A new approach to modal split analysis: Some empirical results

Trmpn. Res:B. Vol. 238. No Prmted in Char Bntain.

1. pp 75-82.

0191-161589 s3.w+ .@I C 1989PeqamonPress plc

1989

A NEW APPROACH TO MODAL SPLIT ANALYSIS: SOME EMPIRICAL RESULTS Department

of Mathematics,

A. REGGIANI University of Bergamo, Via Salvecchio 19, 24100 Bergamo. Italy and

Department

S. STEFANI of Statistics, University of Brescia, Corso Mameli 27, 25122 Brescia. Italy (Received 29 October 1987; in revised form 8 February

1988)

Abstract-This paper describes a new approach (SD-MNL) to analyze modal choice. The SDMNL method allows one to consider travelling modes under uncertainty. In particular, different states of nature for the alternatives are taken into account. In this scenario, Stochastic Dominance rules (SD) will be applied together with Multinomial Logit models (MNL) to describe a flow pattern in Regione Lombardia, Italy. The results obtained fit statistically the observed data and give rise to interesting considerations about travelling modes.

1. INTRODUCTION During the last decade, modal split analysis has received considerable attention via Random Utility models (RUM), which provide an appealing, behavioural frame work for modelling choice processes. These models are mainly based on the principle of utility maximization. It is assumed that the decision-maker’s preferences can be described by a utility function and that the individual will select the alternative with the highest utility. However, these utilities cannot be estimated by the analyst and are therefore treated as random variables. Based on these assumptions, the random approach formalized by Domenchich and McFadden (1975) and McFadden (1974, 1979) then gives the probability that a particular alternative of a finite set will be chosen. In this context, Multinomial Logit models (MNL), which are the most well-known example of RUM, gained much popularity in travel mode analysis, mainly for their computational tractability [see, e.g. Ahsan (1982), Anas (1982), Anas and Duann (1985), Ben-Akiva and Lerman (1985), Horowitz (198.5), Stopher and Meyburg (1976), and Train (1986)]. However, the main limitation is the lack of a criterion identifying a threshold for selecting the efficient (in the economic sense) alternatives. Furthermore, states of nature for the system cannot be studied through RUM: in other words, alternatives can only be certain. It has been shown that Stochastic Dominance (SD) rules can help in overcoming both these limitations (Reggiani and Stefani, 1986). SD rules belong to decision theory under risk, that is, SD can be applied when alternatives are uncertain and their probability distributions are known (Bawa, 1975, 1982; Levy and Sarnat, 1977; Whitmore and Findlay, 1978). In fact, the choice of a travelling mode involves taking into account different states of nature for the system, like weather and road conditions, strikes, etc. SD rules are based on the maximization of expected utility? and have been applied successfully to financial and economic problems [see, among others, Hanoch and Levy (1969), Hadar and Russell (1971), and Doherty (1977)j. They allow also one to perform a preliminary screening of the (random) alternatives, by assuming rationality in the decision-maker’s choice behaviour. Furthermore, if the attitude toward risk is specified, the set of “efficient” alternatives can be further enlarged. By efficient we mean any fFor expected utility maximization, see Bernoulli (19.54). Von Neumann and Morgenstern (1944), Ftiedman and Savage (1948). Hernstein and Milnor (19U), Tobin (19.58). Mossin (1972), Drtze (1974). Fishburn (1977). Daboni (1982). and Schoemaker (1982). 75

A. REGGIANIand S.STEFAXI

76

alternative that would not be preferred to other alternatives in the set. The final result is an efficient set, to which all the efficient alternatives belong. As SD rules refer to a large group of decision makers with some common characteristics, the final choice is not given: in fact, SD rules just indicate the efficient set. It follows then that the connection to RUM can give interesting insights toward the final choice, at least in an aggregated context. The theoretical connections between SD rules and RUM have already been discussed (Reggiani and Stefani, 1986). Therefore, the application to transportation modelling seems particularly appealing: the empirical application will show that travellers assume rationality criteria and the ‘noneconomic” alternatives are in fact avoided. 2.THE

