Methodology To Analyze Driver Decision Environment During Signal Change Intervals: Application of Fuzzy Set Theory

Transcription

1 TRANSPORTATON RESEARCH RECORD Metodology To Analyze Driver Decision Environment During Signal Cange ntervals: Application of Fuzzy Set Teory SHNYA KKUCH AND JEFFREY R. REGNER During a signal cange interval, most drivers must decide weter to stop or go based on uncertain information on speed, te remaining yellow time, and distance from te intersection. Te decision process under fuzzy information, suc as tis case, is suited for analysis by fuzzy set teory. Fuzzy set teory and its logic as been used to analyze te drivers decision between "stop" and "clear" during te signal cange interval. Fuzzy sets of " not safe stopping" and not safe clearing" are defined along te approac roadway. Depending on te way te two sets intersect, te dilemma, indecision, or option zones can be represented. Possibility and necessity measures of "safe stopping" and "safe clearing" were defined, and tey were assumed to represent te decision criteria of aggressive and conservative drivers, respectively. Based on tese measures, a new approac to determine signal cange intervals is suggested. t states tat an interval sould be suc tat, at any point along te approac, te possibility of "safe stopping" or "safe clearing" are always one, and te necessity measure of te two sets sould exceed a certain minimum value. Te signal cange interval as been a topic of traffic engineering studies since its formulation by Gazis et al. (J) in te early 196s. Traditionally, tis interval is computed to eliminate te dilemma zone. Te dilemma zone is known to exist if te signal cange interval is suc tat te driver cannot stop or go witout violating te traffic rules. f a driver is in te dilemma zone at te onset of te yellow pase and decides to proceed, te veicle will still be in te intersection wen te all-red period ends. Conversely, if te driver decides to stop, te veicle will not be able to stop witout entering te intersection. n te traditional formula, signal cange intervals are computed based on te idea tat, at any point along te approac, at least one of te actions (stop or clear) is possible. f all te caracteristics of te approacing veicles and drivers are te same as te assumed input values, and if te driver knows is exact location at te time of signal cange, te signal cange intervals set by te traditional formula sould be adequate to allow time for te driver to complete one of te actions. Hence, at te end of a signal cange interval, te intersection sould be cleared for crossing traffic. However, because te information available to a driver concerning location, te amount of yellow time remaining, approac speed, and oter parameters is fuzzy, te result is driver indecision wen te yellow indication appears. Fur- S. Kikuci, Civil Engineering Department, University of Delaware, Newark, Del J. R. Riegner, TetraTec Ricardson, 56 West Main Street, Cristiana, Del termore, because not all drivers are approacing at te same speed and teir decision criteria are different, some drivers always experience a certain degree of indecision of dilemma. Tus, te traditional formula is not applicable for te analysis of te decision process of individual drivers. Rater, it is suited for diagnostic analysis of te signal timing as stated by Maalel and Prasker (2). n tis paper, we recognize tat some of te information available to te driver is fuzzy; in oter words, te values known to te driver are only approximate. We ten attempt to analyze te decision-making environment using te logic of fuzzy set teory. Two basic fuzzy sets are defined: one representing te minimum distance from wic stopping is possible and te oter te maximum distance from wic te "clearing" maneuver is possible. Bot distances are measured from te intersection stop line. Knowing tese fuzzy distances, we determine te possibility and necessity measures of completing te stopping and clearing maneuvers safely at any point along te approac roadway. Te possibility and necessity measures are believed to represent te decisions of two types of drivers: aggressive (or risk-taking) and conservative (or risk-averting), respectively. We ten introduce a fuzzy set tat represents te approximate location as perceived by a driver and evaluate te drivers likeliood of completing eac of te actions using te possibility and necessity measures. Te level of te drivers anxiety and te zones were te dilemma and indecision occur can also be illustrated using te intersection of te membersip functions of tese