Multi-objective Optimal Synthesis and Design of Froth Flotation

An adaptation, inspired by the concept of jumping genes in biology, is developed for the binary-coded elitist nondominated sorting genetic algorithm (...
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Ind. Eng. Chem. Res. 2005, 44, 2621-2633

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Multi-objective Optimal Synthesis and Design of Froth Flotation Circuits for Mineral Processing, Using the Jumping Gene Adaptation of Genetic Algorithm Chandan Guria Department of Polymer Engineering, Birla Institute of Technology, Ranchi, Mesra 835 215, India

Mohan Verma Department of Space Engineering and Rocketry, Birla Institute of Technology, Ranchi, Mesra 835 215, India

Surya P. Mehrotra† National Metallurgical Laboratory, Jamshedpur 831 007, India

Santosh K. Gupta* Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208 016, India

An adaptation, inspired by the concept of jumping genes in biology, is developed for the binarycoded elitist nondominated sorting genetic algorithm (NSGA-II). This helps in obtaining globaloptimal solutions faster, particularly for problems involving networks. This is because the optimal values of some decision variables in such problems may be 0 or 1, e.g., some streams may be nonexistent in the optimal configuration. It is difficult to generate such chromosomes in the binary-coded NSGA-II (or the unmodified version of the real coded NSGA-II) using the three conventional operations of reproduction, crossover, and mutation. The algorithm developed is used to solve a few sample simple problems involving froth flotation circuits, which represent an important problem in mineral beneficiation. A two-species, two-cell flotation circuit is studied. Both single-objective as well as multi-objective optimizations are performed. The two important objectives used are as follows: (i) the maximization of the recovery of the concentrated ore and (ii) the maximization of the valuable-mineral content (grade) in the concentrated ore. A constraint of a fixed total flotation cell volume is also used. Because these objectives are conflicting, Pareto sets of nondominated solutions are obtained. The algorithm also can be used for the optimization of other networks. Introduction Froth flotation1 is a beneficiation process in which the valuable minerals are separated from their finely ground ores. The fine particles of the ground ore are comprised not only of the valuable, naturally occurring minerals (e.g., Cu2S (for Cu)), but also of other associated minerals (e.g., PbS, FeS, ZnS, etc.), as well as gangue (such as clay, SiO2, etc.). The composition and spatial distribution of the components present in different particles may vary randomly. The separation of valuable minerals by froth flotation is dependent on the differences in the surface properties of the materials involved. When a mixture of minerals is suspended in an aerated liquid (water), the gas (air) bubbles have a tendency to adhere preferentially to one of the constituents, which is more difficult to wet by the liquid. As a result, the effective density of the bubble-particle aggregates is reduced to such an extent that they rise to the surface, from where they are discharged through an overflow weir as a “concentrate” stream. The efficient operation and sepa* To whom correspondence should be addressed. Tel.: 91-512-259 7031. Fax: 91-512-259 0104. E-mail address: [email protected]. † On leave from Department of Metallurgical Engineering, Indian Institute of Technology, Kanpur 208 016, India.

ration of the valuable minerals in a flotation cell are dependent on several factors, e.g., the use of flotation reagents that affect the kinetics of separation, the percent solids loading, the pH of the aerated liquid, and the rate of aeration. In a single flotation cell, there is one feed stream of raw ground-ore slurry and two exit streamssthe mineral-rich floated concentrate and the gangue-rich “tailings”. This is shown schematically in Figure 1. Two important variables characterize the performance of froth flotation cells: the recovery (Rc), which is the ratio of the flow rates of the solid in the concentrate stream to that in the feed stream, given (for the case shown in Figure 1) by

Rc ≡ recovery ≡

C MF

(1)

and the grade (G), which is the fraction of the valuable mineral in the concentrate stream, given (for the case in Figure 1) by

G ≡ grade ≡

C×w )w C

(2)

The separation is seldom complete in a single flotation cell, and several interconnected flotation cells (or banks

10.1021/ie049706i CCC: $30.25 © 2005 American Chemical Society Published on Web 12/09/2004

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Figure 1. Schematic of a single-species, single-cell froth flotation unit.

