Управление в социально-экономических системах PRINCIPLE OF COORDINATED PLANNING IN CONTROL OF DISTRIBUTED PROJECTS AND PROGRAMS

The paper deals with the problem of managing distributed projects and programs. These programs consist of subprograms distributed functionally, administratively or geographically. For in-stance, a program of regional development includes a subprogram of environmental safety. In this regard, the main problem of managing distributed programs is the problem of interests' coordination for all persons concerned. We propose the principle of coordinated planning for designing implementation plans of distributed programs. The variations meet the inequalities  2 +  3 > 0 (for the first pair) and  2 –  3 > 0 (for the second pair). Choose  2 = 0 and  3 > 0. Interestingly, under  3 > 0 we obtain a new pair of


Introduction
Distributed projects (programs) are projects (programs) consisting of subprojects (subprograms) distributed functionally, administratively or geographically. Functional distribution means that there exist different functional directions of a project (program) having a dedicated subproject (subprogram) with a separate manager and team. Among examples, we mention a regional development program which includes several functional directions such as social development, economic development, environmental safety and others. In the case of administrative distribution, there are subprojects (subprograms) in the interests of different administrative or economic institutions. For instance, a regional development program includes development subprograms of member cities, municipalities, etc. with separate managers and teams. The main feature of functionally and administratively distributed projects (programs) is the presence of noncoinciding interests pursued by the managers of associated subprojects (subprograms). Therefore, the major problem in managing functionally and administratively distributed projects (programs) lies in interests' coordination for all persons concerned (basically, the managers of subprojects and subprograms). Geographically distributed projects (programs) can be functionally and administratively distributed and, moreover, have another essential peculiarity. While designing implementation plans of such projects (programs), one should take into account the transfer time of different resources (personnel, equipment, materials): this time is often comparable with (or even exceeds!) the execution time of a job. The reparation and construction of motor roads, railway tracks and bridges are the examples of such projects. 1 We study the problem of interests' coordination among the sub-projects (subprograms) of a functionally or administratively distributed project (program) using the example of a functionally distributed program. All results can be easily applied to geographically or administratively distributed projects and programs.

The principle of coordinated planning in distributed projects (programs)
Consider a functionally distributed program composed of m subprograms covering different directions. In the sequel, the program manager will be called the Principal (P), whereas subprogram managers will be called agents (A).
Suppose that there is a state assessment of each direction (in a qualitative or quantitative scale). Denote by F i , the state assessment of direction i (the goal function of agent i) and by F the goal function of the Principal. The goal function of the Principal depends on the goal functions of agents: (1) This can be a linear, additive or matrix convolution. The Principal has to design a program (a set of projects) in order to maximize the goal function F under limited resources R allocated to the program. On the other hand, each agent / strives to design a subprogram maximizing its goal function F i .
Imagine that the Principal ignores the interests of agents during program design. This would cause a series of negative consequences such as hiding or misrepresentation of information provided by agents to the Principal, non-fulfillment of program measures, etc. For interests' coordination between the Principal and agents, the theory of active systems proposes the principle of coordinated planning [1]. The fundamental idea of this principle is to optimize the Principal's goal function on the set of coordinated plans (i.e., plans such that the goal functions of agents are not smaller than a given threshold). For formal statement of the coordinated planning problem, designate by 0 i F the existing state assessment of direction i. The interests' coordination condition can be a guaranteed increment

Problem statement
There are n candidate projects for inclusion in the program. Implementation of each project i incurs the costs c i and yields the economic effect a ij for direction j (we comprehend effect as the increment of the goal function F j ). Set 3.1. SPECIAL CASE. SINGLE-PURPOSE PROJECTS Consider the following special case of the problem. For each direction j, there exists a set Q j of projects contributing to it; the sets Q j do not intersect. In this case, the problem is treated in two stages. Stage 1. Solve m knapsack problems: maximize subject to the constraints For this, solve the standard knapsack problem (7), (8) under R j = R.
As is well-known, solution of the knapsack problem under R j = R yields optimal solutions for all R j < R. Denote by Y j (R j ) the value of the goal function (7) in the optimal solution as a function of R j . Find the minimum value R j = d j such that Y j (d j ) ≥ b j . As a result, we obtain a relationship Y j (R j ), where d j ≤ R j ≤ R.

