ࡱ> hbjbj kk{! %! 1!1!1!4e!e!e!h!#e!i=&1*(Y*Y*Y*4+N+ Z+ggggggg$lo;gy1!V24+4+V2V2;g1!1!Y*Y*iIIIV2d1!Y*1!Y*gIV2gII a%dY*e!Db*gi0ib& pFZ pT%d p1!%db+`-lI.$/fb+b+b+;g;gG$b+b+b+iV2V2V2V2 pb+b+b+b+b+b+b+b+b+ :The Iterative multiobjective method in optimization process planning Iterativna viekriterijalna metoda u optimiranju tehnolokog procesa Cosic Predrag1, Lisjak Dragutin1, and Antolic Drazen2 1Department of Industrial Engineering, FAMENA, University of Zagreb Zagreb, 10 000, Croatia 1Department of Industrial Engineering, FAMENA University of Zagreb Zagreb, 10 000, Croatia 2AD, Ltd. Ilirski trg Zagreb, 10 000, Croatia Abstract Estimation of production time, delivery term, production costs etc., are some of the key problems of unit production. In the previous research strong correlation was discovered between the features of the product drawing and production time, which has resulted with 8 regression equations. They were realized using stepwise multiple linear regression. Since the optimization of these regression equations did not fully define the most frequent requirements, multiobjective optimization was applied. The applied criteria included: minimum production time, maximum work costs/total costs ratio for a group of workpieces. The group was created using specific classifiers that defined similar workpieces. An iterative STEP method with seven decision variables within a group was applied, and the groups with a high index of determination were selected. Independent values that maximize the work costs/total costs ratio and minimize production times were determined. The obtained regression equations of time production parts and work costs/total costs ratio are included in the objective functions to reduce production time and increasing, work costs/total costs at the same time. The values of decision variables that minimize production time and maximize work costs/total costs ratio were determined. As the solution of the described problem, multicriteria iterative STEP method was applied. Sa~etak Procjena vremena izrade, roka isporuke, troakova izrade, itd, neki su od klju nih problema komadne proizvodnje. U prethodnim istra~ivanju uo ena je jaka korelacijska veza izmeu zna ajki nacrta proizvoda i vremena izrade koja je rezultirala s 8 regresijskih jednad~bi. One su realizirirane primjenom postupnom viaestrukom linearnom regresijom. Kako optimiranje tih regresijskih jednad~bi nije u potpunosti definiralo naj eae zahtjeve, primijenjena je viaekriterijalna optimizacija. Kriteriji su bili :minimalno vrijeme izrade, maksimalan omjer troakova rada prema sveukupnim troakovima za grupu izradaka. Grupa je kreirana posebnim klasifikatorima koji su odredili sli ne izratke. Primijenjen je iterativni STEP model od sedam varijabli odluka unutar grupe, a odabrane su grupe sa visokim indeksom determinacije. Odreene su vrijednosti nezavisnih varijabli maksimizirajui omjer troakova rada i ukupnih troakova te minimiziranjem komadnog vremena. Dobijene regresijske jednad~be komadnog vremena izrade pozicija i omjer troakova rada prema ukupnim troakovima uklju eni su u objektne funkcije kako bi se reduciralo komadno vrijeme izrade te istovremeno poveao omjer troakova rada prema ukupnim troakovima. To je odredilo vrijednosti varijabli odlu ivanja koje minimiziraju komadno vrijeme i maksimiziraju omjer troakova rada na prema ukupnim troakovima. Kao rjeaenje opisanog problema primijenjena je viaekriterujalna interaktivna STEP metoda. Key words: production time, regression analysis, multiobjective method for optimization Klju ne rije i: komadno vrijeme, regresijska analiza, viaekriterijalna metoda optimizacije Introduction Uvod Predicting events, fate of individuals, nations, rulers, health, success in warfare  has always been the focus of interest of all cultures and civilizations. If something could not be reached by ratio (reason), attempts were made to reach it in the sphere of irrational. Mystics, religious prophets, charismatic people with exceptional powers or qualities, people who were able to predict the future, either as sorcerers, astrologers, astronomers, palmists or as economic, stock-exchange, political and geo-strategic analysts, futurists were and still are appreciated in society. This is either due to curiosity, the need for decision-making, the desire for economic stability, good health, or due to fear of the future. In the turbulent, global and neo-liberal market there is a pronounced need for predicting