of cells) are used to improve the process efficiency. In a conventional flotation circuit, the feed is introduced in a “rougher” cell, where a crude separation is affected. For improvement of the grade of the product, the concentrate is re-floated in one or more “cleaner” and “re-cleaner” cells, and the tailings from the rougher stage are re-floated in “scavenger” cells, to extract more of the residual mineral from the gangue-rich stream. The synthesis of such circuits is still an art and there is a need to establish a theoretical basis for it. Because typical flotation circuits may process several thousands of tons of ore per year, even a marginal improvement in their process efficiency can have a significant economic impact. Moreover, mineral-rich ores are becoming depleted, and the treatment of progressively lowerquality ores is becoming necessary, with attendant increases in the processing costs. Hence, the optimal design and synthesis of large circuits is assuming importance. It may be added that the computational complexity of this problem increases considerably with an increase in the number of flotation cells as well as with the number of “species” (described later) in the feed, and robust optimization algorithms must be developed and used. One such algorithm is developed in this paper and tested on some sample circuits to show its efficacy. The design of large-scale flotation plants involves the synthesis of the flotation circuit, the sizing of the cell banks, and the operating conditions of the individual cells. A simple and empirical model has been used for the froth flotation tanks in this work; this model is based on pilot-plant experimentation. This model is commonly used in industrial practice and has been used in several earlier optimization studies. Conventional deterministic search techniques have been applied earlier to obtain optimal circuit configurations. Most of these consider the overall recovery of the concentrate as the objective function while ensuring a specified grade (a constraint). In the early 1970s, Mehrotra and Kapur2 tried to synthesize and design optimal and suboptimal flotation circuits, using the random search method3-5 and the integrated approach (modified complex method) of Umeda and co-workers.6,7 The random search method is very slow, whereas the modified complex method is extremely sensitive to the initial guesses provided for the decision variables, and extensive numerical experimentation is required before reasonable results are obtained. Both of these techniques exhibit serious problems of convergence in the vicinity of the bounds of the decision variables and converge to local optima. Green8 attempted to synthesize optimal flotation circuits using linear programming. Reuter and van Deventer9 refined Green’s work. The approaches based on linear programming are, unfortunately, limited in scope, because they cannot be used for nonlinear objective functions or constraints. The direct search method with a systematic

reduction of the size of the search region, popularly known as the Luss-Jaakola (LJ) method,10 was used by Dey et al.,11 who considered generalized circuit configurations in the absence of self-recycle streams. The resulting solution was dependent largely on the values specified for the different weightings for the recovery and the grade of the concentrate. These studies have been reviewed by Yingling12 and Mehrotra.13 Yingling14 proposed an optimization technique, using the potential theory of Markov chains to obtain optimal solutions considering the cost (reward function) as an objective function. He assumed that the recovery of the desired grade of concentrate is invariant with changes in the total mass flow rate to the vessel. This is not true for optimally synthesized flotation circuits. Loveday and Brouckaert15 analyzed flotation circuit design principles based on the minimization of the volume of the flotation cells. Schena and co-workers16-18 studied the optimal synthesis and design of flotation circuits, considering profit as the objective function with simplified circuit configurations, and solved the optimization problem by decomposing it into subproblems solved sequentially. Empirical design methods and practitioner’s rules were implemented to generate feasible starting solutions as well as to post-process the concentration matrixes for improving the objective function. More recently, AbuAli and Abdel Sabour19 optimized flotation circuits, based on an economic approach. This study enables one to generate the optimal number of flotation cells in a bank and the volume of each flotation cell, to achieve the desired recovery for a specified grade of the concentrate. Their method fails to give optimal circuit configurations. It is observed that conventional techniques discussed previously fail to produce the global optimal circuit configuration. Such solutions are important, not only from a mathematical point of view, but also because of the immense economic implications that they have for the separation of highly valuable and rarely available minerals. Over the last several years, the AI-based genetic algorithm20-22 (GA) has been applied successfully to many optimization problems. It uses a population of several solutions simultaneously along with probabilistic operators, viz, reproduction, crossover, and mutation. In addition, it uses only the values of the objective functions (and not gradients) and does not need initial guesses. Several workers have adapted the basic algorithm (simple GA, or SGA) to solve more-meaningful problems that involve multiple objectives using different techniques developed over the years, each associated with its own advantages and disadvantages. These have been reviewed by Deb23 and Coello Coello et al.24 A popular algorithm for such problems is the nondominated sorting genetic algorithm (NSGA) that was developed by Deb and co-workers.23,25 Two versions of this technique are available: NSGA-I23,25 and NSGA-II.23,26 Bhaskar et al.27 have reviewed a variety of multiobjective optimization problems that have been solved in chemical engineering using NSGA-I and other nonevolutionary techniques. NSGA-II introduces the concept of elitism23 and has been applied recently to solve two highly computationally intense problems in chemical engineering, namely, the multi-objective optimization of an industrial fluidized-bed catalytic cracker unit28 (FCCU) and the unsteady operation of a steam reformer.29 An important feature of NSGA-II is that better members from the parent chromosome pool

