> > : f p P 0: ðo;dÞ
ð2Þ
qk ¼ argminq2½0;1 z½f k þ qðf k f k Þ:
One can note that, the link cost functions are monotonically increasing with respect to the flows and they are assumed to be separable, the function zðf Þ is strictly convex and because the constraints are also convex, it can be proved that there exists one unique solution for this problem (see e.g., Ceylan and Bell, 2004, for details). Applying Lagrangian relaxation technique, the equilibrium solution of this model can be obtained as follows (Damberg et al., 1996): p ¼ d ðo;dÞ Probpðo;dÞ fðo;dÞ
8ðo; dÞ 2 C 8p 2 P ðo;dÞ ;
ð3Þ
where Probpðo;dÞ ¼ P
expðhcpðo;dÞ Þ q2P ðo;dÞ
expðhcqðo;dÞ Þ
8ðo; dÞ 2 C 8p 2 P ðo;dÞ ; ð4Þ
which leads us to a Logit route choice model (Sheffi, 1985). Damberg et al. (1996) have also proposed a method, which hereafter is named Damberg algorithm, in order to construct Logit flows by alternating between two main phases; generating reasonable paths and a descent method for obtaining equilibrium flows on the restricted problem. It seems that, if the set of initial paths consists of almost all of the reasonable paths, it is not necessary to alternate between these phases. On the other hand Damberg algorithm cannot consider all of the paths that may be used by travelers in real problems. Thus, we modify Damberg algorithm so that heuristically applies one iteration with a sufficient number of reasonable paths generated by different path enumeration techniques such as K-shortest paths, K-similar paths, dissimilar paths, etc. In kth iteration of this modified algorithm, ðf p;k ðo;dÞ Þp2P ðo;dÞ ; ðo;dÞ2C are the amount of flows which are assigned to paths, say as-is Logit flows, while ðf p;k ðo;dÞ Þp2P ðo;dÞ ; ðo;dÞ2C are the amount of flows which should be assigned to paths with respect to the costs and Logit route choice model, say to-be Logit flows. The algorithm terminates when the difference between to-be flows and as-is flows is less than a predetermined tolerance in which the as-is flow satisfies the Logit route choice model. Algorithm 2.1. ½flow; fval ¼ MDambergðfia ga2A ; fP ðo;dÞ gðo;dÞ2C ; fd ðo;dÞ gðo;dÞ2C ; ; hÞ p;0 1. Choose initial path flows f 0 ¼ ðfðo;dÞ Þp2P ðo;dÞ ; k k 0 2. Let k ¼ 0, f ¼ 1 and f ¼ f . p;k 3. While maxðo;dÞ2C;p2P ðo;dÞ jf p;k ðo;dÞ f ðo;dÞ j >
P P p (a) Calculate link flows xa ¼ ðo;dÞ2C p2P ðo;dÞ dp;a ðo;dÞ f ðo;dÞ a a a a and link costs t ¼ t ðx ; i Þ for each a 2 A. P p k (b) Determine path costs cp;k a2A ðo;dÞ ,cðo;dÞ ðf Þ ¼ p;a a dðo;dÞ t for all p 2 P ðo;dÞ ; ðo; dÞ 2 C. (c) Compute the auxiliary path flows f k according to the Logit formulas (3 and 4). p;k (d) Terminate if maxðo;dÞ2C;p2P ðo;dÞ jf p;k ðo;dÞ f ðo;dÞ j 6 (e) Solve the following line search problem for minimizing performance function zðf Þ in (1):
ðo;dÞ2C .
(f) Let f kþ1 ¼ f k þ qk ðf k f k Þ and k ¼ k þ 1. 4. Let flow ¼ f k and fval ¼ zðflowÞ as the outputs. Although this scheme consists of a time-consuming iteration, this procedure iterates only once, this variant usually decreases the computation time. In Section 5, the efficiency of this scheme which plays an essential role in next methods, is implicitly represented. The most important part in Algorithm 2.1 is to determine the Logit flows according to disutility measure (4). Our numerical examination shows that this measure may be instable for some very small or large numbers. Because the speed of variation in exponential function is very high, its values with respect to very large positive or negative numbers are close to zero and infinity, respectively. For such cases, computer programs terminated due to Dividing by zero. Therefore, without loss of generality, we utilize the following simple approximated and normalized formula in place of (4) in which the value of proportion of travel cost for a path to the maximal travel cost between the reasonable paths for one OD is applied as disutility measure. Note that since, the costs in (4) have only comparing roles, we can divide them to a fixed parameter. c
Probpðo;dÞ
p
exp h maxfcm ðo;dÞ jm2P ðo;dÞ g ðo;dÞ ¼ cqðo;dÞ P exp h q2P ðo;dÞ maxfcm jm2P ðo;dÞ g
8ðo; dÞ 2 C
ðo;dÞ
ð5Þ 8p 2 P ðo;dÞ : By using this formula, the mentioned instability essentially decreases. But note that, because the standard of disutility as a maximal value is determined depending on the external condition given before the estimation, the solutions are influenced by the initial condition of model. The following discussion shows the practical use of this formula. Proposition 2.2. If we denote the probability of choosing of each path p 2 P ðo;dÞ with respect to formulas (4) and (5) with p;2 Probp;1 ðo;dÞ and Probðo;dÞ ; respectively, for all p; q 2 P ðo;dÞ , we have: q;1 p;2 q;2 Probp;1 ðo;dÞ 6 Probðo;dÞ () Probðo;dÞ 6 Probðo;dÞ :
Proof. We can write directly: Lemma 2.3. With the notation of Proposition (2.2), for all p; q 2 P ðo;dÞ , we have
M. Ghatee, S.M. Hashemi / European Journal of Operational Research 194 (2009) 432–451
435
q;1 p q p q Probp;1 ðo;dÞ 6 Probðo;dÞ () expðhcðo;dÞ Þ 6 expðhcðo;dÞ Þ () hcðo;dÞ 6 hcðo;dÞ
()
hcpðo;dÞ maxfcmðo;dÞ jm 2 P ðo;dÞ g
hcqðo;dÞ
6
maxfcmðo;dÞ jm 2 P ðo;dÞ g ! ! hcpðo;dÞ hcqðo;dÞ () exp 6 maxfcmðo;dÞ jm 2 P ðo;dÞ g maxfcmðo;dÞ jm 2 P ðo;dÞ g
q;2 () Probp;2 ðo;dÞ 6 Probðo;dÞ :
Probp;2 ðo;dÞ Probq;2 ðo;dÞ
( ¼
Probp;1 ðo;dÞ Probq;1 ðo;dÞ
)1= maxfcmðo;dÞ jm2P ðo;dÞ g
Proof. Straightforward.
in :
h
ðo; dÞ increase, we have maxfcmðo;dÞ jm 2 P ðo;dÞ g ! 1. Since, p;1
p;2
Probðo;dÞ Probq;2 ðo;dÞ
q;1
Probðo;dÞ
> 0, with respect to Lemma 2.3, we have
! 1 for all paths p; q 2 P ðo;dÞ and this shows the
closeness of probabilities of choosing paths p and q for traveling between ðo; dÞ with respect to formula (5). On the other hand, by using formula (4) as route choice function, for path p 2 P ðo;dÞ in which cpðo;dÞ ¼ maxfcm ðo;dÞ jm 2 P ðo;dÞ g, we have Probp;1 ðo;dÞ ! 1 while for path q 2 P ðo;dÞ that its cost is much less than cpðo;dÞ , we have Probq;1 ðo;dÞ ! 0. This reveals the farness of probabilities of choosing paths with respect to formula (4). h
q1 ¼ jp
maxfcm jm2P ðo;dÞ g1 ðo;dÞ ðo;dÞ jþ
P
q
q2P ðo;dÞ
expðhcðo;dÞ Þ
expðhcqðo;dÞ Þcpðo;dÞ jP ðo;dÞ j q
q2P ðo;dÞ
Proof. Note that when the travel costs for an OD pair Probðo;dÞ
q2P ðo;dÞ
P
Proposition 2.4. As much as the travel costs increase, the probabilities of choosing paths for traveling between OD pairs with respect to formula (5) become smooth and close together while these probabilities with respect to (4) become essentially different and far each other.
