На рассматриваемом множестве приоритетных проектов после нахождения решения выпуклой задачи распределения ресурсов в соответствии с общим объёмом ассигнований , достигнутой вероятностной оценкой реализуемости вместе с экспертными приоритетами Adv Syst Sci Appl 2019; 04; 66-78 Published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/827 A Planning and Scheduling Method for Large-Scale Innovation Projects Vladimir Topka*, Anatoliy D. Tsvirkun Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Rus- sia E-mail: topka3@mail.ru, tsvirkun@ipu.ru Received May 16, 2019; Revised December 12, 2019; Published December 31, 2019 Abstract. We consider the problem of uniform allocation according to the minimax criterion of a non-renewable resource of an innovative project's activities. A lexicographic method is used for ordering these minimax criteria To solve the problem - one objective function - developed a greedy heuristic for finding the maximum path on an acyclic digraph with double weights on arcs. The greedy heuristic gives approximate solution of the problem mentioned. The obtained solution allows building a project resources plan and schedule with four types of temporary slacks with inaccurate initial data. The proposed approach allows performing an analysis of the project Risk of Performance parameter, for which has not developed the apparatus for its analy- sis. These include the risks that the project when complete fails to perform as intended or fails to meet the mission or business requirements that generated the justification for the project. Perfor- mance risks are estimated on the probability distribution function of a favorable/unfavorable out- come. Proposed lexicographic disintegration is a decomposition procedure and is suitable for planning and scheduling large-scale projects. Practical calculations for the convex problem in the initial statement may be performed with the help of ready-made software, and the obtained greedy solution of the problem-consequence has theoretical significance in graph theory. Keywords: large-scale innovative project, planning, scheduling, reliability index of project activi- ties, greedy algorithm, the maximum path with double weights on arcs. 1. INTRODUCTION In existing project management software systems, some modules perform Risk of Cost and Risk of Schedule assessment. On the other hand, an approach based on the concepts of the theory of reliability can be proposed for modeling uncertainty and risk of innovative projects. In the theory of reliability, the indicator of network reliability is determined by the probability of its connectivity. This requires that at least one path is found connecting the beginning and the end of the network. However, in managing projects in the PERT network, in order to ensure reliable execution of the project, it is necessary to perform all activities included in the project. In this case, the task is to assess not only the connectivity of the be- ginning and the end of the network, but also the completeness of the whole project. In present project management software, a quantitative assessment of Risk of Cost and Risk of Schedule is made on the basis of Monte Carlo simulation. The proposed approach allows one the analysis of the Risk of Performance [1] of a pro- ject for which the apparatus of its analysis has not been developed. The execution risk, upon completion, accomplish the required mission - to achieve the specified technical characteris- tics, is based on the probability distribution function of a favorable/unfavorable outcome. For her: the probability of success of the project is the probability that the execution of this pro- ject will not fail, and the probability of reliable execution is taken as an index of the reliabil- * Corresponding author: topka3@mail.ru mailto:tsvirkun@ipu.ru A PLANNING AND SCHEDULING METHOD FOR LARGE-SCALE INNOVATION PROJECTS 67 Copyright ©2019 ASSA Adv. in Systems Science and Appl. (2019) ity of implementation. The advantage of the proposed approach is that it provides a tool for working in an unexplored area. For activities of innovative projects with a high degree of uncertainty, to obtain the