- Email: [email protected]
Flight graph based genetic algorithm for crew scheduling in airlines
Information Sciences 133 (2001) 165±173
www.elsevier.com/locate/ins
Flight graph based genetic algorithm for crew scheduling in airlines H. Timucin Ozdemir, Chilukuri K. Mohan
*
Department of Electrical Engineering and Computer Science, Syracuse University, 2-177 CST, Syracuse, NY 13244-4100, USA
Abstract Crew scheduling is an NP-hard constrained combinatorial optimization problem, very important for the airline industry [G. Yu (Ed.), Operations Research in Airline Industry, Kluwer Academic Publishers, Dordrecht, 1998]. We solve this problem using a genetic algorithm applied to a ¯ight graph representation that represents several problem-speci®c constraints, unlike previous attempts [D. Levine, Application of a hybrid genetic algorithm to airline crew scheduling, Ph.D. dissertation, Computer Science Department, IIT , Chicago, USA, 1995; J.E. Beasley, P.C. Chu, A genetic algorithm for the set covering problem, Eur. J. Oper. Res. 94 (1996) 392±404; P.C. Chu, J.E. Beasley, A genetic algorithm for the set partitioning problem, Technical report, Imperial College, UK, 1995]. In extensive experimental comparisons on ¯ight data of several airlines, the new approach performed better than other approaches in 17 out of 24 data sets. Ó 2001 Elsevier Science Inc. All rights reserved.
1. Introduction Airline crew scheduling is the assignment of the ¯ight and training activities schedule for some period of time to various crew members, important because crew costs constitute the largest direct operating cost of airlines next to fuel costs [1,3]. Crew scheduling is a dicult combinatorial optimization problem, generally solved by transformation to the set cover problem (SCP) or the set partition *
Corresponding author. E-mail address: [email protected] (C.K. Mohan).
0020-0255/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 0 - 0 2 5 5 ( 0 1 ) 0 0 0 8 3 - 4
166
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
problem (SPP). These approaches use a binary matrix, whose rows represent the ¯ights and columns represent collections of ¯ights that can be allocated to the same crew. Columns (conforming to FAA regulations, company policies, and labor union's requirements) are generated by a complete enumeration or using heuristic methods. Then, the primary problem to be solved is to select a feasible set of columns covering all the ¯ights while minimizing cost. Such an approach is computationally unsatisfactory, especially when the number of ¯ights is large. Section 2 describes the problem constraints, our representation, and the function being optimized. Section 3 presents details of the evolutionary algorithm we used. Section 4 presents experimental results and concludes. 2. Problem and representation For the optimization task, it is preferable to use the ¯ight schedule as input data, rather than pre-processed columns. Some algorithms that rely on column generation techniques attempt to keep the best columns, but some suboptimal columns may be needed to ®nd better solutions [4]. Therefore, our proposed technique starts from the ¯ight schedule and builds the ¯ight graph to take care of some problem-speci®c constraints. Instead of using nodes in a graph to represent cities and edges to represent ¯ights, we propose a much more suitable graph representation that embeds problem constraints by using nodes to represent ¯ights, and edges to represent dependency constraints among ¯ights: an edge exists from the node representing ¯ight X to the node representing ¯ight Y i (i) Y leaves from the destination city of X, and (ii) Y leaves after a prespeci®ed delay following the arrival of X. Each path from a source node of the graph to a sink node represents a feasible sequence of ¯ights that may be assigned to a single crew, capturing some of the essential static constraints of the crew scheduling problem. Other dynamic constraints are enforced by algorithms using this representation. The directed edge i; j denotes that flightj can be ¯own after flighti , enforcing time and city constraints. Table 1 and Fig. 1 illustrate this representation, using the constraint that there is a rest period of at least 20 min between successive ¯ights to which the same crew is assigned. We assume that any crew members may be assigned to any ¯ights, i.e., this is a single ¯eet problem. This assumption does not excessively oversimplify the actual problem since our approach can be applied separately to each ¯eet type, if necessary. The goal of the algorithm is to produce solutions that respect constraints, minimize number of crews, maximize crew time utilization, and balance workloads (between crews).
