The plan for empty container reposition

CHAPTER 3 DESCRIPTION OF PROBLEM

3.6 Geographical Regions with the Sea Transportation Network

3.7.2 The plan for empty container reposition

Factors affecting the plan for empty container reposition are summarized below (see Figure 3.12).

1) Container movement: The difference between I/B and O/B containers results in empty container reposition at each port.

2) Safety stock of empty containers: Container carriers store empty containers to meet customer demand and they attempt to minimize safety stock of empty containers at each port. Safety stock affects the amount of R/I empty containers and R/O empty containers.

3) Service routes: All the service routes form a sea transportation network to make channels for empty container reposition.

4) Geographical region: To avoid the occurrence of empty containers occupying slots for a long-distance, thereby costing the containership freight revenue, the sea transportation network is partitioned into several geographical regions. Empty containers are repositioned within a single region.

5) Cost: Handling costs at port are major and indispensable expenditure. The cost of transportation is divided into three kinds: the cost of owned slot, the cost of charted slot, and the cost of inland drayage by truck.

6) Slot on containership: Empty containers are allocated on unsold slot. If the slot is not available, either the plan for repositioning empty containers is suspended or container carrier charter slots are from other carriers.

Figure 3.12 Influence factors of empty container reposition

CHAPTER 4 THE PLAN FOR CONTIANERSHIP SLOT ALLOCATION

This study uses revenue management modeling as a decision-support tool in forming the containership slot allocation plan for a loaded container. The proposed model, which incorporates the expected cost of empty container reposition, was formulated through mathematical programming to maximize operational profit, subject to the constraints of containership capacity, containership deadweight, and container demand (as seen in Figure 4.1). The proposed model uses a Taiwan shipping company as a case study and a strategy has been developed by means of computational analysis.

Figure 4.1 A concept chart for containership slot allocation

4.1 Assumptions

The following assumptions are made in this research：

(1) The freight of various port-pairs was given. Container carriers charge ocean freight and surcharges including currency adjustment factor (CAF), fuel adjustment factor (FAF), terminal handling charge in loading port (L/THC), terminal handling charge in unloading port (D/THC) and document fee from shippers. Occasionally, container carriers offer all-in tariffs (including surcharges). Tariffs are determined based on quantity of containers, container specification, and customer classification. In this study, freight revenue is estimated as average freight revenue and surcharge for each port-pair (Appendix A).

(2) The variable cost of various port-pairs was given. The fixed cost includes port charges, bunker costs, containership cost and administration. As fixed cost is not affected by variations in shipment, only variable cost is factored into the model (see Appendix B).

(3) The maximum and minimum container demands at various port-pairs are given.

Demand uncertainty is a function of market size; competition; and the ability of shipping agencies in seeking cargo (see Appendix C).

(4) Strategic alliances have grown in significance in recent decades, in an effort to increase market coverage, decrease overheads, share the cost of capital equipment and improve market control (Ryoo et al., 1999). Containership capacity is therefore shared with partners and the container carrier gets operational capacity through joint service, slot-exchange and slot-charters. In this study, operational containership capacity and deadweight tonnage are given.

(5) Safety stock of empty containers, empty container stock, and probability of reposition empty container are given.

(6) From a miscellany of different container types and sizes used by container shipping industry, only

20 ' × 8 ' × 8 ' 6 "

dry container (20’DC),

40 ' × 8 ' × 8 ' 6 "

dry container (40’DC), and

40 ' × 8 ' × 9 ' 6 "

(40’HQ) are considered in this proposed model.

