ࡱ> ebjbj .,VBB8}!\x9>"s&&&&'''78989898989898$;X>]8''''']8&&429C-C-C-'&&78C-'78C-C-V45&G=Q(m5 #8H90x9y5T>1+>55>6H''C-''''']8]8C,'''x9''''>'''''''''B K:ICEIRD2010: Using Multicriteria Optimization in Process Planning Predrag Cosic1, Dragutin Lisjak2 , Drazen Antolic3 1 Faculty of Mechanical Engenering, Zagreb, Croatia,  HYPERLINK "mailto:predrag.cosic@fsb.hr" predrag.cosic@fsb.hr 2 Faculty of Mechanical Engenering, Zagreb, Croatia,  HYPERLINK "mailto:dragutin.lisjak@fsb.hr" dragutin.lisjak@fsb.hr 3AD, Ltd, Ilirski trg 6, Zagreb, Croatia,  HYPERLINK "mailto:drazen@adpp.hr" drazen@adpp.hr 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, multicriteria 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. A STEP model 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 interactive STEP method was applied. Keywords multicriteria optimization, STEP method, process planning 1. Introduction 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 numerous experiences and experience 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 [1] 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 [4, 5]. 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. 2. Results of regression analysis One of the authors was for some time the technical director of INAS company, a successful producer of machine tools in Croatia. Thus, the used technological documentation for conventional machining tools (420 positions) is from that source. By classification of workpieces, determined by BTP form, 8 regression equations for 8 groups of products were obtained. The main grouping criteria were the features (geometrical, tolerance, hardness) from technical drawings and for each workpiece production time was used (technological and auxiliary time). 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. New classifiers were defined W (1-5), based on 5 basic features: W1-material: 1(polymers)-5(alloy steel), W2-shape: 1(rotational)-5(complex), W3- max. workpiece dimension: 1(mini V<120mm)-5(V>2000 mm), W4- complexity, BA  number of dimension lines: 1(very simple BAd"5)-5(5  very complex BA>75), W5- treatment complexity: 1(very rough)-5) very fine). 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 1 Minimum and maximum values of selected variables PRODUCT TYPE - 41113min2.900.1001.0011.210.220.01320.0016.000.92max100.000.4005.0019.6312.500.39720.82033.001.00variablex1x2x3x4x5x6x7Z1Z2Variable descriptionWorkpiece outer diameter Narrowest tolerance of measuresScale of the drawingMaterial mass/strength ratioWall thickness/len-gth ratioProduct surface areaMaterial massProduction timeRatio of Work costs/total costsunit of measuremmmmnumbernumbernumber104 mm2kgh/100number 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 2 Results of stepwise multiple linear regression Regression StatisticsDependent variable -production time Z1Regression StatisticsDependent variable- work costs/ultimate costs ratio Z2Multiple R0.92212166Multiple R0.99207R Square0.85030835R Square0.984202Adjusted R Square0.78481826Adjusted R Square0.977291Standard Error4.09742037Standard Error0.002725Observations24.0Observations24.0Z1CoefficientsZ2CoefficientsIntercept-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 7-0.085513. Description of the model The general multiciteria optimization problem with n decision variables, m constraints and p objectives is [6]:  EMBED Equation.DSMT4  (1)  EMBED Equation.DSMT4  (2) where  EMBED Equation.DSMT4 is the multiciteria objective function and EMBED Equation.DSMT4 , EMBED Equation.DSMT4 , EMBED Equation.DSMT4 are the p individual objective functions. Benayoun (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  (3)  EMBED Equation.DSMT4  (4)  EMBED Equation.DSMT4  (5) 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 (5) The weights  EMBED Equation.DSMT4 in (4) are defined as:  EMBED Equation.DSMT4  (6) where  EMBED Equation.DSMT4  (7) 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  (8) The solution of (3) to (5) with Fd in (5) yields a non-inferior solution  EMBED Equation.DSMT4 , which is closest, given the modified metric in (6), 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  (9)  EMBED Equation.DSMT4  (10)  EMBED Equation.DSMT4  (11) For the new problem  EMBED Equation.DSMT4 , and the other EMBED Equation.DSMT4 are recomputed from (6) for  EMBED Equation.DSMT4 . The problem in (3) to (5) is then resolved with EMBED Equation.DSMT4 , and since EMBED Equation.DSMT4 , (7-78) 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 analysis On the basis of considerations of regression functions in previous sections, the problem of multiciteria optimization with minimization of the objective functions Z1 and Z2 with related constraints (equations (12) to (14)) 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 (12) 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 (13) 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 (14) x1, x2, x3, x4, x5, x6, x7 e" 0 In equations (12) and (13) 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 (13)) 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 3 Values of the decision variables and the objective functions Extreme pointDecision variablesObjective functionsx1x2x3x4x5x6x7Z1(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 3. the data for the first payoff table (Table 4.) have been selected, which is necessary for the calculation of the first compromise solution, Table 4 First payoff table Point of optimal solution XkIdeal values (Mk) of objective functions (Zk) for XkM1=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 (6) and (7) coefficients of equation (4) are calculated as follow: 1=0.0197, 2=10.1974, 1=0.0019 and 2=0.9981. Arranging the obtained equations, the problem of multiciteria optimization has been practically reduced to the problem of single-objective optimization where the variable  EMBED Equation.DSMT4  is minimized according to equation (3). The set of equations for the calculation of the first compromise solution of the given problem is shown in Table 5., and the results of decision variables (x1,...x7) and objective functions Z1 and Z2 are given in Table 6. Table 5 Set of equations of the first compromise solution  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;  Table 6 Results of the first compromise solution 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  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 (15) and (16) 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 (15) -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 (16) 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 7 Second payoff table Point of optimal solution XkIdeal values (Mk) of objective functions (Zk) for XkM1=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 (6) and (7), coefficients of equation (4) are calculated as follow: 1=0.0199, 2=10.1974, 1=0.0019 and 2=0.9981. Since only the value of variable M1 has been changed, the values of 2 and 2 remain the same as in the case of calculation of the first compromise solution. As in the case of the first compromise solution, by arranging the obtained equations, the problem of multiciteria optimization has been reduced to the problem of single-objective optimization where the variable  EMBED Equation.DSMT4  is minimized according to equation (3). The set of equations for the calculation of the second compromise solution of the given problem is shown in Table 8., and the results of decision variables (x1,...x7) and objective functions Z1 and Z2 are given in Table 9. Table 8 Set of equations of the second compromise solution  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 9 Results of the second compromise solution 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)= 0.99154. Conclusion 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. 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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 multiciteria 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. The following can be concluded: it is cost-effective to manufacture products with minimum outside diameter (x1), maximum (wider range) tolerance (x2), maximum scale (x3), maximum strength/mass ratio (x4), minimum of wall thickness/length ratio (x5), maximum product surface area (x6) and minimum mass of material (x7). ACKNOWLEDGMENTS This project is a part of the scientific projects (2007-2009) titled Process Production Impacts to the Competitive and Sustainable Development 120-1521781-3116 financed by the Ministry of Science and Technology of the Republic of Croatia. We express gratitude for the financial support for these projects. References Antolic, D.: 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. Volarevic N., Cosic P.Shape Complexity Measure Study, DAAAM 2005, Opatija, Croatia, 2005. 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. Kovacic, I.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. 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. Cohon, Jared L. Multiobjective programming and planning, Academic Press, Inc. New York , 1978.     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