Improved Genetic Algorithms by Means of Fuzzy Crossover Operators for Revenue Management in Airlines

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1 World Appled Scences Journal (6): 88-86, ISSN IDOSI Publcatons, DOI:.589/dos.wasj Improved Genetc Algorthms by Means of Fuzzy Crossover Operators for Revenue Management n Arlnes M. Sadegh, R. Sadegh, S.A. Mousav and S. Khanmohammad Department of Electrcal Engneerng, Naghshejahan Hgher Educaton Insttute, Isfahan, Iran Department of Electrcal Engneerng, Najafabad Branch, Islamc Azad Unversty, Isfahan, Iran Department of Mechancal Engneerng, Najafabad Branch, Islamc Azad Unversty, Isfahan, Iran Control and Electrcal Engneerng, Unversty of Tabrz, Tabrz, Iran Abstract: Revenue Management s an economc polcy that ncreases the earned proft by adjustng the servce demand and nventory. Revenue Management n arlnes correlates wth nventory control and prce levels n dfferent fare classes. We focus on prcng and seat allocaton problems n arlnes by ntroducng a constraned optmzaton problem n Bnary Integer Programmng (BIP) formulaton. Two BIP problems are represented. Moreover, some mproved Genetc Algorthms (GAs) approaches are used to solve these problems. We ntroduce new crossover operators that assgn a Fuzzy Membershp Functon to each parent n GAs. We acheve better outputs wth new methods that take lower calculaton tmes and earn hgher profts. Three dfferent test problems n dfferent scales are selected to evaluate the effectveness of each algorthm. Ths paper defnes new crossover operators that help to reach better solutons that take lower calculaton tmes and more earned profts. Key words: Genetc Algorthms (Gas) Operatonal Research (OR) Bnary Integer Programmng Revenue Management Fuzzy Logc INTRODUCTION Inventory s pershable; Supply s fxed or can not be adjusted n the short run; Demand can be segmented Ar Transportaton has been faced a contnuous based on a number of marketng factors; Demand s growth n number of arlnes passengers, n recent years. stochastc; Inventory can be sold n advance. The Federal Avaton Admnstraton (FAA) predcts that Servce ndustres, partcularly those n the ths demand wll ncrease even wth hgher rates n future. travel and transportaton markets, often satsfy all Large arlne companes may have approxmately or most of these condtons. Although ts mpact daly flghts. If the revenue from each flght could be depends on the sze of the company and the complexty ncreased by only $, ths would result n an annual of ts operatons, an estmate of -% n revenue revenue ncrease of 9,5,$. Ths smple example ncrease has been drectly attrbuted wth revenue emphaszes revenue management potental to acheve management [,]. hgh revenue []. Revenue management has also taken hold wdely Revenue management s a busness prncple that throughout the rest of the Travel ndustry as well. balances supply and demand to control prce and/or Almost all major Hotels, Car Rental Agences, Cruse nventory avalablty n order to maxmze revenue and Lnes and Passenger Ralroad frms have revenue proft growth. In other words, Revenue Management s management systems []. the allocaton of lmted resources between some Revenue or Yeld Management technques are customers. It uses to make hgher profts wth a partcular comprsed of some basc parts that are grouped n basc nvestment n capacty. The mpact of revenue categores: Product Desgn, Prcng and Capacty management s most sgnfcant n busness envronments Allocaton. These parts are n a completely close relaton where the followng condtons exst: wth each other. Correspondng Author: M. Sadegh, No.6- Sadee Al- khoram and Kashan Sdeway St- Jomhoor Sq-Isfahan, Iran. Postal Code: , Cell: (+98)969, Fax: (+98)

2 World Appl. Sc. J., (6): 88-86, In Ar Transportaton, travel programs mpacts of varous market condtons on the payoffs, (passengers routes from source to ther fnal bookng lmts and prcng strateges of the competng destnatons nclude one or more non-stop flght at a arlnes. A multstage stochastc programmng approach specfc departure tme) determne the nventory or seat to arlne network revenue management s presented n allocaton capacty and arlne customers demonstrate the []. The objectve s to determne seat protecton levels demand. for all tnerares, fare classes, ponts of sale of the arlne The total passenger demand for each travel program network and all bookng horzons such that the expected and combnaton