In. J. Pue Appl. Sci. Technol., 4 (211, pp. 23-29 Inenaional Jounal of Pue and Applied Sciences and Technology ISS 2229-617 Available online a www.ijopaasa.in eseach Pape Opizaion of he Uiliy of a Sucual Model of he eand fo Muli-esinaion on-ok Tavel Using Maxu Enopy Mehod Kakali Kaaka (Su 1,* and Sana Kua Majude 2 1 epaen of Maheaic B.P.Podda Insiue of Manageen and Technology, Podda Viha,137,V.I.P. oad,kolkaa-52, es Bengal, India. 2 epaen of Maheaic Bengal Engineeing and Science Univesiy, Shibpu, Howah-71113, es Bengal, India. *Coesponding auho eail: (kakali_ks@ediffail.co (eceived: 2-2-211; Acceped: 21-3-211 Absac: In his pape we axize he uiliy of a sucual odel of he deand fo uli-desinaion non-wok avel using axu enopy ehod inoduced by Shanon. A uiliy value is assigned when only paial infoaion is available abou he decision ake s pefeences. The axu enopy uiliy soluion ebeds a lage faily of uiliy funcions ha includes he os coonly used funcional fo. The odel pesened hee incopoae avel fequency, desinaion choice and ode choice fo boh single and uli-desinaion avel ino a unified uiliy-axizing faewok. Keywods: Opizaion, Uiliy, Enopy, on-wok avele esinaion, Mode of avel. 1. Inoducion In ecen yeas uiliy axizaion has eeged as a fundaenal behavio pinciple of uban passenge avel deand odeling. Accoding o his pinciple, as individual s pefeences fo he avel opions he face can be descibed by a uiliy funcion, and each individual chooses he opion ha chooses he opion ha axizes his uiliy. In his pape we use he axu enopy ehod o axize his uiliy of avel opions [4,5].
In. J. Pue Appl. Sci. Technol., 4 (211, 23-29 24 The planning of his pape is as follows: 2. [A].evelopen of he odel. [4] 2. [B]. Maxizaion of uiliy using enopy axizaion ehod [5] 2. Modeling and Opizaion 2. [A]. evelopen of he odel The developen of saisfacoy uiliy axiziing he odel [4] of he deand fo ulidesinaion non-wok avel has poved o be consideably oe difficul han he developen of wok ip ode choice odel. This is due ainly o he vaiey and coplexiy of he avel opions available o non-wok aveles. These opions ypically include he fequency, desinaion and ode of avel, aong ohe facos. The odels of oencich and Mc Fadden (1975, Adle and Ben-Akiva (1976 and Chales ive Associaes(1976 pei non-wok(coplee hoe o hoe ound ips o non-wok desinaions pe household pe day and do no pei uli-desinaion ous all ae subsued in his odel.[3]. e A be he se of non-wok ips ha ae available o he ebes of a household. The ip is an eleen of A if iand j is non-wok locaions (possibly including hoe available o he household, and is a ode ha is available o he household fo avel beween iand j.i is no equied ha iand j be disinc. e be he e of day and le be a e ineval sufficienly sho ha he ebes of he household can begin a os one ip duing he e peiod o +. e A A be he se of ips ha household ebes can sa duing o +. The avel opions available o he household duing o + ae Opion 1: Begin a peson ip fo oigin i o desinaion j by ode ( a ou fo hoe o one o oe non-wok desinaion o hoe. A as pa of Opion 2: o no begin non-wok avel ha is pa of a hoe non-wok hoe ou. A uiliy values is associaed wih each of hese opions. The household is assued o choose he opion wih he highes uiliy. The uiliies ae given he following funcional epesenaions: Opion 1: U z, + ε Opion 2: U ( + ε hee: Udeeinisic coponen of uiliy. x a veco of anspoaion level-of-sevice vaiables elevan o he choice. s a veco of household chaaceisics. z a veco of desinaion chaaceisic ohe han anspoaion level-of-sevice ha ae elevan o he choice. nube of peson ips. ε ando coponen of uiliy. is included in he uiliy funcion in ode o capue he effecs of pas avel decisions and fuue avel plans on cuen avel decisions.
In. J. Pue Appl. Sci. Technol., 4 (211, 23-29 25 To deduce he sucue of a deand odel fo he uiliy funcions and he pinciple of uiliy-axizaion i is necessay o specify he pobabliy disibuion of he ando uiliy coponenε. F (ε exp [-exp (-ε ]. hee F is he cuulaive pobabliy disibuion funcion of he ando vaiableε. The pobabliy ha a household ebe chooses o begin he ip + is: pq A A duing o exp( U P( /, A, (2 [exp( U + exp( U ] as appoaches zeo hen he pobabliy of avel duing o + also appoaches zeo and fo sall, he pobabliy of he ip A is popoional o p( /, A P( / A. (3 This plies ha fo sall he deeinisic coponens of uiliy can be epesened as: pq U U z, (,, V V ( z,,, + log (4 Thus fo sall P( / A [exp(v -V ]. (5 o, P( / A [exp(v ] (absobing V inov ow he aginal pobabliy of choosing : P( /, P( / A P ( A P( /, (expv P ( A P ( /, should be a deceasing funcion of if i is hoe and also i is no known a pio. P ( A / i ho e B( [1 k( ]. (7 B and k ae funcions of he household chaaceisics and e of day. ow le i be a non- hoe locaion hen, P, (8 ( A / i hoe, P(/ P / { ( (expv }; i hoe (9 (6 (expv ( + 1+ [exp( V V ( ] ; i hoe Since a os one non-wok ip can begin duing o +,P(/ is equal o he aveage nubes of ips fo i o j by ode ha sa duing o +.The aveage nubes of ips pe day fo i o j by ode, heefoe can be obained by inegaion P( / d.