always
which P
expðhcðo;dÞ Þþmaxfcm jm2P ðo;dÞ gjP ðo;dÞ j ðo;dÞ
and
q2 ¼
. Note that q2 is strictly
less than 1. Therefore, the difference between Probp;2 ðo;dÞ and Probp;1 is less than q . h 1 ðo;dÞ In what follows, we utilize fuzzy numbers to exhibit uncertain number of traveler in TA model. 3. Constructing fuzzy TA model The aim of TA problem is to predict the amount of flows on network links, while it depends on the route choice models. The vast majority of classic TA models are deterministic as they assume that the travel demand between the pairs of OD nodes to be known a priori for the entire planning. These hypotheses aid the solution process and highlight a fundamentally complex characteristic of this problem. As it reflects the different provision of traffic information, the different user equilibrium models might be deduced. Usually uncertain events can be taken by stochastic and fuzzy viewpoints. In the following, a brief review on the advanced results on fuzzy TA is presented. 3.1. Fuzzy TA models
TA models adopted with fuzzy relations and quantities can contribute stochastic versions in uncertainty modeling. 2 Corollary 2.5. By previous notations, when In other word, since the classical random utility-based P ðo;dÞ g > 1, for each path p 2 P ðo;dÞ we approximately have approach is not sufficient for uncertainty handling, some maxfcmðo;dÞ jm 2 P ðo;dÞ g 1 p;2 p;1 alternate solution to probability-based models may be P Probðo;dÞ Probðo;dÞ < : expðhcqðo;dÞ Þ jpðo;dÞ j þ obtained with possibility-based methods, see e.g., Hoogenq2P ðo;dÞ doorn et al. (1993), Kikuchi (1998), Henn and Ottomanelli (2006) and Kikuchi and Chakroborty (2006). But as historProof. For each path p 2 P ðo;dÞ , by using first-order Taylor ical background, fuzzy logic tools initially has been applied approximation, we have to TA models by Teodorovic and Kikuchi (1990), who p m maxfc jm 2 P g þ c designed a set of fuzzy rules and used fuzzy inference techðo;dÞ ðo;dÞ ðo;dÞ P ; niques to study the binary route choice problem. Later, Probp;2 q ðo;dÞ ’ m maxfcðo;dÞ jm 2 P ðo;dÞ gjP ðo;dÞ j þ expðhcðo;dÞ Þ q2P ðo;dÞ Mamdani inference method on drivers’ route choice behavp ior was pursued by Lee et al. (2003). 1 þ cðo;dÞ P Probp;1 ; From a high level categorization, in urban networks we q ðo;dÞ ’ jP ðo;dÞ j þ expðhcðo;dÞ Þ may face with the following three uncertainty sources: q2P ðo;dÞ maxfcmðo;dÞ jm
thus, we can write p;1 Probp;2 ðo;dÞ Probðo;dÞ ¼ q1 :q2 ;
Inexact travel cost (a network with fuzzy link cost), Unsure network topology (a network with fuzzy nodes and fuzzy links),
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Imprecise travel demand (a network with fuzzy excess node or deficit node). Akiyama and Yamanishi, 1993 and also, Liu et al., 2003, have illustrated some imaginary interpretation of fuzzy travel costs in TA models. Also, Lotan and Koutsopoulos, 1993 and Akiyama and Yamanishi, 1993, have developed some models for route choice behavior due to approximate reasoning concepts and fuzzy control. For comparing imprecise costs which independently have been defended by Henn, 2005, possibility and necessity measures were typically pursued. For example Akiyama and Nomura, 1999, took a fuzzy goal G as an upper bound for travel time between and applies the possibility measures to compare the travel time with G. Due to this approach, the degree of acceptance for the drivers who choose that path can be obtained. In addition, Akiyama and Nomura (1999), utilized the convex combination of possibility and necessity as ranking index. Also, he combined the standard fuzzy travel time approach with successive average method which requires to find the alternative paths between OD pairs before path-flow estimation. Then Akiyama (2000a,b) utilized Okada and Soper’ shortest path algorithm (2000), together with the fuzzy goals to find the equilibrium flows. Almost in the same time, Henn (2000) developed a fuzzy route choice model for TA by various comparison indices. He represented the different natures of travelers such as risk taking or risk averting in their route choice decisions. With some special shapes of membership functions, his model finds similar results to the Logit model ones. This model is a generalization of Logit model which provides more flexibility on the modeling of drivers’ travel time perception. Nevertheless, this model cannot be implemented in all of the cases as he noticed. Moreover, Ridwan (2004), utilized a decision structure in route choice, which was not only relied on optimization techniques such as shortest path and least time path, but also, it consists of fuzzy preference relations to yield the actual choice of the travelers. With respect to the network topology viewpoint, Lee et al. (1998), have recommended a fuzzy logic-based model to detect lane-blocking incidents whose effects are manifested by patterns of deterioration in traffic conditions that require adjustments in signal control strategies. Also, Liu et al. (2003) suggested a fuzzy dynamic TA model due to the fuzzy network statues such as normal or congested modes. They converted fuzzy graph into two corresponding crisp graphs to find the representative shortest paths and their membership degrees through solving the K-shortest paths algorithm. In concept of fuzzy quantities and fuzzy relations which are appeared in fuzzy travel demand category, Wang and Liao (1999), have derived some results on fuzzy equilibrium, employing variational inequalities. Also, Chang and Chen (2000) used the variational inequality approach to formulate a link-based fuzzy user-optimal route choice problem embedding link interactions. But these established works are not seen enough. Since, this concept can contribute to handle incomplete and imperfect data in urban problems,
in the next sections, we focus on imprecise number of travelers in TA models and represent them as triangular fuzzy numbers. 3.2. Fuzzy level of OD matrix To explain the advantages of using the fuzzy or stochastic instruments for modeling the OD matrix, first we need to compare the fuzzy and the stochastic mathematical programming. Inuiguchi and Ramik, 2000, presented the following important differences between these two concepts: 1. For a general distribution, a stochastic programming problem cannot usually be solved easily, but a fuzzy mathematical programming problem can be solved easily even when the possibilistic vector is restricted by any unimodal distribution. 2. Suppose that the uncertain variables are independent. Then only a small number of decision variables takes non-zero values in the optimal solution of the fuzzy mathematical programming problem while a large number of decision variables takes non-zero values in the optimal solution of the stochastic programming problem. On the other hand, in the fuzzy mathematical programming, the fuzzy demand is the center of many researches. According to the theoretical view point, fuzzy demands and fuzzy supplies were introduced for example by Yao and Wu (1999). But its importance lays in its exclusive interpretation, i.e., since, in practice, the demands of customers, are not deterministic, their representation with uncertain tools seems to be appropriate. Note that although the stochastic demands of customers have been adopted and tallied with the facts in widespread cases, but stochastic assumption is not reasonable in a vast range of situations and it is not sufficient to describe many states where the probability distribution of demands may be unknown or partially known (Wen and Iwamura, 2008). It is usually very hard to present the precise demands of customers and for many cases, the estimations of probability distributions for demands are not easy due to the lack of data. Instead, expert’s opinion can be used to provide appropriate estimations (Yao and Wu, 1999; Zhou and Liu, 2007). Some examples of fuzzy demand which are mentioned in recent papers, are as follows: In inventory problems the place of fuzzy demand has been deeply introduced (Hseih, 2002; Dutta et al., 2007). For example in estimating of annual demand, since, the demand probably has a little disturbance due to various uncertainties, the fuzzy annual demand was expressed as the neighborhood of the fixed annual demand (Ouyang and Yao (2002)). In economic production quantity problem, the demand quantity has some little disturbances per day which can be exhibited as fuzzy demand quantity as the neighbor of the fixed economic demand quantity (Lee and Yao, 1998).
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437
In many cases of facility location allocation problem, the amount of demand for each facility cannot represented with stochastic parameters and so the fuzzy demand was introduced (Wen and Iwamura, 2008; Zhou and Liu, 2007).
whom want to trip between each OD pair. This is a motivation to introduce a model to exhibit with fuzzy demand. We take this important aspect with fuzzy relations. Consider the following perturbed TA problem:
It is well known that for forecasting the demand in developing TA models, several models assume stochastic demand. In these cases, the demand distribution is derived from evidence recorded in the past. But it is not always possible to forecast the exact information either due to the lack of evidence or uncertainty in judgment. It is difficult to decide the exact demand, i.e., how many travelers will purchase during the whole season/period (Ouyang and Yao, 2002). As another example, we want to build some new highways in one region to service some urban travelers whose demands can neither be given precisely nor from history data. But those demands can be described by the natural language such as large, little or general, etc. Also, when we want to change the place of one absorbing point, e.g., a hospital or a market, it is not easy to predict statistically how many peoples will choose the paths to that point. In these cases, fuzzy set theory may do better in dealing with ambiguous information. Considering this viewpoint, it is interesting to develop fuzzy TA models for situations where the traveler demand is described linguistically, like demand is about d. However, either due to lack of evidence or uncertainty in judgment, to model a realistic decision-making assignment problem the traveler demand can be described by a knowledge-based uncertainty. Note that, although fuzzy concepts are applied to deal with problems in which no statistical data can be used, when the statistical data are available, we can easily combine the statistics and fuzzy techniques to deal with such problem, see e.g. Ouyang and Yao (2002). Now we show that how the travel demand between one OD pair can be represented as fuzzy number. It is important to note that the average of travelers between each OD pair is embedded as the center of triangular fuzzy number, while the spread of triangular numbers can be considered as the data deviation. Note that if one take different spreads, he can regulate the possibility for coming closer or going farther from the average amount. With the help of these spreads, a generalization of confidence interval can be concluded in which network planner can implement his/her perspective about extreme events. For example, if one is risk-avert, he may consider more possibility for events on the right side of the average amount, while a risk-seeker one may act conversely. In the next section, we adopt the fuzziness of traveler demand into the Logit TA model. We shall focus on the advantages of utilizing triangular fuzzy numbers to represent the imprecise number of travelers.
