; Mirdamadi

; Mohammadi

; Malmiry

; Antoine

; Siadat

; Dantan

; Tavakkoli

Cost Engineering for Variation Management during the Product & Process Development

Keywords: Integrated Product and Process Design, Cost, Variation management, Tolerancing, Inspection planning

Variation Management during the Product & Process Development can profoundly impact the quality, the cost of the product and the number of scraps in mass production… Designers want tight tolerances to ensure product performance; manufacturers prefer loose tolerances to reduce manufacturing and assembly cost. To analyse compromise solutions, the primary aim is to establish an objective function. This paper presents a model for the key indicators assessment to the relevance of variation management: cost, and investigates which model used in decision analysis is the most appropriate to prioritize and aggregate the predetermined performance measures. The applications of this model are demonstrated through an industrial case study where tolerance allocation, product development, problem is firstly addressed. Once optimized tolerances are attained, inspection planning, process development, problem is approached to ensure the optimized awaited quality level for the least cost.

1-Introduction

In recent decades, the egregious importance of total quality management has been completely clarified to all industries. In order to maintain profitable and stay in a competitive edge, reaching to high quality level of products, processes and services has been nowadays a vital issue in many organizations, while they cannot survive without providing high quality products. For this aim, manufacturers are applying a variety of tools to improve quality throughout the production process such as Six Sigma, statistical process control (SPC), process improvement, inspection, robust design, etc. Variation/Uncertainty is ubiquitous in any engineering system at all the stages of product development. Variation from different sources affects the product lifecycle and thus, the intended performance of the product. Variation management therefore, during product design is a crucial part of the design process. A review of the general design processes however reveals that the activities intended to study and to minimize the effect of the variation in the design process are mostly addressed in the later phases of the product development phase once the design parameters of the products are fixed. Moreover, many papers focus on the impact of the variation on the product functionalities. Variation management can profoundly impact the quality, the cost of the product and the number of scraps in mass production… Designers want tight tolerances to assure product performance; manufacturers prefer loose tolerances to reduce cost. Variation management is a key element in industry for improving product quality and decreasing the manufacturing cost. Therefore, the development of models for the key indicators assessment to the relevance of a variation management (customer satisfaction, manufacturing cost, inspection cost…) is key issue. In fact, the cost assessment becomes a key activity to improve the tolerance allocation, to select fittest manufacturing resources, inspection allocation planning, or ... A number of cost assessment methods and techniques were developed with reference to particular applications:

-Parametric models for tolerance allocation [1], [2], [3], -Analytical methods for tolerance verification [4], -Analytical methods for the assessment of the economic impact of metrology in manufacturing [5],

[6] -… A significant amount of research has been devoted; two essential inconveniences in the parametric models for tolerance allocation are costly evaluation of their required parameters (mainly with experimental designs) and also their limited validity and generality in the industrial framework. Moreover, the development of specific cost models for each step of the product development increase the modelling cost [START_REF] Mirdamadi | Cost Estimation Method for Variation Management[END_REF]. Therefore, this paper presents a key indicator: Quality Weighted Cost that could be used in several stages of a product design cycle: Tolerance design, CAPP (Computer Aided Process Planning), CAIP (Computer Aided Inspection Planning)… This paper is divided into four main sections. The next section discusses the context: redesign or adaptive design which is the most common design problem undertaken in industry. In this context one indicator is proposed in the section 3: the cost impacted by the variation/uncertainty. The applications of this indicator are demonstrated through an industrial case study in the last sections.

2-Context

This section provides a brief overview of the context of variation management: design process context, main aim of variation management (quality assessment and cost reduction), and the description of variations which affect the customer satisfaction. Most of the design literature focuses on original design problems. But, most design problems encountered in the industry are redesign, variant design or adaptive design of an existing product design [START_REF] Ullman | The Mechanical Design Process[END_REF]. Redesign, Variant Design and adaptive design is a common and widely practiced design task whereas starting from an existing solution, the designer creates a product to meet new requirement, needs and constraints while keeping an existing product design. The resulting product may be adapted for different requirements, or be an improved version of the existing requirement; redesign is also used for different versions of the product to address different segments in the market according to their variable demands [START_REF] Beitz | Engineering design: a systematic approach[END_REF]. The proposed indicators are used for redesign. In this context effective reuse of industry knowledge about variation effects and causes is a key strategic component of Integrated Product and Processes Development (IPPD). For variation management in the context of IPPD, process capability approaches have been developed at most manufacturing companies to enable the manufacturability assessment, to predict the Product designs end quality and to improve design robustness. Process capability approach allows for an understanding of the capability of machines, tools, and operators to manufacture a particular feature of a particular dimension using a specific process [START_REF] Tata | Process capability database usage in industry: Myth vs. reality[END_REF]. These approaches are needed for robust design, optimal tolerance allocation, and variation simulation analysis. In the context of variation management, traditional performance assessments focus on the customer satisfaction. These assessments do not provide information to evaluate the impact of variations on the manufacturing productivity; on the cost … In the following the classification and the impacts of variation are discussed. V. SRINIVASAN [START_REF] Srinivassan | An integrated view of geometrical product specification and verification[END_REF] argues that inevitable variations in production and inspection influence product specification. This inevitability of variations is embodied in the following two axioms: "Axiom of manufacturing imprecision: All manufacturing processes are inherently imprecise and produce parts that vary." "Axiom of measurement uncertainty: No measurement can be absolutely accurate and with every measurement there is some finite uncertainty about the measured attribute or measured value." An important part of quality costs are failure costs that include scraps which are due to the manufacturing imprecisions. Scrap can be defined as the percentage of system quantity that does not meet required production quality standards; it could be evaluated by process capability approaches, and could be measured. Another important part of quality costs are failure costs that include warranty, product liability claims and recall costs, which are due to the measurement uncertainty. In fact, measurement results are affected by measurement uncertainty, which leads to technical and economic risks in industrial companies [6]. By assessing the risks and the connected consequences of the decisions (conformity verification), the significance of the measurement result can be evaluated. Moreover, a brief discussion of this issue is given by Weckmann et al. [6]: "In production metrology workpieces are inspected by measuring their specified characteristics. The generated measurement results are used as a basis for decisions for conformity assessment, process evaluation and statistical process control. Regardless of the economic value of measurement results as a decision base and due to the fact that the benefits through quality inspections in industrial enterprises are mostly hidden, production metrology is often considered merely as source of expenses.

A verifiable proof of the economic value of measurements and the optimization of the ratio between value and expenses of measurement systems failed until now because of the missing monetary evaluation of the value of measurements."

As preliminary conclusion of this section, variations affect the customer satisfaction and the cost.

In the context of interactive design and manufacturing, it is important to ensure the interaction for improving decision making in the design process and the manufacturing. The variation management during the product development is an important issue: decisions can profoundly impact the customer satisfaction, the quality, the cost… It calls for the formation of a cross-functional product development team; which includes people from a wide range of departments, such as: product planning, design, manufacture, assembly, quality assurance ... ; which includes people with different points of view, requirements … In this context, Performance assessment involves multi-dimensional attributes: Quality, Time, Cost, Customer satisfaction … -Quality is traditionally defined in terms of conformance to specifications. According to (ISO 9000: 2000), quality is "the degree to which the set of inherent characteristics fulfil the requirements". This definition of quality prompts quality measures such as the number of defects produced and the cost of quality. The true cost of quality is the function of the prevention, appraisal and failure cost.

Manufacturing process improvement initiatives, such as six sigma, lean manufacturing and statistical process control, have introduced new ranges of quality related measures. For example, in the six sigma initiative, the highest quality is to achieve six sigma capability meaning 3.4 defects per million. Statistical process control measures consist of capability indices, i.e. Cp, Ca, Cpk, etc. -Cost-based performance measures have their origin in accounting management. An accountingbased measure called return on investment (ROI) has been developed to serve as an indicator of the efficiency of their decentralized business units. -… Due to the context and the objective, we focus on the cost and the quality, indirectly customer satisfaction. Moreover, Value in the product context is defined as: a judgment made on product by users on the basis of their expectations and motivations. More specifically, value increases when customer satisfaction increases or the incurred cost on product decreases. This definition can be mathematically symbolized as:

(1)

This performance indicator combines the Customer satisfaction and the Cost which are impacted by the manufacturing variations. In the same way, we propose a new cost model which takes into account the impacts of the manufacturing imprecisions and the measurement uncertainty.

3-Cost models for variation management

As explained in the previous sections of this paper, variation management strongly requires indicators which asses the relevance of several solutions. Later in this article, in accordance to the literature, one of these relevance assessment indicators on which we are focused is called "cost". Among the several cost assessment methods available in the literature, a classification into three main categories could be carried out: parametric, analytical and analogical methods [START_REF] Dantan | Cost estimation for tolerance allocation[END_REF].

Approaches comparisons

The first category, the parametric approaches, gathers all technical solutions aiming at costs assessment by taking into account mathematical relationships linking parameters considered as influential ones. Both the selection of these key parameters and their mathematical relationships with costs are strongly based on the Knowledge of experts (mainly of products design and manufacturing system design) supported by a lot of set of experiments, data gathering and their analysis. The CEF (Cost Estimation Formulae) is the most famous methodology. The goal of this approach is to Cost on Satisfacti alue  V guide the definition of parametric models by analysing continuously the relevance of each parameter in order to remove the useless ones. As an output, the parameters and the mathematical link is validated and can be directly used to the assessment of the costs of new products. A lot of parametric cost models are available in the literature and, by extension; several propose to assess the cost due to tolerance allocation [13, 14 and 15]. This huge number of models can be easily explained by the fact that these formulae are easy to use, give directly fast and relevant results. However they are strongly contextual and, consequently, have a very short domain of validity. Indeed, modifying the location of the factory or the type of product material can impact have strong impacts, not only on value of parameters but on the model itself! Moreover designing these models is quite expensive, because it needs large set of data and experiments to identify the influence and weight of parameters supposed to be cost drivers.

