This paper deals with the problem of maintenance and production planning for randomly failing multiple-product manufacturing system. The latter consists of one machine which produces several types of products in order to satisfy random demands corresponding to every type of product. At any given time, the machine can only produce one type of product and then switches to another one. The purpose of this study is to establish sequentially an economical production plan and an optimal maintenance strategy, taking into account the influence of the production rate on the system’s degradation. Analytical models are developed in order to find the production plan and the preventive maintenance strategy which minimizes sequentially the total production/inventory cost and then the total maintenance cost. Finally, a numerical example is presented to illustrate the usefulness of the proposed approach.
Manufacturing companies must manage several functional capacities successfully, such as production, maintenance, quality, and marketing. One of the keys to success consists in treating all these services simultaneously. On the other hand, the customer satisfaction is one of the first objectives of a company. In fact, the nonsatisfaction of the customer on time is often due to a random demand or a sudden failure of production system. Therefore, it is necessary to develop maintenance policies relating to production, reducing the total production and maintenance cost. One of the first actions of decision-making hierarchy of a company is the development of an economical production plan and an optimal maintenance strategy.
It is necessary to find the best production plan and the best maintenance strategy required by the company to satisfy customers. This is a complex task because there are various uncertainties due to external and internal factors. External factors may be associated with the inability to precisely define the behaviour of the application during periods of production. Internal factors may be associated with the availability of hardware resources of the company. In this context, Filho [
Establishing an optimal production planning and maintenance strategy has always been the greatest challenge for industrial companies. Moreover, during the last few decades, the integration of production and maintenance policies problem has received much research attention. In this context, Nodem et al. [
This work examined a problem of the optimal production planning formulation of a manufacturing system consisting of one machine producing several products in order to meet several random demands. This type of problem was studied by Kenne et al. [
Looking at the literature on integrated maintenance policies, we noticed that the influence of the production rate on the degradation system over a finite planning horizon was rarely addressed in depth. Recently, Zied et al. [
Motivated by the work in the Zied et al. [
This paper is organized as follows: Section
This study treated an industrial case. The problem concerns a textile company located in North Africa specialized in clothing manufacturing. The company’s production system consists of a conversion of three types of fiber into yarn, then fabric, and textiles. These are then fabricated into clothes or other artefacts. The production machine is called the loom, and it uses a jet of air or water to insert the weft. The loom ensures pattern diversity and faultless fabrics by a flexible and gentle material handling process. Fabrics can be in one plain color with or without a simple pattern or they can have decorative designs.
Based on the industrial example described, this study was conducted to deal with the problem of an optimal production and maintenance planning for a manufacturing system. The system is composed of a single machine which produces several products in order to meet corresponding several random demands. The problem is presented in (Figure
Problem description.
The considered equipment is subject to random failures. The degradation of the equipment increases with time and varies according to the production rate. The machine is submitted to a preventive maintenance policy in order to reduce the occurrence of failures. In the literature, the influence of the production rate on the material degradation is rarely studied. In this study, this influence was taken into consideration in order to establish the optimal maintenance strategy.
The model developed in this study is based on the works of Zied et al. [
Firstly, for a randomly given demand, an optimal production plan was established to minimize the average total storage and production costs while satisfying a service level. Secondly, using the obtained optimal production plan and considering its influence on the manufacturing system failure rate, an optimal maintenance schedule is established to minimize the total maintenance cost.
In this paper, we shall as far as possible use the notation summarized as follows:
In this section, we developed an analytical model which minimizes the total cost of production and storage. The decision variables are the production quantities
The state of the stock is determined at the end of each subperiod. Figure
Production plan.
To develop this section, the following assumptions are specifically made: holding and production costs of each product are known and constant; only a single product can be produced in each subperiod; as described in (Figure the standard deviation of demand the inventory balance equation, the service level, the admissibility of production plan, the maximum production capacity.
The model has the following basic structure:
This equation shows that the stock of product
Formally, the function becomes
However, the term
We recall that, in this study, we assume that the horizon is divided into
The mathematical formulation of the proposed problem is based on the extension of the model described by Zied et al. [
Their problem is defined as follows:
Formally, our stochastic production problem is defined as follows:
Under the following constraints:
The constraints below should also be taken into account:
For each subperiod
We admit that a function
Thus, the problem formulation can be presented as follows:
The resolution of the stochastic problem under these assumptions is generally difficult. Thus, its transformation into a deterministic problem facilitates its resolution.
We suppose that the means and variance of demand are known and constant for each product
Therefore,
And
Consider the following:
We know that
Hence,
The distribution function is invertible because it is an increasing and differentiable function.
Hence,
The expression of the total cost of production is presented as follows.
Consider the following:
See Appendix
The maintenance strategy adopted in this study is known as preventive maintenance with minimal repair. The actions of preventive maintenance are practiced in the period durations of maintenance actions are negligible; preventive maintenance actions are always performed at the end of the subperiods of production.
The aim of this maintenance strategy is to find the optimal number of preventive maintenance actions
Before determining the analytical model minimizing the total cost of maintenance, we need first to develop the expression of the failure rate
Recall that the key of this study is the influence of the variation of the production rates on the failure rate.
Figure
Degradation rate.
As presented in Figure
Thus, the expression of the failure rate depending on time and production rate can be written as follows:
The term
Therefore,
Note that
Consider the following:
See Appendix
In order to reduce the complexity of the generation of the optimal number of preventive maintenance, we assume that interventions are made at the end of subperiods.
