Integrated Production-Maintenance Strategy Considering Energy Consumption and Recycling Constraints in Dry Machining

Integrated Production-Maintenance Strategy considering energy consumption and recycling constraints in dry machining

El Mehdi Guendouli  Lahcen Mifdal 

International University of Agadir: Universite Internationale d'Agadir So ene Dellagi 

El Mehdi Kibbou  Abdelhadi Moufki 

Research Article

Keywords: maintenance, production, dry machining, raw material recycling, energy consumption Posted Date: March 5th, 2024

DOI: https://doi.org/10.21203/rs.3.rs-3982933/v1

License:  This work is licensed under a Creative Commons Attribution 4.0 International License.  

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1

Integrated Production-Maintenance Strategy considering energy consumption and recycling constraints in dry machining

El Mehdi Guendouli

^a,b

, Lahcen Mifdal

^b*

, Sofiene Dellagi

^a

, El Mehdi Kibbou

b

, Abdelhadi Moufki

^c

a Laboratoire de Génie Informatique, de Production et de Maintenance, LGIPM, (EA 3096), Université de Lorraine, Metz, France ; b Laboratoire InterDisciplinaire de Recherches Appliquées, LIDRA, Université Internationale d’Agadir- Universiapolis, Agadir, Maroc;

c Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux, LEM3, UMR CNRS 7239 , Université de Lorraine, Metz, France ;

--- Corresponding author : * Lahcen Mifdal : mifdal@e-polytechnique.ma

--- The correct author names and emails are listed below:

• El Mehdi Guendouli: el-mehdi.guendouli@univ-lorraine.fr

•Sofiene Dellagi: sofiene.dellagi@univ-lorraine.fr

•El Mehdi Kibbou: elmehdi.kibbou@e-polytechnique.ma

•Lahcen Mifdal: mifdal@e-polytechnique.ma

•Abdelhadi Moufki: abdelhadi.moufki@univ-lorraine.fr

---

2 Abstract

The current challenge for industrial companies, involved in improving CNC - Computer Numerical Control - mechanical manufacturing machines, consists in integrating production decision aid adapted to the constraints associated with dry machining processes. This tool provides the best choice of appropriate production parameters for dry machining, which has a direct impact on productivity, system degradation and the quality of the final product. The proposed study develops an integrated production-maintenance policy which considers simultaneously several parameters related to production process and manufacturing system environment. In fact, our goal consists in determining an economical integrated production maintenance strategy that minimize the total cost including raw materials, production, recycling and maintenance. Considering two types of raw materials (steel and aluminum) and requesting to a random demand over a finite horizon dissociated to subperiods, the economical integrated maintenance-production, which is obtained by minimizing the total cost, is illustrated by the processing time of each type of raw materials for each subperiod and the optimal preventive maintenance time. The impact of the type of raw material with regard to some parameters like cutting speed, recycle cost, and the degradation system is taken into account in the determination of the economic plan. An analytical model expressing the objective function, the total cost, according to the variable decisions is developed. A numerical solving procedure, a numerical example and a sensitivity study are proposed in order to prove the developed analytical model.

Keywords: maintenance, production, dry machining, raw material recycling, energy consumption

3 I. Introduction and literature review

Manufacturing companies have to manage several functional aspects, such as production, maintenance, and sales. Success lies in managing these areas simultaneously. To ensure effective coordination between these functions, managers and decision-makers need to consider a systemic approach that integrates the interactions between all or some of these complementary functions. Many researchers have been looking into ways of integrating maintenance and production, after decades during which these two functions were studied separately. Work on maintenance policies began with Barlow and Hunter [1] and has been followed by numerous contributions, as highlighted by a study of maintenance models by Rosmaini and Shahrul [2], [3], [4], [5]. Over the last two decades, many companies have become aware of the ineffectiveness of strategies that separated maintenance from production.

Recently, Larbi Rebaiaia and Ait-kadi [6] have developed an approach that enables a numerical comparison to be made between three distinct maintenance strategies. These strategies include minimal repair on failure, full replacement only after the first failure, and full replacement on every failure. These strategies have been integrated into a modified block replacement policy, which encompasses both corrective and preventive interventions.

However, to select the most cost-effective maintenance strategy, the authors provided a concrete example based on an industrial system made up of several components.

In considering operational and environmental constraints, companies in the industrial sectors are constantly looking for strategies to increase efficiency while meeting their customers’

requirements in terms of service, deadlines, quality, and costs. This applies specifically to the machining industry, which is the focus of our paper. Indeed, given the complexity of machining and the need to manage both costs and quality, it is imperative to adopt an integrated approach that take into consideration not only production aspects, but also those related to reliability and maintenance, environment, and recycling activities. When considering production parameters, productivity, system degradation, and energy consumption by manufacturing systems, it is important to also take into account the recycling process of materials. At the outset of this study, apart from the specific case of machining production systems, there are various contributions in the literature dealing with integrated production-maintenance strategies for all types of production systems. Therefore, there is a need for new integrated strategies for maintenance and production. We can see in literature several studies developing integrated production maintenance strategies. For example, Brandolese et al. [7] developed a maintenance strategy for a production system with several machines. This involves scheduling the date and machine responsible for each task, integrating preventive maintenance activities as close as possible to optimal maintenance periods. Chelbi and Rezg [8] developed and optimized a mathematical model in order to determine both the size of the buffer stock and the preventive maintenance period for a production unit subject to regular preventive maintenance of random duration. In addition, Rezg et al. [9] developed both an analytical model and a numerical method to determine a joint optimal inventory management and age-based preventive maintenance strategy for a production system that is subjected to random failures.

4 In the same context, several researchers have considered various external constraints. For example, Dellagi et al. [10] have examined integrated maintenance-production strategies that consider aspects of subcontracting. They developed and optimized a maintenance policy considering the constraints associated with subcontracting. Through a case study, they highlighted the impact of these constraints on the optimal integrated maintenance-production strategy. In the same vein, Dahane et al. [11] carried out an analytical analysis of the problem of integrating subcontracting activities, thus determining the optimal number of subcontracting tasks to be carried out during a maintenance cycle.

In the case of unreliable multi-product production systems, Mifdal et al. [12] have developed a production-integrated maintenance strategy to cope with a number of random breakdowns.

The authors determined the economic production plan for each product, which minimizes setup, production, and storage costs. They determined an optimal maintenance strategy, considering the influence of production rate on system degradation. Dellagi et al. [13]

considered a production system that has to satisfy a random demand over a finite planning horizon under a required service level. This study consists of developing an analytical model to determine a quasi-optimal integrated production and maintenance plan that considers the influence of the production rate variation on the system degradation while at the same time attempting to smooth the production plan by controlling the production rate between periods of the planning horizon.

Reviewing the relationship between the production plan, the system degradation and the quality of the output products, we can cite Gouiaa-Mtibaa et al. [14], who proposed two types of non-defective products : high quality items taken as first-rate products, and substandard quality items as second-rate products. Rework activities are proposed for second- rate and non-conforming products in order to improve their quality conditions and the selling price. Facing the increased system degradation, an improved imperfect preventive maintenance policy is suggested. In that study, the authors developed and optimized an analytical model in order to determine simultaneously the number of produced batches before performing the imperfect PM, and the number of imperfect PM actions to undertake before applying a perfect one by maximizing the total profit integrating selling price and production, maintenance, and reworking costs.

Pal et al. [15] addressed the problem of optimizing the preventive maintenance period and buffer stock size in the context of a production system that is subject to imperfections and may enter an "out of control" state after a random period of time. They also take into account the non-conforming cost and the rework activities improving the profit. In their study, they also examined the possibility of varying buffer size and production rate as decision variables.

