E L S E V I E R European Journal of Operational Research 110 (1998) 342-367 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH T h e o r y a n d M e t h o d o l o g y

**A weight-assessing method with habitual domains **

G w o - H s h i u n g T z e n g *, T i n g - Y u C h e n , J i h - C h a n g W a n g

*Energy and Environmental Research Group and Institute of Traffic and Transportation, College of Management, *
*National Chiao Tung University, 114, 4F, Sec. 1, Chung Hsiao IV.. Rd., Taipei 100, Taiwan, ROC *

Received 7 September 1996; accepted 6 June 1997

**Abstract **

Weights of criteria are the important factors in decision-making. However, from a behavioral perspective, traditional weighting methods account for too few factors to deal with decisions properly. Based on the behavior mechanism and the theory of habitual domains, this study is undertaken to develop a new weight-assessing model that treats decision- making as a dynamically adjusting process proceeding from the ideal to actual states. The new model is built upon the dynamic analysis for the connectivities between criteria instead of the static analysis of traditional models. Finally, we have studied Taipei City motorcycle users, mode-choice behavior through questionnaire in order to show the applica- bility of the new model. From the empirical results, it is found that our weight-assessing method has significant appli- cation potential in practice. © 1998 Elsevier Science B.V. All rights reserved.

*Keywords: *Weight; Criteria; Decision-making; Behavior mechanism; Habitual domains; Connectivity

**1. Introduction **

In the process o f decision-making, trade-offs between the criteria that influence the final decision must be made. These trade-offs can be computed in terms o f a relative ratio o f their importance, which can be pre- sented in a "weight" form. F r o m the viewpoint o f behavior, the influential factors determining the weight o f a decision-making criterion include: the difference between the ideal and actual values o f the criterion (i.e., level o f charge), the diversification and intensification o f other ideas which can activate the criterion (i.e., connectivity), the duration (or tenure) o f the criterion belonging to the core of habitual domains (i.e., the frequency o f input stimuli), the decision-maker's personality and social-economic attributes, the intrin- sic value o f the weight, and the interaction among other criteria. Traditional weighting methods (Hwang and Yoon, 1981), such as the eigen-vector method, weighted least-square method, entropy method, utility function method, consider only the "interaction among other criteria" and forego consideration o f the other factors.

* Corresponding author. E-mail: ghtzeng@ccsun5.nctu.edu.tw.
0377-2217/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved.
*P I l S 0 3 *7 7 - 2 2 1 7 ( 9 7 ) 0 0 2 4 6 - 4

*G.-H. Tzeng et al. I European Journal of Operational Research 110 (1998) 342-367 * 343
In general, most traditional weighting methods are based on static analysis, and their results usually only
reflect the intuition or perception of decision-makers at the time of analysis. In fact, weights of criteria must
be variable in different situations including input information, time, learning process, environment, etc.
Thus, it is not easy to clearly delineate the absolute weightings for decision criteria. On the other hand,
our model treats decision-making as a dynamically adjusting process from the ideal state to the actual state,
allowing us to realize the dynamic change of weights depending on different situations.

This research considers most of these factors simultaneously using the concept of habitual domains. The habitual domains concept was first introduced by Yu in 1980. It claims that a human being's decision-mak- ing process is gradually fixed by habit. The main idea of habitual domains is that the set of ideas and con- cepts that are encoded and stored in the brain tend to progressively stabilize in the absence of an extraordinary destabilizing event. Thus, thinking processes will reach some steady state or may even be- come fixed. Once the habitual domain is extended, it could greatly enhance the quality of decision-making. In order to express the changes in weights throughout the decision-making process, we use connectivi- ties between criteria to set up a fuzzy directed graph that is a collection of crisp criterion sets. According to the connectivity network and neighborhoods identified by the connectivities, this research develops the weight-assessing method with habitual domains as an alternative to traditional weighting methods. Our mod- el can assess weights based on behavior mechanisms and overcome many of the disadvantages of traditional weighting methods.

This paper is organized as follows: in Section 2, we review traditional weighting models. In Section 3, we introduce human behavior mechanisms and describe four hypotheses that capture the basic workings of the brain. We then introduce the concept of habitual domains. Section 4 presents a model of activation pro- pensity and connectivity. We have started with two very practical examples before doing the mathematical work, including the mode choice behavior and the house-purchasing decision-making. Then we use the con- nectivities between criteria to establish the network structure in Section 5. Section 6 introduces an algorithm for weight assessing and provides an empirical study of motorcycle travelers. We end the paper with con- cluding remarks in Section 7.

**2. Review of weight-assessing method **

There are five primary components in any decision-making process, including: (i) decision alternatives; (ii) decision objectives or criteria; (iii) decision outcomes; (iv) preference structure; (v) information inputs (Yu, 1990). Weight is the most general form of preference structure. There are certain benefits in defining a preference structure by a set of weights, such as: (i) it allows the importance of each objective to be repre- sented as a set of numbers; (ii) the ratio of two objectives is equal to their "relative importance"; (iii) the sum of all weights is equal to 1 (Saaty, 1980; Hwang and Yoon, 1981).

There is an abundance of research on weighting characteristics which might be applied to the decision- making process including: rating method (Eckenrode, 1965); utility function method (Keeney and Raiffa, 1976; Keeney and Nair, 1977); entropy method (Zeleny, 1974; Nijkamp, 1977); extreme weight approach; random weight approach (Voogd, 1983); LINMAP (Srinivasan and Shocker, 1973); analytic hierarchy pro- cess (AHP) (Saaty, 1977, 1980); least-square method; logarithmic least-square method; geometric mean method (Krovak, 1987; Cook and Kress, 1988). From a structural viewpoint, there are two types of weight- ing criteria: subjective and objective.

The objective weight can be computed from the outcomes without asking the perceptions of the deci- sion-makers. For example, the extreme weight approach, random weight approach, and entropy method are all objective weight-assessing methods. The entropy method is one of the best objective weight-assessing methods. Entropy is a physical measurement of the second law of thermal-dynamics and has become an important concept in the social sciences as well as in the physical sciences.

*344 * *G.-H. Tzeng et al. I European Journal o f Operational Research 110 (1998) 342-367 *

In the information theory of Shannon, the entropy is used to measure the expected information content of certain messages. Entropy in information theory is a criterion for the amount of "uncertainty" repre- sented by a discrete probability distribution. The entropy method assumes that those criteria with less un- certainty are more important. Moreover, the method computes the anti-uncertainty (amount of information) of each criterion based on its possible outcomes, and normalizes them to a set of weights whose summation is equal to 1.

A H P is the well-known type of subjective weight-assessing method. AHP was introduced by Saaty in the 1970s. AHP organizes all objectives into a hierarchical structure. In AHP, the objectives are independent of each other in the parallel level, and the summations of weights in the same level are equal to their direct higher objective. Saaty suggests two techniques for obtaining the information on preference: pairwise com- parison and eigen-vector computing. In fact, we can get these values by a direct-rating process or compute them through the least-square method, without affecting the validity of the AHP model.

These objective methods show that the weight of a criterion is relative to its clearness. Thus, in the en- tropy method, the clear criteria are more important than the fuzzy criteria. Subjective methods show us to obtain the information on preference by asking the decision-maker. Some of these methods provide ratings or pairwise-comparison techniques, while some suggest that we organize the objectives as a hierarchical structure. There is a common assumption in all of these weight-assessing schemes that the preference struc- ture exists; the problem is how to obtain it. Thus, the question arises: if the preference structure is not sta- ble, will a weight-assessing approach be useful? Is the importance of food to a hungry person similar to that for a normal person? Surely it is not. From the viewpoint of habitual domains, the weight comes from the charge structure, and the difference between the perceived actual state and the ideal state is the primary con- sideration within the charge structure. In other words, the distance between "where you want to go" and "where you are" decides the weight.

