# Integrating Flight Scheduling, Fleet Assignment, and Aircraft Routing Problems with Codesharing Agreements under Stochastic Environment

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## Abstract

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## 1. Introduction

## 2. Literature Review

_{2}emission costs associated with cruise speed settings by integrating the three decisions.

- The development of a new third-stage stochastic non-linear programming model that combines FA with operational FS decisions and AR problem using codesharing agreements.
- The consideration of stochastic demand and stochastic non-cruise times simultaneously in the model.
- The proposal of simulation-based metaheuristic (simheuristic) methods for solving the proposed model. The Monte Carlo method is used for simulation, and the Simulated Annealing (SA) and Cuckoo Search (CS) algorithms are used for metaheuristics. Two new solution methods are proposed by integrating simulation and metaheuristics.
- i.
- SA + MC: Simulated Annealing + Monte Carlo
- ii.
- CS + MC: Cuckoo Search + Monte Carlo

Maintaince Planning | Flight Scheduling | Aircraft Routing | Time Window | Stochastic Demand | Spill Passenger Costs | Passenger Connection Time | Delay Costs | Cruise Time Controbility | Stochastic Non-Cruise Time | Codeshare Aggrements | |
---|---|---|---|---|---|---|---|---|---|---|---|

Authors | MP | FS | AR | TW | SD | SPP | PCT | DC | CTC | NCTU | CA |

Levin [13] | √ | ||||||||||

Berge and Hopperstad [11] | √ | √ | |||||||||

Barnhart et al. [41], Sosnowska and Rolim [42] | √ | √ | |||||||||

Rexing et al. [14] | √ | ||||||||||

Yan and Tseng [43] | √ | ||||||||||

Ahuja et al. [44] | √ | √ | |||||||||

Lohatepanont and Barnhart [2] | √ | √ | |||||||||

Li and Wang [45] | √ | √ | |||||||||

Yan et al. [46] | √ | ||||||||||

Yan and Chen [47] | √ | √ | |||||||||

Pilla et al. [26], Jacobs et al. [22] | √ | ||||||||||

Sherali and Zhu [25], Naumann et al. [27] | √ | ||||||||||

Dumas et al. [23] | √ | √ | |||||||||

Haouari et al. [48] | √ | √ | |||||||||

Sherali et al. [49] | √ | √ | |||||||||

Zeghal et al. [16] | √ | √ | √ | ||||||||

Sherali et al. [33] | √ | √ | √ | ||||||||

Liang and Chaovalitwongse [50] | √ | √ | |||||||||

Pita et al. [30] | √ | √ | √ | √ | √ | ||||||

Cadarso and Marin [19] | √ | √ | |||||||||

Atasoy et al. [29] | √ | √ | |||||||||

Shao et al. [51] | √ | √ | |||||||||

Gürkan et al. [34] | √ | √ | √ | √ | |||||||

Dong et al. [52] | √ | ||||||||||

Liu et al. [53] | √ | √ | |||||||||

Cacchiani and Salazar-González [54] | √ | √ | |||||||||

Cadarso and Celis [28] | √ | √ | √ | ||||||||

Jamili et al. [35] | √ | √ | √ | ||||||||

Şafak et al. [36] | √ | √ | |||||||||

Kenan et al. (a) [6] | √ | √ | √ | √ | √ | ||||||

Kenan et al. (b) [31] | √ | √ | |||||||||

Şafak et al. [37] | √ | √ | √ | √ | √ | √ | √ | √ | √ | ||

Şafak et al. [38] | √ | √ | √ | √ | √ | √ | √ | √ | |||

Wei et al. [55] | √ | √ | √ | √ | |||||||

Cacchiani and Salazar-González [38] | √ | √ | √ | √ | |||||||

Xu et al. [39] | √ | √ | √ | √ | √ | √ | |||||

Ahmed et al. [56] | √ | √ | √ | ||||||||

This study | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ |

## 3. Problem Description and Formulation

#### 3.1. Terminology

- Flight leg: Describes a flight of an aircraft from the departure airport to the destination airport.
- Path (itinerary): A sequence of one or more flight legs between a specific origin and destination.
- Fleet type (aircraft type): A certain model of aircraft. All the aircrafts of the same type have the same cockpit configuration, crew qualification requirements, maintenance requirements, and capacity.