fuzzy sets. Analyses similar to te autors are found in prior papers dealing wit te stopping probability function. Among tem are Olson and Rotery (3) and Seffi and Mamassani (4). Te probability function represents te observed frequency of stopping decisions along te approac. t sows te final action of te drivers in an aggregated form, but it does not differentiate te circumstances in wic individual drivers made te coice. At a given location, te same value of stopping probability may be obtained wen te driver is in te dilemma zone or in te option zone. Te existence of te probability function itself, owever, sows tat every drivers judgment and decision pattern is different and tat eac reacts differently to te given information. Te previous work as elped develop our idea for using fuzzy sets to analyze te effect of fuzziness of information on individual drivers decisions of stopping and clearing. Anoter work related to tis study is one by May (5), in wic extensive field measurements of driver beavior and risks were compiled and analyzed. Wat we propose as possibility and necessity measures in

2 so tis paper are peraps related to Mays observation of risk measurements. Te autors suggest two criteria in determining signal cange intervals: one, te possibilities of bot te stopping or going actions at any point along te approac sould be one; two, te necessity measures of stopping and clearing sould be greater tan a given minimum level at any point along te approac roadway. Furter, it is suggested tat te use of possibility and necessity measures provides te teoretical basis for explaining te regions of dilemma, indecision, and option. TERMNOLOGY AND CONVENTON OF SYMBOLS Te following terminology is used to indicate te condition of te driver at te onset of te yellow indication: dilemma, indecision, option, or imperative. Dilemma is te situation in wic te driver can complete neiter te stopping nor clearing action safely. Two types of dilemma can exist. Type 1 dilemma is te situation in wic te driver can perform neiter action at all. Type 2 dilemma is te situation in wic te driver can perform one or bot of te actions but wit difficulty. Te dilemma zone is te area in wic te driver faces a dilemma of eiter type. ndecision is te situation in wic te driver can complete one of te actions safely and te oter action wit some level of difficulty. Option is te situation in wic te driver can complete bot actions safely. mperative is te condition in wic te driver must coose eiter stopping or clearing; if te driver is far from te intersection, e or se must stop, and if te driver is very close to te intersection, e or se must clear it. A bold letter indicates a fuzzy number set. Te following are symbols of sets and parameters: 1. SD: fuzzy stopping distance (distance from te intersection) wen te approac speed is fuzzy; 2. CD: fuzzy clearing distance (distance from te intersection) wen approac speed (V) and te remaining signal cange interval (t) are fuzzy; 3. V: fuzzy set of approximate speed; 4. S: fuzzy set of safe stopping distances; 5. C: fuzzy set of safe clearing distances; 6. NS: fuzzy set of distances from wic te veicle is not able to stop safely; 7. NC: fuzzy set of distances from wic te veicle is not able to clear safely; 8. L: fuzzy set of te location of te driver at te onset of te yellow indication; 9. t: fuzzy signal interval perceived by te driver; and 1. E: universal set. All distances are measured from te intersection stop line. TRADTONAL MODEL OF SGNAL CHANGE NTERVALS Te traditional model of signal cange intervals is based on equating two distances tat are bot measured from te in- TRA NSPORTA TON RESEARCH RECORD 1368 tersection stop line: one to stop (stopping distance, D,), and te oter to clear te intersection during te signal cange interval (clearing distance Dg). Finding te signal cange interval by equating te two distances means tat, at any point along te approac roadway, eiter te stopping or te clearing action can be completed witin te signal cange interval. Te stopping distance and te clearing distance are, respectively, D, = V 2 /2b + Vd and Dg = Vt - (w + e) + a(t - d) 2 /2 were d = driver perception and reaction time, V = approac speed, b = deceleration rate, w = intersection widt, e veicle lengt, and a = acceleration rate. f it is assumed tat te veicle will not accelerate upon seeing te yellow ligt, t = d + ( V/2b) + ( w + e)!v (3) f it is assumed tat te veicle will accelerate upon seeing te yellow ligt, = d + [ - V + (r- + 2tt(V 2 /2b + V + )]fa (4) Equation 3 is te most commonly quoted expression, and