Numerical Solution of Non-Linear Algebraic ...

Chemical Product and Process Modeling Volume 3, Issue 1

2008

Article 17

Numerical Solution of Non-Linear Algebraic Equations by Modified Genetic Algorithm Mohammad Danish, Department of Chemical Engineering, Aligarh Muslim University, Aligarh - 202002 , U.P. , India Shashi Kumar, Chemical Engineering Department, Indian Institute of Technology Roorkee, Roorkee - 247667 , Uttarakhand, India Surendra Kumar, Chemical Engineering Department , Indian Institute of Technology, Roorkee, Roorkee - 247667, Uttarakhand , India

Recommended Citation: Danish, Mohammad; Kumar, Shashi; and Kumar, Surendra (2008) "Numerical Solution of NonLinear Algebraic Equations by Modified Genetic Algorithm," Chemical Product and Process Modeling: Vol. 3: Iss. 1, Article 17. DOI: 10.2202/1934-2659.1122

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Numerical Solution of Non-Linear Algebraic Equations by Modified Genetic Algorithm Mohammad Danish, Shashi Kumar, and Surendra Kumar

Abstract Numerous unit operations in chemical and process engineering can be represented as a system of non-linear algebraic equations, when modeled for steady state operation ,e.g. isothermal and non-isothermal operations of a series of CSTRs, batteries of evaporators, networks of various separation operations (flash drum, mixers), distillation, extraction and absorption columns, and pumps and piping networks etc. These governing equations are sometimes very difficult to solve due to the nonlinear and uneven nature associated with them. The difficulty level increases when the resulting set of equations become flat near their zeros and thus derivatives based schemes mostly diverge or give poor results. Many a times, even a good initial guess in conventional numerical techniques do not guarantee to have a true solution and problem specific methods have to be designed. This whole scenario can also be viewed as an optimization problem having equality constraints only and casting the equations in the form of norm of function vector, which formulates an objective function to be minimized. The true minimum thus found gives us the correct solution vector. Recently, Genetic Algorithms have been quite effectively used to solve many complex engineering optimization problems. In continuation of our earlier research work where an elitist genetic algorithm was developed for the solutions of various difficult MINLP problems (Danish et al. 2006a and b), this research work extends its application for the solution of difficult non-linear algebraic equations. A novel scheme of dynamic mutation parameter as a function of fitness along with dynamic penalty has been proposed. The small value of mutation parameter in initial stages enables the algorithm to search globally, and the solution thus found is refined by keeping its value higher in later generations. This new scheme is found to be very effective in the sense that the algorithm requires very small population size and comparatively lesser number of generations to give reasonably good solutions. To test the efficacy of algorithm we have solved five sets of difficult non-linear algebraic equations (Dennis and Schnabel, 1983). It is worthwhile to mention that one of these equations was having both its Jacobian and Hessian as zero, at its true solution. Applicability of the developed GA was also demonstrated by simulating an industrial case study of triple effect evaporator used for concentrating the caustic soda solution (Zain and Kumar, 1996), which also poses difficulty during numerical simulation by NewtonRaphson method. KEYWORDS: algebraic equations, numerical solution, optimization, genetic algorithm


Danish et al.: Numerical Solution of Non-Linear Algebraic Equations

1.

Introduction

Algebraic equations are the basic and the most commonly encountered form of any mathematical relation which arise in many areas of engineering and science. Among them, non-linear equations are very interesting because of their peculiar characteristics like existence of multiple solutions and non-convexities etc. In chemical engineering too, a variety of non-linear algebraic equations are obtained whenever any process is modelled in steady state condition and without any space variation (Himmelblau and Bischoff, 1968; Bird et al., 2002). For example, modelling of an assembly of CSTRs in isothermal or non-isothermal conditions, sequence of evaporators, various mass transfer equipments like flash-drums, mixers, distillation, extraction and absorption columns and pumps and piping networks etc all yield non-linear algebraic or transcendental equations. Efforts have been directed for devising fast and efficient numerical algorithm for solving such equations. Different numerical techniques exist for their solutions e.g. multivariable Newton-Raphson method and its variants and Broyden class of methods etc (Dennis and Schnabel, 1983; Ferraris and Tronconi, 1986; Gupta, 1995); several system specific and optimization based approaches are also available (Holland, C.D., 1975; Reklaitis et al., 1983; Floudas, 1995; Beigler et al., 1997; Edgar et al., 2001). The choice of technique depends on the problem and its difficulty level. Though, quick and accurate, these techniques require auxiliary information such as existence and continuity of functions and their derivatives, continuous updating of functions and derivatives, along with a good initial guess, for the proper convergence of the method. Even then, in some situations the equations may pose several difficulties in obtaining their solutions because of the involvement of nonlinearities and unevenness. Moreover, the flatness of equations at their zeros increases the complexity level and thus derivatives based schemes mostly diverge or give poor results. Many a times, even a good initial guess in these numerical techniques do not guarantee to have a true solution and the problem specific methods have to be designed. Recently, many optimization techniques based on probabilistic approaches have been successfully employed e.g. Genetic Algorithms (Holland, J.H., 1975; Goldberg, 1989; Michalewicz, 1992; Deb, 1995, 1999b, 2000a, 2001; Coello Coello et al., 2002), Simulated Annealing (Kirkpatrick et al., 1983), Tabu search (Glover, 1986; Hansen, 1986) and Ant Colony Optimization (Dorigo et al., 1996; Wodrich and Bilchev, 1997; Dorigo and Stützle, 2005) etc. Of these, GAs have been quite popular because of their stochastic approach and versatile nature. Many engineering optimization problems especially Chemical Engineering problems have been successfully solved using them (Androulakis and Venkatasubramanian, 1991; Upreti and Deb, 1997; Nandsana et al., 2003; Hilbert et al., 2006). Recently several multi-objective optimization problems of Chemical 1