min zðf Þ ¼
3.3. TA model with imprecise level of travel demand In many real TA problems, particularly in network planning, we face with the uncertain individual travelers
8 ðaÞ > > > < s:t: ðbÞ > > > : ðcÞ
1 X X p p fðo;dÞ ðLnðfðo;dÞ Þ 1Þ h ðo;dÞ2C p2P ðo;dÞ X Z xa ta ðwÞdw; þ P P
a2A
p p2P ðo;dÞ fðo;dÞ ðo;dÞ2C
p fðo;dÞ
ð6Þ
0
P
d ðo;dÞ
p2P ðo;dÞ
8ðo; dÞ 2 C;
p fðo;dÞ dp;a ðo;dÞ
¼ xa
8a 2 A;
ð7Þ
P 0:
This problem is similar to TA model (1) except the soft constraints (7.a). Because of uncertain daily number of travelers between each pair of OD nodes of urban network, such soft constraint may be appeared. In such status, from the incomplete and imperfect statistical data or experts’ opinion, the lower and upper bounds of demand and also, the most possible or visible state of demand (the amount which is commonly seen in reality) for OD pair ðo; dÞ can be determined easily. We denote these parameters with pos max d min ðo;dÞ , d ðo;dÞ and d ðo;dÞ , respectively. Thus, the level of travel demand is assumed to be represented as triangular fuzzy min max number ðd pos ðo;dÞ ; d ðo;dÞ ; d ðo;dÞ Þ. Thus, we can define a membership function with respect to each soft constraint in (7.a) as follows:
0 l@
X
1 p fðo;dÞ ¼ d ðo;dÞ A ¼
p2P ðo;dÞ
8 pos min d min ðd > ðo;dÞ Þ=ðd ðo;dÞ d ðo;dÞ Þ; > > ðo;dÞ > > pos min > > < d ðo;dÞ 6 d ðo;dÞ 6 d ðo;dÞ ; max
max
pos
ðd ðo;dÞ d ðo;dÞ Þ=ðd ðo;dÞ d ðo;dÞ Þ; > > > max > > d pos > ðo;dÞ 6 d ðo;dÞ 6 d ðo;dÞ ; > : 0; o:w: ð8Þ
It is important to note that this triangular membership function is appropriate for risk neutral network planner, but by using the nonlinear functions LðÞ and RðÞ instead of linear functions in the definition of triangular numbers and working with LR type fuzzy numbers, we can obtain appropriate models satisfying risk averting and seeking attitude (see Appendix A for definition of LR numbers). The following constraints can be utilized in place of soft constraints in (7.a) to guarantee that the level of certainty becomes more than or equal to ‘k’: 0 l@
X p2P ðo;dÞ
1 p fðo;dÞ ¼ d ðo;dÞ A P k 8ðo; dÞ 2 C:
ð9Þ
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M. Ghatee, S.M. Hashemi / European Journal of Operational Research 194 (2009) 432–451
3.4. Equilibrium state in TA model with fuzzy level of travelers From a game-theoretic point of view, a transportation network is considered at equilibrium when all traffic patterns stabilize and no driver has any incentive to change its current route (Wardrop, 1952). In this case, we will say the system is at a user equilibrium state. The other side of the spectrum is when there is a central decision maker that assigns paths to drivers. In this case, the goal is to collectively optimize the utilization of the network; when this goal is achieved we will say that the system is at a social optimum state. Now we want to study the existence of these states for constructed fuzzy TA model. Again, consider the following mathematical programming problem: 1 X X p p fðo;dÞ ðLnðfðo;dÞ Þ 1Þ min zðf Þ ¼ h ðo;dÞ2C p2P ðo;dÞ X Z xa ta ðwÞdw; ð10Þ þ 0
a2A
P p 8 ðaÞ lð fðo;dÞ ¼ d ðo;dÞ Þ P k 8ðo; dÞ 2 C; > > > p2P ðo;dÞ > < P p p;a P fðo;dÞ dðo;dÞ ¼ xa 8a 2 A; s:t: ðbÞ > ðo;dÞ2C p2P ðo;dÞ > > > : ðcÞ f p P 0: ðo;dÞ
s:t:
a2A
P
l
p2P ðo;dÞ
0
p fðo;dÞ
Pk
P p p;a fðo;dÞ dðo;dÞ ¼ xa ðbÞ > > > ðo;dÞ2C p2P ðo;dÞ > > > : ðcÞ f p P 0: ðo;dÞ P
nðo;dÞ ¼
pos 0; d min ðo;dÞ 6 d ðo;dÞ 6 d ðo;dÞ ; max 1; d pos ðo;dÞ < d ðo;dÞ 6 d ðo;dÞ :
ð15Þ
Therefore, the inequality (9) may be rewritten as follows: 8 > > > > > <
P p2P ðo;dÞ
! p fðo;dÞ
d min ðo;dÞ
min =ðd pos ðo;dÞ d ðo;dÞ Þ P k nðo;dÞ ;
! > > P p pos > max max > > : d ðo;dÞ p2P fðo;dÞ =ðd ðo;dÞ d ðo;dÞ Þ P k ð1 nðo;dÞ Þ; ð16Þ
ð11Þ
! ¼ d ðo;dÞ
(
ðo;dÞ
Since, in solving problems under uncertainty it is necessary to exert maximal efforts in seeking the possibilities for overcoming the uncertainty, we also, maximize the certainty level k. To take these targets into account, max–min method, a common scheme in fuzzy programming, may be utilized as follows: 1 X X p p max min zðf Þ ¼ fðo;dÞ ðLnðfðo;dÞ Þ 1Þ k2½0;1 h ðo;dÞ2C p2P ðo;dÞ X Z xa ta ðwÞdw; ð12Þ þ 8 > > > > ðaÞ > > <
vidually to optimize their own utility functions, which is usually the case in transportation networks. Thus, we say that if flow f is optimal solution of our presented problem, at least with k degree of certainty it satisfies the Wardrop’s second principle (1952) and so such flow satisfies user equilibrium principle with certainty level k . However, the system optimality is essentially depended on fitness of provided paths sets between OD pairs. By this background, let define new zero–one variables nðo;dÞ as follows:
8ðo; dÞ 2 C; 8a 2 A; ð13Þ
Assume that k is the optimal solution of the above bi-level programming problem. Then, we should find a flow f which minimizes X 1 X X p p fðo;dÞ ðLnðfðo;dÞ Þ 1Þ þ zðf Þ ¼ h ðo;dÞ2C p2P a2A ðo;dÞ Z xa ta ðwÞdw: ð14Þ 0
But, by comparing this function with the objective function of Logit route choice problem (1) we understand that the congestion can only occur if users choose their paths indi-
where nðo;dÞ ¼ 0, the first constraint is active while the second is inactive. Conversely, the second constraint is taken into account only when nðo;dÞ ¼ 1. Now, in order to define mathematical expressions to implement logic definition (15), we utilize the following constraints: 8P f p 6 Mnðo;dÞ þ d pos > ðo;dÞ ; > > Pp2P ðo;dÞ ðo;dÞ > > p min > > f P d ðo;dÞ ; > < Pp2P ðo;dÞ ðo;dÞ p pos p2P ðo;dÞ fðo;dÞ þ Mð1 nðo;dÞ Þ > d ðo;dÞ ; > > P > p max > > p2P ðo;dÞ fðo;dÞ 6 d ðo;dÞ ; > > > : nðo;dÞ 2 f0; 1g;
ð17Þ
where M is a sufficiently large number (greater than the number of travelers). To verify equivalence between the definition of nðo;dÞ in (15) and the above constraints in (17), one can P note that if nðo;dÞ ¼ 0, the first constraint p pos of (17) gives p2P ðo;dÞ fðo;dÞ 6 d ðo;dÞ . Similarly if nðo;dÞ ¼ 1, P p pos the third constraint in (17) results p2P ðo;dÞ fðo;dÞ > d ðo;dÞ . P p pos Conversely, when p2P ðo;dÞ fðo;dÞ 6 d ðo;dÞ , although from the first constraint of (17) we have nðo;dÞ 2 f0; 1g, but if nðo;dÞ ¼ 1 thePthird constraint is infeasible, therefore, we p pos prove that p2P ðo;dÞ fðo;dÞ 6 d ðo;dÞ , if and only if nðo;dÞ ¼ 0. The inference is valid in order to prove that P same p pos f p2P ðo;dÞ ðo;dÞ > d ðo;dÞ , if and only if nðo;dÞ ¼ 1. Note that the second and fourth restrictions in (17) guarantee to have feasible flows. Now in order to maximize the level of certainty, with the help of (16) and (17), first let us dep Þ satisfying the folfine UðkÞ as the set of flow f ¼ ðfðo;dÞ lowing constraints for each OD pair ðo; dÞ 2 C and p 2 P ðo;dÞ :
M. Ghatee, S.M. Hashemi / European Journal of Operational Research 194 (2009) 432–451
UðkÞ ,
8 > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > :
f
ðaÞ ðbÞ ðcÞ p ¼ ðfðo;dÞ Þ ðdÞ ðeÞ ðfÞ ðgÞ ðhÞ
9 > > p pos min > > fðo;dÞ d min > ðo;dÞ =ðd ðo;dÞ d ðo;dÞ Þ P k nðo;dÞ > > p2P ðo;dÞ > > ! > > > > P > p pos max max d ðo;dÞ fðo;dÞ =ðd ðo;dÞ d ðo;dÞ Þ P k ð1 nðo;dÞ Þ > > > > p2P ðo;dÞ > > > P p > pos > > fðo;dÞ 6 Mnðo;dÞ þ d ðo;dÞ > > p2P ðo;dÞ > = P p min : fðo;dÞ P d ðo;dÞ > > p2P ðo;dÞ > > > P p > > fðo;dÞ þ Mð1 nðo;dÞ Þ > d pos > ðo;dÞ > > p2P ðo;dÞ > > > P p > max > > fðo;dÞ 6 d ðo;dÞ > > p2P ðo;dÞ > > > > > > nðo;dÞ 2 f0; 1g > > > p ; P0 f P
!
ð18Þ
ðo;dÞ
Then, we find optimal certainty degree of model (12) from the following mathematical programming problem: ð19Þ
max k s.t. k 2 ½0; 1 and
439
UðkÞ–£:
Assume that k is the optimal solution of the above problem. Then the following problem should be solved to obtain the equilibrium flows: ( 1 X X p f p ðLnðfðo;dÞ Þ 1Þ min p2P ðo;dÞ ðo;dÞ h ðo;dÞ2C X Z xa a t ðwÞdw ; ð20Þ þ a2A 0 8 p < fðo;dÞ 2 Uðk Þ 8ðo; dÞ 2 C 8p 2 P ðo;dÞ ; P P p a ð21Þ s:t: fðo;dÞ dp;a 8a 2 A: ðo;dÞ ¼ x : ðo;dÞ2C p2P ðo;dÞ
When the certainty level is assumed to be fixed at maximal degree of constraints fulfilling in (18), the proposed problem produces flows with respect to Wardrop’s user equilibrium principle. In the next subsection we follow a useful scheme for optimizing the level of certainty. Subsequently, we find equilibrium flows satisfying the constraint of the above model. 3.5. Maximizing the degree of certainty Considering the programming problem in model (19) restricted with constraints in definition (18), we can derive some simple inequality results. One first investigate the convexity property of set UðkÞ. Note that if the constraint (18.e) destroys the convexity property of set UðkÞ, the optimization techniques, such as simplex method, for solving model (19) may be failed. For example, it is possible that
the optimal solution is binding in > constraint (18.e) and this means that optimal solution is infeasible. To conquer such drawback, the following proposition is presented. p Þ be an optimal solution for Proposition 3.1. Let f ¼ ðfðo;dÞ model (19) satisfying all conditions (18) except (18.e), and the inequality (18.e) be binding. Then, we can find another solution with the same objective value which satisfies all conditions in (18).