it is recommended by TE (6). [A recently recommended alternative from te TE Tecnical Council includes te impact of grade (7).] f tin Equation 3 is set at Dg > D" bot te stopping and going actions are possible at a particular speed. f, on te oter and, t is set at D, > Dg, a dilemma zone is said to exist, because for section (D, - Dg) neiter te stopping nor clearing action is possible. ANALYSS USNG FUZZY SET THEORY Te use of Equation 3 (or 4) assumes tat all te parnmeters are known to te driver as "crisp" values and tat te driver makes te correct decision depending on location. n reality, te drivers understanding of te situation is not clear. He can only approximate te following: Te remaining amber and all-red time, Te location of te veicle relative to te boundary between te stop zone and clearing zone (relative to D, and Dg), Te speed of te veicle, Te veicles acceleration and deceleration capabilities, and Te widt of te intersection (if te driver is not familiar wit it). (1) (2)

3 Kikuci and Riegner 51 Te lengt of te drivers perception/reaction time depends on ow precise te information is to im or er. An interesting analysis of driver perception/reaction time using a computer simulation model is presented by Can and Liao (8). Te model allows te analyst to test driver reaction wile watcing a veicle approacing te intersection on te screen. Te quantities of te parameters above are normally judged by te driver based on is or er experience, intuition, and familiarity wit te intersection and signal. Furtermore, te driver can control te values of some of te parameters (for example, te braking and acceleration force to be applied). Tus, te values of te variables are not completely random; rater, tey are subjectively judged numbers, given a myriad of factors suc as te drivers personality, pysical condition, veicle caracteristics, te environment and geometric design of te intersection approac, relationsip to oter veicles in te traffic stream, and so fort. Yet, in many cases, after stopping or clearing, te driver still wonders if e or se as made te best decision. Detailed discussions of ow some of te parameters influence te drivers decision are presented in te work of Ceng et al. (9). n tis paper, we treat tese parameters as fuzzy, and introduce fuzzy set teory to analyze te decision-making process. Analysis Procedure Te parameters considered fuzzy ere are tose perceived by drivers. Among tem, we assume te following tree parameters as fuzzy quantities: te approac speed, te remaining signal cange interval, and te drivers location at te time of signal cange. Tese tree parameters are selected only to simplify te presentation; fuzziness of oter parameters can be incorporated witout compromising te generality. Wen te drivers knowledge of speed (V) and signal cange interval (t) is approximate. te stopping distance and te clearing distance can be defined as fuzzy quantities wit fuzzy sets called "stopping distance" (SD) and "clearing distance" (CD), respectively. Next, we determine a set tat represents te distances greater tan SD and call it a set of safe stopping distances. Similarly, we determine a set tat represents te distances smaller tan CD and call it a set of safe clearing distances. f te veicle is witin te safe stopping distance, it can stop safely; if it is witin te safe clearing distance, it can clear te intersection safely. Determining weter a driver can stop or clear safely from a particular distance involves comparing te distance wit te fuzzy sets of safe stopping distance and safe clearing distance. A comparison of a crisp number wit a fuzzy number requires te introduction of possibility and necessity measures, because te term greater or smaller (tan a fuzzy number) can only be stated by a degree. n tis case, te possibility measure represents te optimistic judgment and te necessity measure represents te pessimistic judgment wen comparing two numbers. First we develop possibility distributions of "safe stopping" and "safe clearing" wit respect to te distance from te intersection. Tese define te approac area were a stopping maneuver is possible and te area were a clearing maneuver is possible, respectively. Second, we define te necessity mea- sures of te two sets again in terms of te distance from te intersection. Tus, tese possibility and necessity measures can represent te judgment of aggressive and conservative drivers, respectively, because of teir criteria for comparing two numbers. t must be noted tat te types of drivers represent a state of mind; tus, te same driver can be in an aggressive