MULTIPURPOSE VESSEL FLEET FOR SHORT BLACK SEA SHIPPING THROUGH MULTIMODAL TRANSPORT CORRIDORS

A study about the requirements and cargo transportation demand in the Black Sea as part of a multimodal transportation frame is performed, estimating the potential need of a ship fleet of multipurpose ships. The study performs conceptual multipurpose vessel design and fleet sizing using the long-time experience and statistics in defining main dimensions of the ship and her hull form, resistance and propulsion, weights, stability, free-board, seakeeping and manoeuvrability, capital, operational and decommissioning expenditure, where the optimal design solution is obtained based on the energy efficiency, shipbuilding, operation, and resale costs at the end of the service life. A discussion about possible applications of a different fleet of ship sizes in improving the cargo transportation efficiency considers the vessel's typical operational profile in such a way to maximise the economic impact conditional of the unsteady cargo flow and environmental impact.


Introduction
Models for short sea shipping have been intensively developed in the last two decades in connection to intermodal cargo transportation through the sea and roads, where some recent studies can be found in [1][2][3]. In a series of studies in [4,5], transport chains along the motorway of the Sea of Western Europe and the impact of external costs in short sea shipping services were analysed. The "Glossary for transport statistics" [6] defines multimodal transport as the transport of goods by at least two different modes of transport. Intermodal transport is a particular type of multimodal transport using the same transport unit by successive modes without handling the goods themselves when changing modes. The intermodal transport units are container, swap body or a loaded vehicle travelling on another vehicle.
The EU Directive 92/106/EEC (December 1992) established standard rules for the specific combined transport of goods (considered multimodal transport) between the Member States. According to the Report on Combined Transport (CT) from 2020, approximately twothirds (62%) of the units are containers, 21% semi-trailers and 17% swap bodies. The most used units among the groups are 40' containers -43%; Standard non-craneable semi-trailers-49% and swap bodies, Class A -41%.
Ruse-Varna railway line is an essential part of the transport chain across the countries of 3SI from Port of Varna on the Black Sea to Port of Gdansk on the Baltic Sea. Two transport corridors by train (left) and by truck (right) are presented in Fig. 1 (right). EcoTransIT World 4 (Ecological Transport Information Tool for Worldwide Transports) offers an emission calculator for greenhouse gases and exhaust emissions for different services. Table 1 evaluates two South-North corridors. The amount of cargo is 100 TEU containers (10 t/TEU). The benefits of transporting containers from the Black Sea through the 3SI countries by train are indisputable, and this coincides with the strategic objectives of the European Green Deal.
The objective of the study here is to perform MPV fleet sizing for the transportation flow in the Black Sea region as short shipping through multimodal transport corridors using the longtime experience and statistics in defining the optimal ship design solution obtained based on the energy efficiency, shipbuilding, operation, and resale costs at the end of the service life.

Transportation flow in the Black Sea region
The development of world trade influences the development of ships for the carriage of containers. Specialised container ships are the fastest-growing type of ship, and over 60 years, the container capacity has increased more than 25 times [7]. Despite that, from 2011 to 2016, the demand for container shipments was lower than the available capacity of the vessels. The World Container Index, WCI, has continually been growing with minor exceptions, as shown in Fig. 2.
In the first half of 2019, the Black Sea container terminals of Bulgaria, Georgia, Romania, Russia, and Ukraine handled 1,582,932 TEU containers, including empty containers and excluding transhipped containers (Fig. 2). The number of full containers was 1,176,621 TEU (74.33% from total). The total growth achieved from these five countries in the first half of 2019 was 8.52% compared to the same period from the previous year, 2018. There is an increase in turnover in all countries except Russia. In the first half of 2019, the highest growth was achieved by Georgia and Ukraine, 30.62% and 17.89%, respectively. 57.54% of the processed full containers were imported during this period, and 42.46% were exported.  Eurostat statistics provide the development of container transport by country and for the E-27 as a whole. Fig. 3 presents a comparison between the volumes of container traffic for two EU member states in the Black Sea region. There has been continuous growth for Bulgaria since 2011, while Romania has declined volumes since 2015. Turkey is included in Eurostat's container transport statistics from 2010. Fig. 3 (right) compares the magnitudes of containers transported in Turkey and EU-27 countries. The percentage of Turkey container transport has been 20%-24% for the last eight years.
