Hub-and-Spoke Liner Shipping Network Design

with Demand Uncertainty

Speaker: Wang Tingsong

03/Jun/2013

2

Introduction

Literature Review

Problem Statement

Model Development

Solution Algorithm

Numerical Example

Conclusion

Outline

3

Introduction

Liner Shipping

Regular service

Fixed schedule

Fixed route

A liner container shipping company typically operates a fleet

of heterogeneous ships on a set of liner trade routes at a

regular schedule in order to pick up and delivery cargoes for

shippers

Liner Operator

4

Introduction (cont’d)

Hub-and-Spoke (H&S)

Main line: hub ports allocation

Feeder line: feeder ports allocation

Fleet deployment: ship assignment to main line and feeder line

to provide pickup and delivery liner shipping service for shipper

Uncontrollable and unpredicted factors

Container shipment demand of a port pair

Uncertainty

5

Introduction (cont’d)

Objectives

How to design main line and feeder line?

How to deal with uncertainty of container shipment demand?

Model and solution algorithm

Evaluation by implementing a numerical example

6/26

Literature Review

Representative literature Author (s) Year Model Problem Statement

Maurao et al 2001 IP Assign ships with hub and spoke constraints

Aversa et al 2005 MIP Locate a hub port in the East coast of South America

Imai et al 2006 IP Analyze mega ship viability and propose a model to deploy mega ship onto H&S shipping network

Konings 2006 Analyze the cost, service and

geographical characteristics of H&S networks

Hsu and Hsieh 2007 MIP A two-objective model to design a shipping network by minimizing shipping cost and inventory cost

Takano and Arai 2008 MIP Containerized cargo transport problem with H&S network

Imai et al 2009 MIP Compare multi-ports liner shipping network and H&S

network

7

Problem Statement

Given a set of ports which can be divided into two groups: hub

and spoke, and given a set of ships which can be divided into

two groups: mega ship and feeder ship, it intends to make an

optimal hub-and-spoke shipping network design to minimize the

total cost over a given short-term planning horizon while

satisfies a predetermined service-level.

How to design main line?

How to allocate the feeder ships to each feeder line?

8

Problem Statement (cont’d)

Assumption

The planning horizon is 6 months

Containers are homogeneous, all refer to TEUs

The liner operators are required to maintain a weekly shipping

frequency only on main line

The time of ships travelling between any two pots are known and

fixed

No direct link between any two spokes

Containers can be transferred via at most two hub ports

Model Development

9

A typical hub-and-spoke liner shipping network

hub main line

spoke

feeder line

mega ships

feeder ships

10

Model Development(cont’d)

Decision Variables

11

Model Development (cont’d)

Objective Function

,

intrinsic cost

,

transport

min

Feeder Mega

Feeder Mega

Feeder Feeder Mega Megakhs k vhj v

s h h jk v

Feeder Feeder Mega Megakhs khs vhj vhj

s h h jk v

Z c z c z

c x c x

∗ ∗

∈ ∈ ∈∈ ∈

∈ ∈ ∈∈ ∈

= + +

+

∑∑ ∑ ∑ ∑

∑∑ ∑ ∑ ∑

S H HV V

S H HV V

ing cost

transshiping cost

Feeder Mega

trans Feeder trans Megah khs h vhs

s h s hk v

c y c y∈ ∈ ∈ ∈∈ ∈

+

+∑∑ ∑ ∑∑ ∑

S H S HV V

12

Model Development (cont’d)

Constraints

Ship constraints

Chance constraints

capacity constraints , , ,Feeder Feeder Feeder Feeder

khs k kx z C k h s≤ ∀ ∈ ∈ ∈V H S, , ,Mega Mega Mega Mega

vhj v vx z C v h j≤ ∀ ∈ ∈V H

, , ,Feeder Feeder Feederkhs khsy x k h s≤ ∀ ∈ ∈ ∈V H S

, , ,Mega Mega Megavhs vhj

sy x v h j

∈

≤ ∀ ∈ ∀ ∈∑S

V H

( ) maxˆ δ, , ,Feeder Feeder Feeder Feeder Feederk khs kh k khsz t t t k h s+ ≤ ∀ ∈ ∈ ∈V H S

