A Multi-Objective Ant Colony System Algorithm for Airline Crew Rostering Problem with Fairness and Satisfaction

—The airline crew rostering problem (CRP) is signiﬁcant for balancing the workload of crew and for improving the satisfaction rate of crew’s preferences, which is related to the fairness and satisfaction of crew. However, most existing work considers only one objective on fairness or satisfaction. In this study, we propose a new practical model for CRP that takes both fairness and satisfaction into account simultaneously. To solve the multi-objective CRP efﬁciently, we develop an ant colony system (ACS) algorithm based on the multiple populations for multiple objectives (MPMO) framework, termed multi-objective ACS (MOACS). The main contributions of MOACS lie in three aspects. Firstly, two ant colonies are utilized to optimize fairness and satisfaction objectives, respectively. Secondly, a new hybrid complementary heuristic strategy with three kinds of heuristic information schemes is proposed to avoid ant colonies focusing only on their own objectives. Ant colonies randomly choose one of the three schemes to help explore the Pareto front (PF) sufﬁciently. Thirdly, a local search strategy with two types of local search respectively for fairness and satisfaction is designed to further approach the global PF. The MOACS is applied to seven real-world monthly CRPs with different sizes from a major North-American airline. Experimental results show that MOACS generally outperforms the greedy algorithm and some other popular multi-objective optimization algorithms, especially on large-scale instances.

Abstract-The airline crew rostering problem (CRP) is significant for balancing the workload of crew and for improving the satisfaction rate of crew's preferences, which is related to the fairness and satisfaction of crew.However, most existing work considers only one objective on fairness or satisfaction.In this study, we propose a new practical model for CRP that takes both fairness and satisfaction into account simultaneously.To solve the multi-objective CRP efficiently, we develop an ant colony system (ACS) algorithm based on the multiple populations for multiple objectives (MPMO) framework, termed multi-objective ACS (MOACS).The main contributions of MOACS lie in three aspects.Firstly, two ant colonies are utilized to optimize fairness and satisfaction objectives, respectively.Secondly, a new hybrid complementary heuristic strategy with three kinds of heuristic information schemes is proposed to avoid ant colonies focusing only on their own objectives.Ant colonies randomly choose one of the three schemes to help explore the Pareto front (PF) sufficiently.Thirdly, a local search strategy with two types of local search respectively for fairness and satisfaction is designed to further approach the global PF.The MOACS is applied to seven real-world monthly CRPs with different sizes from a major North-American airline.Experimental results show that MOACS generally outperforms the greedy algorithm and some other popular multi-objective optimization algorithms, especially on large-scale instances.

I. INTRODUCTION
A IRLINE crew scheduling is of great significance to airlines, which mainly consists of two stages: crew pairing and crew rostering.In the crew pairing problem (CPP), several flights are organized into a pairing that starts from one base and eventually returns to the same base.Then the crew rostering problem (CRP) aims at assigning each pairing to the crew under a set of constraints.The crew scheduling problem is divided into these two stages mainly for two reasons [1].For one thing, the integrated problem is too large to be handled simultaneously for large-scale instances, and for another, the objectives and constraints of CPP and CRP are different.Some surveys [2], [3] have introduced the related topic.In this study, we concentrate on the CRP due to its direct impact on the fair assignment of workload and the improvement of crew satisfaction.
To better deal with the CRP, the problem model should be firstly formulated and then the methods are designed to solve the model.For the problem model of CRP, there are different kinds of considerations.Many CRP models focus on minimizing the cost from the perspective of airlines.Zeghal and Minoux [4] considered minimizing the total number of additional flight credits for all crew members, and they also took the cost of overnights into account.Santosa et al. [5] regarded the variable cost of roster paid by airlines as one of the objective functions, which was represented by actual flying hours.Ezzinbi et al. [6] considered minimizing the total cost as the main objective of the crew scheduling problem, and the cost was calculated by the sum of different crew pairing costs.
Different from most existing models that consider the cost from the perspective of airlines, we focus on fairness and satisfaction from the perspective of crew members, which directly affects their motivation.Herein, fairness refers to the balanced allocation of pairings among all the crew members, and many studies have studied the evaluation metrics of the fairness.Doi et al. [7] adopted the sum of deviations from This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/standard working hours for each crew member as a measure of fairness, and the deviation is calculated by the linear penalty.De Armas et al. [8] also used the working time as the measure of work-balancing, and they proposed three criteria to describe the deviation as well, which were the standard deviation, interquartile range, and maximum-minimum range.Lučić and Teodorović [9] measured fairness using the weighted sum of average deviations between the real values and their respective ideal numbers, including monthly flight time, allowances, and the number of weekend days.Boufaied et al. [10] proposed balancing three objectives, i.e., the number of flight hours, layovers, and destinations' occurrences.
As for satisfaction, some researchers proposed considering crew preferences.In the modern work environment, taking employee preferences into account in workload scheduling has an important effect on service quality [11].Gamache et al. [12] considered a series of weighted bids that reflected the crew preferences and the assignment had to be completed under strict qualifications.Dawid et al. [13] suggested generating a set of work lines for each crew member based on their requests like days off and preferred flights, and only one of them can be selected.Maenhout and Vanhoucke [14] considered that the crew could convey their preferences for certain attributes of rosters, such as specific pairings, reserve duties, and even the time when they wanted to be scheduled.Kasirzadeh et al. [15] proposed using constraint functions to limit the minimum number of preferred flights and preferred vacations to be satisfied.
However, there are still some defects in the above works.Firstly, some works only optimized a single objective on fairness or satisfaction.In this case, the solution may be good at one objective but extremely poor at the other one, which may lack comprehensiveness and cannot meet the actual needs of the industry [16].Secondly, many studies only used flight time as the evaluation metric of fairness, which was not comprehensive enough.Thirdly, most researches that considered crew preferences only took it as part of the single objective function.This weighted-sum method for multiple objectives is difficult to determine the weights and only one solution can be obtained.
Based on the considerations mentioned above, we propose a new CRP model from the perspective of crew members, which takes both fairness and satisfaction of crew into account simultaneously.The multi-objective model is solved to get a group of non-dominated solutions so that decision-makers can balance all objectives in different situations.A solution is non-dominated if the other obtained solutions cannot be better than it on both objectives.Specifically, this study uses three attributes to measure fairness comprehensively, including monthly flight time, duty time, and overnight time away from home.Herein, the flight time is the real working time, which is usually regarded as the only measure of fairness in most studies.However, the duty time for the crew to stay in the airport may be also long if the paring is loosely scheduled.Therefore, it is also necessary to consider duty time as another measure of fairness.Moreover, crew members prefer to rest at their own bases rather than in hotels away from home.
Therefore, overnight time away from home (i.e., the rest time in hotels) is also used as a measure of fairness.In terms of satisfaction, our proposed model takes crew satisfaction with the preferences for flights and vacations directly as an objective, rather than as an item in a single objective function.
The formulated multi-objective CRP is an NP-hard problem [7] with many practical constraints, including daily restrictions, weekly restrictions, and monthly restrictions.Different countries or airlines may require different constraints to ensure the adequate rest of crew members.The constraints in our model are based on the latest government document [17] from the ministry of transport in China.These constraints involve the limits on the connection between pairings, accumulative flight time for one calendar month, duty time for seven consecutive days and one calendar month, and an additional requirement for the rest time.These complex constraints make it difficult to find feasible solutions in CRP.
Since our proposed CRP model is a challenging discrete combinatorial optimization problem with complex constraints, it is important to choose a suitable optimizer to solve it.Traditional mathematical methods are inappropriate to such an NP-hard problem or cannot obtain the optimal solution in polynomial time.Ant colony optimization (ACO) [18], [19] is a kind of swarm intelligence approach that has been widely used to obtain the optimal or near-optimal solution in discrete combinational optimization problems [20]- [24].In particular, the ant colony system (ACS) [18] is an efficient and widely used variant of ACO.Hence, we study the ACS-based algorithm for CRP in this paper.The ACS optimizer constructs a solution step by step, i.e., the pairings are assigned in turn.Each pairing is assigned to a crew member satisfying a set of constraints in each step, thus the feasibility of the final solution can be guaranteed.Therefore, the ACSbased algorithm can find feasible solutions easily, even on a large-scale instance.As the original ACS is a single objective optimization algorithm, we combine the ACS optimizer with the multiple populations for multiple objectives (MPMO) framework [25], [26] to handle the multi-objective CRP.
Therefore, based on the MPMO framework, we develop a multi-objective ACS (MOACS) algorithm to solve the multiobjective CRP.MOACS uses two ant colonies to optimize fairness and satisfaction objectives, respectively.As each ant colony only focuses on its own objective, it may reduce population diversity to explore the central part of the Pareto front (PF).Hence, a hybrid complementary heuristic (HCH) strategy with three kinds of heuristic information schemes is put forward to improve the search ability of ant colonies.Ant colonies randomly choose one of the three schemes to help explore the PF sufficiently.Furthermore, we design a local search strategy to further approach the global PF, which contains two types of local search for fairness and satisfaction, respectively.In this study, MOACS is tested on real-world monthly CRP with various sizes from a major North-American airline [15].The experimental results indicate that MOACS outperforms some representative multi-objective optimization algorithms in solving CRP, especially on large-scale instances.To sum up, the contributions of this study are as follows: 1) We propose a new practical multi-objective CRP model from the perspective of crew members that takes both fairness and satisfaction into consideration.
2) An HCH strategy is proposed to develop the MOACS combining the ACS optimizer and the MPMO framework, so as to efficiently solve the multi-objective CRP.
3) A local search strategy is designed to approach the global PF, which contains two types of local search for fairness and satisfaction, respectively.
The rest of this paper is arranged as follows.The definition and formulation of the multi-objective CRP are described in Section II.Section III introduces the details of MOACS and its innovation.Section IV presents the experimental results.Finally, the conclusion of this study is given in Section V.