economic trends either in the microsphere or at the macroeconomic level. Defining comparative criteria for performance evaluation of companies in production strategies is an essential element of strategic considerations of the management of individual companies. Defining of long-term business objectives includes also defining of the range of products that have or will have a place in the market. Optimization of technological parameters in production for the purpose of cost reduction or production time shortening is often the subject of interest of numerous researchers and articles. The use of numerous methods of operational research and artificial intelligence are some of the approaches to the given problem. Of course, these are almost always partial approaches because of the complexity of the problem. The managements of companies on the other hand insist on as exact (comprehensive) as possible assistance in decision making, directing researchers to the area of business intelligence by defining broader areas of interest. In times of crisis, recession, and in the normal business conditions as well, managements are constantly confronted with the same questions: how to reduce production times, delivery, production cycle; how to cut all expenses including the costs of product manufacturing, and how to increase own share of the market pie; how to increase productivity; how to balance the productivity of all jobs during the process, especially when cycle production is concerned; how to increase the ratio of productive/unproductive time or cost; how to increase utilization of capacities, how to increase company profitsSuch questions are a constant nightmare of all managements of manufacturing companies. Our experiences and experience numerous of others as well, and following of economic trends in Croatia and wider have motivated us to start research in this area. Since a considerable number of research works and papers are dealing with optimization of technological parameters, we have decided to focus our attention on the relationship between product features (geometry, complexity, quantity,..) and production times and costs [1,2,3,4,5]. It has been proved that it is possible to make estimation of production time applying classification, group technology, stepwise multiple linear regression as the basis for accepting or rejecting of orders, based on 2D [2,3] drawings, and the set basis for automatic retrieval of features from the background of 3D objects (CAD: Pro/E, CATIA) and their transfer to regression models [6,7]. Of course, certain constraints have been set: application of standardized production times from technical documentation or estimations made using CAM software (CATIA, PRO/E, CamWorks), type of production equipment/technological documentation determines whether it will be single- or low-batch production. Initial steps have been taken regarding medium-batch, large-batch or mass production. It has been assumed (relying on experience) that small companies (SMEs) in Croatia make decision about acceptance of production (based on customers design solution of the product, delivery deadlines and manufacturing costs imposed by the customer - PICOS concept: automotive industry VW, GM) on the basis of free intuitive assessment due to the lack of time and experts. This often results in wrong estimates. Since during the process of privatization in Croatia numerous large companies in the field of mechanical engineering disappeared, the newly created companies are doomed to work mainly for large international companies, providing only their work, without own share in innovativeness, without brand or patents and without transfer of new technologies. If the optimization of regression curves is to be applied (independent variables - product features, dependent variable production time), it is hard to explain what it would mean for the minimum or maximum production time for a given group of products. The minimum production time could mean a higher productivity, but we do not know about the profit. The maximum production time could suggest that a higher occupancy of capacities may mean higher earnings, although it may not be so. This dual meaning has led us to introduce multiple objective optimization for a new class of variables that differently classify our products. A response variable (dependent variable) can assume several meanings: maximum profit per product, minimum delivery time (related to production