- Email: [email protected]
A fuzzy decision model for color reproduction
Int. J. Production Economics 58 (1999) 31—37
A fuzzy decision model for color reproduction Cecilia Temponi *, Farid Derisavi Fard, H.W. Corley Southwest Texas State University, San Marcos, TX 78666, USA Xerox Corporation, Dallas, TX, USA The University of Texas at Arlington, Arlington, TX, USA Received 4 June 1997; accepted 29 December 1997
Abstract Color quality appraisal refers to a visual evaluation of an original color image and an assessment of the degree to which that image is matched in photomechanical or electronic reproduction. The evaluation of reproduction quality is quite subjective since it relies on the individual judgments of color technicians who have different perceptions of color and its component qualities. To reduce the amount of rework due to subjective evaluation of color reproduction, an objective model is proposed here. In this model, a fuzzy set of quality factors is introduced and a numerical quality value calculated. This measure is then compared with an acceptable quality value experimentally established for particular customers and printing processes. Finally, the installed model is tested in actual production, and compared with the conventional evaluation method in a cost analysis. 1999 Elsevier Science B.V. All rights reserved. Keywords: Printing process; Color appraisal; Production; Fuzzy model; Cost analysis
1. Introduction In color quality appraisal, the original color image (color transparency, color print, or other reflection copy [1]) and the color reproduction are compared for hue, lightness, and saturation. They are also appraised in terms of tonal range, contrast, and texture, which affect the overall appearance of the color image. However, such analysis is subjec-
* Corresponding author. Tel.: #1 512 245 3189; #1 512 245 3089; e-mail: [email protected].
fax:
tive and varies from one technician to another. This fact is the primary cause of production rework in the prepress industry today when the color reproduction does not meet the customer’s expectation. Quantitative decision techniques have not been heretofore used, but the situation seems a natural application of fuzzy logic. In this paper we construct a model for the numerical evaluation of the quality level of a color reproduction by introducing a set of factors for a color technician to use in their evaluation and a measure of their respective importance. This approach enables the technician to judge the degree of color match acceptable to a particular customer for a specified printing process.
0925-5273/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved PII: S 0 9 2 5 - 5 2 7 3 ( 9 8 ) 0 0 0 7 8 - 4
32
C. Temponi et al. /Int. J. Production Economics 58 (1999) 31—37
2. Preliminaries 2.1. Fuzzy set theory and fuzzy numbers Introduced by Zadeh [2], the theory of fuzzy sets concerns a subset A of a universe of discourse X, where the transition between full membership and no membership is gradual rather than abrupt. A fuzzy subset has no well-defined boundaries as compared with the universe X that consists of definite objects. Most classes of objects encountered in the real world are of a fuzzy nature, without a precise criteria for membership. In such cases it is not necessary for an object to belong or not belong to a particular class. The object may have an intermediate class of membership [3,4]. Fuzzy numbers provide a way to represent one’s knowledge about real-world numbers. Information or knowledge about a number is often based on someone’s experience. An easy way to define a fuzzy number is to determine its lowest value, its highest value, and the most likely value. For example, from the authors’ experience in the prepress industry, it takes from 5 to 12 days to train someone for a particular position, depending on the trainee’s background. However, most people complete the training program in seven days. A fuzzy number representing the number of days required to train a new hire for this particular position is shown in Fig. 1. This fuzzy number is referred to as a triangular fuzzy number (TFN).
2.2. Fuzzy measure The notion of fuzzy measure introduced by Sugeno [5] can be used to model uncertainty, in particular, the vagueness and uncertainty inherent in natural language. In this paper it is used to quantify subjective and vague terms like “too dark”, “more contrast”, “more punch”, or “match the original” used by designers and print buyers in describing reproduction quality to color experts. Such terms are difficult to translate into precise language. For other applications of fuzzy measure see Refs. [4,6—9]. In this work we implement fuzzy measure as in Wang and Klir [10] and define the fuzzy integral of
Fig. 1. Number of days required to train a new hire.
