1 Introduction

The first part of this paper (see [3]) described and studied the mechanisms for selecting the most promising projects in the field of developing products and technologies for using alternative types of energy for the propulsion of trains. Also, it described and studied the mechanisms for allocating a limited budget among the complex projects and their parts.

After selecting projects and determining the volumes of financial resources for them, it is necessary to establish planning and incentive mechanisms for executing the works within these projects. It should be taken into account that the control authority (Principal), who plans the activities and provides incentives, is less aware of the existing capabilities for implementing the works than their executors, experts in the field. Under these conditions, for planning and incentive purposes, the Principal is compelled to request this information from the executors. Since the Principal and the executors have their own interests, the natural problem is to eliminate the strategic behavior of the executors, i.e., any manipulations with the information reported by them to the Principal [4–6, 19, 20].

The second part of this paper considers the problem to design incentive-compatible organizational mechanisms ensuring such strategy-proofness in the case of a system of interconnected executors (agents) with a sequential technology of their works.

The considerations below are essentially based on [5, 6].

Also, the results established in the first part of this paper (see [3]) are used below to study the additional requirements and possibilities for designing a project management system for the scientific and technical activities of the organization implemented by the Principal.

2 Elements of Innovations Management System

2.1 Model of Incentive Mechanism for Saving Energy

As a rule, the projects selected for further development are complex; they represent a system of sub-projects, business processes, activities, and works with the input-output connections described by a network structure (Fig. 1). Each element of such a complex of works contributes to the energy-saving characteristics of the entire project. Therefore, the problem is to stimulate each element of this structure, which is responsible for a corresponding work of the project.

Fig. 1
figure 1

Network structure of complex project: example.

Full size image

A methodology to design such incentive systems was developed within the theory of organizational control and management [1, 9, 10], mainly under the Principal’s complete awareness of the models of agents involved in the implementation of a complex project. In the case of incomplete awareness, which usually occurs in complex projects management, this problem was studied only the hypothesis of weak contagion (weakly interconnected agents [7]). However, the works of a complex project are interconnected by the sequence of execution: the results of previous works strongly affect the results of subsequent works.

Consider an example of an elementary structure of works interconnected by a strict sequence of execution. (The approach introduced below, as well as the results of analysis, will not fundamentally differ in a more general case when the connections of agents are described by a tree.)

Model description

Let the Principal’s goal function be given by

$$\Phi (x,y,r)={c}_{1}{y}_{1}-{z}_{1}({x}_{1},{y}_{1})-{v}_{1}({y}_{1},{r}_{1})+\mathop{\sum }\limits_{i=2}^{n}[{c}_{i}({y}_{i}-{y}_{i-1})-{z}_{i}({x}_{i},{y}_{i})-{v}_{i}({y}_{i},{r}_{i})],$$

with the following notations: yi as the agent’s state characterizing the result of execution of the ith work by him (for example, the specific level of saving energy compared to the current or established norm); xi as a plan for the ith work, assigned by the Principal; zi(xi yi) as the Principal’s loss due to the non-coincidence of the result yi with the plan xi; vi(yi ri) as the loss due to the deviation of the result yi from some best energy-saving level ri; \({r}_{i}\in [{r}_{i}^{{\rm{low}}},{r}_{i}^{{\rm{upp}}}]\), \({y}_{i}\in [0,{r}_{i}^{{\rm{upp}}}]\), and \({x}_{i}\in [0,{r}_{i}^{{\rm{upp}}}]\). Assume that the values ri are given, but the Principal knows only the ranges of these parameters. Here \({r}_{i}^{{\rm{low}}}\) and \({r}_{i}^{{\rm{upp}}}\) are the lower and upper admissible limits of the level ri; ci is the given parameter characterizing the significance of the result yi for the Principal.

