# Development of an economical Production policy for a dry machining production system considering environmental impact and recycling constraints

Lahcen Mifdal, El Mehdi Guendouli, Sofiene Dellagi, Anis Chelbi

► To cite this version:

Lahcen Mifdal, El Mehdi Guendouli, Sofiene Dellagi, Anis Chelbi. Development of an economical Pro-duction policy for a dry machining production system considering environmental impact and recycling constraints. Journal of Physics: Conference Series, 2025, Technologies and Materials for Renewable Energy, Environment and Sustainability, 10/02/2025 - 13/02/2025, Casablanca, Morocco, 3028 (1), pp.012013. 10.1088/1742-6596/3028/1/012013 . hal-05112176

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https://hal.univ-lorraine.fr/hal-05112176v1

Submitted on 13 Jun 2025

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# Development of an economical Production policy for a dry machining production system considering environmental impact and recycling constraints

To cite this article: Lahcen Mifdal et al 2025 J. Phys.: Conf. Ser. 3028 012013

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# Development of an economical Production policy for a dry machining production system considering environmental impact and recycling constraints

Lahcen Mifdal<fnref n="1" />,<fnref n="2" /> *; El Mehdi Guendouli<fnref n="1" />,<fnref n="2" />, Sofiene Dellagi<fnref n="2" />, Anis Chelbi<fnref n="3" />,<fnref n="4" />

**Abstract:** This paper deals with the production planning of a manufacturing system based on a dry machining process, which is considered a "green process" because it does not require the use of cutting fluids. Two types of raw materials with different prices and environmental impact are considered. The aim is to derive a cost-effective production plan over a specified time frame. Such a plan is achieved through total average cost minimization which incorporates the costs of raw materials, production, energy use, recycling, storage, shortages and CO₂ emissions. Demand is taken to be stochastic during the finite time model, consisting of discrete successive production periods. A mathematical model is established that expresses the average total cost as a function of decision variables, which are the quantities of each of the two types of products to be produced in each period. A numerical solving algorithm is devised for the problem at hand and is put to test. To demonstrate how to use the developed format and its consistency, a numerical example is proposed.

## 1. Introduction and literature review

Sustainable development is becoming a priority in a constantly changing industrial context, driven by global initiatives to combat climate change and reduce the ecological footprint of manufacturing processes. The major challenges are to reduce CO₂ emissions and energy consumption, while maintaining efficient production. Among the innovative solutions, dry machining stands out as a promising alternative. This method breaks with traditional practices that rely on the use of cutting fluids. Although these fluids are effective for cooling and lubrication, they pose environmental and health problems due to the hazardous waste they generate and the fume emissions they produce. By eliminating these fluids, dry machining reduces environmental impact, simplifies waste treatment, facilitates recycling and optimizes energy efficiency by limiting ancillary processes such as fluid management. The balance between CO₂ emissions, energy consumption and the adoption of new machining methods is a key issue, especially at a time when sustainability imperatives are becoming unavoidable for industries. Dry machining offers significant environmental gains without sacrificing operational efficiency, while meeting regulatory requirements, market expectations and growing consumer demand for greener products. To achieve their sustainability objectives, companies need to incorporate

<figure><img src="image_3.png" /></figure>

<footnote n="1">¹ Laboratoire InterDisciplinaire de Recherches Appliquées, LIDRA, Université Internationale d'Agadir- Universiapolis, Agadir, Maroc ;</footnote>

<footnote n="2">² Laboratoire de Génie Informatique, de Production et de Maintenance, LGIPM, (EA 3096), Université de Lorraine, Metz, France ;</footnote>

<footnote n="3">³ Laboratoire de Recherche en Robotique Intelligente, Fiabilité et Traitement de Signal et d'Images LR20ES02 RIFTSI, University of Tunis, Tunis, Tunisia ;</footnote>

<footnote n="4">⁴ Mediterranean School of Business, South Mediterranean University, Tunis, Tunisia;</footnote>

<footnote>*Corresponding author: Lahcen Mifdal: mifdal@e-polytechnique.ma</footnote>
innovative approaches that combine respect for the environment with economic viability. Dry machining is emerging as a practical solution, capable of tangibly reducing energy consumption and greenhouse gas emissions.

Manufacturing industries play a pivotal role towards global sustainability, largely because of their substantial contribution to worldwide CO₂ emissions and energy use. This is backed by upcoming studies like Sambandam et al.[1], which indicated that approximately 20% of the Earth's CO₂ emissions were attributable to manufacturing activities, therefore, there is a great demand for innovative strategies in such areas. The key challenge in meeting this call is the implementation of sustainable practices that enhance efficiency while minimizing resource utilization.