time, and also to organizational waste of time, production balancing...), ratio of the production cost and the costs of product materials, ratio of the production cost and the ultimate production cost. Thus, the problem-solving approach has become more complex, and is no longer a mere result of intuition and heuristics, but more exact assessment of common optimum for more set criteria. Previous research 2. Prethodno istra~ivanje In the first part of the research of possible relationship between 2D product features and production time, regression equations were obtained for the considered groups of geometrically and technologically similar products. The research was limited to the following: workpiece initial shape round bar, classical machine tools, small batch production (based on original technological documentation of the former largest machine tools manufacturer Prvomajska in Croatia and in ex-Yugoslavia until the year 1990), and customary sequence of operations. The values of independent variables (50!) were taken from classical paper drawings and technological standards. Of course, a certain degree of subjectivity is present in defining work norms and setting of norms for machining of some parts. Some subjectivity of the people working in the Department of Time and Work Study in Prvomajska Machine Tool Factory (until 1990) could be assumed, because several employees were dealing with time assessment issues. At the same time, work norms for workers performing certain operations were often very low in order to provide overreaching of the work norms and higher wages for direct workers, proving thus the much proclaimed loyalty to the working class and success of the established system of self-management in the Yugoslav type of socialism with a human face. Therefore, having all this in mind, a systematic error was taken into consideration in the estimation of time standards. One of the co-authors of this paper (Antoli) was for some time the technical director of INAS company, a small successor of  Prvomajska Machine Tool Factory, which finally ceased to exist in 2009. Thus, the used technological documentation for classical milling machines (420 positions) is from that source. By classification of products, according to the BTP, 8 regression equations for 8 groups of products were obtained. The main grouping criteria were the features (geometrical, tolerance, hardness) from the technical drawings and for each product the production time was used (technological and auxiliary time). However, since today is the time of 3D modeling, CNC, and machining centers, the initial research for the development of automatic retrieval of product features from 3D models was conducted. Using CAM software, for these 3D models technological time was calculated in order to obtain regression equations for the estimation of production time. Thus, the following was obtained: Table 1. Presentation of created regression equations 2D Tablica 1. Prikaz sa injenih regresijskih jednad~bi 2D NoShape of product - representative of product group Oblik proizvoda-reprezentant proizvodne grupe Regression equations Regresijska jednad~ba Index of determination Index determinacije r2 Relative error Relativna greaka [%]Comment on regression equation Opaska za regresijsku jednad~bu1Whole sample itav uzorakt = - 11.69 + 16.95x45 + 1.22 x40 + 0.54 x47 + 127.47x22  3.24x18 + 0.15x32 + 0.03x60.73655230.74Model is developed with procedure in advance. Three independent variables are omitted x8, x19 and x33. Model je razvijen procedurom unaprijed. Tri nezavisne varijable su izostavljene x8, x19 i x33. 2Round bars Okrugle ipket = 55.47 + 22.43x45 + 1.162 x40 + 0.43x11 + 1.61x50 5.41x8 3.26x18 + 1.78x420.7428530.95Model is developed with procedure in advance.Two independent variables are omitted x1 and x26. Model je razvijen procedurom unaprijed. Dvije nezavisne varijable su izostavljene x1 i x26.3Shafts Osovinet = 6.13 +0.83x2 +1.27x39 3.30x8 +5.51x46  6.86x18 +0.09 x6 + 124.33x220.80762625.90Model covers more narrow field of rotational parts. It gives better results than No.2. Model pokriva u~e podru je rotacijskih proizvoda. Daje bolje rezultate nego No. 2.4Discs Diskovit = - 5.17 + 0.73x47 + 0.93x40 + 5.25x20 + 0.52x24 + 139.11x30 + 0.23x32  0.51x330.80940524.24Simillar results as in No.3. Sli ni rezultati kao u No- 3.5Discs-with fine machining Diskovi fino obraenit = -60.78 + 0.59x47 +047x9 +0.74x1 + 0.25x10 + 0.84x39 + 291.07x25 + 