This note is devoted to multiperiodically operated complex system with inventory couplings transferring waste products from some subsystems as useful components to other subsystems. The flexibility of the inventory couplings is used to force each of the subsystems with its own period and to exploit its particular dynamic properties. This enhances the performance of the complex system endowed with many recycling loops, which reduce the amount of waste products endangering the natural environment. The subsystems are characterized by generalized populations composed of the individuals (the cycles), each of them encompasses its period, its initial state, its local control, and its inventory interaction. An evolutionary optimization algorithm employing such generalized populations coordinated on the basis of the inventory interaction constraints is developed. It includes the stability requirements imposed on the cyclic control processes connected with particular subsystems. The algorithm proposed is applied to the global multiperiodic optimization of some interconnected chemical production processes.

Optimization of complex production systems composed of cooperating subsystems may essentially enhance their overall performance. Multiperiodically operated complex system with the inventory couplings offers new possibilities of improving the system productivity. In such systems waste products of some subsystems can be transferred to suitably chosen cooperating subsystems utilizing them as useful process components. This concept is connected with the tendency of the rearrangement of complex industrial production systems from an open loop form, yielding many waste products, to a closed loop form recycling waste products, and having desired ecological features [

The optimal periodic control (OPC) problem has reached much attention in the literature, where various periodic optimization methods have been presented exploiting the gradient-type hill-climbing [

In the present paper the evolutionary algorithm described in [

Consider the following globally optimal multiperiodic control (GOMC) problem for systems composed of

The objective function (

The constraints (

The constraints (

To develop an optimization algorithm suitable for the problem discussed we use the time scaling

We convert by this way the GOMC problem to the following normalized and discretized form: minimize the global objective function

Setting

We are aimed at the comparison of the three nested kinds of the complex process operation, namely, the static one, the periodic one, and the multiperiodic one.

The way of coding any optimization problems has a main impact on the effectiveness of the evolutionary algorithm. The wrong choice of individuals can cause that the evolutionary algorithm can stack in local solutions or time of searching can increase to not acceptable value. Owing to complexity of the GOMC problem we think that the best way of coding problem (

The form of the individual

A nonuniform mutation operator can be used together with a uniform crossing operator. It creates a new gene

Crossing or mutation operators can give a new individual, which will not fulfil the periodic constraint (

The periodic constraint (

One has the following steps.

Create set

Create function

Calculate probability

Calculate distribution

Draw

Set

If

Draw

If

If

If

If

Having Algorithm

One has the following steps.

Select a control variable

If you chose the control variable to reconstruction, set

If you chose the control variable, modify an element

If the constraint (

Algorithm

We introduce the following extended performance index used at the selection step [

Having in mind the above remarks, we propose the following.

One has the following steps.

Assume initial data:

Exploiting Algorithm

Exploiting Algorithm

Evaluate the performance index

Let

Repeat the following activities until the current population is empty:

draw from the current population

take from

remove from the current population

Applying the crossing operator (

For each cycle

Theory says that the probability of finding a globally optimal solution is close to 1 when the number of generations and individuals is infinity [

Let the parallel chemical reactions

The evolutionary algorithm is time consuming. Therefore one should first run

Since the nonzero conditions for the product concentrations occur in the solutions obtained (see Figure

Diagram of the parallel chemical reactions from Example

The results of the

The suboptimal multiperiodic control

The suboptimal multiperiodic inventory interaction

The suboptimal multiperiodic state

The suboptimal multiperiodic state

The suboptimal multiperiodic cycle for Example

Let us consider the same GOMC problem as in Example

The results presented below were obtained for the following subsystem parameters:

The results of the

The suboptimal multiperiodic control

The suboptimal multiperiodic control

The suboptimal multiperiodic inventory interaction

The suboptimal multiperiodic inventory interaction

The suboptimal multiperiodic state

The suboptimal multiperiodic state

The suboptimal multiperiodic cycle for Example

Multiperiodically operated complex system with the inventory couplings has been modelled and characterized as a system with a high degree of freedom. The operation period, the initial state, the discretized control, and the discretized inventory interaction have been assumed as the local finite-dimensional optimization argument (the controlled cycle of the subsystem). The set of the cycles connected with all the subsystems has been treated as the controlled multicycle of the complex system and optimized by the evolutionary algorithm dealing with the generalized populations coordinated with the aid of the averaged inventory interaction constraints. The use of different operation periods and different stability margins of the subsystems has been taken into account. It has been shown that the multiperiodic operation of the complex cross-recycled chemical production systems may improve its global performance as compared with the static operation, and the periodic operation. This may enhance the utilization of waste products endangering the natural environment.