or conservative state depending on te circumstances. Because te drivers knowledge of a location is also fuzzy, we compute ow muc te drivers approximate location belongs to eac of te sets ("safe stopping" and "safe clearing"). As a result, we sould be able to measure te possibility and te necessity of completing eac action based on te approximate information. Stopping Maneuver Fuzzy Set of Stopping Distance Given an approximate approac speed (V) in Equation l, te approximate stopping distance, SD, is computed, were SD is a fuzzy set. Te membersip function SD is denoted as s (x). A possible sape of 5 (x) is sown in Figure la. Tis function sows te degree tat a given distance x belongs to te set "stopping distance." n oter words, it sows te possibility distribution of stopping distance wen te approac speed is approximately V. Possibility of Safe Stopping Te possibility of stopping safely from a distance x is determined by comparing x wit SD. f x is greater tan SD, te veicle can stop safely. Te set ofnumbers wic is "possibly" greater tan SD is called "safe stopping distance," and it is denoted by S. Because SD is a fuzzy number, te distance greater tan SD must also form a fuzzy set. Te membersip function of set S is 5 (x) = max s (z) or x, ::::; x x > x, were x, is te value of distance were s (x) becomes 1. Tis membersip indicates te possibility tat te veicle can stop safely from distance x. Tus, it is denoted Poss (x E S) = s(x) (6) Te sape of 5 (x) is similar to a cumulative distribution of 5 (x) as seen by te solid line in Figure lb. t sows te degree (in numbers between and 1) tat te veicle can stop safely along te approac. For a distance close to te intersection, te possibility is, wereas for a distance muc farter from te intersection, it is 1. Tis possibility measure sould represent te judgment of an aggressive, risk-taking, or optimistic driver because it accounts for any evidence tat may indicate tat is location is greater tan SD. For te (5)

4 52 TRANSPORTA TJON RESEARCH RECORD 1368 definition of possibility measure, refer to Klir and Folger (1) and Zimmermann (11). Necessity of Safe Stopping Te necessity measure of safe stopping distance (a degree of "necessarily greater" tan SD) is derived from Nec(x E S) = 1 - Poss(x E NS) (7) were NS is te complement of S; tus, Nec(x E S) is 1 minus te possibility of "not being able to stop safely" (or a set of "unsafe stopping" distances) from distance x. Because Poss(x E NS) can be derived from te membersip function tat represents a number possibly less tan SD (as sown by te dased line in Figure lb), Nec(x ES) is derived by Equation 7 and is sown in Figure le. n contrast to Poss(x E S), it can represent te judgment of a conservative or risk-averting driver because it takes only te sure evidence to justify tat x is less tan SD. For an explanation of necessity measures, refer to Klir and Folger (1). From location p on te approac, for example, a risk-averting driver may not feel it is safe to stop, wereas a risk-taking driver may feel it is safe to stop, as seen by te comparison JO DSTANCE FROM NTERSECTON (a)... SAFE STOPPNG DSTANCE, UNSAFE STOPPNG ~DSTANCE o..., DSTANCE FROM NTERSECTON of values of te possibility and necessity measures of safe stopping in Figure le. Te fact tat te necessity distribution of safe stopping is located to te rigt of te possibility distribution in Figure le indicates tat a risk-taking driver would feel te need to stop before te risk-averting driver as eac approaces te intersection. Te difference between te values of possibility and necessity measures originates from te lack of accurate information available to te driver. f sufficient information is available, and if te driver is normative, te possibility and necessity measures sould be equal and te decision becomes crisp as would result from te traditional equation. Clearing Maneuver Fuzzy Set of Clearing Distance Given te approximate values of approac speed and signal cange interval, te fuzzy clearing distance is derived from Equation 2 and is denoted CD. Te membersip function of te set is denoted co(x) and its ypotetical sape is sown in Figure 2a. Possibility of Safe Clearing Te possibility of clearing te intersection from distance x is determined by examining weter x is smaller tan CD. Te membersip function of te set of numbers tat is possibly smaller tan CD is defined by or x ::5 x, x > x, were x 1 is te value of x were c (x) takes te maximum value (wic is 1 ), and C denotes te set of clearing distances. Te possibility of clearing safely from distance x is now presented