# Container Shipping Scheduling Method Based on the Evidence Reasoning Approach in Fluctuating CCFI and BDI Cycle

Authors: Yisong Xu; Xinyu Zhuang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Due to the COVID-19/Omicron pandemic and the trade war and tariffs between China and America, the supply chain between Asia and America, or Asia and Europe, faced unprecedented challenges. With the outbreak of the Russian-Ukrainian war, the global supply chain has become increasingly unbalanced. In particular, the shortage of container ships and the continuous fluctuation of the BDI and CCFI make shipping scheduling increasingly important for ocean carriers. To solve this problem, in this study, we propose an analytic approach considering the fluctuation of BDI and CCFI based on interval evidence reasoning and the Hungarian algorithm. The proposed approach uses two pairs of nonlinear optimisation models to construct a Nash equilibrium assignment model to compute the shipping company’s scheduling and maximum utility in the CCFI and BDI cycles. Compared to the simulation algorithm, the analytical algorithm can take advantage of stability and efficiency. Finally, a COSCO company’s container shipping scheduling problem was examined to demonstrate the efficiency and effectiveness of the proposed approach.


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## Body

## 1. Introduction

One of the pervading headlines throughout 2020–2021 focused on the global supply chain disruption issues affecting businesses and consumers worldwide. However, these situations are likely to persist and are warnings that these critical problems may last in 2022 and potentially beyond. This would keep significant strains on, as they accept higher prices and costs of dealing with business in such a navigate continued uncertainty. According to the trade analysis unit of the S&P Global Market Intelligence survey, the CCFI(China Containerized Freight Index) and BDI(Baltic Dry Index) fluctuated 52% year on year, whereas the carriers had to deal with the capacity imbalance and shortage of container ships. Therefore, shipping scheduling becomes a critical task.For decades, many scheduling methods have been developed, such as scheduling with Soft Time Window [1–3]; Maritime considering inventory, draft limits, and bunker cost [4–6]; routing problem with berthing time clash avoidance constraints [7–10]; green logistics, green ports [11, 12], and Big Data in Maritime Logistics [13, 14]. In these methods, scheduling problems are modelled using shipping matrices in which an attribute is assessed using a certain number of criteria. Under many decision situations, however, using a certain number to represent an evaluation proves to be difficult and unachievable. Evaluation would have been distorted or lost in the process of preaggregating differences, such as a probability distribution, a subjective judgment, or an incomplete part of the information, into a single number [15].Note that using distributed assessment instead of certain numbers would enable various types of attribute values to be incorporated into an evaluation process without preaggregation [16] and in our study, belief decision matrix (BDM) concepts were adapted to evaluate the complex system. To take advantage of the distributed assessment structure contained in the BDM, the D-S theory was employed for information aggregation, which resulted in the ER approach [17–19].In this study, container ship assessments were evaluated using a belief hierarchy, subsets of adjacent grades were set up and a ship assessment model was aggregated with the route data. An evidence-reasoning algorithm with a Nash equilibrium assignment model, ER-NAP, was developed to support ship evaluation and complex ship scheduling in fluctuating CCFI and BDI cycles. In most situations, the target of the assignment is to maximise the participant’s rationality instead of maximising the collective rationality. For example, in an emergency, the most important problem is whether each unit is equally considered [20, 21]. Another application is scheduling problems arising in satellite communications. While many of the base stations seek their most efficient satellites and assigning the satellites fairly is important for the third-party satellite administrators [22]. Contrary to the former, this is a cooperative assignment because a satellite can be assigned to more than one base station at a time. However, all assignment problems, including the noncooperative and cooperative assignment problems were modelled from the perspective of collective rationality and ignored the participants’ rationality. In addition, it is realized that taking the solution with collective rationality into practice may be difficult and sometimes unacceptable because the participants’ rationality would have been lost or distorted in the process of maximising collective rationality. Although such a choice is in a perfectly competitive market, no attempt has been made to test whether the optimal solution can be effectively conducted. However, if one adopts game theory, it may be argued that the existing model for finding the optimal solution experiences shortcomings in these competitive situations. Our study aims to rectify this critical shortcoming, and the optimal solution is extended for the first time.By introducing the coefficients of CCFI and BDI, the ER-NAP algorithm provides a fusion modelling framework and an attribute aggregation process to deal with both ship and route information. A COSCO scheduling example was examined to demonstrate the implementation process of the ER algorithm and its applications.In the following sections, the ER-NAP model is first explained, followed by the coefficient of CCFI and BDI introduction and a scheduling analytical structure fusion with ship route and CCFI and BDI information. In detail, the interval uncertainty of ship attributes, in both quantitative and qualitative terms, transforms into a standard form. Finally, all normalised data were input into the system, and the results were generated synchronously. Other details are demonstrated using a real example from the COSCO Company.


## 2. Evidence Reasoning-Nash Equilibrium Assignment Scheduling Structure

### 2.1. Macrostructure of the Scheduling Process

First, suppose several ships need to be assessed and routes difference would be represented by the characteristic vector, as shown in Table1.(1)
Assuming the number of ships ism, the number of routes is n.(2)
Each ship hasL attributes, such as load efficiency, the number of standard TEU, speed, tonnage, fuel consumption, and age of the ship. The formation of attribute information includes qualitative data, quantity data, fuzzy data, and interval data.(3)
The characteristic of each route is represented by a weight sector. Every single weight correlate attribute of the ship. Different weight sectors suggest that the various ships competing for the most profitable route have disparate efficiency.Table 1
Route weight vector and ship attribute.
AttributesRouteShipwL1wL2……wLna1a2……am1w11w22……w2ne11e12……e1m2w21w22……w2na21a22……e2m⋮⋮⋮⋮⋮⋮⋮⋮⋮LwL1wL2……w2neL1eL2……eLmBased on the attribute metrics of the routes and ships in Table1, we used the interval efficiency matrix calculation method (shown in Section 2.2) to generate the interval efficiency matrix of Route n’s characteristic vector with Ship m’s attributes in Table 2. The Nash equilibrium assignment model with CCFI and BDI eco-efficiency (shown in Section 2.3) was used to calculate the optimal matching of the ship and route.Table 2
Interval efficiency matrix of Route N’s characteristic vector with Ship M’s attribute.
RouteShipM1M2……MmwL1[c11−, c11+][c12−, c12+]……[c1m−, c1m+]wL2[c21−, c21+][c22−, c22+]……[c2m−, c2m+]…………………………wLn[cn1−, cn1+][cn2−, cn2+]……[cnm−, cnm+]