**3. Habitual domains **

In Section 3.1 we introduce some basic human behavior mechanisms. In addition, we shall describe four hypotheses that capture the basic workings of the brain: circuit pattern, unlimited capacity, efficient restruc- turing, and the analogy and association hypothesis. Then, we introduce the concept of habitual domains in Section 3.2. In Section 3.3, we shall introduce four classes of decision problems that are to be selected for particular situations based on the regularity and availability of skill sets.

*3.1. Behavioral bases for decision-making *

The main idea of the model of Yu (1985) of human decision behavior is that each human being has an endowed internal information-processing and problem-solving capacity that is consciously allocated as needed to various activities and events over time to adapt to, and achieve in, the multi-dimensional human environment. The brain is the human internal information processing center. Moreover, it is recognized that when external stimuli are cognitively attended to a human being, a special sequence of circuit patterns of activated neurons, containing the cognitive function, appears in the brain. This sequence represents one of the many possible cognitive functions that has been engendered by the stimuli. Some of the major cat- egories of cognitive brain function include: encoding, storing, retrieving, and interpretation.

Yu (1990) summarized memory and thought processes according to four basic hypotheses: (i) the circuit pattern hypothesis (that is, human memory and thought can be represented by electrochemical patterns in the brain cells); (ii) the unlimited capacity hypothesis (that is, the capacity of memory that our brain can encode (not retrieve) is practically unlimited); (iii) efficient restructuring hypothesis (that is, our memory can be restructured in an efficient way so as to effectively process the encoded information); (iv) analogy

*G.-H. Tzeng et al. I European Journal of Operational Research 110 (1998) 342-367 * 345
and association hypothesis (that is, our brain interprets incoming information using analogy and associa-
tion with the existing memory).

Among these four hypotheses, the analogy and association hypothesis is one of the most pervasive and important observations concerning human cognitive processes. Most people conjure up an impression in response to receiving an abstract symbol and their cognitive ability then enables them to retrieve informa- tion and perform analysis is conserving the symbol. According to Yu's definition, the analogy and associ- ation hypothesis can be stated as follows:

The perception of new events, subjects, or ideas can be learned primarily by analogy and/or associa- tion with what is already known. When faced with a new event, subject, or idea, the brain first investi- gates its features and attributes in order to establish a relationship with what is already known by analogy and/or association. Once the fight relationship has been established, the whole of the past knowledge (preexisting memory structure) is automatically brought to bear on the interpretation and understanding of the new event, subject or idea.

The main points to be emphasized are: (i) there is a preexisting code or memory structure which can po- tentially alter or aid in the interpretation of an arriving symbol; (ii) a relationship between the arriving sym- bol and the preexisting code must be established before the preexisting code can play its role in interpreting the arriving symbol.

*3.2. Habitual domains *

The concept of habitual domains (HD) was first formulated by Yu (1980). It claims that a human being's decision-making process is gradually fixed by habit. The main idea of habitual domains is that the set of ideas and concepts that are encoded and stored in the brain tend to progressively stabilize with time and in the absence of an extraordinary destabilizing event will approach a steady state (Yu, 1990).

There are two kinds of thoughts stored in human memory: (i) the ideas that can be activated in thinking processes; and, (ii) the operators that transform the activated ideas into other ideas. The operators are re- lated to thinking processes or judging methods. Generally speaking, operators are also ideas. However, be- cause of their ability to transform or generate (new) ideas, they are called operators.

Habitual domains have four primary elements (Yu, 1991): (i) potential domain, PDt, which is the col-
lection of ideas or operators that can be potentially activated at time t (or stage t); (ii) actual domain,
ADt, which is the set of ideas or operators that are actually activated at time t (or stage t); (iii) activated
probability, APt, which is defined for each subset of PDt and is the probability that a subset of PDt is ac-
tually activated or is in ADt; and, (iv) reachable domain, *R(It, Or), *which is the set of ideas or operators that
can potentially be reached from the initial set of ideas, *It, *and the initial set of operators, *Or. *

Given a decision-making problem E that catches a decision-maker's attention at time t (or stage t), the
propensity for an idea vi to be activated is denoted by *Pt(vi, E). *As with probability functions, *Pt(vi, E) *may
be estimated by determining its relative frequency, as well as statistical probability (Yu and Zhang, 1989).
The *cz-core Kt *of the HD for problem E at time t (or stage t) is the collection of the ideas or concepts that can
be activated with a propensity larger than or equal to ~. That is,

*Kt(~,E) = {Yi E *HD [ *Pt(vi, E) *>/ct}. (1)

*3.3. Classification o f decision problems *

Depending on the perception of the availability of skill sets and the or-cores of HD, we can classify de- cision problems into four categories: routine problems, mixed routine problems, fuzzy problems and chal-

346 *G -H. Tzeng et al. I European Journal o f Operational Research 110 (1998) 342-367 *

lenging problems (Shi and Yu, 1987; Yu and Zhang, 1992a, b). We note that what is unknown to the de- cision-maker may be known to another. Therefore, the classifications below depend on each decision-mak- er's HD.

(i) *Routine problems. *For a routine problem, the idea set that is needed to successfully solve problems is
well known to the decision-maker, and the decision-maker has acquired the set. With proper training
and practice, most decision-makers can develop the capability to solve routine problems.

(ii) *Mixed routine problems. *A decision problem is called a mixed routine problem if it consists of a number
of routine subproblems.

(iii) *Fuzzy problems. *The truly needed idea set is only fuzzily known to the decision-maker. Therefore, the
decision-maker has not yet mastered the skills, concepts and operations necessary for successfully solv-
ing these problems.

(iv) *Challenging problems. *The truly needed idea set is unknown or only partially known to the decision-
maker. These problems cannot be successfully solved by the ~-core of HD, no matter how small ~ is.

**4. Activation propensity and connectivity **
*4.1. Measurable space *

In order to specify measurable space in our study, we first introduce the concept of goal setting. For a
decision problem, denoted by E, there exists a set of goal functions to be achieved for its satisfactory so-
lution. The goal functions in the internal information-processing center are used to measure the many di-
mensional aspects of the decision problem. For each goal function there is an ideal state or equilibrium
point to reach and maintain. This process is called goal setting. Goal functions can be measured by a col-
lection of elementary criteria, { v l, v2, *v3,..., vn } (n *is the number of criteria), which is finite. Each goal func-
tion is a subset of the total collection of all elementary criteria.We say that two goal functions are related or
associated if their two corresponding criterion subsets have a non-empty intersection. We can consider the
elementary criteria {Vl, v2, v3,..., v,} to be the discussion universe, HD, for the problem E.

For example, when drivers of private vehicles begin a trip, the first problem is how to choose their route. The objectives which drivers would consider during the route choice process include the minimal travel cost, the fastest driving speed, optimal safety and comfort, the fewest risks, and the most familiar route. It is not necessary to optimize each objective for users, but to acquire only the satisfactory level of each objective. Therefore, for the route choice problem E, its discussion universe HD = {travel time, delay, driving speed, degree of safety, degree of comfort, degree of risk, and familiarity to the route}. Another example: when we desire to purchase a house, the criteria to be taken into consideration would be of price, size of the house, age of the house, distance of the house from the office, and convenience to shopping. Ad- ditionally, these criteria would have made up as the discussion universe for the decision-making of house- purchasing.

