Post on 19-Feb-2018
transcript
Online Resource Allocation Problems, with Applicationsto Revenue Management
David Simchi-Levi (MIT)
David Simchi-Levi (MIT) Online Resource Allocation 0 / 22
Introduction
Papers
1 Tight Weight-dependent Competitive Ratios for Online MatchingProblems
joint work with Will Ma
2 Dynamic Recommendation at Checkout under Inventory Constraint
joint work with Xi Chen, Will Ma, and Linwei Xin
David Simchi-Levi (MIT) Online Resource Allocation 3 / 22
Online Resource Allocation
Papers
1 Tight Weight-dependent Competitive Ratios for OnlineMatching Problems
joint work with Will Ma
2 Dynamic Recommendation at Checkout under Inventory Constraint
joint work with Xi Chen, Will Ma, and Linwei Xin
David Simchi-Levi (MIT) Online Resource Allocation 3 / 22
Online Resource Allocation Problem Definition
A General Online Resource Allocation Problem
Known at start:number of items (resources), nunreplenishable starting inventory of each item i , kimi potential prices of each item i , satisfying r
(1)i < . . . < r
(mi )i
For t = 1, 2, . . .1 Customer t Arrives:
observe p(j)t,i , the probability of customer t buying item i at price j , for
all i and j (can be based on customer features or classes)2 Decision:
offer an item it which has not stocked out, at a price jt , to customer t3 Realization:
customer makes purchase with probability p(jt)t,it
if she purchases, then earn revenue r(jt)it
and decrement inventory of itExtensions:
can allow an assortment of items/prices to be offered to each customercan allow an item to have a continuum of potential pricescan allow for fractional inventory consumption
David Simchi-Levi (MIT) Online Resource Allocation 4 / 22
Online Resource Allocation Problem Definition
A General Online Resource Allocation Problem
Known at start:
number of items (resources), nunreplenishable starting inventory of each item i , kimi potential prices of each item i , satisfying r
(1)i < . . . < r
(mi )i
For t = 1, 2, . . .1 Customer t Arrives:
observe p(j)t,i , the probability of customer t buying item i at price j , for
all i and j (can be based on customer features or classes)2 Decision:
offer an item it which has not stocked out, at a price jt , to customer t3 Realization:
customer makes purchase with probability p(jt)t,it
if she purchases, then earn revenue r(jt)it
and decrement inventory of itExtensions:
can allow an assortment of items/prices to be offered to each customercan allow an item to have a continuum of potential pricescan allow for fractional inventory consumption
David Simchi-Levi (MIT) Online Resource Allocation 4 / 22
Online Resource Allocation Problem Definition
A General Online Resource Allocation Problem
Known at start:number of items (resources), n
unreplenishable starting inventory of each item i , kimi potential prices of each item i , satisfying r
(1)i < . . . < r
(mi )i
For t = 1, 2, . . .1 Customer t Arrives:
observe p(j)t,i , the probability of customer t buying item i at price j , for
all i and j (can be based on customer features or classes)2 Decision:
offer an item it which has not stocked out, at a price jt , to customer t3 Realization:
customer makes purchase with probability p(jt)t,it
if she purchases, then earn revenue r(jt)it
and decrement inventory of itExtensions:
can allow an assortment of items/prices to be offered to each customercan allow an item to have a continuum of potential pricescan allow for fractional inventory consumption
David Simchi-Levi (MIT) Online Resource Allocation 4 / 22
Online Resource Allocation Problem Definition
A General Online Resource Allocation Problem
Known at start:number of items (resources), nunreplenishable starting inventory of each item i , ki
mi potential prices of each item i , satisfying r(1)i < . . . < r
(mi )i
For t = 1, 2, . . .1 Customer t Arrives:
observe p(j)t,i , the probability of customer t buying item i at price j , for
all i and j (can be based on customer features or classes)2 Decision:
offer an item it which has not stocked out, at a price jt , to customer t3 Realization:
customer makes purchase with probability p(jt)t,it
if she purchases, then earn revenue r(jt)it
and decrement inventory of itExtensions:
can allow an assortment of items/prices to be offered to each customercan allow an item to have a continuum of potential pricescan allow for fractional inventory consumption
David Simchi-Levi (MIT) Online Resource Allocation 4 / 22
Online Resource Allocation Problem Definition
A General Online Resource Allocation Problem
Known at start:number of items (resources), nunreplenishable starting inventory of each item i , kimi potential prices of each item i , satisfying r
(1)i < . . . < r
(mi )i
For t = 1, 2, . . .1 Customer t Arrives:
observe p(j)t,i , the probability of customer t buying item i at price j , for
all i and j (can be based on customer features or classes)2 Decision:
offer an item it which has not stocked out, at a price jt , to customer t3 Realization:
customer makes purchase with probability p(jt)t,it
if she purchases, then earn revenue r(jt)it
and decrement inventory of itExtensions:
can allow an assortment of items/prices to be offered to each customercan allow an item to have a continuum of potential pricescan allow for fractional inventory consumption
David Simchi-Levi (MIT) Online Resource Allocation 4 / 22
Online Resource Allocation Problem Definition
A General Online Resource Allocation Problem
Known at start:number of items (resources), nunreplenishable starting inventory of each item i , kimi potential prices of each item i , satisfying r
(1)i < . . . < r
(mi )i
For t = 1, 2, . . .
1 Customer t Arrives:observe p
(j)t,i , the probability of customer t buying item i at price j , for
all i and j (can be based on customer features or classes)2 Decision:
offer an item it which has not stocked out, at a price jt , to customer t3 Realization:
customer makes purchase with probability p(jt)t,it
if she purchases, then earn revenue r(jt)it
and decrement inventory of itExtensions:
can allow an assortment of items/prices to be offered to each customercan allow an item to have a continuum of potential pricescan allow for fractional inventory consumption
David Simchi-Levi (MIT) Online Resource Allocation 4 / 22
Online Resource Allocation Problem Definition
A General Online Resource Allocation Problem
Known at start:number of items (resources), nunreplenishable starting inventory of each item i , kimi potential prices of each item i , satisfying r
(1)i < . . . < r
(mi )i
For t = 1, 2, . . .1 Customer t Arrives:
observe p(j)t,i , the probability of customer t buying item i at price j , for
all i and j (can be based on customer features or classes)
2 Decision:offer an item it which has not stocked out, at a price jt , to customer t
3 Realization:customer makes purchase with probability p
(jt)t,it
if she purchases, then earn revenue r(jt)it
and decrement inventory of itExtensions:
can allow an assortment of items/prices to be offered to each customercan allow an item to have a continuum of potential pricescan allow for fractional inventory consumption
David Simchi-Levi (MIT) Online Resource Allocation 4 / 22
Online Resource Allocation Problem Definition
A General Online Resource Allocation Problem
Known at start:number of items (resources), nunreplenishable starting inventory of each item i , kimi potential prices of each item i , satisfying r
(1)i < . . . < r
(mi )i
For t = 1, 2, . . .1 Customer t Arrives:
observe p(j)t,i , the probability of customer t buying item i at price j , for
all i and j (can be based on customer features or classes)2 Decision:
offer an item it which has not stocked out, at a price jt , to customer t
3 Realization:customer makes purchase with probability p
(jt)t,it
if she purchases, then earn revenue r(jt)it
and decrement inventory of itExtensions:
can allow an assortment of items/prices to be offered to each customercan allow an item to have a continuum of potential pricescan allow for fractional inventory consumption
David Simchi-Levi (MIT) Online Resource Allocation 4 / 22
Online Resource Allocation Problem Definition
A General Online Resource Allocation Problem
Known at start:number of items (resources), nunreplenishable starting inventory of each item i , kimi potential prices of each item i , satisfying r
(1)i < . . . < r
(mi )i
For t = 1, 2, . . .1 Customer t Arrives:
observe p(j)t,i , the probability of customer t buying item i at price j , for
all i and j (can be based on customer features or classes)2 Decision:
offer an item it which has not stocked out, at a price jt , to customer t3 Realization:
customer makes purchase with probability p(jt)t,it