A. Problem Description
A base is a large airport where crew members are stationed.For convenience, the crew usually lives in the city where the base is located.Hence, each crew member is related to one base.A flight is an activity in which an aircraft travels directly from one airport to another, generally in a short period of time.A small portion of the flight schedule of one day is listed in Table I.A pairing is a connection of several flights.It starts from the base corresponding to crew members and finally returns to the base, which lasts usually 3-5 days.The CPP organizes the flights into pairings, while the CRP in our study takes pairings already obtained from the CPP as the input and assigns them to specific crew members.An example with six pairings is shown in Fig. 1, where each pairing has included several flights.The CRP aims at assigning all pairings on the left side to the crew members on the right side.
In our proposed CRP model, (b j , s j , e j , f j , d j , o j ) represents the attributes of pairing p j , where j represents the pairing index.Three of these attributes require preprocessing to obtain, which are the f light time f j , duty time d j , and overnight time away from home o j .These three attributes are used comprehensively to measure the fairness of scheduling.Combined with the flight information of pairings in Fig. 1 and the corresponding flight in Table I, the required attribute values can be calculated.The flight time f j refers to the sum of the actual flying time of all flights included in the pairing p j .The duty time d j usually consists of three parts.The first part is all the flight time f j .The second part is the sum of sitting time, i.e., the sum of intervals between flights that is less than the minimum rest time MinRestTime.The third part is the preparation time Brief before the start of the daily mission and the summary time Debrief after the end of the daily mission.We also consider the special cases where the duty time includes an additional fourth part.That is, in some pairings, a flight that a crew member takes only for relocation, which is called a deadhead, may be required.In such a deadhead situation, the crew member flies as a passenger and does not perform any mission, so the deadhead time is not included in the flight time but is included in the duty time because the crew member is still regarded to be on duty during a deadhead.As for the overnight time away from home o j , it is the layover in hotels after the end of the daily mission, that is, the interval between the daily last flight and the first flight in the next day is larger than or equal to MinRestTime, and the arrival airport is not the home base of the crew member.The diagram of the attribute calculation is shown in Fig. 2, where the duty denotes the schedule for one day.Note that the crew member will locate in the corresponding base after performing duty 2 , so the interval between duty 2 and duty 3 is not included in the overnight time away from home.
In addition to the three attributes mentioned above, the other three attributes are the departure base b j , the start time s j , and the end time e j of pairing p j , respectively.s j is equal to the departure time of the first flight included in pairing p j minus the pre-flight preparation time Brief.Similarly, e j is equal to the sum of the arriving time of the last flight included in pairing p j and the summary time Debrief.
Moreover, each crew member can apply for their preferred flights and vacations in advance.Note that the preferred flights of the crew member are associated with their base.There are two ways to improve the satisfaction of crew.One is assigning the pairing that includes preferred flights to the corresponding crew member.The other is avoiding assignments which are during the preferred vacations of crew.
In our CRP model, (cb i , pf i , pv i , cf i , cd i , co i ) represents the attributes of the crew member c i .The first three attributes cb i , pf i , and pv i denote the crew base, preferred f lights, and preferred vacation of crew member c i , respectively.The remaining three attributes cf i , cd i , and co i respectively refer to the cumulative f light time, duty time, and overnight time away from home of crew member c i during a month.Note that we only consider the assignment of pilots, and assume that only one pilot is required per flight in our model.
To make our paper clearer and easier to read, the basic notations and terminologies for the CRP model mentioned above and used following are listed in Table S.I of the supplementary material.