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
167
Table 1 A sample schedule Flight number
Dep. city
Des. city
Dep. time
Arr. time
0 1 2 3 4 5 6 7 8
0 0 3 1 2 1 0 1 1
1 3 1 2 1 0 1 0 0
08:00 09:30 10:35 09:20 10:40 12:00 13:40 15:20 15:30
09:00 10:10 11:35 10:20 11:40 13:20 15:00 15:50 16:30
Fig. 1. Directed graph representation for the ¯ight schedule in Table 1 with minimum 20 min rest time between consecutive ¯ights.
Each crew assignment contains a set of rotations (pairings) for each crew. Each rotation consists of duty periods whose sequence represents a path in the ¯ight graph. Fig. 2 depicts four possible chromosomes for the schedule in Table 1. The algorithm attempts to minimize the total crew cost by using the following constraints: 1. Each ¯ight must be covered by a crew. 2. Each rotation must start and end at the same city. 3. Each rotation must start and end at the same base city. 4. There must be a minimum time delay between consecutive ¯ight legs in a rotation. 5. If the length of the duty period is less than 8 h, then the next duty period must start at least 8 h later. 6. If the length of the duty period is greater than 8 h and less than 13 h, then the next duty period must start at least 12 h later.
168
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
Fig. 2. Four chromosomes are de®ned for the problem in Table 1; of these, (b) is not preferred since the duty period of the second crew exceeds 8 h.
The cost includes overnight stay, deadheadings (crews ¯ying as passengers), and overtime pay. Each crew gets paid a minimum amount to ¯y a certain amount of time for a given period. If the schedule assigns unbalanced rotations, then the company gets punished from two sides: high overtime and under-utilized personnel time. The goal is to build a schedule which has balanced rotations, utilizes the paid time of personnel, and minimizes overtime and overnight payments. A cost value for a rotation is calculated by adding terms corresponding to the following: · under-utilized time between rotations, · under-utilized time between duties in each rotation, · hotel and per-diem expenses for each rotation, and · required and overtime (if needed) pay for each duty in each rotation. The total cost of the schedule is obtained by summing the costs of all crew schedules. A ®tness value is calculated by using the cost augmented by penalty terms for each violation of constraints 1, 2 and 3, listed earlier.
3. Details of genetic algorithm GraGA is a steady-state genetic algorithm that operates on the ¯ight graph representations. Each individual (chromosome) contains a sequence of rotations (pairings) for each crew (see Fig. 3). Each rotation sequence represents a schedule for a crew. Each rotation contains a sequence of duty periods, containing a sequence of edges. The length of the rotations, duty periods, and rest periods are restricted by the regulations. A new ospring may replace one of its parents, using binary tournament selection, except that the best solution in the population is not replaced.
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
169
Fig. 3. GraGA algorithm.
