Mathematical modelling of dispute proceedings between investors and third parties on allegedly violated third-party rights/Ginco proceso tarp investuotoju ir treciuju asmenu del galimai pazeistu treciuju asmenu teisiu matematinis modeliavimas.
Abstract:
The article analyses the possible influence of third-party rights infringed during construction planning on the implementation of an investment project. In a construction project, judicial disputes are an unwanted risk factor, which may disrupt the entire project. It is therefore necessary to plan and apply preventive measures for the mitigation of such risk in the initial planning stage of a construction project. The article, for that purpose, presents modelling a dispute between investors and third persons on allegedly violated third-party rights with the help of a tree that illustrates the possible actions of the dispute parties. A mathematical model for dynamic programming the dispute on allegedly violated third-party rights has been developed; it helps to determine the optimal investor's strategies for each situation that involves decision-making.

Keywords: construction investment process, defence of third-party rights, judicial defence of rights, decision tree, dynamic programming, recurrent equations, optimal behaviour strategy.

Nagrinejama, kaip treciuju asmenu teisiu pazeidimai, planuojant statybas, gali veikti investicinio projekto igyvendinima. Igyvendinant statybos projekta, teisminio ginco atsiradimas yra nepageidaujamas rizikos faktorius, galintis suzlugdyti visa projekta. Todel vykdant statybos projekta jau pradiniame projekto planavimo etape butina numatyti ir taikyti prevencines priemones tokios rizikos mazinimui. Siekiant sio tikslo straipsnyje atliktas ginco tarp investuotoju ir treciuju asmenu del galbut pazeistu treciuju asmenu teisiu modeliavimas, sudarant ginco saliu elgsenos variantu formavimo medi. Sudarytas ginco proceso del galbut pazeistu treciuju asmenu teisiu dinaminio programavimo matematinis modelis, leidziantis nustatyti optimalias investuotojo strategijas kiekvienoje situacijoje, kai reikia priimti sprendimus.

Reiksminiai zodziai: investicinis statybos procesas, treciuju asmenu teisiu gynimas, teisminis teisiu gynimas, sprendimu medis, dinaminis programavimas, rekurentines lygtys, optimali elgesio strategija.

Article Type:
Report
Subject:
Mathematical models (Research)
Third parties (Law) (Research)
Industrial project management (Laws, regulations and rules)
Project management (Laws, regulations and rules)
Investments (Management)
Investments (Laws, regulations and rules)
Authors:
Sostak, Olga Regina
Vakriniene, Sigute
Pub Date:
03/01/2011
Publication:
Name: Journal of Civil Engineering and Management Publisher: Vilnius Gediminas Technical University Audience: Academic Format: Magazine/Journal Subject: Engineering and manufacturing industries Copyright: COPYRIGHT 2011 Vilnius Gediminas Technical University ISSN: 1392-3730
Issue:
Date: March, 2011 Source Volume: 17 Source Issue: 1
Topic:
Event Code: 310 Science & research; 930 Government regulation; 940 Government regulation (cont); 980 Legal issues & crime Advertising Code: 94 Legal/Government Regulation Computer Subject: Government regulation
Geographic:
Geographic Scope: Lithuania Geographic Code: 4EXLT Lithuania
Accession Number:
269229107
Full Text:
1. Introduction

