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Complexity index as a base for decision making whether to introduce NC or conventional machines
lnternattonalJournalofProductwnEconomws,
187
23 ( 1991 ) 187-196
Elsevier
Complexity index as a base for decision making whether to introduce NC or conventional machines S. Ramani and R. Venkatraman Natwnal lnstttute for Training m lndustrtal Engineering, Bombay, india
Abstract Numerical control (NC) systems are widely used in industry today, especmlly m the metal working industry. In devel. opmg countries, many industries are introducing NC machines and the economics of the change assume great significance. In some cases, even conventional machines (automatic mass production or trah~fer line maohines) are good enou8h. Hence the decision of machining a componenl through an NC machine is in the hands of management. This warrants a d~rect and s~mple approach to decision making for selecting the more economical of the two machining methods. This paper attempts to arrive at a yardstick, termed the complextty tndex, used for deciding ~hether the component should be routed through the conventional machine or the NC machine. A rigorous and quantitative approach ofd~scrimmant analysis ~s adopted for various manufacturing components to obtain the d~scnmmant funcUon. The complexity index for a manufacturing component ~s taken to be ~ts &scrlmmant score which ~s used for assigning the components either to the conventional or to the NC machines Statisucal tests of slgmficancc of such a classlfieatmn procedure are performed. An acceptance level for each group (conventional and NC groups) Js calculated (based on the mean score of each group) to determine the m m t m u m complexity index for arriving at the dec~slon rule In this paper, the approach Is presented w~th the md of a case study.
Introduction Numerical coatro; (NC) systems are widely used in industry. In developing countries, many organizations are introducing NC machines and the economic viability of such a change is of concern to them, The decision of whether or not to introduce NC machines is an important question to the management of such organizations; and if the NC machine is introduced, which of the components should be routed through NC and which through conventional facility. This paper attempts to answer these questions by tackling the problem through the discrimmant analysis technique. The subject of introduction of a new technology or process has been studied by many. Newman [ l ], in his paper, has suggested that the economies of the NC machining process should be evaluated based on the cost oi preparing for production, production costs, and other direct costs. Martin [2 ] deals wi)h the breakeven point,
in terms of the volume of the load, at which NC machining becomes economic. Stocker [ 3 ], in his work, cr~ticises the use of the popular concept of 'number of operations' as the sole basis of introducing NC° Holland [4], in his paper, brings out that introduction of a new process merely to adopt the latest technology ts often uneconomical. Azzone and Bertele [5 ] provide a new but complex approach for evaluating economic and strategic aspects in introduction of flexible automation. Recent works have concentrated more on the economics of introducing flexible manufactruing systems and computer integerated manufacturing systems. This may be of interest to the larger industries. However, there ~s a large ?~laulation of small engineering organisations tn developing countries which arc using the conventional machines and are often caught in the decision of whether to introduce NC machines or not. Simple but effective methods are not available for their problem. Hence, the motivation for this paper.
0925-5273/91/$03 50 © 1991 Elsevier Science Pubhshers B V All rights reserved
188
The method proposed uses a factor called the'
comple.~ity index. The objective of the complexity index approach to decision making is to eliminate the need for a detailed economic analysis every time a decision is to be made. Brewer and Millyard [6] termed this factor as the complexityfactor. For easy evaluation of this factor for a component, they defined the complexity factor to be a unique factor- a single parameter for each component. However, they could not relate any one single factor alone to a decision rule. Balding and Farnworth [ 7 ] defined the calculated value of the 'total number of operations' (including tool changes) to be the complexity factor. The decision making however was based on 'savings in time' and the decision rule was to be used for 'components already in production' and 'already tooled'. In both attempts the complexity factor has been viewed as a component defining foctor. While Brewer and Millyard defined the complexity factor to be a single parameter (mostly describing the number of curves etc.), Balding and Farnworth took a less restricted approach - suggesting an equation for calculating the total number of operations. This definition (by Balding and Farnworth) suggests that the complexity factor need not necessarily be a single variable but could be composed of several parameters. By this definition the complexity factor is thus an index on whic~ decision is made and hence the term complexity index was coined in this study. This study propose~ a method of arriving at the complexity index based on a more rigorous quantitative approach than used in earlier re. search. This is expected to provide a more satinfactory decision rule. Approach proposed The problem of obtaining the decision rule could ~e fitted within the framework of the classification problem. The task is one of classifying (assigning or routing) the new component into one of the two populations (NC or conventional machining facility) on the basis of certain measurements on the individual component. The classification procedure was developed by performing a linear two group discriminant analy-
sis. For performing the discriminant analysis, the following were required: (i) a set of variables as predictors, (ii) measured values ofthese variables on a NC group and a conventional group of components. To obtain these input data, 25 components were formed into an NC a~d conventional group by a detailed e:onomic analysis. The predictor variables were selected applying step-wise linear regression analysis on the following dependent variables: 1. NC non-cutting time, 2. NC program planning and preparation time, 3. NC cutting (machining) time, 4. Conventional cutting (machining) time, 5. Conventional non-cutting time. The flexibility of the step-wise regression procedure was utilized to arrive at different sets of significant ~egression predictor variables for all the dependent variables listed above. Linear two group discriminant analysis was performed with each of the following set of predictor variables: 1. 11 predictor variables, 2. 9 predictor variables, 3. 8 predictor variables, and 4. 7 predictor variables. The classification ability of these discriminant functions (with the different predictor variables) was tested and the percentage of correct classifications made by each obtained. Based on these results, the discriminant function, with eight variables being predictors, was selected for developing the decision rule. The classification procedure developed, using this function, was the decision rule for assigning components to the NC and the conventional drilling facilities. Statistical tests for the significance of group means separation along the discriminant axis were performed. Also the relative importance of the predictor variables in the classification (assignment) procedure was assessed. A definition for the complexity index was obtained The discriminant score for a component was taken as a measure of its complexity index. Linear step-wise regression analysis is a computer oriented technique. The procedure starts with including the single predictor variable that
189 has the highest simple correlation with the criterion variable. Variables are then added sequentially based on their ability to account for residual variation in the criterion variable, i.e. the variable which makes the greatest reduction in the error sum ofsquares. This is also the variable which has the highest partial correlation with the dependent variable after the other variables are added. Thus the number of indepe~ent variables per equation is successively ~.reater as the number of steps progresses. "1tm net effect of all this is to partition the predictor variables into two sets: (a) those that enter the final equation versus (b) those that are excluded By setting up appropriate inclusion values it is possible to control the tightness or the looseness with which the predictor set is split into the retained (significant) versus excluded variables.