The second type of approaches is called analogical: they are based on the hypothesis that two products considered similar have similar costs. Then, in order to assess the cost of a new product these approaches try to find, in a huge database containing all former products designed and manufactured by the company what are the ones that can be considered similar. This similarity evaluation is mainly based on a set of parameters considered relevant and discriminant enough to describe both: the product to design and its manufacturing process [START_REF] Duverlie | Cost estimation during design step: Parametric method versus case based reasoning[END_REF]. The technical solutions based on this principle follow the evolution of computer science: starting from group technology to most advanced systems like Cases based reasoning. This approach is currently able not only to give a similar cost but to generate the draft design of the process needed to manufacture the new product to analyse and consequently improve the cost evaluation. Even if these approaches are interesting and can potentially continue their evolution in the same way than computer science [START_REF] Kim | Comparison of construction cost estimating models based on regression analysis, neural networks, and case-based reasoning[END_REF], these approaches faced several drawbacks. The main one is the setup of these approaches to take into account the specificities of the company. Indeed, everything has to be adapted and defined before obtaining the first results: from the identification of the relevant parameters describing the problem to solve and its solution and the way to evaluate the similarity between the case to solve and the former cases.

The last type of approaches is the analytical ones. In order to assess the cost of a product, these solutions aims to decompose all the lifecycle of this product (from its design to its recycling) in order to identify and analyse all the activities involved. The activities are considered as the cost and time drivers: they consume resources and consequently money. By summing the cost of each activity it is possible to assess the global cost of the product. Several examples of this approach can be enumerated: form features [START_REF] Feng | Cost evaluation in design with form features, Computer aided design[END_REF], manufacturing engineering reference model, activity based costing [START_REF] Johnson | Relevance cost: the rise and fall of management accounting[END_REF][START_REF] Gupta | Activity based costing and management and its implications on operations management[END_REF], cost features… This kind of solution is interesting since several activities are common to several types of products: consequently they are really generic and are flexible. However in order to be as precise as possible, the work becomes very meticulous (in order to identify all activities directly or not involved in the product lifecycle) and, consequently, very long too. Moreover, compare to parametric approach the analytical approach are more computer time consuming.

To compare the different methods, it is interesting to establish the criteria with the features expected from such methods to choose the best solution. In this study we used as discriminating features [START_REF] Mirdamadi | Cost Estimation Method for Variation Management[END_REF]:

-Sensitivity of the assessment (repeatability and robustness): ability of method to integrate and take into account the variations of the input data. -Adequacy of the evaluation (precision): ability of method to give accurate results considering the final product cost.

-Deployment/ Simplicity of use: the difficulty of formalizing data preparatory for the evaluation of cost, usually performed by experts. -Speed / Rapidity of estimation: includes both the computation time than the time required to model a new problem. -Scalability of method allows measuring the performance indicator rather than only financial dimension.

Comparing the three methods available to estimate the cost of a product, the drawbacks can be summarized as [START_REF] Mirdamadi | Cost Estimation Method for Variation Management[END_REF]:

-Parametric: although accurate and fast to use, in their range of validity they are limited by restricted scalability. -Analogical: these methods are interesting but the preparatory phase (enrichment of knowledge base, parameters discriminating…) is long for the results. This approach seems impossible to automate in an optimization loop. -Analytical: although the estimation process of these approaches is quite long because they generate and analyse all the operations necessary for obtaining a product, they remain attractive because of their flexibility and accuracy. To manage the variation, the most important criteria is the accuracy. Therefore we focus on the analytical approach. One of the best known analytical approaches is the Activity-Based Costing (ABC) method. The fundamental difference between an ABC system and a traditional costing system is that the latter assumes that product causes costs, whereas ABC assumes that activity causes costs and the cost objects create demand for activities. The consortium for Advanced Management-International (CAM-I) defines activity based costing as follows: "Activity-Based Costing is a methodology that measures cost and performance of activities, resources and cost objects. Resources are assigned to activities, then activities are assigned to cost objects based on the use of consumption of the relevant activities. Activity-Based Costing recognizes the causal relationships of cost drivers to activities'. This method will be developed in the following section. Performance evaluation in the context of variation management reflects two aspects: cost and quality. Therefore, we propose a cost model which takes into account the impact of the variations. Proposed quality-driven ABC (Activity Based Costing) aims to balance the manufacturing cost and successive expenses, and product satisfaction throughout the tolerance and variations analysis; it is the total cost of a marketable product; it includes four costs:

Proposed Model

the manufacturing cost, the verification cost, the scrap cost (internal failure), the cost of external failure, weighted by the occurrence or/and the efficiency of the activities (Equation 2).

                                    . 1 1 . 1 . 1 1 . 1 1 . 1 1 . . . . 1 1 . Cost Total Product Marketable
P C                      (2)
Where:

- This cost model could be simplified regarding to the granularity of the information. Based on the ABC approach, we add two concepts which are associated to activity:

-Activity occurrence: Probability that an activity appears in the process (for example, the probability of an inspection activity occurs in the manufacturing process). -Activity efficiency: The probability that an activity lead to good products (Pc, α, β). This cost model creates the link from cost to quality (two components of the customer satisfaction). As the concept of value, it could be used at different step of the product development for improving the decision making; it could be the objective function of lots of optimization problems for solution space exploration during the Integrated Product and Process Design. The applications of this cost model as an objective function are demonstrated through an industrial case study in the sections 4 and 5:

-Tolerance allocation during the product design -Planning of the inspection activities during the process design

Illustration

To illustrate this model, a real case study is presented (Figures 2 and3) for the inspection planning in automotive industry. Different scenarios of inspection plans are proposed for the oil pump housing:

-Scenario 1: Product conformity control at the end of process -Scenario 2: Process monitoring for all operations with Cp<1.6 -Scenario 3: Product control conformity at the end of process, and the process monitoring for all operations Cp<1.6 -Scenario 4: Product conformity control at the end of all operations with Pp<1.4 -Scenario 5: Product conformity control at the end of all operations Cp(monitoring)<1.4, and the process monitoring for all operations Cp<1.6 One of the assumptions of this illustration is that there is no rework following failure detection, maintenance and assembly activities. Therefore, the inspection activities are regarded as on-line activities. For these activities, enough information is usually available to estimate the cost attribute affected directly. Therefore, it is necessary to evaluate Manufacturing times (

4-Application 1: Tolerance allocation optimization

In the current industrial framework where high precision products have to be low-cost too, a compromise must be found between designers who want tight tolerances to assure product performance and manufacturers who prefer loose tolerances to reduce production cost.

Figure 2: Gear Pump

The first application is the tolerance allocation of a gear pump (Figure 2) which is performed regarding the tolerance cost. In fact, the objective function is: to minimize the Marketable Product Total Cost, by quantifying the best possible geometrical tolerances (mostly dimensional tolerances). This value evaluation is performed with an integrated point of view: taking into account both the Product characteristics and constraints (mainly from the requirement it has to meet) and the Process capabilities and costs (resources are considered as the cost sources). The performance indicator is consequently a balance between two historical opposite points of view:

-Designers who want to tighten tolerances to insure that products are both functional and can be assembled. -Manufacturer who need loose tolerances to short their production cost and ease the manufacturing of the design product. In this case study, the required function is delivering oil with required pressure and speed. The system converts mechanical energy into hydraulic energy. The main property here is chosen as "oil flow" (Q) by the manufacturer.

-Problem formulation

The manufacture of the current oil pump expects an oil flow of 4.45×10 -4 m 3 /s. The designers know that the efficiency and oil flow of the pump is related to different backlashes. These backlashes are between the gears and the casing as well as between the gears and shafts. Too small backlashes will result in friction and too much of them will result in internal flow loss and therefore performance reduction. Achieving precise backlashes is the result of manufacturing precision to obtain tight tolerances. Cost of tolerance is different for different parts. For instance, gears are made of aluminium so the quality level of IT8 is chosen for it. Therefore, because of the diameter of 31mm, the tolerance interval is 39µm. The casing (CPHF) is made of sintered still. So, the quality level of IT6 is chosen for all the entities shown in Figure 3. The impact of tolerance level on the performance is not clear for the designer. Moreover, this impact on the cost of production should be identified as well. The problem formulation was detailed in [START_REF] Dantan | Variation management by functional tolerance allocation and manufacturing process selection[END_REF].

Figure 3: CPHF and its tolerances.

To carry out the evaluation of the value of a product configuration, the tolerance allocation is coupled with the process generation (one result is displayed in Table 1) and its evaluation regarding both its costs and the impacts of it on the product quality. The assessment of these quality parameters (i.e. the probabilities expressed in the equation 1) is performed by simulations. Several technical solutions were used to carry out these simulations: Monte Carlo simulations (used in [START_REF] Dantan | Cost estimation for tolerance allocation[END_REF][START_REF] Speckhart | Calculation of tolerance based on a minimum cost approach[END_REF][START_REF] Sutherland | Mechanism design: Accounting for manufacturing tolerances and costs in function generating problems[END_REF]) ... For the gear pump, the assessment of the quality characterizes the impact of the geometrical deviation on the oil flow. The quality of an oil pump can be evaluated based on its efficiency which includes volumetric and mechanical efficiencies. Volumetric efficiency is related to internal flow loss. The internal flow loss which is related to the laminar flow behavior can be modelled by the pressure required by the engine hydraulic circuit on one hand and the existing backlashes between moving parts on the other hand. The backlashes are directly related to the geometrical deviations. Similarly, mechanical behavior can be modeled through journal bearing friction model. Dimensions, gaps and applied force lead to eccentricity of shaft and it modifies the distance between gear and casing. Based on this modeling, 13 tolerances are identified. Only some tolerances of the part CPHF is shown in Figure 3. The evaluation of the manufacturing process cost is an easier stage. Indeed, when the manufacturing process is generated, the ABC method assesses this parameter. This approach needs that all resources costs and all parameters are known (not necessarily absolutely, but at least relatively).

When both quality measures and cost assessment are available the evaluation of the relevance of one tolerance allocation can be done. The quality weighted cost of both the product and its manufacturing process is available and can be compared with others solutions. In order to find the product configuration having the best quality weighted cost, an optimisation loop is needed. At each step of this loop new tolerance allocation is generated and it cost is assessed. In [START_REF] Dantan | Variation management by functional tolerance allocation and manufacturing process selection[END_REF], Genetic Algorithm supports this optimisation loop; nevertheless several other solutions are available in the literature.