Chemical Product and Process Modeling, Vol. 3 [2008], Iss. 1, Art. 17

Engineering have also been tackled by using GA (Bhaskar et al., 2000a & b; Sankararao and Gupta, 2007a; Tarafder et al., 2007). Even new variants of GA have been proposed and applied to optimize several industrial units e.g. incorporation of jumping gene operator, a concept borrowed from Biological Sciences (Kasat et al., 2003; Guria et al., 2005; Sankararao and Gupta, 2007 b). Genetic Algorithms (Holland, J.H., 1975) are one of the meta-heuristic global search algorithms. They are based on the concept of natural selection and exploit the idea of survival of the fittest. Due to their simple and universal nature GAs have successfully found wide ranging applications in Biology, Engineering, Computer, Physical and Social Sciences, Pattern Recognition and Parallel Processing etc and therefore a lot of study and research material is available on them. In our earlier work (Danish et al., 2006a and b), an elitist real GA with dynamic penalty was developed for the solutions of difficult MINLP problems (Grossmann and Sargent, 1979) which was then conveniently applied to solve several multi-product batch plant design problems (MPBPD). The results obtained were quite satisfactory. The work presented here, reports our experiences and the pitfalls that arise when the above GA is used for solving difficult non-linear algebraic equations, after formulating them into an optimization problem. After performing many numerical experiments and understanding the behaviour, the previous GA is slightly modified in order to obtain the true solution efficiently. Moreover, it is also desired that this job is accomplished in lesser number of generations with a small population size. The designed GA is then successfully applied to solve five standard test problems and the results are compared with those available in literature and a good agreement between them has been found. Thereafter, to ensure the reliability of the modified GA, an industrial case study of a triple effect evaporator system has been attempted. The obtained results are compared and are found to be commensurable with the one available in literature. This evaporator problem is found to be quite sensitive to the value of initial guess, ordering of various equations and variables. Besides, convergence related difficulties are also encountered when one solves it using Newton-Raphson method (Zain and Kumar, 1996). 2. Modified genetic algorithm From many of the numerical experiments done in our previous work, real GA was found to be much more effective as compared to binary GA. Therefore, a real GA with elitism operator was developed for MINLP problems; elitism operator was incorporated so as to preserve the best individuals found so far. The major steps involved in our earlier GA for the solutions of MINLP problems are briefly discussed below; the details can be found in (Danish et al. 2006a and b). DOI: 10.2202/1934-2659.1122


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(i)

A constant size population of real solution vectors, belonging to their respective domains, is generated randomly. (ii) Each of the generated members is assigned a fitness as per some definition. Dynamic penalty is imposed if the solution does not follow associated constraints or some other conditions. (iii) By having a tournament between any two randomly chosen members, selection of the winner for the mating pool is carried out. (iv) Local search within the population is performed by simulated binary cross-over (SBX) operator which exchanges the information (values) between two randomly selected parent solutions so as to modify the population. (v) Thereafter, the modified population is operated by polynomial mutation operator after some predefined mutation probability has happened. (vi) Thus obtained new population is arranged in descending order of their fitness. A specified percentage of best strings from the original population, replaces the same numbers of strings from bottom in the new population. A constant population size is maintained and the modified population goes through steps (ii) to (vi) for further treatment. This process continues until some terminating criteria are met. With the above proposed algorithm, different MPBPD problems have been fruitfully solved. Though, the algorithm is capable of finding true optima for nonlinear equations, yet it encounters several problems related to true convergence e.g. the values obtained are not precise or it took more generations to obtain satisfactory results. Moreover, it is observed that the algorithm faces difficulties for those non-linear algebraic systems where, (i) the true optimum is lying on a flat surface and the fitness of near by solutions is not differing too much and/or (ii) there are many good quality local optima in the very vicinity of true solution. As shown in later sections, the conventional numerical techniques also come across several difficulties in these kinds of problems e.g. the Jacobian vanishing at the roots. Keeping these points in view, the previous algorithm is slightly modified for successfully finding the solutions of difficult non-linear algebraic equations. The key features of the present method include: (i) Dynamic mutation parameter linked to the fitness of the population, (ii) Imposition of the step wise dynamic penalty and (iii) Existence of high local search by assigning a lower value to simulated binary cross-over parameter ηc . In this way, the developed GA is made suitable for non-linear equations possessing flat profiles at their solutions i.e. zeros lie on an even surface. The detailed descriptions of these changes are presented in the next section and the algorithm is shown in Fig. 1.

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GA Starts Initialization of generation t = 0

Specify GA parameters. Also, specify initial & final values of mutation parameters η mi ,η mf . r r r Population of real solution vectors ( x ) are randomly generated in the defined domain [ X min , X max ] . r Evaluation of objective function F ( x ) , imposition of step-wise penalty P and then obtaining fitness f . Selection for mating pool by tournament selection operator.

Members are sorted & stored as per their fitness. Qualified strings along with their replica are stored in mating pool.

Information is exchanged between the selected strings by SBX-cross-over operator.

Values of the members are altered by polynomial mutation operator. This mutation operator uses dynamic mutation parameter η m .

Members are sorted & stored as per their fitness. In elitism operator, members coming through both the routes are compared and a defined percentage of the best members from previous population directly replaces the same number of poor members of the present population.

