Optimization of Capital Distribution and Composition of a Shipping Company Fleet Through Evolutionary Algorithms

Abstract

The research examines the optimization of fleet management of a shipping company through control algorithms, as finding an algorithm that will reduce a marine company’s exposure to risk by diversifying its fleet composition is one way to make it dominant. The study focused on three companies that, in 2014, invested US$$1 billion in a fleet of tankers, using three different techniques to optimize their fleet composition: equal number of ships of all types, the established risk minimization model and the proposed Risky Asset Pricing maximization model. Seven years of data was used for the synthesis and 4 years of data was used for the evaluation. The research findings show that classic portfolio management through risk minimization is ineffective, as it appears to reduce performance below what is a random or evenly distributed fleet. Comparing the three methods, the superiority of the Risky Asset Pricing model is clear. This algorithm looks for solutions where the demand for ships is low but has enormous fluctuation potential and seeks to identify ships that are at high risk with great potential for price increases to maximize investor returns. The value of the research lies in the identification of methods to optimize capital distribution and composition of a shipping company fleet, which presents valuable insights for the benefit of scholars and maritime companies. Moreover, and contrary to extant works that focus on Markowitz’s theory, this article instead describes how evolutionary algorithms can be used to optimize fleet management.

Keywords

Evolutionary algorithms optimization fleet management shipping risky asset pricing model

Introduction

Maritime cycles are divided into three general categories based on their periodicity: short-term shipping cycles, long-term shipping cycles and seasonal shipping cycles. Short-term shipping cycles are influenced by daily and short-term events but could nevertheless be said to have periodicity. The periodicity is not constant, but there are certain phases, and each cycle goes through all phases. Short-term shipping cycles are immediately visible and cannot be predicted, as their duration has been observed to range from 3 to 18 years (Stopford, 2009). Long maritime cycles, although they have little influence on the decisions of a shipowner, have an average life span of approximately 50 years (Rodrigue et al., 1997). Seasonal shipping cycles are very short in duration and are mainly influenced by the normal fluctuations in the marine goods offered. In this research a maritime cycle will be referred to in the short term, except when explicitly stated otherwise.

There is a direct correlation between maritime risk and maritime cycles. Every time the freight offered increases, it creates a high demand for ship availability. Because shipowners are unable to respond immediately to this requirement as a ship’s order takes more than a year to complete, an imbalance in the system happens so that cargo holders (charterers) pay more to serve. Shipowners, seeing a freight increase, understand that there is more demand and try to increase their fleet, but they cannot predict what the demand will be in 2 years when the new ships will be delivered. Also, the international performance of companies is influenced directly and indirectly by other factors such as institutional drivers (Torkkeli et al., 2019). One of the major goals of many institutional bodies, which deal with marine regulations, is the reduction of the total number of accidents (Lois et al., 2004).

The main method of managing the risk used in shipping is to divide the fleet into different markets with the main focus being to share the risk with the freighters. This happens when the shipowner decides to move to the time charter market to lease his/her ships at a projected fare price. However, shipowners do not wish to co-manage the risk with the charterers, let alone move the spots.

In the present research beyond this strategy, we will study the relevant factors that influence a shipowner’s decisions on the fleet’s composition to try to find an algorithm that does not manage the risk in the spot market by diversifying its fleet.

The Theoretical Foundations of Evolutionary Algorithms

Algorithms are used for marine surface vessels, such as a deterministic path planning algorithm that computes a practical and COLREGS-compliant navigation path to prevent vessels from being on a collision course (Tam & Bucknall, 2013). The algorithm was evaluated with a set of test cases, simulating various traffic scenarios. Different aspects of the algorithm were examined, such as the output consistency from different perspectives, practicality of the navigation path, and computational performance.

Evolutionary algorithms are used in neuroevolution for marine navigation. In neuroevolution, units are treated as individuals in a population of artificial neural networks, which, through environmental sensing and evolutionary algorithms, learn to perform given task efficiently (Lacki, 2016). The main task of the system is to learn continuously from environmental influences and to predict the values of navigational parameters of the vessel, after a certain amount of time. The result of a prediction may occur as a warning to a navigator about an incoming threat. Evolutionary algorithms have been used to search for an optimal set of safe trajectories for all the ships involved in an encounter (Szlapczynski, 2013). The methodology was originally designed for open waters or restricted waters when only the standard Convention on the International Regulations for Preventing Collisions at Sea rules apply. The methodology has been designed for possible application in Vessel Traffic Service centers. The examples included are the results of the computer simulation tests carried out for the Gulf of Gdansk TSS to illustrate the methodology’s effectiveness and functional scope.

Zacharioudakis et al. (2011) formulated the mathematical model of the cycle cost for the liner shipping company and attempted to optimize the operational profile of company assets in regards to a specific network of routes of cargo flows and vessel portfolios. In other words, it attempted to give a practical solution to deployment problem of the modern shipping company fleet. This was achieved by developing a generic cost model methodology that aims to minimize total operating costs by using genetic algorithms to optimize various predefined attributes such as operational speed. The finalized model could be applicable to liner shipping companies for optimizing liner networks, as well as for simulating and examining possible scenarios and what-if analyses. Also examined was an optimization model to define the technical and operative features of fleets, which maximizes the opportunities for success, in terms of cost and time, for multimodal chains against the road (Martinez-Lopez et al., 2015). This model regards the relationships among technical alternatives for fleets, vessels, port facilities, and cargo units and their influence on the activity of “many to many” transport networks through short sea shipping. A multi-objective evolutionary algorithm, the NSGA-II, has been applied to resolve this model as applied to multimodal chains between Spain and France through the Atlantic coast. Its application allows the verification of the utility of the proposed model. Finally, the most suitable fleets for the sea motorways Vigo-St. Nazaire and Gijón-St. Nazaire have been identified.

