SOLVING FUZZY DYNAMIC SHIP ROUTING AND SCHEDULING PROBLEM THROUGH MODIFIED GENETIC ALGORITHM

: Main motto of ship routing and scheduling is to reduce the total transportation cost of each ship or vessel without interrupting the demand and supply. In this study, we have proposed a ship routing and scheduling model for commercial ships where, to ensure unhindered demand and supply of products at various ports in a fixed time frame, the dynamic demand and supply of each port were considered under a fuzzy environment. Additionally, simultaneous loading and unloading and a fixed load factor is used to minimize port time and reduce risks, and this aspect of our work makes it realistically inclined. We also show, in our work, speed optimization to reduce fuel consumption and carbon emission. In practice, cost parameters cannot be always determined, it fluctuates at a certain range from time to time. We have treated the imprecise cost parameters as triangular fuzzy numbers. With a


Introduction
Shipping industries play an important role in the growth of the economy of a country, and forms a core part of its trade.In the 20 th century, the maritime transportation grew exponentially.Bulk cargo is generally moved either by sea, or by road, or, by air.Seaborne trade is the cheapest form, and it has spread to 89.6% of total global trade in terms of volume and 70.1% in terms of value in the recent years.In ship routing, each ship starts its journey from initial ports at the beginning of the planning horizon and visits from one port to another port with its containers.A ship visits a set of ports within any planning horizon, and at each of these ports some loading/unloading operations are carried out.To transport the cargos, a heterogeneous fleet of ships are used.There are a fixed capacity and sailing speed for each individual ship.This model aims to design appropriate shipping routes and schedules to minimizing the costs associated with transportation.The present investigation considered heterogeneous ships with a variety of capacities.Determining the navigation of ships or vessels in the maritime environment is a ship routing problem that is similar to the vehicle routing problem on land.Some of the challenges in ship routing and scheduling are as follows: 1.Because a trip may last for numerous days, each ship has a fixed 'time window' to operate at the port.A fixed time is allotted to each ship to complete the operation (loading/unloading) at the port.2. A linear ship route is not always possible because of several factors.3. Sometimes two captains may not agree on the same route for the same container for several reasons.4. Because of variable weather conditions, maritime transport is highly uncertain.
Calculation of the traveling time and cost is difficult because the travel time is affected by the wind speed, direction, and current.Commercial ship operations are categorized as follows: 1) linear, 2) tramp and 3) industrial.Similar to bus transport, a linear operation in sea involves visiting all ports in its route, and ending at the destination port.Tramp ships are analogous to taxis, and their goal is to maximize profit while ensuring unhindered demand and supply of products.Industrial shipment minimizes the cost of shipping products by using the best route.Cargo is allotted to ships at the source port, which is then transported to the destination.Industrial shipping is of two types, namely cargo routing and inventory routing.The cargo routing problem is specified by the time frame for loading and unloading operations, and the demand and supply at specific ports may not be fixed.By contrast, in the inventory routing problem, the product requirement at a particular port is fixed.In this research, we have focused on the cargo routing problem.
In practice, the demand and supply of products at the port are not fixed, it varies from time to time.To satisfy this condition, the loading/unloading operation of some products is determined spontaneously at the port according to the demand/supply.Because of uncertain market conditions, dynamic demand/supply is typical.Loading and unloading operations are performed simultaneously to optimize the port time as well as port congestion.In this study, we considered those operations simultaneously where there is spontaneous and simultaneous loading/unloading.The traveling costs depend on various factors, such as the geographical areas, the weather conditions during the sailing period, and the product being transported, and hence are considered as vague or fuzzy in this model.We have thus worked to minimize the shipping cost and maintain sustainability keeping uncertainty in mind, at the same time ensuring that all cargoes are lifted from the loading port and discharged at the unloading port.This model used container-based cargo to transport goods.Risk factors such as, thunderstorms, tsunamis, icebergs, and piracy, which depends on the time period of sailing and choosing the routes, are also considered as fuzzy numbers.The total risk is considered to be less than the maximum risk which is assigned for the entire trip of that ship.It is this fuzzy approach renders the problem realistic.

