Economic and Energetic Performance Evaluation of Unreliable Production Lines: An Integrated Analytical Approach

The current context is increasingly driving researchers and industry to focus not only on the economic but also on the energetic performance of manufacturing systems. However, considerable work on the enhancement of energetic performance of complex production systems is still needed. This paper addresses a novel integrated analytical method to evaluate simultaneously the economic and energetic performances of a serial production line composed of unreliable machines and intermediate buffers. This approach is based on a discrete Markov chain formulation of machines states transitions and a birth-death Markov process for buffers states evaluation. It introduces throughput, energy consumption and energy efficiency as key performance indicators for assessing economic and energetic performances. Structural characteristics of the problem are analyzed to establish and evaluate the impact of buffers size, reliability parameters, and production rates of the machines on the energetic performance of the production line. A large experimental study, based on different instances inspired by the literature, is carried out to analyze the behavior and the complex trade-off between throughput and energy efficiency performances.


I. INTRODUCTION
For decades, economies around the world have been able to grow rapidly thanks to the almost unlimited availability of energy and cheap resources. However, starting in the 1970s, a great deal of research has highlighted the consequences of limited resources. Since then, many economies have competed vigorously for access to resources. In addition, energy prices have risen sharply and energy costs have become consequent and human-induced climate change is directly related to global energy consumption. In particular, the industrial sector, represents more than 31% of total world energy consumption [19] (Fig.1).
Thus, improving energy efficiency in manufacturing is becoming an unavoidable necessity for energy conservation, emission reduction, and sustainability. Motivated by financial pressures from increasing energy prices, legislative measures such as eco-design standards for industrial machines, as well as signs of induced climate change, the industrial sector The associate editor coordinating the review of this manuscript and approving it for publication was Javed Iqbal .  [19].
is increasingly interested in the issue of energy efficiency. Unfortunately, little work has been done when it comes to studying the combination of energetic and economic efficiency of industrial production lines. In fact, many research results have been reached separately in each field, especially the field of improving energy efficiency for machine tools.
However, a significant lack of work is noticed in the case of complex production systems. Hence, this study, to the best of our knowledge, aims to establish the first integrated approach to evaluate and analyze the complex trade-off between the economic and energetic efficiency of a complex production system. The main contribution of this paper can be summarized as follows: • To develop an integrated analytical and computational method to evaluate both economic and energetic performance of a serial production line composed of unreliable machines and intermediate buffers.
• To formulate throughput, energy consumption, and energy efficiency as key performance indicators for assessing economic and energetic performance.
• To propose a non-linear programming approach implemented on Lingo solver for solving the problem.
• To carry out a large experimental study based on wellestablished instances from the literature, to evaluated and analyze the impact of buffers size, reliability parameters, and machine production rates on the energy efficiency of the system.
• To propose an important critical analysis of the trade-off between both performances. The remainder of this paper is organized as follows: section II presents a literature review of economic performance evaluation for serial production lines, as well as energy evaluation for industrial systems. Section III introduces the problem formulation and assumptions. In section IV and V the economic and energetic evaluation methods are respectively presented to obtain, eventually, throughput, energy consumption, and energy efficiency formulations for a serial production line. Numerical experiments are presented, discussed, and analyzed in section VI. Finally, section VII summarizes the guidelines of our contribution, and highlights future work possibilities.