of fare classes s assumed to be a revenue s maxmzed. [] represents a sem-markov stochastc process. It s represented by an Exponental Decson Problem that consders a sngle flght-leg wth dstrbuton functon. Exponental functon s a useful multple fare classes, overbookng of the flght, concurrent densty functon to estmate the dstances between arrval demand arrvals of passengers from the dfferent fare tmes of customers to a system f t has a fxed rate, as classes and class-dependent, random cancellatons. Ths followed: problem s solved wth a stochastc optmzaton x technque. In [], maxmum number of average fare data e x () f ( x) = { f s selected and ths problem s formulated usng Bnary otherwse Integer Programmng. [] combnes a stochastc gradent x algorthm and approxmate dynamc programmng deas to e x () F( x) = { f, > mprove mportant ssues lke demand uncertanty, nestng otherwse and the dynamc nature of the bookng process. [] outlnes a Revenue Management approach suted to the prcng polcy of low-cost arlnes, where each flght only has one fare avalable at any pont of tme. Effectve Revenue Management bols down to a flght-by-flght dynamc prce optmzaton. The remander of ths artcle s developed as follow: secton represents some basc assumptons and mathematcal formulatons n order to apply revenue management to arlnes n the Bnary Integer Programmng formulatons. Secton ntroduces some methodologes that are used to solve BIP problems. In secton, Computatonal Results and some comparsons between dfferent knds of crossover operators have been mentoned. Fnally, secton 5 s prepared as concluson of the paper. That s Mean and Scale parameter for ths densty functon [5,6]. Therefore, the number of allocated seats for a fare class n a travel program s determned wth exponental densty functon f(x). f(x) s the correspondng cumulatve dstrbuton functon, too. In other words, s the probablty of ncreasng passengers to a specfc dgt n a fare class for a travel program. Cumulatve dstrbuton functon wll be used to defne Maxmum Predcted Revenue ( jk) n secton. It s to be noted that usng exponental dstrbuton s not a prncpal assumpton n ths artcle and could be replaced by other proper dstrbuton functons such as Normal, Gamma and the other ones. In addton, prce structure and seat allocaton polces must be formulated n consstency wth constrants mposed by the arlnes computer reservaton system (CRS) such as level-of-servce (nonstop, drect, sngle-connect, or double-connect), connecton qualty varables, carrer, carrer market presence, fares, arcraft sze and type varables and tme of day among others, whch lmts the number of fare classes [7,8]. Some of the major CRSs are Apollo, EAASY SABRE and System One. A jont seat allocaton and fare-prcng competton model for stochastc demand s proposed n [9]. A numercal analyss s presented to demonstrate the Basc Assumptons and Problem Formulaton: A wde varety of problems can be represented as dscrete optmzaton models. An mportant area of applcaton concerns the effcent management of a lmted number of resources to ncrease productvty and/or proft. Such applcatons are encountered n Operatonal Research problems such as goods dstrbuton, producton schedulng and machne sequencng [5]. We propose two versons of Prcng and Seat allocaton model n the BIP formulaton n ths paper due to [6]. The followng are the basc varants ntroducng the Prcng & Fleet seat allocaton system: 89

3 World Appl. Sc. J., (6): 88-86, Flght Leg Capacty (A l) s the assgned arcraft capacty of a flght leg. The smlar combnatons of fare classes that are consdered as each flght leg represent selected fare classes n CRS (CRS Capacty (M)). Travel Programs nclude one or more flght legs that should be consdered for customers. Maxmum Predcted Revenue( jk) s estmated for each seat as: j= = T P( j ) jk k k Where T k s the prce for -th travel program and fare class k and P(j k) s the probablty of ncreasng passengers from (j-) seat n travel program and fare class k [6]. Xjks bnary decson varable for travel program, j-th seat and fare class k,w m s bnary decson varable for travel program and prce structure m. Also, () descrbes the smlar combnatons of prce structures for travel program, represents seat allocaton capacty for travel program,i(l)shows travel programs that contan l- th flght leg and K() s a set that ncludes all of fare classes selected for travel program. It s notable that (m) s a set of fare