SD-MNL

MODEL

2.1 The theoretical framework for SD As already stated in Section 1, expected utility maximization is the basis for SD rules. They belong to the class of decision problems under risk, that is, it is assumed that, once the outcome states are specified, the corresponding probabilities are known. As they are based on the comparison between the probability distributions of the alternatives, a common (monetary) random variable X (with values in a set 2) is assumed to describe all the alternatives. They require a complete preordering relation in the set C of alternatives, together with the validity of some axioms, such as continuity, independence, and comparability (Hernstein and Milnor, 19.53; Dreze, 1974; Whitmore and Findlay, 1978. If this is true, it is then possible to associate a real number 8, = JZ V(x) dFi (x) to the ith alternative in the set (provided the integral exists), where V(x) represents the decision-maker’s utility function and F,(x) = P(X I x) is the distribution function of the ith alternative. It then follows i Zj(=)

1

z

V(x) dFi (x) zz i

V(x) dF, (x) i,j E C, Z

(2.1)

where z means “preferred or indifferent to.” SD rules analyze risky decisions when V(x) is not fully known, but is presumed to belong to a class I’(x) of real-valued functions. The First Stochastic Dominance (FSD) rule states (2.1) for the class of increasing utility functions. As a consequence, this dominance relation gives a partial preordering on the set C. The following Theorem also holds: THEOREM

2.1 i?j(=)Fi(x)lF,(x)

forxE2

i,JEC

(2.2)

for at least an x, such that &(x0) < Fj(x,) for strict preference. According to (2.2), FSD is really a “common sense” decision rule. As a matter of fact, increasing utilities are quite general, as they describe properties consistent with an “economic” behaviour of the decision maker. If X is interpreted as a monetary variable, profit for instance, a rational decision maker would agree immediately that his/her utility function (even though he/she does not know it completely) is increasing, in other words. more money is always preferred to less money. By using Theorem (2.1), the efficient alternatives are found, but FSD (and, in general, other SD rules) does not give any information about the final choice. 2.2 The theoretical framework for MNL Random Utility Models (RUM) introduced in Section 1 can according to McFadden (1974). Let C denote the finite set of some decision-maker h (described by a vector db of attributes universe H) and let IY,,~ denote the utility of the ith alternative

be formalized as follows, alternatives available to in the decision-maker’s for the individual h.

A new approach to modal split analysis The

probability

that h, drawn at random from H, chooses the ith alternative

77

from

C is Phi = P(llhi > U,,J i,j E C i # j h E H.

(2.3)

Statement (2.3) corresponds to the maximization of the random utility for RUM. Then, it is usually assumed to represent the random utility rlhi as a sum of two components Uhi = Vhi + 8hi i E C h E H,

(2.4)

where Vhi is the deterministic component and a,,, the random component of U,+ Vhi depends both on the measurable attributes ci of the ith alternative and on those of the decision-maker h. 8hi represents unobserved factors, such as the idiosyncracies of the individual h in his/her tastes for the ith alternative with attributes ci, or measurement errors in the data regarding the attributes, etc. Furthermore, as V,,i takes into account the average tastes in the universe H, it follows [see e.g. Leonardi (1985)] Vh, = Vi i E C h E H.

(2.5)

Then, if it is assumed that the errors terms ?J,;are independent and identically distributed across decision makers and alternatives according to a Gumbel distribution, the analytical expression of the probability (2.3) results in the framework of Multinomial Logit models (MNL) as follows PAi = exp Vh, / /

By using (2.5) it is straightforward written as

c

exp V,, i E C h E H.

k

(2.6)

to see (Reggiani and Stefani, 1986) that (2.6) can be

Pi = exp vi

/

c exp vk i E C

(2.7)

k

under the hypothesis of a deterministic utility linear in the attributes of the alternatives and of the individuals. Finally, it has been shown [see again Reggiani and Stefani (1986)] that RUM are consistent with the decision making under risk framework. 2.3 The theoretical framework for SD-MNL approach As pointed out in Sections 2.1 and 2.2, SD and MNL can be reconducted to the maximum utility principle, so that it is plausible to identify a new approach based on a sequential procedure SD-MNL. To this purpose, it can be defined r/hi = Uhi(Vi(X), 6hi) h E H i E C,

(2.8)

where X is underlying all the alternatives, as it has been stressed in Section 2.1 and shi is a stochastic term representing the individual h. Next, if we introduce first the SD approach, defined in Section 2.1, a set of inefficient alternatives can be disregarded so that the whole set C can be reduced to the subset c, with cardinality E I n. It follows that the expected utility for the ith alternative, conditioned on Ehi, is from (2.8), (2.4), and (2.5): Ei( Uhi ) &hi) = Ei(Vi(X))

were Ei is the expectation

operator

+ 8hi = Vi + 8h; h E H i E Z,

for the ith alternative.