B. Objective Formulation
Different from most existing CRP models that consider the cost from the perspective of airlines, we propose the model for the first time taking both fairness and satisfaction objectives into account from the perspective of crew members.This consideration can greatly enhance the enthusiasm and satisfaction of crew members, thereby increasing the quality of flights.
The fairness objective refers to the balanced allocation of workload among all the crew members.As the standard deviation is the most popular indicator to measure the balance of different members, it is used in our paper to measure the fairness objective.Herein, the workload of a crew member includes three attributes, i.e., monthly flight time, duty time, and overnight time away from home.Note that the fairness of crew in each base should be calculated separately because crew members in different bases cannot perform the same pairing.Therefore, the fairness objective is formulated as the sum of standard deviation of the three workload attributes of the crew members in all the bases, as where X is the schedule of all the crew, BN denotes the number of bases, and E b represents the set of all crew members in base b.cd i and co i represent the monthly duty time and overnight time away from home of member c i , respectively, and they are calculated in a similar manner to cf i as shown in (3).fi, j denotes the flight time of the j th pairing in the schedule of crew member c i .n i is the total number of pairings to be performed by c i .d b and o b represent the average duty time and overnight time away from home of all pairings in base b, respectively, and they are calculated in a similar manner to f b as shown in (4).A smaller g 1 indicates a more balanced allocation of workload among all the crew members.
In terms of the satisfaction objective, a crew member is satisfied with the pairing assignment if their preferred flights are included and the assignment does not conflict with their preferred vacation.Therefore, these two attributes are considered in the satisfaction objective.In detail, the percentages of the preferred flights and preferred vacations of all crew members that are satisfied are calculated separately and the sum of these two percentages is used to represent the satisfaction of the pairing assignment for the crew members.Thus, the satisfaction objective is formulated as where spfn denotes the number of satisfied preferred f lights and PFN denotes the total number of preferred flights required from all the crew.Note that some flights may be preferred by more than one crew member.In such cases, each of these preferred flights is counted only once in the PFN.Similarly, spvn represents the number of satisfied preferred vacations and PVN represents the total number of preferred vacations required from all the crew.A larger g 2 indicates a more satisfying pairing assignment scheme.
In our proposed CRP model, both objectives are optimized at the same time.Note that g 1 (X) is a minimal objective, whereas g 2 (X) is a maximal objective.The goal of our model is defined as (6), which is a bi-objective problem. optimize

C. Constraint Formulation
With regard to constraints, different countries or airlines may have different regulations.Our constraints here are based on the latest government document CCAR121-R5 [17] in China.First of all, the connection between pairings should meet the rationality of time and space.Specifically, the rest time between the connected pairings cannot be less than 10 hours, and the pairing should be assigned to the crew member of the corresponding base.Therefore, the constraints of the reasonable connection between linked pairings in time and space are defined as ( 7) and ( 8), respectively, where ŝi, j , êi, j , and bi, j denote the start time, the end time, and the departure base of the j th pairing in the schedule of crew member c i , respectively.
Second, there are restrictions on cumulative flight time.In any calendar month, the flight time of a crew member cannot exceed 100 hours, which is formulated as where cfd i,d refers to the flight time of the crew member c i on the dth day.
Third, there are limits on the duty time for seven consecutive days and one calendar month.During any seven consecutive calendar days, the accumulative duty time cannot exceed 60 hours.In any calendar month, the cumulative duty time cannot exceed 210 hours.Therefore, the constraints of duty time within two different time ranges are defined as (10) and (11), respectively, where cdd i,d refers to the duty time of the crew member c i on the dth day.The range of k in (10) corresponds to all seven consecutive days in a month.
Finally, there is an additional requirement for the rest time.During the 144-hour period prior to a mission, every crew member should be arranged at least 48 consecutive hours for rest.The constraints of additional requirements are formulated as where checkRest i, j = 0 indicates that the j th pairing of member c i meets the additional requirements.rest i, j,k is used to check all the rest time within 144 hours before the start of the j th pairing of member c i .rest i, j,k = 0 indicates that during the 144 hours before the start of the j th pairing, the rest time between the kth and the (k + 1)th pairing of member c i does not exceed 48 hours.A set of constraints discussed above ensures the feasibility of scheduling and sufficient rest for crew members.

III. MOACS ALGORITHM FOR SOLVING CRP
Based on the MPMO framework [25], the MOACS algorithm adopts ACS [18] as the optimizer for multiple ant colonies, and each colony only optimize one objective.To adapt the characteristic of MOACS, we design suitable heuristic information for different objectives and construct solutions according to the pairings start timeline.In addition, a new HCH strategy is proposed to explore a more complete PF.Lastly, we propose a local search strategy with two types of local search to further approach the PF.The entire MOACS algorithm is described below.

A. Solution Encoding
In ACS, fixed-length binary encoding [27] and integer encoding [28] are commonly used.Specifically, binary encoding in CRP uses the indicator variable y i, j to indicate whether the crew member c i performs the pairing p j .The integer encoding in CRP is an integer vector (z 1 , z 2 , . . ., z P N ), where z j represents the employee index of the crew member performing the pairing p j .Since these two encodings are fixed-length, they are easy to implement and facilitate some operators like crossover and mutation in the genetic algorithm [29].However, the constraints involve the crew members' adjacent pairings, and all objectives and constraints are related to crew members.Therefore, it is necessary to decode the solution into a natural expression such as (15) for fitness calculation.In (15), x i, j denotes the pairing index of the j th pairing in the schedule of the crew member c i whose employee index is i , CN denotes the number of crew members, and n i represents the total number of pairings to be performed by member c i .Each row vector represents all the pairings to be performed by a crew member in chronological order.Since the total number of pairings performed by each crew member is not fixed, the length of the row vectors is not fixed.
Based on the above considerations, we directly adopt the variable-length natural expression shown in (15) as the encoding of solutions.An example of solution encoding is shown in Fig. 3.This encoding scheme is clear and does not require a decoding process.Besides, during the solution construction (details refer to Section III-C), MOACS will handle the pairings one by one and assign only one crew member to each pairing according to the pheromone values and heuristic information, which guarantees that each pairing will be assigned to only one crew member.Moreover, according to the above encoding, it is easy to calculate the three attributes of member c i , i.e., the sum of the flight time cf i , duty time cd i , and overnight time away from home co i .

B. Multiple Colonies Framework
The key to solving multi-objective optimization problems (MOPs) with evolutionary algorithms is how to choose suitable solutions for the next generation.Since it is hard to define an appropriate fitness metric in multi-objective functions, the MOACS algorithm borrows the concept of MPMO framework to consider each objective separately.Based on the MPMO framework, multiple ant colonies are adopted, and each ant colony only needs to focus on optimizing a single objective through a conventional optimizer.Herein, the ACS with the same population size m is adopted as the optimizer for each ant colony.As the CRP model is a bi-objective problem, we use two ant colonies named Fcolony and Scolony that set fairness and satisfaction as their optimization objective, respectively.