Mutation parameter is dynamically updated by linking it to the best fitness of the population & is moderated by taking the geometric mean of the last five values. t η mt = (η mi )1− fitness (η mf ) fitness ; η mt =  Π η mj  j =  t −4 

No

Generations completed ? Yes Stop

Fig. 1 Modified Genetic Algorithm DOI: 10.2202/1934-2659.1122


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3. New solution strategy The adopted methodology is basically based on varying the search level i.e. in the beginning major search is initialized for finding the region of global optima. Thereafter, search is gradually refined and in the end, very fine search is maintained in a confined zone close to the global optima. Mutation is a primary search operator and in the presently employed polynomial mutation operator the search level is controlled by varying the mutation probability index ηm . The magnitude of search level has been adjusted by linking ηm with the fitness of the population such that ηm increases with the fitness. This resulting approach is found to be quite fast and effective as compared to the one with constant ηm . In addition, distance based stepwise increasing penalty is imposed with the intention that a constant search power is maintained even when the solutions are near their true values. In this type of penalty, the amount of penalty during a known number of generations (constituting a step) remains constant and the step size is fixed in advance such that the algorithm hopefully finds a better solution. As the algorithm reaches near optima, the population fitness increases which in turn raises ηm and thereby resulting in more refining. Otherwise, if the algorithm moves away from global optima, then in the next step of dynamic penalty, the increased penalty decreases the fitness which in turn reduces ηm and eventually leads to a crude search. Both are somewhat complementary to each other and are desirable when their overall effect increases the fitness with generations.

3.1 Fitness evaluation The actual values of variables in a solution vector are used to assess the associated fitness. Many versions of fitness definitions are available (Costa and Oliveira, 2001; Deb, 2001; Summanwar et al., 2002; Angira, and Babu, 2006; Shopova and Vaklieva-Bancheva 2006). They are basically a measure of the quality of a member. In case of a system of algebraic r r r equations E ( x ) = 0 , i.e. Ei = 0, i = 1, 2...n , the zeros of equations can also be determined by obtaining the zeros of square of the norm of equation vector i.e. r r r r n E T .E = E , E = ∑ Ei2 . Hence, from the optimization point of view, this situation i =1

can also be viewed as a minimization problem. Magnitude of this norm indicates that how far the solution at that stage lies from zeros, and the fitness of a solution

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r vector x is defined to be inversely proportional to the deviation of its norm from zero. In the present approach, one of the equations is treated as an objective function to be minimized while the rest are assumed to be equality constraints and are penalized. Initially same weightage is given to all the equations so that the region of global optimum may be searched in an unbiased way. Thereafter, only constraint equations are penalized to have some relaxation and when the search is confined to a very small but flat region near true root, all the equations are r penalized equally. Therefore, the value of objective function F ( x ) of a member in population at any generation is given by: r F ( x ) = E12 + P ; (1a)

 ∑n E 2 ; i =2 i  where, P =  c n  a × ℜ  t  × b  ∑ Ei2 ;    b  i=2

if t ≤ 500 else if 500 ηmi . As, the fitness can vary from 0 to 1; in the beginning of algorithm when the fitness is nearly zero, the initial lower valued ηmi dominates resulting in major search while in later generations the fitness is near 1, hence ηmf dictates the refined search. In this way the mutation parameter automatically affects the fitness and vice-versa. Because ηm varies exponentially with fitness, the influence of fitness on it is too large. To reduce this impact of fitness on ηm , the geometric mean of the last five consecutive values of ηm is taken in place of the present one and is used for the next generation; this is demonstrated below: Gen 1 : η m1 ( = ηmi )  → f1 ( fitness at t = 1) Genetic Operations

( (= η (= η (= η

) ) → f ( fitness at t = 3) ) → f ( fitness at t = 4) ) → η ( = (η η η η η ) ) → f

1− f1 Genetic Operations Gen 2 : η m 2 = ηmi × η mff1   → f 2 ( fitness at t = 2)

Gen 3 : ηm3 Gen 4 : η m 4 Gen 5 : ηm 5

1− f 2 mi

×η

1− f3 mi

×η

1− f 4 mi

×η

f2 mf f3 mf

f4 mf

Genetic Operations

3

Genetic Operations

4

1/ 5

m5

Genetic Operations

m1 m 2 m 3 m 4 m 5

5

( fitness at t = 5)

...

(4) The combination of these ideas brings about flexibility in search levels. While, the dynamic mutation sharpens the search as the fitness increases, the step wise dynamic penalty not only helps in distinguishing good and poor solutions but also maintains a constant search power during a step. This adaptable search also decreases the number of iterations required to reach the region of global optima and most of the efforts are utilized in getting refined solutions. 4. Test problems and solutions

The five standard test problems have been considered here to test the efficiency and robustness of the developed algorithm. All the test problems basically constitute systems of non-linear algebraic equations which are either poorly scaled and/or ill-conditioned. In problem 2 it is observed that the true solution lies on a flat surface and is surrounded by equally strong local optima. It is important to note that most of the parameters associated with the modified GA are kept constant to remove any ambiguity in algorithm. The variables’ domains are

DOI: 10.2202/1934-2659.1122


8


specified so as to contain only single true optimum otherwise it may lead to another true solution if it exists. It is also experienced that the performance of modified GA is unaffected with the size of the domain if it contains a lone global optimum. All the test problems have been solved and checked a number of times and the obtained results have been found to be within a reasonable tolerance range. The tolerable variation in the results, in each run, is due to the stochastic approach of the algorithm. The processing rate of the GA code with the specified parameters (population, number of variables and equations etc.) for these test problems is found to be approximately 10-13 generations per second on a machine with Celeron M 1.4 GHz processor with 512 MB RAM. Test problem 1 This is an extended Rosenbrock function with n=4 and results into a poorly scaled objective function. But with the help of present GA the solutions have been easily obtained. E1 = 10( x2 − x12 ) = 0 E2 = (1 − x1 ) = 0 E3 = 10( x4 − x32 ) = 0

E4 = (1 − x3 ) = 0

r r r or in compact vector form E ( x ) = 0 .

