Farzad Niakan

Armand Baboli
email: armand.baboli@insa-lyon.fr

Thierry Moyaux

Valérie Botta-Genoulaz

A new multi-objective mathematical model for dynamic cell formation under demand and cost uncertainty considering social criteria

Keywords: dynamic cell formation, robust optimization, corporate social responsibility, multiobjective optimization, non-dominated sorting genetic algorithm

This paper examines a new multi-objective mathematical model in a Dynamic Cell Formation Problem (DCFP), where social criteria and uncertainty conditions are considered. Although corporate social responsibility is one of the important issues that are increasingly considered by researchers and practitioners, it is largely overlooked in the literature on DCFP. In this paper the first objective function minimizes costs related to a machine (machine fixed and variable costs, machine procurement and relocation costs, intra-cell and inter-cell movement costs) and wages, while social issues are maximized (e.g. potential machine hazards are minimized, while job opportunities are maximized). A robust counterpart of the proposed model is then developed by applying the recent robust optimization theory. Afterwards, due to the NP-hardness of DCFP, a non-dominated sorting genetic algorithm (NSGA-II) as a meta-heuristic method is designed. Finally, two deterministic and robust mathematical formulations are compared by a number of nominal realizations under randomly generated test problems. This serves to assess the robustness of the solution achieved by the proposed robust optimization model. The result obtained demonstrates the ability of the robust model to reach appropriate solutions at all levels of uncertainty, specifically when a feasible solution cannot be found with the deterministic model.

Introduction

To remain competitive, manufacturers and producers have been forced to increase the productivity and flexibility of manufacturing systems. The Cellular Manufacturing System (CMS), one of the wellknown production systems, is applied to this end. CMS based on Group Technology (GT) attempts to classify parts and machines in order to create cells and part families. This is called a Cell Formation Problem (CFP). This classification is based on the similarities in the geometry and process requirements of each part, so as to take advantage of the main benefits such as reduction of Work-In-Process (WIP), flow time and space utilization, while also improving production planning and control. But from another point of view, most of the existing cell formation methods have been developed just for one period of the planning horizon, and some parameters like the decrease of product life cycles and product mix changes under dynamic condition require cells of the cellular manufacturing system to be reconfigured (Fig. 1). This is necessary to maintain a high level of performance. Therefore, optimal cells formed in one period might not be optimal in another period, because this reconfiguration of part families and machine grouping may include either substitution of machines between cells or a change in the number of cells. These reasons provide sufficient motivation to pursue the research in multi-period or Dynamic Cellular Manufacturing System (DCMS) approaches. {Please insert Fig. 1 here.} At the same time, pressure from NGOs, social communities and the media is currently prompting researchers and firms to take Corporate Social Responsibility (CSR) into account. For example, some well-known corporations such as Shell, Nike and Wal-Mart have suffered extensive damage due to media reports and public awareness of their CSR activities [START_REF] Cruz | Multiperiod effects of corporate social responsibility on supply chain networks, transaction costs, emissions, and risk[END_REF], [START_REF] Amaeshi | Corporate Social Responsibility in Supply Chains of Global Brands: A Boundaryless Responsibility? Clarifications, Exceptions and Implications[END_REF]. CSR concerns the effect of corporate activities on different social entities such as environment preservation, human rights, occupational safety, etc. [START_REF] Carter | Social responsibility and supply chain relationships[END_REF]. As CSR attempts to incorporate social factors into economic models to provide more benefits for society as a whole, it has an important role in the sustainable improvement of countries. Despite the importance of CSR, one of the neglected issues involved in CMS is CSR, to the extent that there has been no relevant work on the subject.

The research presented in this paper contributes to the existent literature in two ways. First, a multiobjective mathematical model is formulated to integrate the relative costs of DCFP and social issues. In other words, we try to make a trade-off between economic and social criteria while designing CMS in each period. Job opportunities and potential machine hazards are social issues that we have extracted according to the social accountability standards (SAI 8000 [START_REF] Sai | Social Accountability 8000 International Standards[END_REF], ISO 26000 [START_REF] Iso | Final Draft International standard ISO/FDIS 26000[END_REF]) and ergonomic guideline (GRI [START_REF] Gri | Sustainability Reporting Guidelines[END_REF], NIOSH [START_REF]Criteria for a recommended standard: occupational noise exposure[END_REF]), and applied as a second objective. The robust counterpart of the presented model is then also developed to cope with the uncertainty in demand, machine availability and relevant costs of machines. In order to handle uncertainty in parameters and obtain robust optimal solutions, several numerical examples are solved by the NSGA-II algorithm presented here.

The remainder of this paper is organized as follows. In Section 2, we review the relevant literature and define the research motivation. The mathematical model and main assumptions are then presented in Section 3. Section 4 elaborates the robust counterpart mathematical model, while the proposed NSGA-II algorithm is described in Section 5. Computational results and numerical analysis are presented in Section 6. Finally, Section 8 ends with conclusions and some future research directions.

Literature review

First, we present a brief review of the related literature on dynamic cellular manufacturing systems (DCMS). Due to the large number of researches in this area, we have concentrated only on recent studies. In order to make the research gap clear, we have investigated previous studies with regard to several aspects such as objectives, problem-solving approaches, parts movement (inter/intra cell), and the nature of applied parameters (deterministic, stochastic, etc.). Rheault et al. [START_REF] Rheault | Physically reconfigurable virtual cells: A dynamic model for a highly dynamic environment[END_REF] were the first to introduce the concept of dynamic environment in the cell formation problem. Schaller et al. [START_REF] Schaller | A methodology for integrating cell formation and production planning in cellular manufacturing[END_REF] integrated the cell formation problem with inventory aspects. They proved their presentation model by a multiple heuristics procedure and evaluated several alternative lower bounding methods. Chen and Cao [START_REF] Chen | Coordinating production planning in cellular manufacturing environment using Tabu search[END_REF] proposed a mathematical model for a multi-period cellular manufacturing system (CMS) to minimize the total cost, including: inter-cell material handling, inventory costs, and setting up of manufacturing cells. These authors also developed a Tabu Search method to obtain a sound solution and show the efficiency of their model.

Tavakkoli-Moghaddam et al. [START_REF] Tavakkoli-Moghaddam | Solving a dynamic cell formation problem using metaheuristics[END_REF] presented a nonlinear integer model in DCMS with consideration of machine capacity limitation, machine replication, and inter-cell movements that perform in batches.

They used constant and variable costs as well as reconfiguration and inter-cell movement costs to formulate their objective function. These authors solved their models by several traditional metaheuristics comprising Genetic Algorithm (GA), Tabu Search (TS) and Simulated Annealing (SA), and then compared solutions obtained by means of each method to define the best algorithm. Tavakkoli-Moghaddam et al. [START_REF] Tavakkoli-Moghaddam | Solving a Dynamic Cell Formation Problem with Machine Cost and Alternative Process Plan by Memetic Algorithms[END_REF] applied a new Memetic Algorithm (MA) to solve their DCMS model. Defersha and Chen [START_REF] Defersha | A comprehensive mathematical model for the design of cellular manufacturing systems[END_REF] formulated a new comprehensive model containing dynamic cell configuration, alternative routings, lot splitting, sequence of operations and workload balancing. They also considered machine adjacency and cell-size capacity as constraints in their proposed model. Safaei et al. [START_REF] Safaei | Designing cellular manufacturing systems under dynamic and uncertain conditions[END_REF], [START_REF] Safaei | A fuzzy programming approach for a cell formation problem with dynamic and uncertain conditions[END_REF] presented a DCMS mathematical model with uncertain circumstances, assuming fuzzy demand and fuzzy machine availability. They solved their mixed-integer programming model by developing fuzzy programming to determine optimal cell configuration with maximum satisfaction of the fuzzy objective and constraint. Safaei et al. [START_REF] Safaei | A hybrid simulated annealing for solving an extended model of dynamic cellular manufacturing system[END_REF] proposed a mixed-integer programming model in DCMS with consideration batch inter/intra-cell material handling, sequence of operations, alternative process plans and machine replication. In this study, the authors minimized machine variable/constant costs, inter/intra-cell movements and reconfiguration costs as an objective function. They used a hybrid metaheuristic called MFA-SA (Mean Field Annealing-Simulated Annealing) to solve the proposed model and showed the efficiency of their solving method by comparing it with classical SA. Defesha and Chen [START_REF] Defersha | A linear programming embedded genetic algorithm for an integrated cell formation and lot sizing considering product quality[END_REF] integrated DCMS with production lot sizing and formulated a mathematical model to minimize both production and quality-related costs, such as operation costs, setup costs and inventory costs. They solved by a linear programming embedded GA. Aryanezhad et al. [START_REF] Aryanezhad | Dynamic cell formation and the worker assignment problem: a new model[END_REF] proposed a model to transact with a Simultaneous Dynamic Cell Formation and Worker assignment Problem (SDCWP). The objective function of their model has two components: production costs, such as intercell material handling costs and machine costs in the planning horizon; and human issues consisting of hiring costs, firing costs, training costs and salaries. Safaei and Tavakkoli-Moghaddam [START_REF] Safaei | Integrated multi-period cell formation and subcontracting production planning in dynamic cellular manufacturing systems[END_REF] developed a mathematical model to integrate multi-period cell formation and subcontracted production planning in DCMS. In this study the authors made a trade-off between production and outsourcing costs in the reconfiguration of system. Bajestani et al. [START_REF] Bajestani | A multi-objective scatter search for a dynamic cell formation problem[END_REF] formulated a multi-objective mathematical model in DCMS and minimized the sum of various costs and the total cell load variation simultaneously. They obtained a locally Pareto-optimal frontier by using a new Multi-Objective Scatter Search (MOSS) method. Wang et al. [START_REF] Wang | Optimization of the multi-objective dynamic cell formation problem using a scatter search approach[END_REF] presented a non-linear mixed integer program to model a DCSM with three conflicting objectives (machine relocation cost, the utilization rate of machine capacity, and the total number of inter-cell moves over the entire planning horizon). Deljoo et al. [START_REF] Deljoo | Using genetic algorithm to solve dynamic cell formation problem[END_REF] improved previous models presented in the literature by correcting some essential errors which weakened the efficiency of the model. Mahdavi et al. [START_REF] Mahdavi | Designing a mathematical model for dynamic cellular manufacturing systems considering production planning and worker assignment[END_REF] proposed an integer non-linear programming in DCMS with consideration of labor aspects such as worker assignment, alternative workers, available time of workers, hiring and firing costs and wages. They utilized holding and backordering costs in their model as an inventory aspect to make it more realistic. Javadian et al. [START_REF] Javadian | A multi-objective integrated cellular manufacturing systems design with dynamic system reconfiguration[END_REF] presented a multi-objective problem of cellular manufacturing systems in dynamic and deterministic production environments to minimize total cell load variation and the sum of the miscellaneous costs (machine costs, inter/intra-cellular material handling, back orders, inventory holding and subcontracting) simultaneously. A Non-Dominated Sorting Genetic Algorithm (NSGA-II) method was developed to obtain optimal Pareto-frontier. Rafiee et al. [START_REF] Rafiee | A new approach towards integrated cell formation and inventory lot sizing in an unreliable cellular manufacturing system[END_REF] integrated DCMS and inventory lot sizing problems by formulating a comprehensive mathematical model. The proposed model was considered with several design factors, such as machine procurement, cell reconfiguration, preventive and corrective maintenance, intra/inter-cell material handling, subcontracting, inventory cost, and defective parts replacement costs. Saxena and Jain [START_REF] Saxena | Dynamic cellular manufacturing systems design-a comprehensive model[END_REF] developed a mixed-integer nonlinear programming model to merge machine breakdown effects and DCMS by incorporating reliability modeling. The proposed model seeks to minimize the following: intra/inter-cell movement costs and machine procurement costs; machine variable/constant costs, production costs, part holding costs, subcontracting and reconfiguration costs, and machine breakdown repair costs such as production time loss cost due to machine breakdown. Kia et al. [START_REF] Kia | Solving a group layout design model of a dynamic cellular manufacturing system with alternative process routings , lot splitting and flexible reconfiguration by simulated annealing[END_REF] proposed a new mixed-integer non-linear programming model for DCMS by integrating three major decisions in the design of a CMS (cell formation, group layout and group scheduling) and developed an efficient SA method to solve their model. The objective function of this model is minimizing total costs of intra/intercell movement, machine relocation, machine procurement, machine overhead and machine processing.

