2. Evaluation of the probability characteristics of vehicles coming out of order

STATISTICAL PROBABILITY MODELS FOR VEHICLE FLEET

MANAGEMENT

Adolfas Baublys

Vilnius Gediminas Technical University Transport Research Institute Plytinės g. 27, LT-2040 Vilnius, Lithuania,

Fax: 370-2-31 56 13. E-mail: [email protected]

Some algorithms to forecast damaged vehicle repair time, time losses for lack of drivers, and various accidental hindrances during goods handling were proposed using methods of mathematical statistics and probability theory. The paper presents: 1) Processing results of the flow of vehicles coming out of order; 2) Evaluation of the statistical characteristics of vehicles coming out of order; 3) The stochastic model for the estimation of average number of vehicle repair per time period; 4) The stochastic model for determination of desirable vehicle productivity; 5) The stochastic model for short-term estimating of vehicle operation time losses; 6) The algorithm for a vehicle repair time forecasting.

Samples for checking the reliability of the proposed algorithms are presented.

1. Introduction

Among the essential factors hindering the normal haulage of goods there may be indicated insufficient reliability of transport means (particularly related to the increase of exploitation time) and their damages caused by various reasons. The main point of this factor is that in entirely normal conditions during the process of haulage, separate specimen of transport means come out of order - break down - at accidental time intervals. These transport means may be repaired during the time

p

τ , which will be considered random quantity with a relevant distribution law. This article analyses how to evaluate this quantity, to define the losses of useful time of transport means; it will also deal with the road transport means - vehicles - (however, these models may be applied to other transport means as well).

2. Evaluation of the probability characteristics of vehicles coming out of order

Operation process of a transport means Y(t) is a combination of two not-intersecting sets: A - an operating transport means and B - a damaged transport means.

Thus, in that way, the amount of operation hours of the jth type transport means in the period T, has to be analysed as a random quantity, composed of the sum of productive operation segments:

∑

+ = = 1 1 N i ji j S S ,

here N is the number of intervals in the jth type transport operation; and the sums of idle time:

∑

= = N i ji j x x 1 , i.e. t_j =S_j +x_j,

here tj is the number of the jth type transport means operation hours during period T.

Let us assume that at the beginning of operation the working transport means is ϕ

( )

0 ∈A. Consequently, the operation process of the transport means will consist of status changes A1, B1, A2,

B2,...,Ai, Bi. This is a stationary Markov's process with two possible statuses, because every other process in it is determined only by the present status irrespectively from all the former processes. This Markov's process may be entirely described by the matrix of transitional probabilities.

1 1 * 00 * 01 10 * 11 t t t t P P P P ∆ − ∆ ∆ ∆ − = ν µ λ λ , here * ij

P is the probability of transition during a time interval from ith status into the jth status, as

i=0, j=0; µ - is the intensity of transport means restoration; λ is intensity of transport means coming out of order.

Let us take, that time intervals ϕi, correspond to the statuses Ai, i=1, 2, ... and time intervals qi correspond to the statuses Bi. The laws of distribution of these time intervals will be as follows:

{

}

{

}

( ). ); (    = < = < x G x q P x F x P i i ϕ .

Over the period α

( )

0,t_j a transport means was able to operate during the time α

( )

t_j , and not able to operate during the time β

( )

t_j . It is obvious that:

) ( )

(t_j +β t_j =t_j

α .

Therefore it may be written as follows:

{

}

{

( )

}

1 ( , ). ); , ( ) (    − = < = < x t P x t P x t P x t P j j j j α β

Consequently, we may write:

[

( ) ( )

]

, ) ( ) , ( 0 1 * *

∑

∞ = + − − − = i j i j i i j x G x F t x F t x t P (1) here _G*(_x ) i - the i

th _{contraction of distribution function G}

( )

_{x ; F}*_{(x )}

i -the i

th _{contraction of} distribution function F

( )

x .

This expression allows us to discover the law of distribution of general time of non-operational status of transport means, in existing different flows of failures and restorations.

As investigations show, a single restoration time of separate transport means divides up to the Veibul's and indicative laws, uninterrupted operation time corresponds to the logarithmic-normal and indicative laws.

The analysis of a great number of histograms showed that the following variants are possible: 1) uninterrupted operation time distributed according to the logarithmic-normal law; the restoration time - according to the Veibul's law; 2) uninterrupted operation time distributed according to the indicative law; the restoration time - according the Veibol's law; 3) uninterrupted operation time distributed according to the logarithmic-normal law; the restoration time - in accordance with the indicative law; 4) uninterrupted operation and restoration time distributed according to the indicative law [1].