Suppose that

$$\begin{array}{l}0\le {y}_{i}\le {r}_{i},\\ {z}_{i}({x}_{i},{y}_{i})\ge 0,\\ {z}_{i}({x}_{i},{x}_{i})=0,\\ {v}_{i}({y}_{i},{r}_{i})\ge 0,\end{array}$$

and

$${v}_{i}({r}_{i},{r}_{i})=0.$$

In addition, let

$${v}_{i}({y}_{i},{r}_{i})={a}_{i}({r}_{i}-{y}_{i}),$$

where 0 ≤ ai ≤ ci. Each agent executing the ith work (selecting the values yi) knows the true value ri, and the Principal knows only the admissible limits of this indicator, \({r}_{i}^{{\rm{low}}}\) and \({r}_{i}^{{\rm{upp}}}\). The sequential works are interconnected by the following relations:

$${r}_{i+1}^{{\rm{low}}}={r}_{i}^{{\rm{low}}}+{\Delta }_{i+1}^{{\rm{low}}},\quad {r}_{i+1}^{{\rm{upp}}}={r}_{i}^{{\rm{upp}}}+{\Delta }_{i+1}^{{\rm{upp}}},\quad i=1,\ldots ,n-1.$$

Assume that the goal function of agent 1 has the form

$${f}_{1}({x}_{1},{y}_{1},{r}_{1})={\sigma }_{1}({y}_{1})-{\vartheta }_{1}({x}_{1},{y}_{1})-{\omega }_{1}({y}_{1},{r}_{1}),$$

where σ1(y1) is an incentive function, ϑ1(x1, y1) is a penalty function for the nonfulfillment of his plan, and ω1(y1, r1) is his cost function. Represent the goal function of agent i (executor of the ith work) as

$${f}_{i}({x}_{i},{y}_{i},{y}_{i-1},{r}_{i})={\sigma }_{i}({y}_{i-1},{y}_{i})-{\vartheta }_{i}({x}_{i},{y}_{i})-{\omega }_{i}({\delta }_{i},{r}_{i}),$$

where σi(yi−1, yi) is the incentive function for the result yi, given the state yi−1 of the previous agent; ϑi(xi, yi) is the penalty function for the nonfulfillment of the plan; ωi(δi, ri) is the cost function of agent i, which depends on his contribution to the energy-saving characteristics of the complex project. Here δi+1 = yi+1 − yi, i = 1, …, n − 1, δ1 = y1 ≥ 0. Let δi+1 ≥ 0, i = 1, …, n − 1.

For the sake of simplicity, assume that the cost functions can be approximated with sufficient accuracy by a quadratic function: \({\omega }_{i}({\delta }_{i},{r}_{i})={e}_{i}{\delta }_{i}^{2}/2{r}_{i}\), where ei > 0. The cost functions reflect the following important property: the higher the value of ri is, the lower the costs are. In other words, the greater the reserve for saving energy is, the less effort is required for the agent.

In the model of the incentive system under consideration, the participants have the following order of decision-making.

First, the Principal establishes a mechanism μ = {σ(⋅), ϑ(⋅ , ⋅), x(⋅)}, which includes an incentive function σi(⋅), a penalty function ϑi(⋅ , ⋅) for the nonfulfillment of the plan on saving energy, and a rule x(⋅) = (x1(⋅), …, xn(⋅)) to assign plans xi based on the available information ρi about the parameters ri, i = 1, …, n.

Thus, in the system under consideration, an incentive mechanism for saving energy is understood as the set of interconnected planning procedures, incentive functions, and penalty functions.

Second, the agents report to the Principal their estimates ρi of the parameters ri, which are then used for assigning the plans on saving energy xi = xi(ρi) in accordance with the procedures xi(⋅). Third, the agents implement the activities on saving energy for achieving the level yi. Finally, the incentives σi(yi−1, yi) and penalties ϑi(xi, yi) are assigned to the agent in accordance with the functions σi(⋅ , ⋅) and ϑi(⋅ , ⋅).