by te possibility measure and sown by te dased line in Figure 2b: Poss(x E C) = c(x) (9) (8) POSSBUTY MEASURE OF SAFE STOPPNG Similar to te case of te stopping maneuver, tis function is believed to represent te judgment of a risk-taking or optimistic driver. (c) _ NECESSTY MEASURE OF SAFE STOPPNG DSTANCE FROM NTERSECTON FGURE Stopping maneuvers considered: (a) fuzzy set representing stopping distance, possibility distribution of safe stopping distance and unsafe stopping distance, and (c) possibility and necessity distributions of safe stopping distance. Necessity of Safe Clearing Te corresponding necessity is derived from its definition and sown in Figure 2c: Nec(x E C) = 1 - Poss(x E NC) (1) were Poss(x E NC) represents te possibility of "not able to clear" from distance x, in oter words, te possibility tat

5 Kikuci and Riegner 53 J DSTANCE FROM NTERSECTON (a) UNSAFE CLEARNG DSTANCE ~ SAFE CLEARNG -DSTANCE...,. ~ NECESSTY MEASURE OF SAFE CLEARNG DSTANCE FROM NTERSECTON r POSSBLTY MEASURE OF SAFE CLEARNG L _.;::.. ~ (c) DSTANCE FROM NTERSECTON FGURE 2 Clearing maneuvers considered: (a) fuzzy set representing clearing distance, possibility distribution of safe clearing distance and unsafe clearing distance, and (c) possibility and necessity distributions of safe clearing distance. xis greater tan CD, wic is sown by te solid line in Figure 2b. Again, te necessity measure is believed to represent te judgment of a conservative or risk-averting driver. Te fact tat te possibility distribution of safe clearing extends to te rigt of te necessity distribution in Figure 2c indicates tat a risk-taking driver perceives tat it is safe to clear before te risk-averting driver does as eac approaces te intersection. Dilemma Zone, ndecision Zone, and Option Zone Te dilemma, indecision, and option zones can be illustrated by te way in wic S and C intersect. Te zones are defined based on te intersection of Poss(x E S) and Poss(x E C), and also on Nec(x E S) and Nec(x E C), separately. Respectively, tey represent te decision-making environment for risk-taking and risk-averting drivers. Based on Possibility Measure (for Risk-Taking Drivers) Wen te possibility measures of safe stopping and clearing are superimposed, te possible patterns of overlaps are sown in Figures 3a, 3b, and 3c. n Figure 3a, tere is a section (D 1 ) were neiter safe stopping nor safe clearing is possible. Tis section is te Type 1 dilemma zone. n Figures 3a and 3b, tere are zones (D 2 ) were te possibilities of bot safe stopping and safe clearing are less tan 1. Tese zones correspond to te Type 2 dilemma. Also in Figure 3b, at, one of te actions is possible but te oter is not completely possible. Tis zone corresponds to te indecision zone. n Figure 3c, bot te safe stopping and clearing actions are possible in Section. Tis is te option zone. Te option zone, owever, is located between indecision zones. Based on Necessity Measure (for Risk-Averting Drivers) Similarly, te intersection of two necessity measures of safe stopping and clearing are sown in Figures 3d, 3e, and 3f. Similar to te cases of Figures 3a, 3b, and 3c, te dilemma, indecision, and option zones of risk-averting drivers can be identified using te necessity measures. t is clearly seen by comparing Figures 3a and 3d tat te total area of dilemma is greater wen te necessity measures are used to describe it. Tis indicates tat risk-averting drivers would experience a greater level of uncertainty tan te risktaking drivers. Wen te information is assumed to be crisp, as in te traditional signal cange interval formula, te value of te measure canges from to 1 abruptly. Tus, te zones of indecision and Type 2 dilemma could not be identified for te two types of drivers. Degrees of Dilemma n te Type 2 dilemma zone, te degree of dilemma te driver experiences can be expressed by te intersections of te sets "cannot safely stop" (NS) and "cannot safely clear" (NC). NS and NC are te complements of S and C. Te degrees of dilemma based on possibility and necessity measures, respectively, may be described by te eigt of te intersection of Poss(x E NS) and Poss(x E NC), or Nec(x E NS) and Nec(x ENC). Fuzziness of Veicle Location and ts mpact on Dilemma, ndecision, and Option Next, we incorporate te fact tat te drivers knowledge of is location is usually fuzzy at te onset of te yellow indication. Te fuzzy set of tis location is denoted by a membersip function L(x). Tis function represents a fuzzy set tat states "te drivers location is approximately L feet from te intersection." Given 1-(x), te state of te drivers decision process can be examined from te intersections of Poss(x E S) and Poss(x E C) wit L(x). Te intersection indicates te degree tat te approximate distance L belongs to te safe stopping set or te safe clearing set. Te possibility and necessity measures of safe stopping and safe clearing from approximate distance L are computed as Poss(L E S) = max{min(l(x), Poss(x E S))} for all x (11)