Stage 2. Solve the maximization problem of the function
Here we apply dichotomous programming. Each knapsack problem is solved by the backward method.
Example 1. Consider three directions of a program, see Table 1. Table 1 j Direction 1 Direction 2 Direction 3 i 1 2 3 4 5 6 7 8 9 10 11 12 a i 12 30 50 16 16 15 8 18 24 18 10 7 c i 6 5 10 4 4 3 4 3 12 6 5 7 Solve the knapsack problem for direction 1 using dichotomous programming [2]. Fig. 1 shows the dichotomous representation tree of this problem. Step 1. Solve the problem for projects 1 and 2. The resulting solution is described by Table 2. Here the first value indicates the costs and the second value means the economic effect. The results are combined in Table 3. Actually, this table contains only Pareto optimal variants. For instance, we eliminate variant (6;12) as being dominated by variant (5;30) (under smaller costs, it yields higher effect). Step 2. Solve the problem for projects 3 and 4. The solution is illustrated by Table 4. The results can be found in Table 5.   Step 3. Consider the united projects (1;2) and (3;4). The solution is provided by Table 6 and the results are combined in Table 7. As far as b 1 = 20, we reject variants (0;0) and (4;16).  Solve the knapsack problem for direction 2. The solution is described by Table 8. Solve the knapsack problem for direction 3. The solution is described by Table 9 (13) Step 1. Consider directions 1 and 2. The solution is provided by Table 10. And the results can be found in Table 11.   64 79 80 84 95 99 100 114 115 129 130 142 Step 2. Consider united direction (1;2) and direction 3. The solution is combined in Table 12. In Table 12 find a cell with the maximum second value. This is cell (30;123) associated with the effect 123. Cell (30;123) corresponds to variant 5 in Table 11 and variant 1 in Table 9. This variant corresponds to the solution of the knapsack problem x 9 = 0; x 10 = 1; x 11 = 1; x 12 = 0 with costs 11 and effect 28.
And finally, variant 2 in Table 7 corresponds to the following solution of the knapsack problem for direction 1: x 1 = 0; x 2 = 1; x 3 = 0; x 4 = 1 with costs 9 and effect 46.

GENERAL CASE. MULTI-PURPOSE PROJECTS
In the general case, there exist projects whose implementation contributes to several directions. Such projects are said to be multipurpose projects. If the number q of multi-purpose projects is not large, consider all 21' variants of multi-purpose projects inclusion in the program and choose the best one (perform exhaustive search). Example 2. Take 2 directions and 8 projects with the parameters described by Table 13.    As far as b 1 = 20, eliminate rows 0 and 1 from Table 16. Similarly, eliminate columns 0, 1 and 2 due to b 2 = 25. In the resulting table, identify a cell with the maximum second value. Actually, this is cell (30;88) associated with effect 88. Variant 2. Project is included in the program (x 4 = 1; x 5 = 0). In this case, the residual resource makes up R' = 30 -8 = 22. So long as a 41 = 24 and a 42 = 16, then b 1 ' = 0 and b 2 ' = 25 -16 = 9. Hence, we have to eliminate only column 0 and row 0 from Table 16.
Define a cell with the maximum second value among all cells whose first value does not exceed 22. This is cell (18;64) with effect 64. By adding the effect from project 4 (a 41 + a 42 = 40), we get total effect 104.  Table 16. Find a cell with the maximum second value among all cells whose first value does not exceed 20. This is cell (18; 64) yielding effect 64. By adding the effect from project 5, we get total effect 64 + 25 = 89.
Variant 4. Projects 4 and 5 are included in the program (x 4 = x 5 = l). Then we have that R' = 30 -18 = 12, b 1 ' = 0, and b 2 ' = 0. Identify a cell with the maximum second value among all cells whose first value does not exceed 12. This is cell (11;43) with effect 43. By adding the effects from projects 4 and 5, we get total effect 43 + 40 + 25 = 108. The maximum effect is gained by variant 4. Note that cell (11;43) corresponds to variant 1 in Table 15 and variant 2 in Table 14. On the other hand, variant 1 in Table 15 corresponds to the following solution for direction 2: x 6 = 1, x 7 = 0, x 8 = 0. Variant 2 in Table 14 corresponds to the following solution for direction 1: x 1 = 1, x 2 = 0, x 3 = 1, x 4 = 0. And finally, we establish that the program includes projects 1, 3, 4, 5 and 6 with total effect 108 and total costs 29.

