C Chatras

V Giard

M Sali

METHODOLOGICAL ISSUES OF THE MASTER PLAN SCHEDULE CONSTRUCTION IN MASS CUSTOMIZATION Abstract Id: SCML10356

Keywords: Alternative Modules, Planning Bill of Material, Mass customization, Diversity, Master Production Schedule, Product Configurator

In global mass customization supply chains, the geographical dispersion of plants complicates flow control. Because of the high product diversity, the MPS of the final assembly lines is defined for Alternative Modules (AMs). Analysis of the visible diversity, through a car configurator, shows restrictions in the combinations of AMs. Choices made by the customer from the visible diversity trigger choices in the non-visible diversity. Causes and formal representations of the relations between AMs are analyzed. The MPS construction rests on commercial forecasts, for few sets of AMs, and on production information. Their combination leads to a typology that decreases the number of MPSs to build. These MPSs can be defined either through the use of completely defined products, which implies a deterministic view of the MPS construction, or the direct use of planning bills of materials, which offers the possibility of managing, on stochastic bases, the production beyond the frozen horizon.

INTRODUCTION

Few companies have gone as far as those in the automobile industry in the application of mass customization. Even if the advantages for customers appear to be more or less obvious, the impacts of this strategy on the management of the Supply Chain's production part are not as clear. This paper, which uses the automotive industry as an example, analyzes one of the impacts of this diversity. From a production point of view, diversity is obtained in the assembly line through the combination of alternative (engines, gear boxes…) or optional (sunroof…) modules and through a few physical direct operations of differentiation (color…). Usually, customers are not aware of the whole diversity (alternator…). This diversity has several consequences in the production process. By influencing the products' design (modular design, sharing of platforms and components between different ranges of products), it determines the design and the capacity of the assembly line that must be able to support a certain diversity, even if it occasionally causes variability of operation time on some stations, which creates problems in sequencing orders. This sequencing leads to parts requirements for each station, which are necessarily more complicated for alternative modules. Within a longer time frame, diversity poses a problem of replenishment management in the Supply Chain's productive part, which is pulled by the assembly line. This paper addresses this last problem. Scheduling a lightly diversified production may be accomplished at the end-product level, as the limited number of end products allows the determination of a Master Production Schedule (MPS) from their corresponding demand forecasts. In production with mass customization, the variety of final products prevents a direct forecast at the level of the finished goods; demand forecasting is only possible at a more aggregate level, i.e., at the level of the alternative modules. This involves some methodological issues that will be analyzed in this paper. It is necessary to properly describe this diversity (section 2), which has an impact on the available information (section 3) that can be used to schedule the production of alternative modules and their required components with one of two possible approaches (section 4).

DIVERSITY ANALYSIS

The diversity that is observed by a person who is considering purchasing a car is presented by the configurator of the brand (defined below); this diversity allows him to explicitly choose the AMs that are valuable to him. The customer who goes directly to a car dealer will be implicitly guided in his choices the same way. This apparent diversity is only a part of the diversity offered. First, it is important to distinguish the visible from the nonvisible diversity ( §2.1). Then, types of restrictions that ban the free combinations of AMs have to be clarified before introducing formal representations that describe those restrictions ( §2.2).

Visible and non-visible diversity

A configurator is a web tool for customers that permits them to not only interactively define the end-product in accordance with their desires, but also to do so in a progressive manner because the choices they make at a certain point could be constrained by previous choices. For example, the configurator for T model1 usually orders the choices in this manner, but the order is not rigidly enforced: a level of the equipment, an engine and a gear box, plus various options (color, alloy wheels, air conditioning, parking sensors…). The equipment level indicates the level of complexity of services provided and plays a specific role in the determination of some AMs that are non-visible for customers; the concept of equipment level will be developed later. Visible diversity present in the T model's French configurator leads to a commercial offer that provides eight equipment levels, six engines, eleven colors, fifteen wheel rims and over fifty possible options. The actual diversity is not the product of cardinality of those sets as presented in [START_REF] Fisher | The Impact of Product Variety on Automobile Assembly Operations: Empirical Evidence and Simulation Analysis[END_REF]; in fact, the combinations that are permitted are highly limited. The following table, which is an extraction of available information in the T model's French configurator, illustrates these restrictions. Stablein et al. [START_REF] Vollmann | Manufacturing planning and control systems[END_REF] propose a formula to measure the diversity offered while taking into account the constraints between AMs. determines the AMs of some other AMSs, some of which are visible and some of which are not. This point will be developed further. • The commercial offer for the same range of vehicle changes from region to region. "Region" refers to a set of countries grouped by a same commercial offer that is described in a dedicated configurator. Often, all the offers overlap, but each region can offer some specific AM for commercial or regulatory reasons. For example, English drivers want the driver's side to be on the right, German pollution standards are stricter than French standards, the canvas roof is only available in Germany, and the smoker pack is only available in the United Kingdom. Generally, an assembly line produces vehicles for different markets. The AMSs used in the description of vehicles that can be produced on a line merge the AM lists that are associated with different regions; for example, the AMS "T model's roof" must include the AMs "sheet metal roof", "sun roof" and "canvas roof", which results from the consolidation of German and French AMSs. • The characteristics of the offer evolve with time for three reasons. First, some problematic components may be replaced. Then, range rejuvenating can lead to change in some AMSs. Finally, limitations on combinations between AMSs can evolve with time. For example, it may be decided that, for a certain period and equipment level, an optional air conditioning system should become standard. Therefore the diversity offered by a region is always a dated picture that is usable until a new change. This change in time is identified in the literature by the concept of dynamic diversity (Pil & Holweg, [START_REF] Pil | Linking Product Variety to Order-Fulfillment Strategies[END_REF]). In the following, these temporal aspects will not be taken into account. Later, the implications of the possibility of producing vehicles of different ranges (T and C models for example) in the same assembly line will be introduced. This is not a variety of the commercial offer but a variety of production, which complicates control of replenishments of the line if the production sharing between ranges is not stable. In addition to the visible diversity, there is diversity that is not visible by customers. Without being aware of it, the customer implicitly and progressively determines many AMs through choices in the configurator, especially by selecting the equipment level. For example, the equipment level AMS not only determines certain visible components (interior plastic quality, shape of the seat…), but also, most importantly, many non-visible AMs that are not shown in the configurator, such as wiring kits or alternators that are linked with the selected electric equipment (air conditioning system, sunroof's engine, electrical outside mirror or electric window…). These restrictions result from taking into account the constraints based on technical aspects (geometric, electric…) and commercial aspects (segmentations that permit a coherence of the offer and limit cannibalization effects between ranges). The examination of those constraints is a necessary first step for the analysis of their formulations.

Analysis and formal representation of relations between AMs

Three types of relations between AMs (and thus between AMSs) can be identified.

• Total independency with no restrictions; for example, parking sensors can be assembled on any T-model.