Table 1: Results of test problem 1

S. No.

Results

Scaled Variables xi

Exact Value

x1 1 x2 2 x3 3 x4 4 r r r r x ∈  X min , X max  ; X min

1.0000000000 1.0000000000 1.0000000000 1.0000000000 = [-10,-10,-10,-10];

Value (present work)

% Difference

0.9999999853 0.0000014700 0.9999999705 0.0000029500 1.0000000024 0.0000002400 1.0000000047 0.0000004700 r X max = [10,10,10,10];

4

f best = 0.9999999911 , ∑ Ei2 = 2.2385 × 10 −16 . i =1

GA parameters: Generation = 5 × 103 , Population = 30 , E = 10% , pc = 0.8 , pm = 0.5 , ηc = 0 ,

ηmi = 2 × 103 , ηmf = 108 , a = 0.04, b = 100, c = 2 , c1 = 10 −3 , c2 = 10−3 .

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Test problem 2 This is the Extended Powel Singular problem and is very interesting because both the Jacobian and Hessian are singular matrices at the final root and hence, Newton-Raphson technique for its solution cannot be applied. While running the modified GA, it is also observed that as the population approaches near their zeros, the reduction in value of one variable sharply increases the value of the other and thus to achieve zero norm is very difficult. This problem has also been solved using solver toolbox available in Microsoft EXCEL spreadsheet. As, can be seen from Table 2, the results obtained from the modified GA are better than those obtained from EXCEL software.

E1 = x1 + 10 x2 = 0 E2 = 5 ( x3 − x4 ) = 0 E3 = ( x2 − 2 x3 ) = 0 2

E4 = 10 ( x1 − x4 ) = 0 r r r or E ( x ) = 0 2


Results

Scaled S. Variables No. Exact Value xi 1 2 3 4

x1 x2 x3 x4

0.0000000000 0.0000000000 0.0000000000 0.0000000000

EXCEL Spreadsheet


-0.0001124362 0.0000112593 -0.0001856214 -0.0001856343

0.0001470145 -0.0000147017 0.0000469183 0.0000469070

4

r r r r x ∈  X min , X max  ; X min

4

2 -14 2 −15 ∑ Ei = 4.960503 × 10 ∑ Ei = 1.08266 × 10 i =1 i =1 r = [-1,-1,-1,-1]; X max = [1,1,1,1]; f best = 0.999999913 .

GA parameters: Generation = 5.5 × 103 , Population = 30 , E = 10% , pc = 0.8 , pm = 0.5 ,

ηc = 0 , ηmi = 2 × 103 , ηmf = 105 , a = 0.4, b = 100, c = 2 , c1 = 10 −6 , c2 = 10 −6 . Test problem 3 The following set of equations includes the trigonometric functions. It is important to note that the domain is specified in such a way that only (0, 0) is

DOI: 10.2202/1934-2659.1122


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included in it as there are many other zeros for this system and GA may very likely converge to one of them. E1 = cos x1 + 2 sin x1 − cos x2 = 0 E2 = 2 sin x2 − cos x1 + 3 cos x2 − 2 = 0 r r r or E ( x ) = 0 Table 3: Results of test problem 3

Results

Scaled Variables xi

S. No. 1 2

x1 x2

Exact Value


% Difference

0.0000000000 0.0000000000

0.0000000000 0.0000000000

0.0000000000 0.0000000000

r r r r r x ∈  X min , X max  ; X min = [-0.5,-0.5]; X max = [0.5,0.5]; 2

f best = 1.0000000000 , ∑ Ei2 = 0.0000000000 . i =1


ηmi = 2 × 103 , ηmf = 107 , a = 0.4, b = 100, c = 2 , c1 = 10 −3 , c2 = 10−3 . Test problem 4 This problem is obtained from Helical Valley function with n=3 and comprises of transcendental and trigonometric functions. The very good quality results are obtained in a few numbers of generations. E1 = 10 ( x3 − 10 × θ ( x1 , x2 ) ) = 0 E2 = 10

(

)

x12 + x22 − 1 = 0

E3 = x3 = 0  1 x  tan −1  2  if x1 > 0  2π  x1  where, θ ( x1 , x2 ) =   1 + 1 tan −1  x2  if x < 0   1  2 2π  x1   r r r or E ( x ) = 0

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S. No.

Scaled Variables xi

1 2 3

x2

x1 x3

Results Exact Value


% Difference

1.0000000000 0.0000000000 0.0000000000

1.0000000000 0.0000000000 0.0000000000

0.0000000000 0.0000000000 0.0000000000

r r r r r x ∈  X min , X max  ; X min = [-10,-10,-10]; X max = [10,10,10]; 3

f best = 1.0000000000 , ∑ Ei2 = 0.0000000000 . i =1


ηmi = 2 × 103 , ηmf = 1010 , a = 0.4, b = 100, c = 2 , c1 = 10 −3 , c2 = 10−3 . Test problem 5

This is the well known Wood function. It is mentioned in the literature that it is a difficult problem and is often used to test the algorithms for unconstrained minimization. Table 5 depicts that the satisfactory results have been obtained in less number of generations with a reasonably small population size.

E1 = 100 ( x2 − x12 ) + (1 − x1 ) = 0 2

2

E2 = 90 ( x4 − x32 ) + (1 − x1 ) = 0 2

(

2

E3 = 10.1 ( x2 − 1) + ( x4 − 1) 2

2

)=0

E4 = 19.8 ( x2 − 1)( x4 − 1) = 0 r r r or E ( x ) = 0

DOI: 10.2202/1934-2659.1122


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S. No.

