- Email: [email protected]
Economical Analysis of Product Standardization
Copyright @ IF AC Management and Control of Production IFAC and Logistics, Grenoble, France, 2000
ECONOMICAL ANALYSIS OF PRODUCT STANDARDIZATION
Vincent GIARD
Professor at the IAE in Paris· Pantheon - Sorbonne University (Gi.d.lAE@univ·plrisl.fr) (G;"d.lAE@univ'p.n.l.fr)
Abstract: The standardization of partially substitutable components destined to meet a set of needs is studied. The monocriterion approach to standardization proposed in the last century is highlighted by taking into account several criteria and an economic valorization through modeling by linear programming. The notion of diversity is analyzed in this perspective and some cost problems of this method are discussed. Copyright © 2000 IFAC IFAC. Keywords: standardization, product design, mass production, linear programming.
Calibration was the main standardization effort in the era of craftsmanship. It dealt with the establishment of measures (monetary, weight, length, etc.) in order to determine the exchange of goods and services on compadetennine rable bases and to progressively build the related sciences. One has to wait for the industrial revolution, which came about through the evolution of knowledge, before the use of standardized processes allowed production of substitute products, enabling the production of ever more complex finished products manufactured from identical components made on machines, which are also becoming increasingly sophisticated. This product standardization is compatible with a systematic production of "custom" component most often used in a single product. The design rationalization of a range of homogeneous and partially interchangeable products, designed to meet various needs, presented here as the modem definition of product standardization, is rapidly felt felt. We begin with the relationships between diversity and standardization before proposing a lead to optimize this standardization.
the interval that contains Yk (i.e., in the example on Figure l, I, a rope of diameter x2' X2' for a limit traction before rupture Yt and Y2)' between YI
Yl
Diameter Xl XI
X2 x2
3
4 x4
Fig. 1. Analysis of traction limit before rope rupture. rupture, by Renard This approach to the problem can limit the number of item-types to produce and manage. manage, but the number of intervals and their definition remain to be solved. To do so, empirical work proposed solutions (and gave birth namely to the "Renard series" still used in the industry). Before proposing an economic modeling of this problem under a multicriteria generalization approach, it could be useful to put the problem in a larger perspective of diversity and its cost, in an economic context that further aims at interesting the client through custom production of goods and services.
L DIVERSITY AND STANDARDIZATION 1. After an overview of standardization fundamentals, the diversity concept will be examined in detail
1.1. Standardization fundamentals Standardization efforts are undertaken in all industrialized countries in the first half of the nineteenth century. In France, it seems that modeling rnodeling of this rationalization dates back back. to the mid 1800s with the work work. of Renard on French navy ropes: increasing ship sophistication, with custom-designed rigging, and colonial expansion led to a rapidly unmanageable proliferation of rope stocks. The idea is quite simple. In this problem, the main characteristic of a rope is the maximum tensile strength Y that a rope of diameter X can withstand before rupturing. A test on ropes of various diameters helped establish Figure I. 1. Then, subdivide the axis of ordinates into a definite number of disjoint intervals and associate, to any Yk tension request, the diameter associated with the upper limit of
1.2. Diversity Analysis
The global perception of finished products by the consumer or the potential client is not discussed here. That perception is conditioned by numerous factors; several depend on the mix-media selected in marketing mark.eting and are of no concern to us. We only consider the visible production diversity on components of a finished product satisfying needs of an identical nature, the presence of one of these components being mandatory. Doing so, the sup-
265
plementary diversity obtained by adding optional com-
in one finished product due to design. manufacturing
l question of product and finished distribution synergies. ponents isdiversity not analyzed . in one product due to The design, manufacturing plementary obtained by adding optional comrationalization of aThe company must I . oftherefore product be ponents is not anal yzed distribution synergies. question Three types of custom production must be differenti- anddesign treated from a standpoint of technologically similar of a company must therefore be design rationalization ated (See andproduction Pine, 1997). Some charThree typesAnderson of custom must be product differentiproduct families rather than from each individual prodof technologically similar from a standpoint can beand customized by the client for acharperfect treated atedacteristics (See Anderson Pine, 1997). Some product uct, taking into account that the industrial classification product families rather than from each individual prod(customizable prodsuitability his tastes or acteristics canwith be customized byneeds the client for a perfect of complex products with a same basic comtaking into account that the industrial classification example, the or owner a stereo soundprodsystem uct,analysis ucts). For suitability with his tastes needsof(customizable ponentofiscomplex normallyproducts found inwith several different a same basicaggregate comcanFor setexample, up certain of his tuner so the analysis sound system ucts). the characteristics owner of a stereo products (commonality). The different question aggregate of product is normally found in several of certain his stereo system matches his tuner preferences, of his so the the ponent cansound set up characteristics design rationalization meetingofa set of basic (commonality).deals Thewith question product driver of certain makes matches of automobiles can adjust of his stereo system his preferences, the the products sound functional needs through a range of physically interrationalization deals with meeting a set of basic height and inclination the steering can wheel, or the of certain makes ofofautomobiles adjust the PC design driver changeable components, each component with a limfunctional needs through a range of physically interuserand can inclination modify the of parameters of his work environment. the steering wheel, or the PC height ited spectrum for each each of thecomponent functional needs components, with aselected. limproducts products) self-adapting, changeable userOther can modify the(customizing parameters of his workare environment. approach to standardization makes it one compospectrum for each of the functional needs selected. meaning they(customizing automatically adjust toare theself-adapting, context of their itedThis Other products products) of company flexibility, complementing resources approach to standardization makes it one compoutilization, which is namely to toensure the function meaning they automatically adjust the context of their for Thisnent personnel, of company flexibility,procedures). complementing resources which they are isdesigned. product utilization, which namely This to ensure the category function can for be nent(equipment, personnel, procedures). • This component variety must be detectable by the cliillustrated the electric withcategory floatingcan heads be or (equipment, which they arebydesigned. Thisshaver product and he must consider added value. There variety must itbeandetectable by the cli- are by washing machines fuzzy to determine illustrated by the electricusing shaver withlogic floating heads or the • Thisentcomponent ent numerous and he must consider an addeddetectable value. There are cliexamples ofitdiversity by the washmachines cycle (this is fuzzy progress compared to washing by best washing using logic to determine the ofvalue diversity detectable by the Nissan cliexamples ent but of no added for him. For example, customized with a few dozen towashing bestmachines wash cycle (this is progress compared washingpro- numerous value its for supplies him. Forinexample, but of notoadded rationalize 1993; it Nissan had more few often dozenhas washing machines customized withthea user grams, between which troubleprodecid- ent decided to rationalize itsits supplies in (stated 1993; itinhad more & than 300 ashtrays for vehicles Anderson ing).between Finally,which somethe products are has customized by the decided grams, user often trouble decid& thanPine, 300 p. ashtrays for its vehicles in Anderson 95). Product internal(stated diversity not detectable by the ing). Finally, some customized manufacturer (and,products in some are cases, by the salesperson). Product not detectable by p. the95). client is eveninternal worse diversity (screw-products are a clascases, by salesperson). manufacturer (and, in some Several non-exclusive solutions canthe describe this cus- Pine, by the is even worse areonly a classicalclient example of this type(screw-products of diversity). Not is it of Several non-exclusive solutions can describe this customization. of this type of diversity). Not only is it of sical example no added value for the client, but it often generates tomization. • The best known approach is the use of an appropriate added value for (due the client, but it logistics often generates costs to additional problems, modules, eachuse module being selected no additional is the of an appropriate • Thecombination best known of approach costs (duediversity, to additional increased tooling etc.). logistics problems, of modules, each being selected combination from a limited number of module interchangeable modules. additional tooling diversity, etc.). analysis of the classifiinterchangeable modules. from a solution, limited number More often than not, a technical This largely of used in the automotive industry, increased of thewill classifiThis largelya standardized used in the automotive often than not, a technical analysis cation of production and supply item-types show a notsolution, only implies productionindustry, process (to More of production supply item-types will show notminimize only implies a standardized production (to cation significant portionand of components with identical or avery settings and tool changes) and process a total substiwith identical or has verysevportion of components minimize settings and tool changes) and a total substitutability of modules from a same family during significant similar characteristics. This mushrooming effect of but modules a of same family during This mushrooming effect has sev- of tutability assembly, also thefrom design standardized assembly similar eral characteristics. and well known explanations: misunderstanding of standardized of assembly, also the designplatform andimpact well known explanations:item-types misunderstanding supportsbutcalled product (Meyer assembly and Lehner, eralthe of mushrooming on production the and impact of mushrooming item-types production(Not supports product platform risks" (Meyerof andmalfunction, Lehner, 1998) called to limit the "system product support service, the NIHonsyndrome malfunction, product Here), supportarbitrary service, design the NIH syndrome (Not 1998) to limit the "system risks" assured decisions, misleading of sufficient lastingness andofrobust, i.e. accept- andInvented sufficient lastingness and robust, i.e. acceptassured Here), arbitrary designweight, decisions, argument about the "proper etc.", misleading failure of the of constraints (electrical, mechan- Invented ing a of large spectrum of the of argument aboutsystem the "proper weight, toetc.", failurecreation etc.). information - all leading the faster ingical, a large spectrum of constraints (electrical, mechanto the up faster creationone. of It ical, etc.). system rather - all leading a new item-type than looking an existing • The use of adjustable components with reversible set- information a new item-type rather than looking up an existing one. It is clear that the economic approach to standardization adjustable components with reversible