**5. Conclusion**

In this chapter we have used optimal control theory to derive the optimal production rate in a manufacturing system presenting the following features: the demand rate during a certain period depends on the demand rate of the previous period (dependent demand), the de‐ mand rate depends on the inventory level, items in inventory are subject to deterioration, and the firm adopts either a continuous or periodic review policy. In contrast to most of the existing research which uses time series forecasting models, we propose a new model, namely, the demand dynamics equation. This model approaches realistic problems by inte‐ grating the forecasting component into the production planning problem with deteriorating items and stock dependent demand under continuous-review policy. Simulations were con‐ ducted in order to show the performance of the obtained solution. The theoretical and the simulations results allow gaining insights into operational issues and demonstrating the scope for improving stock control systems.

Of course, as with any research work, this study is not without limitations. The main contri‐ bution of our model is equation (5) where we use the demand from the previous period to predict the demand in the current period. The main limitation of that equation is that it in‐ volves two coefficients. We have assumed in this chapter that the parameters *a* and *b* of the demand state equation are known. However, in real life, that may not be the case. We are currently further investigating this model to estimate these parameters in the case when they are unknown, using self-tuning optimal control.

Another research direction would be to use a predictive control strategy where, given the current inventory level, the optimal production rates to be implemented at the beginning of each of the following periods over the control horizon, are determined. Model predictive control (or receding-horizon control) strategies have gained wide-spread acceptance in in‐ dustry. It is also well-known that these models are interesting alternatives for real-time con‐ trol of industrial processes. In the case where the above parameters *a* and *b* are unknown, the self-tuning predictive control can be applied. The proposed control algorithm estimates online these coefficients and feeds the controller to take the optimal production decision.

Note that our state equations are linear and thus linear model predictive control (LMPC), which is widely used both in academic and industrial fields, can be used. Nonlinear model predictive control (NMPC) can be used in case one of the state equations is nonlinear, for example, if equation (5) were of the form

$$\frac{d}{dt}D(t) = aD(t) + \alpha I(t)^{\beta} \tag{51}$$

NMPC has gained significant interest over the past decade. Various NMPC strategies that lead to stability of the closed-loop have been developed in recent years and key questions such as the efficient solution of the occurring open-loop control problem have been exten‐ sively studied.

The case combining unknown coefficients and a nonlinear relationship between the demand rate and the on-hand inventory yields a very complex, highly nonlinear process for which there is no simple mathematical model. The use of fuzzy control seems particularly well ap‐ propriate. Fuzzy control is a technique that should be seen as an extension to existing con‐ trol methods and not their replacement. It provides an extra set of tools which the control engineer has to learn how to use where it makes sense. Nonlinear and partially known sys‐ tems that pose problems to conventional control techniques can be tackled using fuzzy con‐ trol.

### **Acknowledgement**

The second column of Table 4 shows that the optimal cost increases as *a* increases. The third column of Table 4 shows that the optimal cost increases also as *b* increases. The effect of *b* on *J* is however more significant than the effect of *a*. Finally, column 4 of Table 4 shows that the

In this chapter we have used optimal control theory to derive the optimal production rate in a manufacturing system presenting the following features: the demand rate during a certain period depends on the demand rate of the previous period (dependent demand), the de‐ mand rate depends on the inventory level, items in inventory are subject to deterioration, and the firm adopts either a continuous or periodic review policy. In contrast to most of the existing research which uses time series forecasting models, we propose a new model, namely, the demand dynamics equation. This model approaches realistic problems by inte‐ grating the forecasting component into the production planning problem with deteriorating items and stock dependent demand under continuous-review policy. Simulations were con‐ ducted in order to show the performance of the obtained solution. The theoretical and the simulations results allow gaining insights into operational issues and demonstrating the

Of course, as with any research work, this study is not without limitations. The main contri‐ bution of our model is equation (5) where we use the demand from the previous period to predict the demand in the current period. The main limitation of that equation is that it in‐ volves two coefficients. We have assumed in this chapter that the parameters *a* and *b* of the demand state equation are known. However, in real life, that may not be the case. We are currently further investigating this model to estimate these parameters in the case when

Another research direction would be to use a predictive control strategy where, given the current inventory level, the optimal production rates to be implemented at the beginning of each of the following periods over the control horizon, are determined. Model predictive control (or receding-horizon control) strategies have gained wide-spread acceptance in in‐ dustry. It is also well-known that these models are interesting alternatives for real-time con‐ trol of industrial processes. In the case where the above parameters *a* and *b* are unknown, the self-tuning predictive control can be applied. The proposed control algorithm estimates online these coefficients and feeds the controller to take the optimal production decision.

Note that our state equations are linear and thus linear model predictive control (LMPC), which is widely used both in academic and industrial fields, can be used. Nonlinear model predictive control (NMPC) can be used in case one of the state equations is nonlinear, for

*<sup>β</sup>* (51)

*dt D*(*t*)=*aD*(*t*) + *αI*(*t*)

optimal cost decreases as *θ* increases. The effect of *θ* is however almost negligible.

**5. Conclusion**

136 Decision Support Systems

scope for improving stock control systems.

example, if equation (5) were of the form

*d*

they are unknown, using self-tuning optimal control.

This work has been supported by the Research Center of College of Computer and Informa‐ tion Sciences, King Saud University.