- Fleet family (aircraft family): A set of aircraft types, each having the same cockpit configuration and crew qualification requirements. Thus, the same crew can fly any fleet type of the same family.
- Fare class (FC): Available seats on aircrafts are divided into classes according to their fares. The cost of seats in each class is the same. Seat capacities in these classes may be fixed or variable.
- Turnaround time: It is the time required for a fleet type to prepare for flight. This period includes the cleaning of the aircraft, passenger, and baggage movements.
- Cruise time is the time of flight that falls between climb and descent.
- Non-cruise time: It is the sum of the taxi in and out times between an aircraft’s landing and take-off. In other words, it is the time that the aircraft moves on the ground.
- Tax in: It is the duration of the aircraft moving towards the apron after landing.
- Taxi out It is the duration of the aircraft moving from the apron to the take-off field.

#### 3.2. Problem Description

- The analysis of data obtained from the BTS [57] revealed that non-cruise times follows a normal distribution.
- It is assumed that demand follows a uniform distribution with lower and upper parameters covering the minimum and maximum fleet type capacities.
- The departure and arrival times of flights are also pre-determined.
- A time window is used for departure times, and delays that exceed this time window result in additional costs that are included in the objective function.
- Maintenance planning is conducted at either the first or the last airport.
- Codeshare flights are not included in the airline’s routes, and a specific budget is allocated for CA.
- The cruising speed of the fleet type is allowed to vary at certain rates.
- If there are connecting flights for passengers, a minimum time is given for passengers to switch to the next flight, and if this time is not sufficient, passengers miss their connections.

#### 3.3. Mathematical Model

Sets | ||

S | : | Scenarios $\mathrm{s}\in \mathrm{S}$ |

K | : | Fleet types $\mathrm{k}\in \mathrm{K}$ |

F | : | Flights $\mathrm{i}\in \mathrm{F}$ |

${\mathrm{Y}\mathrm{B}}_{\mathrm{i}}$ | : | Flights which have passenger connections from flight $\mathrm{i}\in \mathrm{F}$ |

BU | : | Connected flights sets $\mathrm{i},\mathrm{j}\in \mathrm{F}$ |

${\mathrm{O}}_{\mathrm{i}}$ | : | Flights before flight $\mathrm{i}\in \mathrm{F}$ |

${\mathrm{H}}_{\mathrm{i}}$ | : | Flights after flight $\mathrm{i}\in \mathrm{F}$ |

${\mathrm{S}\mathrm{U}}_{\mathrm{k}}$ | : | Last flights of fleet type $\mathrm{k}\in \mathrm{K}$ |

${\mathrm{I}\mathrm{U}}_{\mathrm{k}}$ | : | First flights of fleet type $\mathrm{k}\in \mathrm{K}$ |

${\rm Y}$ | : | Different allied airlines $\mathsf{\upsilon}\in {\rm Y}$ |

P | : | Codeshare agreements $\mathrm{p}\in \mathrm{P}$ |

Parameters | ||

$\left[{\mathrm{T}}_{\mathrm{i}}^{\mathrm{m}\mathrm{i}\mathrm{n}},{\mathrm{T}}_{\mathrm{i}}^{\mathrm{m}\mathrm{a}\mathrm{x}}\right]$ | : | Earliest and latest departure time of flight i $\mathrm{i}\in \mathrm{F}$ |

${\mathrm{p}\mathrm{r}}_{\mathrm{s}}$ | : | Probability of scenario s $\mathrm{s}\in \mathrm{S}$ |

${\mathrm{A}\mathrm{N}}_{\mathrm{k}}$ | : | Number of current aircraft of fleet type k $\mathrm{k}\in \mathrm{K}$ |

${\mathrm{K}\mathrm{a}\mathrm{p}}_{\mathrm{k}}$ | : | Number of seats in fleet type k $\mathrm{k}\in \mathrm{K}$ |

${\mathrm{D}\mathrm{Y}}_{\mathrm{i}}$ | : | Opportunity cost of spilled passengers of flight i $\mathrm{i}\in \mathrm{F}$ |