Application to the case study In this problem, the objective was to develop a decision rule for assigning components to the NC and convcntional drilling facilities. This decision rule was the classification procedure developed using the results of the linear two groups discriminant analysis. The decision making (using this rule) was in the form of measuring certain variables for each component and taking a dee~slon on the basis of these variables. As the developed final decision rule must be simple to use, the number of such variables incorporated in the decision rule was to be kept within workable limits. Hence, in the selection of the predictor variables for performing the discriminant analysis, some control or restriction o w r the number of variables to be selected as predictors way to be exercised. Also, there was a need for the process of variables select'.'on to be made systematic and simple. It was with these objectives in view, the step-wise regression analysis was used in this problem as the screemng procedure to aid variables selection.
Step-wise regression analysis for obtaining a list of predictor variables Step-w~¢ ~e~ ess~on analysis was performed on the following dependent variables - criterion variable:
1. NC non cutting time, 2. NC program planning and preparation time, 3. CONVENTIONAL cutting time, 4. CONVENTIONAL non-cutting time. In the computer program used for the step-wise regression analysis, the value of the parameter PCT is the inc/uswn value - the minimum level specified by user. This value is the minimum proportion (of the total variance) to be reduced (specified), for a variable to enter the regression equation as a significant predictor. By merely altering the PCT values - inclusion values - it is possible to control the tightness or the looseness with which the predictor set of variables could be split into the significant and insignificant predictors. For the problem, the PCT values were there, fore varied suitably, so as to systematically arrive at various sets of significant predictor variables (at each PCT level). This was done as follows. Computer programs were run initially (for all dependent variables) for a PCT level of 0.01, i.e. all predictor variables explaining at least or more than 1% of the total variance were included as significant predictors and entered the final regression equation. The computer results obtained at this PCT level also gives the proportion by which each of the significant predictor variables were reduced (entering the regression at different steps). It was therefore possible to arrive at the different PCT levels with which further computer runs are to be made to systematically exclude each of these significant variables and to obtain the various sets of significant predictors (at different PCT levels). It must be mentioned that no useful purpose is served by trying to explain 100% of the total variance (by making PCTffi0), as then all the independent variables would enter the regression as significant predictors. In this case the step-wise analysis would no longer be a screening procedure. The list of variables from which predictor variables were drawn: 1. NC non-cutting time (dependent or criterion variable). 2. Total number of holes. 3. Number of types of holes ~classification into hole types), 4. Total number of tool required (total number of operations).
190 5. 6. 7. 8. 9. 10. 11.
Average number of tools/type of hole. Average depth (in ram). Average depth/type of hole (in ram). Average number of pick-feeds/type of hole. Total number of pick-feeds required. Number of pitch circle diameters. V~itical height between planes of operation (in mm). 12. Vertical distance between reference point and plane on which machinmg is done (in mm). 13. Number of clamps in tool path. t 4. Perimeter along which holes are placed (in mm). 15. NC ?rogram planning and preparation time. 16. Conventzonal non-cutting time. From the result of the stepwise linear regresstun analysis, given earlier, it was possible to arrive at the significant regressmn predictor variables (for all the dependent variables) at various inclusion values (PCT values). These variables were as follows~ • PCT=0.01 (i.e., all predictor variables explaining at least 1% of the total variance included as significant variables) 1. Total number of holes. 2. Total number of tools required (total number of operations). 3. Average number of tools/type of hole. 4. Number ofditferent types of holes 5. Total number of pick-feeds required. 6. Average depth/hole type (in ram). 7. Average diameter/hole type (in ram). 8. Perimeter along which are placed (in mm ). • PCT---0.02 (i.e., a!! predictor variables explaining at least 2% of the total variance ineluded as significant variables) 1. Total number of holes. 2. Total number of tools required. 3. Average number of tools/types of hole. 4. Total pick-feeds required. 5. Average depth/hole type (in ram). 6. Perimeter along which holes are placed (in mm). • PCT=0.04 (i.e., all predictor variables explaining at least 4% of the total variance ineluded as significant variables) 1. Total number of holes. 2. Total number of tools required.
3. 4. 5. 6.
Average number of tools/type of hole. Total pick.feeds required. Average depth/hole type (in ram). Perimeter along which holes are placed (in mm). • PCT=0.06 (i.e., all predictor variables explaining at least 6% of the total variance included as significant variables) 1. Total number of holes. 2. Total number of tools required. 3. Total pick-feeds required. 4. Average depth/hole type (in mm). 5. Perimeter along which holes are placed (in mm ). • PCT=0.07 (i.e., all predictor variables explaining at least 7% of the total variance included as significant variables) 1. Total number of holes. 2. Total number of tools required. 3. Average depth/hole type (in ram). 4. Perimeter along which holes are placed (in mm). From these results it is seen that for the PCT values of 0.02 and 0.04 the significant regression predictor variables remains unchanged. Three variables viz., batch size, total quantity planned for production, and machinability index of the material were chosen and these three, together with the significant variables of the step-wise regression procedure (at the various PCT levels) were selected as the different sets of predictor variables. Using these variables, the linear two group discriminant analysis could be performed. A linear two group discriminant analysis to classify components The problem discussed in this paper could be fitted within the frameworks of a classification problem. The problem is that of developing a decision rule for assigning or routing (on the basis of economic considerations) component~ ~,o the NC and conventional drilling facilities. To perform the discriminant analysis, the following were required: (a) A set of predictor variables. These are also the variables measured for a component in order to assign it to the NC or conventional
191 machining facility u~ing the final classification procedure - the decision rule. (b) Values of these predictor variables on a NC and conventional group of component. The input data for the discrimin,~n! analysis are the values of the predictor variables on an NC and convt;ntional group of components. By performing the linear two group discriminant analysis, we obtain a discriminant fun~ction the line which best separates the NC and conventional groups. This discriminant function is used for developing the classification procedure - the decision rule. The decision ,ule is to assign components to the respective groups (machining facilities) only on economic considerations. hence the input NC and conveI~tional groups were themselves formed on the basis of economic considerations so that the discriminant function could be rightfully used for developing the final decision rule. In the industry where the data collection for the problem was done, the introduction of NC machining was done only recently. Components previously machined by the conventional drilling facility are now assigned t~, the NC drilling facility and machined. Hence for a considerable number of components relevant data on both these types of machining were available. A detailed economic analysis was done on 25 components. The relevant time values - program planning, set up, tooling and cycle times - for these components were converted to their cost -
values; set up costs were apportioned over the batch size, and the tooling costs over the total quantity planned for production. The cost/piece for NC machining and conventional machining of these 25 components were arrived at. Using this, the components were assigned to the NC and conventional groups. Based on ~'ae consensus of the managers of the industry an NC group of components was formed by assigning to that group those components showing a savings of at least 25% over the conventional machining of it. Twelve components were thus routed to NC and formed the NC group and 13 components formed the conventional group. The input data for the discriminant analysis are the values of the predictor variables for these NC and conventional groups of components. The variables batch quantity and total quantity planned for production have a direct beating on a detailed economic analysis. As the machining time (and hence its cost) is also affected by the material machined, the machinability index of the material machined was also included as a predictor variable. It was therefore possible to arrive at the different sets of predictor variables. Linear two group discriminant analysis was performed with each of these different sets of variables vs a predictor set of variables. The input data for performing this discriminant analysis are the values of these predictor variables set for the NC and conventional groups of components.