Let HD (habitual domains) be the discussion universe, i.e. a set consisting of all vertices (i.e. criteria) for the discussion of problem E. The family of sets consisting of all the subsets of HD is referred to as the power set of HD and is indicated by P (HD). a(HD) C P (HD) is a family of subsets of HD, so that:

(i) ~ E a(HD) and HD E a(HD); (ii) if A c a(HD), then A C a(HD);

(iii) a(HD) is closed under the operation of set union; that is, if AI,A2, A3,... E a(HD), then Ui°~=I 2i C a(HD)

The set a(HD) is usually called the Borel field or a-field. We usually take a(HD) to be the smallest a-field containing as numbers all of the sets of particular interest, and a(HD) is called a a-algebra. We treat HD as a measurable space,

*G.-H. Tzeng et aLI European Journal of" Operational Research 110 (1998) 342-367 * 347

H D = (HD, a ( H D ) , p),

### (2)

where a ( H D ) is a a-algebra generated by H D , a n d / t is a meaning measure. Assume the H D is finite and that a ( H D ) = P ( H D ) , i.e., every subset of H D is measurable.

*4.2. Activation propensity and connectivity *

Given an input stimulus

*St *

at time t (or stage t), the propensity for criterion v~ to be activated is denoted
by *Pt(viSt). *

F o r c~ E [0, 1], the s-core o f H D at time t (or stage t), denoted by Kt(~, HD), or simply by
Kt~(HD) when no confusion can occur, is defined as the collection o f criteria that can be activated with
a propensity larger than or equal to cc That is,
Kt(c~,HD) = {v~ E H D

*I Pt(v,,St) >~ o~}. *

### (3)

Take route choice as an example: when road users have received the message of a traffic jam on the road ahead, the difference between the ideal and actual values of travel time will be enlarged. Thus, the activation propensity of travel time will increase and such criterion will become the core of decision-making as trav- elers make their en-route switching. Should road users receive the message o f rock slide on the road ahead, their decision core will be transferred to safety. Similarly, the price of the house is most often not the core of habitual domain for rich people during their house-purchasing decision process, and price is, therefore, not a major factor to be considered.

Note that the s - c o r e *Kt(o~, H D ) is just the closed s-cut o f a fuzzy subset o f H D with membership function *
Pt (Yu and Zhang, 1990). Furthermore, the activation propensity function can be generalized into a func-
tion defined on H D × HD:

Definition 4.1. F o r any v~ and vj in H D ,

*Pt(vi, *

*Yi)*denotes the propensity that criterion vj will be activated when the criterion

*vi *

is presented to the input stimuli at time t (or stage t).
It is trivially known that the higher the activation propensity, the more strongly the criteria are connect- ed and vice versa. Therefore, we can reasonably treat the activation propensity function as an approxima- tion o f the connectivity function and define the connectivity function following the definition of (Yu and Zhang, 1990) as:

**Definition 4.2. A function **

*Ct(vi, vj) *

defined on H D x H D at time t (or stage t) is called a connectivity func-
tion on H D if it satisfies the following axioms:
(i)

*Ct(v~, vj) E *

[0, 1];
(ii)

*Ct(vi, vi) *

= 1,Vvi E HD.
Because the number o f stimuli perceived by the senses o f an individual at any point in time is enormous, selective perception is required. Thus, arriving symbols are processed in an unrefined way for the sake of convenience and efficiency. According to the analogy and association hypothesis, when faced with an ex- ternal stimulus, the brain first investigates its features and attributes in order to establish a relationship with what is already known by analogy and association. The relationship can be treated as the connectivity func- tion. Once the right connectivity relationship has been established, the whole o f the past knowledge (pre- existing memory structure) is automatically brought to bear on the interpretation and understanding of the stimuli, resulting in a decision being made.

According to the definition, if

*Ct *

is a connectivity function, then *Ct(vi, v j) *

is called the connectivity from
v~ to *vj *

at time t (or stage t). If *Ct(Yi, •j) > Ct(Yi, Yk),*then we say that

*V i*is more strongly connected to vj than to vk. In terms of the route choice problem, should the driving speed o f the alternative route be slower, the

348 *G.-H. Tzeng et al. I European Journal of Operational Research 110 (1998) 342-367 *

road user would then have to spend more time to reach his destination. Thus, driving speed can be connect- ed with travel time by analogy and association. Likewise, when the driving speed is slower, the feeling of comfort that the route brings to users would be less; as a result, it can be seen that driving speed is connect- ed with degree of comfort. Generally speaking, driving speed is more likely to be associated with travel time; in a word, the connection between driving speed and travel time is stronger than that of driving speed with comfort.

It should be noted that learning processes are usually directed, i.e., the connectivity from vi to *vj *is not
equal to the connectivity from vj to *vi. *Thus, *Ct *is not necessarily a symmetric connectivity function; in oth-
er words, the activation propensity is non-symmetric. We can summarize the above discussion with Prop-
osition 4.1.

**Proposition **

4.1. *Assume that Pt(vj, vi) is such that Pt(vj, vi) = 1 if vt = vj. Let Ct(vi, vj) = Pt(vj, vi) f o r all vi*

*and vj in the*H D

*at time t (or stage t). Then, Ct(vi, vj) is a connectivity function that satisfies the conditions*

*o f Definition 4. 2.*

It is noted that people usually utilize an approximation of the necessary criteria in solving a particular
decision-making problem E based on the limits of personal charge structure, external information, atten-
tion allocation, self-suggestion or physiological monitoring, etc. Decision-makers are also apt to be affected
by external considerations such as time and cost. Only in the absence of palpable external stimulation, goal
setting, charge structure, and information input will the solution process maintain steady. This indicates
that the decision-makers' perceived connection between criteria is only fuzzily known. In other words,
the connectivity between the criteria for a decision-making problem E is roughly, but not clearly, known.
Consequently, the connectivity function on H D will continue to change as long as the situation, viewpoint,
or physiological state varies. This being the case, we define *Ct(v~, vj) *as a function of time t (or stage t).

**5. Connectivity network **

*5.1. N e t w o r k structure *

We assume that H D is discrete and finite; that is, H D = {vl, v2, v 3 , . . . ,

*Vn} *

is finite. We treat H D as a
vertex (or node) set. Let A denote a subset o f H D × HD, i.e., A c_ H D × H D and A is called an arc (or link)
set. An element in A is called an arc. F o r any *vi*and vj in HD, (vi, vj) E A is called an arc that joins

*vi*and

*vj*

(starting from *vi *and arriving at *v j). *Now we set up a connectivity network G.

A fuzzy directed graph (i.e., digraph) *G ( H D , R t ) * consists of a finite set H D -- {vl, v2, v3,..., vn} and a
fuzzy relation

*Rt *

o n H D at time t (or stage t), where the relation *Rt *

satisfies:
(i) *R,(vi, *

*Vj) E *

[0, 1];
(ii) *Rt(vi, vi) = 1, Vvi ~ *H D (reflexivity).

Let the fuzzy relation *Rt *on H D be the connectivity function *Ct(v~, vj) *on H D at time t (or stage t). There-
fore, the digraph is represented by G(HD, *Ct), *i.e., the connectivity network. Consider a connectivity func-
tion *Ct *defined on a finite set HD. The connectivity Ct is interpreted in terms of a connectivity network.
That is, for *vi, vj c *HD, *Ct(vi, vj) *is the grade o f adjacency from *vi *to vj at time t (or stage t). Furthermore,
the adjacency matrix of G(HD, Q) is defined as *[Ct(vi, vj)], *where *vi, vj E *HD.

