European Journal of Operational Research 79 (1994) 537-548 537 North-Holland

Theory and Methodology

Pseudoconvex optimization for a special problem of paint industry

Tibor Il16s *

Department of Operations Research, Lordnd EiStviSs University, Mdtzeum krt. 6-8, H-1088 Budapest, Hungary

Jfinos M a y e r *,* *

Institute fiir Operations Research, Universitiit Ziirich, Moussonstrasse 15, CH-8044 Ziirich, Switzerland

Tamfis Terlaky *,* * *

Department of Mathematics and Computer Science, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, Netherlands

Received September 1991; revised March 1992

Abstract: A new mathematical programming model is presented for the computer color formulation problem. The model is essentially based on the two-constant Kubelka-Munk theory, that describes most of the necessary physical properties of this problem. The model is a nonconvex programming problem. It has a nonconvex objective function with some nice pseudoconvexity properties. The set of feasible solutions is a relatively simple polyhedron. An algorithm is also proposed to solve the problem.

Keywords: Nonconvex optimization; Pseudoconvexity; Kubelka-Munk theory

1. Introduction

The mixing (blending, estimation, approximation) problem frequently arises in practice when dealing with different technical, economical problems. Mixing different materials, gases, chemicals, liquids and approximating a target material with the mixture often leads to similar mathematical (mathematical programming, statistical, information theory) problems.

* Research has been partially supported by OTKA No. 1044 and by OTKA No. 2116. * * On leave from Computer and Automation Institute, Hungarian Academy of Sciences, Budapest, Hungary.

* * * On leave from Department of Operations Research, Lor~ind E6tv6s University, Budapest, Hungary. Correspondence to: Dr. T. Ill6s, Department of Operations Research, Lor~ind E6tv6s University, Mfizeum krt. 6-8, H-1088 Budapest, Hungary.

0377-2217/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0377-2217(93)E0054-2

538 T. lll#s et al. / Pseudoconvex optimization for a paint industry problem

The general form of the mixing (estimation) problem is as follows:

min O(xllc) x E K

where K ___ R n is the set of the possible mixtures, c ~ R n is the target materials and D(x II c) is a function 'measuring' the discrepancy between the mixture and the target.

The set K can be given in several ways. In most of the cases K is a convex polyhedron defined by a set of linear inequalities. This property is usually due to the basic 'mixing' requirements but sometimes the set of the possible mixtures is approximated by a convex polyhedron in order that the problem could be solved effectively. In general D(x II c) is a 'norm' or a 'statistical distance' (divergence) function.

Csisz~r [7] gave a unified approach to statistical divergences. Their applications are discussed, e.g. in [11,17]. Klafszky, Mayer and Terlaky [11] presented a comprehensive numerical and mathematical study of the different mathematical programming models using different special f-divergences with simple linear mixing constraints. A special case, when the Euclidean norm is used in finding the nearest point of a polyhedron to a fixed point, is discussed in some papers of Wolfe [20,21]. Some typical estimation problems are listed below.

Gasoline blending: A given type (by standard) of gasoline is to be mixed (approximated) from the available set of hydrocarbon distillates. Due to its practical importance, this problem was already studied in the early fifties [5] and later [18] too.

Concrete mixing: From different sand and gravel mixtures with different granule distributions a target sand and gravel mixture is to be approximated that fits to the needs of a specified user. This is an every-day problem in construction industry [11,17]. An analogous, mathematically equivalent problem is the asphalt mixing problem discussed, e.g. by Kelle [10].

A new mathematical programming model is presented here for a completely different application of approximation-type problems. The problem to be considered is as follows.

Color matching: A set of pigments and a target color is given. The paint that is the best approximation of the target color is to be mixed from the available set of pigments.

Our model has linear constraints and a pseudoconvex objective function, therefore it can be solved efficiently on a personal computer. Another advantage of our model, contrary to the previous ap- proaches, is that there is no theoretical restriction for the number of applicable pigments. Our implementation makes it possible to use significantly more pigments in practice than in the best algorithms [1-3,12] published till now.

