Adjustable Assignment Model Examples

Aeronautical Engineering College, Civil Aviation University of China, Tianjin 300300, China

Copyright © 2013 Yaohua Li and Na Tan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Fleet Assignment Problem (FAP) of aircraft scheduling in airlines is studied, and the optimization model of FAP is proposed. The objective function of this model is revenue maximization, and it considers comprehensively the difference of scheduled flights and aircraft models in flight areas and mean passenger flows. In order to solve the model, a self-adapting genetic algorithm is supposed to solve the model, which uses natural number coding, adjusts dynamically crossover and mutation operator probability, and adopts intelligent heuristic adjusting to quicken optimization pace. The simulation with production data of an airline shows that the model and algorithms suggested in this paper are feasible and have a good application value.

1. Introduction

Fleet Assignment Problem (FAP) is to assign an aircraft model for each scheduled flight according to the capability of passengers, running cost, and planned revenue of each fleet. This is an important work of aircraft scheduling and planning in airlines. The results of FAP affect not only the cost and revenue of airlines, but also the continuing works, such as linking problem between flights, aircraft’s maintenance route, crew assigning, and flight gate assigning. The aircraft scheduling is a controlling work of production scheduling in airlines. Because of the importance and complexity of the aircraft scheduling work in the air transport, the in-depth research and application have been carried out in aviation developed countries of Europe and America [1–3]. In China, because airlines had small amount of aircrafts a few years ago, they have not paid more attention to production plan and management and their planning mode was simple and manned. At the same time, the research on civil aviation production planning management is few. Recently, with the number of airlines’ aircrafts increasing, the aviation transport market opening, and the aviation market competition pricking up, airlines wake up to the importance and urgency of production scheduling and planning management gradually. But in general, the theory research on aircraft planning and scheduling is still in the underway phase. Reference [4] presents the notion of flight purity and builds fleet assignment model subjected to flight purity according to the characters of Chinese airline network and flight scheduling. A robust mathematical model for the fleet scheduling problem is put forward [5]. According to the data provided by an airline, the computational experiment performed with the improved Grover’s algorithm shows the effectiveness of the proposed model and improves the robustness of the decision. The aircraft scheduling problem based on cooperative multitask assignment is studied [6], and the approach applies branch-and-price algorithm to the cost optimization model with maintenance constraints, and mathematical model of daily utilization ratio is established. A model of flight-string VRP based on the time unit of week is suggested [7], and a parthenogenetic algorithm is suggested for solving the model. In China, although some scholars have studied the problem of aircraft planning and scheduling, the plan of aircraft scheduling in airlines is completed with manpower or half-manpower mode. The level of automation is not tall, and the information system based on the mature models and algorithms is few.

In this paper, according to Chinese factual situation, FAP in airlines is studied in order to establish a foundation for aircraft planning and scheduling automation. Based on the research result, an optimization model of FAP is proposed, which takes the total revenue maximum as objective and can assign an appropriate aircraft type to each flight. For solving the complicated optimization model, an improved genetic algorithm is suggested, which can find out optimal solution quickly. After deep studying, the model and algorithm can be applied in production scheduling of other countries’ airlines.

2. FAP Optimization Model

2.1. Problem Description

In production scheduling of airlines, Fleet Assignment Planning is to assign the most appropriate aircraft type to each flight. The flying performance of different aircraft model is different, for example, voyage range, flying altitude ceiling, maximum take-off weight, and climbing ability. So, a particular route is not suitable for all models of the aircraft to perform. In addition, different models have different seating layout, and their operating costs are not the same. For instance, the seats number of the B737-300 aircraft is about 144, and its direct operating costs are between 30 and 50 thousands of RMB per hour. But the A340-200 aircraft can seat up to about 380 people, and its direct operating cost is more than 100,000 of RMB per hour. The basis for the development of the work is the airworthiness limitations of flight route on aircraft models, each model's cabin distribution, operational cost analysis of the models in the different routes, as well as forecasts of passenger and freight traffic on each flight. The goal is to optimize the allocation of models to flight, in order to minimize the operating costs to complete the flight running tasks.

