The incentive problem of executors of the new products development project at the industrial enterprise is considered in this article. Mastering of a new product leads to the learning effect, which implies reduction of time spent on performing repetitive tasks by workers, resulting in a dynamic change in the economic performance of production. The project of the new products development is considered as a managed hierarchical dynamic system, consisting of a project management board (principal) and teams of agents. The interaction of project participants is formalized as a hierarchical dynamic game. To solve the problem, a numerical algorithm is developed based on a sequential solution of two optimal control problems that are solved using the Bellman dynamic programming method.
(
where Qimin – minimum production volume, taking into account technological and logistic requirements, Qmax – maximum production volume limited by the production capacity of equipment.
The labor costs of manufacturing the product in period t are defined as the multiplication of the product labour intensity cpt and production volume in this period ut:
Сpt сptut , t 1,T. (
Let us substitute the expression (
The production volumes of the finished product ut are chosen by the principal while planning production activities based on its goal function.
Several options are considered as a goal function of the principal. 1. Minimization of the discounted cumulative labor costs of all agents producing a new product: where rp – discount rate of the principal. 2. Maximization of discounted profit from the production of a new product:
bp
T Pt1ut ap xt1 ut max, J p
t1 (1 rp )t where Pt-1 - price of the new product. 3. Maximization of the production volume of a new product:
J p xT max . 4. Minimization of implementation time of the new product development project:
J p T min .
Let us formulate the dynamic task of planning the production volumes of a new product for the principal.
The dynamic planning task consists of finding optimal production volumes utopt , t 1, n satisfying the constraint (
To solve the formulated optimal control problem, Bellman's dynamic programming method [7], implemented in the Free pascal programming environment, was used. Formulation and solutions for dynamic planning problems are given in [8].
As a result of solving the problem of dynamic planning, the principal determines the optimal production volumes of a new product utopt , t 1, n . To implement the project, it is necessary that the production volumes of the product and parts match. But the choice of the actual production volumes of parts, from which the finished product is assembled, is made by agents upon their own interests. The principal influences the production process through the mechanism of material incentives, encouraging agents economically to fulfill the planned production volumes. 2.2. The decision-making model of an agent
The dynamics of the production activity of the i-th agent who manufactures the parts for a new product is described by a discrete equation: yt yt1 vt , t 1,T , (12) where yt - cumulative production volume by the agent for the t-th time period, vt - production volume by the agent at the period t.
The choice of the production volume vt at the period t is the agent’s management.
In the initial period, the number of parts produced by agent is known: y0 Y0 , (13) in the final period, the cumulative volume of the parts produced by the agent should be equal to the specified volume: yT Y0 R, (14) where R – specified number of parts, which coincides with the number of finished products.
The production volume of parts in each period t is imposed by the following restrictions:
Qmin vt Qmax , t 1,T , (15) where Qimin – minimum parts production volume, considering technological and logistic requirements, Qmax – maximum parts production volume limited by the production capacity of equipment.
Restrictions on the parts production volume coincide with the restriction on the final product production volume.
The agent labor costs in monetary terms in the period t are defined as the multiplication of part labor intensity cat, the cost of the norm-hour at the enterprise s, and the parts production volume in this period vt : Сat sсatvt , t 1,T.
Principal uses a dynamic proportional incentive system for the project implementation: t t , vt tvt , t 1,T , The goal function of the agent is to maximize the discounted income:
Ja T t t ,vt Cat max, (21) t 1 1 ra t where ra – agent’s discount rate.
The agent income is the difference between the financial incentives and his labor costs, expressed in monetary terms. Taking into account (18) and (19), the agent goal function (21) will be as follows:
Ja T tvt saa xtb1a vt max .
t1 1 ra t
The stated problem is the problem of discrete system optimal control for an agent. The solution of the stated problem is an optimal control vtopt, t=1,n, satisfying constraint (15), which transfers the discrete system (12) from the initial state (13) to the final state (14) and maximizes the agent’s discounted income (22). Alongside, the solution of the agent‘s optimization task depends on the payment rates for the production unit αt, t=1,n, which are given by the principal.
To solve the formulated optimal control problem, Bellman's dynamic programming method [7], implemented in the Free pascal programming environment, was used. (18) (20) (22) (23) 2.3. Algorithm for solving the dynamic incentive problem
Let us formulate the problem of agent dynamic incentive:
T J p g p (t, ut , xt1) max(min),
t1 Qxt min xt1ututQ, mtax,1,Tt , 1,T , x0 X 0 , xT X 0 R, J a T tvt saa ytb1a vt max,
t1 1 ra t
T
tvt F ,
t1
Qytmin ytv1tvQt, matx, 1t,T 1,,T ,
y0 Y0 ,
yT Y0 R.
where gp - specific form of the goal function of the principal, is determined by one of the expressions (
The formulated problem (23) represents a dynamic game, the solution of which determines the conditions for coordination between the principal and the agent. The solution of the dynamic game will be the parameters of the incentive system and the production volumes of the parts t*, vt* ut* , t 1,T , that deliver the extremum to the goal functions of the principal and the agent.