${\mathrm{B}\mathrm{O}\mathrm{S}}_{\mathrm{k}}$ | : | Unit idle time cost of fleet type k (per minute) $\mathrm{k}\in \mathrm{K}$ |

${\mathrm{K}\mathrm{Y}}_{\mathrm{i}}$ | : | Cost per passenger for miss-connected passengers on flight i $\mathrm{i}\in \mathrm{F}$ |

${\mathrm{D}\mathrm{C}}_{\mathrm{i}}$ | : | Per minute delay cost of flight i $\mathrm{i}\in \mathrm{F}$ |

${\mathrm{f}\mathrm{a}\mathrm{r}\mathrm{e}}_{\mathrm{i}}$ | : | Flight revenue per passenger $\mathrm{i}\in \mathrm{F}$ |

${\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}}_{\mathrm{i}}$ | : | Flight operating cost per passenger $\mathrm{i}\in \mathrm{F}$ |

${\mathsf{\mu}}_{\mathrm{i}\mathrm{s}}$ | : | Demand of flight i in scenario s $\mathrm{s}\in \mathrm{S},\mathrm{i}\in \mathrm{F}$, stochastic |

${\mathsf{\eta}}_{\mathrm{i}\mathrm{s}}$ | : | Non-cruise time of flight i in scenario s $\mathrm{s}\in \mathrm{S},\mathrm{i}\in \mathrm{F}$, stochastic |

$\left[{\mathrm{C}\mathrm{T}}_{\mathrm{i}\mathrm{k}}^{\mathrm{l}},{\mathrm{C}\mathrm{T}}_{\mathrm{i}\mathrm{k}}^{\mathrm{u}}\right]$ | : | Time window for cruise time of flight i of fleet type k $\mathrm{i}\in \mathrm{F},\mathrm{k}\in \mathrm{K}$ |

${\mathrm{T}\mathrm{A}}_{\mathrm{i}\mathrm{k}}$ | : | Turnaround time required to prepare fleet type k after flight i $\mathrm{i}\in \mathrm{F},\mathrm{k}\in \mathrm{K}$ |

${\mathrm{C}\mathrm{P}}_{\mathrm{i}\mathrm{j}}$ | : | Transit time for connected passengers between flights i, j $\mathrm{i}\in \mathrm{J},\mathrm{j}\in {\mathrm{Y}\mathrm{B}}_{\mathrm{i}}$ |

${\mathrm{p}\mathrm{a}\mathrm{s}\mathrm{s}}_{\mathrm{i}\mathrm{j}}$ | : | Number of passengers from flight i connected to flight j $\mathrm{i}\in \mathrm{J},\mathrm{j}\in {\mathrm{Y}\mathrm{B}}_{\mathrm{i}}$ |

${\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}_{\mathsf{\upsilon}\mathrm{p}\mathrm{i}}$ | : | The cost per passenger in contract type p $\mathrm{p}\in \mathrm{P},\mathsf{\upsilon}\in {\rm Y},\mathrm{i}\in \mathrm{F}$ |

${\mathrm{C}\mathrm{a}\mathrm{p}}_{\mathsf{\upsilon}\mathrm{p}\mathrm{i}}$ | : | Capacity provided under contract type p on flight leg i, $\mathrm{p}\in \mathrm{P},\mathsf{\upsilon}\in {\rm Y},\mathrm{i}\in \mathrm{F}$ |

BUD | : | Available budget for codeshare agreements |

θ | : | The ratio of total codeshare flight capacities to the total capacity of the airline |

M | : | A big number |

Decision Variables | ||

${\mathrm{x}}_{\mathrm{i}\mathrm{j}\mathrm{k}}^{1}$ | : | $\left\{\begin{array}{c}1\hspace{1em}\hspace{1em}\mathrm{If}\mathrm{flights}\mathrm{j}\mathrm{comes}\mathrm{after}\mathrm{flight}\mathrm{i}\mathrm{and}\mathrm{both}\mathrm{are}\mathrm{performed}\mathrm{with}\mathrm{fleet}\mathrm{type}\mathrm{k}\hspace{1em}\hspace{1em}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{j}\in \mathrm{F},\mathrm{i}\in {\mathrm{O}}_{\mathrm{j}},k\in K\hfill \\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{otherwise}\hfill \end{array}\right.$ |