C. Solution Construction
Since each crew member's schedule must satisfy the time constraint, it is natural to construct schedules according to the starting timeline of pairings.Thus, we first index all the pairings in this order.The ACS imitates the foraging behavior of ant colonies in finding the shortest path to food.Ants will release chemicals named pheromones on the paths they pass and they will follow the path with higher pheromones, which forms a mechanism similar to the positive feedback.In each step during solution construction, pheromones and heuristic information are used to determine which crew member should be assigned the corresponding pairing p j .The design of the pheromones and heuristic information will be presented in Sections III-D and III-E, respectively.The solution construction process is shown as where β is a parameter to balance the effect of pheromones and heuristic information, col is the identifier of the ant colony (i.e., the Fcolony or the Scolony), and q 0 is the probability of exploitation.τ col (k, j ) and η col (k, j ) represent the col ant colony's pheromone and heuristic information between crew member c k and pairing p j , respectively.For the pairing p j , we firstly generate a random number q to determine which construction method is adopted.If q does not exceed q 0 , the ant will greedily select the crew for exploitation, that is, select the crew member with the largest τ col (k, j ) × [η col (k, j )] β .Otherwise, the crew member i will be selected through the roulette wheel selection according to the probability defined as The heuristic information is not as fixed as traditional problems because the workload balance is constantly changing during the construction.Hence, the heuristic information should be calculated in each step during the construction.Generally speaking, the number of crew members (i.e., CN) is always fewer than the number of pairings (i.e., PN), that is, a crew member should be assigned several pairings in a month.Therefore, in this case, we only need to calculate the heuristic information corresponding to all crew members, instead of all pairings.In contrast, the crew-based 'pilot-bypilot' solution construction method [30] firstly completes the assignment for the first crew member, then the second, until all pairings are assigned, whose time complexity is O(PN 2 ).However, the time complexity of our pairings-based solution construction method is O(PN × CN), which will reduce the computational complexity.Besides, we need to check all available crew members that can satisfy a set of constraints in Section II-C when they perform the current pairing.

D. Pheromone Update 1) Pheromone Initialization:
In MOACS, the initial pheromone values τ 0 of Fcolony and Scolony are defined as (18) and (19), respectively, where PN is the total number of pairings, FGS is the greedy solution about fairness, and SGS is the greedy solution about satisfaction.
Note that different colonies perform the solution construction process independently according to their different objectives and have their own pheromone values under the MPMO framework.This is an advantage so that different scales of the objective values and pheromone values in two colonies do not lead to objective bias.Herein, objective bias means the algorithm tends to optimize more on one of the objectives.
The selection of the crew member for the pairing p j in FGS is defined as where cf i, j , cd i, j , and co i, j denote the cumulative flight time, duty time, and overnight time away from home of member c i , respectively, when pairings from p 1 to p j have been assigned.c f i, j , cd i, j , and co i, j refer to the average flight time, duty time, and overnight time away from home of the first j pairings, respectively.Note that the calculations of these three average values are limited to the base of the crew member c i .
According to (20), the pairing p j is always assigned to the member with the smallest sum of absolute differences between its three attribute values and the corresponding average values.
As for SGS, the selection of the crew member for the pairing p j is defined as where cv i, j = 1, if pairing p j conflicts with the vacation pv i 0, otherwise (23) fn i, j is the number of preferred flights of the crew member c i included in the pairing p j .Since the selection of the crew member for the pairing p j considers two factors with different scales and cv i, j is an indicator variable, fn i, j needs to be normalized.Min-max feature scaling is used for normalization to eliminate the effect of scales.According to (21), the pairing p j is always assigned to the crew member with the most preferred flights in the pairing p j and the least violation of their preferred vacation.
2) Pheromone Local Update: The MOACS algorithm uses ACS as the optimizer, in which all ants construct solutions in parallel.Once an ant has completed a step during the construction (i.e., assigning a pairing to a crew member), it immediately performs the local update of pheromones.It aims to improve the ability of exploration and is described as where ρ is a parameter in the range of (0, 1).Since τ col 0 is a small value as defined as ( 18) and ( 19), the pheromone value τ col (i , j ) between the crew member c i and pairing p j will be decreased after an ant assigns p j to c i in the ant colony col.Therefore, other ants will be more likely to select the other crew member different from c i for p j .The local update of pheromones helps improve the diversity of solutions and avoid premature convergence.
3) Pheromone Global Update: The conventional ACS for single-objective problems updates global pheromones directly through the best solution found so far.However, a set of non-dominated solutions are obtained in MOPs, and we cannot determine which one is the best solution.A different method needs to be designed to update global pheromones for MOPs.There have been some techniques for globally updating pheromones in previous studies, for example, Zhan et al. [25] proposed selecting a random solution from non-dominated solutions to globally update pheromones.Barán and Schaerer [31] proposed updating all the nondominated solutions.Doerner et al. [32] proposed updating the best solution of respective objectives.Updating a random solution from non-dominated solutions will improve the solution diversity, but it lacks efficient search guidance.In contrast, updating all the non-dominated solutions or the best solution of respective objectives will guide ants to search better.However, the former technique may lead to premature convergence and the latter may get trapped in a narrow region of PF.
Therefore, the global update strategy [33] is adopted due to its ability to balance search efficiency and solution diversity, as described below.Firstly, we introduce a set Archive to save all non-dominated solutions from two ant colonies in all generations, and the global update solutions (GUSs) will be selected from the Archive.Then, we sort all solutions in the Archive in ascending order by the objective value g 1 .Finally, Fcolony randomly selects the GUS Fcolony from the first |Archive| × θ solutions with smaller g 1 , whereas Scolony randomly selects its GUS Scolony from the last |Archive| × θ solutions with better g 2 .θ is a parameter in the range of (0, 1), which controls the balance between search efficiency and solution diversity.A small θ accelerates convergence, while a large θ improves the global search ability.The GUS of an ant colony performs the local search strategy described in Section III-F below, which generates a new neighbor solution GUS new .If GUS new and the original GUS are non-dominated to each other, both are allowed to globally update.Otherwise, only the GUS new is required in the global update.Generally, the local search strategy can improve GUS on one objective, so that GUS new is not dominated by the original GUS.The global update of pheromones by GUSs is shown as where ε is a pheromone decay parameter in the range of (0, 1).After the execution of the global update, the promising solutions GUSs are allowed to leave more pheromones on their paths for increasing their attractiveness.Conversely, the pheromones on the paths of the other solutions will evaporate according to the local update to reduce their desirability.

