Models of Postpress Processes Designing Vsevolod Senkivskyi[0000-0002-4510-540X], Alona Kudriashova[0000-0002-0496-1381], Iryna Pikh[0000-0002-9909-84444], Ivan Hileta[0000-0001-6935-2854], Oleh Lytovchenko [0000-0002-5637-6934] Ukrainian Academy of Printing, 19, Pid Holoskom St., Lviv, 79020, Ukraine senk.vm@gmail.com,kudriashovaaliona@gmail.com, pikhirena@gmail.com, hileta@gmail.com, lytowchenko@gmail.com Abstract. The study has highlighted a set of factors influencing the quality of postpress processes designing. A fuzzy preference relations on a given set of al- ternatives has been formed. The matrices of relations of alternatives by Pareto- oriented factors have been constructed and the non-dominant subset has been defined. Based on a fuzzy preference relation, the corresponding sets of non- dominated alternatives have been distinguished. The optimal variant of the ana- lysed process has been selected according to the maximum value of the utility function. Modelling of key operations and functions of postpress processes de- signing has been implemented through the development of a context diagram and a decomposition diagram of IDEF0 models, one of the elements of which are alternatives for this process implementation. Keywords: postpress process, factor, alternative, fuzzy preference relation, uti- lity function, optimization, model, IDEF0 modelling. 1 Introduction The final stage of the book production technology which includes finishing and bin- ding processes is often mistakenly identified with a set of mechanical, cyclically re- petitive actions, depriving them of a highly intellectual information component. This approach increases the likelihood of partial or total rejection of the print run. A typical mistake is also the mismatch of manufactured products to their functional and opera- tional characteristics. Thus, for example, for an edition that should serve for decades, one can use an adhesive binding of organic origin that is not suitable to meet the re- quirements and choose the wrong finishing material. Computer-aided automation does not presents the expected results, since the pro- cedures used are not integrated into a single, indivisible system. Under such condi- tions, post-operational information support is appropriate and necessary, which will result in a predictive assessment of the future products quality. Such an approach in the presence of uncertainty conditions requires the formation, calculation and multi- criteria assessment of alternative options for the implementation of postpress pro- cesses on the basis of fuzzy preference relations and determining the optimal one, which will result in obtaining the proper quality products. The specified procedure Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0) 2019 DCSMart Workshop. becomes an important component of IDEF0 models of the implementation of post- press processes designing. 2 Formal problem statement An important point in the study of postpress processes is considered the presence of technological characteristics or parameters, on which the effectiveness of the publi- cation in the general cycle of its production depends. Generalizing such factors, we introduce the concepts of factors that become the main elements of models for de- termining the priority of the factors influence on the course of implementation and predictive assessment of the quality of postpress processes. Eventually, a set of the designing factors of the postpress processes will be pre- sented in the form R  R1 , R2 , R3 , R4 , R5 , R6 , R7 , R8  , where R1 are the edition param- eters; R2 are structural features; R3 are operating conditions; R4 is a type of production; R5 are materials; R6 is a type of equipment; R7 are technological and economic cal- culations; R8 is a technological process diagram [12]. The initial step to prioritizing the influence of these factors on the process is to design a semantic network, the es- sence of which is to capture the existing relations between the factors. It becomes the basis for the construction of a matrix of pairwise comparisons, processing of which leads to obtaining conditional weight values that determine the numerical priorities of factors — their importance value for the technological process. Next is the calculation and the determination of the optimal (among alternative) options for the implemen- tation of post-press processes designing. A multicriteria optimization of functions r  x    r1  x  , ..., rn  x   on the set B is to distinguish the maximum value of the utility functions ri  x   max, i  1, n. Accor- xD dingly, by the method of linear minimization of the criteria, combining partial target functionals r1 , ..., rn is carried out according to the formula [13-16]: n R  v, x    vi ri  x   max; v  V , (1) xD i 1  n  V  v   v1 , ..., vn  ; vi  0;  vi  1 , Q  i 1  where vi are weights of the factors of Pareto set. For the factors independent in utility and preference, the following utility function exists [17, 18]: n U  x    vi ui  yi , (2) i 1 where: U  x  is a multicriteria utility function;  0  U  x   1 are alternatives x ; vi n is the weight of the i -th criterion, moreover 0  vi  1,  vi  1 ; ui  yi  ; is the utili- i 1 ty function of the i -th criterion  0  ui  yi   1 ; yi is the value of the alternative x by the i -th criterion. 