Scheduling for a Container Supply Chain to Minimize Costs Using the Meta-Innovation Approach

In this study, a problem of scheduling shipping lines for a container supply chain is addressed in order to minimize the costs of charging ships and the cost of maintaining the inventory of empty containers in the port by considering the time window of the port and the amount of fuel. This is a hard-NP problem and cannot be solved on a large scale with precise methods in a logical time. Therefore, to solve and optimize the model, a meta-innovative algorithm, genetic algorithm, has been used. Also, to increase the effectiveness of the genetic algorithm, the parameters of the algorithm are adjusted using the Taguchi method. Finally, a number of problems have been solved to show the performance of this algorithm and its computational results have been compared with the results obtained from GAMS software.


INTRODUCTION
The world fleet expanded to more than nine billion dollars during 2012, reaching more than 1.5 billion deadweight tons (DWT) in 2012, an increase of over 37 percent in just four years.Container transport is regarded as a crucial part of the world's truly global supply chain.Numerous factors should be taken into account for ship route design, one of the most important of which is port service accessibility.In addition, timeline design plays a role in air pollution due to its impact on ship fuel consumption (Lei, 2022).Containers are moved by container shipping companies over scheduled routes.Several products such as manufactured goods, food, clothing and apparel are shipped in containers.Generally, shipping services operate within a fixed schedule and has an appropriate port arrangement, in a way that arrival and departure time at ports are specified and planned.Port schedules are posted on the APL website, and customers can set the delivery time of their shipments based on the dates announced on the website and determine the time required for reaching the port (Salido et al., 2011;Alharbi et al., 2014;Chang et al., 2010;Qi and Song, 2012).Therefore, container shipping is significant part of the world's global supply chain.
The network of container shipping lines includes many shipping lines that must be assessed for the route of each ship.Scheduling shipping lines is a planned decision at technical level made every three to six months.Port service accessibility is the first factor considered for ship route design, without which the ship route schedule design is not possible.The rotating time decreases when the ship's speed is higher in container shipping lines, which leads to a lower number of ships required for transport.The present study aims to develop a model for scheduling shipping lines in each network of container shipping lines in order to minimize the overall ship costs, including the costs of charging ships and the cost of maintaining the inventory of empty containers in the port by considering the time window of the port and the amount of fuel.The remainder of the article is structured, as follows: Section 2 presents the research background, and Section 3 addresses the proposed mathematical model.Section 4 explains the solution method, and Section 5 focuses on the generation of initial solution.Section 6 presents computational results and section seven concludes and makes suggestions for future studies.

RESEARCH BACKGROUND
There are limited studies on the design of ship route schedule design.The first group of studies is related to designing a schedule at the technical planning level.In a research, Wang and Meng (2012) focused on a tactical-level liner ship route schedule design in a container shipping company with a high number of ports and different routes and a fixed sailing speed on each voyage leg (Wang et al., 2014).In another research, X and Song presented an optimal vessel schedule to minimize the total expected fuel consumption.In addition, the rotation time was randomly considered, meaning that ships arrived at a port of call no later than the announced time (Trappey et al., 2011).Wang and Meng (2012) considered a schedule for the ships existing on the route when uncertainty in port operations and scheduling were considered while assuming that the ships could make up for the delay by sailing across the ocean (Yan et al., 2009).In their previous research, Wang and Meng (2012) improved uncertainty at sea and ports by incorporating speed optimization.These scholars adopted a dynamic planning approach to a practical tactical liner ship route schedule design problem with a port time window.They assumed that each port on a ship route is visited only once in a round-trip journey.In reality, however, there are many routes where a port can be visited twice (Yin et al., 2011).Wang et al. (2015) developed their previous research with the assumption that each port on the route of the ship can be visited twice.In addition, they focused on the route of one ship instead of a container network with several shipping routes (Sun et al., 2012).Yan et al. (2009) developed a container routing model aiming at maximizing operating profit in order to set a schedule at operational level.In another research, Brouer et al. (2013) presented the Vessel Schedule Recovery Problem (VSRP) to evaluate a given disruption scenario and to select a recovery action balancing the tradeoff between increased fuel consumption and the impact on cargo in the remaining network and the customer service level.Other studies, such as those developed by Chang et al. (2010, 2011), He et al. (2010), Du et al. (2011), have focused on port operations.Sun et al. (2012) created a general simulation platform, which aims to provide a flexible modeling system.Yan et al. (2009) introduced a distributed agent system for dynamic port planning and scheduling and examined their hypothesis with a case study.Salido et al. (2011) developed a planning technique for solving the container stacking problem and a set of optimized allocation algorithms for solving the berth allocation problem independently.Moreover, He et al. (2010) postulated a strategy for resolve the issue of sharing internal porters across multiple container terminals and selected a near-optimal solution using an optimization method.
Alharbi et al. (2014) examined a practical liner shipping schedule design problem with port time windows for container supply chain networks.They proposed a mixedinteger nonlinear non-convex model that incorporates the availability of ports to minimize the sum of ship cost and fuel cost while considering the port time window.In the end, the model was reformulated as an integer linear optimization model and was solved applying an iterative optimization approach.