E. Hybrid Complementary Heuristic Strategy
Heuristic information is equivalent to the self-experience of ants, which can be used to better guide the search.We first introduce how to choose reasonable heuristic information for the two ant colonies with different optimization objectives.Note that only crew members who meet a set of constraints need to calculate heuristic information, and the heuristic information of remaining members is directly set to zero.Fcolony aims to optimize the fairness objective.Thus, its heuristic information is devised to reduce the sum of deviations of the three attribute values, which are the flight time, duty time, and overnight time away from home.Since the solution construction restricts the heuristic information to be a maximum objective, the heuristic information uses the inverse of the sum of deviations.The smaller the sum of absolute difference between the three attribute values and their corresponding average values, the greater the heuristic information of the crew member.The definition of heuristic information about fairness objective is shown in (27), whose variables have been given in Section III-D.Notice that since solutions constructed by each ant are different, the corresponding heuristic information may be different for different ants, even if the pairing p j is assigned to the same crew member.Therefore, it is necessary to dynamically calculate heuristic information in each step during each ant's solution construction.
In addition, Scolony aims to obtain the schedule containing as many preferred flights and vacations as possible for the crew.Hence, its design of heuristic information involves the number of crew members' preferred flights included in the current pairing and whether the current pairing conflicts with their preferred vacations.The more a crew member's preferred flights are included in the current pairing and the pairing does not conflict with their preferred vacation, the greater the heuristic information of the crew member.The definition of heuristic information about the satisfaction objective is shown in (28).The variables in (28) have been given in Section III-D.
Since each ant colony only focuses on optimizing a single objective optimization problem in the MPMO framework, it may lead to solutions concentrating on the margin of PF.There are two methods to enhance the exploration of the central part of PF.Chen et al. [33] proposed the ordinary complementary heuristic strategy that utilized the heuristic information of the external objective.The other method is to use aggregated heuristic information combining all objectives [34].Based on the second method, we design the suitable aggregated heuristic information defined in (29), which includes two terms about fairness and satisfaction objectives, respectively.The first term η 1 (i , j ) is in the range of (0, ∞) and the second term ( f n i, j − cv i, j + 1) is in the range of [0, 2].Since the two terms are in different scales, we combine them by multiplication to reduce the impact of scales.
In order to fully explore the PF, we propose a hybrid strategy in which an ant colony randomly utilizes the heuristic information of its own objective, the other objective, or the aggregated heuristic information.These are the three heuristic information schemes in our proposed HCH strategy.The HCH strategy is defined in (30) so that the three schemes will be adopted with same probability.In detail, we firstly generate a random number r with uniform distribution to determine the heuristic scheme.If r is less than 1/3, which corresponds to the heuristic information of their own objectives, Fcolony uses η 1 and Scolony uses η 2 as their heuristic information.If r is greater than 1/3 and less than 2/3, which corresponds to the heuristic information of the external objective, Fcolony uses η 2 and Scolony uses η 1 as their heuristic information.Otherwise, the aggregated heuristic strategy is selected, and both Fcolony and Scolony adopt η 3 as their heuristic information.The HCH strategy with three schemes can balance the search of the whole PF.In detail, the first scheme helps to explore the margin of the PF, while the other two schemes can explore the central part of the PF.Note that the random number r is generated per generation, and therefore the heuristic information scheme used in different generations may be different.

F. Local Search Strategy
To enhance the accuracy of solutions generated by MOACS, we design two types of local search for the two objectives respectively.On the one hand, the local search for improving the fairness adjusts the pairings performed by two crew members with the largest deviation.On the other hand, the local search for improving crew satisfaction adjusts crew members who perform pairings during their preferred vacations.In order to approach the global PF, both Fcolony and Scolony randomly choose one of these two local searches to execute.Taking Fcolony as an example, if it executes the local search designed for fairness, its g 1 value is likely to decrease and the margin of PF can be explored.In the other case, if it executes the local search designed for satisfaction, its g 2 value may increase and the central part of PF can be explored.In addition to the HCH strategy mentioned above, this is another way to overcome the shortcoming that each ant colony only focuses on its own objective.
The local search strategy is executed in each ant colony in every generation after the selection of GUS.As only the GUS of each ant colony executes the local search strategy, it does not take too much time.Note that if all preferred vacations in GUS have been satisfied, the solution will directly perform the local search designed for fairness.Otherwise, the GUS will randomly select one of the two types of local search.That is, a random number r 1 within the range of [0, 1] is randomly generated.If r 1 is smaller than 0.5, the ant colony performs the local search designed for fairness.Otherwise, the other local search designed for satisfaction will be performed.These two types of local search are described in detail below.Firstly, the local search designed for fairness is described below, and its pseudocode is shown in Algorithm 1.

Algorithm 1 Local Search for Fairness
Input: GUS Output: the newly generated solution GUS new  We first calculate the sum of the absolute difference absd between the three attribute values and their corresponding average values for all crew members, as shown in Line 4. The crew member C 1 with the largest absd in base b is selected to conduct the local search.The above steps are carried out from Line 2 to Line 7.Then, the matching member C 2 for C 1 can be found from Line 8 to Line 13.The criterion for finding the matching member C 2 is that the average attribute values of the two members C 1 and C 2 are the closest to the average attribute values of all the members in the entire base.Next, we try to transfer the pairings directly as shown in Line 14.The pairings in the crew member's schedule with a large sum of three attribute values between members C 1 and C 2 are transferred to the other member's schedule.After that, we try to exchange the pairings with overlapping time in the schedules of members C 1 and C 2 , as shown in Line 15.Note that the transfer and exchange of pairings require to satisfy a set of constraints, and if the sum of absd of the two members does not decrease after the transfer and exchange, the original schedules will be restored.After the crew members in all bases have been adjusted, the newly generated neighbor solution will be evaluated at last, as shown in Line 16.
The transfer and exchange of pairings in Line 14 and Line 15 of Algorithm 1 are illustrated with an example in Fig. 4. The three numbers ( f j , d j , o j ) around the pairing p j represent the flight time, the duty time, and the overnight time away from home of pairing p j , which is a part of its attributes.The wired arrows represent the timeline.The sum of absd of the two members is 169 before the local search, and it turns to 47 after the transfer of pairing p 3 , so the transfer succeeds.Similarly, the sum of absd increases to 141 when pairing p 4 is transferred from member C 1 to member C 2 .Since it is larger than 47, the transfer is not performed.Next, we try to exchange pairings that overlap in time.In this example, pairings p 1 and p 2 overlap, and the pairing group including pairing p 5 and p 7 overlaps with pairing p 6 .After the exchange for pairing p 1 and p 2 , the sum of absd slightly increases to 59 which is also larger than 47, thus the exchange is still not executed.When the pairing group and pairing p 6 exchange, the sum of absd decreases to 17, so this exchange is performed successfully.The transfer and exchange of pairings make the The other local search designed for satisfaction is described below, and its pseudocode is shown in Algorithm 2. We first randomly select a crew member C from all crew members in base b whose preferred vacation is not satisfied to make the adjustment, as shown in Line 2. Then, we examine the pairings set ProcessingPool that conflicts with their preferred vacation from the schedule of crew member C, which is carried out from Line 3 to Line 6. Next, for each pairing p in ProcessingPool, we check all idle crew members who can perform this pairing and do not conflict with their preferred vacation.This is performed from Line 7 to Line 11.If there are idle crew members, this pairing p is transferred from the schedule of member C to the schedule of the idle crew member with the most preferred flights in pairing p, as shown from Line 12 to Line 14.After the crew members in all bases have been adjusted, the newly generated neighbor solution will be finally evaluated, as shown in Line 15.
The local search for satisfaction is illustrated with an example in Fig. 5.In the example, the preferred vacation of member C conflicts with pairing p 4 and p 6 in their schedule.Both pairings are stored in ProcessingPool temporarily and then check all idle members for each pairing.In this example, member c 3 and c 5 are able to perform pairing p 4 , and only member c 2 can perform pairing p 6 .Additionally, the number in parentheses on the member circle denotes the number of preferred flights of the corresponding member included in the above pairing.Since member c 5 applies for more preferred flights in pairing p 4 , pairing p 4 is assigned to member c 5 at last.Although the preferred flights of member c 2 are not included in pairing p 6 , pairing p 6 is still assigned to member c 2 because pairing p 6 has only one idle member.The local search for satisfaction may lead to an increase in the satisfaction rate of preferred flights and vacations.