In view of the complexity of machining and the need to manage both costs and quality, it is imperative to adopt an integrated approach that considers not only production aspects but also raw material acquisition, environmental impact, and recycling activities. A considerable number of researchers have expressed interest in the environmental impact of production and energy consumption like Cai et al. [2], Tai et al. [3], Guerra-Zubiaga et al. [4]. In relation to Carbon emissions and remanufacturing, Chang et al. [5] concluded that the price of carbon and the savings in carbon emissions per remanufactured product will influence manufacturers' decisions to remanufacture.

In other studies, Turki et al. [6] integrated carbon emissions, carbon taxation and mandatory emission limit strategies, they developed a sustainable manufacturing and remanufacturing model. It includes two parallel machines, manufacturing and reconditioning inventories, and a recovering inventory of well-defined new and reconditioned products. The objective being to increase the capacity of the inventories to maximize profit under the carbon emission constraints.

Recently, Sun et al. [7] have based their work on the case of a specific energy for the cutting process. Rahman et al. [8] present a model for determining the Energy Consumption Allowance of a workpiece, providing a reference value for each energy consumption step throughout the process. In this context, Abdellaoui et al. [9] reviewed energy-efficient machining systems and discussed the energy consumption associated with machining processes.

In Abellán-Nebot et al. [10], the authors meticulously identify twenty-five factors that exert an influence on surface roughness in turning and milling processes, which are classified into three distinct categories: setup, operational, and processing factors. The studied case illustrates how the control of surface roughness can enhance sustainability, including the elimination of grinding stages and the reduction of CO₂ emissions. This research underscores the criticality of comprehending surface roughness for practitioners seeking to enhance sustainability in machining processes.

The work of Soori et al. [11] looks at CNC – Computer Numerical Control – machining carried out according to sustainable practices, with the aim of reducing the carbon footprint, material waste and energy consumption. The authors highlight the importance of virtual simulations for identifying excessive machining parameters, optimising tool paths and selecting more energy-efficient CNC machines, in order to guide better investment decisions.

The environmental implications of the manufacturing industry necessitate the exploration of sustainable alternatives such as dry machining. Recent studies have demonstrated that dry machining can yield surface finishes and tool life that are comparable to, and in some cases superior to, those achieved through conventional machining techniques. In addition, dry machining contributes to the global sustainability goals, particularly with regard to the reduction of greenhouse gas emissions.

In a recent study by Younas et al. [12], the authors investigate the energy consumption and wear mechanisms of carbide cutting tools during dry machining. A full factorial design was utilized to analyse tool wear rate and specific cutting energy for different cutting conditions. The findings of this investigation revealed that both parameters are affected by machining parameters.

The originality of our work lies in taking into account all phases of the production process, from the choice of raw material characteristics to the final product, in the specific case of dry machining. We consider the impact of the characteristics of the raw materials on the cutting speed and the production cost, the associated recycling activities, as well as the environmental impact, illustrated by CO₂ emissions. We design an economic production plan, optimized by minimizing the expected total cost incorporating raw materials, production, storage, recycling and penalties linked to CO₂ emissions. This production plan consists of the economic quantities to produce for each part (Aluminium and Steel).
In this paper, an economical production strategy for dry machining is developed, in which the production parameters are considered simultaneously. They are mainly the cutting speed, production time, cost and the energy consumption of the machine during machining of two materials, Aluminium and Steel. Recycling activities are also considered, along with the penalty related to CO₂ emissions in the production and recycling phases.

The structure of the present paper is as follows: section 2 is dedicated to the description of the problem under discussion and the overall strategy advocated for the optimization. Section 3 is devoted to the development of the mathematical model, while section 4 presents a numerical example intended to illustrate the application of the analytical model. Section 5 presents the numerical results. Finally, section 6 summarizes the conclusions that have been obtained.

## 2. Integrated strategy and problem definition

We analyse a CNC mechanical manufacturing system subject to random breakdowns, that produces Aluminium and Steel parts. The goal is to establish a production plan that allows for satisfying random demands during a planning horizon made of H periods of equal duration $\Delta t$ (see Figure 1).

Each period is divided into two subperiods. The first sub-period whose length is $\Delta t_{AL}(p)$ is devoted to the production of Aluminium parts in period $p$. The second sub-period of length $\Delta t_{ST}(p)$, is devoted to the production of Steel parts. It should be noted that the duration of these sub-periods evolves from one period to another but the period $\Delta t$ stays constant. The production rates for Aluminium and Steel parts during each period are respectively $Pr_{AL}(p)$ and $Pr_{ST}(p)$. These production rates depend on the durations of subperiods allowed to each type of raw material (Aluminium or Steel) and the cutting speed. The breakdown of the planning horizon is shown in Figure 1 below.