Motivation
In ship routing and scheduling problems, a ship with a fixed capacity starts its voyage and visits a set of ports in a fixed time frame to satisfy the demand and supply of those ports.A port can be visited by only one ship at a particular time.De et al. (2017) considered numerous sub-time window concepts in ship routing and scheduling problems.The travelling time of a ship is the time to travel from one port to another port and its operation time is the time needed mainly for loading and unloading.However, within a fixed time frame, due to lack of proper scheduling of ships, there is a delay in cargo transportation from the origin port to the destination port.A penalty cost is imposed for early/late arrival or loading/unloading delay.If any ship exceeds the fixed time frame, it had to pay a penalty calculated according to the additional hours it operates outside the fixed time frame.In this model, loading and unloading operations of various goods is performed simultaneously but unloading operation from a ship is performed before the loading operation starts.
Here the main research motive is total cost minimization and optimum path determination for an industrial ship routing and scheduling in a fixed time frame, in which risk is minimum in a sustainable environment but in uncertain market conditions.Because of COVID-19, social behavior as well as demand and supply fluctuate randomly, affecting the shipping industry.Imran et al. (2020) designed a ship routing and scheduling model for static demand and supply, but the model is not robust in uncertain or dynamic situations.Our second research motive is the facilitating the transport of cargo whose demand and supply are determined dynamically.Yang et al. (2021) designed a tramp ship routing model to address port congestion because of static demand and supply.The result revealed that dynamic demand and supply reduce port congestion and render the system robust.Next, we focused on container-based cargo shipping to minimize the cost and path of travel.Container-based shipping provides an end-to-end approach to customers and shipping companies.
Finally, we focused on increasing the efficiency and effectiveness of the solution method.In this study, we develop a novel selection technique and generation-dependent mutation to achieve a better solution.A nature-based heuristic algorithm requires less time and computational cost.Considering fuzzy numbers, we introduced uncertainty or dynamic attributes to the model and adopted the possibility and necessity approach by using the graded mean integration value defuzzification method.
Thus, we can once again the novelties of the proposed study are as follows: We developed a dynamic cargo ship routing and scheduling algorithm under uncertain environments.
To maintain sustainability conditions, a novel mathematical function was incorporated to study the fuel consumption corresponding to the total carbon emission and the limitations of the journey considered.
We developed an optimal route design with a maximum allowable risk for that trip.
A novel GA (new selection, crossover, and mutation strategies introduced) was developed to address the proposed model.
We considered traveling cost and risk factors as fuzzy values for uncertain environment.
The rest of this paper is organized as follows: Section 2 describes studies relevant to the present investigation.Section 3 describes mathematical preliminaries.Section 4 discusses the proposed dynamic ship routing model.The defuzzification of the proposed model is presented in Section 5.In section 6, a modified GA (MGA) is presented.Section 7 illustrates the outcome of numerical experiments.Statistical test significance is discussed in Section 8. Section 9 provides results and discussions.Managerial insights are provided in Section 10.Finally, conclusions and future scope are presented in Section 11.

