Tanker scheduling by using optimization techniques and a case study

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BAÜ FBE Dergisi

Cilt:9, Sayı:2, 48-62 Aralık 2007

Tanker scheduling by using optimization techniques

and a case study

Aslan Deniz KARAOĞLAN*

Balıkesir University, Engineering and Architectural Faculty, Department of Industrial Engineering,

Cagis Campus, 10145, Balikesir, Turkey Abstract

This study aims to guide companies and researchers who needs to schedule and optimize their transportation systems which are based on shipping. In this study, the problem, that is tried to solve, is providing profit maximization in transporting the different characterized cargoes to determined ports by the ship fleets which contain different kind of ships. In the first part of the study, planning and scheduling in the transportation industry is introduced and in the second part, materials and techniques to perform the scheduling, which are used by the companies that own and operate tanker or cargo ship fleets, are

given. At the third part of the study the constructed scheduling model and the solution of

the schedule which is computed by optimization software is given and at the following part,

the concluVion is presented. The results have shown that, the presented model that is

refered from the literature, can be applied successfully in real word and gives nearly optimal solutions. The case study is performed in shipping company which is established in Istanbul.

Keywords: Tanker scheduling, optimization.

Optimizasyon tekniklerini kullanarak tanker çizelgeleme ve bir

uygulama

Özet

Bu çalışma, gemicilik üzerine kurulu taşıma sistemlerini çizelgelemek ve optimize etmek isteyen firmalara ve araştırmacılara rehberlik etmeyi amaçlamaktadır. Problem bir çok farklı özelliğe sahip gemilerden oluşan bir filo ile farklı özelliklere sahip kargoların belirli limanlara taşınmasında kar maksimizasyonunun sağlanmasıdır. Bunun sağlanması için tanker çizelgeleme problemleri için literatürdeki modellerden faydalanılarak taşıma problemi modellenmiş ve çözüm sonucuna yer verilmiştir. Çalışmanın ilk bölümünde, taşımacılık endüstrisinde planlama ve çizelgeleme tanıtılmış ve ikinci bölümde tanker veya kargo gemi filosu işleten firmaların çizelgelemede kullandığı materyaller ve teknikler

verilmiştir. Çalışmanın üçüncü bölümünde oluşturulan çizelgeleme modeli ve bir paket

programla hesaplanan çözümü verilmiş ve sonraki bölümde de sonuçlar sunulmuştur. Sonuçlar incelendiğinde, literatürden alınan modelin gerçek hayata başarı ile uygulanabildiği ve elde edilen sonuçların optimale yakın olduğu görülmüştür. Çalışmanın

uygulama bölümü Đstanbul’ da faaliyet gösteren bir denizcilik şirketinde

gerçekleştirilmiştir.

Anahtar Kelimeler: Tanker çizelgeleme, optimizasyon.

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Tanker scheduling by using optimization techniques and a case study

49 1. Introduction

The planning and scheduling models in services and the solution methodologies used tend to be different from those applied in manufacturing environments. This talk goes into four classes of models. The first class includes interval scheduling models and reservation systems. The second class involves timetabling and tournament scheduling. The third class consists of transportation models (tanker scheduling, aircraft routing and scheduling and train timetabling). The fourth and last class are the workforce scheduling models [1].

In the transportation industry planning and scheduling problems abound. The variety in the problem is due to the many modes of transportation, e.g., shipping, airlines and railroads. Each mode of transportation has its own set of characteristics. The equipment and resources involved, i.e.,

(i) ships and ports, (ii) planes and airports,

(iii) trains, tracks, and railway stations,

have different cost characteristics, different levels of flexibilities and different planning horizons [2].

Ship scheduling models optimize the transportation of commodities, so they are vital to world trade and millitary logistics. A ship requires a multi-million dolar capital investment and the daily operation costs of a ship can be tens of thousand dollars. Consequently, improved fleet utilization can yield significant financial benefit [3].

Scheduling tankers presents many interesting and varied problems. Among these problems are, how to determine a program of delivery dates at the respected ports and how to route the tankers in fleet [4].

