*5.2. Mathematical Formulation of the Production Problem*

The inventory level of the problem at any time *t* satisfies the governing differential equations

$$\frac{dq(t)}{dt} + \theta q(t) = P(t) - D \text{ for } 0 \le t \le t\_1 \tag{5}$$

$$\frac{dq(t)}{dt} + \theta q(t) = -D \text{ for } t\_1 < t \le T \tag{6}$$

with the conditions *q*(0) = 0, *q*(*t*1) = *Q* and *q*(*T*) = 0. The solutions of Equations (5) and (6) are given by

$$\begin{aligned} q(t) &= \begin{cases} \frac{\beta \delta k\_0}{\beta \delta k \epsilon A} - \frac{D}{\gamma} - \frac{\eta \delta}{\gamma \delta} \Big{(} \{1 - \exp(-\theta t)\} \\ + \frac{c\_1}{\left(k\_4 - \frac{k\_1}{\gamma} + \theta\right)} \Big{\Big{(} \left(k\_4 - \frac{k\_1}{\gamma}\right) \frac{\delta}{\gamma} + \frac{\beta \delta}{\gamma A}\Big{)} \Big{[}\exp\left\{\left(k\_4 - \frac{k\_1}{\gamma}\right)t\right\} - \exp(-\theta t)\Big{)} \end{cases} \\ &+ \frac{c\_2}{\left(k\_4 + \frac{k\_1}{\gamma} - \theta\right)} \Big{\Big{(} \left(k\_4 + \frac{k\_1}{\gamma}\right) \frac{\delta}{\gamma} - \frac{\beta \delta}{\gamma A}\Big{)} \Big{[}\exp\left\{-\left(k\_4 + \frac{k\_1}{\gamma}\right)t\right\} - \exp(-\theta t)\Big{)} \text{ for } 0 < t \le t\_1 \end{cases} \tag{7}$$
 
$$\text{and} \tag{8}$$

Again, using the continuity of *q*(*t*) at *t* = *t*1, we have

$$\begin{split} T &= \frac{1}{\theta} \log \left[ \frac{\theta}{D} \left( \frac{f \delta k\_{\parallel}}{\gamma \delta k\_{\perp} A} - \frac{D}{\theta} - \frac{\eta \delta}{\gamma \delta} \right) \{1 - \exp(-\theta t\_{1})\} \right. \\ &\left. + \frac{c\_{1}}{\left(k\_{4} - \frac{k\_{1}}{2} + \theta\right)} \left\{ \left(k\_{4} - \frac{k\_{1}}{2}\right) \frac{\delta}{\gamma} + \frac{\delta \delta}{\gamma A} \right\} \left[ \exp\left\{ \left(k\_{4} - \frac{k\_{1}}{2}\right) t\_{1} \right\} - \exp(-\theta t\_{1}) \right] \\ &\left. + \frac{c\_{2}}{\left(k\_{4} + \frac{k\_{1}}{2} - \theta\right)} \left\{ \left(k\_{4} + \frac{k\_{1}}{2}\right) \frac{\delta}{\gamma} - \frac{\beta \delta}{\gamma A} \right\} \left[ \exp\left\{ - \left(k\_{4} + \frac{k\_{1}}{2}\right) t\_{1} \right\} - \exp(-\theta t\_{1}) \right] \right] + 1 \right] + t\_{1} \end{split} (9)$$

#### *5.3. Various Components of the System*

The various components of the system are calculated as follows:

(i). Sales revenue (SR):

$$SR = p \int\_0^T D \, dt = pDT$$


*HC* = *h t* 51 0 *q*(*t*) *dt* + *h* 5*T t*1 *q*(*t*) *dt* = *h βδk*<sup>3</sup> *γθk*2*<sup>A</sup>* <sup>−</sup> *<sup>D</sup> <sup>θ</sup>* <sup>−</sup> *ηαδ γθ* /*t*<sup>1</sup> <sup>−</sup> <sup>1</sup> *<sup>θ</sup>* (<sup>1</sup> <sup>−</sup> exp(−*θt*1))<sup>0</sup> + *hc* <sup>1</sup> *<sup>k</sup>*4<sup>−</sup> *<sup>k</sup>*<sup>1</sup> <sup>2</sup> +*θ* /*k*<sup>4</sup> <sup>−</sup> *<sup>k</sup>*<sup>1</sup> 2 *δ <sup>γ</sup>* <sup>+</sup> *βδ γA* 02 exp/*k*4<sup>−</sup> *<sup>k</sup>*<sup>1</sup> 2 *t*1 0 −1 *<sup>k</sup>*4<sup>−</sup> *<sup>k</sup>*<sup>1</sup> 2 <sup>−</sup> <sup>1</sup> *<sup>θ</sup>* (<sup>1</sup> <sup>−</sup> exp(−*θt*1))<sup>3</sup> + *hc*<sup>2</sup> *<sup>k</sup>*4<sup>+</sup> *<sup>k</sup>*<sup>1</sup> <sup>2</sup> −*θ* /*k*<sup>4</sup> <sup>+</sup> *<sup>k</sup>*<sup>1</sup> 2 *δ <sup>γ</sup>* <sup>−</sup> *βδ γA* 02 <sup>1</sup>−exp/ − *<sup>k</sup>*4<sup>+</sup> *<sup>k</sup>*<sup>1</sup> 2 *t*1 0 *<sup>k</sup>*4<sup>+</sup> *<sup>k</sup>*<sup>1</sup> 2 <sup>−</sup> <sup>1</sup> *<sup>θ</sup>* (<sup>1</sup> <sup>−</sup> exp(−*θt*1))<sup>3</sup> + *hD <sup>θ</sup>*<sup>2</sup> [exp{*θ*(*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>*1)} <sup>−</sup> <sup>1</sup>] <sup>−</sup> *hD <sup>θ</sup>* (*T* − *t*1)

