# Optimal Control of Nonsmooth Production Systems with Deteriorating Items, Stock-Dependent Demand, with or without Backorders

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## Abstract

**:**

## 1. Introduction

- System dependent parameters: Taking into account the effect of the system parameters on an inventory system can lead to an improvement of its performance. The dependence of the demand rate on the stock is, without doubt, the dependence that received the most attention, and the literature on this topic is abundant. Among the most recent references we cite [9,10,11,12,13].
- Items backorders: In many real-life situations, demand is not met on time and shortages occur, the condition that exists when the inventory on hand is not sufficient to cover needs. Shortages are undesirable because they are quite expensive. However, in certain situations, management may find it desirable from a cost point of view not only to allow shortages but to plan for them. This specific shortage is called a backorder. After the exhaustion of inventory, we allow a period of time over which backorders accumulate to some level. When allowing backorders, we have, in addition to the usual costs, the additional cost of backordering. For more works on item backorders, we refer the reader to [18,20,21].

## 2. Model without Backorders

- Case 1:
- $P\left(t\right)-D(t,I(t\left)\right)-\theta (t,I(t\left)\right)=0$ on some subset S of $[0,T]$. This means that the firm has to produce the exact total amount corresponding to the amount consumed plus the amount lost due to deterioration. In this case $\frac{d}{dt}I\left(t\right)=0$ on S and ${I}^{*}$ is obviously constant on S and$${P}^{*}\left(t\right)=D(t,{I}^{*}\left(t\right))+\theta (t,{I}^{*}\left(t\right)),\phantom{\rule{1.em}{0ex}}\mathrm{for}\mathrm{all}t\in S.$$Substituting the Equation (9) into Equation (8), we obtained$$\frac{d}{dt}\lambda \left(t\right)={e}^{-\rho t}\left[\frac{d}{dI}h\left(I\left(t\right)\right)+c\frac{\partial}{\partial I}\theta (t,I\left(t\right))+\frac{d}{dP}K\left(P\left(t\right)\right)\left(\frac{\partial}{\partial I}D(t,I\left(t\right))+\frac{\partial}{\partial I}\theta (t,I\left(t\right))\right)\right].$$To get an explicit form of $\lambda $ and $\beta $, we integrated the previous differential equation. Then, we used Equation (9) to derive an explicit form of the Lagrange multiplier function $\mu $. We pointed out that if the obtained function $\mu $ was not nonnegative, then we did not accept the solutions stated in Equation (10).
- Case 2:
- $P\left(t\right)-D(t,I(t\left)\right)-\theta (t,I(t\left)\right)>0$ for $t\in [0,T]\backslash S$. The firm should produce more than the total amount corresponding to the amount consumed plus the amount lost due to deterioration, in order to avoid a shortage situation. In this case, $\mu \left(t\right)=0$ on $[0,T]\backslash S$, and so the necessary conditions in Equations (3), (8) and (9) become$$\frac{d}{dt}\lambda \left(t\right)={e}^{-\rho t}\left[\frac{d}{dI}h\left(I\left(t\right)\right)-c\frac{d}{dI}D(t,I\left(t\right))\right]+\lambda \left(t\right)\left[\frac{\partial}{\partial I}D(t,I\left(t\right))+\frac{\partial}{\partial I}\theta (t,I\left(t\right))\right],$$$$I\left(0\right)={I}_{0},\phantom{\rule{1.em}{0ex}}I\left(T\right)={I}_{T}\phantom{\rule{1.em}{0ex}}\lambda \left(T\right)=\beta ,\phantom{\rule{1.em}{0ex}}\lambda \left(t\right)={e}^{-\rho t}\left[\frac{d}{dP}K\left(P\left(t\right)\right)+c\right].$$Combining the state equation with these equations yields the following second order differential equation:$$\frac{d}{dt}P\left(t\right)\frac{{d}^{2}}{d{P}^{2}}K\left(P\left(t\right)\right)-\left[\rho +\frac{\partial}{\partial I}D(t,I\left(t\right))+\frac{\partial}{\partial I}\theta (t,I\left(t\right))\right]\left[\frac{d}{dP}K\left(P\left(t\right)\right)+c\right]=\frac{d}{dI}h\left(I\left(t\right)\right)-c\frac{\partial}{\partial I}D(t,I\left(t\right)),$$$$I\left(0\right)={I}_{0},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}I\left(T\right)={I}_{T}\phantom{\rule{4pt}{0ex}}c+\frac{d}{dP}K\left(P\left(T\right)\right)=\beta {e}^{\rho T}.$$These equations are enough to determine the optimal solution of problem $\left(\mathcal{P}\right)$. To be able to push the derivations any further, one needs to have an explicit form for the functions involved. For illustration purposes, let us assume the following forms for the cost rates$$K\left(P\right)=\frac{K{P}^{2}}{2},\phantom{\rule{1.em}{0ex}}h\left(I\right)=\frac{h{I}^{2}}{2},$$$$D(t,I\left(t\right))={d}_{1}\left(t\right)+{d}_{2}I\left(t\right),\theta (t,I\left(t\right))={\theta}_{1}\left(t\right)+{\theta}_{2}I\left(t\right).$$Here K, h, ${d}_{2}$, and ${\theta}_{2}$ are positive constants. For these functions the necessary conditions for $({P}^{*},{I}^{*})$ to be an optimal solution of problem $\left(\mathcal{P}\right)$ become$$\frac{{d}^{2}}{d{t}^{2}}I\left(t\right)-\rho \frac{d}{dt}I\left(t\right)-\left[\frac{h}{K}+({d}_{2}+{\theta}_{2})(\rho +{d}_{2}+{\theta}_{2})\right]I\left(t\right)=\alpha \left(t\right),$$$$\alpha \left(t\right)=(\rho +{d}_{2}+{\theta}_{2})({d}_{1}\left(t\right)+{\theta}_{1}\left(t\right))-\frac{d}{dt}{d}_{1}\left(t\right)-\frac{d}{dt}{\theta}_{1}\left(t\right)+\frac{c{\theta}_{2}}{K},$$$$I\left(0\right)={I}_{0},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}I\left(T\right)={I}_{T}.$$This is a two-point boundary value problem (${\mathcal{P}}_{TPBV}$) that we solved in the next proposition.