As investigation shows, the flow of coming out of rejections, restorations and uninterrupted working moments flow correspond to the Puasson's law with the density:

1 )

(_{x e} x

F = − −λ (2)

Let us analyse the process of restoration of transport means. The contraction of the function (2) will be as follows:

[

( )

]

/ ! ) ( * ) ( e t x i F t x i x = −λ − λ −

After noticing that t−s=S, and inserting the equation (2) into the formula (1), we obtain:

) ( ! ) ( ) , ( * 0 x G i S e x x S P _i i i s λ λ

∑

∞ = − = + . (3)

The Laplas-Stiltjes transformation of function (3) will be found:

{

( , ;

}

e ( , ) exp 1 e ( ) 0 0                 − − = + = +

∫

∫

∞ − ∞ − _{dP S x x S dG x} P x S x P L px _λ px _{. (4)}

As it has been said above, the duration of some transport means repair time distributes according to the indicative law:

x

x

g( )=1−e−µ ,

here µ is the density of restoration flow. In the given case:

P dx x dg px x px + = =∞ − − ∞ −

∫

∫

µ µ _µµ 0 0 e e ) ( e .

{

( , );

}

exp 1             + − − = + P S P x x S P L µ µ λ .

We will determine the mathematics expectation M

[ ]

x of repair time, by finding the first derivative dP dL , as P=0: µ µ λ λ µ µ λ − + + = P s P S dP P dL _s 2 e e ) ( ) ( . (5)

When P=0, after a corresponding transformation of the equation (5) we will obtain:

M S x M dP P dL ] [ ) ( λ = = . (6)

Intending to determinate the dispersion _D

[ ]

_x2 _{we have to find the second derivative:}

2 2 2 0 2 2 2_] ( ) 2 [ M S S dP P L d x D p λ λ + =       − = = .

Let us analyse the case when the time of repair of transport means is distributed according to the Veibul's law, i. e.:

exp ) ( 1               −             = − η η σ σ σ η x x x g , (7)

here σ - is the parameter of scale; η - is the parameter of form. By inserting the function (7) into the (4) equation we will get:

exp e 1 exp } ) ( { 1 0                                   −       − − = + − ∞ −

∫

_ση _σ η _σ η λS dP x x P x x S P L px i . (8)

By differentiating the (8 ) equation we will determine the mathematical probability of the repair general time for such a case, when a single restoration time density is divided up according to the Veibul's law and P=0:

                +       − − − × ×            −      −       = =      

∫

∫

∫

∞       −       − − ∞       − − ∞       − − − = 0 1 2 2 1 0 2 2 2 2 0 2 2 0 e e ) 1 ( e e 1) -( S exp e ] [ ) ( dx x S dx x S dx x S dx x x M dP P dL x x x x s p η η η η σ η σ η σ η σ η λ σ σ η λ σ σ η η λ σ σ η λ σ σ η η λ (9)

In accordance with [1] we find that:

∫

∞ _      − − =       0 2 e η σ σ η σ η dx x x _Γ 1 1 _      − η , (10) here Γ- is gamma-function; η σ σ η σ η =             ∞ −

∫

x dx x e 0 2 2 Γ 2 1_      − η ; (11) η σ σ η σ η =             − ∞ −

∫

x dx x e 0 1 ; (12) η σ σ η σ η =             − ∞ −

∫

x dx x e 0 1 2 . (13)

By using (10) - (13) equations, after a corresponding transformation of (9) equation we get:

( )

1 1 2 д 1 -1 д 1 1 exp e ] [ 2         + +             − −       − ⋅       = − η σ σ η η σ η η η σ η λ η λ S x M s _{. (14)}

Using the (10)-(13) equations and having noticed that:

∫

∞ _      − =       0 e η σ σ η σ η dx x x Γ 1 1_      − η ;

∫

∞ _      − − =       0 2 e η σ σ η σ η dx x x Γ 2 1_      + η ,

after a corresponding transformation we will find the dispersion:

(

)

₍

₎

. 1 2 1 1 1 1 1 2 1 1 1 1 exp e ] [ 2 2 2                  − Γ −       − Γ − +       + + ×     ×             − Γ −       − Γ − = − η σ η η σ η η λ η σ σ η σ η λ η η η σ η λ η λ s s s x D s

Knowing the evaluation of jth type transport means repair general time probability characteristics, we may use them in planning the work of transport enterprise. Therefore, there should be known how many transport means will have to be repaired through the period t, and this will be analysed in the next chapter.

3. Probability model for estimation of average number of vehicle repair

per time period

For the determination of the vehicles which require repair (the number of not operating transport means), we will use the integral equation of restoration density f(t).

According to the restoration theory it may be written:

) ( ) ( ) ( ) ( 0 2 1 t τ f t τ dτ f x f t − + =

∫

, (15)

here f(t)- is restoration (repair) density, expressing the average number of a single transport means restoration within the time period t;f₁

( )

t - is the distribution density of new transport means operating time until the first come out of action; f2(t) - is the distribution of operation time density of repaired transport means until the next break down.

Within any period of time the number of repairs of the jth type transport means may be found out:

) ( = 2 1 2 1,

∫

t t t t N f t dt q (16)

here q_t1,t2 - is the number of repairs within the period since t1 until t2; N is the number of the jth type transport means..