TABLE l DlSCrl~!nant coefficients and dlsCnmlnant functions with 8 oredlCtOrvariables DIscnmlnant coefficient
0 00337185 0.0150239 0 0158955 - 0 00267407 0 0000306023 --0 0112818 0.0000913718 -0A 82944
Predictor variable
X~ X2 Xs X4 X5 .,I"6 X7 Xs
Totalnumber of holes Totalnumber of toolsrequired Totalplckfeedsrequired Avelagc depth/hole type Perimeteralong which holesare required Batch quantity Totalquantityplanned forproduction Machlnabilityindex
Discriminantfunctlon: X~ (0.00337185+X2 (0.0150239)+X3 (0.0158955)-X4 (9.00267407) +Xs (0.0000306023)-X6 (0.0112818) +X7 (0.0000913718)-As (0.182944)
192 TABLE 2
Results ofthe test for the classification ability of the dlscnminant functmn with 8 predictors Comp6aent
Discnminant
no,
score
NC group ( 12 cowpcnents) 18 +0 2861127 17 + 0.2238458 9 +0 1270883 13 +0 122615 23 +0 1013264 15 +0 0947236 8 +0,068321 25 +0.0593347 16 +0.0406284 12 +0 0386875 II - 0 016101 14 - 0 1444462 Conventional groro ( 13 components) 20 - 0 0456826 24 - 0.0737869 19 - 0 109367 3 - 0 1260774 22 - 0 1424148 6 - 0.1638048 I - 0.1860664 5 - 0 2556651 2 - 0 2684343 21 - 0 2938314 I0 - 0.2447742 4 - 0 3450603 7 - 0.4285695
Classification matrix Actual NC
Rank
Group assJ&lment (by classification procedure)
Group identity (by detailed analysis)
1 2 3 4 5 6 7 8 9 10 11
NC NC NC NC NC NC NC NC NC NC NC Conventional
NC NC NC NC NC NC NC NC NC NC NC NC
NC Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventional
Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventmnal Corn entional Conventional Conventional Conventional
I 2 3 4 5 6 7 8 9 i0 I1 17.
Conventional
Total
NC ConvenUonal
11 I
I 12
12 13
Total
12
!3
25
Tile results of the discriminant analysis perforrned with 8 predictor variables are given in Table l, givmg the discrlminant coefficients and their respective discriminant functions. The mean values of the predictor variables for each group are calculated from the input data used for performing the discriminant analysis. Using there data, the means of the predictor variables for the NC group and conventional group were calcu. lated and separately substituted in the respective discriminant functions. The mean scores for the
NC and conventional groups for each discriminant function was thus obtained. The classification procedure using a discriminant function was developed as follows: Classify (i.e. assign) a new observation to one of the groups using a discriminant function, the values of the predictor variables were measured for that observation, These values were then substituted in that discriminant function to obtain the discriminant score for that observation. This score was then compared with the discriminant rune-
193
tions mean scores for the groups. The observation is assigned to that group whose mean score is closest to the calculated discriminant score. For the problem under study, it was therefore possible to arrive at such classification procedures using the various discriminant functions. A component could be assigned (classified) to the NC or conventional drilling facility 11S:p.g a-y of these classification procedures, To assign a component using a classification procedure the predictor variables (measured for the component) are substituted in the same discriminant function and the discriminant score for that component obtained. This score was then compared with the discriminant functions mean score for the groups. If the discriminant score for the component was nearer the mean score for the NC group the component was assigned to the NC group. Otherwise, it was assigned to the conventional group. For the probiem, the classification procedures using the various discriminant functions were each tested on the 25 components, whose group identity had already been established on the basis of a detailed economic analysis. The components were assigned to the NC and conventional groups using each of the classification procedure. The results are given in Table 2, The table shows the discriminant scores of the component arranged in descending order. In addition, the table shows a classification matrix to show clearly the number of correct classifications made in each group using each of the classification procedures. The percentage of correct classifications made was also computed. It was observed that the classification procedures using discriminant functions with 1I, 9 and 8 predictor variables correctly classified 23 of the 25 components to their respective NC and conventional groups. The discriminant function with 7 variables as predictors classified correctly 22 of the 25 components. It is seen that various classification procedures have been developed using the discriminant functions with different sets of predictor variables, It is now possible to select a classification procedure for the final decision rule and to explicitly state this decision rule. There is statistical significance to show that the
group means of the predictor variables are significantly separated along the discriminant axis for all the discriminant functions. Statistical tests performed with each function revealed that the group means of the predictor variables had arisen from two different distinct populations - an NC grottp of components and a conventional group of components. Also, results of the tests for the classification ability of these discriminaut functions revealed that the percentage of correct classifications was higher for the discriminant funcJons with 11, 9 and 8 predictor variables than for the function with 7 predictor variables. In addition to the above results, as the operational simplicity of the final decision rule was also to be taken into consideration, the discriminant function with eight predictor variables (with lesser number of predictor variables) was selected. The classification procedure using this discriminant function with eight variables as predictors was used for arriving at the decision rule. Decision making The discriminant function with eight predictor variable: x,(0.00337185)+X~(0.0150239) -I-X3(0.0158955)-X‘,(O.O0267407) +X,(0.0000306023)-X6(0.01 12818) +X,(0.0000913718)-X*(0.182944) Here X,, X,, ... are the predictor variables. These variables are: XI Total number of holes. X2 Total number of tools. X3 Total pick-feeds required. X, Average depth/hole type (in mm). X, Perimeter along which holes are placed (in mm). X, Batch quantity. X, Total quantity planned for production. X8 Machinability index. The mean score for the NC group is -l-0.0835279. The mean score for the conventional group is - 0.2 141181, The classification procedure uses the above results functions as follows. For the decision making, i.e. assignment of a component to the NC or
194 the conventional ~lrillin$ ~¢~Ity (on econo~i: consideratio~ wsmg ~h¢ Oass,dcation procedure, the eight preOictor v~.~ables of the dise~iminant function are measured for the component. The values of these variables are substituted in the above discriminant function to obtain the olscriminanl score tbr the component, if this dis. criminant score is nearer to the mean score for the NC group, the component belongs to tl~e NC group and is therefore assigned to the NC facility. If the discnminant score is nearer the mean score for the conventional group it is assigned to the conventional facility. The decision rule The above classification procedure could be stated in a siraplified form as a decision rule.