set• Thetings use of leads to immediate and cost-efficient customizathat be thedeveloped economic hereunder approach to standardization that will is of interest only if tings customizaof to a immediate componentand bycost-efficient different means (switch, is clear tionleads of interest if will be hereunder is detected fordeveloped mushrooming are not andonly progresof a component bya set different meansselected (switch, tionsoftware... ) to activate of functions from thatcauses causes for mushrooming are not detected and progressivelyeliminated. functions selected from software ... ) to activate a predetermined list;a set thisofdecision could be reversed. a predetermined this decision reversed. This techniquelist; of often selected could in the be design of elec- sivelyeliminated. Searching to optimize standardization supposes on one This technique of often selected in the design of electronic or electric components. Searching optimize standardization supposes on oneand hand thattofunctional needs are adequately assessed electric components. tronic that functional needs are adequately assessed translated into technical specifications, which isand facili• Theoruse of adjustable components with irreversible hand is facilispecifications, tated byinto the technical QFD approach (Quality which Function Deploy• Theadjustments use of adjustable components with irreversible corresponds to an immediate and cost-effi- translated by2),theand QFDonapproach (Quality Function Deploy-and ment the other hand that a coherent adjustments corresponds to an immediate cost-efficient adaptation of a component to theand needs either by tated 2 ), andreflection on the is other hand a solutions coherent considand ment relevant carried outthat on all of a component to the or needs by cient adaptation additional machining operations, by either a chemical reflection Itisiscarried outthat on the all optimum solutions solution consid- of ered adequate. obvious by a chemical additional machining operations, treatment, both irreversible. Suchorproducts are found in relevant It isproblem obvious is that solution ereda adequate. poorly stated notthe ofoptimum a great interest (itof even treatment, bothtrade, irreversible. Such products are manufacturfound in the cutting eyewear, custom bicycle nottheofmethod a great and interest (it even could stated reflectproblem discreditison not on its misthe ing, cutting trade, eyewear, custom bicycle manufacturor replacement doors and windows. Deferred dif- a poorly could reflect discredit onofthe not on to its examine misuse). Determination themethod solutionand portfolio ing,ferentiation or replacement often doors arises and fromwindows. this logic.Deferred difuse). of eliminating the solution aportfolio to examine is Determination based on a study priori none of the cusferentiation this There areoften thus arises severalfrom ways to logic. reach diversity of a finon a study eliminating priorialready none ofexists the custom modes described above. aWhat (interof a fin-and is based There thus several ways to reach diversity ished are product that avoid referring to systematic tomnal modes described above. Whatmust already exists be (intermanufacturing or supply) of course part of ished product that avoidTwo referring to systematic costly customization. standpoints determineand their course be part of nal the manufacturing supply) must alternatives or studied and the of new products planned costly customization. economical analysis.Two standpoints determine their the must alternatives studied andthe theconstraints new products planned take into account of the existing economical analysis. • A component is not a priori designed to be solely used must take into account the constraints of the existing production system or the one undergoing transforma• A component is not a priori designed to be solely used production system orthis thestandardization one undergoing tion 3. Optimizing nottransformaonly normally 3 . Optimizing this standardization onlybut normally 1. This type of diversity generates additional costs when the prod- tionresults in a decrease of productionnot costs, also in an is assembled an assembly line due to the work I. Thisuct type of diversityongenerates additional costs when thevariability prodresults in a decrease of production costs, but also in an
by theon assembly of optional some work2. The Quality Function Deployment is a value chain optimization uct caused is assembled an assembly line duecomponents to the workonvariability stations, specific line balancing andonscheduling prob- 2. Thetechnique. of needs to product service. It from expression of optional components some workQuality Function Deployment is a value chainsupport optimization caused by thecreating assembly 2(00). It probmay be lems creating (See Danjou, and Le Roy, 20 years by major originated Japan and of hasneeds beentoused for over stations, specificGiard line balancing and scheduling product support service. It technique, frominexpression interesting systematically offer a Itpopular the by approach companies. especially the first step20ofyears originated in Japan and has with been used for over Le Roy, 2000). may option be major to lemseconomically (See Danjou, Giard to and focusestoonsystematically a few alternative it isoption then posand if demand Antooverswitch from functional needs technical specifications. economically interesting offeroptions, a popular companies, especially with the tofirst step of the approach to relate to the describedoptions, here. Inversely, one can of functional these techniques found inspecifications. Revelle. Moran view and sible if demand focuses on method a few alternative it is then posAn and over-Cox, switch from needs toistechnical of least popular options. thetoeconomic interest to relate the method described here. Inversely, one can of these techniques is found in Revelle, Moran and Cox, siblequestion view1997. 1997. question the economic interest of least popular options.
266
266
increase flexibility, the company could then react more rapidly and easily to the conjunctural and structural transformations of demand44 .
2. TOWARD OPTIMIZING STANDARDIZATION Once the general modeling of the problem stated by linear programming, we will examine a few methodological problems stated by this economic rationalization, whether we call on the proposed optimization approach or not.
2.1. Baseline model For simple and non-expensive products (for example screw-products or the above-mentioned Nissan ashtrays), a sophisticated analysis is generally not mandatory as long as the reduction in the number of components for a function becomes self-evident from a classification simplification standpoint as well as for production and supply processes. processes. In other cases, the Renard approach is used to rationalize the range of relatively simple products, generally characterized by a unique quantitative criterion. Pareto analyses on need distribution based on this quantitative criterion (See, for example Anderson & Pine, chap. V, 1997) establish empirical approaches to determine a range of products, but the action is based on intuition and barely takes into account the economic standpoint. Product complexity also ensures that the technical analysis cannot have only one criterion. Formulation of this problem by linear programming using integers brings relevant response elements to these concems55 ; the matrix generators approach tested can easily implement them66 . The model will be presented with an example on engines. From a technical standpoint, we have n possible variants of a product7 , whether these variants are presently produced or only under study. These different variants will be identified with the subscript j. Moreover, we presume the demand analysis identified m segments, identified with the subscript i and characterized by a This category of concern is taken into account in DFA (Design 3. TIlis literature; refer to Nof, Wilhelm and Wameck.e, For Assembly) literature~ Ill, and to Redford and Chal, 1994. 1997, chap. Ill. Tarondeau, 1993 & 1997, Cohendet 4. Company flexibility (See Tarondeau. Lierenna, 1989, and Reix, 1997) is classically seen through and Llerenna, mobilized resources (equipment, tooling, personnel and procedures); product-related flexibility is generally highlighted but, dures)~ however, economic instruments for its improvement have not tbe perspective been studied much to our knowledge, and not in the treated here. 5. In an previous research document (Giard, 1990), we presented s.1n more restrictive perspective. In a briefly a similar model in a rmre (Giard, 1998), we proposed a system approach recent publication (Giant, matbematical programto production process modeling through mathematical ming indicating the mutual connections between various models tbe found in the literature on operational research and explaining the complex model creation mechanisms using a set of base components. Formally. Formally, the !be problem here combines a transposition of a nents. 1998, p. client assignment model to production centers (Giard, 1998. witb fixed costs that 46) and the recognition of non-linear costs with brack.ets (Giant, (Giard, 1998. 1998, p. 97), vary by brackets 97). except that tbe the total production of an item-type here is the tbe sum of productions requested by various segments. Murphy, Stohr and Asthana, Astbana, 6. TIlis This MGs approach is described in Murphy. 6. Rosentbal, 1996, Jacquet-Lagreze, 1997 and Giard, 1998. 1992, Rosenthal, metbod is proven in The technical and economic viability of this method TIle numerous achievements (for example. example, two important industrial tbese methods metbods were carried out at !be applications of these the IAE in Paris contracts; they tbey are described in !be scope of research contracts~ within the Andre, 1997. \997, and in Giard and Triomphe, Giard, Triomphe and Andre. Triomphe. 1996). 1996).
267
demand d d,.t • From an existing situation, we must normally have nsm, i.e. the variety of the commercial offer is lower than the variety of the demand. The problem at hand is first related to an eventual decrease in the number of variants (with no economic vision of the problem). A first technical analysis must be performed to characterize the various engines. Selected criteria relate to technical characteristics that are more or less well perceived by demand (power, pollution, consumption... consumption . .. ) and to characteristics of no interest for the client but essential in managing interfaces (weight, overall dimensions, engine mounting method on the chassis ... .. . ). This technical analysis is performed on a table similar to Table 1, I, with a literal in each table cell to indicate the precise positioning of each engine. Table 1: Technical characteristics of engines under study Characteristics Engines ... under study 1 2 - .. P 1 2 }j n A second analysis is to be carried out on technical characteristics of requested engines (Table 2). For each previously detennined determined criterion, an interval of admissible values (minimum power, for example) is then defined based on the type of characteristic or a list of acceptable occurrences is determined if the characteristic is qualitative (such as mounting mode, for example). A difficulty encountered is determining the "appropriate need" aimed at pleasing, as mentioned above. Table 2: Technical characteristics of requested engines Characteristics Requested Demand ...... engines 1 2 P dl 1 d2 2 d ji i dm m ID
Comparing Tables 1 and 2 establishes Table 3, in which coefficients aij take the value 1 if demand in segment i can be met by engine}, engine j, or 0 if it cannot be met (the numerical illustration being fully arbitrary). Table 3: Possibilities of satisfying the demand with the offer Variants of engines studied Market ... }j . .. n 2 3 segment 1 ... . .. 1 0 0 0 0 1 ... 1 1 0 ... 0 0 2 .. ... ... 0 1 0 0 1 3 '
... .... .. ...... ... ... ... ...... 1 ... 0 1 .. . .. ., . .... ... ... . .. 1 ... 1 ... 1 It seems realistic to impose that the entire demand segment be satisfied by a same variant. Consequently, we define the binary variable Xij' which will have the value 1 if demand of segment ij is met totally by an engine}, engine j, and o if it is not. It is of course useless to create a variable xij Xij ...
i ... m
... 0 ... 0
... 1 ... 0
... ...