2. Theoretical background of color matching

The color of a nontransparent material within the visible region is a function of the material's absorption and scattering parameters. Several theories establish connections between absorption, scatter- ing and reflection. Mie's (1908) theory is based on the assumption that absorption and scattering properties are determined by microscopical, but physically different parts of the material. Chan- drashekar's (1950) theory is based on some astrophysical observations. Both approaches are theoretically complicated and need a tremendous amount of computation. These two facts together make these theories practically inapplicable [12].

A theory that describes this phenomenon theoretically well and is also applicable for practical computations was established by Kubelka and Munk in 1931. Due to its practical efficiency this method is widely applied for producing different color recipes [1-3,9,12,13]. Up to now the Kubelka-Munk theory is known to be the best both from a theoretical and a practical point of view. The Kubelka-Munk formula establishes a relation between the absorption coefficient K, the scattering coefficient S of the diffusely reflecting pigment layer, and the interior diffuse reflectance R= of that layer which can be considered as infinitely thick:

K(A)/S(A) = ½(1 - R~(A))2/R~(A). (2.1)

T. lllds et aL / Pseudoconvex optimization for a paint industry problem 539 Here h denotes the actual wavelength. Note, that theoretically K(A) and S(A) are positive, but the practical data does not satisfy this property. Hence the positivity of this parameters cannot be assumed and will not be used in our analysis.

The Kubelka-Munk formula can be used under certain conditions, which are assumed to be fulfilled in this paper. Several effective computational methods have been developed for computing the parame- ter R~ on the basis of the measured diffuse interior reflectance R M [3,12]. In practice the most frequently used formula is the following one:

R=(A) = (RM(A) - k l ) / (1 - k I + k 2 - k2RM(A)),

where R=(A) is the interior diffuse (theoretical) reflectance and RM(A) is the measured reflectance at wavelength A, k~ is the frontsurface reflectance and k 2 is the internal reflectance of the layer [3].

The Kubelka-Munk formula can be transformed into the following form:

g(/~) [(K(~)12 U(~)]1/2 R = ( h ) = l + S(A-----~-- [l S--7~ ) + 2 S---(- ~ - . (2.2)

Here the relation 0 _< R=(A) _< 1 was assumed which is proved to be true in this physical model. Before discussing other details of the color matching problem we have to mention here that only the

reflectance parameter R M can be measured among the physical parameters. This means that parameters of absorption and scattering are computed numerically. A natural condition for 'good', effective preparation of any paint is that one has to know about the available components (basic colors, pigments), i.e. absorption and scattering parameters. The so-called calibration procedure [12] is developed for computing these parameters. This procedure is based on the theoretical assumption that paints (colors) are mixing additively. This is a basic assumption in our model as well.

For calibration the following paints are prepared and their reflectance values are measured at each of the following wavelengths: )t = 380, 400 . . . . . 740 nm, i.e. at nineteen places. (1) A paint with Cw0 = 100% concentration of white pigment. (2) Two paints produced from black and white pigments with concentration parameters Cal and CB2 of

the black pigment and with concentration parameters Cwj and Cwe of the white pigments (CB, < CB2 and they are within the range 3-10%).

(3) A paint produced from white and colored pigments, where the concentration of the colored pigment is Cc, and the concentration of the white pigment is Cw3 (value of the parameter Cc~ has to be within the interval 15-25%).

(4) A paint produced from black and colored pigments, where the concentration of the colored pigment is Cc2 = 98% and the concentration of the black pigment is CB3 = 2%.

Five fractionally linear equations with six variables can be formulated for these paints. For having a unique solution of these equations, the scattering parameter S w of the white pigment is fixed at a theoretical level, i.e.

Sw= 1.

Denote by R1, R 2 . . . . . R5 the reflectance parameters of these following equations:

K w = ½(1 -R1)Z/R1,

½(1 2 - R i + I ) /Ri+, = (CBK B + Cw, Kw)/(CB,SB + Cw, Sw),

1(1 - R4)Z/R4 = (Cw.K w + Cc~Kc)/(Cw.Sw + Cc~Sc),

½(1 -Rs)2 /R5 = (CB3K a + Cc2Kc)/(CB3SB + Cc Sc).