Rafiei and Ghodsi [START_REF] Rafiei | A bi-objective mathematical model toward dynamic cell formation considering labor utilization[END_REF] presented a bi-objective CMS problem, in which they considered humanrelated issues. The proposed model sought to minimize various costs, including machine procurement, relocation costs, machine variable costs, inter/intra-cell movement costs, overtime costs, and labor shifting costs, as a first objective function, and to maximize labor utilization, as a second objective function. They furthermore suggested a hybridization of the ant colony optimization (ACO) algorithm with GA to solve their models. Majazi-Dalfard [START_REF] Dalfard | New mathematical model for problem of dynamic cell formation based on number and average length of intra and intercellular movements[END_REF] developed a new nonlinear integer programming model, into which he incorporated effects of distance in material flow as an important factor of decision making. In this study he applied a new simulated annealing embedded in branch-and-cut to solve the proposed problem. Javadi et al. [START_REF] Javadi | A hybrid electromagnetism-like algorithm for dynamic inter/intra-cell layout problem[END_REF] developed a mathematical model to minimize the total costs of rearrangement and inter/intra-cell movements. They also proposed a hybrid of an electro-magnetismlike (EM-like) algorithm and GA as a solution approach. Aghajani et al. [START_REF] Aghajani | A multiobjective mathematical model for cellular manufacturing systems design with probabilistic demand and machine reliability analysis[END_REF] presented a multi-objective model for dynamic cell formation where the demand is probabilistic. The proposed model consisted of three conflict objectives; reconfiguration cost, penalty cost of machine underutilization, and system failure rate. The authors also solved the model by developing a NSGA-II algorithm and comparing the result with the ɛ-constraint method to show its efficiency. Bootaki et al. [START_REF] Bootaki | New bi-objective robust design-based utilisation towards dynamic cell formation problem with fuzzy random demands[END_REF] presented a bi-objective model in robust design of DCMS, with a fuzzy, random demand. The objective function of models is the minimization of inter-cell movements while the machine and worker utilization are maximized. A new goal programming method named 'Percentage Multi-Choice Goal Programming' (PMCGP) is also proposed to verify the model. Paydar and Saidi-Mehrabad [START_REF] Paydar | Revised multi-choice goal programming for integrated supply chain design and dynamic virtual cell formation with fuzzy parameters[END_REF] formulated a bi-objective model to integrate cell formation and supply chain (procurement, production and distribution planning) which decreases distribution and procurement costs and increases the customer response rate. They applied revised multi-choice goal programming to solve the case study example and to find a compromise solution. Renna and Ambrico [START_REF] Renna | Design and reconfiguration models for dynamic cellular manufacturing to handle market changes[END_REF] developed a model for reconfiguration and scheduling in DCMS. They assumed demands to be probabilistic in discrete probability scenarios. The authors also applied a simulation environment to test the proposed method compared to the manufacturing system without reconfiguration. Table 1 shows the classification of previous study about DCMS. {Please insert Table 1 here.} Most of the criteria studied in these papers are economic rather than social. There are no studies that focus specifically on social criteria, although some research in supply chain configuration does take social issues into account. Carter and Jennings [START_REF] Carter | Social responsibility and supply chain relationships[END_REF] were the first to incorporate the concept of the CSR into the supply chain. Cruz and Wakolbinger [START_REF] Cruz | Multiperiod effects of corporate social responsibility on supply chain networks, transaction costs, emissions, and risk[END_REF] and Cruz and Liu [START_REF] Cruz | Modeling and analysis of the multiperiod effects of social relationship on supply chain networks[END_REF] studied the role of social responsibility on the supply chain and some of its components such as suppliers, retailers and customers.

The difficulty with measuring social responsibility is generally due to its complexity and scope. It can thus be said that it is in a sense inconceivable to measure all aspects of social responsibility. However, for preparation and allocating corporate social responsibility (CSR) in companies, several standards and guidelines have been developed, such as ISO 26000 [START_REF] Iso | Final Draft International standard ISO/FDIS 26000[END_REF], SAI 8000 [START_REF] Sai | Social Accountability 8000 International Standards[END_REF], GRI [START_REF] Gri | Sustainability Reporting Guidelines[END_REF] and NIOSH [START_REF]Criteria for a recommended standard: occupational noise exposure[END_REF] . Based on these documents, we have identified two social criteria that are very important in DCMS.

Motivation

As noted in the previous section, based on the above literature review we have found no research that considers social responsibility issues in the design of DCMS. This aspect was moreover generally neglected in previous research as well. The first contribution of the research presented here is to develop a new multi-objective mathematical model for DCMS configuration with social aspects. In other words, we try to make a trade-off between economic and social criteria while designing CMS in each period. Due to high levels of interaction between human and manufacturing environments, social issues are particularly important. Among of these issues, stress due to job loss [START_REF] Iacovides | The relationship between job stress, burnout and clinical depression[END_REF] and occupational injuries [START_REF] Aryanezhad | Designing safe job rotation schedules based upon workers' skills[END_REF] caused by ergonomic problems are the most common. In our mathematical model we therefore consider job opportunities and potential machine hazards as social objective functions.

The second contribution of this work is that it develops a robust counterpart of the proposed model to deal with the uncertainty of demand, machine availability and machine costs. It may seem easier to use the deterministic parameter in a DCMS configuration, but in the real world many parameters must be considered as uncertain. Moreover, a feasible solution cannot be found for the deterministic approach. These considerations may slightly affect the economic performance of a system, but they allow us to approach reality and above all to have a feasible solution under different kinds and levels of uncertainty.

Due to the complexity of the model (NP Hard) and in order to deal with the uncertainty of parameters and to obtain robust optimal solutions, one of the most common meta-heuristic methods, the NSGA-II algorithm, has been selected. The biggest challenge is to design this meta-heuristic method for DCMS problems. Finally, several numeric experiments allow us to validate the performance of a robust counterpart of the proposed model by sensitivity analysis of uncertainty levels.

Problem description and formulation

In this section the new multi-objective mathematical model is proposed to trade-off between two important objectives in DCMS problems. The first objective function of the model is to minimize some of the costs, including machine fixed and variable costs, inter-cell movement costs, intra-cell movement costs, machine procurement costs, machine relocation costs and workers' wages. The second objective function is to optimize the social issues of the problem by maximizing job opportunities and minimizing potential machine hazards for labor during the planning phase.

In this study an attempt has been made to choose social measures in such a way that they are closely related to the DCMS decisions, have an important role in the DCMS problem, and can be computed and formulated simply in the mathematical model. To this end, two social issues are considered in the model: (1) the number of job opportunities created and (2) the number of potential hazards of machines. The first measure represents the variation of job opportunities created due to the hiring and firing of labor [START_REF] Sai | Social Accountability 8000 International Standards[END_REF], [START_REF] Iso | Final Draft International standard ISO/FDIS 26000[END_REF] resulting from the buying and selling of machines. The second measure, which reflects the average fraction of potential hazards of each machine [START_REF] Gri | Sustainability Reporting Guidelines[END_REF], [START_REF]Criteria for a recommended standard: occupational noise exposure[END_REF] shows how much injury, illness and damage each machine causes workers. The latter measure is taken into account through a parameter called "average fraction of potential machine hazard". This average fraction is a continuous number between 0 (the most safe) and 1(the most dangerous) and is defined by the Decision Maker (DM) using a safety checklist, talking to workers, reviewing the manufacturer's information, and checking the injury and incident reports of each machine. Because of differences in the units of two considered social measures, we have aggregated social measures to the one objective function by normalized weighting method. This aggregation with weighting factors not only represents the importance of each measure but also decreases the computational complexity of evaluating each solution with social aspects [START_REF] Dehghanian | Designing sustainable recovery network of end-of-life products using genetic algorithm[END_REF], [START_REF] Boggia | Measuring sustainable development using a multi-criteria model: A case study[END_REF]. Other assumptions used for formulating the mathematical model are presented as follows:

3.1

Problem assumptions 1. Each part has an operation sequence and must be processed according to the sequence which is extracted from the route sheet parts.