The integral equation (15) is the second-degree Voltaire's equation. Its solution will be applied for the Laplase's transformation, i. e.:

( )

P F

( ) ( ) ( ) ( )

P f P f P F P

f = ₁ + + ₂

or

( )

P F

( )

P

(

F

( )

P

)

f = ₁ 1− ₂ . (17)

As it was stated before, the duration of the operational time of the majority of transport means distributes according to the indication law with the intensity µ₂. Let us assume that the damage time of transport means distributes also according to the indicational law with the failure intensity

1

µ , then it may be put down as follows:

1 1 1( ) _µ µ + = P P F ; (18) 2 2 2( ) _µ µ + = P P F . (19)

By inserting into (17) the equations (18) and (19), after certain transformations we will get the following: ) /( ) ( / ) (P =µ₂ P+ µ₁−µ₂ P+µ₁ f . (20)

The inverse Laplace's transformation will be as follows: 2 2 / λ µ P= , (21) e ) ( ) /( ) ( 1 2 1 1 2 1 µ P µ µ µ µt µ _{− + = −} − _{. (22)}

After the insertion of inverse Laplace's transformation meanings (21) and (22) into the equation (20), we will get: e ) ( ) ( 1 2 1 2 t t f =µ + µ −µ −µ . (23)

Further, after inserting of the meaning (23) into (16) and by integration, we will get:

[

]

(

)

      − − − − = − + = − − −

∫

12 11 2 1 1 2 1 ( )e ( ) e e 1 2 1 1 2 2 2 1 2 t t t t t t t N dt N t t q µ µ µ µ µ µ µ µ µ µ .

Thus, from the obtained equations it is possible to determine the number of the transport means getting out of order within the period t1, t2, and from (14) equation to determine the mathematical probability of the general time of repaired transport means, being aware of the intensity of the flow of coming out of order a piece and the distribution law of time of single repairs.

4. Stochastic model for the determination of desirable vehicle productivity

While planning the work of transport means the amount of haulage works is not always known in advance, i. e. in the course of time the work amounts change. Then a need occurs to make reference at the statistical data accumulated from previous analogical works.

Let us analyse the probabilities characteristics of jth _{type transport means efficiency of labour N,} which is necessary for making the L volume work within the time period t.

It is known from statistical data that the time tj is an random quantity. As observation results show, the law of division of time tj is the composition of many accidental division quantities. Although the laws of distribution of separate time tj components differ, the composition of such laws approaches the normal distribution law when there is a great number of constituents according to the A. Liapunov central theorem:

(

)

] [ 2 ] [ exp ] [ 2 1 ) ( ₂ 2         ₋ − = j j j t t M t t t f σ σ π .

During the observation it was determined that the amount of haulage works Li - distributes according to the logarithmic-normal law when road construction materials are carried:

(

ln [ ]

)

] [ 2 1 exp ] [ 2 1 ) ( ₂ 2         − − = _{j j} j j l M l t t t f σ σ π .

Efficiency distribution integral function may be presented as a double-measured net function

( )

lj tj

( ) ( , ) ) ( _{j j j j} N j f l t dl dt N F j

∫

∞ = α

Since Nj are variable, the distribution density will be:

( )( ) ) ( ) ( 0 0 j j N j j j j j j _dN f t dt fl dl d dN N dF N f j

∫

∫

∞ ∞         = α . (24) We shall write:

[

( ),

]

'( )

[

( ),

]

) ( ' ) , ( ) , ( ) ( ) ( ) ( ) ( y y f y y y f y dx y y x f dx y x f dy d y y y y α α β β δ δ β α β α − + =

∫

∫

. We shall note: , 0 ) ( ' ; ) ( ; ) ( ' ; ) ( 2 = ∞ = − = = j j j j j j j j N N N l N N l N β β α α then

[ ]

[ ] ] [ ] [ 2 1 exp 2 ) ( _{2 2}         −         − =         =

∫

∞ j j j j j j j j j j N l j j j j j t t M N t l N t l N l f N l dt t f dN d j j σ π σ σ .

From (24) proof we have:

    ×         − − =

∫

∞ ] [ ] [ ] [ 2 1 exp 2 ] [ ) ( 2 0 2 j j j j j j j j j _t t M N t l N t l N f σ σ π σ

(

ln [ ]

)

. ] [ 2 1 exp 2 ] [ 1 _{2 j j} 2 _j j j dl l M l l l _      − − × σ π σ (25)

After a certain transformation (25) we shall write: the density of necessary efficiency of jth type transport means distributes as follows:

(

)

] [ ] [ ] [ 2 1 ] [ 2 ] [ ln exp ] [ ] [ 2 1 ) ( 0 2 2 2 j j j j j j j j j j j j j j dl t t M N t l l l M l l N l t N f

∫

∞                 − − − − = σ σ σ σ πσ . (26)

The obtained function enables the calculation of necessary efficiency of jth _{type transport means,} when the amount of works and the time of their carrying out is indeterminate. It is particularly important that there are no precise data about the future structure and the flows of goods.