• With an injective relation, the choice of an AM in an

AMS that is visible in the configurator implies the choice of an AM in another AMS that may or may not be visible, although the reverse is not true. For example, the choice of equipment level influences the choice of the electric wiring kit required by the vehicle. • Compared to the previous relation, the bijective relation is symmetric. An example is the M2 engine, which requires the use of the automatic gear box that is only compatible with that particular engine. • In the "conjunctive" relation, the choice of a combination of AMs in different AMSs determines or limits the choice of an AM in another AMS. This can be illustrated by the choice of air conditioning system, which depends on the equipment level: the system is not permitted for the lowest equipment levels, is optional for mid-range equipment levels and is standard for higher ranges of equipment levels. The vehicle resulting from these constraints is technically coherent. These constraints can be taken into account only if they are transcribed in the Information System. The next paragraphs will examine how to transcribe the constraints. In the formalism of relational databases, AMs are grouped into entity types, and each AM is associated with a unique AMS that belongs to the AMS entity type. The analysis of the dependency relations identified above leads to first introducing an "is-a" relationship and the notion of role in order to realize functional dependency between two AMs in the case of injective relations (as it is usually done in the description of the relationship that links an "incuding component" with an "included" component). The bijective relation can be solved the same way. Conjunctive relations can be managed with an association type whose key is a concatenation of the keys of AMs that belong to different AMSs; this association leads to a functional dependency that points to a unique AM of a different AMS, or to a multivalued dependency if it concerns a sub-set of AMs that belongs to the other AMS. The possibility that the AMS belongs to several different associations poses the problem of the combination of AMs, which may lead to semantics errors, as in the problem regarding the fifth normal form of the relational databases. In practice, without general rules it is better to take into account integrity constraints through associations that respect the fifth normal form. However, the keys for the different associations that result from taking these constraints into account may be too long for practical use. Bertrand & al. [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] had investigated solutions for this problem of formally representing those integrity constraints using the concept of pseudo-items, defined as a group of interdependent AMSs, to obtain an independency between items (pseudo or not). Searching the independency between all items leads to pseudo-items that are composed of almost all of the items; thus, this approach is not as effective as is needed in the car industry. A second approach can be used to formalize those relations. Based on Knowledge Engineering, it calls for predicates to describe conditions of requirement of the AMs that belong to non-independent AMSs. For example, let's introduce three AMSs { }

= ∧ ∨ 1 7 1 1 2 ( ) C A B B .
This predicate is a reverse Bill of Material and is similar to the pegging approach that is well-known in MRP use. Even if this method to describe restrictions is correct, it is very difficult to use in the studied context because of the length of predicates. These two approaches allow the handling of integrity constraints due to commercial offer restrictions. They are fully usable only at the car level by defining the envelope of the authorized combinations. As explained later, the planning process must use the concept of the Planning Bill of Material (PBOM), which is determined by the structure of MAs observed in a set of vehicles produced previously or to be produced in the near future.

INPUT INFORMATION AVAILABLE FOR PLANNING

As explained previously, production planning cannot be based on commercial forecasts expressed at the endproducts level because of the too large variety of endproducts. Thus, it must be based on forecasts at the EMAs level or at a combination of EMAs level ( §3.1.). This information must be combined with production information about the BOM and some Supply Chain characteristics (3.2.). Combining commercial forecast characteristics and production information leads to the definition of an AMs typology that makes it possible to define an MPS for a decreased number of AMSs ( §3.3).

Commercial forecasts

Sales management, after first agreeing with production management, gives its forecasts in terms of the volume of production for an assembly line and for a period. Then, it states, for the same period, forecast structures of demand that can be regarded as forecasts of the Planning Bill of Material (PBOM). These forecasts only address certain visible AMSs that are considered to be important for customers (practical or emotional value) or for manufacturers (because of value of modules and depth of the Supply Chain used). They take into account trends in the evolution of demand, the launching of new models on the market and the impact of special marketing practices of the manufacturer or its competitors (limited series, special offers…). The forecasts are assumed to respect integrity constraints between AMs considered in the analyses. For other AMSs, which are much more numerous, only historical data and information about the integrity constraints are available to create the corresponding MPS.

Production information

Usually, requirements of a highly diversified production, for the middle term calls for a PBOM, whose usage is not evident in the studied context (A). Taking into account certain Supply Chain characteristics limits the necessity of precise planning (B).

A. Planning Bill of Material

PBOMs are used in MRP to solve problems with alternative modules that require components with a lead time longer than the frozen period. This method is based on the use of a generic BOM, which associates each AMS to a virtual component that points to the list of AMs that it includes; this association has a BOM coefficient that represents the demand part of the AM in the considered AMS. If constraints between AMSs exist, the PBOM cannot be used directly without care. Four remarks should be stated about the determination of the PBOM coefficients: • PBOMs do not take into account integrity constraints that are defined at the vehicle level (as explained at the end of §2.2). However, the BOM coefficients must respect some numerical constraints resulting from the indirect accounting of those integrity constraints. For example, if the choice of the i M engine implies the use of the j GB gear box, the BOM coefficient for j GB cannot be lower than the i M coefficient.

• The PBOM is defined for interchangeable lines (or a set of interchangeable line in a factory) because it depends on the satisfied demand structure. If the same line has to supply the demands of different regions, as it was stated in §2.1, it is better to use consolidated AMSs. Therefore, the PBOM coefficients to be used for an aggregated AMS are necessarily defined for an AMS whose AMs are the merge of all of the AMSs lists of each region ( §2.1) and whose coefficients are the weighted sum of the regional BOM coefficients multiplied by the part of the region in the whole production. • The same production line could assemble vehicles belonging to different ranges (T model and C model) if those vehicles share the same platform and many components. Thus, a station of the line may assemble AMs belonging to two AMSs associated with two ranges, as long as both AMSs have the same main function or share some MAs. If the share of production between the two ranges is stable, those coefficients are easily determined by using the method explained in the previous paragraph. For the contrary case, the problem is more difficult, and the PBOM defined for each range has to be conserved in order to be subsequently used in conjunction with a productionsharing ratio between the two ranges. These hypotheses can be determinist or stochastic (in this case, the Monte Carlo method must be used to determine the random demand of an AM). In addition, in this case of a random share between two ranges, the demand of assembled systematic components that are not used by the two ranges becomes random. In this paper, only the case of a line dedicated to a single model or with a constant share between two ranges will be considered. • Postponement decisions are strategic; they bind shortand middle-term decisions as planning ones. Their impacts on the definition of the PBOM are substantial.

The engine factory can send to the assembly line engines that are customized and already include the alternator chosen for the required electrical power and filters that depend on the regulations of the country where the car is sold. It is also possible for the engine factory to deliver "bare" engines to the assembly factory, which is now responsible for assembling the components of the engine differentiation. In this case, the engine BOM is reduced and the assembly factory has to acquire its supplies of alternators, filters… It increases the number of AMSs managed by the assembly factory and, thus, the BOM length. In order to illustrate this idea, for the T-model, there are four "bare" engines and thirty eight customized engines. If postponement is impossible, it is better if the engine factory defines the "bare" engine, which is a transient stage in the process (it cannot be stocked), as a phantom component in the BOM in order to use the more reliable forecasts that exist at the "bare" engine level.