2. The demand of each part type in each period is an uncertain parameter.

3. The capability of each machine for processing parts is known. The time capacity of each machine is also an uncertain parameter.

4. The fixed cost of each machine is independent of the assigned workload. This cost is considered for each machine whether or not it is used in the planning horizon, and includes the costs of total maintenance and overhead services. This cost is considered as an uncertain parameter.

5. The variable cost of each machine includes the operation cost and depends on the workload assigned to each machine. This parameter of the problem is also uncertain.

6. The machine purchasing and selling costs in each period are considered as an uncertain value and include machine prices and freight charges, which means that removal costs between the place of storage and that of installation are not considered.

7. The relocation cost (installation, un-installation, shifting) of each machine type between two periods is an uncertain parameter.

8. Each machine type can process several operations, in other words, each processing of parts can be performed on different machine types with various processing times.

9. The upper and lower bound of the cell size is known and is fixed in all periods.

10. The inter-intra movements of parts are performed in batches with different sizes and the related cost of this movement is dependent on the distance traveled. For decreasing the complexity of the problem we assumed that the distance between two cells (inter) is the same, and that for each cell, the distance between two machines (intra) is the same. Furthermore, the dimensions of all machine types are equal.

11. The labor leveling rule (hiring -firing) in each period allows the DM to change the labor level to at most a certain percentage ( h λ ) of the labor level in the previous period.

Notations

Mathematical formulation

1 1 1 1 1 1 1 1 1 1 int ( 1) int 1 1 1 1 1 1 int ( 1) int 1 1 1
= = = = = = = = - + = = = = = = + = = = +   + -         + -       ∑∑∑ ∑∑∑∑∑ ∑∑ ∑ ∑ ∑ ∑ ∑∑ 1 ( 1) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ( ) 2 p O C M M M pmch j pmch jpmch j c m m m H M C H M C L mch m mh mh mch mch h m c h m c H M H M mh mh mh mh h m h m x x N L Sa K K I I δ ϕ ω - + = = = = = + - = = = = = = + - = = = =   - -     + + + + - ∑ ∑ ∑ ∑ ∑ ∑∑∑ ∑∑∑ ∑∑ ∑∑ (1) 2 1 1 1 1 1 H H H M C jo h h hd mch m h h h m c Max Z hi fi N S θ θ = = = = =     = - -         ∑ ∑ ∑∑∑ (2)
S.t. 1 1 1 C H jpm jpmch c m a x = = = ∑∑ , , j p h ∀ (3) jpmch jpm x a ≤ , , , , j p c m h ∀ (4) 1 1 p O P ph jpm jpmch mh mch p j D t x T N = = ≤ ∑∑ , , m c h ∀ (5) ( 1) mc h mch mch mch N K K N + - -+ - = , , m c h ∀ (6) ( 1) 1 1 C C mc h mh mh mch c c N I I N + - - = = + - = ∑ ∑ , m h ∀ (7) 1 M mch m N UB = ≤ ∑ , c h ∀ (8) 1 M mch m N LB = ≥ ∑ , c h ∀ (9) 1 M mh m h m I L hi + = ≤ ∑ h ∀ ( 10
) 1 M mh m h m I L fi - = ≤ ∑ h ∀ (11)
( )

( 1) 1 1 1 M M C mh mh m mc h m h m m c I I L N L λ + - - = = =       - ≤           ∑ ∑ ∑ h ∀ (12)
{ }

+ - + - ∈ ≥ (13)
Objective function (1) represents the minimizing of total costs, that includes eight terms: the first term relates to the machine fixed costs; the second term represents variable machine costs; the third term relates to the part's inter-cell movement costs; the fourth term represents the part's intra-cell movement costs; the fifth term includes wages or salaries; the sixth term represents machine relocation costs; the seventh term includes machine procurement costs; and the eighth term relates to revenue from selling the machine. Objective function (2) maximizes the social issues of DCMS during the planning horizon that includes maximizing job opportunities and minimizing potential machines hazards. Constraint (3) ensures that each operation of a part is assigned to just one machine and one cell. Constraint (4) guarantees that the process of each part is assigned to the appropriate machines Constraint [START_REF] Iso | Final Draft International standard ISO/FDIS 26000[END_REF] ensures that the time capacity of each machine is not exceeded. Constraints ( 6) and [START_REF]Criteria for a recommended standard: occupational noise exposure[END_REF] ensure that in each planning period the number of machines is balanced. Constraints ( 8) and (9) determine the maximum and minimum cell sizes according to the defined upper and lower bound. Constraints [START_REF] Chen | Coordinating production planning in cellular manufacturing environment using Tabu search[END_REF] and [START_REF] Tavakkoli-Moghaddam | Solving a dynamic cell formation problem using metaheuristics[END_REF] calculate the number of workers that are respectively hired and fired. Constraint [START_REF] Tavakkoli-Moghaddam | Solving a Dynamic Cell Formation Problem with Machine Cost and Alternative Process Plan by Memetic Algorithms[END_REF] guarantees maximum labor leveling variation in each period. Constraint (13) defines non-negativity, binary and integer necessities of decision variables.

Linearization

The third and fourth terms of equation ( 1) in the proposed model make it a nonlinear equation. In order to transform it to a linear equation for the third term, two non-negative variables, 1 jpch Z and 2 jpch Z are added and then the third term is rewritten as follow:

1 int 1 2 int 1 1 1 1 1 ( ) 2 p O H P C ph er jpch jpch er h p j c p D z z B γ - = = = =   +       ∑∑ ∑ ∑ (14)
Then following constraint must be added to the proposed model.

1 2 ( 1) 1 1
, , ,

M M jpch jpch j pmch jpmch m m z z x x j p c h + = = -= - ∀ ∑ ∑ (15)
Similarity, to linearize the forth term of equation ( 1),

- = = = = =     + - +           ∑∑ ∑ ∑ ∑ (16)
And the following constraint must be added to the proposed model.

1 2 ( 1)
, , , ,

jpmch jpmch j pmch jpmch y y x x j p m c h + - = - ∀ (17)
Equation ( 12) is the other case in which the proposed mathematical model is nonlinear. Hence, for linearization, instead of equation [START_REF] Tavakkoli-Moghaddam | Solving a Dynamic Cell Formation Problem with Machine Cost and Alternative Process Plan by Memetic Algorithms[END_REF], equation ( 18) and ( 19) are added to the original model as follows:

( )

( 1) 1 1 1 M M C mh mh m mc h m h m m c I I L N L h λ + - - = = =       - ≤ ∀           ∑ ∑ ∑ (18)
( )

( 1) 1 1 1 M M C mh mh m mc h m h m m c I I L N L h λ - + - = = =       - ≤ ∀           ∑ ∑ ∑ (19)

Robust counterpart mathematical model

In this section, we provide a quick description of the principles of robust optimization. For further details, readers are referred to [START_REF] Ben-Tal | Robust Optimization[END_REF] and references therein. Consider the following linear program (LP):

. .

min cx s t Ax b ≥ (20)

Where n x ∈ℜ is the vector of decision variables, m b ∈ℜ is the right-hand side parameter vector, n c ∈ℜ is the vector of objective function coefficients, and m n A × ∈ℜ , with elements i j a , is the constraint coefficient matrix.

In a typical problem like (LP), c , A and b are assumed to be deterministic and optimal solution is obtained according deterministic circumstances. Some of the data parameters are considered as uncertain in the Robust Optimization (RO) approach, yet they lie within a set that expresses limits on the uncertainty. The foregoing uncertainty set subsequently defines the limits on uncertainty that a solution will be immunized against. That is, solution x deals with any possible uncertainty lying within the set. In the robust optimization approach the (LP) is transformed into a robust counterpart by replacing each constraint that has uncertain coefficients with a constraint that reflects the incorporation of the uncertainty set. Let j c , i j a  and i b  denote, as an uncertain entry in the objective function coefficients, constraint coefficient and right-hand side parameters, respectively. In the proposed model, each of the uncertain parameters is assumed to vary in a specified closed, bounded box [START_REF] Ben-Tal | Robust Solutions of Uncertain Linear Programs[END_REF], [START_REF] Ben-Tal | Retailer-Supplier Flexible Commitments Contracts: A Robust Optimization Approach[END_REF], [START_REF] Pishvaee | A robust optimization approach to closed-loop supply chain network design under uncertainty[END_REF] The general form of this box can be defined as follows:

{ } : , 1, 2,..., n Box t t t u t n ϑ ϑ ϑ ρζ = ∈ℜ -≤ = (21)
Where t ϑ is the normal value of t ϑ as the t th parameter of vectorϑ , and where the positive number t ζ represents the "uncertainty scale" and 0 ρ  is the "uncertainty level". A particular case of interest

is t t ζ ϑ =
, which corresponds to a simple case where the box contains t ϑ , whose relative deviation from the nominal data is up to ρ .