(2.3)

five paints. Formula (2.1) gives the

i = 1 , 2 ,

(2.4)

(2.5)

(2.6)

(2.7)

540 T. Ill& et al. / Pseudoconvex optimization for a paint industry problem

One has to solve these five equations (2.4)-(2.7) for all of the nineteen wavelengths. The solution of these equations give the necessary absorption and scattering parameters of the white, black and colored pigments. Having the parameters for white and black pigments one has to solve only equations (2.6) and (2.7) for a new colored paint. By repeatedly solving these equations all the necessary absorptional and scattering parameters can be obtained.

3. Convexity properties of the Kubelka-Munk function

Convexity properties of the Kubelka-Munk function are examined in this section. Well-known properties of pseudoconvex functions [15] will be used throughout.

Definition 3.1. [15]. Let f : H ---, • be a function, differentiable at a point ~ ~ H, where H c R n. Function f is said to be pseudoconvex at point ~ (on the set H) if for all x ~ H the following conditions hold:

(i) f ( x ) < f ( ~ ) ~ (x - 2)TVf(x)) < 0; (ii) f ( x ) < f ( $ ) ==, (X --$)TVf(x) < 0.

A function f is called pseudoconcave if ( - f ) is pseudoconvex.

Remark. Function f is called quasiconvex at ~ with respect to H, if only (i) holds [15].

The Kubelka-Munk equation (2.2) is based on the following function:

f ( x ) = l + x - ~ x 2 + 2 x , x ~ R + . (3.1)

Some basic properties of this function are examined below.

Lemma 3.1. I f x > O, the function f in (3.1) is strictly monotonically decreasing and convex.

Proof. Let us compute the first derivative of f :

f ' ( x ) = 1 - (x + 1 ) / ~ x 2 + 2 x . (3.2)

This can be reformulated as follows:

f ' ( x ) = 1 - (x + 1)/~/(x + 1) 2 - 1.

It is obvious that the denominator of the above fraction is smaller than the numerator, so f ' ( x ) < 0 holds for all x > 0, i.e. function f is strictly monotonically decreasing. To prove convexity, compute the second derivative of f :

1 f " ( x ) -- ( x i + 2 x ) ~ > 0 for all x > 0 ,

i.e. function f is convex. []

If we use the (2.1) form of the Kubelka-Munk equation then we get a function q : (0, 1] ~ ~ such that

q ( x ) = 1(1 - x ) Z / x . (3.3)

Some basic properties of function q are summarized in the following lemma.

T. lllds et al. / Pseudoconvex optimization for a paint industry problem 541

Lemma 3.2. Function q in (3.3) is strictly monotonically decreasing and convex. Moreover q is the inverse function o f f .

It is assumed that colors are mixing additively, so variable x in the Kubelka-Munk function f can be expressed as

• ciK i i=1

X = n [J.q') F. i=1

Therefore it is not surprising that the following well known result [15] will be used:

Lemma 3.3. [15]. The fractionally linear function t : H ~ R defined by

t ( y ) = (aTy + a ) / ( b T y + [3) (3.5)

is pseudomonotone (i.e. pseudoconvex and pseudoconcave) at all points y~ ~ H if bTy + [3 > 0 in H or bTy +/3 < 0 in H.

If we assume that

~ c i S i > 0, (3.6) i=l

then x is a pseudomonotone function of concentrations c i in relation (3.4). Since the Kubelka-Munk function is defined on positive numbers, the following (rather technical) assumption is needed:

~ c i K i > 0. (3.7) i=1

Recall from Section 2 that assumptions (3.6) and (3.7) hold for the theoretical model, but do not hold (always) for practical data.

Assumptions (3.6) and (3.7) will be refereed to as positivity assumptions and are assumed in the rest of this paper.

Theorem 3.1. Consider the Kubelka-Munk function f : R ~ ~ given by (3.1) and the fractionally linear function t (y) given by (3.5) on the set

A := {y [aTy + ot > 0 and bTy + fl > 0}.

Then the composite function F = f o t is pseudoconvex on A.