B. Supply Chain characteristics

Requirements' control of the AMs assembled on a line does not automatically involve the use of a MPS for each AM. Taking into account the Order Penetration Point (OPP) and characteristics about the cost and bulk of the AM bulk can lead to certain AMs being excluded from the planning process. The supply of the assembly line is always the result of previous purchase orders that had been taken to a tier one supplier. The vendor's production is called "build to order" if it aims to fulfill a certain demand that is known only from the ordering date and is regarded as "build to stock" in the opposite case. Build to order can be managed by a synchronized supply that often involves an assemble-toorder configuration (seat's supply, for example) or a synchronized production (for example, bumpers' production, if the vendor is near assembly line). The OPP defines both the frozen period, during which production orders are known and cannot be modified, and the lead-time, which is the time between the purchase order date and the delivery date. The supplier necessarily produces with a build to stock configuration if its lead time is longer than its customer's OPP. A tier one vendor often has many customers (even if it belongs to the same company because it may deliver to plants dispersed all over the world; for example, this is the case of engine factories); therefore, it usually simultaneously builds to order and to stock. Purchase orders that can be fulfilled by build to order production may not need the knowledge of precise requirements and be controlled by a periodic order-up-to level policy or by an ordering point inventory model. This choice depends on the optimization of the products' cost and bulk, of packaging constraints and of global cost of stock and data processing. A lot of AMs (stickers, outside mirrors' hulls…) can fall into this category.

AMs typology usable for differentiate requirements' management

The mix of commercial and production information leads to suggest an AMs typology that reduces the scope of requirements planning and, thus, limits the number of AMs MPS needed. Within this framework, it is necessary to reintegrate all the assembled systematic components whose demand only depends on the production's total volume forecasted by the sales management. Then, the following should be distinguished:

• Independent AMs that are directly steered by sales management forecasts or those considered to be equivalent because of their relative independency from the sales forecast. • AMs whose demand is deduced from independent AMs. These AMs could be visible (in the configurator) or not. This information could be combined with information deduced from the OPP and with information about certain SC characteristics to potentially isolate AMs whose requirements management could be autonomous. The average shares in value of the assembled components on the production line that are in the following table have been observed for the T-model.

Table 2: Typology of AMs

The definition of the MPS, necessary a priori for all AMs, can be restricted to subsets B and C, with harder methodological difficulties for subset C. The subset concerned with autonomous management must use an inventory model that limits the shortage to an acceptable level, while considering the imprecision of the parameters used.

POSSIBLE MECHANISMS FOR ALTERNATIVE MODULES PLANNING

Starting from the double observation that AMs do not combine freely and that sales management forecasts are made only for certain AMSs, it is obvious that the construction of the MPS for other AMSs poses a real methodological problem. This is true even if the problem is narrowed to AMSs whose lead time is longer than the frozen period. Two solution methods are possible. The first consists in transforming sales management forecasts to forecasts on the Completely Defined end-Products (CDP) that are in accordance with the integrity constraints of the relations between AMs in order to construct an MPS for each AM ( §4.1.). The second method consists of directly using PBOMs, with some precautions ( §4.2.).

CDP use

With respect to the technical and the commercial dated constraints, a CDP is one of the possible combinations of AMs that represents all of the AMSs, excluding certain accessories of personalization (stickers, roof rails...) that are deemed irrelevant for the supply management. A CDP has no physical existence, but it is the reflection of what could be a real finished product; it may be qualified as a virtual product. By construction, a CDP meets all of the constraints imposed by engineering (technical constraints) and sales management (business constraints). This property gives to the CDP the same consistency as the products that are actually sold.

The coherence properties and the tangibility of the CDP are reassuring for supply chain managers. Their use for middle-term planning is seen as a simple extension of short-term scheduling. The use of a CDP guarantees consistent component requisitions under the product nomenclature through the use of inverse nomenclatures. If the use of CDP is essential for short-term scheduling, a planning with CDP beyond the frozen horizon gives the illusion of dealing with a certain forecasted production. Unfortunately, the stability conditions are not met in the middle-term. Changes in production programs with rolling planning cause disturbances that generate a bullwhip effect in the upstream-supply chain (see Childerhouse et al. [START_REF] Childerhouse | On the impact of order volatility in the European automotive sector[END_REF] and Niranjan et al. [START_REF] Niranjan | Measuring information distortion in real-world supply chains[END_REF]). The instability of production programs, without affecting systematically used components (set A), generates a loss of effectiveness and efficiency in the supply management of components of sets B and C. The information provided by the sales department, in the form of demand forecast patterns on certain AMs (similar to PBOMs), is insufficient to allow supply management for the components of set C. In fact, the use of the components of set C in the assembly line depends on the partial or the finished product configuration. The principle of CDP planning is desegregating the sales department demand by progressively combining the AMs that are directly covered by the PBOMs. The combination of AMs of set B enables the identification of the induced AMs for which there is no initial information. The desegregation of the commercial demand for constructing a CDP is completed in each period of the planning horizon under a strict compliance with the technical and the business dated constraints. The MPS that is described at the zero level of nomenclature is then exploited in a classical MRP calculation. While seeking strict consistency within the meaning of the product nomenclature, CDP planning is not without consequences on the performance of the upstream supply chain. Inherent rules within the desegregating process of the sales demand can cause serious disturbances, which propagate along the upstream supply chain. These disturbances, which ultimately have similar consequences on the flow control beyond the frozen horizon, are twofold. -Sales demand deformation: If the successive steps of the sales demand desegregating process contribute to distort the initial demand, i.e., the baseline information on future demand, this can generate a production schedule that is inconsistent with production orders reassembled by the commercial network. A modification of the MPS is necessary, triggering a vicious circle described by Childerhouse et al. [START_REF] Childerhouse | On the impact of order volatility in the European automotive sector[END_REF]. Sales demand deformation is a "static" distortion observed at time t for the same period of the planning horizon. -Rolling planning instability: It happens when the production program that is generated at different execution cycles of the desegregating process exhibits large differences despite a stable sales demand. Rolling planning instability generates difficulties to suppliers. This "dynamic" distortion is bound to the periodic use of the sales demand desegregating process. The analysis of the sales demand desegregating process for the CDP planning in a European car manufacturer has revealed the existence of the two phenomena described above. Starting from PBOMs provided by the sales department on a limited number of AMs, the solution adopted by the manufacturer to generate CDP planning is the use of mathematical programming methods to solve a series of optimization problems for each planning period. Two properties, inherent to the problem of CDP generation, are causing the deformation of the sales demand and the rolling planning instability. -As illustrated in table 2, two coherent planning BOMs at the Equipment Level and Engine are not sufficient to derive average utilization rates for the Equipment Level / Engine combination. We face an underdetermined mathematical problem with, for this example, 14 equations (number of AMs with a planning bill known) and 16 unknowns (number of possible combinations). To be solved, this problem is transformed into a tradeoff between the planning BOMs compliance (marginal sums) and the forecast rates' proximity to permissible combinations (these rates are generally derived from historical data). This arbitration is intended to distort the coefficients of the PBOM because the strict respect of the sales demand is not a constraint. -In a rolling planning, the robustness of the CDP generating process is measured according to its ability to provide a stable result from one planning cycle to another. The higher the sensitivity to changes of the sales demand desegregating process, the greater planning instability. A small change of input data causes increased nervousness in the MPS.

