Clément Chatras
email: clement.chatras@renault.com

Vincent Giard
email: vincent.giard@dauphine.fr

Mustapha Sali
email: mustapha.sali@dauphine.fr

High variety impacts on Bill of Materials Structure: Carmakers case study

Keywords: mass customization, Planning Bill of Materials, generic Bill of Materials, alternative services, automotive industry

The great variety of offerings from companies engaged in mass customization such as in the automotive sector, implies millions of end products. As in the case of carmakers, this variety proceeds from the combination of dozens of optional or alternative components. A number of technical or commercial constraints prevent most combinations of alternative components (ACs). In these conditions, the creation and updating of the full set of the Bills of Materials corresponding to each end-product is extremely complex. Additionally, customers may not define their product requirements by specifying a list of alternative components because of the number of alternative components (including some entirely unknown to them) and their interdependence. In the past 25 years, scant research has been dedicated to Bills of Materials representation problems in this context. The most exciting ones use predicates to formalize combinatorial constraints between ACs and introduce a 'generic Bill of Materials' concept to avoid an exhaustive description of all end products. Additionally, product description for customer purposes was streamlined through a further concept -the set of alternative services (SAS). This approach, implemented by several carmakers in the last fifteen years, superseded the traditional approach to bills of materials, though it never gave rise to the theoretical exercise described here that involves an analysis of its impact on bills of materials and on operational and tactical decisions. It describes the product based on its market features, before moving on to the actual ACs combinations based on predicates for alternative services (ASs). We show how some carmakers successfully use the AS concept to support sales at configurator level and address operational production issues.

INTRODUCTION

Over five decades, many organizations have switched from mass production to mass customization in order to better meet customer expectations. This has resulted in an explosion of potential offering variety that far outstrips actual annual production volume [START_REF] Pil | Linking Product Variety to Order-Fulfillment Strategies[END_REF]. This variety of end products is taken for granted for purposes of this paper.

The automotive industry is a perfect example of this context because a car is the result of the assembly of hundreds of components and few industries have pushed the envelope further in implementing mass customization [START_REF] Anderson | Agile product development for mass customization: how to develop and deliver products for mass customization[END_REF]. Accordingly, all the examples referred to in our paper are drawn from a simplified but real automotive case study, the data of which is used to illustrate some approaches set forth in the literature ( §2) as well as that adopted by several carmakers ( §3).

From a production point of view, variety is obtained on an assembly line through the combination of alternative components and a few physical postponement operations (such as painting…). We assume that an alternative component (AC) is a variant chosen from a set of alternative components (SAC) that may be used in the same assembly line station. Such variants can differ in terms of technology (engine…), shape (wheels...) or functionality (radio / radio with CD player…). Obviously, end products assembled on a given line share many components but those components that are systematically included do not impact the end-product variety. A variety analysis of a Renault final assembly line shows that vehicle production involves around 700 systematic components and as many SACs. The 700 SACs account for a total of more than 1900 ACs.

Through the commercial information at their disposal, customers cannot be aware of the underlying variety of the cars they consider purchasing. Indeed, a lot of SACs such as wires or alternators cannot be selected directly by the customer; such hidden components are indirectly linked to customer choices such as electrical equipment (air conditioning, heated seats, foldable mirrors, electric sunroof…). Therefore, neither can the customers define the vehicle they wish to acquire nor the sales departments the vehicles they offer on sale on the basis of a list of ACs. This observation determines the substantial difference there is between the approach we describe here and the literature on the subject.

All products that can be assembled are traditionally described with reference to a Bill of Materials. The Bill of Materials describes unambiguously a product's composition and is shared by all departments within the organization (production, design and manufacturing engineering, sales, management control, after-sales…) [START_REF] Garwood | Definition and evaluation of product configurator development strategies[END_REF]. Each separate product therefore, has a specific reference number in the Bill of Materials. The Bill of Materials used in production establishes the list and quantity of all components used by an assembly line to obtain a given end product. For some specific needs (commercial, after sales…), companies may amend Bills of Materials but they are all derived from Bills of Materials used for production.

Both the large variety of products and the existence of constraints between ACs generate two practical questions: i) how to create the exhaustive list of Bills of Materials for all products potentially assembled (and does it make sense to do so?)? and ii) how to link a customer's order (during its definition or during its production) to a unique Bill of Materials? To answer these questions one needs to analyze Bill of Materials structure in the information system in the context of mass customization. The aim of this paper is to answer these questions through a new diversity model representation already used by some carmakers that conciliates both commercial and industrial issues.

In the literature review (section 2) we shall first describe the traditional way of building combinatorial diversity into Bills of Materials; this solution presents problems that some researchers tackled by introducing the concept of generic Bill of Materials. But due to the current scale of diversity, this solution is no longer adequate. Under section 3, we develop an additional step observed in some current practices based on introduction of a different end-product description that reflects customer added-value. This we call "services" using a functional approach, rather than an organic approach directly based on components. This approach, which does not explain the BOMs, is used by several carmakers (e.g., PSA, Renault, Nissan, Daimler) but appears never to have been theorized so far. Because in the production phase, the organic approach is the only valid one, we also described the solutions required to derive the list of ACs needed for a vehicle from the description of the services it delivers. In this paper we show that this approach satisfies both commercial operational requirements satisfactorily and, in a separate communication made to this congress, that, subject to certain conditions, it additionally meets tactical requirements (MPS) better than the traditional approaches in view of the forecasting process employed by sales departments.

LITERATURE REVIEW

Our analysis of extent literature focused on papers dealing with the representation of diversity in BOMs. We broke it up into two parts showing how BOMs developed over time.

The traditional representation of combinatory diversity in bills of materials

The representation of combinatory diversity is traditionally based on planning Bills of Materials, also known as modular Bills of Materials (Stonebraker, 1986). The principle is simple: the Bill of Materials of an end product is a list of component references systematically included in the product and a list of dummy component references corresponding to the SACs used on the assembly line. For both real and dummy components, a Bill of Materials coefficient is used. In this system, every dummy component must point to the ACs of the relevant SAC.

To precisely define a specific car to be built, in each SAC, a single AC has to be selected.

In this context, describing an end product in a relational database (Date, 2012) is quite straightforward. The SACs correspond to entity types, the ACs making up the SACs. Systematic components can be grouped in a kit representing a specific entity type made up of a single entity. In this approach, the Bill of Materials may be described through an association of all entity types of the model whose key is a concatenation of the keys of ACs belonging to different SACs. Note that the combinatory constraints between ACs comprising different SACs introduce additional complexity in representing diversity.