if she purchases, then earn revenue r(jt)it
and decrement inventory of it
Extensions:can allow an assortment of items/prices to be offered to each customercan allow an item to have a continuum of potential pricescan allow for fractional inventory consumption
David Simchi-Levi (MIT) Online Resource Allocation 4 / 22
Online Resource Allocation Problem Definition
A General Online Resource Allocation Problem
Known at start:number of items (resources), nunreplenishable starting inventory of each item i , kimi potential prices of each item i , satisfying r
(1)i < . . . < r
(mi )i
For t = 1, 2, . . .1 Customer t Arrives:
observe p(j)t,i , the probability of customer t buying item i at price j , for
all i and j (can be based on customer features or classes)2 Decision:
offer an item it which has not stocked out, at a price jt , to customer t3 Realization:
customer makes purchase with probability p(jt)t,it
if she purchases, then earn revenue r(jt)it
and decrement inventory of itExtensions:
can allow an assortment of items/prices to be offered to each customercan allow an item to have a continuum of potential pricescan allow for fractional inventory consumption
David Simchi-Levi (MIT) Online Resource Allocation 4 / 22
Online Resource Allocation Problem Definition
Competitive Ratio
compare algorithm’s performance vs. optimum which knows all arrivalinformation at the start
develop algorithms whose competitive ratio
infinstance I (incl. arrivals)
E[ALG(I)]
OPT(I)
is bounded by a constant independent of the number ofitems/customers
algorithm makes no assumption on arrival sequence
David Simchi-Levi (MIT) Online Resource Allocation 5 / 22
Online Resource Allocation Problem Definition
Competitive Ratio
compare algorithm’s performance vs. optimum which knows all arrivalinformation at the start
develop algorithms whose competitive ratio
infinstance I (incl. arrivals)
E[ALG(I)]
OPT(I)
is bounded by a constant independent of the number ofitems/customers
algorithm makes no assumption on arrival sequence
David Simchi-Levi (MIT) Online Resource Allocation 5 / 22
Online Resource Allocation Problem Definition
Competitive Ratio
compare algorithm’s performance vs. optimum which knows all arrivalinformation at the start
develop algorithms whose competitive ratio
infinstance I (incl. arrivals)
E[ALG(I)]
OPT(I)
is bounded by a constant independent of the number ofitems/customers
algorithm makes no assumption on arrival sequence
David Simchi-Levi (MIT) Online Resource Allocation 5 / 22
Online Resource Allocation Problem Definition
Competitive Ratio
compare algorithm’s performance vs. optimum which knows all arrivalinformation at the start
develop algorithms whose competitive ratio
infinstance I (incl. arrivals)
E[ALG(I)]
OPT(I)
is bounded by a constant independent of the number ofitems/customers
algorithm makes no assumption on arrival sequence
David Simchi-Levi (MIT) Online Resource Allocation 5 / 22
Online Resource Allocation Intuition
Key Challenges
1 How do we prioritize between different items to be sold?
Price of : $150
Price of : $200
2 When do we reserve the inventory of an item for customers willing topay higher prices?
Potential prices of : $150, $450
p = 0
David Simchi-Levi (MIT) Online Resource Allocation 6 / 22
Online Resource Allocation Intuition
Key Challenges
1 How do we prioritize between different items to be sold?
Price of : $150
Price of : $200
2 When do we reserve the inventory of an item for customers willing topay higher prices?
Potential prices of : $150, $450
p = 0
David Simchi-Levi (MIT) Online Resource Allocation 6 / 22
Online Resource Allocation Intuition
Key Challenges
1 How do we prioritize between different items to be sold?
Price of : $150
Price of : $200
2 When do we reserve the inventory of an item for customers willing topay higher prices?
Potential prices of : $150, $450
p = 0
David Simchi-Levi (MIT) Online Resource Allocation 6 / 22
Online Resource Allocation Intuition
Key Challenges
1 How do we prioritize between different items to be sold?
Price of : $150
Price of : $200
2 When do we reserve the inventory of an item for customers willing topay higher prices?
Potential prices of : $150, $450
p = 0
David Simchi-Levi (MIT) Online Resource Allocation 6 / 22
Online Resource Allocation Intuition
Key Challenges
1 How do we prioritize between different items to be sold?
Price of : $150
Price of : $200
2 When do we reserve the inventory of an item for customers willing topay higher prices?
Potential prices of : $150, $450
p = 0
David Simchi-Levi (MIT) Online Resource Allocation 6 / 22
Online Resource Allocation Intuition
Challenge 1: Multiple Items
Price of : $150
Price of : $200
Solution: “Inventory Balancing”
balance the depletion of different items, relative to their prices
idea has been used in the:online b-matching problem (Kalyanasundaram/Pruhs ’00)Adwords problem (Mehta et al. ’05, Buchbinder/Jain/Naor ’07)personalized assortment problem(Golrezaei/Nazerzadeh/Rusmevichientong ’14)
David Simchi-Levi (MIT) Online Resource Allocation 7 / 22
Online Resource Allocation Intuition
Challenge 1: Multiple Items
Price of : $150
Price of : $200
Solution: “Inventory Balancing”
balance the depletion of different items, relative to their prices
idea has been used in the:online b-matching problem (Kalyanasundaram/Pruhs ’00)Adwords problem (Mehta et al. ’05, Buchbinder/Jain/Naor ’07)personalized assortment problem(Golrezaei/Nazerzadeh/Rusmevichientong ’14)
David Simchi-Levi (MIT) Online Resource Allocation 7 / 22
Online Resource Allocation Intuition
Challenge 1: Multiple Items
Price of : $150
Price of : $200
Solution: “Inventory Balancing”
balance the depletion of different items, relative to their prices
idea has been used in the:online b-matching problem (Kalyanasundaram/Pruhs ’00)Adwords problem (Mehta et al. ’05, Buchbinder/Jain/Naor ’07)personalized assortment problem(Golrezaei/Nazerzadeh/Rusmevichientong ’14)
David Simchi-Levi (MIT) Online Resource Allocation 7 / 22
Online Resource Allocation Intuition
Challenge 1: Multiple Items
Price of : $150
Price of : $200
Solution: “Inventory Balancing”
balance the depletion of different items, relative to their prices
idea has been used in the:online b-matching problem (Kalyanasundaram/Pruhs ’00)Adwords problem (Mehta et al. ’05, Buchbinder/Jain/Naor ’07)personalized assortment problem(Golrezaei/Nazerzadeh/Rusmevichientong ’14)
David Simchi-Levi (MIT) Online Resource Allocation 7 / 22
Online Resource Allocation Intuition
Challenge 2: Multiple Prices
Potential prices of : $150, $450
p = 0
Solution: “Booking Limits”
stop offering the item at the lower price after some fraction has beensold
idea has been used in the:
one-way-trading problem (Lavi/Nisan ’00, El-Yaniv et al. ’01)single-leg booking problem (Ball/Queyranne ’09)online dynamic pricing problem (Ma/S.-L./Teo ’17)
David Simchi-Levi (MIT) Online Resource Allocation 8 / 22
Online Resource Allocation Intuition
Challenge 2: Multiple Prices
Potential prices of : $150, $450
p = 0
Solution: “Booking Limits”
stop offering the item at the lower price after some fraction has beensold
idea has been used in the:
one-way-trading problem (Lavi/Nisan ’00, El-Yaniv et al. ’01)single-leg booking problem (Ball/Queyranne ’09)online dynamic pricing problem (Ma/S.-L./Teo ’17)
David Simchi-Levi (MIT) Online Resource Allocation 8 / 22
Online Resource Allocation Intuition
Challenge 2: Multiple Prices
Potential prices of : $150, $450
p = 0
Solution: “Booking Limits”
stop offering the item at the lower price after some fraction has beensold
idea has been used in the:
one-way-trading problem (Lavi/Nisan ’00, El-Yaniv et al. ’01)single-leg booking problem (Ball/Queyranne ’09)online dynamic pricing problem (Ma/S.-L./Teo ’17)
David Simchi-Levi (MIT) Online Resource Allocation 8 / 22
Online Resource Allocation Intuition
Challenge 2: Multiple Prices
Potential prices of : $150, $450
p = 0
Solution: “Booking Limits”
stop offering the item at the lower price after some fraction has beensold
idea has been used in the:
one-way-trading problem (Lavi/Nisan ’00, El-Yaniv et al. ’01)single-leg booking problem (Ball/Queyranne ’09)online dynamic pricing problem (Ma/S.-L./Teo ’17)
David Simchi-Levi (MIT) Online Resource Allocation 8 / 22
Online Resource Allocation Example
Combining the Challenges
Potential prices for : $150, $450
Potential prices for : $200
p = .2
Bid-price Control Policy: for each customer t, offer the item i andprice j maximizing the expected “profit”
p(j)t,i (r
(j)i − λi ),
where λi is the “cost”, or bid price, for one unit of inventory of i
Key Question: what is the value of λi?
