Yugoslav Jour na l of Operati on s Research
V( l VVVJ , Nu mber 1, 27-34
UM ENTROPY AND UTILITY IN A TRANSPORTATION
SYSTEMS. K. MAZUMDER
or
Department Mat hematics,
Bengal Engineering Collcge (D. U)
H oiorali - 71 1103, Wcst Bengal, Ind ia
N. C.DAS
Department ofApplied Mathematics
Un ivers ity Calcutta
West Bengal, India
or
Abstract: The object of the present paper is to stu dy the interrelation and equ iva lence
between m ethods of maximum en t ropy and utility in determining the trip distributions
in a t r a nspor ta t ion system .
Keywords: Maximum e nt ro py, ut ility, tri p distribu ti on , t ra nsportatio n syste m, Bose-Ei nstei nand
Fermi-Dirac e ntropies.
•
1. INTRODUCTION
In recent years t here has been a sea rch ing concer n with t ransport a t ion
problems in urban a nd enviro nmenta l modelling. The su bject has become an active
fie ld of interest and scientists of different ba ckgrounds are t rying t o enrich th is field by
diffe rent approaches . A recent and more genera l approach is based on t he concept of
en tropy [Sh a n no n & Wea ver, 1949 1and the principle of maximum entropy estimation
IJaynes, 195 71 . This approach was developed profoundly by Wilson 11967, 1970, 1979 1
a nd others [Webber, 19791.
.
Another rou te to t he problem is t hrough t he concept of the eco nomic model of
u tility . T he interrelation between maximum entropy a nd maximum u t ility was poi nted
. The aut hors wi s h to t ha n k P rof. C.G. Cha krabart.i, S. N. nose Professor of Applied Math ematics,
Calcuu a University, for h is kind hel p in t he preparati on of t he pape r.
:.!t\
S .I<. Ma zumde r. N .C . Das / Maxim um Entropy and Uti lity ill a Tran sportal inn System
out by Wilson 11 9711 . The object of t he present paper is to stu dy t he equ ivalence or
interrelation between t hese t wo pr inciples in the context of t r ip dis tribu tion . The
a ppr oach is, however, differen t from that of Wilso n 119711 a nd is more in line with that
of Beckmann 119741 .
T he paper is planned a s follows : In Section t2) we co nsidcre trip distributio n
ba sed on the maximum entro py (Shan no n) model and derive t he same dis tribu tions
from t he maximurn-u tilit ,v model with different cost fun ctions. In Section (:3) we have a
simila r task with the Bose-Einstein a nd Fermi-Dirac form of entropy.
2. TRIP DISTRIBUTION: THE ENTROPY (SHANNON)
APPROACH
Let us conside r a model citv
. with a Central Business District te BD).. a nd a set
of locations or sites :i. i = 1.2. .... /1 : and a set of working places or sectors
;j. j = 1.2... .. m: of the C.B.U.
•
Let 1i J he t he number of t r ips or igi nating from the ith loca t ion place to the
jth working place (dest ina t ion). Then the en t ropy of the trip distribu tion is give n by
I aualogou s to Sha n non t he entropy):
"
S
111
= - ")
__ "T··
L- U III
i I .i--I
T ~I
( 1)
Let us assume that the tota l t r ips originating from site i and the tota l num bel' of
persons em ployed in t hej-th working place he prescribed :
11/
_
Tij = A i .
(/.. -- 1.-.) ., .. • 11 )
. 1
.I
II
'
Hi :
L.. l I.J =
U. -- 1. .)~ ... .. Ill )
t- I
a nd fur t her t hat the total cost of trans portation he fixed:
11
/1 1
~
"~ Y
··C( J'··)
~ T I)
lJ =C
i- I j - I
where Jl.i is t he distance of t hejth sector in t he CBo from the ith location site a n d
C(J'iJ ) is t he cost fun ction. Constraints l2) a nd l:3) are insufficien t to determine trip
•
dis tribu tion 1ij ' This ca n, however, be estima ted by J aynes' maximum-ent r opy
principle 1.laynos, 19571 .