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(elitism) are incorporated along with those from the daughter pool. In addition, it uses an improved methodology for classifying chromosomes into nondominated23 fronts and for evaluating the degree of crowding of individual members, to give Pareto solutions with a good spread of points. It is associated with lower computational complexity.23 More recently, Kasat et al.,30 inspired by jumping genes (JG or transposons) in biology, developed the jumping gene operator. This macro-macro mutation operation in the binary-coded NSGA-II-JG accelerates the optimization of FCCUs by almost 8-fold and provides the global optimal Pareto for a test problem (ZDT4)23 that could not be obtained using the binary-coded NSGA-II. Another adaptation is the real coded NSGA-II-RC using the simulated binary crossover (SBX) operator.23 This improves the “granularity” of the search space and is expected to work at least as well as the binary-coded NSGA-II-JG and its adaptations. In all these evolutionary algorithms, tuning of the computational parameters is necessary. This requires solving the optimization problem several times over. This disadvantage parallels that associated with earlier (conventional) techniques, similar to those used by Mehrotra and Kapur,2 which must be solved with several choices of the initial conditions before the best solution is obtained. An important feature of the optimization of froth flotation circuits (and similar problems associated with other networks or circuits) is that the optimal configuration may involve some decision variables to lie exactly at their lower or upper bounds. For example, in a circuit problem, the optimal fractions of flows in some streams (decision variables) may be equal to zero or unity. The probability of generating chromosomes having these characteristics is not high in NSGA-II, and some type of a macro, or macro-macro, mutation operation should help. A further adaptation of NSGA-II-JG, called NSGAII-mJG (modified JG), is presented in this paper to overcome this problem. It is found to give better results when applied to the single-objective optimization problem studied by Mehrotra and Kapur2 (froth flotation circuit with two cells, and involving two species), as compared to NSGA-II and NSGA-II-JG. NSGA-II-mJG has considerable potential for solving optimization problems involving other circuits as well, e.g., heattransfer networks, trains of distillation columns, etc. The objectives of the present work are, thus, (i) to develop a modified jumping gene operator for optimizing circuits, using an adaptation of the binary-coded NSGAII, and (ii) to use this adaptation to study, first, the single-objective optimization problem of Mehrotra and Kapur,2 and then to solve a simple, two-cell, two-species, multi-objective optimization problem. It may be emphasized that it is not our intent to compare different techniques (although some of this is, indeed, done) nor to establish the superiority of one methodology over the other, but to develop one technique, namely, the binary-coded NSGA-II-mJG, and demonstrate its use in the optimization of a few simple froth flotation circuits. Other multi-objective optimization algorithms, e.g., NSGA-II-RC, can easily be programmed to exploit the advantages offered by the mJG operator. Formulation Model of Flotation Cells and Circuits. A considerable amount of experimental and theoretical work has

been reported in the literature for semibatch and continuous flotation cells. Fichera and Chudacek31 and Kapur and Mehrotra32 reviewed the detailed modeling of froth flotation networks. Because of the complexity of particulate systems, flotation cells are often described using highly simplified, macroscopic models. A steadystate continuous-flow froth flotation cell is usually modeled31,32 in terms of a continuous-flow perfectly mixed stirred-tank reactor (lumped-parameter) model with first-order kinetics.33-38 The complex distribution of components in the fine ore particles is described in terms of a set of far simpler “flotation rate constants” (Ki). This parameter combines ore particles with different compositions and spatial distributions that are associated with similar (over a small range) flotation characteristics and are referenced as the ith “species”. Obviously, this species will have some average composition that is ascertained using laboratory/pilot-plant flotation data. Interestingly, only a few values of Ki are necessary to characterize the feed for a reasonable representation of the operation of a cell. The assumption of identical rate constants in each cell (having different air flow rates, pulp densities, reagent concentrations, etc.) is possibly an oversimplification but is commonly used in industrial practice. More-detailed models relating Ki,j (for the ith species in the jth tank) to the operating variables can easily be used, and the optimization algorithm also can be applied to such detailed models. Appendix A gives a short summary of the balance equations35,36,39,40 for a general configuration of cells with all possible interconnections (see Figure 2 for a two-cell system). The procedure described in this appendix is that for obtaining the flow rates of the different species in the several tailings and concentrate streams for a circuit that has m cells. The set of coupled equations (eq A.10) are first solved, once for each species, using the F04AAF subroutine41,42 in the NAG library. Equation A.15 is then solved. This is followed by the calculation of the overall recovery and grade for the flotation circuit using eqs A.16 and A.17. These calculations may be performed for a specified feed rate and the mass fraction of the valuable mineral in the different species. The volume (Vi) of the individual flotation cells can then be calculated from eq A.18. The Modified JG (mJG) Operator. An important feature of the optimal synthesis of flotation circuits is that the global optimal values of some of the decision variables (e.g., the fraction of flow in any stream) may lie exactly at their bounds (e.g., either 0 or 1). Because the binary-coded NSGA-II is based on three probabilistic operators (viz., reproduction, crossover, and mutation), it is difficult to generate solutions that have such values (involving a sequence of several zeros or ones). A similar problem is also encountered if we use our earlier algorithm,30 the binary-coded NSGA-II-JG, where a substring of a chromosome, identified randomly, is replaced by a randomly generated new binary substring (JG). This new substring may not coincide with a single decision variable (it may overlap several decision variables). Thus, it is not easy to generate chromosomes with decision variables exactly at their bounds using these algorithms. We develop the modified jumping gene (mJG) operator to solve this problem. A fraction of the chromosomes (pmJG) is selected randomly from the daughter population for the mJG operation. The decision variable to be replaced in these is then identified, using a random number. The set of binaries associated

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Figure 2. Generalized two-cell flotation circuit.

with this variable is replaced either by a sequence of all zeros or all ones, using the probability, p11...1. Replacement of more than one decision variable in a chromosome has also been investigated; however, it has not been observed to be of much use. Details are provided in Appendix B. This methodology can easily be used for similar problems involving networks or circuits, even though only froth flotation circuits have been studied in this paper. Optimization Problems Studied Problem 1. We first consider the simple but general circuit shown in Figure 2 that has two flotation cells (m ) 2), with the feed consisting of only two species (n ) 2), which are referenced as the valuable (val) and the gangue (gang). We maximize the overall recovery Rc of the concentrate and use two constraints: (i) obtain a desired grade (Gd) of the concentrate and (ii) have a desired value of the total volume (Vd) of the cells. The decision variables are the set of cell linkage parameters β and δ, as well as the mean residence times, λ, in the cells: T

