A Branch-and-Price Method for an Inventory Routing Problem in the - - PowerPoint PPT Presentation

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A Branch-and-Price Method for an Inventory Routing Problem in the LNG Business

Marielle Christiansena Guy Desaulniersb,c, Jacques Desrosiersb,d, , Roar Grønhauga

• 18. June 2008

aNorwegian University of Science and Technology, bGERAD, cEcole Polytechnique de Montreal, dHEC Montreal

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Agenda

• The LNG Inventory Routing Problem (LNG-IRP)
• Column generation

– Decomposition – The master problem – The subproblems – Branch-and-price

• Computational results
• Concluding remarks

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The LNG-IRP

Exploitation & Production Liquefaction & Storage Shipping Regasification & Storage Gas Utilities Residential Electric Utilities Industries

• Maximize supply chain profit – 2-3 months planning horizon
• Decide LNG production and sales levels on day to day basis
• Optimal ship routes and schedules with corresponding optimal

unloading quantities

– The ship is fully loaded when it sails from a pick-up port – A ship can visit several consecutive delivery ports unloading a number of cargo tanks before returning to a pick-up port

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Inventory management

Inventory balance Berth constraints Inventory balance Berth constraints

inventory ( )

i i it

S s S ≤ ≤ sales( )

it it it

Y y Y ≤ ≤

Time t i

S

i

S

it

s Liquefaction plant i Regasification terminal i

production( )

it it it

Y y Y ≤ ≤ inventory ( )

i i it

S s S ≤ ≤

Time t

i

S

i

S

it

s

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LNG Ships

• Heterogeneous fleet
• Each ship: 4-6 cargo tanks
• LNG transported at boiling state (-162oC)

– Boil-off from each cargo tank (Fixed % of tank capacity per day) – Used as fuel for the ship – Some LNG needed in tank to keep it cool

• Each tank should be unloaded once before refilling

– Ships’ cargo tanks should be as close as possible to full or empty to avoid sloshing – Need to leave just enough cargo in tanks to cover the boil-off for the rest

• f the trip to a pickup port

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Example with P-D-D-P and Boil-off

The unloading quantity at node j cannot be decided before the ship returns to a pick-up port. Assume the pick-up port is l2. Unloading quantity of a tank =

i P j D k1 D l1 P k2 l2

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Tank capacity (1 ) ( )

l i

B T T ⋅ − ⋅ −

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Ship paths

• Geographical route P1→D2→D1→P2→D1→P2→D2
• Schedule T1 T2 T3 T4 T5 T6 T7
• Quantity Q1

Q2 Q3 Q4 Q5 Q6 Q7

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Constrained inventory and prod. capacity

• LNG production volume
• Ship arrival time
• Loading quantity
• Berth capacity (number of ships)

Inventory management and routing

• Liq. plant
• Regas. terminal
• LNG

Sufficient amount of LNG available

• LNG sale
• Ship arrival time
• Unloading quantity
• Berth capacity (number of ships)

Ships (capacity, cost structure)

• Routing
• Arrival time
• # of waiting days outside port
• Loading/unloading quantity
• Boil-off

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Decomposition for Col. Gen.

• Master Problem

– Sales and production at port i , – Inventory management at port i, – Port capacity,

• Subproblem for each ship v

– Ship routing and scheduling, – Ship inventory management

• Number of tanks unloaded at

the delivery port,

• Volume loaded/unloaded at the

ports including boil-off,

it

s

it

y

ijvtr vr

X λ

CAP i

N

ivtr vr

L λ

ivtr vr

Q λ

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Master Problem

{ }

0,1,...., , , , ,

MX D ivtr vr v

L W i N v V t T λ ∈ ∀ ∈ ∈ ∈

(1)

max ,

D P v

EVit it OSTit it vr vr i N t T i N t T v V r R

R y C y C λ

∈ ∈ ∈ ∈ ∈ ∈

− −

∑∑ ∑∑ ∑∑

( 1)

0, , ,

v

it i t i it i ivtr vr v V r R

s s I y I Q i N t T λ

− ∈ ∈

− + − = ∀ ∈ ∈

∑∑

, , ,

v

CAP ivtr vr i v V r R

Z N i N t T λ

∈ ∈

≤ ∀ ∈ ∈

∑∑

(2) (3) (4) (5) (6) (7)

, , ,

i i it

S s S i N t T ≤ ≤ ∀ ∈ ∈ , , ,

it it it

Y y Y i N t T ≤ ≤ ∀ ∈ ∈ 1, ,

v

vr r R

v V λ

∈

= ∀ ∈

∑

Dual variables

αit βit

0, , .

vr v

v V r R λ ≥ ∀ ∈ ∈

θv

{ }

0,1 , , , , ,

v

ijvtr vr r R

X i N j N v V t T λ

∈

∈ ∀ ∈ ∈ ∈ ∈

∑

(8) (9)

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Valid ineq. - aggregated berth constr.