${\mathrm{f}}_{\mathrm{i}\mathrm{k}}^{1}$ | : | $\left\{\begin{array}{c}1\hspace{1em}\hspace{1em}\mathrm{If}\mathrm{flight}\mathrm{i}\mathrm{is}\mathrm{the}\mathrm{first}\mathrm{flight}\mathrm{with}\mathrm{fleet}\mathrm{type}\mathrm{k}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{i}\in \mathrm{F},\mathrm{k}\in K\hfill \\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{otherwise}\hfill \end{array}\right.$ |

${\mathrm{e}}_{\mathrm{i}\mathrm{k}}^{1}$ | : | $\left\{\begin{array}{c}1\hspace{1em}\hspace{1em}\mathrm{If}\mathrm{flight}\mathrm{i}\mathrm{is}\mathrm{the}\mathrm{last}\mathrm{flight}\mathrm{with}\mathrm{fleet}\mathrm{type}\mathrm{k}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{i}\in \mathrm{F},\mathrm{k}\in \mathrm{K}\hfill \\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{otherwise}\hfill \end{array}\right.$ |

${\mathrm{q}}_{\mathsf{\upsilon}\mathrm{p}\mathrm{i}}^{1}$ | : | $\left\{\begin{array}{c}1\hspace{1em}\hspace{1em}\mathrm{If}\mathrm{the}\mathrm{flight}\mathrm{leg}\mathrm{i}\mathrm{is}\mathrm{performed}\mathrm{with}\mathrm{the}\mathrm{aircraft}\mathrm{in}\mathrm{the}\mathrm{contract}\mathrm{p}\mathrm{of}\mathrm{company}\mathsf{\upsilon}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathsf{\upsilon}\in {\rm Y},\mathrm{i}\in \mathrm{F},\mathrm{p}\in \mathrm{P}\hfill \\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{otherwise}\hfill \end{array}\right.$ |

${\mathrm{a}}_{\mathrm{i}}^{2}$ | : | Announced departure time of flight i $\mathrm{i}\in \mathrm{F}$ |

${\mathrm{b}}_{\mathrm{i}\mathrm{s}}^{3}$ | : | Actual departure time of flight i in scenario s $\mathrm{i}\in \mathrm{F},\mathrm{s}\in \mathrm{S}$ |

${\mathrm{c}}_{\mathrm{i}\mathrm{s}}^{3}$ | : | Actual arrival time of flight i in scenario s $\mathrm{i}\in \mathrm{F},\mathrm{s}\in \mathrm{S}$ |

${\mathrm{d}}_{\mathrm{i}\mathrm{k}\mathrm{s}}^{3}$ | : | Cruise time of flight i with fleet type k in scenario s $\mathrm{i}\in \mathrm{F},\mathrm{s}\in \mathrm{S},\mathrm{k}\in \mathrm{K}$ |

${\mathrm{I}\mathrm{T}}_{\mathrm{i}\mathrm{k}\mathrm{s}}^{3}$ | : | Idle time of flight i with fleet type k in scenario s $\mathrm{i}\in \mathrm{F},\mathrm{s}\in \mathrm{S},\mathrm{k}\in \mathrm{K}$ |

${\mathrm{d}\mathrm{e}\mathrm{l}}_{\mathrm{i}\mathrm{s}}^{3}$ | : | Delay time of flight i in scenario s $\mathrm{i}\in \mathrm{F},\mathrm{s}\in \mathrm{S}$ |

${\mathsf{\pi}}_{\mathrm{i}\mathrm{s}}^{3}$ | : | Number of accepted passengers in flight leg i under scenario s $\mathrm{i}\in \mathrm{F},\mathrm{s}\in \mathrm{S}$ |

${\mathsf{\pi}\mathrm{c}}_{\mathrm{i}\mathrm{s}}^{3}$ | : | Number of accepted passengers in flight leg i under scenario s in codeshare $\mathrm{i}\in \mathrm{F},\mathrm{s}\in \mathrm{S}$ |