According to the above description, the robust counterpart of the LP model ( 22) can be stated as equations [START_REF] Mahdavi | Designing a mathematical model for dynamic cellular manufacturing systems considering production planning and worker assignment[END_REF][START_REF] Javadian | A multi-objective integrated cellular manufacturing systems design with dynamic system reconfiguration[END_REF][START_REF] Rafiee | A new approach towards integrated cell formation and inventory lot sizing in an unreliable cellular manufacturing system[END_REF][START_REF] Saxena | Dynamic cellular manufacturing systems design-a comprehensive model[END_REF]:

Mathematical model:  { } . . , 0,1 min cx dy s t ax b ey f x y + ≥ ≥ ∈   ( 22
)
Robust counterpart mathematical model:

min z (23) . . s t c j j j Box c x dy z c u + ≤ ∀ ∈   (24)   a i j i j j j Box a x b a u ≥ ∀ ∈ ( 25
)
f i i i Box e y f f u ≥ ∀ ∈   (26)
{ } , 0,1 x y ∈ Ben-Tal et al. [START_REF] Ben-Tal | Retailer-Supplier Flexible Commitments Contracts: A Robust Optimization Approach[END_REF] demonstrate that in a closed bounded box, the robust counterpart problem can be converted to a tractable equivalent model where Box u is replaced by a finite set ext u consisting of the extreme points of Box u . To represent the tractable form of the robust mathematical model, equations [START_REF] Mahdavi | Designing a mathematical model for dynamic cellular manufacturing systems considering production planning and worker assignment[END_REF][START_REF] Javadian | A multi-objective integrated cellular manufacturing systems design with dynamic system reconfiguration[END_REF][START_REF] Rafiee | A new approach towards integrated cell formation and inventory lot sizing in an unreliable cellular manufacturing system[END_REF][START_REF] Saxena | Dynamic cellular manufacturing systems design-a comprehensive model[END_REF] should be converted to their tractable equivalents. For equation [START_REF] Javadian | A multi-objective integrated cellular manufacturing systems design with dynamic system reconfiguration[END_REF] we have:

{ } , | : , 1, 2,..., c n c c c j j j j j Box Box c j c c x z dy c u u c c c j n ρ ζ ≤ - ∀ ∈ = ∈ℜ - ≤ =     (27)
The left-hand side of equation ( 27) contains the vector of uncertain parameters, while all parameters of the right-hand side are certain. Thus, the tractable form of the above semi-infinite inequality could be written as follows:

{ } { } ( ) , , 1, 2,..., , , 1, 2,
..., ,

j j j j c c j j j c c c j j j c c x z dy x j n x j n η ρ ζ η ρ ζ η + ≤ - ≤ ∀ ∈ ≤ - ∀ ∈ ∑ ( 28
)
For a constraint ,

 1 n i j j i j a x b = ≥ ∑
, we only need to augment the left-hand side of the equation to reflect the uncertainty set in the formulation. Formally, in the augmented constraint we require, for a given solution 𝑥𝑥, that [START_REF] Ben-Tal | Robust Optimization[END_REF], [START_REF] Baron | Facility Location: A Robust Optimization Approach[END_REF]:

  1 min a Box n i j j i a u j a x b ∈ =   ≥     ∑ (29)
or

   : 1 min a i j i j i j a i j n i j j i a a a j a x b ρ ζ - ≤ =   ≥     ∑ ( 30
)
Given the structure of Box u , the optimal solution of the optimization on the left-hand side is

1 1 n n a i j j a i j j i j j a x x b ρ ζ = = - ≥ ∑ ∑ (31)
Which can be reformulated as:

1 1 n n a i j j a i j j i j j a x b ρ ζ ι = = - ≥ ∑ ∑ (32)
(1, 2,..., )

j j j x j n ι ι -≤ ≤ ∀ (33)
Similarly, for inequality [START_REF] Saxena | Dynamic cellular manufacturing systems design-a comprehensive model[END_REF] we have:

{ } { } , 1, 2,.., , | : , 1, 2,..., f n f f f i i i i Box Box i i f i f e y f i n f u u f f f i n ρ ζ ≥ ∀ ∈ ∀ ∈ = ∈ℜ - ≤ =     (34)
Thus, it can be rewritten as follows:

{ } , 1, 2,...,

f i i f i f e y f i n ρ ζ ≥ + ∀ ∈ (
H H H M C jo h h hd mch m h h h m c Max Z hi fi N S θ θ = = = = =     = - -         ∑ ∑ ∑∑∑ (37)
S.t.

1 1 1 1 1 1 1 1 1 1 int 1 2 int 1 1 1 1 int int 1 1 ( ) ( ) 1
= = = = = = = = - = = = = = = ≥ + + + +   + + +         +       ∑∑∑ ∑∑∑∑∑ ∑∑ ∑ ∑ ∑∑ 1 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 ) ( ) 1 ( ( ) ) 2 ( ) ( ) p O C M D mch jpmch jpch jpch ph j c m H M C H M C L mh mch m mh mch mch mh h m c h m c H M H M mh mh mh mh mh mh h m h m y z z N L Sa K K I I δ ϕ ω η δ η ϕ η ω η - = = = + - = = = = = = + - = = = =   + - + +     + + + + + + - + ∑ ∑ ∑ ∑∑∑ ∑∑∑ ∑∑ ∑∑ (38)
= = = = + ≤ + ∑∑ ∑∑ , , m c h ∀ (41) ( 1) mc h mch mch mch N K K N + - -+ - = , , m c h ∀ (42) ( 1) 1 1 C C mc h mh mh mch c c N I I N + - - = = + - = ∑ ∑ , m h ∀ (43) 1 M mch m N UB = ≤ ∑ , c h ∀ (44) 1 M mch m N LB = ≥ ∑ , c h ∀ (45) 1 M mh m h m I L hi + = ≤ ∑ h ∀ ( 46
) 1 M mh m h m I L fi - = ≤ ∑ h ∀ (47)
( )

( 1) 1 1 1 M M C mh mh m mc h m h m m c I I L N L λ + - - = = =       - ≤           ∑ ∑ ∑ h ∀ (48) ( ) ( 1) 1 1 1 M M C mh mh m mc h m h m m c I I L N L λ - + - = = =       - ≤           ∑ ∑ ∑ h ∀ (49) mh mch mh N α α α ρ ζ η ≤ , , m c h ∀ (50) mh mch mh N α α α ρ ζ η ≥ - , , m c h ∀ (51) ph jpmch ph x β β β ρ ζ η ≤ , , , , j p m c h ∀ (52) ph jpmch ph x β β β ρ ζ η ≥ - , , , , j p m c h ∀ (53) D D D ph jpmch ph x ρ ζ η ≤ , , , , j p m c h ∀ (54) D D D ph jpmch ph x ρ ζ η ≥ - , , , , j p m c h ∀ (55) 1 D D D ph jpch ph z ρ ζ η ≤ , , , , j p m c h ∀ (56) 1 D D D ph jpch ph z ρ ζ η ≥ - , , , , j p m c h ∀ (57) 2 D D D ph jpch ph z ρ ζ η ≤ , , , , j p m c h ∀ (58) 2 D D D ph jpch ph z ρ ζ η ≥ - , , , , j p m c h ∀ (59) 1 D D D ph jpmch ph y ρ ζ η ≤ , , , , j p m c h ∀ (60) 1 D D D ph jpmch ph y ρ ζ η ≥ - , , , , j p m c h ∀ (61) 2 D D D ph jpmch ph y ρ ζ η ≤ , , , , j p m c h ∀ (62) 2 D D D ph jpmch ph y ρ ζ η ≥ - , , , , j p m c h ∀ (63) mh mch mh K δ δ δ ρ ζ η + ≤ , , m c h ∀ (64) mh mch mh K δ δ δ ρ ζ η + ≥ - , , m c h ∀ (65) mh mch mh K δ δ δ ρ ζ η - ≤ , , m c h ∀ (67) mh mch mh K δ δ δ ρ ζ η - ≥ - , , m c h ∀ (68) mh mh mh I ϕ ϕ ϕ ρ ζ η + ≤ , m h ∀ (69) mh mh mh I ϕ ϕ ϕ ρ ζ η + ≥ - , m h ∀ (70) mh mh mh I ω ω ω ρ ζ η - ≤ , m h ∀ (71) mh mh mh I ω ω ω ρ ζ η - ≥ - , m h ∀ (72) 1 2 ( 1) 1 1 M M jpch jpch j pmch jpmch m m z z x x + = = -= - ∑ ∑ , , , j p c h ∀ (73) 1 2 ( 1) jpmch jpmch j pmch jpmch y y x x + - = - , , , , j p m c h ∀ (74) { } 0,1 , , , , , , , 0 int
+ - + - ∈ ≥ 1 2 1 2 , , , 0
jpch jpch jpmch jpmch z z y y ≥ , , , , , 0
D mh ph ph mh mh mh α β δ ϕ ω η η η η η η ≥ (75)

NSGA-II algorithm

Numerous methods have been developed to deal with multi-objective problems. Some of the proposed methods define an integrated objective function based on all objective functions (e.g. weighted sum method) and reach an optimal or near-optimal result. Other methods aim to find a set of solutions named Pareto solutions. The Optimal Pareto set involves a number of solutions, none of which can completely dominate other Pareto solutions. Non-Dominated Sorting Genetic Algorithm II (NSGA-II) is a meta-heuristic which belongs to evolutionary algorithms and is a commonly used method when there is a problem with more than one goal. This method, first proposed by Deb et al. [START_REF] Deb | A fast and elitist multiobjective genetic algorithm: NSGA-II[END_REF], to find a near-optimal Pareto set. In fact, an initial population is randomly generated and then, for a given number of iterations, new solutions are obtained from existing solutions, per iteration, through crossover and mutation. Employing a crossover operator, two parents are chosen and they produce two new children by combining their characteristics. It should be mentioned that parents' selection is done by a binary tournament selection process. In this process two members are randomly selected and the winner can stay as a parent in our parent set. This is done 2N times to choose N parents. In a mutation operation, a member is selected randomly and is changed in some respects.