Results

Scaled Variables xi

Exact Value

x1 1 x 2 2 x 3 3 x 4 4 r r r r x ∈  X min , X max  ; X min


% Difference

1.0000000000 1.0000000000 1.0000000000 1.0000000000

1.0000003949 0.0000394900 1.0000000702 0.0000070200 0.9999999790 0.0000021000 1.0000000463 0.0000046300 r = [-10,-10,-10,-10]; X max = [10,10,10,10];

4

f best = 1.00000000 , ∑ Ei2 = 2.6981 × 10 −21 . i =1


ηmi = 2 × 103 , ηmf = 1010 , a = 0.4, b = 100, c = 2 , c1 = 10 −3 , c2 = 10−3 . 4.1 Adjustment of parameter values Tuning of parameters is a vital part of any GA but there are no general guidelines; even if guidelines are available, they are problem specific (Mitchell, 1996). In the present work, tuning has been done with the aim that the modified GA may have effective and swift convergent properties. It is therefore intended that the proper solution may be achieved in lesser number of generations with small population size meanwhile, it is also desired that most of the parameters be kept constant to avoid empiricism. The following propositions have been considered for fine tuning of different parameters. A reasonable amount of preservation is done by keeping the elitism parameter E =10% of the population, so that the good solutions, found so far, are not lost by various genetic operators (this parameter E should be in accordance with the mutation probability otherwise GA will result in premature convergence). At the same time, high cross-over and mutation probabilities are chosen so as to get variety of solutions in a population at any generation. A lower value of the crossover probability distribution index i.e. ηc = 0 for SBX-operator has been taken since its small value permits far-away solutions in a population to be selected for further processing and vice versa. In later converging stages when the global search is diminishing due to larger ηm , this lower value of ηc keeps a higher local search for the solutions and is very promising when the function is flat near its true solution. This parameter is similar to ηm and the more its value, less is the

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change in a variable. The distribution index for polynomial mutationηm increases from its initial value ηmi to final value ηmf , as the best fitness of the population increases with generations and thereby refining the global search as the population reaches near its true optimum (which is either surrounded by equally competent local optima or the function becomes nearly flat in close neighbourhood of the global optimum). The fitness parameters c1 & c2 , are constants and can be adjusted to magnify or lessen the fitness difference between two individuals (denominator should neither be very small nor equal to zero). Their values should be such that a reasonable fitness difference is present between the initial members, members near global optima and the final true solution. Using the above guidelines for parameter setting, the algorithm is now used to solve a system of triple effect evaporator used for concentrating the caustic soda solution. It is important to mention that this problem faced convergence related difficulties when solved by Newton-Raphson method; moreover, the convergence also depends on the order of variables and equations during simulation and on the value of initial guess (Zain and Kumar, 1996). To avoid these shortcomings Zain and Kumar had proposed an improved solution strategy comprising of five equations only. This was achieved by rearranging the present 12 modelling equations into 7 uncoupled and 5 coupled equations. The seven uncoupled equations were solved sequentially and the rest five coupled equations were solved simultaneously. This system of 5 coupled equations is quite easy to solve. In this way, the complete problem was simulated without any convergence related difficulties. Though not shown here, the present and previous algorithms (Danish et al., 2006a and b) are capable of solving this problem successfully when the suggested strategy of five equations is followed. Using present GA the processing rate for evaporator problem is found to be approximately 7 generations per second on the earlier mentioned machine. 5. Industrial case study

The present evaporator problem consists of twelve nonlinear algebraic equations. The interest in this problem lies because of the following reported observations: (i) The equations are non-linear and coupled and present difficulties in numerical simulations. (ii) It has been reported (Zain and Kumar, 1996) that the order of variables is important otherwise in some cases its Jacobian becomes ill conditioned while solving by the Newton-Raphson technique leading to divergence of the method. (iii) Moreover, care should be taken in evaluating the derivatives numerically. The use of proper weights in the relaxation strategy is of utmost importance for getting the converged solution. DOI: 10.2202/1934-2659.1122


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5.1 Steady state modeling equations for triple effect evaporator system with forward feed arrangement The governing non-linear and coupled algebraic equations have been taken from the paper of Zain and Kumar and are reproduced below; the constitutive relations are given in appendix II. The details of their derivation are given else where (Holland, C.D., 1975). First Effect: F [h(TF , xF ) − h(τ1 , x1 )] + V0 λ0 − ( F − L1 )[ H (τ 1 ) − h(τ1 , x1 )] = 0.0 U1 A1 (T0 − τ 1 ) − V0 λ0 = 0.0 m( x1 )T1 + b( x1 ) − τ 1 = 0.0 L1 x1 − FxF = 0.0

Second Effect: L1[h(τ1 , x1 ) − h(τ 2 , x2 )] + ( F − L1 )[ H (τ 1 ) − h(T1 )] − ( L1 − L2 )[ H (τ 2 ) − h(τ 2 , x2 )] = 0.0 U 2 A2 (T1 − τ 2 ) − ( F − L1 )[ H (τ 1 ) − h(T1 )] = 0.0 m( x2 )T2 + b( x2 ) − τ 2 = 0.0 L2 x2 − FxF = 0.0

Third Effect: L2 [h(τ 2 , x2 ) − h(τ 3 , x3 )] + ( L1 − L2 )[ H (τ 2 ) − h(T2 )] − ( L2 − L3 )[ H (τ 3 ) − h(τ 3 , x3 )] = 0.0 U 3 A3 (T2 − τ 3 ) − ( L1 − L2 )[ H (τ 2 ) − h(T2 )] = 0.0 m( x3 )T3 + b( x3 ) − τ 3 = 0.0 L3 x3 − FxF = 0.0

These modelling equations are either used for designing a new evaporator system or for evaluating the performance of an existing one. Before solving these equations, it is necessary to scale the variables to make them of the same order. The scaling procedure is presented below: (i) All the flow rates are divided by the feed flow rate. (ii) All the temperatures are divided by the steam temperature. (iii) All the equations are rearranged and normalized in the following form by dividing the term with positive sign with the one having negative sign. Ei =

term on LHS with + ve sign − 1 = 0; term on LHS with − ve sign

i.e.