Let the efficiency index of the incentive mechanism be defined as the guaranteed value

$$K(\mu ,r)= \mathop {\rm inf}\limits_{\rho \in R(r)} \quad \mathop {\rm inf}\limits_{y\in Y(x(\rho))}\Phi (x(\rho),y,r)$$

of the Principal’s goal function on the sets R(r) (all rational strategies of the agent when choosing the messages ρ = (ρ1, …, ρn)) and Y(x) (all rational strategies of the agents when choosing the values y = (y1, …, yn) under a given plan x = (x1, …, xn). Further analysis will be confined to the mechanisms for which the set R(r) consists of the dominant strategies of different agents: for each agent, there exist messages maximizing his goal function, regardless of the messages of all other agents. The set Y(x) is determined by the sequence in which the agents choose the results yi according to the numbering: agent i maximizes his goal function under a given plan and the result yi−1 of the previous agent. The sets of rational strategies will be described in detail below when designing the optimal mechanism and formulating the results of this paper.

Optimal incentive mechanism design: problem statement

The problem is to determine a mechanism μ* such that, for all values of the parameter r ∈ [rlow, rupp],

$$K({\mu }^{* },r)\ge \mathop {\rm sup}\limits_{\mu \in M}K(\mu ,r)-\varepsilon ,$$

where M is a given compact set of admissible mechanisms and ε > 0.

Let the set M be defined by the following constraints imposed on the incentive functions, penalty functions and planning procedures:

—The incentive functions σi(yi) are piecewise continuous and 0 ≤ σi(yi) ≤ gi.

—The penalty functions ϑi(xi, yi) are piecewise continuous and satisfy the maximum growth constraint,

$${\vartheta }_{i}({x}_{i},{y}_{i})-{\vartheta }_{i}({x}_{i},y^{\prime}_{i})\le {\theta }_{i}(y^{\prime}_{i} ,{y}_{i}),$$

where \({\theta }_{i}(y^{\prime}_{i},{y}_{i})\) is a given value of the maximum growth rate of penalties [6] that satisfies the triangle inequality:

$${\theta }_{i}(y^{\prime}_{i},{y}_{i})+{\theta }_{i}({y}_{i},{x}_{i})\ge {\theta }_{i}(y^{\prime}_{i},{x}_{i}).$$

—The planning procedures xi(ρi) are continuous functions, i = 1, …, n.

As it was demonstrated in [5, 6], under these conditions the optimal incentive mechanism μ* can be designed by solving the optimization problem

$$\begin{array}{c}K({\mu }^{* },r)=\mathop{\max }\limits_{\mu \in {M}_{{\rm{c}}}}\Phi (x(r),x(r),r)\\ =\mathop{\max }\limits_{\mu \in {M}_{{\rm{c}}}}\left\{{c}_{1}{y}_{1}-{v}_{1}({y}_{1},{r}_{1})+\mathop{\sum }\limits_{i=2}^{n}[{c}_{i}({x}_{i}-{x}_{i-1})-{v}_{i}({x}_{i},{r}_{i})]\right\};\end{array}$$
(1)

here Mc is the compact set of admissible mechanisms determined by additional concordance conditions, under which the planning procedure x(ρ) and the incentive system σ(⋅), ϑ(⋅ , ⋅) stimulate the agents to fulfill plans and report reliable information about their parameters to the Principal as their dominant strategies. The compactness of the sets Mc for the system model under consideration was also proved in [5, 6].

Thus, the design of such a mechanism is reduced to determining the set Mc and solving problem (1). The concordance conditions defining the set Mc, for which the maximum value of the efficiency index K(μ*, r) is achieved, will be presented below when formulating the solution of problem (1). As it was shown in [6], the concordance conditions are satisfied for the optimal mechanism; moreover, they stimulate the agent to fulfill the plan and report reliable information to the Principal.

Concordance conditions for fulfillment of plans and truth-telling

First, consider the constraints imposed on the set of mechanisms under which the agents fulfill their plans.

The fulfillment of plans can be guaranteed by applying penalties for any deviations of the agent’s result from the plan. For the set of admissible penalty functions considered here, the optimal penalties coincide with their maximum growth rate θi(xi, yi). The details can be found in [6].