In most of our cities, some parts undergo intensive transformations related to commercialisation, land use and the density of buildings (Kaklauskas et al. 2007; Bardauskiene 2007; Turskis et al. 2006; Zavadskas et al. 2004; Kaganova et al. 2008). Several examples in European cities show that development can embrace internal urban areas (Mcdonald et al. 2009; Kaklauskas et al. 2009; Miller et al. 2004). Currently, Lithuanian cities also witness concentrated development (Zavadskas et al. 2009; Viteikiene and Zavadskas 2007; Burinskiene 2009; Jakaitis et al. 2009). It allows using the existing infrastructure and abandoned urban territories. Such planning also reduces the amount of used land and creates a lasting environment, the immensely dense population of which is not always able to function properly (Burinskiene and Rudzkiene 2009; Lahdenpera 2009; Petrovic et al. 2009; Greater London Authority 2003; Ribeiro 2008; Lindgren and Castell 2008). On one hand, it is a natural stage related to the renovation of neglected valuable urban areas. On the other hand, the course and outcomes at this stage reveal gaps within the renewal process. We are inclined to blame the drawbacks of laws regulating urban planning and protection of visual identity (investors cannot always be expected to abandon their self-centred ends for the sake of urban values, etc.) (Dringelis 2005; Vrubliauskas 2005; Jakaitis 2004; Mickaityte et al. 2008; Banaitis and Banaitiene 2007; Majamaa et al. 2008). This is in a large part influenced by a confusing, non-effective system for the coordination of constructions with government institutions and the public. The regulation of constructions is confusing; builders breach the introduced requirements; officials are frequently provided with the right to easily choose the requirements necessary to be applied. An inappropriate distribution of functions among government institutions and private subjects raise a number of problems (Sostak and Kutut 2009). One of the outcomes of inappropriate legal regulation is the violation of the third-party rights (i.e. the parties not directly related to the investment construction process: the owners of neighbouring plots, users, communities of residential districts, etc.). The article analyses the influence of third-party rights infringed during construction planning on the implementation of an investment project.

The development of the national economy is impossible without construction: people use construction products--various buildings--to live, work and satisfy other social needs. Construction investment contributes to national economic growth and development extensively (Urbanavieiene et al. 2009; Zavadskas and Kaklauskas 2005, 2008). The investment process in construction is long and complicated; it requires enormous financial, intellectual and other resources. If judicial disputes occur during this process, the investor may incur huge loss, and project implementation may be postponed for an indefinite term. Litigation may continue for several years (The judgement of the Supreme Administrative Court of Lithuania of 19 January 2007 in the administrative case; The judgement of the Supreme Administrative Court of Lithuania of 26 January 2007 in the administrative case). Thus, investors are most concerned to avoid any legal disputes and should pay considerable attention to their prevention.

Violations of third-party rights are of benefit neither to third parties, nor to the parties of the investment process, because, on one hand, such violations might wrongfully cause the deterioration of the conditions for life and other activities of third persons. On the other hand, violations of third-party rights at the stage of construction planning may affect the implementation of the investment project, because all solutions violating third-party rights also violate the provisions of legal acts and can be disputed as stipulated by the Law on Administrative Proceedings (hereinafter LAP 2000), the Law on Territorial Planning (2004) and other legal acts (Mitkus and Sostak 2009).

In a construction project, judicial disputes are an unwanted risk factor, which may disrupt the entire project. It is therefore necessary to plan and apply preventive measures for the mitigation of such risk at the initial planning stage of a construction project. To evaluate and eliminate these risk factors, state-of-the-art technologies for construction project planning and management must be integrated into each step of construction project planning and implementation. It is necessary to employ innovative methods for construction project planning and implementation when the conditions are indeterminate (Kahraman and Kaya 2010; Blaszczyk and Nowak 2009). Risk management strategies and the development of a risk management plan must be improved, risk analysis methods and technologies must be used, and the risk reporting mechanism must be implemented. For a successful construction project, it is worth to employ the functions of project management. It is necessary to analyse the risk using the knowledge of relevant experts and to properly evaluate the scope of possible negative effects and their outcomes to the construction project. The findings should influence the subsequent decision-making process. Risks must be monitored and decision-making must be analysed throughout the project lifecycle. Before launching a project, an investor must be ready for any "surprises". Forecasting is the most important part of any strategy, because the actions recommended for certain situations stem from the forecasts of possible outcomes. Thus, investors must be aware of the defence procedures taking place in administrative courts when third-party rights are violated during territorial planning--they must assess possible actions of third parties.

A peace treaty can be signed at any stage. A compromise is achieved in such case and further litigation is avoided (Mitkus and Sostak 2008a).

If a judicial dispute occurs when the construction project is already launched, the investor must also consider all possible actions of judicial institutions. The lessons learned about risk management during the implementation of construction projects should be used in future projects (Zavadskas et al. 2010; Park et al. 2009; Yang et al. 2009; Antucheviciene et al. 2010). Mathematical modelling of the problem in question and the selection of a proper method for optimisation help with determining the optimal investor's behaviour strategy that would allow expecting a certain average profit irrespective of the strategies of third parties and decisions of judicial institutions. Our research employs the mathematical model for stochastic dynamic programming.

2. Mathematical Modelling of a Dispute between Investors and Third Parties on Allegedly Violated Third-party rights with the Help of Stochastic Dynamic Programming

The analysis of the procedure related to the defence of violated third-party rights in administrative courts leads to a conclusion that a judicial dispute may either ruin a construction investment project completely or to cut the expected profits considerably. Largely, it depends on the decisions of the interested communities (third persons) that object to the construction and on the decisions of judicial institutions hearing the disputes. Naturally, investors are most interested to avoid any legal disputes. A possible preventive measure to mitigate such risk is an assessment and proper analysis of all possible future events related to the occurrence of such risk before the investment project is launched. For that purpose, the investor must come up with the scenarios of actions in possible situations and to plan strategic options. The investor, which is most interested to avoid any legal disputes, should assess all possible risk factors that may affect the implementation of a construction project. To illustrate such assessment, we shall turn to mathematical modelling of a dispute between investors and third parties on allegedly violated third-party rights. The dispute between investors and third parties on possibly infringed third-party rights was modelled by creating a tree of the behaviour variants of dispute parties (Fig. 1).

[FIGURE 1 OMITTED]

Besides, the tree of variants helps in finding mistakes made afterwards and in correcting them (Mitkus and Sostak 2008b; Mitkus 2004; Nollke 2007; Ross Quinlan 1993). The tree of the behaviour variants of dispute parties in Fig. 1 models all possible actions of third persons, judicial institutions and the investor in a certain situation. The outcomes of their actions are assessed.

Dynamic programming will be used to find an optimal behaviour strategy for an investor. Dynamic programming is a method of calculation applied in a solution to the multi-stage problems of optimisation. It means that we need to break a complex problem of optimisation into a string of simpler problems. When these problems are solved, it is easy to find an answer to an original problem. Problems are divided following the Belman's Principle of Optimality: an optimal solution (management) has the property that whatever the initial state and initial solution are, the remaining solutions must constitute an optimal policy with regard to the state resulting from the first solutions. Note that by a dynamic problem we usually mean any process which depends on time (in our case, the court proceedings related to the defence of the infringed rights depend on time) (Ciocys and Jasilionis 1990; Taha 1997).

For further modelling, we shall look at the tree of the behaviour variants of dispute parties and make a tree of behaviour strategies for the investor (Fig. 2). If we want to solve the tree of behaviour strategies for the investor mathematically - to perform mathematical modelling--we need numerical values for our specific research case. The values are shown in Table 1. The numerical values are based on actual cases brought to Lithuanian courts (The judgement of the Supreme Administrative Court of Lithuania of 20 February 2006 in the administrative case; The judgement of the Supreme Administrative Court of Lithuania of 19 January 2007 in the administrative case; The judgement of the Supreme Administrative Court of Lithuania of 26 January 2007 in the administrative case). We base our research on a general (abstract) model. The analysis of specific dispute cases in the future, however, could use corrected values and assess all individual aspects related to the conflict.

[FIGURE 2 OMITTED]

3. The Model for Stochastic Dynamic Programming

Let f ([S.sub.i]) be the likely (expected) average investor's profit ensured by state [S.sub.i] and optimal strategy [x.sub.i]--selected from the set of possible strategies in this situation. The Belman's Principle of Optimality gives us recurrent equations for all situations [S.sub.i] :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is the pool of investor's original strategies in the situation (state) [S.sub.i]. Here [[m.sub.i].summation over (j=1)] [x.sub.ij] = 1 and

[x.sub.ij] [greater than or equal to] 0, j = 1, 2, ... , mi.

When the number of possible strategies is [m.sub.i], breaking the set of strategies using Index 2 satisfies equation [J.sub.is] [union] [J.sub.ib] = {1, 2, ..., [m.sub.i]}.

Generally, the intersection of index sets [J.sub.is] and [J.sub.ib] is a non-empty set, which means that the same strategy can lead to the final state or to a situation when the investor must chose a strategy again.

When j [member of] [J.sub.is], strategy [X.sub.ij] may lead the investor to situation [S.sub.ijk] (probability [p.sub.ijk]). The number of situations enabled by the investor's choice of strategy [x.sub.ij] is marked as [n.sub.ij].

When j [member of] [J.sub.ib] , strategy [x.sub.ij] leads the investor (probability [P.sub.ijk]) to the final state with profit [P.sub.jb].

f([S.sub.ijk]) is the likely (expected) average profit ensured by state [S.sub.ijk] and the optimal strategy selected from the pool of possible strategies in this state.

In our model for stochastic dynamic programming, probabilities [p.sub.ijk] and [p.sub.ijb] depend on the probability of decisions made by other institutions.

We shall proceed with the analysis of possible behaviour strategies for an investor that invests into a construction project, seeks maximum profits but faces the opposition of the community. We shall also look at the problem in which optimal strategies are determined using multi-stage optimisation--dynamic programming.

Here, probabilities [p.sub.ijk] and [p.sub.ijb] depend on the probability of certain decisions made by the community opposing the construction and judicial institutions hearing the disputes.

Let us define possible judicial situations.

Let [T.sub.k] be possible judicial states, k = [bar.1.7] (in our research, there are seven of these states).

In each state [T.sub.k], courts may make one of two decisions: [A.sub.k] or [B.sub.k].

Let the probabilities of events [A.sub.k] and [B.sub.k] be [q.sub.kl] = P([A.sub.k]) and [q.sub.k2] = P([B.sub.k]), [q.sub.kl] + [q.sub.k2] = 1 [q.sub.kl] [greater than or equal to] [q.sub.k2] [greater than or equal to] 0.

The first judicial state [T.sub.1] is possible if breaches are determined in planning and implementing construction investment projects and if the interested community applies with a claim to an advance institution for out-of-court case hearing. There are two possible events in such situation:

1) [A.sub.1]: the advance hearing institution rejects the claim (determines that the solutions of the construction investment project do not violate rights of the interested community).

2) [B.sub.1]: the advance hearing institution satisfies the claim.

Let us assess the probabilities of the events:

[q.sub.11] = P([A.sub.1]) = 0.40 , [q.sub.12] = P([B.sub.1]) = 0.60.

The second judicial state [T.sub.2] is possible if the advance hearing institution rejects the claim of the interested community and the interested community applies to a court of first instance. There are two possible events in such situation:

1) [A.sub.2] : the claim of the interested community is rejected.

2) [B.sub.2] : the claim of the interested community is satisfied.

Let us assess the probabilities of the events:

[q.sub.21] = P([A.sub.2]) = 0.50, [q.sub.22] = P([B.sub.2]) = 0.70.

The third judicial state [T.sub.3] is possible if the court of first instance rejects the claim of the interested community and the interested community applies to a court of appeal. There are two possible events in such situation:

1) [A.sub.3]: the claim of the interested community is rejected and the investor makes profit [P.sub.7].

2) [B.sub.3] : the claim of the interested community is satisfied and the cancellation of the solutions related to the construction investment project is initiated--the investor suffers loss N.

Let us assess the probabilities of the events:

[q.sub.31] = P([A.sub.3]) = 0.50, [q.sub.32] = P([B.sub.3]) = 0.50.

The judicial state [T.sub.4] is possible if the court of first instance satisfies the claim of the interested community and the investor applies to a court of appeal. There are two possible events in such situation:

1) [A.sub.4] : the investor's claim is rejected and the cancellation of the solutions is initiated--the investor suffers loss N.

2) [B.sub.4]: the investor's claim is satisfied and the investor makes profit [P.sub.8].

Let us assess the probabilities of the events: [q.sub.41] = P([A.sub.4]) = 0.50, [q.sub.42] = P([B.sub.4]) = 0.50.

The fifth judicial state [T.sub.5] is possible if the advance institution for out-of-court case hearing satisfies the claim of the interested community and the investor applies to a court of first instance. There are two possible events in such situation:

1) [A.sub.5]: the investor's claim is satisfied;

2) [B.sub.5]: the investor's claim is rejected.

Let us assess the probabilities of the events: [q.sub.51] = P([A.sub.5]) = 0.50 , [q.sub.52] = P(B5) = 0.50 .

The sixth judicial state [T.sub.6] is possible if the court of first instance rejects the investor's claim and the investor applies to a court of appeal. There are two possible events in such situation:

1) [A.sub.6] : the investor's claim is rejected and the cancellation of the solutions is initiated--the investor suffers loss N.

2) [B.sub.6] : the investor's claim is satisfied and the investor makes profit [P.sub.10].

Let us assess the probabilities of the events: [q.sub.61] = P([A.sub.6]) = 0.50, [q.sub.62] = P([B.sub.6]) = 0.50.

The seventh judicial state T7 is possible if the court of first instance satisfies the investor's claim and the interested community applies to a court of appeal. There are two possible events in such situation:

1) [A.sub.7] : the claim of the interested community is satisfied and the cancellation of the solutions is initiated--the investor suffers loss N.

2) [B.sub.7] : the claim of the interested community is rejected and the investor makes profit [P.sub.12].

Let us assess the probabilities of the events:

[q.sub.71] = P([A.sub.7]) = 0.50, [q.sub.72] = P([B.sub.7]) = 0.50.

Let [V.sub.j] be the possible states of the interested community (situations when the community decides), j = [bar.1,5] (in our research, there are five of these states).

The interested community may act in two different ways in each state: either to refrain from applying to a judicial institution (event [C.sub.j]) or to apply (event [D.sub.j]).

There are respective probabilities [p.sub.j1] and [p.sub.j2], where [p.sub.j1] + [p.sub.j2] = 1, p.sub.j1] [greater than or equal to] [p.sub.j2] [greater than or equal to] 0.

The first state of interested community [V.sub.1] is possible if the investment solution violates rights. There are two possible events in such case:

1) [C.sub.1] : the interested community fails to see the violations in the investment solution or fails to submit its suggestions or objections before the deadline, thus the investor makes profit P.

2) [D.sub.1] : the interested community determines the violations in the investment solution and submits its suggestions and/or objections before the deadline.

Let us assess the probabilities of the events:

[p.sub.11] = P([C.sub.1]) = 0.80, [p.sub.12] = P([D.sub.1]) = 0.20.

The second state of interested community [V.sub.2] is possible if the investor rejects the suggestions submitted by the interested community regarding the violations in the investment solution. There are two possible events in such case:

1) [C.sub.2] : the interested community does not apply to an advance institution for out-of-court dispute hearing and the investor makes profit [P.sub.3].

2) [D.sub.2] : the interested community applies to an advance institution for out-of-court dispute hearing.

Let us assess the probabilities of the events:

[p.sub.21] = P([C.sub.2]) = 0.30, [p.sub.22] = P([D.sub.2]) = 0.70.

The third state of interested community [V.sub.3] is possible if the advance institution for out-of-court dispute hearing rejects the claim of the interested community. There are two possible events in such case:

1) [C.sub.3] : the interested community does not apply to a court of first instance and the investor makes profit [P.sub.4].

2) [D.sub.3] : the interested community applies to a court of first instance.

Let us assess the probabilities of the events:

[p.sub.31] = P([C.sub.3]) = 0.25 , [p.sub.32] = P([D.sub.3]) = 0.75.

The fourth state of interested community [V.sub.4] is possible if the court of first instance rejects the claim of the interested community. There are two possible events in such case:

1) [C.sub.4] : the interested community does not apply to a court of appeal and the investor makes profit [P.sub.6].

2) [D.sub.4] : the interested community applies to the court of appeal.

Let us assess the probabilities of the events:

[p.sub.4l] = P([C.sub.4]) = 0.15, [p.sub.42] = P([D.sub.4]) = 0.85.

The fifth state of interested community [V.sub.5] is possible if the court of first instance satisfies the investor's claim. There are two possible events in such case:

1) [C.sub.5] : the interested community does not apply to a court of appeal and the investor makes profit [P.sub.11].

2) [D.sub.5] : the interested community applies to a court of appeal.

Let us assess the probabilities of the events:

[P.sub.51] = P([C.sub.5]) = 0.45, [p.sub.52] = P([D.sub.5]) = 0.55.

Let [S.sub.i] be the possible states of the investor--the situations when the investor decides, i = [bar.0.5] (in our research, there are six of these states).

In each state, the investor can choose from either two or three behaviour strategies. Mixed behaviour strategies are also possible, when each original strategy has a probability assigned:

[x.sub.i1] is the probability the first strategy [s.sub.i1] will be selected; is the probability the second strategy [s.sub.i2] will be selected; [x.sub.i3] is the probability the third strategy [s.sub.i3] will be selected. [x.sub.i1] + [x.sub.i2] + [x.sub.i3] = 1, [x.sub.il] [greater than or equal to] 0, l = 1,2,3.

[S.sub.0] is the zero state of the investor. In this state, the investor contemplates whether the investment project is worth launching.

[S.sub.1] is the first state of the investor. If violations are de termined, the investor has three strategies to choose from:

[x.sub.11] : to accept the suggestions of the interested community and to make profit [P.sub.1].

[x.sub.12] : to sign a peace treaty and to make profit [P.sub.2].

[x.sub.13] : to reject the suggestions of the interested community.

[S.sub.2] is the second state of the investor. If the interested

community applies to a court of first instance, the investor has two strategies to choose from:

[x.sub.21] : to sign a peace treaty with the interested community and to make profit P5.

[x.sub.22] : to reject the peace treaty.

[S.sub.3] is the third state of the investor. If the court of first instance satisfies the claim of the interested community, the investor has two strategies to choose from:

[x.sub.31] : to refrain from an application to a court of appeal and to suffer loss N.

[x.sub.32] : to apply to a court of appeal.

[S.sub.4] is the fourth state of the investor. If the advance institution for out-of-court dispute hearing satisfies the claim of the interested community after the hearing, the investor has three strategies to choose from:

[x.sub.41] : to refrain from applying to a court of first instance and to suffer loss N.

[x.sub.42] : to sign a peace treaty with the interested community and to make profit [P.sub.9].

[x.sub.43] : to apply to a court of first instance.

[S.sub.5] is the fifth state of the investor. If the court of first instance rejects the investor's claim, the investor has two strategies to choose from:

[x.sub.51] : to refrain from applying to a court of appeal and to suffer loss N.

[x.sub.52] : to apply to a court of appeal.

We shall proceed with further mathematical modelling and, using the data from our graph (Fig. 2), shall come up with the recurrent equations (2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

In each state, we need to solve the problem of linear optimisation:

max([c.sub.1][x.sub.i1] + [c.sub.2][x.sub.i2] + ... + [c.sub.mi] [x.sub.imi]), [[m.sub.i].summation over (j=1)] [x.sub.ij] = 1, [x.sub.ij] [greater than or equal to] 0,

j = 1, 2, ..., [m.sub.i], in which one of the optimal plans is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, we can replace the recurrent equations with simplified versions (3):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

In order to determine optimal behaviour strategies for the investor, a programme code for a solution to the recurrent equations (3) has been developed in the EXEL environment (see Table 2). Let us analyse calculations in question.

[FIGURE 3 OMITTED]

We solve the recurrent equations to find the specific values of profit in the final state and to determine specific probabilities that the community and judicial institutions will take one or another action. These values are the expected average profit for each state (situation) f ([S.sub.i]) and the optimal situation management marked as [x.sup.*.sub.t]. Obviously, original strategies are optimal for each state; their probability is equal to one.

Optimal investor's strategies determined using our calculations are shown in Fig. 3. A broad analysis of determining the dependency of the solutions on the parameters is possible. If, for instance, the size of loss N in the final situation varies between 1 and 1000, the optimal investor's behaviour remains the same, only values f ([S.sub.5]) and f ([S.sub.3]) change (1,000 monetary units were used as a measuring unit throughout research).

Further research should focus on the analysis of the investor's possibilities of choosing only the projects that would trigger the most positive reactions of third persons with increasing f([S.sub.1]) values.

Therefore, future research should consider the inclusion of several trees of the behaviour variants of dispute parties with the same starting point So and equivalent to the tree used in this research. An investor, when in state So, could select an optimal project based on the same Belman's Principle (see Fig. 3).

4. Conclusions

1. Violations of third-party rights are of benefit neither to third persons, nor to the parties of the construction investment process, because, on one hand, such violations might wrongfully cause the deterioration of conditions for life and other activities of third persons. On the other hand, violations of third-party rights at the stage of construction planning may affect the implementation of the investment project, because all solutions violating third-party rights also violate the provisions of legal acts and can be disputed as stipulated by the Law on Administrative Proceedings (LAP), the Law on Territorial Planning and other legal acts.

2. Investors may incur, and do incur, huge losses when solving disputes on the infringement of third-party rights.

3. In order to make the relations between investors and third parties more rational, a mathematical model of a dispute on allegedly infringed third-party rights has been developed. It helps with determining optimal investor's strategies for each situation of decision-making and thus ensures a certain average profit to the investor irrespective of the strategies chosen by third persons if the probabilities of selecting these strategies are known.

4. The mathematical model for stochastic dynamic programming (EXEL programme code for recurrent equations (3) is used) enables a broad analysis of the dependencies between the optimal investor's strategy and the probabilities that third parties will select a certain strategy. It also helps in analyzing the possible numerical values of profit (or loss).

doi: 10.3846/13923730.2011.560628

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Olga Regina Sostak (1), Sigute Vakriniene (2)

Department of Law, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania

E-mails: (1) Olga-Regina.Sostak@vgtu.lt (corresponding author); (2) Sigute.Vakriniene@vgtu.lt

Received 14 Jun. 2010; accepted 31 Jan. 2011

Olga Regina SOSTAK. A PhD student at the Department of Law, Vilnius Gedminas Technical University (VGTU). Sauletekio al. 11, LT-10223 Vilnius, Lithuania. MA in Civil Engineering (2006, Vilnius Gediminas Technical University). Research interests: land planning, construction investment, decision making, stochastic programming.

Sigute VAKRINIENE. A graduate in mathematics from Vilnius University (1963). Doctor (1972). Associate Professor at the Faculty of Fundamental Science, the Department of Mathematical Statistic, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania. Research interests: operation research-game theory, stochastic programming.
```Table 1. A numerical expression of behaviour strategies for the
investor shown in the tree

No.   PROFIT               LTL                %

1     P = P3 = P4 =      LTL 1m *            100%
2     P1 =               LTL 500,000 **      50%
3     P2 = P5 = P9 =     LTL 950,000 ***     95%
4     P6 = P11 = P7 =    LTL 980,000 ****    98%
5     P8 = P10 = P12 =   LTL 950,000 *****   95%
6     N =                LTL 1 mi ******     -100%

* successful completion of the construction investment project;

** changed project solutions cut the profits by LTL 500,000;

*** a peace treaty is signed with the interested community,
profit decreases by LTL 50,000;

**** the annual litigation expenditures make up LTL 10,000
(2 years), thus profit decreases by LTL 20,000;

***** the annual litigation expenditures make up LTL 10,000
(3 years), plus LTL 20,000 for forensic examinations; thus
profit decreases by LTL 50,000;

****** the construction investment project is cancelled.

Table 2. Optimal investor's strategies calculated using the
programme code developed in the EXEL environment

Situation     f ([S.sub.5])    [x.sup.*.sub.5]

[S.sub.5]          -25         [x.sub.52] = 1

Situation     f ([S.sub.3])    [x.sup.*.sub.3]

[S.sub.3]          -25         [x.sub.32] = 1

Situation     f ([S.sub.4])    [x.sup.*.sub.4]

[S.sub.4]          950         [x.sub.42] = 1

Situation     f ([S.sub.2])    [x.sup.*.sub.2]

[S.sub.2]          950         [x.sub.21] = 1

Situation     f ([S.sub.1])    [x.sup.*.sub.1]

[S.sub.1]         968.5        [x.sub.13] = 1```