Those constraints are technical and/or commercial. Technical constraints result either from physical interaction between components (constraints regarding interface, volume or performance) or from optimization measures in the purchasing strategy. Commercial constraints result from segmentation to establish consistent offerings and avoid cannibalization between ranges. Both types of constraints may reflect a pack rationale (compulsory association of services) or an exclusion rationale. Chart 1 taken from our case study illustrates the respective roles of both the technical and commercial types of constraints.

In order to describe the problem created by the restrictions, let's use a virtual car model whose diversity results from four engines (E1, E 2 , E 3 and E 4 ), two radiators (R 1 and R 2 ), three alternators (A 1 , A 2 and A 3 ), three heating ventilation and air conditioning systems (HVAC) (H 1 , H 2 and H 3 ) and three air conditioning interface panels (I 1 , I 2 and I 3 ). Excluding the systematic components and the other SACs, free of combinatory restrictions, this diversity of components yields different end-products. Chart 2 reads from bottom to top. It shows how the combinatory restrictions between ACs are taken into account step by step. Level 1 associations correspond to physical constraints, level 2 and 3 associations represent physical and commercial constraints. We note a legacy barred combinations of level n (mapped with 'X') with level n+1 (mapped with a blank cell in grey). As a result, the actual number of end products drops from 216 to 10 (marked with an "O" representing the allowed combinations at level 2) that can be summed up with the following Bills of Materials:

E1R1H1A1I1, E 2 R 1 H 1 A 1 I 1 , E 3 R 1 H 1 A 2 I 1 , E 2 R 1 H 2 A 1 I 2 , E 2 R 2 H 2 A 2 I 2 , E 3 R 1 H 2 A 2 I 2 , E 3 R 2 H 2 A 3 I 2 , E 3 R 1 H 3 A 3 I 3 , E 3 R 2 H 3 A 3 I 3 , E 4 R 2 H 3 A 3 I 3 .
Chart 2 illustrates the need to use the 5 th normal form of relational databases to describe the existing Bills of Materials. When the variety of end-products is low, as it is for Desktop computers, this approach seems valid.

Where variety results from the combination of dozens of SACs, however, this approach poses both the problem of defining hundreds of thousands of Bills of Materials describing product variety and the problem of its use in the definition of the customer's choices. In practice it is unrealistic to have a customer specify his product by choosing dozens of ACs (there are too many of them and customers are not even aware of some): another approach is required.

Recourse to generic bills of materials

Hegges and Wortmann (1991) first devised the solution of generic Bills of Materials, and attempted a summary representation of the set of Bills of Materials comprising an end product family. The generic BOM is based on a graph combining the systematic components and selected ACs from SACs. The SACs are labeled primary generic products, and the ACs are selected from parametered values. The solution cuts information redundancy but requires a prior explanation of the valid combinations between ACs. Van Venn and Wortmann (1992) complemented this approach by a translation of the constraints into predicates that reduce the number of ACs combinations. These predicates build on Romanos' suggested approach (1989). [START_REF] Olsen | A procedure oriented generic Bill of Materials[END_REF] then proposed recourse to programming language to describe the Bill of Materials of model variants thus making Bill of Materials more legible when restrictions are complex.

Before further analyzing the literature on generic Bills of Materials, let us discuss the use of predicates. A predicate is a logical expression combining a set of propositions that are either "true" or "false". The combination can be a conjunction, AND (noted ) or a disjunction, OR (noted ) and its result is also either "true" or "false". This logical sentence serves as a shorthand description of the constraints connecting use of an AC ("true" proposition) to the presence in the vehicle of ACs belonging to other SACs. Let ACi -1 be the predicate "true" where the AC i alternative component may be mounted on a vehicle and whose presence depends on that of a set of other AC j alternative components, chosen from several possible sets.

One shall also write AC j (=true) if component AC j is mounted on the vehicle. One can then write the three following predicates matching the conditions of use of the three alternators out of fifteen possible parts (each having a predicate) in the example taken in Chart 2:

1 1 1 1 1 1 2 2 1 2 3 1 1 1 2 2 2 2 2 2 1 3 3 2 2 2 3 3 3 3 3 1 4 2 (( ) ( )) ( (( ) ( ))) ( ) ( (( ) ( ))) ( ( ( ( ) ))
)

A E R H I H I A E R H I H I E R H I A E R H I H I H I E R E R - - - = ∧ ∧ ∧ ∨ ∧ = ∧ ∧ ∧ ∨ ∧ ∨ ∧ ∧ ∧ = ∧ ∧ ∧ ∨ ∧ ∨ ∧ ∧ ∧ ∨ ∧
In an article that aims to integrate data management of Bills of Materials and routings, [START_REF] Jiao | Generic Bill-of-Materials-and-Operations for High-Variety Production Management[END_REF] extended the Hegges and Wortmann's (1991) approach of generic Bill of Materials. They proposed a graph made up of nodes representing physical or abstract entities and lines between nodes to map conjunction constraints ("AND") and disjunction constraints ("XOR").

The abstract entity called a 'generic item', represents a set of variants (physical items) of the same type as ACs of a SAC. The graph is also made up of nodes between generic and physical items in which variety parameters are defined to select "XOR" the appropriate line. Configuration restrictions precluding free combination of variants are described separately by a set of logical restrictions (predicates) because mapping them in the graph proved too difficult.

∨ ∧ Chart 2 : Example of formal representation using the relational database model

The same year, [START_REF] Bertrand | Using hierarchical pseudo bills of material for customer order acceptance and optimal material replenishment in assemble to order manufacturing of non-modular products[END_REF] used a generic Bill of Materials concept called 'pseudo-item' to describe the relevant AC combinations based on given criteria. They proposed structuring the generic Bills of Materials on the basis of pseudo-items corresponding to a predetermined sequence of customer choices. Each customer choice is a specific value of the criteria related to the pseudo-items. The outcome is a graph describing the valid combinations of ACs but it seems only practicable in straightforward cases. Finally they suggested defining the MPS at pseudo-item level, which, again, is only feasible if there are not too many of them. [START_REF] Lamothe | An optimization model for selecting a product family and designing its supply chain[END_REF] have reengineered the generic Bill of Materials concept based on another set of abstract entities, called logical items. These reduce the variety based on explicit economic considerations. These logical items serve mainly to describe market segments defined by a set of needs and demand volume. Each need is translated into a hierarchy of service levels. A service level is matched by a unique indicator for a functional definition of the product that summarizes a set of criteria and serves to rank them. A market segment requiring at least one variant is characterized by the list {demand volume, minimum service level}. Each service level can be satisfied by one or more variants of the end-product. Authors proposed an economic model allowing to determine production variants meeting all market segment demand at the lowest cost.