2.2. FAP Optimization Model Building

Considering the feature of Chinese flight route net and flight plan, under constraints of determined flight schedules, not considering the flight stopovers, only considering aircraft A check, and enough airport capacity, the FAP model which considers the models match, model flying area, as well as the traffic match conditions is proposed as follows: ;   is overall flight number,;   is the number of aircraft models,;  the revenue of aircraft model to perform the flight ,; , , and are the revenue, fixed operating costs, and the variable operating cost of model to fly flight , where, the suitable flying area code of model  ,, the minimum flying area code required by flight ,, the passenger capacity of model ,, the average traffic of flight ,, model code of the model ,, the model code required by flight .

The objective function (1) means that the total income of all flights is largest, after the aircraft types are assigned to all flights considering the bulk of the flights of global optimization. Constraint (2) is to ensure that an equal number of models are selected for flights. Constraint (3) is to ensure that only one model is assigned to each flight. Constraint (4) is to ensure that the model assigned to the flight meets the flight area requirements. Constraint (5) is to ensure that the model assigned to the flight meets the flight traffic requirement. Constraint (6) is to ensure that the code of the model assigned to the flight is greater than the flight code. In order to calculate conveniently, the models flying area code and model code use a natural number coding according to difference of the actual situation; for example, 1, 2 represent the fly zone, representing the flight area of plains and plateaus. The model codes of 1, 2, 3 represent, respectively, the B737-300, B737-800, B757, and so forth. Thus, in order to ensure the operational feasibility in the model calculations, all of these codings take the downward compatible form. That is, high-grade aircrafts can perform the flight requiring low-grade aircraft model, not the contrary.

3. The Solving Algorithm of FAP Optimization Model

It is difficult to solve the FAP optimization model with mathematical programming methods, because FAP is an NP-hard problem. The genetic algorithm (GA) is an adaptive search algorithm which is based on the natural evolution and selection mechanism. And it has been successfully applied to a variety of optimization problems. In this paper, an improved hybrid heuristic genetic algorithm is constructed to solve the model, considering the limitations of the general genetic algorithm. The algorithm uses natural number coding method and dynamically adjusts the crossover and mutation probability.

3.1. Intelligent Heuristic Adjustment for Infeasible Solutions

The usual method for solving constrained optimization problems is to convert it to unconstrained optimization problem, which incorporated the constrained constraints into the evaluation function using the method of weighting coefficients. Thus, although constrained optimization problems can be solved, infeasible solutions may exist in aviation production scheduling production. In order to guarantee that individuals of each generation are feasible solutions, the algorithm will filter the infeasible solution in each generation solutions for every individual and then adjust the infeasible solutions with intelligent heuristic adjustment method. The heuristic rules of the intelligent heuristic adjustment method are based on expert knowledge and relevant constraints. When the individual does not meet the constraint needed to be adjusted, the algorithm adjusts it according to the individual situation and determines the direction and size of adjustment with the expert knowledge rules. Its goal is to ensure that the adjusted individual is feasible solution and is adjusted along the optimized search direction.

3.2. Dynamic Adjustment of the Crossover Probability and Mutation Probability

In order to avoid genetic algorithm falling into a local optimum value and having rapid convergence, genetic operator probability adjustment method in the algorithm is used to dynamically adjust the crossover and mutation probability after [8] is studied. In this paper, the crossover probability and mutation probability of each generation groups are dynamically adjusted according to the degree of concentration of the fitness value. The adjustment method is to establish a judgment standard with the maximum fitness value , minimum fitness value , and average fitness value . Generally, the initial crossover probability is set as , , and mutation probability is set as , . Thus, and are changed with the evaluation (fitness function) of solutions. When the solution has good performance, let and be small to help the algorithm’s rapid convergence. When the solution is lower than the average fitness value, let and be high to prevent the algorithm from optimal solutions into local solution. The adjustment formulas are as follows:

3.3. Steps of the Improved Genetic Algorithm

(1) Inputting the data required by model solving: read the corresponding data information to be calculated.(2) Algorithm parameters initialization: determine the algorithm population numbers and the end of the maximum cycle algebra, the initial values of crossover probabilities and mutation probabilities are set. Then, initial generation chromosomes are given as the current generation chromosome based on the population numbers given.(3) Heuristic correction of the current generation of chromosomes: check the infeasible solutions in chromosomes, and then correct infeasible solutions using the intelligent heuristic rules until they become the feasible solutions.(4) Calculate the adaptation function value of the current generation of chromosome, and record the best individual as the optimal solution. Then, judge whether to satisfy the end criterion; if the answer is yes, jump to (8), or else, jump to (5).(5) Adaptive dynamics: adjust the current chromosome probability, and calculate the probability of crossover and mutation , .(6)Current chromosome genetic manipulation: Cross is completed with probability , mutation operating is done with probability , and then selecting operation is completed, which selects the best chromosome in the current generation.(7) Generation of chromosomes will be selected as the current generation of chromosome; return to (3).(8) Output current optimal solution as the solution of the algorithm.

4. Simulation Research

In order to validate the model and algorithm supposed in this paper, the data of a medium-sized airline, including 4 aircraft models, 50 flights, is selected to study. Raw data are shown in Tables 1, 2, and 3. In this paper, the basic genetic algorithm (GA) and improved adaptive genetic algorithm (IGA) have been used for a comparative study in order to show that the algorithm suggested in this paper is better.

Table 1: The information of aircraft models.

Table 2: The information of flights.

Table 3: The income and cost statistical data of models perform flights (money unit: ten thousands RMB).

The parameters are selected as follows: the number of population genetic algorithm is 20; the IGA initial crossover probabilities are , ; mutation initial probabilities are ;   the algorithm terminates criteria for successive iterations 1000 generation. Basic genetic algorithm (GA) parameters are selected as follows: the number of population genetic algorithm is 20; crossover probability is set as , mutation probability is set as , the algorithm terminates criteria for successive iterations 1000 generation. The simulation calculating result is given in Figure 1 and Table 2, the best values being 567.7 and 526.4, respectively. Basic genetic algorithm can find out a feasible solution, but the result is bad. The computational results are no longer listed in the paper because it is not the emphasised part of this paper. The simulation example flight information is listed in Table 2

An approach for addressing needs of individuals with disabilities

The Americans with Disabilities Act of 1990, its amendments, and other federal and state legislation require that schools make programs accessible to students with disabilities. Disability-related accommodations are unique to each individual. With appropriate accommodations, students with disabilities can be challenged to reach the same high academicstandards set for students without disabilities. DO-IT's Accommodation Model and Student Abilities Profile are tools that can help teachers work with students to create an optimum learning environment.

The Process

It is helpful to have a process to follow when determining appropriate accommodations for students with disabilities. DO-IT has developed a model process and a Student Abilities Profile form for creating effective accommodations.

The Accommodation Model can be used by any instructor and is composed of the following four steps (Burgstahler, 2006, pp. 49-50; Burgstahler & Nourse, 1998, pp. 65-66).

Step #1: What does the task or assignment require?

Break down all of the components of the experiment, assignment, or exercise. As an educator, you are usually focused on the overall outcome of the project. To accommodate a student with a disability it's important to think about the specific settings, tools, skills, and tasks that are required at each step. By analyzing and evaluating the task thoroughly, you will be able to determine how best to fully and effectively include a student with a specific disability.

Step #2: What physical, sensory, and cognitive skills are needed?

Match the tasks required to the physical, sensory, and cognitive skills needed to successfully complete the project. It is easy to say, "If I had a physical, sensory, or cognitive disability I would not be able to complete this assignment," without really determining what skills are needed for specific aspects of the project. We need to separate the real requirements of a specific task from the fictional or perceived requirements of the overall project. It is impossible to place yourself in the shoes of the student with a disability. They may have learned many ways to solve a specific problem or task and work around the limitations imposed by the disability.

