Study on the Mathematic Model of Product Modular System Orienting the Modular Design

Transcription

1 Natre and Science, 2(, 2004, Zhong, et al, Stdy on the Mathematic Model Stdy on the Mathematic Model of Prodct Modlar Orienting the Modlar Design Shisheng Zhong 1, Jiang Li 1, Jin Li 2, Lin Lin 1 (1. College of Mechatronic ering, Harbin Institte of Technology, Harbin, Heilongjiang , China, lijiang@hit.ed.cn; 2. Third Department of Aerospace Indstry Company, Beijing , China Abstract: Modlar design is a key technology of implementing irtal organization and mass cstomization. Orienting the modlar design a mathematic model of modlar system is presented and is sed to realize the mathematic description of modlar decomposing Top-Down process. Extending Boolean matrix of modlar interface relationship is gotten by reerse reasoning strategy of modlar interface relationship. The application based on the mathematic model of modlar system and extending Boolean matrix of modlar interface relationship is conclded, which proides theory fondation for the researching of modlar design software system. [Natre and Science, 2004,2(:61-67]. Keywords: modlarization; modlar design; modlar interface; mass cstomization 1 Introdction Modlar design technology, as the adanced form of standardization, is a key technology of implementing irtal organization and mass cstomization. At present, the research of modlar design is concentrating on the realization of design, prodction, assembly and maintaining of the specific prodct sing modlar design, and doesn t pay attention to the theory stdy. As a reslt, modlar design is still a philosophy discssion and a thinking perfection, and can t form a general theory that instrct indstry to implementing modlar design. Throgh researching on the process of modlar design, this paper presents a mathematical model of prodct modlar system and realizes the mathematical description of the Top-Down modle partition process. The application based on the mathematic model of modlar system and extending Boolean matrix of modlar interface relationship is conclded, which proides theory fondation for the researching of modlar design software system (Tong, 2000; Hang, Prodct Modlar Modeling 2.1 Prodct modlar system mathematical model Modlar prodct consisted of basic modles, which hae independent fnction, independent strctre and sitable granlarity (Figre. Basic modle refers to the most basic nit that constittes modlar prodct Winged Action Part Power Eqipment Control and Gide Part Filler Detonating Fse Radiate Series Gide Fig. 1. Modlar tree of winged missile and constrctional relationship, or the rote of inpt and otpt of stream, energy, information, and so on. The speriority of modlar design is sing existing standard modles to manfactre new prodct by interchange and combination. From the point of iew of systems engineering if we take the basic modles make p of the prodct as inpt of the modlar design and take the 61 editor@sciencepb.net

2 Natre and Science, 2(, 2004, Zhong, et al, Stdy on the Mathematic Model modlar prodct as the otpt of modlar design, modlar prodct is a system in which basic modles associating and effecting each other in a certain exterior enironment (Chen, We call it prodct modlar system, and its mathematical model can be: P = f ( M n, W n n ( Where P represents the prodct modlar system, M n represents the basic modlar ector which make p of the modlar prodct, M = m, m2,, m. W ( 1 L n n n represents the Boolean matrix of interfaces relationship in basic modles, which shows the relationship of the prodct basic modles. W n n = ( ω kl n n, where k, l = 1,2, L, n, and correlatie coefficient ω kl determines if there is a relationship between m and m, k 0, k = l ω kl=ω lk= 0, k l, m and m hae not relationsh ip k l 1, k l, m and m hae relationsh ip k l We name formla ( as mathematical model of prodct modlar system. The meaning of the model is that it abstracts the prodct modlar system as a modle s netty architectre, which takes the modles selection and the definition of their correlatie as system inpt, and takes the prodct fnction and its strctre as system otpt. And this makes the modlar design easier to nderstand. The mathematical model of prodct modlar system can be sed to describe the process of partitioning prodct modles. The process of partitioning prodct modles can be descript as: prodct first class modle second class modle basic modle, we call the modles that is not the top prodct and can be partitioned again, sch as first class modle, second modle and etc, as middle modle, the process can be described as prodct middle modle basic modle. If mid represents the modle (=1,2, in (=1,2, class, the second modle in the first class, power eqipment, will be showed as n l 1 mid 2. Sppose mid can be partitioned into s(s=1,2, modles, and these modles make p of modle ector Ch M = ch m, ch m, L, ch m. According to s ( 1 2 s formla (, middle modle as: mid can be described mid = f ( Ch M s, Ch Ws s (2 Where Ch W s s = (ω fg s s Throgh formla ( and (2, the process of partitioning prodct modles be described as a series of modle ectors and Boolean matrices of interfaces relationship in modles. Combining with modlar tree, these modle ectors and Boolean matrices describe the strctre and the hiberarchy and netty architectre of prodct system at the same time. Bt throgh the modle ectors and Boolean matrices of interfaces relationship in modles we cannot get the Boolean matrix of interfaces relationship in basic modles directly. The process of partitioning of prodct modles is Top-Down and classified. In each partitioning action we get the son modles of the middle modle or prodct and the relationship among them, for example, action part, power eqipment, control and gide system and shell part are partitioned from winged missile prodct by a partitioning action, and at the same time the relationship among them and Boolean matrix of their interfaces relationship is determined (Figre. 2 (a. Bt we cannot get the correlatie among the modles belong to different father modles bt at the same class, for instance, the correlatie between stabilization system in control and gide system and helm in shell part, or engine rack in power eqipment and missile body in shell part, etc. This paper presents extending Boolean matrix of modlar interface relationship and reerse reasoning strategy of modlar interface relationship to sole the problem. 2.2 Extending boolean matrix of modlar interface relationship and reerse reasoning strategy of modlar interface relationship Modlar interface relationship can be diided into inter interface relationship and oter interface relationship. The inter interface relationship refers to the interface relationship among the modles belong to a same sperior modle, for instance, the relationship between stabilization system and control and gide system; the oter interface relationship refers to the interface relationship among the modles belong to different sperior modle or between the modle and the enironment, for example, the correlatie between stabilization system in control and gide system, and helm in shell part, or engine rack in power eqipment and missile body in shell part. Especially, we define the relationship between, f,g=1,2,,s, and ω fg has the same way as ω kl to get its ale editor@sciencepb.net