Here, members of a population are compared with each other in relation to two criteria: rank and crowding distance. Members are sorted and given their own rank. Remember that the solutions of a same rank cannot dominate each other; this means that if two members of the same rank are selected, neither of them is better in all objective functions. On the other hand, crowding distance is a criterion which investigates the diversity of solutions, and high crowding distance is preferred. Of course, crowding distance is a criterion measured for members of a specific rank. Equation (76) explains the way in which crowding distances are measured. In the equation, di is the crowding distance of solution i where r i f represents the r th objective function value of the i th solution, and min r f and max r f are minimum and maximum values of the r th objective function, respectively. The crowding distance of the first and last points, the points which have optimum value in at least one objective, is moreover assumed to be infinitive.

max

1 1 min r r i i i r r r f f d f f + - - = - ∑ (76)

Parameter setting

The NSGA-II algorithm has four vital parameters which should be appropriately tuned before using them to find best results:

• Number of members in a population (NPop) • Number of iterations to find best results (MaxIt) • Crossover rate (CrR) • Mutation rate (MuR)
As a result, a Taguchi design [START_REF] Chatsirirungruang | Application of genetic algorithm and Taguchi method in dynamic robust parameter design for unknown problems[END_REF] is employed to set a suitable level for the parameters. Here, we assume three different levels for each parameter:

) 1 1 1 N Q i i d d SM N d - = - = - ∑ (77)
Distance to an ideal point (D2P) is another metric which calculates the average Euclidian distance of the Pareto results from an ideal point in the space (e.g. Fig 2). Of course, the ideal point is selected by decision makers. {Please insert Fig. 2 here.} Computational time is also an important criterion of a method. As a result, the table of the Taguchi design is done with Minitab software (version 16), and best levels of parameters are selected as represented in Tables 2 and3 andFig 3 3 here.} {Please insert Fig. 3 here.}

Computational results

Some numeric examples have to be generated in order to evaluate the performance of the proposed robust mathematical model. Three sets of examples are therefore generated, based on the pattern in Table 4, in different sizes named S1, S2 and S3, respectively, shown in Table 5. On the other hand, the uncertainty levels for all parameters are the same (i.e. This action is taken into account to simulate five different possible states of a case and to solve all of them, in order to assess the performance of the proposed models. This operation could actually be called realization. All deterministic and robust problems are solved by the NSGA-II algorithm, using Matlab software (version 2012a) on a Core 2 Duo Notebook, 2.67 GHz with 4 GB RAM. Some of these patterns are taken from Safaei et al. [START_REF] Safaei | A hybrid simulated annealing for solving an extended model of dynamic cellular manufacturing system[END_REF], while others are generated for our problem.

{Please insert Table 4 here.} {Please insert Table 5 here.} {Please insert Table 6 here.} {Please insert Table 7 here.} A set of solutions, called Pareto results, is predictably obtained after running NSGA-II. A Pareto set, as mentioned above, involves a number of non-dominated solutions. This means that each result is better than at least one other result in the Pareto set, considering an objective function, and is worse than that, considering another objective function. In all Pareto sets of bi-objective problems, there are two important points: one with an optimum level for one objective, and one with an optimum level for another objective. Tables 6 and7 report the best solutions obtained for Z1 and Z2, respectively. Fig. 4. and Fig. 5. illustrate the difference between the robust and deterministic models in the second size, S2. Each block presents the mean (horizontal line), the standard deviation (internal rectangle), and the max and min value of the objective function (external rectangle). The results demonstrate that the proposed robust model is worse than the deterministic model because in robust optimization all uncertain parameters are considered as worst cases in practice, in order to obtain lower risks and to limit loss as far as possible. While, the robust model is able to reach appropriate solutions in all sizes and at all considered levels of uncertainty, while the deterministic model is probably unfeasible when the level of uncertainty is increased. Moreover, the deterministic model could find better solutions than the robust one when, with the latter, the first objective function is more important and the second objective function is well maximized. However, based on Fig. 4 and Fig. 5, we find that the results of the robust model have higher values of the objective function but a lower standard deviation in comparison with the deterministic model in all uncertainty levels. In other words, low standard deviation of robust results is another reason to demonstrate the efficiency of the proposed model. In fact, as Fig. 4. and Fig. 5. show, there are six comparisons between deterministic and robust results, by which it could be found that most comparisons represent the lower deviation of robust results. However, there are some comparisons which show that the deterministic model could obtain closer solutions. The preference of the deterministic model in some cases (e.g. for an uncertainty level of 0.75 in Fig. 4 and an uncertainty level of 0.25 in Fig. 5) with lower deviation is explained by the use of meta-heuristics, whereby nearoptimum solutions are reached rather than optimum ones. {Please insert Fig. 4 here.} {Please insert Fig. 5 here.} Table 8 shows the computational times of the solved problems. It is worth mentioning that the robust model contains more constraints, decision variables and parameters. It is therefore expected that computational times in a meta-heuristic algorithm will be longer in the robust model than in the deterministic model. {Please insert Table 8 here.} This paper addresses a new DCMS problem by considering social responsibility issues, and introduces a new robust optimization model for dealing with uncertain parameters that are neglected in the literature. First a multi-objective mathematical model is developed, in which the first objective function minimizes related costs of problems, including machine and labor costs. The social aspect, consisting of job opportunities and occupational safety, is maximized by the second objective function. In the proposed model, in order to reduce its complexity, we take into account only two social criteria, even though this model is able to integrate more criteria, such as job severity, occupational diseases, and labor days lost due to injury, in the second objective function.

In order to increase the capability of the proposed model to deal with uncertain data such as machine costs, demand for parts, and machine time capacity, a robust optimization approach is presented. A number of examples are generated and solved by employing a Non-Dominated Sorting Genetic Algorithm (NSGA-II). In our study, three test problems were examined in different states of uncertainty, which meant that each of the based examples was used to generate various problems under uncertain parameters. The results illustrate the supremacy of the robust model in the handling of uncertain parameters and in the robustness of relevant solutions in comparison with the deterministic model. By way of explanation, in contrast to the deterministic model, the robust model could obtain suitable solutions for all levels of uncertainty. Moreover, the results demonstrate the low deviation of the robust model in comparison with the deterministic one.

For the extension of current research, other social and environmental issues can be considered, such as job severity, occupational diseases, labor days lost due to injury, waste of energy, CO2 emissions, etc. Additionally, various uncertain parameters and possibilistic approaches can be applied in future research.

  for part type p C maximum number of cells M number of machine types ph D demand for part p in period h inter p B batch size for inter-cell movement of part p int ra p B batch size for intra-cell movement of part p inter p γ inter-cell movement cost per batch int ra p γ intra-cell movement cost per batch. It is assumed that machine type m in period h mh β variable cost of machine type m in period h per unit of time mh δ relocation cost of machine type m in period h mh T time-capacity of machine type m in period h UB maximum cell size LB minimum cell size m L amount of labor that is required for machine type m mh ϕ Purchase cost of machine type m in period h mh ω marginal revenue from selling machine type m in period h m S average fraction of potential hazard of machine type m hd θ normalized weighting factor of total number of potential hazard of machine type m jo θ normalized weighting factor of total number of job opportunities created h λ maximum percentage of labor leveling variation in period h L mh sa salary of labor machine type m in period h jpm t time required to process operation j of part p on machine type m jpm a 1, if operation j of part p can be done on machine type m ; 0 otherwise Variables mch N number of machine type m assigned to cell c at the beginning of period h mch K + number of machine type m added to cell c at the beginning of period h mch K - number of machine type m removed to cell c at the beginning of period h mh I + number of machine type m purchased at the beginning of period h mh I - number of machine type m sold at the beginning of period h h hi number of labor hired in period h h fi number of labor fired in period h jpmch x 1, if operation j of part p is performed in cell c in period h by machine type m ; 0 otherwise

•

  NPop: 40, 60 and 80 • MaxIt: 20, 50 and 100 • CrR: 0.2, 0.5 and 0.8 • MuR: 0.1, 0.2 and 0.3 A response should furthermore be defined for the Taguchi design. Spacing Metric (SM) is a metric used to evaluate Pareto results and is measured by equation (77) where Q i d represents the Euclidian distance between two consecutive points (i.e. points i and i+1), d is their average, and N is the number of members in final non-dominated solutions. An algorithm with more SM is preferred.

(

  different values: 0.25, 0.50 and 0.75. On each uncertainty level, five problems are derived from S1, S2 and S3 with different amounts of uncertain data, subject to their range of deviation from nominal levels.

Fig. 2 .

 2 Fig. 2. Distance of Pareto solutions from an ideal point

Fig. 1 .

 1 Fig. 1. A schematic view of dynamic cellular manufacturing

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. Signal to noise ratio from Taguchi experiments

  According to the above explanation, for developing the robust counterpart of the proposed model, the related machine costs include fixed costs ( )

					mh δ ,
	purchasing costs ( ) mh ϕ , and sales revenue ( ) mh ω , which are assumed to be uncertain. The other
	parameters, such as demand of each part (	) D and time capacity of machines, ( ) ph mh T are also of an
	uncertain nature. Hence, the robust counterpart of the presented DCMS model with uncertain
	parameters given by box sets is equivalent to the following:
	Min	1 Z			(36)
		2
		1	1	1	1 1

[START_REF] Cao | A robust cell formation approach for varying product demands[END_REF]

i mh α , variable costs ( ) mh β , relocation costs

( )

  . Table 2 displays responses in the experiments. Regarding the same importance of three responses, normalization is used to convert all values to a number in the range [0 1]. Based on these experiments, Fig 3. serves as a graphic tool to choose the best levels for the considered parameters. In fact, level 1 of NPop, i.e. 40, level 2 of MaxIt, i.e. 50, level 3 of CrR, i.e. 0.8, and level 3 of MuR, i.e. 0.3, have been determined as best levels for the parameters. {Please insert Table 2 here.} {Please insert Table