David Simchi-Levi (MIT) Online Resource Allocation 9 / 22
Online Resource Allocation Example
Combining the Challenges
Potential prices for : $150, $450
Potential prices for : $200
p = .2
Bid-price Control Policy: for each customer t, offer the item i andprice j maximizing the expected “profit”
p(j)t,i (r
(j)i − λi ),
where λi is the “cost”, or bid price, for one unit of inventory of i
Key Question: what is the value of λi?
David Simchi-Levi (MIT) Online Resource Allocation 9 / 22
Online Resource Allocation Example
Combining the Challenges
Potential prices for : $150, $450
Potential prices for : $200
p = .2
Bid-price Control Policy: for each customer t, offer the item i andprice j maximizing the expected “profit”
p(j)t,i (r
(j)i − λi ),
where λi is the “cost”, or bid price, for one unit of inventory of i
Key Question: what is the value of λi?
David Simchi-Levi (MIT) Online Resource Allocation 9 / 22
Online Resource Allocation Example
Combining the Challenges
Potential prices for : $150, $450
Potential prices for : $200
p = .2
Bid-price Control Policy: for each customer t, offer the item i andprice j maximizing the expected “profit”
p(j)t,i (r
(j)i − λi ),
where λi is the “cost”, or bid price, for one unit of inventory of i
Key Question: what is the value of λi?
David Simchi-Levi (MIT) Online Resource Allocation 9 / 22
Online Resource Allocation Example
The Value of Inventory
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Typical definition of λi :
solve a deterministic LP based on the forecasted demand over theremaining time horizon
λi is the “reduced cost” of the inventory constraint on item i
Our definition of λi :
let wi be the fraction of item i ’s starting inventory already sold
λi = Φi (wi ), where Φi is an increasing function
David Simchi-Levi (MIT) Online Resource Allocation 10 / 22
Online Resource Allocation Example
The Value of Inventory
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Typical definition of λi :
solve a deterministic LP based on the forecasted demand over theremaining time horizon
λi is the “reduced cost” of the inventory constraint on item i
Our definition of λi :
let wi be the fraction of item i ’s starting inventory already sold
λi = Φi (wi ), where Φi is an increasing function
David Simchi-Levi (MIT) Online Resource Allocation 10 / 22
Online Resource Allocation Example
The Value of Inventory
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Typical definition of λi :
solve a deterministic LP based on the forecasted demand over theremaining time horizon
λi is the “reduced cost” of the inventory constraint on item i
Our definition of λi :
let wi be the fraction of item i ’s starting inventory already sold
λi = Φi (wi ), where Φi is an increasing function
David Simchi-Levi (MIT) Online Resource Allocation 10 / 22
Online Resource Allocation Example
The Value of Inventory
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Typical definition of λi :
solve a deterministic LP based on the forecasted demand over theremaining time horizon
λi is the “reduced cost” of the inventory constraint on item i
Our definition of λi :
let wi be the fraction of item i ’s starting inventory already sold
λi = Φi (wi ), where Φi is an increasing function
David Simchi-Levi (MIT) Online Resource Allocation 10 / 22
Online Resource Allocation Example
The Value of Inventory
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Typical definition of λi :
solve a deterministic LP based on the forecasted demand over theremaining time horizon
λi is the “reduced cost” of the inventory constraint on item i
Our definition of λi :
let wi be the fraction of item i ’s starting inventory already sold
λi = Φi (wi ), where Φi is an increasing function
David Simchi-Levi (MIT) Online Resource Allocation 10 / 22
Online Resource Allocation Example
The Value of Inventory
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Typical definition of λi :
solve a deterministic LP based on the forecasted demand over theremaining time horizon
λi is the “reduced cost” of the inventory constraint on item i
Our definition of λi :
let wi be the fraction of item i ’s starting inventory already sold
λi = Φi (wi ), where Φi is an increasing function
David Simchi-Levi (MIT) Online Resource Allocation 10 / 22
Online Resource Allocation Example
The Value of Inventory
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Typical definition of λi :
solve a deterministic LP based on the forecasted demand over theremaining time horizon
λi is the “reduced cost” of the inventory constraint on item i
Our definition of λi :
let wi be the fraction of item i ’s starting inventory already sold
λi = Φi (wi ), where Φi is an increasing function
David Simchi-Levi (MIT) Online Resource Allocation 10 / 22
Online Resource Allocation Example
Returning to the Example
Potential prices for : $150, $450
Potential prices for : $200
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p = .2
p(j)t,i (r
(j)i − λi )
= .2(450− 50)= 80
p(j)t,i (r
(j)i − λi )
= 1(200− 150)= 50
value of is low: λi = Φi (15 ) ≈ 50 (most units still remaining)
value of is high: λi = Φi (34 ) ≈ 150 (most units already sold)
David Simchi-Levi (MIT) Online Resource Allocation 11 / 22
Online Resource Allocation Example
Returning to the Example
Potential prices for : $150, $450
Potential prices for : $200
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p = .2
p(j)t,i (r
(j)i − λi )
= .2(450− 50)= 80
p(j)t,i (r
(j)i − λi )
= 1(200− 150)= 50
value of is low: λi = Φi (15 ) ≈ 50 (most units still remaining)
value of is high: λi = Φi (34 ) ≈ 150 (most units already sold)
David Simchi-Levi (MIT) Online Resource Allocation 11 / 22
Online Resource Allocation Example
Returning to the Example
Potential prices for : $150, $450
Potential prices for : $200
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p = .2
p(j)t,i (r
(j)i − λi )
= .2(450− 50)= 80
p(j)t,i (r
(j)i − λi )
= 1(200− 150)= 50
value of is low: λi = Φi (15 ) ≈ 50 (most units still remaining)
value of is high: λi = Φi (34 ) ≈ 150 (most units already sold)
David Simchi-Levi (MIT) Online Resource Allocation 11 / 22
Online Resource Allocation Example
Returning to the Example
Potential prices for : $150, $450
Potential prices for : $200
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p = .2
p(j)t,i (r
(j)i − λi )
= .2(450− 50)= 80
p(j)t,i (r
(j)i − λi )
= 1(200− 150)= 50
value of is low: λi = Φi (15 ) ≈ 50 (most units still remaining)
value of is high: λi = Φi (34 ) ≈ 150 (most units already sold)
David Simchi-Levi (MIT) Online Resource Allocation 11 / 22
Online Resource Allocation Example
Returning to the Example
Potential prices for : $150, $450
Potential prices for : $200
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p = .2
p(j)t,i (r
(j)i − λi )
= .2(450− 50)= 80
p(j)t,i (r
(j)i − λi )
= 1(200− 150)= 50
value of is low: λi = Φi (15 ) ≈ 50 (most units still remaining)
value of is high: λi = Φi (34 ) ≈ 150 (most units already sold)
David Simchi-Levi (MIT) Online Resource Allocation 11 / 22
Online Resource Allocation Example
Returning to the Example
Potential prices for : $150, $450
Potential prices for : $200
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p = .2
p(j)t,i (r
(j)i − λi )
= .2(450− 50)= 80
p(j)t,i (r
(j)i − λi )
= 1(200− 150)= 50
value of is low: λi = Φi (15 ) ≈ 50 (most units still remaining)
value of is high: λi = Φi (34 ) ≈ 150 (most units already sold)
David Simchi-Levi (MIT) Online Resource Allocation 11 / 22
Online Resource Allocation Example
Returning to the Example
Potential prices for : $150, $450
Potential prices for : $200
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p = .2
p(j)t,i (r
(j)i − λi )
= .2(450− 50)= 80
p(j)t,i (r
(j)i − λi )
= 1(200− 150)= 50
value of is low: λi = Φi (15 ) ≈ 50 (most units still remaining)