The set Pi(yi−1, ri) of plans maximizing the goal functions of agents (upon fulfillment) is determined by the expression

$${P}_{i}({y}_{i-1},{r}_{i})=\left\{{x}_{i}| {f}_{i}({x}_{i},{x}_{i},{y}_{i-1},{r}_{i})\ge {f}_{i}({x}_{i},{y}_{i},{y}_{i-1},{r}_{i}),\,{x}_{i},{y}_{i}\in \left[0,{r}_{i}^{{\rm{upp}}}\right]\right\}.$$

The following result was established in [6]: for the penalty functions θi(xi, yi) satisfying the triangle inequality, the set Pi(yi−1, ri) coincides with the set

$${Y}_{i}({y}_{i-1},{r}_{i})={\rm{Arg}}\mathop{\max }\limits_{{y}_{i}\in [0,{r}_{i}]}{f}_{i}({x}_{i},{y}_{i},{y}_{i-1},{r}_{i})$$

of all rational strategies of agent i, i.e.,

$${Y}_{i}({y}_{i-1},{r}_{i})={P}_{i}({y}_{i-1},{r}_{i}).$$

The condition xi ∈ Pi(yi−1, ri) is called the maximum concordance condition.

For each agent, determine the sets \({P}_{i}\left({y}_{i-1},{r}_{i}^{{\rm{low}}}\right)\) of all plans implementable for the minimum value of the uncertain parameter \({r}_{i}^{{\rm{low}}}\) under the penalty functions θi(xi, yi). Assume that for \({r}_{i}^{{\rm{low}}}\), the assigned plan is fulfilled due to the penalties θi(xi, yi) only (the incentive function is equal to 0). The maximum value of the plan \({x}_{i}^{{\rm{c}}}={x}_{i}^{{\rm{c}}}\left({y}_{i-1},{r}_{i}^{{\rm{low}}}\right)\) in the set \({P}_{i}\left({y}_{i-1},{r}_{i}^{{\rm{low}}}\right)\) is determined by the inequality

$${\theta }_{i}\left({x}_{i}^{{\rm{c}}}-{y}_{i-1},{y}_{i}-{y}_{i-1}\right)\ge {\omega }_{i}\left({y}_{i}-{y}_{i-1},{r}_{i}^{{\rm{low}}}\right)-{\omega }_{i}\left({x}_{i}^{{\rm{c}}}-{y}_{i-1},{r}_{i}^{{\rm{low}}}\right)$$
(2)

for all yi from the range \(0\le {y}_{i}\le {r}_{i}^{{\rm{low}}}\).

The agents’ truth-telling is guaranteed by the perfect concordance conditions: each agent is assigned plans for which his goal functions achieve maximum on the set of admissible plans. As is well known [5, 6], the optimal planning procedure can be found among the ones satisfying the perfect concordance conditions. The optimal planning procedures will be determined using the optimal mechanism design method presented in [5, 6].

Conditions (2) and the perfect concordance conditions define the set Mc of incentive-compatible plans.

Optimal mechanism

Note that the function

$$\begin{array}{c}\Phi (x,x,r)={c}_{1}{x}_{1}-{v}_{1}({x}_{1},{r}_{1})+\mathop{\sum }\limits_{i=1}^{n}[{c}_{i}({x}_{i}-{x}_{i-1})-{v}_{i}({x}_{i},{r}_{i})]\\ \hspace{0.5em}=({c}_{1}+{a}_{1}){x}_{1}-{a}_{1}{r}_{1}+\mathop{\sum }\limits_{i=2}^{n}[({c}_{i}+{a}_{i})({x}_{i}-{x}_{i-1})-{a}_{i}{r}_{i}]\end{array}$$

increases in (xi − xi−1). Therefore, given the incentive funds g = (g1, …, gn), it suffices to maximize each term of the Principal’s goal function.