### 2.2. Interval Efficiency Matrix Calculation Method

The expected utility calculation model of ER nonlinear programming is a method for calculating the utility of a container ship using a nonlinear programming model. It has two equivalent algorithms: recursive and analytical. In contrast to the traditional evaluation method, the ER method uses basic credibility to evaluate the properties of attribute information in the efficiency matrix. This has the advantage of being able to model uncertainty information to describe the uncertainty of the subjective data. Existing evidence-based reasoning models require the evaluator to provide the basic credibility of the grade relative to the evaluation level. However, in practical applications, it is difficult for the evaluator to directly provide the basic credibility of a decision-making scheme relative to the evaluation level; however, it is possible to provide the range of the basic credibility of a decision-making scheme relative to the evaluation level.For example, it is often difficult to accurately evaluate the basic credibility of the load factor of the ship relative to the evaluation level asSeiaload factor=H1,0.3;H2,0.2;H3,0.5; however, it can be determined that the range of the basic credibility of the load factor of the ship relative to the evaluation level is as follows: Seiaload factor=H1,0.3,0.4H2,0.2,0.3;H30.4,0.5, that is, the belief degree of attribute ei of the decision-making scheme al relative to the evaluation level H1 is a certain value in the interval [0.3, 0.4]; the belief degree of the relative evaluation level H2 is a certain value in the interval [0.2, 0.3]. Numerical value: The belief degree of the relative evaluation level H2 is a certain value between [0.4, 0.5]. In an uncertain environment, the ER algorithm of the belief degree expressed in the form of intervals can help evaluators express their subjective wishes [27–28]. Therefore, this study used the ER parsing algorithm based on the belief degree of the interval form to construct the matching model.


#### 2.2.1. ER Model Based on Interval Belief Degree

In the multi-attribute ship evaluation process, uncertainty data must be considered. LetA=a1,a2,...,am represent the set of m ships; ai represents ship i; E=e1,e2,...,eL represents each ship with L attributes; W=w1,w2,...,wLT represents the weight vector of the attributes where wi represents the route’s characteristic correlate to the ship’s attribute i, and satisfies 0≤wi≤1 and ∑i=1Lwi=1.Suppose that each attribute is represented by N complete and independent belief degree sets:Hn=H1,...,HN， then the evaluation information of attribute ei based on the interval-belief degree can be expressed in the distribution form as Seial=Hn,βn,i−al,βn,i+al,n=1,...,N， βn,i−al≥0， βn,i−≤βn,i≤βn,i+， where βn,i represents the attribute ei of ship al is a single number between βn,i− and βn,i+ in belief degree Hi. If βn,i−≡βn,i+， ei is a single number instead of the interval in Hi.For example,H=H1,H2,H3， Seial=H1,0.3,0.4;H2,0.2,0.3;H30.4,0.5，(1)βH,i−al=max0,1−∑n=1Nβn,i+al=max0,1−0.4−0.3−0.5=0,βH,i+al=1−∑n=1Nβn,i−al=1−0.3−0.2−0.4=0.1.The ignoranceH is a single number between the interval 0,0.1.The efficiency metrics based on the interval belief degree were obtained until the evaluation of theL basic attributes of M ships:(2)Dg=SeialL×M.If the attribute evaluation is represented by the interval-based basic reliability, the evaluation information of decision planai after the attribute evaluation information is aggregated using the ER method is the interval-based basic reliability. Based on the evaluation information in the form of interval basic credibility, a pair of nonlinear programming models is used to aggregate the basic credibility of the attributes relative to the evaluation level as the basic credibility of the decision scheme relative to the evaluation level, which is used to calculate the interval-based evaluation. The basic credibility is located in the upper and lower bounds of the interval.


#### 2.2.2. Calculation Process Based on Nonlinear Programming

First, according to formulas (1)–(5), the decision matrix is transformed into the form of belief degree, and the undistributed part of belief degree mH,i includes two parts: m¯H,i and m˜H,i; m¯H,i is calculated by ei correlate weight, whereas m˜H,i represents the ei’s evaluation information is incomplete.(3)mn,i=miHn∈mn,i−,mn,i+=wiβn,i−al,wiβn,i+al,n=1,...,L,(4)m¯H,i=m¯iH=1−wi,i=1,...,L,(5)m˜H,i=m˜iH∈m˜H,i−,m˜H,i+=wiβH,i−al,wiβH,i+al,i=1,...,L,(6)∑n=1Nmn,i+m¯H,i+m˜H,i=1,i=1,...,L,(7)∑i=1Lwi=1.Second, the evaluation information of theL attributes of the ships is combined. Because the attribute evaluation information of the ships is expressed by the interval belief degree, the expected utility of the aggregated ship attributes may also be in the interval form. Therefore, it is necessary to use the objective-maximising nonlinear programming model (6) and objective-minimizing nonlinear programming model (7) to solve the upper and lower bounds of the expected utility of the ship, respectively.(8)Maxumaxal=∑n=1N−1uHnβnal+uHnβNal+βHal,s.t. βnal=mn/1−m¯H,(9)βH=m˜H1−m¯H,mn=k∏i=1Lmn,i+m¯H,i+m˜H,i−∏i=1Lm¯H,i+m˜H,i,m˜H=k∏i=1Lm¯H,i+m˜H,i−∏i=1Lm¯H,i,m¯H=k∏i=1Lm¯H,i,k=∑n=1N∏i=1Lmn,i+m¯H,i+m˜H,i−N−1∏i=1Lm¯H,i+m˜H,i−1,mn,i−≤mn,i≤mn,i+,n=1,...,N;i=1,...,L,m¯H,i=1−wi,m˜H,i−≤m˜H,i≤m˜H,i+,i=1,...,L,∑n=1Nmn,i+m¯H,i+m˜H,i=1,i=1,...,L.Minuminal=∑n=1N−1uHnβnal+uHnβNal+βHal,s.t. βnal=mn/1−m¯H,(10)βH=m˜H1−m¯H,mn=k[∏i=1Lmn,i+m¯H,i+m˜H,i−∏i=1Lm¯H,i+m˜H,im˜H=k∏i=1Lm¯H,i+m˜H,i−∏i=1Lm¯H,i,m¯H=k∏i=1Lm¯H,i,k=∑n=1N∏i=1Lmn,i+m¯H,i+m˜H,i−N−1∏i=1Lm¯H,i+m˜H,i−1,mn,i−≤mn,i≤mn,i+,n=1,...,N;i=1,...,L,m¯H,i=1−wi,m˜H,i−≤m˜H,i≤m˜H,i+,i=1,...,L,∑n=1Nmn,i+m¯H,i+m˜H,i=1,i=1,...,L.Models (6) and (7) constitute a complete ER-expected utility analysis algorithm. The uM1− represents the floor of the object, whereas the uM1+ represents the ceiling of the object. Using this algorithm, the expected utility uM1−,uM1+,uM2−,uM2+,...,uMm−,uMm+ of ship al can be calculated using the route characteristic vector wL1T=w11,w21,...,wl1.