Case I if I (Mean score for NC group) - (discriminant score) I < I (Mean score for conventional group) - (discriminant score) I then assign component to NC facility.
Case 2 If I (Mean score for NC group) - (discriminant score) [ > [ (Mean score for conventional group) - (discriminant see:e) I then assign component to conventional facility. In the case under study we have: NC acceptance level= +0.0835279. Conventional acceptance level 0.2141181. The decision rule can be restated: =
-
Case 1 If ] (NC acceptance level) - (complexity index) I < I (conventional acceptance level)(complexity index) I then assign component to NC drilling facility.
Case 2 If { (NC acceptance level) - (complexity in, dex) j > ) (conventional acceptance level)(complexity index) I
then assign component to conventional drilling facility. For the decision making theretbre, we measure the eight predictor variables for each component: and readily obtain the discfiminant score - a number. Using this number the component could be assigned to the NC and conventional facility. From a methodological point of view, we have measured the eight predictor variables for a component and collapsed the 'object point' from an eight dimensional scale to a point (discriminant score) on a single discriminant axis. This discriminant score: (a) is readily (or easily) available for a component, (b) is different for each component and is therefore unique (as the measured values of the predictor variables could be different for each component), and (c) gives us readily the decision to assign the component to the NC or conventional facility. The discriminant score for a component is therefore the index which could be defined as the complexity index for a component. For the decision making, therefore, the complexity index for the component is compared with the mean score for the NC group and the mean score for the conventional group (to find whichever is nearer). The mean score for each group can therefore be regarded as the acceptance level for that group. The mean score for the NC group can therefore be termed as the NC acceptance level and the mean score for the conventional group as the conventional acceptance level. The decision rule can therefore be re-stated as follows:
I f the complexity index for a component is nearer the NC acceptance level, the compo,wnt is assigned w the NC facility. If the complexity index is nearer the conventional acceptance level the component is assigned to the conventional facdity The results of such classification on the 25 components chosen for this study are given in Table 3. The approach presented in this study can easily he extended to provide a decision rule for the introduction of CNC facilities in the place of
195 TABLE 3 Class~ficaUonof the component based on ',he coefficient of the predictor variables Component number
Group identity
Predictor variables 2
3
4
6
7
8
I
I 7
7
2
2 3 4 5
3 24 6 6
4 4 3 2
2 0 0 0
22 05 99 663 5 18 5.064 6,59
5
465.5 291 792 285 444
30 10 I0 lO 30
600 200 200 200 600
01 O.1 1 1.7 O,I
Conventional Conventional Conventional Conventional Conventional
6 7
3 3
5 4
3 0
71 17.68
10 30 10 10 10
200 600 200 200 200
0.1 1 01 0.1 !.7
Conventional Conventional NC NC Conventional
8
18
6
0
8,98
9 10
30 6
8 3
4 0
33.424 5 06
195 492 1782 1449 99 294
11 12 13 14 15
5 19 45 13 61
5 6 5 5 6
0 2 0 0 3
2.514 25.722 20.989 2.544 7.42
375 1121 2115 682 5 1586
10 10 10 10 10
200 200 200 200 200
0.1 0.1 0.1 1 1
NC NC NC NC NC
16 17 18 19 20
36 58 58 15 6
8 6 7 7 3
2 0 3 0 0
60.77 12 799 19.27 7 504 2.914
1392 2784 3335 1059 99 315
10 10 10 10 10
200 200 200 200 200
0! 0.1 01 1 01
NC NC NC Convenuonal Conventional
21 22 23 24 25
4 24 40 15 15
3 4 4 8 7
1 0 0 0 I
27 34 7 17 14 91 3 865 16 496
220 432 1930 1413,75 1458 75
30 10 10 I0 10
600 200 200 200 200
0,1 1 0.1 I 0.1
Conventional Conventional NC Con vent ional NC
c o n v e n t i o n a l facilities. T h i s is possible because the cost ealculatton procedures would be the same for C N C m a c h i n e s also. A relevant set o f predictor variables has to be o b t a i n e d based on the data for C N C facility before the d e t e r m i n a t i o n o f the discriminant function and hence the d e e i s m n rule,
through the c o n v e n t i o n a l facility. A simple decision rule based on e c o n o m i c considerations is presented. T h e p r i m e feature o f the proposed m e t h o d is that the decision is based on the single p a r a m e t e r t e r m e d the complexity index although the n ~ m b e r o f predictor variables are many. T h e ease o f application o f the m e t h o d to a live situation ~s also vividly seen.
Conclusions References
In th~s paper an a t t e m p t is m a d e to propose a m e t h o d using the well.known d i s e r i m i n a n t analysis to decide u p o n the introduction o f N C machining, T h e m e t h o d o l o g y presented here is expected to help the m a n a g e m e n t o f organisations in deciding which o f the c o m p o n e n t s should be assigned to N C facility and which ones to process
1 Newman, A.C., 1966, The production engineer and numerical control. Prod. Eng, 45 (4). 20-22, 2 Martin, S,J., 1970 Numerical Control For Machine Tools. The Ent~lishUmversity Press, London. 3 Stocker, W M , 1961. How to prove profit m NC. Am. Mech., 105(22) 77-120
196 4 Holland, D, 1984 Adopt automation - bu! with some consideration. Prod Eng., 63(5): 40-43 5 Azzone, G. and Bertele, U., 1989. Measuring the e¢onotate effectiveness of flexible automation: a new approach. Int. J. Prod. Res., 25(5): 735-749. 6 Brewer, R.C. and Millyard, P.W., 1960. Some economic
aspects ofnumencally controlled machine tools Prod. Eng. 39(3): 141-146. 7 Balding, A.P. and Farnworth, G.H., 1971. Component complexity factors for NC machining. Machinery and Prod, Eng., I |g~3055): 838-841.