Proof. Suppose that for ðo; dÞ 2 C we have: X p fðo;dÞ þ Mð1 nðo;dÞ Þ ¼ d pos ðo;dÞ : p2P ðo;dÞ
Since, M is a large number, the left hand side is close to infinity while the right hand side is finite, except when nðo;dÞ ¼ 1. In this situation, we obtain: X p fðo;dÞ ¼ d pos ðo;dÞ : p2P ðo;dÞ
But this solution may be produced with nðo;dÞ ¼ 0 which binds in the constraint (18.c) and unbinds in (18.e). Thus, we can obtain this solution with respect to nðo;dÞ ¼ 0 instead of the previous infeasible solution. Since, the objective function (20) is independent of nðo;dÞ s, the objective values with respect to these solutions are equal and the proof is complete. h Proposition 3.1 shows that we can use the following restriction in place of (18.e) without loss of generality: X p fðo;dÞ þ Mð1 nðo;dÞ Þ P d pos ð180 Þ ðo;dÞ þ e; p2P ðo;dÞ
where e > 0 is close to zero. Taking this restriction into account, the set UðkÞ becomes convex. Now we wish to follow a computational scheme for maximizing certainty. First we
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M. Ghatee, S.M. Hashemi / European Journal of Operational Research 194 (2009) 432–451
present the following two lemmas which have essential roles in the proposed method which will be presented soon. Lemma 3.2. If k1 P k2 , then Uðk1 Þ # Uðk2 Þ. Lemma 3.3. If for some k the constraints (18) are infeasible, then for each k P k, those constraints are infeasible, too. Both of these lemmas can be proved directly.Now, for convenience in notation, let pos min F ðo;dÞ ðkÞ ¼ maxfd min ðo;dÞ þ ðd ðo;dÞ d ðo;dÞ Þðk nðo;dÞ Þ; min d pos ðo;dÞ Mð1 nðo;dÞ Þ þ e; d ðo;dÞ g;
Gðo;dÞ ðkÞ ¼
minfd max ðo;dÞ Mnðo;dÞ þ
ðk þ nðo;dÞ
1Þðd max ðo;dÞ
ð22Þ
d pos ðo;dÞ Þ;
max d pos ðo;dÞ ; d ðo;dÞ g:
ð23Þ
Remark 3.4. The constraints on flows satisfying in UðkÞ can be replaced with the following two inequalities: 8 P p f 6 Gðo;dÞ ðkÞ > < p2P ðo;dÞ ðo;dÞ P p 8ðo; dÞ 2 C: ð24Þ > fðo;dÞ P F ðo;dÞ ðkÞ : p2P ðo;dÞ
To explain the physical meaning of these parameters, the following properties are useful. Lemma 3.5. For each k 2 ½0; 1 and each OD pair ðo; dÞ 2 P ðo;dÞ we have 8( pos min > F ðo;dÞ ðkÞ ¼ d min > ðo;dÞ ð1 kÞðd ðo;dÞ d ðo;dÞ Þ > > if nðo;dÞ ¼ 1; > < Gðo;dÞ ðkÞ ¼ d max kðd max d pos Þ; ðo;dÞ ðo;dÞ ðo;dÞ ( pos min > F ðo;dÞ ðkÞ ¼ d min > ðo;dÞ þ kðd ðo;dÞ d ðo;dÞ Þ; > > if nðo;dÞ ¼ 0: > : G ðkÞ ¼ d pos ; ðo;dÞ ðo;dÞ Proposition 3.6. For each OD pair ðo; dÞ and each certainty degree k 2 ½0; 1, the value of F ðo;dÞ ðkÞ is increasing with respect to k while the value of Gðo;dÞ ðkÞ is decreasing with respect to k. Proof. Apply Lemma 3.5, the proof is easy.
h
It is interesting to note that when nðo;dÞ ¼ 1 for OD pair ðo; dÞ, according to Lemma 3.5, with respect to the certainty degree k ¼ 1 we have F ðo;dÞ ðk ¼ 1Þ ¼ d min ðo;dÞ and . With other word, the sum of flows Gðo;dÞ ðk ¼ 1Þ ¼ d pos ðo;dÞ on paths between ðo; dÞ enforces to obtain a value in interpos val ½d min ðo;dÞ ; d ðo;dÞ . It means that, with the highest certainty degree, the flow should be obtained values between the least demand and the most possible one. On the other hand, with the lowest certainty degree pos min k ¼ 0, we have F ðo;dÞ ðk ¼ 0Þ ¼ d min ðo;dÞ ðd ðo;dÞ d ðo;dÞ Þ and max Gðo;dÞ ðk ¼ 0Þ ¼ d ðo;dÞ . Thus, the sum of the flow on paths between ðo; dÞ can be permitted to obtain value in interval pos min max ½d min ðo;dÞ ðd ðo;dÞ d ðo;dÞ Þ; d ðo;dÞ .
In the case of nðo;dÞ ¼ 0, we have F ðo;dÞ ðk ¼ 1Þ ¼ d pos ðo;dÞ , and F ðo;dÞ ðk ¼ 0Þ ¼ Gðo;dÞ ðk ¼ 1Þ ¼ Gðo;dÞ ðk ¼ 0Þ ¼ d pos ðo;dÞ d min ðo;dÞ . Thus, according to highest degree of certainty, P p pos p2P ðo;dÞ fðo;dÞ is equal to the most possible demand d ðo;dÞ while with respect to the lowest degree of certainty, this pos term should be taken value in ½d min ðo;dÞ ; d ðo;dÞ . Lemma 3.7. If k1 P k2 , then F ðo;dÞ ðk1 Þ P F ðo;dÞ ðk2 Þ 8ðo; dÞ 2 C; Gðo;dÞ ðk1 Þ 6 Gðo;dÞ ðk2 Þ 8ðo; dÞ 2 C:
ð25Þ
Proof. With respect to Lemma 3.2 the result is straightforward. Also, this result can be derived from Proposition 3.6. h This lemma concludes the following bisection algorithm which finds the maximal level of certainty for model (19). Algorithm 3.8 (Finding the maximal tainty). ½k ¼ Maximal CertaintyðUðkÞÞ
level
of
cer-
1. Set k ¼ 0 and k ¼ 1 and chose the error threshold > 0. 2. while ðk kÞ > k ¼ ðk þ kÞ=2. exit flag ¼ false. Run sub-procedure exit flag ¼ Test FeasibleðUðkÞ; valueðnÞ; listðnÞÞ to check the feasibility of UðkÞ. If exit flag is true, set k ¼ k: else k ¼ k: 3. Return k ¼ k. It is interesting to note that, since, the restrictions (24) are independent of OD pairs ðo; dÞ 2 C, then if F ðo;dÞ ðkÞ 6 Gðo;dÞ ðkÞ;
ð26Þ p fðo;dÞ
we can P assign some positive numbers to variables in p f satisfies the inequalities in (24). But which p2P ðo;dÞ ðo;dÞ F ðo;dÞ ðkÞ and Gðo;dÞ ðkÞ depend on nðo;dÞ variables, too. Thus, in Test FeasibleðUðkÞÞ we pursue the following recursive procedure which traverse the branches of a binary tree whose links labels are the amount of nðo;dÞ variables (see Fig. 1). The arguments valueðnÞ and listðnÞ are the input of this procedure in which the values of nðo;dÞ variables on each branch of tree are saved in valueðnÞ. What of the nðo;dÞ variables which have not still gotten values of f0; 1g, are saved in listðnÞ. When listðnÞ is empty, the inequalities in (24) are evaluated. At the first, since, the search tree consists no leaf, we set listðnÞ ¼ C and valueðnÞ ¼ £. If for one fnðo;dÞ gðo;dÞ2C the constraint (26) is feasible, exit flag is true and the algorithm terminates. Algorithm ðnÞÞ
3.9. exit flag ¼ Test FeasibleðUðkÞ; valueðnÞ; list-
M. Ghatee, S.M. Hashemi / European Journal of Operational Research 194 (2009) 432–451
441
Fig. 1. A binary search tree for traversing all of the combination of nðo;dÞ values. In this figure, we have k ¼ jCj.