Our review of the literature led us to the conclusion that academics who addressed the representation of diversity always did so with reference to the physical nature of components. Indeed, all of them seem to rely on a tree-like description of BOMs even though some of them introduce virtual levels. As a matter of fact, they implicitly share the assumption that a customer (just like the sales department when drawing up forecasts) defines an AC in each SAC. As already noted in the introduction, this approach does not address the operational and tactical issues of such sectors as the automotive sector due to the great diversity of its offering. [START_REF] Lamothe | An optimization model for selecting a product family and designing its supply chain[END_REF] explicitly introduced the commercial point of view in the definition of valid combinations. Their structuring approach is similar to that of commercial offering since a customer can view it through a configurator.

BILL OF MATERIALS STRUCTURE BASED ON SETS OF ALTERNATIVE SERVICES

A configurator [START_REF] Helo | Integrated Vehicle Configuration System-Connecting the domains of mass customization[END_REF][START_REF] Garwood | Definition and evaluation of product configurator development strategies[END_REF][START_REF] Trentin | Increasing the consumerperceived benefits of a mass-customization experience through sales-configurator capabilities[END_REF]) is a web-based interactive application available to potential customers to make a step by step definition of a customized end-product. The choices made at different stages can be restricted by previous choices. The configurator reflects the diversity that an average customer may be aware of. While one could design a configurator enabling customers to customize a product by choosing all the components, this solution nowadays is impracticable because too tedious and because many components require in-depth technical knowledge (more than 700 SACs used for a single car model).

The solution chosen by most configurators is implicitly based on the concept of alternative services (ASs). The ASs are grouped into sets (SASs) from which customers can only choose one AS. The SASs correspond to the logical items used by [START_REF] Lamothe | An optimization model for selecting a product family and designing its supply chain[END_REF]. ASs describe product functionalities open to customization options in a language customers can readily understand. For many years Renault, like most carmakers, opted for this approach based on commercial language (which customers can understand) rather than using an approach based on GBOM. Very early on, Renault has developed a detailed structure for this break up of BOMs into SASs [START_REF] Astesana | Constraint-Based Modeling and Exploitation of a Vehicle Range at Renault's: New Requests for the CSP Formalism[END_REF].

The French web configurator for Renault Twingo usually works this way, but there is no set sequence such as 'equipment level', then 'motorization and transmission ' to end with the various options (e.g., color, alloy wheels, air conditioning and parking sensors). The motorization is a SAS comprising all the main engine features (described in sales literature for example). It is coupled with a lot of other alternative

Chart 3 : SAC determination by SASs components (engine, turbo, filters, sensors, alternator…). The choice of motorization by a customer does not give access to those component variants as these depend on other parameters. This approach helps closely match customer expectations in terms of services leaving out actual product architecture and technical definition.

SASs are related to technical or commercial constraints similar to those between SACs in Chart 1. These commercial constraints have the same origin as those applying to SACs. Technical restrictions between SASs flow from the technical constraints between SACs. This legacy results from the fact that the choice of AS determines the choice of one or more AC(s) as shown in Chart 3 which is based on our real case study. Obviously, all these restrictions constraints between SASs can also be described by predicates, as further discussed below. Despite these restrictions, the diversity of customer offerings remains huge, for example, in France Renault offers millions of different Twingos.

Before describing the impact of AS choice on the determination of the ACs to be used in vehicle production, it is important to highlight the differences with the example of Chart 1 which illustrates the restrictions applying to alternative components. Chart 4 shows that the choice of AC is determined by the choice of one or more alternative services in as many SASs. We will use the term "determination" to refer to the causal relationship that links the choice of an AS within an SAS to the choice of one or more ACs. In this approach, it is important to remember that each SAC is determined by the choice of at least one SAS.

The determination relation, symbolized by arrows on Chart 3, should be explained. When just a few SASs determine the choice of an AC, it is possible to use tables to deliver an exhaustive description of the determination relationship. For example, the three tables (using our simplified real case) above illustrate the combinatory determination of the Alternator (which is an AC) from two SASs, i.e. motorization and cooling system. In the last table, the dashes stand for the exclusions inherited from the first two tables and the stars stand for additional exclusions flowing from the combination of two SASs in determining the SAC. In contrast with Chart 1, the choice of alternator is not based on physical SACs (Engine and Heating system) but rather on virtual commercial SASs: Motorization and Cooling system The configurator integrates the restrictions between SASs by dynamically restricting the set of possible variants for the SASs to the subset of variants compatible with previously selected ASs. In fact, a vehicle ordered by a customer is completely defined by the choice of a single AS from each SAS. The full list of ASs chosen cannot be seen as a vehicle Bill of Materials because there is not normally one-to-one relation between services and components. But the list explicitly and easily determines the set of level 1 components to use on the assembly line. To do so one may adjust the predicates discussed above to determine components.

So let now ACi -1 be the predicate 'true' if AC i component is used to assemble a vehicle defined by a set of alternative services AS i (phrase also used to define the need (AS j =true) for these services in this car). In these conditions, the three predicates previously used to define the choice of alternators versus other ACs may be replaced by the following predicates that relate component use to the choice of some alternative services:

1 1 1 1 2 2 1 1 2 1 3 2 2 3 4 4 4 3 1 3 2 6 3 5 6 ( ( )) ( ) ( ) ( ( )) ( ) ( ) ( ( )
)

A MO CS CS MO CS A CS MO CS MO MO MO MO MO CS A CS MO CS MO MO - - - = ∧ ∨ ∨ ∧ = ∧ ∨ ∧ ∨ ∨ ∨ ∨ ∧ = ∧ ∨ ∧ ∨
To assemble a vehicle, the assembly line uses alternator i where AC i -1 = true in the relevant SAC. This principle applied to all SACs serves to evaluate as many predicates as there are ACs in each SAC, where a single AC can be true because ACs of a given SAC are mutually exclusive. This exploration is not a problem in practice. The predicates referred to under section 3 describe valid combinations of ACs but, unlike those used here, they do not serve to define the AC to be assembled without referring to the entire Bill of Materials for this particular vehicle. Note that the alternator is one of those components customers are not aware of, although they are keenly interested in the motorization and the cooling system option. Thus, the ability to determine the alternator to be used simply by combining two obvious customer choices is a real benefit of this product description approach combining alternative services.