7. Renault's Cleon C\~on plant, for example, manufactures more !ban than 400 possible engine variants.
8 if the corresponding parameter aij is null . Moreover, if nuns. we decide that this demand can be met by several variants, then variable Xij Xjj can have any value between 0 and 1. I. To force the demand of segment i to be met, we must establish the constraint of Relation 1 (which leads, for binary variables, to a non null value).
S1 j4::s;I Zjl +Zj2+ Zj3+ Zj4
O::s;yjl
n
}: ~ xi.t=l xij=1 ,fori= 1,....,m(meetingthedemand) 1, .... ,m(meetingthedemand) Relation 1 j=1 Under these conditions, production Yj of item-type j is then the sum of productions carried out for each segment (demand d;), this production could be null. This constraint is defined by Relation 2: m YJ-= }: J(productiomitem-typq~elation 2 ~ d{xijlotj= d{xij/orj= 1.. 1.. J(productiomitem-tyw)l?elation ;=1 i=1 To establish the annual production cost, we must introduce explicit hypotheses on the item-type costs function form. First, we will assume the cost functions are independent. This restrictive hypothesis will be removed later on. We will also assume variation in annual fixed costs per bracket, bracket. and that for each bracket the variable unit cost may vary but remains constant over disjoint value intervals; these very general hypotheses lead to a cost function of the type described in Figure 2.
r
_#:---1 ~,~
j3Zj4::S;Yj4
Generalizing the reasoning to all variants leads to the Relation 3 objective function and to the constraints described by Relations 4 and 5:
? (K}:
Relation 3 Relation
}: Zjk::s;l ,for ,jor j = =11... ~ Zjks1 ... n k=l k=1
Relation Relation 4
j
j ]) Min Z, with Z = = n ~ (ckY (ckYjk+KjJiZjk) jk+ K jf<:Zjk) j=1 ]=1 k=l
Kj
=
Mjk_1Zjt:!OYjk
Kj3
I
MjO = =0 MMJIj1
MJ2 M j2
M Mj3 j3
Mj4
Fig. 2. Production cost function This cost function is neither concave (which would imply that the mean production cost never increases Yj increases) nor convex (which would imply that when Yj the mean production cost never decreases when Yj increases) and that in the selected example it takes into account the phenomena of scale diseconomies occurring at the saturation limit. This non-linear cost function can easily be taken into account in the objective function of a linear program by introducing as many dununy dummy Yjk productions as there are Kjj value intervals (comprised Mj . k _1 and Mj,k' Mj •k, the upper threshold only is between Mj,k_1 =0) for which the variincluded in the interval, with MjO = Cjk is constant, and are all null except for the able cost Cjk one encompassing production Yj in its value interval and that is, is. of course, equal to this production (yj = = Yjh)' Yjh). From 2. the objective function becomes, the example in Figure 2, for the section related to production cost Y/ Cj "1Yj2 + K j2zj2 ) + z, with Z = (CjlYjl + Kj1zjl ) + ((CjiYj2 + KKj3 (Cj4Yj4 + K Kj4 j3 ) + (Cj4"Yj4 j3zj3 j4zj4) + ... with the following list of constraints9, added to those of Relations 1 and 2:
Min
(Cj3Yj3 (C/3Yj3
8. This special feature, easy to take into account in the problem description by a GM (see, for example. Giard, 1998. and JaquetLagreze, 1997) helps in significantly limiting the size of the problem. This convention makes Relation I useless in the form
This formulation fonnulation must be adapted to take into account positive or negative synergy effects related to simultaneous production of two or more engines on a production site. To do so, relevant and easy modeling tools have to be implemented 11. Let us examine a few examples. • Let us assume that manufacturing more than K variants translates into an increase r+ of annual fixed costs; we then have to create the y+ binary variable. variable, add to the objective function the term y+r+, and add Relation 6 constraints to force y+ to take the value 1 if at least K variants are produced (the objective function is designed to bring y+ to equal 0).
(K. i i}: (t
)
n Relation 6 y+ Zjk
1).
n
}: air ayx ....=l =l ~ j=l IJ j=1 I) negative, the first double constraint is in real9. As variables are not negative. XI s; s M1YI; MIYI; the selected formulation has the sole ity reduced to xl advantage of allowing rationale generalization.
IO.lf IO.If the cost function is concave (non-
268
(K
j
n ~ Zjk ) >KY~ j=1 k=1
~
Relation 7
(t
Relation Here again, this approach can easily be adapted to a7 Zjk]>KY]=1 k=1of variants or to several variant subsets. Among subset Here again, thisprecludes approachpositive can easily be adapted to a others, nothing or negative synergy or to several variant subsets. Among subset of effects to variants occur simultaneously for different variant orthe negative synergy others, nothing positiveof subsets or not, asprecludes a generalization previous comeffects to occur simultaneously for different variant ments. subsets not, asfinally a generalization the previous • We can or assume that certainof fixed costs varycomper ments. bracket with relation to produced quantities over a set • of Weitem-types, can assumenotwithstanding finally that certain costs vary per thefixed possibility offered bracket relationtotoinclude produced over a set for eachwith item-type in quantities its cost function its of item-types, notwithstanding possibility own variation of fixed costs perthe bracket. In thisoffered case, for each item-type include in its cost function its the Q engine subsetto being affected by variations of own variation fixed have coststoper bracket. In this case, fixed costs, we of simply adapt the formulation as the Q engine subset being affected by variations of follows: costs, we simply have to adapttothe as -fixed create variable w corresponding theformulation total producfollows: tion of the Q engine subset, which leads to Relation - 8: create variable w corresponding to the total production of the Q engine subset, which leads to Relation m 8: w= ~ y/= ~ ~ d(xij Relation 8 m j~Q ~Q i=l W= Y;= d{xij - add to the objective function the impact ofRelation variations8 ~QI=I in ~Q fixed costs K , following the same method used
L
L .L
wk
- add to the objective KO) function the impact of variations in fixed costs Kw!:, following the same method used for item-type ~ Kwtzwk;
KO) k=l
L
for item-type Kro~rok; - then adapt Relations 2, 4 and 5, which leads to Relations 9 to 11 k=l - then adapt Relations 2, 4 and 5, which leads to RelaKO) tions 9 to 11 Relation 9 ws~wk
KO) k=l
LWk
Relation 9
k=1 ~ zwJC'1 KO) k=l
Relation 10
w~ KO)
L zro,tSl
Mwk-lzwJC'wk
k=l
Relation 10 = 1... KW' with
Mwk=O Relation 11 Mrok- 1Zro,tSwk
Mrok=O 11 2.2. Method problems created by using thisRelation optimization approach 2.2. Method problems created by using this optimizaThe relevance of this modeling depends both on its the tion approach
use to describe alternate scenarios and on the costs used relevance of this modeling depends both on its the in The the economic function. use to describe alternate scenarios and on the costs used First, we are working at a detail level too coarse to prein the economic function. tend it accurately represents the processes used, the First, we are working a detail level too coarse to seapreeffective demand on theatproduction system, with its tend itandaccurately represents the processes used, the sonal random fluctuations, or the robustness of the effective demand on the production system, with its seaproduction system when faced with problems. This comsonal can and be random the on robustness the ment made fluctuations, for most helportools strategicof deciproduction system when faced with problems. This comsions: the important aspect is the relevance of the order ment can be made for mostdata help in tools on strategic deciof magnitude of collected volume or in value. sions:brings the important is the relevance This us back toaspect valorization problems.of the order of magnitude of collected data in volume or in value. The most important problems we may meet here relate This brings us back to valorization problems. to time or, more specifically, correctly defining the interThe most important problems we maymade meet here relate dependence through time of decisions within the to time or, more specifically, correctly defining the intercost accounting system. dependence through time of decisions made within the In the rare case where the studies alternatives relate to cost accounting system. new production from new equipment, it is easy to see the Inof thea rare case where alternatives relate to use dynamic versionthe of studies the model proposed, as the new production new equipment, easy to see the decision to meetfrom a demand segment byit aisgiven engine is use offor a dynamic version of hypotheses the model proposed, as but the taken all periods 12; other can be used, decision to meet a demand segment a given engine is seem more difficult to justify. The by separation between l2 ; other hypotheses can be used, but taken for all periods fixed costs and variable costs helps isolate start-up seem more difficult to justify. The separation between fixed costs and variable costs helps isolate start-up
269
269