The relationship between the environmental impact of the production and the energy consumption attracts several researchers. For example, Turki et al., 2018 [16] developed an optimal storage and production strategies for both manufacturers and remanufacturers while examining the influence of carbon trading prices and emission caps on during carbon emissions.

Hajej and Rezg [17] have introduced integrated production-maintenance strategy with regard to the energy consumption. This approach considers random demand and a predefined service level. It involves determining the size of the economic production lot and the required

5 number of machines, with the aim of minimizing the total average cost associated with inventory and production. Moreover, considering how the resulting production plan affects system degradation and energy consumption, an optimal maintenance plan is subsequently derived.

Recently, some researchers have shifted focus to the relationship between the mechanic production activities and the system degradation. For example, Majdouline et al. [18] treated especially the case of dry machining. They suggest an integrated production-maintenance approach that enables the simultaneous consideration of various production parameters related to dry machining. These parameters primarily include cutting speed, production time and cost, preventive maintenance interval, quality standards, and selling prices for the final product. A distinctive aspect of this strategy, which operates within a finite time frame, is its ability to determine an optimal change in cutting speed at a specific moment, in conjunction with the scheduling of preventive maintenance intervals. This optimization is aimed at maximizing the expected total profit per unit of time.

Dry machining is a technology that has developed rapidly in recent years. It is a machining process in which the use of cutting fluids and lubricants is avoided in order to reduce costs and protect the environment, [19]. The use of cutting fluid, in traditional machining, has many adverse effects, for instance the fumes generated during the machining process are dangerous for the operator and also affect the environment. Currently, all industries are trying to increase productivity by maintaining quality and reducing production costs. Dry machining can be considered as a green manufacturing process because it significantly reduces environmental pollution. Thus, industries are moving towards eliminating cutting fluids as much as possible. In dry machining, the chips are clean because they are not mixed with the cutting fluid. Therefore, chips handling and recycling are easy. One eliminates also all coolant-related processes (filtration, coolant/chip separators, transport and storage) reducing the overall production costs, [20].

Concerning specific cutting energy, Rahman et al. [21] present a model aid in determining the Energy-Consumption Allowance (ECA) for a workpiece and offers a reference quantity for each energy-consumption step throughout the entire process. It shows great potential for establishing the ECA of a machining system. The concept of an Energy-Consumption Step (ECS) is introduced to uniformly describe different types of energy-consumption procedures in workpiece machining. it includes various aspects like machining ECS, transportation ECS, storage ECS, various sub-ECSs, and fundamental ECSs. In this frame some years before , Liu et al. [22] developed a predictive model in order to quantify the relationships between material removal rate and specific energy, emissions, and environmental impact.

The study examined the emissions and environmental footprint resulting from both the energy consumed by the machine tool and the embodied energy of the cutting tool.

Xia et al. [23] introduce an energy-focused joint optimization strategy called the Energy- Oriented Joint Maintenance and Tool Replacement (EJMR) policy. This was achieved by integrating the mechanisms of energy consumption and opportunities for simultaneous maintenance in a machine-tool system. The central challenge lies in harmonizing the scheduling of preventive maintenance (PM) for the machine with the optimization of sequential tool polish/preventive replacement (PR), thus creating energy-efficient strategies.

6 Several studies, [24], [25], reviewed energy-efficient machining systems and discussed the energy consumption associated with machining processes. It is noteworthy that the cutting process itself accounts for only a small portion of energy consumption, while the majority is attributed to losses, idle running, and secondary systems.

Upon reviewing the literature on machining production systems, it becomes apparent that less focus has been directed to a comprehensive approach, which would encompass not only the production parameters and configurations but also the reliability and maintenance of the production system, as well as the consumption of energy and recycling. It is noteworthy that these three aspects (production, maintenance, and environment) are intricately interconnected in machining processes, yet they have traditionally been addressed independently in the existing literature.

As outlined earlier in this introduction, our paper focuses on one production system, namely material removal machining. This process is widely used in the mechanical industry, including sectors such as the automotive, aerospace, rail, and many others. Dry machining (without the use of cutting fluid) is emerging environmentally as a friendlier and healthier option. This approach is driven by growing consumer demand for environmentally friendly products and government efforts to reduce pollution, encouraging industries to reduce their impact on the planet. In addition, the turning process is commonly used in sectors such as the automotive and aerospace industries, and the manufacture of dies and molds [26]. Due to global economic competition, manufacturers are faced with the need to improve product quality, increase productivity, and extend tool life. However, under certain cutting conditions, various phenomena such as machine tool chatter and tool wear can increase the deterioration of the output product quality, more precisely of the machined surface.

Consequently, the productivity rate is affected. In this context, the use of a predictive model is extremely beneficial for analyzing the links between cutting conditions, energy consumption, recycling, and productivity.

In the proposed paper, we develop an integrated production-maintenance strategy for dry machining in which we simultaneously consider the production parameters, mainly cutting speed, production time and cost, the preventive maintenance period as well as the energy consumption by the machines during machining using two materials Aluminum and steel.

The recycle activities are taken into account.

As for the design of the paper, in Section II, we introduce Integrated strategy and problem definition, describing the problem in question and the overall strategy advocated for optimization. Section III will be devoted to the development of the mathematical model.

Then, in Section IV, we present a numerical example accompanied by a sensitivity analysis to illustrate the application of the analytical model we have developed. Finally, Section V summarizes our conclusions and outlines some perspectives.

II. Integrated strategy and problem definition

We consider a manufacturing system consisting of a CNC - Computer Numerical Control - mechanical manufacturing machines subject to random failures, which consist of manufacturing products using two different materials (Aluminum and Steel) over a finite horizon 𝐻. The aim is to develop a production plan to meet random demand defined for each

7 fixed periods in the horizon , followed by an economical maintenance strategy.

As illustrated in the Figure 1, the planning horizon is subdivided into 𝐻 equal periods of duration ∆𝑡. Each period is divided into two subperiods. The first sub-period whose length is ∆𝑡𝐴𝐿(𝑝) is devoted to the production of Aluminum parts in period 𝑝. The second sub- period of length ∆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 ∆𝑡 stays constant: meaning that ∆𝑡_𝐴𝐿(𝑝) + ∆𝑡𝑆𝑇(𝑝) = ∆𝑡. The production rates for Aluminum and steel parts during each period are respectively 𝑃𝑟𝐴𝐿(𝑝) and 𝑃𝑟𝑆𝑇(𝑝).

The production rates depend on the durations of subperiods allowed to each type of the raw material (aluminium or steel) and its speed cutting.

The Figure 1 below illustrates the distribution of the planning horizon.

Respecting the proposed random demand for each period, with minimizing a total cost including production, inventory, energy, and recycling costs, we will estimate the economical subperiods of production for each type of raw material over the finite horizon.

Then, taking the impact of the production of every type of raw material on the system degradation and relative preventive and corrective maintenance action, we will establish an economical preventive maintenance strategy.

The originality of our proposed work lies in considering the production process in all phases, from the choice of raw material characteristics to the output product in the special case of dry machining. In fact, the impact of the raw material characteristics on the production process (cutting speed, production cost), the degradation of the system and the possible recycle activities related to the raw material, is considered in order to establish an economical integrated production maintenance plan. The economic plan, obtained by minimizing a total cost including raw materials, production, inventory, recycling and maintenance costs, is illustrated by the optimal production periods allowed to produce each type of raw materials and the optimal date of preventive maintenance over a finite horizon in order to request to the random demand allocated according to predefined subperiods inside the horizon. Two types of raw material (steel and aluminium) of physical, mechanical and economic characteristics, are adopted.

