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The High-Variety, Low-Volume (HVLV) scheduling problem is one of the most arduous and combinatorial optimization problems. This paper presents an analytical scheduling model using a tropical algebra called (max,+) algebra. The aim is to find an allocation for each operation and to define the sequence of operations on each machine, so that the resulti...
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... the above non-linear optimization problem to the example of (6x6) Job-Shop HVLV system shown in section 3 with t = u i = 0 for i = 1 : 6 . Then, the obtained optimal value C ∗ = p 466 ; x 564 + p 564 ; x 663 + p 663 ) = 55 time units. The corresponding schedules on the different machines based on the proposed (max, +) model are showed in Figure 4 that shows the order of each job Ji on each machine k . The completion times C i of the different products i = 1 : 6 are presented in Table 4. As mentioned in Subsection 4.1, the proposed model associated to a non-linear optimization algorithm in (max,+) algebra leads to an optimal value of the makespan C max ∗ = 55 time units. A little comparison between our scheduling technique and the two methods proposed in the literature (Wang and Tang, 2011) to minimize the makespan, shows the same values of the optimal makespan (55 time units) (Figure 5). In (Wang and Tang, 2011), the authors have been used the traditional Genetic Algorithm (GA) and an Improved Adaptive Genetic Algorithm (IAGA) for solving the minimum makespan problem of the job- shop scheduling problem presented in Table 3. In the proposed non-linear (max,+) scheduling model, we have not parameters to tun which is the case in (Wang and Tang, 2011) (initial population chose, crossover probability and mutation probability tuning). With a such model, we must only choose the suitable mathematical programming formulation of the scheduling problem and the appropriate decision variables to generate different feasible schedules. The objective of this section is to minimize the total tardiness criterion for a non-linear optimization using the (max, +) algebra and subject to JIT production (Figure 6) As far as we know, there are few researches about scheduling problems in the literature that deal with the total tardiness minimization. Moreover, all these researches don’t handle the JIT production criterion in the scheduling problems. In this section, the total tardiness is minimized, so that the JIT production is satisfied. Then, let define the following ...
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... Manufacturing/production systems can be modelled from a qualitative or quantitative point of view [49]: qualitative models (i.e., Automata, finite state machines, and Petri Nets) capture logical aspects of a system, while quantitative models (i.e., discrete-event simulation and Max-Plus algebra) highlight the quantitative system performance. Regarding these last, according to [50], discrete-event simulation models are dynamic in nature and are often easier to apply than analytic models, while Max-Plus algebra models are static in nature and express the event timing dynamics in terms of a set of algebraic linear equations analogous to conventional state-space linear equations [51]. ...
... ∈ real numbers R ∪−∞ the Max-Plus operators are defined according to the following [49]: ...
... where transportation and set-up times are negligible, and all processing times are deterministic in minutes. Now, according to [56] there are four methods to derive Max-Plus algebra models for a manufacturing system: (i) Timed event graph or a directed graph representation of the system [49,55]. (ii) Max-plus linear queuing networks [57]. ...
Industry 4.0 aims to ensure the future competitiveness of the manufacturing industry, where one of the major challenges faced by its implementation is the manufacturing/production system robustness (that is, able to perform in the presence of noise), as they may not be able to absorb input disruptions without bending or breaking. In this paper we propose to use the Max-Plus algebra approach to study the propagation of manufacturing disturbances (i.e., processing time variations), presenting a case study and performing a sensitivity analysis, with the idea of understanding under which conditions disturbance propagation takes place. Findings show that the impact propagation depends on where the variation source is located within the manufacturing system. Two are the main original contributions of this paper: the use of Max-Plus algebra to study the impact propagation of processing time variations and a four-step methodology to derive the equations representing the deterministic manufacturing system.
... Ensuite, d'autres thèses encadrées ont permis d'apporter un certain nombre d'améliorations importantes (situations de blocage, assemblage, désassemblage, …), à l'outil développé mais aussi à valider la recherche effectuée avec d'autres approches de modélisation telles que les méthodes analytiques, les règles floues ou certaines méthodes d'optimisation [5,6]. ...