Direct use of PBOM

This section will deal neither with the planning of the component systematically used, nor with the planning of alternative modules in a made to stock configuration. For other AMSs and beyond the frozen period, the MPSs are defined by a direct exploitation of PBOMs without using the determinist vision of CDP. The demand of those AMs is considered random and the requirements' control is based on a periodic replenishment. The AMS PBOMs coefficients are used as requisition probabilities of different AMs of the AMS and, for a given AMS, the daily need of the AMs follows a multinomial distribution. Two cases must be distinguished. For the AMs in group B, for which sales department gives forecasts, the requirement's control of the AMs -and of the components they require -does not pose a problem, as an extension of the MRP approach permits either a make to order or a make to stock configuration to be used to manage the affected upstream part of the Supply Chain (Giard & Sali, [START_REF] Giard | Pilotage d'une chaîne logistique par une approche de type MRP dans un environnement partiellement aléatoire[END_REF]). Different benchmarks used to test this approach in comparison to the CDP approach show relevant gains in efficiency (safety stocks reduction) and effectiveness (stock-out reduction).

For the AMs in group C, a similar control to the one used for group B is difficult. Here, even if the above approach were still coherent to use, the problem is to define which PBOMs to use. It seems possible that if the forecast structural characteristics given by sales department are similar with those observed in the past, it is possible to use a PBOM defined from recent historical data. On the contrary, different solutions are possible: -

The simplest one consists in uprating a PBOM coefficient (used as probability); the limit is that it is impossible to know if the rating is sufficient (with an excessive shortage risk) or too high (with a useless safety stock level). If the value of the concerned AMs is not too high, this solution may be used with sufficient ratings. -Another possible solution consists in using historical data and optimal multiple linear regression techniques to find relations between one deduced coefficient of an AM and the coefficients of potential AM inductors whose sales management forecasts are known. Benchmark studies between this approach and the CDP approach, conducted with approximately ten AMs used on the T-model, show a relevant superiority of this approach in both effectiveness and efficiency. Complementary comparative analyses must be completed in order to test the robustness of the proposed solutions to establish an MPS of the AMs of group C.

CONCLUSION

Requirements' control of enterprises oriented in mass customization is improvable and investigations must be conducted in the framework of the approach of the direct use of PBOM. To succeed in providing a correct offer to the market, the enterprises protect them with a considerable safety stock or by recurring emergency transport solutions. Those actions are in a curative way that is based too often, in arbitrages made, on local data that is disconnected from the initial sales forecasts. The improvement of requirements' control, the object of this article, is a matter of preventive logic that uses the best forecasting information and lessens the importance of curative approaches. The problem analysis is more complicated when an assembly line is shared by different products ranges whose daily production is not well known in advance. It is possible to use a revisable hypothesis about this sharing to establish MPSs. Within the framework of the second solution analyzed above, it is possible to combine this source of hazard with those already taken into account using the Monte Carlo method. Thus, the demand of systematic components used by only one product range becomes random. This increase of complexity militate in favor of an anticipation of the production partition rates between products ranges. In practice, numerous methodological problems to efficiently steer requirements of a plant dedicated to mass customization have not yet been solved.

Table 1 :

 1 Example of the limitation of diversity

			Equipment level
	Engine Gear box E1 E2 E3 E4 E5 E6 E7 E8
	M1	Manual X X X X . X . .
	M2	Auto	. . . X . X . .
	M3	Manual	. . . . X . . .
	M4	Manual	. . . . . . X X
	M5	Manual	. X X X . . . .
	M6	Manual	. . . X X X . .
	At this point, AMs should be defined. An AM is a
	component chosen from an Alternative Modules Set
	(AMS), a group of AMs in which all members have the
	The definitions of AMs and AMSs proposed above are only
	applicable to physical objects. They must be extended to
	permit the description of the production variety as it is on
	an assembly line; this requires three generalizations:
	• The equipment level can be regarded as an AMS,
	because a vehicle can only be associated with a
	single equipment level but it is virtual. Its function is
	two-fold: it limits, for technical and/or commercial
	reasons, combinations of AMs that belong to different
	AMSs (this is what the above table illustrates), and it

same set of functions. AMs of a single AMS are assembled in the same station of the line. Optional components (sunroof, parking sensors, embedded computer…) are particular cases of an AM in which, implicitly, the alternative of the optional module may be a physical object (sheet metal roof, bumper without sensors…) or nothing at all (absence of the embedded computer…); in both cases, from a descriptive point of view and to stay general, it is possible to introduce a fictive AM representing the non-use of an optional module. Physical, direct operations of differentiation, such as color, are not included because they do not have consequences on the upstream part of the Supply Chain.

In order to respect confidentiality this paper uses generic names for products and brands


Vincent Giard
email: vincent.giard@dauphine.fr

Mustapha Sali
email: mustapha.sali@renault.com

Monitoring of the upstream part of a supply chain dedicated to customized mass production with a revisited version of MRP

Keywords: MRP, Make To Stock, Safety Stock, Supply Chain, Order Penetration Point

This article focuses on the monitoring of a supply chain dedicated to the mass production of strongly diversified products. Specifically, we are interested in the part of this chain that contributes to the production of a set of alternative modules assembled on a work station of one or several assembly lines, whose production levels are stable. The MRP approach is adopted for the monitoring of this chain. The distance between the production units leads to a mix between production to stock and production to order. In this article, we establish the relations that allow us to define, in a steady state, the quantities to produce that address the requirements of the Master Production Schedule and that are partially or completely random to limit the stockout risk to a very low predetermined level. We will distinguish two cases by accounting for, or not accounting for, problems that are related to quality.

Introduction

In the real world, for Supply Chains (SCs) dedicated to the mass production of strongly diversified products that are characterized by a certain geographical dispersion of the production units of the SC, production is determined by several distant final assembly lines with a stable daily production (at least in the course of a few weeks). Diversity is provided mainly by the assembly of alternative modules (e.g., engines, gear boxes) on several workstations of the assembly line. Each station is dedicated to a different set of alternative modules, of which one must be mounted on the end product processed by this station. The optional modules (e.g., the sliding roof) can be treated as a specific case of the alternative modules. Several examples of this type of SC can be found in the automotive industry.

When the total daily production volume is certain, the daily requirement for all of the systematically assem-bled parts and the components that they use is also certain. The control of these flows is relatively easy in the absence of risk related to the quality of the components, the transportation time or the production time.

After a brief review of the literature ( §1), a revisited version of the MRP will be proposed ( §2) and then numerically illustrated ( §3). In conclusion ( §4), we will describe the conditions that must be guaranteed for applying this approach efficiently.

Literature review

MRP was initially proposed to be implemented in a deterministic environment; no securing method was initially imagined. The confrontation of this idealized perception of planning with the real world highlighted the necessity of dealing with the uncertainty of certain information that is used within the framework of the MRP model. At the end of the 1970s, attempts to adapt the MRP method to an uncertain environment were suggested.