II. LITERATURE REVIEW
The issue of economic efficiency of serial production lines has been largely treated in the literature. In fact, the main approaches for performance evaluation of flow lines can be summarized in three categories as described in [36]: Markov chain analysis, decomposition and aggregation methods, and simulation. Results with Markov chain analysis being usually difficult to obtain for larger production lines, the approximate methods based on decomposition or aggregation, remain the most used [13]. Many methods are derived from these basic approaches, whereas simulation methods are flexible but time-consuming tools, generally adapted to complex systems.
Hence, for small systems with two machines and a single buffer, exact solutions were derived [12], [25]. For larger systems approximate solutions were proposed, generally based on aggregation [20], [21], [26], [27], decomposition approaches [9]- [11], [15]- [17] or other derived analytical methods such as the Equivalent Machine Method [31]. The latter evaluates the system throughput of a buffered serial production line of unreliable machines with exponentially distributed parameters. The proposed method that based on the analysis of the different states of each buffer using birthdeath Markov processes, reduces significantly the state space cardinality of the Markov chain representation of the system and consequently the computational times.
However, nowadays, it has become fundamental to not only optimize production lines from an economic perspective but also from an energetic one. Yet little work has been done when it comes to evaluating energy efficiency for serial production lines, according to our literature review.
Bernoulli serial production lines were studied by [33], who developed an integrated model for improving energy efficiency for lines composed of two machines. The energy consumption of the line was formulated as shown in equation 1. W i is the total electrical power consumed, W Oi represents the set-up power needed for the machine to reach 'ready' status, and k i ρ i the additional power needed to process parts, where k i is a constant (energy coefficient associated to machine M i ), and ρ i the processing rate of machine M i .
Later, [34] extended the study for larger systems with unreliable machines and finite buffers. An integrated model for minimizing energy consumption under a desired production rate was presented (Eq 2), where ρ i is the machine's processing rate and PR d is the desired production rate. This study only considers the processing energy required for production.
Reference [39] also studied energy consumption optimization in two-machine Bernoulli serial lines. It formulated the problem of minimizing the energy consumed by machines in two-machine Bernoulli lines while maintaining the required productivity. An effective algorithm based on binary search method is developed to solve the problem. Based on results from this study as well as the aggregation method, [40] treated the problem for three-machine Bernoulli serial lines. An effective method and an algorithm were designed to solve the energy consumption optimization problem in the threemachine Bernoulli serial line.
On the other hand, [5] developed new energy savings opportunity strategies to maximize energy savings for a serial production line. The ESO (energy saving opportunity) is an opportunity window calculated from on-line production data. It allows certain machines to be turned off for energy saving, without negatively affecting throughput. New energy efficiency performance indicators were presented. The latter used real-time production data to identify the least energyefficient machine on the line. The energy savings opportunity strategy utilizes the Energy Efficiency Performance Indicators EEPI (Eq. 3 where W i,1 is the static part of the VOLUME 8, 2020 energy consumption for machine i,and W i,2 the dynamic part attributed to random disruptions on the line). It takes the opportunity window for the least energy-efficient machine at opportune times, allowing for improvements to be made for this machine to increase the energy efficiency of the line. Later, [6] expanding upon previous research, incorporated the warm-up time of each machine in the analysis.
Reference [2], with a similar perspective, established performance indices for measuring energy efficiency and also productivity, using available sensor data from the production line. The energy structure and effects of downtime events on the production line were studied. Methods for measuring efficiency and highlighting the areas of inefficiency in the system were also proposed. Moreover, transient performance analysis of serial production lines with geometric machines was also studied by [7].
Solutions for the improvement of serial production lines performances were also proposed. Reference [8] studied feedback control of machine start-up for energy consumption reduction in two machine Bernoulli serial line. The authors considered buffer depletion at the end of each shift and used transient analysis. Reference [22] extended the previous study for larger systems considering warm-up time in both transient and steady state. Further solutions for energy efficiency improvement in manufacturing systems were proposed by [41]. The authors developed a Gaussian mixture model to predict machine idle periods duration for manufacturing systems. They suggested optimal actions for energy saving under throughput constraint.