Literature Study
The ship routing and scheduling problem (SRSP) is similar to the traveling salesman problem (TSP) for cargo on land.Bausch et al. (1998) was the first to develop a short-term ship routing and scheduling model.Subsequently, numerous studies have focused on ship routing and scheduling problems.Psaraftis (2019) described a few conceptual hypotheses for the SRSP.However, the study did not provide any practical implementations derived from these.Alfandri et al. (2019) proposed a weekly demand service between a pair of ports for barge containers using a liner service to maximize the profit.However, transport delays may occur in liner shipping for certain routes because of uncertain weather conditions.Rabbani et al. (2019) described a model to determine the best route to minimize the shipping costs and carbon emissions and maximize job creation in ships and ports.In this study, variable speed was considered.Cost minimization and job creation maximization contrast each other.Noshokaty (2021) used information technology in tramp ship routing and scheduling problems to address commodity forecasting.In this model, the method of solution increases the time complexity, when all constraints are considered in commodities forecasting.Zhao et al. (2019) described how operation measures and objectives depend on fuel prices and vessel loads.To generate profits, the speed of a vessel is decreased for carrying a higher vessel load.This scenario sometimes results in missing the time window because of the low-speed voyage.Homsi et al. (2020) described the generation of elementary routes for tackling routing problems related to industrial and tramp ships.However, load-dependent fuel consumption, sea condition, and emission-related vital points were not considered there.Liqan et al. (2020) considered ocean currents for speed optimization to minimize the total fuel consumption.However, this optimization model had been tested for only one ship, and hence the results cannot be exactly generalized.Numerous algorithms have been proposed to optimize ship routing problems, and each algorithm has some pros and cons.Laura et al. (2016) used algorithms, such as the Dijkstra's algorithm, dynamic programming, and iterative methods, to solve ship route optimization problems.These solution methods are problem-dependent, and each method has advantages and disadvantages.These methods are computationally expensive for NP-hard problems.De et al. (2017) described a hybrid particle swarm optimization (PSO) algorithm to solve timewindow-based ship routing problems with multidimensional features.In this study, PSO marginally increases the computational time.Wang et al. (2018) proposed a hybrid mutation operator to maintain the population diversity in ship routing solutions.This method requires a long time to converge to the optimal solution.Alhamad et al. (2019) developed a tabu search method to solve the ship routing problem and revealed that if the problem size increases, the execution time increases drastically.Fan et al. (2019) used a variable neighborhood search algorithm to solve tramp scheduling, which revealed moderate computation time to generate optimal or near-optimal solutions.Roy et al. (2020) proposed a real-life in-vitro-fertilization (IVF)-based crossover in a GA to provide a solution to the TSP.In this study, the IVF crossover provided the concepts of having three parents and diversified the solutions.Pratap et al. (2019) described their own ship routing and scheduling model without considering the relationship between carbon emissions with fuel consumption in 2019, Fan et al. proposed speed optimization in tramp shipping where they have discussed about how to reduce greenhouse gasses.In 2018, according to UNCTAD, sustainable shipping can reduce 50% of the world's carbon dioxide (CO2) within the year 2050 by using slow steaming and alternate fuel.Zhang et al. (2021) described fuel consumption as a black-box concept.However, this study did not provide a transparent fuel consumption concept.Lan et al. (2020) described various carbon emission policies and presented a comparative analysis of these policies.However, the carbon emission policies sometimes reduce the profit of the shipowner.Fan et al. (2019) described the process of multi-type tramp shop scheduling to minimize the total costs of shipping companies.However, real-time ship scheduling was not considered in this study.Wang et al. (2018) did not consider wave and wind disturbance, which considerably affects the ship route design.Sun et al. (2019) described the uncertain planning stage and demand-supply aspects of customers in real time, but only the logistic network forward flow was considered; reverse flows were not considered.Maity et al. (2018) introduced a rough set-based GA, in which an age-dependent selection technique and age-oriented min point crossover was considered.However, this method works with both a high computational time and a high amount of computational resources.Dong & Bain (2020) described a hybrid A * and GA to solve ship pipe route design.This hybrid algorithm requires considerable execution time.Akter et al. (2020) proposed the use of type-2 fuzzy and fuzzy random data to solve a four-dimensional (4D) TSP.In this study, small-size data sets are used.The use of large size data sets becomes computationally expensive.

Mathematical Preliminaries
In this study, we presented preliminary fuzzy number concepts and their membership values and the defuzzification technique required for the proposed model.