characteristics of random variables of their implementation - the likelihood of achieving the goal or probability of technical success, that is, to build the distribution function of such a random variable, you can use empirical data or data obtained as a result of an active experi- ment in which the Monte Carlo method obtained the probability distribution function of the technical success of the activity, the degree of achievement of the specified performance characteristics of this development, in the form of a monotonously increasing with the satu- ration of the function of expended resources [2]. The likelihood of performing the activity by a particular time characterizes the possibil- ity of attaining specific technical indicators of the result of the activity, provided that the previous activity, ensuring its beginning, has been completed. This probability depends on the amount of those or other resources spent, which have a value expression as cost. The approach developed in the article is aimed at developing quantitative methods for assessing and optimizing the project parameters, taking into account the criterion of uniform allocation of a non-renewable (stored) resource with a restriction on the assessment of pro- ject reliability index. 2. INDICATORS OF THE RELIABILITY OF ACTIVITIES AND THE PROJECT AS A WHOLE Previously, integer, linear and some non-linear models were used to describe innovative pro- jects execution [3]. We will model [4] the dependence of the probability of technical success (reliability index) of the project’s activities on the homogeneous non-renewable resource spent by 2-parameter power-concave functions of the form     α 0 1/ α 0 ( ) ε, 1 ,ε 0, ε, , 1,..., , j j j j j j j j u p u u u u j n          (2.1) where 0 α 1j  is the form parameter, 0 ju > 0 is the scale parameter. Which at the late phas- es of an innovation project, when it is advisable to use quantitative methods to assess the un- certainty of the project, quite visually and precisely approximates the practical [2; 5] func- tions of the activities reliability of the innovation project. Let an acyclic directed graph G (E, Γ) be given, where E is the set of vertices corre- sponding to the project’s activities (Activity-on-Node representation), and Г is the set arcs defining partial order relations of direct technological activity precedence. The network G (E, Γ) has one final and one initial vertex; n - final vertex - the last number among all the vertices of the project and the dummy vertex 0, which means the launch of the project. Thus, we have a network G (E, Γ), at the vertices of which a development reliability index is given that satisfies the relation (2.1). We will set the reliability of the project by the probability of its technical success (the degree of achievement of the specified performance characteristics of the project) by a certain allowable period, the assessment of the reliability of the project in the most unfavorable case. In the theory of reliability, such an assessment is determined by the weakest link, i.e. the worst of the technological chains from the vertex of the lower level to the final vertex. 68 V. TOPKA, A. D. TSVIRKUN Copyright ©2019 ASSA Adv. in Systems Science and Appl. (2019)   α 0μ ( , ) μ 0 0 ( ) min ε, 1 , ε 0, α (0,1), [ε , 1], 0; μ , j i i j n jk j k j j jk j i u P u w u w u          (2.2) where 0 0 [ε , 1], ε 0jkw   is arc (j, k) reliability coefficient, (from the matrix of paths along the arcs of the network G (E, Г)) transfer the result of the j-th activity by the arc (j, k) to per- form the k-th activity (like the transfer function of connecting links in theory of automatic control). We denoteμ i the path with the number 1,...,i m on the network from the initial vertex to the final vertex n. When solving problems of project planning in terms of a power model [6], the time variable is not taken into account. In this article we will consider a deterministic network model of the project with disjunctive input arcs OR, or with conjunctive input arcs AND having the property that for the project reliability indicator is performed assessment of lower bound (2) [6]. Thus, we will