β ≡ [β10, β11, β12, β20, β21, β22] δT ≡ [δF1, δF2, δ10, δ11, δ12, δ20, δ21, δ22] λT ≡ [λ1, λ2]

(3)

The λ terms are related to the volumes Vi of the individual cells. The total feed flow rate MF, the feed composition xi (for each of the two species in the system), the flotation rate constants Ki of the two species, the values of the two wi terms (the mass fractions of the valuable mineral in the two species in the concentrate stream, assumed to be the same for the two cells), and the total number of cells (m ) 2) are specified. In addition to the constraints given previously, the following constraints on the decision variables must also be satisfied: m

δF ) 1.0 ∑ i)1 i

(4)

m

δk + δ k ∑ i)1 i

0

) 1.0

(for k ) 1, 2, ..., m)

(5)

) 1.0

(for k ) 1, 2, ..., m)

(6)

m

βk + β k ∑ i)1 i

0

These represent the fact that, for any stream being split, the sum of fractions should be 1. This single-objective optimization problem is written mathematically as Problem 1:

Max f(β,δ,λ) ≡ Rc

(7a)

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2625 Table 1. Details of the Two-Cell, Two-Species Optimization Problems Studied (Problems 1 and 2) parameter

Table 2. Values of the (Best) Computational Parameters Used in NSGA-II-mJG and NSGA-II-RC

value

Feed Composition mass fraction of valuable, x1 22.0% mass fraction of gangue, x2 78.0% Constraints and Bounds 75.0% (Problem 1 only) 0.5663 m3 g0 min e20 min

Gd Vd λ1 λ2

Data K1 () Kval) K2 () Kgang) MF w1 w2 dp d Fw

1.0 min-1 0.1 min-1 11.34 kg/min 1.0 0.0 20.0% 2500 kg/m3 1000 kg/m3

subject to (s. t.) the constraints

G ) Gd

(7b)

V ) Vd

(7c)

model equations (Appendix A)

(7d)

eqs 4-6

(7e)

Ngmax Np lsubstr nV lchrom pc pm pJG pmJG p11...1 random seed number H1 H2 pmr RSBX Rm bounds of decision variables

λi,L e λi e λi,U

δFi δFi T

m

[ ( )] [ ( )]

P1 ≡ H1 1 -

G Gd

2

P2 ≡ H2 1 -

V Vd

2

(8) (9)

The penalty functions are assumed to be zero when the constraints are satisfied (to within four decimal places of Gd or Vd). If the constraints are not satisfied, the penalties are assumed to be large numbers (compared to the value of the objective function), proportional to the square of the constraint violation, as given in eqs 8 and 9. Table 2 gives the values of H1 and H2, as well as the computational parameters used in NSGA-II-mJG for Problem 1. The other constraints on the linkage parameters (eqs 4-6) are handled differently. The linkage

108

(for i ) 1, 2, ..., m)

δki δki T

(7g)

Even this (two-cell, two-species) relatively simple problem is associated with a considerable degree of freedom. The total number of decision variables is 16 (from eq 3), whereas the total number of constraints (from eqs 4-6) is 5. Thus, the total number of degrees of freedom is 11. The detailed information on the design parameters, constraints, and bounds used are given in Table 1. This problem is selected because it is the same as that solved by Mehrotra and Kapur (see their Table 2B, Stage I).2 The equality constraints on G and V are addressed using penalty functions.22 Penalties P1 and/or P2 are added to (for minimization of objective functions) or subtracted from (for maximization) the objective function (or to all the functions in a multi-objective problem) in eq 7a:

100 000 100 32 16 512 0.99 0.025 0.00 0.050 0.90 0.1234

200 000 50 16 0.90

0.123 108 108 0.10 20 200 rigid

(10a)

δFi ∑ i)1

(7f)

(i ) 1, 2)

100 000 50 32 16 512 0.99 0.050 0.00 0.050 0.90 0.1234 108 108

parameters are first generated within their lower and upper bounds (0 and 1) using random numbers. They are then normalized using

and the following bounds on the decision variables:

0 e β δ e 1.0

NSGA-II-mJG NSGA-II-mJG NSGA-II-RC (Problem 1) (Problem 2) (Problem 1)

parameter

m

(10b)

δki + δk0 ∑ i)1 βki βki T

m

βki + βk0 ∑ i)1 (for i ) 0, 1, 2,..., m; k ) 1, 2,..., m) (10c) to give the values of the decision variables to be used. Normalization of the linkage parameters ensures that the constraints (eqs 4-6) are satisfied. If the mJG operator makes the mass fraction of a stream 1.0, normalization is avoided and the associated streams are deleted (to satisfy eqs 4-6), so that the action of the mJG operation is not negated. Normalization does not affect the action of the mJG operator when it makes the mass fraction of any stream 0.0. After solving Problem 1, we can introduce additional constraints on the linkage parameters, to give simplified, near-optimal flotation circuits that are more useful in industry. One example (Adapted Problem 1) is the elimination of all self-recycle streams. Problem 2. After solving the single-objective optimization Problem 1 and its adaptation, we formulate and solve a problem that involves two objective functions (again, for specified values of MF, xi, Ki, wi, two cells (m ) 2), and two species (n ) 2)), described by

max f1(β,δ,λ) ≡ Rc

(11a)

max f2(β,δ,λ) ≡ G

(11b)