There are two container terminals in the ports of Poti and Batumi in Georgia. These ports serve 18 million people from the Caucasus region (Georgia, Azerbaijan, and Armenia) and another 145 million from landlocked countries in Central Asia (Kazakhstan, Uzbekistan, Turkmenistan, Kyrgyzstan, and Tajikistan). The growth of transported containers was 13% in the first quarter of 2018 compared to 2017 [10].
The Transport Corridor Europe-Caucasus-Asia (TRACECA)5 is an internationally recognised program to strengthen economic relations, trade and transport communication in the Black Sea region, the South Caucasus, and Central. The route includes the transport system of the 13 Member States: Azerbaijan, Armenia, Bulgaria, Georgia, Iran, Kazakhstan, Kyrgyzstan, Moldova, Romania, Tajikistan, Turkey, Ukraine, and Uzbekistan, including river transportation chains in the region [11,12].
A better alternative to the TRACECA corridor is The Trans-Caspian International Transport Route (TITR), as shown in Fig. 4. Transportation of goods along this corridor began in 2013 by the port administrations of Azerbaijan, Kazakhstan, and Georgia. Proof of the vitality of this corridor is the announcement from "Transport and Logstics.bg" about train composition, which has travelled the distance from Khorgos, on the border between China and Kazakhstan to Izmit, Turkey, in 12 days.
The TITR organises the transport consisting of 43 FEUs carrying various goods. The block-train entered Kazakhstan via Altinkol railway station and travelled to the port of Aktau (Caspian Sea). From there, the containers were transported to the port of Baku in Azerbaijan by feeder ship. After crossing the Caspian Sea, the train got back on track, and through Akhalkalaki, Georgia reached the Turkish city of Izmit on the Baku-Tbilisi-Kars railway. A recent study [13] provided statistical data for containerships visiting Black Sea ports. The ports of Varna, Burgas, Constanta, Odessa, Novorossiysk, Poti and Ambarli were visited by all possible container vessels ( Fig. 5-left). Just over 50% of these ships are "Small feeder" and "Feeder" (Fig. 5 -right).

Ship design
A new design of multipurpose ships is performed to complete the fleet needed for the unsteady shipping through multimodal transport corridors across the Black Sea. A specialised container ship equipped with cell guides has the advantage of the container handling in the port, but the multipurpose vessel is universal. It can be used for different cargo flows within short sea shipping.
The ship design assesses the specific requirements, including aspects that relate to ship hull descriptors, capacity and visibility, ship operation, equilibrium, and initial stability, resistance and propulsion, motor match, lightweight, dead weight, cargo capacity, free-board, manoeuvrability, seakeeping, energy efficiency, ship strength, capital, operating and fuel and oil cost leading to required fright rate and cash flow analysis, due to the many variables involved in the design process.
The optimisation techniques can be categorised into three groups: mathematical programming techniques such as the genetic algorithm, stochastic process techniques such as the Markov process, and statistical methods such as the design of experiments [14]. The optimisation procedure generates a feasible region of possible design solutions. Still, not all design solutions are optimal for any given objective function, resulting in a trade-off between the objective functions [15]. To address the problem caused by the multiple objective functions, the Pareto (Frontier) optimality is employed [16], where a set of all Pareto optimal solutions represent the design space. The Pareto optimal solution is defined as the solution for which any improvement in one objective will worsen at least one other goal [17,18].
The design includes several phases and processes to guarantee that the designed ship follows the customer's needs. The design is defined as a compromise multi-criteria decision support problem [19,20] with multiple goal constraints. Given the owner's requirements about cargo capacity, speed, operational range, regulations and if existing data on similar ships to find the main ship dimensions and ship hull form descriptors [21] and satisfy system constraints related to ship cargo capacity and visibility, ship operation [22], equilibrium, and initial stability, resistance and propulsion, [23] ship-propeller-motor match [22,24], cargo capacity, free-board, manoeuvrability [25], seakeeping [26], energy efficiency [27] and ship strength [28]. Safety bounds are imposed by ports and canals where necessary, and block coefficient lies between specific ranges [29,30].