A genetic algorithm, coupled with a greedy heuristic and later with a block-based evolutionary algorithm, was also used to examine an integrated problem of optimizing the operations at a commercial bulk material port terminal (Pratap et al., 2016). They simultaneously optimized the stockyard operations and rake schedule for outbound cargo, in conjunction with the arriving vessels and the status of the stockyards at the port. The study resulted in a significant reduction of turnaround time for rakes at the port terminal, which in turn led to monetary savings. The model also automated the day-to-day operational decision-making at the port. Evolutionary algorithms and especially multi-objective evolutionary algorithms have been used to examine two conflicting objectives: total delay and total financial loss. Liner shipping is vulnerable to many disruptive factors such as port congestion or harsh weather, which could result in delays in arriving at ports, and in both financial and reputation losses. The vessel schedule recovery problem (VSRP) is concerned with different possible actions to reduce the effect of disruption, such as a strategy in VSRP which is called the speed-based vessel schedule recovery problem (S-VSRP). Pratap et al. (2016) evaluated the problem in three scenarios (i.e., scalability analysis, vessel steaming policies, and voyage distance analysis) and statistically validated their performance significance. According to experiments, the problem complexity varies in different scenarios. Liner shipping companies use multi-objective evolutionary algorithms in maritime logistics collaboration in the joint-routing network design. The model is called the ship routing problem and two objectives that are minimized are total cost and deviation in fair cost proportion. The method combines NSGA-II and the principles of effective genetic algorithms found in the literature, and an example of the application with data background was from the Indonesian archipelago. The application of evolutionary algorithms and polynomial interpolation is also used in the ship evolutionary trajectory planning method and its comparison to classic approach, where trajectory is modeled by straight lines. Evolutionary algorithms are groups of methods that allow the computation of a collision-free trajectory in real time, while polynomial interpolation models a smooth trajectory, which keeps continuity of velocity and acceleration values along a path in opposition to straight lines approach.

Fleet Composition Optimization Methods

In modern portfolio theory, models based on Markowitz’s theory (1952, 1959) have prevailed. These models are based on the assumption that the optimal portfolio is the one that maximizes profit for a given risk or minimizes the risk for a given profit. Using Markowitz’s theory to optimize the composition of a fleet, we apply the MV-Model, below, but in the reverse and consistent with the Risky Asset Pricing model (Equations [1]–[3]):

\min i m i z e μ_{ρ} = E (R_{p}) = \sum_{i = 1}^{n} Ε (R_{i}) q_{i}

(1)

\max i m i z e σ_{p} = \sqrt{\sum_{i = 1}^{N} \sum_{j = 1}^{N} q_{i} q_{j} σ_{i j}}

(2)

S u b j e c t t o \sum_{i = 1}^{N} \sum_{j}^{Ν} q_{i} v_{j} = B u d g e t

(3)

In this model, inputs are the risky returns and fluctuations of the fleet, and the outputs are the weights, which represent the percentage of total amount invested, corresponding to each type of ship. All solutions to an MV problem form the MV Efficiency Frontier. But there are restrictions to find the best fleet since a shipping company wishes to invest a certain amount with little divergence.

B u d g e t - A l l o w a n c e \leq \sum_{i = 1}^{N} q_{i} v_{j} \leq B u d g e t + A l l o w a n c e

(4)

where q_i is the number of ships of type I, and v_j is the value of this ship type.

In their research, Lin et al. (2001) developed an evolutionary algorithm based on NSGA-II and GENOCOP techniques to solve restriction problems. The criteria for solving the problem were fixed transaction costs and minimum lot costs. The genes of each individual in the population expressed their weight in multiples of lots, and the coding was intact. SBX techniques for crossing and PM for mutation were used to initialize the population. The method showed that it was possible to find solutions and that it was also an NP-difficult problem that did not significantly increase the complexity of the calculation.

Fieldsend et al. (2004) focused on combining optimization with the MV-model. They added a third objective function that evaluated the number of securities structured by prospective. In this problem, the solutions were represented by weights on the total amount invested. In order to evaluate the optimal portfolio with one factor, taking into account the number of shares for its suitability, all combinations of number of securities should be evaluated with the MV-model. This added great computational complexity to the problem. In their work, they suggested that the evaluation of the two objective functions of the MV-model be carried out in parallel with the third criterion of minimizing participation. In particular, the algorithm they proposed is for each board to use information about the best weights on all fronts at the same time. The algorithm generates an n-set of k candidate portfolios, where k is the number of entries and n is the range of all possible k. In each generation the algorithm chooses one set at random and one individual from that set. Then with a 50% probability of either changing only the weights using the Dirichlet method or changing +/− the number of securities. Then it compares the new portfolio with the rest of the same set of crowd, and if not dominated, it adds and subtracts the dominant ones.

Streichert et al. in 2004 used the Markowitz model in combination with limitations on the number of entries, but also set the limit for the minimum amount of participation to avoid few investments in large amounts, trying to minimize transaction costs. Two variables were used for coding the solutions for each gene: one for the actual numbers to encode the weights and one for the binary coding variable indicating whether or not the securities were included in the portfolio. For the evolution cycle, the algorithm uses the NSGA technique. Due to the strong limitations of this technique, the evaluation of the objective functions is done in such a way that those individuals who violate the first criteria of the algorithm are rejected. At the beginning they are evaluated for the number of shares and for the minimum amount of participation. If they do not meet the second criterion then the weights are reset before moving to the third function that will evaluate the minimum lot size. Other methods for portfolio optimization using evolutionary algorithms are those developed by Subbu et al. (2005), Tsao and Liu (2006), and Branke et al. (2009), which search for an optimal portfolio with other constraints such as Rule 5-10-40 applicable to German Law and the Mean Value at Risk-Model as proposed by J. P. Morgan.