187
23 ( 1991 ) 187-196
Elsevier
Complexity index as a base for decision making whether to introduce NC or conventional machines S. Ramani and R. Venkatraman Natwnal lnstttute for Training m lndustrtal Engineering, Bombay, india
Abstract Numerical control (NC) systems are widely used in industry today, especmlly m the metal working industry. In devel. opmg countries, many industries are introducing NC machines and the economics of the change assume great significance. In some cases, even conventional machines (automatic mass production or trah~fer line maohines) are good enou8h. Hence the decision of machining a componenl through an NC machine is in the hands of management. This warrants a d~rect and s~mple approach to decision making for selecting the more economical of the two machining methods. This paper attempts to arrive at a yardstick, termed the complextty tndex, used for deciding ~hether the component should be routed through the conventional machine or the NC machine. A rigorous and quantitative approach ofd~scrimmant analysis ~s adopted for various manufacturing components to obtain the d~scnmmant funcUon. The complexity index for a manufacturing component ~s taken to be ~ts &scrlmmant score which ~s used for assigning the components either to the conventional or to the NC machines Statisucal tests of slgmficancc of such a classlfieatmn procedure are performed. An acceptance level for each group (conventional and NC groups) Js calculated (based on the mean score of each group) to determine the m m t m u m complexity index for arriving at the dec~slon rule In this paper, the approach Is presented w~th the md of a case study.
Introduction Numerical coatro; (NC) systems are widely used in industry. In developing countries, many organizations are introducing NC machines and the economic viability of such a change is of concern to them, The decision of whether or not to introduce NC machines is an important question to the management of such organizations; and if the NC machine is introduced, which of the components should be routed through NC and which through conventional facility. This paper attempts to answer these questions by tackling the problem through the discrimmant analysis technique. The subject of introduction of a new technology or process has been studied by many. Newman [ l ], in his paper, has suggested that the economies of the NC machining process should be evaluated based on the cost oi preparing for production, production costs, and other direct costs. Martin [2 ] deals wi)h the breakeven point,
in terms of the volume of the load, at which NC machining becomes economic. Stocker [ 3 ], in his work, cr~ticises the use of the popular concept of 'number of operations' as the sole basis of introducing NC° Holland [4], in his paper, brings out that introduction of a new process merely to adopt the latest technology ts often uneconomical. Azzone and Bertele [5 ] provide a new but complex approach for evaluating economic and strategic aspects in introduction of flexible automation. Recent works have concentrated more on the economics of introducing flexible manufactruing systems and computer integerated manufacturing systems. This may be of interest to the larger industries. However, there ~s a large ?~laulation of small engineering organisations tn developing countries which arc using the conventional machines and are often caught in the decision of whether to introduce NC machines or not. Simple but effective methods are not available for their problem. Hence, the motivation for this paper.
0925-5273/91/$03 50 © 1991 Elsevier Science Pubhshers B V All rights reserved
188
The method proposed uses a factor called the'
comple.~ity index. The objective of the complexity index approach to decision making is to eliminate the need for a detailed economic analysis every time a decision is to be made. Brewer and Millyard [6] termed this factor as the complexityfactor. For easy evaluation of this factor for a component, they defined the complexity factor to be a unique factor- a single parameter for each component. However, they could not relate any one single factor alone to a decision rule. Balding and Farnworth [ 7 ] defined the calculated value of the 'total number of operations' (including tool changes) to be the complexity factor. The decision making however was based on 'savings in time' and the decision rule was to be used for 'components already in production' and 'already tooled'. In both attempts the complexity factor has been viewed as a component defining foctor. While Brewer and Millyard defined the complexity factor to be a single parameter (mostly describing the number of curves etc.), Balding and Farnworth took a less restricted approach - suggesting an equation for calculating the total number of operations. This definition (by Balding and Farnworth) suggests that the complexity factor need not necessarily be a single variable but could be composed of several parameters. By this definition the complexity factor is thus an index on whic~ decision is made and hence the term complexity index was coined in this study. This study propose~ a method of arriving at the complexity index based on a more rigorous quantitative approach than used in earlier re. search. This is expected to provide a more satinfactory decision rule. Approach proposed The problem of obtaining the decision rule could ~e fitted within the framework of the classification problem. The task is one of classifying (assigning or routing) the new component into one of the two populations (NC or conventional machining facility) on the basis of certain measurements on the individual component. The classification procedure was developed by performing a linear two group discriminant analy-
sis. For performing the discriminant analysis, the following were required: (i) a set of variables as predictors, (ii) measured values ofthese variables on a NC group and a conventional group of components. To obtain these input data, 25 components were formed into an NC a~d conventional group by a detailed e:onomic analysis. The predictor variables were selected applying step-wise linear regression analysis on the following dependent variables: 1. NC non-cutting time, 2. NC program planning and preparation time, 3. NC cutting (machining) time, 4. Conventional cutting (machining) time, 5. Conventional non-cutting time. The flexibility of the step-wise regression procedure was utilized to arrive at different sets of significant ~egression predictor variables for all the dependent variables listed above. Linear two group discriminant analysis was performed with each of the following set of predictor variables: 1. 11 predictor variables, 2. 9 predictor variables, 3. 8 predictor variables, and 4. 7 predictor variables. The classification ability of these discriminant functions (with the different predictor variables) was tested and the percentage of correct classifications made by each obtained. Based on these results, the discriminant function, with eight variables being predictors, was selected for developing the decision rule. The classification procedure developed, using this function, was the decision rule for assigning components to the NC and the conventional drilling facilities. Statistical tests for the significance of group means separation along the discriminant axis were performed. Also the relative importance of the predictor variables in the classification (assignment) procedure was assessed. A definition for the complexity index was obtained The discriminant score for a component was taken as a measure of its complexity index. Linear step-wise regression analysis is a computer oriented technique. The procedure starts with including the single predictor variable that
189 has the highest simple correlation with the criterion variable. Variables are then added sequentially based on their ability to account for residual variation in the criterion variable, i.e. the variable which makes the greatest reduction in the error sum ofsquares. This is also the variable which has the highest partial correlation with the dependent variable after the other variables are added. Thus the number of indepe~ent variables per equation is successively ~.reater as the number of steps progresses. "1tm net effect of all this is to partition the predictor variables into two sets: (a) those that enter the final equation versus (b) those that are excluded By setting up appropriate inclusion values it is possible to control the tightness or the looseness with which the predictor set is split into the retained (significant) versus excluded variables.