III. The analytical model III.1 Production model

III.1.1 Notation

The following notations are used:

Fig1. The integrated production-maintenance strategy over the finite time horizon H.

8 ∆𝒕 : Length of production periods

𝜶𝑨𝑳 : Percentage of material machined in the production of an Aluminum part 𝜶𝑺𝑻 : Percentage of material machined in the production of a steel part

𝐖_𝑨𝑳 : Weight of one Aluminum part 𝐖_𝑺𝑻 : Weight of one steel part

𝑹𝑴𝑪𝑨𝑳 : Unit cost of raw materials for Aluminum parts 𝑹𝑴𝑪_{𝑹𝒆𝒄𝒚𝑨𝑳} : Cost of recycled raw material for Aluminum parts 𝑻𝑹𝑴𝑪𝑨𝑳 : Average total cost of raw materials for Aluminum parts 𝑹𝑴𝑪_{𝑵𝒆𝒘𝑺𝑻} : Unit cost of raw materials for steel parts

𝑺𝑺𝑷_𝑺𝑻 : Unit selling cost of steel chips

𝑹𝑴𝑪_𝑺𝑻 : Unit cost of raw materials for steel parts 𝑻𝑹𝑴𝑪𝑺𝑻 : Average total raw material costs for steel parts

𝑻𝑹𝑴𝑪 : Average total cost of raw materials (Aluminum and steel) 𝑪_𝒆 : Energy unit cost

𝑷𝒄𝑨𝑳 : Cutting power required for machining Aluminum parts 𝑷𝒄𝑺𝑻 : Cutting power required for machining steel parts 𝒇 : Machining system feed speed [mm/rev]

𝒂_𝒑 : Passing depth [mm]

𝒌_𝒄 : Specific cutting pressure [N/𝑚𝑚²] 𝝊_𝒄 : Cutting speed [m/min]

𝑬𝒏𝒓𝒈𝑪𝑨𝑳(𝒑): Energy cost of machining Aluminum parts over the period 𝑝. 𝑻𝑬𝒏𝒓𝒈𝑪_𝑨𝑳 : Average total energy costs for machining Aluminum parts 𝑬𝒏𝒓𝒈𝑪_𝑺𝑻(𝒑) : Energy cost of machining steel parts over the period 𝑝

𝑻𝑬𝒏𝒓𝒈𝑪𝑺𝑻 : Average total energy costs for machining steel parts

𝑻𝑬𝒏𝒓𝒈 : Average total energy costs for machining Aluminum and steel parts 𝐒_𝐀𝐋(𝒑) : Stock levels of Aluminum parts at end of period 𝑝

𝑺_𝑨𝑳^𝑴(𝒑) : Stock level of Aluminum parts at the end of the first sub-period of the period 𝑝.

𝒅_𝑨𝑳(𝒑) : Average demand for Aluminum parts at the end of the period 𝑝 𝐔𝐂𝐬𝐀𝐋 : Unit cost of storing an Aluminum part

𝐂𝐒_𝐀𝐋(𝒑) : Cost of storing an Aluminum part over the period 𝑝 𝑻𝑺𝒕𝑪_𝑨𝑳 : Average total storage costs for Aluminum parts 𝑪𝑺𝑺𝑻(𝒑) : Cost of storing a Steel part over the period 𝑝 𝑺𝑺𝑻(𝒑) : Stock levels of steel parts at end of period 𝑝 𝐔𝐂𝐬_𝐒𝐓 : Unit cost of storing a steel part

𝒅_𝑺𝑻(𝒑) : Average demand for steel parts at end of period 𝑝 𝑻𝑺𝒕𝑪𝑺𝑻 : Average total cost of stocking steel parts

𝑻𝑺𝒕𝒄 : Average total storage costs for Aluminum and steel parts 𝑺𝑪_𝑨𝑳 : Unit shortage costs for Aluminum parts

𝑻𝑺𝒉𝑪𝑨𝑳 : Average total shortage costs for Aluminum parts 𝑺𝑪𝑺𝑻 : Unit shortage costs for steel parts

𝑻𝑺𝒉𝑪𝑺𝑻 : Average total shortage costs for steel parts

9 𝑻𝑺𝒉𝒄 : Average total shortage costs for Aluminum and steel parts

𝑻𝑷𝑪(. ) : Average total production costs

𝑷𝒓𝑨𝑳(𝒑) : Quantity of Aluminum parts produced during the period 𝑝 𝑷𝒓_𝑺𝑻(𝒑) : Quantity of Steel parts produced during the period 𝑝

∆𝒕_𝑺𝑻(𝒑) : Duration of sub-period of production of Aluminum parts in the period 𝑝 𝑵𝑷𝒓𝑨𝑳 : Nominal quantity of Aluminium parts produced

𝑵𝑷𝒓_𝑺𝑻 : Nominal quantity of Steel parts produced The decision variables:

∆𝑡_𝐴𝐿(𝑝) : Duration of sub-period of production of Aluminum parts in the period 𝑝 III.1.2 Production costs

III.1.2.1 Total cost of raw materials

o Cost of raw materials for Aluminium parts

Aluminum 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 𝛼_𝐴𝐿 of recycled material and the rest of raw material.

This process is illustrated in the Figure 2 below:

Fig2. Machining process for aluminium parts.

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

𝑹𝑴𝑪_𝑨𝑳= (𝑹𝑴𝑪_{𝑹𝒆𝒄𝒚𝑨𝑳}× 𝜶_𝑨𝑳) + (𝑹𝑴𝑪_{𝑵𝒆𝒘𝑨𝑳}× (𝟏 − 𝜶_𝑨𝑳)) (1) Given that the weight of each part is W𝐴𝐿 and that a quantity '𝑃𝑟𝐴𝐿(𝑝)' parts are produced during each period 𝑝, the total cost of raw material that will be consumed to produce Aluminum parts, during the planning horizon 𝐻 × ∆𝑡, is defined by the equation below:

10 𝑻𝑹𝑴𝑪_𝑨𝑳 = ∑[𝑹𝑴𝑪_𝑨𝑳× 𝑷𝒓_𝑨𝑳(𝒑) × 𝐖_𝑨𝑳]

𝑯 𝒑=𝟏

(2) We recall that:

W𝐴𝐿 : Weight of one Aluminum part

𝑃𝑟_𝐴𝐿(𝑝) : Quantity of Aluminum parts produced during the period 𝑝 We note that:

𝑷𝒓_𝑨𝑳(𝒑) = 𝑵𝑷𝒓_𝑨𝑳× ∆𝒕_𝑨𝑳(𝒑) (3) Then

𝑻𝑹𝑴𝑪_𝑨𝑳= ∑[𝑹𝑴𝑪_𝑨𝑳× 𝑵𝑷𝒓_𝑨𝑳× ∆𝒕_𝑨𝑳(𝒑) × 𝐖_𝑨𝑳]

𝑯 𝒑=𝟏

(4)

o Cost of raw materials for steel parts

Steel parts are produced from a single type of material: raw material. During machining, swarf (leftover material) is recovered and sold in bulk at the end of each period at a price of 𝑆𝑆𝑃_𝑆𝑇. We consider that the percentage of chips recovered is 𝛼_𝑆𝑇. This process is illustrated in the Figure 3.

Fig3. Machining process for steel parts.