5.9x150.9850578.01Model covers more narrow field of rotational parts. It gives better results than all the previous models. Model pokriva u~e podru je rotacijskih proizvoda. Daje bolje rezultate nego svi prijaanji modeli.6Rotational parts Rotacijski dijelovit = -37.11 + 0.94x40 +0.03x29 +319.22x26 + 0.13x23 + 114.67x43 80.98x45 - 0.46x60.89332127.06Model is better than No. 2 as a result of higher degree of homogenization of data. Solution is better with omitted variable,x2 and included variables x6, x23, x43 and x45. Model je bolji nego No. 2 kao rezultat vieg stupnja homogenizacije podataka . Rjeenje je bolje sa izostavljenom varijablom x2 i uklju enim varijablama x6, x23, x43 i x45. 7Flat bars Pljosnate aipket = -10.96 + 0.58x40 +34.50x45 +218.42x22  5.48x50 + 185.03x26 +0.39x9 -0.50x490.90033215.92Constraints are greater for all variables so results are better. Narrow field of homogenization. Ograni enja su vea za sve varijable pa su i rezultati bolji. U~e podru je homogenizacije.8Sheet metals Metalni limovit = 0.47 +1.27x40 +137.45x45  13.23x43  0.70x43 + 0.28x4 + 0.05x6 +3.91x160.90082324.04Model is characterized with the presence of complex variables x40, x43, x45 Model je karakteriziran sa prisustvom slo~enih varijabli x40, x43, x45-  Y = 28.77308 + 8.277896x19  0.16359Ks  1.46341fea  50.8704x45 + 0.000324 x44 + 0.002462x43 (1) 2.00 < x19 < 8.00 tolerance of dimension line of the part (2) 13.00 < Ks < 46.00 all dimension lines (3) 9.00 < fea < 25.00 features of 3D model (4) 0.174 < x45 < 0.584 mass of the part (5) 4,063.80 < x44 < 74,724.50 volume of the part (6) 6,660.70 < x43 < 28,131.30 superficial area (7) 45.00 < Y < 111.00 production time. (8) Error between estimation by regression and calculated production time for each part (-5.64%;+ 4.32%). Table 2. Overview of new classifiers of products Tablica 2. Pregled novih klasifikatora proizvoda CLASSIFIERS W1  W5 KLASIFIKATORI W1  W5W1 (material-materijal)W2 (shape-oblik)-W3 (according to max. product dimension-suglasno maksimalnoj dimenziji proizvoda )W4 (complexity-slo~enost) BA  number of dimension lines-broj kotaW5 (treatment complexity-slo~enost finoe obrade)1  POLYMERS -POLIMERI 2  ALUMINIUM AND ALUMINIUM ALLOYS ALUMINIJ I ALUMINIJSKE LEGURE 3  COPPER AND COPPER ALLOYS  BAKAR I BAKRENE LEGURE 4  NON-ALLOY STEEL NELEGIRANI ELIK 5  ALLOY STEEL  LEGIRANI ELIK)1  ROTATIONAL ROTACIJSKI (round bars-okrugle aipke, round tubes-okrugle cijevi, hexagons-heksagonski, plates-ploevina) 2  PRISMATIC  PRIZMATI NI (plates-plo a, flat-, rectangular tubes-kvadratne cijevi) 3  PROFILE-PROFILI (L; U; I; Z; C) 4  SHEET-METAL  LIMOVI (foils-folije, straps-trake, sheets-limovi) 5  COMPLEX-SLO}ENI1  MINI (V<120) 2  MIDI (120<V<400) 3  STANDARD (400<V1<1.000) 4  KILO (1.000 <V<2.000) 5  MEGA (V>2.000 mm)1  very simple-veoma jednostavan BAd"5 2  simple-jednostavan 5>BAd"10 3  average -prosje an 11>BAd"25 4  complex-slo~en 25>BAd"75 5  very complex-veoma slo~en BA>751  VERY ROUGH-VEOMA GRUB 2  ROUGH-GRUB 3  MEDIUM  SREDNJA OBRADA 4  FINE-FINA OBRADA 5  VERY FINE TREATMENT  VEOMA FINA OBRADA  Conditions were determined on the basis of the data range on the number of dimension lines of the considered sample of 415 elements. A classifier which is being developed is based on 5 basic product features: W1- MATERIAL (quality of material) W2 SHAPE (prevailing shape of product) W3 SIZE (according to the product maximum dimension) W4 COMPLEXITY (with respect to the number of tips, edges, surfaces; number of dimension lines in 2D model ) W5 - TREATMENT COMPLEXITY ( requirements regarding surface, roughness, measurement tolerances, shape tolerances and position tolerances) It was found that the optimization of regression equations, in order to obtain minimum or maximum production times was insufficient with respect to the needs in real production. Thus, the aim was to obtain, by considering a series of regression equations, the optimum for multiobjective optimization (minimal production time, labor cost/material cost ratio or labor cost/total cost ratio for the selected group of products. As multiobjective optimization requires the same variables (x1,...x7), it was necessary to make new grouping of the basic set (302 workpieces) using new classifiers. The conditions were defined based on the range of data about the number of dimension lines on the considered sample of 415 elements. A classifier that is being developed is based on 5 basic workpiece features. For the purpose