Research continues to find effective MOEA (multi-objective evolutionary algorithms) methods to solve the problem. Lin and Liu (2008) attempted to integrate the MV-model with the least-lot constraint and created three different models. In the first model, they created an objective function based on a demand performance and limited to integer solutions, attempting to minimize portfolio risk. But it did not appear to perform well due to the limitation of integer solutions. The second model used the technique of measuring the deviation of the tested portfolio from a target portfolio. By minimizing this distance, the optimal portfolio was detected in the search area. The third model used fuzzy logic programming. The coding of the solutions was done by converting the investment amounts into integers with the appropriate technique and the yield was applied as a penalty technique. In the end, the most efficient technique was measured to be that of vague logic.

Aranha and Iba (2007) used their single objective function algorithm to set the goal of continuous optimization of a structured portfolio with a view to minimize transaction costs. They did this by adding expected returns and subtracting the cost of risk. The Euclidean distance of weights from period t-1 to period t was used to measure costs. For the coding of the population atoms each gene was designated with two variables. The first being binary and representing whether or not the securities were included in the portfolio and the second variable being real numbers and representing the weight of the securities. The classical methods of genetic algorithms were used for crossover and mutation, which added the ability to copy material from the previous period to the next and the choice of objective function between Euclidean distance and Sharpe’s measure randomly in each generation. This technique, after evaluating it, appeared to perform well, finding optimal portfolios using the NASDAQ indices but did not perform well with the NIKKEI indices.

Design and Development of a Fleet Optimization System

In applying evolutionary algorithms to the composition of a fleet, the most important thing to determine is the objective function that will evaluate its efficiency and judge the suitability of a fleet’s reconstruction at a given time. The simplest model for generating an objective function is the simple Markowitz’s theoretical model where the objective function evaluates the efficiency of the portfolio structure, which is reversed to be compatible with the Risky Asset Pricing model through a single maximization function. This function evaluates the divergence of the fleet and gives us the maximum divergence portfolio:

\max i m i z e f_{s i m p l e} (q) = \frac{1}{σ^{2} (q)}

(5)

where

q = (q_{1}, q_{2}, . . . . ., q_{N})^{T}

is an N-dimensional vector, with N as the number of ships that build the fleet:

\forall_{i}, q_{i} \geq 0,

\sum_{i = 1}^{N} q_{i} v_{i} = B u d g e t

Through the simplest model we find the portfolio of maximum risk and on that basis we begin to build our objective function, introducing the constraints we want in this specific implementation. To the implementation, we will apply an objective function of third factors:

f (q) = K (f_{1} + f_{2} + f_{3} + f_{4})

(6)

where f₁ = evaluation function of fleet diversification

f₂ = risk assessment function of the fleet

f₃ = function of evaluating the present value of the fleet in relation to the same number of ships at the historically lowest market price

In order to be able to integrate the objective function factors to calculate a specific value, we introduce the concept of net return as follows:

N Q (t) = {\sum_{i = 1}^{n} q_{i}}_{} v_{i}

(7)

where Q is the number of each ship, and V is the value of each ship.

To determine the differentiation of the fleet, we introduce the notion of the total number of ships and the minimum total value. In this way, we lead the algorithm to the preference of choosing many types of ships over a few expensive ones with the possibility of being trapped in a local extreme in the area of possible solutions.

f_{1} (t) = N W (t) \sum_{i = 1}^{n} q_{i}

(8)

where t is the time base of the fleet study.

To determine the cost of the fleet risk, we use historical data to find the variance of each ship type. The function that evaluates the combination is as follows:

f_{2} (t) = N W (t) σ (t)

(9)

where f₂ determines the expected losses over time t.

In our study, we are trying to predict different criteria, where each type of ship is in the current shipping market so that we can make a decision on whether to buy or sell ships (Beenstock, 1985). With the rise of the others, due to the increase in orders, the prices of ships of each category are going up. So, the price increase of a ship in relation to the maximum fluctuation it may have historically contains all the information about where the market is located. Buying a ship whose price is at a historically high level indicates that it will soon decline and offers no opportunities. On the contrary, the price at a historically low level indicates that this type of ship has high opportunities. To evaluate the present value of the fleet, we find the historically lowest price and the historically highest price for each type of ship in the last 15 years. We then find the over price ratio as follows:

Over price ratio = (Current value – Historical minimum value)/(Historical maximum value – Historical minimum value)

Then the function that evaluates the combination is as follows:

f_{3} (t) = N W (t) (𲈏_{i = 1}^{N} o p_{i} + λ)

(10)

where op is the over price ratio of each ship.

λ a factor determining the influence of the over price ratio on the selection of the optimal fleet composition.

Then, the objective function evaluation of the fleet performance takes the following form:

f_{f i t n e s s} = N W (t) (\sum_{i = 1}^{n} q_{i} + σ (t) + (\prod_{i = 1}^{N} o p_{i} + λ))

(11)

Clarkson Research data (2019) as published in Shipping Intelligence Weekly Report (2019) were used to evaluate the proposed evolutionary algorithm. The data selected are as follows: 1.