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Table 3. Optimal Solutions for Problem 1 and Adapted Problem 1 NSGA-II-mJGa Problem 1 (reference)

variable

data from Mehrotra and Kapur2 Problem 1

NSGA-IIb Problem 1

NSGA-II-RC datac Problem 1

NSGA-II-mJGc adapted Problem 1

β10 β11 β12

0.9936 0.0064 0.0000

0.783 0.141 0.076

0.5786 0.4214 0.0000

0.0000 0.0000 1.0000

0.0000 0.0000 1.0000

β20 β21 β22

0.0000 1.0000 0.0000

0.002 0.619 0.379

0.0000 1.0000 0.0000

0.6854 0.0000 0.3146

1.0000 0.0000 0.0000

δ10 δ11 δ12

0.0000 0.2792 0.7208

0.000 0.522 0.478

0.0000 0.3355 0.6645

0.6843 0.1829 0.1328

1.0000 0.0000 0.0000

δ20 δ21 δ22

0.5920 0.0000 0.4080

0.973 0.012 0.015

0.5608 0.0000 0.4392

0.0000 0.8823 0.1177

0.0000 1.0000 0.0000

δF1 δF2

0.0000 1.0000

0.000 1.000

0.0069 0.9931

1.0000 0.0000

0.9984 0.0016

λ1 λ2

0.4757 8.3372

0.996 8.650

0.7920 7.4577

9.2338 0.7516

14.1016 0.6533

V1 V2 total volume

0.0403 0.5260 0.5663

0.0456 0.3446 0.3902

0.0690 0.4973 0.5663

0.5084 0.0579 0.5663

0.5267 0.0396 0.5663

recovery (%) grade (%)

24.8761 74.9998

22.3500 75.0000

24.7826 75.0002

24.4816 75.0000

24.8702 75.0001

a Values obtained using reference values listed in Table 2. b Values obtained with p mJG ) pJG ) 0.0. Remaining parameters as given in Table 2. c Parameters are as given in Table 2.

subject to (s. t.) constraints

V ) Vd

(11c)

model equations (Appendix A)

(11d)

eqs 4-6

(11e)

and the following bounds on the decision variables:

0 e β δ e 1.0 λi,L e λi e λi,U

(for i ) 1, 2)

(11f) (11g)

To the best of our knowledge, multi-objective optimization of froth flotation circuits has not been reported in the open literature. Results and Discussion Several preliminary tests were first made on the code that has been developed, to ensure that it was free of errors. These tests (similar to those developed for NSGA-II) are quite standard28,29 and are not described here. The NSGA-II-mJG code was then used to solve the single-objective and multi-objective Problems 1 and 2. Problem 1. This two-cell, two-species single-objective optimization problem is the same as that solved by Mehrotra and Kapur.2 The multi-objective code NSGAII-mJG has been used to solve this single-objective problem, using identical objectives. This procedure is quite common and well-tested.28,29 It is well-known that the use of a multi-objective code (instead of a singleobjective code) for a problem involving only one objective will lead to extra computations (but give the same results). This is not of much concern for the simple problem being studied here but should be avoided for large circuits. The computational parameters to be used in the code are varied (tuned) until the best results are obtained. This procedure is quite common for the successful implementation of optimization techniques.

Figure 3. Variation of the recovery and the grade with the generation number (Problem 1).

The values of the best parameters are given in Table 2 (the effects of variations in these parameters are discussed later). The corresponding final results, shown in column 2 of Table 3 are being referred to as reference results. The central processing unit (CPU) time required for the NSGA-II-mJG code for 100 000 generations on a Pentium IV (1.7 GHz, 256 MB RAM) computer is ∼4.5 h for this problem. The evolution of the overall recovery, overall grade, volume of cell 1, and the total cell volume over the generations are shown in Figures 3 and 4. The trends observed in the plots of the decision variables are similar. (This information is not shown but is available from the authors upon request.) Figure 3 shows that the overall percent recovery of the concentrate stream converges rather slowly but becomes almost constant after ∼5000 generations, reaching its final value of 24.8761% thereafter. This means that we can reduce the computational time by ∼10-fold by reducing the maximum number of generations. The grade reaches its final constrained value of Gd ) 75.0000% even faster.

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Figure 4. Variation of the total cell volume and the volume of cell 1 with the generation number (Problem 1).