f on X with respect to the fuzzy measure k by
f dk" sup [ak(X5F )], ? Z
(1)
where is the minimum operator and F "+x: ? f (x)*a,. Now consider an object to be evaluated, and let X"+x , x ,2, x , denote the factor space L of the object, that is, the set of all its quality factors of interest. The goal is to obtain a numerical quality evaluation of the given object from the evaluations for each individual factor. Assume that each subset A of the factor space X is associated with a real number s(A) between 0 and 1 that indicates the importance of A. Such a real number should be the maximum possible score that the object can gain relying only on the quality factors in A. Obviously the empty set H, which includes no relevant quality factors, has the minimum importance 0; and the whole factor space X has the maximum importance 1. Moreover, if each factor in a factor set A belongs to another factor set B, then A is at most as important as B. It follows that s(H)"0, s(X)"1, and s(A))s(B) whenever ALB. The number of quality factors for the given object is finite, so the set function s is a regular fuzzy measure on the measurable space (X, o(X)) called the importance measure on X [5]. It represents here a composite expert opinion. Given a particular object to be evaluated, a factor space of the object, and an importance measure, the object is evaluated
C. Temponi et al. /Int. J. Production Economics 58 (1999) 31—37
by experts for each individual quality factor x , x , 2 , x . Their subjective opinions are aggreg L ated into a composite quality evaluation function f to yield the scores f (x ), f (x ), 2 , f (x ), where f may be regarded as a measurable function on (X, o(X)) for which f (x )3 [0, 1] for each x 3X. G G As noted by Wang and Klir [10] and Zhong [7], the fuzzy integral fds on X of the scores f (x ) with G respect to the importance measure s yields a single numerical quality evaluation of the given object. The term synthetic evaluation [11] refers to this process of reducing sets of scores assigned to factors influencing a judgment to a single number. It generalizes the weighted average approach to evaluation since interactions between sets of factors can be modeled using a non-additive fuzzy measure. X, s, and f will next be defined for a color reproduction process.
3. The model Color reproduction is classified as either offset or flexo because these two printing processes are completely different and require different color quality consideration. Furthermore, the offset market is further divided into the three separate categories: good-enough color (GE), mid-range color (MR), and high-end color (HE). We limit our presentation here to mid-range offset color reproduction. Similar results have been obtained for the other cases.
3.1. Quality factors Acceptance or rejection of a color reproduction is presently based on subjective evaluation by a color technician. A trained color technician can distinguish about 10 million colors. However, verbal systems can describe only a few hundred colors, physical sample systems a few thousand, and instrumental systems hundreds of thousands [12]. Color technicians monitor the color quality-control process. They accept or reject a reproduction based on the degree of color match between the original and its color reproduction, as well as their perception of its meeting the customer’s standards.
33
The degree of color match between an original and a reproduction can vary due to numerous factors or steps involved in the color reproduction process. In our experience the three major factors affecting a color technician’s decision are gray balance (G), tone reproduction (¹), and overall appearance (A). Therefore, let x be the quality factor G, x be ¹, x be A, and X"+x , x , x ,.
3.2. Importance measure The importance measure s of a color reproduction gives the contribution of each separate quality factor and combination of factors to the total quality level. To establish s, target transparencies were selected and reproduced as mid-range offset color reproductions. Fifteen color technicians were trained to rate the importance to color reproduction quality of the individual quality factors and combination of factors as triangular fuzzy number. Next they were asked to evaluate each reproduction for the quality factors of gray balance (G), appearance (A), tone reproduction (¹), and then to present the results in the form of a triangular fuzzy number between 1 and 10 by specifying its lowest, highest, and most likely value. The decision to use triangular fuzzy numbers between 1 and 10 was based on experience in the printing field and the authors’ belief that color technicians best relate to this scale. The data collection forms for the 15 experts were analyzed using a fuzzy spread sheet application package called FuzzyCalc2+ for Windows2+ to yield a composite importance measure for midrange offset color reproduction. The resulting values of s are presented below in Table 1. Table 1 Importance measure s(A)"0.569 s(¹)"0.342 s(G)"0.397 s(¹, A)"0.819 s(G, A)"0.908 s(G, ¹)"0.673
34
C. Temponi et al. /Int. J. Production Economics 58 (1999) 31—37
3.3. Quality evaluation functions For a particular color reproduction a similar approach can be used to obtain values for f (x ), f (x ), f (x ). Each expert evaluates the reproduction for gray balance, tonal reproduction, and overall appearance in terms of a triangular fuzzy number. These are then defuzzified and averaged using FuzzyCalc2+ to yield a crisp value for f (x ). AccordG ing to Wang and Klir [10], this average is very close to the true quality level if the number of color experts evaluating the color is large enough. 3.4. Example Given a color reproduction, technicians evaluate the individual quality factors G, ¹, A to yield the composite scores f (G), f (¹), f (A). The numerical synthetic evaluation E of the quality of the color reproduction is then obtained from the fuzzy integral f ds of the scores f (x ) with respect to importG ance measure s. Let f (G)"0.6, f (¹)"0.8, f (A)"0.9 for the mid-range offset importance measure given in Table 1. For this color reproduction the numerical synthetic evaluation E is then calculated as follows, where R is the maximum operator.