In a more general perspective, [18] used a thermodynamic framework to characterize the material and energy resources used in manufacturing processes. Various processes were analyzed, the relevance of thermodynamics was illustrated for the analysis of manufacturing processes, and exergy analysis was used to identify where resources are lost in these processes. The aim was the design of efficient processes (see the energy consumption formulation for a manufacturing system in Eq. 4, whereẆ is total power used by the process equipment,Ẇ 0 the ''idle'' power for the equipment in the ready position,ṁ the rate of material processing in (mass/time), and k a constant (J/mass)).Ẇ Reference [14] proposed a novel generic method to model the energy consumption behavior of machines and plants based on a statistical discrete event formulation.
Other research work considered particular fields to study the issue. For instance, [24] evaluated energy consumption performance and energy-saving potentials of the ceramic production chain using a first-order hybrid Petri net model. The authors proposed a multi-objective linear programming model for solving the problem as well as sensitivity analysis.
The latter aims to find the optimal specific energy consumption of the production chain. Reference [29] developed a calculation model to forecast the energy consumption of ball mills in ceramic industry during the grinding process based on power feature deployment. An integrated framework for energy savings in sheet metal forming was developed by [38]. Energy consumption modeling, energy performance evaluation, and production optimization were studied.
Energy efficiency was widely studied for machine tools, considering models based on machine components or machine tools globally. Reference [43] presented a comprehensive literature review about energy consumption models and energy efficiency of machine tools. They discussed the connotation of energy efficiency of machine tools, their design, scheduling, optimization, and assessment based on energy efficiency. They also presented several perspectives for the energy consumption modelling and decomposition for machine tools. Later, [42] presented a systematic overview of the classification and prediction methods of energy consumption, together with strategies for energy consumption reduction in machining processes. Reference [32] developed the total energy efficiency index. This is a metric that quantifies the design of machine tools regarding energy efficiency based on the respective assembly of components. Reference [35] focused on the design and selection of machine tools. They developed key performance indicators referred to as ''inherent energy performance'' (IEP) indexes. This is a systematic method which consists of simplified measurement of basic data and the calculation of the indexes from the data.
Reference [30] studied the green performance of CNC (Computer Numerical Control) machine tools. In this work, a model of energy efficiency, carbon efficiency, and green degree is established. Energy efficiency η was defined by the ratio of the energy required for the cutting process (material removal energy) E cut and the total energy consumed in the machining process E process (Eq. 5). The latter is decomposed into start-up state energy E start , standby state energy E standby , no-load state energy E no−load , and load state energy E load . Each state energy is defined as the corresponding state power (P start , P standby , P no−load , and P load ), balanced by each state's duration (t start , t standby , t no−load , and t load ) for startup, standby, no-load, and load states respectively (Eq. 6). Whereas the cutting energy E cut is obtained based on cutting power P cut and load state time t load (Eq. 7).
The energy efficiency is therefore defined, and its evaluation depends on machine and process parameters. Reference [23] proposed an energy consumption model for a mechanical machining process based on empirical modeling. This work studied the relationship between processing parameters (spindle speed, feed rate, and depth of cut) and energy consumption. Reference [28] focused on machine tools selection. A novel method, named Potential Efficiency (PE) Method, was established for energy efficiency evaluation. In this novel method, machining tasks were also considered along with energy efficiency to provide efficient machine tool selection.
It is clear from the above illustrations and to the best of our knowledge that present literature lacks sufficient insight into a system where both productivity and energy efficiency of serial production lines are integrated. Although the issue has been widely studied and considerable results have been obtained in the case of machine tools, more effort is still required for the study, evaluation, and optimization of energy efficiency for serial production lines. This paper aims to present an analytical method that allows the evaluation of energy consumption and efficiency for serial production lines taking into account their economic performance. Therefore, an integrated method for evaluating both the economic and energetic performance of unreliable serial production lines is developed. Results from numerical experiments are used to investigate the effect of buffer size and machine reliability parameters on the energy consumption and efficiency of the system. The trade-off between both performances is also analyzed in this study.