investments (and thus avoid the problem of determining the depreciation) from direct fixed costs (especially perinvestments (and thus the problem of determining sonnel) that could varyavoid per bracket with relation to quanthe depreciation) from directdirect fixed costs costs (materials, (especiallyetc.). pertities produced and variable sonnel) could varyused per bracket with relation to quanAs this that information in the economic function is tities 13, produced andhave variable direct costs (materials,toetc.). dated we would to call upon actualization corAs thisweight information used inflows the economic function is rectly the monetary generated at different 13 , we would have to call upon actualization to cordated periods. Classic problems are encountered when this rectly weight the monetary flowsinvestment generated alternatives at different approach is applied to compare periods. problems are encountered when this and, moreClassic specifically, to determine the determination of approach applied to VAN compare investment the interestisrate used in calculus and the alternatives problem of and, more specifically, to determine determination of comparing solutions having differentthe useful lives. theWhen interest rate used in VAN calculus and the problem of the problem relates to a set of item-types that at comparing solutions having different useful lives. least partially exists, when it is the object of external item-types system that at Whenortheanproblem to a set supply internal relates production in aofproduction least partially exists, when is the object ofmodified, external which could eventually be itonly marginally supply or an internal problems production in a production system several methodology arise. which could eventually be only marginally modified, • Certain key components of a certain complexity are several methodology designed to be usedproblems in severalarise. finished products, some • of Certain certain complexity are whichkey do components not yet exist;ofthea term "main sub-assemdesigned to beisused finished products, bly" projects usedininseveral the automotive industry.some The of whichprice do not yet exist; the termcreates "main challenging sub-assemtransfer of such components bly" projects is projects used in 14. theA automotive The methodological decision onindustry. standardizatransfer of such creates challenging tion thatprice questions thecomponents economic hypotheses used to methodological projectsl4.must A decision standardizalaunch such components rely on on a cost function tion that questionsdecision the economic hypotheses usedand to that guarantees coherence in time launch such components must rely on a cost between strategic and tactical decisions (Seefunction Giard, that guarantees 1988 and 1992). decision coherence in time and betweenthis strategic and tactical decisions (See • Within framework and more generally, theGiard, deci1988toand 1992). sion stop production of a item-type (or of a group of • Within this can framework moreangenerally, the deciitem-types) lead to and support "item withdrawal sion toThis stop impact production of a item-type ofaccount a group by of cost". can easily be taken(or into item-types) lead l5 to. support an "item withdrawal the objectivecan function cost". This impact can easily be taken into account by • Existing standard costs are only relevant on a certain the objective function I 5. range of quantities produced or supplied; before apply• ing Existing standardthorough costs areinvestigations only relevantare onneeded a certain the method, to range of quantities supplied; before applyreconstitute the costproduced function or and, for supply, carry out the consultation method, thorough investigations needed to aing prior with suppliers on theare basis of volreconstitute thethat costcould function and, for supply, ume scenarios substantially deviate carry from out the a prior consultation with suppliers on the basis of volcurrent solution. ume standardization scenarios that could substantially the • The problem can occurdeviate within from a multicurrent solution. level bill of materials. Decisions taken at a detailed • The can occur multilevelstandardization then are basedproblem on hypotheses of within direct ademand leveldemand bill of originating materials. Decisions at a detailed and from othertaken item-types (using level then based on that hypotheses of directthedemand MRP type are mechanisms) are themselves object and demand originating from other item-types (using 12.Which can be translated by maintaining Relation 1 and folMRP type mechanisms) that are themselves the the object m lowing adaptation of Relation 2 which becomes: ~ the d, ·xfol.. 12.Which can be translated by maintaining RelationyI '(= and ,
)
;=1 m It
I)'
Relal10ns 3 and 5 being modified by thebecomes: inclusionYjl=.2. of thedu·xij' period lowing adaptation of Relation 2 which index, Relation 3 integrating actualization coefficients. .=1 13.With an 3appropriate sub-division, this helps in to Relations and 5 beingtime modified by the inclusion of take the period account possible3 learning effects on recurring costs, with values index. Relation integrating actualization coefficients. cananbeappropriate considered reasonably stable over timetake interval.. 13.that With time sub-division, thiseach helps in to 14.For a more complete presentation problems related ecoaccount possible learning effects on of recuning costs, withtovalues nomic products stable over their life cycle, refer to that canmanagement be consideredofreasonably over each time interval .. Gautier and Giard. 2000.presentation of problems related to ec014.For a more complete nomic management of products their cycle. refer to 4, forover a cost of life cl> , simply add the IS.Using notations of Relation Gautier and Giard, 2000. 15.Using notations of Relation 4. for a cost of • simply add the term Zjk}cI> in the objective function. Knowing that
{l- i {I- i Zjk} k=1
term K j
~
Kj k=1
in the objective function. Knowing that
k=1
Zjk= 1 if item-type j is produced, the cost cl> will not be sup-
2. Z·k=l if item-typej produced. the cost will not beitemsupported in this case alone.isGeneralization of a set 'I' of'P k=1 J K
ported in this case alone. Generalization ofj a set IJI of 1jl itemtypes is immediate {l-t}cI> , with ~ ~ Z 'k-'P< t. Kj ) jE'I' k=1 types is immediate {I-I}. with Z r1jl< I.
2. 2.
jEW k=1 J
of a standardization optimization. optuTIlzation. An independent analysis of these different problems cause the problem of elementary component standardization to depend on hypothetical aggregate component demands and cause the problem of aggregate component standardization to depend on hypothetical costs of elementary components. Convergence toward a set of coherent solutions can empirically be assured through a certain number of iterations, but this model could also be adapted to take l6 , which could lead into account the Bill Of Materials l6 to a model of undue size. • The creation of new components induces management costs related to increased diversity that are difficult to evaluate. In economic calculations made at the design stage, certain companies such as Intel or Renault apply different cost rates to new components and to reused existing components. This incentive to reduce diversity is astute but must be used with great caution within a complete reconsideration required when applying the approach suggested in this paper.
3. BffiLIOGRAPHY Anderson D. M. , Pine IT II J. (1997). Agile Product Development for Mass Customization : How to Develop and Deliver Products for Mass Customization, Niche Markets, lIT, McGraw-HiII, Build-to-Order and Flexible Manufacturing, McGraw-Hill, New-York. Baldwin C. Y., Clark K. B. (1997). (1997). Managing in an age of modularity, Harvard Business Review, vol. 75 sept oct 1997, p. 84-93.
D. & Meeraus A. (1988). GAMS : A Brooke A., Kendrick D. User's Guide, Scientific Press, Redwood City. Lierena P. P. (1989). (1989). Flexibilite, information, deciCohendet P. , Llerena sion, Economica, Paris.
F., Giard V., Le Roy E. (2000). Analyse de la robustesse Danjou E, des ordonnancements/reordonnancements sur ligne Iigne de production et d'assemblage dans I'industrie automobile, to be published in Revue Franfaise Franr;aise de Gestion Industrielle, O vol. 19, nnOl, vo!. l, Paris. Giard V. (1990). Analyse economique de la diversite en produclAE tion, working paper G 9012,1990, URA CNRS 1257, IAE de Lyon. Lyon.
V., Pellegrin C. (1992). (1992) . Fondements de I'evaluation revaluation Giard v., economique dans les modeles economiques de gestion, Franfaise de Gestion, n088, p. 18-31. Revue Franraise
a
V., Triomphe C. (1996). SIAD visant adefinir les services Giard v., offerts au personnel d' un centre de production et s' appuyant sur la programmation mathematique, working paper lAE de Paris (see http://panoramix.univ-parisl.fr/ http://panoramix.univ-paris l.frl 1996.03, IAE GREGORl96-03.pdf). (1998). Processus productifet productif et programmation lineaire, Giard V. (1998). Economica, Paris. V., Triomphe C., Andre R. (2000a). Organist : un Giard V., Systeme Interactif d' Aide ala Definition du niveau de traitement du courrier des bureaux de poste et des toumees d'acheminement d' acheminement a un centre de tri, to be published in vol. 1, n° 1. l. Cahiers du Genie Industriel, vo!.
a
a
V. (2000b). Evaluation economique des choix Gautier F., Giard V. de conception d'un produit nouveau : I'impact sur les m
(yr}; d{xjj)' which defines pro16.Simply replace Relation 2 (YJ= ~ d{xij)' j=l ;=1 d j related to it duction Yy j , as being equal to the sum of demands d; by binary variables
Relation: xij' with the following Relation: m }; d{xi.J-+ d{xj,-+ ~ }; ah/Yh'xhj' where item-type h is related to Yy,= =~ J i=1 ;=1 h=1 item-type j due to the production of 1I item-type h unit requires ahj unit of item-type j. (This item-type h is related to final 2). demands through Relation 2).
m
270
systemes productifs, accepted in the proceedings of the AFC 2000 Conference, 18-20 may 2000. Jacquet-Lagreze E. (1997). Programmation Lineaire - Modelisation et mise en lEuvre ll'uvre informatique, Economica. Meyer M. H., Lehner A. P. (1998). The power of product platM. l..ehner A. forms - building value and cost leadership, The Free Press, New-York. Murphy F., Stohr E E.,., Asthana A. (1992). Representation schemes for linear programming models, Management Science, vol. vo!. 38, n° 7, pp.964-99l. pp.964-991. Nof S. Y., Wilhelm W. E., Wamecke Warnecke H.-J. (1997). Industrial assembly, Chapman & Hall, London London.. Redford A., Chal J. (1994). aSJ'embly : principles (1994). Design for assembly: and practice, Mac-Graw Hill, New-York. Reix R. (1997). Flexibilite , Encyclopedie de gestion, (Y. Simon & P. Joffre Ed.), 2° edition, Economica, Paris Paris.. Revelle J. W. , Cox C. (1997) The QFD Handbook, ReveUe J. B., Moran 1. Wiley,. Rosenthal R. E. (1996). Algebraic Modeling Languages for optimization, Encyclopedia of Operations Research and Management Science, C.. M M.. Hyarris ed ed.,., Science , S. I. Gass & C Kluwer Academic Publishers, Boston Boston.... Smith P. G., Reinertsen D. G. (1998). Developing Products in Half the Time: Time : new rules, new tools, second edition, Wiley, New-York.
I.-C. (1993). Strategie industrielle, Vuibert, Paris. Tarondeau J.-c. I.-C. (1999). Laflexibilite Tarondeau J.-c. Laftexibilite dans les entreprises, PUF, Paris. Williams WilIiams H. P. (1993). Model Building in Mathematical Programming, 3ee edition, Wiley, New-York.