Introduce the numbers γi > 0 and the corresponding inequalities

$$({c}_{i}+{a}_{i})({x}_{i}-{x}_{i-1})-{a}_{i}{r}_{i}\ge {\gamma }_{i}.$$

From these inequalities, find the set of plans for which the value of the terms in the Principal’s goal function is not smaller than the corresponding value γi. This set is determined by the inequality

$${x}_{i}\ge {q}_{i}({x}_{i-1},{\gamma }_{i},{r}_{i})={x}_{i-1}+({\gamma }_{i}+{a}_{i}{r}_{i})/({c}_{i}+{a}_{i}).$$

Consider the planning procedure

$${\tilde{\pi }}_{i}({\gamma }_{i},{\rho }_{i},{x}_{i-1})=\left\{\begin{array}{ll}{x}_{i}^{{\rm{c}}}={x}_{i}^{{\rm{c}}}\left({x}_{i-1},{r}_{i}^{{\rm{low}}}\right),&{\rm{if}}\quad {r}_{i}^{{\rm{low}}}\le {\rho }_{i}\le {\beta }_{i}\\ {q}_{i}({x}_{i-1},{\gamma }_{i},{\rho }_{i}),&{\rm{if}}\quad {\beta }_{i}<{\rho }_{i}\le {r}_{i}^{{\rm{upp}}},\end{array}\right.$$
(3)

where the value βi is obtained by solving the equation

$${x}_{i}^{{\rm{c}}}\left({x}_{i-1},{r}_{i}^{{\rm{low}}}\right)={q}_{i}({x}_{i-1},{\gamma }_{i},{\beta }_{i}),\quad i=1,\ldots ,n.$$

For the planning procedure (3) determine the incentive functions σi(xi−1, yi) that satisfy the perfect concordance conditions with yi = xi = πi(γi, ρi, xi−1).

In accordance with [6], this incentive function is calculated by the formula

$${\sigma }_{i}({x}_{i-1},{y}_{i})=\left\{\begin{array}{ll}0&{\rm{for}}\,0\le {y}_{i}\le {x}_{i}^{{\rm{c}}}\\ \mathop{\int}\limits_{{x}_{i}^{{\rm{c}}}-{x}_{i-1}}^{{y}_{i}-{x}_{i-1}}{\omega ^{\prime} }_{it}(t,{\tilde{r}}_{i}({\gamma }_{i},t))dt&{\rm{for}}\,{\rm{c}}<{y}_{i}\le {q}_{i}\left({x}_{i-1},{\gamma }_{i},{r}_{i}^{{\rm{upp}}}\right)\\ {\bar{g}}_{i}&{\rm{for}}\,{q}_{i}({x}_{i-1},{\gamma }_{i},{r}_{i}^{{\rm{upp}}})<{y}_{i}\le {r}_{i}^{{\rm{upp}}}.\end{array}\right.$$

In this expression, \({\tilde{r}}_{i}({\gamma }_{i},t)\) denotes the inverse of the function \({\tilde{\pi }}_{i}({\gamma }_{i},{\rho }_{i},{x}_{i-1})\) in the variable ρi; \({\omega ^{\prime} }_{it}(t,{\tilde{r}}_{i}({\gamma }_{i},t))\) denotes the partial derivative of the agent’s cost function ωit(t, ri) with respect to the first variable; the value \({\bar{g}}_{i}\) is given by

$${\bar{g}}_{i}=\mathop{\int}\limits_{{x}_{i}^{{\rm{c}}}-{x}_{i-1}}^{{q}_{i}({x}_{i-1},{\gamma }_{i},{r}_{i}^{{\rm{upp}}})-{x}_{i-1}}{\omega ^{\prime} }_{it}(t,{\tilde{r}}_{i}({\gamma }_{i},t))dt.$$

The maximum value of the Principal’s goal function is achieved under the maximum values of the parameters γi, which are in turn achieved by exhausting the incentive funds g = (g1, …, gn).

Assume that the penalty functions are linear, i.e., θi(xiyi) = ki∣yi − xi∣.

Calculate \({x}_{i}^{{\rm{c}}}={x}_{i}^{{\rm{c}}}\left({y}_{i-1},{r}_{i}^{{\rm{low}}}\right)\) from condition (2). For the penalty functions under consideration, condition (2) is equivalent to

$${\left.\frac{d{\omega }_{i}(x,{r}_{i}^{{\rm{low}}})}{dx}\right|}_{x = {x}_{i}^{{\rm{c}}}}={e}_{i}({x}_{i}^{{\rm{c}}}-{x}_{i-1})/{r}_{i}={k}_{i}.$$

Hence, \({x}_{i}^{{\rm{c}}}={x}_{i-1}+{r}_{i}{k}_{i}/{e}_{i}\).