Mehrotra and Kapur2 have solved the same problem, using the modified complex as well as the random search methods. (See their Table 2B, Stage I.) Their results are also shown in Table 3 (column 3). Their optimal recovery of 22.35% is less than the value of 24.8761% obtained with NSGA-II-mJG. Thus, the superiority of NSGA-II-mJG over the techniques used by Mehrotra and Kapur2 is established. Clearly, the techniques used in the latter study did not converge to the global optimal point. This is further confirmed by the fact that these workers obtained a higher recovery of 24.8692% by introducing additional constraints (small and large linkage parameters of 0.8 were forced to be 0 and 1, respectively), and re-running their program (their Stage 2 calculations). We cannot compare our optimal solution of Problem 1 with their Stage 2 solution, because these problems are different. More recently,43 we have found that the results obtained for the more-complex froth flotation circuit involving four cells and three species, using the binary-coded NSGAII-mJG, are also better than those reported by Mehrotra and Kapur.2 This indicates the usefulness of the present algorithm. We also generated the solution of Problem 1 using pmJG (or pJG) ) 0, i.e., using NSGA-II with no mJG (or JG) operation. The results are shown in Table 3 (column 4). The total number of nonexisting streams is less than that indicated by NSGA-II-mJG. As mentioned previously, this is due to the fact that it is very difficult to generate chromosomes that have linkage parameters exactly equal to zero or unity using the reproduction, crossover, and mutation operators alone. Therefore, the overall recovery also is slightly higher with NSGA-IImJG than with NSGA-II. This reconfirms that the adaptation presented in this paper is an improvement over the binary-coded NSGA-II, at least for the problems studied. The improvement is determined to be even better with the more-complex circuits that have been studied recently.43 Problem 1 was also solved using NSGA-II-RC. (The code used is in C and is available via the Internet at www.iitk.ac.in/kangal/soft.htm.) A few tests were made to ensure that it was free of errors (e.g., simulated results using the optimal values (see ref 2, Table 2B, Stage II) from NSGA-II-RC were identical to those in the original reference; some constrained optimization (only residence times of the cells as the decision variables) from NSGA-II-RC were the same those in ref 2; etc.). Once again, the computational parameters were

tuned to get the best results for Problem 1. These are given in Table 2, and the corresponding results are shown in Table 3 (column 5). As expected, this code was observed to converge almost twice as fast, compared to the binary-coded NSGA-II-mJG. However, it converged to a local optimal solution. The optimal recovery of 24.4816% (in 200 000 generations) was less than the value of 24.8761% obtained with the binary-coded NSGA-II-mJG in 100 000 generations, although the same CPU times were involved (the recoveries from NSGA-II-RC at the 100 000th, 150 000th, and 175 000th generations were lower at 23.5067%, 24.2561%, and 24.4767%, respectively). It may be emphasized that the mJG operator was not used in NSGA-II-RC. We believe that the incorporation of this adaptation in this code would also improve the working of this code. An important point that emerges from a comparison of columns 2-5 in Table 3 is that several near-optimal solutions exist in this problem, and these solutions are associated with widely different values of the decision variables. Surprisingly, the performance (in terms of the final solution) of the real coded NSGA-II-RC is determined to be inferior to even the binary-coded NSGA-II (both without the mJG operator) for the present problem. Such optimization problems are extremely difficult to solve, and the use of the mJG operator seems to be helpful. More problems involving networks must be solved to establish the efficacy of this algorithm fully. The optimal solution obtained using NSGA-II-mJG also involves the use of self-recycle streams. Such streams are not popular in flotation circuits. We have solved the Adapted Problem 1 (i.e., β11, β22, δ11, and δ22 are forced to be zero, by selecting their lower and upper bounds to be zero). Table 3 (column 6) shows the solution obtained using 100 000 generations of NSGAII-mJG (with the same set of computational parameters as that for Problem 1). A worsening of the recovery is observed, as expected, unlike the improvement obtained by Mehrotra and Kapur2 on the introduction of additional constraints in their Stage 2. This indicates that (i) the Stage I results of Mehrotra and Kapur2 are not the global optimal solutions and (ii) one should explore the possibility of using self-recycle streams to improve the overall recovery of the concentrate (of a desired grade). The effects of varying the several computational parameters are now studied, using Problem 1 as the test problem. As mentioned previously, the success of any optimization code is dependent on the tuning of several parameters. This is borne out by the results in Table 4. This table shows the overall recovery when different parameters are changed one by one, keeping all others at their reference values (see Table 2). A worsening of the overall recovery is observed in all cases studied (the reference results are generated using the best values of the parameters). A gradual increase of pmJG from 0 to 0.9 shows no general trend on the optimal recoveries. The effect of varying p11...1, pc, pm, and the random seed number22 on the optimal recoveries is also shown in Table 4. It is observed that varying p11...1 does not influence the recovery. However, a change of the other parameters leads to a reduction in the optimal recovery. Indeed, one must perform computational experiments to determine the best set of parameters. Unfortunately, deciding upon the best set of computational parameters is problem-specific, and this dilemma also is encountered28-30 in several other real-life optimization

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Table 4. Effect of Varying the Parameters One by One, in NSGA-II-mJG, on the Recovery (Problem 1) No.