CONCLUSION

In the context of strongly diversified mass production characterized by multiple combinatory restrictions required by this diversity as regards possible alternative components, Bill of Materials construction and use pose daunting methodological problems. We argue that detailed Bills of Materials of all potentially manufactured products makes little sense due to the issue of finding the exact reference of an end product without the list of the relevant alternative components.

Table 1 : Example of alternator AC determination based on 2 SASs

Additionally, it is not feasible for customers to fulfill this condition when expressing their needs, due to the technical knowledge required and the multiplicity of alternative components of which they are unaware. Accordingly, describing an end-product using a combination of alternative services is the only viable alternative to the traditional Bill of Materials approach. This service-based product description, used by several automakers, involves multiple operational consequences.

-Based on services, the configurator walks the potential customer through the choice of SACs on offer. In these circumstances, the guaranty that the customized vehicle will be able to be built ('ready to buy') is obtained by the fact that upstream choices are guided by downstream choices through the dynamic cross referencing of restrictions between SASs. This approach therefore dispenses with the traditional Bills of Materials both at configurator and sales department levels.

-Describing a car through a set of built in alternative services serves to establish the ACs it is made of, through a number of predicates. This solution satisfies short-term practical needs that no longer require the set of all Bills of Materials one may produce.

-The transcription of the constraints by predicates based on alternative services instead of components simplifies their definition and updating since there are more commercial constraints than technical ones. This new representation of end products well addresses some BOM problems in mass customization context, but it has also a huge impact on the way MPSs can be drawn up beyond the frozen horizon. This particular point needs to be highlighted in a dedicated second INCOM paper [START_REF] Chatras | High variety impacts on Master Production Schedule: a case study from the automotive industry[END_REF]. Indeed, increased diversity results from marketing department pressure which assume that diversity, coupled with pricing differentiation is attractive for customers and profitable for business. This strategic vision of diversity combined with global alternative components procurement to lower costs has long been shared by large automotive companies worldwide. But in the aftermath of the 2008 crisis, one notes a paradigm shift in favor of streamlined offerings based on packages (no automatic air conditioning without CD player) or upgrading strategies (standard fitment of electrical rear window, embedded computer …). The introduction of packages serves to generalize the determination of a SAC through a single SAP, which streamlines the forecasting of demand for particular ACs. The low-cost rationale has also flourished in recent years and involves lowering service levels to customers, this strategy also enables the production of less diversified vehicles. This approach as used in the automotive industry appears perfectly valid for other industries that face similar diversity in their finished product offering, obtained with heavy combinatorial constraints between ACs, due to technical and commercial constraints

  Chart 1 : Commercial and technical constraints between SACsIt highlights the difficulties of taking into account the database integrity issues in connection with the combinatory restrictions between CAs in the description of a vehicle in the database.


Lamsade Lamsade

Clément Chatras
email: clement.chatras@dauphine.fr

Vincent Giard
email: vincent.giard@dauphine.fr

Economic variety control and modularity

Keywords: mass customization, variety control, standardization, diversity, modular product

Since the industry revolution, companies have tried to standardize more their components, in order to allow mass production and to increase components' commonality between products. To remain competitive, companies always have to offer more variety to customers. Manufacturers of complex assembled products that are facing the challenge of mass customization are forced to increase simultaneously commonality of used components and the variety of end products. Variety control is more than ever an issue but seems to lack a complete economic insight. In this paper we propose a systematic economic optimization for variety control taking into account the whole total delivered cost (TDC) including investments and operating costs. It is clear that modular product architecture is a cornerstone to allow end product customization at low cost. Modular architecture facilitates variety control; thanks to interfaces standardization and appropriate architectures, the diversity of alternative modules can easily be controlled. On the other hand the decoupling of modules will always be limited due to inevitable technical combinatorial restrictions between modules. After a description of the main concepts of modular product, we will propose an extent to the model of variety control in order to take into account these constraints between modules.

It is generally admitted that the competitive position of a manufacturing company is mainly based on the diversity of its products, thus making it possible "to stick as closely as possible to the demand" with respect to characteristics such as price, quality and availability. The selling price depends on the cost, which is not independent of diversity, as increased diversity is often a source of an increase in costs. In order to remain competitive, companies must control their variety. The design's rationalization of new products makes it possible to obtain lower costs for the desired diversity that the customer will observe (e.g., the diversity as yielded by a car configurator). The structure's rationalization of a set of components ensuring the same whole of functional needs can be achieved with the techniques of standardization that were developed in the middle of the 19th century; we will bring to them the economic light that they are currently lacking ( §1). The approach of achieving diversity through a combination of alternative modules (AMs) belonging to different sets of AMs (AMSs) mobilizes the concepts of platform and standardized interfaces. It is an approach that has been used for approximately forty years to enable the mass production of strongly diversified products. We will analyze ( §2) the characteristics of this approach in which the economic vision is weak and propose to extend the model developed in §1, to the modular approach.

Rationalization of production by using variety control and product standards

Variety control aims at limiting the diversity of the components assembled in complex products. This approach is devoted to the rationalization of the design of these products and is supplemented by the use of product standards for selecting supplied components and defining the characteristics of a complex manufactured good.

Control of the components variety

1.1 First we will make a recap of the initial method of variety control of parts, then we will consider the economic insight that is missing out and finally, we will discuss some extends of this first model.

The initial approach

The efforts of rationalization of production arose at various times, when attempting to produce a set of identical products such as the production lines of the galleys in Venice at the beginning of the 12th century [START_REF] Voss | Learning from the first Operations Management textbook[END_REF][START_REF] Ciciliot | Les chantiers navals en Ligurie du Moyen Âge à l'époque moderne (XIIe -XVIe siècles)[END_REF]. Manufactories were created to produce components or end products that shared the same morphological and functional characteristics. Within this framework, the produced components were interchangeable; this case may be considered as the first form of standardization.

. The problem of the economic relevance of the diversity of a set of components sharing similar functions quickly emerged. Unfortunately, the design of a new product often leads to the creation of a new component with the exact required specifications rather than to the use of an existing component that offers similar functionalities. The standardization of components constitutes a first stage in the rationalization of the production. To avoid increasing diversity, one can force the engineering center to consider using existing components. However, this solution presents the disadvantage of perpetuating a portfolio of alternative components whose composition may be economically not very efficient. It is then judicious to mobilize the techniques of standardization, defined here as the design rationalization of a set of partially interchangeable products created to meet a set of needs.