### 2.3. Nash-Equilibrium Assignment Model

Nash equilibrium is a game approach in nature that provides a new way to study the rationality of ships in assignment problems. In this study, we briefly introduce the transformation method to pave way for future discussion. Assuming that a ship’s payoff is directly proportional to its utility to the route, each ship pursues its most efficient route to maximise its payoff. To deal with conflicts, ships are viewed as players in a game and utility is viewed as their payoff.To illustrate the Nash equilibrium of the ship assignment problem, a benefit-oriented example with two ships and four routes is presented in Table3. Compared with the traditional objective, from an individual perspective, the question is what the best assignment is, while every ship pursues the most profitable route to itself.Table 3
Two ships with four routes.
ShipRoutesABCDShip 1213154Ship 2104145Note. The elements represent the benefits.In contrast to the classical solving process, the assignment problem was transformed into an equivalent game problem as shown in Table4. If different ships cannot afford the same route, diagonal elements of the efficiency matrix do not exist.Table 4
The equivalent game problem.
Ship 2ABCDShip 1A-, -2, 42, 142, 5B13, 10-, -13, 1413, 5C15, 1015, 4-, -15, 5D4, 104, 44, 14-, -Note. “-” represents that ship 1 and ship 2 cannot take on the same route. The elements represent the benefits.Note that the assigned ship’s utility is transformed to its payoff equivalently, and then we can solve this problem using game theory. Table5 shows the results solved by the Hungarian method and Nash equilibrium.Table 5
Results of two ships with four routes.
Hungarian methodNash equilibrium 1Nash equilibrium 2Ship 1Route BRoute CRoute BShip 2Route CRoute ARoute CTotal utility272527Note. The elements represent the benefits.In contrast to the Hungarian method, the Nash equilibrium represents individual rationality. However, it is noted that the Nash equilibrium is not unique in this example, and may be equivalent to the Hungarian solution. In the next section, we will show a cost-oriented Nash equilibrium assignment problem example with three ships and four routes in which its maximal Nash equilibrium is also equivalent to the optimal solution, to lead to the question of “whether the optimal solution of an assignment problem is a Nash equilibrium point in nature.” Table6 shows the example.Table 6
Three ships with four routes.
ShipRouteABCDShip 11041415Ship 29141613Ship 378119Note. The elements represent the benefits.The example above can also be equivalently transformed into a game problem. In this game problem, the number of payoffs is sixty-four, and the number of effective payoffs (without negative infinity) is four. With the increase in the number of ships and routes, the number of effective payoffs is enormous. It was calculated that the number of payoffs in the assignment problem withm ships and n routes is nm. Among these payoffs, the number of effective payoffs (without the payoffs of negative infinity) is Anm, whereas the number of ineffective payoffs (negative infinity) is nm−Anm. The results of the example are shown and analysed first. The results of the assignment problem example with three ships and four routes are presented in Table 7.Table 7
Results solved by the Hungarian method and game method.
Hungarian methodNash equilibrium 1Nash equilibrium 2Nash equilibrium 3Nash equilibrium 4Ship 1Route B (4)Route B (4)Route a (10)Route B (4)Route C (14)Ship 2Route A (9)Route A (9)RouteD (13)RouteD (13)Route A (9)Ship 3RouteD (9)RouteD (9)Route B (8)Route A (7)Route B (8)Total utility2222∗312431As a result, Nash equilibrium 1 is equivalent to the optimal solution. Next, we would prove that the optimal solution is a Nash equilibrium in the assignment problem.Theorem 1.
The limited assignment problem has, and only has pure-strategy Nash equilibrium.From the Nash equilibrium existence theorem [19], we can conclude that the limited assignment problem must have a pure-strategy Nash equilibrium or mixed-strategy Nash equilibrium. Because the assignment problem can be equivalently transformed into a game problem, the assignment problem must have a pure-strategy Nash equilibrium or mixed-strategy Nash equilibrium. Because the assignment problem and its equivalent game problem include the constraints that different ships cannot afford the same route, it is inferred that the mixed-strategy Nash equilibrium cannot be reached. Therefore, the limited assignment problem has, and only has pure-strategy Nash equilibrium.Theorem 2.
The optimal solution is a Nash equilibrium point in the assignment problem.Proof by contradiction:Assume that a limited assignment problem hasm Ships and n Routes, which can be equivalently transformed into a game. This game can be expressed as G=S1,…,Sm;u1,…,um. The optimal solution strategy is sσ=s1σ,…,siσ,…,smσ, if existing i, uisiσ,s−iσ≤uisi,s−iσ, when the others’ choice is s−iσ=s1σ,…,si−1σ,.si+1σ..,smσ. The total efficiency of strategy sσ=s1σ,…,siσ,…,smσ can be increased by uisi,s−iσ−uisiσ,s−iσ. However, this contradicts the assumption that the strategy is the optimal solution.Therefore, for strategysσ=s1σ,…,siσ,…,smσ, the optimal solution is a Nash equilibrium point.Based on Theorem2, we can conclude that the optimal solution is a subset of game solutions. The solution to the assignment problem is extended for the first time. The proposed approach uses two pairs of nonlinear optimisation models to construct an assignment model to compute the matching order and maximum utility of the shipping company, and proposes a method to choose between different assignment sequences. Compared to the simulation algorithm, the analytical algorithm can take advantage of stability and efficiency. Finally, a cargo ship routing problem was examined to demonstrate the implementation process of the proposed approach.In cases where there was competition among the participants, it should be reappraised that the credibility of an assignment is exempted from market power. To address this issue, each participant was viewed as a player who sought to maximise their efficiency, and the assignment problem was solved using game theory for the first time. Specifically, it is proven that the optimal solution is the Nash equilibrium point of the assignment problem. Finally, relevant issues are discussed.