of the research, a group of workpieces (W1-W5) 41113 was selected for further analysis. The code 41113 means: steel rotational small very simple commonly complex - workpieces. From the available database, the minimum and maximum values for independent variables, and dependent variable (Z1-production time), and derived variable Z2 was taken. Table 3. Minimum and maximum values of selected variables Tablica 3. Minimalne i maksimalne vrijednosti odabranih varijabli PRODUCT TYPE - PROIZVOD TIPA - 41113min minimalna2.900.1001.0011.210.220.01320.001611.09max maksimalna100.000.4005.0019.6312.500.39720.820332,524.33 arithmetic mean aritmeti ka sredina28.750.3881.6311.633.470.11770.10517.75406.88standard deviation standardna devijacija21.870.0611.1 31.713.020.11510.1898.83641.74mode mod36.000.4001.0011.210.07350.04812range rang97.100.3004.008.4112.280.38400.819272.513.24sum zbroj689.909.30039.00279.0683.182.82492.5184269.765.21variable varijableX1X2X3X4X5X6X7Y1Y2Variable description Opis varijableProduct outer diameter Vanjski promjer proizvodaNarrowest tolerance of measures Naju~a dimenzijska toleranca Scale of the drawing Mjerilo crte~aMaterial mass/strength ratio Omjer masene vrstoeWall thickness/len-gth ratio- Omjer debljine stijenke i duljineMaterial surface area- Oploaje materijalaMaterial mass  Masa materijalaProduction time Komadno vrijeme proizvodaWork cost/material cost ratio Omjer troka rada i troka materijalaunit of measure jedinica mjeremmmmnumber broj*number brojdm2kgh 10-2number broj Two regression equations, Z1 (production time) and Z2 (labor cost/total cost ratio), were selected. For them multiobjective optimization was also performed. In order to use the same types of variables, new grouping was made using specifically adjusted classifiers. Workpiece classification according to the criterion of complexity was done semi-automatically by setting conditions on certain features of drawings (basic roughness, the finest roughness requirement, the narrowest tolerance of measures, the narrowest tolerance of shape or position (geometry), number of all roughness and geometry requirements in the drawing. Each of these 6 criteria based on its specific conditions is assigned a value ranging from 1 to 5. The obtained result is rounded to integer (e.g. 3.49 is W=3, and 3.51 is W=4), and this integer (in the range from 1 to 5) becomes complexity criterion coefficient (the fifth digit in the code). Table 4. Results of stepwise multiple linear regression Tablica 4. Rezultati postupne viestruke linerarne regresije Regression Statistics Regresijska statistikaDependent variable -production time Zavisne varijable komadno vrijeme Z1Regression Statistics Regresijska statistikaDependent variable- work costs/ultimate costs ratio Zavisne varijable omjer trokovi rada/kriti ni troakovi Z2Multiple R Viaestruki R0.92212166Multiple R Viaestruki R0.99207R Square R20.85030835R Square R20.984202Adjusted R Square Prilagoen R20.78481826Adjusted R Square Prilagoen R20.977291Standard Error Standardna greaka4.09742037Standard Error Standardna greka0.002725Observations Broj pokusa24.0Observations Broj pokusa24.0Z1CoefficientsZ2CoefficientsIntercept-13.490042Intercept0.990439X Variable 10.86652065X Variable 10.000238X Variable 2-0.1993556X Variable 2-0.0039X Variable 30.75343156X Variable 30.00046X Variable 41.41593567X Variable 40.000794X Variable 5-1.8669075X Variable 5-0.00107X Variable 64.83640676X Variable 6-0.04466X Variable 7-51.274031X Variable 7-0.08551 Conditions were set regarding: basic roughness (common for all surfaces that are not separately specified) unit of measure is Ra (surface roughness) finest roughness requirement (specified in the drawing) - unit of measure is Ra, it was so indicated in 2D drawings (roughness requirement) narrowest tolerance of measures (mm) (measurements requirement) narrowest tolerance of diameter (unit of measure is IT diameter requirement) narrowest tolerance of shape or position (geometry requirement) number of all requirements on roughness and geometry specified in the drawing (i.e. how many surfaces are to be particularly finely treated and how many surfaces have special tolerances concerning the shape or position (in relation to another surface; roughness and geometry requirement. Description of the Objective Model 3. Opis objektnog modela The general multiobjective optimization problem with n decision variables, m constraints and p objectives is [8]:  EMBED Equation.DSMT4  (9)  EMBED Equation.DSMT4  (10) where  EMBED Equation.DSMT4 is the multiobjective objective function and EMBED Equation.DSMT4 , EMBED Equation.DSMT4 , EMBED Equation.DSMT4 are the p individual objective functions. Benayoun [9] (1971) developed the step method as an iterative technique that should converge to the best-compromise solution in no more than p iterations, where p is the number of objectives. The method is based on a geometric notion of best, i.e., the minimum distance from an ideal solution, with modifications of this criterion derived from a decision maker's (DM) reactions to a generated solution. The method begins with the construction of a payoff table. The table is found by optimizing each of the p objectives individually, where the solution to the kth such individual optimization, called  EMBED Equation.DSMT4 , gives by definition the maximum value for the kth objective, which is called Mk (i.e., Zk(xk) = Mk). The values of the other p - 1 objectives implied by xk are shown in the kth row of the payoff table. The payoff table is used to develop weights on the distance of a solution from the ideal solution. The step method employs the ideal solution, which has components Mk for k = 1, 2, ... , p. The ideal solution is generally infeasible. The  EMBED Equation.DSMT4 , metric is used to measure distance from the ideal solution. The distance is scaled by a weight based on the range of objective Zk and the feasible region is allowed to change at each iteration of the algorithm. The basic problem in the step method is:  EMBED Equation.DSMT4  (11)  EMBED Equation.DSMT4  (12)  EMBED Equation.DSMT4  (13) where EMBED Equation.DSMT4  is the feasible region at the ith iteration and  EMBED Equation.DSMT4  is used to indicate that the original metric has been modified. Initially,  EMBED Equation.DSMT4 ; i.e., at the start of the algorithm the original feasible region is used in (13) The weights  EMBED Equation.DSMT4 in (12) are defined as:  EMBED Equation.DSMT4  (14) where  EMBED Equation.DSMT4  (15) where nk is the minimum value for the kth objective; i.e. it is the smallest number in the kth column of the payoff table. The  EMBED Equation.DSMT4  are objective function coefficients, where it is assumed that each objective is linear.  EMBED Equation.DSMT4  (16) The solution of (11) to (13) with Fd in (13) yields a non-inferior solution  EMBED Equation.DSMT4 , which is closest, given the modified metric in (14), to the ideal solution. The decision maker (DM) is asked to evaluate this solution. If it is satisfactory, the method terminates; if it is unsatisfactory, then the decision maker specifies an amount Z*k by which objective k* may be decreased in order to improve the level of unsatisfactory objectives, where objective k* is at a more than satisfactory level. A problem with a new feasible region in decision space is then solved. A solution is feasible to the new problem, x  EMBED Equation.DSMT4  Fdi+1, if and only if the following three conditions are satisfied:  EMBED Equation.DSMT4  (17) (8)  EMBED Equation.DSMT4  (18) (9)  EMBED Equation.DSMT4  (19) (10) For the new problem  EMBED Equation.DSMT4 , and the other EMBED Equation.DSMT4 are recomputed from (14) for  EMBED Equation.DSMT4 . The problem in (11) to (13) is then resolved with EMBED Equation.DSMT4 , and since EMBED Equation.DSMT4 , (12) includes constraints for  EMBED Equation.DSMT4  only. The solution to the new problem yields a new non-inferior solution, which the decision maker evaluates. The method continues until the decision maker is satisfied, which the authors claim occurs in fewer than p iterations. 4. Results of the Multiobjective Analysis 4. Rezultati viekriterijalne analize On the basis of considerations of regression functions in previous sections, the problem of multiobjective optimization with minimization of the objective functions Z1 and Z2 with related constraints (equations (20) to (22)) is defined. Min Z1= -13.49004192+0.866520652*x1-0.199355601*x2+0.753431562*x3+1.415935668*x4- 1.866907529*x5+4.836406757*x6-51.27403107*x7 (20) Min Z2= -0.990438731-0.000238475*x1+0.003897645*x2-0.00045981*x3-0.000794225*x4+ 0.0010738*x5+0.044664232*x6+0.085514412*x7 (21) x1 d" 100; x2 d" 0.4; x3 d" 5.0; x4 d" 19.63; x5 d" 12.50; x6 d" 0.3972; x7 d" 0.820 (22) x1, x2, x3, x4, x5, x6, x7 e" 0 In equations (20) and (21) Z1 represents variable T, and Z2 variable TU/TR. It should be mentioned that for the needs of consistency of the