Proof. Function f is strictly monotonically decreasing (see Lemma 3.1) so f ( x ) < f ( 2 ) implies x > 2, i.e.

aTy + Ot aTy + a

x = bTy +---------~ > bT p +--------~ ~. (3.8)

542 T. Ill& et aL / Pseudoconvex optimization for a paint industry problem

Let us compute the gradient of F(y)=f( t (y) ) at y =y.

F(~) = 1 + bT ~ +~------~ bTy 7f l 2bT~ + ~

VF(p) = a(bTy ' +/3)--b(aTy~ + a)

(bT ~ +/3)2

a(bTy +/3)--b(aT)' +a) (l + - (bT~ +/3) 2

aTy + a 1

l

J( t 2 aTy + ot aTy + a

bT.~ +/3 + 2b-T~ +/3

After reordering we have

VF(~) = a(bTy~ + /3)--b(aT~ +a)

(bT~ +/3)2

l+ (aTy +/3) } + a)/(bTy~ 1--

aT~ + a aT~ + a

bT ~ +/3 + 2 bT~, +------~

therefore

(Y -.v)TVF(.v) = aT(y--Y')(bTy'+/3)--bT(y--~')(aTy +a ) ( b T ~ +/3)2 ( -- J ( a T ~ + a ) 2 1 + (aTy + bTy +/3 a)/(bTy++ 2 b--~- + flaTy'+Ot /3)

The sign of this expression is determined by aT(y --~XbT) ' +/3) - bT(y -~)(aT~ + a) since

(bTy +/3)2 > 0 and 1 - x + l

~ ( x + 1) 2 - 1 <0

(remember that f ' < 0. One has to show that

aT( y -- .V) (bT.v +/3) -- bT( y -- ~)(aT~ + Ol) > 0.

This is equivalent to

(aTy + a)(bT~ + 13) -- (bTy +/3)(aT~ + tZ) > 0.

Using the positivity assumptions we get

( ,@ +,~)/(b~y + ~) > (aT~ +,~)/(b'~ + ~),

which proves the validity of relation (ii) in the definition of the pseudoconvexity. The proof of relation (i) is similar to this. So F = f o t is pseudoconvex on the set A. []

T. lll6s et al. / Pseudoconuex optimization for a paint industry problem 543

Assume that n different pigments are available. Moreover assume that additivity and positivity hold for our color matching problem. Then reflectance R K in relation (2.2) of the current paint can be computed by the following formula:

RK(C ) = 1 + - -

• ciKi i~l

E ciSi i = 1

)2n ciKi E ciKi

i=1 + 2 i z l

E ciai E ciai i = 1 i = 1

(3.9)

that is, R r ( c ) is a pseudoconvex function of the concentration of the pigments. Pseudoconvexity properties of different differentiable divergence functions will be examined in the

sequel. All the differentiable divergence functions (Euclidean, entropy, lp, Hellinger) which were studied by Klafszky, Mayer and Terlaky [11] are convex. The question is what convexity properties are preserved as we make the composition of divergencies with the Kubelka-Munk function. Unfortunately the resulting functions are not convex in general. Divergences are additive according to the coordinates, so it is sufficient to examine a single coordinate.

The following result [15] will be used.

Theorem 3.2. i f function g(y) is pseudoconvex at a point ~ ~ H , function k ( x ) is differentiable at point Y, = g( yl ) and k ' ( Y~) > O, then the function w(y) = k(g(y)) is pseudoconvex at point ~ ~ H.

We would like to measure the divergences between the reflectance values (R~(A)) of the target color and the current paint (RK(A, c)).

Euclidean distance

One has to consider the function Ex(Rr(k, c) - R~(A)) 2. As it was mentioned above it is sufficient to consider the function (Rr (h , c) - 3) 2. (For simplicity the notation 6 = R=(A) is used. We know from the Kubelka-Munk theory that 1 _> 6 >_ 0 [12,13].)