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Example**

**1.**

- 1.
- We illustrated the results obtained by considering a production system with the following characteristics: planning horizon of length $T=5$, initial and terminal inventory levels, ${I}_{0}=0$, $I\left(T\right)=10$, unit costs and discount factor $c=h=0.1$, $K=5$ and $\rho =0$, respectively. The demand rate is such that ${d}_{1}\left(t\right)=sin\left(t\right)+1,{d}_{2}=0.1$ and the deterioration rate is such that ${\theta}_{1}\left(t\right)={e}^{-t},{\theta}_{2}=0.1$.The optimal control and state are displayed in Figure 1.The optimal objective function value is $J=216.67$.
- 2.
- To assess the effect of the deterioration rate on the value of the optimal objective function, we set ${\theta}_{1}\equiv 0$ and varied the value of ${\theta}_{2}$ from $0.0005$ to $0.2560$, and we kept all the other parameters as in Example (1). As shown by Table 1, the resulting optimal cost increases as ${\theta}_{2}$ increases.
- 3.
- Next, we studied the effect of the discount factor on the value of the optimal objective function, we set ${\theta}_{1}\equiv 0$ and varied the value of ρ from 0 to $0.1$, and we keep all the other parameters as in Example (1). As shown by Table 2, the resulting optimal cost increases as ρ increases.

## 3. Model with Backorders Allowed

Regime 1 | : | $I\left(t\right)>0$. |

Regime 2 | : | $I\left(t\right)=0$. |

Regime 3 | : | $I\left(t\right)<0$. |

**Scenario 1.**Regime 2 arises on intervals: For simplicity we assumed that there was only one subinterval $[{t}_{0},{t}_{1}]\subset [0,T]$ on which $0\in \left[a\right(t),b(t\left)\right]$. In this scenario, there are four cases; ${I}_{0}<0<{I}_{T},\phantom{\rule{0.277778em}{0ex}}{I}_{T}<0<{I}_{0},\phantom{\rule{0.277778em}{0ex}}{I}_{0},{I}_{T}>0$, and ${I}_{0},{I}_{T}<0$.**Case 1.**${I}_{0}<0<{I}_{T}:$ In this case we proceed as follows:- ∗
- ∗
- ∗