<figure><img src="image_4.png" /><figcaption>Figure 1. The production strategy over the finite time horizon $H \cdot \Delta t$.</figcaption></figure>

The goal is to establish an economic production plan to satisfy random demands for each period, minimizing the expected total cost including production, inventory, energy, recycling costs and the penalty related to CO₂ emission. In fact, we will determine the economical quantities to be produced for each part over the finite time span $H \cdot \Delta t$.

## 3. The analytical Production model

### 3.1. Notation

The following notations are used:

$TEnrgC$ : Average total energy consumption cost and CO₂ emission impact

$TEnvC_{CO_2}$ : Average total environmental cost of CO₂ emission

$TEnrgC_{recycling}$ : Average total energy recycling cost

$TRMC$ : Average total cost of raw materials for Aluminium and Steel parts

$TStc$ : Average total storage costs for Aluminium and Steel parts

$TShc$ : Average total shortage costs for Aluminium and Steel parts

$I_{carbone}$ : Carbon intensity of electricity (in gCO₂/kWh or kgCO₂/kWh)

$SocialC_{CO_2}$ : Social cost of CO₂ (€/tonne)

$TPC(.)$ : Average total production costs

$\Delta t$ : Length of production periods

$\alpha_{AL}$ : Percentage of material machined in the production of an Aluminium part
<table><tr><td>α<sub>ST</sub></td><td>: Percentage of material machined in the production of a Steel part</td></tr><tr><td>W<sub>AL</sub></td><td>: Weight of one Aluminium part</td></tr><tr><td>W<sub>ST</sub></td><td>: Weight of one Steel part</td></tr><tr><td>RMC<sub>AL</sub></td><td>: Unit cost of raw materials for Aluminium parts</td></tr><tr><td>RMC<sub>RecyAL</sub></td><td>: Cost of recycled raw material for Aluminium parts</td></tr><tr><td>RMC<sub>NewST</sub></td><td>: Unit cost of raw materials for Steel parts</td></tr><tr><td>SSP<sub>ST</sub></td><td>: Unit selling cost of Steel chips</td></tr><tr><td>RMC<sub>ST</sub></td><td>: Unit cost of raw materials for Steel parts</td></tr><tr><td>C<sub>e</sub></td><td>: Energy unit cost</td></tr><tr><td>Pc<sub>AL</sub></td><td>: Cutting power required for machining Aluminium parts</td></tr><tr><td>Pc<sub>ST</sub></td><td>: Cutting power required for machining Steel parts</td></tr><tr><td>f</td><td>: Machining system feed speed [mm/rev]</td></tr><tr><td>a<sub>p</sub></td><td>: Passing depth [mm]</td></tr><tr><td>k<sub>c</sub></td><td>: Specific cutting pressure [N/mm<sup>2</sup>]</td></tr><tr><td>v<sub>c</sub></td><td>: Cutting speed [m/min]</td></tr><tr><td>η</td><td>: The efficiency of CNC machine</td></tr><tr><td>S<sub>AL</sub>(p)</td><td>: Stock level of Aluminium parts at end of period p</td></tr><tr><td>S<sup>M</sup><sub>AL</sub>(p)</td><td>: Stock level of Aluminium parts at the end of the first sub-period of the period p.</td></tr><tr><td>d<sub>AL</sub>(p)</td><td>: Average demand for Aluminium parts at the end of the period p</td></tr><tr><td>UCS<sub>AL</sub></td><td>: Unit cost of storing an Aluminium part</td></tr><tr><td>CS<sub>AL</sub>(p)</td><td>: Cost of storing an Aluminium part over the period p</td></tr><tr><td>CS<sub>ST</sub>(p)</td><td>: Cost of storing a Steel part over the period p</td></tr><tr><td>S<sub>ST</sub>(p)</td><td>: Stock level of Steel parts at end of period p</td></tr><tr><td>UCS<sub>ST</sub></td><td>: Unit cost of storing a Steel part</td></tr><tr><td>d<sub>ST</sub>(p)</td><td>: Average demand for Steel parts at end of period p</td></tr><tr><td>SC<sub>AL</sub></td><td>: Unit shortage costs for Aluminium parts</td></tr><tr><td>SC<sub>ST</sub></td><td>: Unit shortage costs for Steel parts</td></tr><tr><td>Δt<sub>AL</sub>(p)</td><td>: Duration of sub-period of production of Aluminium parts in the period p</td></tr><tr><td>Δt<sub>ST</sub>(p)</td><td>: Duration of sub-period of production of Steel parts in the period p</td></tr><tr><td>Pr<sub>AL</sub>max</td><td>: Maximal production of Aluminium part during each period p</td></tr><tr><td>Pr<sub>ST</sub>max</td><td>: Maximal production of Steel part during each period p</td></tr></table>

The decision variables:

Pr<sub>AL</sub>(p) : Quantity of Aluminium parts produced during the period p

Pr<sub>ST</sub>(p) : Quantity of Steel parts produced during the period p

## 3.2. Production costs

### 3.2.1. Average total production cost

We note that average total production cost is composed of the six costs below:

* ✓ Average total cost of raw materials. (TRMC)

* ✓ Average total storage cost. (TStc)

* ✓ Average total shortage cost. (TShc)

* ✓ Average total energy consumption cost and CO₂ emission impact. (TEnrgC)

* ✓ Average total environmental cost of CO₂. (TEnvC<sub>CO₂</sub>)

* ✓ Average total energy recycling cost. (TEnrgC<sub>recycling</sub>)

This average total production cost can be expressed as follows:
$$ 
\begin{aligned}
TPC(Pr_{AL}(\mathbf{p}), Pr_{ST}(\mathbf{p}))
&= \sum_{p=1}^{H} [RMC_{AL} \times Pr_{AL}(\mathbf{p}) \times W_{AL}] \\
&+ \sum_{p=1}^{H} [RMC_{ST} \times Pr_{ST}(\mathbf{p}) \times W_{ST}] \\
&+ UCs_{AL} \\
&\quad \times \sum_{p=1}^{H} [S_{AL}(\mathbf{p} - \mathbf{1}) \times \Delta t_{AL}(\mathbf{p}) \times \mathbf{1}_{S_{AL}(p-1)>0} \\
&\quad + S_{AL}^{M}(\mathbf{p}) \times \Delta t_{ST}(\mathbf{p}) \times \mathbf{1}_{S_{AL}(p)>0}] \\
&+ UCs_{ST} \times \sum_{p=1}^{H} [S_{ST}(\mathbf{p}) \times \Delta t_{ST}(\mathbf{p}) \times \mathbf{1}_{S_{ST}(p)>0}] \\
&+ SC_{AL} \times \sum_{p=1}^{H} [|S_{AL}(\mathbf{p})| \times \mathbf{1}_{S_{AL}(p)<0}] \\
&+ SC_{ST} \times \sum_{p=1}^{H} [|S_{ST}(\mathbf{p})| \times \mathbf{1}_{S_{ST}(p)<0}] \\
&+ \sum_{p=1}^{H} \left[ \frac{f_{AL} \times a_{pAL} \times k_{CAL} \times v_{CAL}}{\eta} \times \Delta t_{AL}(\mathbf{p}) \right. \\
&\quad \left. + \frac{f_{ST} \times a_{pST} \times k_{cST} \times v_{cST}}{\eta} \times \Delta t_{ST}(\mathbf{p}) \right] \times C_e \\
&+ \sum_{p=1}^{H} \left[ \frac{f_{AL} \times a_{pAL} \times k_{CAL} \times v_{CAL}}{\eta} \times \Delta t_{AL}(\mathbf{p}) \right. \\
&\quad \left. + \frac{f_{ST} \times a_{pST} \times k_{cST} \times v_{cST}}{\eta} \times \Delta t_{ST}(\mathbf{p}) \right] \times I_{carbone} \\
&\quad \times \frac{\text{SocialC}_{CO_2}}{1000} \\
&+ \sum_{p=1}^{H} [\alpha_{AL} \times Pr_{AL}(\mathbf{p}) \times W_{AL} \times Enrg_{Kg}] \times C_e
\end{aligned}
\quad (1) $$

This function will be explained in detail in the following sections.

### 3.2.2. Average total cost of raw materials

Aluminium parts are machined from two types of material: raw material and recycled material. The raw material is purchased from an external supplier. On the other hand, the material recycled internally is obtained from chips (leftover material after machining) from machined parts. It is considered that each part consists of a percentage $\alpha_{AL}$ of recycled material and the rest of raw material. However, Steel parts are produced from a single type of material.

This process is illustrated in figure 2 below:
<figure><img src="image_5.png" /><figcaption>Figure 2. Machining process for Aluminium and Steel parts.</figcaption></figure>

The average total raw material cost function over the planning horizon $H$. $\Delta t$ is expressed as:

$$ TRMC = \sum_{p=1}^{H} [RMC_{AL} \times Pr_{AL}(p) \times W_{AL}] + \sum_{p=1}^{H} [RMC_{ST} \times Pr_{ST}(p) \times W_{ST}] \quad (2) $$

The cost of raw material (per Kg) to produce an Aluminium part is expressed by the equation below:

$$ RMC_{AL} = (RMC_{RecyAL} \times \alpha_{AL}) + (RMC_{NewAL} \times (1 - \alpha_{AL})) \quad (3) $$