S .K. Ma zumder, N .C. Da s I Maximum Entropy and Ut ility in a T ra ns por tatio n Sy stem
:W
According to this principle, the least biased distribu tion will be t hat wh ich
maximizes en tro py S gi ven by (1) su bject to const raints (2) and (:3). The maximiza tion
y ield IWilson , 19G71is
T IJ.. --
0 "
b
I
J
1'
;.C(/i. )
(4 )
-
Where parameters 0; , bj and ). are to be determined by the equat ions:
11/
_
a 1· bJ. c i.c( l'v· ) -_
A.
'J
.n )
(t.. - 1._....
1
./- 1
(5 )
1/
c:
a.b,«
;.CI1;,)
= H
, I
'J . ... 11I )
(j. -- 1._.
./
and
( G)
,
2.1. Trip Distribution: Utility Approach
In the above discussion the distribution was determined as a problem of the
theory of information. Since the information (or constraints ) available to us are
insufficient to uniquely determine the exact distribution , we used the maximum
e ntropy principle of stat ist ica l inference. Let us now try to set up an eco nom ic model of
chuice behaviour tha t underlines tr ip making. Here we are to distinguish the t r ips by
pu r poses. In any city work t r ips are really the result of some basic cho ices. A person
usually accepts a job first or decides to form a bu siness centro and the n looks for a
su ita ble place to live. Now, in looking for a place to live he will always try to maximise
his u tility as far as possible with limited available resou rces,
A person looking at variou s potential residences associates a rating or u tili ty
index with each. Let the probability density function of utility x for a particular
lucatiun i be Ii ( x), x > a > O. Then the probability of some thing in such a loca tion
having utility equa l to tc or greater (better) than u. is given by:
P (II ) =
f l ;(x )dx
II
This utility u includes rent, access to schools, hospi ta ls, gas connection , local
shopping facilities , elect r icity board etc., except the distance from CBD. So the net
utility of a place is the utility minus the transportation cnst, t u - Kr;t. ) where r;J is the
distance of the ith secto r from the j th working place in t he CBD and K is tilt'
conversion factor of distance to utility.
•
S.K Mazu mde r, N.C. Das I Maximum E ntro py a nd Utility in a Trans portati on Syste m
30
•
A person will now t ry to maximize the net u t ility . Since the econom ic power of
each person is different , he will accept a place to live with a sa tisfying level of utility ~
(say) so that:
(u- K rij » s
or
u ~ s+ K rij
t8J
Now if Ai is the tota l number of hou sing su pply in t he it h sector a t distance
l ij
from the jth working place, then t he probability of a pe rson living in th is level is:
A-I P (s + K r 1)·· )
HJ
,
If the total number of per sons employed in the j th working place in the CBD is
then [Beckman n, 1974 1:
•
no)
•
'J'..
- A-IH
. p es + K r 1).. )
IJ
J
Com pa r ing l4) a nd nO ) we observe t hat:
a I·bJ.e "e Cru )
A iB , P( . '
+ K r ij )
or
un
C(rij ) - In P t S + Krij ) + In(A i H J ) - In(a i b j )
this shows that a cost ca n a lways be found so t hat ent ropy maximiza t ion
a nd utility maximizat ion become equ iva lent. Let us now illu stra te the equ iva le nce with
some specific examples by fi nding the a ppropr ite cost functions .
0 ,
Examples:
U)
•
First su ppose the distribu tion of ut ility to be t he negati ve exponent ia l:
X ~
( i ( x) = ear ,
0
l 12)
Then from 0 0):
A , BJ
"
f
l' rzr
s< KI'v
dx
I \ I' B J
e
([(Sf
Kru )
a
ow from ltl) we s ee that if the cost of t ravel C( rU) is of the form :
then t h ' two m it hods lead to the sa me ty p , of distribu ti on.
•
S .I<. Ma zumder,
.u:
I '. ' .
Das I Maximum E nl ropy a nd Ut ility in a Tra n por tnt run ~y. 1.. 111
:II
If
·
('
[, (xl = --
X 11/ I
X
-
-
.
I
X c... O
I ,I I
Then
"
('
.. . It, I
.\
!\"J)( A I
~
A 'I. ' 'IJ +··· ·1 A 11/ ' ,11/
'J
I )
which leads to e n t ropy muximization distribution if we take :
'/I
C(rU )
I(S + K rU ) - In
I
"
( II \)
I
A"
I'I' I
I
( I 5)
I
If
'
'I.
a
/ i( x ) = - .
x
11 G)
x ~o
Then
"
'J'.I) - A-' H ) .