Application to the case study In this problem, the objective was to develop a decision rule for assigning components to the NC and convcntional drilling facilities. This decision rule was the classification procedure developed using the results of the linear two groups discriminant analysis. The decision making (using this rule) was in the form of measuring certain variables for each component and taking a dee~slon on the basis of these variables. As the developed final decision rule must be simple to use, the number of such variables incorporated in the decision rule was to be kept within workable limits. Hence, in the selection of the predictor variables for performing the discriminant analysis, some control or restriction o w r the number of variables to be selected as predictors way to be exercised. Also, there was a need for the process of variables select'.'on to be made systematic and simple. It was with these objectives in view, the step-wise regression analysis was used in this problem as the screemng procedure to aid variables selection.
Step-wise regression analysis for obtaining a list of predictor variables Step-w~¢ ~e~ ess~on analysis was performed on the following dependent variables - criterion variable:
1. NC non cutting time, 2. NC program planning and preparation time, 3. CONVENTIONAL cutting time, 4. CONVENTIONAL non-cutting time. In the computer program used for the step-wise regression analysis, the value of the parameter PCT is the inc/uswn value - the minimum level specified by user. This value is the minimum proportion (of the total variance) to be reduced (specified), for a variable to enter the regression equation as a significant predictor. By merely altering the PCT values - inclusion values - it is possible to control the tightness or the looseness with which the predictor set of variables could be split into the significant and insignificant predictors. For the problem, the PCT values were there, fore varied suitably, so as to systematically arrive at various sets of significant predictor variables (at each PCT level). This was done as follows. Computer programs were run initially (for all dependent variables) for a PCT level of 0.01, i.e. all predictor variables explaining at least or more than 1% of the total variance were included as significant predictors and entered the final regression equation. The computer results obtained at this PCT level also gives the proportion by which each of the significant predictor variables were reduced (entering the regression at different steps). It was therefore possible to arrive at the different PCT levels with which further computer runs are to be made to systematically exclude each of these significant variables and to obtain the various sets of significant predictors (at different PCT levels). It must be mentioned that no useful purpose is served by trying to explain 100% of the total variance (by making PCTffi0), as then all the independent variables would enter the regression as significant predictors. In this case the step-wise analysis would no longer be a screening procedure. The list of variables from which predictor variables were drawn: 1. NC non-cutting time (dependent or criterion variable). 2. Total number of holes. 3. Number of types of holes ~classification into hole types), 4. Total number of tool required (total number of operations).
190 5. 6. 7. 8. 9. 10. 11.
Average number of tools/type of hole. Average depth (in ram). Average depth/type of hole (in ram). Average number of pick-feeds/type of hole. Total number of pick-feeds required. Number of pitch circle diameters. V~itical height between planes of operation (in mm). 12. Vertical distance between reference point and plane on which machinmg is done (in mm). 13. Number of clamps in tool path. t 4. Perimeter along which holes are placed (in mm). 15. NC ?rogram planning and preparation time. 16. Conventzonal non-cutting time. From the result of the stepwise linear regresstun analysis, given earlier, it was possible to arrive at the significant regressmn predictor variables (for all the dependent variables) at various inclusion values (PCT values). These variables were as follows~ • PCT=0.01 (i.e., all predictor variables explaining at least 1% of the total variance included as significant variables) 1. Total number of holes. 2. Total number of tools required (total number of operations). 3. Average number of tools/type of hole. 4. Number ofditferent types of holes 5. Total number of pick-feeds required. 6. Average depth/hole type (in ram). 7. Average diameter/hole type (in ram). 8. Perimeter along which are placed (in mm ). • PCT---0.02 (i.e., a!! predictor variables explaining at least 2% of the total variance ineluded as significant variables) 1. Total number of holes. 2. Total number of tools required. 3. Average number of tools/types of hole. 4. Total pick-feeds required. 5. Average depth/hole type (in ram). 6. Perimeter along which holes are placed (in mm). • PCT=0.04 (i.e., all predictor variables explaining at least 4% of the total variance ineluded as significant variables) 1. Total number of holes. 2. Total number of tools required.
3. 4. 5. 6.
Average number of tools/type of hole. Total pick.feeds required. Average depth/hole type (in ram). Perimeter along which holes are placed (in mm). • PCT=0.06 (i.e., all predictor variables explaining at least 6% of the total variance included as significant variables) 1. Total number of holes. 2. Total number of tools required. 3. Total pick-feeds required. 4. Average depth/hole type (in mm). 5. Perimeter along which holes are placed (in mm ). • PCT=0.07 (i.e., all predictor variables explaining at least 7% of the total variance included as significant variables) 1. Total number of holes. 2. Total number of tools required. 3. Average depth/hole type (in ram). 4. Perimeter along which holes are placed (in mm). From these results it is seen that for the PCT values of 0.02 and 0.04 the significant regression predictor variables remains unchanged. Three variables viz., batch size, total quantity planned for production, and machinability index of the material were chosen and these three, together with the significant variables of the step-wise regression procedure (at the various PCT levels) were selected as the different sets of predictor variables. Using these variables, the linear two group discriminant analysis could be performed. A linear two group discriminant analysis to classify components The problem discussed in this paper could be fitted within the frameworks of a classification problem. The problem is that of developing a decision rule for assigning or routing (on the basis of economic considerations) component~ ~,o the NC and conventional drilling facilities. To perform the discriminant analysis, the following were required: (a) A set of predictor variables. These are also the variables measured for a component in order to assign it to the NC or conventional
191 machining facility u~ing the final classification procedure - the decision rule. (b) Values of these predictor variables on a NC and conventional group of component. The input data for the discrimin,~n! analysis are the values of the predictor variables on an NC and convt;ntional group of components. By performing the linear two group discriminant analysis, we obtain a discriminant fun~ction the line which best separates the NC and conventional groups. This discriminant function is used for developing the classification procedure - the decision rule. The decision ,ule is to assign components to the respective groups (machining facilities) only on economic considerations. hence the input NC and conveI~tional groups were themselves formed on the basis of economic considerations so that the discriminant function could be rightfully used for developing the final decision rule. In the industry where the data collection for the problem was done, the introduction of NC machining was done only recently. Components previously machined by the conventional drilling facility are now assigned t~, the NC drilling facility and machined. Hence for a considerable number of components relevant data on both these types of machining were available. A detailed economic analysis was done on 25 components. The relevant time values - program planning, set up, tooling and cycle times - for these components were converted to their cost -
values; set up costs were apportioned over the batch size, and the tooling costs over the total quantity planned for production. The cost/piece for NC machining and conventional machining of these 25 components were arrived at. Using this, the components were assigned to the NC and conventional groups. Based on ~'ae consensus of the managers of the industry an NC group of components was formed by assigning to that group those components showing a savings of at least 25% over the conventional machining of it. Twelve components were thus routed to NC and formed the NC group and 13 components formed the conventional group. The input data for the discriminant analysis are the values of the predictor variables for these NC and conventional groups of components. The variables batch quantity and total quantity planned for production have a direct beating on a detailed economic analysis. As the machining time (and hence its cost) is also affected by the material machined, the machinability index of the material machined was also included as a predictor variable. It was therefore possible to arrive at the different sets of predictor variables. Linear two group discriminant analysis was performed with each of these different sets of variables vs a predictor set of variables. The input data for performing this discriminant analysis are the values of these predictor variables set for the NC and conventional groups of components.