3 Natre and Science, 2(, 2004, Zhong, et al, Stdy on the Mathematic Model Aero Enironment Gide Control and Gide Detonating Fse Action Part Filler Part Power Eqipment Radiate Series (a Modlar relationship in partitioning Aero Enironment Gide Control and Gide Detonating Fse Action Part Filler Part Power Eqipment Radiate Series Figre 2 (b Real modlar relationship Modlar relationship in partitioning and real modlar relationship modle and the oter system enironment as system interface relationship. This relationship belongs to oter interface relationship. Boolean matrix of modlar interface relationship only describes the inter interface relationship and cannot show the oter interface relationship. In order to describe the process of the modlar partitioning process and the strctre of prodct modlar system, we introdce the extending sbordinate modlar ector ExCh M s + 1 = ( ch m1, ch m 2, L, ch m s, om, om represents oter modle that don t belong to a same sperior modle in the same class or be the oter enironment. Based on the extending sbordinate modlar ector, extending Boolean matrix of interface relationship in the sbordinate modles of the middle modle can be showed as ExCh ψ, W ( s + ( s + = ( ( s + ( s + where b, c = 1,2, L, s + 1. When b, c 1,2, L, s, ψ = and ω hae the same way to get there ales. They fg show if modle ch mb and modle ch mc hae interface relationship, and we call them inter correlatie coefficient. When b = s + 1 or c = s +1, ψ shows that if modle ch mc or ch mb hae the interface relationship with om, we call it oter correlatie coefficient. If an interface relationship is exist between b and c, then ψ = ψ = 1, else cb ψ = ψ cb = 0. If Exmid is sed to represent the middle modle that its oter interface relationship is defined, the extending mathematical model of middle modle can be described as = f ( ExCh M s+ 1, ExCh W( s+ ( s+ Exmid (3 Accordingly, if we se ExM 1 = ( m1, m2, L, mn, om to represent extending basic modlar ector of prodct modlar system, and se ExW ( ( = ( ψ eq ( ( ( e, q = 1,2, L, n + to describe the interface relationship 63 editor@sciencepb.net