The first source of uncertainty that was studied was related to the demand. To address this concern, [START_REF] Büchel | An overview of possible procedures for stochastic MRP[END_REF] models uncertainty with random variables and calculates, on each level of the BOM, the variance of the demand for the components, leading to the possibility of calculating safety stocks. The problem of the components that are common to several end-products is managed by considering that their demands are random variables whose variances are the sums of the variances of the requests. The implicit assumption of demand independence does not hold if it concerns a component that is common to a subset of alternative modules belonging to a given set of alternative modules mounted at the same workstation of the final assembly line. Indeed, it is shown below (see §2) that the probability distribution of the demand for a component common to several components of a subset of alternative modules is a weighted sum of binomial distributions; this relationship invalidates the property of the variance additivity.

Based on a case study, [START_REF] De Bodt | Lot sizing and safety stock decisions in an MRP system with demand uncertainty[END_REF] evaluate the effectiveness of the use of safety stocks in MRP with uncertain demands. Their level and localization result from a simple comparison between several scenarios: no determination method is presented in this article. [START_REF] Winjgaard | MRP and inventories[END_REF] describe the principles and advantages of the use of the safety stock, of the safety lead time and of the increase in the quantities of the MPS. A dimensioning method for the safety stocks is provided for serial, convergent and divergent configurations of the supply chain. The authors only consider the case of components that are related to the end-product by a BOM coefficient equal to 1 without accounting for the commonality. [START_REF] Lagodimos | Optimal positioning of safety stocks in MRP[END_REF] examine the problem of the localization of the safety stocks in the MRP of standardized products. Their objective is to maximize the service level by distributing among the various levels of the BOM a given value of safety stock by using an analytical approach. It is considered here that the service level is not a decision variable and that the holding costs are identical for each level of the BOM. This assumption is strong insofar as the value of a component usually decreases with its BOM level. The proposition goes in the direction of a placement of the safety stock on the level of the end-products for the serial structure of a supply chain. Nevertheless, the authors relativize their results by pointing out the limits imposed by the assumptions of their modeling.

Under the assumption that demands are usually distributed in the steady state, [START_REF] Inderfurth | Safety stocks in multi-stage inventory systems under different service measures[END_REF] are interested in the double problem of the dimensioning and the localization of the safety stocks of components. The goal of this study is to reach a given service level (size and duration of stockout) at each level of the supply chain while minimizing the overall carrying costs. They propose a model of optimization that respects a service level that ensures the decoupling of the supply chain units. The requirements of a component come from the final MPS demand and could be related to one or more different end products. In this last case, correlations between demands are introduced to model the requirements of the studied component. This approach is not valid if that component is mounted in different alternative modules that belong to the same set of alternative modules.

On the basis of the holding cost and service rate, [START_REF] Zhao | Evaluation of safety stock methods in multilevel material requirements planning (MRP) systems[END_REF] compare the performance of three dimensioning methods for calculating the safety stock. Accounting for the results realized by (Carlson & al., 1986), they decided to place the safety stock at the level of the end product. For each of the three methods, a different demand cover time is used for the calculation of the safety stock. No justification is given to defend the choices.

In a more recent article, [START_REF] Persona | Optimal safety stock levels of subassemblies and manufacturing components[END_REF] propose an analytical method for dimensioning safety stocks of components or modules in the case of a production or assembly to order. In their modeling, they consider a steady state in which the procurement lead times are certain and the demands for alternative components are correlated within the same family of alternative modules. The concept of correlation used in this article refers to the statistical link that exists between the demands relating to alternative modules of the same set j E . This relation is the result of a constraint based on the multinomial distribution and is not the result of a correlation between random variables. Correlations exist between alternative components that belong to different sets, but they are only important in the case of a component that is used to link two alternative modules that belong to two different sets of alternatives (junction components). [START_REF] Winjgaard | MRP and inventories[END_REF], [START_REF] Guerrero | The dynamics of hedging the master schedule[END_REF], [START_REF] Buzacott | Safety stocks versuss safety time in MRP controlled Production systemss[END_REF], [START_REF] Molinder | Joint optimisation of lot-sizes, safety stocks and safety lead times in an MRP system[END_REF] and [START_REF] Chang | The interchangeability of safety stocks and safety lead time[END_REF] explored other strategies for protection against uncertainty in the MRP environment such as raising the requirements of the MPS or lengthening the lead times artificially. [START_REF] Buzacott | Safety stocks versuss safety time in MRP controlled Production systemss[END_REF] are interested in the possibility of using safety lead times as a means of protection against uncertainty. They consider that a tradeoff must be made between safety stocks and safety lead times according to the reliability of the forecasts, the production capacities, the holding costs and the stockout costs. They consider in their modeling only one MRP echelon operating in an uncertain environment. The conclusion from this study is that safety stocks reduce the total costs in the event of strong uncertainties on the demand. The safety lead time has the ability to anticipate a known demand to come and necessarily leads to an increase in a stock at a level that depends on the period length.

In a similar study, [START_REF] Molinder | Joint optimisation of lot-sizes, safety stocks and safety lead times in an MRP system[END_REF] proposes rules for decision making to help choose between safety stocks and safety lead times. The parameters of the decision refer to the degrees of uncertainty of demand and lead times and the ratio between the stockout costs and holding costs. Its conclusions are somewhat different from those made by [START_REF] Buzacott | Safety stocks versuss safety time in MRP controlled Production systemss[END_REF] in that he recommends the use of safety lead times in the event of strong uncertainties of both demand and lead times. One can note that the cost that is associated with a safety lead time is induced by an increase in the stocks and that consequently, the opposition between the safety stock and the safety lead time is not very clear.

Exploring other ways for dealing with uncertainty in MRP, (Zhao & al., 1993) propose to freeze the master production schedule in a high proportion according to parameters related to the planning horizon and the replanning periodicity. This solution, conceivable in a make to stock context of standardized products, is unusable in a make to order production of diversified products.

The majority of literature review articles relating to MRP have been published in the 2000s. To clearly highlight fields not yet explored by the specialized literature and to clarify the positioning of our work compared to what has already been accomplished, it is interesting to check the analysis framework used by these literature reviews. [START_REF] Guide | A review of techniques for buffering against uncertainty with MRP systems[END_REF] are interested in the techniques used to counter uncertainty in MRP. The analysis framework comprises the following seven axes: the modeling process, the dimensioning method for the safety stocks, their localization, the definition of the safety lead time, the type of uncertainty accounted for in the model, the nature of the planning horizon (rolling or fixed), and finally, the key performance indicator that is retained. The examination of the literature in this article leads to the conclusion that several gaps still exist even if certain suggested solutions remain applicable, according to the authors, under a set of assumptions that are sometimes debatable. These authors also regret the lack of methodology and the absence of precise rules that allow for the establishment of a compromise between the various suggested alternatives for protection.

Another categorization of the literature is proposed by [START_REF] Koh | Uncertainty under MRP-planned manufacture: review and categoriztion[END_REF], in which the analysis is performed on three axes. In the first axis, the authors distinguish between the inside risks that are related to the processes and the outside risks. The second axis is related to the point of occurrence of the risk, which can be upstream or downstream in relation to the productive process. Last, the third axis relates to the type of solution that is recommended to mitigate the risk through safety stocks or techniques, such as an increase in the quantities of the MPS. The interesting conclusion from this article is that very few studies address the interactions between several sources of risk.