classes that does not exst n travel program and prce structure m. In contnuaton to ths secton, two dfferent models of the combned Prcng & Seat allocaton problems are descrbed. Problem: Wth aggregate the basc model that s represented n [6] over all seats of the travel program (n constrants (7)), we have ths formulaton: Subject to = Z( x) Maxmze X I() l j= k K() Λ() m= I() l j= k K() jk l jk X A l L W = I m X + W I; m =,..., Λ( ); k Γ( m) jk m X & W {,} I; j=,..., ; k K ( ); m=,..., Λ() jk m jk () () (5) (6) (7) (8) The objectve functon () maxmzes the predcted revenue for all seats of travel programs n the consdered fare classes. Constrants (5) establsh that the total number of assgned passengers can not exceed the arcraft capacty of a fght leg. Constrants (6) assure that only one prce structure s selected for each travel program. Also, Constrants (7) nsure that fare classes should select from the defned prce structure of each travel program. Fnally, constrants (8) determne that all decson varables should be bnary. Problem : Wth aggregaton of basc model over all seats of travel program and fare classes that does not exst n prce structure m: j= k Γ( m) Im ; =,..., Λ( ) X + Γ ( m) W Γ( m) jk m The second problem s developed by replacng (7) wth (9) (see [6]). Notce that Problem has one and Problem has two ndependent aggregatons. The optmal solutons n represented BIP problems are the hghest proft that an arlne can reach to t. However, an arlne usually can not calculate the optmal soluton n large-scale companes and then, they should satsfy wth only sub-optmal solutons. Closer soluton to optmal, more proft. Another parameter that s too mportant for an arlne s the requred calculaton tme to provdng a soluton. Ths paper defne new crossover operators that help n order to reach better solutons wth lower calculaton tmes and more profts. In other words, these heurstc algorthms help us to apply more accurate and qucker management n arlnes. Soluton Methodology: Computatonal complexty of Integer programmng problems depends on three basc factors: Number of nteger varables Problem structure The lnear constrants of the problem It s notable that sometmes when the number of nteger programmng constrants s ncreased, computatonal complexty s fallen, due to sgnfcant decrease n the number of feasble solutons. (9) 8

4 World Appl. Sc. J., (6): 88-86, The number of solutons n the nteger programmng Also, these algorthms have remarkable potental to n problems wth n bnary varables s. However, the mprove soluton methods due to ther flexblty and number of solutons would be reduplcated when one robustness n executon process. varable s added. As a result, the completeness of these Improvng the crossover operaton s the man problems wll grow exponentally. contrbuton of ths artcle. In contnuance, some new Branch-and-Bound, Cuttng Planes, Branch-and-Cut heurstc crossover operators are ntroduced. These and Branch-and-Prce are some of the most conventonal operators assgn Fuzzy Membershp Functon to each classc technques used n solvng BIP problems. These parent. These functons are allocated to parents methods have returned acceptable results for these chromosomes. Then, they are combned accordng to ther problems n farly small-scale cases. However, the soluton membershp functons. procedure n large-scale problems faces many dffcultes.e. optmal soluton s not acheved and/or t takes long Crossover Operators for Genetc Algorthms tmes to reach optmal results. Fuzzy-Arthmetc Weghted Mean (FAWM) (Bell Shape Therefore, newer computatonal procedures are needed to Type): Ths operator returns chldren that are overcome these mentoned defcences. In ths way, fuzzy-arthmetc weghted mean of two parents []. In the researchers have tended to stochastc optmzaton crossover procedure, algorthm consders cost functon methods such as evolutonary and Swarm Intellgence and allocates respectvely a fuzzy membershp functon to algorthms. They respectvely model natural evoluton and each parent due to ther ftness functon values. After socal behavors n the form of algorthmc mechansms. that, ntersecton pont values of each membershp These methods revew works done on prevous functon wth the fttest parent one (the elte chromosome) steps, mplement search procedure and attempt to mprove are estmated as weghts. solutons n executon process. These coeffcents (weghts) determne chldren as Genetc Algorthms, Evoluton