3 Literature review The analysis of literary sources shows the necessity of a reasonable selection of fac- tors influencing the quality of printed products [1-7]. In recent years, the simulation of postpress processes with the help of computer equipment and specific software has been used [8-11]. As risk and uncertainty are peculiar to complex processes, quantita- tive parameters for making a sound decision about the implementation of the studied process can be obtained on the basis of the methods of the operations research [12- 16]. The design of the studied processes is performed through the calculation of alter- native options for their implementation [17, 18]. It is important to take into account the fact that the equipment for performing separate operations of finishing and bind- ing processes and the materials used for different types of products are individual [19- 23]. The principle of vertical design is actively applied, which distinguishes between the procedures of analysis and synthesis. The synthesis creates the descriptions of objects that reflect their structure and parameters [25]. The selection of technology and postpress equipment depends on the type of printed matter, its purpose, produc- tion volumes, economic and financial indicators of the printing company activities [26, 27]. A significant problem is the adherence to standards for the edition produc- tion, metrological characteristics related to the quality in printing, modelling of busi- ness processes, which are important factors of planning and effective functioning of printing companies. The performed analysis indicates that there is no information approach to the problem of forming the book quality, the final stage of which is the postpress process. The essence of the new methodology is the use of methods and means of theory of operations research, modelling theory, expert assessment of publi- shing and printing processes, which will ensure the proper quality of printed products. 4 Objectives of the work (problem setting) The formation and multicriteria assessment of alternative options for the implement- tation of postpress processes based on fuzzy preference relationships. The determi- nation of Pareto-optimal alternative on the basis of the results analysis of the non- dominant sets intersection of the relation convolutions and the maximum value of the membership function of a common set. Modelling of designing procedures of post- press process by means of context diagrams for the corresponding decomposition levels of IDEF0 models. Obtaining a model-basis for further predictive assessment of the quality of the postpress process. 5 Materials and methods Making management decisions regarding the alternative implementation of techno- logical processes can be complicated by the lack of information about their priority and the inability to quantify the benefits. Instead, it is possible to pair the alternatives in the segment [0; 1] and to represent the data in numerical form. The assessment is carried out on the basis of multicriteria optimization, where the factors of the tech- nological process are the criteria. According to Pareto principle [17, 18], it is suffi- cient to select only the dominant factors with the highest weight parameters, which form Pareto set P( D) , where D  Ri is a set of valid values. Accordingly, with a fuzzy preference relation, decision-making will be exercised by Pareto-optimal alter- natives for a set of alternatives. Introducing a clear relation of non-strict preference Ri for a set of alternatives X   x1 , ..., xn  allows making one of the following statements for any pair of alterna- tives ( x, y) : x is not worse than y , that is x  y,  x, y   R ; y is not worse than x being written as y  x,  y, x   R , x and y are not comparable,  x, y   R,  y, x   R. This approach makes it possible to narrow down the rational selection class. If there is a strict preference  x, y   Rz the alternative x prevails y , that is x  y . With clear utility functions rj of the set X , the alternative x with a higher assessment rj  x  by the factor j is better than the alternative y whose assessment is rj  y  . The above statement is described by a clear relation of the advantage R j of the set X : R j   x, y  : rj  x   rj  y  , x, y  X  (1) To single out Pareto-optimal alternative, it is necessary to select the alternative x0  X with the highest utility ranking on the set of all factors: rj  x0   rj  y  , j  1, m; y  X (2) The convolution of all the criteria of formed Pareto set into a single scalar is car- ried out by the intersection method [17]. m We denote Q1  R j . Thus, the set of alternatives X   x1 , ..., xn  with pre- j 1 ference relation Q1 corresponds to the set of alternatives with utility functions rj  x  . Identifying non-dominant alternatives by a fuzzy preference relation Q1 is to replace several relations