TABLE l DlSCrl~!nant coefficients and dlsCnmlnant functions with 8 oredlCtOrvariables DIscnmlnant coefficient
0 00337185 0.0150239 0 0158955 - 0 00267407 0 0000306023 --0 0112818 0.0000913718 -0A 82944
Predictor variable
X~ X2 Xs X4 X5 .,I"6 X7 Xs
Totalnumber of holes Totalnumber of toolsrequired Totalplckfeedsrequired Avelagc depth/hole type Perimeteralong which holesare required Batch quantity Totalquantityplanned forproduction Machlnabilityindex
Discriminantfunctlon: X~ (0.00337185+X2 (0.0150239)+X3 (0.0158955)-X4 (9.00267407) +Xs (0.0000306023)-X6 (0.0112818) +X7 (0.0000913718)-As (0.182944)
192 TABLE 2
Results ofthe test for the classification ability of the dlscnminant functmn with 8 predictors Comp6aent
Discnminant
no,
score
NC group ( 12 cowpcnents) 18 +0 2861127 17 + 0.2238458 9 +0 1270883 13 +0 122615 23 +0 1013264 15 +0 0947236 8 +0,068321 25 +0.0593347 16 +0.0406284 12 +0 0386875 II - 0 016101 14 - 0 1444462 Conventional groro ( 13 components) 20 - 0 0456826 24 - 0.0737869 19 - 0 109367 3 - 0 1260774 22 - 0 1424148 6 - 0.1638048 I - 0.1860664 5 - 0 2556651 2 - 0 2684343 21 - 0 2938314 I0 - 0.2447742 4 - 0 3450603 7 - 0.4285695
Classification matrix Actual NC
Rank
Group assJ&lment (by classification procedure)
Group identity (by detailed analysis)
1 2 3 4 5 6 7 8 9 10 11
NC NC NC NC NC NC NC NC NC NC NC Conventional
NC NC NC NC NC NC NC NC NC NC NC NC
NC Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventional
Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventional Conventmnal Corn entional Conventional Conventional Conventional
I 2 3 4 5 6 7 8 9 i0 I1 17.
Conventional
Total
NC ConvenUonal
11 I
I 12
12 13
Total
12
!3
25
Tile results of the discriminant analysis perforrned with 8 predictor variables are given in Table l, givmg the discrlminant coefficients and their respective discriminant functions. The mean values of the predictor variables for each group are calculated from the input data used for performing the discriminant analysis. Using there data, the means of the predictor variables for the NC group and conventional group were calcu. lated and separately substituted in the respective discriminant functions. The mean scores for the
NC and conventional groups for each discriminant function was thus obtained. The classification procedure using a discriminant function was developed as follows: Classify (i.e. assign) a new observation to one of the groups using a discriminant function, the values of the predictor variables were measured for that observation, These values were then substituted in that discriminant function to obtain the discriminant score for that observation. This score was then compared with the discriminant rune-
193
tions mean scores for the groups. The observation is assigned to that group whose mean score is closest to the calculated discriminant score. For the problem under study, it was therefore possible to arrive at such classification procedures using the various discriminant functions. A component could be assigned (classified) to the NC or conventional drilling facility 11S:p.g a-y of these classification procedures, To assign a component using a classification procedure the predictor variables (measured for the component) are substituted in the same discriminant function and the discriminant score for that component obtained. This score was then compared with the discriminant functions mean score for the groups. If the discriminant score for the component was nearer the mean score for the NC group the component was assigned to the NC group. Otherwise, it was assigned to the conventional group. For the probiem, the classification procedures using the various discriminant functions were each tested on the 25 components, whose group identity had already been established on the basis of a detailed economic analysis. The components were assigned to the NC and conventional groups using each of the classification procedure. The results are given in Table 2, The table shows the discriminant scores of the component arranged in descending order. In addition, the table shows a classification matrix to show clearly the number of correct classifications made in each group using each of the classification procedures. The percentage of correct classifications made was also computed. It was observed that the classification procedures using discriminant functions with 1I, 9 and 8 predictor variables correctly classified 23 of the 25 components to their respective NC and conventional groups. The discriminant function with 7 variables as predictors classified correctly 22 of the 25 components. It is seen that various classification procedures have been developed using the discriminant functions with different sets of predictor variables, It is now possible to select a classification procedure for the final decision rule and to explicitly state this decision rule. There is statistical significance to show that the
group means of the predictor variables are significantly separated along the discriminant axis for all the discriminant functions. Statistical tests performed with each function revealed that the group means of the predictor variables had arisen from two different distinct populations - an NC grottp of components and a conventional group of components. Also, results of the tests for the classification ability of these discriminaut functions revealed that the percentage of correct classifications was higher for the discriminant funcJons with 11, 9 and 8 predictor variables than for the function with 7 predictor variables. In addition to the above results, as the operational simplicity of the final decision rule was also to be taken into consideration, the discriminant function with eight predictor variables (with lesser number of predictor variables) was selected. The classification procedure using this discriminant function with eight variables as predictors was used for arriving at the decision rule. Decision making The discriminant function with eight predictor variable: x,(0.00337185)+X~(0.0150239) -I-X3(0.0158955)-X‘,(O.O0267407) +X,(0.0000306023)-X6(0.01 12818) +X,(0.0000913718)-X*(0.182944) Here X,, X,, ... are the predictor variables. These variables are: XI Total number of holes. X2 Total number of tools. X3 Total pick-feeds required. X, Average depth/hole type (in mm). X, Perimeter along which holes are placed (in mm). X, Batch quantity. X, Total quantity planned for production. X8 Machinability index. The mean score for the NC group is -l-0.0835279. The mean score for the conventional group is - 0.2 141181, The classification procedure uses the above results functions as follows. For the decision making, i.e. assignment of a component to the NC or
194 the conventional ~lrillin$ ~¢~Ity (on econo~i: consideratio~ wsmg ~h¢ Oass,dcation procedure, the eight preOictor v~.~ables of the dise~iminant function are measured for the component. The values of these variables are substituted in the above discriminant function to obtain the olscriminanl score tbr the component, if this dis. criminant score is nearer to the mean score for the NC group, the component belongs to tl~e NC group and is therefore assigned to the NC facility. If the discnminant score is nearer the mean score for the conventional group it is assigned to the conventional facility. The decision rule The above classification procedure could be stated in a siraplified form as a decision rule.