PROPOSED MATHEMATICAL MODEL
In order to better design the mathematical model, the symbols, variables and parameters are defined first, followed by presenting the objective function and related constraints.Moreover, the necessary explanations of the details of the mathematical model are provided below.

Symbols and Variables
The symbols and variables used in the mathematical model are presented below: R shows the series of shipping routes and P indicates the set of ports, in a way that the route of a ship r R ∈ is shown, as follows: shows the port corresponding to the i-th port of call, and N r is the number of ports existing on the route of the ship.The set of ports of call on the r route are defined, as follows: (2) rij t : the time (day) required for a ship to move from the i-th port to the j-th port on the r route.W: total days of the week, W = {0,1,2,3,4,5,6}, where zero is indicative of Saturday, one shows Sunday, and… x : the number of empty containers reserved at the i-th port on the r route.
: a binary variable; 1, if the ship arrives at the i-th port of call on the r route in the w day of the week.Otherwise, 0.
: a binary variable; 1, if the ship arrives at the b-th harbor of the i-th port of call on the r route in the w day of the week.Otherwise, 0.
In addition, fuel consumption (tons/miles) is considered as a function shown below: : the maximum ship speed in the r route z + : the set of positive integers According to the mentioned symbols, parameters and variables, the mathematical model proposed encompasses a five-expression objective function and 14 sets of constraints, as described below: Accordingly, the objective is to minimize the total costs of recharging ships, costs of inventory of empty containers in the port, and costs of fuel.0 6 . , 24 , Constraints ( 5) guarantee solution symmetry.Constraints (6) indicates that ship cannot travel at speeds in excess of the speed limit.Constraints (7) defines the relationship between a time of a round trip.Constraints (8) shows the time of return to the first port of call on the return path.Constraints (9) demonstrates that the number of ships does not exceed the maximum number of ships Scheduling for a Container Supply Chain to Minimize Costs Using the Meta-Innovation Approach Vol 20, No 4, December 2021, pp.662-671, © 2021 KIIE 665 and is also a positive integer.The time of arrival at each port of call is a non-negative integer shown in Constraints (11).The second set of constraints show the time of week for arrival at each port of call in the network: 7 , The day of arrival at the port of call on the route is shown by constraints (12), whereas constraints (13) demonstrate that each ship must arrive at a port of call in each week.Constraints ( 14) define the binary variable, and the auxiliary variable is defined as a non-negative integer.The third set of constraints points out the accessibility of ports: According to Constraints ( 16), an available port cannot serve more than one ship in a day.Moreover, Constraints ( 17) and ( 18) indicate that each ship can use each port of call within a week.