There are few studies about tanker scheduling in the literature. Fagerholt [5] studied the problem of evaluating the trade-off between the level of customer service and transportation costs. An evaluation is performed on the background of data from a real ship scheduling problem, where each cargo has time windows on both the loading and the discharging.

Cho and Perakis [6] presented an improved, significantly more efficient formulation of an existing model for bulk cargo or semi-bulk cargo ship scheduling problems with a single loading port. The original model, published by Ronen in 1986, was formulated as a non-linear, mixed integer program. In this work, the authors were able to re-formulate it into a linear one, by eliminating all the non-linearities of the original model. In addition, this model has far fewer integer variables than the original one.

Hwang and Rosenberger [7] presented a set-packing model that limits risk using a

quadratic variance constraint. After generating first-order linear constraints to represent the variance constraint, the authors developed a branch-and-price-and-cut algorithm for medium-sized ship-scheduling problems.

Sambracos et al. [8] investigated the introduction of small containers, an important new technology, in an effort to reengineer coastal freight shipping in the Aegean Sea in Greece.

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A. D. Karaoğlan

Infrastructure problems of island ports are documented and the advantages of introducing small containers are discussed.

Fagerholt [9] considered a real liner shipping problem of deciding optimal weekly routes for a given fleet of ships and proposed a solution method for solving the problem.

Sherali et al. [10] explored models and algorithms for routing and scheduling ships in a maritime transportation system. The authors have constructed a mixed integer programming model for the problems of Kuwait Petroleum Corporation (KPC).

Giziakis et al. [11] modeled the operation of passenger vessels as a linear programming (LP) problem on a network of 37 nodes in the Aegean Sea.

In this study, the problem, that is tried to solve, is providing profit maximization in transporting the different characterized cargoes to determined ports by the ship fleets which contain different kind of ships. The company owns four ships and the ships are located in different ports. There are also six cargoes, which have to be transported to different ports. Each ship, that the company owned, has different characteristics and these differences influence the transportation problem of the company. Because all the ships can not transport all the cargoes, and also they can not enter to all ports because of their dimensions. Also the profits that are provided may not be attractive for all alternative schedules. To optimize the transportation problem of the company, the schedules are constructed by considering the characteristics of ships, cargoes and ports, and than the costs of each schedule is calculated. The model that is developed for tanker scheduling in the literature and optimization package program are used to calculate the optimal solution for the company.

2. Materials and methods

A cargo is the entire content of a ship transported between two ports, and a schedule is a sequence of cargoes delivered by the same ship. Ship scheduling problems are solved by generating a set of feasible delivery schedules for each ship and optimizing a set packing problem [3].

Companies that own and operate tanker fleets typically make a distinction between two types of ships. One type of ship is company owned and the other type of ship is chartered. The operating cost of a company owned ship is different from the cost of a charter that is typically determined on the spot market. Each ship has a specific capacity, a given draught, a range of possible speeds and fuel consumptions and a given location and time at which the ship is ready to start a new trip [2].

Each port also has its own characteristics. Port restrictions take the form of limits on the deathweight, draught, lenght, beam and other physical characteristics of the ship. There may be some additional goverment rules in effect; for example, the Nigerian goverment imposes a so-called 90% rule which states that all tankers must be loaded to more than 90% of capacity before sailing [2].

A cargo that has to be transported is characterized by its type, quantity, load port, delivery port, time window constraints on the load and delivery times, and the load and unload

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Tanker scheduling by using optimization techniques and a case study

51

times. A schedule for a ship defines a complete itinerary, listing in sequence the ports visited within the time horizon, the time of entry at each port and the cargoes loaded or delivered at each port [2].

The objective typically is to minimize the total cost of transporting all cargoes. This total cost consists of a number of elements, namely the operating costs for the company-owned ships, the spot charter rates, the fuel costs and the port charges. Port charges vary greatly between ports and within a given port charges typically vary proportionally with the deadweight of the ship [2].

In order to present a formal description of the problem the following notation is used. Let n denote the number of cargoes to be transported and T the number of company-owned tankers. Let Si denote the set of all possible schedules for ship i. Schedule l for ship i, l ∈ Si, is

represented by the column vector [al

i1,ali2,…,alin]. The constant alij is 1 if under schedule l,

ship i transports cargo j and 0 otherwise. The decision variable xl

i is 1 if ship i follows

schedule l and zero otherwise.

