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International Journal of Naval Architecture and Ocean Engineering 8 (2016) 344—359

http://www.journals.elsevier.com/international-journal-of-naval-architecture-and-ocean-engineering/

Assessment of surface ship environment adaptability in seaways: A fuzzy

comprehensive evaluation method

Jialong Jiao, Huilong Ren, Shuzheng Sun*

College of Shipbuilding Engineering, Harbin Engineering University, Harbin, 150001, China

Received 21 December 2015; accepted 3 May 2016 Available online 15 July 2016

Abstract

Due to the increasing occurrence of maritime accidents and high-level requirements and modernization of naval wars, the concept of ship environment adaptability becomes more and more important. Therefore, it is of great importance to carry out an evaluation system for ship environment adaptability, which contributes to both ship design and classification. This paper develops a comprehensive evaluation system for ship environment adaptability based on fuzzy mathematics theory. An evaluation index system for ship environment adaptability is elaborately summarized first. Then the analytic hierarchy process (AHP) and entropy weighting methods are applied to aggregate the evaluations of criteria weights for each criterion and the corresponding subcriteria. Next, the multilevel fuzzy comprehensive evaluation method is applied to assess the ship integrative environment adaptability. Finally, in order to verify the proposed approach, an illustrative example for optimization and evaluation of five ship alternatives is adopted. Moreover, the influence of criteria weights, membership functions and fuzzy operators on the results is also analyzed.

Copyright © 2016 Society of Naval Architects of Korea. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Ship environment adaptability; Evaluation system; Fuzzy theory; Analytic hierarchy process

1. Introduction

Waterway transportation plays an important role in transporting both goods and passengers all over the world, and it is the proven cheapest mode of goods transportation. However, with the increasing tonnage of ships and the emergence of novel fast ships, problems of maritime safety are emerging day by day (Islam et al., 2015; Psarros et al., 2010; Pak et al., 2015). As a result, more and more maritime accidents have been taken place in the past decades because of the insufficient performances of ships to the specified environments. Furthermore, the integrative performance of warships is of great importance to a country who involved in a naval war, and it largely determines the outcome of the naval war. A sea wave

* Corresponding author.

E-mail address: sunshuzheng@hrbeu.edu.cn (S. Sun).

Peer review under responsibility of Society of Naval Architects of Korea.

well-suited, fast navigational, radar stealthy ship has great advantages during a sea war. All these cases can be attributed to the evaluation of ship environment adaptability and thus it becomes a common concerned topic of both civilian and military ships.

The ship environment adaptability refers to the endurances of ships in all kinds of external disturbances, e.g. wind load, wave-induced motion and load, currents impact, and brine-induced corrosion experienced during their lifetime (Jiao et al., 2014; Sun et al., 2014a,b). The environment adaptability is an important factor in the evaluation of integrative navigation performance of ships, especially ships sailing in severe sea conditions for high demand tasks execution. A ship with good environment adaptability can not only prevent its functions from failure in a certain circumstance, but also take advantage of the specified environment so as to improve its performance. Ship environment adaptability of different kind of ships should be evaluated purposefully. The main subjects

http://dx.doi.org/10.1016/j.ijnaoe.2016.05.002

2092-6782/Copyright © 2016 Society of Naval Architects of Korea. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

in terms of ship performance in naval architecture are: sea-keeping performance, wave load, rapidity, stability, maneuverability, etc. Ship environment adaptability system is a multi-variable, multi-objective and multi-hierarchy system which takes different kinds of ship performance into account at the same time.

Assessment of ship environment adaptability in seaways, like many other evaluation issues, is complex because such evaluations usually have many aspects in respects of index factors. Currently, the conventional environment adaptability of ships are evaluated for separate independent performances, e.g. rapidity, seakeeping performance, and wave load behavior, etc., which is quite different from the real navigational conditions. Olson (2009) proposed a method which takes the percent of time that a ship accomplishes a given navigation task in the specified environment as the ship's seakeeping performance evaluation index. Yang et al. (2003) used the fuzzy theory and the genetic algorithm to evaluate the rapidity and maneuverability of ships in calm water. Li and Wu (2004) set up a model to evaluate ship stability by using the height of center of gravity and a new concept of equivalent height of center of gravity. Zhang et al. (2011) summarized the influencing factors on the combat effectiveness of surface ship's acoustic stealth performance.

Currently, a lot of published papers focus on the comprehensive assessment of green shipping. Niese et al. (2015) carried out a ship design evaluation framework in respect of carbon emission by means of Markov decision process. Ship performance in real seas is quite different from that of in still water. Thus ship performance should be evaluated by taking wind and wave effects into account. Class NK carried out the "development of environmental performance technical services of container ships" from 2007 to 2009, which was intended to develop an index of propulsion performance for container ships in actual seas (Nomura et al., 2009). Research and development on 10 mode index for ships at sea have been carried out by Tsujimoto et al. (2012) to study the evaluation approach of ship energy efficiency. Sasa et al. (2015) evaluated the ship performance and analyzed the relationships between ship motions, speed loss and wave conditions from data collected during one-year voyage trials of a 20,000 DWT bulk carrier.

Only a few publications evaluate ship performance by taking multi-subject and multi-variable into consideration. Multidisciplinary design optimization (MDO) has been developed and used by some scholars for ship design. Pan et al. (2009) established a MDO method includes 37 design variables, nine constraints and two objectives for ship optimal design. A MDO based method "Bi-Level Integrated System Collaborative Optimization (BLISCO)" has been adopted to the conceptual design of an human occupied vehicle (HOV), which includes hull module, resistance module, energy module, structure module, weight module, and the stability module (Zhao et al., 2015).

The evaluation of ship integrative performance in wind and wave environment is a reason of concern in the field of naval architecture. Despite a number of current publications are

related to the evaluation of ship performance, few of them focus on the comprehensive performance evaluation of ships and this issue is still far from being completely solved. Thus, there is a real need for the development of an evaluation system that considers as more aspects as possible. In this paper, a comprehensive evaluation system for ship environment adaptability is proposed. First of all, criteria and sub-criteria with respect to ship environment adaptability are proposed by considering multidisciplinary fields of naval architecture, for example, seakeeping, structural mechanics, resistance, and propulsion. Then Analytic Hierarchy Process (AHP) and entropy weighing method are adopted in weighing each of the criteria and subcriteria. Next, a fuzzy comprehensive evaluation method is established to assess the ship environment adaptability. Lastly, an illustrative example is given to elaborate the method proposed in this paper.