The mutation operators resemble well-known GA crossover operators (1PX or 2PX), applied within a single individual: select two edge lists, cut both of them at some suitable point(s), and splice together the pieces. We have experimented with three recombination operators, applied to rotations. The set based operator attempts to use genes inherited from both parents as much as possible. The time based operator cuts the schedule into two pieces and recombines them. The distance preserving operator attempts to preserve genes common to both parents but ®lls up the rest using alleles not used in any parent. Local search follows the application of recombination and mutation operators. Local search reorganizes the rotations based on time and attempts to build a schedule as tight as possible. Then, it applies 1PX and 2PX operator to the rotations of a schedule. An ospring may replace a parent when this improves the schedule cost. Individuals in the initial population are obtained using the following algorithm: · Let V 0 ; E0 be the current graph, and let Es contain edges in E0 incident on sink nodes (with outdegree 0). · while Es 6 fg and V 0 6 fg, do: produce a feasible path by concatenating edges backwards, beginning from edges in Es ; update E0 by removing edges incident on the covered nodes, V 0 by removing covered nodes, and Es to contain edges connected to sink nodes in the new V 0 ; E0 . · If V 0 6 fg, then repeat the previous step after reinserting some previously removed edges into the current graph, penalizing these to discourage their reuse. · If V 0 6 fg, then cover these nodes (¯ights) while relaxing constraint 1, allowing deadheading (more than one crew per ¯ight). The restoration algorithm is required when the solution does not cover all the ¯ights. Where possible, previously discovered low cost rotations are used to
170
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
modify an individual candidate solution, with restricted backtracking required in some cases. A restart procedure is applied periodically, introducing new randomly generated chromosomes into the population. This procedure sorts the chromosomes in decreasing ®tness order, and starting from the second individual marks a chromosome to be replaced if it is within a `distance' (de®ned below) < d from any of the preceding unmarked chromosomes. If E ci denote the set of edges used by chromosome ci , then the distance between chromosomes ci and cj is de®ned to be d ci ; cj 1 jE ci \ E cj j= jE ci j. Each of our experiments applied the restart procedure 10 times.
4. Results Test data sets were obtained by querying various airline internet sites. For ease of comparison with the SCP solution techniques, we enumerated all columns whose total time duration is at most three days. Computational experiments compared GraGA with greedy algorithms developed based on Chvatal's heuristic [5], Back's heuristic [6] and our new heuristic which multiplies B ack's cost-ratio with the number of overlaps between the current cover and the cover of the candidate column. We also examined results obtained using a GA implemented for the SCP [2]. GraGA performed better than other approaches in many respects: better workload distribution, smaller number of crews, and fewer deadheadings, as illustrated in Table 2. In some problems, GraGA produced satisfactory results whereas the number of columns was too large for column based approaches to be applied. To conclude, the proposed graph representation is very useful for highly constrained transportation problems, reducing the search space and facilitating the application of recombination operators. Our approach does away with the burden of column generation, hopefully making this set of problems GA-solvable. Recent results [7] show success in applying this approach to the vehicle routing problem with time windows (VRPTW).
Appendix A. Fitness function A cost value for a rotation is calculated by adding the amount paid to crew and hotels for overnight stays. Using the variables described in Fig. 4, the following criteria are applied when evaluating each rotation: · No duty can be longer than MaxDutyTime and cannot contain more than MaxFlyingTime minutes of ¯ying.