The tanker scheduling problem can now be formulated as follows [2]: Maximize

∑

= T i1

∑

∈_S_i l ∏li xli (Objective function) Subject to

∑

= T i 1

∑

∈_S_i l al

ij xli≤ 1 j=1,….,n (First set of constraints)

∑

∈_S_i l

xl

i≤ 1 i=1,…..,T (Second set of constraints)

xl

i∈ {0,1} l∈Si, i=1,……,T (Remaining constraints)

The objective function specifies that the total profit has to be maximized. The first set of constraints imply that each cargo can be assigned to at most one tanker. The second set of constraints specifies that each tanker can be assigned at most one schedule. The remaining constraints imply that decision variables have to be binary 0-1. This optimization problem is typically referred a set-packing problem [2].

The algorithm used to solve this problem is a branch and bound procedure. However, before the branch and bound procedure is applied, a collection of candidate schedules have to be generated for each ship in the fleet. As stated before, such a schedule specifies an itinerary for a ship, listing the ports visited and the cargoes loaded or delivered at each port. The generation of an each collection of candidate schedules has to be done by a seperate ad-hoc heuristic that is especially designed for this purpose. The collection of candidate schedules should include enough schedules so that potentially optimal schedules are not ignored, but not so many that the set packing problem becomes intractable. Physical constraints such as ship capacity and speed, port depth and time windows limit the number of feasible candidate schedules considerably. Schedules that have a negative profit coefficient in the objective function of the set packing formulation can be omitted as well [2].

The case study has been performed in a shipping company which is located in Istanbul. The data is collected by e-mail, phone and other instrumental documents downloaded from

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A. D. Karaoğlan

related websites . The tables which are composed from these collected data, are given below and after the presentation of these tables the LP model is composed [5, 6].

There are some assumptions in the tables given below such as the ships/tankers are ready to a new trip. And an another assumption is that the ships don’t delay during the navigation between two ports, so there is no extra cost for delays. Also, the fuel consumptions for the ships according to their deadweights and the navigation distances are assumed fixed values as given in Table 8.

It is important to take notice that we consider the characteristics of ships, ports and cargoes that are given in Table 1 through Table 3, while constructing the schedules which are given in Table 4. Also these four tables and Table 8 are used to construct Table 5 by the contribution of the references [12] and [13].

In Table 1 the general characteristics of the four ships, that the company owned, is given. This data are used in matching the cargoes with the ships. For example, if Table 1 browsed, it is seen that Akıncı is a dry cargo ship. So Ammonium Nitrate, which is labeled as Cargo1 in Table 2, can not be transported by this ship. Akıncı is anchored at Ambarlı port according to Table 1. Its possible cargoes can be Cargo2, Cargo3 and Cargo6 according to Table 3. When the characteristics of ports like as “limits on the dead weight”, “draught” and etc. is matched with the characteristics of the ship Atmaca like as “draught”, “beam” and etc, it is seen that the mentioned ship called Atmaca, can perform loading and unloading cargoes (Cargo2, Cargo3, Cargo6) that are located at or will be transported to the ports called Ambarlı, Bandırma, Haydarpasa, Gemlik, Izmir. The possible schedules, that are given in Table 4 are determined similarly as performed in the ship Atmaca example.

After determining the possible schedules, the cost of each schedule is calculated in Table 5 and then the profits are calculated in Table 7 by using the values that are given in Table 5 and Table 6. While calculating the costs of each schedule in Table 5, the costs of the cost items for loading and unloading ports are calculated by the contribution of the references [12] and [13], which are mentioned above.