Triangular Fuzzy Numbers
Here, ~( , , ) A p q r  is a normalized triangular fuzzy number (TFN), which is a subset of S (real numbers).It's membership function is given as follows: ~( ) = sup min ( ), ( ) , , and z = ( , ) From the aforementioned definitions, the following lemmas can easily be derived: where 1  is a predefined possibility.

Defuzzification of the fuzzy number by using graded mean integration
The fuzzy number graded mean integration method with the integral value of graded mean 1  -level of the generalized fuzzy number was used for defuzzification.

Assuming
B is a generalized fuzzy number, then the graded mean integration value (GMIV) of B is denoted by the following equation:

Proposed Dynamic Ship Routing Model
To develop the proposed model, a set of ships, a group of ports, and a set of products were considered.Typically, a ship visits some of the ports to ensure unhindered demand and supply of those ports within a time frame.In this model, a ship starts its journey from a port, visits several ports, and ends its journey at the destination port, which is neither the starting port nor a visited port.For a ship, exceeding the given time window results in a penalty cost.Loading and unloading are performed at the port, but the unloading operation is performed before loading.In practice, the cost may not be deterministic, is imprecise, and considered as fuzzy parameters.
The model was developed by using various dependent and independent variables and several constraints.The assumption of this model are as follows.

Assumptions of the model
To construct the model mathematically, we assumed the following points: 1.A ship visits a set of ports in its voyage in a fixed time frame.
2. We consider heterogeneous ships for this model.Each ship has a different capacity. 3. A container contains only one product at a time.It has a particular demand port and supply port.In between these two ports, containers cannot be unloaded from the ship.4. A ship sails at a certain speed in which fuel consumption is minimum and it does not violate the time frame.The total carbon emission does not exceed the maximum carbon emission index.5. Costs and risks are not fixed, but they vary in a certain range.However, the total risks do not exceed the maximum allowable risk for that route.6. Ship bunkering, cleaning, and maintenance are performed at the operation time at the port if required.7. On a weekly basis, each ship visits the port.Demand and supply vary dynamically at ports.

Mathematical Model
Mathematical indices and parameters are stated in Table 1.Mathematical formulations are given below.The indices obtained from De et al. (2017) work partially.A pl T

Symbols
The beginning of the time window at port p in the time period l.

B pl T
The ending of the time window at port p in the time period l.

pr C
Operational cost (loading/unloading) for a single unit of product r at port p.

pr T
The loading/unloading time required for single unit of product r at port p.

plr W
The demand of product r at port p in time period l.

pqs C
The transportation cost from port p to port q for ship s.where i=1,2,3,4,5,6,7; k=1.   1 plsr I : Total amount of product r available on ship s after an operation that started in time l while departing from port p; The stock level at port p of product r in time period l; , l L p P  , rR  .pqs V : Velocity of ship s while traveling from port p to port q. , p q P  ; sS  .Rf pq : Risk factor between port p and port q; , p q P  .
max r : Maximum risk on that trip.
Objective function: 1 ,, Equation ( 7) minimizes the traveling cost, operation cost, and penalty cost.The constraints are as follows: Constraint ( 8) represents that at least a single ship can operate in port p at the given time l.
Constraint ( 9) represents that ship s at some port p ended its route at the time l.10) represents the time frame range.
Constraint (11) represents that after finishing the operation at port p, ship s travels some distance between port p and port q with a fixed velocity.
, then the ship is not traveling in the sea, that is, Constraint ( 12) represents the sum of the setup time, starting time of operation, and that the total loading/unloading time is equal to the ending of each operation.( 1) Constraint ( 14) represents that after unloading the product from the ship, the empty space in the ship should be greater than the previous period.Products are unloaded container wise.
Constraint ( 15) represents that the quantity product r in ship s at port p in time period (l-1) is onboard and addition or removal of the quantity of product r loaded/unloaded at port p in time period l should not exceed the empty capacity of ship s in time period l.
Constraint ( 16) represents that an upper bound of ship s of product r while sailing does not exceed the ship capacity.
Constraint ( 17) represents that the demand for each product r is satisfied at time m at port p.
Constraint ( 18) represents that the total fuel consumption for ship s for a trip is less than maximum bunker consumption.Here, fuel consumption is considered as a logarithmic function of velocity.When ships travel from port p to port q, the fuel consumption is log( ) Constraint ( 21) represents the total carbon emission at port and sea, which is always less than the maximum carbon emission index for that trip.In the proposed model, we assumed speed to be a logarithmic function, which is appropriate for considering traveling time and carbon emission.