Value of new ship per year

Five-year ship purchase value

Ten-year ship purchase value

Gross revenue per day on the spot market

From the data selection it is obvious that no bonds such as opex by ship type or timing charter income were used. Opex has virtually no impact on the fluctuation or total divergence of the fleet because they are largely considered inelastic and their variation is common to all types of ships. Time charter revenues were also not used because they did not correctly reflect market trends at the time in question. The data range used is between 2007 and 2017. The fleet was created at the beginning of 2014. The optimization was done with data from 2007 to 2013. The shipping company operated with the proposed fleet from January 1, 2014, until December 31, 2017. The evaluation data for the operation for the period 2014–2017 are as follows (Table 1):

Table 1.

2014–2017 Tanker Market Price and Average Gross Revenue per Type.

		MR Clean	LR Clean	LR Dirty	AFRAMAX	SUEZMAX	VLCC
Current ship price (million $)	2014	37	45	45	54	65	97
	2015	36.8	45	45	53	64.5	95.5
	2016	34	37	37	44	55	85
	2017	34	40	40	44	55	82
5 year (million $)	2014	25	33	33	42	57	77
	2015	28	37.5	37.5	46	61	84
	2016	22	28	28	29	40	60
	2017	22	28	28	32	43	64
10 year (million $)	2014	16	23	23	27	37	52
	2015	20	25	25	33	42	59
	2016	16	17	17	18	28	40
	2017	16	17	19	22	28	41
Average earnings ($)	2014	12,517	12,473	18,858	24,705	27,791	27,315
	2015	22,681	24,975	26,998	38,794	46,126	53,102
	2016	12,124	8,962	15,388	22,965	27,567	41,488
	2017	21,244	22,094	26,548	37,977	46,713	64,846

Source: The authors.

The data for the evaluation of the optimal fleet composition used from 2007 to 2013 are as follows (Table 2):

Table 2.

2007–2013 Tanker Market Price and Average Gross Revenue per Type.

		MR Clean	LR Clean	LR Dirty	AFRAMAX	SUEZMAX	VLCC
Current ship price (million $)	2007	53	61	61	73	90	146
	2008	48	51	51	62	91	150
	2009	35	42	42	49	62	101
	2010	37	48	48	57	67	105
	2011	36	45	45	53	61	99
	2012	34	36	36	48	57	93
	2013	34	41	41	48	57	93
5 year (million $)	2007	52	60	60	70	92	135
	2008	52	61	61	71	94	138
	2009	23	35	35	41	56	79
	2010	26	36	36	40	59	85
	2011	26	31	31	35	47	58
	2012	25	25	25	27	40	57
	2013	29	31	31	32	42	60
10 year (million $)	2007	42	50	50	60	74	110
	2008	42	52	52	61	76	113
	2009	15	22	22	24	43	59
	2010	19	25	25	28	40	60
	2011	16	21	21	20	32	36
	2012	17	17	17	17	26	37
	2013	19	21	21	22	27	41
Average earnings ($)	2007	24,416	22,412	31,162	32,808	44,142	55,744
	2008	23,325	21,422	36,845	49,944	76,634	92,511
	2009	9,349	11,330	17,160	19,034	34,006	37,181
	2010	7,213	13,148	14,956	19,792	31,259	37,929
	2011	7,587	12,644	10,535	13,528	19,217	16,856
	2012	5,234	13,599	15,194	15,285	21,140	27,879
	2013	13,464	10,668	11,127	14,131	15,511	16,217

Source: The authors.

Evaluation of Historical Ship Performance

For the composition of the fleet, the performance of each ship was first evaluated as follows:

E (r_{t}) = V a l u e_{t} - V a l u e_{t - 1} + E a r n i n g s_{t}

(12)

where “Value” is the value of the particular ship at time t, “Earnings” is the annual fare of the particular ship at time t, and prices are in the millions of dollars.

The results about the composition of the fleet and each ship performance are presented in Table 3.

Table 3.

MR, LR Clean, R Dirty, AFRAMAX, SUEZMAX, VLCC, Performance Analysis, and Evaluation for Years 2007–2017.