### 2.4. Calculation Process with CCFI and BDI Coefficiency

Owing to the fluctuation of the CCFI and BDI, the scheduling operator faces increasingly complex price data. To pursue profit maximisation, they must periodically adjust the scheduling plan with the indexes. From the perspective of the operator of scheduling, the route’s characteristic vectorwL1T=w11,w21,...,wl1 includes two opposite parts: benefit variables’ weights and cost variables’ weights. For example, the benefit variables include load factors, deadweight, and a number of standard TEUs, whereas the cost variables include annual M&R and manning costs, sea fuel consumption, and off-hire days. If the route price increases rapidly, the benefit variables should be assigned higher weights. Otherwise, if the route price plummets, the cost variables should be assigned higher weights. Fortunately, all these price changes can be reflected by the CCFI indices as shown in Table 8.Table 8
Index of CCFI.
CCFIRoute 1Route 2Route 3Route 4Route 5Route 6Route 7CC1C2C3C4C5C6C7Source: the official website of CCFI.In Table8, the CCFI index is calculated by all routes’ price indexes with the weighted average methods, and we can use equation (8) to generate a vector of weights for a vector of routes.(11)WRoutei=CiC.First, all the route’s characteristic vectorswL1T=w11,w21,...,wl1 can be divided into two parts: benefit variable weights and cost variables’ weights. The weights of the benefit variables’ weights are multiplied by WRoute i, whereas the cost variables’ weights are divided by WRoute I. Secondly, normalizing the new benefit variables and cost variables constructs the route’s characteristic vector. Finally, we use the route’s characteristic vector, which considers the CCFI indices to match the routes and ships with the methods described above. In addition, if we use the BDI indexes instead of the CCFI indexes, all steps are the same.


## 2.1. Macrostructure of the Scheduling Process

First, suppose several ships need to be assessed and routes difference would be represented by the characteristic vector, as shown in Table1.(1)
Assuming the number of ships ism, the number of routes is n.(2)
Each ship hasL attributes, such as load efficiency, the number of standard TEU, speed, tonnage, fuel consumption, and age of the ship. The formation of attribute information includes qualitative data, quantity data, fuzzy data, and interval data.(3)
The characteristic of each route is represented by a weight sector. Every single weight correlate attribute of the ship. Different weight sectors suggest that the various ships competing for the most profitable route have disparate efficiency.Table 1
Route weight vector and ship attribute.
AttributesRouteShipwL1wL2……wLna1a2……am1w11w22……w2ne11e12……e1m2w21w22……w2na21a22……e2m⋮⋮⋮⋮⋮⋮⋮⋮⋮LwL1wL2……w2neL1eL2……eLmBased on the attribute metrics of the routes and ships in Table1, we used the interval efficiency matrix calculation method (shown in Section 2.2) to generate the interval efficiency matrix of Route n’s characteristic vector with Ship m’s attributes in Table 2. The Nash equilibrium assignment model with CCFI and BDI eco-efficiency (shown in Section 2.3) was used to calculate the optimal matching of the ship and route.Table 2
Interval efficiency matrix of Route N’s characteristic vector with Ship M’s attribute.
RouteShipM1M2……MmwL1[c11−, c11+][c12−, c12+]……[c1m−, c1m+]wL2[c21−, c21+][c22−, c22+]……[c2m−, c2m+]…………………………wLn[cn1−, cn1+][cn2−, cn2+]……[cnm−, cnm+]


## 2.2. Interval Efficiency Matrix Calculation Method

The expected utility calculation model of ER nonlinear programming is a method for calculating the utility of a container ship using a nonlinear programming model. It has two equivalent algorithms: recursive and analytical. In contrast to the traditional evaluation method, the ER method uses basic credibility to evaluate the properties of attribute information in the efficiency matrix. This has the advantage of being able to model uncertainty information to describe the uncertainty of the subjective data. Existing evidence-based reasoning models require the evaluator to provide the basic credibility of the grade relative to the evaluation level. However, in practical applications, it is difficult for the evaluator to directly provide the basic credibility of a decision-making scheme relative to the evaluation level; however, it is possible to provide the range of the basic credibility of a decision-making scheme relative to the evaluation level.For example, it is often difficult to accurately evaluate the basic credibility of the load factor of the ship relative to the evaluation level asSeiaload factor=H1,0.3;H2,0.2;H3,0.5; however, it can be determined that the range of the basic credibility of the load factor of the ship relative to the evaluation level is as follows: Seiaload factor=H1,0.3,0.4H2,0.2,0.3;H30.4,0.5, that is, the belief degree of attribute ei of the decision-making scheme al relative to the evaluation level H1 is a certain value in the interval [0.3, 0.4]; the belief degree of the relative evaluation level H2 is a certain value in the interval [0.2, 0.3]. Numerical value: The belief degree of the relative evaluation level H2 is a certain value between [0.4, 0.5]. In an uncertain environment, the ER algorithm of the belief degree expressed in the form of intervals can help evaluators express their subjective wishes [27–28]. Therefore, this study used the ER parsing algorithm based on the belief degree of the interval form to construct the matching model.