${\mathrm{w}}_{\mathrm{i}\mathrm{j}\mathrm{s}}^{3}$ | : | $\left\{\begin{array}{c}1\hspace{1em}\hspace{1em}\mathrm{In}\mathrm{scenario}\mathrm{s},\mathrm{if}\mathrm{passengers}\mathrm{on}\mathrm{flight}\mathrm{i}\mathrm{miss}\mathrm{flight}\mathrm{j}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{i}\in \mathrm{J},\mathrm{j}\in {YB}_{\mathrm{i}},\mathrm{s}\in \mathrm{S},\hfill \\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{otherwise}\hfill \end{array}\right.$ |

## 4. Solution Methodology

#### 4.1. Simulated Annealing

Algorithm 1: SA + MC Algorithm |

Input flight informations Create initial solution ${s}_{0}\in S:$ Create feasible routes for flights Generate random values $\left({\mathsf{\mu}}_{\mathrm{i}\mathrm{s}},{\mathsf{\eta}}_{\mathrm{i}\mathrm{s}}\right)$ from appropriate distributions for scenarios Calculate the expected demand value for each flight Assign capacity airplanes that match these demand values Calculate the objective function value $f\left({s}_{0}\right)$ with short Monte Carlo simulation of the initial solution Set an initial temperature: $T>0;$ $s={s}_{0};f\left(s\right)=f\left({s}_{0}\right);$ ${s}_{iyi}={s}_{0};f\left({s}_{iyi}\right)=f\left({s}_{0}\right);$ Repeat$\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}=1$ RepeatGenerate a neighbor solution ${s}^{\prime}$ of S Create routes based on new assignment Calculate the objective function value $f\left({s}^{\prime}\right)$ with a short Monte Carlo simulation of ${s}^{\prime}$ solution $\Delta =f\left({s}^{\prime}\right)-f\left(s\right);$ If $\Delta \ge 0$ then $s={s}^{\prime};$ If $\Delta <0$ then generate a random number u in the range (0,1) If $u<{e}^{\frac{\Delta}{T}}$ then $s={s}^{\prime};$ If $f\left({s}^{\prime}\right)<f\left({s}_{iyi}\right)$ then ${s}_{iyi}={s}^{\prime}$ $\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}=\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}+1;$ Until $count>M$ (number of neighbors to search)$i=i+1;$ $T\left(i\right)=\alpha \ast T\left(i-1\right);$ Until (until the stopping criteria is met)${s}_{iyi}$ best solution Calculate objective function value f(s_iyi) with long Monte Carlo simulation |

Algorithm 2: Neighbor Solution Search Algorithm |

$\mathrm{f}\mathrm{l}\mathrm{t}=0,$ y, generate a random value between 0–1 If $y\le 0.3$ then $\mathrm{For}(i=0;i3;i++)$ Assign fleet type assigned to randomly selected flight in the same solution to flight $(flt+i)$ in solution End for If not For $(i=0;i3;i++)$ Assign random fleet type to flight $(flt+i)$ in solution End for End if $flt=flt+1$ |

#### 4.2. Cuckoo Search Algorithm (CS)

Algorithm 3: CS + MC Algorithm |

Input flight informations Generate P initial populations ${x}_{i}$ (i = 1, …, P) Create feasible routes for flights Generate random values $\left({\mathsf{\mu}}_{\mathrm{i}\mathrm{s}},{\mathsf{\eta}}_{\mathrm{i}\mathrm{s}}\right)$ from appropriate distributions for scenarios Calculate the expected demand value for each flight Assign capacity airplanes that match these demand values Calculate the objective function value $f\left({x}_{i}\right)$ with short Monte carlo simulation of ${x}_{i}$ solution While $(t<Maximumiteration)$ doTake a random cuckoo ${x}_{i}$ Local search ${x}_{i}^{\prime}$ and $f\left({x}_{i}^{\prime}\right)$ Randomly select a nest (let it be ${x}_{j}$) in P nests If $(f\left({x}_{i}^{\prime}\right)>f\left({x}_{j}\right))$ thenReplace ${x}_{j}$ with new solution end ifAbandon the worst nests with α ratio of ${p}_{\alpha}$ and create new ones Create routes for flights Assign a random fleet type to each route Calculate the objective function value with a short Monte Carlo simulation of each new solution Keep best solutions (nests with quality solution) Sort the solutions and find the current best solution end whileCalculate objective function value with long Monte Carlo simulation of best solution |