G. Complete MOACS Algorithm
The overall pseudocode of the MOACS is shown in Algorithm 3. Firstly, the greedy solutions FGS and SGS are generated, which are used to initialize the pheromones by ( 18) and ( 19), and the global archive is also initialized.Then in every generation, the heuristic strategy is randomly selected.Two ant colonies calculate their heuristic information according to (30).In each step during the solution construction, a crew member is selected for each pairing by (16) and the pheromones are locally updated according to (24).After all ants have constructed solutions, these newly generated solutions will be evaluated and added into the global archive.All dominated solutions in the archive will be eliminated, and then GUS Fcolony and GUS Scolony are selected from it.If all preferred vacations have been satisfied, the GUS will directly perform Algorithm 1. Otherwise, it will randomly perform Algorithm 1 or Algorithm 2. The global update of pheromones is conducted according to (25) and (26).If the newly generated GUS new and the original GUS are nondominated to each other, both are used in the global update of pheromones.Otherwise, only the GUS new is used.MOACS terminates when the maximum execution time is reached and outputs all non-dominated solutions in the archive.

IV. EXPERIMENTS AND COMPARISONS
In this section, experimental tests are performed on the realworld dataset to verify the performance of MOACS solving CRP.All the algorithms are implemented in C++ and run on a with a Xeon Quad CPU E3-1225 and 4.0GB of RAM.

A. Test Dataset
We use the experimental instances in [15] to test the performance of all algorithms in solving CRP.This dataset comes from a major North-American airline and provides   We use the pairings from the dataset [15] as our input of the CRP.The pairings are the solutions of the CPP.Since our proposed model further involves preferred flights and vacations, we randomly generate these data.The generator for preferred flights from the dataset recommends that each crew member prefers a certain percentage per f of flights associated with their base.Instead of randomly selecting preferred flights from all flights, this way is more reasonable in the actual situation, because crew members generally prefer flights close to their bases.In addition, the generator for preferred vacations from the dataset suggests that a certain percentage per v of all crew members apply for vacation.The vacation periods can occur on any day of the month with lengths between 2 and 10 days with uniform distribution.Their preferred vacations start at 00:00 and end at 23:59 by default.Here we set per f to 10% and per v to 30%.

B. Experimental Settings
The parameter configurations in our proposed MOACS algorithm are shown as follows.The number of ants in each ant colony m is set to 60.The CRP-related parameters in Section II are Brief = 30 minutes, Debrief = 30 minutes, and MinRestTime = 9 hours.The configurations of other ACS-related parameters are the same as in [18] (except for the initial pheromone value τ 0 ), where the parameter β = 2, q 0 = 0.9, ρ = 0.1, and ε = 0.1.The setting of τ 0 for Fcolony and Scolony is given in (18) and (19), respectively.The parameter θ in the global update of pheromone is set to 0.1.
MOACS is compared with three algorithms, greedy heuristic, NSGA-II [29], and P-ACO [35].NSGA-II is a popular algorithm to solve MOPs.Based on the NSGA-II variant, Chen et al. [29] designed crossover and mutation operators for this problem and proposed a special repair operator to improve the feasibility of solutions.The parameters of NSGA-II [29] are set according to the corresponding reference.The population size and offspring size are set to 100 and 80, respectively.The crossover rate is set to 1.0, the mutation rate is set to 0.3, and the parameter related to the population update is set to 0.5.The maximum generations are 2000 in NSGA-II for each instance.Since the execution time of other heuristic algorithms is different from NSGA-II in every generation, all the heuristic algorithms use the same execution time as NSGA-II to ensure the fairness of comparison.P-ACO [35] is a classical multiobjective algorithm based on ACO with multiple pheromone matrices for multiple objectives, but only one heuristic matrix that integrates the information of all objectives.The parameters of P-ACO [35] are the same as MOACS.Since P-ACO adopts only one heuristic matrix, we design the heuristic information to be the same as (29) in Section III-E for P-ACO.All the heuristic algorithms run independently 10 times.

C. Experimental Results
Although the proposed CRP model is a bi-objective problem, common performance metrics such as inverted generational distance cannot be used because the true PF is not known.Hence, the two performance metrics hypervolume (HV) and C(X, Y ) [36] are adopted for performance evaluation.HV represents the volume among a solution set and a reference point.The reference point is selected by the worst objective values about fairness and satisfaction from all solutions obtained by all comparison algorithms in all runs.The larger the HV of the solution set, the better its diversity and convergence efficiency.The HV is calculated independently in each run and the average result of the 10 runs is compared.
In addition, C(X, Y ) concerns the dominant relationship between two sets of solutions and is defined as where x y denotes solution x dominates or equal to y.We collect all non-dominated solutions obtained in 10 runs to calculate the C(X, Y ) value.