4.2 Model Formulation

Maximize

_∑∑∑ [ ( ) ] k

index of container specification,

k ∈ K

n

The number of calling ports in a service route

( ) [ ( ) ] [ ( ) ] { ^mod ¹ ^, ¹ ^mod ¹ ^, ^, ¹ ^mod ¹ }

= + + + + − +

z n z n z n n

T

K

The sequence of calling ports on the service route

( ) ( [ 1 ) mod ] 1

FR

ij Freight revenue including ocean freight and surcharge of

k ∈ K

type container delivered from port

i ∈ P

to port

j

∈ (unit：USD)

P

VC

ij Variable cost of

k ∈ K

type container delivered from port

i ∈ P

to port

P

j

∈ , including handling charges at both ports, commissions, container rental (depreciation) and repair, truck fee and depot stowage costs (unit：

USD)

EC

i Expected cost of empty container reposition of

k ∈ K

type at loading port

P

POR

i Probability of repositioning empty container of

k ∈ K

type at loading port

P

i ∈

EC

j Expected cost of empty container reposition of

k ∈ K

type at unloading port

j

∈ (unit：SD)

P

POR

j Probability of repositioning empty container of

^k ^∈ ^K

type at discharging port

j

∈

P

OC

i The operational capacity on containership when containership leaved from port

i ∈ P

(unit：TEU, twenty-foot equivalent units)

DW

i The operational deadweight tonnage on containership when containership leaved from port

i ∈ P

(unit：ton)

D

i The maximum of deadweight tonnage for all loaded containers at loading port

i ∈ P

(unit：ton)

λk Transferring coefficient of TEU by

k ∈ K

type. 20’DC is referred to as

“Twenty-Foot-Container” which equals to one Twenty-Foot Equivalent Unit (1 TEU). 40’DC and 40’HQ are referred to as “Forty-Foot-Container (FEU)” which equals to two Twenty-Foot Equivalent Unit (2 TEU).

The decision variable is

X

_ij^k(the number of slot allocation for

k ∈ K

type loaded containers delivered from the loading port

i ∈ P

to the unloading port

j

∈ ).

P

The objective function (1) sleeks to maximize operational profit. Constraint (2) of containership capacity requires that total allocated slot of loaded containers do not exceed containership operational capacity. Constraint (3) of containership deadweight requires that total weight of loaded containers do not exceed operational deadweight tonnage. Constraint (4) of container demand requires that slots allocated to various port-pairs be within the min-max boundaries of loaded container demand.

Constraint (5) represents the total deadweight tonnage of loaded slots which could not exceed the upper bound deadweight tonnage in the loading port. Constraint (6) defines the decision variable to be integers.

4.3 Case Study

To discuss the analytical results of the proposed model and its application, this research uses one of Taiwan’s shipping companies (T Line), which has a long history of operation on the intra-Asian service routes.

4.3.1 Background and relevant data

T Line runs one service route, named CHI (China-Hong Kong-Indonesia) service, calling at Qingdao (TAO), Shanghai(SHA), Hong Kong(HKG), Manila(MNL), Jakarta(JKT), Surabaya(SUB), Manila, and Hong Kong again, and then returns to Qingdao for a roundtrip (as shown in Figure 4.1). Four full-container containerships were deployed on this service route to provide weekly service. The containership capacity was 1,100 TEU and 15,400 tons deadweight. To decrease overhead and share the cost of capital equipment, T Line cooperated with other container carriers in launching the CHI service through joint service, slot exchange and slot charter. T Line had an operational capacity of 350 TEUs and 4,900 tons on a containership. The container management department regularly recorded empty container stock of O/B containers and I/B containers, and then classified them into five types: S(surplus),

record, it meant that a large number of empty containers had accumulated at depot, requiring immediate repositioning out. The study supposed:

POR

_i^k

= 1 or POR

^k_j

= 1

. If S was recorded, preparation was made for repositioning empty containers out or in.

The study then supposed:

POR

_i^k

= 0 . 5 or POR

^k_j

= 0 . 5

. If A was recorded, this meant that the amount of empty containers was equal to the safety stock. The study then supposed:

POR

_i^k

= 0 or POR

^k_j

= 0

. The record of empty container stock at each port is displayed in Table 5.1. Although this proposed model included a deadweight constraint, we analyzed the operational capacity of slot allocation without it, as the container deadweight data was unavailable.