Table 2 The cost of best solution obtained using various algorithms; #F denotes the number of ¯ights, #C denotes the number of columns, and #D denotes the number of days spanned by the schedule Problem Name
#D
Chvatal SCP [5]
Back SCP [6]
SCP based CBGA [2]
Greedy SCP(new)
GraGA
16 40 52 76 81 107 122 123 133 182 184 205 295 399 453 857 904 1008 1105 1466 1698 3428 4415 5318 5618 15 250 14 254 21 308
1 1 7 1 1 1 2 2 7 1 1 1 1 2 1 1 1 2 1 2 1 2 2 7 2 2 2 7
3600 9698 22609.7 8340 9636.6 8400 6900 16 070 10441.5 11 400 12 940 14752.8 10 050 39279.7 19 270 16 840 14114 14 984 11 400 20248.9 23806.9 78343.8 89482.1 22042.8 40398.6 16593.5 27444.2 22636.8
3600 9730 17609 7540 8914.1 7054 4500 15 903 9088.5 9444 10 310 12072.8 8750 38020.5 13 680 13 272 9755 11 414 6360 18744.5 20335.9 71984.7 82463.9 18207.3 29178.6 11298.5 23 184 15431.9
3600 9356 18546.3 7120 8790.3 7080 4500 15 598 9756.5 9120 9680 12100.8 8290 35000.7 14 070 12 592 10679 10 784 6380 16 946 20043.1 78 421 88907.5 19356.5 33340.6 11473.5 23854.8 17 085
3600 9730 17609 7553 8914.1 7107 4527 15 903 9118.5 9432 10 310 12072.8 8750 37696.5 14 010 13 310 9740 11 530 6360 18440.5 20365.8 71714.7 82121.7 17593.5 29224.6 11319.5 24 327 19420.9
3600 8912 15992.1 7112 8975.33 7092 4500 15 576 8349.5 9452 10 100 11707.8 8180 39435.5 14 500 12 210 9580 10 678 5800 20732.3 18961.3 77440.8 90 517 16914.2 30006.6 11298.5 30026.2 12321.5
1
7
7
3
17
171
Number of best results
26 43 28 55 54 63 46 85 32 86 75 93 72 173 110 102 118 73 73 86 133 333 380 83 217 97 158 74
#C
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
GBXORD_M_26 Delta_B727_43 MandarinAir GBXSEA_M_55 OrlandoAir GBXDFW_M_63 GBXSFO_M_46 Delta_B737_85 LynxAir GBXDFW_M_86 GBXSEA_M_75 GBXMIA_M_93 GBXDEN_M_72 Delta_MD88_173 GBXSEA_M_110 Delta_102 GBXATL_M_118 GBXORD_MT_73 GBXBOS_M_73 Delta_B727_86 Delta_133 Delta_MD88_333 Delta_MD88_380 OrcaAir Delta_217 GBXSFO_MT_97 GBXDFW_MT_158 THY_74
#F
172
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
Fig. 4. Variables used in the problem to calculate the cost of each schedule, with values used in our simulations.
· If a duty contains less than MaxFlyingTime minutes of ¯ying then a crew get paid at PayForMinPaidFlyingTime (PFMPFT). · If the ¯ying time (s) in a duty exceeds MinPaidFlyingTime (M) then a crew earns P s M E: · If a crew stays overnight in another place dierent from his/her domicile, then for each overnight stay H PDE LengthOfStay will be charged to this rotation. · If the length of a duty (ld) is longer than AveDutyTime (ADT), then the minimum required rest time (MRRT) is calculated as ld ADT MxDT MxR MnR
ADT MnR;
where MxR, MnR, and ADT conforms to the standard values used in the industry. · If the length of rest (RT) between two duties is longer than Mean Required Rest Time (MRRT), then this under-utilized crew time is added as an UUC: UUC RT
MRRT CPMU:
In summary, the components of cost for a sequence of rotations assigned to a crew are: · under-utilized time between rotations, · under-utilized time between duties in each rotation, · hotel and per-diem expenses for each rotation, and · required pay for each duty in each rotation.
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
173
Each rotation starts from and ends at the same base city and there is a minimum time requirement between consecutive ¯ights (30 min). After the cost of schedule is calculated, a ®tness value is assigned to this solution. The ®tness value is de®ned as nef nsf nuf 1 scost 1 1 1 ; nf nf nf where scost 1
cost minCostInPop ; maxCostInPop minCostInPop
minCostInPop is the minimum cost value in the current population; maxCostInPop is the maximum cost value in the current population; nf is the number of ¯ights in the problem; nef is the number of rotations that does not end where it is started; nsf is the number of rotations that does not start from a base city, and; nuf is the number of uncovered ¯ights.