speak about pi from (2.1) as a reliability index of activity, and about Pn from (2.2) - as an assessment of indicator of guaranteed reliability of project per- formance. In the model for representing network constraints, we will use the matrix of paths (along arcs) of the network G (E, Г): which is based on the list of arcs of the network as the characteristic function of the constraint system. In the network model of the project G (E, Г), each path from the initial vertex to the final one is uniquely represented by a line of a spe- cially constructed matrix of paths (along arcs)    , , μj k i l j k  , in which its elements  arcs stand in the i-th line only if (j, k) - arc belongs to i -th path. All other cases when  , μ i j k  there are empty cells. 3. THE PROBLEM OF SUCCESSIVE UNIFORM ALLOCATION OF NON- RENEWABLE RESOURCE BY CHEBYSHEV CRITERION Let  1 2u= , , ..., nu u u be the vector of a non-renewable resource allocated for the execu- tion of all activities of the project G (E, Г). As an objective function for the allocation of re- sources, we will use uniform, or Chebyshev criterion in the form u max inf , 0, 1,..., .j j Uj E u u j n     (3.1) A problem of the form (3.1) is usually called a discrete minimax problem. Therefore, this discrete minimax problem is considered in the form of the following problem of smooth conditional minimization 0 0 ( , ) inf , ju u U u       0 α 00 ( , ) μ 1/ α 0 0 , 1,..., , ε, 1 , μ , 1,..., , ε, , 1,..., . j i j j j jk i j k j j j u u j n u U w p i m u u U u j n                       (3.2) By logarithmizing the second group of constraints of the problem (3.2) and taking into account the first one, we get A PLANNING AND SCHEDULING METHOD FOR LARGE-SCALE INNOVATION PROJECTS 69 Copyright ©2019 ASSA Adv. in Systems Science and Appl. (2019) 0 0 ( , ) min , ju u U u   (3.3) 0 0 ( , ) μ ( , ) μ0 ( , ) μ 0 ln ln ln ln , α . i i i j jk j k j k j j k j u w p u U u U                 (3.4) Let the constraints of the problem be consistent, both here and in the future .U  Due to inequality (3.4), the consequence of the original formulation, taking into account the net- work specifics of the problem, we will seek its solution in the form of the maximum path with double weights, when the optimal solution of the problem (3.3), (3.4) is realized on a path (possibly a set of paths) with a value 0λ such that 0 0 ( , ) μ ( , ) μ0 μ ( , ) μ ln ln ln λ max α i i i i j jk j k j k j j k u w p          ( , ) μ μ ( , ) μ σ max . α i i i jk j k j j k      Wherein, 0 0 0 ( , ) min expλ ju u U u   . Therefore, it is necessary to find (possibly, a set) of maximal paths with double weights on arcs in an acyclic directed graph G (E, Γ).     0* μ * 0 0 0* 0* μ max λ μ , exp λ exp λ μ , μ i i j Arg u u j      We assume that the parameters of the problem are such that here and in the future 0 u U it is satisfied. Received resource critical path, with the maximum consumption of a non-renewable resource. Let  0* μ μ max λ μ i iArg are found. The optimal solution of (3.3), (3.4) we fix:   * 0 0* 0 0* fixed, μ , μ fixed. ju u j E j j       And we consider the problem of allocating a non-renewable resource according to a successively applied minimax criterion, varying uncommitted variables. In this case, the task of the second stage is: 0 u max inf , 0, 1,..., .j j Uj E E u u j n      or     1 1 ( , ) 1 α α* 00 ( , ) μ 0 inf , , 1,..., , ε, 1 , , μ , 1,..., . j j j i u u U j j j jk j k j j i u u u j n u u U w p u u U i m                    70 V. TOPKA, A. D. TSVIRKUN Copyright ©2019 ASSA Adv. in Systems Science and Appl. (2019) By logarithmizing the second group of constraints of the problem and taking into ac- count the first one, we get 0* 0 * 0 ( , ) μ ( , ) μ μ1 ( , ) μ ln ln ln α ln ln α i i i j jk j j j k j k j j j k u w p u u             0* 0 * 0 ( , ) μ ( , ) μ μ1 μ ( , ) μ ln ln ln α ln λ max α i i i i j jk j j j k j k j j j k u w p u                1 ( , ) μ 1* 1 μ μ ( , ) μ σ max , μ max λ μ . α i i i i jk j k i j j k Arg       We also fix the optimal solution to the problem of the second stage:  1* 1 1 1* 1* 0* 1 exp λ μ fixed, μ \ μ = fixed. ju u j E      At the same time, we gradually expand the viewer