Figure 4.2 Service route of CHI service

Table 4.1 Record of empty container stock

Port TAO SHA HKG MNN MNS JKT SUB

Item

20'DC Stock D D A S S SS D

Probability 50% 50% 0% 50% 50% 100% 50%

Handling Cost (USD) 44 56 52 52 52 53 53

22.0 28.0 0.0 (26.0) (26.0) (53.0) 26.5 (22.0) (28.0) 0.0 26.0 26.0 53.0 (26.5)

40'DC Stock DD DD A SS SS S DD

Probability 100% 100% 0% 100% 100% 50% 100%

Handling Cost (USD) 64 83 77 66 66 79 79

64.0 83.0 0.0 (66.0) (66.0) (39.5) 79.0 (64.0) (83.0) 0.0 66.0 66.0 39.5 (79.0)

40'HD Stock D DD D S S S D

Probability 100% 100% 50% 50% 50% 50% 50%

Handling Cost (USD) 64 83 77 66 66 79 79

64.0 83.0 0.0 (66.0) (66.0) (39.5) 39.5 (64.0) (83.0) 0.0 66.0 66.0 39.5 (39.5) Notes: S: Surplus SS: Serious Surplus

D:Shortage DD: Serious Shortage A: Balance

4.3.2 Computational results

Table 4.2 presents an optimal plan of slot allocation solution using the WinQSB 2.0 software. For instance, the shipping agency at TAO has a slot allocation for 21 TEU and 19 FEU to HKG; 23 TEU to MNN; 18 TEU and 6 FEU to MNS; 42 TEU and 5 FEU to JKT; 14 TEU and 4 FEU to SUB. Total O/B cargos from TAO are 118 TEU and 34 FEU (186 TEUs). Total I/B cargos to TAO port are 63 TEU and 53 FEU (169 TEUs). Total loaded cargos are 1,025 TEUs and load factor (L/F) is 2.93 (1,025 TEUs / 350 TEUs) for a roundtrip.

Table 4.2 An optimal plan of containership slot allocation for CHI service

POD TAO SHA HKG MNN MNS JKT SUB MNS HKG

POL (BOX) (TEU)

Figure 4.2, 4.3 and 4.4 illustrate an imbalance between O/B cargo and I/B cargo for 20’DC, 40’DC, 40’HQ. I/B cargo are greater than O/B cargo at MNL (MNN+MNS) port, an import-oriented port, and as a consequence a large number of empty containers accumulated. In contrast, SUB port often faces a shortage of empty containers, because O/B cargo is exported to other regions. Additionally, there is a huge imbalance for 40’DC at SHA port. T line repositions a large number of empty containers of 40’DC into SHA port resulting in a high cost of empty container repositioning.

0 50 100 150

(B OX)

20'DC O/B 20'DC I/B

20'DC O/B 101 67 118 2 60 87 66

20'DC I/B 60 89 85 15 74 140 38

TAO SHA HKG MNN MNS JKT SUB

Figure 4.3 Imbalance between O/B cargo and I/B cargo for 20’DC

0 40'DC O/B 40'DC I/B

40'DC O/B 12 28 32 1 6 11 5

40'DC I/B 24 2 20 5 16 27 1

TAO SHA HKG MNN MNS JKT SUB

Figure 4.4 Imbalance between O/B cargo and I/B cargo for 40’DC

0 40'HQ O/B 40'HQ I/B

40'HQ O/B 22 29 37 0 11 36 23

40'HQ I/B 25 7 47 2 48 23 6

TAO SHA HKG MNN MNS JKT SUB

Figure 4.5 Imbalance between O/B cargo and I/B cargo for 40’HQ

Table 4.3 compares the experimental and actual results. Two voyages were randomly selected in order to calculate actual slot on containerships in the CHI service. The total number of “model” loaded containers (1,007 TEUs) was greater than the actual number in voyage 1 (738 TEUs) and voyage 2 (980 TEUs) (see Appendix D and E). Model operation profit (US$251,052) was also higher than actual profit on voyage 1 (US$171,521) and voyages 2 (US$216,762). The ration of empty container reposition, 57.6% drawn from the proposed model, was less than actual slots of voyage 2 (78.16%).