As you may see, in the proof of (26) function there is an indeterminate integral, thus if we wish to continue the operation with (26) function, we have to investigate its post-integral expression. In the result of investigation of the post-integral expression it may be put down with the exactness of 0.0001:

(

)

(

)

∫

∫

                − − − − ≈ ≈                 − − − − ∞ 5000 0 2 2 2 2 0 2 2 2 2 . ] [ ] [ ] [ 2 1 ] [ 2 ] [ ln exp ] [ ] [ ] [ 2 1 ] [ 2 ] [ ln exp j j j j j j j j j j j j j j j j j j j j dl t t M N t l l t M l l dl t t M N t l l l M l l σ σ σ σ σ σ

Then (26) function will be as follows:

(

)

. ] [ ] [ ] [ 2 1 ] [ 2 ] [ ln exp ] [ ] [ 2 1 ) ( 2 5000 0 2 2 2 j j j j j j j j j j j j j dl t t M N t l l l M l l N l t N f             − −     ₋ − =

∫

σ σ σ σ πσ (27)

The mathematical expectation of the necessary labour efficiency of jth _{group transport means, after} a respective transformation will be:

(

)

. ] [ ] [ ] [ 2 1 ] [ 2 ] [ ln exp ] [ ] [ 2 ] [ ] [ 1 ) ( ' = ] [ 2 2 2 5000 0 2 2 j j j j j j j j j j j j j j j j j j j dl t t M N t l l l M l l t M N l N t l N l t N f dN d N M                 − − − − × ×     −             − =

∫

σ σ σ σ σ πσ

Dispersion of the necessary productivity of jth type transport means will be:

(

)

. ] [ ] [ ] [ 2 1 ] [ 2 ] [ ln exp ] [ ] [ ] [ ] [ ] [ 2 ] [ ] [ ] [ 2 2 3 ] [ ] [ 2 3 ] [ ] [ 1 ) ( ] [ 2 2 2 2 2 2 3 2 2 2 2 3 2 3 2 2 2 5000 0 2 2 2 2 j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j dl t t M N t l l l M l N t l t M N t l l t M N l N t l N t l t M N l N t l N l t M N l N t l N l t N f dN d N D                 − − −     ₋ −         − −         − × × + +         − − −     −             − = =

∫

σ σ σ σ σ σ σ σ σ σ πσ

If necessary, it is possible to find an expression of other central moments as well, although because of their complexity they are not presented here.

5. Probability model dedicated for a short-term forecasts

of vehicle productive time losses

A planned jth group transport means operation time resources for a planned period may be decreased because of the following accidental reasons: 1) idle time of transport means caused by their repair; 2) idle time of transport means caused by the lack of drivers; 3) idle time of transport means caused by different hindrances at the time of loading and unloading (during goods handling). Let us assume that xj - is a random quantity, estimating jth group transport time losses caused by different accidental factors, and is expressed as follows:

z y x

x_j = + + ;

here z - is the duration of repair of the jth group transport means, in case of their accidental failure; y - time losses of jth transport means group for lack of drivers; x - idle time of jth group transport means caused by hindrance during the goods loading - unloading time.

Since x, y and z are random quantities, then xj is also a random quantity, the distribution law of which may be found out as a composition of distribution laws of random quantities x, y and z:

fx(x), fy(y) and fz(z), then the density of their distribution sum may be found out in the following way. The density x1= y + z of sum of two random quantities will be:

∫

∞ ∞ − − = ( ) ( ) ) ( _{1 1} 1 x f y f x y dy f_{x y z} . (28)

Noting that xi= x + x1, we calculate the distribution density in the point x1 in the following way:

∫

∞ ∞ − − = ( ) ( ) ) (x f x f ₁ x x dx f _{i x x i} (29)

or by inserting (28) into (29) we shall get:

∫

∫

∞ ∞ − ∞ ∞ − − − = ( ) ( ) ( ) ) (x f x f y f x x y dydx f _{i x y z i} . (30)

By integration of (30) the density of parameter xi may be obtained. This operation may be carried out only within certain limits. The integration time and the precision of the obtained result depend basically on selected limiting conditions.

Since the densities of distributions fx(x), fy(y), fz(z) differ from zero only in a certain narrow interval, so in the integration - in the intervals, where density does not equal zero. For distributions, the densities of which theoretically continue to infinity, the density different from zero will be in the interval

(

m−K₁σ2 ,m+K₂σ2

)

; here m - average distribution; _σ2_{- dispersion of distribution; K}

1 and K2 - constants (for the symmetrical distribution K1 = K2 = 4, for the distribution with a positive asymmetry K1 = 4; K2 = 6÷8).

Having evaluated the limits, the density formula (30) may be written in this way:

∫

∫

− − = d c z y b a x x f y f x x y dydx f x f( ₁) ( ) ( ) ( ₁ ) here , max 2 2 2 1 _    _      _{+ + +} − − = m_x k _{x x} x_i m_z m_y k _{z y} a σ σ σ , min 2 2 1 2 _    _      _{+ + +} − − = m_x k _{x x} x_i m_z m_y k _{z y} b σ σ σ ;

(

)

(

,

)

maxm_y k_1y x_i x m_z k_{2 z} c= − − − + σ ;

(

)

(

, ,

)

minm_y k_{2y y} x_i x m_z k_{1 z} d = − σ − − + σ .

Let us analyse a concrete example when parameters are distributed as follows:

x - according to the logarithmic normal law

( ) e 2 1 ) ( 2 2 2 ln β α β π − − ⋅ = x x x x f ; (31)

y - according to the normal law

e 2 1 ) ( _2σ2 µ β π − − ⋅ = y y x y f ; (32)

z - according to the indicative law

z z z

f ( )=λe−λ . (33)

By inserting (31), (32) and (33) into (30) we shall obtain:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ₌         = = = =         − − ∞ ∞ ₋ − + ∞ ∞ ₋ − − + − − − + − + − − − ∞ ∞ ₋ −

∫

∫

∫

∫

∫

∫

dx dy x f dydx x f dydx x f x f x x y x x x y y x y x x y x i i i i µλ σ λ λ σ λσ µ σ λ λ σ µ λσ µ λσ µ σ µ λ σ µ σ π λ σ π λ λ σ π 2 0 0 2 0 0 2 2 2 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 e e e 2 1 ) ( e 2 1 ) ( = e e 2 1 ) ( ) ( ( ) ( ) _{1 ,} e e 2 1 e 0 2 2 ln 2 2 2 2 2 dx x x x x i

∫

∞ − − −         −               − Φ − = σ λσ µ β π λ β λ σ µλ σ λ ₍₃₄₎ where e 2 1 ) ( 2 2

∫

∞ − = Φ t t u du π .

In figure 1 there are demonstrated curves of densities of random quantities (31), (32) and (33), as well as the sums (34) of densities provided the meanings of parameters are following:

Figure 1. The curves of densities of distribution of random quantities x, y, z: 1 - indicative distribution (time of repair); 2 - logarithmic-normal distribution (losses of time caused by the lack of drivers); 3 - normal distribution (losses of time caused by hindrances within the time of loading - unloading); 4 - x, y, and z as well as the density of the distribution sum

6. The algorithm for a transport vehicle repair time forecasting

As practice shows, in the cause of exploitation of transport means, starting with the beginning of exploitation to the writing off the obsolete equipment, the following situation may be observed: at the beginning of the exploitation their reliability is of a certain level, which during a certain time is increasing (the time of wear of mechanisms and component parts). Later on it stays at the same level (within a normal exploitation period) and finally there occurs a sudden decrease of reliability, caused by the start of an intense wear of units/assemblies and of separate parts, as well as frequent failures (Figure 2). Therefore the repair time also changes, because in the first stage the time is necessary for various adjustments, required for adjusting of separate units, also for exchange or repair of slight reliability units. In the second stage, the time necessary for repair is stable for a long period, because it is required only for planned repair - maintenance check-ups. In the third stage the time required for repair starts to increase rapidly and consequently it becomes necessary to write off the transport means or to put it on major repairs.

The analysis of the monthly time Lt of vehicles repair shows that it has accidental characteristics, nevertheless the average meaning (within a certain time limits) distributes according to some law. The aim of our investigation is to describe analytically the change of average monthly repair time, depending on a month t.

months

Analytical expression of the function L_tv = f(t ) from the physical point of view is as follows:

(

e e

)

; ) ( ( ) ) 1 ( t ct D t t E A L v − − − ₊ + = β (35)

[

( ) ( )

]

2 1 ) 2 ( _{E A t B C t D t D} L v t = + − + − + − , (36)

here A shows the work of an vehicle in the initial period of exploitation; B - the same period in the end (35); C - the period of intense wear and tear; D - the starting point of intense wear and tear (36); E - an average time of repair in the stationary conditions.

Since the functions (1)and (2)

v

v t

t L

L are non-rectilinear and even non-polinomous, it is impossible to find out coefficients A, B, C, D and E by traditional methods of regressive analysis. Therefore in this case they were found out by minimising quadratic/square function of losses.

[

( )

]

2 1 2 1 ) 2 (

∑

= − = N t t t L t L S v (37)

here N - the number of deductions; Lt - the vehicle repair time observed within t a month period. As it may be seen from (37), the function of coefficients A, B, C, D, E may be multi-extremal. Therefore, we do not use ordinary/usual methods of search of minimum (gradient, the fastest descent), we find the minimum (37) by the method of statistic experiments. For this purpose we have selected the initial intervals (A01, A02), (B01, B02), (C01, C02), (D01, D02), (E01, E02), in which it was meant to find the searched meanings of coefficients A, B, C, D, E, in presents of which the function (37) has the minimum. Further, in the given interval with random numbers (normally distributed) sensing element we generated the collection of the coefficients A, B, C, D, E and calculated the meanings of the function S. We repeated this operation several thousand times, afterwards we analysed that collection of coefficients, at presence of which the average quadratic function of losses had the least meaning. Later on we chose new, much narrower intervals (A11,

A12), ... , (E11, E12) and again we repeated the search procedure.