### 3.2.3. Average total storage cost

Based on figure 3 below, the average total cost of stocking Aluminium parts and Steel parts over the planning horizon $H$. $\Delta t$ is expressed by:

$$ TStc = UCs_{AL} \\ \times \sum_{p=1}^{H} [S_{AL}(p-1) \times \Delta t_{AL}(p) \times 1_{S_{AL}(p-1)>0} \\ + S_{AL}^{M}(p) \times \Delta t_{ST}(p) \times 1_{S_{AL}(p)>0}] \\ + UCs_{ST} \times \sum_{p=1}^{H} [S_{ST}(p) \times \Delta t_{ST}(p) \times 1_{S_{ST}(p)>0}] \quad (4) $$

We note that:

$$ \Delta t_{AL}(p) = \frac{Pr_{AL}(p)}{NPr_{AL}} \quad (5) $$

$$ \Delta t_{ST}(p) = \frac{Pr_{ST}(p)}{NPr_{ST}} \quad (6) $$

We recall that:

$S_{AL}(p)$ : Stock level of Aluminium parts at end of period $p$

$S_{AL}^{M}(p)$ : Stock level of Aluminium parts at the end of the first sub-period of the period $p$.

$S_{ST}(p)$ : Stock level of Steel parts at end of period $p$

These dynamic stock functions will be presented later.
The term "$1_{S_{AL}(p)>0}$" is equal to 1 if the quantity of Aluminium parts stored during $\Delta t_{AL}(p)$ is positive, and equal to 0 otherwise.

The term "$1_{S_{ST}(p)>0}$" is equal to 1 if the quantity of Steel parts stored in period $p$ is positive, and equal to 0 otherwise.

The term "$1_{S_{AL}^M(p)>0}$" is equal to 1 if the quantity of Aluminium parts stored during $\Delta t_{ST}(p)$ is positive, and equal to 0 otherwise.

<figure><img src="image_6.png" /><figcaption>Figure 3. Evolution of the Storage level of the Aluminium and Steel parts</figcaption></figure>

### 3.2.4. Average total shortage cost

Based on figure 3, the average total shortage cost function for Aluminium and Steel parts over the entire planning horizon $H \cdot \Delta t$, is expressed as follows:

$$ TShc = SC_{AL} \times \sum_{p=1}^{H} [|S_{AL}(p)| \times 1_{S_{AL}(p)<0}] + SC_{ST} \times \sum_{p=1}^{H} [|S_{ST}(p)| \times 1_{S_{ST}(p)<0}] \quad (7) $$

### 3.2.5. Average total energy consumption cost and CO₂ emission impact

The average total energy cost is expressed by the following function:

$$ TEnrgC = \sum_{p=1}^{H} \left[ \frac{f_{AL} \times a_{pAL} \times k_{cAL} \times v_{cAL}}{\eta} \times \Delta t_{AL}(p) + \frac{f_{ST} \times a_{pST} \times k_{cST} \times v_{cST}}{\eta} \times \Delta t_{ST}(p) \right] \times C_e \quad (8) $$

We note that the average total energy cost is obtained as follows:

$$ TEnrgC = TEnrg \times C_e \quad (9) $$

Based on [13], the energy consumed by the machine while machining an Aluminium part and Steel part respectively is calculated as follows:
$$TEnrg = \sum_{p=1}^{H} \left[ \frac{Pc_{AL}}{\eta} \times \Delta t_{AL}(\mathbf{p}) + \frac{Pc_{ST}}{\eta} \times \Delta t_{ST}(\mathbf{p}) \right] \quad (10)$$

The functions of the cutting power required during a drying operation are expressed as follows:

$$Pc_{AL} = f_{AL} \times a_{pAL} \times k_{cAL} \times v_{cAL} \quad (11)$$

$$Pc_{ST} = f_{ST} \times a_{pST} \times k_{cST} \times v_{cST} \quad (12)$$

### 3.2.6. Average total environmental cost of CO₂

The average total environmental cost of CO₂ over the entire planning horizon H. Δt is formulated as:

$$TEnvC_{CO_2} = \sum_{p=1}^{H} \left[ \frac{f_{AL} \times a_{pAL} \times k_{cAL} \times v_{cAL}}{\eta} \times \Delta t_{AL}(\mathbf{p}) + \frac{f_{ST} \times a_{pST} \times k_{cST} \times v_{cST}}{\eta} \times \Delta t_{ST}(\mathbf{p}) \right] \times I_{\text{carbone}} \times \frac{\text{SocialC}_{CO_2}}{1000} \quad (13)$$

In order to incorporate an environmental cost related to CO₂ emission, it is necessary to multiply CO₂ emissions by an estimated cost per ton of CO₂. This social cost of carbon is subject to variation in accordance with environmental policies and objectives. [14]

$$TEnvC_{CO_2} = CO_2 \text{ emissions} \times \frac{\text{SocialC}_{CO_2}}{1000} \quad (14)$$

CO₂ emissions associated with energy consumption are determined by the carbon intensity of the electricity utilized, which is frequently expressed in grams or kilograms of CO₂ per (kWh) of energy consumed.