Equation eq. 5 defines the raw material cost (per Kg) to produce a steel part:

𝑹𝑴𝑪_𝑺𝑻 = 𝑹𝑴𝑪_{𝑵𝒆𝒘𝑺𝑻} − 𝑺𝑺𝑷_𝑺𝑻× 𝜶_𝑺𝑻 (5) Given that the weight of each steel part is W_𝑆𝑇 and that 𝑃𝑟_𝑆𝑇(𝑝) parts are produced during each period 𝑝, the total raw material cost for machining steel parts over the entire planning horizon is defined by the following equation:

𝑻𝑹𝑴𝑪_𝑺𝑻 = ∑[𝑹𝑴𝑪_𝑺𝑻 × 𝑷𝒓_𝑺𝑻(𝒑) × 𝐖_𝑺𝑻]

𝑯 𝒑=𝟏

(6)

11 We note that:

𝑷𝒓_𝑺𝑻(𝒑) = 𝑵𝑷𝒓𝑺𝑻× ∆𝒕_𝑺𝑻(𝒑) (7)

∆𝒕𝑺𝑻(𝒑) = ∆𝒕 − ∆𝒕𝑨𝑳(𝒑) (8) Then

𝑻𝑹𝑴𝑪_𝑺𝑻 = ∑[𝑹𝑴𝑪_𝑺𝑻× 𝑵𝑷𝒓_𝑺𝑻× (∆𝒕 − ∆𝒕_𝑨𝑳(𝒑)) × 𝐖_𝑺𝑻]

𝑯 𝒑=𝟏

(9)

The average total raw material cost function over the planning horizon 𝐻 × ∆𝑡 is therefore expressed as:

𝑻𝑹𝑴𝑪 = ∑[𝑹𝑴𝑪_𝑨𝑳× 𝑵𝑷𝒓_𝑨𝑳× ∆𝒕_𝑨𝑳(𝒑) × 𝐖_𝑨𝑳]

𝑯

𝒑=𝟏

+ ∑[𝑹𝑴𝑪𝑺𝑻× 𝑵𝑷𝒓𝑺𝑻× (∆𝒕 − ∆𝒕𝑨𝑳(𝒑)) × 𝐖𝑺𝑻]

𝑯 𝒑=𝟏

(10)

II.1.2.2 Average total energy cost

To develop this model, we considered that the energy cost depends on the cutting power required (𝑃_𝑐) during a drying operation. The equation below will be used to calculate (𝑃_𝑐):

𝑷_𝒄 = 𝒇× 𝒂_𝒑× 𝒌_𝒄× 𝝊_𝒄 (11)

o Energy cost of machining Aluminum parts

A cutting power 𝑃𝑐𝐴𝐿 in [W] is required to produce an Aluminum part on a machine tool.

The power 𝑃𝑐_𝐴𝐿 required during the turning (machining) operation, can be obtained using the formula below:

𝑷𝒄_𝑨𝑳= 𝒇_𝑨𝑳× 𝒂_𝒑𝑨𝑳× 𝒌_𝒄𝑨𝑳× 𝝊_𝒄𝑨𝑳 (12)

The energy cost [W] to produce one Aluminum part during each period 𝑝 is defined as follows:

𝑬𝒏𝒓𝒈𝑪_𝑨𝑳(𝒑) = 𝑷𝒄_𝑨𝑳× ∆𝒕_𝑨𝑳(𝒑) × 𝑪_𝒆 (13) The average total energy cost for manufacturing Aluminum parts over the planning horizon 𝐻 × ∆𝑡 is defined by the following equation:

𝑻𝑬𝒏𝒓𝒈𝑪_𝑨𝑳= ∑[𝑷𝒄_𝑨𝑳× ∆𝒕_𝑨𝑳(𝒑)] × 𝑪_𝒆

𝑯 𝒑=𝟏

(14)

o Energy cost of machining steel parts

To produce a steel part on a machine tool, a cutting power 𝑃𝑐𝑺𝑻 in [kW] is required. The cutting power 𝑃𝑐_𝑺𝑻required during the turning operation, can be calculated by the formula below:

12

𝑷𝒄_𝑺𝑻 =𝒇_𝑺𝑻× 𝒂_𝒑𝑺𝑻× 𝒌_𝒄𝑺𝑻× 𝝊_𝒄𝑺𝑻 (15)

The energy cost [kW] to machine a steel part during each period 𝑝 is defined as follows:

𝑬𝒏𝒓𝒈𝑪_𝑺𝑻(𝒑) = 𝑷𝒄_𝑺𝑻× (∆𝒕 − ∆𝒕_𝑨𝑳(𝒑)) × 𝑪_𝒆 (16) Thus, the average total energy cost for manufacturing steel parts over the planning horizon 𝐻 × ∆𝑡 is calculated as follows:

𝑻𝑬𝒏𝒓𝒈𝑪𝑺𝑻= ∑[𝑷𝒄𝑺𝑻× (∆𝒕 − ∆𝒕𝑨𝑳(𝒑))] × 𝑪𝒆 𝑯

𝒑=𝟏

(17) Based on the two equations eq. 14 and eq. 17, the average total energy cost is expressed by the following function:

𝑻𝑬𝒏𝒓𝒈𝑪 = ∑[𝑷𝒄𝑨𝑳× ∆𝒕𝑨𝑳(𝒑) + 𝑷𝒄𝑺𝑻× (∆𝒕 − ∆𝒕𝑨𝑳(𝒑))] × 𝑪𝒆 𝑯

𝒑=𝟏

(18)

III.1.2.3 Average total storage cost

o Storage costs for Aluminum parts

The Figure 4 illustrates the evolution of production demand, as well as the inventory status of Aluminum parts over the 𝐻 × ∆𝑡 planning horizon.

Fig4. Aluminum parts inventory trends.

Based on the diagram above, for any period 𝑝, the cost of storing Aluminum parts CSAL(p) is defined by the following equation:

𝐂𝐒_𝐀𝐋(𝐩) = 𝑼𝑪𝒔_𝑨𝑳

× [𝐒𝐀𝐋(𝐩 − 𝟏) × ∆𝐭𝐀𝐋(𝐩) × 𝟏𝑺_𝑨𝑳(𝒑−𝟏)>𝟎

+ 𝑺_𝑨𝑳^𝑴(𝐩) × (∆𝒕 − ∆𝒕_𝑨𝑳(𝒑)) × 𝟏_{𝑺𝑨𝑳(𝒑)>𝟎}]

(19)

13 The term "1_{𝑆𝐴𝐿(𝑝)>0}" is equal to 1 if the quantity of Aluminum parts stored in period 𝑝 is positive and equal to 0 otherwise.

Given that the planning horizon consists of 𝐻 periods, the function of the average total cost of stocking Aluminum parts is as follows:

𝑻𝑺𝒕𝑪𝑨𝑳= 𝑼𝑪𝒔𝑨𝑳

× ∑[𝐒𝐀𝐋(𝐩 − 𝟏) × ∆𝐭𝐀𝐋(𝐩) × 𝟏𝑺_𝑨𝑳(𝒑−𝟏)>𝟎 𝑯

𝒑=𝟏

+ 𝑺_𝑨𝑳^𝑴(𝐩) × (∆𝒕 − ∆𝒕_𝑨𝑳(𝒑)) × 𝟏𝑺_𝑨𝑳(𝒑)>𝟎]

(20)

The dynamic equation for the stock condition of Aluminum parts at the end of each period 𝑝 is represented by the following function:

𝑺_𝑨𝑳^𝑴(𝐩) = 𝑺(𝒑 − 𝟏) + (𝑵𝑷𝒓𝑨𝑳× ∆𝒕𝑨𝑳(𝒑)) (21) The dynamic equation for the stock condition of Aluminum parts at the end of each period 𝑝 is represented by the following function:

𝑺𝑨𝑳(𝒑) = 𝑺_𝑨𝑳^𝑴(𝐩) − 𝒅𝑨𝑳(𝒑) (22) o Storage costs for Steel parts

The evolution of demand, production, and stock levels for steel parts over the planning horizon 𝐻 × ∆𝑡 is illustrated in the Figure 5 below.