Strateges, Partcle FAWM of two parents, as below: Swarm Optmzaton are the most conventonal nstances of them. P( + ) j = B P ( ) j j + B P j ( j+ ) () Gas are heurstc methods that help to fnd better solutons n NP-complete problems. They use some In above equaton, P represents generated parents n strngs of symbols called chromosomes. Solutons are crossover process. Its frst entry s the Generaton number represented on these strngs drectly or wth usng a and the second one s the parent number n Genetc defned transformaton functon. Algorthm. Bj s the weght allocated to j-th parent that Selecton, crossover and mutaton are the man s returned n -th teraton. operators of the genetc search. The repeated use of these In ths crossover operaton, Bell shape membershp operators s a flterng process, whch results n the functons are selected (Fgure ). subsequent populatons of ndvduals wth better ftness It s noted that all generated chldren n above values. Ths probablstc nature of GA makes t a unque process are feasble wth respect to lnear constrants of algorthm for convergency towards the global optmum. the optmzaton problem. However, beng a heurstc search method, GA can not guarantee fndng the optmal soluton [7]. Fuzzy-arthmetc Weghted Mean (Bell Shape Type): Snce GAs are usually used n NP-complete problems, Ths type s the same as prevous one except that the the dstance between returned soluton and the optmal fttest parent n each generaton (P ) s selected as a soluton s unknown. common parent of next generaton chldren. The other To fnd more complete nformaton about GAs, refer parents (P (j+) ) play the role of second parents, to [8]. respectvely. Gas are beng appled to a great number of nteger programmng problems n practcal cases such as Vehcle P( + ) j= B P ( ) j + B P Routng Problems (VRP), Travelng Salesman Problems j ( j+ ) () (TSP), Producton Plannng Problems n Flexble Manufacturng Systems (FMS), Transportaton Logstcs In Table, Bell shape functons are used to calculate Problems, Path & Trajectory Plannng for robots and - the Fuzzy-Arthmetc Weghted Mean n numercal Knapsack Problems []. examples: 8

5 World Appl. Sc. J., (6): 88-86, p 5 p p Bellshape Membershp Functon p p Unversal Set Fg. : Bell shape fuzzy membershp functons and ther ntersecton ponts p 5 p p p p Trangle Membershp Functon Unversal Set Fg. : Trangular shape fuzzy membershp functons and ther ntersecton ponts Fuzzy-Arthmetc Weghted Mean (Trangular Shape Type ): Ths methodology s mplemented smlar to the frst operator (secton..) but only membershp functons that are allocated to parents are n trangular shapes (Fgure ). Equaton () determnes the FAWM crossover that s used trangular functons. P = P + ( ) P In above formula, parent n -th rung. ( + ) j T j T ( j+ ) j Tj j () s the assgned weght for j-th Fuzzy-Arthmetc Weghted Mean (Trangular Shape Type ): Fnally, ths method s runed n the smlar way wth secton.. but only trangular functons are selected for parents: Table : Bell shape membershp functons n fawm crossover P W P' + P"+ P= W= P'(+)= P''(+)= P= W=.68 P'(+)= P''(+)=6 P= W=.67 P'(+)= P''(+)= P= W=. P'(+)=7 P''(+)=7 P5=5 W5=. P'(+)5=57 P''(+)5= Notes P: generated parents n -th teraton P' +: generated parents n (+)-th teraton by usng Bell shape Functons (Type ) P"+: generated parents n (+)-th teraton by usng Bell shape Functons (Type ) P, P' +, P"+ are rounded decmal values equal wth chromosomes bnary strngs. 8

6 World Appl. Sc. J., (6): 88-86, Table : Trangular shape membershp functons n fawm crossover P W P' + P"+ P= W= P'(+)= P''(+)= P= W=.8 P'(+)= P''(+)= P= W=.8 P'(+)= P''(+)=6 P= W=.6 P'(+)= P''(+)=5 P5=5 W5=.6 P'(+)5=5 P''(+)5=8 Notes P: generated parents n -th teraton P' +: generated parents n (+)-th teraton by usng Trangular shape Functons (Type ) P"+: generated parents n (+)-th teraton by usng Trangular shape Functons (Type ) P, P' +, P"+ are rounded decmal values equal wth chromosomes bnary strngs. Table : Test problem specfcatons for soluton procedures Test Problems a tp tp (l) L d M k TP TP TP Tp: the -th Test Problem. a: total number of arports tp: total number of travelng programs(wth or Flght Leg(s)) tp(l): number of travelng programs that contan each flght leg L: number of all flght legs d: number of all tme blocks such as the days of week, month,... M: maxmum number of fare classes n CRS (CRS Capacty) k: number of fare classes consdered for tnerary n selected prce