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53 Ta nk er s ch ed ulin g b y u sin g o pti m iza tio n te ch niq ue s a nd a c ase stu dy

Table 1. Data related to the ships/tankers

SHIP

NO NAME OF THE SHIP /

TANKER CAPACITY OF THE (MT) SHIP / TANKER SPEEDS DRAUGHT RANGE OF POSSIBLE FUEL

CONSUMTIONS GIVEN LOCATION TIME AT WHICH THE

SHIP IS READY TO START A NEW TRIP NOTE HIRING COST

(IF THE (GROSS SHIP / TANKER IS HIRED) DEATH WEIGHT TONE) (GT) THE LENGHT OF THE SHIP / TANKER (MT) BEAM (MT) 1 ATMACA 1 6200000 DWT (9641 kg) 4.82 11-12

kts 6 M/T FO & 1 M/T DO SAMSUN READY ABLE TO CARRY

LIQUID AND FREIGHT CONTAINER

- 3500 80 16.2

2 ATMACA 2 20000

Tone 16.6 18-30 kts 10 M/T FO & 1 M/T DO BANDIRMA READY ONLY ABLE TO CARRY

HOUSEHOLD FUEL

- 9200 150 30.3

3 TRITON 20000

Tone 11 16-25 kts 15M/T FO & 1 M/T DO GEMLIK READY DRY CARGO SHIP - 8500 110 22.4

4 AKINCI 15000

Tone 9 14-20 kts 8 M/T FO & 1 M/T DO AMBARLI READY DRY CARGO SHIP - 7100 100 20.1

Table 2. Data related to the ports

PORT

NO NAME OF THE PORT LIMITS ON THE DEADWEIGHT

(GT)

DRAUGHT

(MT) LENGHT CONSTRAINT FOR

THE SHIPS/TANKERS

BEAM COMPANY LAWS IF EXISTS

TIME NEEDED TO ENTER THE PORT FOR THE SHIPS/TANKERS

NOTE

1 AMBARLI N/A 14 N/A N/A N/A ~ 45 Min -

2 SAMSUN 9500 18 150 m N/A N/A 6 Hours Canal + 45 Min to Dock -

3 IZMIT 9000 25 N/A N/A 2 tugboat must

be hired 5 Hours Canal + 45 Min to Dock -

4 GEMLIK 9000 17 N/A N/A 2 tugboat must

be hired 5 Hours Canal + 45 Min to Dock -

5 HAYDARPASA 8600 13 120 m N/A N/A 60 Min -

6 MERSIN N/A 18 N/A N/A N/A 30 Min -

7 ISKENDERUN N/A 23 N/A N/A N/A 30 Min -

8 BANDIRMA N/A 22 N/A N/A N/A 30 Min -

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54 A . D . K ara o ğla n

Table 3. Data related to the cargoes

CARGO

NO TYPE OF THE CARGO QUANTITY LOAD PORT DELIVERY PORT LOAD DATE DELIVERY TIME LOADING TIME UNLOADING TIME NOTE

1 AMMONIUM

NITRATE 6000000 dwt (9330 kg)

SAMSUN MERSIN 11.01.2007 96 HOURS 48 HOURS

96 HOURS -

2 LIVE STOCK 5000 Unit

=1900 Tone AMBARLI BANDIRMA 12.01.2007 12 HOURS 10 HOURS 12 HOURS -

3 BULK

FREIGHT (Solid)

1000 Tone BANDIRMA HAYDARPASA 13.01.2007 13 HOURS 24 HOURS -

4 OIL 10000 Tone IZMIR ISKENDERUN 12.01.2007 96 HOURS 48 HOURS 40 HOURS -

5 OIL 8000 Tone IZMIR ISKENDERUN 12.01.2007 96 HOURS 36 HOURS 30 HOURS -

6 BULK

FREIGHT (Solid)

500 Tone GEMLIK IZMIT 12.01.2007 8 HOURS 24 HOURS 20 HOURS -

Table 4. Schedules

SCHEDULES

SHIP NO:1 SHIP NO:2 SHIP NO:3 SHIP NO:4

TYPE OF CARGO 1. S C H ED U LE 2. S C H ED U LE 3. S C H ED U LE 4. S C H ED U LE 1. S C H ED U LE 2. S C H ED U LE 3. S C H ED U LE 4. S C H ED U LE 1. S C H ED U LE 2. S C H ED U LE 3. S C H ED U LE 4. S C H ED U LE 1. S C H ED U LE 2. S C H ED U LE 3. S C H ED U LE 4. S C H ED U LE