E" f ds "[0.6s(X5F )]R[0.8s(X5F )] R[0.9s(X5F )] "[0.6s(X)]R[0.8s(¹, A)]R[0.9s(A)] "[0.61]R[0.80.819]R[0.90.569] "0.6R0.8R0.569 "0.8. 3.5. Quantifying customer quality expectation Once the importance measure is established, a decision (accept or reject) on a particular color reproduction requires that the customer’s quality expectation be quantified. For nine months three color technicians were instructed to grade the qual-
ity level of factors G, ¹, A with triangular fuzzy numbers between 1 and 10 for any reproduction deemed acceptable for customer review. These numbers were then entered on a quality factor evaluation log form. When a customer accepted a color reproduction, FuzzyCalc2+ averaged the three fuzzy numbers using fuzzy set theory and then defuzzified the result. In this manner, the minimum quality value required for acceptance by a particular customer could be obtained. For example, magazine DM required a synthetic quality level of 0.75 for acceptance.
4. Results Twenty-five mid-range offset color reproductions were selected to test the performance of the model in a color reproduction environment. Each reproduction was reviewed by three color experts and evaluated for the quality factors of G, ¹, A. The scores are shown in Table 2. For each reproduction the synthetic quality value was calculated. The jobs were then rejected or accepted after their synthetic quality value was compared with the acceptable quality values in Table 3, which shows that 20 jobs were accepted and five rejected. To compare these results with the conventional subjective method, a fourth color expert evaluated the 25 reproductions and rejected eight. Regardless of a job’s acceptance or rejection using either approach, each job was sent to the customer. An evaluation of the model’s performance requires that costs be associated with the four possible combinations of rejecting or accepting a job by color expert and customer. A job here is defined as an 8.5 by 11 color reproduction, and every round of color modification involves the following steps (1—6) and factors (7—10): 1. 2. 3. 4. 5. 6. 7. 8.
locating the digital file, loading the file on the color workstation, performing the color modification, archiving the new image, manufacturing a set of film, manufacturing a prepress color proof, disrupting the production schedule, increase in overtime on misplaced jobs,
C. Temponi et al. /Int. J. Production Economics 58 (1999) 31—37
35
Table 2 Quality scores for 25 targeted mid-range color reproductions Job
01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Evaluator C1
Evaluator C2
Evaluator C3
G
A
¹
G
A
¹
G
A
¹
0.9 0.9 0.9 0.9 0.8 0.75 0.8 0.7 0.8 0.7 0.9 0.7 0.75 0.75 0.75 0.75 0.75 0.75 0.9 0.9 0.75 0.75 0.65 0.9 0.75
0.8 0.8 0.8 0.8 0.7 0.75 0.6 0.8 0.7 0.9 0.75 0.8 0.7 0.65 0.8 0.65 0.75 0.75 0.8 0.8 0.8 0.8 0.85 0.75 0.9
0.9 0.85 0.9 0.8 0.7 0.9 0.9 0.6 0.7 0.8 0.8 0.5 0.85 0.65 0.85 0.8 0.75 0.8 0.8 0.8 0.8 0.7 0.75 0.8 0.8
0.85 0.7 0.85 0.85 0.6 0.65 0.6 0.85 0.6 0.85 0.65 0.8 0.7 0.8 0.8 0.6 0.8 0.9 0.85 0.75 0.65 0.85 0.9 0.85 0.9
0.9 0.7 0.9 0.9 0.7 0.8 0.8 0.7 0.7 0.6 0.8 0.6 0.8 0.7 0.7 0.8 0.8 0.75 0.9 0.7 0.9 0.9 0.75 0.8 0.75
0.85 0.75 0.85 0.85 0.75 0.75 0.75 0.7 0.75 0.9 0.9 0.7 0.8 0.7 0.7 0.7 0.7 0.8 0.85 0.85 0.85 0.7 0.9 0.8 0.85
0.7 0.75 0.85 0.85 0.8 0.7 0.7 0.8 0.8 0.9 0.9 0.7 0.8 0.85 0.85 0.8 0.8 0.8 0.7 0.8 0.8 0.85 0.9 0.8 0.85
0.85 0.8 0.85 0.85 0.75 0.7 0.85 0.75 0.75 0.8 0.8 0.7 0.8 0.7 0.85 0.6 0.85 0.9 0.85 0.75 0.8 0.85 0.8 0.8 0.75
0.9 0.8 0.9 0.75 0.8 0.7 0.8 0.7 0.8 0.7 0.7 0.6 0.9 0.7 0.65 0.7 0.85 0.85 0.75 0.7 0.7 0.9 0.8 0.85 0.8
9. possible late delivery to customer, 10. press downtime. The color modification costs for 50 reworked jobs, each dollar amount a crisp number, was taken from company records. These data result from standard company accounting procedures and unfortunately include some subjectivity, particularly for factors 7—10. However, it can be shown using a chi-square goodness-of-fit test that these 50 modification costs fit a triangular probability distribution between $125 and $175 with its mean, median, and mode at $150. If a customer rejects a job that color experts accept, the above modification costs are incurred in the rework but increased by the intangible cost of customer dissatisfaction. Indeed, it is common in the prepress industry for customers to pull jobs and