The modeling of the rationalization of the composition of a set of components that ensure the same functionality was proposed in 1877 by Colonel Charles Renard who was interested in the rationalization among cables of various diameters used for the stowing and constructing of captive military balloons. The excessive number of parts (425) posed serious logistic problems. The approach used made it possible to reduce the number to 17 to satisfy all requirements. This 1 solution rests on a relatively simple concept. In the posed problem, the unique functional characteristic of a cable is the maximum tensile strength Y that a cable of diameter X can withstand before rupturing. A test on cables of different diameters made it possible to construct Figure 1. It is then enough to cut out the y-axis in a certain number of intervals and to associate with any k y tension request the diameter associated with the upper limit of the interval that contains k y (i.e., in our example, a cable of diameter 2

x for any maximum tensile strength ranging between 1 y and 2 y ). ISO 3-1973, ISO 17-1973, SO 497-1973and ANSI Z17.1-1973). This structured approach presents two important limitations as no economic tradeoff is regarded in this rationalization and it uses only one criterion, which is characterized by a continuous variable, in the characterization of the component. [START_REF] Giard | Economical analysis of product standardization[END_REF]Giard ( , 2003) ) proposes a generalization of the approach to address these two restrictions. The term of alternative module (AM) is used here in preference to that of alternative component of similar functionalities. These AMs belong to a unique set (AMS) of AMs. It starts from an AMS J E of J n AMs, such as a set of gasoline engines. This set includes ′ J n existing AMs and ′′ ′ = -J J J n n n new AMs. A set of p criteria corresponding to functional needs is retained. These criteria can be quantitative (weight, power, height, etc.) or qualitative (standard used, assembly interfaces, etc.). For example, the first engine has a weight of 176 kg, it delivers a power of 90 hp and is Euro 6compliant (Euro 6 is an European emission standard for new cars), etc. The list of criteria is the result of a consultation of experts, when defining the needs to meet (see here after). The dynamic evolution of the requirements may lead to some new criteria. To satisfy these new needs, new alternative modules may be required. As they didn't exist yet, their functional characterization should be based on the expected functional request.

Introduction to an economic insight

Table 1: Functional characterization of the AMs of the AMS E j In addition, the set of m needs, which these components must satisfy is known. These needs are characterized by all the criteria introduced previously. For each quantitative criterion, a need is defined by a range of values (e.g., for the first need, the engine must deliver a minimal power of 80 hp and must weigh less than 150 kg). For each qualitative criterion, a need is defined by a list of qualitative items (e.g., for the first need, the engine must be Euro 6 -compliant). This information is the result of a consultation of experts, and can lead to certain revisions of specifications considered to be unnecessarily constraining. i d is the annual demand of the need i that is to be satisfied.

1 2 … l … p 1 2 … j … n J Criteria Alternative Modules E J 1 2 … l … p 1 d 1 2 d 2 … … i d i … … m d m Demands Criteria Needs 1 2 … j … n J 1 2 … i … m
Alternative Modules E J

Needs

The binary variable ij x takes a value of 1 if the AM j satisfies the need i. This need i is assumed to be covered by a unique AM j but this AM j can meet several needs. Note that it is useless to create a variable ij x if the corresponding parameter ij a is null; this limits drastically the number of decision variables. Equation (1) enforces each need i to be satisfied.

= ∀ ∑ 1, ij j x i (1)
The production of the AM j is ⋅

∑ ij i i x d .
The total annual delivery cost of a manufactured AM includes its production costs, the costs of the purchased parts that it includes in case of a new AM in project, an appropriate share 1 for the cost of design studies (which gives an economic advantage for the existing AMs). This total annual cost is easier to define if the AM is bought. In both cases, this total annual delivery cost is generally not proportional to the quantity produced (or bought) and can be regarded as linear for a quantity in the interval j k , and 0 otherwise. The production (or supply) of the AM j may be null and is impossible on more than one interval, which is enforced by equation ( 2).

= = ≤ ∑ 1 1 j j j j k K k k z (2)
The variables j k z and j k u are linked by the constraints (3), preventing more than one positive

j k u . -⋅ ≤ < ⋅ ∀ 1 , j j j j j k k k k k j M z u M z k (3)
The production, possibly nil, of the AM j can then be written as

= = ∑ 1 j j j j k K k k
u , which leads to the equation ( 4)

= = = ⋅ ∑ ∑ 1 j j j j k K k ij i k i u x d (4)
The cost function of a positive quantity j k u , assumed to be linear on the interval

j k is + ⋅ j j j k k k A c y .
As only one interval j k can be used, the cost function of the AM j is defined by relation ( 5).

= = ⋅ + ⋅ ∑ 1 ( ) j j j j j j j k K k k k k k A z c u (5)
The optimal composition of an AMS is that which minimizes the sum of the total delivery cost of the selected AMs, given by relation ( 6).

= = = = ⋅ + ⋅ ∑ ∑ 1 1 ( ) j j J j j j j j k K j n k k k k j k A z c u (6)

Extensions of the model

The previous formulation assumes that optimal decisions can be taken for any AMS independently from decisions taken for others AMSs. This is not always the case. In § 2.2, we will examine the case of physical constraints that may link AMs belonging to two AMSs or more. Here we take into account global constraints that the set of AMSs' optimal solutions must respect. Those environmental constraints may lead to downgrade the optimality of solutions obtained for some AMSs. The first aspect deals with positive or negative cost synergy (Giard 2003). The second issue relates of the concern to avoid a dramatic change in the existing solution. The last point is the introduction of the possibility to cover a need with several AMs.

Let's examine the cost synergies.

-If more than κ AMs, whose index run from 1 j to 2 j , are produced in the same plant κ → > -+ 2 1

( 1) j j , the annual total cost of procurement must be increased by the additional cost + Γ , which may correspond to a share of an investment used by all these AMs. Then the term γ +

+ ⋅ Γ (

) is to add to relation ( 6), where γ is a binary variable; the constraint (7) enforces γ to takes a value of 1 if at least κ of those AMs are produced.

κ γ = = = = < + ⋅ ∑ ∑ 2 1 1 r j i m ir j r j i x n (7)
-Conversely, if this production yields an economy of - Γ , the term γ - -⋅ Γ ( ) has to be added to relation ( 6) and the equation ( 8) enforces γ to takes a value of 1 only if this condition is met.

κ γ = = = = > ⋅ ∑ ∑ 2 1 1 r j i m ir r j i x (8)
The concern for avoiding a dramatic change in the existing situation justifies the presence of the existing components in the list of candidates to the selection. From this perspective, one can decide that based on the set of the ′ < n n AMs currently used (top of the list of the n candidates), one will retain at least ′′ ′ < n n AMs, which leads to the equation ( 9).