set  1 0j E E E    . Next, we consider the minimax problem of the third stage, the optimal solution of the previous stages is fixed, and the unfixed variables vary.  1 0 u max inf , 0, 1,..., .j j Uj E E E u u j n       We continue to allocate resources according to the minimax objective function succes- sively until at some stage q (the set indicator) we get the domain of definition of the problem  0... ... q rE E E E     and   * * * 1 * 0 exp λ μ fixed, μ \ μ fixed, q q q q j q q q r u u E j j               0 q rE E . (3.5) The overall problem is solved by successive optimization of the remaining sub graph by the minimax objective function at each stage, as in the lexicographic method of organiz- ing the solution of multi-criteria optimization problems. When the whole set of vertices is covered in optimal ways, this will solve the problem of a successive, monotonically expand- ing the scanned area, uniform allocation of a non-renewable resource according to a minimax criterion with a restriction on the assessment of reliability index of the project in the worst case. This procedure converges in a finite number of stages. Such a lexicographic disintegra- tion, in essence, is a decomposition procedure and is suitable for the planning of large-scale projects. To find the path of maximum efficiency in [7, p. 13], an algorithm of a search type was proposed (with exponential complexity) that reduces to finding the maximum path in the network. In [8, p. 192], to find in the graph with double weights of the cycle with the mini- A PLANNING AND SCHEDULING METHOD FOR LARGE-SCALE INNOVATION PROJECTS 71 Copyright ©2019 ASSA Adv. in Systems Science and Appl. (2019) mum value, the detection algorithm in the graph of the negative weight cycle is used (pro- vided that for all cycles the sum of weights in the denominator is positive). The solution of this problem with double weights using the proposed algorithm requires 3 1 log η O E       opera- tions, where η the magnitude of the error (weakly polynomial algorithm), E - is the num- ber of vertices in the network. In [9], a general method was proposed for solving the average problem in linear spaces. From it, as a special case, follows a strong polynomial algorithm for finding the minimal average contour in a weakly connected oriented graph, which com- plexity is  3 O E Г . To solve the problem of finding the maximum path with double weights in the acyclic digraph G (E, Г), we propose a greedy heuristic, for which, as will be shown below, the estimate  2 O Г of its computational complexity is valid. The advantage of which, compared with the above, is lower computational complexity. 4. GREEDY ALGORITHM OF FINDING THE MAXIMUM PATH IN DIGRAPH WITH DOUBLE WEIGHTS ON ARCS Construction of a set of paths  sM  , with weights ji and j on arcs lji , (j,i)  Г of the digraph G (E, Γ) covering all its vertices E. The initial step. Watched: arcs :L  , vertices :E  , path number s: = 1. 1). Selection of an arc on a set of unvisited arcs Г =Г\L. In the acyclic digraph G (E, Γ), we choose an arc jil , where  ,j i Г is optimal by the local criterion ( , ) σ max . α ji ji j i Г j l Arg   In the array L we bring the arc jil : : jiL l . 2). Build a path. 2.1). Adding new arcs. Construct a path s from the initial dummy vertex j: 1 jГ   to the final vertex j = n: jГ  by local criterion, i.e. for an arc jil , adjacent arcs kjl and itl are selected from condi- tions 1 ( , ) ( , ) σ σ max , α α σ σ max , α α j i kj ji j i L kj k Г k j j L ji it j i L it t Г j i j L l Arg l Arg                  which are added to the existing arcs : jiL l , forming a path  ... ...s kj itl L l    with double weights. Where ( , ) σ ji j i L  and α j j L  - values for the path found in the previous step. 72 V. TOPKA, A. D. TSVIRKUN Copyright ©2019 ASSA Adv. in Systems Science and Appl. (2019) Putting μ : 1s  , for the constructed in such a way the track μ s , we define the val- ue  λ μs s , s = 1,2, ...  ( , ) μ μ μ ( , ) μ σ max λ μ α ji j i s s s j j i       . List of viewed arcs -  : ... ...kj itL l L l   ; for viewed vertices – Es= {j|j  s}; E' := E'  Es. 2.2). Avoiding algorithm loops on scanned paths. In order to prevent the algorithm from looping along the scanned path μ s , in the future we will calculate the indicator value 1 μ s . If for non-viewed vertices \E E  , then let \k E E We put 1, if , δ 0, if \ . k k E k E E     Then μ μ 0 0 δ 1, ; δ 0, if μ : \ . k k k k k E k k E E             In this case, we set μ1 μ μ , if δ 1, 1, if δ 0, k ks k k           and then in the future, we will sequentially determine  ( , ) μ 1 1 μ 1 μ ( , ) μ σ max λ μ α ji j i s s s j j i           1 1λ μs s     only for new paths 1μ s that do not coincide with μs , s = 1,2, ... 3). Cycle If for unvisited arcs Г   , then cycle through : 1s s  and go to step 1. otherwise: Viewing the constructed set of paths  1μs and determining the optimal path(s) with double weights on arcs 0*μ Argmaxλ (μ )s s s  , s = 1,2, ... For the first stage (3.3), (3.4) of the problem:  0 0*λ λ μ , and for the subsequent ones  *λ λ μr r , r = 1, ..., q. In the algorithm, the review of arcs for constructing a set of paths is carried out from Г to  , which allows you to avoid a complete search of the paths, and the introduction of a counter μ s   for paths that have at least one unwatched arc eliminates looping the algo- rithm on the same old path. Described algorithm is, in essence, greedy heuristic and so it gives approximate solution of the above problem. The complexity of the algorithm described is estimated as follows. The selection of the initial locally optimal arc lji requires Г operations, building a path  from the initial vertex to the final vertex is also requires the order of D Г operations, where D is the maximum de- gree of the vertices of the graph G (E, Γ). The construction of all such paths for each of the Г arcs is repeated about  О Г D Г Г   time plus the successive selection of the λ-op- timal path, which will require order Г operations. Therefore, the complexity makes A PLANNING AND SCHEDULING METHOD FOR LARGE-SCALE INNOVATION PROJECTS 73 Copyright ©2019 ASSA Adv. in Systems Science and Appl. (2019) up    2 О ОГ D Г Г Г Г    of operations. And for a complete vertex coverage, you will need to repeat such an algorithm one more Г time, then the total will be required  3 О Г operations. For clarity, we illustrate (Table 1) finding the first maximal path with double weights on arcs by calculating all the available routes and choosing the maximal one. Table 1. Finding the first maximal values with double weights on the paths for 0lnu path number 0lnu 1 4.0628 2 2.3828 3 2.7416 4 2.3891 5 2.9082 6 2.1519 7 2.3016 8 2.1389 9 2.8276 10 2.5899 11 2.2212 12 2.2322 13 2.2010 14 2.5093 15 2.3978 16 2.2619 17 2.7578 18 3.1007 19 2.3034 20 2.2053 A test network consisting of 32 vertex-activities, from the well-known library of net- work models of projects from http://www.om-db.wi.tum.de/psplib/ provide precedence rela- tions of the selected network project as in the Table 2 below. Table 2. Precedence relations of the test network project. jobnr. - activity number; #modes - the number of execution modes (does not matter); #successors - the number of vertices at which the edges of the relations of precedence go out of the vertex of this work; successors - numbers of activities that go to the edges of the relations of the preceding from this vertex. Table 2. Precedence relations of the test network project. jobnr. #modes #successors successors 1 1 3 2 3 4 2 1 3 6 11 15 3 1 3 7 8 13 4 1 3 5 9 10 5 1 1 20 6 1 1 30 7 1 1 27 http://www.om-db.wi.tum.de/psplib/ 74 V. TOPKA, A. D. TSVIRKUN Copyright ©2019 ASSA Adv. in Systems Science and Appl. (2019) 8 1 3 12 19 27 9 1 1 14 10 1 2 16 25 11 1 2 20 26 12 1 1 14 13 1 2 17 18 14 1 1 17 15 1 1 25 16 1 2 21 22 17 1 1 22 18 1 2 20 22 19 1 2 24 29 20 1 2 23 25 21 1 1 28 22 1 1 23 23 1 1 24 24 1 1 30 25 1 1 30 26 1 1 31 27 1 1 28 28 1 1 31 29 1 1 32 30 1 1 32 31 1 1 32 32 1 0 Above information from Table 2 about relations of technological precedence of select- ed network project will be organized then with the help of matrix "columns - vertices" and "rows - paths by vertices". Which specifies the characteristic function of the constraint sys- tem, and the parameters are specified in the last two columns of Table 3 and Table 