1. If exit flag is true return. 2. Else (a) If listðnÞ is empty (b) With respect to the valueðnÞ for each ðo; dÞ 2 C, compute F ðo;dÞ ðkÞ and Gðo;dÞ ðkÞ using expressions (22) and (23). If for at least one ðo; dÞ 2 C, Gðo;dÞ ðkÞ < F ðo;dÞ ðkÞ – exit flag is false. – Return. Else exit flag is true. Return. (c) Pick up the element ðo0 ; d 0 Þ from top of listðnÞ. (d) Insert nðo0 ;d 0 Þ ¼ 1 in the valueðnÞ. (e) Run exit flag ¼ Test FeasibleðUðkÞ; valueðnÞ; listðnÞÞ. (f) If exit flag is false Insert nðo0 ;d 0 Þ ¼ 0 in the valueðnÞ. Run exit flag ¼ Test FeasibleðUðkÞ; valueðnÞ; listðnÞÞ. 3. Return exit falg. Since, this procedure terminates when the first feasible solution is found, it may be repeated 2c times in which c ¼ jCj is the number of OD pairs. However, each iteration consists of at most c simple inequalities checking which is not very time consuming. The following interesting lemma shows that we need run either (2.e) or (2.g). Clearly, this point accelerate the algorithm at least twice. Lemma 3.10. (i) If for some ðo; dÞ 2 C with respect to nðo;dÞ ¼ 1, the feasible flow set UðkÞ is empty, then for nðo;dÞ ¼ 0 this set is empty, too.(ii) Conversely, by using nðo;dÞ ¼ 0, if UðkÞ is empty, then the same result happens when nðo;dÞ ¼ 1. Proof. Denote the feasible flow satisfying in constraint (18.a) and (18.b) with Aðk; nðo;dÞ Þ and Bðk; nðo;dÞ Þ, respectively. It is easy to prove that
Aðk; fnðo;dÞ ¼ 0gÞ # Aðk; fnðo;dÞ ¼ 1gÞ;
ð27Þ
Bðk; fnðo;dÞ ¼ 1gÞ # Bðk; fnðo;dÞ ¼ 0gÞ:
ð28Þ
Thus, if for nðo;dÞ ¼ 1, UðkÞ is empty, then with respect to (27) for nðo;dÞ ¼ 0, UðkÞ is empty, too. On the other hand according to (28), if nðo;dÞ ¼ 0 implies UðkÞ ¼ ;, then UðkÞ is empty with respect to nðo;dÞ ¼ 1. h Now we return to the subproblem (20) and reshape it as follows when the optimal degree of certainty is k : 8 <1 X X p p min fðo;dÞ ðLnðfðo;dÞ Þ 1Þ :h ðo;dÞ2C p2P ðo;dÞ ) X Z xa þ ta ðwÞdw ; ð29Þ a2A
0
P p 8 F ðo;dÞ ðk Þ 6 fðo;dÞ 6 Gðo;dÞ ðk Þ 8ðo; dÞ 2 C; > > > p2P ðo;dÞ > < p f P 0 8p 2 P ðo;dÞ ; s:t: ðo;dÞ > P P > p p;a > > fðo;dÞ dðo;dÞ ¼ xa 8a 2 A: : ðo;dÞ2C p2P ðo;dÞ
ð30Þ Note that, this problem is similar to Logit model which allows finding equilibrium flows from a convex mathematical programing problem (see e.g., Ceylan and Bell, 2004). This model has nonlinear objective function together with linear constraints. It seems that for solving this problem, Frank– Wolfe method (see e.g. Sheffi, 1985) which iteratively utilizes the approximated linear programming problem, can be used. But, since, the nðo;dÞ variables should be gotten zero–one values, a branch and bound scheme is employed as a sub-procedure. We implemented this scheme on MATLAB 7.0, but unfortunately, we could not find desirable solution. In our tests with respect to the different hs, the resulting solutions were the corners of the feasible solutions polyhedral. This means that some components of resulted flow are zero, while in real status, almost all of the paths are used by urban drivers and so the flow through each of the paths is assumed to be strictly positive number (see
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M. Ghatee, S.M. Hashemi / European Journal of Operational Research 194 (2009) 432–451
e.g., Ceylan and Bell, 2004, for more details). We can explain why some of the components of resulted flow are zero. Note that since, the smallest registrable number in MATLAB codes is eps ¼ 2:2204 10016 , and with respect to this value as the components of flow, the first term in the objective function (29) is very small number, and epsðLnðepsÞ 1Þ ¼ 8:2253 10015 , then this number is ignored and our code cannot forbid such flows. On the other hand, Frank–Wolfe method iteratively uses linear programming schemes such as simplex algorithm which finds a corner point of feasible set as extreme solution. Furthermore, according to getting deviation from function (29), in the objective function of Frank–Wolfe method only p p Þ exists and for fðo;dÞ ¼ eps, this term is equal to Lnðfðo;dÞ 36.0437. This number in compare to the other large transportation costs is easily ignorable. Thus, for our task, the Frank–Wolfe algorithm cannot find the desirable flows with non-zero components. In the next section we present a better scheme for solving this problem with respect to quasi-Logit formulas for fuzzy TA model.
seen in Eq. (44)) instead of link-capacity, this unit capacity does not inhibit flowing more than one unit through the mentioned path. p;þ P0 Now let define two new non-negative variables fðo;dÞ p; p and fðo;dÞ P 0 in relating to the variable fðo;dÞ and use the following simple change of variable: p p;þ p; ¼ fðo;dÞ þ fðo;dÞ fðo;dÞ
X
p;þ fðo;dÞ ¼ F ðo;dÞ
s:t:
a2A
P F ðo;dÞ
8ðo; dÞ 2 C;
ð32Þ
p0
p ðo;dÞ fðo;dÞ þ fðo;dÞ ¼ Gðo;dÞ
8ðo; dÞ 2 C;
ð33Þ
p2P ðo;dÞ p fðo;dÞ
X
P 0 8ðo; dÞ 2 C 8p 2 P ðo;dÞ ; X p p;a fðo;dÞ dðo;dÞ ¼ xa 8a 2 A;
Therefore, we have: X
p fðo;dÞ ¼
p2P ðo;dÞ
X
p;þ fðo;dÞ þ
p2P ðo;dÞ
X
p; fðo;dÞ P
p2P ðo;dÞ
X
p;þ fðo;dÞ
p2P ðo;dÞ
ð38Þ
which satisfies (32). On the other hand, substituting those new variables into Eq. (33) we obtain: X
p;þ fðo;dÞ þ
p2P ðo;dÞ
X
p0
p; ðo;dÞ fðo;dÞ þ fðo;dÞ ¼ Gðo;dÞ ;
ð39Þ
p2P ðo;dÞ
and with respect to Eq. (37), we obtain: X
p0
p2P ðo;dÞ
p; ðo;dÞ fðo;dÞ þ fðo;dÞ ¼ Gðo;dÞ F ðo;dÞ :
ð40Þ
Now extend the set of paths between the OD pair ðo; dÞ as follows: P 0ðo;dÞ ¼ P ðo;dÞ [ fp0ðo;dÞ , o ! w ! dg:
ð34Þ ð35Þ
ðo;dÞ2C p2P ðo;dÞ p0ðo;dÞ
ð37Þ
0
p fðo;dÞ
p2P ðo;dÞ
X
8ðo; dÞ 2 C 8p 2 P ðo;dÞ :
p2P ðo;dÞ
¼ F ðo;dÞ ;
In this section, we show that how it is possible to combine the modified Damberg Algorithm 2.1 and branch and bound scheme to solve efficiently the subproblem (29). Furthermore, we construct some formulas representing equilibrium flows of fuzzy TA model. Hereafter, since, k is assumed to be a fixed number, for simplicity without any ambiguity, we eliminate k from the formulation of models. Thus, we can rewrite our model as follows: ( 1 X X p f p ðLnðfðo;dÞ Þ 1Þ min p2P ðo;dÞ ðo;dÞ h ðo;dÞ2C ) X Z xa þ ta ðwÞdw ð31Þ
ð36Þ
Provide that
4. Finding equilibrium flows in fuzzy TA model
X
8ðo; dÞ 2 C 8p 2 P ðo;dÞ :
where fðo;dÞ for each ðo; dÞ 2 C is a slack variable with respect to the new defined path p0ðo;dÞ with zero cost. By inserting a new node in network, say w, the new path p0ðo;dÞ can be considered as a path through the dummy links ðo; wÞ and ðw; dÞ. Moreover, in order to bound the amount of flow through such links, the capacity of them are taken as one. Note that without this policy, all flows are moving through such path which is not desirable. On the other hand, since, we use link-impedance (one example can be
Furthermore, suppose that p0
ðo;dÞ fðo;dÞ
;
p0
ðo;dÞ ¼ fðo;dÞ ;
and p0
ðo;dÞ fðo;dÞ
;þ
¼ 0:
According to Eqs. (37) and (40), the subproblem (31) can be reformulated as the following two adherent problems:
M. Ghatee, S.M. Hashemi / European Journal of Operational Research 194 (2009) 432–451
8 P P p;þ P R xa;þ a p;þ > min 1h fðo;dÞ ðLnðfðo;dÞ Þ 1Þ þ t ðwÞdw; > 0 > > a2A ðo;dÞ2C p2P ðo;dÞ > > > 8 P p;þ > > > fðo;dÞ ¼ F ðo;dÞ ; 8ðo; dÞ 2 C; > > > > > p2P ðo;dÞ > > > < > > p;þ > > 8p 2 P ðo;dÞ ; s:t: fðo;dÞ P 0 > > > > P P p;þ p;a > > > > a;þ > > fðo;dÞ dðo;dÞ ¼ x 8a 2 A; > : < ðo;dÞ2C p2P ðo;dÞ P p; P R xa; a p; > min 1 P > fðo;dÞ ðLnðfðo;dÞ Þ 1Þ þ t ðwÞdw; > 0 h > > a2A ðo;dÞ2C p2P ðo;dÞ > > > 8 P p; > > fðo;dÞ ¼ Gðo;dÞ F ðo;dÞ 8ðo; dÞ 2 C; > > > > > > p2P 0ðo;dÞ > > > < > > p; > > s:t: fðo;dÞ P0 8p 2 P 0ðo;dÞ ; > > > > P P > > p; p;a > a; > > > 8a 2 A: : : p2P 0ðo;dÞ fðo;dÞ dðo;dÞ ¼ x
443
into account by using a mutually consistent scheme in order p; p;þ Þ and ðfðo;dÞ Þ with respect to each other, see for to find ðfðo;dÞ an example, Ceylan and Bell (2004) who used similar scheme for TA and signal optimization jointly.
ðo;dÞ2C
ð41Þ But note that both of these problems are similar to the traditional Logit model (1) and so Damberg algorithm or modified Damberg algorithm (Algorithm 2.1) may be purp;þ p; sued to obtain the equilibrium flows fðo;dÞ and fðo;dÞ , respectively. The following theorem summarizes the presented discussion. Theorem 4.1. Suppose that the flow on two adherent problems in model (41) are independent of each other. The equilibrium solution of model (29) is as follows: Fig. 2. A network with 13 nodes, 30 links and five junctions.
p ¼ F ðo;dÞ P fðo;dÞ
expðhcp Þ þ ðGðo;dÞ F ðo;dÞ Þ p p2P ðo;dÞ expðhc Þ
Table 1 The capacity and free-flow cost of network links (depicted in Fig. 2)
expðhcp Þ : P expðhcp Þ p2P 0
ð42Þ
ðo;dÞ
Proof. According to similarity between adherent problems in model (41) and traditional Logit model (1), also because of independency assumption of theorem, the equilibrium solutions can be obtained with Eqs. (3) and (4). Thus, we have: expðhcp Þ p p2P ðo;dÞ expðhc Þ
p;þ ¼ F ðo;dÞ P fðo;dÞ
p; ¼ ðGðo;dÞ F ðo;dÞ Þ P fðo;dÞ
8ðo; dÞ 2 C 8p 2 P ðo;dÞ ;
expðhcp Þ expðhcp Þ p2P 0 ðo;dÞ
8ðo; dÞ 2 C 8p 2 P 0ðo;dÞ : Therefore, based on straightforward. h
Eq.
