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40Russian Journal of Logistics and Transport Management, Vol.2, No.1, 2015
©Wieslaw Pasewicz
West Pomeranian University of Technology
CONTROL OF THE TRAFFIC STREAM ON THE ROAD OF THE TRANSPORT NETWORK (Methodological Note)
Abstract
A purpose of the paper is determining the optimum procedure for the process, consisting of the finished number of stages (numbers of nodes of the transport network along with the moving traffic stream), when everyone is characterised by the square loss function. The square loss function is leading decision-making variable based on linear functions, which a decisionmaker can control for establishing optimum procedures.
Keywords: square loss, control variable, stage of the process, optimum sequence, the state of the transport system, traffic stream.
1 Introduction
Denote by
h (a,b)
< (ii, ji X (h, J 2 X -, (in, jn X ..., (V , V ) >
< Pi, P2, ., Pn >
(1)
a road between nodes (junctions) a and b of the transport network, e.g. Jacyna (2001) and Leszczynski (1999). Let xab be the amount of the traffic stream on the way h (a, b). Now we are assuming that the quantity of the stream after particular elements pn (n =1,., N ) of the way h (a,b) in relations of the transport is not known. Let xnb - Xn (n = 1, ., N) be the quantity of the stream on the road pn, which is a random variable. Suppose that values of random variables Xi,X2, ... , X are the states of the stochastic system in N stages sequential process. Thus, X may indicate the initial state of the transport system, and X2, ... , Xn will be successive states in remaining N -1 stages. Besides, let on the n-th stage of the process the distribution of the state Xn+1 depends on the current state Xn (Markov
chain), and from the value un of the real variable called the control variable which the decision-maker can choose (Wang and Saridis, 2005). More precisely, the given process will be proceeding, according to the following set of equations (Arefiev and Pasevich, 2000):
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xn+1 = anxn + Pn +un + Zn ; f^or n = 1,..., N-1 , (2)
where an (an* 0) and pn are constants, un is a chosen value after conducting observationX and Zn ~ N (0, a2), i.e. a random variableZn has a normal distribution with the expected value zero and with equal variance a2. Variables Z, ..., ZN_n are representing random disturbances appearing in the system (we assume their independence), and a knowledge of the initial state of the system is assumed X = x.
2 Analysis process associated with an operation of the traffic stream
The initial state X = x of arrangement is known, and the decision-maker can choose the value u of the control variable. Another parameter X of the system is characterised by the normal distribution with the average a x + Pi + u and the variance a 2,
that is X ~ N (a x +px + u, a 2 ). After observing the state X = X, a decisionmaker is making a choice of the value u of the control variable. Moreover, another state is characterised by the normal distribution with the average a2x + P2 + u2 and the variance a2, i.e.
X ~ N (a x +P2+ X, a 2). The process is proceeding farther into an identical manner all the way to the moment of conducting the observation of the final state X. Let us suppose that on n- th stage of the process (n = 1, ., N-1) a decisionmaker have to choose the control variable so that in the next phase the state of the systemXn+1 is close to the certain value A. We will assume that the loss associated with not achieving the planned target value at the given stage is equal to the square loss function of the form:
sn = Qn(Xn+1 - dn)2, (3)
where qn (qn > 0) is a weight that shows the relative meaning of the given value mistake Xn+1 - dn. If the decision-maker could choose the value of the control variable without limitation and costs, then optimal choice on the n-th stage of the process would consist in the determination such u , which would make the expected value anxn +pn+ un of the next state Xn+1 equal of the target value dn. However, in practical situations, the decision-maker will be seeking such solutions that would provide correct relations between costs of the operations and losses from not achieving the target value.
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3 Results
Let us assume that the cost choice of the control variable un on the n- th stage of the process is equal rnu2, rn > 0. Therefore, the total loss on n- th stage (n = 1, ..., N-1) can be seen from the expressions below:
Sn = qn(Xn+1 - dn)2 + rnu2. (4)
While the loss associated with the whole process is the following sum:
N-1
S = I S„ .
n=1
(5)
The sequence values of the control variable ulv--,uN_i have to be so well-matched in order to minimize the expected value of the above sum. We will be considering the given n-th stage (n = 1, ..., N-1) assuming, that locationXn = xn was just observed and it is necessary to choose the value wn. The total loss in
N-1
remaining stages of the process will amount I St.
i=n
On the base (2) a choice of the optimum un will depend on x„ and will not be directly dependent on previous states of the system, or earlier values of the
N-1
control variable. Let Vn (xn) means the expected value of the sum IS . In
i=n
particular, V1 (x1) is minimal expected value of the total loss for the whole process if the initial state is equal Xi = xi. Let farther En ( n = 1, ., N-1) means any expected value which it is possible to calculate, concerning the conditional
distribution, when Xn =Xn, and the valueUn is given (Roy, 1990). If we will define the function Vn+1(xn+1) as identity equal zero, then functions V1, ., VN , have to fulfil the following relations:
Vn (xn) = inf E [Sn + Vn+1 (Xn+1)], n = 1, ...,N-1.
(6)
The function Vn (xn) is a quadratic function of the form:
Vn (xn) = kn-1 (xn - ln-1 )2 + mn-1. (7)
From the relation (2) results, the constant o is as follows:
EniXn+i o ( (ХцХ-п “I" fin un °)) + an. (8)
Using the expression (4):
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EniSn) qn(^n-^n + fin + un dn) + rnun + (Jn^n-
(9)
Moreover, taking into account (7):
EnWn&n+l)] _ kn(^n-^n + fin + un ln - 1 ) + kn^n + Win - (10)
Based on the relation (6), optimum value w„ have to minimize the sum of the values from the equalities (9) and (10). Since this sum is a quadratic function of the variable wn, the optimum value wn is possible to find on the ground of the basic differentiation, i-e-:
{EJSn + tn+1(*n+1)]llin 9 О 2 JJn(^n-^n + fin + un dn) + 2rnUn + (11)
+ 2 kn(dn%n + fin + un ln—1 ) _ 0-
Therefore:
4, A S X
.)]
qndn+kn ln_ !-(qn+к n)(ccnxn+fin) qn+kn+rn
(12)
4 Remarks
Two following properties of optimum solutions are important. Firstly, the optimum value of the control variable at every stage of the process is a linear function of the state of the system in this stage. Secondly, in this case, we did not use normal distributions of random variables ZZN_j. Therefore, the optimum sequence of the value, which we received, will be optimal for the any described process with relations (2), if only deviations Z,...,Z^will be independent and will have distributions with zero mean and finite variance. These facts are justifying the wide applicability of the described model.
References
Arefiev, I.B. & Pasevich, B. (2000). Management of the Multyransport System by the
quadratic loss function, Procedings of the International Conference APSU. NorthWestern Correspondence Polytechnic Institute, St. Petersburg, 14 -17.
Wang, F. Y. & Saridis, G. N. (2005). On successive approximation of optimal control of stochastic dynamic systems. In Modeling Uncertainty. Springer US, 333-358.
Jacyna, M. (2001). Modeling Wielokryterialne in Application of Functional Assessment of Transportation Systems, Transport, Warsaw University of Technology Publishing House, 47, 3-139.
Leszczynski, J. (1999). Modeling Systems and Processes Transport. Warsaw University of Technology Publishing House, Warsaw, Poland.
Roy, B. (1990). Decision-Aid and Decision-Making. In: Carlos A. Bana e Costa (Ed.). Readings in Multiple Criteria Decision Aid. Springer Berlin Heidelberg, Berlin, Germany.
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