(iv). Production cost (PC):

PC = *cp t* 51 0 *P*(*t*) *dt* = *cp <sup>δ</sup> B t* 51 0 *y*(*t*) *dt* = *cp <sup>δ</sup> γ* 2 *c*1 *<sup>k</sup>*4<sup>−</sup> *<sup>k</sup>*<sup>1</sup> <sup>2</sup> <sup>+</sup> *<sup>β</sup> A <sup>k</sup>*4<sup>−</sup> *<sup>k</sup>*<sup>1</sup> 2 exp/*k*<sup>4</sup> <sup>−</sup> *<sup>k</sup>*<sup>1</sup> 2 *t*1 0 − 1 <sup>+</sup> *<sup>c</sup>*<sup>2</sup> <sup>−</sup>*k*4<sup>−</sup> *<sup>k</sup>*<sup>1</sup> <sup>2</sup> <sup>+</sup> *<sup>β</sup> A <sup>k</sup>*4<sup>+</sup> *<sup>k</sup>*<sup>1</sup> 2 <sup>1</sup> <sup>−</sup> exp/−*k*<sup>4</sup> <sup>−</sup> *<sup>k</sup>*<sup>1</sup> 2 *t*1 0<sup>3</sup> + *cp <sup>δ</sup> γ βk*<sup>3</sup> *Ak*<sup>2</sup> <sup>−</sup> *ηα t*1

(v). Preservation cost: *CP* = *ξT*.

Therefore, the profit per unit time of the system is given by

$$TP(t\_1, p, \xi) = \frac{1}{T} [SR - PC - HC - \mathcal{C}\_{\mathcal{O}} - CP]^{-1}$$

Now, the corresponding maximization problem of the system is given by

$$\begin{aligned} \text{Maximize } TP(t\_1, p, \xi) \\ \text{subject to } t\_1 > 0, \; 0 < p < \frac{a}{b} \end{aligned} \tag{10}$$

## **6. Solution Methodology**

The corresponding optimization problem (10) of the proposed production system is clearly highly non-linear in nature with respect to the decision variables *t*1, *p*, *ξ*. It is difficult to solve (10) by any analytical method, such as the gradient-based technique, Lagrange's multiplier method, Newton's method, saddle point optimization techniques, and so on. Thus, in order to solve the mentioned optimization problem (10), the following algorithms built in MATHEMATICA software are used:


The discussions of the above-mentioned algorithms are done based on the following generalized optimization problem:

$$\begin{aligned} \text{Maximize } f(u) \\ \text{subject to } t \in \mathcal{S} \subseteq \mathbb{R}^n \\ \text{where } f: \mathcal{S} \to \mathbb{R} \end{aligned}$$

(i) Differential Evolution (DE)

Differential evolution is one of the popular search techniques in the area of optimization. The algorithm of this optimizer has the following attributes:


Generally, this process is converged if deviation in between the best functional values in the new position and old population as well as the deviation between the new best point and the old best point are less than the tolerances.

The values of parameters used in the Differential Evolution are given in Table 2.


**Table 2.** The values of parameters used in the Differential Evolution.

(ii) Simulated Annealing (SA)

Simulated annealing is another random search-based meta-heuristic maximizer. The algorithm of this maximizer is inspired by physical activity of annealing, in which a metallic object is warmed up to an extreme temperature and allowed to cool gently. In this process, the atomic structure of metal reaches the lower energy level from the upper, and thus becomes a tougher metal. Exploring this concept in optimization, the algorithm of simulated annealing allows to move away from a local minimizer, and to traverse and settle on a better position and, ultimately, on the global maximizer.

During the iterative process, a new point *unew* is created in the neighboring point *u*. Thus, the radius of the neighborhood is decreased from iteration to iteration. The best-found point *ubest* obtained so far is tracked as follows:

If *f*(*unew*) > *f*(*ubest*),*unew* replaces *ubest* and *u*.

Otherwise, *unew* replaces *u* with a probability *eb*(*i*,Δ*<sup>f</sup>* , *<sup>f</sup>*0), where *b* is the Boltzmann exponent, *I* is the current iteration, Δ*f* is the change in the objective value, and *f*<sup>0</sup> is the last iteration objective function value.

The default function for *b* is taken as <sup>−</sup>Δ*<sup>f</sup>* log(*i*+1) <sup>10</sup> .

Simulated annealing is used for multi-initial points and obtains an optimizer among them. In general, the default number of initial points is taken as min{2*n*, 50}.

The starting points is repeated until achieving of the maximum number of iterations and this method converges to a point.

The values of the parameters of the Simulated annealing are given in Table 3.

**Table 3.** The values of parameters used in the Simulated annealing.


Solution Procedure

To solve the optimization Problem (10), the following steps are followed:

**Step 1:** Set the initial values of all input inventory parameters.

**Step 2:** Define the objective Function (10) in MATHEMATICA.

**Step 3:** Use the following comments:

"NMaximize [objective, decision variables, Method → "SimulatedAnnealing"] "NMaximize [objective, decision variables, Method → "DifferentialEvaluation"] **Step 4:** Compile and execute.

**Step 5:** Check the result.

**Step 6:** If the program is convergent and the results are feasible, go to **Step 8**, otherwise go to **Step 7**.

**Step 7:** Repeat **Steps 1** to **6**.

**Step 8:** Print the optimal results.

**Step 9:** Stop.