Defuzzification technique for the proposed model
The traveling costs and the risk factors are fuzzy numbers, which are denoted as ~( , ) C i j and ~( , ) r i j respectively, where ~max r is the maximum risk level of a particular path.In the proposed model, the total traveling cost and risk are expressed as follows: M = total number of nodes on that route.

Possibility Approaches (optimistic)
By using Equation (2), we obtain the following objective and constants: minimize M = total number of nodes on that route.where
Traveling fuzzy cost for a path is , and Subject to Where , ( , ) M = total number of nodes on that route.The objective function in Equation (31) becomes:  are possibility levels (predefined).

Necessity Approaches (pessimistic)
By using Equation (3), we obtain the following objective and constants: M = total number of nodes on that route.
Traveling fuzzy cost for a path is , and 1 Where , ( , ) M = total number of nodes on that route.

GMIV Approach
By applying GMIV for the defuzzification on SRSP using Equation (29), we obtain the following equation: To minimize Here, we consider w = 0.5, for an optimal solution.

MGA
In 1975, Holland was the first to propose natural and artificial system GA, which was subsequently used in natural evolution and optimization problems.Adaptive heuristic search algorithms evolved from GA.This algorithm used a random search technique to obtain an optimized result.Although random search is used, it is not random, and the previous information is used to direct the search in an appropriate search space to obtain optimum results.provides superior optimization when high dimensional search spaces and numerous variables are present.

Proposed Bird's nest building material selection technique
To solve the ship routing and scheduling model through MGA, we proposed a novel Bird's nest building material selection technique (BBMST) for selection.Not all birds excel at making nests.Hamerkop (Scopus umbretta), ruby-throated hummingbird (Archilochus colubirs), sociable weaver (Philetairus socius), baya weaver (Ploceus Philippinus), etc make good nests.Among these birds, "sociable weaver" birds select the best material for nest building, and they check the building material with their beak.We selected the best solution from all possible solutions set to obtain the optimum result in ship routing.
In the BBMST algorithm, first, the fitness values of all chromosomes are calculated, and the minimum cost path, which is the main objective of this model, is implemented.In each iteration, we selected the minimum cost of the chromosomes, and based on the cost of other chromosomes, we calculated the threshold value.The probability of selection was randomly generated, which is between 0 and 1.If the threshold value is less than the probability of selection, then the current chromosome is selected, otherwise the minimum threshold valued chromosome is considered as fittest for the IVF crossover process.
Let C[i][j] represent the traveling cost between i th port to j th port, N is the set of ports, and pv is the total number of chromosomes.

Algorithm1: BBMST selection process
Require: Set of given port N and the size of the population is pv.Ensure: A set of fittest chromosomes.

