The object of this study is a long-term capital-intensive investment megaproject with many participants and high risks of implementing their projects. The length of procedures for harmonizing investment projects of participants and the di culty of coordinating their actions to achieve the megaproject's goals necessitate the creation of an adequate resource management tool. A step-by-step algorithm is proposed. The algorithm was tested on real economic information of the East Siberian Oil and Gas Complex megaproject and showed its ability to work. The general applicability of the algorithm is demonstrated.
and the upper boundary bij (pessimistic time) of the execution time are given. The works of the megaproject Gij are grouped according to the projects Gk, 1 k n ,where n- is the number of projects in the model. Each of the Gk projects has a deadline Dk of its end, as well as the minimum and maximum permissible probability of completion of the project during this period (P and P accordingly).
The main idea of the problem of allocating resources between n projects is to increase the probability of their completion in the target time frame Dk for given initial volumes of the megaproject Gk investment resources C. If any of the projects Gk at the time t 0 can not be completed within the prescriptive time with an acceptable probability, then a redistribution of the remaining investment n resources P Ck(t) between the projects of Gk.
k=1 2
Formal Description of the Network Model We introduce the notation: C - the initial volume of investment resources of the megaproject for the implementation of all projects; Ckt - investment resources (budget) allocated to the k-th project at the moment t 0, Ck0 = Ck; Tk(Ckt) -the random duration of the k-th project on the basis of the budget allocated to it Ckt; Dk - the target date for the implementation of the k-th project; (i; j)k 2 Gk - work(i; j) included in the k-th project; cijk - budget allocated for the implementation of work (i; j)k of the project Gk; cijk min -the minimum budget value, allowing to perform the work (i; j)k; cijk max -the maximum budget value, allowing to perform the work (i; j)k; k - priority coe cient (degree of importance) of the k-th project.
As the objective function, the sum of the products of the priority project
coe cients and the probabilities of their completion in the appropriate legislative
terms is used. It is necessary to determine the values at which the objective
function is maximal:
under conditions
where
C
(
Dk) Pk
Pk(Ckt) n X Ckt = k=1
Pk ; 1 n X Ck(t) k=1 k n n P Ck - initial volume of investment resources of the megaproject for k=1 realization of all n projects; Pk(Ckt) = P (t + Tk(Ckt) Dk) - the probability of completion of the k-th project on time Dk with the allocated budget Ckt; Pk = Pk(Ckt) - the probability of completion of the k-th project on time Dk with the allocated investments Ckt; Pk = Pk(Ckt ) - the probability of completion of the k-th project on time Dk with the allocated investments Ckt ; Ck(t) - the remaining unused investment resources for the k-th project at time t 0.
The proposed algorithm for solving the problem (
The traditional PERT method uses only the activity time means to calculate the critical path, reducing the stochastic model to a deterministic model. In PERT, three time estimates are required for each activity. The time estimates represent a pessimistic time, an optimistic time, and a most likely time for duration of the activity. The method assumes that the sum of the mean completion times of activities on the critical path is normally distributed. This allows the calculation of the probability of completing the project within a given time period. A single critical path is thus calculated and relied upon, where in reality, there may be numerous possible critical paths that exist. For a large network plan the probability that any given path could be the critical path may be very small.
PERT method yields results which are biased high. If network has multiple parallel paths with relatively equal means, PERT calculations will be considerably biased [6]. As a result, the time to complete a project calculated by the traditional PERT method is almost always too short [11].
In accordance with the PERT method, the solution of problem (
Resolving the Problem of Resource Allocation for Network Projects 3.1
Approval For all projects Gk, 1 k n, we solve the auxiliary problem with p = Pk and p = Pk [2]. As a result, we obtain the values of Pk , Pk and the budget values Ckt and Ckt . Let us write the dependent between Pk(Ckt) and Ckt : Pk(Ckt)
Pk
Pk
Pk =
Ckt
Ckt and expressing Pk(Ckt), we substitute in the objective function in the form n X k=1 We denote by
Pk Ckt Pk Ckt
Pk Ck Pk Ck k Ckt + k = k;
Pk Pk
Ckt
Ckt Ckt Ckt
Pk
Ck Pk Ck
Ckt Ckt k
! max
k =
k
and we transform the objective function of the original model (
Algorithm for Solving the Main Problem
The problem (
At the rst step, we allocate for each remaining un nished project the corresponding minimum budgetary volume Ckt . Let's designate the remaining budget Ct. We order the sequence f kg in decreasing order. We denote the new ordinal numbers of this sequence by symbols f1; : : : ; fn. In the second step, we set j = 1 and compute j = minf(C fj;t C fj;t); Ctg. At the third step for the project, we determine its nal budget C fj;t. We are correcting the remaining budget Ct in the fourth step.