References [1] G. Yu (Ed.), Operations Research in Airline Industry, Kluwer Academic Publishers, Dordrecht, 1998. [2] J.E. Beasley, P.C. Chu, A genetic algorithm for the set covering problem, Eur. J. Oper. Res. 94 (1996) 392±404. [3] R. Anbil, E. Gelman, B. Patty, R. Tanga, Recent advances in crew-pairing optimization at American airlines, Interfaces 21 (1) (1991) 62±74. [4] J.-M. Rousseau, J. Desrosiers, Results obtained with Crew-Opt: a column generation method for transit crew scheduling, in: Proceedings of the Sixth International Workshop on ComputerAided Scheduling of Public Transport, Springer, New York, 1995, pp. 349±358. [5] V. Chvatal, A greedy heuristic for the set covering problem, Math. Oper. Res. 4 (3) (1979) 233± 235. [6] T. B ack, M. Schutz, S. Khuri. A comparative study of a penalty functions, a repair heuristic, and stochastic operators with the set-covering problem, in: Proceedings of the European Conference on Arti®cial Evolution, Springer, New York, 1995, pp. 320±332. [7] H.T. Ozdemir, Graph based evolutionary algorithms for transportation problems, Ph.D. dissertation, Department of EECS, Syracuse University, 2001.
www.elsevier.com/locate/ins
Flight graph based genetic algorithm for crew scheduling in airlines H. Timucin Ozdemir, Chilukuri K. Mohan
*
Department of Electrical Engineering and Computer Science, Syracuse University, 2-177 CST, Syracuse, NY 13244-4100, USA
Abstract Crew scheduling is an NP-hard constrained combinatorial optimization problem, very important for the airline industry [G. Yu (Ed.), Operations Research in Airline Industry, Kluwer Academic Publishers, Dordrecht, 1998]. We solve this problem using a genetic algorithm applied to a ¯ight graph representation that represents several problem-speci®c constraints, unlike previous attempts [D. Levine, Application of a hybrid genetic algorithm to airline crew scheduling, Ph.D. dissertation, Computer Science Department, IIT , Chicago, USA, 1995; J.E. Beasley, P.C. Chu, A genetic algorithm for the set covering problem, Eur. J. Oper. Res. 94 (1996) 392±404; P.C. Chu, J.E. Beasley, A genetic algorithm for the set partitioning problem, Technical report, Imperial College, UK, 1995]. In extensive experimental comparisons on ¯ight data of several airlines, the new approach performed better than other approaches in 17 out of 24 data sets. Ó 2001 Elsevier Science Inc. All rights reserved.
1. Introduction Airline crew scheduling is the assignment of the ¯ight and training activities schedule for some period of time to various crew members, important because crew costs constitute the largest direct operating cost of airlines next to fuel costs [1,3]. Crew scheduling is a dicult combinatorial optimization problem, generally solved by transformation to the set cover problem (SCP) or the set partition *
Corresponding author. E-mail address: [email protected] (C.K. Mohan).
0020-0255/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 0 - 0 2 5 5 ( 0 1 ) 0 0 0 8 3 - 4
166
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
problem (SPP). These approaches use a binary matrix, whose rows represent the ¯ights and columns represent collections of ¯ights that can be allocated to the same crew. Columns (conforming to FAA regulations, company policies, and labor union's requirements) are generated by a complete enumeration or using heuristic methods. Then, the primary problem to be solved is to select a feasible set of columns covering all the ¯ights while minimizing cost. Such an approach is computationally unsatisfactory, especially when the number of ¯ights is large. Section 2 describes the problem constraints, our representation, and the function being optimized. Section 3 presents details of the evolutionary algorithm we used. Section 4 presents experimental results and concludes. 2. Problem and representation For the optimization task, it is preferable to use the ¯ight schedule as input data, rather than pre-processed columns. Some algorithms that rely on column generation techniques attempt to keep the best columns, but some suboptimal columns may be needed to ®nd better solutions [4]. Therefore, our proposed technique starts from the ¯ight schedule and builds the ¯ight graph to take care of some problem-speci®c constraints. Instead of using nodes in a graph to represent cities and edges to represent ¯ights, we propose a much more suitable graph representation that embeds problem constraints by using nodes to represent ¯ights, and edges to represent dependency constraints among ¯ights: an edge exists from the node representing ¯ight X to the node representing ¯ight Y i (i) Y leaves from the destination city of X, and (ii) Y leaves after a prespeci®ed delay following the arrival of X. Each path from a source node of the graph to a sink node represents a feasible sequence of ¯ights that may be assigned to a single crew, capturing some of the essential static constraints of the crew scheduling problem. Other dynamic constraints are enforced by algorithms using this representation. The directed edge i; j denotes that flightj can be ¯own after flighti , enforcing time and city constraints. Table 1 and Fig. 1 illustrate this representation, using the constraint that there is a rest period of at least 20 min between successive ¯ights to which the same crew is assigned. We assume that any crew members may be assigned to any ¯ights, i.e., this is a single ¯eet problem. This assumption does not excessively oversimplify the actual problem since our approach can be applied separately to each ¯eet type, if necessary. The goal of the algorithm is to produce solutions that respect constraints, minimize number of crews, maximize crew time utilization, and balance workloads (between crews).