parameter changed

2 3 4 5 6 7 8 9 10-14 15 16 17 18 19 20

reference values: pmJG ) pm ) 0.05; p11...1 ) 0.90; pc ) 0.99; random seed number ) 0.1234 pmJG ) 0.00 pmJG ) 0.01 pmJG ) 0.10 pmJG ) 0.20 pmJG ) 0.30 pmJG ) 0.40 pmJG ) 0.50 pmJG ) 0.90 p11...1 ) 0.1, 0.2, 0.5, 0.7, 0.8 pc ) 0.97 pc ) 0.98 pm ) 0.02 pm ) 0.005 random seed number ) 0.5565 random seed number ) 0.8565

1

overall recovery (%) 24.8761 24.7826 24.7251 24.7510 24.5588 24.7897 24.8537 24.8062 24.7906 24.8761 24.8273 24.7659 24.6829 23.9060 24.7281 24.5584

Table 5. Effect of pJG (NSGA-II-JG) on the Recovery (Problem 1) (for pmJG ) 0) No.

pJG

recovery (%)

1 2 3 4 5 6 7

reference values: pmJG * 0; pJG ) 0 pJG ) 0.15 pJG ) 0.25 pJG ) 0.50 pJG ) 0.65 pJG ) 0.75 pJG ) 0.95

24.8761 24.6037 24.6327 24.1151 24.6054 24.6022 23.7951

Table 6. Effect of pJG (NSGA-II-JG and NSGA-II-mJG) on Recovery (Problem 1) (for pmJG ) 0.05) No.

pJG

recovery (%)

1 2 3 4 5 6 7

reference value: pJG ) 0 pJG ) 0.05 pJG ) 0.15 pJG ) 0.25 pJG ) 0.50 pJG ) 0.75 pJG ) 0.85

24.8761 24.6657 24.6822 24.7830 24.8442 24.6459 24.7669

problems. The increase in the computational time associated with the multiple application of the algorithm is akin to that associated with the multiple use of traditional algorithms of optimization, where one must explore different starting points. By setting pmJG ) 0 (i.e., using NSGA-II-JG and not NSGA-II-mJG) and increasing pJG while keeping the other parameters at their reference values given in Table 2, we obtain lower optimal recoveries (see Table 5). This confirms the inadequacy of the binary-coded NSGA-II-JG for solving this problem. Table 6 (with pmJG ) 0.05 and with different values of pJG; i.e., using a mixture of NSGA-II-JG and NSGA-II-mJG) shows that this combination of techniques also is inadequate, possibly because one operation counteracts the other. We expect similar inferences for other network optimization problems in which optimal values of the decision variables may lie at their bounds (nonexistent streams). Problem 2. The NSGA-II-mJG code is now used to solve the multi-objective optimization Problem 2 (see Table 1). The CPU time required for the NSGA-II-mJG code for 100 000 generations on a Pentium IV (1.7 GHz, 256 MB RAM) computer is ∼4.5 h for this problem also. Both the overall recovery and the grade of the concentrate (for a specified total cell volume of 0.5663 m3) are maximized. Several sets of computational parameters

Figure 5. Pareto set of optimal solutions for Problem 2.

are tried, and the best solutions are obtained using the values given in Table 2. These are almost identical to those for Problem 1, except for pm, which is 0.025, instead of 0.050. Optimal solutions obtained using pm ) 0.050 are only very slightly inferior to those obtained with pm ) 0.025. Figure 5 shows the final set (after 100 000 generations) of nondominated Pareto optimal solutions. As one goes from any point on this plot to any other, one objective function improves (increases) while the other worsens (decreases), which is a characteristic of Pareto sets of nondominating solutions.22,23 The present results cannot be compared with existing results, because multi-objective optimization of froth flotation circuits has not been reported in the open literature. However, a (standard) check on the correctness of the Pareto set can be made using the -constraint method.44 In this, one of the two objective functions is put as a constant, . We select the point G ) 75.9741, Rc ) 24.2654% (a point that is near the unique solution for Problem 1) for checking. We solved the single-objective optimization (SOO) problem using the constraint  ) Gd ) 75.9741%, and obtained values of G ) 75.9740 and Rc ) 24.2698%. This is very similar to the Pareto point selected for the MOO problem. This match gives us confidence that the Pareto set is correct. The Pareto optimal solutions obtained with pm ) 0.025, 0.050, and 0.1 (all others being the same as those for Problem 1), as well as with pm ) 0.025, pmJG ) 0.5 (instead of with pm ) 0.050, pmJG ) 0.050 used for Problem 1) are quite similar to each other. The decision variables associated with the several Pareto points in Figure 6 are shown in Figures 6-12. Relatively smooth plots (with minor amounts of scatter) are observed for several variables. However, Figures 7 and 8 show that there is considerable scatter in the optimal values of β21, β22, δ11, and δ12. The differences in these four variables clearly compensate for each other and do not affect the Pareto set much. Such insensitivity of the Pareto set to scatter in a few decision variables has been encountered in earlier, real-life studies27,45,46 and can possibly be eliminated by assuming,46 a priori, parametrized smooth functions for these, giving nearoptimal solutions that are more useful. The evolution of the Pareto set over the generations is shown in Figure 13. It is observed that the results do not change significantly after ∼1000 generations. Again, one can reduce the computational time by using a lower value of the maximum number of generations as the stopping criterion. Pareto sets were also obtained for

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2629

Figure 6. Optimal linkage parameters β associated with the first cell for the Pareto points in Figure 5 (Problem 2).