### 2.2.1. ER Model Based on Interval Belief Degree

In the multi-attribute ship evaluation process, uncertainty data must be considered. LetA=a1,a2,...,am represent the set of m ships; ai represents ship i; E=e1,e2,...,eL represents each ship with L attributes; W=w1,w2,...,wLT represents the weight vector of the attributes where wi represents the route’s characteristic correlate to the ship’s attribute i, and satisfies 0≤wi≤1 and ∑i=1Lwi=1.Suppose that each attribute is represented by N complete and independent belief degree sets:Hn=H1,...,HN， then the evaluation information of attribute ei based on the interval-belief degree can be expressed in the distribution form as Seial=Hn,βn,i−al,βn,i+al,n=1,...,N， βn,i−al≥0， βn,i−≤βn,i≤βn,i+， where βn,i represents the attribute ei of ship al is a single number between βn,i− and βn,i+ in belief degree Hi. If βn,i−≡βn,i+， ei is a single number instead of the interval in Hi.For example,H=H1,H2,H3， Seial=H1,0.3,0.4;H2,0.2,0.3;H30.4,0.5，(1)βH,i−al=max0,1−∑n=1Nβn,i+al=max0,1−0.4−0.3−0.5=0,βH,i+al=1−∑n=1Nβn,i−al=1−0.3−0.2−0.4=0.1.The ignoranceH is a single number between the interval 0,0.1.The efficiency metrics based on the interval belief degree were obtained until the evaluation of theL basic attributes of M ships:(2)Dg=SeialL×M.If the attribute evaluation is represented by the interval-based basic reliability, the evaluation information of decision planai after the attribute evaluation information is aggregated using the ER method is the interval-based basic reliability. Based on the evaluation information in the form of interval basic credibility, a pair of nonlinear programming models is used to aggregate the basic credibility of the attributes relative to the evaluation level as the basic credibility of the decision scheme relative to the evaluation level, which is used to calculate the interval-based evaluation. The basic credibility is located in the upper and lower bounds of the interval.


### 2.2.2. Calculation Process Based on Nonlinear Programming

First, according to formulas (1)–(5), the decision matrix is transformed into the form of belief degree, and the undistributed part of belief degree mH,i includes two parts: m¯H,i and m˜H,i; m¯H,i is calculated by ei correlate weight, whereas m˜H,i represents the ei’s evaluation information is incomplete.(3)mn,i=miHn∈mn,i−,mn,i+=wiβn,i−al,wiβn,i+al,n=1,...,L,(4)m¯H,i=m¯iH=1−wi,i=1,...,L,(5)m˜H,i=m˜iH∈m˜H,i−,m˜H,i+=wiβH,i−al,wiβH,i+al,i=1,...,L,(6)∑n=1Nmn,i+m¯H,i+m˜H,i=1,i=1,...,L,(7)∑i=1Lwi=1.Second, the evaluation information of theL attributes of the ships is combined. Because the attribute evaluation information of the ships is expressed by the interval belief degree, the expected utility of the aggregated ship attributes may also be in the interval form. Therefore, it is necessary to use the objective-maximising nonlinear programming model (6) and objective-minimizing nonlinear programming model (7) to solve the upper and lower bounds of the expected utility of the ship, respectively.(8)Maxumaxal=∑n=1N−1uHnβnal+uHnβNal+βHal,s.t. βnal=mn/1−m¯H,(9)βH=m˜H1−m¯H,mn=k∏i=1Lmn,i+m¯H,i+m˜H,i−∏i=1Lm¯H,i+m˜H,i,m˜H=k∏i=1Lm¯H,i+m˜H,i−∏i=1Lm¯H,i,m¯H=k∏i=1Lm¯H,i,k=∑n=1N∏i=1Lmn,i+m¯H,i+m˜H,i−N−1∏i=1Lm¯H,i+m˜H,i−1,mn,i−≤mn,i≤mn,i+,n=1,...,N;i=1,...,L,m¯H,i=1−wi,m˜H,i−≤m˜H,i≤m˜H,i+,i=1,...,L,∑n=1Nmn,i+m¯H,i+m˜H,i=1,i=1,...,L.Minuminal=∑n=1N−1uHnβnal+uHnβNal+βHal,s.t. βnal=mn/1−m¯H,(10)βH=m˜H1−m¯H,mn=k[∏i=1Lmn,i+m¯H,i+m˜H,i−∏i=1Lm¯H,i+m˜H,im˜H=k∏i=1Lm¯H,i+m˜H,i−∏i=1Lm¯H,i,m¯H=k∏i=1Lm¯H,i,k=∑n=1N∏i=1Lmn,i+m¯H,i+m˜H,i−N−1∏i=1Lm¯H,i+m˜H,i−1,mn,i−≤mn,i≤mn,i+,n=1,...,N;i=1,...,L,m¯H,i=1−wi,m˜H,i−≤m˜H,i≤m˜H,i+,i=1,...,L,∑n=1Nmn,i+m¯H,i+m˜H,i=1,i=1,...,L.Models (6) and (7) constitute a complete ER-expected utility analysis algorithm. The uM1− represents the floor of the object, whereas the uM1+ represents the ceiling of the object. Using this algorithm, the expected utility uM1−,uM1+,uM2−,uM2+,...,uMm−,uMm+ of ship al can be calculated using the route characteristic vector wL1T=w11,w21,...,wl1.


## 2.2.1. ER Model Based on Interval Belief Degree

In the multi-attribute ship evaluation process, uncertainty data must be considered. LetA=a1,a2,...,am represent the set of m ships; ai represents ship i; E=e1,e2,...,eL represents each ship with L attributes; W=w1,w2,...,wLT represents the weight vector of the attributes where wi represents the route’s characteristic correlate to the ship’s attribute i, and satisfies 0≤wi≤1 and ∑i=1Lwi=1.Suppose that each attribute is represented by N complete and independent belief degree sets:Hn=H1,...,HN， then the evaluation information of attribute ei based on the interval-belief degree can be expressed in the distribution form as Seial=Hn,βn,i−al,βn,i+al,n=1,...,N， βn,i−al≥0， βn,i−≤βn,i≤βn,i+， where βn,i represents the attribute ei of ship al is a single number between βn,i− and βn,i+ in belief degree Hi. If βn,i−≡βn,i+， ei is a single number instead of the interval in Hi.For example,H=H1,H2,H3， Seial=H1,0.3,0.4;H2,0.2,0.3;H30.4,0.5，(1)βH,i−al=max0,1−∑n=1Nβn,i+al=max0,1−0.4−0.3−0.5=0,βH,i+al=1−∑n=1Nβn,i−al=1−0.3−0.2−0.4=0.1.The ignoranceH is a single number between the interval 0,0.1.The efficiency metrics based on the interval belief degree were obtained until the evaluation of theL basic attributes of M ships:(2)Dg=SeialL×M.If the attribute evaluation is represented by the interval-based basic reliability, the evaluation information of decision planai after the attribute evaluation information is aggregated using the ER method is the interval-based basic reliability. Based on the evaluation information in the form of interval basic credibility, a pair of nonlinear programming models is used to aggregate the basic credibility of the attributes relative to the evaluation level as the basic credibility of the decision scheme relative to the evaluation level, which is used to calculate the interval-based evaluation. The basic credibility is located in the upper and lower bounds of the interval.