-Results

The Figure 4 illustrates the evolution of the Marketable Product Total Cost (which represents in fact the opposite of the value) of the product configuration. The Figure 5 illustrates the evolution and consequently the optimisation of tolerance allocation. In the top of this figure, the evolution of the three tolerances (Figure 3) is plotted: these three converges to the optimum value.

5-Application 2: Inspection planning optimization

Quality management in logistics reduces costs and enhanced customer satisfaction. An effective quality management should be carried out only on the basis of reliable measurements. Obtaining this information depends on unconditional observance of the unity of the initial measures and on correct measurement procedures at all steps of the product life-cycle. Production metrology is the fundamental tool to gain information and knowledge in all phases of the life-cycle of any product to help linking the separate processes. Therefore, however, it must be productive in an economic way, both cost efficient and relevant to satisfy the single process requirements of information. In production metrology products are inspected by measuring their specified characteristics. The results of measurement are used as a basis for planning an inspection process (IP) to make decision on conformity assessment, process monitoring and statistical process control. Since the benefits through quality controls in industries are mostly hidden, production metrology is often considered merely as source of expenses. On the other hand, the proof of the economic value of measurements and the ratio determination between value and expenses of measurement systems have failed while the monetary evaluation of the value of measurements is missing. Pfeiffer [START_REF] Pfeiffer | Production metrology[END_REF] and Zhao et al. [START_REF] Zhao | Computer-Aided Inspection Planning-The state of the art[END_REF] defined IP planning (IPP) as an activity that determines which quality characteristics of a product should be inspected, where and when. In almost all manufacturing organisations, inspections are used during the processes to reach quality specifications instead of having an acceptance or a rejection inspection at the end. To achieve this goal, effective inspection planning should efficiently be integrated with the production logistics [START_REF] Mohammadi | Mathematical modelling of a robust inspection process plan: Taguchi and Monte Carlo methods[END_REF]. The application 2 tries to integrate production logistics and quality control and design an effective IPP to makes decision regarding to which quality characteristics of the product need what kind of inspections (i.e., which-what decision); and when these inspections should be performed through the production process (i.e., when decision) in order to minimize manufacturing and inspection cost and maximize customer satisfaction. Through this problem, two kinds of product conformity (CI) and process monitoring (MI) inspections are considered [START_REF] Mohammadi | Mathematical modelling of a robust inspection process plan: Taguchi and Monte Carlo methods[END_REF].

-Problem formulation

In production metrology products are inspected by measuring their specified characteristics and the results of measurement are used as a basis for designing an IPP to decide which quality characteristics need what kind of CI and/or MI and when. For this aim, consider a serial multi-stage production system (MPS) with N stages, in which in-process parts pass sequentially from stage 1 to stage N and inspections of parts are performed at m (m≤N) locations. It should be noted that each stage can be an operation and a set of operations can be performed on the same machine. At each stage, a part (output of the immediately preceding stage) enters the processing station where a manufacturing operation is performed on it. Output of this operation is transferred to an inspection station or to the next processing stage. Suppose that a part consists of K quality characteristics and all characteristics of the part are simultaneously operated throughout the system. A part is 'nonconforming' if any quality characteristic does not meet design requirement. If a CI is performed between the i-th and (i+1)-th processing stations, non-conforming parts originated at the ith operation or at some of the earlier stages are detected and scrapped and no rework is considered. Besides, If an MI is performed between the ith and (i+1)-th processing stations, the processing features are monitored after a specific number of parts. An inspection operation may involve errors of two types: misclassification of a conforming component as non-conforming (type I error) and nonconforming one as conforming (type II error). This application proposes a bi-objective mixed-integer programming (BOMIP) model in order to simultaneously minimize manufacturing cost and maximizing customer satisfaction. The proposed cost model (Section 3) is divided in two components: Objective function (1) minimize sum of production cost, scrap cost, fixed and variable costs of CI and MI and fixed space cost of inspections, respectively. Objective function (2) attempts to minimize total warranty cost or indirectly minimize the number of undetected nonconforming parts that are transferred to the customers, while the lower the number of undetected nonconforming parts, the higher the satisfaction of customers. Lack of information about production processes and several environmental factors imposes a degree of uncertainty to the planning parameters, which directly affect other decisions of inspection process. In most of manufacturing industries, a minimum level of uncertainty is inevitable. There are several parameters in the proposed BOMIP model that are affected by environmental factors and may fluctuate over the time. These parameters include production and inspection times, error types I and II of the inspection activities, and dispersion and misadjustment of the production processes. Hence, manufacturers are interested in less sensitive manufacturing processes. These manufacturing processes are robust processes, which are relatively insensitive to alteration of uncertain parameters. Objective functions are transformed into two new objective functions that minimize both expected value and variance of each objective function under variation to search the robust optimal solutions.

-Result

In order to validate the correctness of the proposed robust BOMIP, an industrial case related to the main part of the gear pump with 15 quality characteristics is studied in this section. First, some information about the industrial case is presented such as production time, operation capability, failure rate and allowable places to perform inspection for each quality characteristic (Table 3, in which the first to sixth columns explain name of the operations, production time, process capabilities Cp and Pp and the allowable places (AP) that inspections (i.e., CI and MI) of each quality characteristic can be stationed). After solving the proposed global robust BOMIP model, the Pareto frontier of the problem can be illustrated as Figure 6, in which the dash and solid lines represent Pareto frontiers of the problem with deterministic and uncertain parameters, respectively. In order to better understanding the which-what and when decisions, the structure of a sample nondominated solution from the global robust Pareto frontier (i.e., shown in Figure 6) is depicted as Figure 7. In Figure 7, circles show the operations and squares represent quality characteristics. Quality characteristics that need CI or/and MI, have been illustrated in blue and yellow squares. For example in Figure 4, quality characteristics number 2, 4, and 6 to 12 need MI and quality characteristics number 7 and 11 to 15 need CI. It can be seen that quality characteristics number 7, 11 and 12 need both CI and MI, simultaneously. In addition, MI for quality characteristics number 2, 4, and 6 to 12, is performed after operations 6, 6, 6, 15, 15, 10, 10, 15, and 15, respectively. Similarly, CI for all quality characteristics number 7 and 11 to 15 is performed after operations number 15 (i.e., at the end of production line). It is noteworthy that CI is performed for characteristics with lower value of capabilities, while the lower the value of Cp and Pp are, the higher the number of scraps is.

Figure 1 :

 1 Figure 1: Comparisons of the three main approaches.

  Cmanu: Cost of manufacturing activities of a product, -Cmonit & Cinspctn: Costs of monitoring and inspection activities, -OccPmonit & OccPinspctn: Occurrences or frequencies of the monitoring and inspection activities -Cprd scrpng: Cost of product scraping and recycling, -Cprd maintnc: Cost of warranty, product liability claims and recall -PC: Occurrence probability of conform product, which depends on manufacturing imperfections and component tolerances α: Occurrence probability of non-detection of non-conformity which is due to the measurement uncertainty. β: Occurrence probability of non-detection of conformity (false alarm) which is due to the measurement uncertainty. -PC.(1-α): Percentage of Marketable conform products -Β.(1-PC): Percentage of Marketable non-conform products -((1-PC).(1-β): Percentage of Detected non-conform products -PC.α: Percentage of Undetected non-conform products

Figure 4 :

 4 Figure 4: Evolution of fitness values during the Genetic Algorithm iterations.

Figure 5 :

 5 Figure 5: Evolution of tolerance values during the Genetic Algorithm iterations.

Figure 6 :

 6 Figure 6: Pareto frontier of the proposed BOMIP model.

Table 1 -

 1 MT), Process capabilities (Table 1 -Cp and Pp), the non-conformity rates with monitoring (Table 1 -NC Rate 1) and without monitoring (Table 1 -NC Rate 2) of each manufacturing operation.

Table 1 :

 1 Details of the process plan.

		Parameter value
	Process operation	MT	Cp	Pp	NC rate 1 NC rate 2
		(min)
	Rough milling PL100	.148 2	1.50 0 ppm	7 ppm
	Rough milling PL100	.166 2	1.50 0 ppm	7 ppm
	Rough milling PL101	.133 2	1.66 0 ppm	1 ppm
	Boring CY110	.154 1.60 1.33 2 ppm	66 ppm
	Rough drilling CY108 & CY109	.09	2	1.66 0 ppm	1 ppm
	Chamfering CY108 & CY109	.25	2	1.66 0 ppm	1 ppm
	Chamfering CY100 & CY101	.257 1.50 1.20 7 ppm	318 ppm
	Boring CY100	.257 1.50 1.20 7 ppm	318 ppm
	Boring CY101	.122 1.66 1.30 1 ppm	96 ppm
	Rough drilling CY102 & CY103	.109 1.66 1.40 1 ppm	27 ppm
	Rough drilling CY111	.134 1.66 1.40 1 ppm	27 ppm
	Boring CY108 & CY109	.122 1.30 1.10 96 ppm	967 ppm
	Boring CY102 & CY103	.122 1.30 1	96 ppm	2700ppm
	Boring CY111	.117 1.66 1.33 1 ppm	66 ppm
	Finish milling PL100	.129 1.66 1.33 1 ppm	66 ppm
	The numerical values for each scenario are obtained as reported in Table 2:

Table 2 :

 2 Details of cost.

	Scenario	1	2	3	4	5
	Cmanu	10,09 10,09 10,09 10,09 10,09
	PC	0,9954 0,9994 0,9994 0,9966 0,9996
	C	    1 1 . Inspctn . c Monit P OccP   . Inspctn C OccP . c Monit P    	2,01	0,8	2,81	5,34	5,64

81 12,10 12,97 15,80 15,77

Table 3 :