The parameter βi in expression (3) can be obtained from the condition

$${x}_{i}^{{\rm{c}}}\left({x}_{i-1},{r}_{i}^{{\rm{low}}}\right)={x}_{i-1}+({\gamma }_{i}+{\alpha }_{i}{\beta }_{i})/({c}_{i}+{a}_{i}),$$

which gives

$${\beta }_{i}=\left[\left({x}_{i}^{{\rm{c}}}({x}_{i-1},{r}_{i}^{{\rm{low}}})-{x}_{i-1}\right)({c}_{i}+{a}_{i})-{\gamma }_{i}\right]/{a}_{i}={r}_{i}{k}_{i}({c}_{i}+{a}_{i})/({e}_{i}{a}_{i})-{\gamma }_{i}/{a}_{i}.$$

As a result, expression (3) for the planning procedure takes the form

$${\tilde{\pi }}_{i}({\gamma }_{i},{\rho }_{i},{x}_{i-1})=\left\{\begin{array}{ll}{x}_{i-1}+{r}_{i}^{{\rm{low}}}{k}_{i}/{e}_{i},&{\rm{if}}\,{r}_{i}^{{\rm{low}}}\le {\rho }_{i}\le {\beta }_{i}\\ {x}_{i-1}+({\gamma }_{i}+{a}_{i}{r}_{i})/({c}_{i}+{a}_{i}),&{\rm{if}}\,{\beta }_{i}\le {\rho }_{i}\le {r}_{i}^{{\rm{upp}}}.\end{array}\right.$$
(4)

Determine \({\tilde{r}}_{i}({\gamma }_{i},t)\), the inverse of the function \({\tilde{\pi }}_{i}({\gamma }_{i},{\rho }_{i},{x}_{i-1})\) on the interval

$$\left({x}_{i}^{c}\left({x}_{i-1},{r}_{i}^{{\rm{low}}}\right)-{x}_{i-1},({\gamma }_{i}+{a}_{i}{r}_{i}^{{\rm{upp}}})/({c}_{i}+{a}_{i})\right).$$

Obviously,

$${\tilde{r}}_{i}({\gamma }_{i},t)=(t-{x}_{i-1})({c}_{i}+{a}_{i})/{a}_{i}-{\gamma }_{i}/{a}_{i}.$$

Hence, the planning procedure (4) satisfies the perfect concordance conditions [6] if the incentive function is given by

$${\sigma }_{i}({x}_{i-1},{y}_{i})=\left\{\begin{array}{l}0\quad {\rm{for}}\quad 0\le {y}_{i}\le {x}_{i-1}+{r}_{i}^{{\rm{low}}}{k}_{i}/{e}_{i}\\ ({y}_{i}-{x}_{i-1}-{r}_{i}^{{\rm{low}}}{k}_{i}/{e}_{i}){a}_{i}{e}_{i}/({c}_{i}+{a}_{i})\\ {\rm{for}}\quad {x}_{i-1}+{r}_{i}^{{\rm{low}}}{k}_{i}/{e}_{i}<{y}_{i}\le ({\gamma }_{i}+{a}_{i}{r}_{i}^{{\rm{upp}}})/({c}_{i}+{a}_{i})\\ {g}_{i}\quad {\rm{for}}\quad ({\gamma }_{i}+{a}_{i}{r}_{i}^{{\rm{upp}}})/({c}_{i}+{a}_{i})<{y}_{i}\le {r}_{i}^{{\rm{upp}}}.\end{array}\right.$$

Find the optimal value of the parameter \({\gamma }_{i}^{* }\) from the exhaustion condition of the incentive fund:

$${g}_{i}=\left[({\gamma }_{i}^{* }+{a}_{i}{r}_{i}^{{\rm{upp}}})/({c}_{i}+{a}_{i})-{r}_{i}^{{\rm{low}}}{k}_{i}/{e}_{i}\right]{a}_{i}{e}_{i}/({c}_{i}+{a}_{i}).$$