In. J. Pue Appl. Sci. Technol., 4 (211, 23-29 26 hee he inegal exends ove a day. oe ha and ae elaed by. (11 j h hee he subscip h signifies hoe. Fo equal o o 1,define ( [ log exp( V d]. (12 ( ( by ( is he e inegaed uiliy of avel fo i o j by ode,given ha ohe ips o non-wok desinaions ae ade duing he day. Using equaion, (11, (12 and inegaion of (9 ove a day yields (exp ; i hoe (13 ( ( (exp {1 + [exp( ]}. (14 If a household has j non-wok desinaions and M odes available o hen (11, (13, (14 define a syse of MJ(J+1+1 equaion in he MJ(J+3+1 unknown quaniies,,, and whee j and ange ove he available desinaions and odes. To obain a unique soluion fo he unknown quaniie i is necessay o add 2Mj equaion o he syse epesened by (11.(13,(14 we ge ; i ho e, (15 j j ; j ho e. (16 i i Equaion(11&(13,hough (16 consiue a syse ha is solvable fo,,, and. ow we define 1+ [exp( ( k ] (17a and 1 ; i hoe ( 1, (17b j whee p,q, o be he pq eleen of aix. ( ( [exp + pq, exp( hp + p pq, q h pq, exp( hp + ( ]/ (18 and
In. J. Pue Appl. Sci. Technol., 4 (211, 23-29 27 (exp k ; i h. Equaions (17a,(18,(19 can be splified gealy if i is assued ha he uiliy funcion ( can be wien in he fo [1] & [4] (19 ( * F + H ( s; j h F + G ( s; i h (2 F + H ( s; j h The e inegaed uiliy funcion can be epesened as he depends on anspoaion levels of sevice, household chaaceisics and desinaion chaaceisics bu ha independen of pas avel decisions and fuue avel plans and a coponen G o H ha epesens he effec on e-inegaed uiliy of cuen avel of non-wok ips ha have been o will be ade a ohe es of day. Because avel behavio depends only on he diffeence beween he uiliies of avel opion hee is no loss of genealiy in assuing ha H * ( s is zeo fo all and s. Using he uiliy specificaion (2 equaion (17a, (18 and (19 becoe 1+ [expg kj (21 ( s expg1 ( s] j. k, exp[ F ( {exp[ F + G ( s] k 1 exp[ F + H ( s]} k + G ( s]; j h exp[ F + G ( s]; i h. (22 (23 (24 In addiion, heoal no. of sojouns a non-wok locaion which can be obained by suaion fo equaion (22 and (23, is exp[ ( kj, F + G s (25 Equaion (22 o (25 ogehe wih he definiional elaions (17b and (21 consiue he desied odel of non-wok avel deand. ow we have o opize (2 subjec o consain equaions (24 and (25. [1,2] 2. [B]. Maxizaion of uiliy: ow we axize ( F ; j h
In. J. Pue Appl. Sci. Technol., 4 (211, 23-29 28 F + G ( s; i h (26 Subjec o he consain * F + H ( s; j h ( ( s H and s {exp[ F + H1( s]} k exp[ F + G ( s]; i h And exp[ ( kj, F + G s ] (27 e use enopy axizing ehod o opize he uiliy The agangian of he above odel is as follows: [5] (x,z, λ, λ ;( 2 j h { F 1 z + λ 1( {exp[ F + H1( s]} k exp[ F + G ( s]; i h + λ 2 ( kj, exp[ F + G ( s} ow x Gives he value of ( x ( {( x k + x x x λ2 k } + { + x λ x } F x and gives he value of s 1 x x x (28 λ [{ s }{ + + s 1 k s s λ1[ ] [ s λ2[ { s F s ] s s kh s ] }] (29
In. J. Pue Appl. Sci. Technol., 4 (211, 23-29 29 Also fo gives z [ z λ1 [ λ z 2.(3 { z z + z z + z z }{ ] F( x, z F z + F z }] Since G and H is independen of pas avel decision, hen hee cases of e inegaed uiliy funcion (26 can be expessed by he fis one. Thus fo he above equaions (28, (29 and (3 we ge he values of,, which opizes he uiliy funcion.[1,2,4] 3. Conclusion: In his pape we opize he uiliy funcion using enopy opizaion ehod insead of uiliy axizing ehod. ow consideable aenion has been paid in he anspoaion eseach lieaue o he inepeaion of anspoaion levels of sevice, household chaaceisics and desinaion chaaceisics fo he applicaion of above odel. Since his objecive funcion is inaely elaed o enopy axizaion, soe geneal conclusions ae dawn on he elaion beween he popeies of odel based on enopy axizaion. 4. Acknowledgen: The auho hanks he Maheaics epaen of he Bengal Engineeing and Science Univesiy, Shibpu, Howah fo he hospialiy duing he e when his eseach was caied ou. efeences [1] Abba Ali E, Maxu enopy uiliy, Opeaion eseach, 54 (26, 277-29. [2] Abba Ali E,An enopy appoach fo uiliy assignen in decision analysis. C.illiaed.Poc.22 nd Inena.okshop on Bayesian Infeence and Maxu Enopy Mehods in Sci. Engg.Moscow,I,22 [3] M. Foian, Uiliy, enopy and a PAAOX of affic flow, Tanspoaion eseach 15(1981, 327-33. [4] Hoowiz, Joel, Tanspoaion eseach-b; A uiliy axizing odel of he deand fo uli desinaion non-wok avel, 14B(198, 369-386. [5] J.. Kapu, Maxu-Enopy Models in Science and Engineeing, iley Easen ied, 1989, 37-7.