2. Background and evaluation index system

2.1. Background

The shipping industry has witnessed an enormous progress in the last decades for the sake of the rapid growth in waterway transportation needs. However, fatal maritime accidents are the nightmares of seafarers, passengers and the public. Since ships sailing in unknown severe and high-risk environments, many ship accidents occur at seas as well as in river waters. Ship accidents can be categorized into various classification such as collision, capsizing, grounding, material fatigue, hull damage, sinking, etc (Toffoli et al., 2005; Nielsen, 1999; Yip, 2008). All these kinds of accidents can be attributed to the insufficient performances of ships to the specified environments. From a ship accident database collected by Wang (2010) about 677 incidents all over the world were extracted during the period from 1978 to 2008. The classification of these ship accidents is summarized in Fig. 1.

explosion Water

29.99% ingress

Fig. 1. Classification of ship accidents.

Shipping industry is also one of the stakeholders in environmental protection issues. According to a statistics data, almost 90% of the global trade goods are transported by ships. As shown in Fig. 2, the Deadweight Tonnage (DWT) of global merchant ships increased rapidly in the past two decades. The fuel consumption of ships has great influence on the global emissions. As a result, shipping industry is responsible for 3% of global CO2 emissions, 14—15% of global NOX emissions, and 16% of global SOX emissions (Sherbaz and Duan, 2014; Westerlund et al., 2015). The gas emissions by river ships of China in the year 2013 are shown in Fig. 3. The gas emitted by river ships is of great harm for the citizens lived around the rivers and ports.

The International Maritime Organization (IMO) stipulated that Energy Efficiency Design Index (EEDI) be applied to new ships since 2013 (Cheng and Li, 2012; Liu et al., 2012). This is the first mandatory legal document for ships more than 400 DWT aimed to reduce the emissions of greenhouse gases. As is known, the velocity term in the EEDI calculation formula denotes the velocity achieved of a ship in calm water. However, the velocity achieved in calm water, even though useful, is questionable in this respect since ships travel in real seas with the interactions of winds, waves and currents. As a result, some scholars proposed and added a speed reduction coefficient of ship in wind and wave environment in the formula as a temporary expedient to address this problem. On the other hand, the increase of index requirements for ship integrative navigation performance brings both challenges and opportunities for shipyards, i.e., those who build ships with better resistance and rapid performance will receive more orders. Many other integrative sailing performance criteria involved in the EEDI calculation, such as engine efficiency, propulsion performance and seakeeping, are also criteria of ship environment adaptability. Moreover, the mass reduction of light ship can spare more weights for goods carrying, which can be optimized by using the knowledge of ship structural strength. In a word, all these issues fall within the scope of ship comprehensive performance. Thus, the evaluation system proposed in this paper is also applicable to the assessment of green shipping, which is helpful for the improvement of the EEDI index.

18 16 14

o10 £ 6

1998 2000 2002 2004 2006 2008 2010 2012 2014

Fig. 2. DWT of global merchant ships.

w 20 0

PM2.5 Others

Fig. 3. Components of gas emissions by river ships of China in 2013.

The assessment of ship environment adaptability is also widely used in naval ships. A ship with good integrative performance is extremely competitive during a war. One example would be the Zumwalt-class destroyer DDG-1000 of America, as shown in Fig. 4. DDG-1000 has a greatly reduced crew size (less than half crew compared with the legacy ships) and is equipped with a number of automated systems (Quintana et al., 2007). Usability testing and usability assessments of the user-centered system have been conducted by its design verification integrated product team. Good seakeeping ability improves the fire accuracy of the ship-borne weapon systems and also provides comfortable working conditions for both the navies and onboard equipments. Good rapidity and maneuverability which provide the ship's flexible navigation are also important during the war. The residual stability and flood-ability make the ship's vitality stronger even when attacked by an enemy. Excellent stealth ship ensures that it is harder to be detected by the enemy.

2.2. Evaluation index system

According to modernized navy requirements, the development of a set of evaluation system for naval ships is of great necessity, and quite a few countries are devoted to investigating the operational platforms of naval ships. Thus this paper mainly focuses on the wind and wave environment adaptability of naval surface ships. As mentioned before, a ship encounters complex environments during its lifetime. Ship environment adaptability should be evaluated by taking as many subjects as possible into consideration. According to the classical ship theory (Sheng and Liu, 2003), disciplines in naval architecture can be generally categorized into ship hydrostatics, ship resistance, ship propulsion, ship maneuverability and seakeeping performance. Additionally, due to the fact that the increasing of ship size and the reduction of light ship weight, ship overall and local structural strength must be considered, especially for naval ships serve in rough seas. The stealth is also one of the significant factors in terms of naval ships that involved in naval wars. Thus enough attentions should also be paid to the wave load and stealth characteristics of ships besides the five basic disciplines.

In summary, seakeeping, wave load, rapidity, maneuverability, stability, floodability and stealth are selected as the first-level criteria of environment adaptability. Then several representative sub-subjects are chosen in each of the subject as the second-level subcriteria. The typical evaluation criteria and subcriteria are described below. In addition, the hierarchy evaluation index system for ship environment adaptability established is summarized in Fig. 5.

2.2.1. Seakeeping performance

Seakeeping performance is a measurement of a ship to the specified wave condition (Li, 2003). A ship of good sea-keeping performance is said to be operated effectively and without pronounced speed reduction even in severe sea states. The ship's six degrees of freedom (6-DOF) motions, vertical accelerations at bow, middle and aft areas, comfort criteria such as seasickness, and observable phenomena such as bow impact and green water on the deck are the classical sea-keeping performance evaluation indexes. In this study, the ship 6-DOF motions, vertical accelerations, and frequency of green water on deck and bow emergence are selected as the

subcriteria of seakeeping subject, since they are the most representative and accessible indexes in both experimental and numerical seakeeping studies. The seakeeping performance can be obtained by tests (e.g. tank model test and full-scale sea trial) and numerical calculations (e.g. strip theory, three-dimensional nonlinear approach).

2.2.2. Wave load

In order to ensure the safety of large ships in rough seas, providing enough structural strength and preventing the structure/material from damage/fatigue are of particular importance. Generally, according to wave load basic theory (Dai et al., 2007) more attention should be paid to the longitudinal strength of a ship, especially for a ship that travels at high speeds in severe seas. A ship will experience enormous vertical sectional bending moment and shearing force when it travels in rough head seas. Horizontal sectional loads and torsion loads should also be considered when sailing in oblique waves. Additionally, bow slamming events which cause whipping loads and harmonic vibrations which cause springing loads should be considered in corresponding

Ship environemntal adaptability

Ship 6-DOF motions; Vertical acceleration; Frequency of green water;

frequency of bow emergence; Comfort criteria...

Vertical/horizontal load;

Sectional bending moment; Shearing force; Torsion load; Shipping loads; /mpact pressure; .armonic vibrations; Springing loads...

Seakeeping Wave load Rapidity Maneuverability Stability Floodability Stealth

Viscous resistance;

^ave-making

resistance;

Wind resistance;

Wave added

resistance;

Speed loss in waves;

Propulsion

efficiency;

Cavitation

characteristics;

.Engine power

capacity;

Strength of

propellers.

Tactical diameter; Maximum advance; Transfer at 90° change ofheading; 7ime to change heading from 90° to 180°;

Transfer loss of steady speed; .initial turning time; Time from the rudder reversion to the maximum heading; Overshoot angle and period ofthe first heading oscillation; Stopping time and distance.

.flighting moment; lighting arm; initial metacenter height;

.Maximum righting lever;

Wind heeling lever; ^ngle of heel under action of steady wind;

^ngle of roll to windward due to wave action; Wind resistance level;

Stability criterion number.

New parameters of stability after damage; Factor of subdivision; Freeboard height...

.adar cross-section (RCS);

Visibility and noise; Radar absorbing material (RAM); Stealth efficiency.

Fig. 5. Typical evaluation index system for ship environment adaptability.

circumstances. To conclude, vertical, horizontal and torsion sectional loads, slamming loads, springing loads and impact pressure are chose as the subcriteria of wave load.

2.2.3. Rapidity

Rapidity is one of the most important characteristics of both naval and merchant ships. The rapidity of a ship refers to the ability of achieving a certain navigational speed under minimum engine power. Usually the rapidity includes two aspects: ship resistance and ship propulsion. A ship of excellent rapidity is said to be with well-optimized streamline which produce less drag and with high efficiency propulsion system which consume less fuel when it sails at sea.

Generally speaking, ship viscous resistance, wave-making resistance and wind resistance account for majority of the total resistance. Since a ship often travels in waves, the added resistance and speed loss in waves are also important factors of assessing the ship resistance behavior.

On the other hand, more efficient propulsion systems are adopted by ships to overcome the resistance. Since majority ships are propelled by screw propellers, propulsion efficiency (shaft transmission efficiency, open water efficiency, hull efficiency and relative rotation efficiency), cavitation characteristics, engine power capacity and strength of propellers are selected as the subcriteria of propulsion characteristics.

2.2.4. Maneuverability

Ship maneuverability refers to the ability of changing or keeping movement states or course of a ship. Usually, ship maneuverability is investigated according to the following three aspects: course stability (straight line/direction/position stability), turning ability, brake stability.

According to Kornev (2013), the main maneuvering experiments performed by free running models are: turning circle test, zigzag maneuver, spiral maneuver and stop maneuver. During the turning circle test, parameters such as tactical diameter, maximum advance, transfer at 90° change of heading, times to change heading from 90° to 180°, and transfer loss of steady speed are measured. Zigzag maneuver tests are usually performed by 10°/10° and 20°/20° styles of rudder angle. As a post voyage analysis, initial turning time, time from the rudder reversion to the maximum ship heading angle, overshoot angle and period of the first heading oscillation are the representative parameters analyzed. The maneuvering diagram is obtained by spiral maneuver, from which the turning ability and yaw stability of the ship can be estimated. The stopping time and distance—from stopping the engine and reversing full astern to the ship motion and speed become zero—are regarded as the representative parameters of stop maneuver. In addition, the acceptable scope of the parameters can be referred according to the recommended values by IMO regulations (IMO Committee, 2002).

2.2.5. Stability

Ship stability refers to the ability of restoring balance after the combination effects of winds, waves, currents and moving loads whether intact or damaged (Ying et al., 2009). Basic

parameters such as righting moment can be obtained based on center of gravity, center of buoyancy, righting arm and metacenter etc. IMO (2008) has specified the standard of parameters such as criteria regarding righting lever curve properties (the righting lever GZ, the maximum righting lever and the initial metacentric height) and severe wind and rolling criterion (wind heeling lever, angle of heel under action of steady wind and angle of roll to windward due to wave action). Li and Wu (2004) proposed a ship stability evaluation method by using the height of gravity centre, and the concept of equivalent height of gravity centre was established. Criteria such as wind resistance level, stability criterion number (the ratio of minimum overturning moment to maximum wind tilting moment) etc. can also be selected as the criteria of ship stability.

2.2.6. Floodability

Ship floodability (unsinkability) is achieved by dividing the volume of the ship into watertight compartments by its decks and bulkheads. If a ship's hull is made up of watertight compartments, any flooding resulting from a breach of the hull can be contained in the compartments where the flooding occurs. The stability loss of damaged vessel can be calculated by the stability theory and then obtain the new stability behavior. Generally, there are three kinds of damaged cabin, and two approaches, i.e. buoyancy loss approach and weight add approach, are used to calculate the floodability. According to Lloyd's rule, the survival probability of a damaged ship should be higher than 95% within 96 h after damage. The new parameters of stability after damage, factor of subdivision and freeboard height can be selected as the floodability criteria.

2.2.7. Stealth

The stealth of ships refers that architects using stealth technology construction techniques to ensure that ships harder to be detected by radar, visual, sonar, or infrared methods, or several of these methods. It is of special importance for warships. Yang et al. (2010) concluded that reduction of radar cross-section (RCS), visibility and noise, using of radar absorbing material (RAM), and improvement of stealth efficiency are the common used techniques for stealth ships currently.

3. Evaluation index weight analysis

The determination of index weight is one of the key steps during the comprehensive evaluation. Generally, there are two different approaches of determining the weight: subjective weighing method and objective weighing method. The former, the result of which mostly depends on the decision-makers' individual opinions, is an important and widely used method. It is usually determined by AHP method (Sun et al., 2014a,b; Wu et al., 2010; Su et al., 2013; Zhao et al., 2012). The Entropy weighing method, as an objective weighing method, is also used to calculate the weights in this paper. Entropy is an important notion measuring uncertainty of information systems in the information theory (Zhang et al., 2014; Du et al.,

2014; Liu et al., 2009). The two weighing methods, i.e., AHP method and entropy weighing method, are introduced respectively as follows.

3.1. Analytic hierarchy process

where R.I. is the average random index which depends on the value of n and can be found from Li et al. (2014).

When C.R. < 0.1, it is considered that the consistency of judgment matrix is acceptable. And then the relative weight outputs W.

AHP is a systematic and hierarchical multiple factors assessment analysis method. It is very convenient in the multiobjective weighing condition. The steps to calculate the index weight by AHP are described as follows:

Step 1. Constructing the judgment matrix. The judgment matrix is expressed as follows:

aii ai2 ••• ain a2i a22 ••• a2n

ani an2 • ' ' ann

To obtain the performance score, the 1—9 scale method, which is specified in Table 1, is used to indicate the relative strength of each pair of elements in the same hierarchy. There are n(n-1)/2 pairs need to be compared in a n dimensional matrix. The judgment matrix elements should have the following characteristics:

i; i = i;2; "V

1= i/aji, i;j = i; 2, •••, n. i = aikjajk ; i,j, k = i; 2, •••, n.

Step 2. Solving the eigenvalues and eigenvectors. After solving the eigenvalues of judgment matrix A, 1max, the largest characteristics root, can be obtained. And W is the corresponding standardized eigenvector of 1max. Step 3. Consistency checking. The following condition should be satisfied during consistency check:

AW = lmaxW. (5)

The consistency check index C.I. is defined as follows:

^max n

Calculation consistency ratio C.R. is defined as follows:

3.2. Entropy weighing method

Shannon first proposed the concept of information entropy in the year 1948. Information entropy was regarded as the uncertainty of a stochastic event or metric of information content. Suppose there are t ship samples taken to evaluate the environment adaptability. Each sample has m evaluated indices. Then establish a judgment matrix X of size t x m, which can be expressed as:

Xii Xi2 x2i x22

xti xt2

xit x2t

The procedure of calculating the index weight by entropy weighing method is described as follows:

Step 1. The equation to calculate the index entropy is expressed as follows:

Ej = -kJ2pjln pij; j=i; 2

where k is a positive value, and k = 1/ln t, Pj is the specific gravity value for each x,j, and

^ xij; i i ; 2

Step 2. Calculating degree of variation coefficient of the indices using the following formula:

Dj = i -Ej; j = i;2, m.

Step 3. Calculating the entropy weight coefficient of the indices using:

T,Dj; j = i; 2

3.3. The combination weight

Table i

The scale method of i—9 in comparison.

Scale Meaning

i Same importance

3 Moderate importance

5 Strong importance

7 Very strong importance

9 Extremely strong importance

2,4,6,8 Median in two adjacent judgments

The index weights calculated by AHP and entropy weighing method are A1 = (a], a^, ■■■a]n) and A2 = (a2, a2, —am), respectively. Where, i refers to the ith criterion in the firstlevel, m is the number of subcriteria in the ith criterion. Then the combination weight can be obtained by:

A, = bAj + (1 - b)A2 . (13)

where b is the preference coefficient.

The preference coefficient is usually decided according to the attitude of decision-makers. If b = 0, it means that the weight is fully made according to the objective weighing method; else if b = 1, it means the weight is fully decided according to the subjective weighing method. In the former circumstance, the experts may not care about the decision. Conversely, the latter extreme choice can be adopted when the experts want to fully grasp the decision according to their desire. The influence of human consciousness can be checked by changing the value of preference parameter.

4. Multilevel fuzzy comprehensive evaluation

4.1. Multilevel fuzzy comprehensive evaluation process

The multilevel fuzzy comprehensive evaluation method, combining fuzzy theory and mathematical model, is a useful method in multiobjective assessment (Chiu et al., 2014; Xiong and Xian, 2003). The methods for obtaining evaluation index set and weight set have been described above. For the sake of clarity, they are summarized below.

The first-level layer index set is constructed as follows:

U ={ui, u2,..., un}. (14)

Assume that in the first-layer index set there are n criteria ui (i = 1,2,...,n), and each criterion includes several subcriteria. The second-level layer subcriteria of the ith criterion are constructed as follows:

ui = { un, ui2; ...; uim} . (15)

The weight set of the first-level layer index set is expressed as follows:

A = {a1,a2,..., a„}.

The weight set of second-level layer index set of the ith index set in first-level layer is expressed as follows:

Ai = {aH; ai2,..., aim}. (17)

According to the evaluation standard and grade, the remark set can be defined as:

V = {vi, V2,..., Vp} .

where p is the number of remark set elements.

Usually there is a membership degree of the element to a fuzzy set. A fuzzy set is defined by a membership function that maps elements to degrees of membership within a certain interval, which is usually the value in interval [0, 1]. If the degree of membership is zero, it means that the element does not belong to the set. If the degree of membership is one, it means that the element belongs fully to the set. If the degree of membership lies within the interval [0, 1], it means that the element has a certain degree of membership to the set. According to the multilevel fuzzy comprehensive evaluation method, the ship environment adaptability can be assessed from bottom to top layer by layer, and the evaluation index of the second-level layer is degree of membership of the firstlevel layer. Define the degree of membership of index utj in remark set element vk as:

rijk, i = 1,2, •••,n;j = 1,2, •••,m;j = 1,2,•••,p. (19)

Then the relation matrix of the second-level layer is expressed as:

Rn " rt11 ri12 • ' ri1p

Rt = Ri2 = rt21 ri22 • ■ r-i2p

_ Rim _ rim1 rim2 • rimp

i = 1,2, •••,n.

Table 2

The common used membership functions.

Function

Parametric representation

Graphical representation

Triangular function

Gaussian function

Shape r function

Cauchy function

r(x) = {

(x — a)=(b — a), a <x < b (c — x)/(c — b), b < x < c'

r(x) = e—k(x — a)2

, ek(x—a), x < a

r(x) = { e—k(x—a) x > a ■

r(x) =

1 + a(x — a)

Synthesize the weight A, with the second-level fuzzy relation matrix Ri, and then the membership matrix Bt is obtained:

Definition 5. M(«v) model:

Bi = Ai+Ri; i = 1; 2; n .

where + denotes a fuzzy synthesis operator. Membership matrix Bi is the fuzzy comprehensive evaluation matrix of first-level.

Due to the hierarchical structure of the evaluation index system, the relation matrix of first-level layer is expressed as:

R =[B1 B2 ••• Bn]. (22)

Then the fuzzy comprehensive evaluation of second-level can be obtained by the following equation:

B = A-R . (23)

4.2. Membership functions

In many cases, the judgment of membership is a fuzzy and vague state because there is no obvious and clear grading mark in the actual status. To accurately determine the degree of membership, it is important to choose a suitable membership function. The common used membership functions of fuzzy logic are shown in Table 2.

4.3. Fuzzy operators

In order to solve Eq. (23), a fuzzy algorithm is adopted. The common used fuzzy operators and their calculation rules are defined as follows:

Definition 1. Zadeh algorithm A: a Kb = min(a; b) .

where, a and b are arbitrary constant numbers, and have the same meaning in Definitions 2 and 3 below.

Definition 2. Zadeh algorithm V:

aVb = max(a, b) .

Definition 3. Bounded algorithm 4: a4b = min(1, a + b).

Definition 4. M(A,V) model:

bj = (aiArj).

where ai denotes the weight, rij denotes the degree of membership, and they have the same meaning in the following definitions.

bj = V aiTij. i=1

Definition 6. M(a,4) model:

; X (aiArij^.

bj = min 1 ; ai rij

Definition 7. M(>,®) model:

; X>2aij.

bj = min 1; airij

Definition 8. M(«,+) model:

bj = airij

5. Case study and discussion

In this section, the proposed method is applied to select the best candidate for ship integrative performance optimization. The project was conducted with the collaboration of Harbin Engineering University (HEU), 701 and 702 Institutes of China Shipbuilding Industry Corporation (CSIC). The main objective of the project is to develop a kind of high seakeeping performance stealth hybrid monohull. In this project, deep-V section and UV section ships were proposed based on the conventional round bilge ship owing to their excellent navigation performance. It is noted that an elaborately optimized streamlined semi-submerged bow (SSB) (Patent license number ZL200810075176.8) was installed at the bow bottom of the deep-V monohulls, which could further improve their seakeeping performance. It has already been verified that the novel ships have unique advantages over each other. For example, the vertical movement performance of deep-V ships is largely improved compared with the round bilge ship. The UV-shaped ships have better stability performance than round bilge ship.

There are five ship alternatives which need to be evaluated. The traditional round bilge ship (R0, see Fig. 6 (a)) is the 'mother' ship designed. The comprehensive performance of the proposed V-shaped ships (V1 and V2, see Fig. 6 (b) and (c)) and UV-shaped ships (UV1 and UV2, see Fig. 6 (d) and (e)) were compared with R0. The main parameters of ship prototypes are shown in Table 3.

Fig. 6. Body plans of the ship alternatives.

Table 3

Main dimensions of the ships.

Alternative R0 V1 V2 UV1 UV2

Length (m) 180 180 180 180 180

Breadth (m) 20 23 23 21 21

Depth (m) 14 14 14 14 14

Draft (m) 7 7 7 7 7

Displacement (kTon) 12 14 14 13 13

5.1. The procedure of evaluation

In summary, the procedure of assessing ship environment adaptability based on fuzzy theory is shown in Fig. 7, and the definite process is summarized below.

Step 1. Establish evaluation index system. Since this project mainly concerns the wind and wave environment adaptability of the novel ships, and for the sake of simplification,

only some typical representative criteria, i.e. seakeeping, rapidity and stability, were chosen in this case study. However, the evaluation method established can be generalized to a wide application in terms of ship comprehensive performance evaluation. The first-level index set is expressed as follows:

U = {u1, u2, u3}, where, u1 denotes seakeeping performance, u2 denotes rapidity, u3 denotes stability.

The second-level index sets are expressed as follows:

u1 = {u11, u12, u13, u14}, where, u11 denotes heave, u12 denotes pitch, u13 denotes bow vertical acceleration, u14 denotes green water on deck frequency; u2 = {u21, u22}, where, u21 denotes speed in waves, u22 denotes surface roughness;

u3 = {u31, u32}, where, u31 denotes wind resistance level, u32 denotes stability criterion number.

Establish evaluation index system

Raw data Percentile score

Fig. 7. Flowchart of evaluation procedure.

The data of seakeeping performance and rapidity of the ship in the level 7 sea state was obtained by tank model experiments. The values of heave, pitch and bow vertical acceleration are the statistical significant amplitude values in the head wave condition. During the tests, pitch and heave motions were measured by a seaworthiness instrument (see Fig. 8(a)); vertical acceleration was measured by the acceler-ometer mounted on the deck; the frequency of green water on deck was counted by the playback of video recordings. The speed in waves is the average sailing speed achieved of ship when acting a specified known towing force. The data of stability was obtained by numerical calculation with the help of Hydromax module of Maxsurf software, and example of the created geometric models is shown in Fig. 8(b). The resistance level denotes maximum transverse wind speed endured by the ship in calm water, and stability criterion number was the value of ship during the hurricane case. The values of representative criteria are shown in Table 4, and all the values have been converted into the data corresponding to ship prototype by using of the similitude law.

Step 2. Calculate the weights of the criteria. The relative weights were made by a team of experts from HEU, 701 and 702 Institutes of CSIC who involved in the project. The decision-makers—who clearly know the ultimate goal of the project—gathered together and debated in terms of the

(a) Experimental model setup (V2) (b) Geometric model by Maxsurf (RO and V2)

Fig. 8. Photograph of model setup.

Table 4

Values of representative criteria.

Criteria Subcriteria R0 Vi V2 UVi UV2 Min Max

Seakeeping Heave (m) 2.54 2.72 2.68 3.03 2.74 2.54 3.03

Pitch (deg) 3.56 2.38 2.47 2.95 2.68 2.38 3.56

Acceleration (m/s2) 6.22 4.63 4.52 4.97 4.49 4.49 6.22

Green water (frequency/min) 2.i2 2.26 i.98 i.56 i.84 i.56 2.26

Rapidity Speed in waves (knot) 22.6 23.4 23.5 22.9 23.3 22.6 23.5

Surface roughness (mm) 8.0 7.8 7.5 8.i 7.9 7.5 8.i

Stability Wind resistance level (m/s) i43 i68 i76 i88 i76 i43 i88

Stability criterion number i.58 i.84 i.79 i.76 i.9i i.58 i.9i

Table 5

Weights of criteria.

Criteria (u) Weights (a) Subcriteria (ui) Weights a1 a2 Ai

Ui 0.3 U11 0.35i 0.202 0.277

U12 0.35i 0.215 0.283

U13 0.189 0.194 0.i9i

U14 0.109 0.389 0.249

U2 0.6 U21 0.666 0.378 0.522

U22 0.333 0.622 0.478

U3 0.i U31 0.750 0.494 0.622

U32 0.250 0.506 0.378

characteristic value, characteristic vector and random consistence ratio of matrix P1 are 1 = 4.01, W = [0.351, 0.351, 0.189, 0.109] and C.R. = 0.0037, respectively. The characteristic value, characteristic vector and random consistence ratio of matrix P2 are 1 = 2, W = [0.666, 0.333] and C.R. = 0, respectively. The characteristic value, characteristic vector and random consistence ratio of matrix P3 are 1 = 2, W = [0.75, 0.25] and C.R. = 0, respectively.

Entropy weighing method is used to analyze the objective weights using the data in Table 4. The calculated weights are shown in Table 5. In this case study, the preference coefficient b = 0.5 is selected.

issues, and finally reached a consensus. According to the experts' remarks, the judgment matrix of the first-level is:

1 1/2 3'

2 1 6 1/3 1/6 1

The judgment matrices of the second-level are:

i i 2 3

i i 2 3

i/2 i/2 i 2

i/3 i/3 i/2 i

and P3

By AHP calculation, the characteristic value of matrix P is 1 = 3, the corresponding characteristic vector is W = [0.3, 0.6, 0.1], and the random consistence ratio C.R. = 0, which is less than 0.1, meets the consistence requirements. Similarly, the

Step 3. Establish remark set. In this study, V = {v1, v2, — , v10} = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100}. The

0 0 60

Criterion value

k=0 k=1

Fig. 9. Graph of semi sine shape function.

elements represent the percentile scores. A value of 100 corresponds to the best performance. This remark set is used for both of the evaluation layers. Step 4. Preliminary data processing. Usually most of the criterion values are in a given reference range. However, some outliers will distort the weights. In order to overcome this defect, a semi sine shape function, whose evaluation characteristics is similar to the evaluation mind of human beings, is adopted. The semi sine shape function, which also converts the original data into percentile scores, is expressed as follows:

N = 50

1 + (-1)k sin

where N denotes the grading score, umax denotes the specified maximum value; umin denotes the specified minimum value; u denotes the value of criterion. k is a constant number alternative from 0 and 1. When k = 0, it denotes that the bigger the value is, the better the adaptability is. Or when k = 1, it denotes that the smaller the value is, the better the adaptability is. The graphical representation of the semi sine shape function is shown in Fig. 9.

In this illustrative case, the smaller the significant pitch, heave, bow acceleration, green water frequency, and surface roughness, the better the environment adaptability of ship is. On the contrary, the bigger the sailing speed in the waves, wind resistance level, and stability criterion number, the better the environment adaptability of ship is. Formula (32) is used to transform the original data in Table 4 into percentile scores, which are used for the calculation of degree of membership in the following step. In the formula, for each of the subcriterion, umin is the minimum value among the five alternative ships (column 8 in Table 4); umax is the maximum value among the five alternative ships (column 9 in Table 4); u is the value of the criterion of the specified ship for calculation. As a result, the converted percentile scores by using Formula (32) are shown in Table 6.

Step 5. Fuzzy comprehensive evaluation. It is important to select an accurate membership function to determine the membership degree first. As mentioned above, the common used membership functions are summarized in Table 2. The

linear membership functions, such as trapezoidal and triangular functions, are intrinsically linked to linear philosophy. So it is questionable to assess nonlinear and complex events using such linear and simple membership functions. The curve of Shape G and Cauchy functions is not smooth enough and the relative larger degree centralizes around the peak value with a narrow-banded distribution form. On the other hand, the Gaussian function is a useful tool in describing membership function of multiparameter nonlinear issue, since it is similar with the vagueness of human knowledge. In this case Gaussian function is used to calculate the degree of membership of each criterion score in the remark set. The membership function is expresses as follows:

-0.005 (vk -N)2

where vk denotes the element in remark set, N denotes the percentile score in Table 6. The relation matrices in the second-level layer calculated by Formula (33) are summarized in Table 7.

By using fuzzy algorithm of Formula (31), fuzzy comprehensive evaluation sets can be obtained. The second level and first level fuzzy comprehensive evaluation sets are shown in Table 8 and Table 9, respectively.

Step 6. Evaluation result. By multiplying the first level fuzzy comprehensive evaluation set with the remark set, the evaluation result can be obtained as:

The evaluation results of the five ships are shown in Fig. 10. Obviously, the environment adaptability of V2 is relatively better than the others. The optimized four ships all have better performance than the traditional round bilge ship. Moreover, the scores of deep-V form ships are much higher than those of UV form ships in this case. However, it has also been verified that the stability of UV form ships is better than that of deep-V form ships. It is refer to mention that there is a pronounced difference between the SSB (i.e. the bulb bow) of ships UV1 and UV2; and the relative better seakeeping and resistance performance of UV2 is largely attributed to the dedicated SSB.

5.2. The sensitivity analysis

Table 6

Percentile scores.

Criteria Subcriteria R0 V1 V2 UV1 UV2

Ui Uli 100 70 81 0 64

U12 0 100 99 53 85

U13 0 98 100 82 100

u14 10 0 35 100 65

U2 U2I 0 97 100 25 88

U22 7 50 100 0 25

U3 U3I 0 59 83 100 83

U32 0 89 71 57 100

Since only one condition, i.e., the specified weighing preference coefficient, membership function, and fuzzy operator, was considered in the above case study, this section is aimed at analyzing the results of the sensitivity for the illustrative case.

The sensitivity analyses are conducted corresponding to four different control strategies: the weighing set, the preference coefficient, the membership function, and the fuzzy operator. They are introduced as follows.

In the comparative analysis of weighing set, assume that the importance of each criterion and subcriterion are equal, i.e., P = [1]3*3, Pi = [1]4*4, P2 = [1]2*2, P3 = [1]2*2. Other

Table 7

Relation matrices in the second-level layer.

Sample

Symbol

Relation matrix

"0 0 0 0 0 0 0.011 0.135 0.607 1

0.607 0.135 0.011 0 0 0 0 0 0 0

0.607 0.135 0.011 0 0 0 0 0 0 0

1 0.607 0.135 0.011 0 0 0 0 0 0

0.607 0.135 0.011 0 0 0 0 0 0 0

0.956 0.430 0.071 0.004 0 0 0 0 0 0

"0.607 0.135 0.011 0 0 0 0 0 0 0 "

0.607 0.135 0.011 0 0 0 0 0 0 0

"0 0 0 0.011 0.135 0.607 1 0.607 0.135 0.011"

0 0 0 0 0 0 0.011 0.135 0.607 1

0 0 0 0 0 0.001 0.020 0.198 0.726 0.980

0.607 0.135 0.011 0 0 0 0 0 0 0

0 0 0 0 0 0.001 0.026 0.236 0.783 0.956"

0 0.011 0.135 0.607 1 0.607 0.135 0.011 0 0

"0 0 0.015 0.164 0.667 0.995 0.546 0.110 0.008 0

0 0 0 0 0 0.015 0.164 0.667 0.995 0.546

"0 0 0 0 0.008 0.110 0.546 0.995 0.667 0.164"

0 0 0 0 0 0 0.015 0.164 0.667 0.995

0 0 0 0 0 0 0.011 0.135 0.607 1

0.044 0.325 0.882 0.882 0.325 0.044 0.002 0 0 0

0 0 0 0 0 0 0.011 0.135 0.607 1 "

0 0 0 0 0 0 0.011 0.135 0.607 1

0 0 0 0 0.004 0.071 0.430 0.956 0.783 0.236"

0 0 0 0.008 0.110 0.546 0.995 0.667 0.164 0.015

"0.607 0.135 0.011 0 0 0 0 0 0 0

0 0.004 0.071 0.430 0.956 0.783 0.236 0.026 0.001 0

0 0 0 0 0.006 0.089 0.487 0.980 0.726 0.198

0 0 0 0 0 0 0.011 0.135 0.607 1

0.325 0.882 0.882 0.325 0.044 0.002 0 0 0 0 "

0.607 0.135 0.011 0 0 0 0 0 0 0

0 0 0 0 0 0 0.011 0.135 0.607 1 "

0 0.001 0.026 0.236 0.783 0.956 0.430 0.071 0.004 0

"0 0 0.003 0.056 0.375 0.923 0.835 0.278 0.034 0.002"

0 0 0 0 0.002 0.044 0.325 0.882 0.882 0.325

0 0 0 0 0 0 0.011 0.135 0.607 1

0 0 0.002 0.044 0.325 0.882 0.882 0.325 0.044 0.002

0 0 0 0 0.001 0.020 0.198 0.726 0.980 0.487"

0.325 0.882 0.882 0.325 0.044 0.002 0 0 0 0

0 0 0 0 0.004 0.071 0.430 0.956 0.783 0.236"

0 0 0 0 0 0 0.011 0.135 0.607 1

control strategies are the same as those in the above illustrative case. The comparison of final evaluation results are shown in Fig. 11. As seen from the results, the influence of weighing on the result is not pronounced. An interesting phenomenon can be found from the results: the score of V2 evaluated by the combination weight—decided with the experts' efforts—is higher than the unit—and arbitrary—weighting coefficients; also V2 is the demonstrated best candidate; so the experts' efforts are helpful for finding the potential alternative.

The influence of weighing preference coefficient b on the results is shown in Fig. 12. As seen from the results, the scores

change linearly as the preference coefficient varies from 0 to 1. It is clear that the influence of different b is not obvious for R0, V2 and UV1. It can be concluded that b = 0.5 is a reasonable choice, since it is a compromise between the two extreme conditions.

In order to investigate the influence of membership functions on the evaluation results, the result obtained by using the triangular function, which is defined in Table 2, is compared with the result obtained by the Gaussian function. In the triangular function, a = 0 and c = 110 are chose in this calculation. And the membership function is expresses as follows:

Table 8

Fuzzy comprehensive evaluation sets of second level.

Sample Symbol Evaluation sets

b, [0.419 0.168 0.030 0.002 0 0 0.002 0.029 0.131 0.217]

r0 b2 [0.708 0.253 0.036 0.002 0 0 0 0 0 0 ]

b3 [0.805 0.180 0.015 0 0 0 0 0 0 0 ]

b, [0.086 0.019 0.002 0.002 0.021 0.096 0.163 0.140 0.199 0.271]

VI b2 [0 0.002 0.029 0.129 0.213 0.130 0.035 0.057 0.182 0.222]

b3 [0 0 0.004 0.042 0.169 0.254 0.163 0.130 0.155 0.084]

bi [0.005 0.037 0.101 0.101 0.038 0.019 0.073 0.160 0.226 0.239]

V2 b2 [0 0 0 0 0 0 0.006 0.077 0.346 0.570]

b3 [0 0 0 0.001 0.018 0.101 0.258 0.340 0.220 0.061]

bi [0.092 0.021 0.013 0.067 0.149 0.130 0.089 0.125 0.159 0.157]

UV1 b2 [0.279 0.320 0.283 0.103 0.014 0.001 0 0 0 0 ]

b3 [0 0 0.005 0.044 0.145 0.177 0.083 0.054 0.186 0.305]

bi [0 0 0.001 0.011 0.079 0.208 0.232 0.184 0.164 0.121]

UV2 b2 [0.064 0.173 0.173 0.064 0.009 0.005 0.042 0.156 0.210 0.104]

b3 [0 0 0 0 0.001 0.020 0.123 0.293 0.325 0.238]

Table 9

Fuzzy comprehensive evaluation sets of first level.

Sample_Evaluation sets_

RO [0.631 0.220 0.032 0.002 0 0 0.001 0.009 0.039 0.065] vi [0.026 0.007 0.018 0.082 0.151 0.132 0.086 0.089 0.185 0.223] V2 [0.002 0.011 0.030 0.031 0.013 0.016 0.052 0.128 0.297 0.420] uv1 [0.195 0.198 0.174 0.086 0.067 0.057 0.035 0.043 0.066 0.078] UV2_[0.038 0.104 0.104 0.042 0.029 0.067 0.107 0.178 0.208 0.123]

r(Vk) = {(U0 - x)/(110 - N) • (35)

The comparative results of different membership functions are shown in Fig. 13. It is seen from the figure that membership function has pronounced influence on the results. Although the ranking of the alternatives remains unchanged for different membership functions, the scores obtained by Gaussian function distribute in a large range, i.e. from 22.6 to 86.6, which is beneficial for the identification of ship comprehensive performance.

To investigate the influence of fuzzy operators on the evaluation results, fuzzy operators I and II, respectively expressed as Formulas (27) and (28), are adopted. The evaluation results are compared with the result obtained by using fuzzy operator III, which is expressed as Formula (31). The comparative results are shown in Fig. 14. The results show that the fuzzy operators have moderate influence on the results. And the ranking of the scores of five alternatives by fuzzy operator III is relative wider than the other two cases.

In summary, as seen from Figs. 11 —14, the results obtained by using different weight sets, different preference coefficients, different membership functions and different fuzzy operators indicate that the algorithm selection has a certain influence on the evaluation results. However, the rankings obtained from evaluation scores for the five ships are almost

the same, e.g., V2 is the best choice and R0 is the least one. It is noteworthy to mention that the influence of criteria weighings on the results is not as pronounced as the influence of membership functions and fuzzy operators; and the preference parameter has a moderate influence on the evaluation result.

6. Prospective of ship comprehensive performance evaluation

The Environment Adaptability Research Center of Ships (EARCS) of HEU is devoted to studying integrative sailing performance of surface ship by both experimental and numerical approaches from the early 21st century. The development of ship environment adaptability evaluation system is still in progress.

In the above example analysis, the representative index values of ships were obtained by laboratory tank measurement or numerical simulation rather than in real wind and sea wave conditions. In order to obtain the most reliable data regarding ship integrative sailing performance, the EARCS has carried out a series of experimental measurements by using large scale models in real sea conditions in the recent years. The testing models are usually manufactured more than 10 m long in order to reduce the scale effects. Superstructures and appendages are equipped onboard the model during the measurements so that the interactions of winds, waves and currents effects can be

R0 V1 V2 UV1 UV2 Fig. 10. Evaluation results.

a 60 o

By combination weight By equal weight

Fig. 11. Evaluation results by different weighings.

taken into account. The models are usually designed to be segmented so that load behavior as well as floodability are involved and investigated. Moreover, the self propulsion system allows for the study of rapidity and maneuverability. Photographs of the large-scale UV-shaped (LUV-01) model and the testing scene during measurement are illustrated in Fig. 15.

The trial data obtained lays a solid foundation for the further evaluation of ship environment adaptability. Almost all

By Gaussian function By triangular function

Fig. 13. Evaluation results by different membership functions.

kinds of standard ship performance tests can be performed by using these large scale models, such as seakeeping, hydro-elasticity, resistance, propulsion, maneuverability, stability and floodability tests, etc. This testing approach not only allows for the investigation of independent disciplines, but also provides a platform for the analysis of multidiscipline comprehensive sailing performance of ships. Moreover, some dangerous testing schemes that not easy be performed by full-scale measurement, such as tests in extreme sea states and underwater explosion tests, can be also conducted by this kind of models. During the tests, the model can be designed to fulfill a series of simulation tasks in a specified natural condition. Then the measured values of criteria in Section 2.2 as well as the time and power consumed during the simulations can be used for the comprehensive evaluation.

7. Conclusions

In this paper, the ship environment adaptability assessment system based on fuzzy comprehensive evaluation is proposed. With the example of assessing the comprehensive performance of five ships, the following conclusions can be made:

0I-,-1-,-1-,-1-,-1-,-1

0.0 0.2 0.4 0.6 0.8 1.0

Preference coefficient p

-ro -vi -v2

-uvl -uv2

Fig. 12. Influence of weighing preference coefficients.

100 |-

R0 V1 V2 UV1 UV2

By fuzzy operator I By fuzzy operator 11 By fuzzy operator 111

Fig. 14. Evaluation results by different fuzzy operators.

(a) Overview of the large model (b) Experimental scene

Fig. 15. Field testing measurement for ship comprehensive performance.

(1) The evaluation index system established by combining the seven subjects of naval architecture is well representative of ship environment adaptability.

(2) The combination weighing method, which considers both the subjective as well as the objective effects, is reasonable enough and it lays a foundation for the fuzzy comprehensive evaluation.

(3) The results of fuzzy comprehensive evaluation of five ships in the case study show that the deep-V shape hybrid monohull V2 stands out from the others for its excellent performance.

(4) According to the sensitivity analyses, the membership functions and fuzzy operators have moderate influence on the evaluation results. However, the influence caused by criteria weighing and preference parameter is much weaker in this case study.

(5) The results obtained by using different weight parameters, different membership functions and different fuzzy operators show that the algorithm selection has a certain influence on the evaluation results. However, it is reliable enough for the qualitative analysis of ship integrative performance. Therefore, the evaluation system proposed is an effective tool in the qualitative analysis and comparison in naval architecture.

A fuzzy comprehensive evaluation system has been developed in this study. Although only typical criteria are identified for the comprehensive performance evaluation of naval ships, the fuzzy comprehensive evaluation method established in this paper provides a general application for other kinds of ships. However, there are still challenges regarding the proposed evaluation approach that need further improvement. The index values were obtained by corresponding unique aspect of tested or calculated method and the influence of different subjects is not considered.

Acknowledgments

The authors would like to thank the reviewers for their valuable remarks and comments. The authors would also like to thank Doctor Christiaan Adika Adenya for checking the grammar of this paper. This work was supported by the

National Natural Science Foundations of China (no. 51079034 and no. 51209054).

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