Dynamic Type Matching
Ming Hu
with Yun Zhou
Rotman School of Management, University of Toronto
May 16, 2016Symposium on the Sharing Economy
University of Minnesota
1
Emerging Applications
Car Hailing
When is the greedy matching optimal?
2
Emerging Applications
Car Hailing
When is the greedy matching optimal?
2
Model Features
m
n
Centralized matching by a platform
Inter-temporal uncertainty
3
Model Features
m
n
Centralized matching by a platformInter-temporal uncertainty
3
Emerging Applications: e-Commerce
Amazon: inventory commingling program
4
Emerging Applications: e-CommerceAmazon: inventory commingling program
m
n
Supply owned by Amazon or third party merchants
Online demandAmazon
Types: geographic locations (horizontally differentiated)“idiosyncratic” preference
5
Emerging Applications: e-CommerceAmazon: inventory commingling program
m
n
Supply owned by Amazon or third party merchants
Online demandAmazon
Types: geographic locations (horizontally differentiated)“idiosyncratic” preference
5
Emerging Applications: Organ TransplantKidney allocation Zenios et al. 2000, Su and Zenios 2005
Liver allocation Akan et al. 2014
m
n
Harvested organs Patients in need of transplantation
United Network for Organ Sharing
(UNOS)
Types: health status (vertically differentiated)“uniform” preference
blood/tissue (horizontally differentiated)
6
Emerging Applications: Organ TransplantKidney allocation Zenios et al. 2000, Su and Zenios 2005
Liver allocation Akan et al. 2014
m
n
Harvested organs Patients in need of transplantation
United Network for Organ Sharing
(UNOS)
Types: health status (vertically differentiated)“uniform” preference
blood/tissue (horizontally differentiated)
6
Emerging Applications: Organ TransplantKidney allocation Zenios et al. 2000, Su and Zenios 2005
Liver allocation Akan et al. 2014
m
n
Harvested organs Patients in need of transplantation
United Network for Organ Sharing
(UNOS)
Types: health status (vertically differentiated)“uniform” preference
blood/tissue (horizontally differentiated)
6
The Model
An intermediary firm matches:Demand types D = {1, 2, . . . ,n}, indexed by iSupplier types S = {1, 2, . . . ,m}, indexed by j
Random arrivals in a period with arbitrary distributionsDemand D = (D1, . . . ,Dn)Supply S = (S1, . . . ,Sm)
Decisions, revenue and costsDecisions: matching quantity qij (Q)Unit reward rij (R)Unit holding cost c and h for unmatched demand andsupply, resp.
Unmatched demand and supply carry over to the nextperiod with rates α and β, resp.
7
The Model
An intermediary firm matches:Demand types D = {1, 2, . . . ,n}, indexed by iSupplier types S = {1, 2, . . . ,m}, indexed by j
Random arrivals in a period with arbitrary distributionsDemand D = (D1, . . . ,Dn)Supply S = (S1, . . . ,Sm)
Decisions, revenue and costsDecisions: matching quantity qij (Q)Unit reward rij (R)Unit holding cost c and h for unmatched demand andsupply, resp.
Unmatched demand and supply carry over to the nextperiod with rates α and β, resp.
7
The Model
An intermediary firm matches:Demand types D = {1, 2, . . . ,n}, indexed by iSupplier types S = {1, 2, . . . ,m}, indexed by j
Random arrivals in a period with arbitrary distributionsDemand D = (D1, . . . ,Dn)Supply S = (S1, . . . ,Sm)
Decisions, revenue and costsDecisions: matching quantity qij (Q)Unit reward rij (R)Unit holding cost c and h for unmatched demand andsupply, resp.
Unmatched demand and supply carry over to the nextperiod with rates α and β, resp.
7
Stochastic Dynamic Program
State variables (x,y), after arrival before matchingx: demand levelsy: supply levels
Post matching levels (u,v), after matchingu = x− 1mQT and v = y− 1nQ
Optimal recursion
Vt(x,y) = maxQ∈{Q≥0|u≥0,v≥0}
Ht(Q, x,y),
Ht(Q, x,y) = R ◦Q− c1nuT − h1mvT
+γEVt+1(αu + D, βv + S)
VT+1(x,y) = 0
8
Classic Settings
Capacity management with upgrading Shumsky and Zhang (2009), Yu et
al. (2015)
Centralized matching market e.g., medical residence
Inventory rationingAssignment/transportation problemType mating Duenyas et al. (1997)
9
Overview of Results
◦ Build a general dynamic matching framework◦ Derive distribution-free structural results
General priority properties under modified Mongecondition
Sufficient, and robustly necessaryVertically differentiated types
Quality-based priorityHorizontally differentiated types
Distance-based priority
Bounds and heuristics
10
Overview of Results
◦ Build a general dynamic matching framework◦ Derive distribution-free structural results
General priority properties under modified Mongecondition
Sufficient, and robustly necessary
Vertically differentiated typesQuality-based priority
Horizontally differentiated typesDistance-based priority
Bounds and heuristics
10
Overview of Results
◦ Build a general dynamic matching framework◦ Derive distribution-free structural results
General priority properties under modified Mongecondition
Sufficient, and robustly necessaryVertically differentiated types
Quality-based priorityHorizontally differentiated types
Distance-based priority
Bounds and heuristics
10
When to Prioritize One Pair over Another?
j
j’’
i
i’
1
2
4
Greedy matching is not optimal!
11
When to Prioritize One Pair over Another?
j
j’’
i
i’
1
2
4
Greedy matching is not optimal!11
A Relation of Neighboring Arcs
Definition (Modified Monge Condition)We say (i, j) � (i, j′), if
(i) rij ≥ rij′
(ii)rij + ri′j′ ≥ rij′ + ri′j (D)
for all i′ ∈ D.
+
j
j’
j
j’
≥
i
i’ ‘
i
i’ ‘
12
A Partial Order between Arcs
Definition (Arcs without common nodes)For i 6= i′ and j 6= j′, we say (i, j) � (i′, j′) if there exists a decreasingsequence of neighboring arcs connecting the two.
j
j’
i
i’
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Greedy Matching for a Perfect Pair
Theorem (When Greedy Matching is Optimal)If (i, j) � (i, j′) for all j′ ∈ S and (i, j) � (i′, j) for all i′ ∈ D,
q∗ij = min¶
xi, yj©.
14
Priority Hierarchy
TheoremThere exists an optimal decision Q∗ such thatfor any (i, j) � (i′, j′),
(i, j) has a higher priority to be matched over (i′, j′).
The proof generalizes the augmenting path approach to DPWe do not require all neighboring arcs are comparableFor horizontal and vertical cases, all neighboring arcs areindeed comparable
15
Priority Hierarchy
TheoremThere exists an optimal decision Q∗ such thatfor any (i, j) � (i′, j′),
(i, j) has a higher priority to be matched over (i′, j′).
The proof generalizes the augmenting path approach to DP
We do not require all neighboring arcs are comparableFor horizontal and vertical cases, all neighboring arcs areindeed comparable
15
Priority Hierarchy
TheoremThere exists an optimal decision Q∗ such thatfor any (i, j) � (i′, j′),
(i, j) has a higher priority to be matched over (i′, j′).
The proof generalizes the augmenting path approach to DPWe do not require all neighboring arcs are comparable
For horizontal and vertical cases, all neighboring arcs areindeed comparable
15
Priority Hierarchy
TheoremThere exists an optimal decision Q∗ such thatfor any (i, j) � (i′, j′),
(i, j) has a higher priority to be matched over (i′, j′).