Hence,

$${\gamma }_{i}^{* }=\left[{g}_{i}({c}_{i}+{a}_{i})/{a}_{i}{e}_{i}+{r}_{i}^{{\rm{low}}}{k}_{i}/{e}_{i}\right]-{a}_{i}{r}_{i}^{{\rm{upp}}}$$

and the optimal value of the Principal’s goal function is

$$\Phi ({x}^{* },{x}^{* },r)=\mathop{\sum }\limits_{i=1}^{n}{\gamma }_{i}^{* }.$$

It should be emphasized that in this optimal mechanism, the agents are assigned the plans beneficial for them based on the information they actually report. Therefore, the Principal can receive from the agents the plans xi instead of the estimates ρi = ri, under the assumption that the agents can calculate the plans themselves. In this case, the optimal mechanism under consideration acquires the features of a counter-planning mechanism [8].

Optimal counter-planning mechanism

Since the mechanism under consideration satisfies the maximum and perfect concordance conditions, the levels of saving energy chosen by the agents coincide with the plans assigned to them, yi = xi, and the agents are interested in truth-telling, ρi = ri. In addition,

$${r}_{i}=({x}_{i}-{x}_{i-1})({c}_{i}+{a}_{i})/{a}_{i}-{\gamma }_{i}/{a}_{i}.$$

Substitute this result into the incentive function formula and take into account the equality yi = xi to arrive at the following expression for the incentive function on the interval defined by the inequalities

$${x}_{i-1}+{r}_{i}^{{\rm{low}}}{k}_{i}/{e}_{i}<{y}_{i}\le ({\gamma }_{i}+{a}_{i}{r}_{i}^{{\rm{upp}}})/({c}_{i}+{a}_{i}):\,{A}_{i}({x}_{i}-{x}_{i-1})-{k}_{i}| {y}_{i}-{x}_{i}| ,$$

where

$${A}_{i}=[{a}_{i}{e}_{i}-{k}_{i}({c}_{i}+{a}_{i}+{\gamma }_{i}^{* })]/({c}_{i}+{a}_{i}).$$

The incentive function Ai(xi − xi−1) − ki∣yi − xi∣, in combination with the planning procedure in which the agents report the plans instead of the parameters ρi, corresponds to the counter-planning mechanism [8]. In counter planning, the agents report to the Principal the plans that are beneficial for them (instead of the parameters ρi), and the Principal stimulates the agents to ensure the profitability of the "most intense” plans under given constraints on the incentive funds.

Thus, the optimality of the counter planning mechanism has been established.

2.2 Elements of Management Structure

In practice, the mechanisms for selecting priority projects, allocating financial resources among them, and stimulating energy-saving activities require some changes in the innovations management structure of the organization.

Consider two modes of management, in which these mechanisms are used. They will be called the modes of strategic and operational management. In the organizational structure, the blocks of strategic and operational management are formed accordingly.

The mode of strategic management covers the following problems:

—classification and selection of the most promising projects (some methods for solving these problems were discussed in Section 2 of the first part; see [3]);

—allocation of financial resources for the most promising projects and formation of orders on the development of financial models and business plans for their implementation, which are sent to the block of operational management (this problem can be solved using the modification of cost-effectiveness analysis described in Section 3 of the paper [3]);

—development and approval of incentive mechanisms for implementing the complex of projects (an incentive mechanism for a chain of project works has been described in Section 2.1 of this paper).

The mode of operational management covers the following problems:

—monitoring of current projects and development of corrective actions;

—provision of relevant information on any changes and the appearance of innovations for current projects, which is intended for the block of strategic management;

—development of financial models and business plans for further implementation of the projects selected;

—network planning and scheduling;

—implementation of incentive mechanisms.

The mechanisms described above can be applied under some necessary changes in the existing management structure of the organization.

Here are two aspects that can be taken into account in the management structure.

The first aspect is that the structure related to strategic management should include units (departments or persons) responsible for different levels of the integrated assessment procedure described above; see Fig. 2.