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
167
Table 1 A sample schedule Flight number
Dep. city
Des. city
Dep. time
Arr. time
0 1 2 3 4 5 6 7 8
0 0 3 1 2 1 0 1 1
1 3 1 2 1 0 1 0 0
08:00 09:30 10:35 09:20 10:40 12:00 13:40 15:20 15:30
09:00 10:10 11:35 10:20 11:40 13:20 15:00 15:50 16:30
Fig. 1. Directed graph representation for the ¯ight schedule in Table 1 with minimum 20 min rest time between consecutive ¯ights.
Each crew assignment contains a set of rotations (pairings) for each crew. Each rotation consists of duty periods whose sequence represents a path in the ¯ight graph. Fig. 2 depicts four possible chromosomes for the schedule in Table 1. The algorithm attempts to minimize the total crew cost by using the following constraints: 1. Each ¯ight must be covered by a crew. 2. Each rotation must start and end at the same city. 3. Each rotation must start and end at the same base city. 4. There must be a minimum time delay between consecutive ¯ight legs in a rotation. 5. If the length of the duty period is less than 8 h, then the next duty period must start at least 8 h later. 6. If the length of the duty period is greater than 8 h and less than 13 h, then the next duty period must start at least 12 h later.
168
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
Fig. 2. Four chromosomes are de®ned for the problem in Table 1; of these, (b) is not preferred since the duty period of the second crew exceeds 8 h.
The cost includes overnight stay, deadheadings (crews ¯ying as passengers), and overtime pay. Each crew gets paid a minimum amount to ¯y a certain amount of time for a given period. If the schedule assigns unbalanced rotations, then the company gets punished from two sides: high overtime and under-utilized personnel time. The goal is to build a schedule which has balanced rotations, utilizes the paid time of personnel, and minimizes overtime and overnight payments. A cost value for a rotation is calculated by adding terms corresponding to the following: · under-utilized time between rotations, · under-utilized time between duties in each rotation, · hotel and per-diem expenses for each rotation, and · required and overtime (if needed) pay for each duty in each rotation. The total cost of the schedule is obtained by summing the costs of all crew schedules. A ®tness value is calculated by using the cost augmented by penalty terms for each violation of constraints 1, 2 and 3, listed earlier.
3. Details of genetic algorithm GraGA is a steady-state genetic algorithm that operates on the ¯ight graph representations. Each individual (chromosome) contains a sequence of rotations (pairings) for each crew (see Fig. 3). Each rotation sequence represents a schedule for a crew. Each rotation contains a sequence of duty periods, containing a sequence of edges. The length of the rotations, duty periods, and rest periods are restricted by the regulations. A new ospring may replace one of its parents, using binary tournament selection, except that the best solution in the population is not replaced.
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
169
Fig. 3. GraGA algorithm.