Table 4.3 Comparing optimal containership slot allocation with actuality

Item Opeational

Optimal slot allocation 350 1,007 2.93 264,933 13,881 251,052 580 57.60%

Actual slot Voyage 1 350 738 2.11 184,727 13,206 171,521 376 50.95%

Voyage 2 350 980 2.80 235,150 18,388 216,762 766 78.16%

Notes: 1) Load Factor (L/F): Total loaded container / Operational capacity

2) Ratio of empty container reposition: the number of empty container reposition / Total loaded container

Figure 4.6 Comparing optimal containership slot allocation with actuality in total loaded container and operational profit

4.4 Strategy Analysis

It was proposed that the T Line develop both short-term and long-term strategies to improve their management of slot allocation (as seen in Figure 4.8).

4.4.1 The short-term strategy

Further analysis compared the actual number of slots on a containership with that of the proposed model, and the actual number of slots at the point of departure from each port (as seen in Figure 4.7). A containership received many slots when it departed from SHA port on voyage 2. Total loaded cargo on the containership exceeded operational capacity (350 TEUs), indicating that containers which were loaded at the TAO and SHA occupied too many slots. As a result, the shipping agency at HKG did not have enough space to load its own containers. Such a situation often creates friction among shipping agencies at the TAO, SHA and HKG.

T Line needs to take action to solve this problem.

One such action is to unload cargo loaded at TAO or SHA to provide space to the shipping agency at HKG. Consequently, T Line will be made to bear the

additional costs of discharging and of a second loading. Also, by doing so, it might jeopardize its reputation with customers whose cargo was discharged.

An alternative action would be to charter slots from alliance partners.

In the other situation pertaining to voyage 1, the containership was not fully loaded and freight revenue was lost from unsold slots. Also, there were dramatic swings in both voyages (voyage 1 and voyage2) that did not occur in the proposed model.

Figure 4.7 Comparing total slots on containership with optimal containership slot allocation and actual slots on yoage1 and voyage 2

4.4.2 The long-term strategy

According to model results, a loaded containership departed from SHA and HKG at T Line’s 350 TEU operational capacities. At subsequent ports the containership was not fully loaded. Based on analyzed parameter data, cargo demand was less than maximum cargo demand

( DU

_ij^k

)

at some ports.

Strategies to solve this problem:

The first strategy would be to request shipping agencies at MNL, JKT, SUB

and HKG to increase their marketing effort and solicit additional cargo to achieve

full operational capacity.

A second strategy would be to adjust its alliance strategy in order to reduce operational capacity, overhead and risk through slot-exchange or slot-charter with other carriers.

Given this second strategy, freight revenue would be reduced for shipping agencies at TAO, SHA, and HKG because of the reduced slot allocation. However this adjustment strategy has the potential to increase the utilization rate and load factor, and achieve increased performance. For example, if the operational capacity was reduced to 300 TEUs, the containership would be fully loaded after departing from SHA, HKG, MNN and SUB. If the operational capacity was reduced to 250 TEUs, then the containership would be fully loaded with cargo for the roundtrip voyage (as seen in Figure 4.9) .

In order to recover market activity at TAO and SHA, a new service route might be designed for short distance.

Figure 4.8 Total loaded cargos on containership vs. operational capacity

4.4.3 Summary

Container carriers face problems of excess operational capacity, meaning capacity that is not fully utilized. Figure 4.8 illustrates the action that T Line might conduct to solve these problems. Additional, strategies might be developed to improve the management of slot allocation.

Containerships might exert an influence on shipping agencies through a fine system, slot allocation reduction, or cancellation of shipping agency authority.

More importantly, they might set up a booking system to control loaded cargo in advance and communicate closely with sales departments in order to adjust slots at each port, thereby eliminating unsold slots.

Problem Action Result

Excess operational

capacity

Unloaded cargo which were loaded at TAO or SHA

already

Charter slots from alliance partners

Increase cost

Not fully utilize operational

capacity

Lost freight revenue for unsold slot Create friction among

shipping agencies at TAO, SHA, and HKG

Request shipping agencies at MNL, JKT, SUB, and HKG

to seek additional cargo

Figure 4.9 Problem, action and result for actual slots on voyage1 and voyage 2

CHAPTER 5 THE PLAN FOR EMPTY CONTAINER REPOSITION

Container carriers often follow a rule of thumb when repositioning empty containers. As a consequence, a large number of empty containers occupying slots on a containership tend to accumulate, and they are frequently repositioned throughout one voyage. The problems resulting from this practice pertain to a loss of freight revenue and the occurrence of storage expenses at empty container depots. This study addresses these problems by grouping them into two categories: the upper problem and the lower problem. The upper problem is concerned with identifying and estimating empty container stock for each port. The lower problem or transportation problem pertains to the cost of empty container reposition (as seen in Figure 5.1). The proposed model provides an effective plan for empty container reposition. In addition, it offers the possibility of providing container carriers with a strategy to improve management of slot allocation.

Figure 5.1 A concept chart for empty container reposition

5.1 Assumptions

The following assumptions are imposed for this proposed model：

(1) Owned containers and long-term leased containers are considered. Short-term leased containers are leased in from leasing company for temporary using, such as helping ports having a serious shortage of empty containers to meet demand requirements. Because of high rental cost, short-term containers are leased off to leasing company when the target is achieved. Container carriers do not schedule efficient short-term containers in the same way as owned containers.

(2) The cost of transportation mode for various origin-destination port pairs is given (see Appendix E).

(3) The number of I/B containers and O/B containers at each port is known.

(4) The number of safety stock for empty containers at each port is known.

(5) No limits on the number of slots to allocate empty containers.

(6) The plan for empty container reposition is scheduled in a certain time period and empty containers are split into several voyages to reposition.

5.2 Model Formulation

The problem consists of two parts. One part is the upper-problem, which identified and estimated empty container stock at each port. The other is the lower-problem, which modeled empty container reposition planning as the Transportation Problem by Liner Problem. The upper-problem 【UP】and the lower-problem 【LP】may be drawn up as follows:

【UP】

H set of port within sea transportation network,

{ TYO NGO KEL SUB }

H = , ,..., ,...,

K set of container specification,

{ ^DC ^DC ^HQ }

K

= 1:20' , 2:40' , 3:40'

h

index of port within sea transportation network,

h ∈ H t

index of time period

RI

ht quantity of repositioned-into empty container of

k ∈ K

type in t period at port

h ∈ H

RO

ht quantity of repositioned-out empty container of

k ∈ K

type in t period at port

h ∈ H

Empty container stock, supply of empty containers and demand of empty containers at each port are given. The objective function is minimizing total transportation cost of repositioning empty containers within the sea transportation network. 【LP】may be formulated as follows:

【LP】

F number of port group within sea transportation network H set of port within sea transportation network,

{ ^TYO ^NGO ^KEL ^SUB }

G

s set of ports having a surplus of empty containers within sea transportation network

G

f set of ports having a surplus of empty containers within

f

∈ group

F G

d set of ports having a shortage of empty containers with sea transportation

network

G

f set of ports having a shortage of empty containers within

f

∈ group

F f

index of port group within sea transportation network,

f

∈

F

α index of loading port within sea transportation network, α

∈ H

β index of unloading port within sea transportation network, β∈

H

ρ =1, if transportation mode

αβ ^m ^∈ ^M

, repositioning empty containers from port

G

α∈ to port β∈

G

^d, was selected =0, otherwise

C _αβ

mk cost of repositioning an empty container of

k ∈ K

type from port α∈

G

^s to port β∈

G

^d by transportation mode

m ∈ M

αβ

=1, if it had direct sailing from port α∈

G

^s to port β∈

G

^d within sea transportation network

=0, otherwise

S _α

k supply number of

k ∈ K

type empty containers at port α∈

G

D _β

k demand number of

k ∈ K

type empty containers at port β∈

G

The decision variable is

E _αβ

^k (the number of

k ∈ K

type empty container

reposition from port α

∈ G

^s_f to portβ

∈ G

^d_f ). The objective function (11) is minimizing total transportation cost of empty container reposition in variable transportation modes. Constraint (12) guarantees that just one transportation mode is selected to reposition empty containers from port α

∈ G

^s_f to port β

∈ G

^d_f . Constraint (13) ensures

k ∈ K

type empty containers repositioned out from port

G

∈

to portβ

∈ G

^d_f are equal to the supply of

k ∈ K

type empty containers at

port α

∈ G

^s_f for each group

p

∈ . Constraint (14) ensures the number of

P K

k ∈

type empty containers repositioned into from port α

∈ G

^s_f to portβ

∈ G

^d_f is

the same as the demand of

k ∈ K

type empty containers at port β

∈ G

^d_f for each group

f

∈ . Constraint (15) and constraint (16) stipulates symbols must equal 0 or

F

1. Finally, constraint (17) is integer constraint.

5.3 Case Study

To explain the application and results of the proposed model, this study uses a Taiwan Shipping Company (T Line) as an example.

5.3.1 Background and relevant data

The sea transportation network of T Line is composed of 17 service routes.

Service coverage is between Japan, Korea, China, Taiwan, Hong Kong, Philippines, Vietnam, Thailand, Malaysia, Singapore, and Indonesia. The main service routes provide long distance shipping in the sea transportation network, and major types of sailing schedules ranging from type 1 to type 4 have been designed. Meanwhile, rapid and short service routes are run to provide high sailing-frequency service with types ranging from type 5 to type 7 (as seen in Table 5.1).

Table 5.1 Types of sailing scheduling for T Line

Types Sailing Scheduling

Type 1 service routes were designed between Japan, Taiwan, Hong Kong, and Thailand

Type 2 service routes were designed between Korea, North China, Taiwan, Hong Kong, Singapore, and Indonesia

Type 3 service routes were designed between North China, Hong Kong, Philippines, and Indonesia

Long Distance

Type 4 service routes were designed between Middle China, Hong Kong, Philippines, and Thailand

Type 5 service routes were designed between Taiwan and Hong Kong Type 6 service routes were designed between Taiwan and Philippines Short

Distance

Type 7 service routes were designed between Taiwan and China

The sea transportation network was partitioned into five geographical regions because of sailing distance, service routes, and practical operation (as seen in Figure 5.2 and Table 5.2). The two main geographical regions are between East-North Asia and Taiwan and the region between East-South Asia and Hong Kong. The three subordinate regions are between Taiwan and Hong Kong, Kaohsiung and Manila, and Manila and North China. HKG port, KEL port and KHH port provide the highest calling-frequency with eleven service routes in one week (i.e., eleven containerships called ports during one week). OIT port, ICN port, SGN port, PKG port, and PGU port are called only once a week. It is convenient to reposition empty containers at ports with high calling-frequency and a challenge to reposition them at ports with low calling-frequency. T Line looks forward to reducing the cost of empty container reposition by adopting transportation modes of owned slot within the sea transportation network. The transportation mode of inland drayage by truck has been adopted in channels between UKB port and OSA port in Japan, among KEL port,

TXG port, and KHH port in Taiwan, between BKK port and LCH port in Thailand, and between JKT port and SUB port in Indonesia. In this case study, we collected statistics of O/B containers and I/B containers for one month and classified them as either supply or demand (as seen in Table 5.3, Figure 5.3, Figure 5.4, and Figure 5.5).