IVF crossover
In the proposed MGA, IVF is used for the crossover process.In the IVF process, three parents, namely one father (Pr1), one mother (Pr2), and a surrogate mother (Sr) are selected randomly, depending on their crossover probability (Pc) and bring them in mating pool, and two children were created, child1 and child2.First randomly generate a node (for example: ai) and place it at the first position at chid1 and update the parents with ai.Next, the least cost valued node is selected from three parents in mating pool.Repeat the previous step until all the nodes are not selected and always check whether the same node is selected previously or not.Similarly, the same steps are performed for child2.In the matting pool of GA, two new offspring are created with superior information, which is inherited from parents.The IVF is a parallel flow process, which receives the input portion of various parents from the GA population.The individuals are supplied by the IVF process.Figure 3 displays the IVF process in a pictorial form.The algorithm steps of IVF crossover are given below: Algorithm2: IVF crossover 1. Begin 2. Choose three parents (Pr1, Pr2 and Sr) randomly from mating pool depending on crossover probability Pc. 3. Nodes in three parents arePr1: (a1, a2, …, aN), Pr2: (r1, r2, …., rN) and Sr (s1, s2, ….., sN ). 4. Randomly choose a node ai. 5. Place that ai node at the beginning of child1 and update the three parents.6.In child1 place ai at the first position.7. Find the minimum cost between (ai, a1), (ai, r1) and (ai, s1).8. Choose the minimum cost node and place it to the next position of child1 and update parents.9. Repeat steps 7 and 8 until all nodes are not selected without repetition of any node.10.Repeat the same steps for child2.11.End

Generation-dependent mutation
After the crossover process, the algorithm goes through the mutation process.In this algorithm, we used generation-dependent mutation.The mutation prevents the solution to be trapped in a local minimum and premature convergence.The mutation maintains genetic diversity in the population.In a population, a random change in some of genes result in one chromosome.This phenomenon produces a novel offspring with a new genetic structure.Mutation helps escape from local minima, find the global minima, and maintain the diversity of the population.The probability of generation-dependent mutation ( m p ) is expressed as follows: When the number of generations increases, m p decreases normally.

Mutation Process
The proposed ship routing and scheduling model is a node-dependent problem.To mutate the chromosome X = ( 12 , ,...., Set g= number of the current generation.
end for

End
Because the process is an NP-hard problem, the exact approach may cause high computational time.Meta-heuristic approaches provide an optimal result in a reasonable time.In this study, we proposed a modified GA-based BBMST, in vitro fertilization, and generation-dependent mutation.

Complexity Analysis
We considered P as the initial population, N as the number of nodes, g as the number of generations,

Space complexity
In the proposed MGA, the total space is considered as the population multiplied by the number of nodes, that is, PN .As p > N, the space complexity of the proposed algorithm is O( PN ).

Numerical Experiments
The optimum value is obtained by running this algorithm up to 500 generations considering the crossover probability as 0.61 and considering generation-dependent mutation.We used a 2.40 GHz i7 processor with 12GB of RAM with Windows 10 for our numerical calculations.The use of fuzzy theory renders the model robust.
In this model, we considered 3 ships and 15 ports.We collected data from the Haldia Port and traveling cost in 10k of each cell value in matrix, which is expressed as follows: The loading and unloading operation cost for each product and penalty cost for the outside time from timeframe for each port and cost in 1 k of each cell of the matrix is presented in Table 3.By using the deterministic traveling cost presented in Table 2, operation cost and penalty cost presented in Table 3, we obtained the results for the MGA and GA to obtain the optimum cost of the proposed algorithm.The final path for the four ships and their optimum cost for both the algorithms are presented in the given table: To simulate practical applications, we obtained the fuzzy traveling cost matrix.Here, the triangular cost fuzzy matrix is used for the proposed model as follows: Solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 351   By using the fuzzy triangular cost matrix presented in Table 5, we calculated the final path for three ships and their optimum cost for our MGA and GA.The aforementioned table reveals the results for the fuzzy data.
Table 7 presents the deterministic risk matrix between every two ports.We considered a value between 0 and 1 as the deterministic risk factor.The table below displays the risk achieved by the deterministic risk factor and fuzzy risk factor from Tables 7 and 8, respectively, at a particular path for both the algorithms.

Statistical Test
We compared the statistical test quantitative decision of the proposed algorithm with other those of other conventional algorithms.The analysis of variance (ANOVA) was performed to indicate the statistical significance of the proposed algorithm with the conventional algorithms (RWGA and PBGA).