value of is high: λi = Φi (34 ) ≈ 150 (most units already sold)
David Simchi-Levi (MIT) Online Resource Allocation 11 / 22
Online Resource Allocation Example
Returning to the Example
Potential prices for : $150, $450
Potential prices for : $200
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p(j)t,i (r
(j)i − λi )
= 1(150− 50)= 100
p = .2
p(j)t,i (r
(j)i − λi )
= .2(450− 50)= 80
p(j)t,i (r
(j)i − λi )
= 1(200− 150)= 50
value of is low: λi = Φi (15 ) ≈ 50 (most units still remaining)
value of is high: λi = Φi (34 ) ≈ 150 (most units already sold)
David Simchi-Levi (MIT) Online Resource Allocation 11 / 22
Online Resource Allocation Example
Exact Value Function Φi for i =
0 1
Fraction Sold, wi
$450
Φi (wi )
$150
α
Φi increases from 0 to themaximum price of $450 over [0, 1]
Φi is piecewise-convex
the value α at whichΦi (α) = $150 is the “bookinglimit” for the lower price
the algorithm is highlydisincentivized to offer the itemat $150 as wi approaches α, andstops completely once wi ≥ α
α = ln 2(r−1)√1+4r(r−1)/e−1
, where r is the
ratio of high price to low price
different than the optimal single-itembooking limit σ = r
2r−1of
Ball/Queyranne1 5
r0.5
1
σ = r2r−1
α = ln2(r−1)√
1+4r(r−1)/e−1
David Simchi-Levi (MIT) Online Resource Allocation 12 / 22
Online Resource Allocation Example
Exact Value Function Φi for i =
0 1
Fraction Sold, wi
$450
Φi (wi )
$150
α
Φi increases from 0 to themaximum price of $450 over [0, 1]
Φi is piecewise-convex
the value α at whichΦi (α) = $150 is the “bookinglimit” for the lower price
the algorithm is highlydisincentivized to offer the itemat $150 as wi approaches α, andstops completely once wi ≥ α
α = ln 2(r−1)√1+4r(r−1)/e−1
, where r is the
ratio of high price to low price
different than the optimal single-itembooking limit σ = r
2r−1of
Ball/Queyranne1 5
r0.5
1
σ = r2r−1
α = ln2(r−1)√
1+4r(r−1)/e−1
David Simchi-Levi (MIT) Online Resource Allocation 12 / 22
Online Resource Allocation Example
Exact Value Function Φi for i =
0 1
Fraction Sold, wi
$450
Φi (wi )
$150
α
Φi increases from 0 to themaximum price of $450 over [0, 1]
Φi is piecewise-convex
the value α at whichΦi (α) = $150 is the “bookinglimit” for the lower price
the algorithm is highlydisincentivized to offer the itemat $150 as wi approaches α, andstops completely once wi ≥ α
α = ln 2(r−1)√1+4r(r−1)/e−1
, where r is the
ratio of high price to low price
different than the optimal single-itembooking limit σ = r
2r−1of
Ball/Queyranne1 5
r0.5
1
σ = r2r−1
α = ln2(r−1)√
1+4r(r−1)/e−1
David Simchi-Levi (MIT) Online Resource Allocation 12 / 22
Online Resource Allocation Example
Exact Value Function Φi for i =
0 1
Fraction Sold, wi
$450
Φi (wi )
$150
α
Φi increases from 0 to themaximum price of $450 over [0, 1]
Φi is piecewise-convex
the value α at whichΦi (α) = $150 is the “bookinglimit” for the lower price
the algorithm is highlydisincentivized to offer the itemat $150 as wi approaches α, andstops completely once wi ≥ α
α = ln 2(r−1)√1+4r(r−1)/e−1
, where r is the
ratio of high price to low price
different than the optimal single-itembooking limit σ = r
2r−1of
Ball/Queyranne1 5
r0.5
1
σ = r2r−1
α = ln2(r−1)√
1+4r(r−1)/e−1
David Simchi-Levi (MIT) Online Resource Allocation 12 / 22
Online Resource Allocation Example
Exact Value Function Φi for i =
0 1
Fraction Sold, wi
$450
Φi (wi )
$150
α
Φi increases from 0 to themaximum price of $450 over [0, 1]
Φi is piecewise-convex
the value α at whichΦi (α) = $150 is the “bookinglimit” for the lower price
the algorithm is highlydisincentivized to offer the itemat $150 as wi approaches α, andstops completely once wi ≥ α
α = ln 2(r−1)√1+4r(r−1)/e−1
, where r is the
ratio of high price to low price
different than the optimal single-itembooking limit σ = r
2r−1of
Ball/Queyranne1 5
r0.5
1
σ = r2r−1
α = ln2(r−1)√
1+4r(r−1)/e−1
David Simchi-Levi (MIT) Online Resource Allocation 12 / 22
Online Resource Allocation Example
Exact Value Function Φi for i =
0 1
Fraction Sold, wi
$450
Φi (wi )
$150
α
Φi increases from 0 to themaximum price of $450 over [0, 1]
Φi is piecewise-convex
the value α at whichΦi (α) = $150 is the “bookinglimit” for the lower price
the algorithm is highlydisincentivized to offer the itemat $150 as wi approaches α, andstops completely once wi ≥ α
α = ln 2(r−1)√1+4r(r−1)/e−1
, where r is the
ratio of high price to low price
different than the optimal single-itembooking limit σ = r
2r−1of
Ball/Queyranne1 5
r0.5
1
σ = r2r−1
α = ln2(r−1)√
1+4r(r−1)/e−1
David Simchi-Levi (MIT) Online Resource Allocation 12 / 22
Online Resource Allocation Example
Exact Value Function Φi for i =
0 1
Fraction Sold, wi
$450
Φi (wi )
$150
α
Φi increases from 0 to themaximum price of $450 over [0, 1]
Φi is piecewise-convex
the value α at whichΦi (α) = $150 is the “bookinglimit” for the lower price
the algorithm is highlydisincentivized to offer the itemat $150 as wi approaches α, andstops completely once wi ≥ α
α = ln 2(r−1)√1+4r(r−1)/e−1
, where r is the
ratio of high price to low price
different than the optimal single-itembooking limit σ = r
2r−1of
Ball/Queyranne1 5
r0.5
1
σ = r2r−1
α = ln2(r−1)√
1+4r(r−1)/e−1
David Simchi-Levi (MIT) Online Resource Allocation 12 / 22
Online Resource Allocation Example
Exact Value Function Φi for i =
0 1
Fraction Sold, wi
$450
Φi (wi )
$150
α
Φi increases from 0 to themaximum price of $450 over [0, 1]
Φi is piecewise-convex
the value α at whichΦi (α) = $150 is the “bookinglimit” for the lower price
the algorithm is highlydisincentivized to offer the itemat $150 as wi approaches α, andstops completely once wi ≥ α
α = ln 2(r−1)√1+4r(r−1)/e−1
, where r is the
ratio of high price to low price
different than the optimal single-itembooking limit σ = r
2r−1of
Ball/Queyranne
1 5r0.5
1
σ = r2r−1
α = ln2(r−1)√
1+4r(r−1)/e−1
David Simchi-Levi (MIT) Online Resource Allocation 12 / 22
Online Resource Allocation Example
Exact Value Function Φi for i =
0 1
Fraction Sold, wi
$450
Φi (wi )
$150
α
Φi increases from 0 to themaximum price of $450 over [0, 1]
Φi is piecewise-convex
the value α at whichΦi (α) = $150 is the “bookinglimit” for the lower price
the algorithm is highlydisincentivized to offer the itemat $150 as wi approaches α, andstops completely once wi ≥ α
α = ln 2(r−1)√1+4r(r−1)/e−1
, where r is the
ratio of high price to low price
different than the optimal single-itembooking limit σ = r
2r−1of
Ball/Queyranne1 5
r0.5
1
σ = r2r−1
α = ln2(r−1)√
1+4r(r−1)/e−1
David Simchi-Levi (MIT) Online Resource Allocation 12 / 22
Online Resource Allocation Algorithm and Analysis
General Value Function Φi for 2 Prices
when mi = 2, Φi is piecewise-convex with 2 pieces, separated by α:
0 1wi
r(2)i
Φi
r(1)i
α
r(1)i
ewi−1eα−1
r(1)i + (r
(2)i − r
(1)i ) ewi−α−1
e1−α−1
α is given by α(r(2)i /r
(1)i ), where α(r) = ln 2(r−1)√
1+4r(r−1)/e−1
the competitive ratio associated with Φi , CRi , is then 1− e−α
David Simchi-Levi (MIT) Online Resource Allocation 13 / 22
Online Resource Allocation Algorithm and Analysis
General Value Function Φi for 2 Prices
when mi = 2, Φi is piecewise-convex with 2 pieces, separated by α:
0 1wi
r(2)i
Φi
r(1)i
α
r(1)i
ewi−1eα−1
r(1)i + (r
(2)i − r
(1)i ) ewi−α−1
e1−α−1
α is given by α(r(2)i /r
(1)i ), where α(r) = ln 2(r−1)√
1+4r(r−1)/e−1
the competitive ratio associated with Φi , CRi , is then 1− e−α
David Simchi-Levi (MIT) Online Resource Allocation 13 / 22
Online Resource Allocation Algorithm and Analysis
General Value Function Φi for 2 Prices
when mi = 2, Φi is piecewise-convex with 2 pieces, separated by α:
0 1wi
r(2)i
Φi
r(1)i
α
r(1)i
ewi−1eα−1
r(1)i + (r
(2)i − r
(1)i ) ewi−α−1
e1−α−1
α is given by α(r(2)i /r
(1)i ), where α(r) = ln 2(r−1)√
1+4r(r−1)/e−1
the competitive ratio associated with Φi , CRi , is then 1− e−α
David Simchi-Levi (MIT) Online Resource Allocation 13 / 22
Online Resource Allocation Algorithm and Analysis