An a-core of G -- (HD, Ct) is defined to be a crisp digraph *K t ( G ), *when

*KT(C) = * *v: E 6 I * *>- * (4)

*G.-H. Tzeng et al. I European Journal of Operational Research 110 (1998) 342~67 * 349
The concept of a connectivity network G can be considered to be a collection of its ~-cores. In particular,

*K°(G) *

is always a complete digraph; that is, every pair of criteria in K~t(G ) is adjacent.
*5.2. Assessment of connectivity *

In this section we discuss a method for assessing connectivity functions based on max-min operators. Before further discussion, we know that the arc (vi, vy) is active at the level ~t if

*Ct(vi, *

vj) >~ e; that is,
*(vi, vj) *

is present in *K~(G) *

at time t (or stage t). Thus, *Ct(vi, vj) *

is equal to the maximum level at which
the arc *(vi, vy) *

is active.
A walk in a connectivity network G(HD, Ct) from

*vii *

to vik is a sequence of criteria v~l, vi2,..., v~k that
are connected by the arcs *(Vil,Vn),..., (v~k-l,vik) *

(these arcs are usually considered to be part of the
walk). A walk is called simple if each criterion appears in it only once at most, and a simple walk is
called a path. Hence a path in G is a sequence of distinct criteria such that for all *(vi, vy), *

*Ct(v~, *

vj) >/e; in other words, a path in G is also called active at the level ~ if the path is present in
*KT(G ). *

The strength of the path is min *{Ct(v~, *

vj)} for all criteria contained in the path. The length of a
path is the number of criteria contained in the path. The directed path also represents a learning sequence.
It should be noted that some learning processes perhaps express a directed walk, not a path. However, as we
know, if the walk is active at some level, the path contained in the walk is also active at the same level. Such
being the case, in our study we consider only those situations in which learning processes are represented by
the paths for simplicity.
A criterion vj of G is called at-reachable from another criterion v~ if there is a path from v~ to vj in the crisp digraph Kt~(G). Furthermore, any criterion

*vi *

is ~-reachable from Yi itself, for any ~ E [0, 1]. G is or-con-
nected if and only if all pairs of criteria of G are ~-reachable; that is, G is called or-connected if *Kt(G ) *

is
connected (Miyamoto, 1990).
Given a connectivity network G(HD,

*Ct), *

a connectivity function C~ is defined by *C2t(vi, *

vj) = ~ if and
only if there is a path of length 2 from criterion v~ to criterion vy in *KT(G), *

and for any e > 0 there is no
path of length 2 from *vi *

to v] in *Kt+~(G) *

at time t (or stage t). Therefore, *Ct(vi , vy) *

is the maximum level
of the ~-core so that there is a path of length 2 starting from vi and arriving at *vj. *

The discussion concerning Ct ~, k > 2, can be carried out in a similar manner. First,

*Ck,(vi, vj)= *

• if
and only if there is a path of length k from criterion *vi *

to criterion vy in *K~(G) *

at time t (or stage t). More-
over, if *Ckt(vi, vj) *

= ~, then there is no path of length k from *vi *

to vj in *K~'+'(G) *

for any e > 0 at time t (or
stage t). That is, *Ckt (v~, v j) *

is the maximum level of the ~-core so that there is a path of length k starting from
v~ and arriving at vj. Consequently, the connectivity Ct k, k/> 2 is calculated by the max-min composition
rule:
Proposition 5.1.

*Given a connectivity network *

G(HD, *Ct), then the connectivity Cgt at time t (or stage t) is cal- *

*culated by the following iterative formula: *

*C2t (vi, Yj) = *

max *min[Ct(vi, v), C,(v, *

vj)],
vcHD

### (5)

*Ckt(vi, v j) *

= max min[Ct k-l *(vi, *

v), *Ct(v, *

vs) ]
vcHD
*where n & the number of criter& & G. *

(3 ~<,k ~<,n), (6)

Proof. Find out v' such that v' is the element belonging to H D that achieves max

*min[Ct(vi, *

v), *Ct(v, *

vy)] = *min[Ct(vi, v'), Ct(v', *

vj)].
350 *G.-H. Tzeng et al. I European Journal o f Operational Research 110 (1998) 342-367 *

Let ~' =

*min[Ct(vi, *

v'), *Ct(v', vj)]. *

Then both a r c *(vi, v t) *

and arc *(v', vj) *

are active at the level ~t'. This means
that there is a path of length 2 at the level ~' (i.e., in Kt ~' (G)) starting from v; and arriving at

*vj. *

Therefore, we
have the following inequality:
*C2(vi, vj) >~ min[Ct(vi, *

v'), *Ct(v t, *

•j)]
= max *min[Ct(vi, v), Ct(v, vj)]. *

vEHD

On the other hand, for any e > 0 there is no path of length 2 from vi to

*vj *

in *K~'+~(G) *

at time t (or stage t).
That is, for any e > 0, *Ct(vg, v) < ~' + e, Ct(v, *

vj) < ct' + e, for all v E HD. Furthermore, from the definition
of *Cet (vi, vj), *

we have *C2(vg, vg) < ~' + ~. *

This means that

*C2(vi, *

vj) > max *min[Ct(vi, v), Ct(v, *

vfl]
vcHD
does not hold for any

*vi, vj E *

HD.
Hence, we have proved the first equation of the proposition. Moreover, the second equation can be proved in the same way except for the "path of length k" condition. For simplicity, we omit the details. The connectivity function Ct ~ can be calculated by the iterative formula described in Proposition 5.1. [] Let Ct*

*(vi, *

vfl = ~ if and only if the criterion vj is reachable from *Yi *

in K 7 (G), and vj is not reachable from
*vi *

in *K~+~(G) *

for any e > 0 at time t (or stage t). Then, *Ct(vi, vfl *

is the maximum level such that vj is reach-
able from *vi; *

that is, *vj *

is or-reachable from *vi *

if and only if *C;(vi, vj) >~ ~. *

The connectivity C t on H D de-
scribed above is called a connectivitiy index. The connectivity index is calculated from Ctk,, k = 1,2, 3 , . . . , n
(n is the number of criteria), as described in Proposition 5.2.
**Proposition **

5.2. *Given a connectivity network *

G(HD, *Ct), then the connectivity index C t at time t *

(or stage t)
*is calculated from Ckt, where k = 1,2, 3 , . . . , n: *

*C;(v,, vj) = max{ Ct(vi, vj), C2(v,, *

v f l , . . . , *Ckt (v,, *

v j ) , . . . , *Ct (v,, *

vj)}. (7)
*Furthermore, the connectivity index C~ is also given by: *

*C~ (vi, vj) = max{min{ Ct(x,y)l(x,y ) E Path(vi, vj) } l Path(v~, *

vj)}, (8)
*where Path(vi, vy) is a path in G from vi to vj. *

Proof. From the definition it is obvious that the connectivity index C t (vi, vj) is equal to the maximum level of all the paths of arbitrary length (it is noted that the length of the path is always less than or equal to the number of criteria in HD) from

*vi *

to *vj, *

which in turn is equal to the maximum of *Ckt(vi, vj) *

the maximum
level of paths of length k, for all k. This implies the first equation. As for the latter equation, that is simply
another expression of the first equation. []
Remark 5.1. Given a connectivity network G(HD, Ct),

*Ct(l~i, 1Jj) *

is the connectivity index from *vi *

to vj at
time t (or stage t) which satisfies the following axioms:
(i)

*Ct(v,,vg) E *

[0.1];
(ii) *Ct (vi.vt ) = 1, Vvg E *

HD.
Consequently, we can assess the connectivity index C~ by the max-min extension (according to the m a x - min composition rule) discussed above. Moreover, the max-min extension of the connectivity function Ct can be called the "conservative extension" (Yu and Zhang, 1990).