ECONOMICAL ANALYSIS OF PRODUCT STANDARDIZATION
Vincent GIARD
Professor at the IAE in Paris· Pantheon - Sorbonne University (Gi.d.lAE@univ·plrisl.fr) (G;"d.lAE@univ'p.n.l.fr)
Abstract: The standardization of partially substitutable components destined to meet a set of needs is studied. The monocriterion approach to standardization proposed in the last century is highlighted by taking into account several criteria and an economic valorization through modeling by linear programming. The notion of diversity is analyzed in this perspective and some cost problems of this method are discussed. Copyright © 2000 IFAC IFAC. Keywords: standardization, product design, mass production, linear programming.
Calibration was the main standardization effort in the era of craftsmanship. It dealt with the establishment of measures (monetary, weight, length, etc.) in order to determine the exchange of goods and services on compadetennine rable bases and to progressively build the related sciences. One has to wait for the industrial revolution, which came about through the evolution of knowledge, before the use of standardized processes allowed production of substitute products, enabling the production of ever more complex finished products manufactured from identical components made on machines, which are also becoming increasingly sophisticated. This product standardization is compatible with a systematic production of "custom" component most often used in a single product. The design rationalization of a range of homogeneous and partially interchangeable products, designed to meet various needs, presented here as the modem definition of product standardization, is rapidly felt felt. We begin with the relationships between diversity and standardization before proposing a lead to optimize this standardization.
the interval that contains Yk (i.e., in the example on Figure l, I, a rope of diameter x2' X2' for a limit traction before rupture Yt and Y2)' between YI
Yl
Diameter Xl XI
X2 x2
3
4 x4
Fig. 1. Analysis of traction limit before rope rupture. rupture, by Renard This approach to the problem can limit the number of item-types to produce and manage. manage, but the number of intervals and their definition remain to be solved. To do so, empirical work proposed solutions (and gave birth namely to the "Renard series" still used in the industry). Before proposing an economic modeling of this problem under a multicriteria generalization approach, it could be useful to put the problem in a larger perspective of diversity and its cost, in an economic context that further aims at interesting the client through custom production of goods and services.
L DIVERSITY AND STANDARDIZATION 1. After an overview of standardization fundamentals, the diversity concept will be examined in detail
1.1. Standardization fundamentals Standardization efforts are undertaken in all industrialized countries in the first half of the nineteenth century. In France, it seems that modeling rnodeling of this rationalization dates back back. to the mid 1800s with the work work. of Renard on French navy ropes: increasing ship sophistication, with custom-designed rigging, and colonial expansion led to a rapidly unmanageable proliferation of rope stocks. The idea is quite simple. In this problem, the main characteristic of a rope is the maximum tensile strength Y that a rope of diameter X can withstand before rupturing. A test on ropes of various diameters helped establish Figure I. 1. Then, subdivide the axis of ordinates into a definite number of disjoint intervals and associate, to any Yk tension request, the diameter associated with the upper limit of
1.2. Diversity Analysis
The global perception of finished products by the consumer or the potential client is not discussed here. That perception is conditioned by numerous factors; several depend on the mix-media selected in marketing mark.eting and are of no concern to us. We only consider the visible production diversity on components of a finished product satisfying needs of an identical nature, the presence of one of these components being mandatory. Doing so, the sup-
265
plementary diversity obtained by adding optional com-
in one finished product due to design. manufacturing
l question of product and finished distribution synergies. ponents isdiversity not analyzed . in one product due to The design, manufacturing plementary obtained by adding optional comrationalization of aThe company must I . oftherefore product be ponents is not anal yzed distribution synergies. question Three types of custom production must be differenti- anddesign treated from a standpoint of technologically similar of a company must therefore be design rationalization ated (See andproduction Pine, 1997). Some charThree typesAnderson of custom must be product differentiproduct families rather than from each individual prodof technologically similar from a standpoint can beand customized by the client for acharperfect treated atedacteristics (See Anderson Pine, 1997). Some product uct, taking into account that the industrial classification product families rather than from each individual prod(customizable prodsuitability his tastes or acteristics canwith be customized byneeds the client for a perfect of complex products with a same basic comtaking into account that the industrial classification example, the or owner a stereo soundprodsystem uct,analysis ucts). For suitability with his tastes needsof(customizable ponentofiscomplex normallyproducts found inwith several different a same basicaggregate comcanFor setexample, up certain of his tuner so the analysis sound system ucts). the characteristics owner of a stereo products (commonality). The different question aggregate of product is normally found in several of certain his stereo system matches his tuner preferences, of his so the the ponent cansound set up characteristics design rationalization meetingofa set of basic (commonality).deals Thewith question product driver of certain makes matches of automobiles can adjust of his stereo system his preferences, the the products sound functional needs through a range of physically interrationalization deals with meeting a set of basic height and inclination the steering can wheel, or the of certain makes ofofautomobiles adjust the PC design driver changeable components, each component with a limfunctional needs through a range of physically interuserand can inclination modify the of parameters of his work environment. the steering wheel, or the PC height ited spectrum for each each of thecomponent functional needs components, with aselected. limproducts products) self-adapting, changeable userOther can modify the(customizing parameters of his workare environment. approach to standardization makes it one compospectrum for each of the functional needs selected. meaning they(customizing automatically adjust toare theself-adapting, context of their itedThis Other products products) of company flexibility, complementing resources approach to standardization makes it one compoutilization, which is namely to toensure the function meaning they automatically adjust the context of their for Thisnent personnel, of company flexibility,procedures). complementing resources which they are isdesigned. product utilization, which namely This to ensure the category function can for be nent(equipment, personnel, procedures). • This component variety must be detectable by the cliillustrated the electric withcategory floatingcan heads be or (equipment, which they arebydesigned. Thisshaver product and he must consider added value. There variety must itbeandetectable by the cli- are by washing machines fuzzy to determine illustrated by the electricusing shaver withlogic floating heads or the • Thisentcomponent ent numerous and he must consider an addeddetectable value. There are cliexamples ofitdiversity by the washmachines cycle (this is fuzzy progress compared to washing by best washing using logic to determine the ofvalue diversity detectable by the Nissan cliexamples ent but of no added for him. For example, customized with a few dozen towashing bestmachines wash cycle (this is progress compared washingpro- numerous value its for supplies him. Forinexample, but of notoadded rationalize 1993; it Nissan had more few often dozenhas washing machines customized withthea user grams, between which troubleprodecid- ent decided to rationalize itsits supplies in (stated 1993; itinhad more & than 300 ashtrays for vehicles Anderson ing).between Finally,which somethe products are has customized by the decided grams, user often trouble decid& thanPine, 300 p. ashtrays for its vehicles in Anderson 95). Product internal(stated diversity not detectable by the ing). Finally, some customized manufacturer (and,products in some are cases, by the salesperson). Product not detectable by p. the95). client is eveninternal worse diversity (screw-products are a clascases, by salesperson). manufacturer (and, in some Several non-exclusive solutions canthe describe this cus- Pine, by the is even worse areonly a classicalclient example of this type(screw-products of diversity). Not is it of Several non-exclusive solutions can describe this customization. of this type of diversity). Not only is it of sical example no added value for the client, but it often generates tomization. • The best known approach is the use of an appropriate added value for (due the client, but it logistics often generates costs to additional problems, modules, eachuse module being selected no additional is the of an appropriate • Thecombination best known of approach costs (duediversity, to additional increased tooling etc.). logistics problems, of modules, each being selected combination from a limited number of module interchangeable modules. additional tooling diversity, etc.). analysis of the classifiinterchangeable modules. from a solution, limited number More often than not, a technical This largely of used in the automotive industry, increased of thewill classifiThis largelya standardized used in the automotive often than not, a technical analysis cation of production and supply item-types show a notsolution, only implies productionindustry, process (to More of production supply item-types will show notminimize only implies a standardized production (to cation significant portionand of components with identical or avery settings and tool changes) and process a total substiwith identical or has verysevportion of components minimize settings and tool changes) and a total substitutability of modules from a same family during significant similar characteristics. This mushrooming effect of but modules a of same family during This mushrooming effect has sev- of tutability assembly, also thefrom design standardized assembly similar eral characteristics. and well known explanations: misunderstanding of standardized of assembly, also the designplatform andimpact well known explanations:item-types misunderstanding supportsbutcalled product (Meyer assembly and Lehner, eralthe of mushrooming on production the and impact of mushrooming item-types production(Not supports product platform risks" (Meyerof andmalfunction, Lehner, 1998) called to limit the "system product support service, the NIHonsyndrome malfunction, product Here), supportarbitrary service, design the NIH syndrome (Not 1998) to limit the "system risks" assured decisions, misleading of sufficient lastingness andofrobust, i.e. accept- andInvented sufficient lastingness and robust, i.e. acceptassured Here), arbitrary designweight, decisions, argument about the "proper etc.", misleading failure of the of constraints (electrical, mechan- Invented ing a of large spectrum of the of argument aboutsystem the "proper weight, toetc.", failurecreation etc.). information - all leading the faster ingical, a large spectrum of constraints (electrical, mechanto the up faster creationone. of It ical, etc.). system rather - all leading a new item-type than looking an existing • The use of adjustable components with reversible set- information a new item-type rather than looking up an existing one. It is clear that the economic approach to standardization adjustable components with reversible set• Thetings use of leads to immediate and cost-efficient customizathat be thedeveloped