More recently, [START_REF] Dolgui | Supply planning under uncertainties in MRP environments:A state of the art[END_REF] proposed an analysis of articles that relate to the management of the risk in MRP. The articles selected are classified into three categories based on the type of the risk taken: the demand, the lead time and a combination of both. The authors briefly present some of the techniques that are commonly used to control the consequences of uncertainty, namely the frozen horizon, safety stock and safety lead times. Aside from the limitations highlighted by this study and those identified by [START_REF] Guide | A review of techniques for buffering against uncertainty with MRP systems[END_REF] and [START_REF] Koh | Uncertainty under MRP-planned manufacture: review and categoriztion[END_REF], they emphasized that there are very few articles that address the MRP replannification in multi-echelon configurations.

The risk that is taken in the productive part of SC, treating the procurement of optional or alternative components (and the components that they use) for assembly lines dedicated to the mass production of customized products, presents several specific characteristics that have not yet been considered in the literature and that we will attempt to develop.

The MRP model revisited

Treatment uncertainty in the traditional approach of the MRP

The frozen horizon F H is the initial part of the plan- ning horizon P H on which the demands of the first periods of the MPS (from t to 1 F t H + -) are intangible. This concept is related to that of the Order Penetration Point (OPP), which delimits what can be made to order from what is to make to stock [START_REF] Vollmann | Manufacturing Planning and Control Systems, 4th Edition[END_REF]. This separation, obvious in the SC convergent network, is less clear in a divergent network that characterizes an SC that has several distant assembly lines. In this case, a part mounted on those lines could be partly made-toorder (MTO) and partly made-to-stock (MTS), depending on the distance between the assembly lines and the plant that produces that component.

The application in the cascade of the BOM explosion results in finding ik a units of reference i, of the BOM level n, included in the reference k pertaining to the subset ik E of the MPS (level 1 of the BOM). In addition, the application in the cascade of the mechanism of the lead times offsets leads to a lag ik λ between the period t of launching into production, the reference i and the period ik t λ + for the use of the reference k in the MPS. This scenario makes it possible to bind the gross requirement it GR to a reference i of level n of the nomen- clature at period t, with the demands kt MPS ′ of refer- ence k (level 1 of the nomenclature) at periods t t ′ > and not with the planned orders of the references of level n n ′ < of the nomenclature, which use that refer- ence i directly (which is typical in an MRP).

To establish this relation, it is advisable to notice that, in a deterministic universe (non-revisable MPS, absence of quality problems, deterministic lead times), the on-hand balance it OHB at the end of the period t for any reference i must be null because, in this deterministic context, a stock generates a cost without any useful counterpart. Consequently, the planned order it PO for the reference i at the beginning of the period t is necessarily equal to the quantity , i i t L q + , which is delivered at the beginning of period i t L + , taking into account the lead time i L of this reference i. In its turn, , i i t L q + is neces- sarily equal to the net requirement , i i t L NR + for this period, which is equal to the gross requirement

, i i t L GR +
for this same period. These various values are linked to the requirements of the MPS by the relation (1). , , ,

i i i it i t L i t L i t L PO q NR GR + + + = = = , ik ik ik k t a MPS λ + = ⋅ ∑ E (1)
The reference i is entirely made-to-order if Max ( ) . The latter case, which is not treated, to our knowledge, in the literature on the OPP or on MRP, is important to consider because it is quite common in the modern world SC, such as in the automotive industry.

A company can use an MTO policy to control the production of a set of references that are manufactured on various sites belonging to its SC only if it has an OPP that is defined on a frozen horizon, such as , Max ( )

F i k ik H λ ≥
. By increasing the lags ik λ , the replacement of nearby suppliers by remote suppliers can lead insidiously to the non-observance of that condition and, thus, can trigger a loss of flow control in the SC, unless this reference is partially (or completely) madeto-stock with appropriate rules to avoid stockouts.

In this type of SC, the diversity of the end-products is too important to allow the MPS to be defined at the level of the end-product. The MPS must be defined at the level of the alternative modules. The planning BOM c  of a set of K alternate modules describes the average structure for the use of these alternative modules by a set of end-products that are assembled on the same line. It is necessary to define as many planning BOMs as there are assembly lines using these alternative components because of the difference in the structure of the demand that must be met. The elements of this vector c  correspond to the average proportions k c of the consumption of the modules K;

1 K c c c     =         with 1 1 k K k k c = = = ∑ . c 
can be regarded as the vector of probabilities for the use of the alternative modules in the steady state to characterize the demands X   of the alternative modules beyond the frozen horizon. The random demand of the alternative modules

X   follows the Multinomial distri- bution ( , ) N c  M
, in which N is the known daily production of the line.

When we are interesting in a specific module k with a probability of use k c , we can consider the presence of a complementary fictitious module k' with a probability of use k c ′ such as

| 1 k h k h K h k c c c ′ ≤ ≠ = = - ∑ . Regarding the module k, the Multinomial distribution ( , ) N c  M becomes equivalent to ( , ) N c′  M where 1 k k k k c c c c c ′     ′ = =     -      , which in turn is equivalent to the Binomial distribution ( , ) k N c

B

. In these conditions, the daily demand k X of the module k is considered as a random variable that follows the Binomial distribution ( ,

) k N c B .

MRP in an MTO/MTS environment without a quality problem

The flow synchronization in the type of SC studied here can depend on the MPR if its rules can be adapted to calculate the planned orders of a component i, which depends on MPS requirements going beyond the frozen horizon ( , ). , ,

i ik ik H ik F i t L ik k t GR a MPS λ λ < + + = ⋅ ∑ E , ik ik H ik F ik k t a X λ λ ≥ + + ⋅ ∑ E (2)
In the logic of the MRP, the planned delivery , + i i t L q (equal to the planned order it LP ) corresponds to the net requirement , i i t L NR + of this reference i for this period

+ i t L ,
i t L + -: 1 , 1 , 1 , , i i t t L i t L i t i t i t t t AI OHB q GR ′= + - ′ ′ + - - ′=   = + - ∑   (3)
Then, in the determination of the planned delivery

, + i i t L q , it
i ik i ik H ik F it i t L ik k t i t L PO q a MPS R λ λ < + + + = = ⋅ + ∑ E 1 , 0 [ i h L it i t h it h OHB q GR = - + = - + - ∑ 1 , 1 ] i ik ik F ik h L H h ik k t h h a MPS λ λ = - < + + - = - ⋅ ∑ ∑ E (5)
The production of the entire made-to-stock is a special case of relation ( 5), in which , 1

ik ik F i k H L λ ∀ ∈ ≥ + -

E

, to make random all of the gross requirements for the period 1 t + to i t L + . In that case, no deterministic requirement of MPS stands, and the relation ( 6) is to be used. 1 , , , 0

i i i h L it i t L it i t L it i t h h PO q GR R OHB q = - + + + =   = = + - -∑     with 1, ik F i ik H L k λ ≥ + -∀ ∈E (6)
In the steady state, ,

it i R R t ′ ′ = ∀ . i R can be drawn from relation (5), for period t , with , , i it i t L q LP t ′ ′ - ′ =
∀ and can be replaced by that expression in that relation for period 1 t + . After simplification, one obtains relations (7) and ( 8), which are more convenient to use in the steady state than the relations ( 5) and ( 6). Because the deterministic part of the MPS requirements vary with the used horizon, a superscript is necessary to notate precisely at what initial period the analysis of the future requirements is performed.