Figure 3.Duration of vehicle repair, depending on its exploitation time:

1 - experimental time curve; 2 - calculated/computed curve according to (36); 3 – curve of forecast according to average quantities; 4 - curve of forecast according to (51)

On the basis of the experiments the discrimination of models was performed. Namely, exactly being

N=245 first model S(1)=1,816, and S(2)=0,375 (Table 1). TABLE 1

Results of coefficients counting for nine cargo-carrying vehicles

Vehicle number N A B C D E S 1 241 -0,221 33,6 0,186 167,7 5,16 0,3775 2 246 -0,251 34,4 0,363 170,7 5,00 0,868 3 242 -0,160 47,6 0,458 206,7 5,03 0,753 4 225 -0,158 57,9 0,565 192,4 5,42 0,851 5 250 -0,163 56,1 0,325 201,5 5,24 0,864 6 235 -0,250 38,1 0,390 187,9 5,23 0,749 7 224 -0,178 43,0 0,396 177,5 4,80 1,283 8 216 -0,336 29,8 0,490 177,2 5,77 0,510 9 220 -0,264 31,6 0,632 180,3 4,92 0,587

By using the criterion F [1]:

) /( ) /( 2 2 ) 2 ( 1 ) 1 ( F K N S K N S < − − ,

here K1 and K2 - quantity of parameters in the models (35) and (36), in our case K1=K2=5 we shall get: 3 , 23 ) 375 , 0 /( ) 816 , 1 ( / (2) 2 2 ) 1 ( _{S = =} S ,

and this very much exceeds the level of dependency F even when Q=0,05%, which in this case is less than 1.75.

Thus, when the coefficients A, B, C, D, E are known, the repair time of transport means within a month t may be calculated according to (36). Then average quadratic deviation will be S .

Forecasting. In practice the coefficients A, B, C, D, E are not known. If the work of automobiles has been analysed for a certain time, then the coefficients A, B, E may be found from the sum of minimum condition:

[

( ) ( )

]

2 1 1 1 1 1

∑

      _{− + − + −} = N L_t E At B t B N S (38)

Then the coefficients C and D are defined by principle of Bayas considering that the searched parameters are as much random quantities, the distribution of which is known before. Let us assume that the coefficients A, B, C, D, E are random quantities, distributed according to the normal law, the density of which we shall put down by the equation:

) ( )' ( 2 1 exp ) 2 ( ) ( /2 1/2      _{− − −} = π − P x µ P x µ x f P , here ,                 = E D C B A

x µ -mathematical expectance vector; P-1 - co-variation matrix. The print shows the

operation of transformation.

We will take the aprioritative average distribution as normal:

) ( )' ( 2 exp ) 2 ( ) ( 1/2

∑

1/2 0

∑

     _{− − −} = − n _{m m} m f π µ µ .

Wirschaft's matrix P in the n0 degree:

2 1 exp 2 ) ( 0 ₀ 2 _{, 0} 0 0 0      − =

∑

∑

− P t n C P f _rn P n .

If the sample of vector's coefficients was x1, x2, ..., xn, then posterior vector's coefficient would distribute as follows: 2 ) ( )' ( 1 1 2 2 2 ) 2 ( )) , , /( ( 1 1 1 1 1 2 1 1 1 1 2 / 1 1 1 2 / 1 n x x n n P n n n x x x f P m       − Σ − + +       − Γ       Γ Σ = − − − µ µ π K , (39) here

(

)

[

]

. ' 1 1 ; 1 ; ' ' ) 1 ( ' ; ; 1 1 ' 1 1 1 1 0 0 0 1 1 0 1 0 1

∑

∑

= =      − − = = − + − + + Σ = Σ + = + = n i n i i x x nxx n S x n x n n nxx S n n n n x n n n n n µµ µµ µ µ

We shall put down the conditional distribution of unknown parameters C and D, if A, B and E

are known (defined by (38)).

We shall divide the vector x into two parts:

        =             =         = D C x C B A x x x x (1) (2) ) 2 ( ) 1 ( and here , .

Analogically we shall divide the vector 12 11 1 _       = M M

M and the matrix

122 121 112 111 1       Σ Σ Σ Σ = Σ N ,

here M - vector-column from three elements (matrix Σ11 is of current 3×3, and Σ22 - of current 2×2.

Then the quadratic form of the expression (39) may be written as follows:

(

)

), ( ) ( ) ( )' ( ) ( )' ( ) ( )' ( ) ( ' 22 2 22 1 12 2 11 1 21 1 12 2 12 2 12 1 11 1 11 1 11 1 11 1 1 1 1 1 M x M x M x M x M x M x M x M x M x M x K_F − Σ − + + − Σ − + − Σ − + + − Σ − = − Σ − = −

here Σ11_{, Σ}12_{, Σ}21_{, Σ}22 _{- are component parts of inverted matrix} 22 1 Σ , i. e.: 22 1 21 1 12 1 11 1 1 1       Σ Σ Σ Σ = Σ− _, then 11 1 Σ - is of current 3×3, and 22 1 Σ - of current 2×2.