Let us denote

• ciKi t ( c ) i = l , f ( x ) = 1 + x - ~ x 2 + 2 x , h ( u ) = ( u - 6 ) 2,

~-~ ciS i i = 1

so R(A, c) = f ( t ( c ) ) . By properly choosing functions k and g in Theorem 3.2, the following proposition is derived:

Proposition 3.1. Assume that (3.6), (3.7) and 0 < 6 < 1 hoM. Then the function F(c) = h( f ( t ( c ) ) ) (i) is pseudoconvex on the set {c [ f ( t ( c ) ) > 6}, or

(ii) /s pseudoconvex on the set {c I f ( t ( c ) ) < 3}.

Proof. To prove the claims we use Theorem 3.2. First we choose k = h and g = f o t. By Theorem 3.1, the function f o t is pseudoconvex. To prove (i) we have to show that k ' (u ) = h ' (u) > 0 if f ( t ( c ) ) > 3.

Obviously,

h ' ( u ) -- 2 ( u - a ) = 2( f ( t ( c ) ) - 3 ) >O ,~ f ( t ( c ) ) > 3,

hence part (i) is proved.

544 T. lllds et al. / Pseudoconvex optimization for a paint industry problem

In proving (ii) we choose k = h o f and g = t. It is known that t is a pseudoconvex function; therefore we look for the set where k ' ( x ) > 0. By differentiating k we have

( X+l 1 k ' ( x ) = 2 ( l + x - ~ - + 2 x - 8 ) 1 - / ( x + l ) 2 _ l .

Obviously k ' ( x ) > 0 if and only if 1 + x - ~ 2x - 3 < 0 because 1 < (x + 1)/~/(x + 1) 2 - 1. Both

1 + x - 3 and ~x2+ 2x are positive, hence (1 + x - 8 ) 2 < x 2 + 2x must hold. This is equivalent to (1 - 8) 2 + 2(1 - 6)x < 2x, from which

1 ( 1 - 3 ) 2 / ~ < x = t(c).

Because f is strictly monotonically decreasing (Lemma 3.1), (ii) follows. []

Remark. Remind, that 8 denotes the reflectance of the target color. Hence condition (i) says: If the reflectance of the mixture is greater than the reflectance of the target, then the Euclidean distance function is pseudoconvex. This means that this function is pseudoconvex in the reflectance space on the positive orthant which is translated to the reflectance point associated with the target.

Entropy, lp norm

If the discrepancy of the target and the mixture is measured by entropy or lp norm, then select function h as h ( u ) = u log(u/8) or h(u)= ( 1 / p ) ( u - 6) p respectively. By similar computations as performed above (Proposition 3.1) it is easy to see that the following statements are sufficient to guarantee pseudoconvexity of the resulting functions:

Proposition 3.2. Assume that (3.6), (3.7) and 0 < 3 < 1 hold. Then the function F(c) = h( f ( t (c)) ) (i) is pseudoconvex on the set {c [ f ( t (c ) ) > 8/e}, or

(ii) is pseudoconvex on the set {c I f ( t (c ) ) < 6/e}, if h denotes the entropy function and e is the base of the natural logarithm.

Proposition 3.3. Assume that (3.6), (3.7) and 0 < 3 < 1 hold. Then the function F(c) = h( f ( t (c)) ) (i) is pseudoconvex on the set {c I f ( t (c ) ) > 3}, or

(ii) /s pseudoconvex on the set {c I f ( t (c ) ) < 8}, if h denotes the lp norm and the number p is even.

Proposition 3.4. Let g = f o t, where f is the Kubelka-Munk function and t is the fractionally linear function with the positivity assumption. Then the function F ( c ) = k(g(c)) is pseudoconvex on the set {c I g(c) < 8}, if k denotes the Ip norm and the number p is odd.

Proofs. Simple computation. []

Based on these observations and utilizing the results established by Klafszky, Mayer and Terlaky [11] we decided to use the Euclidean distance in our model.

4. A new mathematical programming model for color matching

Sections 2 and 3 provide the physical and mathematical background of the new mathematical programming model for the color matching problem.