Fig5. Steel parts inventory trends.

The cost of storing steel parts will be determined for each period. For any period 𝑝, the storage cost CSST(p) is defined by the following equation:

𝐂𝐒_𝐒𝐓(𝒑) = 𝑼𝑪𝒔_𝑺𝑻× 𝐒_𝐒𝐓(𝐩

− 𝟏) × (∆𝒕 − ∆𝒕𝑨𝑳(𝒑)) × 𝟏𝑺_𝑺𝑻(𝒑−𝟏)>𝟎 (23) Given that the stock is fed by the machined quantities 𝑃𝑟_𝑆𝑇(𝑝) at the end of each period 𝑝 and the random demands 𝑑𝑆𝑇(𝑝) are subtracted from the stock at the end of the same period

14 𝑝, the expression of the dynamic equation of the state of the stock of steel parts during each period 𝑝, can be formulated as follows:

𝑺_𝑺𝑻(𝒑) = 𝑺𝑺𝑻(𝒑 − 𝟏) + (𝑵𝑷𝒓𝑺𝑻× (∆𝒕 − ∆𝒕_𝑨𝑳(𝒑))) − 𝒅𝑺𝑻(𝒑) (24) Given that the planning horizon is subdivided into 𝐻 periods, the expression of the average total cost of stocking steel parts will be as follows:

𝑻𝑺𝒕𝑪_𝑺𝑻 = 𝑼𝑪𝒔_𝑺𝑻× ∑[𝑺_𝑺𝑻(𝒑)^×(∆𝒕 − ∆𝒕_𝑨𝑳(𝒑)) × 𝟏𝑺_𝑺𝑻(𝒑−𝟏)>𝟎]

𝑯 𝒑=𝟏

(25) The term "1_{𝑆𝑆𝑇(𝑝)>0}" is equal to 1 if the quantity of steel parts stored in period 𝑝 is positive and equal to 0 otherwise.

Thus, the average total cost of stocking Aluminum parts and steel parts over the planning horizon 𝐻 × ∆𝑡𝑆𝑇 is expressed by:

𝑻𝑺𝒕𝒄 = 𝐔𝐂𝐬_𝐀𝐋

×∑[𝐒_𝐀𝐋(𝐩 − 𝟏) × ∆𝐭_𝐀𝐋(𝐩)× 𝟏_{𝑺𝑨𝑳(𝒑−𝟏)>𝟎}

𝑯

𝒑=𝟏

+ 𝑺_𝑨𝑳^𝑴(𝐩) ×(∆𝒕 − ∆𝒕𝑨𝑳(𝒑))× 𝟏_{𝑺𝑨𝑳(𝒑)>𝟎}]

+ 𝑼𝑪𝒔_𝑺𝑻× ∑[𝑺_𝑺𝑻(𝒑)×(∆𝒕 − ∆𝒕_𝑨𝑳(𝒑)) × 𝟏_{𝑺𝑺𝑻(𝒑−𝟏)>𝟎}]

𝑯 𝒑=𝟏

(26)

III.1.2.4 Average total shortage cost

o Shortage costs for Aluminium parts

We assume that for the two cases of raw materials (Aluminium and steel) the units short are lost (no backorders) and a shortage cost is incurred. In fact, the shortage cost is taken into consideration when demand cannot be met, in other words, when the stock situation becomes negative. Since the unit shortage cost for Aluminum parts is 𝑆𝐶_𝐴𝐿, the expression for the average shortage cost for each period 𝑝 is formulated as follows:

𝑺𝒉𝑪𝑨𝑳(𝒑) = 𝑺𝑪𝑨𝑳× |𝑺𝑨𝑳(𝒑)| × 𝟏𝑺_𝑨𝑳(𝒑)<𝟎 (27) The term "1_{𝑆𝐴𝐿(𝑝)<0}" is equal to 1 if the quantity of Aluminum parts stored in period 𝑝 is negative and equal to 0 otherwise.

We recall that:

𝑺_𝑨𝑳^𝑴(𝐩) = 𝑺(𝒑 − 𝟏) + (𝑵𝑷𝒓_𝑨𝑳× ∆𝒕_𝑨𝑳(𝒑)) (28) With the number of periods in the planning horizon equal to 𝐻, the average total shortage cost for Aluminum parts is expressed as follows:

𝑻𝑺𝒉𝑪𝑨𝑳= 𝑺𝑪𝑨𝑳× ∑[|𝑺𝑨𝑳(𝒑)| × 𝟏𝑺_𝑨𝑳(𝒑)<𝟎]

𝑯 𝒑=𝟏

(29)

o Shortage costs for Steel parts

If the unit shortage cost of steel parts is 𝑆𝐶𝑆𝑇, the average shortage cost for each period 𝑝 can be expressed as:

15 𝑺𝒉𝑪_𝑺𝑻(𝒑) = 𝑺𝑪_𝑺𝑻× |𝑺_𝑺𝑻(𝒑)| × 𝟏𝑺_𝑺𝑻(𝒑)<𝟎 (30) With, "1_{𝑆𝑆𝑇(𝑝)<0}" is equal to 1 if the stocked quantity of steel parts in period 𝑝 is negative and equal to 0 otherwise.

Thus, the average total shortage cost for steel parts is expressed as follows:

𝑻𝑺𝒉𝑪𝑺𝑻 = 𝑺𝑪𝑺𝑻× ∑[|𝑺𝑺𝑻(𝒑)| × 𝟏𝑺_𝑺𝑻(𝒑)<𝟎]

𝑯 𝒑=𝟏

(31) Based on the two equations eq. 29 and eq. 31, the average total shortage cost function for Aluminum and steel parts over the entire planning horizon 𝐻 × ∆𝑡, is expressed as follows:

𝑻𝑺𝒉𝒄 = 𝑺𝑪𝑨𝑳× ∑[|𝑺𝑨𝑳(𝒑)| × 𝟏𝑺_𝑨𝑳(𝒑)<𝟎]

𝑯

𝒑=𝟏

+ 𝑺𝑪_𝑺𝑻× ∑[|𝑺_𝑺𝑻(𝒑)| × 𝟏𝑺_𝑺𝑻(𝒑)<𝟎]

𝑯 𝒑=𝟏

(32)

III.1.2.5 Average total production cost

The average total cost of production policy is simply the sum of the four costs defined above.

This cost can therefore be expressed as follows:

𝑻𝑷𝑪(∆𝒕𝑨𝑳(𝒑))

= ∑[𝑹𝑴𝑪𝑨𝑳× 𝑵𝑷𝒓𝑨𝑳× ∆𝒕𝑨𝑳(𝒑) × 𝐖𝑨𝑳]

𝑯

𝒑=𝟏

+ ∑[𝑹𝑴𝑪_𝑺𝑻× 𝑵𝑷𝒓_𝑺𝑻× (∆𝒕 − ∆𝒕_𝑨𝑳(𝒑)) × 𝐖_𝑺𝑻]

𝑯

𝒑=𝟏

+ ∑[𝑷𝒄𝑨𝑳× ∆𝒕𝑨𝑳(𝒑) + 𝑷𝒄𝑺𝑻× (∆𝒕 − ∆𝒕𝑨𝑳(𝒑))] × 𝑪𝒆 𝑯

𝒑=𝟏

+ 𝑼𝑪𝒔𝑨𝑳

× ∑[𝐒𝐀𝐋(𝐩 − 𝟏) × ∆𝐭𝐀𝐋(𝐩) × 𝟏𝑺_𝑨𝑳(𝒑−𝟏)>𝟎 𝑯

𝒑=𝟏

+ 𝑺_𝑨𝑳^𝑴(𝐩) × (∆𝒕 − ∆𝒕_𝑨𝑳(𝒑)) × 𝟏_{𝑺𝑨𝑳(𝒑)>𝟎}]