 3 Details of industrial case for the inspection planning.

	Process operation	Parameter value MT (min) Cp	Pp	AP
	Rough milling PL100	.148	2	1.50	1→13
	Rough milling PL100	.166	2	1.50	2→14
	Rough milling PL101	.133	2	1.66	3→15
	Boring CY110	.154	1.60 1.33	4→10
	Rough drilling CY108 & CY109	.09	2	1.66	5→10
	Chamfering CY108 & CY109	.25	2	1.66	6→6
	Chamfering CY100 & CY101	.257	1.50 1.20	7→15
	Boring CY100	.257	1.50 1.20	8→15
	Boring CY101	.122	1.66 1.30	9→12
	Rough drilling CY102 & CY103	.109	1.66 1.40	10→12
	Rough drilling CY111	.134	1.66 1.40	11→15
	Boring CY108 & CY109	.122	1.30 1.10	12→15
	Boring CY102 & CY103	.122	1.30 1	13→15
	Boring CY111	.117	1.66 1.33	14→15
	Finish milling PL100	.129	1.66 1.33	15→15

6-Conclusion

Variation management is typically done in the factory during production. It can be defined as the resources allocation to reduce or/and mitigate the impact of the manufacturing imprecisions based on cost and risk. During the past two decades variation management efforts have been moved upstream to the design stages of products and processes: Design for Six Sigma (DFSS) Robust Design Methodology... Therefore, this paper presents models for the assessment of the key indicator of the relevance of a variation management: cost affected by the variations and the uncertainty. This indicator is very powerful: due to its expression and design, it is a very flexible way to evaluate the relevance of product based on multiple parameters. Several other characteristics could be added to the Marketable Product Total Cost such as its environmental footprint for instance. This addition of several other parameters, taking into account several points of views and their complete satisfactions can evolve this performance evaluation to value assessment [26 and 27]. Regarding the complexity of the stakeholders involved in both design and production of a product, the use of value network concept is an axis to explore in order to generalize and improve the way to evaluate and optimize the numerous products' performances the designers have to handle. The application of models is demonstrated through an industrial case study: the tolerance allocation and the inspection process selection of the gear pump. Other applications [START_REF] Mousavi | A new intuitionistic fuzzy grey model for selection problems with an application to the inspection planning in manufacturing firms[END_REF][START_REF] Hassan | Conceptual process planning an improvement approach using QFD, FMEA, and ABC methods[END_REF][START_REF] Hassan | A quality/cost-based improvement approach for conceptual process planning[END_REF] illustrate the scalability of the proposed model.

7-Acknowledgments

The authors would like to acknowledge France embassy in Iran and Campus France for their financial supports.

8-References


G G ‡

A mode-matching model for the impedance of perforated plate liners

A mode-matching model applicable to perforated liners in the linear regime is presented. It complements semi-empirical models not suitable for the high frequency regime, for liner featuring thick plate and micro perforated liner. The mentioned models are compared to measurements. From these comparisons, it appears that removing the short tube approximation is not enough to capture completely the losses in the linear regime. Further investigations have been carried out using a finite element method model based on compressible Linearised Navier-Stokes equations. This method showed to be in good agreement with experimental data. Thus, the dissipation due to viscosity was mapped around the aperture. This approach is supposed to provide a more precise understanding of the physical phenomenon of interest. Based on those results, the next step is to propose an upgraded mode-matching model computationally efficient that overtakes empirical models thanks to physical-based modeling.

I. Introduction

P acoustics liners are a widespread technology used to reduce noise emissions from turbofan. Extensive research has already been conducted to optimize liners impedance in order to reach the maximum sound attenuation (see for instance Spillere [START_REF] Spillere | An improved impedance eduction technique based on impedance models and the mode matching method[END_REF] and Seo [START_REF] Seo | Design of silencer using resonator arrays with high sound pressure and grazing flow[END_REF]). The linear physics phenomenon responsible for sound absorption are well known. However, the associated semi-empirical models such as Guess [START_REF] Guess | Calculation of perforated plate liner parameters from specified acoustic resistance and reactance[END_REF] or Maa [START_REF] Maa | Potential of microperforated panel absorber[END_REF], are applicable to a limited range of geometric configurations such as macro perforated liners. On the other hand, no reliable and time effective model appears to exist for micro perforated liners, presenting a percentage open area, named porosity, below 5 % and a perforation diameter inferior to 1 mm. Therefore, a Mode-Matching (MM) model based on Helmholtz equation is proposed. This approach allows to remove the short tube approximation, outlined further below. The model also includes the velocity profile through an aperture and the interaction effect between the perforations. This strategy is more comprehensive than the named semi-empirical models and is expected to be in better agreement with experimental data. A Finite Element Method model (FEM) based on compressible Linearised Navier-Stokes (LNS) equations is used as a subsequent comparison. As we shall see further, the results comparison tends to point out a missing dissipation mechanism in the current MM modeling. The corresponding phenomenon is highlighted through the post processing of FEM data, involving the analysis of the rate of dissipation due to viscosity near the perforation. The losses scheme highlighted on the aperture edges complete the existing visco-thermal modeling proposed by Zwikker & Kosten [START_REF] Zwikker | Sound absorbing materials[END_REF] and Stinson [START_REF] Stinson | The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape[END_REF] for cylindrical perforation. As a first step, the three models used in the study are detailed, beginning with Guess model, the mode-matching model and the numerical model. Thereafter, a comparison with measurements is presented. Finally, the dissipation analysis is described to provide further modeling perspectives. The radius of the exterior domain, the neck and the cavity domain are written, respectively R ext , R neck and R cav . The plate porosity is noted σ. An e jωt time convention is used.

II. Semi empirical model

Guess model [START_REF] Guess | Calculation of perforated plate liner parameters from specified acoustic resistance and reactance[END_REF] was chosen as a comparison basis for this study. The term accounting for viscosity is a common results derived by Zwikker & Kosten [START_REF] Zwikker | Sound absorbing materials[END_REF] averaging the velocity v across a tube and considering a linear pressure gradient along the tube, i.e. ∂p/∂z = (p exitp entr y )/T. These assumptions are also known as the short tube approximation. Building this model on established results, in particular Morse & Ingard [START_REF] Morse | Theoretical Acoustics[END_REF] for radiation impedance and Ingard [START_REF] Ingard | Absorption Characteristics of Nonlinear Acoustic Resonators[END_REF] for non linear impedance, Guess gives explicit expressions for the resistance and the reactance:

Θ = (z ν ) + 2π 2 σ R neck λ 2 where K 2 = -jω ν , (1)
and,

χ = (z ν ) -cot (k L) + ωδ σc 0 , (2)
where z ν is the non-dimensional impedance resulting from viscous and mass effects in the apertures:

z ν = - jωT σc 0 2J 1 (K R neck ) K R neck J 0 (K R neck ) -1 -1 . (3)
K is the Stokes wave number, ν is the kinematic viscosity, λ is the wavelength, k = ω/c 0 is the wave number, ω is angular frequency, c 0 is the sound speed, δ the end correction and J 0 and J 1 are the zeroth and first order Bessel functions, respectively, of the first kind (of complex argument). Thecot(k L) term corresponds to the cavity reactance. The present paper deals only with the linear regime, therefore, for the following comparisons to be consistent the non linear impedance is neglected. The end correction is defined as:

δ = 16R neck 3π 1 - √ σ 1 + 5 • 10 3 | u 0 | c 0 2 1 + 10 4 | u 0 | c 0 2 . (4)
The 16R neck /(3π) term is derived from Rayleigh's integral [START_REF] Pierce | Acoustics: An Introduction to its Physical Principles and Applications[END_REF] and model the radiation effect. The factor 1 -√ σ account for the interaction between multiple perforations. Finally the term featuring the acoustic velocity u 0 account for sound amplitude effects. Guess approximate equation (3) for the high and low frequency regime, namely for |K R neck > 10|:

z ν z νH = √ 2νωT σc 0 R neck + j ωT σc 0 + √ 2νωT σc 0 R neck , (5)
and for |K R neck < 1|:

z ν z νL = 8νT σc 0 (R neck ) 2 + j 4ωT 3σc 0 . (6)
T = T + 2R neck is a corrected length accounting for the viscous effects occurring outside the perforation. It may be noted that equation ( 5) is suited for macro perforated configuration. In addition, approximations ( 5) and ( 6) are not valid for the micro perforated configurations studied further below, for which 1 < |K R neck | < 10. Lack of information is given to model the impedance in this range of parameters in Guess [START_REF] Guess | Calculation of perforated plate liner parameters from specified acoustic resistance and reactance[END_REF]. Therefore, in this set of parameters, the model is completed by implementing the following linear law:

z ν z ν M = z νL (ω L ) + z νH(ω H ) -z νL(ω L ) ω H -ω L ω, (7)
where ω H = 100ν/(R neck ) 2 and ω H = ν/(R neck ) 2 .

III. Mode-matching model

The mode-matching model is based on three circular ducts, one interface between the exterior and the neck and one interface between the neck and the cavity. The exterior duct is semi-infinite and a unit plane wave with a normal incidence to the plate is considered. The propagation is modeled using Helmholtz equation in the absence of background flow and the following boundary conditions: ∆ + k 2 p = 0, ∇p • n = 0 on S ext , S cav , S ext,lat , S cav,lat and S bc , [START_REF] Ingard | Absorption Characteristics of Nonlinear Acoustic Resonators[END_REF] with p the pressure field and n the ingoing unit vector. The corresponding solutions in the (r, z) plane to Eq. ( 8) in the exterior, the neck and the cavity, respectively I, I I and I I I, is of the form:

p(r, z) h = ∞ m=0 A h m e jk h z m z + B h m e -jk h z m z Φ h m , with h = I, I I, I I I. (9)
m is the mode index and Φ m is a shape function. The number of modes in domain I and I I I is truncated to N. When modes are added on both sides of the plate, the radiation impedance is accounted for naturally. The impact of this parameter will be discussed further below in section V. As we are making the plane wave hypothesis in the neck, one single mode is accounted for in domain I I. The complex wave number used to model the attenuation in the neck reads:

k I I z = ω ρ(ω)C(ω), (10)
where ρ(ω) is the equivalent complex density and C(ω) is the equivalent complex compressibility defined by Zwikker & Kosten [START_REF] Zwikker | Sound absorbing materials[END_REF]. The velocities in domain I and I I I are determined using the Euler equation:

v = ∇p -jρ 0 ω , (11)
where ρ 0 is the air density. The velocity profile in the neck is taken into account in the MM model, the expression of v neck projected on the axial unit vector e z , derived from Stinson [START_REF] Stinson | The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape[END_REF], is:

v neck (r, z) • e z = ∂p ∂z 1 jρ 0 ω J 0 (Kr) J 0 K R neck -1 . (12)
The mode-matching method is implemented by enforcing the continuity of pressure and normal velocity at both interfaces [START_REF] Sergent | Optimisation géométrique du contrôle actif dans les gaines de ventilation[END_REF]. When matching the exterior and the neck domain, the weak formulation for the pressure continuity can be expressed as:

∀n ∈ N ∫ S nec k 0 p neck Ψ neck n dS neck = ∫ S ne c k 0 p ext Ψ neck n dS neck , (13)
with p neck the pressure field in the neck, p ext the pressure field in the exterior domain and Ψ neck n the n th test function corresponding to the neck modal basis. The normal velocity continuity condition needs to respect

v ext (r, z = 0) • e z = v neck (z = 0) • e z on S neck and v ext (r, z = 0) • e z = 0 on S ext \ S neck .
The corresponding weak formulation is expressed as follow:

∀l ∈ N ∫ S e x t 0 v ext • e z Ψ ext l dS ext = ∫ S ne c k 0 v neck • e z Ψ ext l dS neck , (14)
with v ext the velocity field in the exterior domain, v neck the velocity field in the neck and Ψ neck l the l th test function corresponding to the exterior modal basis. The same equations were used when matching the cavity and the neck domains. As mentionned above, a plane wave is implemented in the exterior domain. The ingoing plane wave with amplitude A I 0 = 1 is implemented as:

∇p • n = 2jk A I 0 e -jk L T + jk p on S in . ( 15
)
Using the rigid boundary condition on the back cover, a linear system is composed, giving access to the modal amplitudes A l m and B l m . Thus, the reflection coefficient is derived as

R M M = B I 0 /A I 0 .
Finally, the impedance is determined using the following expression:

z ω = 1 + R M M 1 -R M M . ( 16
)
The stream line confinement driven by the periodic layout of the perforations (also known as the interaction effect) is modeled implicitly by adjusting the exterior radius to fit the plate porosity. Figure 2 illustrates this phenomenon. This effect was also modeled by Melling [START_REF] Melling | The acoustic impendance of perforates at medium and high sound pressure levels[END_REF] and Fok [START_REF] Fok | Theoretical research of the conductivity of a circular aperture in a partition across a pipe[END_REF].

IV. Finite element method model

A finite element method model based the linearised compressible Navier-Stokes equations is introduced. We are modelling the impedance of the geometry illustrated in Figure 1. According to Roche [START_REF] Roche | Aircraft Fan Noise Absorption: DNS of the Acoustic Dissipation of Resonant Liners[END_REF], a 2D axi-symmetrical FEM domain is sufficient to capture correctly the physical phenomenon occurring in the linear regime without grazing flow.

In the following, L r e f is the physical distance which corresponds to a unitary distance in the mesh. We use L r e f = 2R cav . The physical quantities are marked by * and we define the non-dimensional as:

x = x * L r e f , u = u * c 0 , c = c * c 0 , ρ = ρ * ρ 0 , p = p * ρ 0 c 2 0 , T e = T * e c p c 2 0 , e = e * c 2 0 .
u is the velocity, c is the sound speed, ρ is the density, p is the pressure, T e is the temperature, c p is the specific heat capacity at constant pressure and e the specific internal energy. The following dimensionless numbers are introduced as well:

Re a = ρ 0 c 0 L re f µ 0 , Pr = c p µ 0 k T e . ( 17
)
Re a is the acoustic Reynolds number, Pr is the Prandtl number, µ 0 is the dynamic viscosity and k T e the air thermal conductivity. The following governing equations derived from [START_REF] Wendt | Computational Fluid Dynamics -An introduction: With contributions by John D. Anderson[END_REF] are linearised around ρ 0 , u 0 , p 0 , T e0 and µ 0 : where c v is the specific heat capacity at constant volume. The deviatoric stress tensor τ can be expressed as:

τ = 1 Re a ∇u + (∇u) T + µ B - 2 3 µ ∇ • uI , (18)
where µ B = 2/3 is the non-dimensional bulk viscosity and I is identity matrix.

An adiabatic natural boundary condition ∇T e • n = 0 with n the unit normal vector is enforced on every boundary. The free slip boundary conditions u • n = 0 are natural boundary conditions as well. Finally, an ingoing plane wave of amplitude W is enforced using [START_REF] Geuzaine | Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities[END_REF]. The reflection coefficient is derived from ( 15) as:

R F E M = e -jk L T W p -W e -jk L T , ( 19
)
where p is the pressure averaged over the upper boundary of the domain. We will now focus on the numerical aspects of this model. The mesh is composed of Delaunay triangular elements, generated using Gmsh [START_REF] Geuzaine | Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities[END_REF]. Figure 3 represents the mesh of configuration 3, of which the parameters are detailed in Table 2.

Figure 3 Mesh near the neck for configuration 3.

The mesh is refined on the domain boundaries and especially in the neck and its edges, i.e. where most of the dissipation mechanism are expected to occur. The numerical implementation is performed using GetFEM++ [START_REF] Pommier | GetFEM++. An Open Source Finite Element Library[END_REF]. A second order Lagrange polynomial is chosen to interpolate the velocity u. A first order polynomial is used for the density ρ and the temperature T e .

V. Model comparison

Four configurations have been chosen to perform a comparison between the three models described previously and experimental data. The corresponding parameters are summarized in Table 2. The measurements were carried out using a 29 mm diameter impedance tube with a 120 dB Sound Pressure Level (SPL). The frequency range is between 1000 Hz and 5000 Hz with a white noise source. The measured resonance frequency is noted f res .

Configuration Porosity σ (%) Radius R neck (mm) Thickness T (mm) Cavity height L (mm) f res (Hz) The Helmholtz number H e is introduced as:

1
H e = L re f c 0 f * , (20)
where f * is the physical frequency. The upper graphs correspond to the resistance and the lower graphs to the plate reactance. In the following comparisons, "Guess" refers to Guess model using equation (3) and "Guess approximation" refers to Guess model using equations ( 6), ( 5) and ( 7) depending on the value of |K R neck |. Results for macro perforated configurations are plotted on Figure 4 and 5. In this case, |K R neck | is between 16 and 37. Thus, we're using Guess high frequency approximation, applicable to macro perforated liners. When no approximation are made on equation (3) and no corrected length is added to account for viscous effects at the entrances of the neck, the Guess and MM predictions are alike. When the Guess approximation is used, a good agreement is obtained between model and measurements, showing the impact of the corrected length T . The resistance is underestimated by the FEM model and by the MM model. The plate reactance is overestimated on the upper part of the Helmholtz number range by Guess approximation, Guess and the FEM. The MM slightly underestimates the plate reactance. The convergence of the MM method is shown by the correspondence of the impedance using 10 modes and 50 modes. Results for micro perforated configurations are displayed on Figure 6 and7. |K R neck | is now between 3 and 7. Once again, the mode-matching provides similar results to Guess without approximations and underestimates the resistance for both configurations. Guess undervalues the resistance for configuration 3 while it is very accurate for configuration 4, highlighting the validity limit of the corrected length T . The finite element method is correctly predicting the resistance for both configuration 3 and 4. In the same way as the macro perforated configurations, the reactance is overestimated in the high half of the Helmholtz number range by the three models. A dissipation mechanism seems to be absent from the MM model. Therefore, we investigate the velocity profile in the middle of the perforation and at its edges for configuration 3. We compare the axial velocity obtained with the mode-matching and the norm of the velocity obtained with the FEM. This comparison is carried out for two locations, namely at z = 0 and z = -T/2. This is conducted for a 10 modes computation on Figure 8 and a 50 modes computation on Figure 9. The profiles are consistent with Stinson [6] narrow tube classification, i.e. when the viscous boundary layer is large compared to the tube radius. The profiles are plotted at 1270 Hz near the measured resonance frequency, i.e. 1392 Hz. The agreement between mode matching and FEM is globally good except near the discontinuity due to the corner of the perforation at r = 0.15 mm. This zone requires a certain number of modes to be described correctly by the MM model. Even though oscillations are still visible in the 50 modes case, their impact on impedance results appeared to be negligible when observing Figure 4. A velocity gradient is noticeable between z = 0 and z = -T/2 in the FEM. Indeed, at r = 0, we have

| v ext/neck M M • ez | | v ext/neck F EM | | v neck-mid M M • ez | | v neck-mid
| v ext/neck M M • ez | | v ext/neck F EM | | v neck-mid M M • ez | | v neck-mid
| v next/neck F E M |= 0.36 m/s and | v neck-mid F E M
|= 0.32 m/s. Such a velocity gradient is not observed in the mode-matching. Furthermore, it may be noted that Guess neglects this effect when using the short tube approximation.

VI. Dissipation rate

The dissipation mechanism about the neck appear to be modeled incompletely by the mode matching. With the intention of identifying the remaining losses, the rate of dissipation per unit mass due to viscosity is estimated through the previous FEM. With our choice of µ B , it's dimensionless expression is derived from Batchelor [START_REF] Batchelor | An introduction to fluid dynamics, Cambridge mathematical library[END_REF] as:

D (u) = 2 Re a 1 4 ∇u + (∇u) T : ∇u + (∇u) T - 1 3 Tr (∇u) 2 , ( 21
)
where Tr is the trace. The dissipation resulting from the acoustic boundary layer in the neck is clearly observable, and is modelled by MM through the Zwicker & Kosten losses. The dissipation mechanism in the corner zone seems to be significant though it is not explicitly accounted for in any model. On the other hand, the length correction proposed by Guess to account for viscous effects outside the perforation appears to be accurate, and to implicitly account for the named corner effect. A similar phenomenon is reported by Roche [START_REF] Roche | Simulation numérique de l'absorption acoustique de matériaux résonnants en présence d'écoulement[END_REF]. This cannot be taken into account in the MM plane wave modeling in the neck. That may explain the mismatch in terms of resistance between MM with both FEM and Guess.