CARGO1 AMMONIUM NITRATE X X

CARGO2 LIVESTOCK X X X

CARGO3 BULK FREIGHT (Solid) X X X

CARGO4 OIL X

CARGO5 OIL X

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55 Ta nk er s ch ed ulin g b y u sin g o pti m iza tio n te ch niq ue s a nd a c ase stu dy Table 5. Costs SHIP NO NAME OF THE

SHIP/TANKER COSTS ACCORDING TO THE 1. SCHEDULE ($) COSTS ACCORDING TO THE 2. SCHEDULE ($) COSTS ACCORDING TO THE 3. SCHEDULE ($) COSTS ACCORDING TO THE4. SCHEDULE ($)

1 ATMACA 1 (4500 GT) LOCATION= SAMSUN FUEL COSTS=SAMSUN-MERSIN =3000 LOADING=(SAMSUN) (DANGEROUS CONTAINER)= 110 TERMINAL= 42

WAREHOUSING CHARGE (FIRST 15 DAYS)= 3.6

CHARGES OF THE PORT WHICH THE LOADING IS PERFORMED= (OTHER CARGO VESSEL) - PILOTAGE= 350 - TUG BOAT= 303 - WARP= 65 - SHELTERING= 40 - DUMPING OF WASTE (SOLID-LIQUID)(BY LAND)= 20+30

- FRESH WATER= (BY VALVE)= 5 UNLOADING=(MERSIN) (DANGEROUS CONTAINER)= 114 TERMINAL= 42 WAREHOUSING CHARGE (FIRST 15 DAYS)= 4.8 CHARGES OF THE PORT WHICH

THE UNLOADING IS PERFORMED= (OTHER CARGO VESSEL)

- PILOTAGE= 380 - TUG BOAT= 336 - WARP= 75 - SHELTERING= 40 - DUMPING OF WASTE

FUEL COSTS= BANDIRMA= 1500 + 500 LOADING =(AMBARLI ) (TARIFF OF SAMSUN HOURBOUR)

(LIVESTOCK)= 1.5 TERMINAL= 0

WAREHOUSING CHARGE (FIRST 15 DAYS)= 0

CHARGES OF THE PORT WHICH THE LOADING IS PERFORMED= (OTHER CARGO VESSEL) - PILOTAGE= 350 - TUG BOAT= 303 - WARP= 65 - SHELTERING= 40 - DUMPING OF WASTE (SOLID-LIQUID)(BY LAND)= 20+30 - FRESH WATER= (BY VALVE)= 5 UNLOADING= (BANDIRMA) LIVESTOCK= 1.5

TERMINAL= 0

WAREHOUSING CHARGE (FIRST 15 DAYS) = 0

CHARGES OF THE PORT WHICH THE UNLOADING IS PERFORMED= (OTHER CARGO VESSEL) - PILOTAGE= 350 - TUG BOAT= 303 - WARP= 65 - SHELTERING= 40 - DUMPING OF WASTE

(SOLID-LIQUID)(FROM SEA BEING AT ANCHOR)= 20+30

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56 A . D . K ara o ğla n (SOLID-LIQUID)(FROM SEA BEING AT ANCHOR)= 20+30

- FRESH WATER= (BY VALVE)= 6 GRATUITY TO PILOT= 1000 TOTAL= $ 6016.4 TOTAL= $ 5647.5 GRATUITY TO PILOT= 1000 SHIP NO NAME OF THE

2 ATMACA 2 (9200 GT)

LOCATION= BANDIRMA

FUEL COSTS= ISKENDERUN= 700+4000 LOADING= (IZMIR )

(BULK FREIGHT (LIQUID)) TERMINAL= 4

- FRESH WATER= (BY VALVE)= 5 UNLOADING= (ISKENDERUN) (BULK FREIGHT (LIQUID))= 0 TERMINAL= 2

CHARGES OF THE PORT WHICH THE UNLOADING IS PERFORMED= (OTHER CARGO VESSEL)

- PILOTAGE= 800

FUEL COSTS MERSIN= 1700+ 4000 LOADING= ( SAMSUN )