E1 = ( F [h(TF , xF ) − h(τ1 , x1 )] + V0 λ0 ) / ( ( F − L1 )[ H (τ 1 ) − h(τ 1 , x1 )] ) − 1 = 0; E2 = ( L1[h(τ 1 , x1 ) − h(τ 2 , x2 )] + ( F − L1 )[ H (τ 1 ) − h(T1 )]) /(( L1 − L2 )[ H (τ 2 ) − h(τ 2 , x2 )]) − 1 = 0; E3 = U 2 A2 (T1 − τ 2 ) /(( F − L1 )[ H (τ1 ) − h(T1 )]) − 1 = 0;

15



E4 = ( L2 [h(τ 2 , x2 ) − h(τ 3 , x3 )] + ( L1 − L2 )[ H (τ 2 ) − h(T2 )]) /(( L2 − L3 )[ H (τ 3 ) − h(τ 3 , x3 )]) − 1 = 0; E5 = U 3 A3 (T2 − τ 3 ) /(( L1 − L2 )[ H (τ 2 ) − h(T2 )]) − 1 = 0; E6 = U1 A1 (T0 − τ 1 ) /(V0 λ0 ) − 1 = 0; E7 = (m( x1 )T1 + b( x1 )) / τ 1 − 1 = 0; E8 = L1 x1 /( FxF ) − 1 = 0; E9 = (m( x2 )T2 + b( x2 )) / τ 2 − 1 = 0; E10 = L2 x2 /( FxF ) − 1 = 0; E11 = (m( x3 )T3 + b( x3 )) / τ 3 − 1 = 0; E12 = L3 x3 / ( FxF ) − 1 = 0;

Operating conditions: Feed (kg/hr) : 25000.000 Feed concentration (% weight fraction) : 8.000 o Feed temperature ( C ) : 110.000 Area of 1st, 2nd and 3rd effect (m2) : 85.34, 77.45 & 69.74 o Input steam temperature ( C ) : 165.01496 o Last effect temperature ( C ) : 33.545586 5.2 Results of triple effect evaporator system for different parameter values For the above operating conditions, the evaporator problem has been solved using proposed GA for various parameter settings. The results obtained have been compared with the one obtained by Zain & Kumar (shown in Table 6) and the effects of different schemes on the results are shown and discussed below. Different schemes include: dynamic mutation vs. constant mutation (both in the presence of dynamic penalty), dynamic mutation vs. constant mutation (both in the absence of dynamic penalty) and the effect of elitism on the convergence. The simulation results have been summarized in Table 7 and depicted in the form of two types of plots (shown in Figs. 2-7). The two types of plots are fitness vs. generations and log ηm vs. generations. Table 7 shows the results of triple effect evaporator system solved by the present algorithm. The obtained results have been categorized into different cases for various parameters’ values. For all the cases, parameters namely Population=30, pc = 0.8 , pm = 0.5 ,ηc = 0 , a = 0.2, b = 100, c = 2 , c1 = 1 , and c2 = 1 have been kept constant and mentioned above the Table 7. From Table 7 and Figs. 2-7, following observations may be made: 1. Cases 1(a) & 1(c) represents the successful runs when ηm is changed dynamically. Case 1(b) is just a replica of Case 1(a) and has been included to DOI: 10.2202/1934-2659.1122


16


show one of those very few runs which failed (due to probabilistic nature of algorithm). Although case 1 (b) is taken as unsuccessful but values of variables obtained after 5,000 generations are quite close to true solution values. If one continues above 5,000 generations, he is likely to approach to the solution vector. This is evident from case 1(c) where number of generations have been increased to 20,000 and for this case best fitness f best =1 12

and ∑ Ei2 = 10−18 and the run is considered as extremely successful. i =1

Table 6: Comparison of results for triple effect evaporator system

S. No.

Scaled Variables

xˆi 1 2 3 4 5 6 7 8 9 10 11 12

Results (Zain and Kumar)

Results (Present Study) Case: 1(a)

Case: 1(c)

xˆ12 = Vo / F

0.7630796087486242 0.8317171073593507 0.5041591682625670 0.6364335950739348 0.2260926694242861 0.1048383406957922 0.8524147665486763 0.1586800459777299 0.6711675626252620 0.3538372128725376 0.3220719571752862 0.3023499143374576

0.763108 0.831658 0.504201 0.636369 0.226139 0.104835 0.852427 0.158668 0.671123 0.353765 0.322027 0.302324

0.763079 0.831717 0.504158 0.636434 0.226092 0.104838 0.852415 0.15868 0.671168 0.353838 0.322073 0.30235

f best =

-

0.990448

1

3.53718 × 10-11

9.64 × 10-9

3.01 × 10-18

xˆ1 = L1 / F xˆ2 = T1 / To xˆ3 = L2 / F xˆ4 = T2 / To xˆ5 = L3 / F xˆ6 = x1 xˆ7 = τ 1 / To xˆ8 = x2 xˆ9 = τ 2 / To xˆ10 = x3 xˆ11 = τ 3 / To

12

2 ∑ Ei =

i =1

2a. For cases 2-4, ηm is kept constant with generations and as evident from the value of norm shown in Table 7, the runs for cases 2-4 are declared as unsuccessful. Though the fitness curve (Fig. 2) for case 4 seems to reach unity, yet the norm value is unsatisfactory, this is because there was no penalization and it can be seen from equations 1 & 2 that in the absence of penalty fitness will be very near to unity for this value of norm. An important point to note is that constant ηm has been used with and without dynamic

17



penalty and the presence or absence of dynamic penalty does not significantly improve the results if ηm is kept constant. 2b. Cases 4, 5 & 6 indicate that dynamic ηm has been helpful in obtaining the true solution within 5,000 generations even without dynamic penalty. 3. Comparing cases 1(a) & 5, one can find that the inclusion of dynamic penalty improves the quality of results. r further r r r xˆ ∈  X min , X max  ; X min = [0, 0.1, 0, 0.1, 0, 0, 0.1, 0, 0.1, 0, 0, 0.1]; r X max = [1,1,1,1,1,1,1,1,1,1,1,1]; Table 7: Results of triple effect evaporator system for different cases (GA parameters: Population = 30 , pc = 0.8 , pm = 0.5 , ηc = 0 , a = 0.2, b = 100, c = 2 , c1 = 1 , c2 = 1 ) Case Number

Parameters Used

Best fitness

Square of norm

f best

∑ Ei

12

2

Remarks

i =1

Dynamic mutation with step wise dynamic penalty has been used. Run is taken as Successful.