Jn. r
s+ L ' v
'J
A , H) 0 ~
a2
- dx - - ---..:::-,,-'J
x
( S+ K r IJ )-
which is the ge neralised gravity model . We observe tha t if we take th ' cost fun ction :
( I7)
G(r,j ) - In (8 + Kr,) )
the maximum-entropy distribution leads to the utility distribu tion .
a.TRIP DISTRIBUTION: THE ENTROPY (BOSE-E INSTE IN
AND FERMI-DIRAC) APPROACH
In th is sectio n we sha ll try to obtain trip distribution bas sd on en tropies oth sr
than that of Shan non a nd exam ine the ro le of the utility fun ction in ge ne rat ing' these
types of distributions.
Let T ij be the number of trips from the i th secto r (or igin: to th 'jth working
place , and the constraints or information available be t he sa me as those o f (~ ) a nd (al .
Now the problem is to estim ate
7:, on t he basis of information tco ust ru iuts ( ~ l
and (3)) . We apply Jaynes' maximum e nt ropy principle wit h quantum mea sure of
en t ropy :
:~ ~
S .K. Ma zu mder . N.C. Das I Maximum Entropy and Ut ility in a Tran spo rta t ion System
..
"'"
"
",
S = - ,L-L.
~ T I}
·· In T IJ- +a Y
~ 'L. (1
;= 1j = 1
;=J j = 1
us)
- sr:(/ j ln (1 +a T --)
I)
where a = +1 for the Bose-Einstein en t ropy and a = - 1 for t he Fermi-Dirac en t ropy .
,
The maximization of en t ropy S su bject to co nstraints (2) and (3 ) leads to the
distribution [Kapur, 1990 1:
1
TI}-- = ----,-::--' C(
(19 )
b
,{
,;,
)
a I· .I-c
- -a
T he value a = 1 in (19) corresponds to the Fermi-Dirac distribu tion of t r ips . In t h is
ca se at most one t rip end is permit ted per destination (job ). The value (I = - 1 in (19)
corresponds to the Base-Einstein distribu tion of trips. This corresponds to unlimited
ends per destinat ion [Fisk, 198 5 1.
We s ha ll discuss the fea sibility of distribution (19 ) in reality la ter
O I l.
3.1. Trip Distribution: Utility Approach
We have seen in sect ion (2) t ha t :
1'..(/ - AB
-P('s+ K r IJ-- )
I
J
\ :W )
a nd
,
P Ut) =
f 1";( x) clx
/I
where 1"; ( x) is some kind of u tility fu nction for t he i th loca tion. This may be till'
potential fu nction of thejth work place or may he some function wh ich depends on tilt'
utility of t he ith origin (sector -living place ) and the at traction of t hejth work place . A ,
-
lind B j lire some prescribed values relat ed to t he i t h or igin (living place ) a nd j t h
destinatio n (work place ). Aga in we observe fro m UO ) a nd (19 ):
AI IJ .I-P; S + K,.--)
u
1
a Isol .I.c
,le l ,. -)
'.I - a
II I"
a ,b -c
'.1
1
'+a
A I H .I-}J(S + K ,.-)
(/
,lC ( /;. )
-
S .K. Mazumcler, N.C. Das I Maximum Entropy a ncl Utility in a Transportati on System
or ,
a
C( rij ) _ 111I- - +
1
I
a.b
ko
·· )
1 .1.
11·H .1·b.1·P (S + K r I.J
::J::J
(21)
So. costs ca n abo be found which can make entropy maximization and u tili ty
equ ivalent. The equ iva lence will abo be illu strated with some specific exam ples.
Examples:
rj l
-
Let u s consider utilitv function :
(Ie
.r
( i (x l =
'J
.r
•
> (/
(e.( +a ) -
Then we have:
".
'I': - A H ·
I.J
I
J
f
(leX
s+ K J ! (e
x
l
a
'J
+ 0)-
oA ·H ·
dx - kH . _ --;;-__ _
I
J
I
J
" . Kr.
~ ..·K,.·
e
'J + 0
e
u +0
We observe t ha t, fur a = + 1 , t he distribution (23) so obtained resembles the quan tum
distr ibu t ion of trips with C( rii) - S + Krij .
