ࡱ> VwUz7 vLbjbjUU )7|7|zDl~~~~dT|P    3I\8::::::$q ^QY/3YY^&~~  &&&Y~8  8&Y8&&9,& D ˤ5"\<d4ŝ0d#$#&~~~~Decision Making Model Based on Grading Approach Drazan Kozak1*, Zeljko Ivandic2, Milan Kljajin3 1 PhD, Assistant Professor, Josip Juraj Strossmayer University of Osijek, Mechanical Engineering Faculty in Slavonski Brod, Trg Ivane Brlic-Mazuranic 18, HR-35000 Slavonski Brod, Croatia, Tel. +385 35 446 188, Fax: +385 35 446 446, E-mail: dkozak@sfsb.hr 2 PhD, Teaching Assistant, Josip Juraj Strossmayer University of Osijek, Mechanical Engineering Faculty in Slavonski Brod, Trg Ivane Brlic-Mazuranic 18, HR-35000 Slavonski Brod, Croatia, Tel. +385 35 446 188, Fax: +385 35 446 446, E-mail: zivandic@sfsb.hr 3 PhD, Associate Professor, Josip Juraj Strossmayer University of Osijek, Mechanical Engineering Faculty in Slavonski Brod, Trg Ivane Brlic-Mazuranic 18, HR-35000 Slavonski Brod, Croatia, Tel. +385 35 446 188, Fax: +385 35 446 446, E-mail: mkljajin@sfsb.hr Keywords: Product development, conceptual design, decision theory, grading design parameters Abstract: Complex customer requirements (CR) enforce the companies to make cheaper products with better performances on the global market in the shortest time. In that sense, a large number of the companies try to improve their products through anew, variable, innovative or adaptive design. However, old fashion sequential approach to product development is avoiding today. Modern product development concept is based on the product as a technical system. Such a development demands integrated solution and logistic approach through whole product life cycle. Therefore, it is necessary to map specified customer requirements on the particular design parameters (DP). This is arbitrary for the success of the product on the global market. Developed model in this paper connects the product as a technical system and designer as the member of the development team in the environment. Proposed decision-making model based on the mathematical formalisms analyses the acceptability level of n variant solutions (VS) related to CR as functional requirements (FR) by two different grading models. It is possible to estimate the level of the fulfilment of particular DP for any variant solution referred to different hierarchic structures of the evaluators team. One team of evaluators is organised on the centralised model and the second is based on the antagonistic approach. Through this analysis is it possible to rank particular variant solutions and determinate which solution is most acceptable comparing to CR. This model presents how to map CR via DP into VS with the aim that developed product satisfied the market demands identificated in DP extracted from CR. * corresponding author 1 Introduction Evaluations of different design options for a complex engineering product often need to take into account many performance attributes so that the economic and technical aspects of the product can be comprehensively assessed. Design process includes all phases of design task, from initial idea of possible design solution to final solution using needed analysis and synthesis procedures. Design solution presents generated answer of the design process on given design task inside some technical system. In this process transformation, from idea to principally variant solution, interaction between technical system characteristics and process of solution generating is present. Technical system is defined with exterior and interior properties, which are determining the complexity of the influence on the design solution (1, 2, 3(. Interior properties are determining the domain of principled variant solutions, which are managed by designer. External properties present the domain of definition of all design solutions and they are in interaction with interior properties. Design process is directed toward connection between functional products structure and its shape. The basis for interact connection between function and shape of the product make demands and desires, which are described by words and graphically. Described requirements are defined in the rule as a function and/or functionality. Graphical requirements are defined as a shaped model or as known technical system. Functional demands are accepted and defined as axioms in axiomatic design. Therefore, influence parameters of the design process are defined by axioms. Design parameters are defined as design properties. These design parameters have the influence on the design effectiveness and product development through next properties: minimising of all parts of design solutions, modul development and modularity princip use, using of standard components, designed parts have to be multi-functional, product design for multi-purpose use, design of the product has to be technological, simple assembly and disassembly, maximising of compatibility and single application. Axiomatic design is presented by three domains, which take completely the part in the flow of the design process. Design process in axiomatic design is usually presented by transformation of customer requirements (CRs) to functionality domain requirements (FRs), domain of physical requirements (DPs) and process variables domain (PVs), as is schematically depicted in Fig. 1 (4(. Axiomatic design enables mathematical formalism of design process.  EMBED Word.Picture.8  Fig. 1 Design process according to axiomatic theory (4( The product structure as a design solution is determined by relations between functionality and physical domain, while design process is defined by relations between physical and process domain. Defined axioms are usually given in matrix form using vector algebra. Design process as interaction between defined domains is given by matrix equation  EMBED Equation.3  according to (4(. Axiomatic design theory offers systematic approach to design. In this way formal description of design process is good tool for product development as a technical system. Design process itself begins with design goals defining, which have to satisfy customer requirements set. Continuously information processing between different domains (Fig. 1) creates variants of principally solutions (VS) associating requirements in one domain to the same in the other domain. Each functional requirement (FR) is independent inside its domain. These FRs should be associated to design parameters (DP) in physical domain. The number of possible solutions in creative phase of design for each set of FRs depends on designer knowledge and experience. 2 Criteria preference model and grades assignment Most decision problems involve multiple attributes (criteria). Weights may also be assigned to attributes to represent their relative importance. To assess an alternative on attributes, numerical values or subjective judgements can be used to differentiate one alternative from another. In this section, we use a belief structure to describe subjective assessment information. To this purpose, the model for procedural evaluation the importance of some criteria in conceptual phase of design has been developed. Criteria set (Kk) is conditioned by set of design properties (DPs). The grade of any criteria gives the answer about utility level of some design property. Preference of one variant solution to another is based on the criteria assessment. The evaluation model includes complete analysis of all alternatives assessing any particular solution according to the same criteria. This overall assessment should enable us to choose the best quality solution with the highest total grade value. The multiattribute decision analysis demands clear and argued mathematical procedure for criteria selection and grades assignment. Interactive model between variant solutions and criteria set may be presented as in Fig. 2.  EMBED Word.Picture.8  Fig. 2 Relations between variant solutions and criteria set Each principally variant solution  EMBED Equation.3  has own relations r with each criteria  EMBED Equation.3  from design properties set:  EMBED Equation.3  (2.1) where relation matrix is given by:  EMBED Equation.3  (2.2) The mapping  EMBED Equation.3  enables us to write equation (2.1) in a matrix form:  EMBED Equation.3  (2.3) what is proven in (5, 6, 7(. 3 Evaluation grades scale and preferences definition This chapter defines and describes the scale of grades and preferences as a set of qualitative and quantitative attributes. The assignment of grades and preferences to attributes leads to assessment of goodness of any variant solution. Quantitative attributes may be measured by using exact numbers, while qualitative attributes are those that may not be readily assessed by using exact numbers in the first instance due to their subjective nature. The preference of any criteria is derived from its importance for variant solution. There are more than one evaluation grades scale and according to (8( 26 different scales are tested and compared. Generally, the scale of quantitative grades may be given as a set:  EMBED Equation.3  (3.1) where particular numerical value from the set S for one criteria and one variant solution means the grade of its relation. The scale of evaluation grades may be defined as a set of qualitative attributes S ' also:  EMBED Equation.3  (3.2) The mapping of quantitative values from the set S into set of qualitative values S' is a trivial proof of set elements equality if the next condition is fulfilled:  EMBED Equation.3  (3.3) The matrix of relations R has the components as the relations between criteria and variant solutions. Any grade component (Hij) is conditioned by elements of S i S'( EMBED Equation.3 ), where  EMBED Equation.3 . All grades composes the  EMBED Equation.3  matrix of grades  EMBED Equation.3  or  EMBED Equation.3 . Key role in this evaluation grades model plays the procedure of assignment the grades to attributes, what is usually done by one or more evaluators. Evaluator (E) is in the rule widely educated expert from the multidiscipline team, who is deeply involved in the development of the evaluated product. Each evaluator from the set of n evaluators ( EMBED Equation.3 ) should assign its grade to each particular criteria of each variant solution. In that case, we have the set of evaluators and the set of grades, whit relations between. In this work, two decision models for grade assignment and preferences definition are compared, according to hierarchic organisation structure: centralised decision making model (one alone or one before all unit structure), antagonistic decision making model (everybody for itself or one related to all quasi-satisfied structure). 4 Grading of attributes in centralised model Evaluators belong to the team with central organised structure, which is composed from two levels: - presentation level, which is presented by the leader of evaluators team and - level of grade analysing, which is presented by evaluators analysts for grade assignment. The presentation level has the task to unite the common grade of hierarchic structure of evaluators for each evaluation criteria related to some known solution. The level of grade analysing presents substantial character of the model, i.e. all of that what analysts have to know about evaluation problem. Characteristic of this model is an obligation of the evaluator analyst to clear and explain assigned grade to the leader of the team. Between different decision levels an interaction is permanent present with the aim to control are the evaluation rules are following. The validity of the model has to be checked before its application to determinate the consistency in evaluating process. The relation between grades and evaluators has been performed as a surjection relation, not the bijection. Note that evaluators have not the constraint by choosing the grade type according to proposed scale related to considered evaluation criteria of given variant solution. It means that is possible to have the case where certain number of evaluators give the same grade to some alternative. The final criteria grade is composed from different grade values with different weights. Grade weight factor presents the frequency of appearing of the same grade in evaluation procedure. Generalising such a weight formulation, we can assume a set n of grades  EMBED Equation.3  for k-th criteria of r-th variant solution assigned by n evaluators and n real numbers