(2.9)

78

A. REGGIANIand S. STEFANI

Note that from (2.9) we implicitly assume that 8,,, are not state-of-the-world dependent. In other words, that means that the decision process is affected by the states of nature on the average only. This hypothesis allows, as can be seen in the following, to attain an analytically tractable and easily testable model. At this point, the utility maximizing hypothesis can be applied, as defined in (2.3) so that: Phi = P(Vj + Shi > V, + S,)

h E H i,j E C i # j.

(2.10)

Then, if we consider the particular assumptions leading to MNL models the choice probability can be formulated as follows, for the reduced set c: Pi = exp V;

/

C exp V, i E C.

(2.11)

k

In synthesis, this combined method DS-MNL allows to identify the efficient alternatives on which it is then possible to apply a MNL model for predicting the choice probabilities. On the contrary, if we assume that & depends on the states-of-the-world of the alternatives, it follows that (2.9) becomes: Ej(Uh;) = Ej(V,(X))

+ Ej(&;)

= Vi + Fih,i h E H i E %.

(2.12)

It turns out that in this case MNL no longer holds, since &, is not Gumbel distributed. It is worth mentioning that if we assume Z$,,normally distributed, &,; is still normal, so that the Probit Model may come through. On the other hand, even though (2.12) may represent a more realistic behaviour, it is known that the Probit Model is very difficult to handle empirically. An empirical application of this approach will be illustrated in the next sections with reference to the modal choice in a real transportation system. 3. THE

EMPIRICAL

RESULTS

3.1 States of nature and utility To investigate empirically SD-MNL procedure, travel mode choices have been analyzed for one-way trips for the purpose of work and study. The data originate from the interview information compiled as part of the 1981 census of population for the area Bergamo-Milano, Italy. The individual information has been aggregated in four levels of income: Hi, Hz, H,, and H4 arranged in order of increasing income. In particular, H, defines the class of students, Hz the employees, H3 the ruling staff, and H, the professionals. The available alternatives are: train, bus, and car. The related attributes have been incorporated in the utility function for each income class as follows:

Vi = Kf exp

(-bti

-cpi

+ df)

i = 1, 2, 3,

(3.1)

where Ki, ti, pi, and fi represent the explanatory variables capacity, time, cost, and frequency, respectively, for the modal choice i and a, 6, c, and d are the positive relevant parameters for each income set. By capacity we mean the maximum number of available seats. We assume that commuters care about capacity, as, ceteris paribus, modes with more capacity are likely to be less crowded. Furthermore, the exponential form (3.1) has been adopted instead of the usual linear form, in order to take more into account the variables time, cost, and frequency with respect to capacity. At this point, as all the attributes are supposed to be random variables (see Section 2.1) we assume four possible states of nature depending on usual travel conditions (s,), unfavourable conditions due to bad weather (s?), strikes (sJ, and favourable conditions (sJ.

A new approach to modal split analysis

79

Table 1. Probabilities of the states of nature

P 01) P (rz)

Train

Bus

Car

,893 .033 ,066 .008

.952 .033 ,013 .002

.951 .033 .014 .002

Table 2. Weights for each income class

: :

HI

H?

H,

H,

.16 .14 60 .lO

.16 .42 .32 .lO

.29 .32 .28 .ll

.32 .33 .20 .15

Then it has been possible to define the following probabilities P(Si), i = i, . . . , 4, of the states of nature for each travel mode? (see Table 1). The following matrix of the parameters associated with the attributes for each socioeconomic class (see Table 2) has been estimated according to some surveys, while the numerical values of the attributes have been derived from census data. In particular, all the attributes have been converted according to a common monetary value (thousands of Italian lira). 3.2 The results The tables for the three travel modes have been computed according to the scheme given in Section 3.1; then, the utilities for each mode appearing in Tables 3-5 have been obtained, via a log transformation, using the linear algebra package GAUSS/PC. Furthermore, the expected utilities have been obtained according to Ei(Vi) = C Vj P(Sj) i

i = 1, 2, 3,

where P(Si) are the probabilities for the states of nature associated to each mode (see Table 1). The results are in Table 6. Then, FSD rule has been used to select the efficient alternatives for each income

Table 3. Utilities for mode TRAIN

H,

2.95 .21 4.05 9.15

.47 .OO .09 .16

.65 .OtJ .24 .40

4.80 .55 944 23.74

Table 4. Utilities for mode BUS

H, HZ H, H,

tInformation

SI

sz

3)

s*

.66 .14 2.78 12.42

.23

.66 .14 2.78 12.42

.66 .14 2.78 12.42

:: 1.79

provided by Ferrovie dello Stato and Societa Autostrade.