ANOVA for the efficiency test
The performance of MGA, RW selection-based GA (RWGA), and probabilistic selection-based GA (PBGA) for solving the standard ship routing problem was compared.We obtained various parametric values, such as the different number of vessels, V3, V4, and V5; different number of containers, C7, C10, C15, C20, and C23, and four instances, namely instance 1 (I1), instance 2 (I2), instance 3 (I3), and instance 4 (I4).Various instances obtained various source ports and destination ports for the container, which changed the path of ships.In this model, nine types of data sets were considered for the ANOVA test.These nine data sets were categorized into two, short sea and deep-sea categories.Short sea data sets are represented as three ships and seven containers (V3C7), three ships and ten containers (V3C10), three ships and fifteen containers (V3C15), four ships and fifteen containers (V4C15), four ship and 20 containers (V4C20), five ship and 20 containers (V5C20).Deep sea data sets are represented as three ships and 15 containers (V3dC15), three ships, and 20 containers (V3dC20), four ships, and 23 containers (V4dC23).We obtained the result The total sample size is nine for each algorithm, and the number of the algorithm is three.The critical value of V ≈ 3.40 and p is extremely smaller than α = 0.05 for all the cases.Because the F critical value is smaller than F, we rejected the null hypothesis, which provides statistical significance to the result.A significant difference was observed between the algorithms.Therefore, the proposed modified algorithm is more efficient than the other two algorithms.

Result discussion
We considered various sets of input values for dynamic ship routing and scheduling problems to achieve appropriate numerical outcomes.Table 2 provides the crisp data input for the traveling cost.Table 3 provides the operation cost of each product and the penalty cost for violating the time frame in case of each port.Table 4 displays the path and cost as the output from our ship routing model with a crisp cost matrix obtained by using classical GA and MGA.Table 5 provides the triangular fuzzy input data for traveling cost.Table 6 presents the output of our ship routing model considering fuzzy data obtained using classical GA and MGA.In Tables 4 and 6, we can see that the proposed algorithm provided the optimum result for crisp data input as well as the fuzzy data input.The use of crisp risk factors is presented in Table 7, whereas the use of fuzzy risk factors is exhibited in Table 8.Table 9 displays the risk achieved for the deterministic as well as the fuzzy models for both GA and our MGA.Here, Rmax for a route represents the maximum risk value on that route.Table 9 reveals that the proposed MG exhibited the best result for both the crisp and fuzzy data sets.The comparative results for the ship routing benchmark instances for the MGA, RWGA, and PBGA are displayed in Table 10, which reveals that the proposed MGA provides the best results for benchmark instances.In Table 11, we have compared the winning runs of our algorithm and those of two other algorithms.Table 12 displays the ANOVA test results for the three algorithms working on four instances of the data input for the standard ship routing problem.From Table 12, it seems we can conclude that the proposed modified algorithm works more efficiently and provides better results than the other two algorithms.

Managerial Insights
The substitution of the new model gives more space to the manager for making decisions regarding which approach to adopt for incurring maximum more profit for the organization.With the implementation of our model, a manager has the freedom to design the shipping route under an uncertain environment, a situation which can be easily applied to the shipping industry in real life.Our proposed model can handle dynamic demand and supply, and also generate optimum routes and minimize the overall costs of the transportation by ship, which is the principal motto of any ship routing and scheduling algorithm.Risk factors are considered in calculating optimum routes, which provides a realistic approach to the shipping industry.The use of cost minimization and time window reduces overall costs and maintain shipping discipline.That is why the proposed model can generate more profit than other previous models.The demand and supply of products at the port may vary because of market conditions or social issues and use of this model leads to less congestion at the port.Considering the cost and risks parameter as the fuzzy number makes the model realistic because costs and risks factor are changed time to time.This dynamic environment makes ship routing and scheduling problems more suitable for practical use.Therefore, management should adopt this realistic model, which is cost effective and manages the risks well.