One of the most unstable governing parameters in the ship design is the voyage costs associated with the fuel costs, ports duties, tugs, pilotage, and canal charges. The vessel's operation at lower speeds results in fuel savings because of the reduced water resistance. The ship's revenue can be earned by transporting the cargo from one port to another port. However, the price to be charged to these services may vary where for the linear operator, it is tradition to include bunker and port handling costs in the freight rates. Defining the freight rate for cargo transportation on the assumed route requires numerous factors such as the real market demand of the region, more precise cost calculations, and others. However, for the present study, a fixed freight rate is assumed.
Two models in the ship design, related to the ship and voyage descriptions, are typically accounted for [31,32]. The first includes the main ship dimensions, hull form generation, hydrostatics, free-board, resistance, propulsion, energy efficiency, lightship weight, cargo capacity, stability, and capital and decommissioning cost. The voyage model includes port sequence, voyage legs data, port/terminal data, cargo handled/port, round trip time/cost, annual cargo, and operational cost. The design solution has to satisfy the constraints related to entire ship systems.
The objective functions defined are to minimise the required fright rate, minimise fuel consumption, and determine the best ship-propeller-motor match. The design governing parameters are ship length, breadth, draft, depth, number of propeller blades, propeller speed, expanded area ratio, pitch ratio, and satisfy the energy efficiency design index, EEDI. The NSGA -II optimisation algorithm [33] satisfying a set of constraints is employed in the present study to ensure that the optimal solution can be obtained quickly with sufficient quantity and accuracy. Several studies related to ship design accounting for the limitations associated with the small and medium enterprises, SME end EEDI employing NSGA-II have been performed recently and can be seen in [34,35].
The conceptual ship design performed here consists of a set of modules. It is considered a process of solving a system of equations that describes the ship performance using consistent approximations. The design solution is obtained by satisfying the constraints and requirements by modifying the controllable variables [32]. The concept design and optimisation methodology adopted here is shown in Fig. 6. The design process starts with defining the governing parameters in developing the ship model where several mathematical modules are implemented, and design variables, objective functions and constraints are determined. The initial hull offset is based on a sister ship. A Lackenby [39] transformation is used to adjust the original offset through the design parameters to calculate the hydrostatic and sectional area curve.
Several modules are used to generate the complete ship model, including operation, capacity, visibility, LW and DW, free-board, equilibrium and stability, seakeeping, bow design, resistance and propulsion, cavitation control, hull-propeller-motor design selection and match, strength assessment, control and manoeuvrability, cash-flow and required fright rate. The Energy Efficiency Design Index expresses the Greenhouse Gas emission footprint. The multiobjective optimisation is performed using the Genetic Algorithm NSGA-II [33].
The current study design a series of ships to be built in SME shipyards and operating in the Black Sea for transporting 20, 40 and 45-foot containers, identifying the main dimensions and ship form obtained from systematic form series input data transformation concerning Lpp, B, D, T and block coefficient, Cb [36], capacity using semi-empirical mathematical formulae from statistics, regression analyses of data of similar vessels [37][38][39] to estimate the structural weight regression equations based on a statistical analysis of existing ships can be used [40][41][42], visibility, free-board [43], initial stability [30], bow, and stern design [30,44], resistance and propulsion [45], and propeller system engine match [22,24], control and manoeuvrability [46][47][48], seakeeping [28], strength [49] energy efficiency index [27,50], capital expenditure CAPEX and operational expenditures OPEX [40,51] that leads to the required fright rate.
The operational and capital expenditure costs are related to changes that appear to be fixed costs, i.e., inflation, policy changes, and others. Still, the voyage costs are variable as a function of the activities occurring during the voyage service. The variable costs depend on every specific voyage, especially on the ports, distance, and cargo.
Parameter estimates of cost are based on design parameters such as ship size, weight, horsepower, etc., using a mathematical relationship between the input parameter and the cost historically determined through the regression analysis.