*G.-H. Tzeng et al. I European Journal of Operational Research 110 (1998) 342-367 * 351

We have already shown that learning processes are usually directed, i.e., the connectivity from vi to

*vj *

may not be equal to the connectivity from vj to v~. Hence, *Ct *

is not necessarily a symmetric connectivity
function. Furthermore, the connectivity function may be non-transitive. Ct is (max-min) transitive if and
only if
*Ct('gi, Vk) *

*~*max

*min[Ct(vi, vj), Ct(vj, *

vk)] (9)
*vjEHD *

is satisfied for each pair (v~, vk) E H D x HD. A

*Ct *

failing to satisfy the above inequality for some criteria of
H D is called non-transitive; that is,
*Ct(vi, *

vk) < max *min[Ct(vi, vj), Ct(vy, *

vk)] (10)
v/EHD

for some

*(vi, vk) E *

H D x HD. For example, if there is no arc connecting vk directly from v~, this implies that
it is practically impossible to learn vk from v~ directly. Therefore, the connectivity function *Ct *

does not re-
quire symmetry and transitivity.
*5.3. Generalized current domain *

As mentioned before, there exists a set of goal functions that must be achieved for the satisfactory so- lution of a decision problem E. Goal functions can be measured by finite elementary criteria,

*{Vl,V2, V3,...,vn} (n *

is the number of criteria). The ideal values of criteria are denoted by
q* = {q~, q~, q ~ , . . . , q*}. In parallel with goal setting, goal state evaluation is constantly being performed
in the brain. For the external stimuli, we continuously investigate, measure and attempt to detect any
and all current deviations from ideal goal states. This process is called state evaluation. The actual values
of criteria are denoted by q = {q l, q2, q 3 , . . . , qn }. Goal setting and state evaluation are dynamic, interactive
processes that are affected by physiological forces, self-suggestion, external information forces, current
memory and information-processing capacity (Yu, 1990).
Each stimulus is related to a set of goal functions. When there is an unfavorable deviation of the per- ceived value from the ideal, each goal function will produce a corresponding level of charge. Take the dy- namic route choice problem as example: when users receive information of the real-time traffic jam on the road ahead, their expected travel time and delay will increase if they proceed straight ahead. Furthermore, their driving speed will also be slowed. Thus, the charge of travel time, delay, and driving speed directly related to the real-time information will, as well, increase. The totality of the charge by all goal functions is called the charge structure. The charge structure can change dynamically since, at any point in time, our attention will be drawn to the event that has the greatest influence on charge structure. The difference be- tween the ideal and actual values of each criterion is calculated by [Iq* - q[[, which also measures the level of charge. We can summarize the above discussion with Definition 5.1.

Definition 5.1. For any criterion vi in HD, q* and

*qi *

respectively denote the ideal and actual values of criteria
*vi. *

Furthermore, the level of charge, denoted by *Qi, *

o f *vi *

is measured by the difference between ideal and
actual values
Q, = Ilq* -q~ll vi E [1,n], (11)

where

### 11 II

is a meaning norm such that 0 ~<*Qi *

~< 1 for all *Yi E*HD.

If St \ H D ~ ~b, it indicates that the stimulus St is unknown or only partially known to our existing HD. This situation implies that

*St *

contains some elementary criteria outside of the existing discussion universe
(HD) for the decision problem E. It also implies that *St *

cannot be completely contained in any or-core, no
352 *G.-H. Tzeng et al. I European Journal o f Operational Research 110 (1998) 342-367 *

matter how small ~ is. For example, if road users do not have enough experience about ramp control of the highway, they would not be sure whether everything goes well as they enter the highway. Additionally, if users have no information about the traffic situation of other substitute path, it is very difficult for them to conduct route choice decision-making. Furthermore, even if the decision has been made, the result, such as total travel time, would be not quite desirable. In this case, the decision problem E will be challenging to the decision-maker. However, we notice that what is unknown to one person may possibly be known to some- one else; that is, what is a challenging problem to one person may be a fuzzy or routine problem to another. Even so, E is still a challenging problem to all decision-makers. Generally speaking, challenging problems are very difficult to solve unless decision-makers can expand and/or restructure their HD. For simplicity, we will not discuss the case of St \ H D # q~.

On the other hand, even though what is not in H D may be more important than what is in it, people usually ignore those criteria belonging to St \ HD. This observation can be attributed to the property of habitual domains that is an indication of the way that people process arriving stimuli. In this context of arriving stimuli, absorption is defined as the possibility that information input will be accepted. A sugges- tion is more easily accepted if it strikes a consonance in the receiver's memory; therefore, the degree of ab- sorption will depend on a decision-maker's memory, goal setting, state evaluation, charge structure, and charge release. For example, sales advertisements about houses would often publicize such public facilities as swimming pool, exercise facilities, etc. However, this kind of information would be ignored by those who cannot swim or hate sports. Generally speaking, people will actively or progressively learn to accept those ideas, concepts and experiences which can help them reach their goals; that is, they accept the stimuli which are related to individual goal functions. If the external information is not related to a decision-maker's goal setting, charge structure and charge release, it is likely to be rejected. Therefore, it is not necessary to con- sider criteria outside of the existing discussion universe (HD) of decision problem E.

Given an external stimulus

*St *

at time t (or stage t) of the problem E, we assume that the arriving stimulus
can be broken down into several elementary criteria belonging to HD; that is, *St c_ *

HD. The corresponding
level o f charge for each criterion *vi *

is denoted by *Qi. *

For ~ E [0, 1], the a-core o f *St *

at time t (or stage t),
denoted by S t, is defined as the collection of criteria that can be activated with a level of charge larger than
or equal to ~. That is,
*S t = {Vi *E S i n H D I Qi ~>z~}. (12)

S t is the actual domain in a narrow sense at the time t (or stage t) concerned with an external stimulus

*St. *

Before illustrating the actual domain in a broad sense and the reachable domain, we define the connectivity from the existing domain to a particular criterion. Let us first give the following axiomatic definition: Definition 5.2. The mapping cgt : a(HD) x H D ~

*Ctt *

is called a connectivity function of criteria with sub-
sets of *HD *

at time t (or stage t) if it satisfies the following axioms:
(i) cg/> 0 (non-negativity); (ii)

*c~t(Aj, vi) *

= 1,Vv, E Aj;
(iii)

*cgt(Aj, vi) <~ *

cgt(Ak, *vi), Vvi *

E HD; if Aj C_ Ak (monotonicity), where a(HD) is a a-algebra generated by
HD.
As mentioned before, S t is the existing domain which can represent a criterion set activated by an ex- ternal stimulus at time t (or stage t) for the problem E; that is, S~' is the collection of criteria that can be activated with a level of charge larger than or equal to ~. Now, since c~, is a connectivity function, we can denote

*¢gt(St, vi) *

as the connectivity of a criterion *vi *

with the existing domain (the actual domain in
a narrow sense) S~'. When there is no confusion, we treat a connectivity function of criteria with the actual
domain as a connectivity function. That is, a function *Ct(S~', vi) *

defined on a(HD) x H D at time t (or stage
t) is called a connectivity function. The proof o f Proposition 5.3 demonstrates that *~g,(St, *

v;) E [0, 1].
*G.-H. Tzeng et al. / European Journal of Operational Research 110 (1998) 342-367 * 353
**Proposition 5.3. **

*(~t(St, vi) is *

*bounded by 1.*

**Proof. For any **V i in H D and any S t in a(HD), we can obtain the following inequality according to the
*monotonicity axiom of (gt and the condition that S t C_ St c_ H D , *

*(~2t(at, Yt) ~ (~2,(St, v,) ~ *

(gt(HD, v;) -- 1. []
As for the relationship between the connectivity function of a criterion to a criterion and that of a do- main to a criterion, the latter can be considered to be an extension of the former. To make this concept clearer, let us give Proposition 5.4.