+ 𝑼𝑪𝒔𝑺𝑻× ∑[𝑺𝑺𝑻(𝒑) × (∆𝒕 − ∆𝒕𝑨𝑳(𝒑)) × 𝟏𝑺_𝑺𝑻(𝒑−𝟏)>𝟎]

𝑯

𝒑=𝟏

+ 𝑺𝑪_𝑨𝑳× ∑[|𝑺_𝑨𝑳(𝒑)| × 𝟏_{𝑺𝑨𝑳(𝒑)<𝟎}]

𝑯

𝒑=𝟏

+ 𝑺𝑪𝑺𝑻× ∑[|𝑺𝑺𝑻(𝒑)| × 𝟏𝑺_𝑺𝑻(𝒑)<𝟎]

𝑯 𝒑=𝟏

(33)

16 III.1.3 Economic production planning

To define the economic production plan, we need to minimize the average total production cost function in order to determine the optimal production subperiods for every type of raw materials. The problem will be formulated as follows:

𝑴𝒊𝒏 [∑[𝑹𝑴𝑪𝑨𝑳× 𝑵𝑷𝒓𝑨𝑳× ∆𝒕𝑨𝑳(𝒑) × 𝐖𝑨𝑳]

𝑯

𝒑=𝟏

+ ∑[𝑹𝑴𝑪𝑺𝑻× 𝑵𝑷𝒓𝑺𝑻× (∆𝒕 − ∆𝒕𝑨𝑳(𝒑)) × 𝐖𝑺𝑻]

𝑯

𝒑=𝟏

+ ∑[𝑷𝒄_𝑨𝑳× ∆𝒕_𝑨𝑳(𝒑) + 𝑷𝒄𝑺𝑻× (∆𝒕 − ∆𝒕_𝑨𝑳(𝒑))] × 𝑪𝒆 𝑯

𝒑=𝟏

+ 𝑼𝑪𝒔_𝑨𝑳

× ∑[𝐒_𝐀𝐋(𝐩 − 𝟏) × ∆𝐭_𝐀𝐋(𝐩) × 𝟏_{𝑺𝑨𝑳(𝒑−𝟏)>𝟎}

𝑯

𝒑=𝟏

+ 𝑺_𝑨𝑳^𝑴(𝐩) × (∆𝒕 − ∆𝒕𝑨𝑳(𝒑)) × 𝟏𝑺_𝑨𝑳(𝒑)>𝟎]

+ 𝑼𝑪𝒔_𝑺𝑻× ∑[𝑺_𝑺𝑻(𝒑) × (∆𝒕 − ∆𝒕_𝑨𝑳(𝒑)) × 𝟏_{𝑺𝑺𝑻(𝒑−𝟏)>𝟎}]

𝑯

𝒑=𝟏

+ 𝑺𝑪𝑨𝑳× ∑[|𝑺𝑨𝑳(𝒑)| × 𝟏𝑺_𝑨𝑳(𝒑)<𝟎]

𝑯

𝒑=𝟏

+ 𝑺𝑪_𝑺𝑻× ∑[|𝑺_𝑺𝑻(𝒑)| × 𝟏𝑺_𝑺𝑻(𝒑)<𝟎]

𝑯 𝒑=𝟏

]

(34)

Under the following constraints:

{

𝑁𝑃𝑟_𝐴𝐿× ∆𝑡_𝐴𝐿(𝑝) ≤ 𝑃𝑟𝐴𝐿𝑚𝑎𝑥 𝑁𝑃𝑟_𝑆𝑇× ( ∆𝑡 − ∆𝑡_𝐴𝐿(𝑝)) ≤ 𝑃𝑟𝑆𝑇𝑚𝑎𝑥 𝑆_𝐴𝐿^𝑀(p) = 𝑆(𝑝 − 1) + 𝑁𝑃𝑟_𝐴𝐿× ∆𝑡_𝐴𝐿(𝑝)

𝑆_𝐴𝐿(𝑝) = 𝑆𝐴𝐿𝑀(p) − 𝑑𝐴𝐿(𝑝)

𝑆_𝑆𝑇(𝑝) = 𝑆𝑆𝑇(𝑝 − 1) + 𝑁𝑃𝑟𝑆𝑇× ( ∆𝑡 − ∆𝑡_𝐴𝐿(𝑝)) − 𝑑𝑆𝑇(𝑝) 0 < ∆𝑡_𝐴𝐿(𝑝) ≤ ∆𝑡

We recall that the decision variables that will minimize the average total cost of production is: ∆tAL(p).

The first two constraints require the satisfaction rate to exceed the maximum production for each type of raw materials. The others concern the evolution of the level of the stock of each type of raw materials. The last constraint requires that the sub-periods ∆𝑡𝐴𝐿(𝑝) be limited between 0 and ∆𝑡.

17 III.2 Maintenance model

III.2.1 Description of maintenance strategy

The maintenance strategy developed in this paper is known in literature as Perfect preventive maintenance with minimal repair at failures [27]. This strategy is characterized by adopting perfect PM preventive maintenance actions at constant intervals (T, 2T…NT), and only minimal repair is practiced for failures that occurred between PM actions. In order to not disturb the production plan, the PM actions will be realized at the end of the period of production.

We recall that the planning horizon is divided into 𝐻 periods of duration ∆𝑡. Each period is subdivided into two production sub-periods ∆𝑡_𝐴𝐿(𝑝) and ∆𝑡_𝑆𝑇(𝑝). PM preventive maintenance actions are applied after 𝑛 periods. The time interval 𝑇 between two preventive maintenance actions can be obtained using the equation below:

𝑇 = 𝑛 × ∆𝑡 (35)

The Figure 6 below illustrates the distribution of maintenance actions throughout the 𝐻 × ∆𝑡 planning horizon.

Fig6. Distribution of maintenance actions.

The horizon is divided into 𝑁𝑐 PM preventive maintenance periods of equal duration T.

These maintenance actions are applied at periods 𝑖 × 𝑇, (𝑖 = 1, … , 𝑁𝑐). This number is also called Number of cycles 𝑁_𝑐 and is represented by the equation eq. 36 :

𝑵𝑪= 𝑰𝒏𝒕 [𝑯 × ∆𝒕

𝑻 ] (36)

We specify that after a perfect PM action, the condition of the machine is considered as good as brand new. In the event of system failure between preventive maintenance actions, only minimal repair is applied. The duration of PM and CM are assumed negligible.

We note that part of the originality of this study consists in taking into account the impact of the evolution of the production rate of such type of raw materials on the failure rate of the system. This impact is illustrated in the mathematical model developed in next section.

18 III.2.1 Maintenance strategy development

III.2.1.1 Notation

The notations below are used to develop the maintenance strategy model:

𝒏 : Number of production periods before each PM action 𝑵_𝑪 : Number of PM actions (cycles)

𝚽_𝑻 : Total average number of failures over the horizon H

𝝀_𝑨𝑳^𝑷 (𝒕) : Failure rate according to the production of aluminum raw material in period P 𝝀_𝑺𝑻^𝑷 (𝒕) : Failure rate according to the production of Steel raw material in period P 𝑷𝑴𝒄 : Unit cost of PM actions

𝑪𝑴𝒄 : Unit cost of CM actions

𝑻𝑴𝑪(𝑻) : Average total maintenance costs The decision variable:

𝑻 : Time required to perform preventive maintenance

III.2.1.2 The failure rates 𝝀_𝑨𝑳^𝒑 (𝒕) and 𝝀_𝑆𝑇^𝒑 (𝒕)

The evolution of the failure rate over the planning horizon 𝐻 × ∆𝑡 is illustrated in the Figure 7, below:

Fig7. Evolution du taux de défaillance.