6 54 TRANSPORTATON RESEARCH RECORD 1368 Poss (xec Poss (xe S) Nee (xes) (a) DSTANCE FROM NTERSECTON (d) DSTANCE FROM NTERSECTON Poss (xec Poss (xes) Nee (xes) DSTANCE FROM NTERSECTON (e) DSTANCE FROM NTERSECTON Poss (xec) {Nee (xe C). (c) DSTANCE FROM NTERSECTON OL-~~..:::;..~~~~~~~~-~,---~~~- DSTANCE FROM NTERSECTON (f) FGURE 3 Dilemma, option, and indecision zones considered: (a) Type 1 and 2 dilemma zones based on possibility distribution, Type 2 dilemma zone based on possibility distribution, (c) option and indecision zones based on necessity distribution, (d) Type 1 and 2 dilemma zones based on necessity distribution, (e) Type 2 dilemma zone based on necessity distribution, and (j) option and indecision zones based on necessity distributions. Nec(L E S) = 1 - Poss(L E NS) (12) Poss(L E C) = max{min(l(x), Poss (x E C))} for all x (13) Nec(L E C) = 1 - Poss(L E NC) (14) For an approximate distance L, te possibilities of safe stopping and safe clearing according to Equations 11 and 13 are illustrated in Figures 4a and 4b. Tey are given by te eigts of a and b, respectively, in te figures. Criteria for Determining Signal Cange ntervals Determination of signal cange intervals sould account for te fuzziness of perceived values of te parameters and te difference in decision criteria of different types of drivers. Te following criteria may be suggested to determine te signal cange intervals: 1. Te possibility of taking at least one action safely must be guaranteed at any point x along te approac: Min (Poss(x E S), Poss(x E C)) = 1 (15) Tis criterion accounts for safe completion of actions by aggressive drivers. 2. Te necessity measures of taking one of te two actions must be greater tan a given level u at any point along te approac: Min (Nec(x E S), Nec(x E C)) ;;::: u (16) Tis criterion guarantees te minimum level of safe completion of action. t is an attempt to offer a certainty level tat a risk-averting driver can take at least one of te actions safely. Te signal cange interval derived from te traditional formula satisfies te first criterion so tat at least one of te actions is possible along te approac. t does not allow for te maximum certainty of safe stopping or safe clearing wen measured on te basis of necessity.

7 Kikuci and Riegner 5.5 APPROXMATELY L SAFE STOPPNG DSTANCE (a~)-----l--~l d-s_t_a-nc_e_f_r_o_m_l_nte_r_s-ec-t-o~n l7 t a-p_p_r_o-xmately SAFE CLEARNG DSTANCE L DSTANCE FROM NTERSECTON FGURE 4 Possibility measures for safe stopping and clearing from a distance of approximately L: (a) possibility of safe stopping and possibility of safe clearing. EXAMPLE ANALYSS n tis section, using a set of example values, we present two procedures: one to identify te dilemma and indecision zones and a second to develop signal cange intervals based on te metod just presented. Te following values are used for te examples: approac speed (V) = 4 mp, deceleration rate = 9 ft/sec 2, intersection widt ( w) = SO ft, veicle lengt ( f) = 2 ft, and perception/reaction time (d) = 1.S sec. Te signal cange interval computed wit te traditional formula Equation 3 is S.3 sec. Dilemma and ndecision Zones We now analyze te dilemma and indecision zones for te same intersection, assuming tat te perceived approac speed and signal cange interval are approximately 4 mp and approximately S sec, respectively. Te fuzzy sets for te approac speed (V) and perceived signal cange interval (r) are defined using a triangular fuzzy number (TFN) of te form (fl, f2, f3), wic respectively represent minimum, most likely, and maximum values for te approximate number. "Approximately 4 mp" can be represented by (3, 4, SO) mp and te "approximately S sec" by (4, S, 6) sec. Tis assumption of a TFN for approximately 4 mp is reasonable wen compared to te observed distribution of approac speed presented by Olson and Rotery (3). ntroducing tese TFNs for V and t, we compute te fuzzy sets for "stopping distance" and " clearing distance" as SD = (174, 282, 46) ft and CD = (138, 282, 447) ft using Equations 1 and 3. Tese fuzzy sets are sown in Figures Sa and Sb. For te aritmetic operations of te fuzzy numbers, refer to Kaufmann and Gupta (12). Te