1) Comparison With Non-Metaheuristic:
The greedy heuristic is an efficient non-metaheuristic algorithm to deal with single-objective problems.The greedy algorithms for optimizing fairness and satisfaction are termed GreedF and GreedS, respectively, both of which have been described in Section III-D.The comparisons of MOACS with GreedF and GreedS are presented in Table III, where the best results are in boldface.Since MOACS is a multi-objective algorithm, its result is a solution set.Here, we use the best solution for the corresponding single objective in each run and use their median of the 10 runs for comparison.In other words, the solutions with minimum g 1 and maximum g 2 in the median run, i.e., the margin of the PF, are used to compare with the greedy algorithms, which are shown in the left and right side of Table III, respectively.
As can be seen from Table III, MOACS performs much better than the greedy algorithms on g 1 on most instances.Although the average of MOACS cannot reach the best value on fairness in I5-757 and I7-320, MOACS can still obtain the better g 1 in some runs on these two instances.As for the performance in terms of satisfaction, MOACS can get the best result in I1-727, and obtain the solutions close to GreedS on g 2 on other instances.The reason for the good performance of greedy algorithm in optimizing g 2 is that the selection for a pairing has less impact on the selection for the other pairings.Thus, it tends to obtain a fairly high satisfaction rate.However, the optimization of g 1 is more complex so that the greedy algorithm may easily get trapped in a local optimum.Moreover, since MOACS optimizes both objectives g 1 and g 2 at the same time rather than simply focuses on g 2 , it is reasonable that MOACS cannot obtain the best g 2 on some instances.Although MOACS cannot exceed GreedS on g 2 , it has a significant advantage on another objective g 1 .In a word, MOACS can get the solutions as good as or even better than that of the greedy algorithms on the corresponding objective, while it is far better than greedy algorithms on the other objective.
2) Comparison With Multi-Objective Optimization: The multi-objective algorithms adopted for comparison are the NSGA-II variant [29] and P-ACO [35].Since the two comparison algorithms do not contain the local search, we use a MOACS variant without the local search for fair comparisons, which is termed MOACS-noLS.
Since the CRP is an NP-hard problem with various constraints, it is difficult to find a feasible solution.Firstly, we focus on the feasible rate (FR) of all comparison algorithms, which is the ratio of the number of feasible runs to the total number of runs.The comparisons of MOACS-noLS with NSGA-II and P-ACO in terms of FR are shown in Table IV, where the best results are in boldface.We can find that ACO-based algorithms have a significant advantage in finding feasible solutions.Both P-ACO and MOACS-noLS are able to obtain feasible solutions in all 10 runs on all instances because a set of constraints is checked and satisfied in each step during the solution construction.In contrast, NSGA-II only has a 50% probability of finding a feasible solution in I2-DC9 and even can hardly obtain a feasible solution on the last four large-scale instances.As the solutions obtained by NSGA-II are randomly generated at the beginning, it is hard to satisfy a set of constraints in such high dimensional search space.
Table IV also compares MOACS-noLS with NSGA-II and P-ACO in terms of HV, where the best results are in boldface.Wilcoxon rank-sum tests with significance level 0.05 are used for the significance tests.The symbols '+', '≈', and '−' mean that MOACS-noLS is significantly better than, similar to, and significantly worse than the corresponding algorithm, respectively.As can be seen from Table IV, MOACS-noLS outperforms the other two comparison algorithms on HV.Note that NSGA-II can hardly find a feasible solution on large-scale instances.MOACS-noLS performs better than NSGA-II obviously on seven instances.Compared with P-ACO, MOACS-noLS has a significantly large HV and the gap is more distinct with the growth of instance size.Moreover, the comparison results on another performance metric C(X, Y ) are shown in Table V.The larger values between C(MOACS-noLS, -) and C(-, MOACS-noLS) are in boldface.We can find that C(MOACS-noLS, -) values are significantly larger than C(-, MOACS-noLS) values on all seven instances compared with NSGA-II and P-ACO.It indicates that MOACS-noLS outperforms the other two comparison algorithms.Specifically, most C(MOACS-noLS, -) values are very close to one, which denotes most solutions obtained by the other algorithms are dominated by at least one solution of MOACS-noLS.The C(NSGA-II, MOACS-noLS) values on most instances extremely approximate zero except for I1-727, which means very few solutions of MOACS-noLS are dominated by or equal to the solutions of NSGA-II.Although some solutions of MOACS-noLS are dominated by or equal to those found by P-ACO, C(MOACS-noLS, P-ACO) values are still obviously larger than C(P-ACO, MOACS-noLS) values on all instances.
The two performance metrics discussed above focus on different aspects of PF, and MOACS-noLS outperforms the other algorithms on both metrics.HV considers the solution quality in terms of diversity and convergence, and C(X, Y ) reflects the dominant relationship between solution sets.Combining these two performance metrics can provide a more convincing result, but lacks an intuitive comparison.In order to visualize the solution sets obtained by different algorithms, we plot all Note that satisfaction g 2 is a maximum objective, we convert the ordinate to 2-g 2 to observe the solution set clearly, which denotes the sum of dissatisfaction rates and is a minimum objective.
These figures further confirm that MOACS-noLS has overall better performance than the other two compared algorithms.It can be seen from the previous experiments that the solutions of NSGA-II in I2-DC9 and the last four large-scale instances are basically infeasible.Therefore, the PFs of NSGA-II on these instances are quite different from the other algorithms.Compared with P-ACO, MOACS-noLS obtains a broader PF due to its pheromone update rule and HCH strategy.The pheromone update rule increases the diversity of solutions, which helps explore the whole PF.Meanwhile, MOACS-noLS adopts the HCH strategy with three kinds of heuristic information schemes, which also leads to the exploration of the whole PF.By contrast, P-ACO merely uses one integrated heuristic matrix, which results in the solution set focusing only on the central part of the PF.

D. Effectiveness of Local Search Strategy
We propose a local search strategy with two types of local search for the two objectives respectively to help MOACS further approach the global PF.To investigate the effectiveness of the local search strategy, we compare the performance of MOACS-noLS and MOACS using the HV and C(X, Y ) metrics.Both algorithms run independently 10 times.The comparison results containing two types of experiments are shown in Table S.II of the supplementary material, where the larger results are in boldface.Since both algorithms involve stochastic factors, Wilcoxon rank-sum tests with significance level 0.05 are used to statistically evaluate the results.The symbols '+', '≈', and '−' mean that the performance of MOACS is significantly better than, similar to, and significantly worse than that of MOACS-noLS on HV, respectively.
From the left side of Table S.II, we can see that MOACS outperforms MOACS-noLS with regard to HV on all seven instances.As for the C(X, Y ) metric, the values of C(MOACS, MOACS-noLS) are larger than C(MOACS-noLS, MOACS) on all instances, which further confirms that MOACS performs better than MOACS-noLS.Both results reflect the effectiveness of the local search strategy.Also taking I2-DC9 and I5-757 as examples, we curve all the solutions obtained in 10 runs to compare both algorithms intuitively and the results are shown in Fig. 7.The PF distribution of MOACS is better than that of MOACS-noLS on both small-scale instance I2-DC9 and large-scale instance I5-757.Since two types of local search are performed randomly, the solutions are able to approach the global PF with better g 1 or g 2 , which helps improve their quality.In conclusion, all the above results validate the effectiveness of the local search strategy.