objective functions Z1 and Z2, for the objective function Z2 (equation (21)) the signs of the coefficients of variables and of the free member have been changed. The values of objective functions Z1 and Z2 in the extreme points of the set of possible solutions (feasible region) are given in Table 3. It is visible from the table that that there is no common set of points (x1,... x7) where both functions Z1 and Z2 have extreme (maximum) values, and thus the need for optimization of the given problem is justified. Table 5. Values of the decision variables and the objective functions Tablica 5. Vrijednosti varijabli odlu ivanja i objektnih funkcija Extreme point Ekstremna to kaDecision variables Varijable odlukeObjective functions Objektne funkcijex1x2x3x4x5x6x7Z1(x1...x7)Z2(x1...x7)A10000000073.1620-1.0143B00.400000-13.5698-0.9889C0050000-9.7229-0.9927D00019.6300014.3048-1.0060E000012.5000-36.8264-0.9770F000000.39720-11.5690-0.9727G0000000.820-55.5347-0.9203 On the basis of the data given in Table 5. the data for the first payoff table (Table 6.) have been selected, which is necessary for the calculation of the first compromise solution. Table 6. First payoff table Tablica 6. Prva payoff tablica Point of optimal solution To ka optimalnog rjeaenja XkIdeal values (Mk) of objective functions (Zk) for Xk Idealne vrijednosti (Mk) objektne funkcije (Zk) za Xk M1=Z1(Xk)M2=Z2(Xk)X1=(100,0,0,0,0,0,0)73.1620-1.0143X2=(0,0,0,0,0,0,0.820)-55.5347-0.9203 where k=1...2. In accordance with equations (14) and (15) coefficients of equation (12) are calculated, which is shown by the expressions (23) through (26).  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The set of equations for the calculation of the first compromise solution of the given problem is shown in Table 6., and the results of decision variables (x1,...x7) and objective functions Z1 and Z2 are given in Table 8. Table 7. Set of equations of the first compromise solution Tablica 7. Set jednad~bi prvog kompromisnog rjeaenja  EMBED Equation.DSMT4  - EMBED Equation.DSMT4 -0.016463892*x1+0.003787756*x2-0.014315200*x3-0.026902778*x4+0.035471243*x5-0.091891728*x6+ 0.974206590*x7  EMBED Equation.DSMT4 -1.6465 - EMBED Equation.DSMT4 +0.000238022*x1-0.003890239*x2+0.000458936*x3+0.000792716*x4-0.001071760*x5-0.044579370*x6- 0.085351935*x7 EMBED Equation.DSMT4 -0.070005466 x1 d" 100; x2 d" 0.4; x3 d" 5.0; x4 d" 19.63; x5 d" 12.50; x6 d" 0.3972; x7 d" 0.820;  Since in the given problem there are two objective functions, it is necessary to make calculation of the second compromise solution, and thus the previous equations for Z1 and Z2 become new constraints shown in equations (27) and (28) Table 8. Results of the first compromise solution Tablica 8. Rzultati prvog kompromisnog rjeenja x1=100; x2=0.4; x3=1.0; x4=12.0428; x5=12.5; x6=0.3962; x7=9999998E-4;  EMBED Equation.DSMT4 =7.128304E-2; Min Z1(x1,...x7) = 69.4161 Min Z2(x1,...x7) = -0.9915 Max Z2(x1,...x7) = 0.9915  0.866520652*x1-0.199355601*x2+0.753431562*x3+1.415935668*x4-1.866907529*x5+4.836406757*x6-51.27403107*x7 EMBED Equation.DSMT4  82.90614192 (27) -0.000238475*x1+0.003897645*x2-0.00045981*x3-0.000794225*x4 + 0.0010738*x5 + 0.044664232*x6 + 0.085514412*x7  EMBED Equation.DSMT4  -0.001061269 (28) Since the value Min Z1(x1,...x7)= 69.4161, it has been decided that the previous value for M1 =73.1620 is to be reduced for the value of 33.1620, and thus the new value for M1=40. The second payoff table is given below. Table 9. Second payoff table Tablica 9. Druga payoff tablica Point of optimal solution To ka optimalnog rjeaenja XkIdeal values (Mk) of objective functions (Zk) for Xk Idealne vrijednosti (Mk) objektne funkcije (Zk) za Xk M1=Z1(Xk)M2=Z2(Xk)X1=(100,0,0,0,0,0,0)73.1620-33.1620=40-1.0143X2=(0,0,0,0,0,0,0.820)-55.5347-0.9203 where k=1...2. In accordance with equations (14) and (15), coefficients of equation (12) are calculated, which is shown by the expressions (29) through (32). Since only the value of variable M1 has been changed, the values of equations (30) and (32) remain the same as in the case of calculation of the first compromise solution.  