4 Natre and Science, 2(, 2004, Zhong, et al, Stdy on the Mathematic Model among basic modles or between basic modle and system enironment, and tilize ExP to represent the prodct system, then extending mathematical model of prodct modlar system can be described as ExP = f n ExW (4 ( ExM +1, ( ( The extending mathematical model of prodct modlar system not only shows the relationship of system basic modles, bt also defines the correlatie of basic modles and the enironment. The model frther ncoers the essence of prodct modlar system, and can be sed to describe the process of partitioning prodct modles, which is the coming into being process of the modles and the extending Boolean matrices of their relationship. This paper presents the reerse reasoning strategy of modlar interface relationship to get the relationship of basic modles, that is to say, the extending Boolean matrices of interface relationship in basic modles. Reerse reasoning strategy of modlar interface relationship is that sing the modlar tree and the extending Boolean matrices of modlar interface relationship formed in partitioning process, and throgh interface matching to sperpose the extending Boolean matrices of modlar interface relationship of eery class to the last class by breaking p the interface relationship in middle modles and transforming it to the interface relationship in basic modles, in the end obtaining the extending Boolean matrix of interface relationship in prodct basic modles. In order to pt in force the reerse reasoning strategy of modlar interface relationship, and realize the coming into being of the extending Boolean matrix of interface relationship in basic modles atomatically, the prodct modlar design system mst hae the ability: ( The oter interface relationship can be transferred and their information is eqal. That is to say, the oter interface relationship can be diided and distribted to the oter interface relationship in the sbordinate modles, and parameters of the interface will not change. (2 Descript and definition of modlar interface information can be qeried and distingished. So we can confirm if the modlar interface relationship is matching according to the qalitatie and qantitatie information. The reerse reasoning strategy of modlar interface relationship can be described as next fie steps: ( According to principle of partitioning modles, the prodct is broken p to sitable classes. Partitioning process of middle modles incldes confirming sbordinate modles nmber, describing basic information of sbordinate modles, defining interface relationship (inclding inter interface relationship and oter interface relationship of sbordinate modles and describing interface information of sbordinate modles. We get the extending Boolean matrices of interface relationship in the sbordinate modles of prodct and middle modles, and at same time we get the modlar tree that shows the hiberarchy of prodct system. Sppose that the max class of modlar tree is z, if array nm( (=1,2,,z represents the modle nmber of the class, ( represents the sbordinate modle nmber of the modle (=1,2,,nm( in class, then according to formla (3, the middle modle of eery class can be showed as mid = f ( ExCh M chn ( ( + 1, ExCh W ( chn ( ( + ( chn ( ( + where ExCh M ( + 1 = ( ch m1, ch m2, L, ch m (, om ExCh W ( chn ( ( + ( chn ( ( + = ( ψ ( row, col ( chn ( ( + ( ch( ( + row, col = 1,2, L, ( + 1.Then the sbordinate modle of class is the modle of modle +1, and ch m = mid ( i = ( x, j = 1,2, L, (. j i+ j x= 1 Especially, when a basic modle appears in middle classes, it will be taken as the sbordinate modle of itself. (2 Sppose that the modle nmber of last class is n, then nm ( z = n. We arrange the basic modles of the prodct to the basic ector of the prodct modlar system according to their top-down seqence in modlar tree. Combing with the oter modle, the basic ector is extended to the extending basic ector ( m1, m2, L, mn, om (where z m, and i = mid i i = 1,2, L,n.The extending basic ector determines the colmns and the rows of the extending Boolean matrix of interface relationship in basic modles. Let 64 editor@sciencepb.net

5 Natre and Science, 2(, 2004, Zhong, et al, Stdy on the Mathematic Model correlatie coefficient be zero, that is to say, ExW. ( ( = ( ψ eq ( ( = (0( ( Table 1 is an extending Boolean matrix of interface relationship in winged missile basic modles. 1 (3 For mid z ( = 1,2, L, nm( z, if z 1 mid is a middle modle, then its sbordinate modles mst be basic modles. Sppose that row, col = 1,2, L, z (, then ψ in ( row, col z 1 Exch W and ψ in (i +row, i+ col ExW( ( n + ( z ( + ( z ( + are eqal. According to inter correlatie coefficient of the extending Boolean matrices of interface relationship in the sbordinate modles of middle modles, we may get inter correlatie coefficient of the extending Boolean matrix of interface relationship in basic modles, and these inter correlatie coefficients are close to the diagonal of the extending Boolean matrix of interface relationship in basic modles, see the part of Figre 1. For example, we can get inter correlatie coefficients of basic modle missile body, missile wings and helm in shell part throgh the extending Boolean matrix of interface relationship in the sbordinate modles of modle shell part. (4 For modle mid z 2 ( = 1,2, L, nm ( z 2 z 2 in class z-2, if mid is a middle modle and its sbordinate modles are basic modles, then we replace inter correlatie coefficients of the basic modles with inter correlatie coefficients of the sbordinate modles of modle z 2 mid sbordinate modles of following the step 3. If the z 2 mid are middle modles, we sppose that row, col = 1,2, L, z 2( seeing abot the middle modle ch 2 m Exch W ch 2 m z row z z 1 col in ( z 2( + ( z 2( + 1 where ψ 1. ( row, col = and and According to the characters of modlar design, the inter interface relationship in the father modles is the oter interface relationship in their son modles, so if modle z ch 2 m row and ch z 2 m col hae a inter interface relationship, it means that there are oter interface relationships in their son modles. In this sitation, in the extending Boolean matrices of interface relationship in their son modles we shold find ot the basic modles that their oter correlatie coefficients are 1. According to the ability that the interface information can be qeried and distingished, and the character that the interface relationship can be transferred and its qantity is eqal, we can find the basic modles that are the sbordinate modles and hae interface relationship by the type of interface relationship, the main parameters of interface relationship and the condition that the middle modles are partitioned. And we define the correlatie coefficients accordingly as 1. For example, the confirming of ψ and 2, 6 ψ, and so on in Table 1. 6, 10 z 2 When sbordinate modles of mid are middle modles or basic modles, the management is same as aboe, and at this time, if basic modle and the sbordinate modles of middle modle hae a interface relationship, that is to say, the relationship is the inter interface relationship of basic modle and the oter interface relationship of the sbordinate modle. (5 Repeat step 4, and transfer the inter interface relationship in eery middle modles to oter interface relationship in their sbordinate modles, so interface relationships are transferred downwards and finally trn into the interface relationship of basic modles. And the system interface relationship of prodct also can be transferred to the oter interface relationship in basic modles. See the oter interface relationship of and m 8 m 5 in Table 1. The whole extending Boolean matrix of interface relationship in prodct basic modles is established. Reerse reasoning strategy of modlar interface relationship realizes establish of the extending Boolean matrix of interface relationship in prodct basic modles. The matrix shows the relationship in basic modles and in basic modles and the enironment. Combining with the modlar definition model, we can realize the ato form of the extending Boolean matrix of interface relationship in basic modles editor@sciencepb.net