The mutation operators resemble well-known GA crossover operators (1PX or 2PX), applied within a single individual: select two edge lists, cut both of them at some suitable point(s), and splice together the pieces. We have experimented with three recombination operators, applied to rotations. The set based operator attempts to use genes inherited from both parents as much as possible. The time based operator cuts the schedule into two pieces and recombines them. The distance preserving operator attempts to preserve genes common to both parents but ®lls up the rest using alleles not used in any parent. Local search follows the application of recombination and mutation operators. Local search reorganizes the rotations based on time and attempts to build a schedule as tight as possible. Then, it applies 1PX and 2PX operator to the rotations of a schedule. An ospring may replace a parent when this improves the schedule cost. Individuals in the initial population are obtained using the following algorithm: · Let V 0 ; E0 be the current graph, and let Es contain edges in E0 incident on sink nodes (with outdegree 0). · while Es 6 fg and V 0 6 fg, do: produce a feasible path by concatenating edges backwards, beginning from edges in Es ; update E0 by removing edges incident on the covered nodes, V 0 by removing covered nodes, and Es to contain edges connected to sink nodes in the new V 0 ; E0 . · If V 0 6 fg, then repeat the previous step after reinserting some previously removed edges into the current graph, penalizing these to discourage their reuse. · If V 0 6 fg, then cover these nodes (¯ights) while relaxing constraint 1, allowing deadheading (more than one crew per ¯ight). The restoration algorithm is required when the solution does not cover all the ¯ights. Where possible, previously discovered low cost rotations are used to
170
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
modify an individual candidate solution, with restricted backtracking required in some cases. A restart procedure is applied periodically, introducing new randomly generated chromosomes into the population. This procedure sorts the chromosomes in decreasing ®tness order, and starting from the second individual marks a chromosome to be replaced if it is within a `distance' (de®ned below) < d from any of the preceding unmarked chromosomes. If E ci denote the set of edges used by chromosome ci , then the distance between chromosomes ci and cj is de®ned to be d ci ; cj 1 jE ci \ E cj j= jE ci j. Each of our experiments applied the restart procedure 10 times.
4. Results Test data sets were obtained by querying various airline internet sites. For ease of comparison with the SCP solution techniques, we enumerated all columns whose total time duration is at most three days. Computational experiments compared GraGA with greedy algorithms developed based on Chvatal's heuristic [5], Back's heuristic [6] and our new heuristic which multiplies B ack's cost-ratio with the number of overlaps between the current cover and the cover of the candidate column. We also examined results obtained using a GA implemented for the SCP [2]. GraGA performed better than other approaches in many respects: better workload distribution, smaller number of crews, and fewer deadheadings, as illustrated in Table 2. In some problems, GraGA produced satisfactory results whereas the number of columns was too large for column based approaches to be applied. To conclude, the proposed graph representation is very useful for highly constrained transportation problems, reducing the search space and facilitating the application of recombination operators. Our approach does away with the burden of column generation, hopefully making this set of problems GA-solvable. Recent results [7] show success in applying this approach to the vehicle routing problem with time windows (VRPTW).
Appendix A. Fitness function A cost value for a rotation is calculated by adding the amount paid to crew and hotels for overnight stays. Using the variables described in Fig. 4, the following criteria are applied when evaluating each rotation: · No duty can be longer than MaxDutyTime and cannot contain more than MaxFlyingTime minutes of ¯ying.