The function for calculating CO₂ emissions is as follows: [15]

$$CO_2 \text{ emissions} = TEnrg \times I_{\text{carbone}} \quad (15)$$

### 3.2.7. Average total energy recycling cost

The average total energy recycling cost is formulated by the formula below:

$$TEnrgC_{recycling} = \sum_{p=1}^{H} [\alpha_{AL} \times Pr_{AL}(\mathbf{p}) \times W_{AL} \times Enrg_{Kg}] \times C_e \quad (16)$$

### 3.2.8. Economic production plan

To establish the economic production plan, we need to minimize the average total production cost function in order to determine the optimal quantities to produce for each part (Aluminium and Steel) during each period.
$$
\begin{equation}
\begin{aligned}
& \text{Min} \left[ \sum_{p=1}^{H} [\mathbf{RM}C_{AL} \times \mathbf{Pr}_{AL}(\mathbf{p}) \times \mathbf{W}_{AL}] + \sum_{p=1}^{H} [\mathbf{RM}C_{ST} \times \mathbf{Pr}_{ST}(\mathbf{p}) \times \mathbf{W}_{ST}] \right. \\
& \qquad \left. + \mathbf{UC}s_{AL} \right. \\
& \qquad \left. \times \sum_{p=1}^{H} [\mathbf{s}_{AL}(\mathbf{p}-\mathbf{1}) \times \Delta t_{AL}(\mathbf{p}) \times \mathbf{1}_{S_{AL}(p-1)>0} \right. \\
& \qquad \left. + \mathbf{s}_{AL}^{M}(\mathbf{p}) \times \Delta t_{ST}(\mathbf{p}) \times \mathbf{1}_{S_{AL}(p)>0}] \right. \\
& \qquad \left. + \mathbf{UC}s_{ST} \times \sum_{p=1}^{H} [\mathbf{s}_{ST}(\mathbf{p}) \times \Delta t_{ST}(\mathbf{p}) \times \mathbf{1}_{S_{ST}(p)>0}] \right. \\
& \qquad \left. + \mathbf{SC}_{AL} \times \sum_{p=1}^{H} [|\mathbf{s}_{AL}(\mathbf{p})| \times \mathbf{1}_{S_{AL}(p)<0}] \right. \\
& \qquad \left. + \mathbf{SC}_{ST} \times \sum_{p=1}^{H} [|\mathbf{s}_{ST}(\mathbf{p})| \times \mathbf{1}_{S_{ST}(p)<0}] \right. \\
& \qquad \left. + \sum_{p=1}^{H} \left[ \frac{f_{AL} \times \mathbf{a}_{pAL} \times \mathbf{k}_{CAL} \times \mathbf{v}_{CAL}}{\eta} \times \Delta t_{AL}(\mathbf{p}) \right. \right. \\
& \qquad \left. \left. + \frac{f_{ST} \times \mathbf{a}_{pST} \times \mathbf{k}_{cST} \times \mathbf{v}_{cST}}{\eta} \times \Delta t_{ST}(\mathbf{p}) \right] \times C_e \right. \\
& \qquad \left. + \sum_{p=1}^{H} \left[ \frac{f_{AL} \times \mathbf{a}_{pAL} \times \mathbf{k}_{CAL} \times \mathbf{v}_{CAL}}{\eta} \times \Delta t_{AL}(\mathbf{p}) \right. \right. \\
& \qquad \left. \left. + \frac{f_{ST} \times \mathbf{a}_{pST} \times \mathbf{k}_{cST} \times \mathbf{v}_{cST}}{\eta} \times \Delta t_{ST}(\mathbf{p}) \right] \times I_{carbone} \right. \\
& \qquad \left. \times \frac{\mathbf{SocialC}_{CO_2}}{1000} \right. \\
& \qquad \left. + \sum_{p=1}^{H} [\alpha_{AL} \times \mathbf{Pr}_{AL}(\mathbf{p}) \times \mathbf{W}_{AL} \times \mathbf{Enrg}_{Kg}] \times C_e \right]
\end{aligned}
\tag{17}
\end{equation}
$$

To establish the economic production plan, the following constraints must be considered.

$$
\left\{
\begin{array}{l}
\Delta t_{AL}(p) + \Delta t_{ST}(p) = \Delta t \\
0 \le \Delta t_{AL}(p) \le \Delta t \\
0 \le \Delta t_{ST}(p) \le \Delta t \\
S_{AL}^{M}(p) = S(p-1) + NPr_{AL} \times \Delta t_{AL}(p) \\
S_{AL}(p) = S_{AL}^{M}(p) - d_{AL}(p) \\
S_{ST}(p) = S_{ST}(p-1) + NPr_{ST} \times S_{ST}(p) - d_{ST}(p) \\
NPr_{AL} \times \Delta t_{AL}(p) \le Pr_{AL} \max \\
NPr_{ST} \times \Delta t_{ST}(p) \le Pr_{ST} \max
\end{array}
\right.
$$

4. Numerical example

A solving procedure has been developed and implemented using Mathematica software. We submitted
it to extensive testing using numerous numerical examples, varying input parameters.
Presented herein are the obtained results, highlighting some sets of input data that were specifically chosen while still adhering to practical settings.