Fig. 2
figure 2

Efficiency assessment elements included in project management structure: example.

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The left-hand side of this figure shows the structure of the integrated assessment procedure considered in Section 2 of the paper [3]. The right-hand side of this figure presents the elements of the hierarchical management structure responsible for the integrated and intermediate assessments of projects and their monitoring. For example, element 1 is responsible for calculating the final assessments (ratings) of projects and observing their dynamics over time; element 2 is subordinate to element 1, being responsible for the intermediate assessment and monitoring of projects in accordance with the intermediate assessment F. The responsibilities of elements 3 and 4 in this hierarchical structure are distributed by analogy, in accordance with the intermediate estimates D and E.

The second aspect of corrections in the management structure is determined by the resource allocation mechanism for different groups of projects; see Section 3 of the paper [3].

Large organizations such as JSC RZhD, as a rule, have complex distributed structures, by territory and functions. Therefore, when forming an innovative development program, it is important to consider the interests of all the main structural units. Otherwise, in the long run the lop-sided innovation strategy will reduce the efficiency of the entire system. Consider the design problem of a concordant innovative program, in which each main structural unit is guaranteed some degree of participation. Two types of such guarantees (concordance conditions) will be considered as follows.

1. Concordance by resource, when each structural unit surely receives a minimum amount of funds for innovative development.

2. Concordance by effectiveness, when each structural unit is surely included in the program of projects with a total effect exceeding some lower bound.

Note that concordance by effectiveness has a significantly greater degree of manipulation (strategic behavior for distorting the actual information) than concordance by resource. Really, a structural unit surely included in the program of projects with a total effect exceeding some lower bound will be interested in overestimating the project costs.

On the contrary, in accordance with the resource concordance condition, the most efficient projects are included in the program. Therefore, further analysis will be confined to the case of concordance by resource.

In general, the allocation method of financial resources is similar to the one described in Section 3 of the first part (see [3]). It consists of two stages as follows.

Stage I. For each structural unit participating in the program, select the minimum set of projects with the volume of financing not smaller than a guaranteed threshold, using the algorithm from Section 3.

Stage II. Apply the algorithm described in Section 3 of the first part (see [3]) to the entire set of projects.

The allocation of financial resources among the complex projects of the network multi-project system under consideration is then used to determine the composition of the project offices (POs) of the hierarchical management structure (Fig. 3).

Fig. 3
figure 3

Project offices in management structure: example.

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In Fig. 3, CO means the central project office; the numbers 1, 2, 3, …, n indicate different projects in a multi-project system. As an illustration, projects 1, 2, 5, and 6 are combined into a group of projects with the first priority for financial resources, in accordance with the cost-effectiveness analysis procedure discussed in Section 3 of the first part (see the paper [3]); projects 3, 4, and 7 form the group with the second priority; finally, the last group of projects to receive the residual financial resources contains four projects, including project n.

The proposals on possible corrections in the management structure of the organization are intended to improve the effectiveness of research, development, and implementation of the most promising projects. They can serve as additional tools for the implementation of the existing standards [15–18] and management system [11–14] of scientific and technical activity adopted in the organization.

3 Conclusions

The integrated assessment system and the allocation mechanism of investments proposed in this paper can be used to form a responsibility matrix for implementing complex projects within the organization. The models for assessing and identifying priority projects, as well as the allocation mechanism of financial resources based on the choice of the most effective projects, contribute to the concentration of financial resources on the most promising areas of scientific and technological development.

The system of projects selection and grouping into multi-project complexes can be employed for refining the process-based life cycle management models of scientific-technical and innovative developments.

The models studied in this paper can be used to form project offices for managing the implementation of complex projects; the incentive mechanisms can be embedded into the statutes for stimulating the scientific, R&D, and implementation activities of working groups.

Considered in the aggregate, the mechanisms presented in this paper give an example of an integrated mechanism for managing complex projects.

4 Funding

This work was supported in part by the Russian Foundation for Basic Research, and OJSC RZhD, project no. 17-20-05216.