III. PROBLEM FORMULATION
The objective of this study is to develop an analytical and computational method that allows the evaluation of both economic and energetic performance of a serial production line. The system under consideration consists of K machines connected by intermediate storage areas (K − 1) buffers (Fig. 2).
The system is subject to non-availabilities due to the limited capacity of the buffers and/or the failure and repair rates of the machines. The following assumptions are frequently used in the literature. They address the machines, the buffers, the energy consumption and their mutual interactions: • The failure state of the machines depends on the operations. A machine cannot fail if it is starved or blocked.
• The first machine cannot be starved and the last machine cannot be blocked.
• The failure and repair times are independent and distributed according to an exponential law.
• Buffer B i has a finite capacity N and cannot be down. The transition times between machines and buffers are zero.
• Energy consumption of each buffer B i is neglected. We introduce the following notations: Production line throughput

IV. ECONOMIC EVALUATION APPROACH
The economic evaluation (throughput evaluation) is based on the Equivalent Machine Method developed by [31]. This method evaluates the system throughput of a buffered serial production line of unreliable machines with exponentially distributed parameters. Based on the analysis of the different states of each buffer using birth-death Markov processes, each original machine is replaced by an equivalent one taking into account the probabilities of blockage and starvation.
The simplest system of two machines separated by one buffer is used as a building block for the general model. To analyze the steady states of the buffer, a birth-death Markov process with (N + 1) states is considered. N is the capacity of the intermediate buffer, ω 1 and ω 2 respectively the birth and death transition rates, and α the processing rates ratio related to the buffer B. The differential equations for the probability that the system is in state j at time t are: Considering all the differential terms equal to zero at the steady state, and after simplifying the system above and considering the normalization equation: N j=0 P j = 1, the steady probabilities of each buffer state are obtained (Eq. 9). Further VOLUME 8, 2020 formulation details can be found in [31].
The probabilities of empty and full buffer states are given by the following equations: The effective production rate of each machine M i is therefore defined as follows: Such that: . . , K − 1} are subject to starvation and blockage and their effective processing rates are affected by the availabilities of the buffers B j−1 and B j+1 . The first buffer and the last are particular cases since the first machine cannot be starved and the last machine cannot be blocked. The different states of the (K − 1) buffers are modeled by (K − 1) related birth-death Markov processes. Each stochastic process is defined by its processing rates ratio α j defined as follows: Based on the previous analysis for the simple system composed of two machines and one buffer, the probabilities of empty and full states of each buffer B j are given by Eq. 14 and Eq. 15 respectively.
The effective production rates of each machine considering the previous results are given by the following equation: where: Finally, the throughput of the line in defined as the bottleneck between the effective production rates of all the machines (Eq. 17).
The relevance of this method has been established in the literature. In fact, it allows the reduction of the state space cardinality of the system through analyzing only full and empty buffer states. Therefore, the computational times are significantly reduced with very accurate results when compared to other existing methods in the literature [1], [31].

V. ENERGY EVALUATION APPROACH
The literature review has allowed us to develop an accurate approach for energy evaluation for the specificities of the problem and the research needs.
The approach to evaluating energy consumption is to assess it per part. The energy consumption is usually evaluated per part in the literature. Several segmentation methods have been introduced by [43]. The latter are summarized in Fig. 3. Depending on the needs of the study, the energy consumption of a machine can be evaluated according to several axes. For the conducted study, we have chosen to assess energy consumption according to the machine states, since we aim at developing an evaluation method valid for all types of serial production lines. Therefore, we could not develop a model based on components or details specific to a certain machine type.
For this study, energy consumption is evaluated according to machine states. We will therefore have a specific energy consumption for each state in which a machine M i could be. The states: Operating, Down, Starved, Blocked, as well as Starved and Blocked at the same time are the states in which a machine M i of the line could be found. First, the state probabilities for each machine M i are calculated by modeling the system using a Markov chain with discrete time and states.

A. MARKOVIAN REPRESENTATION OF MACHINE STATES
A Markov chain with discrete time and states is considered. The set of states S i consists of the different states of each machine: Operating, Down, Starved, Blocked, and Starved&Blocked. For presentation concerns, we refer to these states as 1, 2, 3, 4, and 5 respectively. The set of states S i is therefore given as follows: 185334 VOLUME 8, 2020  We consider the graph of states (Fig. 4) which presents the different transitions between each state of the machine. The transition probabilities matrix A i is also given for each machine M i , i = 2 . . . K − 1 (equation 19), as shown at the bottom of the page.
The probabilities of empty and full buffer states are calculated using the formulations presented in the previous section in the context of the throughput evaluation. However, there are two special cases, the case of the first machine (i = 1) that cannot be starved, and the case of the last machine (i = K ) that cannot be blocked according to the assumptions developed previously. The graph of states is illustrated by Fig. 5 and the transition probabilities matrix is given as follows: The set of states S consists of: Operating, Down, Starved (S i = {1, 2, 3}). The last machine of the line cannot be blocked.
VOLUME 8, 2020 The graph of states is illustrated by Fig. 6 and the transition probabilities matrix is given as follows: The following step is to calculate the steady probabilities for each machine M i to be in each of the possible states. The stationary state is defined by the solution to the system: where π i and A i are respectively the stationary probabilities vector and the transition probabilities matrix presented earlier for each machine M i , i = 1 . . . K .
The following results are obtained (21)- (25), as shown at the bottom of the page.