(36),
the
result
is
Corollary 4.2. The flow in adherent problems in (41) may be influenced on each other, because of sharing some links in paths consisting in P ðo;dÞ and P 0ðo;dÞ . But this influence is scarce. However, the decision maker can take this influence
Ind.
Tail-N.
Head-N.
Free-F.-Cost
Link-cap
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
1 3 3 4 6 4 6 5 7 6 3 7 3 2 2 8 9 8 10 8 8 7 11 7 12 6 12 11 13 11
3 1 4 3 4 6 5 6 6 7 7 3 2 3 8 2 8 9 8 10 7 8 7 11 6 12 11 12 11 13
4 19 13 22 13 12 21 14 8 17 21 4 17 11 20 14 18 12 10 8 8 17 10 15 7 18 11 21 21 16
252 454 415 326 413 334 175 149 174 271 367 158 423 288 189 355 277 226 351 314 401 482 265 444 90 494 139 130 442 372
444
M. Ghatee, S.M. Hashemi / European Journal of Operational Research 194 (2009) 432–451
Since, the formula (42) is similar to Logit formula (4) for traditional stochastic user equilibrium problem, we name Eq. (42) as quasi-Logit formula for TA model with fuzzy level of travel demand. Again, note that since, the model (29) consists of zero– one variables nðo;dÞ , we should utilize a branch and bound mechanism together with quasi-Logit formulas to obtain equilibrium flows. The following algorithm may be employed to find the equilibrium flow ‘‘fl” and its objective value ‘‘fv” in TA model (29). In addition, this algorithm returns exit flag as true if the objective function value is finite. The branch and bound tree is also implemented with
listðnÞ and valueðnÞ as for Algorithm 3.9 are stated. Also, PL has a tabu-list role and is used as evidence recorded of leaves which are previously processed and we should not take these leaves into account in the next iterations, i.e., what of the different leaves processed in one iteration of recursive algorithm is saved in ‘‘PL” in order to ban reprocessing that setting of parameters in next iterations of recursive algorithm. We use the following initial values: listðnÞ ¼ C, valueðnÞ ¼ ;,
Table 2 The fuzzy travel demands and some reasonable paths based on the penalty paths approach, together with the Logit flows with respect to the minimal, most possible and maximal amount of demands (o,d)
Tra-Dem
Paths
(1,5)
(100,80,140)
1!3!4!6!5 1!3!7!6!5 1 ! 3 ! 2 ! 8 ! 7 ! 11 ! 12 ! 5
39.994 39.964 20.042
31.995 31.972 16.034
55.991 55.95 28.059
(1,13)
(150,100,160)
1 ! 3 ! 7 ! 11 ! 13 1 ! 3 ! 4 ! 6 ! 12 ! 11 ! 13 1 ! 3 ! 2 ! 8 ! 8 ! 7 ! 11 ! 13
59.272 56.339 34.39
39.514 37.559 22.927
63.223 60.095 36.683
(1,7)
(200,160,250)
1!1!3!7 1!3!4!6!7 1!3!2!8!7
83.294 77.819 38.886
66.636 62.256 31.109
104.12 97.274 48.608
(9,7)
(40,30,45)
9!8!7 9!8!2!3!7
23.551 16.449
17.663 12.337
26.495 18.505
(9,13)
(160,120,180)
9 ! 8 ! 7 ! 11 ! 13 9 ! 8 ! 2 ! 3 ! 4 ! 6 ! 12 ! 11 ! 13 9 ! 8 ! 2 ! 3 ! 7 ! 6 ! 12 ! 11 ! 13
63.375 48.341 48.284
47.531 36.256 36.213
71.297 54.384 54.319
(9,5)
(130,100,190)
9!8!7!6!5 9!8!2!3!4!6!5 9 ! 8 ! 7 ! 11 ! 12 ! 6 ! 5
52.046 43.61 34.343
40.036 33.546 26.418
76.068 63.738 50.194
(13,1)
(120,110,150)
13 ! 11 ! 7 ! 3 ! 1 13 ! 11 ! 12 ! 6 ! 4 ! 3 ! 1 13 ! 11 ! 7 ! 8 ! 2 ! 3 ! 1
51.433 27.754 40.812
47.147 25.441 37.411
64.292 34.693 51.016
(13,5)
(120,100,140)
13 ! 11 ! 7 ! 6 ! 5 13 ! 11 ! 12 ! 6 ! 5 13 ! 11 ! 7 ! 3 ! 4 ! 6 ! 5
46.686 34.47 38.843
38.905 28.725 32.369
54.467 40.216 45.317
(13,9)
(50,40,60)
13 ! 11 ! 7 ! 8 ! 9 13 ! 11 ! 12 ! 6 ! 7 ! 3 ! 2 ! 8 ! 9 13 ! 11 ! 12 ! 6 ! 4 ! 3 ! 2 ! 8 ! 9
24.832 13.678 11.489
19.866 10.943 9.1915
29.799 16.414 13.787
(5,7)
(80,70,120)
5!6!7 5 ! 6 ! 12 ! 11 ! 7 5!6!4!3!7
37.707 25.127 17.166
32.993 21.987 15.02
56.56 37.691 25.749
(5,9)
(100,90,120)
5!6!7!8!9 5!6!4!3!2!8!9 5 ! 6 ! 12 ! 11 ! 7 ! 8 ! 9
44.156 19.565 36.279
39.74 17.609 32.651
52.987 23.478 43.535
Obj. value
(Mode)
114875.38
In final column, the equilibrium flows obtained by Ghatee Algorithm 4.3 are represented. The notation ‘‘ ” is used for the flows on slack paths.
(Min)
77671.11
(Max)
216324.58
(Ours) 36.74 35.75 23.54 ‘‘3.97” 55.15 42.61 41.22 ‘‘11.02” 82.72 56.54 52.39 ‘‘8.35” 23.87 13.68 ‘‘2.45” 68.31 42.43 40.87 ‘‘8.38” 48.86 39.06 35.54 ‘‘6.54” 51 30.14 36.84 ‘‘2.03” 42.92 36.12 36.39 ‘‘4.56” 22.15 13.5 12.32 ‘‘2.04” 34.18 23.63 20.16 ‘‘2.03” 39.94 27.27 30.66 ‘‘2.13” 114918.65
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maximal acceptable error , the processed list PL ¼ ;, the equilibrium flow fl ¼ ;, the equilibrium flow value fv ¼ 1. Algorithm 4.3. ½fl; fv; exit flag ¼ Ghateeði; k ; fP ðo;dÞ gðo;dÞ2C ; listðnÞ; valueðnÞ; ; h; PL; fl; fvÞ If listðnÞ is not empty 1. Pick up the element ðo0 ; d 0 Þ from the top of listðnÞ and eliminate it from list. 2. Insert nðo0 ;d 0 Þ ¼ 1 in the valueðnÞ. 3. Run ½fl1; fv1; exit flag1 ¼ Ghateeði; k ; fP ðo;dÞ gðo;dÞ2C ; listðnÞ; valueðnÞ; ; h; PL; fl; fvÞ. 4. If exit flag1 is true and fv1 < fv – fl=fl1; – fv=fv1; – exit flag ¼ exot flag1; 5. If exit flag1 is true – Insert nðo0 ;d 0 Þ ¼ 0 in the valueðnÞ instead of nðo0;d0Þ ¼ 1 – Run ½fl0; fv0; exit flag0 ¼ Ghateeði; k ; fP ðo;dÞ gðo;dÞ2C ; listðnÞ; valueðnÞ; ; h; PL; fl; fvÞ. – If exit flag0 is true and fv0 < fv fl ¼ fl0; fv ¼ fv0; exit_flag=exit_flag0; 6. Return Else – According to valueðnÞ create the vector nðo;dÞ for each of the ðo; dÞ belongs to C. – If ðnðo;dÞ Þðo;dÞ2C does not exist in PL list 1. Insert ðnðo;dÞ Þðo;dÞ2C in PL. 2. With respect to ðnðo;dÞ Þðo;dÞ2C and k , determine the two vectors F ðo;dÞ and Gðo;dÞ using expressions (22) and (23). 3. If for at least one ðo; dÞ 2 C, Gðo;dÞ < F ðo;dÞ then set exit flag as false and return, otherwise exit flag is true. 4. Insert a dummy node w in the network. 5. For each ðo; dÞ 2 C set uðo;wÞ ¼ uðw;dÞ ¼ 1 and iðo;wÞ ¼ iðw;dÞ ¼ 1 or M (a positive number). The flow on these dummy links can be controlled with the amount of the free-flow costs iðo;wÞ and iðw;dÞ . 6. For each ðo; dÞ 2 C define slack path p0ðo;dÞ , o ! w ! d. 7. ½fl1; fv1 ¼ M Dambergðfiga2A ; fP ðo;dÞ gðo;dÞ2C ; fF ðo:dÞ gðo;dÞ2C ; ; hÞ. 8. ½fl2; fv2 ¼ M Dambergðfiga2A ; fP ðo;dÞ [ p0ðo;dÞ gðo;dÞ2 C; fGðo;dÞ F ðo;dÞ gðo;dÞ2C ; ; hÞ. 9. Set flnew ¼ fl1 þ fl2 and fvnew ¼ fv1 þ fv2. 10. if fvnew < fv fl=flnew. fv=fvnew. – Else
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Return. End of Algorithm This algorithm finds the equilibrium flow of model (29). In the next section we demonstrate the efficiency of algorithm on a sample network. But it is important to note that this algorithm may be repeated 2c times (c ¼ jCj). Thus, the minimization problem in (41) should be solved many times as equal to the number of combination of installing for each OD pair. This means that, the consuming time of calculation should be increased rapidly according to the number of OD pairs. For extending the algorithm to the real scale problem we propose the following heuristic based on the mathematical expected values of fuzzy numbers. Property 4.4. Let the fuzzy level of travel demands between OD pairs ðo; dÞ 2 C are represented as LR fuzzy numbers min max fðd pos ðo;dÞ ; d ðo;dÞ ; d ðo;dÞ ÞLR gðo;dÞ2C . Then the Eq. (15) can be approximated as the follows: 8 R d pos d pos x R d max xd pos
> < 0; d min L d pos dx; dx > d pos R d max d pos d min nðo;dÞ ¼ > 1; R d pos L d pos x dx 6 R d max R xd pos dx: : d min d pos d max d pos d pos d min ð43Þ By the other word, when the expected amount of travel demand between OD pair ðo; dÞ along with the lower and most possible values is greater than such amount along with the most possible and upper values, we perceive that the real travel demand is less than its most possible value with high possibility, and therefore, we set nðo;dÞ ¼ 0. Same explanation seems to be true for other case. In addition, employing metaheuristic schemes such as genetic algorithm, simulated annealing or tabu search, for solving the constructed model, may also produce interesting results. But this requires further studies.