, i it i t L it PO q GR + = = 1 0 , 1 i ik ik F ik h L t H h ik h k t h a MPS λ λ = - < + = -+ - + ⋅ ∑ ∑ E 1 1 0 , 1 i ik ik F ik h L t H h ik h k t h a MPS λ λ = - - < + = -+ - - ⋅ ∑ ∑ E (7) , i it i t L it PO q GR + = = (Steady state and , 1 ik ik F i k H L λ ∀ ∈ ≥ + - E ) (8)
A change of the steady state at period Max ( )

k ik t λ +
modifies the order-up-to level. This change is progressive and lasts for Max ( )

k ik F H λ - periods.
During that transitory state, the difference between the new value for the order-up-to level and the previous value is to add to the relations ( 7) and ( 8) for calculating the planned order.

MRP in an MTO/MTS environment with a quality problem

Let us assume that no quality problem occurs for all of the components that include component i of level n, until the modules k of level 0, and that the quality is checked before the departure of the deliveries.

i ik ik H ik F i t L ik k t i Z a MPS λ λ π < + + ∼ ⋅ ∑ BN E , i i i t L TS U + → = with , ,

P( )

i i i t L i t L Z U α + + ′ > = (9)
In a rolling planning, it is the variation of this target-stock

, , 1 + + - - i i i t L i t L U U
that is added to the gross requirement 10) is to be used, as follows:

, i i t L GR + to calculate the planned delivery , + i i t L q , if 1 > i L ; otherwise, , 1 i t AOH -replaces , 1 i i t L U + -. Then, relation (
, ,

i ik ik H ik F it i t L ik k t PO q a MPS λ λ < + + = = ⋅ ∑ E , , 1
i i i t L i t L U U + + - + - (10)
The name "target-stock" is used here in preference to that of safety stock, which is defined classically as the difference between an order-up-to level and an average demand (which is not the case here). In the absence of defective pieces in the production that are launched in the period t, the projected inventory

, i i t L AI + at the end of period i t L + (after control and before expedition) is equal to this target-stock , i i t L U + .
The use of these relations implies that a calculation of the target-stock is made at each decision-making point. If the probability of nonconformity is weak, then a value for the stock-target will be valid for a relatively broad range of values of Gross Requirements, which allows an easy construction of a decision table that avoids repeating the calculation each time.

If the gross requirement

, i i t L GR + is completely or par- tially random, then the first argument of the Negative binomial distribution becomes random. The risk of nonconformity must be taken into account in the definition of the order-up-to level , + i i t L R because only one safety stock can be used. This risk sharing, by simultaneously accounting for the variability of the demand and the variability of the quality, does not lead to separating the origin of the stockout. Then, the order-up-to level , + i i t L R is defined by relation ( 11) and is calculated by using the Monte Carlo method, as follows: , ,

P( )

i i i t L i t L W R α + + > = where , , , i i i i t L i t L i t L W Y Z + + + = + , 1 , , 0 i i ik ik F ik h L i t L H h ik k t h h Y a X λ λ = - + ≥ + + - = = ⋅ ∑ ∑ E and , , (
i i i t L i t L Z Y + + ∼BN , 0 , ) i ik ik F ik h L H h ik k t h i h a MPS λ λ π = < + + - = + ⋅ ∑ ∑ E (11)
The exact solutions of relations ( 10) and ( 11) oblige us to compute the probability distributions

, + i i t L Z or , i i t L
W + at every decision-making point. In the steady state, if i π is weak, then it is simpler to regard i Z or i W as random variables in which all of the requisitions of the MPS intervening in the calculation of the gross requirement are random; then, relation ( 12) can be used, as follows:

( )

i i P W R α > = with i i i W Y Z = + ( , ) i i i i Z Y Y π ′ ∼ + BN 1 0 i ik ik F h L i H h ik k h Y a X λ = - ≥ + = = ⋅ ∑ ∑ E 0 i ik ik F h L i H h ik k h Y a X λ = < + = ′ = ⋅ ∑ ∑ E (12)
The generalization of the approach that accounts for the problems of quality in the production of the components j of level 1 n + used by component i of level n is rather simple at the level of the principles. This approach is, however, complicated to implement in practice because it does not lead to analytical relations as simple as those found above. It is necessary to continue the calculation of planned orders it PO ′ for the periods t t ′ > using the same approach as for the calculation of it PO by distin- guishing the deterministic part from the random part. After that, it is necessary to calculate planned orders of the components j by adapting the approach that led to the relation (11).

Numerical example

Let us illustrate the use of these analytical relations through a fictitious example. We consider the productive part of an automotive supply chain that leads to two cars' assembly factories (A and B) belonging to the same company. These two factories have a common engine supplier, which does not have other customers. The engine plant is supplied by a unique supplier of pistons and is its unique customer. The piston plant buys the piston crowns from a unique supplier, and it is also the supplier's unique customer. It is assumed that these different plants use an MRP-I approach that shares a common periodicity (a time bucket of two days) for launching production and receiving deliveries. The MPS of factories A and B are defined at the level of the alternative modules (engines), in other words, at level 1 of the BOM. These data, as well as the initial onhand balance and planned deliveries, are provided in table 1 (to see at the end of this article). In this example, it is easy to calculate the λ ik values for the pistons, for example, for the piston crowns (see table 2). All of the calculations in the table below were performed by considering that the frozen horizon length is 9 periods, with the result that one is under the normal conditions of use for MRP. Table 2: λ ik values for the pistons and piston crowns

The analytical solutions suggested for determining the order-up-to level or the target-stock will be illustrated with a very low level of risk. This choice is justified by the seriousness of the economic consequences of stockouts, which encourages taking expensive measures for meeting the demands to avoid the propagation of disturbances along the supply chain. . With a frozen horizon of 7 periods, pistons are made-to-order, and piston crowns are made partly to order and partly to stock (see table 3).

Production partly to stock in the absence of a quality problem

In table 1, information on the MPS corresponds to the expected values of the engine demands, beyond this 7th period. Table 3 links the gross requirements of the piston crowns to the engine demands. PO q = = is the sum of the deterministic part of the gross requirement of period 3 (516) and of 3 2 6548 1014 5534

i i R AI - = - = .
After having illustrated the use of relation ( 5), let us next illustrate relation ( 7), which is valid in the steady state. By performing the calculations for MRP one period later (see table 5), the decision to be made at the beginning of period 2 depends now on the engine demands known for the periods 2 to 8. Let us suppose that, for this last period, the demands for the engines are

Production in the presence of a quality problem

To illustrate relation ( 9), it will be assumed that problems of quality are encountered in the manufacturing of the pistons, with a risk of 0.1% π = that a piston is regarded as defective after quality control. With a gross requirement of 6050 units for period 3 (see table 1), the target-stock is 17, as defined by the fractile of ,3 pistons Z ∼ (6050;0,1%) BN for the risk 0.01% α = .