Table 4

 4 Pattern of sample generation

		S1	S2	S3
	|Op|×|P|×|M|×|C|×|H|	3×3×4×3×2	6×6×7×5×4	8×8×9×6×5

Table 5

 5 Different sizes of problems

				Deterministic	Robust
				Z1	Z2	Z1	Z2
			1	2060152.047	2.298	3969828.543	5.576
			2	2817647.19	6.1729	3509775.811	8.472
		S1	3	2210973.77	2.792	3858171.512	5.035
			4	3328183.943	2.568	4951873.559	9.731
			5	3227828.209	6.308	4079282.307	7.638
			1	52397144.58	11.112	51645075.02	12.444
	ρ = 0.25	S2	2 3 4	50929235.04 43446369.88 46511081.63	13.349 10.725 11.492	49585007.78 51015775.08 56762119.88	11.376 13.964 15.304
			5	51324191.58	14.484	52640809.24	12.878
			1	Infeasible	Infeasible	130682131.9	26.795
			2	Infeasible	Infeasible	127053558.4	26.109
		S3	3	Infeasible	Infeasible	139439766.5	29.318
			4	158126369.4	31.328	133304620.6	27.144
			5	145392597.4	25.644	129728057.2	24.946
			1	1533608.95	2.009	3481689.56	2.068
			2	3384264.888	4.847	5025683.047	6.759
		S1	3	2783557.22	3.302	5353712.626	5.087
			4	1824289.798	3.291	3231658.229	7.844
			5	1643472.424	1.439	5049564.677	10.615
			1	33447983.78	9.366	58145649.45	15.771
	ρ = 0.50	S2	2 3 4	41522784.35 44879080.15 42390396.25	10.483 11.941 9.517	57491016.65 57229791.32 55506872.72	14.984 15.743 14.359
			5	35466965.22	8.799	55233404.07	14.511
			1	Infeasible	Infeasible	137158768.8	26.207
			2	158828320.2	28.383	136221613.4	26.245
		S3	3	Infeasible	Infeasible	160363246.6	30.289
			4	172310552.5	32.35	144875052.9	28.409
			5	152058111.5	27.463	140746973.3	27.12
			1	3103390.005	2.488	5309200.308	9.064
			2	2256637.887	3.214	4692328.174	2.415
		S1	3	2299605.933	3.949	4387337.881	8.488
			4	2944939.769	4.776	6721649.424	13.588
			5	1842341.824	2.982	3445643.678	2.267
			1	54402305.04	11.289	68658803.71	19.284
	ρ = 0.75	S2	2 3 4	Infeasible 54493461.49 36131036.71	Infeasible 13.346 7.866	63567388.11 66275690.62 51240167.22	16.685 17.785 13.064
			5	46308241.75	10.08	74419317.06	17.68
			1	Infeasible	Infeasible	162653027.6	31.354
			2	Infeasible	Infeasible	161786933.5	31.758
		S3	3	Infeasible	Infeasible	150016398.3	31.526
			4	Infeasible	Infeasible	164223732.7	31.244
			5	Infeasible	Infeasible	147613662	32.015

Table 6

 6 Objective functions value with best Z1 in final Pareto results

				Deterministic	Robust
				Z1	Z2	Z1	Z2
			1	8754495.995	22.466	8052327.457	20.838
			2	6908261.096	19.23	7315716.184	21.061
		S1	3	7866827.803	19.794	12274043.99	24.567
			4	9520855.238	20.985	8325079.542	21.862
			5	6756378.563	18.934	8954907.277	24.052
			1	77909945.1	195.836	79277216.27	211.592
	ρ = 0.25	S2	2 3 4	77111769.43 66841833.63 75987695.73	192.337 179.589 197.336	84780633.65 72011777.22 81542885.38	228.818 209.990 231.031
			5	67570577.73	199.618	63376425.69	197.675
			1	Infeasible	Infeasible	179016979.1	396.233
			2	Infeasible	Infeasible	186232041.9	397.4183
		S3	3	Infeasible	Infeasible	180777164.1	410.16735
			4	182781896.4	380.913	167776446.6	377.61048
			5	198087733	382.255	171878867.6	364.0492
			1	7947074.394	19.764	7902448.414	21.681
			2	11575765.43	29.852	11950406.63	25.483
		S1	3	8944049.478	16.654	12333551.09	23.192
			4	4595541.273	15.388	7024440.88	21.841
			5	4396903.496	14.431	9833181.338	26.866
			1	74385259.9	188.367	85857450.43	239.501
	ρ = 0.50	S2	2 3 4	67801283.87 71363586.18 64996306.38	174.3761 183.426 153.374	93747776.57 82361563.93 83582231.93	243.903 232.719 239.931
			5	64455603.07	183.208	89842825.5	241.261
			1	Infeasible	Infeasible	192315126.2	408.985
			2	224139966.6	428.385	196541829.3	423.739
		S3	3	Infeasible	Infeasible	215277385.5	468.27
			4	192048328.9	353.183	195250678	420.575
			5	215198051.6	372.967	181121899.1	391.303
			1	8925452.973	24.941	10563626.35	29.854
			2	7319703.219	20.395	11361483.47	27.281
		S1	3	12143392.35	28.438	8937980.168	29.798
			4	9339241.583	20.563	14584946.98	35.799
			5	5212238.84	12.288	8663060.137	19.786
			1	72335438.41	162.589	87826885.85	258.516
	ρ = 0.75	S2	2 3 4	Infeasible 86370932.99 62184179.47	Infeasible 221.959 145.705	100359058.4 96011762.26 81592600.49	264.211 281.821 226.414
			5	86072370.65	204.311	110391114	279.322
			1	Infeasible	Infeasible	223630224	483.155
			2	Infeasible	Infeasible	211832468.2	451.861
		S3	3	Infeasible	Infeasible	226999065	494.931
			4	Infeasible	Infeasible	203310318	444.501
			5	Infeasible	Infeasible	203836403.6	448.986

Table 7

 7 Objective functions value with best Z2 in final Pareto results

		ρ = 0.25		ρ = 0.50		ρ = 0.75
		Deterministic	Robust	Deterministic	Robust	Deterministic	Robust
	S1	2.845	5.377	2.851	5.66	2.836	5.808
	S2	72.143	127.145	71.831	125.077	72.356	124.255
	S3	368.323	614.43	352.78	617.599	-	607.002

Table 8

 8 Computational times (Second)

Acknowledgment

The authors gratefully acknowledge the scientific and financial supports of ARC 8 of Region Rhône-Alpes, France in developing this paper.


Claire Boulesteix
email: claire.boulesteix@univ-lr.fr

Fernando Pedraza
email: fpedraza@univ-lr.fr

Suitable Sealants for Cracked Aluminized Austenitic Steels and Their Oxidation Behaviour

Keywords: Sealants, Phosphate sealing, Aluminide coatings, Austenitic stainless steels, Oxidation

Aluminizing of steels may result in the appearance of tensile cracks from the coating surface to the substrate due to the difference in thermal expansion coefficients. The cracks often need to be sealed to make the coatings impervious to chemical attack and to limit their mechanical degradation. In this work, dipping of the cracked aluminide coatings elaborated on IN-800HT in 0.5, 1 and 3.75 mol.L -1 concentrations of H3PO4 in acetone and increasing the dipping time are investigated. The three concentrations resulted in the formation of P-O compounds and of Al2O3. However, the 0.5 mol.L -1 allowed better wetting of the surface.

Further reduction of the concentration from 0.5 mol.L -1 to 0.2 mol.L -1 brought about the complete sealing of both large and fine cracks. The efficiency of the sealing process with 0.2 mol.L -1 and 0.5 mol.L -1 was thereafter evaluated by cyclic oxidation in air at 650°C for more than 1000 cycles. The overall microstructure and composition of the aluminide coatings remained identical after sealing and the oxidation behaviour comparable to the untreated counterparts. The oxide scale formed on the untreated samples appeared thicker compared to the sealed ones. The latter were covered by a thin layer of AlPO4 which evolved towards a thin protective alumina scale. However, for the sealed samples, some minor cracks were detected after oxidation but they appeared completely healed with Al2O3 and no propagation of them beyond the additive layer/interdiffusion zone was noticed.

INTRODUCTION

Aluminide coatings have proven their ability to reduce the oxidation rate of Fe-base alloys components in different atmospheres through the formation of an adherent, stable and protective alumina scale [1][2][3][4]. For instance, slurry aluminide coatings were reported to increase the lifetime of austenitic steels at high temperatures in different atmospheres [5,6]. However, the appearance of tensile cracks in the aluminide coatings are often observed due to the differences of thermal expansion coefficients between the AlxFey intermetallic layers and the substrate [3,4,7]. For instance, Xu et al. estimated that the thermal expansion coefficient of the IN-800 HT austenitic stainless steel is 15 ppm.°C -1 at 840°C [8] whereas the FeAl phase one is about 22-23 ppm.°C -1 [9]. This mismatch explains the formation of tensile cracks in the coating. Moreover, the greater the Al content, the harder the intermetallic phase but the more brittle they become [7,10]. Therefore, the cracks are often blunted at the interface between the additive layer (Al-rich) and the interdiffusion zone where the Al and Fe contents are equivalent or are even Fe-rich. These cracks are normal to the coating surface and become preferential paths for the environmental degradation of the coating [11]. Sealing of the cracks can either occur naturally by interdiffusion but this requires relatively thin cracks [START_REF] Pedraza | Internal report, European project[END_REF] or by oxidation [4].

The latter can occur in relatively wide cracks as demonstrated by Aguero et al. when subjecting their cracked Fe2Al5 slurry coatings on ferritic-martensitic P92 steel for very long periods in steam at 650°C and 1 bar [4].

In plant conditions where the temperatures and pressures are particularly harsh austenitic stainless steels are preferred to ferritic-martensitic steels from the mechanical standpoint [START_REF] Viswanathan | Advanced Materials and Processes[END_REF].

Also, the aluminide coatings contain a lower density of cracks than in ferritic alloys due to a smaller mismatch between the thermal expansion coefficients [7]. In this work, IN-800HT was chosen as material of study since its high Cr content (around 21 wt%) allows its use in wet atmospheres [START_REF] Gan | [END_REF]. However, G.R Holcomb anticipated that in ultrasupercritical steam conditions, some Cr evaporation could occur [15] and the Cr flow to form a protective Cr2O3 scale is too slow in austenitic steels at such moderate temperatures [16]. To overcome such problems, slurry aluminide coatings were applied on austenitic stainless steels and their resistance to steam was proven under isothermal conditions despite the presence of tensile cracks due to the formation of a protective -Al2O3 [5]. However, since the steam power plants will need to operate under cyclic conditions [17], the cracks of the coatings shall be sealed prior to operation. Chemical sealing of cracks can be employed to improve the hot corrosion resistance and the mechanical properties of porous, thick thermal barrier [18] or to enhance the anti-corrosion properties of the alumina-forming coating [19]. Sealing is commonly achieved by impregnating the coating with organic or inorganic solutions [20]. Orthophosphoric acid (H3PO4) and commercial phosphate solutions have been successfully used as sealants for porous oxide coatings [20][21][START_REF] Chraska | Thermal spray: Meeting the Challenges of the 21 th Century[END_REF] while mixed H3PO4 and acetone were employed to seal γ-TiAl [START_REF] Brou | [END_REF][24][25]. With the latter, a gel-like deposit was left over the surface and the high temperature oxidation resistance in air was greatly improved. Also, phosphate-based sealants have demonstrated their ability to penetrate into the coating and to reduce the cyclic oxidation rate at high temperatures [18,26]. However, the effect of concentration and of dipping time on the resulting surface product and properties is yet unknown on aluminide coatings.