<caption>**Table 1. Numerical data.**</caption>

**Data related to the production activity:**

<table><tr><td>Machining process</td><td>Turning</td></tr><tr><td>Work material</td><td>42CrMo4 Steel (a medium carbon Steel close to an AISI 4142 Steel) et AlSi12Cu Aluminium</td></tr><tr><td>Tool</td><td>An uncoated CERMET insert without chipbreaker</td></tr><tr><td>I<sub>carbone</sub></td><td>0.4 Kg CO<sub>2</sub>/kWh</td></tr><tr><td>Social C<sub>CO<sub>2</sub></sub></td><td>50 €/tonne</td></tr><tr><td>Enrg<sub>Kg</sub></td><td>9 kWh/Kg</td></tr><tr><td>η</td><td>80%</td></tr><tr><td>α<sub>AL</sub></td><td>20%</td></tr><tr><td>RMC<sub>AL</sub></td><td>150 €</td></tr><tr><td>RMC<sub>ST</sub></td><td>100 €</td></tr><tr><td>W<sub>AL</sub></td><td>2</td></tr><tr><td>W<sub>ST</sub></td><td>5</td></tr><tr><td>f<sub>AL</sub></td><td>0.1 mm/tr</td></tr><tr><td>a<sub>pAL</sub></td><td>4 mm</td></tr><tr><td>k<sub>AL</sub></td><td>800 Mpa</td></tr><tr><td>υ<sub>AL</sub></td><td>900 m/min</td></tr><tr><td>f<sub>ST</sub></td><td>0.15 mm/tr</td></tr><tr><td>a<sub>pST</sub></td><td>1.5 mm</td></tr><tr><td>k<sub>CST</sub></td><td>2 000 Mpa</td></tr><tr><td>υ<sub>CST</sub></td><td>400 m/min</td></tr><tr><td>C<sub>e</sub></td><td>0.2276 €/kWh</td></tr><tr><td>UCS<sub>AL</sub></td><td>2 €</td></tr><tr><td>UCS<sub>ST</sub></td><td>3 €</td></tr><tr><td>SC<sub>AL</sub></td><td>15 €</td></tr><tr><td>SC<sub>ST</sub></td><td>50 €</td></tr><tr><td>S<sub>AL</sub>(0)</td><td>12</td></tr><tr><td>S<sub>ST</sub>(0)</td><td>10</td></tr><tr><td>NPr<sub>AL</sub></td><td>45</td></tr><tr><td>NPr<sub>ST</sub></td><td>60</td></tr><tr><td>Pr<sub>AL</sub>max</td><td>87</td></tr><tr><td>Pr<sub>ST</sub>max</td><td>100</td></tr><tr><td>Δt</td><td>2 months</td></tr><tr><td>H</td><td>30 months</td></tr></table>

**Random demand**

<table><tr><td>dAL<sub>1</sub> = 55</td><td>dAL<sub>11</sub> = 21</td><td>dAL<sub>21</sub> = 40</td><td>dST<sub>1</sub> = 14</td><td>dST<sub>11</sub> = 24</td><td>dST<sub>21</sub> = 32</td></tr><tr><td>dAL<sub>2</sub> = 24</td><td>dAL<sub>12</sub> = 10</td><td>dAL<sub>22</sub> = 11</td><td>dST<sub>2</sub> = 51</td><td>dST<sub>12</sub> = 9</td><td>dST<sub>22</sub> = 51</td></tr><tr><td>dAL<sub>3</sub> = 32</td><td>dAL<sub>13</sub> = 8</td><td>dAL<sub>23</sub> = 48</td><td>dST<sub>3</sub> = 28</td><td>dST<sub>13</sub> = 40</td><td>dST<sub>23</sub> = 30</td></tr><tr><td>dAL<sub>4</sub> = 16</td><td>dAL<sub>14</sub> = 39</td><td>dAL<sub>24</sub> = 20</td><td>dST<sub>4</sub> = 8</td><td>dST<sub>14</sub> = 30</td><td>dST<sub>24</sub> = 18</td></tr><tr><td>dAL<sub>5</sub> = 22</td><td>dAL<sub>15</sub> = 57</td><td>dAL<sub>25</sub> = 8</td><td>dST<sub>5</sub> = 33</td><td>dST<sub>15</sub> = 13</td><td>dST<sub>25</sub> = 27</td></tr><tr><td>dAL<sub>6</sub> = 38</td><td>dAL<sub>16</sub> = 18</td><td>dAL<sub>26</sub> = 29</td><td>dST<sub>6</sub> = 20</td><td>dST<sub>16</sub> = 40</td><td>dST<sub>26</sub> = 20</td></tr><tr><td>dAL<sub>7</sub> = 42</td><td>dAL<sub>17</sub> = 30</td><td>dAL<sub>27</sub> = 38</td><td>dST<sub>7</sub> = 41</td><td>dST<sub>17</sub> = 19</td><td>dST<sub>27</sub> = 32</td></tr><tr><td>dAL<sub>8</sub> = 17</td><td>dAL<sub>18</sub> = 49</td><td>dAL<sub>28</sub> = 15</td><td>dST<sub>8</sub> = 10</td><td>dST<sub>18</sub> = 50</td><td>dST<sub>28</sub> = 9</td></tr><tr><td>dAL<sub>9</sub> = 54</td><td>dAL<sub>19</sub> = 19</td><td>dAL<sub>29</sub> = 60</td><td>dST<sub>9</sub> = 33</td><td>dST<sub>19</sub> = 27</td><td>dST<sub>29</sub> = 38</td></tr><tr><td>dAL<sub>10</sub> = 21</td><td>dAL<sub>20</sub> = 50</td><td>dAL<sub>30</sub> = 22</td><td>dST<sub>10</sub> = 42</td><td>dST<sub>20</sub> = 11</td><td>dST<sub>30</sub> = 56</td></tr></table>
## 5. Numerical results