80

A. REGGIANIand S. STEFANI Table 5. Utilities for mode CAR

H, HZ H, H,

$1

s:

s3

34

.02 .15 3.50 74.11

.oo .Ol .47 6.29

.Ol .Ol .48 9.61

.02 .16 3.74 79.77

class, according to Theorem 2.1. For two income classes, Hz and H3,all modes turned out to be efficient, while, in classes H, and H4,car and train respectively resulted to be inefficient and were therefore eliminated, as Table 7 shows. Results are quite clear and very reasonable: car is preferred (and considered “economic”) by the highest income class, while the lowest class (students) prefer train and bus. These considerations may be very interesting in a planning context, where evaluating the more reasonable way of travelling, according to some rationality criteria, turns out to be crucial. Table 8 shows the probabilities computed from (2.11) and Tables 6 and 7. By converting in absolute frequencies the results reported in Table 8, we obtain the theoretical distribution of travellers, per income and mode under SD-MNL, as Table 9 shows. We also report the observed data, according to the 1981 Census, per income and mode between Bergamo and Milan0 in Table 10. Thus, from the last rows in Tables 9 and 10 (totals per mode), we obtain the theoretical and observed distributions of 100 travellers per mode, as Table 11 shows. At this point, we perform a goodness-of-fit test to assess a statistical significant difference between the theoretical and observed distributions, as from Table 11, under the null hypothesis of no difference. The x2 test gives as a result x’ = .88. This value, with 3 - 1 = 2 degrees of freedom, is such that p(x* 5 .SS) < .05, while p(x’ 5 5.99) = .95. Thus, we conclude that the null hypothesis cannot be rejected.

Table 6. Expected utilities per mode and income class

HI H? HJ H,

Train

Bus

Car

2.73 .19 3.72 8.41

.65 .13 2.72 12.11

.02 .14 3.38 71.43

Table 7. The efficient set

H,

HZ H,

H,

tDenotes

Train

Bus

t

t

t t

t t t

Car

:

t

an efficient alternative.

Table 8. Final probabilities under SD-MNL Train

Bus

H,

.89

HZ H,

.35 .48

.ll .32 .18

H,

.OO

Car .33 .34 1.00

81

A new approach to modal split analysis Table 9. Absolute

frequencies under SD-MNL and Milan0 Train

between

Bergamo

Bus

Car

Total

HHI

4.537 3.064

4,149 379

4.2780

12,964 3,443

H: H, Total

3.028 0 10,629

1,135 0 5,663

2,145 1,022 7,445

6,308 1,022 23,737

Table 10. Observed freauencies between Bereamo and Milan0

H: :: K Total

Total

2,694

500

249

3,443

4,477 2,862 98 10,131

4,783 1,368 63 6,714

3,704 2,078 861 6,892

12,964 6,308 1,022 23,737

Table 11. Theoretical

SD-MNL Observed

Car

Bus

Train

(under SD-MNL) per mode

and observed distributions

Train

Bus

Car

Total

45 43

24 28

31 29

100 100

4. CONCLUSION

In conclusion, the observed probabilities do not differ from the theoretical ones obtained through SD-MNL at 5% significance level; furthermore, SD-MNL also give interesting insights about the characteristics of the travel modes. First, as can be seen for the income classes Hz and H3, the alternatives that give rise to almost uniform distribution of commuters are all classified efficient; while, as for H, and H4,a threshoid has been determined by means of SD-MNL procedure under which the “unlikely” alternatives are eliminated. It should be noted that in this approach FSD rule, based on increasing utilities, has been used: in fact, FSD is the less questionable criterion among other SD rules and can be applied to very general settings where there is no sufficient information about decision-maker’s utilities. Acknowledgemenrs-This work has been supported by Progetto Finalizzato Trasporti CNR Sottoprogetto II Tema 2 Ricerca “Modelli matematici di equilibrio del traffic0 su una rete di trasporto” and by the National Grant M.P.I. 40% 1985/6. Although the work is attributable to both the authors, Reggiani has written Sections 2.2, 2.3, and 3.1, and Stefani has written Sections 2.1,3.2, and 4. The authors would like to thank L. Peccati. S. Biffignandi, and an anonymous referee for the helpful theoretical comments.

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