		2007	2008	2009	2010	2011	2012	2013	2014	2015	2016	2017
New	MR	53.00	48.00	35.00	37.00	36.00	34.00	34.00	37.00	36.80	34.00	34.00
	LR Clean	61.00	51.00	42.00	48.00	45.00	36.00	41.00	45.00	45.00	37.00	40.00
	R Dirty	61.00	51.00	42.00	48.00	45.00	36.00	41.00	45.00	45.00	37.00	40.00
	AFRAMAX	73.00	62.00	49.00	57.00	53.00	48.00	48.00	54.00	53.00	44.00	44.00
	SUEZMAX	90.00	91.00	62.00	67.00	61.00	57.00	57.00	65.00	64.50	55.00	55.00
	VLCC	146.00	150.00	101.00	105.00	99.00	93.00	93.00	97.00	95.50	85.00	82.00
5 years old	MR	52.00	52.00	23.00	26.00	26.00	25.00	29.00	25.00	28.00	22.00	22.00
	LR Clean	60.00	61.00	35.00	36.00	31.00	25.00	31.00	33.00	37.50	28.00	28.00
	R Dirty	60.00	61.00	35.00	36.00	31.00	25.00	31.00	33.00	37.50	28.00	28.00
	AFRAMAX	70.00	71.00	41.00	40.00	35.00	27.00	32.00	42.00	46.00	29.00	32.00
	SUEZMAX	92.00	94.00	56.00	59.00	47.00	40.00	42.00	57.00	61.00	40.00	43.00
	VLCC	135.00	138.00	79.00	85.00	58.00	57.00	60.00	77.00	84.00	60.00	64.00
10 years old	MR	42.00	42.00	15.00	19.00	16.00	17.00	19.00	16.00	20.00	16.00	16.00
	LR Clean	50.00	52.00	22.00	25.00	21.00	17.00	21.00	23.00	25.00	17.00	17.00
	R Dirty	50.00	52.00	22.00	25.00	21.00	17.00	21.00	23.00	25.00	17.00	19.00
	AFRAMAX	60.00	61.00	24.00	28.00	20.00	17.00	22.00	27.00	33.00	18.00	22.00
	SUEZMAX	74.00	76.00	43.00	40.00	32.00	26.00	27.00	37.00	42.00	28.00	28.00
	VLCC	110.00	113.00	59.00	60.00	36.00	37.00	41.00	52.00	59.00	40.00	41.00
Earnings	MR	8.91	8.51	3.41	2.63	2.77	1.91	4.91	4.57	8.28	4.43	7.75
	LR Clean	8.18	7.82	4.14	4.80	4.62	4.96	3.89	4.55	9.12	3.27	8.06
	R Dirty	11.37	13.45	6.26	5.46	3.85	5.55	4.06	6.88	9.85	5.62	9.69
	AFRAMAX	11.97	18.23	6.95	7.22	4.94	5.58	5.16	9.02	14.16	8.38	13.86
	SUEZMAX	16.11	27.97	12.41	11.41	7.01	7.72	5.66	10.14	16.84	10.06	17.05
	VLCC	20.35	33.77	13.57	13.84	6.15	10.18	5.92	9.97	19.38	15.14	23.67
Return new	MR		7%	−20%	13%	5%	0%	14%	22%	22%	4%	23%
	LR Clean		−4%	−10%	26%	3%	−9%	25%	21%	20%	−11%	30%
	R Dirty		6%	−5%	27%	2%	−8%	25%	27%	22%	−5%	34%
	AFRAMAX		10%	−10%	31%	2%	1%	11%	31%	24%	−1%	32%
	SUEZMAX		32%	−18%	26%	2%	6%	10%	32%	25%	1%	31%
	VLCC		26%	−24%	18%	0%	4%	6%	15%	18%	5%	24%
Return 5 years	MR		16%	−49%	24%	11%	4%	36%	2%	45%	−6%	35%
	LR Clean		15%	−36%	17%	−1%	−3%	40%	21%	41%	−17%	29%
	R Dirty		24%	−32%	18%	−3%	−1%	40%	29%	43%	−10%	35%
	AFRAMAX		27%	−32%	15%	0%	−7%	38%	59%	43%	−19%	58%
	SUEZMAX		33%	−27%	26%	−8%	2%	19%	60%	37%	−18%	50%
	VLCC		27%	−33%	25%	−25%	16%	16%	45%	34%	−11%	46%
Return 10 years	MR		20%	−56%	44%	−1%	18%	41%	8%	77%	2%	48%
	LR Clean		20%	−50%	35%	2%	5%	46%	31%	48%	−19%	47%
	R Dirty		31%	−46%	38%	−1%	7%	47%	42%	52%	−10%	69%
	AFRAMAX		32%	−49%	47%	−11%	13%	60%	64%	75%	−20%	99%
	SUEZMAX		41%	−27%	20%	−2%	5%	26%	75%	59%	−9%	61%
	VLCC		33%	−36%	25%	−30%	31%	27%	51%	51%	−7%	62%
Average MR return	MR		14%	−42%	27%	5%	7%	30%	11%	48%	0%	36%
	LR Clean		10%	−32%	26%	2%	−3%	37%	24%	37%	−15%	35%
	R Dirty		20%	−28%	28%	−1%	−1%	38%	33%	39%	−8%	46%
	AFRAMAX		23%	−30%	31%	−3%	2%	36%	51%	47%	−13%	63%
	SUEZMAX		35%	−24%	24%	−3%	4%	18%	55%	40%	−9%	47%
	VLCC		29%	−31%	23%	−18%	17%	16%	37%	34%	−4%	44%
Over price ratio new	MR	1.00	0.74	0.05	0.16	0.11	0.00	0.00	0.16	0.15	0.00	0.00
	LR Clean	1.00	0.60	0.24	0.48	0.36	0.00	0.20	0.36	0.36	0.04	0.16
	R Dirty	1.00	0.60	0.24	0.48	0.36	0.00	0.20	0.36	0.36	0.04	0.16
	AFRAMAX	1.00	0.56	0.04	0.36	0.20	0.00	0.00	0.24	0.20	−0.16	−0.16
	SUEZMAX	0.93	1.00	0.14	0.21	0.11	0.00	0.00	0.07	0.04	−0.14	−0.19
	VLCC	0.93	1.00	0.14	0.21	0.11	0.00	0.00	0.07	0.04	−0.14	−0.19
Over price ratio 5 years	MR	1.00	1.00	0.00	0.10	0.10	0.07	0.21	0.07	0.17	−0.03	−0.03
	LR Clean	0.97	1.00	0.28	0.31	0.17	0.00	0.17	0.22	0.35	0.08	0.08
	R Dirty	0.97	1.00	0.28	0.31	0.17	0.00	0.17	0.22	0.35	0.08	0.08
	AFRAMAX	0.98	1.00	0.32	0.30	0.18	0.00	0.11	0.34	0.43	0.05	0.11
	SUEZMAX	0.96	1.00	0.30	0.35	0.13	0.00	0.04	0.31	0.39	0.00	0.06
	VLCC	0.96	1.00	0.27	0.35	0.01	0.00	0.04	0.25	0.33	0.04	0.09
Over price ratio 10 years	MR	1.00	1.00	0.00	0.15	0.04	0.07	0.15	0.04	0.19	0.04	0.04
	LR Clean	0.94	1.00	0.14	0.23	0.11	0.00	0.11	0.17	0.23	0.00	0.00
	R Dirty	0.94	1.00	0.14	0.23	0.11	0.00	0.11	0.17	0.23	0.00	0.06
	AFRAMAX	0.98	1.00	0.16	0.25	0.07	0.00	0.11	0.23	0.36	0.02	0.11
	SUEZMAX	0.96	1.00	0.34	0.28	0.12	0.00	0.02	0.22	0.32	0.04	0.04
	VLCC	0.96	1.00	0.30	0.31	0.00	0.01	0.06	0.21	0.30	0.05	0.06

Source: The authors.

The final step in generating input data for the evolutionary algorithm is to calculate the ship’s co-fluctuation for each possible combination as given in Table 4.

Table 4.

Fleet Covariance.

Covarience Asset	MR Clean New	MR Clean 5 Year	MR Clean 10 Year	LR Clean New	LR Clean 5 Year	LR Clean 10 Year	LR Dirty New	LR Dirty 5 Year	LR Dirty 10 Year	AFRAMAX New	AFRAMAX 5 Year	AFRAMAX 10 Year	SUEZMAX New	SUEZMAX 5 Year	SUEZMAX 10 Year	VLCC New	VLCC 5 Year	VLCC 10 Year
MR Clean New	0.013	0.031	0.037	0.013	0.025	0.035	0.012	0.025	0.034	0.012	0.024	0.039	0.013	0.031	0.037	0.013	0.025	0.035
MR Clean 5 Year	0.037	0.073	0.087	0.028	0.060	0.082	0.027	0.059	0.081	0.025	0.058	0.092	0.037	0.073	0.087	0.028	0.060	0.082
MR Clean 10 Year	0.044	0.087	0.113	0.035	0.071	0.101	0.033	0.071	0.102	0.035	0.067	0.119	0.044	0.087	0.113	0.035	0.071	0.101
LR Clean New	0.016	0.034	0.042	0.026	0.031	0.042	0.024	0.029	0.040	0.018	0.028	0.048	0.016	0.034	0.042	0.026	0.031	0.042
LR Clean 5 Year	0.030	0.072	0.085	0.031	0.064	0.084	0.030	0.064	0.084	0.024	0.062	0.098	0.030	0.072	0.085	0.031	0.064	0.084
LR Clean 10 Year	0.042	0.099	0.122	0.042	0.084	0.114	0.040	0.083	0.114	0.037	0.080	0.133	0.042	0.099	0.122	0.042	0.084	0.114
LR Dirty New	0.015	0.032	0.040	0.024	0.030	0.040	0.023	0.029	0.040	0.017	0.029	0.048	0.015	0.032	0.040	0.024	0.030	0.040
LR Dirty 5 Year	0.029	0.070	0.085	0.029	0.064	0.083	0.029	0.065	0.085	0.024	0.064	0.099	0.029	0.070	0.085	0.029	0.064	0.083
LR Dirty 10 Year	0.041	0.097	0.122	0.040	0.084	0.114	0.040	0.085	0.116	0.038	0.082	0.136	0.041	0.097	0.122	0.040	0.084	0.114
AFRAMAX New	0.014	0.030	0.042	0.018	0.024	0.037	0.017	0.024	0.038	0.019	0.023	0.044	0.014	0.030	0.042	0.018	0.024	0.037
AFRAMAX 5 Year	0.029	0.069	0.080	0.028	0.062	0.080	0.029	0.064	0.082	0.023	0.065	0.095	0.029	0.069	0.080	0.028	0.062	0.080
AFRAMAX 10 Year	0.047	0.110	0.143	0.048	0.098	0.133	0.048	0.099	0.136	0.044	0.095	0.162	0.047	0.110	0.143	0.048	0.098	0.133
SUEZMAX New	0.018	0.041	0.054	0.013	0.033	0.048	0.015	0.036	0.052	0.020	0.036	0.059	0.018	0.041	0.054	0.013	0.033	0.048
SUEZMAX 5 Year	0.024	0.056	0.073	0.020	0.048	0.067	0.023	0.052	0.072	0.025	0.052	0.084	0.024	0.056	0.073	0.020	0.048	0.067
SUEZMAX 10 Year	0.025	0.059	0.073	0.017	0.052	0.069	0.021	0.056	0.074	0.022	0.056	0.085	0.025	0.059	0.073	0.017	0.052	0.069
VLCC New	0.017	0.041	0.053	0.011	0.032	0.046	0.013	0.035	0.050	0.018	0.035	0.056	0.017	0.041	0.053	0.011	0.032	0.046
VLCC 5 Year	0.024	0.057	0.084	0.018	0.049	0.071	0.021	0.053	0.077	0.026	0.050	0.094	0.024	0.057	0.084	0.018	0.049	0.071
VLCC 10 Year	0.027	0.066	0.099	0.018	0.058	0.083	0.021	0.063	0.090	0.027	0.058	0.111	0.027	0.066	0.099	0.018	0.058	0.083

Source: The authors.

Fleet Composition and Assessment

The evaluation was made of three companies that chose to invest US$1 billion in 2014 in a fleet of tankers. The three companies optimized the fleet composition with three different techniques:

• An equal number of ships of all types

• The established risk minimization model

• The proposed Risky Asset Pricing maximization model

The optimization was done with the Microsoft Excel’s Solver tool, and the equilibrium model is as follows (Tables 5 –7):

Table 5.

Equal Fleet Fitness Results.

Year of fleet build	2014
Year of evaluation	2017
Over price ratio factor	1
Fitness value	2,314.73
Fleet deviation	1.41
Total budget	1,000

Source: The authors.

Table 6.

Equal Fleet Composition.

Type	Vessels	Price	Over Price Ratio Normalized	Total Value	Fleet
MR Clean New	1.45	34	1.00	49.3	6%
MR Clean 5 Year	1.45	29	0.99	42.0	6%
MR Clean 10 Year	1.45	19	0.99	27.5	6%
LR Clean New	1.45	41	0.99	59.4	6%
LR Clean 5 Year	1.45	31	0.99	44.9	6%
LR Clean 10 Year	1.45	21	0.99	30.4	6%
LR Dirty New	1.45	41	0.99	59.4	6%
LR Dirty 5 Year	1.45	31	0.99	44.9	6%
LR Dirty 10 Year	1.45	21	0.99	30.4	6%
AFRAMAX New	1.45	48	1.00	69.6	6%
AFRAMAX 5 Year	1.45	32	0.99	46.4	6%
AFRAMAX 10 Year	1.45	22	0.99	31.9	6%
SUEZMAX New	1.45	57	1.00	82.6	6%
SUEZMAX 5 Year	1.45	42	1.00	60.9	6%
SUEZMAX 10 Year	1.45	27	1.00	39.1	6%
VLCC New	1.45	93	1.00	134.8	6%
VLCC 5 Year	1.45	60	1.00	87.0	6%
VLCC 10 Year	1.45	41	1.00	59.4	6%
Total	26.09		0.91	1,000

Source: The authors.

Table 7.

Equal Fleet Annual Return and Gross Revenue.

Return	2014	2015	2016	2017
MR Clean New	10.97	11.71	2.36	11.24
MR Clean 5 Year	0.82	16.35	−2.28	11.24
MR Clean 10 Year	2.27	17.80	0.62	11.24
LR Clean New	12.40	13.21	−6.85	16.04
LR Clean 5 Year	9.50	19.73	−9.03	11.69
LR Clean 10 Year	9.50	16.11	−6.85	11.69
LR Dirty New	11.42	15.73	9.59	15.49
LR Dirty 5 Year	15.77	14.28	−3.45	18.39
LR Dirty 10 Year	12.87	20.80	−5.63	14.04
AFRAMAX New	21.76	19.07	−0.90	20.09
AFRAMAX 5 Year	27.56	26.32	−12.49	24.44
AFRAMAX 10 Year	20.31	29.22	−9.59	25.89
SUEZMAX New	26.30	23.68	0.81	24.71
SUEZMAX 5 Year	36.44	30.20	−15.85	29.06
SUEZMAX 10 Year	29.19	31.65	−5.71	24.71
VLCC New	20.25	25.92	6.73	29.95
VLCC 5 Year	39.09	38.24	−12.84	40.10
VLCC 10 Year	30.39	38.24	−5.59	35.75
Total	336.82	408.23	−76.96	375.75
Grand total	1043.85

Source: The authors.

An analysis of equal fleet fitness results and composition is taking place, and equal fleet annual return and gross revenue are calculated. The results are presented in Tables 5 –7, which indicate that, from the creation of the fleet in 2014 until the end of 2017, the total return was US$1,043.45 million. Then, the minimum risk fleet results, composition and annual return and gross revenue are calculated and presented using minimum risk model, in Tables 8 –10.

Table 8.

Minimum Risk Fleet Results.

Year of fleet build	2014
Year of evaluation	2017
Over price ratio factor	1
Fitness value	4,012.42
Fleet deviation	0.29
Total budget	1,000

Source: The authors.

Table 9.

Minimum Risk Fleet Composition.

Type	Vessels	Value	Over Price Ratio Normalized	Total Value	Fleet
MR Clean New	2	34	1.00	68	12%
MR Clean 5 Year	0	29	1.00	0	0%
MR Clean 10 Year	0	19	1.00	0	0%
LR Clean New	1	41	0.99	41	6%
LR Clean 5 Year	0	31	1.00	0	0%
LR Clean 10 Year	0	21	1.00	0	0%
LR Dirty New	1	41	0.99	41	6%
LR Dirty 5 Year	0	31	1.00	0	0%
LR Dirty 10 Year	0	21	1.00	0	0%
AFRAMAX New	4	48	1.00	192	24%
AFRAMAX 5 Year	0	32	1.00	0	0%
AFRAMAX 10 Year	0	22	1.00	0	0%
SUEZMAX New	4	57	1.00	228	24%
SUEZMAX 5 Year	0	42	1.00	0	0%
SUEZMAX 10 Year	0	27	1.00	0	0%
VLCC New	4	93	1.00	372	24%
VLCC 5 Year	1	60	1.00	60	6%
VLCC 10 Year	0	41	1.00	0	0%
Total	17		0.97	1002

Source: The authors.

Table 10.

Minimum Risk Fleet Annual Return and Gross Revenue.