T. lll~s et al. / Pseudoconvex optimization for a paint industry problem 545

First we discuss the constraints of our model. For the concentrations of any mixture the following constraints must hold:

~ C i = 100, (4.1) i = 1

c i>_O, i = 1 , 2 . . . . . n, (4.2)

where c i is the concentration of pigment i in the mixture. We gave the positivity conditions (3.6) and (3.7) in Section 3. For any feasible solution of the color

matching problem inequalities (3.6) and (3.7) must hold, i.e. n

~, ciSi(h) > 0, )t = 380,400 . . . . . 740 nm, (4.3a) i = 1

and

Y~. ciKi(h ) > 0, A = 380 ,400 , . . . , 740 nm, (4.3b) i=l

where Si(A) is the scattering coefficient and Ki(A) the absorption coefficient of pigment i at wavelength h.

Using statement (i) of Propositions 3.1 and 3.3 we get another constraint:

f ( t(c)) > 8, (4.4a)

where 8 = R~o(h), c is the concentration vector, t is the fractional linear function and f is the Kubelka-Munk function defined by (3.1).

Using function q from (3.4) and Lemma 3.2, we get

t(c) < q ( 8 ) . (nAb)

Let /~®(h)= q(R®(h))= ½(1 - R®(A))2/R®(A) at wavelength h. Using (4.3a), (4.3b) and (4.4b) may be rewritten in the following way:

tl

ci(Ki(A ) - / ~ ( A ) S i ( A ) ) < 0, h = 380, 400 . . . . . 740 nm. (4.4c) i = 1

If we use statement (ii) from Propositions 3.1 and 3.3, then instead of (4.4c) we easily get

~ci(Ki(A)-I~(A)Si(A))>O, A = 3 8 0 , 4 0 0 . . . . . 740nm. (4.4d) i = 1

If we would use Proposition 3.2 or 3.4, we would get another constraint similar to (4.4c) or (4.4d). Assumptions (4.1) and (4.2) are generally referred to as mixing conditions and assumptions (4.3a) and

(4.3b) are the positivity requirements introduced in Section 3. Constraints (4.4c) guarantee that case (i) holds in Proposition 3.1 and Proposition 3.3 for all h.

Let

M = {c ~ R n Isuch that (4.1), (4.2), (4.3a), (4.3b) and (4.4c) hold},

N = {c ~ R n [such that (4.1), (4.2), (4.3a), (4.3b) and (4.4d) hold}.

Obviously M and N are convex polyhedrons. The mathematical programming model is the following:

(M1)

min IIRK(c)-Ro~II cEM

546 T. Illds et aL / Pseudoconvex optimization for a paint industry problem

where c is the concentration vector (decision variable), RK(c) is the reflectance vector of the mixture that is computed by the Kubelka-Munk formula (3.9), and R= is the theoretical reflectance vector of the sample (target) color, M is the set of feasible solutions, and tl" II is the Euclidean norm in this case. Similar models can be derived with the use of the other convex differentiable divergences.

If we replace the feasible set M with the set N then we get a similar model with the same pseudoconvexity properties.

Using Euclidean distance in •n, one can show that the objective function of the model (M~) is again pseudoconvex as is given in the next theorem.

Theorem 4.1. The function F(c ) = Y'.~(Rk(A, c) - RM(A)) 2 is pseudoconvex on the sets M or N.

Proof. Obvious from Theorems 3.1 and 3.2, using Proposition 3.1. []

This result suggests the following approach to solve problem (M~). If a good starting point (that is in the pseudoconvexity region) is available, any gradient type algorithm is applicable to generate a 'nearly optimal' or optimal solution for this model.

The advantage of this model is that the feasible set is a simple polyhedron defined by linear constraints. The objective function is pseudoconvex on an orthant which is translated to the 'ideal' point. If the necessary pigments are available, then the ideal target point is always the optimal solution. In this case theoretically the correct receipt can be found by our approach. These properties are utilized in our computer code. Therefore a nearly optimal solution is effectively generated in spite of the unpleasant fact that the problem may have several local optima.