Case I if I (Mean score for NC group) - (discriminant score) I < I (Mean score for conventional group) - (discriminant score) I then assign component to NC facility.
Case 2 If I (Mean score for NC group) - (discriminant score) [ > [ (Mean score for conventional group) - (discriminant see:e) I then assign component to conventional facility. In the case under study we have: NC acceptance level= +0.0835279. Conventional acceptance level 0.2141181. The decision rule can be restated: =
-
Case 1 If ] (NC acceptance level) - (complexity index) I < I (conventional acceptance level)(complexity index) I then assign component to NC drilling facility.
Case 2 If { (NC acceptance level) - (complexity in, dex) j > ) (conventional acceptance level)(complexity index) I
then assign component to conventional drilling facility. For the decision making theretbre, we measure the eight predictor variables for each component: and readily obtain the discfiminant score - a number. Using this number the component could be assigned to the NC and conventional facility. From a methodological point of view, we have measured the eight predictor variables for a component and collapsed the 'object point' from an eight dimensional scale to a point (discriminant score) on a single discriminant axis. This discriminant score: (a) is readily (or easily) available for a component, (b) is different for each component and is therefore unique (as the measured values of the predictor variables could be different for each component), and (c) gives us readily the decision to assign the component to the NC or conventional facility. The discriminant score for a component is therefore the index which could be defined as the complexity index for a component. For the decision making, therefore, the complexity index for the component is compared with the mean score for the NC group and the mean score for the conventional group (to find whichever is nearer). The mean score for each group can therefore be regarded as the acceptance level for that group. The mean score for the NC group can therefore be termed as the NC acceptance level and the mean score for the conventional group as the conventional acceptance level. The decision rule can therefore be re-stated as follows:
I f the complexity index for a component is nearer the NC acceptance level, the compo,wnt is assigned w the NC facility. If the complexity index is nearer the conventional acceptance level the component is assigned to the conventional facdity The results of such classification on the 25 components chosen for this study are given in Table 3. The approach presented in this study can easily he extended to provide a decision rule for the introduction of CNC facilities in the place of
195 TABLE 3 Class~ficaUonof the component based on ',he coefficient of the predictor variables Component number
Group identity
Predictor variables 2
3
4
6
7
8
I
I 7
7
2
2 3 4 5
3 24 6 6
4 4 3 2
2 0 0 0
22 05 99 663 5 18 5.064 6,59
5
465.5 291 792 285 444
30 10 I0 lO 30
600 200 200 200 600
01 O.1 1 1.7 O,I
Conventional Conventional Conventional Conventional Conventional
6 7
3 3
5 4
3 0
71 17.68
10 30 10 10 10
200 600 200 200 200
0.1 1 01 0.1 !.7
Conventional Conventional NC NC Conventional
8
18
6
0
8,98
9 10
30 6
8 3
4 0
33.424 5 06
195 492 1782 1449 99 294
11 12 13 14 15
5 19 45 13 61
5 6 5 5 6
0 2 0 0 3
2.514 25.722 20.989 2.544 7.42
375 1121 2115 682 5 1586
10 10 10 10 10
200 200 200 200 200
0.1 0.1 0.1 1 1
NC NC NC NC NC
16 17 18 19 20
36 58 58 15 6
8 6 7 7 3
2 0 3 0 0
60.77 12 799 19.27 7 504 2.914
1392 2784 3335 1059 99 315
10 10 10 10 10
200 200 200 200 200
0! 0.1 01 1 01
NC NC NC Convenuonal Conventional
21 22 23 24 25
4 24 40 15 15
3 4 4 8 7
1 0 0 0 I
27 34 7 17 14 91 3 865 16 496
220 432 1930 1413,75 1458 75
30 10 10 I0 10
600 200 200 200 200
0,1 1 0.1 I 0.1
Conventional Conventional NC Con vent ional NC
c o n v e n t i o n a l facilities. T h i s is possible because the cost ealculatton procedures would be the same for C N C m a c h i n e s also. A relevant set o f predictor variables has to be o b t a i n e d based on the data for C N C facility before the d e t e r m i n a t i o n o f the discriminant function and hence the d e e i s m n rule,
through the c o n v e n t i o n a l facility. A simple decision rule based on e c o n o m i c considerations is presented. T h e p r i m e feature o f the proposed m e t h o d is that the decision is based on the single p a r a m e t e r t e r m e d the complexity index although the n ~ m b e r o f predictor variables are many. T h e ease o f application o f the m e t h o d to a live situation ~s also vividly seen.
Conclusions References
In th~s paper an a t t e m p t is m a d e to propose a m e t h o d using the well.known d i s e r i m i n a n t analysis to decide u p o n the introduction o f N C machining, T h e m e t h o d o l o g y presented here is expected to help the m a n a g e m e n t o f organisations in deciding which o f the c o m p o n e n t s should be assigned to N C facility and which ones to process
1 Newman, A.C., 1966, The production engineer and numerical control. Prod. Eng, 45 (4). 20-22, 2 Martin, S,J., 1970 Numerical Control For Machine Tools. The Ent~lishUmversity Press, London. 3 Stocker, W M , 1961. How to prove profit m NC. Am. Mech., 105(22) 77-120
196 4 Holland, D, 1984 Adopt automation - bu! with some consideration. Prod Eng., 63(5): 40-43 5 Azzone, G. and Bertele, U., 1989. Measuring the e¢onotate effectiveness of flexible automation: a new approach. Int. J. Prod. Res., 25(5): 735-749. 6 Brewer, R.C. and Millyard, P.W., 1960. Some economic
aspects ofnumencally controlled machine tools Prod. Eng. 39(3): 141-146. 7 Balding, A.P. and Farnworth, G.H., 1971. Component complexity factors for NC machining. Machinery and Prod, Eng., I |g~3055): 838-841.