as  EMBED Equation.3 , which mean weight factors of some attribute. Total weight factor is:  EMBED Equation.3  (4.1) It follows by normalising of all evaluators weight factors:  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  (4.2)  EMBED Equation.3  (4.3) Real numbers  EMBED Equation.3  are normalised weight factors of criteria grade for particular variant solutions. Regarding aforementioned expressions the real grade weight of any particular criteria  EMBED Equation.3  for any variant solution  EMBED Equation.3  may be written as:  EMBED Equation.3  (4.4) For  EMBED Equation.3  and  EMBED Equation.3 :  EMBED Equation.3  (4.5) For  EMBED Equation.3  and  EMBED Equation.3 :  EMBED Equation.3  (4.6) Grades matrix for r-variants and k-criteria created from n evaluators and n ( EMBED Equation.3 ) normalised weight factors is determined as:  EMBED Equation.3  (4.7) that is:  EMBED Equation.3  (4.8) Upper equation could be shortly written as:  EMBED Equation.3  (4.9) One can conclude that equation (4.9) presents mathematical formalisms in matrix form needed to describe evaluating of k criteria for r variant solutions by n evaluators. Taking equations (2.1) and (2.2) into consideration, it is possible to identify the evaluation procedure for any particular criteria and variants as:  EMBED Equation.3  (4.10)  EMBED Equation.3  (4.11) It is necessary to encircle all decimal numbers by analytical calculation of matrix components. 5 Antagonistic decision making model When evaluators belong to the team, which applies antagonistic model of decision making, every member of the team determinate the grades absolutely independent for itself. Are the grades good estimated might be visible assessing the goodness of the variant solution. In this way one incoherent and unconnected team with opposite means is created. Characteristic of such an antagonistic model is the freedom of decision making without constraints and mid-phase control. The reasons for incoherency might be different interests to the assessment of evaluated variant solutions, but also cultural, educational, personal etc. reasons inside the evaluators team. Each evaluator defines two numerical values for each criteria grade independently: - the first numerical value presents so-called minimal level (reservation) of grades factor, pointed as  EMBED Equation.3 , where  EMBED Equation.3  is numerical value  EMBED Equation.3  for  EMBED Equation.3  criteria on the reservation level - the second numerical value, which presents maximal level (aspiration) of grades factor. This value has to be converged as one desired value, pointed as  EMBED Equation.3 , where  EMBED Equation.3  is value  EMBED Equation.3  for EMBED Equation.3 criteria on the aspiration level. The matrix of grades on the reservation level is given by:  EMBED Equation.3  (5.1) The matrix of grades on the aspiration level is given by:  EMBED Equation.3  (5.2) Derived grade value presents the arithmetical meanvalue:  EMBED Equation.3  (5.3) (where are j the number of criteria ( EMBED Equation.3 ) and i the evaluators number  EMBED Equation.3 ). This solution is quasi-satisfied solution, because it presents the compromise in variant solutions evaluation between minimal and maximal grades values. 6 Conclusions In this work mathematical formalisms of the goodness of some variant solution toward given criteria in the conceptual phase of product development is presented. Proposed model assures clearly defined procedure for grades assignment, with the aim to rank all variant solutions. Highest value of the grade obtained by evaluation procedure means the best solution. Both presented models of grading make more easily the choice of the acceptable alternative, make the time of product development in the conceptual phase shorter and accept the different structures of evaluators team. Working examples of applying these grading models may be found in the reference (9(. References (1(Hubka, V.: Theorie der Konstruktionsprozesse, Springer Verlag, Berlin, 1976(2(Hubka, V., Eder W.E.: Theory of technical systems, Springer Verlag, Berlin, 1988(3(Duhovnik, J., Tav ar, J.: Elektronsko poslovanje in tehni ni informacijski sistemi, Ljubljana, 2000(4(Suh, N.P.: The Principles of Design, Oxford University Press, 1990(5(Veljan, D.: Kombinatorna i diskretna matematika, Algoritam, Zagreb, 2001(6(Kurepa, S.: Uvod u lineranu lagebru, `kolska knjiga, Zagreb, 1975(7(Blanuaa, D.: Viaa matematika, I dio, 1 svezak, Tehni ka knjiga, Zagreb, 1989(8(Saaty, T.L.: The Analytic Hierarchy Process, RWS Publications, Pittsburgh, 1996(9(Ivandic, Z.: The conceptual evaluation of design parameters, Dissertation (in Croatian), University of Zagreb, 2002 01=>?OP_`abcababcn IKL]^_`YZmnopҼҴī jEHUj7=B CJUV jUj3=B CJUV6 j6U j]mH sH  j[mH sH  j] j[ 6mH sH mH sH  6mH sH  5mH sH mH sH  H*mH sH  CJmH sH H*5H*5CJ601abaabcn " IJKLcdefghi$a$$a$sLuLijklmno~:` $$Ifa$$ & Fa$$`a$$a$fg_"`"|"}"z1$$IfTlK4 la $$Ifa$$a$$`a$$a$$a$/$$Ifl6$$4 la p~fg`"a"x"y"z"{"}"~""""""""$$*$,$R$T$V$X$$$$$$$$ %"%H%J%L%N%%%%ƿ j(EHUjb)@ CJUV js%EHUj5@ CJUV ju#EHUjb=B CJUV 6OJQJ jy!EHUjb=B CJUV 6mH sH mH sH  j UjYd=B CJUV jU5H*5CJ$ j] j[3}"~"""$$$% %f%h% &/&L&M&N&&& )P)Q)u)v)))N*O*$a$$a$$a$$a$$a$%%%% &&!&"&#&$&A&B&I&J&L&M&N&&&((((Q)R)e)f)g)h)u)))D*E*F*O*P*c*d*e*f*****++++,+Ŀĺĥĥ~j:B CJUVmH sH 6 j2EHUj=B CJUV 56CJ56 jZ0EHUj}=B CJUV j]5 j[55CJCJ$ j] j[ jC.EHUjG@ CJUVmH sH  jU j+EHUj[E@ CJUV0O*q*r*++9+:++,,.;/EHUj=B CJUV j<EHUj @ CJUV j9EHUj4=B CJUVmHnHu j7EHUj=B CJUVmHnHu6H*65 jU jP5EHU/---/)03s6t6666666666666666667777O7P7Q7d7e7f7g7r7s7777777777»|j@ CJUV jLEHUj{ҳ@ CJUV CJmH sH mH sH  CJ mH sH  j/JEHUj@ CJUV jHEHUj@ CJUV j EEHU%jd@B 5CJUV\^JmH sH  56\]5\5 jU jBEHU.r7s7777!8"8K9k9999:::::::::$;%;D;E;;;<$a$$a$$a$$a$7777777778888!8"8/808C8D8E8F8889999 9!949596979J9K9L9_9`9ƻԸԙԀuejd@B CJUV^JmH sH j6YEHUmHsHj>Գ@ CJUVmHsHjVEHUmHsHj Գ@ CJUVmHsH jTEHUjӳ@ CJUVCJ jREHUmHsHjVӳ@ CJUVmHsHmHsHjUmHsH jOEHUj"@ CJUV jU jNEHU%`9a9b9o9p9999999999999999999999999999999:濴椙拀rgj?iEHUmHsHjɳ@ CJUVmHsHjigEHUmHsHjP@ CJUVmHsHj;cEHUmHsHjd@B CJUV^JmH sH jaaEHUmHsHjs@ CJUVmHsHj_EHUmHsHj7@ CJUVmHsHmHsHjUmHsHjk[EHUmHsH#:::::.:/:=:>:U:V:f:i:j:}:~:::::::::::::::::::$;%;&;9;ž֧֕t֏l5CJmHsHj+xEHUmHsHjd@B CJUV^JmH sH  5mHsH CJmHsHj\qEHUmHsHjd@B CJUV^JmH sH  CJmHsH jKoEHUjӳ@ CJUV jU6mHsHjUmHsHjkEHUmHsHjd@B CJUV^JmH sH %9;:;;;<;D;;;;;;;;;<<<<<<<<<<<<<<</=V=?@@@@@@@@@@@@@֝ւwij@ CJUVmHsHjZEHUmHsHj2e@B CJUV^JmH sH jEHlUmHsHj$e@B CJUV^JmH sH CJ jEHUje@B CJUV^JmH sH  jU55CJ6]mHsHjUmHsHjh}EHlUmHsHje@B CJUV^JmH sH )<<<<<<</=0=1=V=W=>???@@@HAIAxByBBBB$`a$d^$`a$$`a$$a$$a$$a$@@@@@AAA AAA"A#A$A%AAAAAAAAABBBBBB#B$B7B8B9B:B>B?BRBSB潲梗~ncjEHUmHsHj^e@B CJUV^JmH sH jEHUmHsHj@ CJUVmHsHj"EHUmHsHjSe@B CJUV^JmH sH j>EHUmHsHj@ CJUVmHsHjΓEHUmHsHjIe@B CJUV^JmH sH mHsHjUmHsHjEHUmHsH&SBTBUBBBBBBBBBCC*C+C,C-C`!=^JDeejQ@'DVxZklTE>3mіƮ *AT R ŵᇖkw*ȫb-T֒( ?h"DLJhhg}mnsg{;g:<l_>@6L P #/EPFb5-%v#U6&dAtRK(-]kih-/..G'QG[HF@䪀yPV^F΋7 P# ɓ.(|+pqoA.2[ϡڿL=QfY3GK:_KH(§L Gw5:“d%Dڋ }Ĺل6&IVUM3h9k\{ȟL2nG_D~}@D~ݳ"+YVRi q~eJYO`>&aQ "/nX n(GP̅ᬲ׍u.k28Y.l5Kv:T,2:E1H}sUƈr:M NZBpPK 6O⑳N\ecqWK@ϰȁ^Eι9&#l n]N"b)q1@m}#"BR-KgL+)YX |B}!q%PY+|ʑ|6izb/&쫰\z?Z4 \z̪պ1mFeuf#MX)}Ynu-+ag2a%lf7 W43*uSlh#t,K}̣ VPm4MWӪh1 A?FYgS:VXV"ӱ1EV1x6e "?ّktHjEnԊs-K}>M7]z6;iHN>:2QVGetk^ƫVL!G +9+C iY ;cEpQ+γLf?275NV+UPRECR+V2}Xt"TUToZqimwVɰVǒq_ǖQX\+3e0RֱmZ;b'; ,K}>M7}v;iHN>Vj^G]Z9Qz5:Zq R*XzV"zVmlByo6-CrVz9(VVf9:iTvύ,Wfَq^|eJڏlUE?[c_a#MZM :o=jg gF ^#6c0'z];>qmumu|ؔ\D_ӄ}:2v a%vS2g:1aFK+bǜAVD XA۵9W|5I=isAlw=>`d/צdz~X#g=cec-3 ö=&dE-{l]/h%D5Z.\KQ|j'}+lԂObԲvkēon|vrJq[bvh!K|%⑾ Yė!N ?sSDd@TB + S A&? 2B֜tjFf|>`!B֜tjFf| pXJ_xcdd``~ @c112BYL%bpu`!Uc,^63f:05%xxU$HPRTj( (]i}uuu{lߧ@ʢ "XHŠɽ7ɝ$r;̙sU(5HUi*ՠqDL_͇dN;An8fJqY<q>DMJ!:k{۫40sjNѶ9ڨUk:j2Fw7N8*]3k?(0>#UFWpLgT猄+4K{5a ([}it>#5) lm $)|MkسNCE m1 Rclu(#3eCvjosG k#Z·ՉkZG^fkjDkޯ#EWՁ'ZgN!%߯ HZF-1:Ѫ 3 Ȫ753}Ž&0*7J-DrGK +8;3#Ȏkodvd1^pm\>үzU8SB#(z.-}G2|.-%wzM/z2\˒sWr^"w:T0 *5] ݤpMw:{k"שNVwn21k,ʙ핉W[ڌg12E﷣wZ)39GTʓQtL |GJ 9 _𕈳#!r3b 3v{P0R=7n ".w.+v^ Wk2M*沅^t~ dμ7?󩹯7G{.ܣ瓣}Xs,GW(6G+%͘vɖepMgs#w\=#G's#H-zzJ֕^u&k*35w?cG+|/k^ֱ$T_^,6G7Vx.=+iw <(r]7>̣Wd]XFoLAȯQKkIssZҜTh߾K3E  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLNOPQRSTWyYZ[\]^_abcdefghijklmnopqrstuv|}~Root Entry F ߤxData M6WordDocument)ObjectPool | ߤ_1111307166C F||Data X1Table` ,CompObjhDdTTJ 1 C A? 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