6 Natre and Science, 2(, 2004, Zhong, et al, Stdy on the Mathematic Model 3 Application of the Model Throgh the mathematics of prodct modlar system we can know the interface relationship of eery two basic modles. If we se to ( k = 1,2, L, n express the basic modles aggregate which hae R k relationship with the basic modle R m k, then = { r r = m ω and r 0,l = 1,2, L n} (5 k l l l lk l, 0, ω = 0 lk Where r. l = ml ω lk = ml, ω lk = 1 Table 1 The extending Boolean matrix of interface relationship in prodct basic modles of Figre 1 m 1 m 2 m 3 m 4 m 5 m 6 m 7 m 8 m 9 m 10 m 11 m 12 Om M M M M M M M M M M M M om Note: the space in the Table is 0. And we can know if basic modle has relationship with oter enironment throgh jdging the ale of ( n +k m k ω and ω ( +. The basic modles k n aggregate that relates with oter enironment is R { r r = m, and r 0, l = 1,2, n} n + 1 = l l l ω l( l L, Gide Fig. 3. Relationship figre of For example, R 10 is a basic modles aggregate that relate to the basic modle m 10. According to Table 1 and formlation (5, we can get { m, m, m, m, m, m m } R = , Specific relationship is showed in Figre 3. According to the relationship in basic modles, the hierarchy and inclsie relationship between basic modles and middle modles, we can jdge if there is a relationship between random two modles (middle modles or basic modles. The jdgment of the relationship in modles proides a possibility of partitioning the task of modlar design and demonstrates which two companies hae relationship in the process of prodct concrrent design. The application of the mathematic model of prodct modlar system also can be researched and discssed as follow: ( Realize the atomatization and intelligentization of prodct modlar synthesis sing the editor@sciencepb.net

7 Natre and Science, 2(, 2004, Zhong, et al, Stdy on the Mathematic Model technology of neral net. We can take prodct basic modlar ector and the extending Boolean matrix of interface relationship in basic modles as inpt and take prodct fnction and strctre as otpt, sing the technology of neral net to realize modlar synthesis. (2 Jdge if the modlar partition is reasonable by the extending Boolean matrix of interface relationship in basic modles. The degree of the prodct modlarization relates to the interface nmbers of prodct basic modles directly. We can Jdge if the modlar partition is reasonable by compting interface nmbers in the extending Boolean matrix of interface relationship. Denmark scholar Jliana Hsan has done some research (Hsan, (3 Application of the extending Boolean matrix s ariation. The extending Boolean matrix of interface relationship in basic modles describes the common interface relationship in basic modles. We can map the matrix to different departments of the company, sch as design, prodct, assemble, and so on. 4 Conclsion Orienting the modlar design a mathematic model of modlar system is presented and is sed to realize the mathematic description of modlar decomposing Top-Down process. Base on the model, mathematic expression of the prodct interface relationship is discssed. Extending Boolean matrix of modlar interface relationship is gotten by reerse reasoning strategy of modlar interface relationship. The application based on the mathematic model of modlar system and extending Boolean matrix of modlar interface relationship is conclded, which proides theory fondation for the researching of modlar design software system. Acknowledge This research is spported by the High-Tech Research and Deelopment Program of China (No. 2001AA References [1] Chen L, L J. Theory and application of systems engineering. Academic Magazine Pblishing Company, Beijing, China, [2] Hang C. Oeriew of modlar prodct deelopment. Proc Natl Sci Conc ROC(A, 2000, 24: [3] Hsan J. Impacts of spplier byer relationships on modlarization in new prodct deelopment, Eropean Jornal of Prchasing and Spply Management, 1999, [4] Tong S. Modlar principle, design method and application., Standard Pblishing Company of China,Beijing, China, editor@sciencepb.net