( ) max

,

ˆ δ,Mega Mega Mega Mega Megav vhj vj vhj v

h jz t t t v

∈

+ ≤ ∀ ∈∑H

V

δ1, ,Feeder Feederkhs

sh k

∈

≤ ∀ ∈ ∈∑S

H V

δ1, ,Feeder

Feederkhs

k

h s∈

≥ ∀ ∈ ∈∑V

H S

,

δδ7, , ,Mega

Mega Mega Mega Megavhj vhj vhj

h jv

t v h j∈∈

≥ ∀ ∈ ∀ ∈∑ ∑HV

V H

Prξ, , 1Feeder

Feeder Feederk k hs

k

z C h s∈

≥ ∀ ∈ ∈ ≥ −α

∑V

H S

Prξ, , 1Mega

Mega Megav v hj

v

z C h j∈

≥ ∀ ∈ ≥ −α

∑V

H

Transshipment constraints

Time constraints

13

Solution Algorithm

Difficulties of Solving the Model

Nonlinear

Probabilistic form

( ) max

,

ˆ δ,Mega Mega Mega Mega Megav vhj vj vhj v

h jz t t t v

∈

+ ≤ ∀ ∈∑H

V

Prξ, , 1Feeder

Feeder Feederk k hs

k

z C h s∈

≥ ∀ ∈ ∈ ≥ −α

∑V

H S

Prξ, , 1Mega

Mega Megav v hj

v

z C h j∈

≥ ∀ ∈ ≥ −α

∑V

H

14

Solution Algorithm (cont’d)

Linearization It is noticed that if decision variables is removed, then

the problem is reduced to a mixed integer linear programming

(MILP) model with chance constraints.

If the mega ship v serves only two ports and does not visit any

other port enroute, it makes the maximum number of trips; if it

serves all ports enroute, it makes the minimum number of

trips

For example, if 2 ≤ ≤ 5, then has 4 values: 2,3,4,5.

For each problem with a corresponding value of , it is a

MILP model with chance constraints.

Megavz

Megavz Mega

vzMegavz

15

Solution Algorithm (cont’d)

Sample Average Approximation (SAA)

Let

( ) ( )( ): Pr , 0Mega ML MegaMegap = ∆ >z zξ

( ),

, maxξFeeder

FL Feeder Feeder FeederFeeder hs k kh s

k

z C∈ ∈

∈

∆ = −

∑zξ

H SV

( ),

, maxξMega

ML Mega Mega MegaMega hj v vh j

v

z C∈

∈

∆ = −

∑zξ

HV

We then define ( ) ( )( ): Pr , 0Feeder FL FeederFeederp = ∆ >z zξ

Rewrite the chance constraints

Prξ, , 1Feeder

Feeder Feederk k hs

k

z C h s∈

≥ ∀ ∈ ∈ ≥ −α

∑V

H S

Prξ, , 1Mega

Mega Megav v hj

v

z C h j∈

≥ ∀ ∈ ≥ −α

∑V

H

( )Feederp ≤ αz

( )Megap ≤ αz

16

Solution Algorithm (cont’d)

SAA(Atlason et al., 2008; Luedtke and Ahmed, 2008)

Let ( ) ( ) ( )( )0,1

,Feeder

NN Feeder FL Feeder n

np N∞

=

= ∆∑z zξ 11

( ) ( ) ( )( )0,1

,Mega

NN Mega FL Mega n

np N∞

=

= ∆∑z zξ 11

( ) ( )0,

1, if 0,:

0, if 0.y

yy∞

>= ≤

11

Then ( )Feederp ≤ αz

( )Megap ≤ αz

( )N Feederp ≤ βz

( )N Megap ≤ βzβ can be different with α

Law of Large Number Theory

17

Solution Algorithm (cont’d)

MILP

{ }β min intrinsic cost transporting cost transshiping costNZ = + +s.t.

Ship constraints + Chance constraints

capacity constraints + Transshipment constraints +

ξξ, 1, , ; ,Feeder

Feeder n Feeder Feeder nn hs k k hs

k

z C n N h s∈

ϑ + ≥ ∀ = ∀ ∈ ∈∑ V

H S

1

NFeedern

nN

=

ϑ ≤ β∑

ξξ, 1, , ; ,Mega

Mega n Mega Mega nn hj v v hj

v

z C n N h j∈

ϑ + ≥ ∀ = ∀ ∈∑ V

H

1

NMegan

nN

=

ϑ ≤ β∑ { }, 0,1 NFeeder Megan nϑ ϑ ∈

( )N Feederp ≤ βz

( )N Megap ≤ βz

equivalent

equivalent

Time constraints +

18

Solution Algorithm (cont’d)

Solution quality(Luedtke and Ahemd, 2008)

Lower bound

Let , is a lower bound of Z with a significance level ε Lβα > LNZβ

Upper bound

Let , is a upper bound of Z with a significance level ε Uβα < UNZβ

19

Numerical Example

20

Results

Sensitivity Analysis of SAA Parameters

Results(cont’d)

21

Shanghai

Hong Kong

Singapore Penang

Port Kelang

Shekou

Xingang Dalian

Qingdao

Results(cont’d)

22

23

Conclusion

A H&S shipping network design problem

Demand uncertainty

Linearization

SAA

24