Step #3: What components of the task require accommodation?

Once the task has been analyzed and the skills needed are identified, determine what accommodations may be required or how the learning experience might be altered to make it more accessible. Determine the level of difficulty of the project and determine how best to make an accommodation to create an inclusive environment for a student with a disability. It is very important to consult with the student, to determine what they perceive as aspects of a project that they may need an accommodation or assistance.

Step #4: What accommodation options exist?

Now that the tasks that need accommodation have been determined, identify what resources exist for providing the needed accommodation(s). The student may have some good ideas. However, this is a time when other staff and professionals may have expertise in specific areas and be called on to provide input. In some cases, having students work in groups where each person is assigned a task that they have the ability to complete, provides a reasonable alternative.

The Student Abilities Profile

The Student Abilities Profile form is designed to help you determine a student's abilities and break down the individual components of an assignment. The form asks you to briefly describe the student; the classroom or laboratory environment; equipment or supplies needed; physical, sensory, and cognitive skills needed for the task; possible accommodations; and available professional and external resources. Examples of completed forms for specific students can be found at the DO-IT Student Abiliy Profile page.

Student Abilities Profile

Narrative Regarding Student and Accommodation Issues:


Equipment: (lab equipment, computer, manipulatives)

Environment: (fumes, odors, dust, temperature, noise, group work)

Physical Challenges

Accommodations Needed

Options and Resources

Sensory Challenges

Accommodations Needed

Options and Resources

Cognitive Challenges

Accommodations Needed

Options and Resources

Physical, Sensory, & Cognitive Issues and Challenges 

Physical Issues

Think of the required physical aspects of the task. What will make the environment accessible, keep the student safe and allow him/her to be an active participant? What equipment must be manipulated?

Physical Challenges

  1. lift / carry
  2. stamina / endurance
  3. push / pull
  4. knee / squat
  5. reach
  6. repetitive tasks
  7. fine motor: pinch / grasp
  8. fine motor: manipulate / maneuver
  9. gross motor
  10. sit in chair
  11. walk / stand
  12. balance
  13. bend / twist
  14. stoop / crouch
  15. other

Sensory Issues

Think of room temperature, noise, fumes, dust, odors, and allergies. Also consider the ability to speak and/or communicate, and the visual aspects of the task or assignment.

Sensory Challenges

  1. vision
  2. hearing
  3. touch
  4. smell
  5. taste
  6. oral communication
  7. temperature
  8. fumes
  9. external stimuli
  10. lighting
  11. other

Cognitive Issues

What memory and communication skills are needed? What is the level of complexity of the task.

Cognitive Challenges

  1. short term memory
  2. long term memory
  3. task complexity
  4. reading
  5. writing
  6. spelling
  7. string of numbers (math)
  8. paying attention
  9. visual, auditory, or kinesthetic learner.
  10. self-esteem / advocacy issues
  11. behavior issues / acting out
  12. other

Universal Design

Building accessibility features into an environment or service, such as a science or math class, can make it more usable by all students. For example, having one workstation in a science lab that is adjustable in height makes the space accessible to future students who might use wheelchairs, or be large or small in stature. More information about and examples of universal design can be found at the Applications of Universal Design web page.

One Science Teacher's Experiences with the Accommodation Model andStudent Abilities Profile

So what do l do when a student with a disability actually walks or rolls through the doorway? I use DO-IT's Accommodation Model andStudent Abilities Profile. I take time at the beginning of the year to sit down with the student and the student's special education teacher. This meeting is well worth the effort.

I consult with the expert, the student, when filling out the Profile. Some of the accommodations that the student and I discuss may be as simple as photocopying lecture notes, creating a lap desk for a student who uses a wheelchair to allow them to use a microscope, providing access to safety equipment, or having handouts enlarged. Using this method, I am better able to understand how the student learns best and what specific accommodations will be most effective.