General Value Function Φi for 2 Prices
when mi = 2, Φi is piecewise-convex with 2 pieces, separated by α:
0 1wi
r(2)i
Φi
r(1)i
α
r(1)i
ewi−1eα−1
r(1)i + (r
(2)i − r
(1)i ) ewi−α−1
e1−α−1
α is given by α(r(2)i /r
(1)i ), where α(r) = ln 2(r−1)√
1+4r(r−1)/e−1
the competitive ratio associated with Φi , CRi , is then 1− e−α
David Simchi-Levi (MIT) Online Resource Allocation 13 / 22
Online Resource Allocation Algorithm and Analysis
General Value Function Φi for Multiple Prices
for any number mi of prices, Φi is designed, based on r(1)i , . . . , r
(mi )i , to
maximize CRi
we derive Φi by solving a differential equation arising from a primal-dualanalysis
in general, Φi is piecewise-convex with mi pieces, separated by mi − 1
booking limits α(1)i , . . . , α
(mi−1)i
α(1)i , . . . , α
(mi−1)i correspond to roots of a degree-mi polynomial, which can
be computed using bisection search
David Simchi-Levi (MIT) Online Resource Allocation 14 / 22
Online Resource Allocation Algorithm and Analysis
General Value Function Φi for Multiple Prices
for any number mi of prices, Φi is designed, based on r(1)i , . . . , r
(mi )i , to
maximize CRi
we derive Φi by solving a differential equation arising from a primal-dualanalysis
in general, Φi is piecewise-convex with mi pieces, separated by mi − 1
booking limits α(1)i , . . . , α
(mi−1)i
α(1)i , . . . , α
(mi−1)i correspond to roots of a degree-mi polynomial, which can
be computed using bisection search
David Simchi-Levi (MIT) Online Resource Allocation 14 / 22
Online Resource Allocation Algorithm and Analysis
General Value Function Φi for Multiple Prices
for any number mi of prices, Φi is designed, based on r(1)i , . . . , r
(mi )i , to
maximize CRi
we derive Φi by solving a differential equation arising from a primal-dualanalysis
in general, Φi is piecewise-convex with mi pieces, separated by mi − 1
booking limits α(1)i , . . . , α
(mi−1)i
α(1)i , . . . , α
(mi−1)i correspond to roots of a degree-mi polynomial, which can
be computed using bisection search
David Simchi-Levi (MIT) Online Resource Allocation 14 / 22
Online Resource Allocation Algorithm and Analysis
General Value Function Φi for Multiple Prices
for any number mi of prices, Φi is designed, based on r(1)i , . . . , r
(mi )i , to
maximize CRi
we derive Φi by solving a differential equation arising from a primal-dualanalysis
in general, Φi is piecewise-convex with mi pieces, separated by mi − 1
booking limits α(1)i , . . . , α
(mi−1)i
α(1)i , . . . , α
(mi−1)i correspond to roots of a degree-mi polynomial, which can
be computed using bisection search
David Simchi-Levi (MIT) Online Resource Allocation 14 / 22
Online Resource Allocation Algorithm and Analysis
General Value Function Φi for Multiple Prices
for any number mi of prices, Φi is designed, based on r(1)i , . . . , r
(mi )i , to
maximize CRi
we derive Φi by solving a differential equation arising from a primal-dualanalysis
in general, Φi is piecewise-convex with mi pieces, separated by mi − 1
booking limits α(1)i , . . . , α
(mi−1)i
α(1)i , . . . , α
(mi−1)i correspond to roots of a degree-mi polynomial, which can
be computed using bisection search
David Simchi-Levi (MIT) Online Resource Allocation 14 / 22
Online Resource Allocation Algorithm and Analysis
Overall Algorithm
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Having derived Φ1, . . . ,Φn, two different algorithms are possible:1 Algorithm 1:
λi = Φi (wi ), where wi is the fraction of item i ’s starting inventory soldwi is dynamically incremented as sales are realized
2 Algorithm 2:
more applicable when the starting inventories ki are smallassume WOLOG that ki = 1 for all ithen λi = Φi (Wi ), where Wi ∼ Unif[0, 1]Wi is a random seed which is initialized independently for each i
David Simchi-Levi (MIT) Online Resource Allocation 15 / 22
Online Resource Allocation Algorithm and Analysis
Overall Algorithm
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Having derived Φ1, . . . ,Φn, two different algorithms are possible:
1 Algorithm 1:
λi = Φi (wi ), where wi is the fraction of item i ’s starting inventory soldwi is dynamically incremented as sales are realized
2 Algorithm 2:
more applicable when the starting inventories ki are smallassume WOLOG that ki = 1 for all ithen λi = Φi (Wi ), where Wi ∼ Unif[0, 1]Wi is a random seed which is initialized independently for each i
David Simchi-Levi (MIT) Online Resource Allocation 15 / 22
Online Resource Allocation Algorithm and Analysis
Overall Algorithm
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Having derived Φ1, . . . ,Φn, two different algorithms are possible:1 Algorithm 1:
λi = Φi (wi ), where wi is the fraction of item i ’s starting inventory soldwi is dynamically incremented as sales are realized
2 Algorithm 2:
more applicable when the starting inventories ki are smallassume WOLOG that ki = 1 for all ithen λi = Φi (Wi ), where Wi ∼ Unif[0, 1]Wi is a random seed which is initialized independently for each i
David Simchi-Levi (MIT) Online Resource Allocation 15 / 22
Online Resource Allocation Algorithm and Analysis
Overall Algorithm
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Having derived Φ1, . . . ,Φn, two different algorithms are possible:1 Algorithm 1:
λi = Φi (wi ), where wi is the fraction of item i ’s starting inventory soldwi is dynamically incremented as sales are realized
2 Algorithm 2:
more applicable when the starting inventories ki are smallassume WOLOG that ki = 1 for all ithen λi = Φi (Wi ), where Wi ∼ Unif[0, 1]Wi is a random seed which is initialized independently for each i
David Simchi-Levi (MIT) Online Resource Allocation 15 / 22
Online Resource Allocation Algorithm and Analysis
Overall Algorithm
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Having derived Φ1, . . . ,Φn, two different algorithms are possible:1 Algorithm 1:
λi = Φi (wi ), where wi is the fraction of item i ’s starting inventory soldwi is dynamically incremented as sales are realized
2 Algorithm 2:
more applicable when the starting inventories ki are small
assume WOLOG that ki = 1 for all ithen λi = Φi (Wi ), where Wi ∼ Unif[0, 1]Wi is a random seed which is initialized independently for each i
David Simchi-Levi (MIT) Online Resource Allocation 15 / 22
Online Resource Allocation Algorithm and Analysis
Overall Algorithm
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Having derived Φ1, . . . ,Φn, two different algorithms are possible:1 Algorithm 1:
λi = Φi (wi ), where wi is the fraction of item i ’s starting inventory soldwi is dynamically incremented as sales are realized
2 Algorithm 2:
more applicable when the starting inventories ki are smallassume WOLOG that ki = 1 for all i
then λi = Φi (Wi ), where Wi ∼ Unif[0, 1]Wi is a random seed which is initialized independently for each i
David Simchi-Levi (MIT) Online Resource Allocation 15 / 22
Online Resource Allocation Algorithm and Analysis
Overall Algorithm
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Having derived Φ1, . . . ,Φn, two different algorithms are possible:1 Algorithm 1:
λi = Φi (wi ), where wi is the fraction of item i ’s starting inventory soldwi is dynamically incremented as sales are realized
2 Algorithm 2:
more applicable when the starting inventories ki are smallassume WOLOG that ki = 1 for all ithen λi = Φi (Wi ), where Wi ∼ Unif[0, 1]
Wi is a random seed which is initialized independently for each i
David Simchi-Levi (MIT) Online Resource Allocation 15 / 22
Online Resource Allocation Algorithm and Analysis
Overall Algorithm
Bid-price Control Policy:
maxi ,j
p(j)t,i (r
(j)i − λi )
Having derived Φ1, . . . ,Φn, two different algorithms are possible:1 Algorithm 1:
λi = Φi (wi ), where wi is the fraction of item i ’s starting inventory soldwi is dynamically incremented as sales are realized
2 Algorithm 2:
more applicable when the starting inventories ki are smallassume WOLOG that ki = 1 for all ithen λi = Φi (Wi ), where Wi ∼ Unif[0, 1]Wi is a random seed which is initialized independently for each i
David Simchi-Levi (MIT) Online Resource Allocation 15 / 22
Online Resource Allocation Algorithm and Analysis
The Competitive Ratio
largestarting
inventories
Algorithm 1achieves mini CRi
deterministiccustomer
purchasing
Algorithm 2achieves mini CRi
Algorithm 1= Algorithm 2
counterexample counterexample counterexample
We construct a counterexample showing our results are tight, i.e. noonline algorithm can achieve a competitive ratio better than mini CRi
David Simchi-Levi (MIT) Online Resource Allocation 16 / 22
Online Resource Allocation Algorithm and Analysis
The Competitive Ratio
largestarting
inventories
Algorithm 1achieves mini CRi
deterministiccustomer
purchasing
Algorithm 2achieves mini CRi
Algorithm 1= Algorithm 2
counterexample counterexample counterexample
We construct a counterexample showing our results are tight, i.e. noonline algorithm can achieve a competitive ratio better than mini CRi
David Simchi-Levi (MIT) Online Resource Allocation 16 / 22
Online Resource Allocation Algorithm and Analysis
The Competitive Ratio
largestarting
inventories
Algorithm 1achieves mini CRi
deterministiccustomer
purchasing
Algorithm 2achieves mini CRi
Algorithm 1= Algorithm 2
counterexample counterexample counterexample
We construct a counterexample showing our results are tight, i.e. noonline algorithm can achieve a competitive ratio better than mini CRi
David Simchi-Levi (MIT) Online Resource Allocation 16 / 22
Online Resource Allocation Algorithm and Analysis
The Competitive Ratio
largestarting
inventories
Algorithm 1achieves mini CRi
deterministiccustomer
purchasing
Algorithm 2achieves mini CRi
Algorithm 1= Algorithm 2
counterexample counterexample counterexample
We construct a counterexample showing our results are tight, i.e. noonline algorithm can achieve a competitive ratio better than mini CRi
David Simchi-Levi (MIT) Online Resource Allocation 16 / 22
Online Resource Allocation Algorithm and Analysis
The Competitive Ratio
largestarting
inventories
Algorithm 1achieves mini CRi
deterministiccustomer
purchasing
Algorithm 2achieves mini CRi
Algorithm 1= Algorithm 2
counterexample counterexample counterexample
We construct a counterexample showing our results are tight, i.e. noonline algorithm can achieve a competitive ratio better than mini CRi
David Simchi-Levi (MIT) Online Resource Allocation 16 / 22
Online Resource Allocation Algorithm and Analysis
The Competitive Ratio
largestarting
inventories
Algorithm 1achieves mini CRi
deterministiccustomer
purchasing
Algorithm 2achieves mini CRi
Algorithm 1= Algorithm 2
counterexample counterexample counterexample
We construct a counterexample showing our results are tight, i.e. noonline algorithm can achieve a competitive ratio better than mini CRi
David Simchi-Levi (MIT) Online Resource Allocation 16 / 22
Online Resource Allocation Algorithm and Analysis
The Competitive Ratio
largestarting
inventories
Algorithm 1achieves mini CRi
deterministiccustomer
purchasing
Algorithm 2achieves mini CRi
Algorithm 1= Algorithm 2
counterexample counterexample counterexample
We construct a counterexample showing our results are tight, i.e. noonline algorithm can achieve a competitive ratio better than mini CRi
David Simchi-Levi (MIT) Online Resource Allocation 16 / 22
Online Resource Allocation Algorithm and Analysis
The Optimal Competitive Ratio
largestarting
inventories
Algorithm 1achieves mini CRi
deterministiccustomer
purchasing
Algorithm 2achieves mini CRi
Algorithm 1= Algorithm 2
counterexample counterexample counterexample
We construct a counterexample showing our results are tight, i.e. noonline algorithm can achieve a competitive ratio better than mini CRi
David Simchi-Levi (MIT) Online Resource Allocation 16 / 22
Online Resource Allocation Algorithm and Analysis
The Optimal Competitive Ratio
largestarting
inventories
Algorithm 1achieves mini CRi
deterministiccustomer
purchasing
Algorithm 2achieves mini CRi
Algorithm 1= Algorithm 2
counterexample
counterexample counterexample
We construct a counterexample showing our results are tight, i.e. noonline algorithm can achieve a competitive ratio better than mini CRi
David Simchi-Levi (MIT) Online Resource Allocation 16 / 22
Online Resource Allocation Algorithm and Analysis
The Optimal Competitive Ratio
largestarting
inventories
Algorithm 1achieves mini CRi
deterministiccustomer
purchasing
Algorithm 2achieves mini CRi
Algorithm 1= Algorithm 2
counterexample
counterexample
counterexample
We construct a counterexample showing our results are tight, i.e. noonline algorithm can achieve a competitive ratio better than mini CRi
David Simchi-Levi (MIT) Online Resource Allocation 16 / 22
Online Resource Allocation Algorithm and Analysis
The Optimal Competitive Ratio
largestarting
inventories
Algorithm 1achieves mini CRi
deterministiccustomer
purchasing
Algorithm 2achieves mini CRi
Algorithm 1= Algorithm 2
counterexample counterexample
counterexample
We construct a counterexample showing our results are tight, i.e. noonline algorithm can achieve a competitive ratio better than mini CRi
David Simchi-Levi (MIT) Online Resource Allocation 16 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14][ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16][ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14][ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16][ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14][ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16][ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14][ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16][ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14][ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16][ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14]
[ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16][ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14]
[ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16]
[ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14]
[ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16]
[ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14]
[ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16]
[ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14]
[ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16]
[ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14]
[ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16]
[ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14]
[ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16]
[ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14]
[ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16]
[ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14]
[ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16]
[ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14]
[ 12,≈ .621][MP12][ 1
2,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16]
[ 14,↗][CMSX16][ 1
4,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387][↙, e−1
2e−1≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14]
[ 12,≈ .621][MP12]
[ 12,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16]
[ 14,↗][CMSX16]
[ 14,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387]
[↙, e−12e−1
≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Algorithm and Analysis
Illustration of Competitive Ratios with Two Prices per Item
∞
r
1
r = maxir
(2)i
r(1)i
General Stochastic Setting
1k ∞
k = mini ki
StochasticPurchasing
DeterministicPurchasing
Multiple Items
Single Item
tight results
non-tight results
increasingcompetitive ratios
1− 1e≈ .632
[KP00,MSVV05,BJN07,GNR14]
1− 1e≈ .632
[KVV90,AGKM11,DJK13]
1− 1e≈ .632