structure(prce Structure Capacty) : mean and scale parameter for exponental densty functon. P = P + ( ) P ( + ) j T T ( j+ ) j j () In Table, Trangular shape Membershp functons are appled to Fuzzy-Arthmetc Weghted Mean process n some numercal scenaros. Computatonal Results: At frst, we brng forward dfferent scales Test Problems that ther specfcatons, are summarzed n Table. The number of bnary varables (X jk & W m) and equalty & nequalty constrants are compared n table, for each Test Problem. We determne results n the vewpont of calculaton tmes and average profts for each problem. The effectveness of ntroduced crossover operators has been examned by means of Test Problems. CPU tmes are n seconds and have been calculated snce the start of soluton process. Numbers are average of ten ndependent smulatons after complete generatons. All procedures are smulated by a Laptop computer wth an Intel Core Duo. GHz processor. Table : Dfferent dmensons of the test problems Test Number Number Equalty Inequalty Problems of Wm of Xjk constrants constrants TP 6 8 TP 6 8 TP Table 5: Classcal genetc algorthm specfcatons GA Parameters Type or Quantty Populaton Sze Selecton Functon Unform Creaton Functon Unform Elte Number Mutaton Functon Adaptve Feasble Crossover Fracton.5 Crossover Functon Scattered Generatons Table 6: Calculaton tmes for crossover types (Problem ) Crossover Types TP TP TP Scattered Sngle Pont Two Pont Intermedate FAWM Bell ) type : ) FAWM Bell ) type : ) FAWM Tr ) type : ) FAWM Tr ) type : ) Table 7: Average proft for crossover types (Problem ) Crossover Types TP TP TP Scattered Sngle Pont Two Pont Intermedate FAWM Bell ) type : ) FAWM Bell ) type : ) FAWM Tr ) type : ) FAWM Tr ) type : ) Table 8: Calculaton tmes for crossover types (Problem ) Crossover Types TP TP TP Scattered Sngle Pont Two Pont Intermedate.5..6 FAWM Bell ) type : ) FAWM Bell ) type : ) FAWM Tr ) type : ) FAWM Tr ) type : )

7 World Appl. Sc. J., (6): 88-86, Table 9: Average proft for crossover types (Problem ) Crossover Types TP TP TP Scattered Sngle Pont Two Pont Intermedate FAWM Bell ) type : ) FAWM Bell ) type : ) FAWM Tr ) type : ) FAWM Tr ) type : ) Basc Genetc Algorthm that soluton procedures started wth, are specfed n Table 5. Tables 6 to 9 represent the calculaton tmes and average profts for Problem and Problem, respectvely. The followng results are obtaned wth consderng of Best Soluton Tmes and Average Profts between Crossover Types for each Test Problem. Green ellpses show low Soluton Tmes, Blue ellpses ndcate hgh average Profts and the Red ones represent low Soluton Tmes and hgh Average Profts n one case, smultaneously. Problem : In above tables, not only fuzzy crossover methods enhanced the qualty of solutons sgnfcantly n most cases, but also they had farly good calculaton tmes. Repeatng the fttest parent n all chldren, leads to great cancellaton of GA random capablty, n second and forth operators. As a result, soluton ablty of these methods was decreased. Then, less than our expectaton, these methods could not have sutable solvablty, especally for problem. It s predctable that wth reducng crossover rate and ncreasng mutaton rate n the populaton of chromosomes, ths defcency can be easly compensated. In other words, algorthms wth hgh degree of randomzaton return better resultants n soluton process. Membershp Functon Wdth Effect on Algorthm Qualty Bell Shape Functons: Fgure shows the effect of Bell shape wdth on the soluton process speed and the solutons ftness value. Problem Problem Fg. : Bell shape Functon wdth effect on the calculaton tme and the qualty of solutons 8

8 World Appl. Sc. J., (6): 88-86, Problem Problem Fg. : Trangular Shape Functon wdth effect on the calculaton tme and the qualty of solutons Calculaton tmes were approxmately fxed for consderng hgher weghts for ftter parents. However, dfferent wdths of membershp functons. Algorthms exceptonal cases also can occur wth neglgble were speeded up a lttle wth ncreasng the wdth probabltes. Hence, the lnear weghts are more sutable (decreasng the ). and consequently they have better performances. In most cases, the qualty of returned solutons was ncreased usng wder membershp functons. Due to CONCLUSION stochastc characterstc of GA, procedures were also encountered wth exceptonal cases. In ths paper, som new heurstc algorthms have been In other words, algorthms were sent back ftter developed to solve