economic hereunder approach to standardization that will is of interest only if tings customizaof to a immediate componentand bycost-efficient different means (switch, is clear tionleads of interest if will be hereunder is detected fordeveloped mushrooming are not andonly progresof a component bya set different meansselected (switch, tionsoftware... ) to activate of functions from thatcauses causes for mushrooming are not detected and progressivelyeliminated. functions selected from software ... ) to activate a predetermined list;a set thisofdecision could be reversed. a predetermined this decision reversed. This techniquelist; of often selected could in the be design of elec- sivelyeliminated. Searching to optimize standardization supposes on one This technique of often selected in the design of electronic or electric components. Searching optimize standardization supposes on oneand hand thattofunctional needs are adequately assessed electric components. tronic that functional needs are adequately assessed translated into technical specifications, which isand facili• Theoruse of adjustable components with irreversible hand is facilispecifications, tated byinto the technical QFD approach (Quality which Function Deploy• Theadjustments use of adjustable components with irreversible corresponds to an immediate and cost-effi- translated by2),theand QFDonapproach (Quality Function Deploy-and ment the other hand that a coherent adjustments corresponds to an immediate cost-efficient adaptation of a component to theand needs either by tated 2 ), andreflection on the is other hand a solutions coherent considand ment relevant carried outthat on all of a component to the or needs by cient adaptation additional machining operations, by either a chemical reflection Itisiscarried outthat on the all optimum solutions solution consid- of ered adequate. obvious by a chemical additional machining operations, treatment, both irreversible. Suchorproducts are found in relevant It isproblem obvious is that solution ereda adequate. poorly stated notthe ofoptimum a great interest (itof even treatment, bothtrade, irreversible. Such products are manufacturfound in the cutting eyewear, custom bicycle nottheofmethod a great and interest (it even could stated reflectproblem discreditison not on its misthe ing, cutting trade, eyewear, custom bicycle manufacturor replacement doors and windows. Deferred dif- a poorly could reflect discredit onofthe not on to its examine misuse). Determination themethod solutionand portfolio ing,ferentiation or replacement often doors arises and fromwindows. this logic.Deferred difuse). of eliminating the solution aportfolio to examine is Determination based on a study priori none of the cusferentiation this There areoften thus arises severalfrom ways to logic. reach diversity of a finon a study eliminating priorialready none ofexists the custom modes described above. aWhat (interof a fin-and is based There thus several ways to reach diversity ished are product that avoid referring to systematic tomnal modes described above. Whatmust already exists be (intermanufacturing or supply) of course part of ished product that avoidTwo referring to systematic costly customization. standpoints determineand their course be part of nal the manufacturing supply) must alternatives or studied and the of new products planned costly customization. economical analysis.Two standpoints determine their the must alternatives studied andthe theconstraints new products planned take into account of the existing economical analysis. • A component is not a priori designed to be solely used must take into account the constraints of the existing production system or the one undergoing transforma• A component is not a priori designed to be solely used production system orthis thestandardization one undergoing tion 3. Optimizing nottransformaonly normally 3 . Optimizing this standardization onlybut normally 1. This type of diversity generates additional costs when the prod- tionresults in a decrease of productionnot costs, also in an is assembled an assembly line due to the work I. Thisuct type of diversityongenerates additional costs when thevariability prodresults in a decrease of production costs, but also in an
by theon assembly of optional some work2. The Quality Function Deployment is a value chain optimization uct caused is assembled an assembly line duecomponents to the workonvariability stations, specific line balancing andonscheduling prob- 2. Thetechnique. of needs to product service. It from expression of optional components some workQuality Function Deployment is a value chainsupport optimization caused by thecreating assembly 2(00). It probmay be lems creating (See Danjou, and Le Roy, 20 years by major originated Japan and of hasneeds beentoused for over stations, specificGiard line balancing and scheduling product support service. It technique, frominexpression interesting systematically offer a Itpopular the by approach companies. especially the first step20ofyears originated in Japan and has with been used for over Le Roy, 2000). may option be major to lemseconomically (See Danjou, Giard to and focusestoonsystematically a few alternative it isoption then posand if demand Antooverswitch from functional needs technical specifications. economically interesting offeroptions, a popular companies, especially with the tofirst step of the approach to relate to the describedoptions, here. Inversely, one can of functional these techniques found inspecifications. Revelle. Moran view and sible if demand focuses on method a few alternative it is then posAn and over-Cox, switch from needs toistechnical of least popular options. thetoeconomic interest to relate the method described here. Inversely, one can of these techniques is found in Revelle, Moran and Cox, siblequestion view1997. 1997. question the economic interest of least popular options.
266
266
increase flexibility, the company could then react more rapidly and easily to the conjunctural and structural transformations of demand44 .
2. TOWARD OPTIMIZING STANDARDIZATION Once the general modeling of the problem stated by linear programming, we will examine a few methodological problems stated by this economic rationalization, whether we call on the proposed optimization approach or not.
2.1. Baseline model For simple and non-expensive products (for example screw-products or the above-mentioned Nissan ashtrays), a sophisticated analysis is generally not mandatory as long as the reduction in the number of components for a function becomes self-evident from a classification simplification standpoint as well as for production and supply processes. processes. In other cases, the Renard approach is used to rationalize the range of relatively simple products, generally characterized by a unique quantitative criterion. Pareto analyses on need distribution based on this quantitative criterion (See, for example Anderson & Pine, chap. V, 1997) establish empirical approaches to determine a range of products, but the action is based on intuition and barely takes into account the economic standpoint. Product complexity also ensures that the technical analysis cannot have only one criterion. Formulation of this problem by linear programming using integers brings relevant response elements to these concems55 ; the matrix generators approach tested can easily implement them66 . The model will be presented with an example on engines. From a technical standpoint, we have n possible variants of a product7 , whether these variants are presently produced or only under study. These different variants will be identified with the subscript j. Moreover, we presume the demand analysis identified m segments, identified with the subscript i and characterized by a This category of concern is taken into account in DFA (Design 3. TIlis literature; refer to Nof, Wilhelm and Wameck.e, For Assembly) literature~ Ill, and to Redford and Chal, 1994. 1997, chap. Ill. Tarondeau, 1993 & 1997, Cohendet 4. Company flexibility (See Tarondeau. Lierenna, 1989, and Reix, 1997) is classically seen through and Llerenna, mobilized resources (equipment, tooling, personnel and procedures); product-related flexibility is generally highlighted but, dures)~ however, economic instruments for its improvement have not tbe perspective been studied much to our knowledge, and not in the treated here. 5. In an previous research document (Giard, 1990), we presented s.1n more restrictive perspective. In a briefly a similar model in a rmre (Giard, 1998), we proposed a system approach recent publication (Giant, matbematical programto production process modeling through mathematical ming indicating the mutual connections between various models tbe found in the literature on operational research and explaining the complex model creation mechanisms using a set of base components. Formally. Formally, the !be problem here combines a transposition of a nents. 1998, p. client assignment model to production centers (Giard, 1998. witb fixed costs that 46) and the recognition of non-linear costs with brack.ets (Giant, (Giard, 1998. 1998, p. 97), vary by brackets 97). except that tbe the total production of an item-type here is the tbe sum of productions requested by various segments. Murphy, Stohr and Asthana, Astbana, 6. TIlis This MGs approach is described in Murphy. 6. Rosentbal, 1996, Jacquet-Lagreze, 1997 and Giard, 1998. 1992, Rosenthal, metbod is proven in The technical and economic viability of this method TIle numerous achievements (for example. example, two important industrial tbese methods metbods were carried out at !be applications of these the IAE in Paris contracts; they tbey are described in !be scope of research contracts~ within the Andre, 1997. \997, and in Giard and Triomphe, Giard, Triomphe and Andre. Triomphe. 1996). 1996).
267
demand d d,.t • From an existing situation, we must normally have nsm, i.e. the variety of the commercial offer is lower than the variety of the demand. The problem at hand is first related to an eventual decrease in the number of variants (with no economic vision of the problem). A first technical analysis must be performed to characterize the various engines. Selected criteria relate to technical characteristics that are more or less well perceived by demand (power, pollution, consumption... consumption . .. ) and to characteristics of no interest for the client but essential in managing interfaces (weight, overall dimensions, engine mounting method on the chassis ... .. . ). This technical analysis is performed on a table similar to Table 1, I, with a literal in each table cell to indicate the precise positioning of each engine. Table 1: Technical characteristics of engines under study Characteristics Engines ... under study 1 2 - .. P 1 2 }j n A second analysis is to be carried out on technical characteristics of requested engines (Table 2). For each previously detennined determined criterion, an interval of admissible values (minimum power, for example) is then defined based on the type of characteristic or a list of acceptable occurrences is determined if the characteristic is qualitative (such as mounting mode, for example). A difficulty encountered is determining the "appropriate need" aimed at pleasing, as mentioned above. Table 2: Technical characteristics of requested engines Characteristics Requested Demand ...... engines 1 2 P dl 1 d2 2 d ji i dm m ID
Comparing Tables 1 and 2 establishes Table 3, in which coefficients aij take the value 1 if demand in segment i can be met by engine}, engine j, or 0 if it cannot be met (the numerical illustration being fully arbitrary). Table 3: Possibilities of satisfying the demand with the offer Variants of engines studied Market ... }j . .. n 2 3 segment 1 ... . .. 1 0 0 0 0 1 ... 1 1 0 ... 0 0 2 .. ... ... 0 1 0 0 1 3 '
... .... .. ...... ... ... ... ...... 1 ... 0 1 .. . .. ., . .... ... ... . .. 1 ... 1 ... 1 It seems realistic to impose that the entire demand segment be satisfied by a same variant. Consequently, we define the binary variable Xij' which will have the value 1 if demand of segment ij is met totally by an engine}, engine j, and o if it is not. It is of course useless to create a variable xij Xij ...
i ... m
... 0 ... 0
... 1 ... 0
... ...