(DANGEROUS CONTAINER)= 110 TERMINAL= 12

CHARGES OF THE PORT WHICH THE LOADING IS PERFORMED= (OTHER CARGO VESSEL) - PILOTAGE= 740 - TUG BOAT= 591 - WARP= 153 - SHELTERING= 100 - DUMPING OF WASTE (SOLID-LIQUID)(BY LAND)= 30+45 - FRESH WATER= (BY VALVE)= 5 UNLOADING=(MERSIN)

DANGEROUS CONTAINER= 114 TERMINAL= 42

- PILOTAGE= 800 - TUG BOAT= 654 - WARP= 165 - SHELTERING= 100

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57 Ta nk er s ch ed ulin g b y u sin g o pti m iza tio n te ch niq ue s a nd a c ase stu dy - TUG BOAT= 654 - WARP= 165 - SHELTERING= 100 - DUMPING OF WASTE (SOLID-LIQUID)(BY LAND)= 30+45

- FRESH WATER= (FROM SEA)= 5

GRATUITY TO PILOT= 1000 GRATUITY TO PILOT = 1000+1000+1000 TOTAL= $ 9275.5 TOTAL= $ 12450.4

- DUMPING OF WASTE

- FRESH WATER= (FROM SEA)= 6

3 TRITON (8500 GT) LOCATION= GEMLIK

FUEL COSTS= GEMLIK-AMBARLI-BANDIRMA= 700 + 700

LOADING= ( AMBARLI ) (TARIFF OF SAMSUN PORT)

CHARGES OF THE PORT WHICH

THE LOADING IS PERFORMED= (OTHER CARGO VESSEL) - PILOTAGE= 675 - TUGBOAT= 543 - WARP= 139 - SHELTERING= 90 - DUMPING OF WASTE (SOLID-LIQUID)(BY LAND)= 30+45

- FRESH WATER= (BY VALVE)= 5 UNLOADING=(BANDIRMA)

- PILOTAGE= 675 - TUGBOAT= 543 - WARP= 139

FUEL COSTS =GEMLIK-BANDIRMA-IZMIT-HAYDARPASA= 700+ 700 + 700 LOADING =( GEMLIK + BANDIRMA ) TARIFF OF BANDIRMA HOURBOUR (BULK FREIGHT (LIQUID))= 4 TERMINAL= 1.5

CHARGES OF THE PORT WHICH THE LOADING IS PERFORMED= (OTHER CARGO VESSEL)

- PILOTAGE= 675 - TUG BOAT= 543 - WARP= 139 - SHELTERING= 90

- DUMPING OF WASTE LIQUID)(BY LAND)= 30+45 - FRESH WATER= (BY VALVE)= 5 TARIFF OF GEMLIK PORT= 6004 UNLOADING= (IZMIT + HAYDARPAŞA) CHARGES FOR IZMIT HOURBOUR= 11363

CHARGES FOR HAYDARPASA HOURBOUR=

(BULK FREIGHT(SOLID))= 4 TERMINAL= 1.5

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58 A . D . K ara o ğla n - SHELTERING= 90 - DUMPING OF WASTE (SOLID-LIQUID)(FROM SEA BEING AT ANCHOR)= 30+45

- FRESH WATER= (FROM SEA)= 5 TOTAL= $ 4457 - PILOTAGE= 675 - TUG BOAT= 543 - WARP= 139 - SHELTERING= 90 - DUMPING OF WASTE

- FRESH WATER= (FROM SEA)= 5 TOTAL= $22535.15

SHIP NO

NAME OF THE

4 AKINCI (7100 GT)

LOCATION= AMBARLI

FUEL COSTS= BANDIRMA= 700 LOADING= (AMBARLI) (SAMSUN PORT TARIFF)

- FRESH WATER= (BY VALVE)= 5 UNLOADING= (BANDIRMA) LIVESTOCK= 1.5 TERMINAL= 0 WAREHOUSING CHARGE (FIRST 15 DAYS)= 0

CHARGES OF THE PORT WHICH THE UNLOADING IS PERFORMED=

FUEL COSTS BANDIRMA- HAYDARPASA= 700+700+700+700 LOADING= ( GEMLIK+BANDIRMA ) CHARGES FOR GEMLIK PORT= 6004 CHARGES FOR BANDIRMA PORT= (BULK FREIGHT (SOLID))= 4 TERMINAL= 1.5