(ii )
We have already seen that the utility distribution follows a negati ve
exponentia l as
T·.
IJ
Fi(x ) = C
"" , X
>
O. '}ii turns ou t to be of t he form :
- 11 : s+ K ,; . )
.
.e
A B
I
J
J
(241
ex
which leads to the quantum distribution of trips if:
(25)
4. CONCLUSION
The paper is concerned with two approaches to the problem of decision making
ill a tr a nspor t ation system. The two approaches are based on the concept s uf ent ropy
and utility. In the present paper we have tried to show the interrelation between till.'
above two approaches mathematically in the case of trip distribution. In this sense the
pa pe r is of a theoretical ty pe. But it is not purely hypothetical because S OIlW of tho
fun ctions u sed in this paper have been su ccessfu lly applied in some other models. For
exam ple the exponentia l utility function has been used in the risk-sharing problem
IKapur, 19901 . The choice uf the utility fun ctions ((x) is somewha t of an ad hoc
nature. It will depend un different econom ic problems or situat ions an d its success will
also be based un the proper choice of the utility function [T'rihu s, 19691. In t his paper
•
:34
S.K Mazu mder, N.C. Das I Maxi mum E ntropy a nd Uti lity in a T ra ns portation Syste m
we have selected a number of utility fun ctions ( (x ) and have shown how the
maximum-utility method which is of great commercial importance can be converted
into a well-established stat ist ical decision theory based on the maximum-entropy
pr inciple [Wilson , 1970, Jumarie, 1990J. Regarding t he applicability of the BoseEin stein and Fermi-Dirac en tropies we state that though the Shannon en t ro py has a
wide range of applica bility , Bose-Einstein and Fermi-Dirac entropies have also been
applied successfu lly in the case of work- trip distribu tion and commodity distribu tion
respectively [Fisk, 19851 . Ours is t he first a pproach to show the equ iva lence between
the maximum u tility and maximum-entropy principle based un the Bose-Einstein and
Fermi-Dirac entru pies.
A large number uf papers have been written applying the maximum-entropy
technique and maximum utility techn iques but ou r aim is to show that buth ent ro py
a nd u tility can be adopted by skillfu l proponents to explain almost any form of
tr anspur ta t ion problem by u se of either an appropriate ent r opy 01' an utility functiun .
•
REFERENCES
•
III
Bec kma nn, M.J , "E nt ropy, gra vi ty a nd utility in transportation modelling", in: G. Men ge
(ed.) , Inform ation , Interference and Decision , D. Re idel tra n I. to Dordrecht , Hollan d, HJ74.
121 E vans, A W., "Some proper t ies or trip d istr ibu t ion models", T ran sportation R esearch , 4 ( HJ7 0)
19-36.
131 Fisk, C., "E n t ropy a nd in fo r mat ion t heory : Are we missi ng so met hing'!", Euu. and p lannuig;
14A (1985 ) 679-7 10.
141 J ayn es, E .T., "In forma t ion t heory a nd statistica l mechani cs", Physical Reoicuis , 106 ( H157J
620-630 .
15 1 J um a r ie, G., R elative lnformation and Application , S pringe r-Ver lag, Berlin , 195 7.
161 Kapur, J .N., M aximum Entropy Models in S cience and Engineering , Wil ey Easter n, Ne w
Delh i, H190 .
17 1 S ha n no n, C.E., a nd Wea ve r, W., Math em atical T heory of Com m unication, U nive rsi ty 01"
Illinois P ress, Urbana, Chicago, U149.
1 I Tri bu s, M., R ational Description , Decision and Desig ns , Pargamon Press, New-Yor k, 1969.
191 Wilson, A.G ., "A stat istical t heo ry 01" s pat ial dist ribut ion model s", T ransportation Research . 1
(1 967 ) 253-269.
II0j Wi lson, AG ., Entropy in Urban and R egional M odelling , Pio n, London , H170.
II 11 Wilsnn, AG ., Optimization in Localion al and Tran sport A nalysis , J oh n Wiley , Ch iches tel',
1979.
II 21 Web ber, M.J., lnformation Theory an Urban Spatial Structures , Croo m He lm, Lo ndo n, UJ7!J.
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