Table 2 The cost of best solution obtained using various algorithms; #F denotes the number of ¯ights, #C denotes the number of columns, and #D denotes the number of days spanned by the schedule Problem Name
#D
Chvatal SCP [5]
Back SCP [6]
SCP based CBGA [2]
Greedy SCP(new)
GraGA
16 40 52 76 81 107 122 123 133 182 184 205 295 399 453 857 904 1008 1105 1466 1698 3428 4415 5318 5618 15 250 14 254 21 308
1 1 7 1 1 1 2 2 7 1 1 1 1 2 1 1 1 2 1 2 1 2 2 7 2 2 2 7
3600 9698 22609.7 8340 9636.6 8400 6900 16 070 10441.5 11 400 12 940 14752.8 10 050 39279.7 19 270 16 840 14114 14 984 11 400 20248.9 23806.9 78343.8 89482.1 22042.8 40398.6 16593.5 27444.2 22636.8
3600 9730 17609 7540 8914.1 7054 4500 15 903 9088.5 9444 10 310 12072.8 8750 38020.5 13 680 13 272 9755 11 414 6360 18744.5 20335.9 71984.7 82463.9 18207.3 29178.6 11298.5 23 184 15431.9
3600 9356 18546.3 7120 8790.3 7080 4500 15 598 9756.5 9120 9680 12100.8 8290 35000.7 14 070 12 592 10679 10 784 6380 16 946 20043.1 78 421 88907.5 19356.5 33340.6 11473.5 23854.8 17 085
3600 9730 17609 7553 8914.1 7107 4527 15 903 9118.5 9432 10 310 12072.8 8750 37696.5 14 010 13 310 9740 11 530 6360 18440.5 20365.8 71714.7 82121.7 17593.5 29224.6 11319.5 24 327 19420.9
3600 8912 15992.1 7112 8975.33 7092 4500 15 576 8349.5 9452 10 100 11707.8 8180 39435.5 14 500 12 210 9580 10 678 5800 20732.3 18961.3 77440.8 90 517 16914.2 30006.6 11298.5 30026.2 12321.5
1
7
7
3
17
171
Number of best results
26 43 28 55 54 63 46 85 32 86 75 93 72 173 110 102 118 73 73 86 133 333 380 83 217 97 158 74
#C
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
GBXORD_M_26 Delta_B727_43 MandarinAir GBXSEA_M_55 OrlandoAir GBXDFW_M_63 GBXSFO_M_46 Delta_B737_85 LynxAir GBXDFW_M_86 GBXSEA_M_75 GBXMIA_M_93 GBXDEN_M_72 Delta_MD88_173 GBXSEA_M_110 Delta_102 GBXATL_M_118 GBXORD_MT_73 GBXBOS_M_73 Delta_B727_86 Delta_133 Delta_MD88_333 Delta_MD88_380 OrcaAir Delta_217 GBXSFO_MT_97 GBXDFW_MT_158 THY_74
#F
172
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
Fig. 4. Variables used in the problem to calculate the cost of each schedule, with values used in our simulations.
· If a duty contains less than MaxFlyingTime minutes of ¯ying then a crew get paid at PayForMinPaidFlyingTime (PFMPFT). · If the ¯ying time (s) in a duty exceeds MinPaidFlyingTime (M) then a crew earns P s M E: · If a crew stays overnight in another place dierent from his/her domicile, then for each overnight stay H PDE LengthOfStay will be charged to this rotation. · If the length of a duty (ld) is longer than AveDutyTime (ADT), then the minimum required rest time (MRRT) is calculated as ld ADT MxDT MxR MnR
ADT MnR;
where MxR, MnR, and ADT conforms to the standard values used in the industry. · If the length of rest (RT) between two duties is longer than Mean Required Rest Time (MRRT), then this under-utilized crew time is added as an UUC: UUC RT
MRRT CPMU:
In summary, the components of cost for a sequence of rotations assigned to a crew are: · under-utilized time between rotations, · under-utilized time between duties in each rotation, · hotel and per-diem expenses for each rotation, and · required pay for each duty in each rotation.
H.T. Ozdemir, C.K. Mohan / Information Sciences 133 (2001) 165±173
173
Each rotation starts from and ends at the same base city and there is a minimum time requirement between consecutive ¯ights (30 min). After the cost of schedule is calculated, a ®tness value is assigned to this solution. The ®tness value is de®ned as nef nsf nuf 1 scost 1 1 1 ; nf nf nf where scost 1
cost minCostInPop ; maxCostInPop minCostInPop
minCostInPop is the minimum cost value in the current population; maxCostInPop is the maximum cost value in the current population; nf is the number of ¯ights in the problem; nef is the number of rotations that does not end where it is started; nsf is the number of rotations that does not start from a base city, and; nuf is the number of uncovered ¯ights.
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