- Email: [email protected]
Scheduling the sequence of aircraft landings for a single runway using a fuzzy programming approach
Journal of Air Transport Management 25 (2012) 15e18
Contents lists available at SciVerse ScienceDirect
Journal of Air Transport Management journal homepage: www.elsevier.com/locate/jairtraman
Note
Scheduling the sequence of aircraft landings for a single runway using a fuzzy programming approach Reza Tavakkoli-Moghaddam*, Mojtaba Yaghoubi-Panah, Farzad Radmehr University of Tehran, Tehran 11155-4563, Iran
a b s t r a c t Keywords: Runway use Aircraft landing sequencing Fuzzy programming approach
This paper examines the ways of landing aircraft with the least waiting time in time windows under critical conditions, such as the closest time of landing to the target times for each aircraft or the minimum time of landing the planes. Therefore, we face with two conflict objectives, namely minimizing the total cost of the deviation from the target times and minimizing the completion time of the landing sequence. To solve such a problem, we use a fuzzy programming approach and an estimator for landing the sequence of planes. The results are compared with actual landings. 2012 Elsevier Ltd. All rights reserved.
1. Introduction Before landing, an aircraft must go through an approach stage directed by air traffic controllers. When entering the airport radar range, the aircraft’s flight number, altitude and speed are transmitted to the air traffic control tower. Based on this information, controllers give instructions regarding the approach corridor to use and to the speed and altitude of the aircraft to align it with the allocated runway. During peak hours, controllers handle the landings of a continuous flow of aircraft entering their radar range. The capacity of runways is constrained making the scheduling of landings difficult. Because of environmental, political and geographical constraints, physical capacity cannot easily be increased, hence requiring improved decision support tools if more traffic is to be handled. The immediate problem confronting controllers is deciding which aircraft to land next and when. When an aircraft enters radar range, a target landing time is defined in terms of the time an aircraft could land flying straight to the runway at its cruising speed. This target is bounded by an earliest and a latest landing time, with the former being as the time the aircraft could land flying directly to the runway at its fastest speed with no holding, and the latest landing time by the time it could land if it is held for its maximum permitted time before landing. The time between an aircraft’s earliest and latest time is the time window. Hence, scheduled landing times as decided by air traffic controllers must lay within aircraft time windows. To maintain an aircraft’s aerodynamic stability,
* Corresponding author. Tel.: þ98 21 82084183; fax: þ98 21 88013102. E-mail address: [email protected] (R. Tavakkoli-Moghaddam). 0969-6997/$ e see front matter 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jairtraman.2012.03.004
separation distances based on the preceding aircraft types must be respected during landing.1 These separation times are the main limiting factors on runway usage and these constraints mainly apply for aircraft landing on the same runway. If several runways are available for landing the application of such constraints for aircraft landing on different runways depends upon the relative positions of the runways. In this paper we consider the static, or off-line, version of the problem, where the set of aircraft that are waiting to land is known. This contrasts with the dynamic, or on-line, case where decisions about each aircraft must be made as time passes, aircraft land and new aircraft appear (Tang et al., 2008). 2. The problem One of the issues involved is to determine the timing for landing aircraft within an airline’s scheduling as early as while simultaneously minimizing the completion time of the entire landing sequence. We look at this problem assuming; the aircraft waiting to land is known, that there is only one runway with one plane landing and clearing the runway at a time. We make use of the following notation: jxi landing time for aircraft i i ði˛PÞ ai how soon aircraft i lands before Ti ði˛PÞ bi how late aircraft i lands after Ti ði˛PÞ gi unit costs for aircraft i landing earlier than the target time
1 Here separation distances are dealt with by converting them into separation times using a fixed landing speed depending on the aircraft type.