Let us formulate an algorithm for solving the dynamic problem of proportional incentive.
1. The optimal control problem for the principal is solved using Bellman's dynamic programming method, the new product optimal planned volumes ut, t=1,T are found.
2. The principal management is being set – the parameters of the incentive function αt,, t=1,T should meet the condition of not exceeding the agent's payroll budget: T tut ( t ) F.
t 1 3. For the given parameters of the incentive function αt, t=1,T the optimum control problem for the agent is solved using the Bellman dynamic programming method, and the agent's optimal response is determined as an actual part production volumes vt(αt), t=1,T.
4. The coincidence condition of the planned and actual production volumes that the agent chooses is checked:
T
ut vt ( t )2 ,
t 1
where ε- predetermined small value. If the condition is satisfied, then the dynamic proportional incentive problem is solved. If
not, then the parameters of the incentive function αt, t=1,T are changed and the step 2 is repeated. The principal goal functions
(
1. Minimization of discounted costs for agent incentives:
J p tT1 1trvpt t min . (24) 2. Maximization of discounted income from the agent's production activity:
T ( pt t )vt max, J p
t 1 1 rp t
Qytmin ytv1tvQt, matx, 1t,T 1,,T , where pt – part price produced by the agent.
In this case, the task of the agent dynamic incentive is as follows:
T J p g p (t, vt , t ) max(min), t1
T tvt saa ytb1a vt max .
J a t1 1 ra t (25) (26) where gp - specific form of the goal function of the principal, is determined by one of the expressions (24) - (26).
The solution of the dynamic game will be the parameters of the incentive function and the parts production volumes t*, vt* , t 1,T , that deliver the extremum to the goal functions of the principal and the agent. Let us formulate an algorithm for solving the dynamic problem of proportional incentive in the case when the goal function of the principal depends on the incentive costs (26): 1. The parameters of the principal incentive function αt, t=1,T are given.
2. For the given parameters of the incentive function αt, t=1,T the optimum control problem for the agent is solved using the Bellman dynamic programming method and the agent's optimal response is determined - the actual part production volumes vt(αt), t=1,T.
3. The found agent's response vt(αt) is substituted into the optimal control problem for the principal, which is solved by Bellman's dynamic programming method. Thus the optimal parameters of the incentive function αt(vt(αt)), t=1,T are found.
4. The condition of coincidence of the parameters of the incentive function at the given iteration and the previous one is checked:
T t (vt ( t )) t 2 , t 1 where ε- predetermined small value. If the condition is satisfied, then the dynamic proportional incentive problem is solved. If not, the parameters of the incentive function must be changed and the step 2 repeated.
The problem of dynamic incentive with the following initial data is considered.
12 42,64 xt01,7ut min J p
t1 1 rp t xt xt1 ut , t 1,12, 0 ut 40, t 1,12, x0 1, xT 241, J a T tvt 3837 ,6 yt01,1vt max,
t1 1 ra t T tvt 960000 , t1 yt yt1 vt , t 1,12, 0 vt 40, t 1,12, y0 1, yT 241.
The numerical solution of the problem was obtained using the proposed algorithm.
The planned trajectory corresponds to the trajectory, which minimizes labor costs of agents and coincides with the rate of a product mastering bp=-0,7. At a constant parameters of an incentive function the agent chooses an actual trajectory of the production cumulative volume, which corresponds to an agent’s learning rate ba=-0,1. Figure 1 shows the planned and actual trajectory of the cumulative volume of production of a new product.
1 2 3 4 5
6 7 8 periods, months 9 10 11 12
4. Applying a payment rate in the form of a linearly decreasing function from the cumulative production volume leads to the agent selection of trajectories with a lower learning rate. The agent moves from "fast" trajectory to "slow", less "convex" one. The lager modulo value of the control parameter k corresponds to the agent’s chose of a less "convex" trajectory (Fig. 3). actual planned k=0 k=0,05 k=0,15 1
Dynamic decision-making models for the principal and agent in the project for the production of a new product have been developed. The problems of agent dynamic incentive for various goal functions of the principal are formulated. Two options are considered: the goal functions include the costs of material incentives for the agent and do not include them.
For both variants, a numerical algorithm is proposed, based on a sequential solution of two optimal control problems, which are solved using Bellman's dynamic programming method.
A numerical example of the problem solution for the principal goal function, which does not depend on the agent incentive costs, is given. It is shown that the application of the agent salary rate in the form of a linear function, that depends on the cumulative production volume, ensures that the agent selects the planned trajectory of the principal.
The reported study was funded by RFBR and Samara region according to the research project № 17-46-630606.