A Branch-and-Price Approach for a Ship Routing Problem with - - PowerPoint PPT Presentation

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A Branch-and-Price Approach for a Ship Routing Problem with Multiple Products and Inventory Constraints

Rutger de Mare1 Dennis Huisman2,3

• 1. ORTEC, Gouda
• 2. Econometric Institute and ECOPT, Erasmus Univ. Rotterdam
• 3. Dept. of Logistics, Netherlands Railways

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Outline

• Practical Application
• Problem description
• Mathematical Formulation
• Column generation algorithm

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Practical Application

Oil Company

• 1 refinery (no inventory constraints)
• ±12 depots (one per harbor)
• ± 8 grades
• ± 7 ships
• compartments (differ in size, 3-6)

40 30

15

10 10

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40 30

15

10 10

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Practical Application

• There is a schedule
• Question: Can we adjust the schedule due

to changes in the data?

• We construct an “optimal” schedule from

scratch

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Problem Description

• Ships

– Compartments with capacities – Travel times for each possible trip – Fixed (un)loading times – Wait at harbor is possible

• Harbor

– Inventory constraints

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Inventory levels

Hard Minimal Inventory Level (Unpumpable Level) Soft Maximal Inventory Level (Alarm Level +) Soft Minimal Inventory Level (Alarm Level -) Hard Maximal Inventory Level (tank top/capacity)

Upper Alarm Zone Preferred Inventory Zone Lower Alarm Zone Unpumpables

Actual Inventory Level

Ullage

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Inventory levels

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Notation (Christiansen, 1998)

• (i,m): harbor i, number of arrival m
• (i,m,j,n): traveling from (i,m) to (j,n)
• V: set of ships
• H: set of harbors
• G: set of grades
• C: set of compartments

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Decision Variables

• λcr: compartment c travels schedule r
• θigs: harbor i follows for grade g

sequence s

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Mathematical Formulation (1)

∑∑ ∑ ∑∑

∈ ∈ ∈ ∈ ∈

+

S ig c

H i G g S s igs HG igs C c R r cr C cr

C C θ λ min

Sailing cost route r (only for c= 1) Penalties for violating alarm levels

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Mathematical Formulation (2)

g m i Q Q g m i Y B

S s igs HG imgs cr R r C imcgr C c S s igs imgs V v r c R r r imc

c ig v c v

, , , , 1

* *

∀ = − ∀ = +

∑ ∑ ∑ ∑ ∑∑

∈ ∈ ∈ ∈ ∈ ∈

θ λ θ λ

1 if (i,m) is in schedule r to serve g 0 if (i,m) is in sequence s to (un)load g Quantity of g (un)loaded at (i,m) in sequence s Quantity of g (un)loaded at (i.m) in schedule r

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Mathematical Formulation (3)

c m i v T T g m i T T

cr R r C imcr r c R r C r imc S s igs HG imgs V v r c R r C r imc

c v v c v ig v v c v

, , , , ,

* * * * * *

∀ = − ∀ = −

∑ ∑ ∑ ∑ ∑

∈ ∈ ∈ ∈ ∈

λ λ θ λ

Time (un)loading starts at (i,m) in schedule r, first compartment Time (un)loading starts at (i,m) in sequence s Time (un)loading starts at (i,m) in schedule r, first compartment Time (un)loading starts at (i,m) in schedule r, compartment c

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Mathematical Formulation (3)

) , , , ( , } 1 , { , , , , 1 1 n j m i c X s g i r c g i c

c ig c

R r cr imjncr igs cr S s igs R r cr

∀ ∈ ∀ ≥ ∀ ≥ ∀ = ∀ =

∑ ∑ ∑

∈ ∈ ∈

λ θ λ θ λ

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Branch-and-Price Framework

Add initial columns to RMP Solve LP relaxation of the RMP Solve HGP for each HG Solve CP for each C Find integer solution RMP Add new columns to RMP Find good columns for each Harbour-Grade Find good columns for each Compartment Good columns found? Yes No

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Compartment Sub problem

• Node: harbor arrival plus quantity per grade
• Reduced cost per arc:

– Fixed – Time dependent

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Compartment Sub problem

• Shortest path from s to any other node (taking

into account time aspect)

• Can be solved with Dynamic Programming

(Christiansen, 1998/9)

s

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Branching

• Arcs in the network
• Aggressive 1-branch

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Implementation Issues

• Pricing problem: exact or heuristically?
• Difficult to get a feasible solution of the

model: relax and penalize constraints?

• Cycles in the compartment sub problem

(generates redundant columns)

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Results

• Will follow

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Questions?