j , n, r .
Constraints: µijnr = 0
−Mnr ≤ Uinr − Ujnr ≤ 0, ∀i , j , n, r .
Uinr ≤ Ujnr , ∀i , j , n, r .
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Integrated framework
Comparing utilities
(µijnr − 1)Mnr ≤ Uinr − Ujnr ≤ µijnrMnr , ∀i , j , n, r .
Equivalence if no tie
µijnr = 1 =⇒ Uinr ≥ Ujnr
µijnr = 0 =⇒ Uinr ≤ Ujnr
Uinr > Ujnr =⇒ µijnr = 1
Uinr < Ujnr =⇒ µijnr = 0
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Integrated framework
Accounting for availabilities
Motivation
If yi = 0, alternative i is not available.
Its utility should not be involved in any constraint.
New variables: two alternatives are both available
ηij = yiyj
Linearization:
yi + yj ≤ 1 + ηij ,
ηij ≤ yi ,
ηij ≤ yj .
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Integrated framework
Comparing utilities of available alternatives
Constraints
Mnrηij − 2Mnr ≤ Uinr − Ujnr −Mnrµijnr ≤ (1− ηij)Mnr , ∀i , j , n, r .
ηij = 1 and µijnr = 1
0 ≤ Uinr − Ujnr ≤ Mnr , ∀i , j , n, r .
ηij = 1 and µijnr = 0
−Mnr ≤ Uinr − Ujnr ≤ 0, ∀i , j , n, r .
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Integrated framework
Comparing utilities of available alternatives
Constraints
Mnrηij − 2Mnr ≤ Uinr − Ujnr −Mnrµijnr ≤ (1− ηij)Mnr , ∀i , j , n, r .
ηij = 0 and µijnr = 1
−Mnr ≤ Uinr − Ujnr ≤ 2Mnr , ∀i , j , n, r ,
ηij = 0 and µijnr = 0
−2Mnr ≤ Uinr − Ujnr ≤ Mnr , ∀i , j , n, r ,
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Integrated framework
Comparing utilities of available alternatives
Valid inequalities
µijnr ≤ yi , ∀i , j , n, r ,
µijnr + µjinr ≤ 1, ∀i , j , n, r .
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Integrated framework
The choice
Variables
winr =
{
1 if n chooses i in scenario r ,0 otherwise
Maximum utility
winr ≤ µijnr , ∀i , j , n, r .
Availability
winr ≤ yi , ∀i , n, r .
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Integrated framework
The choice
One choice∑
i∈C
winr = 1, ∀n, r .
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Integrated framework
Demand and revenues
Demand
Wi =1
R
n∑
n=1
R∑
r=1
winr .
Revenues
Ri =1
R
N∑
n=1
pi
R∑
r=1
winr .
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A simple example
Outline
1 Introduction
2 Demand
3 Supply
4 Integrated framework
5 A simple example
A linear formulationExample: one theaterExample: two theaters
6 Summary7 Appendix: dealing with capacities
Example: two theaters
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A simple example
A simple example
Data
C: set of movies
Population of N individuals
Utility function:Uin = βinpin + f (zin) + εin
Decision variables
What movies to propose? yi
What price? pin
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A simple example
Demand model
Logit model
Probability that n chooses movie i :
P(i |y , pn, zn) =yie
βinpin+f (zin)
∑
j yjeβjnpjn+f (zjn)
Total revenue:∑
i∈C
yi
N∑
n=1
pinP(i |y , pn, zn)
Non linear and non convex in the decision variables
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A simple example Example: one theater
Example: programming movie theaters
Data
Two alternatives: my theater (m) andthe competition (c)
We assume an homogeneouspopulation of N individuals
Uc = 0 + εc
Um = βcpm + εm
βc < 0
Logit model: εm i.i.d. EV
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A simple example Example: one theater
Demand and revenues
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Dem
and
Revenues
Price
RevenuesDemand
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A simple example Example: one theater
Optimization (with GLPK)
Data
N = 1
R = 100
Um = −10pm + 3
Prices: 0.10, 0.20, 0.30, 0.40,0.50
Results
Optimum price: 0.3
Demand: 56%
Revenues: 0.168
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A simple example Example: one theater
Heterogeneous population
Two groups in the population