Following is an example of how I used the Accommodation Model and Student Abilities Profile with a student named Alice, who has Attention-Deficit/Hyperactivity Disorder (AD/HD). The specific task is alloying zinc and copper. Notes on the four questions presented in theAccommodation Model are provided below (Burgstahler, 2006, pp. 52-53; Burgstahler & Nourse, 1998, p. 96-98).

Step 1: The task requires the ability to:

  • use small letters
  • follow directions
  • follow safety procedures
  • use a scale
  • use a hot plate
  • use a Bunsen burner
  • write observations

Step 2: The physical, sensory, and cognitive skills needed for the task include the following:

Physical challenges:

  • fine motor grasp/manipulation
  • sitting
  • walking/standing

Sensory challenges:

Cognitive challenges:

  • complex thinking
  • reading
  • writing
  • attention span
  • behavior that ensures safety

Step 3: Components of the task that require accommodation for Alice include the following:


  • Alice needs no physical accommodations.


  • There will need to be some accommodations for sensory issues, such as dimming lights, limiting external stimulation, and sitting next to quieter students.


  • Alice will need accommodations to help breakdown complex tasks and focus on the activities.

Step 4: Accommodation options include the following:

  • I will make sure that Alice is near me when I explain directions.
  • If the lab is set up in advance, I will not put the items in front of Alice because she will play with the equipment rather than pay attention.
  • I will make sure she has a responsible partner for the lab and give her the directions two or three steps at a time. When she finishes a set of steps, she can have the next set. This way I know she is following the directions and not jumping around just to get done. It also means that she is checking in with me even if I can't check on her every ten minutes.
  • When the students begin to work, I will check with Alice and make sure she has heard all of the safety issues before starting.
  • Every time Alice does a good job in the lab, I will make sure to tell her.

These notes can be used while completing the Student Abilities Profile for Alice. Her form, along with more examples of Student Abilities Profiles created by DO-IT staff and collaborators (Burgstahler, 2006, pp.55-80; Burgstahler & Nourse, pp. 69-94). The completed Student Abilities Profile provides a record with a focus on what the student's abilities are and what he or she may need assistance with. It can be kept in the student's file and used for reference throughout the year to make sure that the student participates to the fullest extent possible in all activities.


The content of this handout has been replicated in other DO-IT publications that include the following:

Burgstahler, S. (Ed.) (2006). Making math, science, and technology instruction accessible to students with disabilities. Seattle: DO-IT, University of Washington.

Burgstahler, S. & Nourse, S. (Eds.) (1998). Accommodating students with disabilities in math and science classes. Seattle: DO-IT, University of Washington.

Additional Resources

More examples of accommodations for students with disabilities can be found in the DO-IT publications. These include:

For information on a proactive approach to serving students with disabilities, consult The Center for Universal Design in Education.

About DO-IT

DO-IT (Disabilities, Opportunities, Internetworking, and Technology) serves to increase the successful participation of individuals with disabilities in challenging academic programs such as those in science, engineering, mathematics, and technology. Primary funding for DO-IT is provided by the National Science Foundation, the State of Washington, and the U.S. Department of Education. DO-IT is a collaboration of UW Information Technology and the Colleges of Engineering and Education at the University of Washington.

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To order free publications or newsletters use the DO-IT Publications Order Form; to order videos and training materials use the Videos, Books and Comprehensive Training Materials Order Form.

For further information, to be placed on the DO-IT mailing list, request materials in an alternate format, or to make comments or suggestions about DO-IT publications or web pages contact:

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DO-IT Funding and Partners


Creation of these materials was funded by the Dwight D. Eisenhower Professional Development Program through the Washington State Office of the Superintendent of Public Instruction (Grant #GR-91915) and the National Science Foundation (Grant #CNS-0540615, #9800324, #9950003). Any questions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the federal government.

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