16
R. Tavakkoli-Moghaddam et al. / Journal of Air Transport Management 25 (2012) 15e18
hi unit costs for aircraft i landing later than the target time Sij required separation time between aircraft i and j, where i lands before j Ti target time for aircraft i Ei earliest possible time of landing aircraft i Li latest possible time of landing aircraft i dij 1 if aircraft i lands before, 0 otherwise Cmax completion time ½Ei ; Lj predetermined landing time window for aircraft i. We consider two possibility, that, first, the landing time of planes is essentially fixed and second, that this is not the case and each plane has to be treated separately so as to minimize the movements over the entire landing bank. The objective is to minimizing the weighted sum of deviations of landing time from the target time; in other words, aircraft should land close to the target time.
min Z ¼
P X
ðgi ai þ hi bi Þ
(1)
ci˛Ps
(2)
i¼1
s.t.
Ei xi Li
xj xi þ Sij Li þ Sij Ej dji
ci; j
(3)
ai Ti xi ci˛Ps
(4)
0 ai Ti Ei
(5)
ci˛Ps
bi xi Ti ci˛Ps
(6)
0 bi Li Ti
(7)
xi ¼ Ti ai þ bi dij þ dji ¼ 1 dii ; Sii ¼ 0
ci˛Ps ci˛Ps
ci; j
(8)
isj
(9)
ci
(10)
xi 0 and dij ˛f0; 1g
(11)
xi ; ai ; bi 0
(12)
ci˛Ps
Constraint 2 ensures that each plane appearing in Ps lands within its time window. Constraint 3 is the time of landing each plane. Constraints 4 and 5 ensure that ai is at least zero and the time difference between Ti and xi, and at most the time difference between Ti and Ei. Constraints 6 and 7 are similar equations for bi. Eq. (8) relates the landing time (xi) to the time plane i lands before (ai), or after (bi) the target time (Ti), Eq. (9) ensures that a plane can land in each sequence. Eqs. (10) and (11) are required constraints. Unlike the first model, the second minimizes the overall time over which the planes land; this can cause an increase in deviations between the landing times and target times for aircraft.
min Z ¼ Cmax
dij þ dji ¼ 1 dii ; Sii ¼ 0
ci; j
isj
ci
(18)
xi 0 and dij ˛f0; 1g
(19)
Eq. (13) is the objective function minimizing the costs of landing planes. Eq. (14) satisfies the landing time frame for plane i with earliest and latest time of landing. Eq. (15) expresses the time of landing each respective plans. Eq. (16) ensures that the landing time should be lower than the completion time. Eq. (17) ensures that the plane can land one time in each sequence. Eqs. (18) and (19) are required for estimation. 3. Fuzzy goal programming When vague information related to the objective functions exists and there is inherent uncertainty in the relative importance of goals, the problem can be formulated as a fuzzy goalprogramming problem. The time of aircraft landing is treated as a concept of fuzzy. Fuzzy goal programming is a methodology that allows analysis when less than full information is available (Bellman and Zadeh, 1970) A typical fuzzy mixed-integer goal programming problem is: Find Xi. s.t.
~ Zi Xi yZ l
cl
hj ðXi Þ dj
cj
Sk ðXi Þ ¼ Ck
(20)
ck
Xi 0 and integer
ci
where; Z1 ðXi Þ is the lth objective function. jj ðXi Þ is the jth left hand side (LHS) of the inequality constraint. Sk ðXi Þ is the kth LHS of the equality constraint. Z~ l is the target value of lth objective function. dj is the available resource of the jth equality constraint. Ck is the available resource of the kth equality constraint. The symbol y in Eq. (20) indicates the fuzziness of the goal; it reflects the linguistic term ‘about’ and it means that Zl (Xi) should
(13)
s.t:
Ei xi Li
ci˛Ps
xj xi þ Sij Li þ Sij Ej dij Cmax xi þ Sij
ci
(14) ci; j
(15) (16)
(17)
Fig. 1. Membership function related to objectives.
R. Tavakkoli-Moghaddam et al. / Journal of Air Transport Management 25 (2012) 15e18
Step 1: Construct the following f-MIGP for the multi-objective aircraft landing sequencing formulation:
Table 1 Obtained results for small-sized problems. Planes
Z1*
Z2*
a
Sequence of landing
3 4 5 6 7 8 9
26 50 90 146 182 246 310
5 7 9 11 13 15 17
1 1 1 1 1 0.96 0.9
3e2e1 4e3e2e1 3e4e2e1e5 3e4e2e1e6e5 7e3e4e2e1e6e5 7e3e2e4e8e1e6e5 7e3e4e9e2e1e8e6e5
be in the vicinity of the aspiration Z l . The lth fuzzy goal signifies that the decision maker will be satisfied even for values Zl (Xi) y Z l , slightly greater than (or less than) up to a stated deviations signified by the tolerance limit. The jth system constraint hj (Xi) dj and kth system constraint Sk (Xi) ¼ Ck are assumed to be crisp. Definition: a fuzzy set A in X is defined by A ¼ {x, uA (x) jx ˛ X} where x, uA (x):X / [0, 1] is called the membership function of A, and uA(x) is the degree of membership to which x belongs to A. Let pl be the maximum tolerance limit to gl determine by the decision-maker. Thus, using the concept of fuzzy sets, the membership function of the objective functions can be represented as (Javadi et al., 2008):
8 1 > > < Z ðX gl Þ mZI ðXÞ ¼ 1 I i > pl > : 0
n
P P i¼1
gi ai hi bi yZ1
Cmax yZ2 s.t.Constraits: 2e12,14e19 Step 2: Solve the lth objective function with an optimization technique and set gl to the objective function value of the minimum solution according Fig. 1. Step 3: Determine the values of the other objective functions of the sequence in obtained the previous step and set pl equal to the maximum among the values obtained; gl Step 4: Repeat Steps 2 and 3 for all objective functions. Step 5: Define the membership function of each fuzzy goal in the f-MIGP. Step 6: Construct the c-MIGP (i.e., crisp mixed-integer goal programming problem) formulation of the f-MIGP (Yager, 1977): max a s.t.
a mZi ðXÞ if Zl ðXi Þ < gl
hj ðXi Þ dj
if gl Zl ðXÞ gl þ pl
(21)
if Zl ðXÞ gl þ pl
Sk ðXi Þ ¼ Ck
cj ck
Xi 0; integer
The term mZ ðXÞ indicates the desirability of DM to solution X in term of the lth objective and the decision space is defined as the intersection of a membership function of objectives (Eq. (20)) and system constraints as:
Z* ¼
17
o XLl¼1 Zl* XG; 0 a 1
where G represents system constraints, or feasible space, and a is the aspiration level which indicates that mZ ðXÞ is a nonnegative real number whose value is finite, and is usually in the interval of [0,1]. Also the membership function of Z is described as (Yager, 1977):
mZ ðXÞ ¼ minfmZl ðxÞg l¼1
To solve the f-MIGP for the aircraft landing a sequencing approach is used:
(22)
ci
4. Results The effectiveness of the FGP approach designed for the multiobjective aircraft landing sequencing problem is examined using a set of experiments for small and large-sized problems. The first experiment is carried out on a set of small-sized problems involving three to nine planes. Table 1 gives the results for the seven test problems, increasing the number of planes from three to nine. As the table shows the degree of satisfaction is unity with three to seven planes; but by increasing the number of waiting planes, the problem can be more complicated and the degree of satisfaction is less than unity. Our large-sized problem is solved for 10e20 planes with the fuzzy goal programming based algorithm is applied to each of them. Table 2 gives the results for these seven test problems. Compared with the small-sized problem, the degree of satisfaction again generally decreases with increasing the number of
Table 2 Obtained results for large-sized problems. Planes
Z1*
Z2*
a
Sequence of landing
10 11 12 13 14 15 16 17 18 19 20
386 472 540 628 712 826 898 994 1082 1183 1278
19 21 23 25 27 29 31 33 35 39 41
0.67 0.92 0.71 0.72 0.65 0.73 0.97 0.76 0.69 0.64 0.67
9e7e4e8e3e1e10e2e6e5 4e9e7e10e3e2e8e1e5e11e6 4e12e3e9e10e2e1e6e8e7e5e11 12e7e3e11e13e4e1e8e10e9e5e2e6 12e13e14e1e7e3e9e2e8e10e6e4e11e5 7e10e12e5e3e13e9e4e14e15e1e2e8e11e6 14e12e3e2e13e7e15e9e16e10e8e1e11e4e6e5 12e15e16e11e7e17e14e8e10e3e1e2e4e13e9e5e6 15e7e12e18e13e16e11e17e4e8e14e10e9e2e5e1e6e3 4e19e18e16e15e7e9e12e17e13e10e11e2e14e3e6e8e1e5 19e14e15e17e12e18e8e16e4e2e9e13e3e10e7e1e6e11e20e5
18
R. Tavakkoli-Moghaddam et al. / Journal of Air Transport Management 25 (2012) 15e18
planes but the descending order in a is not smooth and sometimes rises when additional planes are added. Therefore, increasing the number planes can means a better situation over a range of additions. 5. Conclusion We have considered the sequence of landing planes when minimizing the sum of the costs of deviation from the target times and minimizing the completion time of the landing sequence using a fuzzy programming approach. We have also presented an estimator for comparing and analyzing the answerers and finally presented the experimental results for small and large-sized problems with three to 20 waiting aircrafts.
Finally, we have analyzed the related results while increasing the waiting planes.
References Bellman, R.E., Zadeh, L.A., 1970. Decision making in a fuzzy environment. Mngm. Sci. 17, 141e164. Javadi, B., Rahimi-Vahed, A., Rabbani, M., Dangchi, M., 2008. Solving a multiobjective mixed-model assembly line sequencing problem by a fuzzy goal programming approach. Int. J. Adv. Manuf. Tech. 39, 975e982. Tang, K., Wang, Z., Cao, X., Zhang, J., 2008. A multi-objective evolutionary approach to aircraft landing scheduling problems. In: Proceedings of the IEEE Congress on Evolutionary Computation, pp. 3651e3657. Yager, R.R., 1977. Multiple objective decision-making using fuzzy sets. Int. J. Man-Machine Studies 9, 375e382.
Contents lists available at SciVerse ScienceDirect
Journal of Air Transport Management journal homepage: www.elsevier.com/locate/jairtraman
Note
Scheduling the sequence of aircraft landings for a single runway using a fuzzy programming approach Reza Tavakkoli-Moghaddam*, Mojtaba Yaghoubi-Panah, Farzad Radmehr University of Tehran, Tehran 11155-4563, Iran
a b s t r a c t Keywords: Runway use Aircraft landing sequencing Fuzzy programming approach
This paper examines the ways of landing aircraft with the least waiting time in time windows under critical conditions, such as the closest time of landing to the target times for each aircraft or the minimum time of landing the planes. Therefore, we face with two conflict objectives, namely minimizing the total cost of the deviation from the target times and minimizing the completion time of the landing sequence. To solve such a problem, we use a fuzzy programming approach and an estimator for landing the sequence of planes. The results are compared with actual landings. 2012 Elsevier Ltd. All rights reserved.
1. Introduction Before landing, an aircraft must go through an approach stage directed by air traffic controllers. When entering the airport radar range, the aircraft’s flight number, altitude and speed are transmitted to the air traffic control tower. Based on this information, controllers give instructions regarding the approach corridor to use and to the speed and altitude of the aircraft to align it with the allocated runway. During peak hours, controllers handle the landings of a continuous flow of aircraft entering their radar range. The capacity of runways is constrained making the scheduling of landings difficult. Because of environmental, political and geographical constraints, physical capacity cannot easily be increased, hence requiring improved decision support tools if more traffic is to be handled. The immediate problem confronting controllers is deciding which aircraft to land next and when. When an aircraft enters radar range, a target landing time is defined in terms of the time an aircraft could land flying straight to the runway at its cruising speed. This target is bounded by an earliest and a latest landing time, with the former being as the time the aircraft could land flying directly to the runway at its fastest speed with no holding, and the latest landing time by the time it could land if it is held for its maximum permitted time before landing. The time between an aircraft’s earliest and latest time is the time window. Hence, scheduled landing times as decided by air traffic controllers must lay within aircraft time windows. To maintain an aircraft’s aerodynamic stability,
* Corresponding author. Tel.: þ98 21 82084183; fax: þ98 21 88013102. E-mail address: [email protected] (R. Tavakkoli-Moghaddam). 0969-6997/$ e see front matter 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jairtraman.2012.03.004
separation distances based on the preceding aircraft types must be respected during landing.1 These separation times are the main limiting factors on runway usage and these constraints mainly apply for aircraft landing on the same runway. If several runways are available for landing the application of such constraints for aircraft landing on different runways depends upon the relative positions of the runways. In this paper we consider the static, or off-line, version of the problem, where the set of aircraft that are waiting to land is known. This contrasts with the dynamic, or on-line, case where decisions about each aircraft must be made as time passes, aircraft land and new aircraft appear (Tang et al., 2008). 2. The problem One of the issues involved is to determine the timing for landing aircraft within an airline’s scheduling as early as while simultaneously minimizing the completion time of the entire landing sequence. We look at this problem assuming; the aircraft waiting to land is known, that there is only one runway with one plane landing and clearing the runway at a time. We make use of the following notation: jxi landing time for aircraft i i ði˛PÞ ai how soon aircraft i lands before Ti ði˛PÞ bi how late aircraft i lands after Ti ði˛PÞ gi unit costs for aircraft i landing earlier than the target time
1 Here separation distances are dealt with by converting them into separation times using a fixed landing speed depending on the aircraft type.
16
R. Tavakkoli-Moghaddam et al. / Journal of Air Transport Management 25 (2012) 15e18
hi unit costs for aircraft i landing later than the target time Sij required separation time between aircraft i and j, where i lands before j Ti target time for aircraft i Ei earliest possible time of landing aircraft i Li latest possible time of landing aircraft i dij 1 if aircraft i lands before, 0 otherwise Cmax completion time ½Ei ; Lj predetermined landing time window for aircraft i. We consider two possibility, that, first, the landing time of planes is essentially fixed and second, that this is not the case and each plane has to be treated separately so as to minimize the movements over the entire landing bank. The objective is to minimizing the weighted sum of deviations of landing time from the target time; in other words, aircraft should land close to the target time.
min Z ¼
P X
ðgi ai þ hi bi Þ
(1)
ci˛Ps
(2)
i¼1
s.t.
Ei xi Li
xj xi þ Sij Li þ Sij Ej dji
ci; j
(3)
ai Ti xi ci˛Ps
(4)
0 ai Ti Ei
(5)
ci˛Ps
bi xi Ti ci˛Ps
(6)
0 bi Li Ti
(7)
xi ¼ Ti ai þ bi dij þ dji ¼ 1 dii ; Sii ¼ 0
ci˛Ps ci˛Ps
ci; j
(8)
isj
(9)
ci
(10)
xi 0 and dij ˛f0; 1g
(11)
xi ; ai ; bi 0
(12)
ci˛Ps
Constraint 2 ensures that each plane appearing in Ps lands within its time window. Constraint 3 is the time of landing each plane. Constraints 4 and 5 ensure that ai is at least zero and the time difference between Ti and xi, and at most the time difference between Ti and Ei. Constraints 6 and 7 are similar equations for bi. Eq. (8) relates the landing time (xi) to the time plane i lands before (ai), or after (bi) the target time (Ti), Eq. (9) ensures that a plane can land in each sequence. Eqs. (10) and (11) are required constraints. Unlike the first model, the second minimizes the overall time over which the planes land; this can cause an increase in deviations between the landing times and target times for aircraft.
min Z ¼ Cmax
dij þ dji ¼ 1 dii ; Sii ¼ 0
ci; j
isj
ci
(18)
xi 0 and dij ˛f0; 1g
(19)
Eq. (13) is the objective function minimizing the costs of landing planes. Eq. (14) satisfies the landing time frame for plane i with earliest and latest time of landing. Eq. (15) expresses the time of landing each respective plans. Eq. (16) ensures that the landing time should be lower than the completion time. Eq. (17) ensures that the plane can land one time in each sequence. Eqs. (18) and (19) are required for estimation. 3. Fuzzy goal programming When vague information related to the objective functions exists and there is inherent uncertainty in the relative importance of goals, the problem can be formulated as a fuzzy goalprogramming problem. The time of aircraft landing is treated as a concept of fuzzy. Fuzzy goal programming is a methodology that allows analysis when less than full information is available (Bellman and Zadeh, 1970) A typical fuzzy mixed-integer goal programming problem is: Find Xi. s.t.
~ Zi Xi yZ l
cl
hj ðXi Þ dj
cj
Sk ðXi Þ ¼ Ck
(20)
ck
Xi 0 and integer
ci
where; Z1 ðXi Þ is the lth objective function. jj ðXi Þ is the jth left hand side (LHS) of the inequality constraint. Sk ðXi Þ is the kth LHS of the equality constraint. Z~ l is the target value of lth objective function. dj is the available resource of the jth equality constraint. Ck is the available resource of the kth equality constraint. The symbol y in Eq. (20) indicates the fuzziness of the goal; it reflects the linguistic term ‘about’ and it means that Zl (Xi) should
(13)
s.t:
Ei xi Li
ci˛Ps
xj xi þ Sij Li þ Sij Ej dij Cmax xi þ Sij
ci
(14) ci; j
(15) (16)
(17)
Fig. 1. Membership function related to objectives.
R. Tavakkoli-Moghaddam et al. / Journal of Air Transport Management 25 (2012) 15e18
Step 1: Construct the following f-MIGP for the multi-objective aircraft landing sequencing formulation:
Table 1 Obtained results for small-sized problems. Planes
Z1*
Z2*
a
Sequence of landing
3 4 5 6 7 8 9
26 50 90 146 182 246 310
5 7 9 11 13 15 17
1 1 1 1 1 0.96 0.9
3e2e1 4e3e2e1 3e4e2e1e5 3e4e2e1e6e5 7e3e4e2e1e6e5 7e3e2e4e8e1e6e5 7e3e4e9e2e1e8e6e5
be in the vicinity of the aspiration Z l . The lth fuzzy goal signifies that the decision maker will be satisfied even for values Zl (Xi) y Z l , slightly greater than (or less than) up to a stated deviations signified by the tolerance limit. The jth system constraint hj (Xi) dj and kth system constraint Sk (Xi) ¼ Ck are assumed to be crisp. Definition: a fuzzy set A in X is defined by A ¼ {x, uA (x) jx ˛ X} where x, uA (x):X / [0, 1] is called the membership function of A, and uA(x) is the degree of membership to which x belongs to A. Let pl be the maximum tolerance limit to gl determine by the decision-maker. Thus, using the concept of fuzzy sets, the membership function of the objective functions can be represented as (Javadi et al., 2008):
8 1 > > < Z ðX gl Þ mZI ðXÞ ¼ 1 I i > pl > : 0
n
P P i¼1
gi ai hi bi yZ1
Cmax yZ2 s.t.Constraits: 2e12,14e19 Step 2: Solve the lth objective function with an optimization technique and set gl to the objective function value of the minimum solution according Fig. 1. Step 3: Determine the values of the other objective functions of the sequence in obtained the previous step and set pl equal to the maximum among the values obtained; gl Step 4: Repeat Steps 2 and 3 for all objective functions. Step 5: Define the membership function of each fuzzy goal in the f-MIGP. Step 6: Construct the c-MIGP (i.e., crisp mixed-integer goal programming problem) formulation of the f-MIGP (Yager, 1977): max a s.t.
a mZi ðXÞ if Zl ðXi Þ < gl
hj ðXi Þ dj
if gl Zl ðXÞ gl þ pl
(21)
if Zl ðXÞ gl þ pl
Sk ðXi Þ ¼ Ck
cj ck
Xi 0; integer
The term mZ ðXÞ indicates the desirability of DM to solution X in term of the lth objective and the decision space is defined as the intersection of a membership function of objectives (Eq. (20)) and system constraints as:
Z* ¼
17
o XLl¼1 Zl* XG; 0 a 1
where G represents system constraints, or feasible space, and a is the aspiration level which indicates that mZ ðXÞ is a nonnegative real number whose value is finite, and is usually in the interval of [0,1]. Also the membership function of Z is described as (Yager, 1977):
mZ ðXÞ ¼ minfmZl ðxÞg l¼1
To solve the f-MIGP for the aircraft landing a sequencing approach is used:
(22)
ci
4. Results The effectiveness of the FGP approach designed for the multiobjective aircraft landing sequencing problem is examined using a set of experiments for small and large-sized problems. The first experiment is carried out on a set of small-sized problems involving three to nine planes. Table 1 gives the results for the seven test problems, increasing the number of planes from three to nine. As the table shows the degree of satisfaction is unity with three to seven planes; but by increasing the number of waiting planes, the problem can be more complicated and the degree of satisfaction is less than unity. Our large-sized problem is solved for 10e20 planes with the fuzzy goal programming based algorithm is applied to each of them. Table 2 gives the results for these seven test problems. Compared with the small-sized problem, the degree of satisfaction again generally decreases with increasing the number of
Table 2 Obtained results for large-sized problems. Planes
Z1*
Z2*
a
Sequence of landing
10 11 12 13 14 15 16 17 18 19 20
386 472 540 628 712 826 898 994 1082 1183 1278
19 21 23 25 27 29 31 33 35 39 41
0.67 0.92 0.71 0.72 0.65 0.73 0.97 0.76 0.69 0.64 0.67
9e7e4e8e3e1e10e2e6e5 4e9e7e10e3e2e8e1e5e11e6 4e12e3e9e10e2e1e6e8e7e5e11 12e7e3e11e13e4e1e8e10e9e5e2e6 12e13e14e1e7e3e9e2e8e10e6e4e11e5 7e10e12e5e3e13e9e4e14e15e1e2e8e11e6 14e12e3e2e13e7e15e9e16e10e8e1e11e4e6e5 12e15e16e11e7e17e14e8e10e3e1e2e4e13e9e5e6 15e7e12e18e13e16e11e17e4e8e14e10e9e2e5e1e6e3 4e19e18e16e15e7e9e12e17e13e10e11e2e14e3e6e8e1e5 19e14e15e17e12e18e8e16e4e2e9e13e3e10e7e1e6e11e20e5
18
R. Tavakkoli-Moghaddam et al. / Journal of Air Transport Management 25 (2012) 15e18
planes but the descending order in a is not smooth and sometimes rises when additional planes are added. Therefore, increasing the number planes can means a better situation over a range of additions. 5. Conclusion We have considered the sequence of landing planes when minimizing the sum of the costs of deviation from the target times and minimizing the completion time of the landing sequence using a fuzzy programming approach. We have also presented an estimator for comparing and analyzing the answerers and finally presented the experimental results for small and large-sized problems with three to 20 waiting aircrafts.
Finally, we have analyzed the related results while increasing the waiting planes.
References Bellman, R.E., Zadeh, L.A., 1970. Decision making in a fuzzy environment. Mngm. Sci. 17, 141e164. Javadi, B., Rahimi-Vahed, A., Rabbani, M., Dangchi, M., 2008. Solving a multiobjective mixed-model assembly line sequencing problem by a fuzzy goal programming approach. Int. J. Adv. Manuf. Tech. 39, 975e982. Tang, K., Wang, Z., Cao, X., Zhang, J., 2008. A multi-objective evolutionary approach to aircraft landing scheduling problems. In: Proceedings of the IEEE Congress on Evolutionary Computation, pp. 3651e3657. Yager, R.R., 1977. Multiple objective decision-making using fuzzy sets. Int. J. Man-Machine Studies 9, 375e382.