corresponding possibility and necessity measures of "safe stopping" and "safe clearing" are computed in Equations S, 7, 8, and 1 and sown in Figures Sc and Sd. Te indecision and dilemma zones are presented in Figures Se and Sf. Figure Se sows tat te possibility measure of at least one of te actions is 1 along te approac; tus, at least one action is possible. Tis is expected, because te values of f2 for SD and CD are te same as te original crisp values. Te necessity measures of te two sets, illustrated in Figure Sf, sow Type 2 dilemma zones. Te value of te necessity measure is less tan 1. Tese two figures indicate tat, for a S-sec signal cange interval, no dilemma exists for risk-taking drivers, but a Type 2 dilemma exists for risk-averting drivers. Te lengt and te location of te indecision zone or Type 2 dilemma zone in Figures Se and Sf are compared wit te observed and derived stopping probabilities presented by Olson and Rotery (3), Seffi and Mamassani ( 4), and Zegeer and Deen (13). Our example sows tat te Type 2 dilemma zone lies between 138 and 447 ft from te intersection as seen in Figure Sf. Te observations reported by Olson and Rotery (3) sow te range in wic stopping probability is between and 1 as 2 to 38 ft at approac speed SO mp, and 8 to 2 ft at 3 mp speed; Zegeer and Deen (12) sow 1 to 3 ft at approac speed 4 mp; Seffi and Mamassanis ( 4) derived probability sows approximately 6 to 3SO ft at 4 mp. Figure 6 compares teir stopping probability functions wit our possibility and necessity measures of "safe stopping" and "safe clearing." Te saded area is bounded by te possibility of "safe stopping" and te necessity of "not safe to clear," te former representing te aggressive drivers stopping criterion and te latter te conservative drivers clearing criterion. Lines 1 and 2 represent te stopping probability curves sown by Seffi and Mamassani ( 4) and Zegeer and Deen (13), respectively. Lines 3 and 4 and Lines S and 6 are te observed stopping probability frequencies for approac speed of approximately 3 mp and SO mp, respectively, reported by Olson and Rotery (3). Te saded area is very close to Lines 1 and 2, and it also lies between line pairs of 3 mp approac speed and SO mp. Line 7 sows te necessity measure of safe stopping-a risk-averting drivers stopping criterion. Te caracteristics of te intersections presented in te previous papers are probably not identical. However, te lines derived by te fuzzy measure closely matc wit te ones tat were surveyed or matematically derived previously. Tis suggests tat te fuzzy measures can be an alternative metod to identify te zones of dilemma and indecision, and to examine te adequacy of te signal cange interval. Signal Cange nterval Te autors next used te criteria presented earlier to suggest a signal cange interval tat accounts for te drivers fuzzy perception of V and t. Because te first criterion is satisfied by te S.3 sec, te interval computed at te beginning of tis section, te signal cange interval tat satisfies te second criterion is computed. Tis is accomplised by determining r so tat te slope of te decreasing section of Nec(x E C) intersects wit Nec(x E S) in Figure 7 at a eigt greater tan ex. n oter words, te following condition must be satisfied at te intersection of te Nec(x E C) and Nec(x E S) lines:

8 1 2 (a) 5 feet o. ~~... ~.. ~~... ~~~.._... ~,. ~~~ O 5 feet Poss (xes) 1 2 (c)..//, ~ ~ feet. Nec(xeC)_) Poss (xec)..., o.._~~... ~~----~... ~~~""-"--~----~~~ (d) 4 5 feet Poss(xeS) 1 2 (e).. ndecision zone "" ( Poss(xeC) feet ,.. """ (~C)_! Type 2 dilemma zone o. ~~... ~~.. ~--... ~~~... ~~~. ~~~ reel (f) -1 FGURE S Fuzzy sets and possibility and necessity measures for example problem: (a) fuzzy stopping distance (SD), fuzzy clearing distance (CD), (c) safe stopping distance (S), (d) safe clearing distance (C), (e) indecision zone, and (/) Type 2 dilemma zone..5 ol-~~~l-..:::;;...::;...jl.oell: 1 2 Distance from te intersection (feet) 3 /7 4 Notes - Saded area: likeliood of stopping derived from te fuzzy measures Nae (xe NC) and Poss (xe S) at approac speed 4 "1l; Lina 1: Stopping probability (approac speed 4 mp) by Zegaer and Deen (13); Line 2: Stopping probability (approac speed 4 mp) by Saffi and Mamassani (4); Lines 3 and 4: Stopping probability (approac speed 3 mp) by Olson and Rotery (3); Lines 5 and 6: Stopping probability (approac speed 5 mp) by Olson and Rotery (3); Lina 7: Necessity maaaure of stopping (Nae (xe S) at approac speed 4 mp. FGURE 6 Comparison of stopping probabilities and likeliood of stopping derived from fuzzy measures.