R j  j  1, m  by the intersection between them. We will assume that  j  x, y  is a membership function of a clear preference relation rj . We form the condition: 1, if rj  x   rj  y  , then  x, y   R j  j  x, y    (3) 0, if  x, y   R Accordingly, the membership function of the convolution Q1 is written as follows: Q  x, y   min 1  x, y  , 2  x, y  ,..., n  x, y  1 (4) The convolution of the criteria, taking into account the weight values of the process factors v j and the corresponding utility functions, will be: Q  x   min v j rj  x  (5) j The convolution of the initial relations Q2 is also formed by the weight values of the analysed factors v j and the corresponding utility functions: m m Q2   v j rj  x  , where  v j  1, v j  0 (6) j 1 j 1 It corresponds to the following membership function: [15-18] m Q  x, y    v j  j  x, y  2 (7) j 1 The methodology of IDEF0 modelling has been used to model the studied process, which involves the construction of context diagrams of a tree structure, created on the principle of decomposition. Next, we denote the context diagram A-0, and the de- composition diagram of the first level — A-1. The arrows of the input type (what is being processed) will be the set of values I  I1 , ..., I n  , the arrows of the control type (procedures and management strategies) will be the set C  C1 , ..., Cn  , the ar- rows of the output type (result) will be the set O  O1 , ..., On  , and the arrows of the mechanism type (required resources) will be the set M  M1 , ..., M n  . 6 Experiment We determine the quality of postpress processes designing by assessing fuzzy prefe- rence relations Ri on the set of alternatives X  x1 , x2 , x3  : R1 (the edition parame- ters) — x1  x2 , x2  x3 ; R2 (operating conditions) — x1  x3 , x2  x3 ; R3 (structu- ral features) — x1  x2 , x2  x3 ; R4 (a type of production) — x1  x2 , x2  x3 . We form the matrices of relations for the factors R1 , R2 , R3 and R4 . We use two types of numeric visualizers: 0 and 1, where 0 is the absence of preference. x1 x2 x3 x1 x2 x3 R  xi , x j  = R  xi , x j  = x1 1 1 0 x1 1 0 0 1 x2 1 1 0 2 x2 1 1 1 x3 1 1 1 x3 1 0 1 x1 x2 x3 x1 x2 x3 R  xi , x j  = R  xi , x j  = x1 1 1 1 x1 1 1 1 3 2 x2 0 1 1 x2 0 1 1 x3 0 1 1 x3 0 1 1 We construct the convolution of relations Q1  R1 R2 R3 R4 . The absence of preference of the xi -th alternative is considered to be zero by the factor analysed. x1 x2 x3 Q  xi , x j  = x1 1 0 0 1 x2 0 1 0 x3 0 0 1 According to the convolution of relations Q1 , a subset of non-dominant alterna- tives will look like this [14-17]:  4  Qнд  x   1  sup  Q  y, x   Q  x, y  (8)  j 1  1 1 1 y X Having used these matrices of relations and (8), we get: Qнд  x1   1  sup  x2 x1  x1 x2 ; x3 x1  x1 x3  ; 1 y X  нд Q1  x2   1  sup  x1 x2  x2 x1 ; x3 x2  x2 x3  ; y X Qнд  x3   1  sup  x1 x3  x3 x1 ; x2 x3  x3 x2  . 1 y X Now we set the fuzzy preference relation Q2 , j  1, 4 : 4 Q2   v j rj  x  (9) j 1 We form the membership functions for the convolution of relations Q2 and the corresponding sets of non-dominant alternatives: 4 4 Q  x, y    w j  j  x, y  ,  w j  1, w j  0 2 (10) j 1 j 1 Q  x1 , x2   v1  R  x1 , x2   v2  R  x1 , x2   v3  R  x1 , x2   v4  R  x1 , x2  2 1 2 3 4 Q  x1 , x3   v1  R  x1 , x3   v2  R  x1 , x3   v3  R  x1 , x3   v4  R  x1 , x3  2 1 2 3 4 Q  x2 , x1   v1  R  x2 , x1   v2  R  x2 , x1   v3  R  x2 , x1   v4  R  x2 , x1  2 1 2 3 4 Q  x2 , x3   v1  R  x2 , x3   v2  R  x2 , x3   v3  R  x2 , x3   v4  R  x2 , x3  2 1 2 3 4 Q  x3 , x1   v1  R  x3 , x1   v2  R  x3 , x1   v3  R  x3 , x1   v4  R  x3 , x1  2 1 2 3 4 Q  x3 , x2   v1  R  x3 , x2   v2  R  x3 , x2   v3  R  x3 , x2   v4  R  x3 , x2  2 1 2 3 4  4  Qнд  x   1  sup  Q  y, x   Q  x, y  (11)  j 1  2 2 2 y X  2 2  Qнд  x1   1  sup  Q  x2 , x1   Q  x1 , x2   ;  Q  x3 , x1   Q  x1 , x3   2 2 2   x   1  sup    x , x     x , x   ;    x , x     x , x   нд Q2 2 Q2 1 2 Q2 2 1 Q2 3 2 Q2 2 3   x   1  sup    x , x     x , x   ;    x , x     x , x   нд Q2 3 Q2 1 3 Q2 3 1 Q2 2 3 Q2 3 2 To determine Pareto-optimal alternative, we perform the intersection of sets Q1нд and Q2нд and define the membership function of a common set: Qнд  Q1нд Q2нд (12) нд  x   min Qнд  x  , Qнд  x  1 2 (13) The selection of the most efficient alternative is made by the maximum numeric value of the membership function Qнд  xi  . Since the alternatives for postpress pro- cesses designing are of the control type (procedures and management strategies), we obtain the following context diagram of IDEF0 model (Fig. 1). C1 C2 C3 Regulatory technical Opera- Implementation and technological ting con- alternatives documentation ditions Quality level of postpress processes Edition parameters designing I1 Postpress processes O1 I 2 Printed sheets designing Finished O2 project Hardware and Staff, the subject area software, other experts, interested people tools M1 M2 Fig. 1. Context diagram of IDEF0 model of the postpress processes implementation 7 Results As a result of the corresponding calculations, we get a subset of non-dominant al- ternatives of the following form: Qнд  x1   1  sup 0  0;0  0  1; 1 y X  нд Q1  x2   1  sup 0  0;0  0  1; y X Qнд  x3   1  sup 0  0;0  0  1. 