CHARGES OF THE PORT WHICH THE LOADING IS PERFORMED= (OTHER CARGO VESSEL) - PILOTAGE= 610 - TUG BOAT= 495 - WARP= 125 - SHELTERING= 80 - DUMPING OF WASTE (SOLID-LIQUID)(BY LAND)= 30+45 - FRESH WATER= (BY VALVE)= 5 UNLOADING=(IZMIT+HAYDARPASA) CHARGES FOR IZMIT PORT= 11363 CHARGES FOR HAYDARPASA PORT=

(BULK FREIGHT (SOLID))= 4

TERMINAL= 1.5

WAREHOUSING CHARGE (FIRST 15

FUEL COSTS= BANDIRMA-HAYDARPASA= 700+700 LOADING= ( BANDIRMA ) BULK FREIGHT(LIQUID)= 4 TERMINAL= 1.5 WAREHOUSING CHARGE (FIRST 15 DAYS)= 0.15

- FRESH WATER= (BY VALVE)= 5

UNLOADING= (HAYDARPASA) (BULK FREIGHT (SOLID)= 4 TERMINAL= 1.5

FUEL COSTS= GEMLIK-IZMIT= 700+700 LOADING=( GEMLIK ) (BULK FREIGHT (SOLID) CHARGES FOR GEMLIK PORT= 6004

UNLOADING= (IZMIT) BULK FREIGHT(SOLID) CHARGES FOR IZMIT PORT= 11363

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59 Ta nk er s ch ed ulin g b y u sin g o pti m iza tio n te ch niq ue s a nd a c ase stu dy

(OTHER CARGO VESSEL) - PILOTAGE= 610 - TUGBOAT= 495 - WARP= 125 - SHELTERING= 80 - DUMPING OF WASTE (SOLID-LIQUID)(FROM SEA BEING AT ANCHOR)= 30+45 - FRESH WATER= (BY VALVE)= 5

TOTAL= $ 3483

DAYS)= 0.15

CHARGES OF THE PORT WHICH THE UNLOADING IS PERFORMED= (OTHER CARGO VESSEL) - PILOTAGE= 610 - TUG BOAT= 495 - WARP= 125 - SHELTERING= 80 - DUMPING OF WASTE

- FRESH WATER= (BY VALVE)= 10

TOTAL= $ 22963.3 = 10

CHARGES OF THE PORT WHICH THE UNLOADING IS PERFORMED= (OTHER CARGO VESSEL) - PILOTAGE= 610 - TUG BOAT= 495 - PLAMAR= 125 - SHELTERING= 80 - DUMPING OF WASTE (SOLID-LIQUID)(FROM SEA BEING AT ANCHOR)= 30+45 - FRESH WATER= (BYVALVE)

TOTAL= $4196.3

TOTAL= $ 18767

Table 6. Income

SHIP NO NAME OF THE SHIP/TANKER INCOME ACCORDING TO THE

1. SCHEDULE($) INCOME ACCORDING TO THE 2. SCHEDULE($) INCOME ACCORDING TO THE 3. SCHEDULE($) INCOME ACCORDING TO THE 4. SCHEDULE($)

1 ATMACA 1 10000 6000 - -

2 ATMACA 2 10000 12000 - -

3 TRITON 6000 20000 - -

4 AKINCI 6000 20000 10000 20000

Table 7. Profit

SHIP NO NAME OF THE SHIP/TANKER PROFIT ACCORDING TO THE

1. SCHEDULE($) PROFIT ACCORDING TO THE 2. SCHEDULE($) PROFIT ACCORDING TO THE 3. SCHEDULE($) PROFIT ACCORDING TO THE 4. SCHEDULE($)

1 ATMACA 1 3983.6 352.5 - -

2 ATMACA 2 724.5 -450.4 - -

3 TRITON 1543 -2535.15 - -

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A. D. Karaoğlan

Table 8. Fuel consumption assumptions for the ships according to their deadweights

Fuel Consumption($) DEADWEIGHT 0-5000 GT 10000-15000 SHORT 500 1000 MIDDLE 1500 2000 D is ta nc e LONG 3000 5000 3. Results and discussion

Xij= i. Ship/Tanker, j. Schedule (i=1,2,3,4) (j=1,2,3,4)

Max P= 3983.6X11+352.5X12+724.5X21-450.4X22+1543X31-2535.15X32+2517X41

-2963.3X42+5803.7X43+1233X44

s.t.