1− e−α(r)[MS17]
1− 1√e≈ .393
12
[MP12,GNR14][ 12,≈ .621][MP12]
[ 12,≈ .621]
r2(2r−1)
[MS17]
14
[CMSX16][ 14,↗][CMSX16]
[ 14,↗]
1−1/e
(1+k)(1−e−1/k )[MS17]
1−e−α(r)
(1+k)(e1/k−1)[MS17]
12
1
r2r−1
[BQ09]
1− 1e
[MST17][↙, e−1
2e−1≈ .387]
[↙, e−12e−1
≈ .387]
David Simchi-Levi (MIT) Online Resource Allocation 17 / 22
Online Resource Allocation Simulations
Simulations on Hotel Data Set
publicly-accessible hotel data set from MSOM journal(Bodea/Ferguson/Garrow ’09)
multiple items (room categories) with multiple prices (rates)
Room Category Discounted Rate Rack Rate Number of Rooms
King $307 $361 440Queen $304 $361 130Suite $384 $496 110
Two Double $306 $342 170
data is given for check-in dates in March–April 2007
we assume that each check-in date is a separate problem instance
the arrival sequence for each check-in date is given by the transactionhistory for that date (over the year before check-in)
on average, 1340 total arrivals per check-in date (scaled by 10)
David Simchi-Levi (MIT) Online Resource Allocation 18 / 22
Online Resource Allocation Simulations
Simulations on Hotel Data Set
publicly-accessible hotel data set from MSOM journal(Bodea/Ferguson/Garrow ’09)
multiple items (room categories) with multiple prices (rates)
Room Category Discounted Rate Rack Rate Number of Rooms
King $307 $361 440Queen $304 $361 130Suite $384 $496 110
Two Double $306 $342 170
data is given for check-in dates in March–April 2007
we assume that each check-in date is a separate problem instance
the arrival sequence for each check-in date is given by the transactionhistory for that date (over the year before check-in)
on average, 1340 total arrivals per check-in date (scaled by 10)
David Simchi-Levi (MIT) Online Resource Allocation 18 / 22
Online Resource Allocation Simulations
Simulations on Hotel Data Set
publicly-accessible hotel data set from MSOM journal(Bodea/Ferguson/Garrow ’09)
multiple items (room categories) with multiple prices (rates)
Room Category Discounted Rate Rack Rate Number of Rooms
King $307 $361 440Queen $304 $361 130Suite $384 $496 110
Two Double $306 $342 170
data is given for check-in dates in March–April 2007
we assume that each check-in date is a separate problem instance
the arrival sequence for each check-in date is given by the transactionhistory for that date (over the year before check-in)
on average, 1340 total arrivals per check-in date (scaled by 10)
David Simchi-Levi (MIT) Online Resource Allocation 18 / 22
Online Resource Allocation Simulations
Simulations on Hotel Data Set
publicly-accessible hotel data set from MSOM journal(Bodea/Ferguson/Garrow ’09)
multiple items (room categories) with multiple prices (rates)
Room Category Discounted Rate Rack Rate Number of Rooms
King $307 $361 440Queen $304 $361 130Suite $384 $496 110
Two Double $306 $342 170
data is given for check-in dates in March–April 2007
we assume that each check-in date is a separate problem instance
the arrival sequence for each check-in date is given by the transactionhistory for that date (over the year before check-in)
on average, 1340 total arrivals per check-in date (scaled by 10)
David Simchi-Levi (MIT) Online Resource Allocation 18 / 22
Online Resource Allocation Simulations
Simulations on Hotel Data Set
publicly-accessible hotel data set from MSOM journal(Bodea/Ferguson/Garrow ’09)
multiple items (room categories) with multiple prices (rates)
Room Category Discounted Rate Rack Rate Number of Rooms
King $307 $361 440Queen $304 $361 130Suite $384 $496 110
Two Double $306 $342 170
data is given for check-in dates in March–April 2007
we assume that each check-in date is a separate problem instance
the arrival sequence for each check-in date is given by the transactionhistory for that date (over the year before check-in)
on average, 1340 total arrivals per check-in date (scaled by 10)
David Simchi-Levi (MIT) Online Resource Allocation 18 / 22
Online Resource Allocation Simulations
Simulations on Hotel Data Set
publicly-accessible hotel data set from MSOM journal(Bodea/Ferguson/Garrow ’09)
multiple items (room categories) with multiple prices (rates)
Room Category Discounted Rate Rack Rate Number of Rooms
King $307 $361 440Queen $304 $361 130Suite $384 $496 110
Two Double $306 $342 170
data is given for check-in dates in March–April 2007
we assume that each check-in date is a separate problem instance
the arrival sequence for each check-in date is given by the transactionhistory for that date (over the year before check-in)
on average, 1340 total arrivals per check-in date (scaled by 10)
David Simchi-Levi (MIT) Online Resource Allocation 18 / 22
Online Resource Allocation Simulations
Simulations on Hotel Data Set
publicly-accessible hotel data set from MSOM journal(Bodea/Ferguson/Garrow ’09)
multiple items (room categories) with multiple prices (rates)
Room Category Discounted Rate Rack Rate Number of Rooms
King $307 $361 440Queen $304 $361 130Suite $384 $496 110
Two Double $306 $342 170
data is given for check-in dates in March–April 2007
we assume that each check-in date is a separate problem instance
the arrival sequence for each check-in date is given by the transactionhistory for that date (over the year before check-in)
on average, 1340 total arrivals per check-in date (scaled by 10)
David Simchi-Levi (MIT) Online Resource Allocation 18 / 22
Online Resource Allocation Simulations
Algorithms Compared
Algorithm decides which room/fare combinations should be offered todifferent customers from different channels (website, CRO/CRS, GDS):
Forecast-dependent Algorithms
update these assortments based on remaining inventory,relative to the forecasted distribution of customers still to come (thisdepends on the number of days until check-in);
Forecast-independent Algorithms
optimize these assortments for the worst case,dynamically hedging against the worst case as inventory is sold;
Hybrid Algorithms
follow the forecast-dependent algorithm during each time step,however, if the decision prescribed is suboptimal in the worst-casebeyond a certain threshold,then it uses the decision of the forecast-independent algorithm instead.
David Simchi-Levi (MIT) Online Resource Allocation 19 / 22
Online Resource Allocation Simulations
Algorithms Compared
Algorithm decides which room/fare combinations should be offered todifferent customers from different channels (website, CRO/CRS, GDS):
Forecast-dependent Algorithms
update these assortments based on remaining inventory,relative to the forecasted distribution of customers still to come (thisdepends on the number of days until check-in);
Forecast-independent Algorithms
optimize these assortments for the worst case,dynamically hedging against the worst case as inventory is sold;
Hybrid Algorithms
follow the forecast-dependent algorithm during each time step,however, if the decision prescribed is suboptimal in the worst-casebeyond a certain threshold,then it uses the decision of the forecast-independent algorithm instead.
David Simchi-Levi (MIT) Online Resource Allocation 19 / 22
Online Resource Allocation Simulations
Algorithms Compared
Algorithm decides which room/fare combinations should be offered todifferent customers from different channels (website, CRO/CRS, GDS):
Forecast-dependent Algorithms
update these assortments based on remaining inventory,relative to the forecasted distribution of customers still to come (thisdepends on the number of days until check-in);
Forecast-independent Algorithms
optimize these assortments for the worst case,dynamically hedging against the worst case as inventory is sold;
Hybrid Algorithms
follow the forecast-dependent algorithm during each time step,however, if the decision prescribed is suboptimal in the worst-casebeyond a certain threshold,then it uses the decision of the forecast-independent algorithm instead.