E. Parameters Analysis
The MOACS parameters include m, β, q 0 , ρ, ε, and θ .The settings of β, q 0 , ρ, and ε are the typical scheme of ACS, so only m and θ are analyzed , there seems to be no significant difference in the performance of all settings.The medium-sized m such as "40" has a slightly large HV on the first three instances, but its performance is slightly worse than the other settings on most large-scale instances.On the one hand, smaller ant colonies can perform more generations in the same execution time, so the local search strategy will be executed more times, which is helpful for exploitation.On the other hand, an increase in the number of ants can also help increase the search area [37], but it will reduce the number of times that the local search strategy is executed.Since these two factors are in conflict, m is insensitive in MOACS.Finally, we select the setting "60" because it has the overall best performance.
The other parameter we study is the selection rate θ proposed in the global update of pheromones in Section III-D.We set θ from 0 to 0.4 with a step size of 0.1.Note that θ = 0 represents that MOACS always selects the best solution of each objective as the GUS. the setting "0.1" for θ obtains the best performance among other settings on most instances.On the first three smallscale instances, the larger θ can get better HV for it provides more diversity.However, an excessively large θ brings a worse HV on the large-scale instances I4-D95 and I6-319 due to the reduction in convergence efficiency.The results show that the setting "0.1" for θ is suitable for balancing the solution diversity and search efficiency.Moreover, the setting "0.1" is better than "0" on most instances, which also confirms the effectiveness of the global update strategy.

V. CONCLUSION
Different from most existing models that consider the cost from the perspective of airlines, this study proposes a new multi-objective CRP model from the perspective of crew members.Specifically, the fairness and satisfaction of crew are taken into account at the same time.Combining the ACS optimizer and the MPMO framework, a novel MOACS algorithm is developed to solve the multi-objective CRP efficiently.Under the MPMO framework, two ant colonies are employed to optimize fairness and satisfaction objectives, respectively.Besides, to prevent ant colonies from concentrating only on their own objectives, we propose a new HCH strategy with three kinds of heuristic information schemes.It randomly uses the heuristic information of two ant colonies and the aggregated heuristic information, which helps explore the global PF.Furthermore, we design a local search strategy with two types of local search for fairness and satisfaction objectives, respectively, which helps approach the global PF.
Experiments are conducted on monthly CRP with different sizes from the real world.The results verify the effectiveness of MOACS in solving multi-objective CRP.It generally outperforms the greedy algorithm and other multiobjective evolutionary-based algorithms, especially on largescale instances.In conclusion, MOACS is more effective than the compared algorithms in dealing with the multi-objective CRP.In future work, the number and qualifications of crew members required for flights will be further considered in the CRP model.

Fig. 2 .
Fig. 2. Schematic diagram of attribute calculation for the pairing.

Fig. 4 .
Fig. 4. Schematic diagram of the local search for fairness.

Algorithm 2 3 ProcessingPool = ∅; 4 foreach 5 ifthen 6 ProcessingPool = ProcessingPool ∪ p; 7 foreach 8 I 9 foreach member mem in base b do 10 if 11 I
Local Search for Satisfaction Input: GUS Output: the newly generated solution GUS new 1 for b = 1: BN do 2 Randomly select a member C from all crew members in base b whose preferred vacation is not satisfied; pairing p in the schedule of member C do pairing p conflicts with the preferred vacation of member C pairing p ∈ Pr ocessi ng Pool do dleMembers Pool = ∅; member mem can perform pairing p and pairing p does not conflict with his preferred vacation then dleMembers Pool = I dleMembers Pool ∪ mem; 12 if I dleMembers Pool = ∅ then 13 Select the idle crew member idlemem with the most preferred flights in pairing p; 14 TransferSinglePairing(C, idlemem, p); 15 Evaluate the newly generated neighbor solution GUS new ; three attribute values of member C 1 and C 2 close to the actual average of their base b.

Fig. 5 .
Fig. 5. Schematic diagram of the local search for satisfaction.

Algorithm 3 6 HeurSchNo 7 for col = 1 : 2 do 8 for j = 1 :do 9 for k = 1 : m do 10 Calculate 20 Perform
Procedure of MOACS 1 t = 0; 2 Generate greedy solutions FGS and SGS; 3 Archive = ∅; 4 Initialize the pheromone τ ) and (19), respectively; 5 while t ≤ Max T ime do = rand()%3; // 0, 1, 2 represent three different heuristic schemes PN heuristic information of the kth ant in the colth colony based on HeurSchNo by (30); 11 Select a crew member by (16) and add the pairing p j to their schedule; 12 Perform the local update of pheromone according to (24); 13 for k = 1 : m do 14 Evaluate the kth ant in the colth colony; 15 Add the kth ant in the colth colony to Archive; 16 Eliminate all dominated solutions in Archive; 17 for col = 1 : 2 do 18 Select the global update solution(GU S) for the colth colony; 19 if all preferred vacations in GUS have been satisfied then update of pheromone for GUS new according to (25) and (26); 24 if GU S new and the original GUS are non-dominated to each other then 25 Perform the global update of pheromone for the original GUS according to (25) and (26); 26 Update the elapsed time t; Output: All non-dominated solutions in Archive
. The investigation results are plotted in Fig. S.2 of the supplementary material.The x-axis of the figure denotes the corresponding parameter settings and the y-axis denotes the average HV among 10 independent runs.Note that when testing a parameter, all other parameters are set the same as stated in Section IV-B.Firstly, we analyze the influence of the parameter m.We set m from 20 to 60 with a step of 10.The comparison results of average HV on all seven instances with respect to the parameter m are shown in Fig. S.2(a).From Fig. S.2(a) The comparison results of average HV on all seven instances in regard to the parameter θ are shown in Fig. S.2(b).From Fig. S.2(b), we can find that

TABLE I EXAMPLE
OF FLIGHTS INFORMATION Fig. 1.Diagram of crew rostering scheme.

TABLE II SIZE
OF SEVEN TEST INSTANCES seven instances of different sizes as shown in TableII.The last four instances are large.In our proposed CRP model, we consider monthly schedules that have more guiding value in the practical application.

TABLE III COMPARISONS
OF MOACS WITH NON-METAHEURISTIC ALGORITHMS ON TWO OBJECTIVES It represents the ratio of the number of solutions in Y dominated by any solution in X and is in the range of [0, 1].If C(X, Y ) equals to one, it means each solution in Y is dominated by or equal to at least one solution in X.If C(X, Y ) equals to zero, it represents that no solutions in Y can be dominated by any solution in X.Note that the sum of C(X, Y ) and C(Y , X) is generally not equal to one.If C(X, Y ) is larger than C(Y , X), it represents the solution set X is better than Y .