**4. Simulation results**

#### **4.1. Simulation of continuous review model**

To illustrate numerically the results obtained, firstly we present some simulations for opti‐ mal control of the continuous-review integrated production-forecasting system with stockdependent demand and deteriorating items. The data used in this simulation is presented in Table 1.

Using the MATLAB software, we implemented the results of the previous section and ob‐ tained the graphs below. Figures 1, 2 and 3 show the variations of Δ*I*(*t*), Δ*D*(*t*), and Δ*P*(*t*). We observe that they all converge toward zero, as desired.

Using equation (3), the optimal cost is found to be *J* = 3047.93. A sensitivity analysis is per‐ formed in order to assess the effect of some of the system parameters on the optimal cost. The analysis is conducted by keeping the values of the parameters at the base values shown in Table 1 and varying successively one parameter at a time. We were interested in the effect on the value of the optimal objective function *J* of the parameters *a*, *b*, and *θ*, that we varied from 0.1 to 0.9. Table 2 summarizes the results of the sensitivity analysis.

**Figure 2.** Variations of the optimal demand rate

Optimal Control of Integrated Production – Forecasting System 131

**Figure 3.** Variations of the optimal production rate

**Figure 1.** Variation of the optimal inventory level

**Figure 2.** Variations of the optimal demand rate

**4. Simulation results**

130 Decision Support Systems

Table 1.

**4.1. Simulation of continuous review model**

**Figure 1.** Variation of the optimal inventory level

We observe that they all converge toward zero, as desired.

from 0.1 to 0.9. Table 2 summarizes the results of the sensitivity analysis.

To illustrate numerically the results obtained, firstly we present some simulations for opti‐ mal control of the continuous-review integrated production-forecasting system with stockdependent demand and deteriorating items. The data used in this simulation is presented in

Using the MATLAB software, we implemented the results of the previous section and ob‐ tained the graphs below. Figures 1, 2 and 3 show the variations of Δ*I*(*t*), Δ*D*(*t*), and Δ*P*(*t*).

Using equation (3), the optimal cost is found to be *J* = 3047.93. A sensitivity analysis is per‐ formed in order to assess the effect of some of the system parameters on the optimal cost. The analysis is conducted by keeping the values of the parameters at the base values shown in Table 1 and varying successively one parameter at a time. We were interested in the effect on the value of the optimal objective function *J* of the parameters *a*, *b*, and *θ*, that we varied

**Figure 3.** Variations of the optimal production rate


**Parameters Values planning horizon length** *T* 10 **number of subintervals** *N* 51 **coefficient for demand dynamics** *a* 0.1 **coefficient for demand dynamics** *b* 0.2 **deterioration rate θ** 0.01 **deviation cost for inventory level** *qI* 20 **deviation cost for demand rate** *qD* 15 **deviation cost for production rate** *r* 0.01

For the periodic review case, the simulation results are shown in the graphs below. Figure 4 shows the variations of the optimal inventory level and the inventory goal level. We observe

**I(k) and Ihat(k)**

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> <sup>45</sup> <sup>50</sup> 3.5

*Time k*

^

(*k*)very closely.

Optimal Control of Integrated Production – Forecasting System 133

I(k) Ihat(k)

**Table 3.** Data for periodic-review model

4

**Figure 4.** Optimal and inventory goal levels

4.5

5

5.5

6

6.5

7

7.5

that except for the early transient periods, *I*(*k*) follows *I*

**Table 2.** Sensitivity analysis

As can be seen, the objective function decreases as any of the three parameters increases.