David Simchi-Levi (MIT) Online Resource Allocation 19 / 22
Online Resource Allocation Simulations
Algorithms Compared
Algorithm decides which room/fare combinations should be offered todifferent customers from different channels (website, CRO/CRS, GDS):
Forecast-dependent Algorithms
update these assortments based on remaining inventory,relative to the forecasted distribution of customers still to come (thisdepends on the number of days until check-in);
Forecast-independent Algorithms
optimize these assortments for the worst case,dynamically hedging against the worst case as inventory is sold;
Hybrid Algorithms
follow the forecast-dependent algorithm during each time step,however, if the decision prescribed is suboptimal in the worst-casebeyond a certain threshold,then it uses the decision of the forecast-independent algorithm instead.
David Simchi-Levi (MIT) Online Resource Allocation 19 / 22
Online Resource Allocation Simulations
Algorithms Compared
Algorithm decides which room/fare combinations should be offered todifferent customers from different channels (website, CRO/CRS, GDS):
Forecast-dependent Algorithms
update these assortments based on remaining inventory,relative to the forecasted distribution of customers still to come (thisdepends on the number of days until check-in);
Forecast-independent Algorithms
optimize these assortments for the worst case,dynamically hedging against the worst case as inventory is sold;
Hybrid Algorithms
follow the forecast-dependent algorithm during each time step,however, if the decision prescribed is suboptimal in the worst-casebeyond a certain threshold,then it uses the decision of the forecast-independent algorithm instead.
David Simchi-Levi (MIT) Online Resource Allocation 19 / 22
Online Resource Allocation Simulations
Results
—Greater Fare Differentiation
David Simchi-Levi (MIT) Online Resource Allocation 20 / 22
Online Resource Allocation Simulations
Results
—Greater Fare Differentiation
our algorithm extracts alarge fraction of optimumon all instances
the Hybrid of our algorithmand LP Resolving performseven better
the forecast-dependentalgorithms exhibit a lotmore variance
we also tested additionalforecast-independentalgorithms
David Simchi-Levi (MIT) Online Resource Allocation 20 / 22
Online Resource Allocation Simulations
Results
—Greater Fare Differentiation
our algorithm extracts alarge fraction of optimumon all instances
the Hybrid of our algorithmand LP Resolving performseven better
the forecast-dependentalgorithms exhibit a lotmore variance
we also tested additionalforecast-independentalgorithms
David Simchi-Levi (MIT) Online Resource Allocation 20 / 22
Online Resource Allocation Simulations
Results
—Greater Fare Differentiation
our algorithm extracts alarge fraction of optimumon all instances
the Hybrid of our algorithmand LP Resolving performseven better
the forecast-dependentalgorithms exhibit a lotmore variance
we also tested additionalforecast-independentalgorithms
David Simchi-Levi (MIT) Online Resource Allocation 20 / 22
Online Resource Allocation Simulations
Results
—Greater Fare Differentiation
our algorithm extracts alarge fraction of optimumon all instances
the Hybrid of our algorithmand LP Resolving performseven better
the forecast-dependentalgorithms exhibit a lotmore variance
we also tested additionalforecast-independentalgorithms
David Simchi-Levi (MIT) Online Resource Allocation 20 / 22
Online Resource Allocation Simulations
Results
—Greater Fare Differentiation
our algorithm extracts alarge fraction of optimumon all instances
the Hybrid of our algorithmand LP Resolving performseven better
the forecast-dependentalgorithms exhibit a lotmore variance
we also tested additionalforecast-independentalgorithms
David Simchi-Levi (MIT) Online Resource Allocation 20 / 22
Online Resource Allocation Simulations
Results—Greater Fare Differentiation
David Simchi-Levi (MIT) Online Resource Allocation 20 / 22
Online Resource Allocation Simulations
Results—Greater Fare Differentiation
our algorithm performsbetter than the Hybrid inthis situation
David Simchi-Levi (MIT) Online Resource Allocation 20 / 22
Online Resource Allocation Simulations
Results—Greater Fare Differentiation
our algorithm performsbetter than the Hybrid inthis situation
David Simchi-Levi (MIT) Online Resource Allocation 20 / 22
Conclusion
Summary
We derive the “worst-case-optimal” function for the value of inventory, whenthere are both multiple items and multiple prices for each item
We extend the applicability of the worst-case approach in revenuemanagement, providing a forecast-independent benchmark which can alwaysbe referenced while making decisions
Complements to Competitive Ratio Analysis:
Heuristics when the stochastic process generating future arrivals is given(Chan/Farias ’09, Ciocan/Farias ’12, Chen/Farias ’13, Jasin/Kumar ’12,Gallego et al. ’15)
Improved algorithms/bounds assuming the arrivals appear in a random order(Kesselheim et al. ’13)
Analyze difference instead of ratio (Reiman/Wang ’08,Badanidiyuru/Kleinberg/Slivkins ’13, Ferreira/S.-L./Wang ’16,Cheung/S.-L. ’16)
David Simchi-Levi (MIT) Online Resource Allocation 21 / 22
Conclusion
Summary
We derive the “worst-case-optimal” function for the value of inventory, whenthere are both multiple items and multiple prices for each item
We extend the applicability of the worst-case approach in revenuemanagement, providing a forecast-independent benchmark which can alwaysbe referenced while making decisions
Complements to Competitive Ratio Analysis:
Heuristics when the stochastic process generating future arrivals is given(Chan/Farias ’09, Ciocan/Farias ’12, Chen/Farias ’13, Jasin/Kumar ’12,Gallego et al. ’15)
Improved algorithms/bounds assuming the arrivals appear in a random order(Kesselheim et al. ’13)
Analyze difference instead of ratio (Reiman/Wang ’08,Badanidiyuru/Kleinberg/Slivkins ’13, Ferreira/S.-L./Wang ’16,Cheung/S.-L. ’16)
David Simchi-Levi (MIT) Online Resource Allocation 21 / 22
Conclusion
Summary
We derive the “worst-case-optimal” function for the value of inventory, whenthere are both multiple items and multiple prices for each item
We extend the applicability of the worst-case approach in revenuemanagement, providing a forecast-independent benchmark which can alwaysbe referenced while making decisions
Complements to Competitive Ratio Analysis:
Heuristics when the stochastic process generating future arrivals is given(Chan/Farias ’09, Ciocan/Farias ’12, Chen/Farias ’13, Jasin/Kumar ’12,Gallego et al. ’15)
Improved algorithms/bounds assuming the arrivals appear in a random order(Kesselheim et al. ’13)
Analyze difference instead of ratio (Reiman/Wang ’08,Badanidiyuru/Kleinberg/Slivkins ’13, Ferreira/S.-L./Wang ’16,Cheung/S.-L. ’16)
David Simchi-Levi (MIT) Online Resource Allocation 21 / 22
Conclusion
Summary
We derive the “worst-case-optimal” function for the value of inventory, whenthere are both multiple items and multiple prices for each item
We extend the applicability of the worst-case approach in revenuemanagement, providing a forecast-independent benchmark which can alwaysbe referenced while making decisions
Complements to Competitive Ratio Analysis:
Heuristics when the stochastic process generating future arrivals is given(Chan/Farias ’09, Ciocan/Farias ’12, Chen/Farias ’13, Jasin/Kumar ’12,Gallego et al. ’15)
Improved algorithms/bounds assuming the arrivals appear in a random order(Kesselheim et al. ’13)
Analyze difference instead of ratio (Reiman/Wang ’08,Badanidiyuru/Kleinberg/Slivkins ’13, Ferreira/S.-L./Wang ’16,Cheung/S.-L. ’16)
David Simchi-Levi (MIT) Online Resource Allocation 21 / 22
Conclusion
Thanks!
David Simchi-Levi: dslevi@mit.edu
Papers
1 Tight Weight-dependent Competitive Ratios for OnlineEdge-weighted Matching, with Application to Revenue Management,with Will Ma (available on SSRN, 2017)
2 Dynamic Recommendation at Checkout under Inventory Constraint,with Xi Chen, Will Ma, and Linwei Xin (available on SSRN, 2016)
David Simchi-Levi (MIT) Online Resource Allocation 22 / 22