We now integrate the dissipation rate in four different domain for all four configurations near their respective resonances frequencies. Figure 12 map the mentioned zones, situated around the aperture. Ω 1 corresponds to a domain surrounding the aperture. Ω 2 stands for the edges of the neck, where the dissipation mechanism displayed on Figure 10 is taking place. Finally, Ω 3 coincide with the neck region and Ω 4 corresponds to a region close to the wall, where a significant part of the losses occurs. For a better interpretation of the dissipation mechanism in the perforation, we introduce the acoustic viscous boundary layer thickness δ V (see Bruneau [START_REF] Bruneau | Fundamentals of acoustics[END_REF] for reference), which can be written as follows:

δ V = 2ν ω , (22)
Table 2 presents the relative contributions from each domain as a percentage of the total viscous losses in the FEM domain. For each configuration, the remaining losses apart from the four domains, are inferior or equal to 0.5 %.

Configuration The corners contributions is quite low when considering the micro perforated configurations 3&4, for which the 2R neck /T ratio is low. As the boundary layer thickness is large compared to the aperture radius, the dissipation is significant both in domain Ω 3 and Ω 4 . When 2R neck /T is closer to 1, i.e. for macro perforated configurations 1&2, the corners contributions is much more significant. In that case, the boundary layer thickness is small compared to the aperture radius, consistent with the fact that the dissipation is substantial in domain Ω 4 and minor in domain Ω 3 . Finally, the dissipation in Ω 1 constitutes a quarter of the losses of the micro perforated configurations. This value drop to 14.5 % and 7.7 % for macro perforated configurations, showing the impact of the porosity on the dissipation surrounding the aperture.

Ω 1 Ω 2 Ω 3 Ω 4 Frequency 2R neck /T δ V /R neck 1

VII. Perspectives

The edges effects appear to be more pronounced for macro perforated configurations than for micro perforated configurations. Therefore, the Zwikker & Kosten theory seem to be more adapted for the micro perforated case. In order to properly account for the dissipation on the corners, the modeling effort might need to focus on high order viscous modes in the neck of circular cross section. Alternatively, the losses can be completed using the empirical length corrections currently available in the literature, which is a reliable way to model the impedance of liners in a given range of parameters. However, this type of corrections are not adapted yet to unconventional liners, such as micro perforated liners or conical apertures liners. In recent years, theoretical studies have been conducted by Bendali et al. [START_REF] Bendali | Mathematical Justification of the Rayleigh Conductivity Model for Perforated Plates in Acoustics[END_REF] and Laurens et al. [START_REF] Laurens | Effective conditions for the reflection of an acoustic wave by low-porosity perforated plates[END_REF] providing the Rayleigh's conductivity for the named non standard configurations in the linear regime. Regarding conical perforations, Honzík et al. [START_REF] Honzík | Transfer function of small convergent-divergent (C-D) nozzles and opposite (D-C) devices filled with thermo-viscous fluid at rest[END_REF] has proposed a transfer matrix accounting for the visco-thermal losses in the neck. In a forthcoming work, it is intended to compare these theoretical results with measurements.

Figure 1

 1 Figure 1 introduce the modeled geometry composed of three cylindrical domains, namely the exterior (I), the neck (I I) and the cavity (I I I). A 2D axi-symmetrical problem is considered.

Figure 1

 1 Figure 1 Scheme of the modeled geometry.

Figure 2

 2 Figure 2 Scheme of the hole interaction effect.

Figure 4 Figure 5

 45 Figure 4 Impedance results for configuration 1, a macro perforated liner with a low porosity.

Figure 6 Figure 7

 67 Figure 6 Impedance results for configuration 3, a micro perforated liner with a low porosity.

Figure 8

 8 Figure 8 Velocity profile obtained through mode matching (10 modes) and FEM at the interface exterior/neck (z = 0) and in the middle of the neck (z = -T/2).

Figure 9

 9 Figure 9 Velocity profile obtained through mode matching (50 modes) and FEM at the interface exterior/neck (z = 0) and in the middle of the neck (z = -T/2).

  Figures 10 and 11 are a representation of D(u) through a colormap focusing on the upper corner and the neck at 1270 Hz near the measured resonance frequency, i.e. 1392 Hz.

Figure 10

 10 Figure 10 Dimensionless dissipation rate near the corner for configuration 1.

Figure 11

 11 Figure 11 Dimensionless dissipation rate near the neck for configuration 1.

Figure 12

 12 Figure 12 Scheme of the dissipation zones.

Table 1 Perforated plate configurations.

 1

		6	0.8	1.5	19	1712
	2	14	0.8	2.1	19	2464
	3	1.4	0.15	0.6	19	1392
	4	4.2	0.15	0.6	19	2512

Table 2 Dissipation rate contributions from the four domains.

 2

		14.5 % 46.7 % 3.7 % 34.7 %	1870 Hz	0.94	0.03
	2	7.7 % 38.6 % 1.0 % 52.5 %	2379 Hz	0.76	0.03
	3	25.3 % 8.7 % 30.2 % 35.2 %	1270 Hz	0.5	0.41
	4	25.3 % 10.5 % 25.6 % 38.2 %	2275 Hz	0.5	0.31

VIII. Acknowledgement

The authors gratefully acknowledge the support from ANR Chaire industrielle MACIA (ANR-16-CHIN-0002). This study is done in the context of a CIFRE Ph.D. Thesis financed by the ANRT and Safran.


L A Shah

A Etienne

A Siadat

F B Vernadat
email: vernadat30@gmail.com

Performance Visualization in Industrial Systems for Informed Decision Making

Keywords: Information Visualization, Value, Risk measures, Performance Measurement Systems, Decision Support

Information visualization is a key component of many decision support tools in sciences and engineering.

Graph is a visual construct that is widely used to model information for visualization. In this paper, a value-risk graph is proposed to visualize the results of value-risk based performance measurement systems (PMSs). The proposed graph for PMS divides the overall performance of industrial systems into distinct zones to facilitate the decision-making process. The upper bound, lower bound, and target value of each measure are decided by the performance evaluator and, then, transformed into normalized values using value theory principles. The aggregation of normalized measures along value and risk lines when combined defines "highly desirable", "feasible", "risky", and "unacceptable" zones. Scenario performance data when plotted on the graph visualize the overall performance of the system in terms of value and risk. The proposed decision-making value-risk graph is illustrated with an example dealing with manufacturing process design but it can be applied to any kind of system evaluation.

INTRODUCTION 1.

The last three decades have witnessed considerable development in the field of performance measurement and management. Models, frameworks, and methodologies have been developed to effectively measure the performance of organizations [START_REF] Bititci | Managerial processes: business process that sustain performance[END_REF][START_REF] Cocca | A framework to assess performance measurement systems in SMEs[END_REF][START_REF] Kaplan | The Balance Scorecard: Translating Strategy into Action[END_REF]. Their use was not limited to a particular discipline but employed across different disciplines such as manufacturing (Berrah and Clivillé, 2007;[START_REF] Vernadat | VR-PMS: a new approach for performance measurement and management of industrial systems[END_REF], civil engineering [START_REF] Bassioni | Performance measurement in construction[END_REF][START_REF] Kagioglou | Performance management in construction: a conceptual framework[END_REF], healthcare [START_REF] Shankar Purbey | Performance measurement system for healthcare processes[END_REF], public sector [START_REF] Van Dooren | Performance management in the public sector[END_REF], and others. This wide spectrum demonstrates the interest in performance measurement systems (PMSs). However, many surveys point out the disappointment of organizations with their PMSs [START_REF] Fletcher | Appraisal: Improving Performance and Developing the Individual[END_REF][START_REF] Unit | The new look of corporate performance measurement[END_REF]. There is still a feeling among academics and practitioners that the potential of PMSs is rarely exploited. [START_REF] Bourne | Reflections on the role, use and benefits of corporate performance measurement in the UK[END_REF] conducted a survey regarding the use of PMSs in the private sector. Conversely to prevailing belief, they found that PMSs are not yet widespread. From their conclusion, KPIs are by far the most commonly used measurement tools in the industry. A [START_REF] Kpmg | Achieving Measurable Performance Improvement in a Changing World: The Search for New Insights[END_REF] survey reported that the most common disappointment with regard to PMSs is the lack of data integrity and their inability to produce meaningful information to support decision making. The report then identifies better information quality and communication as a potential area for improvement. The issue of communication and comprehension of performance measures throughout the organization is often reported in the literature (Aki Jääskeläinen and Sillanpää, 2013;[START_REF] Wade | Corporate performance management: how to build a better organization through measurement-driven strategic alignment[END_REF].

To tackle the issue of performance measure communication and comprehension, visualization techniques can play a significant role. Visualization is a powerful tool to present elegantly multifaceted measures and to facilitate the decisionmaking process. Visualization refers to both visual or graphical representation and the cognitive process of understanding an image [START_REF] Mazza | Introduction to information visualization[END_REF]. In performance management and decision contexts, visualization is about the representation of performance related data, information, concepts, and knowledge in a graphical way to gain insights to make easier decision-making processes. Suitable visualization can simplify complexity of the perception process, accelerate perception and, thus, achieve a cognitive relief [START_REF] Adamantios | Management Information Systems for Enterprise Applications: Business Issues[END_REF]. Moreover, the visualization process can be applied to improve understanding over large data sets without complex quantitative methods (Aki-Jääskeläinen and Juho-Matias, 2016).

Despite clear potential, the use of visualization methods in management is still limited [START_REF] Al-Kassab | Information visualization to support management decisions[END_REF]. Although, it is gaining increasing attention in the field of performance measurement, the topic is still quite ambiguous and new in this field. Visualization is usually perceived as synonymous to dashboards. Much of the research in the field is focused on presentation formats of dashboards devised for balanced scorecards (BSC). For instance, Cardinaels and van Veen-Dirks (2010) investigated the various effects of tabular data in BSC on managerial evaluation. [START_REF] Banker | The judgmental effects of strategy maps in balanced scorecard performance evaluations[END_REF] analyzed the influence of presenting graphical information in BSC on evaluators' decision quality and concluded that the graph did influence evaluators' decision as predicted. Moreover, the addition of graph with tabular data increases the accuracy of performance evaluation [START_REF] Kagioglou | Performance management in construction: a conceptual framework[END_REF]. However, little effort is made so far to reduce the cognitive burden of performance evaluators by reducing the display measures without sacrificing the dimensions of performance. So, many measures on display will definitely require more cognitive effort to synthetize performance information.