Table 3 Some parameters with respect to the equilibrium flow of model (31) are presented P P p p Gðo;dÞ (o,d) F ðo;dÞ p2P ðo;dÞ fðo;dÞ p2P 0 fðo;dÞ ðo;dÞ
(1,5) (1,13) (1,7) (9,7) (9,13) (9,5) (13,1) (13,5) (13,9) (5,7) (5,9)
90 125 180 35 140 115 115 110 45 75 95
100 150 200 40 160 130 120 120 50 80 100
96.03 138.98 191.65 37.55 151.61 123.46 117.98 115.43 47.97 77.97 97.87
100.00 150.00 200.00 40.00 159.99 130.00 120.01 119.99 50.01 80.00 100.00
In last two columns, the sum of the flow on paths between each of the OD pairs, with and without considering dummy path, respectively, are represented.
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5. Implementation and numerical results In this section, we briefly illustrate the efficiency of the proposed algorithm by studying on a sample network with 13 nodes, 30 links and five junctions (depicted in Fig. 2. In Table 1, the capacity and free-flow cost of network links are represented. According to the well known cost function presented by the US Bureau of Public Roads, 1964, we consider the link costs as follows: a b ! x a a a a ; ð44Þ t ðx ; i Þ ¼ i 1 þ a a u
where ua and ia are the capacity and free-flow cost of link a 2 A and parameters a and b are fixed values (usual values are a ¼ 0:15 and b ¼ 4). It is well known that, in route choice estimation, the set of reasonable paths is predetermined and the different values of TA flows on the network links might be estimated comparing to the TA without any assumption for the reasonable paths. In this examination, for convenience, to produce the reasonable paths, among a lot of path enumeration techniques, we utilize the following three approaches, but note that in real problems, if the number of reasonable paths is small, the result may be not adopted as well with actual environment. For such cases,
Table 4 The 3-shortest paths and dissimilar paths with their free-flow costs and also, equilibrium flows by using Ghatee algorithm 3-Shortest paths
Free-F. cost
1!3!4!6!5 1!3!7!6!5 1 ! 3 ! 7 ! 11 ! 12 ! 6 ! 5
50 54 89
1 ! 3 ! 7 ! 11 ! 13 1 ! 3 ! 4 ! 6 ! 12 ! 11 ! 13 1 ! 3 ! 4 ! 6 ! 7 ! 11 ! 13
56 74 77
1!3!7 1!3!4!6!7 1 ! 3 ! 4 ! 6 ! 12 ! 11 ! 7
25 46 68
9!8!7 9!8!2!3!7 9!8!2!3!4!6!7
26 64 85
9 ! 8 ! 7 ! 11 ! 13 9 ! 8 ! 7 ! 6 ! 12 ! 11 ! 13 9 ! 8 ! 2 ! 3 ! 7 ! 11 ! 13
57 79 95
9!8!7!6!5 9!8!7!3!4!6!5 9!8!2!3!4!6!5
55 76 89
13 ! 11 ! 7 ! 3 ! 1 13 ! 11 ! 12 ! 6 ! 7 ! 3 ! 1 13 ! 11 ! 7 ! 8 ! 2 ! 3 ! 1
54 89 92
13 ! 11 ! 7 ! 6 ! 5 13 ! 11 ! 12 ! 6 ! 5 13 ! 11 ! 7 ! 3 ! 4 ! 6 ! 5
60 70 81
13 ! 11 ! 7 ! 8 ! 9 13 ! 11 ! 7 ! 3 ! 2 ! 8 ! 9 13 ! 11 ! 12 ! 6 ! 7 ! 8 ! 9
60 84 95
5!6!7 5 ! 6 ! 12 ! 11 ! 7 5!6!4!3!7
31 53 70
5!6!7!8!9 5 ! 6 ! 12 ! 11 ! 7 ! 8 ! 9 5!6!7!3!2!8!9
60 82 84
Objective value The notation ‘‘ ” is used for the flows on slack paths.
Opt. flows 35.14 34.38 26.24 ‘‘4.23” 55.52 41.69 41.7 ‘‘11.09” 84.66 62.58 45.09 ‘‘7.67” 17.55 11.63 8.91 ‘‘1.92” 62.15 47 42.32 ‘‘8.54” 47.27 39.61 36.49 ‘‘6.63” 49.99 33.46 34.39 ‘‘2.15” 42.02 38.68 34.83 ‘‘4.47” 19.46 15.15 13.27 ‘‘2.12” 33.91 23.53 20.51 ‘‘2.05” 38.98 29.12 29.69 110449.02
Dissimilar paths
Free-F. cost
1 ! 3 ! 7 ! 11 ! 12 ! 6 ! 5 1!3!4!6!5 1!3!2!8!7!6!5
89 97 78
1 ! 3 ! 2 ! 8 ! 7 ! 11 ! 13 1 ! 3 ! 4 ! 6 ! 12 ! 11 ! 13 1 ! 3 ! 7 ! 11 ! 13
80 74 56
1 ! 3 ! 4 ! 6 ! 12 ! 11 ! 7 1 ! 3 ! 4 ! 6 ! 12 ! 11 ! 7 1!3!2!8!7
85 68 49
9!8!2!3!7 9!8!7
64 24
9 ! 8 ! 2 ! 3 ! 7 ! 11 ! 13 9 ! 8 ! 7 ! 3 ! 4 ! 6 ! 12 ! 11 ! 13 9 ! 8 ! 7 ! 11 ! 13
95 100 57
9!8!2!3!4!6!5 9!8!7!6!5 9 ! 8 ! 7 ! 11 ! 12 ! 6 ! 5
89 102 90
13 ! 11 ! 7 ! 6 ! 4 ! 3 ! 1 13 ! 11 ! 12 ! 6 ! 7 ! 3 ! 1 13 ! 11 ! 7 ! 3 ! 1
93 89 103
13 ! 11 ! 7 ! 6 ! 5 13 ! 11 ! 7 ! 8 ! 2 ! 3 ! 4 ! 6 ! 5 13 ! 11 ! 12 ! 6 ! 5
109 119 70
13 ! 11 ! 7 ! 3 ! 2 ! 8 ! 9 13 ! 11 ! 12 ! 6 ! 7 ! 8 !! 9 13 ! 11 ! 7 ! 8 ! 9
84 95 107
5!6!4!3!7 5!6!7 5 ! 6 ! 12 ! 11 ! 7
70 94 53
5!6!4!3!2!8!9 5 ! 6 ! 12 ! 11 ! 7 ! 8 ! 9 5!6!7!8!9
98 82 60
Opt. flows 27.77 31.96 35.89 ‘‘4.39” 40.42 39.86 58.9 ‘‘10.82” 50.31 62.05 79.39 ‘‘8.25” 22.96 14.5 ‘‘2.54” 48.3 38.82 64.75 ‘‘8.13” 46.45 42.16 34.91 ”6.48” 44.45 32.58 40.84 ‘‘2.13” 38.71 38.16 38.42 ‘‘4.71” 16.97 12.58 18.38 ‘‘2.07” 27.19 26.26 24.28 ‘‘2.27” 27.18 28.06 42.63 ‘‘2.13” 139232.75
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the decision maker, by help of network experts, live statistics and results of simulation softwares, should be found enough number of paths who gratifies from the sufficiently of reasonable paths, for more discussion see Akgun et al. (2000), Zijpp and Catalano (2005) or Ghatee (2005), Section 3. 1. Penalty paths approach, which imposes a cumulative penalty on the impedance of all links in the previous resulting shortest path, (see e.g., Barra et al., 1993). At the first, we find eight paths between each OD pair nodes and select disjoint paths between them. 2. Dissimilar paths idea, which presents some paths with acceptable lengths by few common links as possible (Akgun et al., 2000). For generating the initial set of paths to measure dissimilarity between paths, we utilize gateway paths approach (Lombard and Church, 1993) which finds specially different alternatives by selecting gateway nodes and computing shortest paths that pass through those gateways. 3. K-shortest paths scheme, which produces the set PK ¼ fp1 ; . . . ; pK g of paths, such that costðpi Þ 6 cost ðpiþ1 Þ for any i 2 f1; . . . ; K 1g and costðpK Þ is less than the cost of any path q R PK . Lawler, 1972, has presented an Oðn3 Þ algorithm for finding acyclic K-shortest paths. This algorithm may be stopped with less than K paths. In Appendix B, we modify this algorithm to obtain exactly K paths (if they exist). We set h ¼ 1 and ¼ 0:5 and consider 11 pairs of OD nodes. In the first column of Table 2 the travel demands for all of these pairs are shown. Also, in that table, some disjoint reasonable paths by using ‘‘addition” penalty approach (Akgun et al., 2000), are presented. The maximal certainty that is obtained using Algorithm 3.8 is k ¼ 0:5. As we see through this paper, by ignoring from certainty degree, the objective function (1) exists in all of the presented models. Thus, we agree to compare the results based on this function. Now we compare the equilibrium flows
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obtained from Algorithm 4.3 with three equilibrium flows with respect to the minimal demands, the most possible demands (most visible ones) and the maximal demands. In Table 2 the equilibrium flows with respect to these levels of demands are recorded. It is important to note that the optimal value of objective function (1) provided by our method is less than that one with respect to the most possible case which is also, the most visible case in uncertain region. Thus, our solution seems to be most appropriate between the results of random real scenarios. Also ‘‘k ¼ 0:5” means that with more than 50% of certainty we confide that the flow with respect to our method is better than the results with respect to the most possible demands. In Table 3 some details of obtained solution are represented, too. This table appears that the sum of flows on paths through OD pairs ðo; dÞ 2 C is exactly between F ðo;dÞ and Gðo;dÞ , which justifies the accuracy of our method by computation viewpoint. Also, as would be expected, the slack variables have gotten small values according to the used strategy that increases the impedance of congested paths. In the second numerical test, we try to find three disjoint dissimilar paths and also, 3-shortest paths for each pair of OD nodes. Unfortunately, for fourth pair, ð9; 7Þ, dissimilar paths algorithm with dissimilar function presented by Akgun et al. (2000), could find only two paths. These mentioned paths with their free-flow costs together with the equilibrium flows obtained by Algorithm 4.3 are represented in Table 4. In Fig. 3 we show the obtained optimal value of objective function (31) in each leaf of branch and bound tree (a vector of ns or in more familiar each leaf corresponds to one n vector) when one of the three methods is opt to use as path enumeration technique. It is clear that for all of the leaves in this sample network, the obtained optimal value according to K-shortest path scheme is the best which is coincided with drivers’ appetite to use shortest paths, while the result of dissimilar paths is close to that of penalty paths. But note that, although in this sample network, for path generating, the K-shortest paths scheme
Fig. 3. The comparison of three approach, K-shortest paths, dissimilar paths and penalty paths.