THE PROPOSED GA SOLUTION METHOD
Since the problem in the present study is an NP-hard problem, it is impossible to use precise methods for solving the problem in reasonable time.Therefore, heuristic or metaheuristic methods should be used to solve the problem.In this research, the genetics algorithm (GA) is applied to solve the problem.In addition, GAMS and GA are used to exhibit the solvability of the model at different dimensions.The GA implementation stages and the solution process are explained below.

INITIAL SOLUTION GENERATION
An initial population of chromosomes must be generated after determining the coding system and specifying the method of turning each solution into chromosome.In most cases, the initial population is created randomly.
However, heuristic methods are sometimes used to increase algorithm quality and speed to generate the initial population.The size of the initial population often depends on the other coded string.For instance, the initial population selected must be definitely larger when there are 32-bit chromosomes in the problem, compared to 16bit chromosomes.The cutoff point is often between 80 to 95%, whereas the probability of mutation is considered between 0.5 and 1 percent and the population size regarded in the range of 20-30.Based on the fit function, the selected chromosomes are allocated a real amount, which shows their value, and the GA stages continue.If there is a very low number of chromosomes, the GA will be less able to perform less combination operations, and only a small part of the search space will be discovered.On the other hand, the process of the GA algorithm will be slow if there is a very high number of chromosomes.According to the results, the use of the population will not be extremely effective as a result of some constraints that mostly depend on coding and the problem itself.

Coding
This is probably the most difficult problem solution stage of the algorithm method.Binary coding is one of the simplest coding techniques and the best conversion for genetic operators.In binary conversion, the members of the population become strings of zeros and ones.Figure 1 shows the chromosome coding structure.

Population
In the GA, the concept of population is similar to what actually exists in normal life.As the first stage of the GA, it is required to generate a set of doable solutions as the initial population.The members of this set are often selected randomly.However, some conditions are used in the optimization algorithms so that there is not an exces- sively scattered population.In addition, the number of population members depend on the type of problem.In fact, the number of members is a parameter that can be changed by improving the accuracy of solutions and search convergence speed.While an eight-member population might be completely suitable for some problems, a population with more than 100 members might not be sufficient for other problems.Therefore, it is best to specify the population within the range of 10-160 members.

Population Size
Equation ( 19) is used to calculate the population size: For instance, if each chromosome has a length of 25, then we will have: , or if the length of each chromosome is equal to 25, the population size will be equal to 270.

Roulette Wheel Selection
The roulette wheel is one of the most suitable random selection methods, in which the probability of selection corresponding to each chromosome is estimated based on its fitness.In this regard, if f k shows the fit value of the k-th chromosome, the probability of survival corresponding to that chromosome will be as shown in Equation (20): Afterwards, the chromosomes are sorted based on P k , and q k , which is the cumulative values of P k , are obtained using Equation (21): In fact, the roulette wheel creates a random number between zero and one for selection of each chromosome, and the chromosome corresponding to the mentioned number is selected whatever the range of the number is.

Mutation
In nature, some factors, such as ultraviolet radiation, cause unpredictable changes in chromosomes.Since GAs follow the law of evolution, these algorithms use the lowprobability mutation operator.The mutation causes a search in the intact spaces of the problem, and the most important responsibility of mutation is to avoid convergence to local optimization.Figure 2 shows mutation and its function.
In general, mutation prevents the GA from locating itself in local extremes.Each member is a value determined by the user that depends on the mutation possibility (P m ).