Uin = βnpi + cn
Young fans: 2/3
β1 = −10, c1 = 3
Others: 1/3
β1 = −0.9, c1 = 0
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A simple example Example: one theater
Demand and revenues
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Dem
and
Revenues
Price
RevenuesDemand
Young fansOthers
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A simple example Example: one theater
Optimization
Data
N = 3
R = 100
Um1 = −10pm + 3
Um2 = −0.9pm
Prices: 0.3, 0.7, 1.1, 1.5, 1.9
Results
Optimum price: 0.3
Customer 1 (fan): 60% [theory:50 %]
Customer 2 (fan) : 49%[theory: 50 %]
Customer 3 (other) : 45%[theory: 43 %]
Demand: 1.54 (51%)
Revenues: 0.48
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A simple example Example: two theaters
Two theaters, different types of films
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A simple example Example: two theaters
Two theaters, different types of films
Theater m
Expensive
Star Wars Episode VII
Theater k
Cheap
Tinker Tailor Soldier Spy
Heterogeneous demand
Two third of the population is young (price sensitive)
One third of the population is old (less price sensitive)
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A simple example Example: two theaters
Two theaters, different types of films
Data
Theaters m and k
N = 6
R = 10
Umn = −10pm + 4 , n = 1, 2, 4, 5
Umn = −0.9pm, n = 3, 6
Ukn = −10pk + 0 , n = 1, 2, 4, 5
Ukn = −0.9pk , n = 3, 6
Prices m: 1.0, 1.2, 1.4, 1.6, 1.8
Prices k: half price
Theater m
Optimum price m: 1.6
4 young customers: 0
2 old customers: 0.5
Demand: 0.5 (8.3%)
Revenues: 0.8
Theater k
Optimum price m: 0.5
Young customers: 0.8
Old customers: 1.5
Demand: 2.3 (38%)
Revenues: 1.15
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A simple example Example: two theaters
Two theaters, same type of films
Theater m
Expensive
Star Wars Episode VII
Theater k
Cheap
Star Wars Episode VIII
Heterogeneous demand
Two third of the population is young (price sensitive)
One third of the population is old (less price sensitive)
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A simple example Example: two theaters
Two theaters, same type of films
Data
Theaters m and k
N = 6
R = 10
Umn = −10pm + 4 ,n = 1, 2, 4, 5
Umn = −0.9pm, n = 3, 6
Ukn = −10pk + 4 ,n = 1, 2, 4, 5
Ukn = −0.9pk , n = 3, 6
Prices m: 1.0, 1.2, 1.4, 1.6, 1.8
Prices k : half price
Theater m
Optimum price m: 1.8
Young customers: 0
Old customers: 1.9
Demand: 1.9 (31.7%)
Revenues: 3.42
Theater k
Closed
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A simple example Example: two theaters
Extension: dealing with capacities
Demand may exceed supply
Not every choice can beaccommodated
Difficulty: who has access?
Assumption: priority list isexogenous
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Summary
Outline
1 Introduction
2 Demand
3 Supply
4 Integrated framework
5 A simple example
A linear formulationExample: one theaterExample: two theaters
6 Summary7 Appendix: dealing with capacities
Example: two theaters
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Summary
Summary
Demand and supply
Supply: prices and capacity
Demand: choice of customers
Interaction between the two
Discrete choice models
Rich family of behavioral models
Strong theoretical foundations
Great deal of concrete applications
Capture the heterogeneity of behavior
Probabilistic models
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Summary
Optimization
Discrete choice models
Non linear and non convex
Idea: use utility instead of probability
Rely on simulation to capture stochasticity
Proposed formulation
General: not designed for a specific application or context.
Flexible: wide variety of demand and supply models.
Scalable: the level of complexity can be adjusted.
Integrated: not sequential.
Operational: can be solved efficiently.