7. Renault's Cleon C\~on plant, for example, manufactures more !ban than 400 possible engine variants.
8 if the corresponding parameter aij is null . Moreover, if nuns. we decide that this demand can be met by several variants, then variable Xij Xjj can have any value between 0 and 1. I. To force the demand of segment i to be met, we must establish the constraint of Relation 1 (which leads, for binary variables, to a non null value).
S1 j4::s;I Zjl +Zj2+ Zj3+ Zj4
O::s;yjl
n
}: ~ xi.t=l xij=1 ,fori= 1,....,m(meetingthedemand) 1, .... ,m(meetingthedemand) Relation 1 j=1 Under these conditions, production Yj of item-type j is then the sum of productions carried out for each segment (demand d;), this production could be null. This constraint is defined by Relation 2: m YJ-= }: J(productiomitem-typq~elation 2 ~ d{xijlotj= d{xij/orj= 1.. 1.. J(productiomitem-tyw)l?elation ;=1 i=1 To establish the annual production cost, we must introduce explicit hypotheses on the item-type costs function form. First, we will assume the cost functions are independent. This restrictive hypothesis will be removed later on. We will also assume variation in annual fixed costs per bracket, bracket. and that for each bracket the variable unit cost may vary but remains constant over disjoint value intervals; these very general hypotheses lead to a cost function of the type described in Figure 2.
r
_#:---1 ~,~
j3Zj4::S;Yj4
Generalizing the reasoning to all variants leads to the Relation 3 objective function and to the constraints described by Relations 4 and 5:
? (K}:
Relation 3 Relation
}: Zjk::s;l ,for ,jor j = =11... ~ Zjks1 ... n k=l k=1
Relation Relation 4
j
j ]) Min Z, with Z = = n ~ (ckY (ckYjk+KjJiZjk) jk+ K jf<:Zjk) j=1 ]=1 k=l
Kj
=
Mjk_1Zjt:!OYjk
Kj3
I
MjO = =0 MMJIj1
MJ2 M j2
M Mj3 j3
Mj4
Fig. 2. Production cost function This cost function is neither concave (which would imply that the mean production cost never increases Yj increases) nor convex (which would imply that when Yj the mean production cost never decreases when Yj increases) and that in the selected example it takes into account the phenomena of scale diseconomies occurring at the saturation limit. This non-linear cost function can easily be taken into account in the objective function of a linear program by introducing as many dununy dummy Yjk productions as there are Kjj value intervals (comprised Mj . k _1 and Mj,k' Mj •k, the upper threshold only is between Mj,k_1 =0) for which the variincluded in the interval, with MjO = Cjk is constant, and are all null except for the able cost Cjk one encompassing production Yj in its value interval and that is, is. of course, equal to this production (yj = = Yjh)' Yjh). From 2. the objective function becomes, the example in Figure 2, for the section related to production cost Y/ Cj "1Yj2 + K j2zj2 ) + z, with Z = (CjlYjl + Kj1zjl ) + ((CjiYj2 + KKj3 (Cj4Yj4 + K Kj4 j3 ) + (Cj4"Yj4 j3zj3 j4zj4) + ... with the following list of constraints9, added to those of Relations 1 and 2:
Min
(Cj3Yj3 (C/3Yj3
8. This special feature, easy to take into account in the problem description by a GM (see, for example. Giard, 1998. and JaquetLagreze, 1997) helps in significantly limiting the size of the problem. This convention makes Relation I useless in the form
This formulation fonnulation must be adapted to take into account positive or negative synergy effects related to simultaneous production of two or more engines on a production site. To do so, relevant and easy modeling tools have to be implemented 11. Let us examine a few examples. • Let us assume that manufacturing more than K variants translates into an increase r+ of annual fixed costs; we then have to create the y+ binary variable. variable, add to the objective function the term y+r+, and add Relation 6 constraints to force y+ to take the value 1 if at least K variants are produced (the objective function is designed to bring y+ to equal 0).
(K. i i}: (t
)
n Relation 6 y+ Zjk
1).
n
}: air ayx ....=l =l ~ j=l IJ j=1 I) negative, the first double constraint is in real9. As variables are not negative. XI s; s M1YI; MIYI; the selected formulation has the sole ity reduced to xl advantage of allowing rationale generalization.
IO.lf IO.If the cost function is concave (non-
268
(K
j
n ~ Zjk ) >KY~ j=1 k=1
~
Relation 7
(t
Relation Here again, this approach can easily be adapted to a7 Zjk]>KY]=1 k=1of variants or to several variant subsets. Among subset Here again, thisprecludes approachpositive can easily be adapted to a others, nothing or negative synergy or to several variant subsets. Among subset of effects to variants occur simultaneously for different variant orthe negative synergy others, nothing positiveof subsets or not, asprecludes a generalization previous comeffects to occur simultaneously for different variant ments. subsets not, asfinally a generalization the previous • We can or assume that certainof fixed costs varycomper ments. bracket with relation to produced quantities over a set • of Weitem-types, can assumenotwithstanding finally that certain costs vary per thefixed possibility offered bracket relationtotoinclude produced over a set for eachwith item-type in quantities its cost function its of item-types, notwithstanding possibility own variation of fixed costs perthe bracket. In thisoffered case, for each item-type include in its cost function its the Q engine subsetto being affected by variations of own variation fixed have coststoper bracket. In this case, fixed costs, we of simply adapt the formulation as the Q engine subset being affected by variations of follows: costs, we simply have to adapttothe as -fixed create variable w corresponding theformulation total producfollows: tion of the Q engine subset, which leads to Relation - 8: create variable w corresponding to the total production of the Q engine subset, which leads to Relation m 8: w= ~ y/= ~ ~ d(xij Relation 8 m j~Q ~Q i=l W= Y;= d{xij - add to the objective function the impact ofRelation variations8 ~QI=I in ~Q fixed costs K , following the same method used
L
L .L
wk
- add to the objective KO) function the impact of variations in fixed costs Kw!:, following the same method used for item-type ~ Kwtzwk;
KO) k=l
L
for item-type Kro~rok; - then adapt Relations 2, 4 and 5, which leads to Relations 9 to 11 k=l - then adapt Relations 2, 4 and 5, which leads to RelaKO) tions 9 to 11 Relation 9 ws~wk
KO) k=l
LWk
Relation 9
k=1 ~ zwJC'1 KO) k=l
Relation 10
w~ KO)
L zro,tSl
Mwk-lzwJC'wk
k=l
Relation 10 = 1... KW' with
Mwk=O Relation 11 Mrok- 1Zro,tSwk
Mrok=O 11 2.2. Method problems created by using thisRelation optimization approach 2.2. Method problems created by using this optimizaThe relevance of this modeling depends both on its the tion approach
use to describe alternate scenarios and on the costs used relevance of this modeling depends both on its the in The the economic function. use to describe alternate scenarios and on the costs used First, we are working at a detail level too coarse to prein the economic function. tend it accurately represents the processes used, the First, we are working a detail level too coarse to seapreeffective demand on theatproduction system, with its tend itandaccurately represents the processes used, the sonal random fluctuations, or the robustness of the effective demand on the production system, with its seaproduction system when faced with problems. This comsonal can and be random the on robustness the ment made fluctuations, for most helportools strategicof deciproduction system when faced with problems. This comsions: the important aspect is the relevance of the order ment can be made for mostdata help in tools on strategic deciof magnitude of collected volume or in value. sions:brings the important is the relevance This us back toaspect valorization problems.of the order of magnitude of collected data in volume or in value. The most important problems we may meet here relate This brings us back to valorization problems. to time or, more specifically, correctly defining the interThe most important problems we maymade meet here relate dependence through time of decisions within the to time or, more specifically, correctly defining the intercost accounting system. dependence through time of decisions made within the In the rare case where the studies alternatives relate to cost accounting system. new production from new equipment, it is easy to see the Inof thea rare case where alternatives relate to use dynamic versionthe of studies the model proposed, as the new production new equipment, easy to see the decision to meetfrom a demand segment byit aisgiven engine is use offor a dynamic version of hypotheses the model proposed, as but the taken all periods 12; other can be used, decision to meet a demand segment a given engine is seem more difficult to justify. The by separation between l2 ; other hypotheses can be used, but taken for all periods fixed costs and variable costs helps isolate start-up seem more difficult to justify. The separation between fixed costs and variable costs helps isolate start-up
269
269