GA parameters: Generation = 5 × 103 , E = 10% , η mi = 2 × 103 , η mf = 1010 Penalty structure: if t ≤ 200 r 12 F ( xˆ ) = E12 + ∑ Ei2 ; i =2

1(a)

else if 200 < t ≤ 500 c r   t   12 2 2 F ( xˆ ) = E1 +  a × ℜ   × b  ∑ Ei ;  b   i =2  else c r   t   12 F ( xˆ ) =  a × ℜ   × b  ∑ Ei2 ;  b   i =1  .

0.990448

9.64E-09

1(b)

Every thing same as Case 1(a)

0.018113

5.42E-05

1(c)

Every thing same as Case 1(a) except Generation = 20 × 103

1

3.01E-18

DOI: 10.2202/1934-2659.1122


Dynamic mutation with step wise dynamic penalty has been used. Run is taken as Unsuccessful. Dynamic mutation with step wise dynamic penalty has been used. Run is taken as extremely successful.

18


Table 7 continued

Case Number

Parameters Used

Best fitness

Square of norm

f best

∑ Ei

12

2

i =1

2

Every thing same as Case 1(a) except η mi = ηmf = 2 × 103 = constt.

0.011566

8.55E-05

3

Every thing same as Case 1(a) except η mi = η mf = 5 × 103 = constt.

0.049236

1.93E-05

0.999987

1.3E-05

4

Every thing same as Case 1(a) except η mi = η mf = 5 × 103 = constt. and

Remarks

with out any penalty.

5

Every thing same as Case 1(a) but with out any penalty.

0.999999 553

4.47E-07

6

Every thing same as Case 1(a) except η mi = ηmf = 10 × 103 = constt.

0.255146

2.92E-06

7

Every thing same as Case 1(a) except E = 20%

0.909465 382

9.95E-08

8


0.648286 531

5.43E-07

9


0.477512 096

1.09E-06

Constant mutation with step wise dynamic penalty has been used. Run is taken as unsuccessful. Constant mutation with step wise dynamic penalty has been used. Run is taken as unsuccessful. Constant mutation without any penalty has been used. Run is taken as unsuccessful. Dynamic mutation without any penalty has been used. Run is taken as successful. Constant mutation with step wise dynamic penalty has been used. Run is taken as successful. Dynamic mutation with step wise dynamic penalty has been used. Run is taken as successful. Dynamic mutation with step wise dynamic penalty has been used. Run is taken as successful. Dynamic mutation with step wise dynamic penalty has been used. Run is taken as successful.

19



Table 7 continued

Case Number

10

Parameters Used


Best fitness

Square of norm

f best

∑ Ei

1.314E-07

12

2

Remarks

i =1

7.609473982

Dynamic mutation with stepwise dynamic penalty has been used. Run is taken as failed.

4. From the results of cases 7-10 one concludes that enhancement of E (from 20%-75%) has adverse effect on the convergence of the solution with in prespecified generations. 5. In fitness vs. generation plots of all the cases shown in Figs. 2-4 (except for cases 4 and 5), the sudden decrease in fitness signifies the abrupt increase in penalty while increase in fitness indicates that the algorithm has found some equally good solutions. In cases 4 and 5 there was no penalization and thus the fitness never decreased. All these behaviours are also visible in log ηm vs. generation plots shown in Figs. 5-7 for respective cases. Due to exponential dependency of ηm on fitness the plot of log ηm vs. generations bear the same pattern as shown in Figs. 2 - 4. In the light of above findings one can conclude that algorithm with dynamic mutation outperformed the algorithm with constant mutation irrespective of the amount of penalty. Inclusion of dynamic penalty further boosts the performance. Moreover, higher value of elitism is undesirable as it renders the algorithm to converge prematurely. It can be seen that the result of case 1(c) were much better than those of Zain & Kumar respectively, though it took 20000 generations.

DOI: 10.2202/1934-2659.1122


20


1

0.9

0.8

Best fitness

0.7 case:1a case:1b case:2 case:3 case:4 case:5

0.6

0.5

0.4

0.3

0.2

0.1

0 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Generations

Fig. 2 Best fitness vs. generation plots

1

0.9

0.8

Best fitness

0.7

0.6

case:6 case:7 case:8 case:9 case:10

0.5

0.4

0.3

0.2

0.1

0 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Generations

Fig. 3 Best fitness vs. generation plots 21



1

0.9

0.8

Best fitness

0.7

0.6

0.5

case:1c

0.4

0.3

0.2

0.1

0 0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Generations

Fig. 4 Best fitness vs. generation plots 1.00E+01

9.00E+00

Log(ηm)

8.00E+00

case:1a case:1b case:2 case:3 case:4 case:5

7.00E+00

6.00E+00

5.00E+00

4.00E+00

3.00E+00 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Generations

Fig. 5 Log(ηm) vs. generation plots DOI: 10.2202/1934-2659.1122


22


10

9

Log(ηm)