9 Kikuci and Riegner 57 a...,,. l *" ORGNA~ Nec(xeC),: ~ Nec(xeS) S3 C2 DSTANCE FROM NTERSECTON te degree of safety for completing tem for two extreme types of drivers (risk-taking and risk-averting). Te normative drivers beavior peraps lies between te two extreme types of drivers; tus te proposed metod can identify te ranges of te dilemma and indecision zones. Te approac presented ere could be extended to te analysis of oter riskmeasurement problems in traffic engineering. FGURE 7 Determination of C, and C 2 for given a. (17) were C,, C 2, S 2, and S 3 are as sown in Figure 7. Because C, and C 2 are functions of, once tey are determined by Equation 17, t can be obtained. For an a value of.5, for example, t becomes 7.8 sec. Tis is 2.5 sec longer tan te value computed by Equation 3. However, te 7.8 sec of yellow and all-red time for te given condition is still witin te range of maximum signal cange interval demand surveyed by Lin and Vijaykumar (14) for similar conditions. CONCLUSONS Recognizing te fact tat drivers must decide weter to stop or clear based on fuzzy information, te autors propose te use of fuzzy set teory for te analysis of te drivers decisionmaking environment at signalized intersections. Using fuzzy measures of te safe stopping and safe clearing sets, te dilemma, indecision, and option zones are defined along te approac. Te possibility and necessity measures of te two sets identify tese zones for risk-taking and riskaverting drivers, respectively. Te intersection of te complements of te two sets identifies te level of dilemma for te two types of drivers. Criteria for setting signal cange intervals are also suggested wen te information available to te driver on speed, location, and te remaining time of te signal cange interval are vague. Te difference between te possibility and necessity measures narrows as more accurate information becomes available to te driver. Eventually, if te information is totally crisp to te driver, te two measures coincide. Under tis environment, te traditional equation for te signal cange interval is justified. Providing accurate information to te driver reduces dilemma and indecision and, terefore, elps to reduce te signal cange interval. Any measures resulting in more accurate driver-perceived information is essential to reduce driver indecision and sorten te signal cange interval. Te stopping probability functions studied by many in te past represent only te consequences of decisions made. Te possibility and necessity measures tat we propose, on te oter and, can indicate te availability of te coices and ACKNOWLEDGMENT Te autors are grateful to Seiici Kagaya of Hokkaido University, Japan, and Vijaykumar Perincerry of te University of Delaware for teir valuable suggestions during te course of tis study. REFERENCES 1. D. Gazis, R. Herman, and A. Mmadudin. Te Problem of te Amber Signal Ligt in Traffic Flow. Traffic Engineering, July 196, pp D. Maalel and J. N. Prasker. A Beavioral Approac to Risk Estimation of Rear End Collisions at Signalized lntersections. n Transportation Researc Record 1114, TRB, National Researc Council, Wasington, D.C., pp P. L. Olson and R. W. Rotery. Deceleration Levels and Clearance Times Associated wit te Amber Pase of Traffic Signals. Traffic Engineering, April 1972, pp Y. Seffi and H. Mamassani. A Model of Driver Beavior at Hig Speed Signalized ntersections. Transportation Science, Vol. 15, No. 1, Feb. 1981, pp A. D. May. Clearance nterval at Traffic Signals. n Higway Researc Record 221, HRB, National Researc Council, Wasington, D.C., 1968, pp nstitute of Transportation Engineers. Transportation and Traffic Handbook (W. S. Homberger, ed.). Prentice Hall, Englewood Cliffs, N.J., TE Tecnical Council Committee 4A-16. Determining Veicle Signal Cange ntervals.!te Journal, July 1989, pp Y. Can and T. Liao. Setting Cange ntervals at Signalized ntersections.!te Journal, Feb. 1987, pp M.-S. Cen, C. J. Messer, and A. J. Santiago. Timing Traffic Signal Cange ntervals Based on Driver Beavior. n Transportation Researc Record 127, TRB, National Researc Council, Wasington, D.C., 1985, pp G. J. Klir and T. A. Folger, Fuzzy Sets, Uncertainty and nformation. Prentice Hall, Englewood Cliffs, N.J., 1988, pp H.-J. Zimmermann. Fuzzy Set Teory and ts Applications (2nd edition). Kluwer Academic Publisers, Norwell, Mass., 199, pp A. Kaufmann and M. M. Gupta. introduction to Fuzzy Matematics Teory and Applications. Van Nostrand Reinold, New York, C. V. Zegeer and R. C. Deen. Green Extension Systems at Hig Speed ntersections. /TE Journal, Nov. 1978, pp F.-B. Lin and S. Vijaykumar. Timing Design of Signal Cange nterval. Traffic Engineering and Control, Oct. 1988, pp Publication of tis paper sponsored by Committee 11 Traffic Control Devices.