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Fig. 4. The results of equilibrium flows for K-shortest paths approach by three level of certainty k ¼ 0:5; 0:25; 0:1.
Fig. 5. An enlarged version of Fig. 4.
shows suitable result when the aim is to minimize the travel time, it may be provided better results by using dissimilar paths scheme in order to minimize congestion. Thus, it is not easy to present a generalized assertion about which of techniques should be utilized for generating the reasonable paths. Also, it is very important to note that for real size networks, the number of alternatives which is used in this examination, is too short and these paths are provided to show the efficiency of the proposed algorithm. For real networks, a combination of several path enumeration techniques may be used to produce the reasonable paths. However, as much as the number of reasonable paths increases, the computational time increases. One point which may be useful in analysis, is that the drivers usually use particular
paths for travel and the flows on complicated and complex paths are very small, see e.g., Zijpp and Catalano (2005). But again we mention that the accuracy degree is directly depended on the number of reasonable paths. In another examination we try to compare the results of K-shortest paths in equilibrium model (31) with some level of certainty. For this aim, we suppose that k ¼ 0:5; 0:25; 0:1, and run our algorithm with K-shortest paths which are represented in Table 4. The optimal value of objective function (31) for different ns with respect to the tree leaves for these three ks are shown in Fig. 4 and Fig. 5. These figures clarify that as much as the level of certainty decreases, the probability of finding the better objective value enhanced. However the equilibrium solution is not essentially changed. In
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other word, when certainty degree is chosen small, the uncertain decision region extends and because of searching on an extended space, we can find better optimal values but the equilibrium solutions are not essentially different in compare with equilibrium solutions with high certainty degree. Furthermore, these figures reveal that the map of values of objective function (31) vary harmonically and these values regularly change with respect to the level of certainty. In familiar way their increasing and decreasing are as the same for different certainty degrees. To explain the reason of this tendency, note that the objective function (31) is independent of the degree of certainty and this degree is only influenced on feasible set. But F ðo;dÞ ðkÞ and Gðo;dÞ ðkÞ according to Lemma 3.5 linearly vary with respect to k and so the constraints related to F ðo;dÞ ðkÞ and Gðo;dÞ ðkÞ with respect to different ks are parallel and therefore the set of feasible points are usually similar except of one scaling parameter. Thus, the optimal solutions with respect to different ks are close to each other. This useful point shows that, the obtained solution with respect to one fixed certainty degree, is optimistically good even for higher degrees of certainty.
6. Conclusion and future directions With respect to the uncertainty as a crucial issue in TA models and insufficiency of traditional random utility to take into account the uncertainty due to randomness of traffic patterns and its important role in ATIS and ATMS, a TA model with fuzzy level of travel demand is introduced in this paper. We utilize triangular fuzzy numbers to exhibit the imprecise number of traveler whom want travel between the OD pairs. Such numbers reflects the perception of individuals to travel. Then we transform the soft constraints of our model into some linear constraints consisting of zero–one variables and pathflow ones. Such basic paths are reasonable or preferred paths which can be provided by path enumeration techniques such as K-shortest paths, K-similar paths or dissimilar paths. Then we maximize the degree of certainty and obtain some formulas for the equilibrium flow of simplified version of fuzzy TA that is named as quasiLogit formulas. An example is addressed to illustrate the efficiency of our algorithms for maximizing certainty and finding equilibrium flows. Our results reveal that with K-shortest path scheme, we can find reasonable solution with the different level of certainty, but it is not a general result. In next works, we incorporate fuzzy random variable as traveler demand in order to develop the model in a mixed imprecise and uncertain environment. Research on the uncertain costs and capacities together with the concepts of this paper, also, examining this scheme on real urban networks are left to the next works. Employing metaheuristic schemes for solving the constructed model in this paper may also produce intersecting results.
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Acknowledgement The authors would like to express particular thanks to the anonymous referees and honorable editors for their valuable comments, which led us to improvements in this paper. Also, the support from Tehran Control Traffic Organization is gratefully acknowledged. Appendix A. Fuzzy numbers A fuzzy number is a convex normalized fuzzy set of the real line R, whose membership function is piecewise continuous. A fuzzy number ~a ¼ ðapos ; amin ; amax ÞLR is said to be an LR type fuzzy number, if its membership function satisfies ( pos
a x ; x 6 apos ; L apos amin xa
l~a ðxÞ ¼ pos R amax apos ; x P apos ; where the symmetric non-increasing function L : ½0; 1#½0; 1 is the left shape function, that Lð0Þ ¼ 1. Naturally, a right shape function RðÞ is similarly defined as LðÞ. If LðÞ and RðÞ are linear, the LR type fuzzy number is said triangular. Appendix B. K-Shortest path algorithm For finding acyclic K-shortest paths, Lawler’s algorithm (1972) may be terminated with less number of paths than K. The following MATLAB pseudo-code employs a simple mechanism to find more paths in these cases. Note that according to Lawler’s definition, ‘‘include” is a subpath which every path should start with that segment. In our examination include is initially taken as empty set. A is incident matrix of network while C is a matrix consists of links costs. The returned arguments of algorithm are the set of paths between o and d, say pred, and the cost of K-shortest paths, say distance. Algorithm 7.1. ½pred; distance ¼ LawlerðA; C; o; d; K; includeÞ ½pred1; distance1; d include initial ¼ find pathðA; C; include; o; dÞ. Set ¼ pred1; Value Set ¼ distance1; A ¼ A; include ¼ include; k ¼ 0 While k < K 1. If isemptyðValue SetÞ includenew ¼ include , and (a) Anew ¼ A , d includenew ¼ d include initial. (b) For k new ¼ 1 : k i. pred1 ¼ predðk new; :Þ, qk link ¼ ½, j ¼ d. ii. While j ¼ d includenew - i=pred1(j). - qk link ¼ ½i; j; qk link. - j ¼ i. iii. For qk i ¼ 1 : sizeðqk link; 1Þ – For q ¼ 1 : ðqk i 1Þ i ¼ qk linkðq; 1Þ, j ¼ qk linkðq; 2Þ.
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includeðjÞ ¼ i, d include ¼ j. – Anewðqk linkðqk i; 1Þ; qk linkðqk i; 2ÞÞ ¼ 0. – ½p1 n; d1 n ¼ find path includeðAnew; C; include; s include; o; dÞ – If isemptyðp1 nÞ test ¼ 0. For k path ¼ 1 : k,
empty set. Also ½pred; distance; d include ¼ find path ðA; C; include; o; dÞ, finds the subpath from the head of ‘‘include” (starting at o) to d include. Then this function finds a path from d include to d and concatenate these subpaths and return d include together with found path. References
If isequalðp1 n; predðk path; :ÞÞ; test ¼ 1
and
break:
If test=0 Set ¼ ½Set; p1 n;
Value Set
¼ ½Value Set; d1 n: (c) If isemptyðValue SetÞ break. d include ¼ d include initial, A ¼ A ; include ¼ include 3. ½valmin; indmin ¼ minðValue SetÞ. 4. k ¼ k þ 1. 5. predðk; :Þ ¼ Setðindmin; :Þ, and Setðindmin; :Þ ¼ ½ . 6. distanceðkÞ ¼ Value SetðindminÞ, and Value Set ðindminÞ ¼ ½. 7. If k ¼ K break, else pred1 ¼ predðk; :Þ 8. qk link ¼ ½ , j ¼ d. 9. While j ¼ d include initial – i=pred1(j). – qk link ¼ ½i; j; qk link. – j=i. 10. For qk i ¼ 1 : sizeðqk link; 1Þ (a) for q ¼ 1 : ðqk i 1Þ – i ¼ qk linkðq; 1Þ, j ¼ qk linkðq; 2Þ. – includeðjÞ ¼ i. – d include ¼ j. (b) Aðqk linkðqk i; 1Þ; qk linkðqk i; 2ÞÞ ¼ 0. (c) ½pred1; distance1 ¼ find path includeðA; C; include; d include; o; dÞ. (d) If isempty(pred1) continue. (e) test = 0; (f) for k path ¼ 1 : k – If isequalðpred1; predðk path; :ÞÞ test = 1. break. (g) If test==0 – For i ¼ 1 : sizeðSet; 1Þ If isequal(pred1,Set(i)) . test=1. . break. (h) If test=0 – Set ¼ ½Set; pred1; – Value Set ¼ ½Value Set; distance1; End of Algorithm 2.
In the above Algorithm ½pred; distance ¼ find path include ðA; C; include; d include; o; dÞ is a function which finds a path from o to d and sharing with ‘‘include” from node o to node d inlude. If such path does not exist pred is redefined as
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