The obtained results are shown below. They represent the quantities to be produced of each part in Aluminium and Steel in each period. Such that the expected total cost is minimized.

<table><caption>Table 2. Numerical results of production model.</caption><thead><tr><th>H</th><th>Pr<sub>AL</sub>(p)</th><th>Pr<sub>ST</sub>(p)</th><th>H</th><th>Pr<sub>AL</sub>(p)</th><th>Pr<sub>ST</sub>(p)</th><th>H</th><th>Pr<sub>AL</sub>(p)</th><th>Pr<sub>ST</sub>(p)</th></tr></thead><tbody><tr><td>1</td><td>27</td><td>83</td><td>11</td><td>15</td><td>100</td><td>21</td><td>78</td><td>16</td></tr><tr><td>2</td><td>67</td><td>30</td><td>12</td><td>87</td><td>5</td><td>22</td><td>71</td><td>26</td></tr><tr><td>3</td><td>16</td><td>99</td><td>13</td><td>15</td><td>100</td><td>23</td><td>79</td><td>15</td></tr><tr><td>4</td><td>87</td><td>4</td><td>14</td><td>19</td><td>94</td><td>24</td><td>83</td><td>9</td></tr><tr><td>5</td><td>61</td><td>39</td><td>15</td><td>85</td><td>7</td><td>25</td><td>80</td><td>14</td></tr><tr><td>6</td><td>15</td><td>100</td><td>16</td><td>15</td><td>100</td><td>26</td><td>82</td><td>10</td></tr><tr><td>7</td><td>75</td><td>21</td><td>17</td><td>83</td><td>9</td><td>27</td><td>78</td><td>16</td></tr><tr><td>8</td><td>21</td><td>92</td><td>18</td><td>24</td><td>87</td><td>28</td><td>87</td><td>5</td></tr><tr><td>9</td><td>78</td><td>17</td><td>19</td><td>15</td><td>100</td><td>29</td><td>76</td><td>19</td></tr><tr><td>10</td><td>27</td><td>84</td><td>20</td><td>86</td><td>6</td><td>30</td><td>69</td><td>28</td></tr></tbody></table>

## 6. Conclusion

This paper tackled the problem of production planning in the context of dry machining that has a rather low environmental impact. In this context, the aim of this study was to derive a production plan that is both cost-effective and good for the environment. Two types of products with a different material, Aluminium and Steel, are to be produced to meet a random demand over a finite time horizon made of equal periods. Production periods are broken down into smaller chunks, focusing separately on Aluminium or Steel, with the goal of lowering overall costs including raw materials, production, energy use, recycling, storage, shortages and CO₂ emissions. By looking at economic, technical, and environmental factors together, the proposed approach allowed achieving a balance that supports both profitability and sustainability. This approach promotes the circular economy by reducing the need for new raw materials, boosting energy efficiency, and maximizing resource recovery. It is meant to contribute to providing a roadmap for manufacturers aiming to shift toward more sustainable systems. Practically speaking, the approach adopted in this work can help reduce the production costs of manufacturers and the environmental impact. The installation of robotic sorters in the production process can also help lower production costs by enabling better raw material recovery, improving recycling efficiency, and reducing energy requirements. Direct extensions of this work include multi-stage production planning and the use of AI-driven demand forecasting to increase forecast accuracy.

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