The aim of this paper is to simplify performance evaluation and presentation by aggregating the measures along value and risk lines in the first place and, then, to propose a valuerisk graph for the purpose of performance visualization. To develop the indicators and the value-risk graph, the paper relies on the standard information visualization process.

INFORMATION VISUALIZATION PROCESS 2. IN PMSs

The process of information visualization [START_REF] Al-Kassab | Information visualization to support management decisions[END_REF] can be segmented into: (a) raw data collection; (b) data transformation; (c) data warehousing; and (d) visual transformation (Fig. 1).

Raw data collection

Data transformation

Data warehouse

Data visualization

Fig. 1. Information visualization process

This visualization process is used as a framework in the paper to guide performance indicators and value-risk graph development. The graph acts as a tool for performance display of the industrial system to meet the managerial requirement. Moreover, tools, methods, and techniques in each segment of Fig. 1 are identified and employed to support the performance indicator development in the visualization process. The following sub-sections briefly explain the segments of the process.

Raw Data Collection

First, raw data about a system or process need to be collected.

In the PMS context, raw data refer to variables or parameters of interest within the system understudy. For instance, in manufacturing and supply chain settings, the parameters of interest and to be measured can be "number of orders dispatched" or "number of orders fulfilled". However, they are variables and not performance measures per se, because they do not communicate any performance information unless they are placed in a business context. By combining the two variables in a ratio form, a performance measure of "perfect order fulfilment" can be developed.

To collect data for performance measurement, two methods can be employed: ECOGRAI [START_REF] Doumeingts | ECOGRAI-a method to design and to implement performance measurement systems for industrial organizations-concepts and application to the maintenance function[END_REF] and value focused thinking (VFT) [START_REF] Keeney | Value-focused thinking: a path to creative decision-making[END_REF]. The former method uses the triplet objective-variable-performance indicator and searches variables on which the decision maker can act to reach their objectives as well as performance indicators that measure the efficiency of the objectives attainment process. The VFT method, on the other hand, is an objective-driven performance measure determination and evaluation process, which uses value theory to compute the value of a system. In the current study, the VFT approach is used to model objectives and risks of the system.

Objective modeling: By definition, value is the degree of satisfaction and fulfilment of stakeholders' objectives. Data regarding value are identified by means of the VFT technique. VFT identifies objectives 𝑂 𝑖 ( 𝑖 = 1, … 𝑛) suitable for the process considered and defines measures 𝑀 𝑗 ( 𝑗 = 1, … , 𝑚) in-line with the objectives. For instance, in a supply chain context, a high level objective can be "To satisfy customer order". The lower level objectives are identified next by asking a simple question "What do you mean by that?". For example, the customer order satisfaction means "on-time delivery", "minimum cost", and "high quality". This process of objective decomposition continues until the upper level objective cannot be further decomposed. The identified objectives are then organized in a hierarchical form (Fig. 2).

Fig. 2. Objective hierarchy

The sub-objectives on the lowest level of the hierarchical structure are quantifiable and, hence, performance measures (or criteria) can be defined for each of them. For instance, for the refined objectives of maturity and recoverability, "the number of process errors occurred" and the MTBF (Mean Time between Failures) can be defined as performance measures, respectively. In the same manner, measures of interest are identified and listed as raw data of the visualization process.

Risk modeling: Similar to objective modeling, risk modeling starts by identifying a global risk which is the opposite of a global objective. For instance, if the global objective of a process is to satisfy customer orders, then the corresponding global risk would be: "failure to satisfy customer order". The global risk is refined and then arranged in the form of a risk hierarchy as shown by Fig. 3.

Fig. 3. Risk event hierarchy

Once the risks of the system have been identified, they are then analyzed with the failure mode, effect, and criticality analysis (FMECA) method as shown in Table 1. The critical risks in terms of RPN in the FMECA table are then considered for further processing and low RPN risks are eliminated from the visualization process.

In short, the outcome of this segment of the performance visualization process is a list of performance and risk measures needed to compute the value created and the extent to which it is exposed.

Data Transformation

By data transformation, we mean the normalization of performance measures into commensurate performance expressions. Performance measures are heterogeneous in nature due to the multidimensionality of the performance concept and it is often hard for an evaluator to reach a logical conclusion. For instance, metrics such as "order fill rate", "order cycle time", "on-time delivery", "cost per order", "productivity", and "system utilization" are measured in different units because they refer to different dimensions of a manufacturing system performance. The list can be more exhaustive if more aspects of the system are considered. A Skandia IC report [START_REF] Tan | The evolving research on intellectual capital[END_REF] uses 112 performance metrics to measure five areas of focus including financial, customer, process, renewal, and development as well as human resources. Making a decision based on so many measures is beyond the cognitive capacity of human agents.

To remedy this situation, one way is to normalize the measures using multi-attribute value theory (MAVT). The MAVT is a structured methodology to handle trade-off among multiple objectives (and thus performance measures/criteria) and to assign a utility value to each measure by rescaling them onto a 0-1 scale, where "0" represents the worst preference and "1" represents the best one. This transformation (mapping) of physical measure onto a [0, 1] scale is also known as value elicitation process and can be performed as follows [START_REF] Berrah | Monitoring the improvement of an overall industrial performance based on a Choquet integral aggregation[END_REF].

𝑃: 𝑂 × 𝑀 → 𝐸 (𝑜, 𝑚) → 𝑃(𝑜, 𝑚) = 𝑃
where 𝑂, 𝑀 𝑎𝑛𝑑 𝐸 represent universes of discourse, respectively of the set of objectives o, of the set of measures m and of the performance expressions 𝑃.

To obtain the performance expressions 𝑃, two finite sets M = {m 1, m 2 … m n } and A = {a 1, a 2 … a n }, representing measures and alternatives respectively, are considered. Then, a ∈ A and m ∈ M can be associated with a profile p i a = (p m 1 a , p m 2 a … p m n a ) where p i a represents a partial score of measure 𝑖 (performance/risk expression) in alternative 𝑎 on a [0, 1] scale. This partial score determination can be carried out using the MACBETH methodology. Details of the methodology for the value elicitation process can be found in the literature [START_REF] Cliville | Quantitative expression and aggregation of performance measurements based on the MACBETH multi-criteria method[END_REF][START_REF] Shah | Value-Risk based evaluation of industrial systems[END_REF].

Data Warehousing

One of the objectives of a PMS is to provide comprehensive and timely information on the performance of a process/system. To this end, data and information related to performance measurement and measures need to be centrally stored in a database; to be made accessible by different stakeholders of the system such as analysists or departmental managers either for decision making or performance improvement initiatives [START_REF] Kueng | A holistic process performance analysis through a performance data warehouse[END_REF]. In this paper, data warehouse refers simply to a collection of performance and risk expressions of the performance and risk measures identified in the raw data collection phase of the process. The expressions are structured in a tabular form and easily made accessible to evaluators for the purpose of decision making.

The authors don't apply any data modeling tool such as entity diagrams or UML classes to model the performance data.

Visual Transformation

Visuals are commonly used in lean manufacturing to visualize workflows, performance results, and opportunities for improvement. For this purpose, scoreboards are developed where key performance indicators are communicated in real time. However, one scorecard for individual performance measure requires space to present performance data and effort to make a decision based on scorecards. They are imperative at operational level. However, at the tactical and strategic levels, one needs a graph or two to encompass the whole performance visualization of the system. This is only possible if the unit performance expressions have been aggregated.

Aggregation:

The aggregation is formalized by the following mapping [START_REF] Berrah | Monitoring the improvement of an overall industrial performance based on a Choquet integral aggregation[END_REF].

𝐴𝑔 ∶ 𝐸 1 × 𝐸 2 × • • • × 𝐸 𝑛 → 𝐸 (𝑃 𝑚 1 , 𝑃 𝑚 2 , . . . , 𝑃 𝑚 𝑛 ) → 𝑃 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 = 𝐴𝑔(𝑃 𝑚 1 , 𝑃 𝑚 2 , . . . , 𝑃 𝑚 𝑛 )
𝐸 𝑖 is the universe of discourse of the performance expressions (𝑃 1 , 𝑃 2 , . . . , 𝑃 𝑛 ) and 𝐸 denotes the aggregated (𝐴𝑔) performance.

In our case, the aggregated function 𝐴𝑔 in the above mapping is a 2-additive Choquet Integral (CI) [START_REF] Grabisch | A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid[END_REF]. Mathematically,

C u (x) = ∑ v i p i - 1 2 n i=1 ∑ I ij |p i -p j | n i=1 (1)
where C u models vectors of performance expressions p i , v i denotes a Shapley index that represents importance of expression v i relative to all other expressions (with

∑ v i = n i=1
1), and I ij is the interaction between expressions (x i , x j ), ranging in [-1,1].

Construction of the Value-Risk Graph:

By computing the global value and risk indicators, it is now convenient to develop a two dimensional value-risk graph in the range of [0, 1] on the x-axis and y-axis.

To this end, the x-axis is divided into the three following ranges: Value indifference-It refers to the range of values which are not significant and for which the company will avoid pursuing the process; Value tolerance/acceptable-This range remains between the upper bound of value indifference, until the point where the value starts becoming desirable. In this range, the company may pursue the process; and Value desirable-Beyond the acceptable range is the desirable range, i.e., the company is willing to pursue the process. These ranges are drawn on the x-axis line as shown in Fig. 4.