Thèmes-Productique-Automatique-Sciences pour l'Ingénieur-Mécanique-Informatique Résumé-« Apollo » est un outil dédié à la simulation des systèmes manufacturiers. Conçu pour des utilisateurs potentiels non spécialistes des techniques de simulation, à partir de concepts simples et proches des éléments réels du système de production. Du fait qu'il est orienté composants, la programmation au sens informatique est complètement absente et l'utilisation est largement simplifiée sans pour autant réduire les capacités de représentation des systèmes. L'analyse des résultats est aidée par l'exploitation d'indicateurs de performance pertinents et normalisés ainsi que des tableaux de bord associés aux composants. « Apollo » permet de simuler un système afin de valider, invalider ou modifier sa conception ; évaluer une politique de contrôle ; comparer différentes alternatives de production (flux poussé, flux tiré) ; comparer différentes stratégies de pilotage, …. Même si « Apollo » est largement utilisé en entreprise pour simuler des systèmes réels et en recherche pour valider des approches et méthodes nouvelles, il a été dédié initialement à l'enseignement en formation à Bac+5 et principalement dans les formations IUT à Bac+2 et +3.
... Generally, production scheduling is done to allocate a limited set of resources to a limited number of jobs thus optimizing the system's performances according to one or more criteria where various constraints are taken into consideration. In order to deal with sequencing decisions, control variables must be introduced in the model [25]. Moreover, most research dealing with job-shop problems assume that machines are always available. ...
High-Variety, Low-Volume (HVLV) manufacturing systems are built to produce parts of several types in small quantities and under multiple production objectives. They relate to job-shop systems well known by researchers. One of the most studied assumptions of HVLV systems scheduling is considering that machines may be periodically unavailable during the production scheduling. This article deals with an analytical integrating method using (max, +) algebra to model HVLV scheduling problems subject to preventive maintenance (PM) while considering machines availability constraints. Each machine is subject to PM while maintaining flexibility for the start time of the maintenance activities during the planning period. The proposed model controls the placement of maintenance activities along the production operations. Indeed, the sequencing of maintenance activities on the machines depends on the criteria to minimize and may be different for each criteria value. For preventive maintenance, the proposed model aims to generate the best sequencing between activities while respecting the planning program that satisfy the optimal criteria values. In order to illustrate the performance of the proposed methodology, a simulation example is given.
... Generally, production scheduling problems are done to allocate a limited set of resources to a limited number of jobs optimizing the system performances according to one or more criteria where various constraints are taken into consideration (Kusiak and Ahn (1992)). In order to deal with sequencing decisions, control variables have been introduced in the model (Nasri et al. (2011c)). In this context, the dioid algebraic model has been developed to generate all feasible schedules by choosing different values for decision variables. ...
... The focus of this section concerns a review for the HVLV systems scheduling using (max, +) algebra. For more details, readers are invited to read (Nasri et al. (2011c)). ...
... The approach presented in (Nasri et al. (2011c)) is a direct systematic procedure, relevant to a wide class of manufacturing systems, especially HVLV systems. The technique used to solve the scheduling problem of HVLV systems considers the problem as a mathematical programming formulation while using decision variables. ...
... For tackling this problem, several researchers have included recently unavailability periods like maintenance MOSIM'12 -June 06-08, 2012 -Bordeaux -France tivities in their studies (Zribi et al., 2008) and (Sbihi and Vernier, 2008). In this framework, to deal with sequencing decisions, control variables have been introduced in the scheduling model (Nasri et al., 2011). A dioid algebraic model has been developed to generate all feasible schedules by choosing different values for decision variables. ...
... Two kinds of maintenance tasks are incorporated to the proposed model: repetitive periodic maintenance operations with equal periods on each machine and flexible periodic maintenance activities with different time intervals between two consecutive maintenance tasks. The remainder of this paper is organized as follows: Section 2 gives a short review for the state-space HVLV systems scheduling modeling (Nasri et al., 2011 ). In Section 3, PM is considered. ...
... The focus of this section concerns a review for the HVLV systems scheduling using (max, +) algebra. For more details, readers are invited to read (Nasri et al., 2011). ...
... Alors les systèmes HVLV non libres sont non linéaire dans cette structure. Nous montrons tout d'abord que la modélisation des systèmes HVLV à décision libre par l'algèbre (max, +) se ramène à un modèle mathématique couramment utilisé par l'automaticien : c'est une représentation d'état linéaire similaire à celle utilisée pour représenter les systèmes linéaires invariants continus [63][64][65][66]. Partant de cette représentation d'état, nous proposons un enrichissement de ce modèle par l'introduction des nouvelles variables de contrôle et de décision afin de pouvoir exhiber la dynamique non linéaire d'un système HVLV non libre. ...