EMBED Equation.DSMT4  (29)  EMBED Equation.DSMT4  (30)  EMBED Equation.DSMT4  (31)  EMBED Equation.DSMT4  (32) As in the case of the first compromise solution, by arranging the obtained equations, the problem of multiobjective optimization has been reduced to the problem of single-objective optimization where the variable  EMBED Equation.DSMT4  is minimized according to equation (11). The set of equations for the calculation of the second compromise solution of the given problem is shown in Table 10., and the results of decision variables (x1,...x7) and objective functions Z1 and Z2 are given in Table 11. Table 10. Set of equations of the second compromise solution Tablica 10. Set jednad~bi drugog kompromisnog rjeaenja  EMBED Equation.DSMT4  - EMBED Equation.DSMT4 -0.001646389*x1+0.000378776*x2-0.001431520*x3-0.002690278*x4+0.003547124*x5-0.009189173*x6+ 0.097420659*x7  EMBED Equation.DSMT4  -0.101631080 - EMBED Equation.DSMT4 +0.000238022*x1-0.003890239*x2+0.000458936*x3+0.000792716*x4-0.001071760*x5-0.044579370*x6- 0.085351935*x7  EMBED Equation.DSMT4  -0.070005466 x1 d" 100; x2 d" 0.4; x3 d" 5.0; x4 d" 19.63; x5 d" 12.50; x6 d" 0.3972; x7 d" 0.820; 0.866520652*x1-0.199355601*x2+0.753431562*x3+1.415935668*x4-1.866907529*x5+4.836406757*x6- 51.27403107*x7  EMBED Equation.DSMT4  82.90614192 -0.000238475*x1+0.003897645*x2-0.00045981*x3-0.000794225*x4+0.0010738*x5 +0.044664232*x6+ 0.085514412*x7  EMBED Equation.DSMT4  -0.001061269  Table 11. Results of the second compromise solution Tablica 11 Rezultati drugog kompomisnog rjeenja x1= 3.37147; x2= 0.3711865; x3= 4.553035; x4= 18.92068; x5= 0.2269908; x6= 0.2826709; x7= 2.965111E-2;  EMBED Equation.DSMT4 = 7.682257E-2; Min Z1(x1,...x7)= 19.0013 Min Z2(x1,...x7)= -0.9915 Max Z2(x1,...x7)= 0.9915  5. Conclusion 5. Zaklju ak The paper presents research on the development of a model for the estimation of production time for unit production or medium size batch production. As a result, eight regression equations were obtained. They show estimation of the production time as a function of geometrical and technological characteristics of a homogeneous group of products that were grouped using logical operators. Using specifically developed 5 classifiers at 5 levels, on the sample taken from the real production a homogenous group was formed which resulted in a regression equation showing dependence between production time (Z1) and 7 independent variables (x1,...x7). After that, the dependence between the work costs/total costs ratio (Z2) and independent variables (x1,...x7) is shown in another regression equation. The optimization part of the work considers the possibility of application of standard STEP method as multiobjective optimization approach in optimization of production problems, where the objective functions are obtained by regression model. The results obtained by application of STEP method indicate that its application is possible in the optimization of decision variables of the given objective functions. It is evident that the results of both objective functions are within the statistical range, i.e. Min Z1(x1,...x7) = 19.0013 and Max Z2(x1,...x7) = 0.9915, and thus it is not necessary to introduce a new payoff table to find a new compromise (feasible) solution. 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References 6. Literatura [1] G. Simunovi, T. Saric, R. Lujic: Application of Neural Networks in Evaluation of Technological Time, Strojniaki vestnik-Journal of Mechanical Engineering 54 (2008)3, 179-188. [2] P. Cosic, D. Antolic and I. Milic: Web Oriented Sequence Operations, 19th International Conference on Production Research, ICPR-19, July 29-August 2, 2007, Valparaiso, Chile, on CD, 2007. [3] D. Antolic: Estimation of production times by regression models (in Croatian language), Masters thesis, University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, FSB, Zagreb, 2007. [4] Volarevic N., Cosic P.: Shape Complexity Measure Study, DAAAM 2005, Opatija, Croatia, 2005. [5] Volarevic N., Cosic P.: Improving Process Planning through Sequencing the Operations, 7th International conference on AMST '05(Advanced Manufacturing Systems and Technology), Udine, Italy, 2005. [6] I. Kovacic: An overview of fast estimation of production times and delivery deadlines (in Croatian), Graduation thesis, University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture FSB, Zagreb, 2007. [7] Cosic, P, Milcic, D., Kovacic, I. Production Time Estimation as the Part of Collaborative Virtual Manufacturing, International Centre for Innovation and Industrial Logistics -. ICIL 2008, International, March 9 15, 2008, Tel Aviv, Israel pp. 93-100. [8] Cohon, Jared L: Multiobjective programming and planning, Academic Press, Inc. New York , 1978. 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