**Case 2.**${I}_{T}<0<{I}_{0}$. This case is unlikely to happen in practice, but if it does, then we proceed as follows:- ∗
- ∗
- ∗

**Case 3.**${I}_{0},{I}_{T}<0$. In this case we solved Equation (24) twice, once with the boundary conditions ${I}_{0}<0$ and $I\left({t}_{0}\right)=0$ and once with the boundary conditions $I\left({t}_{1}\right)=0$ and ${I}_{T}<0$. The optimal level ${I}^{*}$ is the function given by the solution of Equation (24) over the interval $[0,{t}_{0}]$ and by the solution of Equation (24) on the interval $[{t}_{1},T]$, and ${I}^{*}\equiv 0$ on $[{t}_{0},{t}_{1}]$.

**Scenario 2.**Regime 2 does not arise on intervals: As in the previous scenario, we again had to consider the four cases; ${I}_{0}<0<{I}_{T},\phantom{\rule{0.277778em}{0ex}}{I}_{T}<0<{I}_{0},\phantom{\rule{0.277778em}{0ex}}{I}_{0},{I}_{T}>0$, and ${I}_{0},{I}_{T}<0$.**Case 1.**${I}_{0}<0<{I}_{T}:$ For simplicity we assumed that there is only one point ${t}_{0}\in [0,T]$ with $I\left({t}_{0}\right)=0$. In this case we proceed as follows:**Step 1.**- Solve Equation (24) with the boundary conditions ${I}_{0}$ and $I\left(T\right)$, and determine the value ${t}_{0}$ for which $I\left({t}_{0}\right)=0$.

**Step 2.****Step 3.**The optimal level ${I}^{*}$ is the one with the smallest objective function value.

**Case 2.**${I}_{T}<0<{I}_{0}$. This case is unlikely to happen in practice, but if it does, then we proceed as in case 1.**Case 3.**${I}_{0},{I}_{T}<0$. In this case we solved the differential Equation (24) with the boundary conditions ${I}_{0}<0$ and ${I}_{T}<0$.**Case 4.**${I}_{0},{I}_{T}>0$. In this case we solved the differential Equation (22) with the boundary conditions ${I}_{0}>0$ and ${I}_{T}>0$.

**Remark**

**1.**

**Example**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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${\theta}_{2}$ | 0.0005 | 0.001 | 0.002 | 0.004 | 0.008 | 0.016 | 0.032 | 0.064 | 0.128 | 0.256 |

J | 490.49 | 491.19 | 492.61 | 495.45 | 501.15 | 512.64 | 536.02 | 584.22 | 685.60 | 902.52 |

$\rho $ | 0 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 | 0.1 |

J | 216.67 | 664.06 | 664.36 | 664.86 | 665.55 | 666.42 | 667.48 | 668.71 | 670.11 | 671.68 | 673.41 |

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**MDPI and ACS Style**

Bounkhel, M.; Tadj, L.; Benhadid, Y.; Hedjar, R.
Optimal Control of Nonsmooth Production Systems with Deteriorating Items, Stock-Dependent Demand, with or without Backorders. *Symmetry* **2019**, *11*, 183.
https://doi.org/10.3390/sym11020183

**AMA Style**

Bounkhel M, Tadj L, Benhadid Y, Hedjar R.
Optimal Control of Nonsmooth Production Systems with Deteriorating Items, Stock-Dependent Demand, with or without Backorders. *Symmetry*. 2019; 11(2):183.
https://doi.org/10.3390/sym11020183

**Chicago/Turabian Style**

Bounkhel, Messaoud, Lotfi Tadj, Yacine Benhadid, and Ramdane Hedjar.
2019. "Optimal Control of Nonsmooth Production Systems with Deteriorating Items, Stock-Dependent Demand, with or without Backorders" *Symmetry* 11, no. 2: 183.
https://doi.org/10.3390/sym11020183