As shown in the Figure 7, the system failure rate varies according to the material processed (Aluminum or steel). As mentioned before in our study, we take into account the impact of the type of raw material used on the evolution of the failure rate of the production system. It should be noted that the first study dealt with the influence of certain production-related parameters on the degradation of the production system and, therefore, on the optimal preventive maintenance plan to be implemented, is presented in the work of Zied et al. [28].

They consider the impact of the production rate variation on the failure rate and the economic

19 maintenance plan established. In the way, in our proposed study we take into consideration the impact of the raw material type (steel or aluminium) on the nominal failure rate. It should be noted that this failure rate is also influenced by the amount of material removed during each period. In order to illustrate these impacts mathematically we put forward the following equations representing respectively the failure rates according to aluminium and steel for very period:

𝝀_𝑨𝑳^𝒑 (𝒕) =^{∆𝒕𝑨𝑳}(𝒑)

∆𝒕 × 𝝀𝒏𝑨𝑳(𝒕) (37)

𝝀_𝑺𝑻^𝒑 (𝒕) =^{∆𝒕 − ∆𝒕𝑨𝑳}(𝒑)

∆𝒕 × 𝝀𝒏𝑺𝑻(𝒕) (38)

We not that 𝜆𝑛_𝐴𝐿(𝑡) and 𝜆𝑛_𝑆𝑇(𝑡) are respectively the nominal failure rates according to the production of aluminum and steel raw material.

III.2.1.3 Average number of failures 𝜱_𝑻

As previously mentioned, the failure rate changes from one sub-period to the next, depending on the material processed and the quantity of material machined. Based on the evolution of the failure rate illustrated in the Figure 7, the average number of failures over the horizon H×∆t is expressed as follow:

𝚽𝑻 = ∑ ∑ [∫ ^{∆𝒕𝑨𝑳(𝑷)}𝝀_𝑨𝑳^𝑷 (𝒕)𝒅𝒕

𝟎 + ∫ ^{∆𝒕𝑺𝑻(𝒑)}𝝀_𝑺𝑻^𝒑 (𝒕)𝒅𝒕

𝟎 𝒏×𝒄

𝒑=(𝒄−𝟏)×𝒏+𝟏 𝑵𝒄

𝒄=𝟏

+ ∑[𝝀_𝑨𝑳^𝒊 (∆𝒕𝑨𝑳(𝒊)) + 𝝀_𝑺𝑻^𝒊 (∆𝒕𝑺𝑻(𝒊))]

𝒑−𝟏 𝒊=𝟏

× ∆𝒕𝑨𝑳(𝒑)

+ ( ∑ [𝝀_𝑨𝑳^𝒊 (∆𝒕𝑨𝑳(𝒊)) + 𝝀_𝑺𝑻^𝒊 (∆𝒕𝑺𝑻(𝒊))]

𝒑−𝟏 𝒊=(𝒄−𝟏)×𝒏+𝟏

+ 𝝀_𝑨𝑳^𝒑 (∆𝒕𝑨𝑳(𝒑))) × ∆𝒕𝑺𝑻(𝒑)]

+ ∑ [∫ ^{∆𝒕𝑨𝑳(𝒑)}𝝀_𝑨𝑳^𝑷 (𝒕)𝒅𝒕

𝟎 + ∫ ^{∆𝒕𝑺𝑻(𝒑)}𝝀_𝑺𝑻^𝒑 (𝒕)𝒅𝒕

𝟎 𝑯

𝒑=𝑵𝒄×𝒏+𝟏

+ (∆𝒕_𝑨𝑳(𝒑) + ∆𝒕_𝑺𝑻(𝒑))

× ∑ [𝝀_𝑨𝑳^𝒊 (∆𝒕𝑨𝑳(𝒊)) + 𝝀_𝑺𝑻^𝒊 (∆𝒕𝑺𝑻(𝒊))]

𝒑−𝟏

𝒊=𝑵𝒄×𝒏+𝟏

+ 𝝀_𝑨𝑳^𝒊 (∆𝒕𝑨𝑳(𝒑)) × ∆𝒕𝑺𝑻(𝒑)]

(39)

20 Proof:

o Average number of failures in the first cycle

For any period 𝑝 of the first cycle (before the first preventive maintenance PM), the functions that make up the function of the average number of failures Φ_𝑝 are shown below:

S-P 1

Φ_𝑝¹ = ∫ ^{∆𝑡𝐴𝐿(𝑃)}𝝀_𝐴𝐿^𝑷 (𝑡)𝑑𝑡

0 + ∑[𝝀_𝐴𝐿^𝒊 (∆𝑡_𝐴𝐿(𝑖)) + 𝝀𝑆𝑇𝒊 (∆𝑡_𝑆𝑇(𝑖))]

𝑝−1 𝑖=1

× ∆𝑡_𝐴𝐿(𝑝) S-P 2

Φ_𝑝² = ∫ ^{∆𝑡𝑆𝑇(𝑝)}𝜆_𝑆𝑇^𝑝 (𝑡)𝑑𝑡

0

+ [∑[𝜆_𝐴𝐿^𝑖 (∆𝑡_𝐴𝐿(𝑖)) + 𝜆𝑆𝑇𝑖 (∆𝑡_𝑆𝑇(𝑖))]

𝑝−1 𝑖=1

+ 𝜆_𝐴𝐿^𝑝 (∆𝑡_𝐴𝐿(𝑝))]

× ∆𝑡_𝑆𝑇(𝑝)

We recall that each cycle consists of 𝑛 periods. Based therefore on the two functions Φ𝑝1 and Φ_𝑝², below, the average number of failures in any period 𝑝 of the first cycle Φ_𝐶1is expressed by the function below:

𝚽_𝑪𝟏 = ∑[𝚽_𝒑^𝟏+ 𝚽_𝒑^𝟐 ]

𝒏

𝒑=𝟏

Then:

(40)

𝚽_𝑪𝟏 = ∑ [∫ ^{∆𝒕𝑨𝑳(𝑷)}𝝀_𝑨𝑳^𝑷 (𝒕)𝒅𝒕

𝟎 + ∫ ^{∆𝒕𝑺𝑻(𝒑)}𝝀_𝑺𝑻^𝒑 (𝒕)𝒅𝒕

𝟎 𝒏

𝒑=𝟏

+ ∑[𝝀_𝑨𝑳^𝒊 (∆𝒕_𝑨𝑳(𝒊)) + 𝝀_𝑺𝑻^𝒊 (∆𝒕_𝑺𝑻(𝒊))]

𝒑−𝟏 𝒊=𝟏

× ∆𝒕_𝑨𝑳(𝒑)

+ (∑[𝝀_𝑨𝑳^𝒊 (∆𝒕_𝑨𝑳(𝒊)) + 𝝀_𝑺𝑻^𝒊 (∆𝒕_𝑺𝑻(𝒊))]

𝒑−𝟏 𝒊=𝟏

+ 𝝀_𝑨𝑳^𝒑 (∆𝒕_𝑨𝑳(𝒑)))

× ∆𝒕_𝑺𝑻(𝒑)]

(41)

o Average number of failures from the beginning to the last PM action

The difficulty in formulating the function for the average number of failures in this study lies in the fact that it changes from one cycle to another. This is, of course, due to the influence of the system failure rate by the amount of material machined in each sub-period.