Algorithm 4: Local Search Algorithm |

For $(fl=0;flflightnumber-2;fl++)$ y, generate a random value between 0–1 If $y\le 0.3$ then For $(i=0;i3;i++)$ Assign fleet type assigned to randomly selected flight in the same solution to flight $(fl+i)$ in solution ${x}_{i}$ Create routes based on new assignment Calculate the objective function value $f\left({x}_{i}^{\prime}\right)$ with short Monte Carlo simulation of new solution $\left({x}_{i}^{\prime}\right)$ End for If not For $(i=0;i3;i++)$ Assign random fleet type to flight $(fl+i)$ in solution ${x}_{i}$ Create routes based on new assignment Calculate the objective function value $f\left({x}_{i}^{\prime}\right)$ with short Monte Carlo simulation of new solution $\left({x}_{i}^{\prime}\right)$ End for End if End for |

#### 4.3. Small Example

## 5. Computational Experiments

#### 5.1. Data Descriptions

- A 10 min tolerance period is added to the departure times.
- The turnaround time of each fleet type is multiplied by the complexity coefficient of the airport for each flight and fleet type to obtain the turnaround time for that flight.
- The connection time for the passengers is set at 30 min.
- The highest and lowest cruise times are calculated by subtracting 20 min from the total flight time. The lowest cruise times are then compressed by 15% of the highest cruise time.
- The profit from one passenger is defined as the basic spill cost of a lost passenger. The passenger spill cost for each flight has been calculated by multiplying the basic spill cost by the airport complexity coefficient.
- The cost of a missed connection is set at USD 50 per passenger.

#### 5.2. Computational Results

#### 5.3. Effect of Codeshare Flights

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Comparison of profits of algorithms (

**a**) for 30 flights, (

**b**) for 78 flights, (

**c**) for 127 flights, (

**d**) for 180 flights.

**Figure 5.**The effect of codesharing on profits with (

**a**) SA + MC simheuristic algorithm and (

**b**) CS + MC simheuristic algorithm.

Min. Departure Time | Max. Departure Time | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Flights (i) | Departure Airports | Arrival Airports | Departure Time | Arrival Time | ${\mathbf{T}}_{\mathbf{i}}^{\mathbf{m}\mathbf{i}\mathbf{n}}$ (min) | ${\mathbf{T}}_{\mathbf{i}}^{\mathbf{m}\mathbf{a}\mathbf{x}}$ (min) | ||||

1 | LAX | ATL | 08:00 | 12:30 | 470 | 490 | ||||

2 | ATL | LAX | 13:10 | 17:50 | 780 | 800 | ||||

3 | LAX | DFW | 18:40 | 21:55 | 1110 | 1130 | ||||

4 | LAX | MIA | 05:00 | 10:10 | 290 | 310 | ||||

5 | MIA | LAX | 11:30 | 16:50 | 680 | 700 | ||||

6 | LAX | OGG | 18:00 | 23:40 | 1070 | 1090 | ||||

SpilledPassengersCost | MisconnectedPassenger Cost | Aircraft Delay Cost | Flight Fare perPassenger | Flight Operating Cost per Passenger | ||||||

Flights (i) | ${\mathrm{D}\mathrm{Y}}_{\mathrm{i}}$ (USD) | ${\mathrm{K}\mathrm{Y}}_{\mathrm{i}}$ (USD) | ${\mathrm{D}\mathrm{C}}_{\mathrm{i}}$ (USD) | ${\mathrm{f}\mathrm{a}\mathrm{r}\mathrm{e}}_{\mathrm{i}}$ (USD) | ${\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}}_{\mathrm{i}}$ (USD) | |||||

1 | 102 | 50 | 4.85 | 255 | 153 | |||||

2 | 102 | 50 | 7.03 | 255 | 153 | |||||

3 | 88 | 50 | 4.29 | 220 | 132 | |||||

4 | 163.6 | 50 | 3.84 | 409 | 245.4 | |||||

5 | 124 | 50 | 7.21 | 310 | 186 | |||||

6 | 89.6 | 50 | 5.51 | 224 | 134 | |||||

Codesharing Costs per Passenger | Codesharing Capacity | Min. Cruise Time | Max. Cruise Time | Turnaround Time | ||||||

${\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}_{\mathsf{\upsilon}\mathrm{p}\mathrm{i}}$ (USD) | ${\mathrm{C}\mathrm{a}\mathrm{p}}_{\mathsf{\upsilon}\mathrm{p}\mathrm{i}}$ | ${\mathrm{C}\mathrm{T}}_{\mathrm{i}\mathrm{k}}^{\mathrm{l}}$ (min) | ${\mathrm{C}\mathrm{T}}_{\mathrm{i}\mathrm{k}}^{\mathrm{u}}$ (min) | ${\mathrm{T}\mathrm{A}}_{\mathrm{i}\mathrm{k}}$ (min) | ||||||

Flights (i) | p = 1 | p = 2 | p = 1 | p = 2 | k = 1 | k = 2 | k = 1 | k = 2 | k = 1 | k = 2 |

1 | 76.5 | 65.03 | 80 | 110 | 213 | 213 | 250 | 250 | 42 | 39 |

2 | 77 | 65 | 80 | 110 | 221 | 221 | 260 | 260 | 46 | 43 |

3 | 66 | 56.1 | 80 | 110 | 149 | 149 | 175 | 175 | 44 | 42 |

4 | 122.7 | 104.3 | 80 | 110 | 247 | 247 | 290 | 290 | 50 | 47 |

5 | 93 | 79.05 | 80 | 110 | 255 | 255 | 300 | 300 | 46 | 43 |

6 | 67.2 | 57.12 | 80 | 110 | 272 | 272 | 320 | 320 | 26 | 25 |

Number of Aircrafts | Capacity of Fleet Type | Aircraft Idle Time Costs | |
---|---|---|---|

Fleet type (k) | ${\mathrm{A}\mathrm{N}}_{\mathrm{k}}$ | ${\mathrm{K}\mathrm{a}\mathrm{p}}_{\mathrm{k}}$ | ${\mathrm{B}\mathrm{O}\mathrm{S}}_{\mathrm{k}}$ (USD/min) |

1 | 1 | 172 | 40 |

2 | 1 | 187 | 55 |

GAMs/BARON | SA + MC | CS + MC | |
---|---|---|---|

Objective function values (USD) | 89,422.62 | 89,422.62 | 89,422.62 |

Assignments | 1-codeshare 4-A321-200 2-codeshare 5-A321-200 3-codeshare 6-A321-200 | ||

Route | 4-5-6 | ||

Stations | Codeshare: LAX-ATL Codeshare: ATL-LAX Codeshare: LAX-DFW Route: LAX-MIA-LAX-OGG |

Flights | Actual Departure Time | Actual Arrival Time | Cruise Time (min) | İdle Time (min) | Delay Time (min) | Non-Cruise Time (min) | Demand | Passenger Number | |
---|---|---|---|---|---|---|---|---|---|

First scenario results | 1 | 147 | 110 | ||||||

2 | 74 | 74 | |||||||

3 | 220 | 110 | |||||||

4 | 05:44 | 10:58 | 290 | 0 | 34 | 24 | 129 | 129 | |

5 | 11:45 | 17:07 | 300 | 0 | 5 | 22 | 201 | 187 | |

6 | 17:50 | 22:46 | 272 | 0 | 24 | 172 | 172 | ||

Second scenario results | 1 | 83 | 83 | ||||||

2 | 136 | 110 | |||||||

3 | 65 | 65 | |||||||

4 | 05:45 | 10:57 | 290 | 0 | 35 | 22 | 138 | 138 | |

5 | 11:44 | 17:07 | 300 | 0 | 4 | 23 | 159 | 159 | |

6 | 17:50 | 22:43 | 272 | 0 | 21 | 170 | 170 |

Fleet Type | B737-800 | A321-200 | A319-100 | A321-NEO | B787-8 | ERJ-175 |
---|---|---|---|---|---|---|

Capacity | 172 | 187 | 128 | 196 | 234 | 76 |

Base Turntime | 25.47 | 25.47 | 23.87 | 31.05 | 42.39 | 16.47 |

Idle Time Costs (USD/min) | 140 | 142 | 136 | 144 | 147 | 125 |

SA + MC | CS + MC | ||||||||
---|---|---|---|---|---|---|---|---|---|

Problems | Scenario Numbers | Flight Numbers | Aircraft Numbers | Obj. Function Values (USD) | Code Flight Numbers | Times (s) | Obj. Function Values (USD) | Code Flight Numbers | Times (s) |