4. Table 3. Upper the border ub for 0lnu № j= Upper the border ub for 0lnu ub = form parameter j  scale parameter 0 ju  0 0.3000 0.5100 1 4.3944 0.2500 0.6000 2 4.1650 0.2200 0.5000 3 0 0.2400 0.5000 4 0 0.2500 0.5000 5 0 0.2800 0.4800 6 4.3565 0.2300 0.4780 7 0 0.2330 0.4750 8 0 0.2350 0.4600 9 0 0.2380 0.4550 10 0 0.2400 0.4500 11 0 0.3000 0.4480 A PLANNING AND SCHEDULING METHOD FOR LARGE-SCALE INNOVATION PROJECTS 75 Copyright ©2019 ASSA Adv. in Systems Science and Appl. (2019) 12 0 0.3200 0.4460 13 0 0.3500 0.4450 14 0 0.3800 0.4420 15 0 0.4000 0.4400 16 0 0.4200 0.4300 17 0 0.4300 0.4200 18 0 0.4400 0.4000 19 0 0.4420 0.3800 20 0 0.4450 0.3500 21 0 0.4460 0.3200 22 0 0.4480 0.3000 23 0 0.4500 0.2400 24 0 0.4550 0.2380 25 0 0.4600 0.2350 26 0 0.4750 0.2330 27 0 0.4780 0.2300 28 0 0.4800 0.2800 29 0 0.5000 0.2500 30 4.4742 0.2800 0.7000 31 0 0.5000 0.3200 32 4.1759 0.3000 0.7000 Table 4. Numerical solution to the problem (3.2) as a whole № j= ln ju = Restrictions c = Upper the border ub = form parame- ter j  scale parameter 0 ju  0 3.1203 25.000 3,1203 0,300 0,510 1 4.3944 4.1021 4,3944 0,250 0,600 2 4.1650 3.7607 4,1650 0,220 0,500 3 3.8179 3.4393 3,8179 0,240 0,500 4 3.6652 3.2914 3,6652 0,250 0,500 5 0.2924 0.0444 3,1267 0,280 0,480 6 0.2924 0.0444 4,3565 0,230 0,478 7 0.2924 0.0394 3,7124 0,233 0,475 8 0.2924 0.0444 3,5443 0,235 0,460 9 0.2924 0.1276 3,4537 0,238 0,455 10 0.2924 0.0444 3,3789 0,240 0,450 11 0.2924 0.0444 2,6883 0,300 0,448 12 0.2924 0.0444 2,5063 0,320 0,446 13 1.7583 1.4512 2,2850 0,350 0,445 14 0.2924 0.0444 2,0868 0,380 0,442 15 0.2924 0.0444 1,9711 0,400 0,440 16 0.2924 0.0444 1,8225 0,420 0,430 17 1.2483 0.9752 1,7254 0,430 0,420 18 1.5753 1.2756 1,5753 0,440 0,400 76 V. TOPKA, A. D. TSVIRKUN Copyright ©2019 ASSA Adv. in Systems Science and Appl. (2019) 19 0.2924 0.0444 1,4522 0,442 0,380 20 1.2576 0.9652 1,2576 0,445 0,350 21 0.2924 0.0307 1,0538 0,446 0,320 22 0.9051 0.6127 0,9051 0,448 0,300 23 0.4052 0.1128 0,4052 0,450 0,240 24 0.3823 0.0899 0,3823 0,455 0,238 25 0.3506 0.0582 0,3506 0,460 0,235 26 0.3215 0.0291 0,3215 0,475 0,233 27 0.2924 0 0,2924 0,478 0,230 28 0.7010 0.4086 0,7010 0,480 0,280 29 0.4463 0.1539 0,4463 0,500 0,250 30 4.4742 4.1818 4,4742 0,280 0,700 31 0.9400 0.6476 0,9400 0,500 0,320 32 4.1759 3.8835 4,1759 0,300 0,700 0u 1.3396. If we solve the problem in question (3.2) as a whole, as the problem of mathematical programming using ready-made software, we obtain the numerical solution presented in Ta- ble 4 above. For the 0 0 [ , 1], 0jkw    – arc reliability coefficient (j, k), (from the matrix of paths along the arcs of the network G (E, Г):) transfer the result of the j-th activity through the arc (j, k) to perform the k-th activity (like the transfer function of the connection links in the the- ory of automatic control) we have the following Table 5 values. Table 5. Arcs reliability coefficient 0Г arcs value jkw 1 (5,20) 0.95 2 (11,20) 0.95 3 (18,20) 0.95 4 (16,22) 0.95 5 (17,22) 0.95 6 (18,22) 0.95 7 (10,25) 0.95 8 (15,25) 0.95 9 (20,25) 0.95 For all other arcs of the network 1 01 \ .jk jkw w Г Г Г    5. DETERMINATION OF THE SCHEDULE WITH INACCURATE SOURCE DATA Since the values of the scheduling model parameters 0 0 0 (0,1), [ , 1], 0, 0, , 1,..., , ( , ) ( , ) j jk jw u j k n j k Г G Е Г          are known with a certain error, the resource allocation algorithm, as well as the physical vol- ume of work v [USD  hour] and the amount of a non-renewable resource, found as a re- sult of solving the considered problem, which has the value expression u [USD], are approx- imate, the project’s time parameters - the duration of the activity * */ r j j jt v u , r = 0,1, ..., A PLANNING AND SCHEDULING METHOD FOR LARGE-SCALE INNOVATION PROJECTS 77 Copyright ©2019 ASSA Adv. in Systems Science and Appl. (2019) q 1,...,j n ; and the activities slacks are determined by processing the numbers known with an error. The obtained solution of problem (3.1), (3.5) together with reliability constraints, vector       * * *u , 0, 1,..., , 0,1,..., , , r r j ju u j n r q r j q n      such that (omitting the index r)  * * * * * 0 0( ) ( )j j j j ju u u u u          . (5.1) By denoting