• By use of problem characteristics (inventory limits, production and sale limits,

berth constraints, ship capacities, shortest round trip for a ship), we can calculate

the upper and lower limits on the number of visits to a port for all time intervals

Ship visits Time intervals 0 5 10 15 20 25 30 35 40 45

7 6 5 4 3 2 1

Upper limits Lower limits Not redundant limits

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The Subproblems (1:2)

• Heterogeneous fleet → One subproblem for each ship
• Reduced cost for a ship route variable

( )

Max .

vr vr ivtr it i ivtr it v i N t T

C C Z I Q β α θ

∈ ∈

= − − ∑ ∑ − −

Ship route sailing cost Port capacities Loading/unloading quantities Convexity constraint

• Longest path subproblems with side constraints caused by

unloading restrictions in number of tanks and boil-off

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The Subproblems (2:2)

• A node: Feasible combination of time and port

– Unloading in number of cargo tanks at delivery ports

• The boil-off complicates the problem

– Do not know the exact amount of cargo unloaded at the delivery ports before the ships return to a pick-up port – DP where partial paths can only be compared in pick-up nodes

Time 1 2 4 3 5 6 8 7 P1 P2 D1 D2 D3 9 10 12 11 13 14 16 15

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Accelerating strategies in col. gen.

• Greedy Heuristic for solving the subproblems

– Assume full unloading and does not consider boil-off – Post calculate boil-off – Topological sorted acyclic network without any complicating side constraints – When the greedy heuristic stops generating improving columns, switch to the exact DP algorithm

• Remove all berth constraints and add violated once during B&P
• Add several columns between each call to RMP

– Several runs of the greedy heuristic – Manipulate the cost between each run to give incentive to find columns which traverse different arcs

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Branch-and-Price

Depth-first B&B strategy with backtracking for the column generation

Four branching strategies

1. Branch on berth constraints in RMP (and aggregated berth constraints – valid inequalities) 2. Branch on the sum of all ships sailing from a specific port in a given time period (nodes in the subproblem) 3. Branch on the arcs in the subproblem, 4. Branch on deliveries (tanks), { }

0,1

v

ijvtr vr r R

X λ

∈

∈

∑

{ }

0,1,....,

MX ivtr vr v

L W λ ∈

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Computational Results –

based on real world planning problems

15/ 364 576 69 8454/ 36000 43 8724/ 36000 5613 5 /6/60 9 30/ 709 518 44 2348/ 36000 16 13/ 36000 2815 5 /6/45 8 14/ 1 089 1/ 14 0/ 39 859 5 /6/30 7 45/ 435 875 34 114/ 36000 28 223/ 36000 2110 3 /4/60 6 32/ 43 391 65/ 1219 0/ 13625 1213 3 /4/45 5 9/ 1 116 0/ 10 0/ 14 429 3 /4/30 4 22/ 8 283 2/ 338 27 70/ 36000 1144 2 /5/60 3 8/ 516 0/ 9 4/ 973 647 2 /5/45 2 7/ 117 0/ 0 0/ 0 257 2 /5/30 1 Gap 1.MIP/Total (s) Gap 1.MIP/Total (s) Arcs s/p/t Id. #MIPsol/ BB-nodes B-P Path flow

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More results solved by B&P

323 / 5273 29793 255.6 14 36000 10826 3010 3/4/75 13 24 / 384 2437 30.2 12 2889 2 1681 2/5/75 12 135 / 8980 10394 76.5 13 19707 1877 4834 2/4/75 11 81 / 7601 18 333 305.2 29 26527 4 2744 2/3/75 10 11 / 169 1010 43.4 32 1219 65 1213 3/4/45 5 gSP / eSP Sec. RMP Sec. BB Node (1000) # MIP Sol. Total Sec. 1.MIP Sec. Arcs s/p/t Id.

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Concluding Remarks

• New type of problem

– Extension of the maritime inventory routing problem

• Both master problem and subproblems are complicated
• Real sized instances are solved to optimality by col. gen.
• Future research

– Improve B&P by reducing the size of the search tree and the time spent in the master problem – Different decomposition – Developing more valid inequalities – Developing solution methods for extended LNG-IRP’s

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A Branch-and-Price Method for an Inventory Routing Problem in the LNG Business

Marielle Christiansena Guy Desaulniersb,c, Jacques Desrosiersb,d, , Roar Grønhauga

• 18. June 2008

aNorwegian University of Science and Technology, bGERAD, cEcole Polytechnique de Montreal, dHEC Montreal