Péter Várlaki István Pályi Lajos Tóth Department of MathematicsSzéchenyi István University

Széchenyi István UniversityEgyetem tér 1., H-9026 Gyor

Department of Logistics and Forwarding

Budapest University of Technologyand Economics

Hungary Széchenyi István UniversityEgyetem tér 1., H-9026 Gyor

Hungary Hungaryvarlaki@kme.bme.hu tothl@sze.hu

Abstract - Improving transportation infrastructure raisesseveral questions requiri ng complex analysis. Creating the mathematic model for transportation reconstruction needed developing a quick, dynamic programming process whichconsiders the costs of single reconstruction phases and the direct or indirect profit emerging after completingreconstruction works. In the study we examined theapplication possibilities of the highly prestigious Bellman-theorem. We found that the combinatorial method foroptimal distribution of investments is not very efficient in the current issue, although it applies the sequential method and the Bellman-theorem in a remarkable way. The methodproposed in the study applies exponential partial profit-functions. The primary advantage is that it solves theproblem of the complicated nonlinear dynamic programming on the basis of direct relations, without numeric analysis.Consequently, results are supplied quickly and the method allows solving enormous problems without any constraints.

I INTRODUCTION

Transportation infrastructure in Hungary requiresconsiderable developing. Out of date infrastructure is a barrier for economic development as some regions aredifficult to reach, and the road system is over centralised.Gauge of transit transportation leads to overloaded road networks at certain sections, impedes local transportation and has bad effect on living circumstances in settlements.28 percent of main transit roads run through settlements causing frequent traffic congestions, slow transportationand increasing negative effects on the environment. Long distance transportation runs on roads of low quality and capacity which cannot meet the requirements forcontinuous transportation flow.Constructing a modern, up to date infrastructure should contribute to economic development in out of dateHungarian settlements. Modern infrastructure improvestransportation connections between regions and changes distribution of transportation by decreasing transittransportation and potential number of accidents. Regionaldevelopment can improve multiple effects of economic development by improving settlement reach ability; overmore, roads of improved quality with higher transportcapacity contribute to building a stable, well constructed,reliable road system which is a basis for raising regional economic power. Improving transportation infrastructure raises severalquestions requiring complex analysis. Some of them are as follows:

– What are the average costs of reconstructing 1 km of roads and what are the cost components?

– What national values get wasted or emergewithout or after reconstruction works?

– What are the effects of reconstruction on tourism, nostalgia journeys and other technical-industrialrelated issues?

– What are the potential reconstruction strategies? (Road sections to be improved? Which one is the best to start with? Which one should it be followed by? Costs are of key importance and cannot be avoided. How much profit is expected afterreconstruction phases and complete reconstruction works (that must be reconsidered when making a strategy)?

– Which mathematical mo dels are applicable tomake reconstruction optimal?

Reconstruction works require modern computationalmodels e.g. creating easily operable and extendabledatabases for storing big quantities of data on theinfrastructure system. Creating the mathematic mo del for transportation reconstruction needed developing a quick, dynamic programming process which considers the costs of single reconstruction phases and the direct or indirect profit emerging after completing reconstruction works. We can apply an effective computational calculation method which impressingly shows the efficiency of the optimal strategy.

II OPTIMISING THEOREM

Reconsidering the processes applied in the actual field, itcan be stated that the dynamic programming methods,applying graph theory mo dels and based on the RichardBellman theorem, are widespread. The Americanmathematician Bellman created the most efficient method for finding optimum of goal functions occurring ineconomy related phenomena. It is of key importance to mention that this method is also efficient by researches and analyses in the fields of physics and mathematics.In practice, Bellman’s method is known as ’optimizingtheorem’, as in mathematics the proper terms are ’axioms’ or ’theorems’, instead of theories. The significance and efficiency of sequential optimizing methods, derived from the theorem, are experienced in various fields which is rooted in the sequential character of most economicproblems. When it comes to discrete variables, thus, graphs are applied in most cases. The basic point of the method is the ’optimising theorem’ which is possible to present in various aspects, generally as a theory.

Reconstruction Decision Model for Trasportation Infrastructure Systems

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Ildikó GombaszögiBudapest Tech

III OPTIMIZING THEOREM BY DISCRETE AND DETERMINISTIC SYSTEMS

Consider the system, of which state, due to a decision, can change in the single k-phases. In the single k-phases (k = 0, 1, 2, ..., N) the system exists in finite or infinite butcountable state. A certain sequence of decisions, with k ranging from 0 to N (k = 0 – N), is defined as ’policy’. The term ’sub-policy’ is used for a sequence of related decis ionsbeing part of any policy. If we take a value-functiondepending on state changes and intend to optimise it, the following statement is true of the Bellman-theorem: Anyoptimal policy must include only optimal sub-policies. (It is easy to prove: consider a subpolicy – part of any optimal policy. If this subpolicy is not optimal, there should exist a better subpolicy which, with the rest of the policy, could improve the policy. This notion opposes the hypothesis .The theorem can also be extended for cases with continuous variables defining the actual states.)Bellman formulated the previous theorem in general form as follows: ’Any policy is optimal if, in a certain period –independently from the character of the previous decisions– the forthcoming decisions make an optimal policy taking the results of the previous decisions into account.’ This theorem is easily applicable in solving sequential problems.

IV IN WHICH SPECIAL AREAS ARE THE MATHEMATICALMODELS FOR OPTIMAL RECONSTRUCTION STRATEGY

APPLICABLE?

A Applying sequential optimisingMotorway constructions: Suppose that a motorway is built between points A and N. The motorway runs near some towns and is divided in five sections. Cost varieties for the single sections are estimated and analysed. The costsinclude road construction works, buildings, occupiedgrounds and social costs etc. If starting point is A, and final point is N, we have to calculate the minimal costs. Calldecision variables of the single road sections as x0, x1, x2, x3, x4 and x5. These x0,x1,x2,x3,x4,x5 variables do not have a numeric value but required states. The followingpoints belong to the single s ets of points:

x0:{A}xl:{B,C,D}x2:{E,F,G}x3:{H,I,J,K}x4:{L, M}

x5:{N}Call costs of section I. as vI(x0, x1). This cost is dependent of the ’values’ taken by x0 and x1. In this case x0 can take only A. It could also take a different one provided thestarting point of the road is optional to select. Call costs of section II. as vII(x1, x2). These costs are dependent of x1 and x2. The way of definition is the same for vIII(x2, x3),vIV(x3, x4) and vV(x4, x5). After that the total costs of the motorway accounts to:F(x0,x1,x2,x3,x4,x5) = vI (x0, x1)+vII(x1, x2) +…+ vV(x4, x5). (3)

According to the Bellman-theorem, optimizing must bemade by finding the minimal costs of section I consideringthe final points as B, C and D. Let minimal costs be marked

by fI(x1). Similarly, minimal costs altogether for sections I and II are marked by fI,II(x2), for sections I,II-III byfI,II,III(x3), for sections I,II-III, IV by fI,II,III,IV(x4), andfor sections I,II-III, IV,V by fI,II,III,IV,V(x5).If sequential optimizing is carried out in a more convenient direction or starting from a given phase, calculation of problems could be done in a simpler way. All that depends on the functional relations connecting single state variables in the analysed system. Sequential optimising can becarried or by any method, provided the idea is based on theoptimising theorem.

B Combinatorial problem – distribution of investmentsIn this case sequence of phase analysis is optional. In the four economic regions, marked by: I, II, III and IV, the aim is to improve sales. A certain amount of ’A’ is available for investments which is to be shared between the four regions.Regarding the investments, the potential profit in the single regions should be considered as available information, (Table 1). Suppose, sum A accounts to 100 million, or 10 –provided 10 million makes a unit.This way the following policy can be applied: 3 in region I,1 in II, 5 in III, 1 in IV. Profit in this case accounts to:0.651+0.251+0.620+0.201=1.723This problem is of combinatorial character so it could be solved by listing the solution possibilities. This way the number of policies to be analysed is considerable, that is exactly : 286.

Table 1

Investment and profit

Investment

Profit

I II III IV0 0.000 0.000 0.000 0.0001 0.281 0.251 0.153 0.2012 0.452 0.419 0.255 1.0003 0.651 0.552 0.401 0.4224 0.783 0.651 0.500 0.4815 0.910 0.752 0.620 0.5356 1.024 0.809 0.732 0.5657 1.132 0.852 0.822 0.5878 1.234 0.887 0.901 0.6019 1.327 0.901 0.962 0.603

10 1.381 0.901 1.000 0.604

Name the investment of tens of millions as xl, x2, x3, x4 (indivisible unit) in regions I, II, III and IV. The profit inthe single regions is v1(x1), v2(x2), v3(x3) and v4(x4), and the following equation defines the total profit:

F(x1, x2, x3, x4) = v1(x1)+v2(x2)+v3(x3)+v4(x4)

bounded by: Sxi = 10, being aware of the fact that the four variables can be only 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10. Within this range the Bellman theorem is well applicable. If the indivisible investment unit is considered 1 million, the huge number of potential policies would be impossible to handle.

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Due to the great number of roads it is obvious that the above described methods – with a wide range of application possibilities, though, - do not contribute to a fast optimizingproblem solution.

V NONLINEAR DYNAMIC PROGRAMMING FOR OPTIMAL DISTRIBUTION OF INVESTMENTS

Consider a purpose oriented method of enormous efficiency that we primarily suggest applying because of its quickness.The number of roads selected for reconstruction is marked by n. Reconstruction costs based on technical assessment for the single cases are named K1, K2, K3… Kn,. Availablefunds for reconstruction are named A, where the total sum necessary exceeds the available funds:

A < K1 + K2 + K3 +… + Kn

We have to find the optimal distribution policy, where the variables x1, x2, x3… xn represent the unknown optimal costs related to the actually executed reconstructions. Totalprofit from reconstructions is defined in the function f(x1,x2, x3… xn), which is also the goal function to beoptimised. Since profit emerges after the singlereconstruction works, f is the linear combination of the unit-constants of the functions on the profit emerging from the single reconstructions:

:= ( )f , ,x1 x2 xn ∑ = i 1

n

( )f i xi

Accurate and precise defining of the partial profit-functionsfi(xi) is of basic significance by each reconstruction, as expectedly the required total funds are not available. It is important to have precise previous information on thepotential results of partial reconstruction and define theminimal reconstruction costs necessary to startreconstruction works. Obviously, partial profit-functionsare monotonic ascending top-bounded functions as it is superfluous to exceed reconstruction costs Ki, since costs exceeding Ki do not result in any profit. Suppose that the function fi(xi) can be differentiated according to xi(i=1,2,…,n).

Function g(x) represents bounded resources:

:= ( )g x − ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n

xi A

x is a vector with coordinates x1, x2, x3… xn: x[x1, x2, x3… xn]. We form the function h(x,?) and apply themultiplication Langrange-method as part of optimalprogramming tasks to define conditional minimum and maximum values:

:= ( )h ,x λ − ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n

( )fi xi λ ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟ − ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n

xi A

After deriving h according to variables xi the equation i=1,2,…,n is :

:= ei = ∂∂xi

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟ −

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n

( )fi xi λ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟ −

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n

xi A 0

Deriving h according to ? leads to the equation n+1, whichserved as bounding condition before:

:= e + n 1 = ∂∂λ

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟ − ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n

( )fi xi λ ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟ − ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n

xi A 0

After deriving, the equations i=1,2,…,n are as follows:

:= ei = − ⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟∂

∂xi

( )fi xi λ 0

Equation n+1 is:

:= e + n 1 = − + ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n

xi A 0

The following exponential function, which meets therequirements above, is applicable to describe nonlinearpartial profit-functions. Component 0=ßi is a parameter for representing utility.

:= ( )fi xi β i Ki ( ) − 1 e( )−ci x i

Substituting the above functions in the equation and makingderivations:

:= ei = − βi Ki ci e( )−ci xi

λ 0The unknown xi in each equation marked by i:

:= xi − − ( )ln λ ( )ln β i Ki ci

ciExpression relating xi is substituted into equation n+1:

:= e + n 1 = − + ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n 1ci

( )ln λ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n ( )ln β i Ki ci

ciA

Equation n+1 results in correlation relating ln(?):

:= ( )ln λ −− +

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n ( )ln β i Ki ci

ciA

∑ = i 1

n 1ci

Correlation relating ln(?) is resubstituted in equation for xi,which results in a direct formula for each optimal xi:

:= xi

− + + ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n ( )ln βi Ki ci

ciA ( )ln βi Ki ci

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n 1ci

ci⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟∑

= i 1

n 1ci

The most appropriate way to select the constant ci in the function is that, considering xi =Ki, the function fi(xi)approximates Ki by mistakes of maximum 1%, where ßi=1:

:= ci( )ln 100Ki

The greatest advantage of the method is obviously that it solves the problem of the complicated nonlinear dynamic programming on the basis of direct relations, by using a closed formula. Consequently, results are available soon and enormous problems can be solved. This method is based on the application of exponential partial profit-functions. Applying any different exponential partial profit-

P. Várlaki, I. Pályi, L. Tóth, • Reconstruction Decision Model for Trasportation Infrastructure Systems

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functions for the problem requires solving a nonlinearfunction-equation containing n+1 equations by usingnumeric methods.In case ofK1 + K2 + K3 +… + Kn = Athe method provides the values trivially :

x1 = K1, x2= K2, x3= K3, … , xn= Kn

but in case of actual constraints:A < K1 + K2 + K3 +… + Kn

ßi and cost components Ki are of great significance when optimising profit-functions.

VI CONCLUSION

In the study we examined the application possibilities of the highly prestigious Bellman-theorem. We found that thecombinatorial method for optimal distribution ofinvestments is not very efficient in the current issue,although it applies the sequential method and the Bellman-theorem in a remarkable way. A minor problem is that it uses ’whole number’ units. A considerable disadvantage is, however, that due to the great number of roads solution processes and methods are not fast enough. In practice, overmore, the total sum necessary for reconstruction exceedsthe available funds, which requires the application of aconditional optimum calculation method.The method proposed in the study applies exponentialpartial profit-functions. The primary advantage is that it solves the problem of the complicated nonlinear dynamic programming on the basis of direct relations, withoutnumeric analysis. Consequently, results are suppliedquickly and the method allows solving enormous problemswithout any constraints .

ACKNOWLEDGMENT

This work was supported by the Hungarian NationalScience Research Found (OTKA) under grants T042826 and T042896.

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