effcently Prcng & Seat allocaton outputs by ncreasng Bell shape functon wdth (closng problem n arlnes. These new methods use fuzzy to lnear form (Trangular Shape)). membershp functons to mprove crossover operator n Genetc Algorthm. We also search to fnd the best wdth Trangular Functons: Fgure shows the effect of n Bell shape and Trangle shape Fuzzy Membershp trangular shape wdth on the soluton process speed and Functons. the solutons ftness value. Our smulatons confrm that the wdest Bell shape Wth narrowng trangular membershp functons functon and the narrowest Trangular one are the best (decreasng the ), calculaton tmes closed to selectons for solvng represented problems. mnmum value. On the other hand, average profts of Two dfferent problems are formulated. Problem are nearly fxed values and even unbelevably, Reachng to Maxmum Revenue s the object of cost the average proft of Problem reaches to maxmum functon and system constrants such as Flght Leg values! Capacty, CRS Capacty, the number of defned fare So, the narrower shapes are more approprate n classes and the other ones are also assumed n model solvng mentoned problems wth GA, because of defnton. 85

9 World Appl. Sc. J., (6): 88-86, Each soluton s a convex hull of BIP problems.. Maoller, A., W. Raomsch and K. Weber, 8. Arlne network revenue management by multstage Fnally, the followng suggestons are recommended stochastc programmng. Computatonal Management for future works: Scence. 5-: Gosav, A., N. Bandla and Tapas K. Das,. A Usng more completed models wth other effcent renforcement learnng approach to a sngle leg arlne parameters n prcng and seat allocaton process revenue management problem wth multple fare such as Qualty Level, Safety Level and Satsfactory classes and overbookng. Sprnger Netherlands - IIE Degree that s delneated for all arlnes. Transactons, Applyng search methods that can contrbute to. Xu, J., A. Lm and M. Sohon, 8. Solvng the speed up convergence of GAs such as Messy GA. herarchcal data selecton problem arsng n arlne Utlzaton of Hybrd Algorthms n soluton process revenue management systems. Internatonal Journal that are used to combne the abltes of classcal and of Revenue Management, heurstc methods to return faster and overprecse. Bertsmas, D. and S. De Boer, 5. Smulaton-Based outputs. Bookng Lmts for Arlne Revenue Management. Operatons Research, REFERENCES. Klophaus, R., 6. Arlne Revenue Management n a Changng Busness Envronment. Proceedngs. Smth, B.C., J.F. Lemkuhler and R.M. Darrow, of the 5th Internatonal Conference RelStat Yeld management at Amercan arlnes Interfaces, : Laskar, E.C., K.E. Parsopoulos and M.N. Vrahats.. Feldman, J.M., 99. Gettng serous on prcng. Ar Partcle Swarm Optmzaton for Integer Programmng; Transport World. (): Unversty of Patras: Artfcal Intellgence Research. Shumsky, R., Coordnatng Revenue Management Centre; Greece. Decsons n Arlne Allances. Unversty of 6. Kuyumcu, A. and A. Garca-Daz,. A polyhedral Cncnnat: Cncnnat: USA. graph theory approach to revenue management n the. Integrated Decsons and Systems, Inc. The Bascs of arlne ndustry. PERGAMON Computers & Industral Revenue Management; ID-MK--v-YMBasc Engneerng, 8: Keyvan, S., Mrrazav, Dylan F. Jones and M. Tamz, 5. Henk, C., Tjms.. A Frst Course n Stochastc. Theory and Methodology: A comparson of Models, John Wley and Sons, Ltd; Great Brtan. genetc and conventonal methods for the soluton of 6. Sadegh, M. and S. Khanmohammad, 8. Modelng nteger goal programmes. European Journal of of Revenue Management procedure n Ar Operatonal Research, : Transportaton. IAMOT Conference on Management 8. Randy, L. Haupt and Sue Ellen Haupt,. Practcal of Technology, Genetc Algorthms. nd Ed. John Wley and Sons, 7. Feldman, J.M., 99. To ren n those CRSs. Ar Inc.; Hoboken: New Jersey;. Transport World. 8(): Gunther, R. Radl, An Improved Genetc Algorthm 8. Gregory, M., Coldren, Frank S. Koppelman, for the Multconstraned - Knapsack Problem. K. Kasturrangan and A. Mukherjee,. Ar Travel Venna Unversty of Technology: Venna: Austra. tnerary share predcton: Logt model development. Tmothy J. Ross, 997. Fuzzy Logc wth Engneerng at a major U.S. arlne. 8nd Annual Meetng of the Applcatons. McGraw-Hll, Inc.; Sngapore; Transportaton Research Board, Washngton D.C. USA. 9. Asf, S. Raza and A. Akgunduz, 8. An arlne revenue management prcng game wth seat allocaton. Internatonal Journal of Revenue Management, -: