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

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Emerging Applications

Car Hailing

When is the greedy matching optimal?

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Emerging Applications

Car Hailing

When is the greedy matching optimal?

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Model Features

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n

Centralized matching by a platform

Inter-temporal uncertainty

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Model Features

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Centralized matching by a platformInter-temporal uncertainty

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Emerging Applications: e-Commerce

Amazon: inventory commingling program

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Emerging Applications: e-CommerceAmazon: inventory commingling program

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Supply owned by Amazon or third party merchants

Online demandAmazon

Types: geographic locations (horizontally differentiated)“idiosyncratic” preference

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Emerging Applications: e-CommerceAmazon: inventory commingling program

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Supply owned by Amazon or third party merchants

Online demandAmazon

Types: geographic locations (horizontally differentiated)“idiosyncratic” preference

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Emerging Applications: Organ TransplantKidney allocation Zenios et al. 2000, Su and Zenios 2005

Liver allocation Akan et al. 2014

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Harvested organs Patients in need of transplantation

United Network for Organ Sharing

(UNOS)

Types: health status (vertically differentiated)“uniform” preference

blood/tissue (horizontally differentiated)

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Emerging Applications: Organ TransplantKidney allocation Zenios et al. 2000, Su and Zenios 2005

Liver allocation Akan et al. 2014

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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)

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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)

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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.

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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.

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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.

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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

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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)

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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

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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

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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

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When to Prioritize One Pair over Another?

j

j’’

i

i’

1

2

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Greedy matching is not optimal!

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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’ ‘  

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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©.

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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

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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

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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

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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

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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

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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 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

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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

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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

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Vertically Differentiated Types

Decomposable reward:

rij = rdi + rs

j

Centralized medical residency assignment Agarwal (2015)

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Vertical Model: Optimal Policy

Top-down matching:Line up demand and supply from high to lowMatch up from the top (to some level)

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Vertical Model: Optimal Policy (Dynamic View)

Type

iDemand

Type

jSupply aij

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Vertical Model: Optimal Policy (Dynamic View)

Type

iDemand

Type

jSupply aij

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Vertical Model: Optimal Policy (Dynamic View)

Type

iDemand

Type

jSupply aij

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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

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Logistics with Fixed Routes in the Same Direction

UberPool

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More Emerging Applications

Load Matching

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Horizontal Model: Car Pooling

j

j’

i

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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.

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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

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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)

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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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