′ = = = = ′′ ≥ ∑ ∑ 1 1 j n i m ij J j i x n (9)
One can also replace this relation ( 9) by a constraint on a total volume of AMs to be kept or on a valorization of the total volume.

It may be interesting to cover the need i with several AMs rather than with a unique AM. Two remarks are pertinent to the scope of this approach.

-The reduction in the variety of an AMS is without interest when that set is created to offer a visible differentiation of the end product (design problem). This is the case, for example, regarding the hubcaps or the seats of a car. Thus, it is useful for those AMSs that the customer does not see or is not interested in, such as radiators.

-Some AMs can differ from others based solely on their interfaces with some AMs that pertain to other AMSs with which they will be assembled (gearboxes and engines, for example). The problem of the standardization of the interfaces is not sufficiently addressed in the approach described above. Indeed, this approach treats the rationalization of the composition of an AMS independent of other AMSs used in the same product range. The optimality of the decisions proposed pursuant to this approach with various AMSs must be reconsidered due to the need to create some junction components or because it is physically impossible to connect the considered AMs. Taking into account this interfacing is explicit in the modular architecture (see §3) and will lead to a broader approach of this economic rationalization.

The standardization 1.2

The International Standard Organization defines (ISO, 2004) standardization as an activity that "consists of the processes of formulating, issuing and implementing standards", and it defines a standard as "a document, established by consensus and approved by a recognized body, that provides, for common and repeated use, rules, guidelines or characteristics for activities or their results, aimed at the achievement of the optimum degree of order in a given context". The ISO adds two additional definitions, however. A product standard is defined as a "standard that specifies requirements to be fulfilled by a product or a group of products, to establish its fitness for purpose", and an interface standard is a "standard that specifies requirements concerned with the compatibility of products or systems at their points of interconnection".

These two last definitions explain the interest of standardization in the rationalization of the design of the products. Indeed, a complex product includes components or AMs designed and manufactured by the company (or by a subcontractor) as well as components or AMs that have been purchased. The technical description of the latter is accompanied by the mention of standards of products and/or of interfaces. Accordingly, the research department can then more easily select the components or AMs to be integrated in a new product, thereby limiting the risks of dysfunction of the end product. If the end products are sold as components to be integrated in an end product by another company and if the information that this component adheres to certain standards constitutes a selling point, these standards define a set of constraints that restrict the space of design and speed up the product design.

Rationalization from a modular perspective

Industries with scholars seek to manage efficiently the design, production, sales and updates of products that incorporate a degree of diversity to attain mass customization. In this way, the modular approach is a powerful tool for companies:

-First, because the division into independent modules allows for parallel designs, the total time for a design is reduced and modifications or upgrades of products can be implemented more often and at lower costs, thus facilitating innovation [START_REF] Baldwin | Managing in an age of modularity[END_REF][START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF][START_REF] Dahmus | Modular product architecture[END_REF], Pandremenos et al. 2009).

-Because the division can simplify the management associated with the diversity in production, the standardization between products may improve the robustness of assembling factories as they will be able to produce a greater variety of products on the same line and they will be less impacted by frequent product changes over time). Furthermore, independency between modules simplifies forecast management [START_REF] Lamouri | The two level master production schedule and planning bills in a just in time MRP context[END_REF][START_REF] Sali | Exploitation de la demande prévisionnelle pour le pilotage des flux amont d'une chaîne logistique dédiée à la production de masse de produits fortement diversifiés[END_REF].

In this section, we first introduce the concept used by the modular approach, and we then propose a generalization of our method of rationalization within the particular framework of modular architecture.

Fundamental Concepts

We define a modular system as a set of interdependent sub-systems or modules that are smaller than the entire system and can therefore be managed (designed and/or produced and/or sold and/or updated) independently [START_REF] Baldwin | Managing in an age of modularity[END_REF]. This independency results from a prior division [START_REF] Baldwin | Managing in an age of modularity[END_REF][START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF] ad hoc (unique for each system and each organization) of the system and the definition of decoupled interfaces [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF] between modules. [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF] adds that a module must correspond to, at the most, a unique function (but a function may be the result of a set of modules) to have an efficient division. Modular architecture permits the generation of alternative modules (AM) that will offer customization to the customer (Pine II 1993[START_REF] Baldwin | Managing in an age of modularity[END_REF]. A component, defined as a separable sub-system [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF], is a module only if it addresses a unique function and if its interfaces are decoupled, that is, if they permit the generation of alternative modules. In the following situation, we reserve the term "component" to sub-systems that are not modules.

The functions are intrinsic characteristics of a product that contribute to the global service provided by the product [START_REF] Ulrich | Product Design and Development[END_REF][START_REF] Eggen | Modular product development[END_REF]. The definition of a unique but complex function rather than a set of functional elements allows us to define modules at a more economically pertinent aggregate level. For example, in automobile one of the most complex modules is the powertrain. We can associate a unique function to the powertrain as it: permits the autonomous moving of a vehicle. In fact, this function could be decomposed through a functional tree whose first level would be composed of creating energy and transforming this energy into motion. This functional decomposition leads to precise functional elements that are traduced in the component or module (e.g., to create the spark in the cylinder, spark plugs).

Thus, there are certain relevant points to be made:

-Going deeper into the tree, the functional study is already making technical choices (localization of the engine in the vehicle, alimentation type, etc.).

-A consequence of the first point is that it reduces the possibilities for innovation (at level 0 or 1 it is possible to develop the electrical engine, but it is impossible to do so at the level of functional elements).

-These choices arise partially from choices the company has made in the past. That is, we can define a brand-new car using a new powertrain that is based on an existing engine.

-The boundary level between the component made and the components purchased defines the functional element level. That level evolves over time and is specific to an organization and sometimes to a specific product/manufacturer couple.

-Therefore, the definition of the modules is not a simple objective functional division, as it takes necessarily into account different life cycles, such as those of the product and the architecture, as well as the make or buy frontiers in the supply chain [START_REF] Campagnolo | The concept of Modularity in Management Studies: A literature Review[END_REF].

The definition of modules calls for the definition of standard interfaces. We define interfaces as any interactions between two components [START_REF] Chen | Interface strategies in modular product innovation[END_REF], Erens & Verhuslt 1997) -Environmental interfaces (effects between components such as thermal and magnetic)

The standardization of interfaces is a concept older than that of modularity. For example, the standardization has been used for decades in the construction of costless custom-made electrical and plumbing networks. Two objectives can lead a company to standardize its component (modules or not) interfaces:

-At the product range level, the objective is to reuse the modules developed for one product on other products [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF]. This approach, which aims to reduce diversity, is often proposed in the literature design as it reduces development time and the need for additional resources. This approach, however, runs the risk of cannibalization between products if we reuse a module that is visible to the customer [START_REF] Eggen | Modular product development[END_REF]. On the other hand, this can also be a marketing strategy, as is the case with the Ikea group.