A cost breakdown divides costs into material, labour, overhead, and profit. The material involves all shipyard purchases: materials, equipment, subcontracted work, outside engineering services, etc. labour includes wages and benefits paid to shipyard employees whose work is directly connected with a ship. Overhead is the sum of all internal shipyard costs that cannot be directly attributed to any given contract [51,52] Estimating the cost of the overhead, O is generally approximated as a percentage of the labour cost. The profit, Pr is calculated as a percentage of the summation of all the material, labour, and overhead costs.
The CAPEX cost is estimated as: where wi and Ci are the weight and cost estimation of the hull structure (i=1) and equipment and fitting (i=2), C3 is the cost of machinery and MHi and Cmi are the man-hour estimation and the cost of man-hour of the hull structure (i=1), equipment and fitting (i=2) and propulsive machinery (i=3). An essential step in the ship design is the trade route analysis, a restricted draft to define the cargo availability and the necessary fuel oil etc. and the total annual cost.
Many cargo ship operators figure on To=350 operating days per year, the remaining days devoted to shipyard repairs. Ships with faster port turnaround have less time for dockside repairs and may require an additional ten days' repair time per year.
The annual operating cost, OPEX, is given as: where Cport is the port duties, nv is the voyages per year, is the average annual crew cost, including benefits, is the annual cost of the crew member insurance, is the annual administrative cost, is the annual docking cost, is the annual fuel and oil cost, and ℎ is the annual handling cost.
The ship's revenue can be earned by transporting the container from one port to other ports, either door to door services or port to port. However, the price to be charged to these services may vary depending on several factors. For the liner operator, it is tradition to include bunker and port handling costs in the freight rates. However, for the case study purposes, a fixed freight rate is assumed.
To define the freight rate for the transportation of cargo on the assumed route requires to involve numerous factors such as the real market demand of the region, more precise cost calculations, and others. The decision making of investment is made through the standard discounted cash flow approach, which is based on the net present value, which is the sum of expected future cash flows minus the first investments. The required net profitability rate is assumed as = 2%, and the years of ship life operation are = 25 years. The capital recovery factor, , is: The average annual inflation rate is assumed as = 3%, and the income tax rate is = 15%. The tax correction, is defined as: The discounted annual average cost of the investment, , , is: where is the annual fuel cost. The economic feasibility analysis covers the cash flow, which contains the costs of the ship from the construction to operating costs throughout the service life. On the other hand, with the cargo transported by the ship during the entire operation time, it is possible to calculate the total cost/cargo transported ratio, whose value equals the costs dispensed with the revenue obtained. This value is known as the required fright rate, , , defined for the th ship in th route is defined as: where , is the annual cargo capacity. The fleet design meets annual demand for cargo transportation, so the objective is to estimate the number of ships needed (dimensioning a fleet) and the dimensions of each of them and the number of trips made by each vessel over a year of operation.
For long journeys, the larger the ship (capacity to carry cargo), the better, because therefore, in one trip, more cargo is transported. With this, a smaller number of ships would be necessary for the composition of the fleet. Also, the ratio denominator that makes up the required freight rate decreases. On the other hand, when the size of the ship is increased excessively, the need arises to strengthen the ship structure (the larger the ship, the more flexible it becomes) with the use of more steel or high tensile steel (increasing the weight of steel and therefore the construction cost), also, the larger the ship, the higher the time spent in the ports and terminals for loading and unloading tends to increase the costs, that is, the numerator of the Required Freight Rate ratio. Thus, the solution is to find the set of main ship dimensions, which allows the vessel to transport the maximum load at a more competitive freight rate, lower total cost.
To minimise the required fright rate, , the governing parameters are the main dimensions of the ship, service speed, block coefficient, service life of the ship, price of steel used in the construction of the ship per tonne, distance navigated, unloading, and unloading times of ports and terminals where the ship is expected to operate, etc.
A systematic variation of the ship's main dimensions is employed to obtain the best freight rate. This is done as one of the dimensions was fixed, and the others were varied until all had been varied. From this set of main dimensions, the one that provided the lowest required shipping rate is adopted.