NETWORK PROGRAMMING METHOD
Under a large number of multi-purpose projects, program design based on their exhaustive search becomes inefficient. Consider the branch-and-bound method with estimation using network programming [3]. Let us illustrate this method for the inverse problem: minimize the costs required for obtaining a given total effect. In other words, the problem is to minimize the goal function We provide a simple example below. Example 3. There are 4 projects with the parameters described by Table 17. The number of directions equals 2. Set b 1 = 10, b 2 = 8 and B = 30. According to Table 17, projects 2 and 3 are multi-purpose. Fig. 2 shows the network representation of the associated constraints. Theory of network programming prescribes splitting arbitrarily the costs с 2 and с 3 of multi-purpose projects into two components s 21 , s 22 and s 31 , s 32 , respectively (since vertices 2 and 3 have 2 outgoing arcs, see Fig. 2). For instance, take s 21 = s 22 = 1, s 31 = 1, s 32 = 3. This leads to two estimation problems for each direction. The estimation problem for direction 1: minimize   . Denote by Z 1 (B 1 ) the optimal value of С 1 (x). The solution is described by Table 18.
  . Designate by Z 2 (B 2 ) the optimal value of С 2 (x). The solution is defined by Table 19.  Consider Table 20 and choose a cell with the minimum first value among all cells whose second value is not smaller than B = 30. These are cells (8;35) and (8;33) with costs 8. According to the fundamental theorem of network programming, in our example costs 8 provide a lower estimate of the costs in the original problem. Define the corresponding optimal solutions by the backward method. Cell (8;35) corresponds to variant 2 in Table 19 and variant 4 in Table 18. Next, variant 2 in Table 19 answers the solution of the estimation problem for direction 2: x 2 = 0, x 3 = 0, x 4 = 1. Variant 5 in Table 18 answers to the solution of the first estimation problem: x 1 = 1, x 2 = 1, x 3 = 1. The obtained pair of solutions does not define an admissible solution.
Cell (8;33) corresponds to variant 3 in Table 19 and variant 3 in Table 18. Variant 3 in Table 19 answers to the solution x 2 = 1, x 3 = 0, x 4 = 1 of the second estimation problem, whereas variant 3 in Table 18 corresponds to the solution x 1 = 1, x 2 = 0, x 3 = 1 of the first estimation problem. Again, this pair of solutions does not define an admissible solution of the original problem (it represents a lower estimate only).
To proceed, we may either improve the derived estimates (using other costs splitting for multipurpose projects) or apply the branch-and-bound method with the derived estimates. Let us illustrate the branch-and-bound method. Choose direction 2 for branching. Decompose the solution set into two subsets: x 2 = 1(subset 1) and x 2 = 0 (subset 2).
Note that the above pair of solutions defines an admissible, ergo optimal solution on the subset x 2 = 1 c with costs 8.
Estimation on subset 2 (x 2 = 0). Solve the estimation problem for direction 1: 3x 1 + x 3  min subject to the constraint 12x 1 + 9x 3 ≥ B 1 ', where 10 ≤ B 1 ≤ 30. The solution is shown by Table 24.   The solution answers to cell (10;35) with costs 10. Choose subset 1 (x 2 = 1). The corresponding optimal solution is x 1 = 1, x 2 = 1, x 3 = 0, x 4 = 1 with costs 8. Fig. 3 shows the branching tree. The second solution method of the problem consists in maximum increase of the lower estimate via optimal split of the costs c 2 and c 3 under the constraints s 21 + s 22 = c 2 , s 31 + s 32 = c 3 . This is the so-called generalized dual problem (GDP).
According to [3], the GDP represents a convex programming problem. However, one should have in mind a couple of important aspects. First, numerical experiments have demonstrated that, generally, computational time required for lower estimate improvement is not compensated owing to smaller branching in the branch-and-boundary method. Second, in many cases the GDP possesses a non-integer solution; as is well-known, non-integer parameters make the knapsack problem NP-complex. Therefore, it is strongly recommended to obtain estimates under a given initial costs split of multipurpose projects.
We endeavor to improve the derived estimate. For s 21 = s 22 = 1, s 31 = 3, there are two pairs of solutions to the estimation problems. The first pair of solutions has the form x 1 = 1, x 2 = 1, x 3 = 1, x 2 = 0, x 3 = 0, x 4 = 1. And the second pair of solutions is defined by x 1 = 1, x 2 = 0, x 3 = 1, x 2 = 1, x 3 = 0, x 4 = 1. Designate by  2 and  3 the variations of the estimates s 22 and s 32 , respectively. Note that the optimal solutions of the estimation problems remain same under small values of  2 and  3 . To increase the lower estimate, we should increase the lower estimate for each pair of solutions. 8 8 10 Using such norm , the Principal participates in joint financing of all projects with nonnegative profits. Note that the Principal can choose a norm  >  0 . This leads to the problem of program design with the maximum total effect under a guaranteed effect of each agent. Actually, the problem has been studied above.
To solve inequality (17), for each project defines the norm 1 ji ji ji c a    (here с ji ≥ а ji , otherwise, the project is beneficial to the agent without additional financing). Renumber all projects in the ascending order of  ji , i.e.,  1 ≤  2 ≤ ••• ≤  q , where q means the number of projects. Determine maximum number k such that.
In this case, subprogram 1 includes all projects 1-4, whereas subprogram 2 consists of project 5 only. However, if the guaranteed effects of the subprograms d 1 = d 2 = 50, then the plan coordination condition breaks for subprogram 2. Therefore, we choose  k =  6 = 0,6. As a result, subprogram 2 includes projects 5 and 6, with the total effect a 5 + a 6 = 70 > 50 and costs 80 of agent 2.
Note that if we reduce the guaranteed effect of subprogram 2 to 40, then the norm  goes down to 0.5. This allows increasing appreciably the total effect (from 330 to 250).