8

case:6 case:7 case:8 case:9 case:10

7

6

5

4

3 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Generations

Fig. 6 Log(ηm) vs. generation plots 10

9

Log(ηm)

8

7 case:1c 6

5

4

3 0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Generations

Fig. 7 Log(ηm) vs. generation plots

23



6. Conclusions

A modified real elitist GA with dynamic mutation parameter and stepwise dynamic penalty has been developed for the solutions of non-linear algebraic and transcendental equations with the aim that the solutions may be obtained in lesser generations with small population size. This problem is attacked by transforming the system of non-linear equations into an optimization problem and thereafter, the associated global optima is found with the help of modified GA. To make the GA adaptable for such situations, following changes have been incorporated: (i) The mutation parameter has been forced to update continuously with generation by linking it to the fitness of the best individual of a population in a generation. (ii) Step wise dynamic penalty is imposed on the solutions, which do not satisfy the system. It is observed that with the above modifications the performance of the GA got improved manifolds; also, there is no trouble of proper ordering of equations/variables and of specifying an appropriate initial guess. The so designed GA is then employed for solving several difficult test problems and the results are compared and are found to be in good agreement with those available in literature. The usefulness of the modified GA has further been illustrated by applying it to solve an industrial problem of triple effect evaporator system used for concentrating the caustic soda solution. As mentioned by Zain and Kumar (1996), this evaporator problem is found to be difficult when solved using NewtonRaphson method. Here also, the modified GA seems to be quite efficient in obtaining the good quality results. Different result Tables and Figures support this observation. In future, efforts should be done to make the algorithm more efficient so as to get precise results in a lesser time with the use of lesser population. Besides, better programming of the code can further reduce the computational time. General guidelines for parameters setting can also be drawn for its execution to solve various problems belonging to different fields, and thus designing a GA to be more versatile and robust. Acknowledgement: Authors are grateful to the reviewers for their valuable suggestions and comments. Nomenclature A heat transfer area [m2] (used in evaporator problem). a , b, c penalty parameters used in equation (1). c1 , c2 fitness parameters used in equation (2). E elitism parameter. Ei ith equation. f fitness. DOI: 10.2202/1934-2659.1122


24


objective function; feed flow rate [kg/hr] in evaporator problem. liquid enthalpy [kJ/kg] (used in evaporator problem). vapour enthalpy [kJ/kg] (used in evaporator problem). liquid flow rate [kg/hr] (used in evaporator problem). number of equations. cross-over and mutation probabilities. penalty. generation number. saturation temperature [oC] (used in evaporator problem). over all heat transfer coefficient [kJ/hr m2 oC] (used in evaporator problem). V steam rate [kg/hr] (used in evaporator problem). x1 , x2 , x3 weight fraction of caustic soda in effect 1, 2 and 3 respectively (used in evaporator problem). xˆi ith scaled variable (used in evaporator problem). r x solution vector. r r X min , X max specified minimum and maximum values of solution vectors (domain). λ latent heat of vapourization [kJ/kg] (used in evaporator problem). ηc cross-over probability distribution index. ηm mutation probability distribution index. ηmi ,η mf specified initial and final values of mutation probability distribution indices (mutation parameter). τ solution temperature [oC] (used in evaporator problem). F h H L n pc , pm P t T U

1,2,3

i, j F 0

effect number (used in evaporator problem). index variables. feed (used in evaporator problem). input stream (used in evaporator problem).

Appendix I Offspring calculation in SBX crossover operator (Deb, 2001) Spread factor βi is calculated by generating random number ui ∈ (0,1) as:

25


Chemical Product and Process Modeling, Vol. 3 [2008], Iss. 1, Art. 17 1  η +1 if u i ≤ 0.5;  (2 u i ) c ,  1 βi =   η c +1 1  , otherw ise.    2(1 − u i )  

Components of offspring solution vectors are found from the components of parents’ solution vectors: xi(1,1+ t ) = 0.5*  (1 + βi ) xi(1,t ) + (1 − βi ) xi(2,t )  , xi(2,1+ t ) = 0.5* (1 − βi ) xi(1,t ) + (1 + βi ) xi(2,t )  .

Appendix II Constitutive Relationships for Caustic Soda Solution (Zain and Kumar, 1996): 1. Boiling point of solution. τ = (1.0 + 0.1419526 x)T + 150.75706 x 2 − 2.0795138 x ; 2. Overall heat transfer coefficient.

( x)

U = 977.66 τ

0.2823

; error ±3.53%

3. Latent heat of vaporization of steam. λ = −80.345T − 21035.87 / T + 2049.123 T − 4213.519 ln(T ) + 0.0918T 2 − 1.04 × 10−4 T 3 + 8597.953 ;

32 ≤ T ≤ 180o C

;

error ±0.85% ; 4. Enthalpy of saturated and superheated steam. H = 4.154T + 2.0125 × 10−4 T 2 + 1.62(τ − T ) + 2.0285 × 10−4 (τ 2 − T 2 )

− 0.3747 × 10−7 (τ 3 − T 3 ) + λ ;

T ≤ 200o C

;

error ±1.2% ; 5. Enthalpy of condensate at saturation temperature. h = 0.103527 + 4.18625T ; 5 ≤ T ≤ 160o C ; error ±0.66% ; 6. Enthalpy of caustic soda solution. h = 2.596971 + 158.896827 x + 3.74576τ − 2594.5098 x 2 − 3.758577 xτ + 0.004723τ 2 + 9164.489089 x 3 + 11.005268 x 2τ − 0.002463xτ 2 − 0.000031τ 3 − 5913.313486 x 4 − 12.344381x 3τ − 0.010289 x 2τ 2 + 0.000046 xτ 3 ; T ≤ 204o C , 0.02 ≤ x ≤ 0.56

error ±2.2% ;

DOI: 10.2202/1934-2659.1122


26


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