switch suppliers. When a job is rejected by the customer, standard accounting procedures add $25 to the modification costs. Hence, in this case the total cost follows a triangular probability distribution between $150 and $200. When a color expert rejects a job that would have been accepted by the customer, the actual color modification costs are incurred, as well as opportunity-lost costs in both the rework and the initial work for which too much effort was taken. Unfortunately, actual data on these costs are virtually impossible to obtain. Jobs rejected by color experts would have to be sent to the customer until a sufficient number had been accepted. However, even then costs would be subjective — the cost of the unnecessary effort, for example. Instead, the authors talked with the management and accounting, who deemed the cost in this case as the
36
C. Temponi et al. /Int. J. Production Economics 58 (1999) 31—37
Table 3 Accepted or rejected targeted mid-range color reproductions
Table 4 Crisp cost matrix C
Job
Quality value
Result
Cust(A)
01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0.818595 0.766667 0.85 0.85 0.716667 0.75 0.75 0.75 0.716667 0.766667 0.783333 0.7 0.766667 0.7 0.783333 0.683333 0.783333 0.8 0.8 0.75 0.783333 0.816667 0.8 0.783333 0.8
A A A A R A A A R A A R A R A R A A A A A A A A A
CX(A) CX(R)
critical one in the industry. Being too good is expensive in color reproduction. From such discussions we concluded that a reasonable cost model was a triangular probability distribution between $300 and $400 with a mean, median, and mode being $350. Since there is no cost associated with acceptance of a job by both the color expert and the customer, the previous information suggests the following matrix C of mean costs (also the median and modal costs). In Table 4, CX(A) and CX(R) denote the color expert accepting and rejecting a job, respectively, while C(A) and C(R) refer to the customer’s acceptance and rejection. Recall that all 25 jobs were sent to the customer. Of the 20 accepted by the model, two were rejected by the customer, who also accepted one of five that the model rejected. Of the 17 accepted by the color expert using the usual approach, one was rejected by the customer, who also accepted two of eight
Cust(R) $0 $350
$175 $150
that the expert rejected. These results may be represented by the matrices C(A) C(R)
C(A) C(R) CX(A) CX(R)
18 2 1
4
and
CX(A) CX(R)
16 1 2
6
.
The cost C associated with the model is then given + by
18 2 $0 $175 C " "$1300, + 1 4 $350 $150 where denotes the matrix multiplication (18)(0)#(2)(175)#(1)(350)#(4)(150). Similarly C 1 for the standard or conventional approach is
16 1 $0 $175 "$1775. C" 1 2 6 $350 $150 These numbers show a performance by the model superior to that for the usual approach. The main reason is that the model rejected fewer acceptable jobs than the usual approach. Based on this small sample and cost estimates, our model seems well-suited to make decisions for mid-range color reproduction.
5. Conclusion The fuzzy approach to color quality evaluation proposed here produced better acceptance/rejection results on a small sample of mid-range color reproductions than the conventional method. The limitations of the work are the cost estimates, which are difficult to obtain, and the size of the sample. Unfortunately, both are difficult to obtain, and one
C. Temponi et al. /Int. J. Production Economics 58 (1999) 31—37
of the authors has recently change industry so that data is no longer available. Further research should address these points. In addition, such additional factors as the quality of the original and a measure of the customer’s standards might be considered. References [1] ANSI, Color Print,Transparencies, and Photomechanical Reproductions Viewing Conditions, PH2.3, 1989. [2] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338—353. [3] A. Kandel, Fuzzy statistics and forecast evaluation, IEEE Transactions in Systems, Man and Cybernetics 8 (1978) 396—401. [4] M.T. Lamata, S. Moral, Classification of fuzzy measures, Fuzzy Sets and System 33 (1989) 243—253.
37
[5] M. Sugeno, Theory of Fuzzy Integral and Its Application, Ph.D. Dissertation, Tokyo Institute of Technology, 1974. [6] G. Banon, Distinction between several subsets of fuzzy measure, Fuzzy Sets and Systems 5 (1981) 291—305. [7] Q. Zhong, On fuzzy measure and fuzzy integral on fuzzy set, Fuzzy Sets and Systems 37 (1990) 77—92. [8] P. Wakker, A behavioral foundation for fuzzy measures, Fuzzy Sets and Systems 37 (1990) 327—350. [9] H. Suzuki, On fuzzy measures defined by fuzzy integrals, Journal of Mathematical Analysis and Applications 132 (1985) 87—101. [10] Z. Wang, G.J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992. [11] J.R. Sims, Z. Wang, Fuzzy measures and fuzzy integrals: an overview, International Journal of General Systems 17 (1990) 157—189. [12] G.G. Field, Color scanning and imaging systems, Research Project Report No. 1431, Graphic Arts Technical Foundation, Pittsburgh, PA, 1990.