-At a module level, it is easy to create diversity with respect to performance for a given function [START_REF] Sanchez | Strategic product creation: managing new interactions of technology, markets, and organizations[END_REF], 2002[START_REF] Baldwin | Managing in an age of modularity[END_REF], Pine II 1993). This approach, which aims to simplify the management and the creation of strong diversity, is present in the literature focused on mass customization. Our work regarding rationalization of the number of alternative modules is consistent with this use of modularity. Thus, a module can be described as a collection of AM for a sub-system associated with a function. Each AM represents a specific level of performance or a peculiar technical solution to accomplish the considered function, and accordingly, the standardization of the interfaces is the cornerstone of a good modular vision [START_REF] Sanchez | Modularity, flexibility, and knowledge management in product and organization design[END_REF].

The very early definition of the interfaces in a new project generally does not include their concrete design. In fact, the definition of the standard interfaces is the explanation of design rules and specifications that will serve as input constraints in the further designs of AM. The early definition of those standards, which must be stable over time, implies a huge expertise regarding the products and a solid understanding of their forward evolutions [START_REF] Erens | Architectures for product families[END_REF][START_REF] Chen | Interface strategies in modular product innovation[END_REF][START_REF] Eggen | Modular product development[END_REF][START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF].

The definition of the modules, based on their functional division and by the description of their interfaces, is a strategic activity for companies that is usually referred to as the architecture phase. [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF] defines the architecture as:

-the arrangement of functional elements, -the mapping from functional elements to physical components, and the specification of the interfaces among interacting physical components.

The definition of the modular architecture of a product is the moment when the functions that will be realized by the modules are identified. Indeed, it is incorrect to consider that a product is necessarily either modular or integral with respect to its architecture [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF]. Thus, the architecture defines the frontiers between components that will be developed using modular architecture from those that will be developed using integral architecture [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF][START_REF] Chen | Interface strategies in modular product innovation[END_REF].

As [START_REF] Erens | Architectures for product families[END_REF] emphasize, the architecture of a product facilitates the identification of the components (which may be modules) that will be chosen as stable, that is, without diversity. For example, when Renault develops a new car, despite the huge diversity of products generated by the customization decisions made by the customer, a substantial number of used components is common. This approach gives a first definition of the concept of platform. Indeed, the platform can be defined as the set or sub-set of stable modules or components within a range of products, including the interfaces [START_REF] Chen | Interface strategies in modular product innovation[END_REF][START_REF] Dahmus | Modular product architecture[END_REF]. Once the platform is built, it represents both the architecture of the product and a certain number of stable modules or components. Thus, a product family is defined as the set of products belonging to different market segments but sharing the same platform. [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF] describes 3 possible types of modular architectures:

-Bus type: The platform includes a physical support element on which all of the modules are plugged. The interfaces are standardized within the range of products and are common for all modules. While this approach is often used in personal computers, it seems unusable in the automobile industry.

-Slot type: The platform includes a physical support element on which all modules are plugged. The interfaces are standardized within the range of products but are specific for each module. This type of architecture is the historical automobile platform form of architecture. For us, the modular approach aims to exceed this vision.

-Sectional type: The physical support no longer exists. The modules are assembled to obtain the overall complex system. The interfaces are standardized within the range of products but are generally specific for each module. This architecture seems to be the one most commonly used in the automobile industry with the modular approach (for example, the gearbox is plugged on the engine that is plugged on the chassis).

With the first type, it seems easier to reach an independency between AMSs, because modules are essentially only interfaced with the platform. Thus, this architecture may be more compliant for the economic standardization model proposed above ( §1.1). With the second type, modules often have more interfaces (with more AMSs) and in practice remain technically dependent. For example all gearboxes will not be assembled with all the engines. Thus this approach calls for a generalization of our model which takes into account those constraints. We will propose it in §2.2.

Given the modular approach, the platform can appear at the level of a given product family, as a set of modules without diversity, or at the level of the range of products, as a group of alternative module sets. The concept of product family is then defined by the finite diversity of interchangeable variants for the rest of the alternative modules [START_REF] Dahmus | Modular product architecture[END_REF].

Modular architecture is based on module independency. This independency simplifies the use of the method of rationalization at the level of each set of variants of modules. In practice, this independency is difficult to achieve, and at best, it is limited to the application cases anticipated [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF]. Thus, the independency between modules is real only for a delimited range of use. Given that the technical and economical optimization is opposed to the absolute independency between modules, firms rightly do not seek to reach optimization [START_REF] Ulrich | The role of product architecture in the manufacturing firm[END_REF][START_REF] Eggen | Modular product development[END_REF]. All gearboxes are not compatible with all engines of a car manufacturer. Therefore, the rationalization of diversity within each module must, in practice, take into account the interdependency between modules.

Rationalization of the modular architecture

The approach described in § 1.1 is used without issue to rationalize the composition of an AMS having no dependent links [START_REF] Chatras | Methodological issues of the master plan schedule construction in mass customization[END_REF]) with other AMSs. When two AMs j and h pertaining to two different AMSs J E and H E are physically dependent (engine and gearbox, for example), these AMs can be assembled only if the interface between these modules allows for it. One can note that the AMS H E can correspond to a set of alternative platforms on which the AMs can be mounted. It is preferable to start with a set of elementary AMs (engine rather than powertrain) and take into account the postponement possibilities in the definition of these AMs. This interdependence implies an adaptation of the concept of need that cannot be defined any further at the level of an AMS, but rather at the level of a set of interdependent AMSs. For example, an independent list of 20 needs can be associated with the engines and another independent list of 4 needs can be associated with the gearboxes. The simultaneous determination of the engines and gearboxes to be used can be based on a combined list of 31 needs (less than the combinations without restrictions, 4x20), making it possible to obtain by aggregation the two independent lists. One preserves index i for this broader list with which the needs requirements i d are associated.

a . + ≤ + ∀ 1 , , , ij ih jh x v b i j h (11)
The cost function to be minimized is then the sum of all of the cost functions, defined by relation ( 6), of the AMS selected in these two AMSs.