This thesis deals with the development of a flow scheduling/optimization approach applied to the field of high-variety, low-volume production systems called HVLV (HighVariety, Low-Volume) systems. In this context, the flow of parts is represented by a discreet flow model. The discontinuous behavior of HVLV systems can be characterized by the knowledge of the starting and ending times of its activities. (Max, +) algebra is used to represent these kinds of systems where relationships between the starting times of the activities require both the maximum and addition operators. In order to use (max, +) algebra for HVLV systems scheduling, it is necessary to solve into this algebra an optimization problem subject to conflicts and constraints. In this research, we have first of all developed a scheduling (max, +) model for HVLV systems where decision variables are introduced to solve the conflict problem between operations carried out on the machines. Then, we have improved the proposed model to deal with preventive maintenance. Two kinds of maintenance are considered: Repetitive Periodic Maintenance (RPM) and Flexible Periodic Maintenance (FPM). In both cases, a non-linear optimization problem with constraints is solved to minimize some performance criteria. Lastly, simulation results on some complex HVLV job-shop systems are presented to illustrate the feasibility of the proposed methodology..
... For tackling this problem, several researchers have included recently unavailability periods like maintenance MOSIM'12 -June 06-08, 2012 -Bordeaux -France tivities in their studies (Zribi et al., 2008) and (Sbihi and Vernier, 2008). In this framework, to deal with sequencing decisions, control variables have been introduced in the scheduling model (Nasri et al., 2011). A dioid algebraic model has been developed to generate all feasible schedules by choosing different values for decision variables. ...
... Two kinds of maintenance tasks are incorporated to the proposed model: repetitive periodic maintenance operations with equal periods on each machine and flexible periodic maintenance activities with different time intervals between two consecutive maintenance tasks. The remainder of this paper is organized as follows: Section 2 gives a short review for the state-space HVLV systems scheduling modeling (Nasri et al., 2011 ). In Section 3, PM is considered. ...
... The focus of this section concerns a review for the HVLV systems scheduling using (max, +) algebra. For more details, readers are invited to read (Nasri et al., 2011). ...
Most production scheduling problems, including High-Variety, Low-Volume (HVLV) scheduling problems assume that machines are continuously available. However, in most actual situations, machines become unavailable during certain periods when preventive maintenance (PM) is scheduled. In this paper, a HVLV scheduling problem is proposed while considering machines availability constraints. Each machine is subject to PM while maintaining flexibility in the start time of maintenance activities during the planning period. In this paper, two situations are investigated. First, the maintenance tasks are periodically scheduled: maintenance is required after a periodic time (all periods are equals on each machine). Second, time intervals between two consecutive maintenance activities are not equals (flexible periodic maintenance). However, time intervals are known in advance. Consequently, the maintenance operations are controllable. The jobs and the maintenance activities are scheduled simultaneously. Also, the maintenance tasks are scheduled between them, such that a regular criterion is optimized. In order to illustrate the performance of the proposed methodology, a simulation example is given.
... Generally, production scheduling problems are done to allocate a limited set of resources to a limited number of jobs optimizing the system performances according to one or more criteria where various constraints are taken into consideration (Kusiak and Ahn (1992)). In order to deal with sequencing decisions, control variables have been introduced in the model (Nasri et al. (2011c)). In this context, the dioid algebraic model has been developed to generate all feasible schedules by choosing different values for decision variables. ...
... The focus of this section concerns a review for the HVLV systems scheduling using (max, +) algebra. For more details, readers are invited to read (Nasri et al. (2011c)). ...
... The approach presented in (Nasri et al. (2011c)) is a direct systematic procedure, relevant to a wide class of manufacturing systems, especially HVLV systems. The technique used to solve the scheduling problem of HVLV systems considers the problem as a mathematical programming formulation while using decision variables. ...
The High-Variety, Low-Volume (HVLV) scheduling problem is one of the most ardu- ous combinatorial optimization problems. This paper considers an interesting formulation of the HVLV scheduling problem using (max, +) algebra while periodic Preventive Maintenance (PM) is considered. Maintenance is time based since activities are periodically fixed: maintenance is required after a periodic time interval (all periods are equals on each machine). In this paper, the maintenance tasks of machines are controllable.The jobs and the maintenance operations are scheduled simultaneously. Also, the maintenance operations are scheduled between each other, so that a regular criterion is optimized. To generate feasible schedules, constrained decision variables are incorporated into the (max, +) model. The validity of the proposed approach is illustrated by simulation examples.