Another advantage of our model is that the reflectance curve of the target sample is approximated by using the available n different pigments. This is a great advantage compared with the previous models, since there is no theoretical bound for n (for the number of pigments), while Allen's [1,2] algorithm works only for 4 pigments, and Allen and Kuehni [3,12] claim that no method is known that can handle more than 12 different pigments. With our experimental PC implementation we could handle up to 20 pigments but there is no theoretical limit for the number of pigments.

5. Computational experiences

Model (M 1) has been solved by a reduced gradient type algorithm proposed by Murtagh and Saunders [16]. The algorithm was coded in FORTRAN language, for IBM compatible PC's. In our experience the true composition of the sample was recovered for several different samples with at most ten trial mixtures. In any case some trial mixtures are necessary since the reflectance curve does not determine sufficiently well all the chromatic properties of the paint. These properties are examined in the so-called CIELAB color space [9,12,13]. This is a three-dimensional space. Reflectance values have to be transformed to the CIELAB space by the color matching function [9,12,13]. These computational

Table 1 Number of iterations required to solve the test problems

Trials No. of pigments Iterations Objective value

K 1 2 2 0.636 K 2 5 6 0.636 K 3 4 3 0.636 K 4 4 2 0.636 K 5 4 4 0.696 K 6 6 8 0.636 K 7 4 5 0.636

T. Ill& et al. / Pseudoconvex optimization for a paint industry problem

Table 2 Concentrations of the pigments in the trials

547

Trials White Black Redl Red2 Yellow Blue

K 1 - 2.00 98.00 - - - K 2 - 2.00 98.00 0.00 0.00 0.00 K 3 - 2.00 98.00 0.00 0.00 - K 4 - 2.00 98.00 - 0.00 0.00 K s - - 0.00 96.57 3.41 0.02 K 6 0.00 2.00 98.00 0.00 0.00 0.00 K 7 0.00 1.99 97.97 0.04 - -

Table 3 Analysis of the results in the CIELAB color space

Trials A Ea~ A Ca~ A L* A Ha~

K t 0.07 0.06 0.01 0.04 K 2 0.07 0.06 0.01 0.04 K 3 0.07 0.06 0.01 0.04 K 4 0.07 0.06 0.01 0.04 K 5 4.71 2.94 - 1.32 3.43 K 6 0.07 0.06 0.01 0.04 K 7 0.08 - 0.02 - 0.01 0.03

Concentrations of the pigments are unknown. A E ~ is the CIELAB color difference, AC~ is the CIELAB chroma difference, AHa~ is the CIELAB hue-difference, and AL* is the CIE 1976 lightness difference [14]

methods are well known; since they are not used in our model, we do not discuss them in detail. This is an evaluation procedure after optimization.

A simple example is presented to illustrate our computational results.

Example. A sample consisting of 2% black and 98% red pigments is given. Seven trial mixtures were computed using different sets of pigments. These pigments were: two different red pigments, white, black, yellow and blue pigments. The number of pigments is equal to the number of variables. In our model we have 57 inequality constraints and 1 equality constraint.

First we show in Table 1 that in all cases just a few iterations may be necessary to solve the problem. The concentration of the different pigments in our solutions are shown in Table 2. (Note that ' - '

indicates that the pigment was not in that pigment set.) These results demonstrate the power of our new approach. We were able to locate the optimal

solution ( K 1 , K 2 , K3 , K4 , K 6) or to get a solution very close to it ( K 7 ) if the desired pigments were available.

Assume that the combination and concentrations of the pigments in the given sample are unknown. In this case we have to analyze our results (Table 2) using other methods than mathematical programming techniques. The method of analysis is based on the international standards [9,12,13]. The analyses of our results (Table 3) are done in the CIELAB color space. Some types of differences between the given sample and the computed mixtures in CIELAB color space are determined.

Tables 2 and 3 show that the optimal solution of model ( M 1 ) depends on the set of pigments. To our best knowledge, the paints of most chemical companies consist of 2 -6 different colored

pigments. Our method may handle an arbitrary number of pigments together.

References

[1] Allen, E., "Basic equations used in computer color matching", Journal of the Optical Society of America 56/9 (1966) 1256-1259.

548 T. Ill~s et al. / Pseudoconvex optimization for a paint industry problem

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