#### **4.2. Simulation of periodic review model**

In this second part of the simulation, we illustrate the results obtained on the optimal con‐ trol of the periodic review integrated production-forecasting system with stock-dependent demand and deteriorating items. Thus, consider the production planning problem for a firm for the next *T* units of time. Divide this interval into *N* subintervals of equal length. Assume the product in stock deteriorates at the rate *θ*. Assume also the variations of the demand rate occur according to the dynamics (10). The firm has set the following targets. For *k* =1, ⋯, *N* , the goal inventory level and goal demand rate are assumed to be as follows:

*I* ^ (*k*)=5 <sup>+</sup> 1.5 *sign*(*sin*( <sup>2</sup>*π<sup>k</sup>* <sup>40</sup> )) and *D* ^ (*k*)=2 <sup>+</sup> 0.5 *sign*(*sin*( <sup>2</sup>*π<sup>k</sup>* <sup>15</sup> ))

where the sign function of a real number *x* is defined by

$$
\operatorname{sign}(\mathbf{x}) = \begin{vmatrix}
\cdot \mathbf{1} & \text{if } \mathbf{x} \le \mathbf{0}, \\
\mathbf{0} & \text{if } \mathbf{x} = \mathbf{0}, \\
\mathbf{1} & \text{if } \mathbf{x} \ge \mathbf{0}.
\end{vmatrix}
$$

We have to note that the goal inventory level and the goal demand rate were constant in the continuous review case.

The goal production rate is then computed using

$$
\stackrel{\frown}{P}(k) = \stackrel{\frown}{D}(k) + \stackrel{\frown}{\partial}\stackrel{\frown}{I}(k),
$$

where we assume that the inventory goal level is constant over a certain range. The penalties for deviating from these targets are *qI* for the inventory level, *qD* for the demand rate, and *r* for the production rate. The data are summarized in Table 3.


**Table 3.** Data for periodic-review model

*a* / *b* / θ *J*(*a*) *J*(*b*) *J*(θ)

0.1 5822.40 23426.21 3047.93 0.2 489.77 7648.73 463.79 0.3 485.62 5560.00 461.53 0.4 481.57 740.12 459.30 0.5 477.58 659.34 457.09 0.6 473.67 600.85 454.91 0.7 469.84 556.02 452.76 0.8 466.08 520.24 450.63 0.9 462.38 490.83 448.52

As can be seen, the objective function decreases as any of the three parameters increases.

In this second part of the simulation, we illustrate the results obtained on the optimal con‐ trol of the periodic review integrated production-forecasting system with stock-dependent demand and deteriorating items. Thus, consider the production planning problem for a firm for the next *T* units of time. Divide this interval into *N* subintervals of equal length. Assume the product in stock deteriorates at the rate *θ*. Assume also the variations of the demand rate occur according to the dynamics (10). The firm has set the following targets. For *k* =1, ⋯, *N* , the goal inventory level and goal demand rate are assumed to be as follows:

^ (*k*)=2 <sup>+</sup> 0.5 *sign*(*sin*( <sup>2</sup>*π<sup>k</sup>*

We have to note that the goal inventory level and the goal demand rate were constant in the

where we assume that the inventory goal level is constant over a certain range. The penalties for deviating from these targets are *qI* for the inventory level, *qD* for the demand rate, and *r*

<sup>15</sup> ))

**Table 2.** Sensitivity analysis

132 Decision Support Systems

(*k*)=5 <sup>+</sup> 1.5 *sign*(*sin*( <sup>2</sup>*π<sup>k</sup>*


continuous review case.

^ (*k*) <sup>+</sup> *<sup>θ</sup> <sup>I</sup>* ^ (*k*)

*I* ^

*P* ^(*k*)=*<sup>D</sup>*

*sign*(*x*)={

**4.2. Simulation of periodic review model**

<sup>40</sup> )) and *D*

where the sign function of a real number *x* is defined by

The goal production rate is then computed using

for the production rate. The data are summarized in Table 3.

For the periodic review case, the simulation results are shown in the graphs below. Figure 4 shows the variations of the optimal inventory level and the inventory goal level. We observe that except for the early transient periods, *I*(*k*) follows *I* ^ (*k*)very closely.