*Proposition 5.4. (1) Let H D be finite and C t be the connectivity index with C t (v~, v j) the strongest connectivity *

*from criteria vi to vj. Given an external stimulus St at time t (or stage t), the ~-core o f St is denoted by S t, where *

*S t E a(HD). For any criterion vj in HD, tf we define the connectivity o f vj with the existing domain S t as *

*(gt(St~,vj) = max{Ct(vi, vj) [ vi E St~}, * (13)

*then (gt satisfies the axioms as defined in Definition 5.2. *

*(2) The power set o f l i D is indicated by P(HD). Assume that a ( H D ) = P(HD). Given that (gt is a con- *
*nectivity function o f criteria with the existing domain as defined in Definition 5.2 then *

*C t (vi, vj) = ~t({v,}, vj) * (14)

*is the connectivity index starting from v, arriving at vj as described in Remark 5.1. *

Proof. (1) According to Proposition 5.2, we know that the connectivity index C t is given by

*C~(v,, vj) + max{min{Ct(x,y)l[ (x,y) E Path(v,, b ) } I Path(vi, vj)}, *

*where Ct(vi, vj) is the connectivity from v; to vj. *

*(i) For any vi E S t and any vj c HD, the non-negativity of ~, is implied by the relationship *

*~,(St, ~j) >1 C;(v,, vj) >1 C,(v,, ~j) >>. O. *

*(ii) If vj E S t, then *

### vj) >1

*C;(vj, vj) =*

By Proposition 5.1, we know we can see that (~,({$7, b ) =

(iii) Let S t E a(HD); if S t

*max{Ct(v,, vj)lvi E St} *

.

that

*(~t(St, *

v j ) ~ 1. By combining this relationship with the above condition,
1.
*c_ St fl, then Vvj E H D we have the following inequalities: *

*<~ m a x { Ct (vi, vj)[vi E Stfl},cgt(St, vj) <~ (rt, (Sflt ,vj). *

It is obvious that there exists monotonicity of (gt. Thus (1) is proved.

*(2) (i) By the non-negativity of Ct and Proposition 5.3. we see that 0 <~ ~gt({vi}, vj) ~< 1 corresponding *

**o **

**o**

**C;(v),, vs) <. 1. **

**C;(v),, vs) <. 1.**

*(ii) If v, = v2, then vj E {vi} and Ct(vi, vj) = (gt({vi}, vy) = 1. Consequently, by (i) and (ii), Ct(v,, vj) sat- *
*isfies the following axioms: (i) Ct(v, , vj) E [0, 1]; (ii) Ct(vi, vj) = 1 if v, = b- This completes the proof. * []
**Remark 5.2. For any vi c S t and any vj c HD, Path*(vi, Vg) is the optimal path from vi to vj such that ***(gt(S t, vi) is a maximum. In other words, Path* (vi, vj) is the strongest path because its connectivity index is *
the maximal of all possible paths.

### 354

*G.-H. Tzeng et al. I European Journal of Operational Research 110 (1998) 342-367*

According to the analogy and association hypothesis, we can conclude that new things can be more easily learned if they are similar to some things that are already known. Because people associate arriving stimuli with preexisting memory, arriving stimuli tend to be initially treated as either positive or negative. If the ar- riving stimulus is perceived to be relevant to a decision-makers goal function, then it is more carefully ex- amined and composed to fit preexisting codes. Also, we may say that if a newly arriving stimulus is very similar to a preexisting code, it will be quickly processed in a positive or favorable way. Conversely, a stim- ulus that is perceived as irrelevant or quite dissimilar to preexisting codes will be filtered or ignored. It should be noted that frequently repeated events will have a stronger influence on analogy and association. However, those events preexisting in weak codes, which are stored in remote areas of the brain, will have little impact on the analogy and association process. Therefore, we must first specify the influential domain from the pre- existing memory through external stimuli. We can restrict the neighborhood of the actual domain in the nar- row sense to be the reachable domain. In order to figure out the neighborhood of the actual domain in this case, we facilitate our discussion by using the connectivity of criteria with the existing domain.

Definition 5.3. Given a connectivity network G(HD,

*Ct) *

and an external stimulus *St *

at time t (or stage t), the
a-core of *St *

is denoted by S 7 and cgt is a connectivity function of the criteria with the existing domain. Let
2 HD denote the collection of all non-empty compact subsets of HD. The e-neighborhood o f S~' for S t E 2 HD

is defined by

*Nt(St, e) *

*Nt(St, e ) = {vj E *

H D \ S, I 3vi ~ S t , ~ *~,(ST, v j) >1 *

e}, *VS 7 ~ *

2 riD. (15)
S 7 can be considered as the actual domain that contains the set of criteria that are actually activated. Moreover,

*Nt(S t, e) *

represents the reachable domain that contains the collection of criteria that are reach-
able from the existing domain through external stimuli. Therefore, St ~ tO N,, (S~', e) is the actual domain in a
broad sense, and we call it the *generalized current domain. *

When external stimuli are repeated, the corresponding circuit patterns will be reinforced and strength- ened. Furthermore, the stronger the circuit patterns become, the more easily the corresponding circuit pat- terns are retrieved in the learning processes. Therefore, it seems reasonable to assume that the connection between a pair of criteria in the actual domain and its e-neighborhood will be reinforced as the learning process progresses. In other words, the connectivity between pairs of criteria in the generalized current do- main will increase. Hence, the connectivity function

*Ct *

must be updated after each learning iteration.
The uncertainty that arises from human thought processes and the randomness associated with exper- iments is often confused by social scientists (Kaufmann and Gupta, 1991). Some o f the data obtained in this manner are hybrid; that is, their components are not homogeneous, but a blend of precise and fuzzy infor- mation. Furthermore, a fuzzy relation, such as the connectivity function, is not a measurement. That is, the connectivity function is a subjective valuation assigned by one or more human operators. To simplify mat- ters, we suppose that

*Ct(vi, vj) *

is a continuous random variable, uniformly distributed within the interval
### [(C(vi, vj)) 1/fl,

*(C(vi,*

### !)j))fl] ,

where

*-C(vi, D) *

is the mean of *Ct(vi, vj). fl *

is called the determinate index and its value is in the unit interval
[0,1]. fl characterizes the degree o f certainty since the higher the fl value is the less change is performed by
the connectivity function. That is, when fl approaches 1, the connectivity is rather stable. On the other
hand, when fl approaches 0, we do not have sufficient evidence to point out the exact value of the connec-
tivity function. Let U(0,1) represent a continuous random variable that is uniformly distributed over the
interval [0,1]. F r o m simple proportionality, we can write
*G.-H. Tzeng et aL I EuropeanJournalofOperationalResearch 110 (1998) 342-367 *

355
Thus, it is very simple to generate *Ct(vi, vj) *

from a given U(0, 1) provided the lower bound and upper
bound are known. In order to reflect the fact that the connectivity between each pair of criteria is enhanced
through the learning processes, we define an index parameter *It (vi, v j) *

for each pair *(vi, v j) *

belonging to H D at
time t (or stage t) and a concentration parameter 6 as follows. The initial values o f / f o r pairs of criteria are set
to zero. If vi is activated when vj is presented to the input stimuli, the value of I increases by 1. The concen-
tration parameter, 6, represents the change in size of the definition domain, and 0 < 6 < 1. Consequently, the
connectivity function is calculated within the adjustment interval
*[ ( C(vi, vj) + It(vi, *

vj)3)'/#, (C(vi, *vj) + I,(vi, vj)6)# 1 *

and now

*Ct(vi, vj) *

is given by
*Ct(vi, vj) *

= min{ 1, (C(vi, *vj) + It(vi, vj)3)1/# + *

[(C(vi, *vj) + It(vi, *

vj)6) #
*-(-C(vi, vj) + It(vi, *

vj)6)1/#] U(0,1)). (17)
To avoid the condition where the connectivity exceeds 1, we utilize a " m i n " operator. That is, we use the index parameter, I, and the concentration parameter, 6, to indicate the reinforcement change of circuit pat- terns.