1 y X Qнд  x   1;1;1 1 The membership functions of the additive convolution of relations Q2 for each planned alternative, when the values of the factors weights are set v1  0,53; v2  0, 27; v3  0,13; v4  0,07 have such values: Q2  x1 , x2   0,73 ; Q2  x1 , x3    0, 2 ; Q2  x2 , x1   0,8 ; Q2  x2 , x3   0, 47 ; Q2  x3 , x1   0,8 ; Q2  x3 , x2   0,73 . We present the values of the membership functions with the help of the matrix of relations: x1 x2 x3 Q  xi , x j  = x1 1 0,73 0,2 2 x2 0,8 1 0,47 x3 0,8 0,73 1 We find the elements of the subset of non-dominant alternatives for the relation Q2 : Qнд  x1   1  sup  0,8  0, 73 ;  0,8  0, 2   0, 4 2 Qнд  x2   1  sup  0, 73  0,8  ;  0, 73  0, 47   0, 74 2 Qнд  x3   1  sup  0, 2  0,8  ;  0, 47  0, 73  1, 26 2 After calculations, we get: Qнд  xi   0, 4; 0,74; 1, 26. 2 According to the intersection of sets Q1нд and Q2нд the maximum value will have the membership function Qнд  x3   0, 4; 0,74; 1, 26 , i.e. the third option is consi- dered the optimal one. The process of functional decomposition of the context diagram shown in Fig. 1 consists in its division into lower order functions and setting of the direction of the boundary arrows, which contributes to the detailing of activities within the studied process. Based on the above statements, we construct the resulting decomposition diagram of the context diagram (Fig. 2). C1 C2 C3 Regulatory technical Implementation Operating and technological conditions alternatives documentation Edition pa- DED Structural rameters I1 features I2 Printed sheets DRE DSO Formalized re- Quality level of quirements postpress processes designing DPM O1 Hardware and Technological O2 Finished software, other process diagram project tools Staff, the subject area ex- perts, interested people M1 M2 Fig. 2. Diagram of the first decomposition of IDEF0 model of the postpress processes designing The diagram of the first decomposition of IDEF0 model of the postpress processes designing contains the following functional blocks: DED — the determination of the edition design, DRE — the determination of the requirements for the finished edition, DSO — the determination of the sequence of technological operations, DPM — the determination of the processing modes [12, 19]. We will analyse the components information load of the sets of boundary arrows in IDEF0 model of the postpress processes designing: – I1 (the edition parameters). The key parameters of the book editions are a kind, type, format and volume. Editions are divided into kinds according to a number of typological features: production method, periodicity, material structure, composition of the main text, language feature, purpose, frequency of issue, structure, etc. The format determines the size of the print sheets and the finished book block. The vol- ume indicates the number of paper sheets or pages within a single copy [19]. – I 2 (printed sheets). The result of prepress processing of the originals and prin- ting of the print run is the printed paper sheets that arrive at the postpress section. – C1 (regulatory technical and technological documentation). Regulatory technical documents include technical requirements and legal regulations, in particular: laws, standards, specifications, codes of established practice, etc. – C2 (operating conditions). Operating conditions include the lifespan and opera- ting intensity of the finished edition [19]. – C3 (implementation alternatives). Pareto-optimal alternatives, determined by the assessment of fuzzy relations on a given set of alternatives [18]. – O1 (quality level of postpress processes designing). The result of the post-press processes designing is an appropriate level of the project quality. – O2 (finished project). The project determines the progress of all technological actions aimed at the implementation of postpress processes. – M 1 (hardware and software, other tools). The postpress processes designing is done using the computer technology and specific, narrow-profile software. – M 2 (staff, the subject area experts, interested people). The participation of pro- duction workers in the design. If necessary, the authors and customers of the book edition are involved [19, 20]. 8 Conclusions Matrices of relations for certain factors influencing the quality of postpress processes designing have been formed. Membership functions for the convolution of relations and corresponding sets of non-dominant alternatives have been received. The selec- tion of the most efficient alternative according to the maximum value of the member- ship function of the convolutions has been implemented. Modelling of key operations and functions provides the basis for predictive as- sessment of the quality of the studied process. References 1. Denison, E.: Print and production finishes for sustainable design. RotoVision, 192 p. (2009). 2. Kamath, H. N., Rodrigues, L. L.: Influence of overall equipment effectiveness on print quality, delivery and cost: A system dynamics approach. 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