A fuzzy decision model for color reproduction Cecilia Temponi *, Farid Derisavi Fard, H.W. Corley Southwest Texas State University, San Marcos, TX 78666, USA Xerox Corporation, Dallas, TX, USA The University of Texas at Arlington, Arlington, TX, USA Received 4 June 1997; accepted 29 December 1997
Abstract Color quality appraisal refers to a visual evaluation of an original color image and an assessment of the degree to which that image is matched in photomechanical or electronic reproduction. The evaluation of reproduction quality is quite subjective since it relies on the individual judgments of color technicians who have different perceptions of color and its component qualities. To reduce the amount of rework due to subjective evaluation of color reproduction, an objective model is proposed here. In this model, a fuzzy set of quality factors is introduced and a numerical quality value calculated. This measure is then compared with an acceptable quality value experimentally established for particular customers and printing processes. Finally, the installed model is tested in actual production, and compared with the conventional evaluation method in a cost analysis. 1999 Elsevier Science B.V. All rights reserved. Keywords: Printing process; Color appraisal; Production; Fuzzy model; Cost analysis
1. Introduction In color quality appraisal, the original color image (color transparency, color print, or other reflection copy [1]) and the color reproduction are compared for hue, lightness, and saturation. They are also appraised in terms of tonal range, contrast, and texture, which affect the overall appearance of the color image. However, such analysis is subjec-
* Corresponding author. Tel.: #1 512 245 3189; #1 512 245 3089; e-mail: [email protected].
fax:
tive and varies from one technician to another. This fact is the primary cause of production rework in the prepress industry today when the color reproduction does not meet the customer’s expectation. Quantitative decision techniques have not been heretofore used, but the situation seems a natural application of fuzzy logic. In this paper we construct a model for the numerical evaluation of the quality level of a color reproduction by introducing a set of factors for a color technician to use in their evaluation and a measure of their respective importance. This approach enables the technician to judge the degree of color match acceptable to a particular customer for a specified printing process.
0925-5273/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved PII: S 0 9 2 5 - 5 2 7 3 ( 9 8 ) 0 0 0 7 8 - 4
32
C. Temponi et al. /Int. J. Production Economics 58 (1999) 31—37
2. Preliminaries 2.1. Fuzzy set theory and fuzzy numbers Introduced by Zadeh [2], the theory of fuzzy sets concerns a subset A of a universe of discourse X, where the transition between full membership and no membership is gradual rather than abrupt. A fuzzy subset has no well-defined boundaries as compared with the universe X that consists of definite objects. Most classes of objects encountered in the real world are of a fuzzy nature, without a precise criteria for membership. In such cases it is not necessary for an object to belong or not belong to a particular class. The object may have an intermediate class of membership [3,4]. Fuzzy numbers provide a way to represent one’s knowledge about real-world numbers. Information or knowledge about a number is often based on someone’s experience. An easy way to define a fuzzy number is to determine its lowest value, its highest value, and the most likely value. For example, from the authors’ experience in the prepress industry, it takes from 5 to 12 days to train someone for a particular position, depending on the trainee’s background. However, most people complete the training program in seven days. A fuzzy number representing the number of days required to train a new hire for this particular position is shown in Fig. 1. This fuzzy number is referred to as a triangular fuzzy number (TFN).
2.2. Fuzzy measure The notion of fuzzy measure introduced by Sugeno [5] can be used to model uncertainty, in particular, the vagueness and uncertainty inherent in natural language. In this paper it is used to quantify subjective and vague terms like “too dark”, “more contrast”, “more punch”, or “match the original” used by designers and print buyers in describing reproduction quality to color experts. Such terms are difficult to translate into precise language. For other applications of fuzzy measure see Refs. [4,6—9]. In this work we implement fuzzy measure as in Wang and Klir [10] and define the fuzzy integral of
Fig. 1. Number of days required to train a new hire.