C. ENERGY CONSUMPTION FORMULATION
The energy evaluation is considered over a horizon, in which, a machine M i is: • Down with a probability P 2,i ; thus, consuming a constant amount of energy that is represented by E down,i .
• Running without load, i.e. starved, blocked or starved and blocked with a probability equal to (P 3,i + P 4,i + P 5,i ); thus, consuming a constant amount of energy noted E no−load,i .
• Operating/processing parts, and thus consuming a constant amount of energy noted E cload,i with a probability P 1,i and a variable part of energy which depends on the number of parts operated. This variable energy is equal to the energy e op,i required to operate one part, times the number of parts actually operated by the machine M i . The constant energies E down,i , E no−load,i , and E cload,i are evaluated in units of energy. While e op,i , which is the specific operation energy, is evaluated in energy units per part manufactured.
After calculating the steady probabilities, the energy consumption E i of a machine M i is obtained. The latter is evaluated for the general case i = {2, . . . , K − 1} as well as for the case of the first machine (i = 1) and the last machine (i = K ) as follows:  (26) Case i = 1: Case i = K : The energy consumption of a production line made of K machines and (K − 1) buffers is the sum of energy consumption of the K machines of the serial line (Eq. 29).
However, the energy consumption does not completely describe the energy efficiency. The key is the proportion of the energy consumed in effectively producing products. The energy efficiency η of the production line is therefore defined by the ratio of the energy consumed while operating effectively (production) E Op and the total energy consumed by the production line E (Eq. 30). where: A specific energy efficiency indicator is introduced, which is the energy efficiency of each machine M i :

D. INTEGRATED EVALUATION METHOD IMPLEMENTATION
In addition to being a novel integrated method for evaluating both economic and energetic performance of unreliable production lines, this method presents a substantial originality. The computational complexity is lower than the approaches based on other methods, such as decomposition or aggregation. In fact, the computational complexity of the proposed method in this paper is independent of the buffers sizes. Considering a production line of K machines and K − 1 buffers, we need to solve: • 2(K − 1) non-linear equations to obtain empty and full buffer state probabilities (P 0 j and P N j j , j = 1 . . . K − 1). • (K − 1) non-linear equations to obtain the processing rate ratio α j for each buffer B j , j = 1 . . . K − 1.
• K non-linear equations to obtain the equivalent throughput ρ i for each machine M i , i = 1 . . . K .
• 5K non-linear equations to obtain the steady state probabilities for each machine M i .
• K linear equations for calculating the energy consumption of each machine M i , i = 1 . . . K . VOLUME 8, 2020 • 3 equations for calculating the throughput ψ, energy consumption E and energy efficiency η of the production line. Therefore, this problem requires solving a system of 10K equations including 9K − 1 non-linear equations. The procedure of the proposed analytical method for evaluating both economic and energetic performances is described by Algorithm 1. For its resolution, this mixed integer non-linear program has been implemented on a LINGO 18.0 solver.

VI. NUMERICAL EXPERIMENTS
In order to analyze the behavior and the trade-off between the economic and the energetic performance of serial production lines, we tested our evaluation model on a LINGO 18.0 solver using existing instances in the literature. We assume, as in the literature [33], [34], and without loss of generality, that an operating machine consumes the same amount of energy while running without load E no−load,i in addition to the energy required to manufacture each part e op,i * ρ i . We therefore have: E cload,i = E no−load,i . The energy consumption E i for each machine M i becomes: A simplified version of the evaluation model is then obtained. Moreover, for all tests, we have chosen to keep energy parameters constant for all machines. Due to the complexity of the procedure and the unavailability of energy benchmarks (instances and data), their influence will be studied in future work.
The choice of these parameters was inspired from the literature. Reference [2] in the numerical experiments section, chose to evaluate the idling power as a fraction of the operating power, the ratio is approximately of 50%. For our study, the same ratio is maintained. The fraction of the no-load energy and the operating energy is approximately equal to 50% for an isolated system. E down usually neglected in the literature, is considered in our study, equal to 10% of the no-load energy. The energy parameters are therefore fixed as follows: Thereafter, the performance of the evaluation model is studied. The economic evaluation has already been validated [1], [31] and its relevance has been proven. However, it is not possible to assess performance in terms of energy evaluation, due to the unavailability of a benchmark in the literature.
Moreover, several tests are conducted in order to analyze the behavior of serial production lines regarding their economic and energetic performance. We start by analyzing the simplest system with two-machines and one buffer, for which we study correlations with energy consumption and energy efficiency. Afterwards, larger systems with K machines and (K − 1) buffers are analyzed. Test procedures and results are described in the following section.