inaccurately set time values as  δt , using actions on approximate num- bers [5], one can obtain approximate parameters of the optimal schedule. To do this, we cal- culate the deterministic project schedule for one pass in the forward and reverse directions based on the obtained optimal (under given error) duration of the project * 0( )it  , starting from the initial project activities. The Critical Path Method, with values, specified inaccurately (5.1), using actions on approximate numbers [5], also allows you to calculate late deadlines for performing project activities in a back pass through the network, starting from the project completion date (cal- culated by direct passing through the network). And also to calculate slacks of activities [5]: total (full) slack, free (local) slack, safe slack, independent slack. The total (full) slack tSL of the project - the period of time by which you can postpone the activity without violating the limitations and deadlines for the project t ( ) ( ) ( ) ( ) ( ).lb eb lc ec i i i i iSL t t t t         Free (local) slack fSL of the project - the period of time by which you can postpone the activity without violating the deadline for the subsequent work – f ( ) min( ( ) ( )) ( ). i eb ec i j ij i j Г SL t d t         The safe slack sSL of the project activity is the period of time for which activity j can be prolonged when all its predecessors 1 ji Г  started working at the latest date so that the short- est possible time to complete the project does not increase – 1 s ( ) ( ) max ( ). j lb lc j j i i Г SL t t       An independent slack i jSL of the project activity j is equal to the time for which the duration of activity j can be prolonged regardless of the time of completion of its predeces- sors - 1 ji Г  and the time of the beginning of its followers jk Г  1 i *( ) max 0;min ( ) max ( ) ( ) . j j eb lc j k i j k Г i Г SL t t t               The following relations are valid (under 0  ): t f i t s i( ) ( ) ( ); ( ) ( ) ( ). j j j j j jSL SL SL SL SL SL          CONCLUSION The article discusses a deterministic network model of a project in which, for a project's overall reliability indicator, it is fair its assessment from below in the form of the weakest link – in the worst technological chain from the initial vertex of the project to the final one. 78 V. TOPKA, A. D. TSVIRKUN Copyright ©2019 ASSA Adv. in Systems Science and Appl. (2019) For the indicator of the reliability of the activity-vertex of the project, a power-concave mod- el is used depending on the spent non-renewable (stored) resource. The peculiarity of the model is weighted network arcs, which set the arc reliability factor (j, k), transferring the re- sult of the j-th activity by the arc (j, k) to perform the k-th activity. Under these conditions, the problem of planning and scheduling projects is considered on the basis of a successively applied minimax criterion according to the type of lexico- graphic ordering of criteria in multi-criteria optimization problems. Such a lexicographic dis- sociation, in essence, is a decomposition procedure and is suitable for planning large-scale projects. To solve the problem of uniform allocation of a non-renewable resource, a greedy heu- ristic has been developed for finding the maximum path with double weights on arcs of an acyclic digraph, computational complexity that is quadratic in the number of arcs. The greedy heuristic gives approximate solution of the problem under consideration. Which al- lows you to get a resource critical path, with the maximum consumption of a non-renewable resource. After that, the project schedule is determined, including its temporary critical path, with inaccurate data. Practical calculations for the considered convex problem (3.2) can be performed using ready-made software, and the resulting greedy solution of the problem- consequence (3.3), (3.4) has theoretical significance in graph theory. REFERENCES 1. The Owner’s Role in Project Risk Management. (2005) Washington, D.C.: The Na- tional Academies Press, www.nap.edu. 2. Elkjaer M. (2000). Stochastic Budget Simulation, Int. J. 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