**6. Weight-assessing method with habitual domains **

*6.1. Architecture *

The architecture of the weight-assessing method with habitual domains is shown in Fig. 1. All elements within the network are fully interconnected.

The weight-assessing method with habitual domains is based on competitive learning. As we described above, the elementary criteria are the basic elements of our discussion universe. Each criterion has routes that connect it to the neighboring ideas. In the presence of external stimuli, some criteria can be "fired" and "lit up" sequentially through the learning processes, while others will remain "dark". The external stimuli are registered and processed using the circuit patterns or sequences of circuit patterns in our discussion uni- verse. In other words, external stimuli are encoded as digraphs, routes or paths of lit ideas, and activation occurs when attention is paid. This in turn triggers the appropriate interconnecting arcs to fire or the ap- propriate criteria to discharge. Therefore, the output ideas of the digraph compete with each other and only some of them are activated or fired at any one time. Such being the case, our weight-assessing method is based on the concept of competitive learning.

In the most general case, competitive learning belongs to unsupervised learning (Fausett, 1994). In un- supervised learning, the network is presented with a set of training patterns, but is not given a target answer for each input pattern. Thus, we can modify the weights of the network without specifying the desired output for any of the input patterns.

In competitive learning, the output ideas of the network compete among themselves to be active (or fired). Competitive learning begins with a random arrangement of weights and gives all output ideas a chance to compete. It also limits the strength of the weights. Each criterion has a temporally fixed amount of weight. The weights are limited to values between 0 and 1; that is,

n

*wt(vi) >1 0 *

for each *vi, Z w t ( v i ) *

= l, (18)
356 *G.-H. Tzeng et al. / European Journal o f Operational Research 110 (1998) 342-367 *

Fig. 1. Architecture of weight-assessing method with habitual domains.

where

*wt(vi) *

is the weight of the criterion vi (competitive layer) from the input stimulus St.
An internal mechanism creates a competition among the ideas for the right to respond to a given subset of input stimuli, so that only one output idea, or only one idea per group, is active (i.e., " o n " ) at a time. The idea that wins the competition is called the "winner-take-all" idea (Haykin, 1994). In this study, the criteria in the generalized current domain are chosen as the winners during the learning process. This is different from the traditional concept of winner-take-all because there is not necessarily only one criterion in the win- ner group when the competition is completed. This form of competition among a group of criteria is called

*generalized winner-take-all, *

and our weight-assessing method performs a generalized winner-take-all com-
petition.

Let S t denote the actual domain of all winning criteria, and its e-neighborhood (i.e., the reachable do- main) be

*Nt(S~, 5). *

The output signals of the generalized current domain are set equal to one; the output
signals of all the criteria that lose the competition are set equal to zero. The output signal is also called
the index parameter *It. *

We use the winning set and its neighborhood to update the weights of the network.
Then, we can form a new weight vector that is a linear combination of the old weight vector and the current
input vector. Weight corrections are accumulated over an entire epoch of training patterns (i.e., batch up-
dating). This updating procedure has a smoothing effect on the correction terms. The learning rule of the
weight correction is thus
*G.-H. Tzeng et al. I European Journal of Operational Research 110 (1998) 342-367 * 357

*Wt+l (Vi) = Wt(~i) ÷ A w t ( v t ) . * (19)

We can define the adjustment factor r/~ for each criterion vi through the level of charge and the connec-
tivity function. For each *vi E Nt(S t, ~), vk *is the precedent criterion of v~ so that

*C;(vk, vi) : max{Ct(vj, vi ) I vj E St}. *

The level of charge of vk is denoted by Qk. The adjustment factor r/i is computed by:

*Qi * if *vi E s~, *

*~li = * *QkC~t(St, vi) * if *vi G N t ( S ~ , e), *

0 otherwise.

(20)

Assume that the input stimulus contains a set of p vectors. Let ~1 be the average of all r/is during the particular learning iteration; then ~ is

**~p rti **

r/-- 7 = (21)

P

Using the average adjustment factor, we can obtain the change *Awt(vi) *

Aw,(vi) = ~, x ~7 - w , ( v i ) *, * (22)

where it is the learning-rate parameter and its value is chosen by users. Note that the values of it' s must be between 0 and 1.

Remark 6.1. The learning rule of the weight correction, as described by (19)-(22), guarantees that the sum of the weights for all criteria in our discussion universe is always equal to one.

Proof. For all criteria in our discussion universe H D at time t (or stage t), the sum of weight changes is equal to zero; that is,

**Z, **

**Z,**

**A w , ( v i ) =****× L, **

**× L,**

= ( t × [ 1 - 1 ] = 0 . (22)

Thus, the sum of the weights for all criteria in H D at time t + 1 (or stage t + 1) is

*wt+l (vi) = y ~ wt(vi) + ~ Awt(vi) = 1 + 0 = 1. * *[] *

*i * *i * *i *

We now want to discuss equilibrium states of the routine and mixed routine systems. From Proposi- tion 6.1, the weights of criteria clearly are only related to the level of charge and the connectivities with the existing domain in the equilibrium state.

Proposition 6.1. *For routine and mixed routine problems, the weight o f criterion vi is only related to the level o f *
*charge and the connectivity o f vi with the existing domain in the equilibrium state. Moreover, ceteris paribus, *
*the weight o f vi is also fixed and stable. *

358 *G.-H. Tzeng et al. I European Journal o f Operational Research 110 (1998) 342-367 *

Proof. Let

*hk *

be the probability that stimulus Sk is presented on any trail and *gk(Vi) *

be the probability that
criterion v~ wins (i.e., is in the generalized current domain) when stimulus Sk is presented. We consider the
case in which
**Awt(vi)hkgk(vi) = O, ***k *

that is, the case in which the average change in the weights is zero. We refer to such a state as the equilib- rium state (Rumelhart and McClelland, 1986). Thus, using the learning rule and averaging over all of the stimulus patterns, we have

*it Z ['--'~ hkgk(vi) -- ~t Z[wt(vi)hkgk(vi)] *

= 0,
*k [ j,Tj *

and thus

** - khkgk(vi) **

**- khkgk(vi)**

**rl-- i **

**rl-- i**

*wt(vi) = * *E k h k g k ( v i ) * *--- *

The average adjustment factor for criterion vi is calculated using the charge and the connectivity func- tion, so the weight of

*vi *

is only related to the level of charge and the connectivity of *vi *

in the existing do-
main. Let us consider the situation in which the level of charge and the connectivity corresponding to the
input stimuli remain stable. In this case, the average adjustment factor does not make significant changes;
thus, the weight is a constant. Whenever the system is in the state in which, on average, the weights are not
changing, we say that the system has reached an equilibrium state. When this happens, the system always
responds the same way to a particular stimulus pattern. Note that the equilibrium state only holds for rou-
tine and mixed routine problems.
According to Yu (1990), a routine problem means that the needed idea set (i.e., the truly needed com- petence set) is well known, and the decision-maker has mastered the set. Because the needed idea set is well- known, ceteris paribus, the difference between the ideal and actual values of each decision criterion does not change. Furthermore, the decision-maker has acquired and mastered the truly needed set, so his con- nectivity network is quite stable. Since a mixed routine problem consists of a number of routine subprob- lems, the discussion concerned with mixed routine problem is similar. Therefore, the weights of criteria remain unchanged in response to any single stimulus in the equilibrium state in routine and mixed routine problem. []

From Proposition 6.1, we know that for routine and mixed routine problems, the weights of criteria are stable in the equilibrium state. However, it is possible that weights will be pushed out of equilibrium by an unfortunate sequence of stimuli. In this case, the system can move toward a new equilibrium state (or pos- sibly back to a previous one).