Return	2014	2015	2016	2017
MR Clean New	15.14	16.16	3.25	15.51
MR Clean 5 Year	0.00	0.00	0.00	0.00
MR Clean 10 Year	0.00	0.00	0.00	0.00
LR Clean New	8.55	9.12	−4.73	11.06
LR Clean 5 Year	0.00	0.00	0.00	0.00
LR Clean 10 Year	0.00	0.00	0.00	0.00
LR Dirty New	7.88	10.85	6.62	10.69
LR Dirty 5 Year	0.00	0.00	0.00	0.00
LR Dirty 10 Year	0.00	0.00	0.00	0.00
AFRAMAX New	60.07	52.64	−2.47	55.45
AFRAMAX 5 Year	0.00	0.00	0.00	0.00
AFRAMAX 10 Year	0.00	0.00	0.00	0.00
SUEZMAX New	72.57	65.34	2.25	68.20
SUEZMAX 5 Year	0.00	0.00	0.00	0.00
SUEZMAX 10 Year	0.00	0.00	0.00	0.00
VLCC New	55.88	71.53	18.57	82.68
VLCC 5 Year	26.97	26.38	−8.86	27.67
VLCC 10 Year	0.00	0.00	0.00	0.00
Total	247.07	252.02	14.63	271.25
Grand total	784.97

Source: The authors.

Results about minimum risk fleet, composition, annual return, and gross revenue indicate that, from the 2014 when the fleet was created until the end of 2017, the total return was US$784.95 million (Tables 8 –10). Finally, using the maximum risk from the Risky Asset Pricing model, we calculate the maximum risk fleet results and composition as well as the maximum risk fleet annual return and gross revenue (Tables 11 –13).

Table 11.

Maximum Risk Fleet Results.

Year of fleet build	2014
Year of evaluation	2017
Over price ratio factor	1
Fitness value	12,5014.80
Fleet deviation	3.35
Total budget	1,000

Source: The authors.

Table 12.

Maximum Risk Fleet Composition.

Type	Vessels	Value	Over Price Ratio Normalized	Total Value	Fleet
MR Clean New	0	34	1.00	0	0%
MR Clean 5 Year	4	29	0.98	116	11%
MR Clean 10 Year	4	19	0.98	76	11%
LR Clean New	0	41	1.00	0	0%
LR Clean 5 Year	4	31	0.98	124	11%
LR Clean 10 Year	4	21	0.99	84	11%
LR Dirty New	0	41	1.00	0	0%
LR Dirty 5 Year	3	31	0.99	93	8%
LR Dirty 10 Year	4	21	0.99	84	11%
AFRAMAX New	0	48	1.00	0	0%
AFRAMAX 5 Year	3	32	0.99	96	8%
AFRAMAX 10 Year	4	22	0.99	88	11%
SUEZMAX New	0	57	1.00	0	0%
SUEZMAX 5 Year	0	42	1.00	0	0%
SUEZMAX 10 Year	4	27	1.00	108	11%
VLCC New	0	93	1.00	0	0%
VLCC 5 Year	0	60	1.00	0	0%
VLCC 10 Year	3	41	0.99	123	8%
Total	37		0.88	992

Source: The authors.

Table 13.

Maximum Risk Fleet Annual Return and Gross Revenue.

Return	2014	2015	2016	2017
MR Clean New	0.00	0.00	0.00	0.00
MR Clean 5 Year	2.27	45.11	−6.30	31.02
MR Clean 10 Year	6.27	49.11	1.70	31.02
LR Clean New	0.00	0.00	0.00	0.00
LR Clean 5 Year	26.21	54.46	−24.92	32.26
LR Clean 10 Year	26.21	44.46	−18.92	32.26
LR Dirty New	0.00	0.00	0.00	0.00
LR Dirty 5 Year	32.65	29.56	−7.15	38.07
LR Dirty 10 Year	35.53	57.42	−15.53	38.76
AFRAMAX New	0.00	0.00	0.00	0.00
AFRAMAX 5 Year	57.05	54.48	−25.85	50.58
AFRAMAX 10 Year	56.07	80.64	−26.47	71.45
SUEZMAX New	0.00	0.00	0.00	0.00
SUEZMAX 5 Year	0.00	0.00	0.00	0.00
SUEZMAX 10 Year	80.57	87.34	−15.75	68.20
VLCC New	0.00	0.00	0.00	0.00
VLCC 5 Year	0.00	0.00	0.00	0.00
VLCC 10 Year	62.91	79.15	−11.57	74.01
Total	385.76	581.74	−150.76	467.62
Grand total	1,284.36

Source: The authors.

Findings using the maximum risk from the Risky Asset Pricing model indicate that at the end of 2017 the total return is US$1,284.36 million since the creation of the fleet in 2014 (Tables 11 –13).

Results and Discussion

This research examines the management and optimization of fleet management of a shipping company through the control algorithms. Finding an algorithm that will reduce a marine company’s exposure to risk by diversifying its fleet composition is one way to make it dominant. Classic portfolio management through risk minimization is clearly ineffective, as it seems to reduce performance below what is a random or evenly distributed fleet. By comparing three methods, it is clear that the Risky Asset Pricing model is superior, as it optimizes the fleet over the excess present value of ships with an effective 25% over an accidental fleet and 60% over the conventional minimum risk method. In essence, this algorithm looks for solutions where the demand for ships is low but has enormous fluctuation potential. It is basically looking to find ships that are at high risk with great potential for price increases to give the investor possible big returns on which the most successful players in shipping operate.

Portfolio management through risk minimization is ineffective as it appears to reduce performance. Finally, it is clear that using the expected performance to optimize a fleet is ineffective and offers nothing due to the large variation.

Results are important for investors and companies. In the present study, due to a lack of access to a large number of historical data, perhaps data may be within the same shipping cycle. In order to draw safer conclusions, in future research, the composition should be at least 17 years old with current prices at present value.

Also, optimization should be done again for a time period at least 17 years so that a company has experienced at least one full shipping cycle. In such a model, the fleet should also be reorganized at regular intervals.

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

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