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Summary
Ongoing research
Revenue management
Airlines, train operators, etc.
Decomposition methods
Scenarios are (almost) independent from each other (except objectivefunction)
Individuals are also loosely coupled (except for capacity constraints)
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Summary
Thank you!
Questions?
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Appendix: dealing with capacities
Outline
1 Introduction
2 Demand
3 Supply
4 Integrated framework
5 A simple example
A linear formulationExample: one theaterExample: two theaters
6 Summary7 Appendix: dealing with capacities
Example: two theaters
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Appendix: dealing with capacities
Dealing with capacities
Demand may exceed supply
Not every choice can beaccommodated
Difficulty: who has access?
Assumption: priority list isexogenous
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Appendix: dealing with capacities
Priority list
Application dependent
First in, first out
Frequent travelers
Subscribers
...
In this framework
The list of customers must be sorted
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Appendix: dealing with capacities
Dealing with capacities
Variables
yin: decision of the operator
yinr : availability
Constraints
N∑
n=1
winr ≤ ci
yinr ≤ yin
yi(n+1)r ≤ yinr
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Appendix: dealing with capacities
Constraints
ci (1− yinr ) ≤n−1∑
m=1
wimr + (1− yin)cmax
yin = 1, yinr = 1
0 ≤n−1∑
m=1
wimr
yin = 1, yinr = 0
ci ≤n−1∑
m=1
wimr
yin = 0, yinr = 0
ci ≤n−1∑
m=1
wimr + cmax
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Appendix: dealing with capacities
Constraints
n−1∑
m=1
wimr + (1− yin)cmax ≤ (ci − 1)yinr +max(n, cmax)(1− yinr )
yin = 1, yinr = 1
1 +n−1∑
m=1
wimr ≤ ci
yin = 1, yinr = 0
n−1∑
m=1
wimr ≤ max(n, cmax)
yin = 0, yinr = 0
n−1∑
m=1
wimr + cmax ≤ max(n, cmax)
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Appendix: dealing with capacities Example: two theaters
Two theaters, different types of films
Data
Theaters m and k
Capacity: 2
N = 6
R = 5
Umn = −10pm + 4, n = 1, 2, 4, 5
Umn = −0.9pm, n = 3, 6
Ukn = −10pk + 0, n = 1, 2, 4, 5
Ukn = −0.9pk , n = 3, 6
Prices m: 1.0, 1.2, 1.4, 1.6, 1.8
Prices k: half price
Theater m
Optimum price m: 1.8
Demand: 0.2 (3.3%)
Revenues: 0.36
Theater k
Optimum price m: 0.5
Demand: 2 (33.3%)
Revenues: 1.15
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Appendix: dealing with capacities Example: two theaters
Example of two scenarios
Customer Choice Capacity m Capacity k
1 0 2 22 0 2 23 k 2 14 0 2 15 0 2 16 k 2 0
Customer Choice Capacity m Capacity k
1 0 2 22 k 2 13 0 2 14 k 2 05 0 2 06 0 2 0
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Appendix: dealing with capacities Example: two theaters
Two theaters: all prices divided by 2
Data
Theaters m and k
Capacity: 2
N = 6
R = 5
Umn = −10pm + 4, n = 1, 2, 4, 5
Umn = −0.9pm, n = 3, 6
Ukn = −10pk + 0, n = 1, 2, 4, 5
Ukn = −0.9pk , n = 3, 6
Prices m: 0.5, 0.6, 0.7, 0.8, 0.9
Prices k: half price
Theater m
Optimum price m: 0.5
Demand: 1.4
Revenues: 0.7
Theater k
Optimum price m: 0.45
Demand: 1.6
Revenues: 0.72
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Appendix: dealing with capacities Example: two theaters
Example of two scenarios
Customer Choice Capacity m Capacity k
1 0 2 22 0 2 23 0 2 24 k 2 15 k 2 06 0 2 0
Customer Choice Capacity m Capacity k
1 k 2 12 k 2 03 0 2 04 m 1 05 0 1 06 m 0 0
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