f on X with respect to the fuzzy measure k by
f dk" sup [ak(X5F )], ? Z
(1)
where is the minimum operator and F "+x: ? f (x)*a,. Now consider an object to be evaluated, and let X"+x , x ,2, x , denote the factor space L of the object, that is, the set of all its quality factors of interest. The goal is to obtain a numerical quality evaluation of the given object from the evaluations for each individual factor. Assume that each subset A of the factor space X is associated with a real number s(A) between 0 and 1 that indicates the importance of A. Such a real number should be the maximum possible score that the object can gain relying only on the quality factors in A. Obviously the empty set H, which includes no relevant quality factors, has the minimum importance 0; and the whole factor space X has the maximum importance 1. Moreover, if each factor in a factor set A belongs to another factor set B, then A is at most as important as B. It follows that s(H)"0, s(X)"1, and s(A))s(B) whenever ALB. The number of quality factors for the given object is finite, so the set function s is a regular fuzzy measure on the measurable space (X, o(X)) called the importance measure on X [5]. It represents here a composite expert opinion. Given a particular object to be evaluated, a factor space of the object, and an importance measure, the object is evaluated
C. Temponi et al. /Int. J. Production Economics 58 (1999) 31—37
by experts for each individual quality factor x , x , 2 , x . Their subjective opinions are aggreg L ated into a composite quality evaluation function f to yield the scores f (x ), f (x ), 2 , f (x ), where f may be regarded as a measurable function on (X, o(X)) for which f (x )3 [0, 1] for each x 3X. G G As noted by Wang and Klir [10] and Zhong [7], the fuzzy integral fds on X of the scores f (x ) with G respect to the importance measure s yields a single numerical quality evaluation of the given object. The term synthetic evaluation [11] refers to this process of reducing sets of scores assigned to factors influencing a judgment to a single number. It generalizes the weighted average approach to evaluation since interactions between sets of factors can be modeled using a non-additive fuzzy measure. X, s, and f will next be defined for a color reproduction process.
3. The model Color reproduction is classified as either offset or flexo because these two printing processes are completely different and require different color quality consideration. Furthermore, the offset market is further divided into the three separate categories: good-enough color (GE), mid-range color (MR), and high-end color (HE). We limit our presentation here to mid-range offset color reproduction. Similar results have been obtained for the other cases.
3.1. Quality factors Acceptance or rejection of a color reproduction is presently based on subjective evaluation by a color technician. A trained color technician can distinguish about 10 million colors. However, verbal systems can describe only a few hundred colors, physical sample systems a few thousand, and instrumental systems hundreds of thousands [12]. Color technicians monitor the color quality-control process. They accept or reject a reproduction based on the degree of color match between the original and its color reproduction, as well as their perception of its meeting the customer’s standards.
33
The degree of color match between an original and a reproduction can vary due to numerous factors or steps involved in the color reproduction process. In our experience the three major factors affecting a color technician’s decision are gray balance (G), tone reproduction (¹), and overall appearance (A). Therefore, let x be the quality factor G, x be ¹, x be A, and X"+x , x , x ,.
3.2. Importance measure The importance measure s of a color reproduction gives the contribution of each separate quality factor and combination of factors to the total quality level. To establish s, target transparencies were selected and reproduced as mid-range offset color reproductions. Fifteen color technicians were trained to rate the importance to color reproduction quality of the individual quality factors and combination of factors as triangular fuzzy number. Next they were asked to evaluate each reproduction for the quality factors of gray balance (G), appearance (A), tone reproduction (¹), and then to present the results in the form of a triangular fuzzy number between 1 and 10 by specifying its lowest, highest, and most likely value. The decision to use triangular fuzzy numbers between 1 and 10 was based on experience in the printing field and the authors’ belief that color technicians best relate to this scale. The data collection forms for the 15 experts were analyzed using a fuzzy spread sheet application package called FuzzyCalc2+ for Windows2+ to yield a composite importance measure for midrange offset color reproduction. The resulting values of s are presented below in Table 1. Table 1 Importance measure s(A)"0.569 s(¹)"0.342 s(G)"0.397 s(¹, A)"0.819 s(G, A)"0.908 s(G, ¹)"0.673
34
C. Temponi et al. /Int. J. Production Economics 58 (1999) 31—37
3.3. Quality evaluation functions For a particular color reproduction a similar approach can be used to obtain values for f (x ), f (x ), f (x ). Each expert evaluates the reproduction for gray balance, tonal reproduction, and overall appearance in terms of a triangular fuzzy number. These are then defuzzified and averaged using FuzzyCalc2+ to yield a crisp value for f (x ). AccordG ing to Wang and Klir [10], this average is very close to the true quality level if the number of color experts evaluating the color is large enough. 3.4. Example Given a color reproduction, technicians evaluate the individual quality factors G, ¹, A to yield the composite scores f (G), f (¹), f (A). The numerical synthetic evaluation E of the quality of the color reproduction is then obtained from the fuzzy integral f ds of the scores f (x ) with respect to importG ance measure s. Let f (G)"0.6, f (¹)"0.8, f (A)"0.9 for the mid-range offset importance measure given in Table 1. For this color reproduction the numerical synthetic evaluation E is then calculated as follows, where R is the maximum operator.