We recall that the number of cycles 𝑁𝑐 during the planning horizon 𝐻 × ∆𝑡 is represented by the equation eq. 36.

21 Based on the equation eq. 41, the average number of failures from the beginning to the last PM (all cycles except the last one, which is between the last PM and the end of the planning horizon) is defined by the function below:

𝚽_𝑪 = ∑ ∑ [∫ ^{∆𝒕𝑨𝑳(𝑷)}𝝀_𝑨𝑳^𝑷 (𝒕)𝒅𝒕

𝟎 + ∫ ^{∆𝒕𝑺𝑻(𝒑)}𝝀_𝑺𝑻^𝒑 (𝒕)𝒅𝒕

𝟎 𝒏×𝒄

𝒑=(𝒄−𝟏)×𝒏+𝟏 𝑵𝒄

𝒄=𝟏

+ ∑ [𝝀_𝑨𝑳^𝒊 (∆𝒕_𝑨𝑳(𝒊)) + 𝝀_𝑺𝑻^𝒊 (∆𝒕_𝑺𝑻(𝒊))]

𝒑−𝟏 𝒊=(𝒄−𝟏)×𝒏+𝟏

× ∆𝒕_𝑨𝑳(𝒑)

+ (∑[𝝀_𝑨𝑳^𝒊 (∆𝒕_𝑨𝑳(𝒊)) + 𝝀_𝑺𝑻^𝒊 (∆𝒕_𝑺𝑻(𝒊))]

𝒑−𝟏 𝒊=𝟏

+ 𝝀_𝑨𝑳^𝒑 (∆𝒕_𝑨𝑳(𝒑)))

× ∆𝒕𝑺𝑻(𝒑)]

(42)

o Average number of failures in the last cycle

As shown in the Figure 7 illustrating the evolution of the failure rate, we note that the last cycle (between the last PM action and the end of the planning horizon) is not necessarily a complete cycle, i.e. it does not necessarily consist of 𝑛 periods.

Therefore, the function of the average number of failures in the last cycle Φ_𝐿𝑐, is represented by the equation below:

𝚽𝑳𝑪 = ∑ [∫ ^{∆𝒕𝑨𝑳(𝒑)}𝝀_𝑨𝑳^𝑷 (𝒕)𝒅𝒕

𝟎 + ∫ ^{∆𝒕𝑺𝑻(𝒑)}𝝀_𝑺𝑻^𝒑 (𝒕)𝒅𝒕

𝟎 𝑯

𝒑=𝑵𝒄×𝒏+𝟏

+ (∆𝒕_𝑨𝑳(𝒑) + ∆𝒕𝑺𝑻(𝒑))

× ∑ [𝝀_𝑨𝑳^𝒊 (∆𝒕_𝑨𝑳(𝒊)) + 𝝀_𝑺𝑻^𝒊 (∆𝒕_𝑺𝑻(𝒊))]

𝒑−𝟏

𝒊=𝑵𝒄×𝒏+𝟏

+ 𝝀_𝑨𝑳^𝒊 (∆𝒕_𝑨𝑳(𝒑)) × ∆𝒕_𝑺𝑻(𝒑)]

(43)

Finaly, to determine the function for the average number of failures Φ_𝑇over the total horizon 𝐻 × ∆𝑡, simply sum the two functions Φ_𝐶and Φ_𝐿𝐶. The function Φ_𝑇 is represented by the equation (39).

III.2.1.4 Average total maintenance cost

The average total maintenance cost of the proposed strategy is composed of two costs:

corrective maintenance cost 𝐶𝑀𝑐𝑇and preventive maintenance cost 𝐶𝑀𝑝𝑇.

22 III.2.1.4.1 Corrective maintenance costs

The cost of corrective maintenance can be obtained by multiplying the average unit cost of a corrective maintenance action 𝐶𝑀_𝑐 , by the average number of failures over the finite horizon. The function of this cost is shown below:

𝑪𝑴_𝒄^𝑻 = 𝑪𝑴_𝒄 × 𝚽_𝑻 (44)

III.2.1.4.2 Preventive maintenance costs

On the other hand, the total PM cost can be calculated by multiplying the average unit cost of a preventive maintenance action 𝑃𝑀𝑐 , by the number of preventive maintenance actions.

This number corresponds perfectly to the number of cycles 𝑁𝑐 (Equation 36). This cost can be formulated as follows:

𝑪𝑴_𝒑^𝑻 = 𝑷𝑴_𝒄 × 𝑵𝒄 (45)

III.2.1.4.3 Total maintenance cost

Based on the equations eq. (35, 39, 44 & 45), the average total maintenance cost of the proposed strategy, including PM and CM actions, is therefore formulated as follows:

𝑻𝑪𝑴 (𝑻) = 𝑪𝑴_𝒄

× [

∑ ∑

[

∫ ^{∆𝒕𝑨𝑳(𝑷)}𝝀_𝑨𝑳^𝑷 (𝒕)𝒅𝒕

𝟎 + ∫ ^{∆𝒕𝑺𝑻(𝒑)}𝝀_𝑺𝑻^𝒑 (𝒕)𝒅𝒕

𝟎 𝑻

∆𝒕×𝒄

𝒑=(𝒄−𝟏)×_∆𝒕^𝑻+𝟏 𝑰𝒏𝒕[^{𝑯 × ∆𝒕} _𝑻 ]

𝒄=𝟏

+ ∑[𝝀_𝑨𝑳^𝒊 (∆𝒕_𝑨𝑳(𝒊)) + 𝝀_𝑺𝑻^𝒊 (∆𝒕_𝑺𝑻(𝒊))]

𝒑−𝟏 𝒊=𝟏

× ∆𝒕_𝑨𝑳(𝒑)

+ (

∑ [𝝀_𝑨𝑳^𝒊 (∆𝒕_𝑨𝑳(𝒊)) + 𝝀_𝑺𝑻^𝒊 (∆𝒕_𝑺𝑻(𝒊))]

𝒑−𝟏

𝒊=(𝒄−𝟏)×_∆𝒕^𝑻+𝟏

+ 𝝀_𝑨𝑳^𝒑 (∆𝒕𝑨𝑳(𝒑)) )

× ∆𝒕𝑺𝑻(𝒑) ] + ∑ [∫ ^{∆𝒕𝑨𝑳(𝒑)}𝝀_𝑨𝑳^𝑷 (𝒕)𝒅𝒕

𝟎 + ∫ ^{∆𝒕𝑺𝑻(𝒑)}𝝀_𝑺𝑻^𝒑 (𝒕)𝒅𝒕

𝟎 𝑯

𝒑=𝑰𝒏𝒕[^{𝑯 × ∆𝒕} _𝑻 ]×_∆𝒕^𝑻+𝟏

+ (∆𝒕_𝑨𝑳(𝒑) + ∆𝒕𝑺𝑻(𝒑))

× ∑ [𝝀_𝑨𝑳^𝒊 (∆𝒕_𝑨𝑳(𝒊)) + 𝝀_𝑺𝑻^𝒊 (∆𝒕_𝑺𝑻(𝒊))]

𝒑−𝟏

𝒊=𝑰𝒏𝒕[^{𝑯 × ∆𝒕} _𝑻 ]×_∆𝒕^𝑻+𝟏

+ 𝝀_𝑨𝑳^𝒊 (∆𝒕𝑨𝑳(𝒑)) × ∆𝒕𝑺𝑻(𝒑)]

]

+ 𝑷𝑴𝒄 × 𝑰𝒏𝒕 [𝑯 × ∆𝒕

𝑻 ]

(46)