## 2.2.2. Calculation Process Based on Nonlinear Programming

First, according to formulas (1)–(5), the decision matrix is transformed into the form of belief degree, and the undistributed part of belief degree mH,i includes two parts: m¯H,i and m˜H,i; m¯H,i is calculated by ei correlate weight, whereas m˜H,i represents the ei’s evaluation information is incomplete.(3)mn,i=miHn∈mn,i−,mn,i+=wiβn,i−al,wiβn,i+al,n=1,...,L,(4)m¯H,i=m¯iH=1−wi,i=1,...,L,(5)m˜H,i=m˜iH∈m˜H,i−,m˜H,i+=wiβH,i−al,wiβH,i+al,i=1,...,L,(6)∑n=1Nmn,i+m¯H,i+m˜H,i=1,i=1,...,L,(7)∑i=1Lwi=1.Second, the evaluation information of theL attributes of the ships is combined. Because the attribute evaluation information of the ships is expressed by the interval belief degree, the expected utility of the aggregated ship attributes may also be in the interval form. Therefore, it is necessary to use the objective-maximising nonlinear programming model (6) and objective-minimizing nonlinear programming model (7) to solve the upper and lower bounds of the expected utility of the ship, respectively.(8)Maxumaxal=∑n=1N−1uHnβnal+uHnβNal+βHal,s.t. βnal=mn/1−m¯H,(9)βH=m˜H1−m¯H,mn=k∏i=1Lmn,i+m¯H,i+m˜H,i−∏i=1Lm¯H,i+m˜H,i,m˜H=k∏i=1Lm¯H,i+m˜H,i−∏i=1Lm¯H,i,m¯H=k∏i=1Lm¯H,i,k=∑n=1N∏i=1Lmn,i+m¯H,i+m˜H,i−N−1∏i=1Lm¯H,i+m˜H,i−1,mn,i−≤mn,i≤mn,i+,n=1,...,N;i=1,...,L,m¯H,i=1−wi,m˜H,i−≤m˜H,i≤m˜H,i+,i=1,...,L,∑n=1Nmn,i+m¯H,i+m˜H,i=1,i=1,...,L.Minuminal=∑n=1N−1uHnβnal+uHnβNal+βHal,s.t. βnal=mn/1−m¯H,(10)βH=m˜H1−m¯H,mn=k[∏i=1Lmn,i+m¯H,i+m˜H,i−∏i=1Lm¯H,i+m˜H,im˜H=k∏i=1Lm¯H,i+m˜H,i−∏i=1Lm¯H,i,m¯H=k∏i=1Lm¯H,i,k=∑n=1N∏i=1Lmn,i+m¯H,i+m˜H,i−N−1∏i=1Lm¯H,i+m˜H,i−1,mn,i−≤mn,i≤mn,i+,n=1,...,N;i=1,...,L,m¯H,i=1−wi,m˜H,i−≤m˜H,i≤m˜H,i+,i=1,...,L,∑n=1Nmn,i+m¯H,i+m˜H,i=1,i=1,...,L.Models (6) and (7) constitute a complete ER-expected utility analysis algorithm. The uM1− represents the floor of the object, whereas the uM1+ represents the ceiling of the object. Using this algorithm, the expected utility uM1−,uM1+,uM2−,uM2+,...,uMm−,uMm+ of ship al can be calculated using the route characteristic vector wL1T=w11,w21,...,wl1.


## 2.3. Nash-Equilibrium Assignment Model

Nash equilibrium is a game approach in nature that provides a new way to study the rationality of ships in assignment problems. In this study, we briefly introduce the transformation method to pave way for future discussion. Assuming that a ship’s payoff is directly proportional to its utility to the route, each ship pursues its most efficient route to maximise its payoff. To deal with conflicts, ships are viewed as players in a game and utility is viewed as their payoff.To illustrate the Nash equilibrium of the ship assignment problem, a benefit-oriented example with two ships and four routes is presented in Table3. Compared with the traditional objective, from an individual perspective, the question is what the best assignment is, while every ship pursues the most profitable route to itself.Table 3
Two ships with four routes.
ShipRoutesABCDShip 1213154Ship 2104145Note. The elements represent the benefits.In contrast to the classical solving process, the assignment problem was transformed into an equivalent game problem as shown in Table4. If different ships cannot afford the same route, diagonal elements of the efficiency matrix do not exist.Table 4
The equivalent game problem.
Ship 2ABCDShip 1A-, -2, 42, 142, 5B13, 10-, -13, 1413, 5C15, 1015, 4-, -15, 5D4, 104, 44, 14-, -Note. “-” represents that ship 1 and ship 2 cannot take on the same route. The elements represent the benefits.Note that the assigned ship’s utility is transformed to its payoff equivalently, and then we can solve this problem using game theory. Table5 shows the results solved by the Hungarian method and Nash equilibrium.Table 5
Results of two ships with four routes.
Hungarian methodNash equilibrium 1Nash equilibrium 2Ship 1Route BRoute CRoute BShip 2Route CRoute ARoute CTotal utility272527Note. The elements represent the benefits.In contrast to the Hungarian method, the Nash equilibrium represents individual rationality. However, it is noted that the Nash equilibrium is not unique in this example, and may be equivalent to the Hungarian solution. In the next section, we will show a cost-oriented Nash equilibrium assignment problem example with three ships and four routes in which its maximal Nash equilibrium is also equivalent to the optimal solution, to lead to the question of “whether the optimal solution of an assignment problem is a Nash equilibrium point in nature.” Table6 shows the example.Table 6
Three ships with four routes.
ShipRouteABCDShip 11041415Ship 29141613Ship 378119Note. The elements represent the benefits.The example above can also be equivalently transformed into a game problem. In this game problem, the number of payoffs is sixty-four, and the number of effective payoffs (without negative infinity) is four. With the increase in the number of ships and routes, the number of effective payoffs is enormous. It was calculated that the number of payoffs in the assignment problem withm ships and n routes is nm. Among these payoffs, the number of effective payoffs (without the payoffs of negative infinity) is Anm, whereas the number of ineffective payoffs (negative infinity) is nm−Anm. The results of the example are shown and analysed first. The results of the assignment problem example with three ships and four routes are presented in Table 7.Table 7
Results solved by the Hungarian method and game method.
Hungarian methodNash equilibrium 1Nash equilibrium 2Nash equilibrium 3Nash equilibrium 4Ship 1Route B (4)Route B (4)Route a (10)Route B (4)Route C (14)Ship 2Route A (9)Route A (9)RouteD (13)RouteD (13)Route A (9)Ship 3RouteD (9)RouteD (9)Route B (8)Route A (7)Route B (8)Total utility2222∗312431As a result, Nash equilibrium 1 is equivalent to the optimal solution. Next, we would prove that the optimal solution is a Nash equilibrium in the assignment problem.Theorem 1.
The limited assignment problem has, and only has pure-strategy Nash equilibrium.From the Nash equilibrium existence theorem [19], we can conclude that the limited assignment problem must have a pure-strategy Nash equilibrium or mixed-strategy Nash equilibrium. Because the assignment problem can be equivalently transformed into a game problem, the assignment problem must have a pure-strategy Nash equilibrium or mixed-strategy Nash equilibrium. Because the assignment problem and its equivalent game problem include the constraints that different ships cannot afford the same route, it is inferred that the mixed-strategy Nash equilibrium cannot be reached. Therefore, the limited assignment problem has, and only has pure-strategy Nash equilibrium.Theorem 2.
The optimal solution is a Nash equilibrium point in the assignment problem.Proof by contradiction:Assume that a limited assignment problem hasm Ships and n Routes, which can be equivalently transformed into a game. This game can be expressed as G=S1,…,Sm;u1,…,um. The optimal solution strategy is sσ=s1σ,…,siσ,…,smσ, if existing i, uisiσ,s−iσ≤uisi,s−iσ, when the others’ choice is s−iσ=s1σ,…,si−1σ,.si+1σ..,smσ. The total efficiency of strategy sσ=s1σ,…,siσ,…,smσ can be increased by uisi,s−iσ−uisiσ,s−iσ. However, this contradicts the assumption that the strategy is the optimal solution.Therefore, for strategysσ=s1σ,…,siσ,…,smσ, the optimal solution is a Nash equilibrium point.Based on Theorem2, we can conclude that the optimal solution is a subset of game solutions. The solution to the assignment problem is extended for the first time. The proposed approach uses two pairs of nonlinear optimisation models to construct an assignment model to compute the matching order and maximum utility of the shipping company, and proposes a method to choose between different assignment sequences. Compared to the simulation algorithm, the analytical algorithm can take advantage of stability and efficiency. Finally, a cargo ship routing problem was examined to demonstrate the implementation process of the proposed approach.In cases where there was competition among the participants, it should be reappraised that the credibility of an assignment is exempted from market power. To address this issue, each participant was viewed as a player who sought to maximise their efficiency, and the assignment problem was solved using game theory for the first time. Specifically, it is proven that the optimal solution is the Nash equilibrium point of the assignment problem. Finally, relevant issues are discussed.