The depreciation is also analysed in the present study, which reflects the loss of performance due to age, higher maintenance expenses, a level of technical obsolescence, and expectations about the economic life of the vessel. It represents the amount the ship was paid in cash when the project started. It is just bookkeeping, so profit will be lower than cash flow by that amount. The interest, amortisation, cost of interest and depreciation and depth balance, calculated for a ship of L=150 m, B=28.3 m, D=13.04 m, T=7.86 m, is given in Fig. 7.   Fig. 7. Interest, amortisation, cost of interest and depreciation and depth balance The parameters that are included in the economic analysis are the required net profitability rate, 2%, the number of years of ship operation 25, years, the average annual inflation rate, 3%, the residual value, 15%, the depreciation time, ten years, the contract assignment, 7/1/2020, the ship delivery, 7/1/2021, the time of construction, 12 months, and the days of operation per year, 350 days.
The Greenhouse gas emission footprint is expressed by the Energy Efficiency Design Index, EEDI and the optimisation procedure guarantees that the acceptable levels are satisfied. EEDI is defined in [53] and its required values in [54], adjusted by a reduction factor relative to EEDI baseline as a function of the primary and auxiliary engine power deadweight and service speed.
To avoid vessels becoming underpowered, IMO published a guideline related to the minimum propulsion power to maintain the manoeuvrability of ships in adverse conditions [25,54]. Two IMO assessment levels have been developed: Level 1, which is a simple, generic method based on DWT, and ship type only input, leading to a crude, conservative approach, and Level 2 is based on individually estimated ship resistance component relating to the power prediction.
The simplified assessment procedure is based on the principle that if the ship has sufficient installed power to move with a particular advanced speed in head waves and wind, the ship will keep the course in waves and wind from any other direction. The minimum ship speed of advancement in head waves and wind is thus selected depending on ship design, in such a way that the fulfilment of the ship speed of advance requirements means fulfilment of course-keeping requirements. The energy efficiency design index, EEDI, calculated for the analysed multipurpose vessels is given in Fig. 8. The recent development of short sea shipping in the Black Sea region has shown that intermodal transportation will play an important role. At the same time, the coaster fleet of the Black Sea region is of considerable age, and the increased freight rates enforce new orders of ships [55].
The designed ships in the present study are in the range of 4,000 to 14,000 DWT. The design governing parameters are the length between perpendiculars, moulded breadth, moulded depth, moulded draft, service speed, number of blades, propeller speed, expanded area ratio, pitch ratio and the constraints assumed for the present study are shown in Table 2, some of which have also been used in [35]. An additional constrain about the minimum engine power is included to guarantee the engine power is sufficient to perform satisfactory manoeuvring in severe weather conditions [56]. The Black Sea routes and assumed cargo demand used in the present study are given in Table 3.
The ship design and fleet sizing formulated here is for a given cargo to be transported at a constant speed between specified ports. Following the IMO regulations and data on similar ships to find the principal dimensions of ships and the size of the fleet. The objective functions defined are to minimise the required fright rate, minimise fuel consumption, and determine the best ship-propeller-motor match. The design governing parameters are ship length, breadth, draft, depth, number of propeller blades, propeller speed, expanded area ratio, pitch ratio, and satisfy the energy efficiency design index, EEDI. The NSGA -II optimisation algorithm [33] satisfying a set of constraints is employed in the present study to ensure that the optimal solution can be obtained quickly with sufficient quantity and accuracy.  The principal objective is finding an optimal fleet ship size, considering the possible negative impact of a drop in the volume in the cargo flow. When the fleet size is defined, the cargo traffic flow is assumed at the beginning of the year. When the need for transporting cargo is greater than the ship cargo capacity, the excess cargo transportation will be subcontracted to an alternative transportation (charter) deliverer and returned by the end of the year. If the cargo demand requires fewer ships than the owned transportation cargo capacity, no chartering will occur.
In many cases, the cargo flow is unstable. It needs to be described in a probabilistic manner and to determine the optimal size of the ship fleet, the expected total cost per unit time needs to be minimised.
The present study compares the transportation of cargoes using different sizes of containers. The demand for transported cargo is given in tonne instead of the number of containers needed for this purpose. It is assumed that only 60% of the cargo capacity of containers is used for cargo transportation. This is explained by the fact that the total weight of the container should not exceed the maximum gross weight and depends on local road and rail transportations capacities. If the container is loaded to its maximum capacity, the weight needs to be evenly distributed over the floor area. If the cargo lays over about 50 % of the floor-length, its weight cannot exceed 66 % of the container max payload in the case of 66% of the length is 75% of the weight and for 75% is 80%.
In terms of cost, the price of containers depends on the size and the condition. The cost of the 20-foot is assumed to be approximately 75% of the 40-foot container. For new 20 -foot containers is in the range of $5,000 -$6,000 and for the 40-foot containers from $6,500 to $7,000, The price is different for used containers where for 20-foot containers is between $1,200 and $2,500, where for 40-foot containers is between $2,600 and $3,300 as can be seen in [57].
Each of these container sizes is designed to carry specific types of cargo, where 20-foot containers are related to more weight than bulk cargo, contrary to the 40-foot containers used to transport bulky loads. The 40-foot containers can't transport twice the 20-foot cargo weight, but at the same time, the 40-foot containers can transport more than twice the cargo volume of the 20-foot containers. However, cargo space is not always the most critical factor in choosing the size of containers for transporting specific cargoes.
The 40-foot containers provide more space for loading and storage, proportional to the length of the containers used. The 20-foot containers are best for transporting dense objects and heavy loads. For light cargoes, the 40-foot containers are preferable, and they can load more than twice the volume of the 20-foot containers.
There is a loading time issue with the 20-foot containers as they can take more time when being shipped by train or truck to reach the cargo ship since the 20-foot containers are transported on the rail in pairs. However, the shorter 20-foot containers are more adaptable to the limited cargo hull space, especially for small ships with a shorter cylindrical part of the ship hull.
The Intermodal Association of North America's ETSO database provides a monthly view of movements of 20-foot, 40-foot, and 45-foot containers in the Inland Point Intermodal market [58]. A long-term evolution has been shown in Fig. 9 wherein the year 2020, the 40-foot containers comprise over 51 per cent, 20-foot containers take 34 per cent, and 45-foot units are 15 per cent.  Table 4 shows the main characteristics of eleven designed ships, i=1, …, 11, of different sizes, that are part of a fleet for transporting cargo between Varna and Poti ( =1), including the ship units needed to cover the transportation demand and transported by 20, 40 and 45-foot containers for any ship. The ship's length is step-wise changed by five meters in the range between 100 and 150 m. The ratio of Lpp/B ranges from 5.1 to 5.3, Lpp/D ranges from 9.6 to 11.5, B/T ranges from 2.5 to 3.2, and B/D ranges from 1.6 to 2.5. The vessel speed Vs and Cb were constant at 14 knots and 0.63, respectively, varying the ship's breadth, depth, and draft. It can be noticed from Table 4 that for transporting similar cargo weight, the shorter ship, due to the shorter cylindrical part of the ship hull, are more efficient in transporting 20-foot containers compared to 40 and 45-foot containers. From a length of 150 of the ship, as an example here, a similar efficiency of transporting 20, 40 and 45-foot containers are observed.

Ship fleet sizing
The optimisation of fleet composition or completing the fleet is a complex problem that involves many variables and constraints. This task can be resolved in two stages [32]. In the first stage, after analysing the possible cargo flows, available financial resources, the number of the owned ships and other data determines the number of ships to be built. At the second stage, completing the fleet leads to defining the technical specification for the designed ship. This type of problem was solved in [59] in optimising the fleet composition.
The ship fleet sizing is based on the formal voyage description of the ship mission, for a round trip voyage of a set of ports and the inter-connecting routes, where for a given ship, the times and costs spent during the voyage to be defined and for a round trip detailed estimates of the OPEX are estimated, and a more elaborated technical-economical assessment of the ship as an investment is performed. The ship's optimal main dimensions and characteristics are defined by exploring ships' characteristics and running systematic variations of the main dimensions and an optimisation procedure by satisfying the imposed constraints [5,60,61].