For a fuzzy problem, the ideas, concepts and skills needed to successfully solve the decision problem are roughly, but not clearly, known. This implies that the decision-maker has not mastered the ideas and skills necessary for solving these problems (Yu, 1990). Such being the case, the connectivity network of the de- cision-maker does not reach a stable state. However, we can obtain a

*temporally *

fixed amount of weight
through the algorithm introduced in Section 6.2 even though the weight does not remain stable throughout
the decision-making process.
The architecture and algorithm introduced in the next section for the connectivity network can be used in routine, mixed routine, and fuzzy problems.

*G.-H. Tzeng et al. I European Journal of Operational Research 110 (1998) 342-367 *

359
*6.2. Algorithm *

The algorithm given here is suitable for routine, mixed routine, or fuzzy problems. Note that weights for routine or mixed routine problems are rather stable, but weights for fuzzy problems are in a stable state only when there has been a convergence that satisfied the stopping rule. When the stopping condition is false, we cannot obtain the approximate amounts of weights.

*Step O. *Initialize weights *wt(vi) *for each vi belonging to H D so that *~i"=l wt(vi) = 1. *

Initialize the continuous random variate, U(0, 1), that is uniformly distributed over the interval [0,1].
Initialize the index parameters *It(vi, *vj) = 0 for each pair (vi, vj) belonging to digraph G.

Initialize the e-neighborhoods *Nt(St, *e) = ~b for all S t E 2 HD.

Obtain the ideal values of criteria q*, the actual values of criteria q, and the initial mean connec-
tivity matrix Ct -- *[-fft(vi, vj)] *for each pair (vi, vj) belonging to the digraph G through the ques-
tionnaire survey.

Set the concentration parameter 6, 0 < 6 < 1. Set the determinate index/3, 0 ~</3 ~< 1.

Set the learning rate parameters i r Set the threshold parameter ~, 0 ~< ~ ~< 1.

*Step 1. * Compute the initial connectivity for each pair *(vi, v i) *belonging to digraph G.

*CI(Yi, Yj) ~ (-C(Yi, Yj))l/fl"~ [(-C(Yi, Yj)) fl- (-C(Yi, Yj))l/fl]U(O, 1). *

*Step 2. * When the stopping condition is false, do steps 3-13.

*Step 3. * For each stimulus vector St, do steps 4-9.

*Step 4. * Specify the actual domain in the narrow sense, S~':

*S t = {Yi ~ St t~ *

H D I *Qi >~*~}, where

*Qi*=

### IIv*

- v i i i ,*vi E*[1,n].

*Step 5. * For each criterion v-i ~ H D \ S t, compute the connectivity of v-i with the existing domain St:
c~/(St, *v-i) *= max{Ct*(vi, *vj) [ vi c *St},

where *Ct (vi , v i) = max{min{Ct(x,y)] * (x,y) E *Path(vi, vj) }lPath(vi, *v-i)}.

*Step 6. * Find the e-neighborhood of S~':

*Nt(St, e) = {v-i C *H D \ *Stl3vi c S t, ~ cgt(St, vj) *>~ e}, VSt ~ E 2 HD.

*Step *7. For each *vi E S t, vj c Nt(S t, e), Path* (vi, v-i) *is the optimal path from vi to *v-i *such that

*~gt(S t, vi) *is a maximum.

Update the index parameters for all criteria x,y within *Path* (vi, v j): *

gnew) (x,y) = gold)(x,y) + 1.

*Step 8. * For each *vi, *compute the adjustment factor *qi: *

*Qi * if *vi E S~, *

*qi = * *Qk%(St, vi) * if *vi ~ Nt(S t, E), *

0 otherwise.

where vk is the precedent criterion of vi.

360

*G.-H. Tzeng et al. / European Journal of Operational Research 110 (1998) 342-367 *

Ci new)" _{(~i, v j) = }

*min(1,(c}°ld)(vi, *

_{vj) + }

*I~new)(vi, *

*vj)~)l/l~+[(c~°ld)(vi, vj)+l{neW)(vi, vj)~)l~ *

*Step 10. *

Let ~ be the average of all r/~s.
**_ ** **~ p **r/i

r/i =

P

*Step 11. *

Compute the weights wt+l (vi) for each criterion vi within HD at stage t + 1:
**w,+l (v,) = w,(v,) + **

**Awt(v,), **

**Awt(v,),**

*whereAwt(vi):'tx [ ( ~ ) - w t ( v i ) ] *

*• *

*Step 12. *

Update the learning rate.
*Step 13. *

Test for the stopping condition:
Let Owt(vi) = Iw'+l (vi) - wt(vi)l
If maxi Owt(vi) is smaller than a specified tolerance Vi E [1, n], then stop; otherwise, continue.

*6.3. Empirical study *

In this section, we will discuss an application that employs the weight-assessing method with habitual domains. In the following, we will introduce the background of the problem and the questionnaire content of our empirical study.

In Taiwan, a great number of people use motorcycles as their primary commuting mode because of low car-ownership and low car-operating costs, comparatively small size, high mobility, and ease of operation. Take Taipei City, for instance: registered motorcycles amounted to 54% of all motor vehicles at the end of 1995, and the percentage of growth has risen 10% in the last two years. Furthermore, such statistics do not account for the number of motorcycles coming from other cities, which has not yet been taken into account. The massive amount of mixed traffic flow created by motorcycles has had severe effects in urban areas. And it has also posed a serious threat to safety and driving in urban traffic, not to mention air and noise pol- lution as well as the high risk of damages resulting from the inferior stability and fragile structure of mo- torcycles. Therefore, how to promote public transportation in motorcycle users habitual domains becomes more and more important for traffic authorities.

This empirical study is undertaken to test the applicability of our weight-assessing method through ques- tionnaires to motorcycle travelers in Taipei. We investigated the decision attributes as well as the grade of importance considered by motorcycle travelers during their mode choice process. It is possible to encourage motorcycle users to shift to public transit if the latter possesses those attributes. Such being the case, the traffic authorities can realize how to attract motorcycle travelers to use transit under the specific attribute stimulus. Therefore, the empirical results for motorcycle travelers will help to set up transportation market- ing strategies and corresponding policies favoring public transportation usage in the future.

A two-stage approach was used for the empirical research of motorcycle users in Taipei. We first ob- tained the influential factors of mode choice for motorcycle users through the first stage questionnaire with open questions. Nine criteria of high membership are selected, which includes walking time (vl), waiting time (v2), in-vehicle time (v3), transfer time (V4) , accessibility (vs), travel c o s t (v6) , punctual arrival (V7) ,