## 2.4. Calculation Process with CCFI and BDI Coefficiency

Owing to the fluctuation of the CCFI and BDI, the scheduling operator faces increasingly complex price data. To pursue profit maximisation, they must periodically adjust the scheduling plan with the indexes. From the perspective of the operator of scheduling, the route’s characteristic vectorwL1T=w11,w21,...,wl1 includes two opposite parts: benefit variables’ weights and cost variables’ weights. For example, the benefit variables include load factors, deadweight, and a number of standard TEUs, whereas the cost variables include annual M&R and manning costs, sea fuel consumption, and off-hire days. If the route price increases rapidly, the benefit variables should be assigned higher weights. Otherwise, if the route price plummets, the cost variables should be assigned higher weights. Fortunately, all these price changes can be reflected by the CCFI indices as shown in Table 8.Table 8
Index of CCFI.
CCFIRoute 1Route 2Route 3Route 4Route 5Route 6Route 7CC1C2C3C4C5C6C7Source: the official website of CCFI.In Table8, the CCFI index is calculated by all routes’ price indexes with the weighted average methods, and we can use equation (8) to generate a vector of weights for a vector of routes.(11)WRoutei=CiC.First, all the route’s characteristic vectorswL1T=w11,w21,...,wl1 can be divided into two parts: benefit variable weights and cost variables’ weights. The weights of the benefit variables’ weights are multiplied by WRoute i, whereas the cost variables’ weights are divided by WRoute I. Secondly, normalizing the new benefit variables and cost variables constructs the route’s characteristic vector. Finally, we use the route’s characteristic vector, which considers the CCFI indices to match the routes and ships with the methods described above. In addition, if we use the BDI indexes instead of the CCFI indexes, all steps are the same.


