Robust Repositioning in Large-scale Networks

Robust Empty Reposi.oning in Large-‐Scale Freight Consolida.on

Networks Alan Erera1, Antonio Carbajal1,

Mar.n Savelsbergh2 1 School of Industrial and Systems Engineering, Georgia Tech 2 University of Newcastle, Australia

Odysseus 2012

What to remember

1.  Robust models for empty mobile resource management pragma.c and effec.ve

2.  Empty resource hubs useful for very large-‐scale reposi.oning networks

3.  Rolling-‐horizon deployments of two-‐stage robust op.miza.on models should u.lize:

–  short robust horizons –  rolling robust constraints

Customer Satellite terminal Hub terminal

destination

origin

Consolida.on networks

Satellite terminal Hub terminal

Consolida.on networks

+5

-3

-5

-1

-1

+1 +4 +1

-3

+1

-2

+3

+3 +3

0

0 -6

-4

+1 +1

+2

+4

-3 +1 -1

+2

Net weekly surplus of empty trailers

Dynamic trailer reposi.oning

•  Large-‐scale terminal network – 250+ satellites and hubs

•  Dynamics – Several decision epochs daily – Today’s projected demand for trailers accurate – Tomorrow’s (and beyond) significantly uncertain

•  Goal – Best empty reposi.oning plan each epoch

Modeling approaches

•  Determinis.c rolling-‐horizon network flow LP – Assume that trailer demands tomorrow (and beyond) behave as expected

•  Stochas.c models – Minimize expected costs given probabilis.c model of demand

– Powell (87), Frantzeskakis and Powell (90), Cheung and Powell (96), Godfrey and Powell (02a, 02b)

– Crainic (93), Di Francesco, et al. (09)

Modeling approaches

•  Robust op.miza.on models – Bertsimas and Sim (03), Atamturk and Zhang (07) – Morales (06), Erera et. al. (09)

•  Two-‐stage model •  Explicit focus on future feasibility •  Minimize cost of planned movements such that a feasible set of recovery movements exists for each non-‐extreme scenario

Two-‐stage robust reposi.oning

t=1 t=0 t=2 t=3 t=4 t=5 A

B

C

D

E

First stage decisions


t=1 t=0 t=2 t=3 t=4 t=5 A

B

C

D

E

First stage net supply bi

Initial trailers

+2

+6

Known and expected future loaded moves

-2

+2

-1

+1


t=1 t=0 t=2 t=3 t=4 t=5 A

B

C

D

E

Second stage “recovery” decisions


t=1 t=0 t=2 t=3 t=4 t=5 A

B

C

D

E

Second stage uncertain demand

Intervals on future loaded moves

[0, 2]

[�a, �a]

•  First stage network flow

•  Second stage “recovery flow” for each scenario


�

a∈δ+(i)

xa −�

a∈δ−(i)

xa = bi ∀ i ∈ N

min�

a

caxa

xa + wa(ω) ≥ 0

∀ a ∈ A

∀ a ∈ A

xa ≥ 0 and integer

�

a∈δ+(i)

wa(ω)−�

a∈δ−(i)

wa(ω) = bi(ω)− bi ∀ i ∈ N

C

B

A

•  Key result: Existence of Recovery Flow


�

a∈δ+(U)∩I

xa ≥ ν(U) ∀ U is inbound-closed

t=1 t=0 t=2 t=3 t=4 t=5

C

B

A

•  Inbound closed set – node set with no incoming recovery transporta.on arcs


t=1 t=0 t=2 t=3 t=4 t=5

C

B

A

•  Inbound closed set –  this example not inbound-‐closed


t=1 t=0 t=2 t=3 t=4 t=5

C

B

A

•  Worst-‐case vulnerability of inbound-‐closed set


t=1 t=0 t=2 t=3 t=4 t=5

�

a∈δ+(U)∩I

xa ≥ ν(U) ∀ U is inbound-closed

ν(U) =�

a∈δ+(U)

�a −�

a∈δ−(U)

�a +�

i∈U

bi

[0, 2]

[1, 5]

Challenges

(1) Smart recovery network –  Low-‐cost moves (since costs not modeled) –  Opera.onally simple

(2) Appropriate use of two-‐stage model –  Controlling conserva.sm pragma.cally –  Special considera.ons for rolling horizon

implementa.on –  Solvable (but very large scale) MIPs

Smart recovery network

Region 1

Region 2

Region 3

Empty hubs

Controlling conserva.sm

•  Exclude extreme scenarios – Narrow the width of intervals – Limit to k the number of uncertain quan..es that may simultaneously take on an extreme quan.ty

•  Challenges for large .me-‐expanded networks – Very large numbers of inbound-‐closed sets and associated robust constraints:

– Difficult to judge in advance which robust constraints will be ac.ve

[�a, �a]

> O(τnS )

C

B

A

•  Bounded vulnerability of inbound-‐closed set


t=1 t=0 t=2 t=3 t=4 t=5

[0, 2]

[1, 5]

maxz

�

a∈δ+(U)

(�a − �a)za +�

a∈δ−(U)

(�a − �a)za |�

za = k

Appropriate use of two-‐stage model Terminal limit

A

B

C

D

known future

–  inbound-‐closed sets with L+1 terminals or fewer

Appropriate use of two-‐stage model Robust horizon

A

B

C

D

known future

robust horizon

–  inbound-‐closed sets include no nodes aher RH

Appropriate use of two-‐stage model Rolling-horizon robust constraints

A

B

C

D

known future

–  add constraints now for future horizon rolls •  assume that demand intervals do not change

robust horizon

Tes.ng the ideas

•  Givens – Historical data from a na.onal consolida.on trucking carrier

– Loaded moves involve 264 terminals – Reposi.oning moves (truck and rail) – 10 empty hubs – At most 4 daily dispatch .mes per terminal – Wide forecast intervals on loaded demands (+/-‐ 50% of actual)

Tes.ng the ideas

•  Horizons – 14 weeks of data – Planning horizon of 7 days for each model

•  Network size for 7-‐day planning horizon – 5,000 .me-‐space nodes – 300,000 arcs

•  Primarily reposi.oning arcs •  Limited connec.ons

Tes.ng the ideas

•  Simulate – Assume today’s loaded demands known – Solve model, implement today’s decisions

•  Assume trailer deficits covered by an outsourced trailer

– Draw realiza.on of tomorrow’s demands – Repeat

Results Figure 18: Unmet demands on a given day - Scenario 2

Figure 19: Cumulative unmet demands - Scenario 2

104

Results Figure 20: Execution costs on a given day - Scenario 2

Figure 21: Cumulative execution costs - Scenario 2

105

Short planning horizons

Figure 22: Cumulative unmet demands with di!erent planning horizons - Scenario 1

106

Next steps

•  Refinements – More reasonable model of true uncertainty in demand

– Understand sources of cost escala.on, including if and where excessive conserva.sm introduced

•  Empty hub selec.on •  Fleet size versus reposi.oning cost for robust plans

What to remember

1.  Robust models for empty mobile resource management pragma.c and effec.ve

2.  Empty resource hubs useful for very large-‐scale reposi.oning networks

3.  Rolling-‐horizon deployments of two-‐stage robust op.miza.on models should u.lize:

–  short robust horizons –  rolling robust constraints

Robust Repositioning in Large-scale Networks

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