GA Parameter Tuning
The efficiency of metaheuristic algorithms is directly related to tuning its parameters so that the correct choice of parameter values increases the efficiency of the algorithm.The improper tuning of parameters could lead to improper solutions for the problem under study.While various statistical methods have been proposed for designing test, increasing the number of factors studied, complex and time-consuming calculations are performed in using a comprehensive approach such as complete factor experiments.Taguchi introduced a set of fractional factor experiments, which specifically reduce the number of tests required for assessment while maintaining the required information (1).In this project, control factors of Taguchi method include parameters of GA, population size and number of iterations, probability of crossover performance and probability of mutation performance.Three levels are used for analyzing the optimal values of parameters in Taguchi method, as shown in Table 1: For the factors considered from the standard tables of orthogonal arrays, the most suitable array is L9, which is presented in Table 2.The integer values in each column   Vol 20, No 4, December 2021, pp.662-671, © 2021 KIIE 671 M. Kavitha is currently working in Saveetha university, India, and completed PhD.degree in maharishi University, Lucknow.Her research interests include embedded system and bioinformatics.He has published research out-comes in several IEEE conferences and peer-reviewed international journals and has about 12 research publications to his credit.He is a member of IEEE and ACM.
Several parameters are used in the proposed mathematical model, as shown below:α : Ship fuel prices (unit/ton) cost of maintaining the inventory of empty containers at the i-th port in the r route ri L : the distance between the i-th port and the j-th port in the r route port ri t : the time passed by a ship at the i-th route in the r route min ri t : the shortest time expected for shipping to the i-th port in the r route max r m : the maximum number of ships existing in the r route hmax ri x : the maximum number of empty containers reserved at the i-th port in the r route max r v

Table 1 .
Genetics algorithm factors

Table 4 .
Computational results of GAMS and genetics algorithm Professor at the State Islamic University of Sumatera Utara.He finished his Ph.D in Mathematic from University of Sumatera Utara and participated in several high profile conferences.In addition to his academic career Dr. Ismail Husein, M.Si held several managerial positions in his university.Currently, He focus in Mathematical modelling and Operations Research.Arif Suhada, He Is As A Representative Of Application Development and Lab in Andalas Prima Teknologi.Study Field of Production System Engineering, Information Systems, Business Intelligence and Analytics (BIA), Artificial Intelligence, Data Science.Dr. Paitoon Chetthamrongchai is the director of BBA International program, Faculty of Business Administration, Kasetsart University, Thailand.Dr. Chetthamrongchai's research interests in data analytic in marketing, supply chain and consumer and consultancy work have focused on developing virtual customer assistants as a strategic framework for organisation.His work demonstrates how firms' performance is linked to the strategic marketing.These findings are based on extensive empirical work with services for the government and private companies in Thailand.Vice-rector for the implementation of strategic development programs candidate of pedagogical sciences, assistant professor, Born in Stavropol on May 25, 1971.He has over 70 published scientific papers, including 21 papers in journals indexed by international databases Scopus and Web of Science.Research interests: functional-semiological and cognitive linguistics; linguocultural synergy; university science management Anis Siti Nurrohkayati is a Lecturer at the Department of Mechanical Engineering, Universitas Muhammadiyah Kalimantan Timur, Samarinda Indonesia.She has extensive experience managing and resolving industrial minds, particularly in the manufacturer industry.Her areas of interest in research include operational management, engineering quality, supply chain engineering, material quality, and manufacturing design.Vo Hoang Ca, Ho Chi Minh City Open University, Vietnam, Research Areas Language, tourism, economy Languages Vietnamese, English Huynh Tan Hoi, FPT university, Vietnam Degree Ph.D. in education management, M.A in education Research Areas Language, tourism, medication, economy Languages Vietnamese, English, Japanese, Chinese John William Grimaldo Guerrero is a professor at the Department of Energy, Universidad de la Costa, Colombia.He received degrees in Petroleum Engineering, MSc Electrical Engineering, MBA, and Specialist in Energy and Mining Colombian Law, and he is doing his Doctorate in Energy Engineering.His research interest includes data science, business analytics, improvement of productivity, mining-energy policies, and energy markets.Scheduling for a Container Supply Chain to Minimize Costs Using the Meta-Innovation Approach Dr. Ismail Husein, M. Si is currently an Assistant