**Figure 4.** Optimal and inventory goal levels

Figure 5 shows the variations of the optimal demand rate and the demand goal rate. We ob‐ serve that except for the early transient periods, *D*(*k*) follows *D* ^ (*k*) very closely.

Finally, Figure 6 shows the variations of the optimal production rate and the production goal rate. We again observe that except for the early transient periods, *P*(*k*) follows *P* ^ (*k*) very closely.

The optimal cost is found to be *J* =10.3081. Here also a sensitivity analysis is performed in order to assess the effect of some of the system parameters on the optimal cost. The analysis is conducted by keeping the values of the parameters at the base values shown in Table 3 and varying successively one parameter at a time. We were interested in the effect on *J* of the parameters *a*, *b*, and *θ*, that we varied from 0.1 to 0.9. Table 4 summarizes the results of the sensitivity analysis.

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> <sup>45</sup> <sup>50</sup> 1.4

**a/b/θ J(a) J(b) J(θ)**

**0.1** 10.3081 10.0807 10.3072

**0.2** 10.3173 10.3081 10.3064

**0.3** 10.3339 10.6868 10.3056

**0.4** 10.3601 11.2166 10.3050

**0.5** 10.4004 11.8973 10.3044

**0.6** 10.4630 12.7284 10.3036

**0.7** 10.5651 13.7098 10.3040

**0.8** 10.7444 14.8413 10.3033

**0.9** 11.0880 16.1226 10.3032

**Table 4.** Effect of the parameters *a*, *b* and θ on the optimal cost *J*

**Time k**

P(k) and Phat(k)

P(k) Phat(k)

Optimal Control of Integrated Production – Forecasting System 135

1.6

**Figure 6.** Optimal and production goal rates

1.8

2

2.2

2.4

2.6

2.8

**Figure 5.** Optimal and demand goal levels

**Figure 6.** Optimal and production goal rates

Figure 5 shows the variations of the optimal demand rate and the demand goal rate. We ob‐

Finally, Figure 6 shows the variations of the optimal production rate and the production goal rate. We again observe that except for the early transient periods, *P*(*k*) follows *P*

The optimal cost is found to be *J* =10.3081. Here also a sensitivity analysis is performed in order to assess the effect of some of the system parameters on the optimal cost. The analysis is conducted by keeping the values of the parameters at the base values shown in Table 3 and varying successively one parameter at a time. We were interested in the effect on *J* of the parameters *a*, *b*, and *θ*, that we varied from 0.1 to 0.9. Table 4 summarizes the results of

**D(k) and Dhat(k)**

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> <sup>45</sup> <sup>50</sup> 1.4

**Time k**

^ (*k*) very closely.

^ (*k*)

D(k) Dhat(k)

serve that except for the early transient periods, *D*(*k*) follows *D*

very closely.

134 Decision Support Systems

the sensitivity analysis.

1.6

**Figure 5.** Optimal and demand goal levels

1.8

2

2.2

2.4

2.6

2.8


**Table 4.** Effect of the parameters *a*, *b* and θ on the optimal cost *J*

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

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‐

Optimal Control of Integrated Production – Forecasting System 137

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‐

This work has been supported by the Research Center of College of Computer and Informa‐

1 King Saud University, College of Computer and Information Sciences, Department of

2 Dalhousie University, Faculty of Management, School of Business Administration, Halifax,

3 American University in Dubai, College of Business Administration, Department of Man‐

[1] Strijbosch, L.W.G., Syntetos, A.A., Boylan, J.E., and Janssen, E. On the interaction be‐ tween forecasting and stock control: The case of non-stationary demand. Internation‐

al Journal of Production Economics2011; 133:1, 470-480.

sively studied.

trol.

**Acknowledgement**

**Author details**

Nova Scotia, Canada

agement, Dubai, UAE

**References**

R. Hedjar1

tion Sciences, King Saud University.

, L. Tadj2\* and C. Abid3

\*Address all correspondence to: Lotfi.Tadj@dal.ca

Computer Engineering, Riyadh, Saudi Arabia