Hence, the function of the period of intervention is presented as follows:
To determine the average number of failures expression
The evolution of the failure rate during the interval
Therefore, the average number of failures expression during the interval
Thus, the average number of failures expression
Consider the following:
Note that
We recall that the initial expression of the total cost of maintenance presented in (
The following equation determines analytically the optimal solution:
We start by proving the existence of a local minimum.
We have the following.
Limits at the terminals of
Moreover,
Consider the following:
The resolution of this maintenance policy, using a numerical procedure, is performed by incrementing the number of maintenance intervals until an
From the industrial example presented in Section
Demands | ||||||||
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Trim. 1 | Trim. 2 | Trim. 3 | Trim. 4 | Trim. 5 | Trim. 6 | Trim. 7 | Trim. 8 | |
Product 1 | 201 | 199 | 198 | 199 | 201 | 202 | 200 | 199 |
Product 2 | 111 | 119 | 108 | 111 | 112 | 110 | 110 | 119 |
Product 3 | 321 | 322 | 323 | 319 | 321 | 317 | 320 | 319 |
The other data are presented as shown in Table
Initial stock level |
Nominal production quantities |
Unit production costs |
Unit holding costs |
Satisfaction rates |
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Product 1 | 110 | 750 | 13 | 3 | 87 |
Product 2 | 85 | 530 | 17 | 5 | 95 |
Product 3 | 145 | 1150 | 9 | 2 | 90 |
the law of failure characterizing the nominal conditions is Weibull. It is defined by scale parameter shape parameter position parameter the initial failure rate:
These parameters provide information on the evolution of the failure rate in time.
This failure rate is increasing and linear over time. Thus, the function of the nominal failure rate is expressed by
The economic production plan obtained is presented in Table
The optimal production plan.
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0.85 | 0.71 | 1.44 | 1.19 | 1.20 | 0.61 | 0.81 | 1.18 | 1.01 | 0.43 | 0.74 | 1.83 | |
Product 1 | 0 | 169 | 0 | 388 | 0 | 0 | 0 | 321 | 0 | 0 | 151 | 0 |
Product 2 | 150 | 0 | 0 | 0 | 185 | 0 | 134 | 0 | 0 | 0 | 0 | 312 |
Product 3 | 0 | 0 | 507 | 0 | 0 | 230 | 0 | 0 | 387 | 158 | 0 | 0 |
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1.82 | 0.87 | 0.31 | 0.56 | 0.55 | 1.89 | 1.36 | 0.51 | 1.13 | 1.05 | 0.77 | 1.18 | |
Product 1 | 0 | 212 | 0 | 0 | 138 | 0 | 272 | 0 | 0 | 130 | 0 | 0 |
Product 2 | 0 | 0 | 52 | 58 | 0 | 0 | 0 | 0 | 92 | 0 | 81 | 0 |
Product 3 | 554 | 0 | 0 | 0 | 0 | 422 | 0 | 202 | 0 | 0 | 0 | 135 |
As described in Figure
Sequential production and maintenance optimization.
Figure
The total cost of maintenance depending to
We recall that the specificity of this study is that it considered the impact of the production rate variation on the system degradation and consequently on the optimal maintenance strategy adopted in the case of multiple product. In order to show the significance of our study we will consider, in this section, the case of not considering the influence of the production rate variation on the system’s degradation. That is to say, we assume that the manufacturing system is exploited at its maximal production rate every time. Analytically, we will consider the nominal failure rate which depends only on time. The results of this study are presented in Table
The sensitivity study based on the variation of production rate.
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Case 1: considering variation of production rate | 3 316 | 2 |
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Case 2: not considering the variation of production rate | 3 704 | 3 |
The optimal number of preventive maintenance obtained in the case when we did not consider the variation of production rate is
Several studies have addressed issues related to production and maintenance problem. But, the consideration of the materiel degradation according to the production rate in the case of multiple-product has been rarely studied.
This study was conducted to deal with the problem of an optimal production and maintenance planning for a manufacturing system. The significance of the present study is that we took into account the influence of the production plan on the system degradation in order to establish an optimal maintenance strategy. The considered system is composed of a single machine which produces several products in order to meet corresponding several random demands.
In this paper, we have discussed the problem of integrated maintenance to production for a manufacturing system consisting of a single machine which produces several types of products to satisfy several random demands. As the machine is subject to random failures, preventive maintenance actions are considered in order to improve its reliability. At failure, a minimal repair is carried out to restore the system into the operating state without changing its failure rate.
At first we have formulated a stochastic production problem. To solve this problem, we have used a production policy to achieve a level of economic output. This policy is characterized by the transformation of the problem to a deterministic equivalent problem in order to obtain the economic production plan. In the second step, taking into account the economic production plan obtained, we have studied and optimized the maintenance policy. This policy is defined by preventive actions carried out at constant time intervals. The objective of this policy is to determine the optimal number of preventive maintenance and the optimal intervention periods over a finite horizon. This policy is characterized by a failure rate for a linear degradation of the equipment considering the influence of production rate variation on the system degradation and on the optimal maintenance plan in the case of multiple products represents.
The promising results obtained in this thesis can lead to interesting perspectives. A perspective that we are looking for at the short term, is to consider maintenance durations. We recall that, throughout our study, we neglected the durations of actions of preventive and corrective maintenance. It is clear that the consideration of these durations impacts the optimal maintenance plan and the established production plan. In the medium term, it is interesting to concretely consider the impact of logistics service on the study. It is clear that the in-maintenance logistics are absent in most researches. The combination of maintenance logistics and production represents a motivating perspective in this field of study.
Another interesting perspective specifying the manufactured product can be explored.
We have for for for any value of for any value of
On the other hand,
Equation ( for for any value of for any value of
The authors declare that there is no conflict of interests regarding the publication of this paper.