Conclusions and Future Scope
In our model, we have assumed that ships start their sailing from starting ports, visit other ports to ensure the unhindered demand and supply of products at those ports, and end their routing in their destination ports.We obtained the optimum paths and traveling costs for various ships, and hence the optimal cost is calculated by adding the cost of traveling, loading-unloading operations, and penalties (levied to ensure port discipline).Cost efficiency and environmental friendliness are prioritized by considering a few aspects such as the dynamic demand and supply models followed in ports, container-based shipping, and techniques for saving fuel.Fuzzy cost and fuzzy risk factor are introduced to represent the impreciseness of the parameters.Therefore, in our model, a real-time ship routing approach is proposed in an uncertain environment.The advantages of this model are its ability to the determined optimal path and cost when various real-life parameters are uncertain.This model maintains the port time as well as the traffic, and handles the container congestion very smoothly.We have used BBMST selection and IVF crossover techniques in our work.A generation-dependent mutation was used in the proposed algorithm in which, when the number of generations increased, the algorithm provided a better result.The advantages of the proposed methodology are that the novel BBMST selects the better solution for further steps, which gives optimal solution in less computational time.The IVF-based crossover method has been used on very few instances of work before.The proposed MGA clearly revealed itself as being superior other RWGA and PBGA in our numerical experiments.The ANOVA test revealed that the proposed algorithm works efficiently.
However, the model has some limitations.The maximum risk (Rmax) is selected randomly to ensure the total risk does not exceed the maximum risk.Dynamic weather routing may define various risks of a route for a ship for a particular time, which is not considered.We did not consider the container type, so our models are based only on standard containers.There also are some limitations in the solving method: as the approach is heuristic, a high computational complexity exists.When numerous constraints are considered, the MGA falls short in solving the problem.
The futures cope of this model is actually quite wide.Various types of containers (foldable container and smart container) can be introduced in the future.We can predict the dynamic risks of each route using Artificial Intelligence and modern technology in the future.Various ranking-based fuzzy approaches, including the type-2 fuzzy approach, can be used to solve complex uncertain routing problems.We may use rough numbers, fuzzy systems, or even neutrosophic numbers in the future.

Figure 2 .
Figure 2. Graphical representation of the proposed model p and port q.
) represents penalty time at port p. 1 are interchanged.The same process repeated times to obtain the best offspring.
the time complexity of three operations, selection, crossover, and mutation are O( PN ), O( respectively.For the proposed algorithm, complexity is as follows: 6.4.1.Time complexity Because the proposed MGA consists of BBMST selection, IVF crossover, and generation-dependent mutation, the time complexity of this algorithm is the maximum time complexity of these three processes.O( PN ), O( complexity of three processes.Therefore, O( 2 PN ) is the time complexity of the proposed algorithm.

:
: Total operating time outside the time frame in time period l at port p; Total quantity of product r loaded/unloaded at port p from ship s in time l; sl E Storage capacity of ship s in time period l. pr Y Total storage capacity associated with the depot at port p for the product r.prUThe setup time required for operation (loading/unloading) at port p for product r.plCPenalty cost per hour, associated with operation delay in time l at port p. s H Carbon emission per hour for ship s. g Constant (Relationship constant of speed and fuel consumption).sfFuel consumption rate per hour for ship s.max e Maximum carbon emission for that trip.maxB Maximum bunker consumption for that trip in crushing speed.plN Solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 341 Constraint (20) represents that the total risk achieved by the port is less than equal to the maximum risk on that trip.
Constraint (19)represents the storage capacity of port p for product r.

Table 2 .
Input Data: Deterministic Traveling Cost Matrix

Table 3 .
Operation Cost and Penalty Cost Matrix

Table 5 .
Input Data: Fuzzy Cost Matrix

Table 6 .
Result for Fuzzy Model

Table 8 .
Fuzzy Risk Matrix

Table 9 .
Result for Risk Factor

Table 12 .
ANOVA Test Result