Figure 5.2 Geographical groups in sea transportation network

Table 5.2 A classified catalogue of geographical group

Group Region

1 Japan, Korea, North China, Middle China, and Taiwan

2 Hong Kong, Middle China, South China, Philippines, Vietnam, Thailand, Malaysia, Singapore(SIN), and Indonesia.

3 Taiwan and Hong Kong(HKG) 4 Kaohsiung and Manila

5 Manila, Qingdao, and Shanghai.

Group 1

Group 2 Group 3 Group 4

Group 5

Table 5.3 Monthly data of supply and demand

Port Amount Port Amount Port Amount Port Amount Port Amount Port Amount

(box) (box) (box) (box) (box) (box)

TYO 511 TKY 62 TYO 196 UKB 3 TYO 137 TKY 18

TOTAL 3,016 3,016 880 880 1,146 1,146

Demand Supply Demand

Figure 5.3 Supply port and demand port for 20’DC

Supply

Demand

(300)

Figure 5.4 Supply port and demand port for 40’DC

(600)

Figure 5.5 Supply port and demand port for 40’HQ

Supply

Demand

5.3.2 Computational results

The computational results of the optimal solution are shown in Table 5.4 which presents a plan that T Line might implement for empty container reposition. For example, 511 boxes of 20’DC empty containers and 138 boxes of 40’DC empty container from TYO port to KHH port were slotted for reposition during one month.

There was an uncertain number of available slots on the containership, which meant that empty containers might be repositioned over several voyages. Two service routes were sailed between TYO port and KHH port and provided eight voyages during one

在文檔中近洋貨櫃航商艙位配置及空櫃調度之最適化研究 (頁 61-0)

CHAPTER 3 DESCRIPTION OF PROBLEM

3.6 Geographical Regions with the Sea Transportation Network

3.7.2 The plan for empty container reposition

CHAPTER 4 THE PLAN FOR CONTIANERSHIP SLOT ALLOCATION

4.1 Assumptions

20 ' × 8 ' × 8 ' 6 "

40 ' × 8 ' × 8 ' 6 "

40 ' × 8 ' × 9 ' 6 "

4.2 Model Formulation

∑∑∑ [ ( ) ] k

k ∈ K

n

( ) [ ( ) ] [ ( ) ] { mod 1 , 1 mod 1 , , 1 mod 1 }

= + + + + − +

z n z n z n n

T

K

( ) ( [ 1 ) mod ] 1

FR

k ∈ K

i ∈ P

j

P

VC

k ∈ K

i ∈ P

P

j

EC

k ∈ K

P

POR

k ∈ K

P

i ∈

EC

k ∈ K

j

P

POR

k ∈ K

j

P

OC

i ∈ P

DW

i ∈ P

D

i ∈ P

k ∈ K

X

k ∈ K

i ∈ P

j

P

4.3 Case Study

4.3.1 Background and relevant data

POR

= 1 or POR

= 1

POR

= 0 . 5 or POR

= 0 . 5

POR

= 0 or POR

= 0

Port TAO SHA HKG MNN MNS JKT SUB

Item

20'DC Stock D D A S S SS D

Probability 50% 50% 0% 50% 50% 100% 50%

Handling Cost (USD) 44 56 52 52 52 53 53

22.0 28.0 0.0 (26.0) (26.0) (53.0) 26.5 (22.0) (28.0) 0.0 26.0 26.0 53.0 (26.5)

40'DC Stock DD DD A SS SS S DD

Probability 100% 100% 0% 100% 100% 50% 100%

Handling Cost (USD) 64 83 77 66 66 79 79

64.0 83.0 0.0 (66.0) (66.0) (39.5) 79.0 (64.0) (83.0) 0.0 66.0 66.0 39.5 (79.0)

40'HD Stock D DD D S S S D

Probability 100% 100% 50% 50% 50% 50% 50%

Handling Cost (USD) 64 83 77 66 66 79 79

_∑∑∑ [ ( ) ] k

( ) [ ( ) ] [ ( ) ] { ^mod ¹ ^, ¹ ^mod ¹ ^, ^, ¹ ^mod ¹ }

^k ^∈ ^K

{ ^DC ^DC ^HQ }