We shall put down the expression (40) in the shape of quadratic form from x2:

[

]

[

] [

]

[

( ) ( )

] [

' ( ) ( )

]

) ( ) ( ' ) ( ) ( ) ( ) ( 2 2 ) ( 2 ) ( 2 ) ( )' ' ( 11 1 21 1 1 22 1 12 22 1 11 21 1 1 22 1 12 11 1 21 1 1 22 1 12 22 1 11 21 1 1 22 1 12 11 1 22 1 12 22 1 1 22 1 22 2 2 22 1 2 12 12 1 12 12 22 1 2 2 22 12 11 1 21 1 12 11 1 21 1 12 11 1 11 1 11 + − Σ Σ − Σ − Σ Σ − − − − Σ Σ − Σ − Σ Σ − + + − Σ − Σ Σ Σ − Σ = = Σ + Σ − Σ + − Σ − − − Σ + − Σ − = − − − − − M x M M x M M x M M x M M x M x x x M M M x x x M x M M x x M x M x K_F . ) ( 2 ) ( )' (x1−M₁₁ Σ11₁ x1−M₁₁ − M₁₂' Σ₁21 x1 −M₁₁ +M₁₂' Σ22M₁₂ + (41)

After noticing that the first three members of the right part of the expression (41) make quadratic form and after the performance of algebraic operations with matrices, we shall obtain:

[

]

{

− − Σ Σ −

}

Σ − = − 12 2 22 1 11 1 21 1 1 22 1 12 2 _{M ( ) (x M ) {x [M} x K_F

[

( )

]

( ). )' ( )]} ( ) ( 22 1 21 1 ₁₁ 1 12 11 1 11 1 11 1 21 1 1 22 1 Σ x −M + x −M Σ −Σ Σ Σ x −M Σ − − − ₍₄₂₎

Further, before inserting (42) into (39), we shall insert the ( ) , , , ( 22) 1 21 ,

1 12 11 1 22 1 21 1 1 22 1 Σ Σ Σ −Σ Σ Σ Σ − − _of

the matrix, which are inverted matrices of the functions of the sub-matrices , , , 22 1 21 21 1 11 1 Σ Σ Σ Σ in the shape of 1 1− Σ functions of sub-matrices Σ111, Σ112, Σ121, Σ122.

Having noticed that Σ1(Σ1)-1=1 (unity matrix), we have the following:

0 121 22 1 111 21 1 Σ +Σ Σ = Σ ; (43) 1 121 12 1 111 11 1Σ +Σ Σ = Σ ; (44)

1 122 22 1 112 21 11Σ +Σ Σ = Σ . (45)

From (43) we shall find:

( )

(

)

( )

21 ₁₁₁. 1 1 22 1 121 1 111 121 21 1 1 22 1 Σ Σ Σ − = Σ Σ Σ − = Σ Σ − − − (46)

After insertion of (46) into (44) we shall get:

( )

21 ₁₁₁ 1 1 1 22 1 12 1 111 11 1 Σ −Σ Σ Σ Σ = Σ − or

( )

1 . 111 21 1 1 22 1 12 1 11 1 − − Σ = Σ Σ Σ Σ From (43) we have: 1 111 121 22 1 21 1 =−Σ Σ Σ− Σ .

After having inserted the latter expression into (45), after transformation of matrices, we shall get:

(

)

1 112 1 111 122 21 1 − − _Σ Σ − Σ = Σ . (47)

Consequently the quadratic form (42) may be put down as follows:

( ) (

)

[

]

{

}

'

(

1 ₁₁₂

)

1 111 121 121 11 1 111 121 12 2_{− +Σ Σ − Σ −Σ Σ Σ ×} = _{x M} − _{x M} − − K_F

(

) (

)

[

]

{

'

}

(

'

)

' 1

(

' ₁₁

)

. 11 11 11 1 111 121 12 2 _{M x M x M x M} x − +Σ Σ − + − −Σ − × − − ₍₄₈₎

By inserting (48) into (47) we shall get the expression of the aposterioric conditional density:

( )

[

[

(

−

)

]

}(

Σ −Σ Σ Σ

) {

−

[

+ Σ Σ + −     ₊ +       − Γ Σ       Γ = − − − − 12 2 1 112 1 111 121 122 11 1 111 121 12 2 2 1 1 2 / 1 1 2 / 1 1 1 1 2 1 2 ' + 1 1 2 2 2 2 )) ' , , , , /( ( M x M x M x n n p n n n x x x x x f _n π K

(

'

)

]

}

[

(

'

)

1

(

' ₁₁

)

]

/2. 11 1 11 11 1 111 121 m M x M x M x − − − _{− + − Σ −} Σ Σ + (49)

Wishing to evaluate the parameters C and D, we have to find the most probable meanings of the vector x2. Of course, the maximum of aposteriorical conditional density (49) will be obtained when the first member of the quadratic form KF will equal 0. Thus, we have the following estimation:

( )

(

'

)