If Ct = 0, then nish the calculation. Otherwise, then j = j + 1 and the condition j n is satis ed, then we continue the calculations with j . Until we completely disassemble all the cases.
The optimality of the algorithm follows from the presence of a monotonically decreasing sequence f fj g, and also from the allocation to each next project fj at the step of the maximum possible additional budget j from the Ct budget left at the disposal of the company. 3.3
The Formulation and Algorithm for Solving the Auxiliary
Problem
Let consider an auxiliary problem for the case of one project:
for the given "reliability" p of the project G, 0 < p < 1, nd the minimum value
(
X cij fi;jg P (T (C)
D) = p;
cij min
cij
cij max
with constraints
In the rst step, we determine the values of C1, C2 and C3. We solve problem
(
As a result of solving this problem, we obtain the project reliability values p1, p2 and p3. Compare the values of p1 and p2: if 8" 0, p1 p2 < ", then nish the execution of the algorithm, otherwise go to the next step. Analyzing the inequality p1 < p < p3, if it is satis ed, then go to step 2, otherwise to step 3. At the nal stage of the algorithm, the value C = C3 is the minimum budget that ensures the "reliability" p of this project. 4
Implementation of the Algorithm To solve the problem, we developed a step-by-step algorithm that is implemented using C ++ methods in the programming environment of Microsoft Visual Studio. The algorithm was tested on real economic information and showed its ability to work. The distribution of investment resources for the construction of the seven sectors of the East Siberia-Paci c Ocean (ESPO) oil pipeline, which is the basis for the formation of the ESOGC megaproject was considered. 5
The network graph of the construction and operation the ESPO pipeline is presented in the form of projects the construction of seven pipeline sections 1.11.7 (Fig. 1). Each of pipeline sections is described by the uni ed module of the technological sequence of work and directive events. Triangles denote the uni ed network modules of sections of the oil pipeline (1.1 Taishet - Ust-Kut, 1.2 Lensk - Ust-Kut, 1.3 Lensk - Olekminsk, 1.4 Olekminsk - Aldan, 1.5 Aldan-Tynda, 1.6 Tynda - Skovorodino, 1.7 Skovorodino - Nakhodka).
The uni ed module of the pipeline is given in Fig. 2. The uni ed pipeline
module re ects ve stages of the project: 1. development of feasibility study
(feasibility study) - work (
The universal method presented in this paper was used to calculate a more realistic duration for the construction of the seven sectors East Siberia-Paci c Ocean oil pipeline. Basic input data and calculated results of the resource allocation problem for network projects are presented in Table 1.
The initial volumes of investment resources of the megaproject were adopted in the amount of 11.861 bn rubles. The optimal allocation of resources for the pipeline section projects is determined on condition that the project are completed within the deadlines. The work schedule and the dynamics of investment distribution have been constructed. The problem of allocating investment resources between the pipeline construction projects is being solved. This problem is to increase the probability of completion of projects in the target dates for initial volumes of resources. To solve the problem, a step-by-step algorithm is proposed. Based on the technological sequence of the pipeline construction, an appropriate network schedule was formed. The volume of investments for each section of the pipeline is calculated.
Given the initial volumes of investment resources of the megaproject in the amount of 11.861 bn rubles. The optimal allocation of resources between the pipeline sections is determined, provided that they are completed within the target dates. The schedule of work execution and the dynamics of investment distribution have been constructed.
The obtained results show the e ective of the proposed method to calculate the resource allocation between subprojects and realistic completion time for speci ed project implementation deadlines. The suggested method can be recommended for use by project managers in order to allocate resources and prevent possible abandonment of project completion deadlines.