The proof generalizes the augmenting path approach to DPWe do not require all neighboring arcs are comparableFor horizontal and vertical cases, all neighboring arcs areindeed comparable
15
Monge Sequence
By Gaspard Monge in 1781
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Monge Sequence
By Gaspard Monge in 1781
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Comparison with Monge Sequence
Monge sequence (1781) Modified Monge conditiona sequence pairs
static, deterministic and balanced dynamic, stochastic and unbalancedtransportation problem transportation problemsufficient and necessary sufficient, and robustly necessary
a greedy algorithm: Our result:(1) priority property (1) priority property
(2) match as much as possible (2) match-down-to policy
18
Comparison with Monge Sequence
Monge sequence (1781) Modified Monge conditiona sequence pairs
static, deterministic and balanced dynamic, stochastic and unbalancedtransportation problem transportation problem
sufficient and necessary sufficient, and robustly necessarya greedy algorithm: Our result:
(1) priority property (1) priority property(2) match as much as possible (2) match-down-to policy
18
Comparison with Monge Sequence
Monge sequence (1781) Modified Monge conditiona sequence pairs
static, deterministic and balanced dynamic, stochastic and unbalancedtransportation problem transportation problemsufficient and necessary sufficient, and robustly necessary
a greedy algorithm: Our result:(1) priority property (1) priority property
(2) match as much as possible (2) match-down-to policy
18
Comparison with Monge Sequence
Monge sequence (1781) Modified Monge conditiona sequence pairs
static, deterministic and balanced dynamic, stochastic and unbalancedtransportation problem transportation problemsufficient and necessary sufficient, and robustly necessary
a greedy algorithm: Our result:(1) priority property (1) priority property
(2) match as much as possible (2) match-down-to policy
18
Vertically Differentiated Types
Decomposable reward:
rij = rdi + rs
j
Centralized medical residency assignment Agarwal (2015)
19
Vertical Model: Optimal Policy
Top-down matching:Line up demand and supply from high to lowMatch up from the top (to some level)
20
Vertical Model: Optimal Policy (Dynamic View)
Type
iDemand
Type
jSupply aij
21
Vertical Model: Optimal Policy (Dynamic View)
Type
iDemand
Type
jSupply aij
21
Vertical Model: Optimal Policy (Dynamic View)
Type
iDemand
Type
jSupply aij
21
Horizontal Model: 2-to-2 Case
n = m = 2rii ≥ max{ri,−i, r−i,i} for {i,−i} = {1, 2}
Horizontally Differentiated Types
perfect pair
perfect pair
imperfect pair
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Horizontal Model: Optimal Policy of 2-to-2 Case
PropositionStep 1. Greedy matching for the perfect pair: Match type idemand with type i supply as much as possible, i = 1, 2Step 2. Match-down-to policy for the imperfect pair: Match typei demand with type −i supply only when ηi ≡ xi − yi > 0 andη−i ≡ x−i − y−i < 0
The remaining quantity of type i demand and type −i supplyafter Step 1: ηi and −η−i, resp.; q∗−i,i = 0The optimal protection level ait(η) ≥ 0 (η ≡ ηi + η−i)
If ηi ≥ η+ + ait(η), then reduce type i demand to η+ + ait(η), type−i supply to η− + ait(η)If ηi < η+ + ait(η), do not match type and set q∗i,−i = 0
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Horizontally Differentiated Types
j
i
rij = f (dij), where dij is the clockwise distance between i and j
24
Logistics with Fixed Routes in the Same Direction
UberPool
25
More Emerging Applications
Load Matching
26
Horizontal Model: Car Pooling
j
j’
i
27
Horizontal Model: Priority by Distance
Theorem (Greedy Match of Perfect Pair)Suppose that type i demand and type j supply are closest to each other.If f is nonincreasing and convex, q∗ij = min{xi, yj}.
Theorem (Distance-Based Priority of Imperfect Pairs)If f is nonincreasing and linear, for any given type i demand,
the closer its distance to a type j supply, the higher the priority inmatching the demand-supply pair (i, j);Along the priority hierarchy, the optimal matching is amatch-down-to policy.
28
Deterministic Heuristic for the General Problem
The deterministic model provides an upper bound for thestochastic modelSuccessively resolving the deterministic model isasymptotically optimal for the stochastic model
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Extensions
Time-dependent parametersType-dependent parameters, e.g., c1 ≥ · · · ≥ cn,h1 ≥ · · · ≥ hm
Random abandonmentsForbidden arcsForced maxing-outA continuum of typesInfinite horizon with discounted or long-run averagepayoffOther forms of rij
rij = min¶
rdi , r
sj
©rij = max
¶rd
i , rsj
©Endogenized supply process and pricing
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Summary
m
n
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A New Form of Matching Supply with Demand
Operations Management manages the process of matchingsupply with demandFoundations
Inventory management (e.g., base-stock policy)Revenue management (e.g., protection level)
New form of business process
Matching in a two-sided market with crowdsourced supply(sharing economy)
32
Summary: Distribution-Free Structural Results
General priority properties under modified Mongecondition
Sufficient, and robustly necessaryVertically differentiated types
Quality-based priority+match-down-to policyHorizontally differentiated types
Distance-based priority+match-down-to policy
Bounds and heuristics
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Q & A
Thank you!
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