E" f ds "[0.6s(X5F )]R[0.8s(X5F )] R[0.9s(X5F )] "[0.6s(X)]R[0.8s(¹, A)]R[0.9s(A)] "[0.61]R[0.80.819]R[0.90.569] "0.6R0.8R0.569 "0.8. 3.5. Quantifying customer quality expectation Once the importance measure is established, a decision (accept or reject) on a particular color reproduction requires that the customer’s quality expectation be quantified. For nine months three color technicians were instructed to grade the qual-
ity level of factors G, ¹, A with triangular fuzzy numbers between 1 and 10 for any reproduction deemed acceptable for customer review. These numbers were then entered on a quality factor evaluation log form. When a customer accepted a color reproduction, FuzzyCalc2+ averaged the three fuzzy numbers using fuzzy set theory and then defuzzified the result. In this manner, the minimum quality value required for acceptance by a particular customer could be obtained. For example, magazine DM required a synthetic quality level of 0.75 for acceptance.
4. Results Twenty-five mid-range offset color reproductions were selected to test the performance of the model in a color reproduction environment. Each reproduction was reviewed by three color experts and evaluated for the quality factors of G, ¹, A. The scores are shown in Table 2. For each reproduction the synthetic quality value was calculated. The jobs were then rejected or accepted after their synthetic quality value was compared with the acceptable quality values in Table 3, which shows that 20 jobs were accepted and five rejected. To compare these results with the conventional subjective method, a fourth color expert evaluated the 25 reproductions and rejected eight. Regardless of a job’s acceptance or rejection using either approach, each job was sent to the customer. An evaluation of the model’s performance requires that costs be associated with the four possible combinations of rejecting or accepting a job by color expert and customer. A job here is defined as an 8.5 by 11 color reproduction, and every round of color modification involves the following steps (1—6) and factors (7—10): 1. 2. 3. 4. 5. 6. 7. 8.
locating the digital file, loading the file on the color workstation, performing the color modification, archiving the new image, manufacturing a set of film, manufacturing a prepress color proof, disrupting the production schedule, increase in overtime on misplaced jobs,
C. Temponi et al. /Int. J. Production Economics 58 (1999) 31—37
35
Table 2 Quality scores for 25 targeted mid-range color reproductions Job
01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Evaluator C1
Evaluator C2
Evaluator C3
G
A
¹
G
A
¹
G
A
¹
0.9 0.9 0.9 0.9 0.8 0.75 0.8 0.7 0.8 0.7 0.9 0.7 0.75 0.75 0.75 0.75 0.75 0.75 0.9 0.9 0.75 0.75 0.65 0.9 0.75
0.8 0.8 0.8 0.8 0.7 0.75 0.6 0.8 0.7 0.9 0.75 0.8 0.7 0.65 0.8 0.65 0.75 0.75 0.8 0.8 0.8 0.8 0.85 0.75 0.9
0.9 0.85 0.9 0.8 0.7 0.9 0.9 0.6 0.7 0.8 0.8 0.5 0.85 0.65 0.85 0.8 0.75 0.8 0.8 0.8 0.8 0.7 0.75 0.8 0.8
0.85 0.7 0.85 0.85 0.6 0.65 0.6 0.85 0.6 0.85 0.65 0.8 0.7 0.8 0.8 0.6 0.8 0.9 0.85 0.75 0.65 0.85 0.9 0.85 0.9
0.9 0.7 0.9 0.9 0.7 0.8 0.8 0.7 0.7 0.6 0.8 0.6 0.8 0.7 0.7 0.8 0.8 0.75 0.9 0.7 0.9 0.9 0.75 0.8 0.75
0.85 0.75 0.85 0.85 0.75 0.75 0.75 0.7 0.75 0.9 0.9 0.7 0.8 0.7 0.7 0.7 0.7 0.8 0.85 0.85 0.85 0.7 0.9 0.8 0.85
0.7 0.75 0.85 0.85 0.8 0.7 0.7 0.8 0.8 0.9 0.9 0.7 0.8 0.85 0.85 0.8 0.8 0.8 0.7 0.8 0.8 0.85 0.9 0.8 0.85
0.85 0.8 0.85 0.85 0.75 0.7 0.85 0.75 0.75 0.8 0.8 0.7 0.8 0.7 0.85 0.6 0.85 0.9 0.85 0.75 0.8 0.85 0.8 0.8 0.75
0.9 0.8 0.9 0.75 0.8 0.7 0.8 0.7 0.8 0.7 0.7 0.6 0.9 0.7 0.65 0.7 0.85 0.85 0.75 0.7 0.7 0.9 0.8 0.85 0.8
9. possible late delivery to customer, 10. press downtime. The color modification costs for 50 reworked jobs, each dollar amount a crisp number, was taken from company records. These data result from standard company accounting procedures and unfortunately include some subjectivity, particularly for factors 7—10. However, it can be shown using a chi-square goodness-of-fit test that these 50 modification costs fit a triangular probability distribution between $125 and $175 with its mean, median, and mode at $150. If a customer rejects a job that color experts accept, the above modification costs are incurred in the rework but increased by the intangible cost of customer dissatisfaction. Indeed, it is common in the prepress industry for customers to pull jobs and