The route established for the operation of the ships is Porti → Porti+1 → Porti and using only one type of ship of different sizes, transporting the maximum cargo of Lψ, max. The number of cranes in ports that any ship can use is defined as a function of the length of the ship as The waiting time (days) in ports in a round trip is defined as: The transportation efficiency, Etr of any fleet configuration composed of n identical ships that operate in m ports is capable of transporting Li, cargo per round trip and performing , round trips in satisfying the cargo demand, D em is defined as: Fleet sizing aims to define the number of ships and sizes that need to be involved in cargo transportation producing a maximum benefit for the fleet. The fleet may consist of n different sizes of operating ships. There are several approaches to estimate the unit cost accounting for various factors. For example, the sum of operating, voyage, cargo handling, and capital costs per year is divided by the dead weight of the ship [63]. Estimating the freight rate for cargo transportation in the analysed route requires many factors, including the market demand, which is not the objective of the present study. The assumed freight rate reduces the shipping cost and makes a satisfactory level of earning in the present study. The freight rates for transporting cargo by the i th ship in  th route is estimated as FRi.
In addition to owned vessels, more vessels on a time-charter basis can be leased within a considered period, , ℎ − of the i th ship in  th route at the rate, , ℎ − . For each ship and route of transportation, the constant annual and variable costs , are defined. The constant annual cost is associated with the capital expenditure and related required net profitability rate, a number of years of ship operation, average annual inflation rate, LIBOR (London Interbank Offered Rate), bank premium, capital recovery factor, tax correction, discounted average yearly cost of the investment, capital recovery period, depreciation time own investment, which lead to the constant depreciation payment. The variable cost is associated with any other operational cost, which in general varies with time.
The arrival rate of ships to ports may be assumed to follow the Poisson distribution with an arrival rate. The annual downtime cost for ships in the fleet, waiting in the port and handling cargo, − , is defined as: The total costs, Cfleet includes the cost of operation, constant cost, leasing of ships on a time-charter basis. The cost of the ship-OFF, out of use, because of the lack of cargo leading to a reduction of the demand to be transported, , is assumed as 15 % of the operational cost.
A decrease in cargo transportation demand also reduces the fleet's revenue and operating costs. Each vessel of the fleet has to contribute to the revenue of the fleet during the service life. Otherwise, it is more efficient to be leased on a time-charter basis. The time-charted-OUT ships contribute to the revenue without considering the constant costs of the vessels.
The optimal fleet ship size is based on the maximum profit of the fleet cargo transportation, considering the possible negative impact of a drop in the volume in the cargo flow. When the fleet size is defined, the cargo traffic flow is assumed at the beginning of the year. When the need for transporting cargo is greater than the ship cargo capacity, the excess cargo transportation will be subcontracted to an alternative transportation (charter) deliverer and returned by the end of the year. If the cargo demand requires fewer ships than the existing transportation cargo capacity, no chartering will occur.
In many cases, the cargo flow is unstable, and to determine the optimal size of the ship fleet, the expected total cost per unit time needs to be minimised. The demand per unit time, as a simplification of the problem here, is assumed to be distributed by a  The variable fleet cost related to charter-OUT ships is considered zero since renting the ships is expected to be covered, and the total variable fleet cost is defined as: The cargo-handling capacity of the cranes is assumed as Ccran =360 Cont./day. The operation time of the ship is considered as 350 days per year. The port route distances and geographical locations have already been shown in Table 3. At defining the fleet size, a critical parameter is the cargo flow between analysed ports. For each cargo flow direction, the lower and upper levels of transported cargoes, used as an example here, are shown in Table 5.
L,min 1,000,000 500,000 250,000 250,000 L,max 2,000,000 1,000,000 500,000 500,000 In addition to owned vessels, more vessels of a similar tonnage on a time-charter basis can be leased within a considered period , is assumed as 1.5 times the annual variable cost of the i th ship in the ψ th route. For each ship and route of transportation, the constant yearly cost, and variable cost, are calculated based on the assumption that the required net profitability rate is 2%, for 25 years of ship operation, an average annual inflation rate of 3% and LIBOR (London Interbank Offered Rate) of 8% [18,34].
The fleet number of ships transported a cargo of different designed ships as a part of the fleet composed of identical vessels as a function of ship length are shown in Fig. 10, and Fig.  11 shows the associated annual round voyages and fleet-annual average cost.