1 | 100 | 30 | 6 | 319,314.3 | 0 | 209.26 | 322,102.7 | 0 | 216.01 |

2 | 100 | 78 | 14 | 767,161.46 | 25 | 665.96 | 874,041.44 | 2 | 758.32 |

3 | 100 | 127 | 20 | 1,327,546.52 | 28 | 1467.69 | 1,436,292.3 | 12 | 2406.63 |

4 | 100 | 180 | 25 | 1,731,738.91 | 56 | 2097.55 | 1,947,504.96 | 12 | 3432.55 |

SA + MC | CS + MC | |||||
---|---|---|---|---|---|---|

Run | Profit (USD) | Route Number | Deadhead Flights | Profit (USD) | Route Number | Deadhead Flights |

1 | 1,270,180.8 | 44 | 21 | 1,605,883.39 | 46 | 14 |

2 | 1,048,609.36 | 44 | 21 | 1,553,091.89 | 45 | 15 |

3 | 1,138,399.14 | 44 | 21 | 1,572,453.71 | 46 | 10 |

4 | 1,000,286.11 | 45 | 17 | 1,477,181.86 | 46 | 12 |

5 | 1,240,760.95 | 45 | 17 | 1,618,113.63 | 45 | 17 |

6 | 1,282,708.31 | 44 | 20 | 1,479,553.91 | 45 | 15 |

7 | 1,183,632.79 | 45 | 13 | 1,542,911.92 | 46 | 10 |

8 | 1,173,466.25 | 43 | 26 | 1,623,962.93 | 46 | 10 |

9 | 1,015,405.99 | 46 | 10 | 1,485,155.01 | 46 | 10 |

10 | 1,137,581.98 | 43 | 25 | 1,591,569.03 | 46 | 10 |

SA + MC | CS + MC | |||||
---|---|---|---|---|---|---|

Run | Profit (USD) | Route Number | Codeshare Flights | Profit (USD) | Route Number | Codeshare Flights |

1 | 1,641,346.89 | 38 | 52 | 1,616,784.05 | 44 | 24 |

2 | 1,561,180.42 | 36 | 62 | 1,885,131.11 | 42 | 28 |

3 | 1,605,142.62 | 36 | 62 | 1,807,812.28 | 43 | 30 |

4 | 1,626,210.07 | 36 | 54 | 1,912,243.16 | 45 | 14 |

5 | 1,731,738.91 | 37 | 56 | 1,825,287.15 | 40 | 37 |

6 | 1,659,333.69 | 39 | 50 | 1,947,504.96 | 46 | 12 |

7 | 1,548,412.45 | 36 | 54 | 1,812,634.21 | 43 | 33 |

8 | 1,571,768.08 | 37 | 55 | 1,815,737.61 | 43 | 26 |

9 | 1,591,618.95 | 36 | 57 | 1,819,234.24 | 44 | 20 |

10 | 1,501,147.29 | 35 | 64 | 1,870,894.85 | 42 | 28 |

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**MDPI and ACS Style**

Kızıloğlu, K.; Sakallı, Ü.S.
Integrating Flight Scheduling, Fleet Assignment, and Aircraft Routing Problems with Codesharing Agreements under Stochastic Environment. *Aerospace* **2023**, *10*, 1031.
https://doi.org/10.3390/aerospace10121031

**AMA Style**

Kızıloğlu K, Sakallı ÜS.
Integrating Flight Scheduling, Fleet Assignment, and Aircraft Routing Problems with Codesharing Agreements under Stochastic Environment. *Aerospace*. 2023; 10(12):1031.
https://doi.org/10.3390/aerospace10121031

**Chicago/Turabian Style**

Kızıloğlu, Kübra, and Ümit Sami Sakallı.
2023. "Integrating Flight Scheduling, Fleet Assignment, and Aircraft Routing Problems with Codesharing Agreements under Stochastic Environment" *Aerospace* 10, no. 12: 1031.
https://doi.org/10.3390/aerospace10121031