investments (and thus avoid the problem of determining the depreciation) from direct fixed costs (especially perinvestments (and thus the problem of determining sonnel) that could varyavoid per bracket with relation to quanthe depreciation) from directdirect fixed costs costs (materials, (especiallyetc.). pertities produced and variable sonnel) could varyused per bracket with relation to quanAs this that information in the economic function is tities 13, produced andhave variable direct costs (materials,toetc.). dated we would to call upon actualization corAs thisweight information used inflows the economic function is rectly the monetary generated at different 13 , we would have to call upon actualization to cordated periods. Classic problems are encountered when this rectly weight the monetary flowsinvestment generated alternatives at different approach is applied to compare periods. problems are encountered when this and, moreClassic specifically, to determine the determination of approach applied to VAN compare investment the interestisrate used in calculus and the alternatives problem of and, more specifically, to determine determination of comparing solutions having differentthe useful lives. theWhen interest rate used in VAN calculus and the problem of the problem relates to a set of item-types that at comparing solutions having different useful lives. least partially exists, when it is the object of external item-types system that at Whenortheanproblem to a set supply internal relates production in aofproduction least partially exists, when is the object ofmodified, external which could eventually be itonly marginally supply or an internal problems production in a production system several methodology arise. which could eventually be only marginally modified, • Certain key components of a certain complexity are several methodology designed to be usedproblems in severalarise. finished products, some • of Certain certain complexity are whichkey do components not yet exist;ofthea term "main sub-assemdesigned to beisused finished products, bly" projects usedininseveral the automotive industry.some The of whichprice do not yet exist; the termcreates "main challenging sub-assemtransfer of such components bly" projects is projects used in 14. theA automotive The methodological decision onindustry. standardizatransfer of such creates challenging tion thatprice questions thecomponents economic hypotheses used to methodological projectsl4.must A decision standardizalaunch such components rely on on a cost function tion that questionsdecision the economic hypotheses usedand to that guarantees coherence in time launch such components must rely on a cost between strategic and tactical decisions (Seefunction Giard, that guarantees 1988 and 1992). decision coherence in time and betweenthis strategic and tactical decisions (See • Within framework and more generally, theGiard, deci1988toand 1992). sion stop production of a item-type (or of a group of • Within this can framework moreangenerally, the deciitem-types) lead to and support "item withdrawal sion toThis stop impact production of a item-type ofaccount a group by of cost". can easily be taken(or into item-types) lead l5 to. support an "item withdrawal the objectivecan function cost". This impact can easily be taken into account by • Existing standard costs are only relevant on a certain the objective function I 5. range of quantities produced or supplied; before apply• ing Existing standardthorough costs areinvestigations only relevantare onneeded a certain the method, to range of quantities supplied; before applyreconstitute the costproduced function or and, for supply, carry out the consultation method, thorough investigations needed to aing prior with suppliers on theare basis of volreconstitute thethat costcould function and, for supply, ume scenarios substantially deviate carry from out the a prior consultation with suppliers on the basis of volcurrent solution. ume standardization scenarios that could substantially the • The problem can occurdeviate within from a multicurrent solution. level bill of materials. Decisions taken at a detailed • The can occur multilevelstandardization then are basedproblem on hypotheses of within direct ademand leveldemand bill of originating materials. Decisions at a detailed and from othertaken item-types (using level then based on that hypotheses of directthedemand MRP type are mechanisms) are themselves object and demand originating from other item-types (using 12.Which can be translated by maintaining Relation 1 and folMRP type mechanisms) that are themselves the the object m lowing adaptation of Relation 2 which becomes: ~ the d, ·xfol.. 12.Which can be translated by maintaining RelationyI '(= and ,
)
;=1 m It
I)'
Relal10ns 3 and 5 being modified by thebecomes: inclusionYjl=.2. of thedu·xij' period lowing adaptation of Relation 2 which index, Relation 3 integrating actualization coefficients. .=1 13.With an 3appropriate sub-division, this helps in to Relations and 5 beingtime modified by the inclusion of take the period account possible3 learning effects on recurring costs, with values index. Relation integrating actualization coefficients. cananbeappropriate considered reasonably stable over timetake interval.. 13.that With time sub-division, thiseach helps in to 14.For a more complete presentation problems related ecoaccount possible learning effects on of recuning costs, withtovalues nomic products stable over their life cycle, refer to that canmanagement be consideredofreasonably over each time interval .. Gautier and Giard. 2000.presentation of problems related to ec014.For a more complete nomic management of products their cycle. refer to 4, forover a cost of life cl> , simply add the IS.Using notations of Relation Gautier and Giard, 2000. 15.Using notations of Relation 4. for a cost of • simply add the term Zjk}cI> in the objective function. Knowing that
{l- i {I- i Zjk} k=1
term K j
~
Kj k=1
in the objective function. Knowing that
k=1
Zjk= 1 if item-type j is produced, the cost cl> will not be sup-
2. Z·k=l if item-typej produced. the cost will not beitemsupported in this case alone.isGeneralization of a set 'I' of'P k=1 J K
ported in this case alone. Generalization ofj a set IJI of 1jl itemtypes is immediate {l-t}cI> , with ~ ~ Z 'k-'P< t. Kj ) jE'I' k=1 types is immediate {I-I}. with Z r1jl< I.
2. 2.
jEW k=1 J
of a standardization optimization. optuTIlzation. An independent analysis of these different problems cause the problem of elementary component standardization to depend on hypothetical aggregate component demands and cause the problem of aggregate component standardization to depend on hypothetical costs of elementary components. Convergence toward a set of coherent solutions can empirically be assured through a certain number of iterations, but this model could also be adapted to take l6 , which could lead into account the Bill Of Materials l6 to a model of undue size. • The creation of new components induces management costs related to increased diversity that are difficult to evaluate. In economic calculations made at the design stage, certain companies such as Intel or Renault apply different cost rates to new components and to reused existing components. This incentive to reduce diversity is astute but must be used with great caution within a complete reconsideration required when applying the approach suggested in this paper.
3. BffiLIOGRAPHY Anderson D. M. , Pine IT II J. (1997). Agile Product Development for Mass Customization : How to Develop and Deliver Products for Mass Customization, Niche Markets, lIT, McGraw-HiII, Build-to-Order and Flexible Manufacturing, McGraw-Hill, New-York. Baldwin C. Y., Clark K. B. (1997). (1997). Managing in an age of modularity, Harvard Business Review, vol. 75 sept oct 1997, p. 84-93.
D. & Meeraus A. (1988). GAMS : A Brooke A., Kendrick D. User's Guide, Scientific Press, Redwood City. Lierena P. P. (1989). (1989). Flexibilite, information, deciCohendet P. , Llerena sion, Economica, Paris.
F., Giard V., Le Roy E. (2000). Analyse de la robustesse Danjou E, des ordonnancements/reordonnancements sur ligne Iigne de production et d'assemblage dans I'industrie automobile, to be published in Revue Franfaise Franr;aise de Gestion Industrielle, O vol. 19, nnOl, vo!. l, Paris. Giard V. (1990). Analyse economique de la diversite en produclAE tion, working paper G 9012,1990, URA CNRS 1257, IAE de Lyon. Lyon.
V., Pellegrin C. (1992). (1992) . Fondements de I'evaluation revaluation Giard v., economique dans les modeles economiques de gestion, Franfaise de Gestion, n088, p. 18-31. Revue Franraise
a
V., Triomphe C. (1996). SIAD visant adefinir les services Giard v., offerts au personnel d' un centre de production et s' appuyant sur la programmation mathematique, working paper lAE de Paris (see http://panoramix.univ-parisl.fr/ http://panoramix.univ-paris l.frl 1996.03, IAE GREGORl96-03.pdf). (1998). Processus productifet productif et programmation lineaire, Giard V. (1998). Economica, Paris. V., Triomphe C., Andre R. (2000a). Organist : un Giard V., Systeme Interactif d' Aide ala Definition du niveau de traitement du courrier des bureaux de poste et des toumees d'acheminement d' acheminement a un centre de tri, to be published in vol. 1, n° 1. l. Cahiers du Genie Industriel, vo!.
a
a
V. (2000b). Evaluation economique des choix Gautier F., Giard V. de conception d'un produit nouveau : I'impact sur les m
(yr}; d{xjj)' which defines pro16.Simply replace Relation 2 (YJ= ~ d{xij)' j=l ;=1 d j related to it duction Yy j , as being equal to the sum of demands d; by binary variables
Relation: xij' with the following Relation: m }; d{xi.J-+ d{xj,-+ ~ }; ah/Yh'xhj' where item-type h is related to Yy,= =~ J i=1 ;=1 h=1 item-type j due to the production of 1I item-type h unit requires ahj unit of item-type j. (This item-type h is related to final 2). demands through Relation 2).
m
270
systemes productifs, accepted in the proceedings of the AFC 2000 Conference, 18-20 may 2000. Jacquet-Lagreze E. (1997). Programmation Lineaire - Modelisation et mise en lEuvre ll'uvre informatique, Economica. Meyer M. H., Lehner A. P. (1998). The power of product platM. l..ehner A. forms - building value and cost leadership, The Free Press, New-York. Murphy F., Stohr E E.,., Asthana A. (1992). Representation schemes for linear programming models, Management Science, vol. vo!. 38, n° 7, pp.964-99l. pp.964-991. Nof S. Y., Wilhelm W. E., Wamecke Warnecke H.-J. (1997). Industrial assembly, Chapman & Hall, London London.. Redford A., Chal J. (1994). aSJ'embly : principles (1994). Design for assembly: and practice, Mac-Graw Hill, New-York. Reix R. (1997). Flexibilite , Encyclopedie de gestion, (Y. Simon & P. Joffre Ed.), 2° edition, Economica, Paris Paris.. Revelle J. W. , Cox C. (1997) The QFD Handbook, ReveUe J. B., Moran 1. Wiley,. Rosenthal R. E. (1996). Algebraic Modeling Languages for optimization, Encyclopedia of Operations Research and Management Science, C.. M M.. Hyarris ed ed.,., Science , S. I. Gass & C Kluwer Academic Publishers, Boston Boston.... Smith P. G., Reinertsen D. G. (1998). Developing Products in Half the Time: Time : new rules, new tools, second edition, Wiley, New-York.
I.-C. (1993). Strategie industrielle, Vuibert, Paris. Tarondeau J.-c. I.-C. (1999). Laflexibilite Tarondeau J.-c. Laftexibilite dans les entreprises, PUF, Paris. Williams WilIiams H. P. (1993). Model Building in Mathematical Programming, 3ee edition, Wiley, New-York.