switch suppliers. When a job is rejected by the customer, standard accounting procedures add $25 to the modification costs. Hence, in this case the total cost follows a triangular probability distribution between $150 and $200. When a color expert rejects a job that would have been accepted by the customer, the actual color modification costs are incurred, as well as opportunity-lost costs in both the rework and the initial work for which too much effort was taken. Unfortunately, actual data on these costs are virtually impossible to obtain. Jobs rejected by color experts would have to be sent to the customer until a sufficient number had been accepted. However, even then costs would be subjective — the cost of the unnecessary effort, for example. Instead, the authors talked with the management and accounting, who deemed the cost in this case as the
36
C. Temponi et al. /Int. J. Production Economics 58 (1999) 31—37
Table 3 Accepted or rejected targeted mid-range color reproductions
Table 4 Crisp cost matrix C
Job
Quality value
Result
Cust(A)
01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0.818595 0.766667 0.85 0.85 0.716667 0.75 0.75 0.75 0.716667 0.766667 0.783333 0.7 0.766667 0.7 0.783333 0.683333 0.783333 0.8 0.8 0.75 0.783333 0.816667 0.8 0.783333 0.8
A A A A R A A A R A A R A R A R A A A A A A A A A
CX(A) CX(R)
critical one in the industry. Being too good is expensive in color reproduction. From such discussions we concluded that a reasonable cost model was a triangular probability distribution between $300 and $400 with a mean, median, and mode being $350. Since there is no cost associated with acceptance of a job by both the color expert and the customer, the previous information suggests the following matrix C of mean costs (also the median and modal costs). In Table 4, CX(A) and CX(R) denote the color expert accepting and rejecting a job, respectively, while C(A) and C(R) refer to the customer’s acceptance and rejection. Recall that all 25 jobs were sent to the customer. Of the 20 accepted by the model, two were rejected by the customer, who also accepted one of five that the model rejected. Of the 17 accepted by the color expert using the usual approach, one was rejected by the customer, who also accepted two of eight
Cust(R) $0 $350
$175 $150
that the expert rejected. These results may be represented by the matrices C(A) C(R)
C(A) C(R) CX(A) CX(R)
18 2 1
4
and
CX(A) CX(R)
16 1 2
6
.
The cost C associated with the model is then given + by
18 2 $0 $175 C " "$1300, + 1 4 $350 $150 where denotes the matrix multiplication (18)(0)#(2)(175)#(1)(350)#(4)(150). Similarly C 1 for the standard or conventional approach is
16 1 $0 $175 "$1775. C" 1 2 6 $350 $150 These numbers show a performance by the model superior to that for the usual approach. The main reason is that the model rejected fewer acceptable jobs than the usual approach. Based on this small sample and cost estimates, our model seems well-suited to make decisions for mid-range color reproduction.
5. Conclusion The fuzzy approach to color quality evaluation proposed here produced better acceptance/rejection results on a small sample of mid-range color reproductions than the conventional method. The limitations of the work are the cost estimates, which are difficult to obtain, and the size of the sample. Unfortunately, both are difficult to obtain, and one
C. Temponi et al. /Int. J. Production Economics 58 (1999) 31—37
of the authors has recently change industry so that data is no longer available. Further research should address these points. In addition, such additional factors as the quality of the original and a measure of the customer’s standards might be considered. References [1] ANSI, Color Print,Transparencies, and Photomechanical Reproductions Viewing Conditions, PH2.3, 1989. [2] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338—353. [3] A. Kandel, Fuzzy statistics and forecast evaluation, IEEE Transactions in Systems, Man and Cybernetics 8 (1978) 396—401. [4] M.T. Lamata, S. Moral, Classification of fuzzy measures, Fuzzy Sets and System 33 (1989) 243—253.
37
[5] M. Sugeno, Theory of Fuzzy Integral and Its Application, Ph.D. Dissertation, Tokyo Institute of Technology, 1974. [6] G. Banon, Distinction between several subsets of fuzzy measure, Fuzzy Sets and Systems 5 (1981) 291—305. [7] Q. Zhong, On fuzzy measure and fuzzy integral on fuzzy set, Fuzzy Sets and Systems 37 (1990) 77—92. [8] P. Wakker, A behavioral foundation for fuzzy measures, Fuzzy Sets and Systems 37 (1990) 327—350. [9] H. Suzuki, On fuzzy measures defined by fuzzy integrals, Journal of Mathematical Analysis and Applications 132 (1985) 87—101. [10] Z. Wang, G.J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992. [11] J.R. Sims, Z. Wang, Fuzzy measures and fuzzy integrals: an overview, International Journal of General Systems 17 (1990) 157—189. [12] G.G. Field, Color scanning and imaging systems, Research Project Report No. 1431, Graphic Arts Technical Foundation, Pittsburgh, PA, 1990.