Using 20-foot containers to transport the same amount of cargo requires a smaller number of ships. The difference between the number of ships when 40 and 45-foot containers are used is only seen for a very short ship of L=100 m.
The using 40 and 45 containers transport more cargo for a ship with a length greater than 140m. The fleet annual cargo round voyages and the average cost for the ship using 20-foot containers are less than 40 and 45-foot containers (see Fig. 11).
The fleet-cargo transportation efficiency, calculated as the ratio between used and existing cargo space, is shown in Fig. 12, where the fleet is composed of n identical ships that operate in m ports capable of transporting Li, cargo per round trip and performing , round trips in satisfying the cargo demand, using different sizes of containers.
It has to be stressed that the number of containers that any ship may transport is not only based on the maximum container capacity of any particular ship, but it is also conditional to the static equilibrium and stability, which can be made much easier with a small container since they present more degree of freedom to be placed in different location of the dedicated cargo place and require less ballast.
In the case of unstable cargo flow, the expected total cost per unit time needs to be described in a probabilistic manner to determine the optimal size of the ship fleet. The defined ship fleet annual costs and ship fleet profit are shown in Fig. 13 to Fig. 14     The distribution of the unit of different ship sizes for transporting cargo in the  th route, conditional on the own used ships and cargo transported only by 20-foot containers, is shown in Fig. 15 and for 40-and 45-foot containers in Fig. 16. The total distributed units of ships of different sizes should equal the own ships (Fig. 15, left and Fig. 16, left) and chartered-in (Fig.  15, right and Fig. 16, right). The x-axis in Fig. 15 and Fig. 16 shows the capacity of i th ship-size used in all routes. The y-axis shows the transported cargo in any  th route by all sizes of ships, and the z-axis shows the units of different ship sizes used in different routes. In some cases, one ship size serves in several routes. However, due to the lower utilisation, some ship sizes will not be utilised, and the cargo transported by the lower used ship size will be taken from the ship sizes that are more utilised. The diversity of ship sizes in the fleet will be reduced as a function of the number of owned ships.
However, it has to be pointed out that the theoretically estimated optimal fleet size solution and its usage in transporting the demand cargo contain partial ship units of different ship sizes, as can be seen in Fig. 15 and Fig. 16. The actual, real ship size units will be defined by rounding up the theoretically estimated ones, which will lead to diverse and most conservative ship size unit utilisation. The rounding up process should start from the most dominant used ship sizes down to the less prevalent used ship sizes keeping the fleet's total capacity equal or bigger to the cargo demand needed to be transported in different routes.  The cargo flow is assumed to be unstable, and to determine the optimal size of the ship fleet, the expected total cost per unit time needs were minimised. For each fleet, the average waiting and handling time and yearly voyage time was analysed. In Case 6, the optimal number of ships and size from the total cost viewpoint is six ship units. The fleet's average waiting and handling time is 44.5%; that is, the voyage time is 55.5 % of the service time. A high utilisation for the fleet is achieved only when it is being operated efficiently. For all analysed fleet ship configurations, the traffic intensity is less than one, which leads to no queue being built up because ships are arriving slower than processing time.

Conclusions
The study presented here performed multipurpose vessel fleet sizing for the Black Sea short shipping through multimodal transport corridors using the long-time experience and statistics in defining the optimal design solution obtained based on the energy efficiency design index, shipbuilding, operation, and resale costs at the end of the service life. Cargo transportation demand in the Black Sea as a part of a multimodal transportation frame was performed, estimating the potential need for ship fleet of multipurpose ships. A discussion about possible applications of a different fleet of ship sizes in improving the cargo transportation efficiency to maximise the economic impact conditional of the unsteady cargo flow and environmental impact. The study has shown that using 40-foot containers has some advantages compared to 20 and 45-foot containers. The encountered design solutions demonstrated a high utilisation for the fleet and operated efficiently without creating a queue since ships are arriving slower than processing time. A formulation for ship fleet sizing was used, simulating the transportation of only one size of containers at a time, but it can be easily extended for more realistic scenarios. Different assumptions were made related to the costs and cargo demand that are not essential for the formulation but needed for the example calculation.