Fig. 4. Ranges of values on the x-axis

In a similar way, the y-axis is divided into: Risk appetite-It refers to the risk an organization is willing to accept in the pursuit of process objectives; Risk tolerance-It specifies the maximum risk that the organization is willing to take in the pursuit of the process objectives; and Risk intolerance-It corresponds to the risk level not acceptable. The ranges have been drawn on Fig. 5 Fig. 5. Range of risks on the y-axis

In addition, it is proposed to develop a value-to-risk curve to model the acceptability of a process when the value progresses with regard to risk. In reality, organizations may take risk beyond the risk tolerance in pursuit of value creation for their stakeholders. For this purpose, a value/risk ratio can be defined that will be restricted within a minimum acceptable value and a point of risk beyond which an organization cannot afford to pursue its objectives in any respect. The aim of the value/risk ratio is to determine the upper bound for the acceptability of a process in pursuit of objectives fulfillment. However, its determination is still subjective and depends on the company's attitude towards pursuing its objectives (value creation) and risk taking.

By defining the ranges for measures and value/risk ratio, a Value-Risk Graph is developed as illustrated in Fig. 6. To determine the value and risk ranges quantitatively, a target, lower and upper bounds for each performance, and risk measure are defined. For instance, a process cost in the range of 10, 12, and 16 units will have a lower bound of 10, a target of 12 and an upper bound of 16 units, respectively. In this context, the lower bound is the ideal scenario while the upper bound is the worst case. Once the ranges for each measure have been defined, they are then normalized in the range of [0, 1] using the value elicitation technique and aggregated to obtain global indicators [START_REF] Shah | Value-Risk based evaluation of industrial systems[END_REF].

Fig. 6. Value-Risk Graph for decision-making APPLICATION 3.

To illustrate the process of performance visualization in a manufacturing company, a manufacturing case study is presented. The company designs and fabricates products on make-to-order (MTO) basis. The reference product for this case study is a mechanical locator (cf. Fig. 7), i.e., a work holding device used for centering a part on a machine-tool.

Fig. 7. Mechanical locator

To illustrate the use of the proposed visualization process within the company, a manufacturing scenario is defined.

Case study: Manufacturing Scenario

The company under study receives an order for 200 high quality mechanical locators from a customer with a lead time of two weeks (10 working days). The price of the product is kept at $14 per unit. The technical performance of the product is measured in terms of satisfaction index (q). Moreover, the company has sufficient resources at its disposal, therefore part axles, bodies, and caps are machined at the facility while springs and bolts are purchased from the market. We assume that all raw materials and the purchased parts are available whenever needed. Failing to satisfy the customer order will cause penalty cost. A product having satisfaction index below 0.8 is rejected. In case of delay after the tolerance period (2 days), the company has to pay $2/unit time tardiness up to 5 working days for each product; beyond this period, the order is cancelled and backlog cost of $10 per unit is paid.

Application of the process to the mfg. scenario

Let us apply the proposed visualization process to this scenario. First of all, raw data regarding performance and risk measures are obtained using the objective and risk models (cf. Fig. 2 and Fig. 3). The objective model is used to derive the performance measures: manufacturing cycle time (C 1 ), manufacturing total cost (C 2 ), technical performance (C 3 ), and employee satisfaction (C 4 ). From the risk model, major risk events relevant to time, cost, and quality dimensions that correspond to schedule risk (R1), cost overrun (R 2 ), and performance risk (R 3 ), respectively, are identified.

Before carrying out the normalization step, an upper bound (UB), lower bound (LB), and target score for each measure must be determined to define the ranges for desirability, acceptability, and indifference for the value axis as well as risk appetite, risk tolerance, and risk intolerance for the risk 2). For the cost and time objectives, the lowest limit is the desirable one while for the measures of employee satisfaction and technical performance, the upper limit is desired. In the case of risk measures, the lower limit is always preferred. [START_REF] Shah | Value-Risk based evaluation of industrial systems[END_REF]Clivillé et al., 2007).

The expressions and the identified parameters v i and I ij are put in Equation ( 1) to obtain a global score for both value and risk.

 Minimum acceptable global value = 0.39  Maximum acceptable global risk = 0.59 These values will be used as points of reference (cf. Fig. 8) to appraise the actual scenario understudy. Now, in the real scenario, three process plans for the mechanical locator are generated and evaluated in the simulation environment against the performance measures.

The results of the simulation experiments are obtained and tabulated in Table 3. To obtain the value and risk expressions with MACBETH, the process plans are ranked on the basis of desirability and strengths of preference for each criterion are provided as follows:

𝐶 1 ⇒ 𝐺𝑜𝑜𝑑 > 1 𝑃𝑃2 > 2 𝑃𝑃1 > 4 𝑃𝑃3 > 1 𝑁𝑒𝑢𝑡𝑟𝑎𝑙 𝐶 2 ⇒ 𝐺𝑜𝑜𝑑 > 2 𝑃𝑃3 > 1 𝑃𝑃2 > 2 𝑃𝑃1 > 1 𝑁𝑒𝑢𝑡𝑟𝑎𝑙 𝐶 3 ⇒ 𝐺𝑜𝑜𝑑 > 1 𝑃𝑃2 > 1 𝑃𝑃1 > 2 𝑃𝑃3 > 3 𝑁𝑒𝑢𝑡𝑟𝑎𝑙 𝐶 4 ⇒ 𝐺𝑜𝑜𝑑 > 0 𝑃𝑃3 > 4 𝑃𝑃2 > 1 𝑃𝑃1 > 1 𝑁𝑒𝑢𝑡𝑟𝑎𝑙
In the same way, information is provided on alternatives for each risk measure. This ordinal preference modeling is then transformed into normalized value and risk expressions with MACBETH and presented in Table 4.

To consolidate the expressions, the Shapley indices 𝑣 𝑖 and the interaction parameters 𝐼 𝑖𝑗 have already been determined in the calculation of global minimum value and maximum risk and presented in Table 5. The 2-additive Choquet integral model (cf. Equation 1) is then employed to aggregate the value and risk expressions as tabulated in Table 5. From the Value-Risk Graph, it is clear that only process plan PP1 falls in the desirable region for the scenario understudy. Therefore, the process plan PP1 is chosen to manufacture the product, while the scenarios PP2 and PP3 are dropped.

CONCLUSIONS AND FUTURE WORK 4.

The work reported in the paper addresses performance visualization and identifies the tools and techniques of the decision theory to be used for each phase of the process to develop global value and risk indicators. Based on the global indicators, the Value-Risk Graph is constructed that defines zones representing "highly desirable", "feasible", "risky", and "unacceptable" situations to make the decision-making process easier and clearly visible.

Future work should include the extension of the current Value and Risk approach by adding the Cost dimension to the existing Value-Risk Graph. To develop the global cost indicator, the same approach will be employed, and the cost dimension, then, will be added to the graph along z-axis making a cubic graph in order to provide all the essential elements for a robust and informed decision-making as requested by the BCVR (Benefit-Cost-Value-Risk) Methodology [START_REF] Li | Performance evaluation and decision support for industrial system management: a benefit-cost-value-risk based methodology[END_REF], which is partly based on this work.

Table 1 :

 1 Excerpt of FMECA process for "Schedule risk"

	Process Fail. mode Causes	Effects	P C D RPN
	Activity	Duration	Process	Uncertain	5 7 6 210
	i	estimation	-novelty	Lead time
		error	-complexity

Table 2 :

 2 Bounds and target on measuresValues of the measures (Table2) are transformed into normalized measures (also called expressions) via the MACBETH method, where the upper and lower bounds provide the good and neutral reference points in the value elicitation process. For details, see[START_REF] Shah | Value-Risk based evaluation of industrial systems[END_REF]. The outputs of the MACBETH tool are the following expressions for the measures: C 1 = 0.33, C 2 =0.4, C 3 =0.43, C 4 = 0.48, R 1 =0.57, R 2 = 0.67, and R 3 =0.55. Next, Shapley indices v i and the interaction parameters I ij I ij are determined as presented in Table5(For details, see

		C 1	C 2 C 3	C 4	R 1	R 2 R 3
	UB	24	18 100	100 4	5	1
	Target	20	14 80	75	2	2	0.3
	LB	18	12 75	60	1	1	0.2

Table 3 :

 3 Simulation results for three process plan models

		C 1	C 2	C 3	C 4	R 1	R 2	R 3
	PP1	20.5 16.6 0.87 0.94 0.715 5.57 3.96
	PP2	20.4 13.8 0.90 0.91 0.28 2.46 4.12
	PP3	22.4 12.4 0.78 0.74 3.6	0.415 3.94

Table 4 :

 4 Value and risk expressions for the mfg. scenario

		x 1	x 2	x 3	x 4	r 1	r 2	r 3
	PP1	0.63 0.1	0.5	0.14 0.12	0.98	0.66
	PP2	0.83 0.5	0.6	0.20 0.38	0.57	0.78
	PP3	0.18 0.7	0.11 1.0	0.88	0.14	0.56

Table 5 :

 5 Value and risk indicators for the mfg. scenario The interaction, for example, between C 1 and C 4, i.e. I 14 , indicates that mfg. time and employee satisfaction positively affect each other; however, their interaction is minimal. For the index, i.e. 𝑣 𝑖 , technical performance carries more weight (0.3) followed by mfg. time (0.26) while employee satisfaction is the least important of all (0.18) in the Value category. The aggregated scores for the process plans are plotted on the Value-Risk Graph as shown in Fig.8. Fig. 8. Value Risk Graph for the manufacturing scenario

		x 1 x 2	x 3	x 4	𝓥	r 1	r 2	r 3 𝓡
	PP1	0.1 0.63 0.5 0.14 𝟎. 𝟒𝟓 0.98 0.66 0.12 𝟎. 𝟓𝟑
	PP2	0.5 0.83 0.6 0.20 𝟎. 𝟑𝟏 0.57 0.78 0.38 𝟎. 𝟓𝟗
	PP3	0.7 0.18 0.11 1.0 𝟎. 𝟏𝟔 0.14 0.56 0.88 𝟎. 𝟓𝟔
	𝑣 𝑖	0.26 0.24 0.30 0.18		0.34 0.18 0.48
		I 12 I 13	I 14 I 23		I 12 0.090
	𝐼 𝑖𝑗	0.211 0.21 0.11 0.2 I 24 I 34		I 13 0.13 I 23 0.090
		0.159		0.158
		1
						Risky
						Adventure
						Feasable
						Zone
						Desirable
						Zone