Therefore, this work investigates first the formation of the phosphating treatment as a sealant of aluminide coatings and evaluates afterwards the efficiency of the sealing treatment on the cracked aluminide coatings subjected to cyclic oxidation in air at 650°C for more than 1000 cycles. Cracking of the aluminide coatings for different applications can occur. Therefore, air was employed instead of steam as a more general oxidizing medium. In addition, Pedraza et al.

showed that the aluminide coatings degrade faster in air than in pure steam under isothermal conditions at 650°C and 1 bar [5]. Furthermore, the oxidation tests were carried out in more accelerated cyclic conditions (1h at 650°C + 15 min cooling) than in service to verify the resistance of the sealed cracks to consecutive thermal shocks.

EXPERIMENTAL PROCEDURE

Material of Study

IN800HT coupons of 16 mm diameter and 2 mm thick were ground with SiC (180) paper then rinsed in deionized water and cleaned with ethanol under ultrasonic agitation before drying with hot air. The composition (nominal and determined by EDS analysis) of this alloy is given in table 1. The samples were then aluminized by applying the aluminum slurry containing 43wt% of Al micro-particles (4-5 μm average size, Hermillon, France) dispersed in 57wt% of binder (1/10 polyvinyl alcohol (PVA)/deionised water). After drying in ambient air for 1h at least [27], a heat treatment was conducted in argon atmosphere (3h at 400°C to evaporate water and to decompose the PVA, then a second step at 650°C for 3h in order to allow the diffusion of Al and substrate compounds and a final step at 1100°C to homogenize the coating) resulting in the formation of a B2-(Fe,Ni)Al diffusion coating coat containing tensile cracks [5] and in its preoxidation [28]. After the thermal treatment, the surface of samples was grit blasted with a 220 mesh Al2O3 to remove the residues of the slurry ("bisque").

Sealing

The phosphoric acid solution was obtained by diluting orthophosphoric acid solution (85wt% H3PO4, d =1.685 g.cm -3 ) with acetone to get different concentrations (0.2; 0.5; 1 and 3.75 mol.L - 1 ) of H3PO4. The aluminized IN800HT samples were immersed up to 5 min in the solutions at room temperature. The samples were then let dry for 1 h and overnight prior to further analysis.

Cyclic oxidation tests in air

The oxidation tests realized on aluminized and aluminized/sealed IN-800HT samples were performed in a tubular furnace. Like in the study of Aguero et al. [4], the samples were cyclically exposed for more than 1000 cycles to 650°C for 1 h and subsequent cooling for 5 min in natural air [4]. Air was employed instead of steam as it degrades faster the slurry aluminide coating under isothermal conditions [5].

Characterization

The surfaces and the cross-sections of the aluminized and aluminized/sealed samples were characterized by scanning-electron microscopy (SEM) coupled to energy-dispersive spectrometry (EDS) using a FEI Quanta 200F Field Emission Gun Scanning Electron Microscope (FEG-SEM) at 20 kV with an EDAX detector operating under low vacuum.

Moreover, X-ray maps were realized on the sealed samples in order to observe the spatial distribution of the elements in a wider area. The cross sections were mounted in resin and gently polished till colloidal silica finishing. The crystal structures of aluminized and aluminized/sealed samples were determined by X-ray Diffraction (XRD) using a Bruker AXS D8 Advance (CuKα radiation) in θ-2θ mode for the coatings after removal by grit blasting of the slurry residue left over them. Raman microspectrometry was also performed with a Horiba Jobin Yvon LabRam HR apparatus using a He-Ne laser (632.8 nm). The analyses were performed every 500 µm from the centre to the edges to map the likely evolutions in composition at local scale. Water contact angle measurements were performed by the sessile drop method using water drops of 4µL at room temperature using a SEE System.

RESULTS AND DISCUSSIONS

Slurry aluminide coatings characterization

After annealing of the slurry, the coating displayed three different sublayers (Fig. 1). The layer at the top was composed of the residues of the slurry ("bisque"), which was subsequently removed by grit blasting for the further investigation (Fig. 1b). The intermediate ("additive") layer grew by major outward diffusion of the substrate elements at high temperature while the inner ("interdiffusion") layer resulted from interdiffusion between Al and Fe [28] after previous Fe dissolution in molten Al and subsequent Al inward diffusion at intermediate temperatures [29]. The additive layer contained thin and thick cracks before and after removal of the bisque.

The cracks are believed to originate from the difference of the thermal expansion between the intermetallic phases of the coating and the austenitic substrate [3,4,7,9] and appeared blunted at the interdiffusion layer. Indeed, the additive and the interdiffusion layers were respectively

Sealing of aluminide coatings

Fig. 2 shows the mass gains with dipping time for the 0.5, 1 and 3.75M solutions of H3PO4 and the two drying processes. It appeared that the greater the concentration of H3PO4, the greater the specific mass gain, which is indicative of a thicker phosphate layer. The increase was sharp for the first 2 minutes but drastically slowed down afterwards. The same trend was observed after natural evaporation of the solvent after the overnight dry. In spite of the long overnight drying, some blisters formed at the surface of the 3.75M sealed surfaces as shown in Fig. 3.

The formation of the blisters can be ascribed to the phosphating reaction (equation <1>) that results in aluminium phosphate and hydrogen [20]. The greater the concentration of H3PO4, the lower the pH and the faster the formation of the phosphated layer but the greater the production of hydrogen [31,32]. Depending on the phosphating rate (i.e. concentration of H3PO4), hydrogen may get entrapped in the conversion layer. The coalescence of the trapped hydrogen results in blistering and bursting as observed with the 3.75M and to a much less extent in the 1M sealing solutions, i.e. when the concentration is almost four-fold decreased.

Al (s) + H3PO4 (aq)  AlPO4 (s) + 1.5 H2 (g) <1>

Regardless of the concentration of H3PO4, the X-ray patterns of Figure 4 only displayed the major peaks of the B2-(Fe,Ni)Al phase of the aluminide coatings in addition to some very weak signal of AlOOH. No specific peak corresponding to the AlPO4 was identified even with the 3.75M concentration, which suggest that the thickness of this phosphate layer shall be thin.

Since the surface of the 3.75M samples was very inhomogeneous and the large probed area with XRD did not provide any specific compound, the 3.75 M concentration was dropped out for any further investigation. Indeed, the Raman analysis provide a very local information so that disparate results can result in the blistered surfaces of the 3.75M treated samples.

Fig. 5 shows the Raman spectra in different areas of the samples with 0.5 and 1 mol.L -1 concentrations and dipping time. Clearly, all the Raman peaks corresponded to the P-O vibration of H3PO4 [START_REF] Brou | [END_REF]24,33] while -Al2O3 was detected in the fluorescence region.

However, the latter is often characterized by a doublet [34,35] that could not be identified here anywhere at the surface. Therefore, it appears that either soaking of the surface with H3PO4 impeded the adequate resolution of the peak or that Al2O3 was insufficiently crystallized and/or contained too many defects [34]. The potential transformation of the Al oxy-hydroxide identified by XRD in Fig. 4 into an oxide due to the laser heating may also occur but we have no clear proof for this. Irrespective of this, increasing the concentration of H3PO4 from 0.5 to 1 mol.L -1 did not seem to change drastically neither the composition of the surface nor their relative amount (note that the spectra are given in normalized intensity). In contrast, the relative content of Al2O3 with respect the phosphate groups seems to increase with increasing dipping time, in particular between 2 and 5 min for both concentrations. However, the appearance of the surface was quite different under the electron microscope (Fig. 6). The greater the concentration the greater the surface coverage but also the greater the appearance of a dry-mud morphology. Similarly, the longer the immersion time, the more cracked the deposit appeared.

The surface coverage is though quite chemically even from 1 min onwards for both concentrations as demonstrated by the EDS analysis of Fig. 7. Therefore, both the concentration and time increase shall thus result in thicker phosphate layers that shrink upon drying to result in the dry-mud morphology as also demonstrated by Brou et al. on TiAl surfaces [24]. In spite of the cracked appearance of the deposits, the evolution of the wettability with time was the same (see Fig. 8). This seems to indicate that the surface roughness achieved by the grit blasting is preponderant than the surface composition as Mahadik et al. demonstrated in cracked CeO2 electrodeposited coatings [36]. However the 1 mol.L -1 concentrated sealing solutions resulted in pitting of the aluminide coating (Fig. 6). Such phenomena were also reported in the initial stages of soft chemical stripping of aluminide coatings in H3PO4 with in an organic solvent [37] and can be ascribed to the partial dissolution of alumina passive films in H3PO4 and further progress of the local attack [38]. Therefore, the 1 mol.L -1 solution was also disregarded for sealing of cracks.

In summary, the 1 and the 3.75 mol.L -1 concentrations were disregarded to avoid defects on the surface. In the oxidation tests, only 0.5 mol.L -1 concentration will be compared with 0.2 mol.L - 1 . The latter was visually less viscous and supposed to penetrate more readily in the thin cracks of the coating. However, it is assumed that the AlPO4 formed according to equation <1> since no Raman or XRD signal of the reaction products between H3PO4 and Al [20] were observed.

Indeed, dipping for 2 min in very low concentrations of H3PO4 in acetone (0.2 and 0.5 mol.L - 1 ) resulted in a very light coverage of the surface since the roughness the grit blasting performed after the aluminization to remove the "bisque" is mainly highlighted (Fig. 9). Nevertheless, the surfaces are more mat after dipping in the phosphate solutions compared to the untreated one.

According to the EDS analyses (not shown), the surfaces contained between 1.5 and 5 at% P with increasing sealant concentration. In the cross-sections, the X-ray maps of Fig. 10 clearly indicated that both the surface and the cracks (thin and thick) were rich in P and O. Therefore, it clearly appeared that a 2 minutes dipping time was enough to seal thin and thick cracks into the coating.