If the quality control, performed before sending the delivery of 5780 pistons at the beginning of period 1, rejects 7 pistons, then the available inventory at the end of period 2 is decreased by 7, and the initial planned order (5812) is increased by 17 7 24 + = to reach 5836 units. It is interesting to note that the target-stock of 17 units is associated with a range of gross requirements from 5507 to 6269, which suggests the possibility of avoiding calculations by using decision tables that are easy to establish.

To illustrate relation (11), we suppose now that the production of engines and pistons are without defects and that the defect risk for a piston crown is still 0.1% This scenario yields the new order-up to level that protects from the two sources of randomness.

Conclusions

Thus, it is possible to guarantee a good performance in piloting a supply chain dedicated to mass-customized production using a revisited approach of MRP that combines MTO and MTS, accounting for the risks of the demand fluctuation and the quality and allowing the use of different frozen horizons depending on the knowledge of the customers' demands. Two observations must be made with regard to the underlying assumptions that place conditions on the effectiveness of this approach.

The use of the planning BOM c  of a set of K alternative modules assumes its stability. In the opposite case, it is necessary to anticipate the dated transformations of this structure that are induced by changes in the customers' expectations or by marketing actions (limited series, promotions) of the company or its competitors. A regular and rigorous analysis of these upcoming changes is essential; otherwise, the piloting of the supply chain will not remain under control. The costs of this informational monitoring are low, taking into consideration those induced by incoherent decisions in the supply chain. The organization and the follow-up of this vigilance place a condition on the performance, even perennity, of the supply chain.

On the principles level, the mechanisms proposed are relatively easy to implement because it is a question of transmitting order-ups to levels and target-stocks. The implementation is more difficult when the concerned plants do not belong to the same company. A contractual agreement must then be proposed, accepted by all and implemented. An additional difficulty is related to the fact that a supplier of the supply chain can belong to several supply chains (Renault and PSA supply chains share many suppliers). Technically, the generalization of the suggested approach could be immediate if the same principles of piloting are retained by all, which makes it possible to pool the risk; however, this strategy then poses some problems in the contractual formulation because the safety stocks are calibrated for aggregated demands. A less efficient solution consists of regarding the customers of the same supplier as independent. In addition, it is advisable to not underestimate the problems of confidentiality that are involved by the sharing of information that is coming from several supply chains.

  to the "regular" case of MRP use. The reference i is entirely made-tostock if Min ( ) the MPS requirements used in this relation are unknown. Last, a mix of MTO and MTS occurs when the reference i is linked with at least a reference k such as ik F

Figure 1 :

 1 Figure 1: BOM and lead time definitionFigure1describes the relevant extract of BOM for the studied subset and shows the lead times. At level 0, one finds that the vehicles manufactured by the two distant assembly plants are dedicated to different markets that do not have the same structure of demand. At level 1, one finds the planning BOM that is associated with each one of these lines; among the 5 alternative modules of the set E, only the engines E 1 and E 5 use the pistons that are of interest here. The piston crowns are specific to each type of piston. Lead times integrate times of production and transportation to the customer. The distances between the assembly factories and the engines plant are such that it is necessary to isolate a transportation lead time in the mechanism of MRP.

Y

  corresponds to the same sum of the random re- quirements of periods 2 to 3. Then, the distribution of

  which is equal to the gross requirement

				, i i t L GR +
	for this period, minus the projected available inventory
	, i t L i AI + -at the end of the previous period 1	+ -i t L	1	.
	This projected inventory	, i t L i AI + -depends on the on-1
	hand balance	, 1 i t OHB -at the end of the period	1 t -and
	on the gross requirements and planned deliveries of the
	periods t to	1

  is necessary to integrate uncertainty relating to

												-The deterministic requirement for period	i t L + , i.e.,
												ik ∑ E	ik λ	<	H	F	ik a MPS ⋅	, k t	+	ik λ	,
												-The difference between , i i t L R + and	, i t L i AI + -1 ′	.
												Then, the planned order	it PO is defined by relation (5),
												as follows:
												,	,	,
	the gross requirement	, i i t L GR + and to the projected in-
	ventory	, i t L i AI + -. The quantity to be delivered at the 1
	beginning of the period is then the sum of a known
	quantity	ik ∑ E	ik λ	<	H	F	ik a MPS ⋅	, k t	+	ik λ	and a quantity
	defined as the difference between an order-up-to level
	, + i i t L R	and a deterministic projected available invento-
	ry	, i t L i AI + -1 ′	, which is derived from relation (3) in
	which only the deterministic part of	, i t GR ′ is used.
	Then, , + i i t L R	corresponds to the fractile, which is asso-
	ciated to the stockout risk α of a probability distribution
	of a demand , + i i t L Y	, i t L i Y (P( +	>	, i t L i R +	) ) α = . The de-
	mand , + i i t L Y	corresponds to the sum of the random
	requirements of the periods	1 t + to	i t L + , with the
	gross requirement	it GR of the first period being neces-
	sarily known. Given that the MPS requirements of the
	module k for the period	t	+	λ ik	-	h are unknown, if
	ik λ -≥ h H	F	, then , + i i t L Y	is defined by relation (4), as
	follows:
	, i t L i Y + = 0 h L i h = = ∑	1 -	ik ik λ ∑ E	≥	F H h ik a X + ⋅	, k t	ik λ + -	h	(4)
	Therefore, the planned order
												i t L q	at period	i t L + ,
	is the sum of the following:

it PO of reference i, which is equal to the planned delivery , + i

Table 3 :

 3 Decomposition of the requirements of the piston crown by engines MPSThe deterministic parts of the gross requirements of piston crowns (reference i) for periods 2 and 3 are 5284 and 516, respectively; is completely deterministic (a firm order sent by the pistons factory).

	According to the relation (4), with	2 L = , the demand i
	3 Y corresponds to the sum of the random requirements i
	of	the	periods		2	to	3:
	3 4 Y = × i B	(960;0,2) 4 + × B	(1840;0,54)
	4 + × B	(960;0,2) 6 + × B	(960;0,1)	. This distribution is
	a weighted sum of Binomial distributions and does not
	have an analytical solution. It is determined numerically
	without difficulty using the Monte Carlo simulation
	method.	With		a	risk	of	0.01%,
	3 Y > P(	6548) 0.0001 = and the order-up-to level 3 i R is
	6548.
			5 796			6 376
			1,0%		98,0%			1,0%
	0,007
	0,006
	0,005
	0,004
	0,003
	0,002
	0,001
	0,000
		5 600	5 800	6 000		6 200	6 400	6 600	6 800
			Figure 2: Distribution of 3 i Y
	The use of 3
	i AI =		at the end of period 2 (and thus at the be-

1 ( 5812) i GR = i R is illustrated in table 4

(see the end of this article), which shows, for a horizon passing from 9 to 7 periods, the calculation of the planned order of the piston crowns, the planned orders of the pistons and the engines that remain unchanged. The available inventory 2 1014

MPS of Engines E1 et E5 defined at the beginning of period 1