= = = = ⋅ + ⋅ ∑ ∑ 1 1 ( ) j j J j j j j j k K j n k k k k j k A z c u = = = = + ⋅ + ⋅ ∑ ∑ 1 1 ( ) H h H h h h h h h n k K k k k k h k A z c u (12)
The coupling between the modules j and h not having the same interface can be possible due to a junction component with a unit cost jh e . Thus,

1 2 … j … n H 1 2 … h … n H 1 d 1 2 d 2 … … i d i … … m d m
Alternative Modules

= = = = ⋅ + ⋅ ∑ ∑ 1 1 ( ) j j J j j j j j k K j n k k k k j k A z c u = = = = + ⋅ + ⋅ ∑ ∑ 1 1 ( ) H h H h h h h h h n k K k k k k h k A z c u = = = = = + ∑ ∑ ∑ 1 1 1 J H j n h n m jh ijh j h i e u ( 15
)
The generalization of this approach to several AMSs physically dependent does not pose a particular problem of formalization as long as the number of dependent AMSs is low. The main problem lies within the definition of the needs as it must be common to this dependent AMS.

Conclusion

The mass customization is the paradox of modern manufacturing companies. Firms succeed in offering increasingly more products within an increasingly shorter time to market at an always lower cost. To overcome this challenge, a good understanding of the different levels of diversity is necessary. Indeed, diversity has different meanings because it can be measured at different levels of needs or at different levels in the bill of material. Therefore, companies want to find ways to maximize the offered diversity of the end product while they rationalize the diversity of components or modules that constitute the products. This rationalization goes through a product standardization approach for a given function. Modular architecture, which is based on a precise functional division of the products, permits not only the simplification of the generation and management of the needed diversity, but it also offers a strong framework for portfolio analysis, thus accentuating standardization. Given that modules are complex components, standardization based on a single parameter is no longer acceptable. In addition, the complexity of the overall products prevents companies from reaching the desired independency. As a result, the standardization of alternative modules cannot be performed freely without taking into account the choices of standardizations in other alternative module sets. In this paper, we first proposed a model of diversity rationalization that allows the optimization of the total delivery cost regarding a set of technical parameters. This approach takes into account the architectures and components developed in the past and can be easily generalized to take into account the dynamic evolution of demand. We propose an approach that extends the product standardization approach to a range of alternative modules sets with explicit compatibility restrictions.

This variety control approach should be introduced in the design phase during the range renewal. It requires economic information that is available only during a late stage in the development of alternative modules. Generally, especially with respect to the increase in modularity, at this later stage, each cell of the organization is highly specialized. This contributes to complicate the global approach that takes into account the interdependencies between modules. Therefore, the question of optimal organization, which permits greater control regarding relevant variety among the overall range of products, seems to be a fundamental question that future research should address. Numerous studies have emphasized that one of the main impacts of the modular product architecture is the functional division of the organization [START_REF] Baldwin | Managing in an age of modularity[END_REF][START_REF] Sako | Modular strategies in cars and computers[END_REF], Galvin & Morkel 2001, etc.). If, in a certain way, modular architecture facilitates product standardization, the consequences of the organizational trends on product standardization remain unclear.

Figure 1 :

 1 Figure 1: Rationalization of the composition of a set of components (Col. Renard)The implementation of this relatively simple idea poses the problem of the determination of the number of intervals and of their upper bounds. Empirical studies of Renard led him to split the same range of values (10 to 100) in different predetermined numbers of intervals whose upper bounds were defined in a geometric progression. Known as the Renard series, they are still used in industry under the name of the Internationally Standardized Series(Standards ISO 3-1973, ISO 17-1973, SO 497-1973 and ANSI Z17.1-1973). This structured approach presents two important limitations as no economic tradeoff is regarded in this rationalization and it uses only one criterion, which is characterized by a continuous variable, in the characterization of the component.

  of production (or supply) of the AM j. This maximum capacity may be increased ( → new maximum capacity (machines…). Its impact on the total annual delivery cost is a fixed cost to bear when the first j K M unit is produced. This fixed cost is a share of the investment cost, calculated as suggested above.

Figure 2 :

 2 Figure 2: Example of a total annual cost of procurement Let

  . To simplify the exhaustive determination of the interfaces, since 1994, Sanchez has differentiated five main types of interactions: -Attachment interfaces (how components are connected/plugged) -Transfer interfaces (how energy is transferred) -Control interfaces (how information signals are transferred) -Spatial interfaces (packaging constraints and precise localization)

Table 2 : Functional characterization and importance of demands to be covered by the AMS E j

 2

	The Boolean ij a takes a value of 1 only if the
	component j is usable to satisfy need i, and 0,
	otherwise. In our example, the engine 1 meets all
	the requirements of need 1 → (	11 a	= 1) only if the
	three criteria introduced previously are used

Table 3 :

 3 Ability of the AMs to meet the needs (Boolean parameters)

  Then,

	ij x becomes continuous	(0	≤	x	ij	≤	1)	because of
	certain constraints on production or supply
	capacities.
	At last, this formulation can also be adapted easily
	to go beyond the annual average requirements by
	introducing the forecast evolution for these
	demands, based on trade or technical reasons.
	Then	i d becomes it d ( < ≤ 1 t T ) and the cost
	function to be minimized becomes a sum of the discounted annual costs 2 . The AMS for the
	selection evolves over time, which makes it
	possible to introduce AMs that are not immediately
	available ( ij x becomes	) x and to gradually relax ijt
	the constraint on a set of current AMs to retain ( ′′ n
	becomes ′′′ t n where ′′′ 2 t n	≥	′′′ 1 t n , for	t	2	>	1 t ).

Table 3 :

 3 Ability of the AMs EJ and EH to meet the needs' (Boolean parameters) and needs' annual demandsThe introduction of the new AMS H E implies the creation of the binary variable ih v ,which takes a value of 1 if the AM h satisfies the need i, assumed to be covered by only one AM h (it can meet several needs). As previously noted, it is useless to create a variable ih v

								if the corresponding
	parameter ih a	is null, to limit the number of
	decision variables. Equation (10) enforces each
	need i to be satisfied.
		∑	h v	ih	= ∀ 1, i	(10)
	The production of the AM h is	⋅ ∑ ih i h v d .
	The Boolean parameter	b	jh	formalizes the
	dependence between J E and H E ; it takes a value
	of 1 if the AMs j and h can be directly coupled, and
	0 otherwise. To take into account the problems of
	interfaces, it is necessary to prevent the variables
	ij x and ih v (introduced only if	ij a	= 1	and	ih a	= 1	)
	to take a value of 1, if	b	jh	= 0	, it is enough to add
	the constraint (11), constraint, created only if
	ij a	= 1	and	ih	= 1

The discount rate must be the one used for calculating the annual share of investment in development studies of a new module or in capacity extension.