* * * 1 ₁₁ 111 121 12 2 _{M x M} D C x _= +Σ Σ −      = − ₍₅₀₎

Thus, having estimated the observations L1, L2,...,LN parameters A, B and E (from conditional distribution - the parameters C* and D*), we make the following equation for forecasting of vehicles monthly repair time:

[

( ) ( ) *( *) *( * . 2 1 ) (t E At B At B C t D C t D L_pr = + − + − + − + − (51)

For verification of the working capacity of the method an experiment was made, by estimation of parameters C and D while intending to forecast the time of vehicle monthly repair. From the data of the Table 1 the estimations M2, Σ11 and M11 were obtained, which were necessary to calculate

C and D according to (50). We shall indicate xi=(xAi, xBi, xCi, xDi, xEi)′ five- measured vector of the coefficients A, B, C, D, E for a vehicle i. If xi were divided into two parts, and

xi=

(

x_Ai,x_Bi,x_Ci

)

is three-measured vector, and x₂'_i =

(

x_Ci,x_Di

)

- is a two measurement vector, then estimations M2, Σ11 and M11 may be put down as follows:

(

)(

)

(

ˆ

)(

ˆ

)

, 1 1 ˆ ;' ˆ ˆ 1 1 ˆ ; 1 ˆ ; 1 ˆ 11 1 1 11 1 111 11 1 1 12 2 121 1 1 12 1 2 2 M x M x N M x M x N x N M x N M i N i i i N i i N i i N i i − − − = Σ − − − = Σ = =

∑

∑

∑

∑

= = = =

here N - sample quantity (in our case N - 9).

Further, for all the meanings of A, B, E coefficients, i. e. vectors x1i, according to the formula (50), the forecasted meanings were defined:

(

ˆ

)

. ˆ ˆ ˆ 11 1 1 111 12 12 * * M x M D C i i i _{= −Σ Σ −}       ₋

The results of calculation of quantities *

i

C and *

i

D , as well as the real meanings of C* and D*, are presented in the Table 2.

TABLE 2

Results of calculation of quantities C* and D*

Vehicle number Ci* Ci Di* Di 1 0,255 0,186 168,3 161,7 2 0,461 0,363 178,2 170,7 3 0,351 0,458 187,7 206,7 4 0,442 0,565 200,7 192,4 5 0,485 0,325 201,3 201,5 6 0,461 0,390 181,2 187,9 7 0,405 0,396 186,1 177,5 8 0,463 0,490 169,9 171,2 9 0,482 0,632 176,5 180,3 Average meaning 0,423 0,423 183,3 183,3

As may be seen from the Table 2, the forecasted quantities C* and D* are proximate to real ones. But in that case the deviations of the real meanings from those forecasted are lesser than from the average meanings. Therefore, while planning repair works, it is purposeful to forecast their duration according to the proposed algorithm.

7. Conclusions

1. When the estimations of probability characteristics of transport means general operation time are known, then we may use them in planning of transport enterprise work. Therefore, it is necessary to know how many transport means will have to be repaired within a relevant time period. For this purpose there is proposed the probability model, which estimates an average number of transport means repairs within a time period.

2. While planning the operation of transport means the amount of works to do is not always known beforehand, i. e. in the course of time the amounts of works change. Therefore, there is proposed stochastic model for definition of the necessary productivity of transport means.

3. The time of transport means operation may decrease for a certain period because of their idle time caused by repair, by lack of drivers, by different hindrances during the loading - unloading time. Seeking to evaluate the influence of these reasons to the planning of transport means work - the probability model is proposed, which enables a short-term forecasting of operation time losses for transport means.

4. Due to certain economic conditions, in Lithuania the road transport means - vehicles - are in exploitation for 15 - 25 years. While the age of transport means increases, correspondingly increases the time required for their repair. So there is proposed the transport means repair duration forecasting algorithm, which estimates the time of their exploitation.

Reference

1. Baublys. Introduction to the theory of transport systems. Vilnius, Technika, 1997 (in Lithuanian language)

Author

ADOLFAS BAUBLYS

Doctor habil, Professor (1980) of department of Transport Management, Director of Transport Science Institute (1998). Vilnius Gediminas Technical University Transport Research Institute, Plytinės g. 27, LT-2040 Vilnius, Lithuania, Fax: 370-2-31 56 13. E-mail: [email protected]

Doctor habil of technical science, Vilnius Civil Engineering Institute (VISI, now VGTU), 1978. Doctor of Technical sciences, VISI, 1969. First degree in Mechanical Engineering, Kaunas Polytechnic Institute (KPI, Vilnius Branch, later VISI), 1967. Membership: member and expert of Lithuanian Academy of Sciences. Publications (textbooks/manuals): “Transport Policy” (1996), “Passenger and goods transportation by road transport” (1994, 1995), “Goods transportation railway, sea and air transport” (1995), “Transport systems” (1995, 1996), “International Transportation by Road Transport” (1996), “Introduction to Transport systems theory” (1997), “Cargo Transportation” (1998). Research interests: transport policy, modelling of transport systems, multimodal transport development. Author of 197 scientific articles. Chief and scientific editor of “Transport” the prestige journal of Lithuania.