A. TWO MACHINES-ONE BUFFER SYSTEM ANALYSIS
In order to study certain behaviors and correlations, the simplest system with two machines and one buffer is first analyzed. Interesting results have been formulated in this section.
First, we consider a system composed of two machines separated by an intermediate stock. We study the correlation between the buffer size, the energy consumption E and energy effciency η respectively. The buffer size N is therefore varied from 1 to 100 and tests on [16] instances are performed (The parameters of each configuration are presented in Table 4). The first configuration represents a homogeneous line with identical machines, the second represents a homogeneous line with non-identical machines, while the last one deals with the general case of non-identical machines in a nonhomogeneous line.
Then, to establish the relationship between the reliability parameters of the two machines of the production line and respectively the line's energy consumption and energy efficiency, tests are conducted on a homogeneous line and then on a non-homogeneous line (the parameters of each configuration are presented in Table 5). For each of the two lines, four scenarios are tested by varying the size of the buffer (1, 10, 25, and 100).
Additional tests are conducted on [37] instances. The latter consider several configurations varying the reliability parameters of the machines, their production capacities, and the size of the buffer. The parameters of each configuration are presented in Table 6.
The parameters of the first machine are kept constant, whereas for the second machine, they are chosen in such a way that for each configuration, the machines have the same isolated production rate as indicated in the Table 6. Each configuration has three different instances varying the failure and repair rates and production capacities of the second machine. The size of the intermediate stock varies from 1 to 100 for each instance. For each configuration, the first two instances consider the case of a homogeneous line, whereas the third considers the non-homogeneous case. Subsequently, the impact of the positioning of the machines in a line with two machines and an intermediate stock on energy consumption and efficiency is studied. For this purpose, evaluation tests are carried on the two configurations represented in Table 7. For each configuration two instances are tested, in which the positions of the two machines are inverted. The energy consumption and energy efficiency of the line are then evaluated by varying the size of the buffer from 1 to 100. The parameters of each configuration are presented in Table 7.

RESULTS ANALYSIS
The significant findings obtained from the experimental study are highlighted and summarized in Table 1.  Buffer Size Correlation: the increase in the buffer size N leads to greater energy consumption and better energy efficiency for all configurations (Fig. 7). For energy consumption, we notice an increase of 16.88%, 21.82%, and 39.96% from a buffer size of 1 to 100, for the first, second, and last configuration respectively. The increase ratio of energy consumption is more significant for greater machine availabilities (e.g. configuration 3).
From the point of view of energy efficiency, the increase is of 45.06%, 49.70%, and 39.96% from a buffer size of 1 to 100 for the respective configurations.
Machine Reliability Parameters Correlation: we notice a decrease of 23.9% in energy consumption and 1.58% in energy efficiency between a value of 0.01 and a value of 1    of the ratio λ 1 µ 1 . This supports the result in Table 1 for the negative correlation of the ratio λ i µ i with the energy consumption (Fig. 8) and energy efficiency of the line respectively.
Additional Analysis: for all configurations, the first and second instances consume the same amount of energy and VOLUME 8, 2020    are characterized by an identical energy efficiency. However, the third instance of each configuration is distinguishable.    Configuration 1 demonstrated almost no gap for energy consumption and efficiency of the three instances, it is of 0.24% and 0.01% between instance 3 and 2 for energy consumption and energy efficiency respectively ( Fig. 9a and  Fig. 9b).
The gap beginning with the second configuration, where it is 1.05% and 0.14% for energy consumption and energy efficiency respectively (Fig. 10a and Fig. 10b). It continues to increase attaining 6.16% for energy consumption    Ranking for non-homogeneous line with non-identical machines and buffers and λ i = λ and µ i = µ. and 3.94% for energy efficiency in configuration 4 ( Fig. 12a and Fig. 12b).
This illustrates the result summarized in Table 1, claiming that a change in the reliability parameters has no impact either on the energy consumption or the energy efficiency (corresponding graphs of instances 1 and 2 overlap for all four configurations). However, an increase in machine capacity leads to less energy consumption and better energy efficiency.