## 3. Application of the ER-Nash Equilibrium Assignment Approach to a COSCO Container Ship Scheduling Problem

In this section, the ship scheduling problem of COSCO Company is examined to demonstrate the application of the evidence reasoning-Nash equilibrium assignment approach with interval belief degrees. One of the important reasons for choosing COSCO as the research object is that it assumes the first place in container carrier areas worldwide. The corporation has invested in 58 terminals, including 51 container terminals, and the annual throughput of its container terminals amounts to 115.1 million TEU (data source: COSCO official website).In this paper, we choose sixteen representative cargo ships to show the application of the method. The assessment grades is shown in Table9; the original assessment data of ships is shown in Tables 10 and 11; the route’s characteristic is shown in Table 12; and the index of CCFI is shown in Table 13. In the reality, the maximum draft of Panama Canal is 13.41 meters, whereas the maximum draft of Suez Canal is 18.9 meters. And each route has one ship at least, each ship pursues its profit maximisation. Finally, the result showed in Table 14.Table 9
Assessment grades for eight quantitative attributes.
GradePAGVEExpected utility of each grade00.40.60.81Number of standard TEU500500080001500025000Deadweight (tonnage)50005000080000150000250000Speed (knots)811141720Maximum draft (meter)16.515.2414.93107Age (year)25151050Annual M&R and manning costs（million dollars/year）0.511.522.5Sea fuel consumption (tons per day)320160804020Off-hire (days per year)30211470Table 10
Original assessment data for COSCO’s sixteen cargo ships.
Attribute ship nameCOSCO PLANETCOSCO SAKURACOSCO AMERICACSCL SATURNIMO：9795660IMO：8795794IMO：9345427IMO：9467299Load factors{(V, [0.2,0.4]),(E, [0.7, 0.8])}{(V, [0.1, 0.3]), (E, [0.75, 0.9])}{(A, [0.3, 0.5]), (G,[0.5, 0.6])}{(A, [0.2, 0.4]), (G,[0.8, 0.95])}Number of standard TEU21237138001006214074Deadweight (tonnage)198195146507109950155426Speed(knots)18.115.910.113.8Maximum draft (Meter)161614.52515.5Date of built (year/month)201909201812200803201112Annual M&R and manning costs（million dollars/year）1.571.172.122.35Sea fuel consumption (tons per day)[280,316][217,249][163,188][207,232]Off-hire (days per year)[9,12][10,15][18,20][15,17]TIAN XIANG HECOSCO BOSTONCOSCO COLOMBOXIN SHAN TOUIMO：9300300IMO:9335173IMO:9484285IMO：9309942Load factors{(A, [0.3, 0.4]), (E,[0.5, 0.6])}{(P, [0.5, 0.65]), (A,[0.4, 0.5])}{(A, [0.4, 0.5]), (G, [0.55, 0.65])}{(P, [0.5, 0.65]), (A, [0.4, 0.5])}Number of standard TEU5816508942534250Deadweight (tonnage)67209682404997152157Speed (knots)17.79.618.510.7Maximum draft (meter)14.0213.512.612.8Date of built (year/month)200502200707201206200509Annual M&R and manning costs（million dollars/year）1.581.521.131.47Sea fuel consumption (tons per day)[92, 98][81, 99][65, 89][67, 92]Off-hire (days per year)[19, 22][17, 19][10–13][15, 18]Data source： COSCO（SH601919 and HK01919）operation report and IHS Maritime’s report.Table 11
Original assessment data for COSCO’s sixteen cargo ships.
Attribute ship nameCOSCO SANTOSCOSCO WELLINGTONXIN NAN TONGSONG YUN HEIMO：9484376IMO: 948441792621329160700Load factors{(A, [0.4, 0.5]), (G, [0.55, 0.65])}{(A, [0.4, 0.5]), (G, [0.55, 0.65])}{(P, [0.5, 0.65]), (A, [0.4, 0.5])}{(P, [0.5, 0.65]), (A, [0.4, 0.5])}Number of standard TEU4253425340511432Deadweight (tonnage)49959499595015124237Speed (knots)9.216.215.613.5Maximum draft (meter)111112.810Date of built (year/month)201303201306200311199809Annual M&R and manning costs（million dollars/year）1.491.331.591.81Sea fuel consumption (tons per day)[68, 87][69, 91][69, 87][22, 31]Off-hire (days per year)[13, 15][12, 15][16, 19][16, 18]LING YUN HECOSCO KIKUCOSCO AQUARIUSCOSCO ASHDODIMO：9223772IMO：9247871IMO：9789623IMO：9518335Load factors{(P, [0.5, 0.65]), (A, [0.4, 0.5])}{(P, [0.5, 0.65]), (A, [0.4, 0.5])}{(V, [0.2, 0.4]), (E, [0.7, 0.8])}{(A, [0.4, 0.5]), (G, [0.55, 0.65])}Number of standard TEU1702542192734253Deadweight (tonnage)25723929419704949956Speed (knots)14.39.118.615.3Maximum draft (meter)10.77.811611Date of built (year/month)200009200201201906201308Annual M&R and manning costs（million dollars/year）2.021.781.561.31Sea fuel consumption (tons per day)[29, 42][11, 16][282, 325][76, 80]Off-hire (days per year)[17, 19][18, 21][11, 13][13, 15]Data source: COSCO（SH601919 and HK01919）operation report and IHS Maritime’s report.Table 12
Route’s characteristics with route’s weights.
Attribute/Weights (Route)Trans-Pacific lineEurope lineEurope (Mediterranean Sea) and Atlantic lineAsia-Pacific lineLatin America and Africa lineCentral America lineSoutheast Asia and South Asia lineLoad factors0.120.180.10.10.140.10.12Number of standard TEU0.150.140.120.120.170.120.12Deadweight (tonnage)0.10.080.150.080.120.090.14Speed (knots)0.10.080.10.20.120.090.1Maxmiun draft (meter)0.120.180.10.10.140.10.06Date of built (year/month)0.150.120.120.120.10.170.09Annual M&R and manning costs（million dollars/year）0.10.080.10.10.060.150.1Sea fuel consumption (tons per day)0.10.080.150.10.080.120.17Off-hire (days per year)0.060.060.060.080.070.060.15Data resource: https://lines.coscoshipping.com/home/Services/route/11 (COSCO’s official website)Table 13
The index of CCFI in 5th June.
CCFITrans-Pacific lineEurope lineEurope (Mediterranean Sea) and Atlantic lineAsia-Pacific lineLatin America and Africa lineCentral America lineSoutheast Asia and South Asia line3186.653024.655024.216065.342682.43003.441885.951692.98Source: the official website of Shanghai Shipping Exchange(SSE).Table 14
Results (based on the index of CCFI in 5th June).
Trans-Pacific lineEurope lineEurope (Mediterranean Sea) and Atlantic lineAsia-Pacific lineLatin America and Africa lineCentral America lineSoutheast Asia and South Asia lineCOSCO SANTOSCOSCO PLANETCOSCO KIKUXIN SHAN TOUXIN NAN TONGSONG YUN HELING YUN HECOSCO WELLINGTONCOSCO COLOMBOCSCL SATURNTIAN XIANG HECOSCO AMERICACOSCO AQUARIUSCOSCO BOSTONCOSCO SAKURACOSCO ASHDOD


## 4. Conclusions

In this study, we propose a ship scheduling method for fluctuating price routes based on evidence reasoning and the Nash equilibrium assignment model in fluctuating CCFI and BDI cycles. Compared to the current simulation algorithm, this method is an analytical algorithm and has better stability, accuracy, and computational efficiency. In particular, it is suitable for the ship scheduling problem of correlated route information, including quantitative, qualitative, and interval data. With the strong fluctuation of the BDI or CCFI indexes, dynamic scheduling can be implemented by programming this method to solve the large number of problems. An example of the COSCO company scheduling problem shows that the method can achieve overall optimal matching according to the fuzzy data. In addition, according to the nature of the assignment problem, the method can be extended to a matching problem, where the number of routes is not equal to the number of ships. When the route price increases, the virtual route is set up in the Nash equilibrium assignment problem, whereas the virtual route cuts down with its plummet.



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*Source: 3997361-2022-07-18.xml*