Cyclic oxidation tests on aluminized and aluminized/sealed samples

After more than 1000 cycles at 650°C in air, the three coatings exhibited the same microstructure (Fig. 11). Moreover, the thickness of the coatings was quite similar (about 45 µm) to the unoxidized ones. As shown in Fig. 11.a, no cracks were detected on the Al coated samples, which is indicative of a self-healing mechanism by interdiffusion and by oxidation of the cracks [4]. The sealed cracks remained unaltered and did not propagate either upon cycling (Figs. 11b and11c). The EDS profiles of Fig. 12 show that the Al concentration increased in the external 20 µm which indicates that the three coatings provided a sufficient Al supply to grow and maintain a very thin Al2O3 scale at the surface and into the cracks. The concentration of the other major elements was proportionally very alike before and after oxidation. Therefore, the B2-(Fe,Ni)Al phase remained stable over the duration of the cyclic oxidation test at 650°C.

After oxidation, the X-ray maps of Fig. 13 revealed that the thickness of the oxide scale on the untreated coatings was about 5 µm thick and seemed to have grown over the surface of the coating. In contrast, such oxide scale is much thinner (between 1 to 2 µm) in the sealed coatings is continuous to the coating. Another significant feature is that both the surface and the cracks are very O-rich with respect the rest of the coating and that P can still be somewhat detected at the surface of the 0.5M sealed coating. Indeed, spot EDS analyses at the surface and in the cracks allowed to reveal very small amounts (≈ 1-2 at%) of P, and therefore cannot be clearly distinguished in the X-ray maps. Interestingly, the X-ray patterns let appear weak peaks of α-Al2O3 and major reflections of the aluminide coating in the three coatings but not any Pcontaining compound in the sealed ones (Figure 14a). Similarly, the Raman spectra only provide the signal of α-Al2O3 whose intensity increases with sealant concentration (Fig. 14b).

This corroborates the findings of the very thin oxide scales observed in the SEM cross-sections (Fig. 13).

The oxidation of the untreated aluminide coating follows a classic mechanism in which Al diffuses outwardly while oxygen diffuses inwardly. Here, the Al-rich outermost layer of the coating enhances the cation diffusion even at such low temperatures [5]. Therefore, the alumina layer grows at the expense of the Al consumption of the coating, hence there is a marked enrichment of the Al content at the outermost part of the coating (Fig. 12).

At such low oxidation temperature one could expect -Al2O3 to form but instead, the stable -Al2O3 is identified in the X-ray patterns of Fig. 14a and in the Raman spectra of Fig. 14b. Sirota and Shokhina demonstrated by XRD that 1670 h were required to transform 50% of the amorphous layer into -Al2O3 at 600°C [39]. However, at 800°C, the time needed to obtain the same result dropped to just 0.5h. Clearly, an increase of temperature stabilises the -Al2O3.

However, -Al2O3 simultaneously transforms into other metastable aluminas or even in the -Al2O3 stable alumina phase depending on the conditions. In this work, the aluminide coatings are pre-oxidized with -Al2O3 upon the slurry aluminizing process under Ar (g) flow [F. Therefore, upon the subsequent cyclic oxidation at 650°C, the -Al2O3 can be sustained through the third element effect provided by Cr [40]. As reported by Brumm and Grabke in the oxidation of NiAl with different Cr contents the reaction of metastable θ-Al2O3 or -Al2O3 with Cr2O3 leads to formation of hexagonal (Al,Cr)2O3 crystals which can act as nuclei for the formation of hexagonal -Al2O3 [41].

Conversely, the sealed samples seem to follow a different oxidation mechanism. Upon sealing, AlPO4 and Al2O3 have been shown to form (Fig. 5). Vippola et al. demonstrated that the reaction of alumina with H3PO4 can lead to the formation of both the orthophosphate Al(PO)4

and the metaphosphate Al(PO3)3 [20]. Since their decomposition temperatures are well above 650°C, these compounds shall remain stable at our oxidation temperature of 650°C. Moreover, Brou et al. [START_REF] Brou | [END_REF][24][25] showed that the formation of metal pyrophosphates [AlxPyOz] could also occur. Depending on the temperature and time, the pyrophosphates evolve into Al2O3 and P2O5 [24]. Diffusion of P in the aluminide coating could occur but we have no experimental evidence for this. However, the transformation of the Al phosphates into α-Al2O3 is clearly demonstrated by XRD and Raman spectroscopy. However, and in agreement with the findings of Brou et al.

on TiAl but at higher temperature [START_REF] Brou | [END_REF][24][25], the phosphate layer seems to block diffusion, hence limiting oxidation while the growth of alumina would be delayed by the transformation of the aluminium phosphates. As a result, the oxide scales are much thinner than in the untreated specimens (Fig. 13) In contrast, sufficiently low concentrations (0.2 and 0.5 mol.L -1 ) covered fully the surfaces though the 0.5 mol.L -1 solutions resulted in cracked layers by drying of the phosphate layer.

CONCLUSIONS

The 0.2 and 0.5 mol.L -1 concentrations allowed also a complete sealing of thin and thick cracks of the aluminide coatings.

After more than 1000 cycles of exposure to air at 650°C, the microstructure and the composition of the sealed aluminide coatings remained similar to the untreated ones. The cracks observed were in fact sealed with -Al2O3 but no P-containing compound was detected with our analytical techniques. The oxide scales were thinner in the sealed than in the untreated ones.

This appears to result from the delayed transformation of pyrophosphates to a protective and stable alumina layer. The cracks in the untreated coatings were healed by interdiffusion. In all cases, the stable α-Al2O3 scale formed upon the coating process was maintained at 650°C over the 1000 cycles.

  composed of the B2-(Fe,Ni)Al and -(Fe,Ni)3Al phases according to the EDS concentration profiles of Fig. 1c and the Al-Fe-Ni ternary diagram at 1050°C [30]. The B2-(Fe,Ni)Al was also probed by XRD (not shown) where all the diffracting peaks corresponded to the JCPDS pattern 01-073-8858 of the Al(Fe0.5Ni0.5) phase.

  In addition, the aluminide coatings contain up to 12 at% Cr.

  Sealing of the surface and of cracks of an aluminide coating on IN-800HT stainless steel with different concentrations of H3PO4 in acetone and different dipping times. P-O (phosphate) compounds and of the -Al2O3 layers formed. High concentrations (3.75 and 1 mol.L -1 ) resulted in, respectively, blistering by hydrogen evolution and pitting by breakdown of the passive layer.

Figure 1 .

 1 Figure 1. Back-scattered SEM cross-sections morphology of the aluminized IN-800HT sample by slurry technique before (a) and after (b) grit blasting and corresponding EDS profiles (c).

Figure 2 .

 2 Figure 2. Evolution of deposited mass of sealant with concentration and dipping time at room temperature. Drying overnight brought about evaporation of the solvent.

Figure 3 .

 3 Figure 3. Surfaces of the aluminide coatings after dipping in 0.5, 1 and 3.75 mol.L -1 H3PO4/acetone with time after 24-h drying in open air at room temperature.

Figure 4 .

 4 Figure 4. X-ray patterns of the surfaces of the aluminide coatings sealed for 5 min in 0.5, 1 and 3.75 mol.L -1 H3PO4/acetone (black, red and blue curves, respectively) and subsequently dried naturally in open air at room temperature.

Figure 5 .

 5 Figure 5. Raman spectra of the surfaces of the aluminide coatings treated with 0.5 and 1 mol.L -1 H3PO4/acetone at different dipping times and subsequently dried naturally in open air at room temperature.

Figure 6 .

 6 Figure 6. SEM images of the surfaces of the aluminide coatings treated with 0.5 and 1 mol.L -1 H3PO4/acetone at different dipping times and subsequently dried naturally in open air at room temperature.

Figure 7 .

 7 Figure 7. Al to (O+P) ratios calculated from the EDS analyses of the surfaces treated with 0.5 and 1 mol.L -1 H3PO4/acetone and different times.

Figure 8 .

 8 Figure 8. Water contact angle measurements of the surfaces treated with 0.5 and 1 mol.L -1 H3PO4/acetone and different times compared to the as-received aluminide coating surface. N.B. "GB" corresponds to "grit blasted" surface, i.e. prior to sealing.

Figure 9 .

 9 Figure 9. Aluminized IN-800HT after grit blasting and before sealing tests (a), after sealing with 0.2 mol.L -1 (b) and 0.5 mol.L -1 (c) H3PO4/acetone for 2 minutes.

Figure 10 .

 10 Figure 10. X-ray maps after sealing of the aluminide coatings of IN-800 HT with 0.2 mol.L -1 (a) and 0.5 mol.L -1 (b) H3PO4/acetone for 2 minutes.

Figure 11 .

 11 Figure 11. SEM cross-section of Al coated samples after oxidation without sealing (a) and with a 0.2 mol.L -1 (b) and 0.5 mol.L -1 (c) sealing treatment.

Figure 12 .

 12 Figure 12. EDS profiles of Al coated and oxidized IN-800HT without or with a 0.2 mol.L -1 and 0.5 mol.L -1 sealing step.

Figure 13 .

 13 Figure 13. X-ray maps realized, after oxidation, on Al coated IN-800HT samples without (a) and with 0.2 mol.L -1 (b) and 0.5 mol.L -1 (c) sealants.

Figure 14 .

 14 Figure 14. Analyses of the surfaces of Al coated samples with or without an initial sealing treatment, after 1000 cycles. (a) X-ray patterns and (b) Raman spectra.

TABLE 1 . Composition (nominal and by EDS) of IN800HT Fe Ni Cr Co Al Si Ti Mo Mn Cu

 1

	Nominal
	composition	Bal 30-35	19-23	-	0.25-0.6	-	0.25-0.6	-	-	0.1-0.3
	(%mass)
	EDS (%mass)	45	29	21	< 1	< 1	1-2	< 1	< 0.5	< 1	< 0.5

ACKNOWLEDGEMENTS

The authors gratefully acknowledge European Union for the funding of FP7 project "POEMA: Production of Coatings for New Efficient and Clean Coal Power Plant Materials" (Grant agreement no. 310436). Dr. B. Bouchaud is also acknowledged for his contribution to this paper.