B. ANALYSIS OF A SYSTEM WITH K MACHINES AND (K-1) INTERMEDIATE BUFFERS
In this part, instances of [9] for homogeneous and nonhomogeneous production lines are considered.   Instances studied by [4] for homogeneous lines and by [3] for non-homogeneous lines are also used. However, we only use one failure mode. The examples consist of 2 configurations with production lines composed of 15 and 20 machines respectively. The first configuration considers the case of a homogeneous line and the case of a non-homogeneous line, while the last one only considers a non-homogeneous line. The different parameters of these application examples are shown in Tables 11 and 12. Numerical results are presented in Table 2.
We notice from the previous numerical results that producing more does not necessarily lead to greater energy consumption. This is illustrated by the five machines configuration, where the homogeneous line (case A) produces less that the non-homogeneous line (case B) but consumes more than the later. We conduct further analysis on several cases of serial production lines in support of this result as well as to highlight other findings. These tests' results are summarized in Table 3.
It is important to note that these results (for both cases with two machines and with K machines) were presented after a large variety of conducted tests on instances from the literature as well as appropriately generated instances.

VII. CONCLUSION
In this paper, a new integrated analytical and computational method to evaluate both economic and energetic performance of unreliable serial production lines was developed. The economic performance is characterized by the throughput, while for the energy performance, energy consumption and energy efficiency indicators are introduced. This evaluation method is based on a decomposition of energy consumption according to machine states. A Markov formulation is used to obtain stationary probabilities for each machine in each of the possible states and then to formulate the energy consumption and energy efficiency for each machine M i , as well as for the serial production line. The proposed analytical model with its two components, economic and energetic, is formulated as a non-linear programming method implemented on LINGO 18.0 solver.
Thereafter, numerical tests were conducted using instances taken from the literature, for the elementary system with two machines and one buffer as well as for the general case of K machines and (K − 1) buffers. Results were used to analyze the behavior of the economic and energetic performance of serial production lines as well as the trade-off between the two performances. The impact of buffers size, reliability parameters and machines production rates on the energy consumption and efficiency was analyzed.
The study of the trade-off between economic and energetic performance has demonstrated that they are characterized by a very complex inter-dependency. Numerical results have shown that the correlation is far from obvious. Indeed,      tests demonstrated that producing more does not necessarily involve consuming more energy. Both performances are also strongly influenced by buffers size, machine reliability parameters, and production line structure.
We believe that this work makes a major contribution to the field of production systems engineering and has the potential to be the reference method for the evaluation of both economic and energetic performances of serial production lines. Moreover, since the proposed approach is derived for the unreliable serial line model, researchers and practitioners can easily adapt and extend it to more complex systems (e.g. assembly lines, series-parallel systems, re-entrant systems, etc.). This work introduces several future research opportunities: • The proposed method could be adapted for other production systems configurations such as assembly lines or series-parallel systems.
• The study could be extended for lines with nonexponential machines.
• Future analysis could include energy parameters influence on energy performances.
• The method could be applied on practical case studies for more accurate energy analysis.
YASSINE OUAZENE received the Engineer degree in industrial engineering from the Polytechnic School of Algiers, Algeria, in 2009, and the master's and Ph.D. degrees in systems optimization and safety from the University of Technology of Troyes, in 2010 and 2013, respectively. He is currently an Associate Professor with the Industrial Systems Optimization Laboratory, University of Technology of Troyes. He is author or coauthor of more than 44 scientific publications, including 12 articles from international journals. His research interests include production systems planning and scheduling, manufacturing systems design, energy management issues in production systems, and optimization methods and algorithms.
FAROUK YALAOUI received the engineer degree in industrial engineering from the Polytechnic School of Algiers, Algeria, in 1995, the master's degree in industrial system engineering from the Polytechnic Institute of Lorraine, Nancy, France, in 1997, and the Ph.D. degree in production management from the University of Technology of Troyes (UTT), France, in 2000.
He has been the Scientific Director of the Industrial Chair Connected Innovation, UTT, since 2016. He is currently a Full Professor with UTT, where he is also the Senior Vice President of Research. His research interests include scheduling problems, system design, operations research, modeling, analysis and optimization of logistic and production systems, reliability and maintenance optimization, and optimization problems in general. He is author or coauthor of more than 440 contributions, publications, or communications with one patent, three books (Ellipses, Hermes-Lavoisier, Willey, and Sons), three edited books (Springer), 12 book chapters, and 80 articles in journals, such as IIE Transactions, European Journal of Operational Research, International Journal of Production Economics, IEEE TRANSACTIONS ON RELIABILITY, Reliability Engineering and System Safety, Computers and Operations Research, and Journal of Intelligent Manufacturing. He also published more than 270 articles in conference proceedings and presented 42 invited speeches (seminaries or conferences plenary sessions). VOLUME 8, 2020