X11+X22 ≤1 Constraint related to the 1. cargo

X12+X31+X41 ≤1 Constraint related to the 2. cargo

X21 ≤1 Constraint related to the 4. cargo

X21 ≤1 Constraint related to the 5. cargo

X11+X12 ≤1 Constraint related to the ship no:1

X41+X42+X43+ X44≤1 Constraint related to the ship no:4

Xij=0,1 (i=1,2,3,4) (j=1,2,3,4)

The notation of model according to LINDO package programme which is used in computing the solutions and the solution report are given below:

Max 983.6X11+352.5X12+724.5X21-450.4X22+1543X31-2535.15X32+2517X41-2963.3X42 +5803.7X43+1233X44 subject to X11+X22<=1 X12+X31+X41<=1 X32+X42+X43<=1 X21<=1 X32+X42+X44<=1 X11+X12 <=1 X21+X22 <=1 X31+X32 <=1 X41+X42+X43+X44<=1 END INTE X11

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Tanker scheduling by using optimization techniques and a case study 61 INTE X12 INTE X21 INTE X22 INTE X31 INTE X32 INTE X41 INTE X42 INTE X43 INTE X44

LP optimum found at step 6 and the objective value is $9054.79980. Integer solution is found same as in the solution of the expanded model of branch and bound. There is no branches, the optimum is found at the initial solution of the branch and bound algorithm. If we used Xij≥0 constraint instead of Xij=0,1 (i=1,2,3,4) (j=1,2,3,4) constraint, to expand the integer

model and solve the model by branch and bound algorithm by manualy we calculate the below report. The integer solution and the same optimal profit value is computed at the initial solution and we have zero branches. The results are given below as summarized.

OBJECTIVE FUNCTION VALUE 1) 9054.800

VARIABLE VALUE REDUCED COST X11 1.000000 0.000000 X12 0.000000 631.099976 X21 1.000000 0.000000 X22 0.000000 1174.900024 X31 1.000000 0.000000 X32 0.000000 4078.149902 X41 0.000000 3286.699951 X42 0.000000 8767.000000 X43 1.000000 0.000000 X44 0.000000 4570.700195

The optimal schedule is found by assigning 1. ship to 1. schedule, 2. ship to 1. schedule, 3. ship to 1. schedule and 4. ship to 3. schedule and the optimum value of the objective function is $ 9054,8. The results have shown that, the presented model that is refered from the literature, can be applied successfully in real word and gives nearly optimal solutions.

4. Conclusion

In the transportation industry planning and scheduling problems are used commonly. The variety in the problem is due to the many modes of transportation, e.g., shipping, airlines, and railroads. Each mode of transportation has its own set of characteristics. Ship scheduling models optimize the transportation of commodities, while the tanker scheduling models optimize the transportation of oil. So they are vital to world trade and millitary logistics and also scheduling the transportation in shipping industry has its own linear model and solution

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A. D. Karaoğlan

techniques which takes an important place in optimization. In this study a scheduling application in shipping industry is performed to a fleet composed of cargo ships and tankers and the method and the solution is presented. The schedules are constructed intuitively and selecting the best schedule is performed by using branch and bound algorithm. It is clear that a ship requires a multi-million dolar capital investment and the daily operation costs of a ship can be tens of thousand dollars. Consequently, improved fleet utilization can yield significant financial benefit. So this study guides companies and researches who needs to schedule and optimize their transportation systems which are constructed on shipping.

Acknowledgement

I want to thank Associate Professor Dr. Ramazan YAMAN, Professor Dr. Đrem

ÖZKARAHAN and Investment Development and Application Expert Assistant N. Özge TUNÇAY for their great support and constructive comments to this study.

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