Figure 7. Optimal linkage parameters β associated with the second cell (Problem 2).

Figure 8. Optimal linkage parameters δ associated with the first cell (Problem 2).

Problem 2, with no self-recycle streams. The solutions were slightly inferior to those shown in Figure 5, although the scatter was lower (details can be supplied by the authors upon request). Conclusions An adaptation of the binary-coded elitist nondominated sorting genetic algorithm (GA)23,26 (NSGA-II) has

Figure 9. Optimal linkage parameters δ associated with the second cell (Problem 2).

Figure 10. Optimal feed distribution parameters δF associated with the Pareto points in Figure 5 (Problem 2).

Figure 11. Optimal mean cell residence times, λi (Problem 2).

been suggested in this paper, inspired by the concept of jumping genes in biology. This adaptation improves the earlier algorithm and provides better (global optimal) solutions, at least for the froth flotation circuits that have been studied. It is anticipated that this code would also give improved results for other network optimization problems (where some streams may not exist in the optimal configuration). The modified jump-

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least one of us (S.K.G.). Partial financial support from the Department of Science and Technology, Government of India, New Delhi (through Grant No. III-5(13)/2001ET) is gratefully acknowledged. Nomenclature

Figure 12. Optimal cell volumes (Problem 2).

A ) defined in eq A.10 b ) defined in eqs A.10 and A.12 Ci ) mass flow rate of solids in the concentrate stream from the ith cell, kg min-1 d ) density of solids, kg m-3 dp ) mass percent of solids in the tailing fi ) ith objective function G ) overall grade of the concentrate stream Hi ) ith penalty parameter (eqs 8 and 9) Ki ) overall flotation rate constant of the ith species, min-1 lchrom ) total length (number of binaries) of a chromosome lsubstr ) length (number of binaries) of a substring representing a decision variable m ) total number of cells in the circuit MF ) mass flow rate of solids in the fresh feed, kg min-1 n ) total number of “species” in the feed nV ) number of decision variables Ngen ) generation number Ngmax ) maximum number of generations Np ) total number of chromosomes in the population pc ) crossover probability pJG ) jumping probability for the jumping gene (JG) operator pmJG ) modified jumping probability for the modifed jumping gene (mJG) operator pm ) mutation probability pmr ) real parameter mutation probability p11...1 ) probability for changing all binaries of a selected decision variable to zero Pi ) ith penalty function (eqs 8 and 9) r ) random number RC ) overall recovery of the concentrate stream Rm ) real parameter mutation parameter RSBX ) real parameter simulated binary crossover parameter Ti ) mass flow rate of solids in the tailings stream from the ith cell, kg min-1 u ) defined in eq A.11 Vi ) volume of the ith cell in the flotation circuit, m3 V ) total volume of all m cells in the flotation circuit, m3 wi ) mass fraction of the valuable mineral in the ith species in the concentrate xi ) mass fraction of the ith species in the feed Greek Symbols

Figure 13. Evolution of the Pareto set (Figure 5) over the generations.

ing gene (mJG) adaptation can also be used in other GA-based techniques. Acknowledgment It is a pleasure to dedicate this paper to Professor W. Harmon Ray, who has been a great inspiration to at

Rij ) defined in eq A.4 βij ) fraction of the concentrate going from the ith cell to the jth cell δij ) fraction of the tailings going from the ith cell to the jth cell δFj ) fraction of fresh feed going to the jth cell φij ) functionality (eq A.13) λi ) mean residence time of the ith cell, min Fw ) density of pure water, kg m-3 Subscripts/Superscripts d ) desired value gang ) gangue L ) lower bound of a decision variable

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2631

One may note that all of the m × n equations in eq A.9 are not coupled together. In fact, these equations can easily be rewritten in terms of several independent smaller subsets, one for each value of j. For the jth species, we can write

U ) upper bound of a decision variable val ) valuable mineral

Appendix A. Model of Continuous Froth Flotation Cells If C(K), T(K), and MF(K) represent the flow rates of a solid species having the rate constant K in the concentrate, tailings, and feed streams, respectively, we have2

Kλ C(K) ) MF(K) 1 + Kλ

(

)

Ajuj ) bj

(A.10)

where

(A.1)

and

(for j ) 1, 2, ..., n)

ujT ≡ [Tj,1, Tj,2, Tj,3, ..., Tj,m]

(A.11)

with the rth element in bj given by

T(K) ) MF(K)

(1 +1Kλ)

where λ is the mean residence time in the cell.2 We consider the generalized circuit configuration (see Figure 2 for an example involving two cells), in which all the streams go to all the cells. βij and δij represent the fractions of the (mass) flow rates of the solids in a stream issuing from cell i (or the feed, F), for i ) 1, 2,..., m, and going to another cell j (j ) 0 represents a product stream). A simple mass balance gives the total mass flow rate of the jth species (for j ) 1, 2,..., n) entering the ith cell as m

Fj,i ) MF,jδFi +

∑ k)1

m

Tj,kδki +

bj(r) ) MF,jδFrRj,r

(A.2)

∑ Cj,kβki k)1

(A.3)

and the (r, s)th elements of Aj being easily written. One can obtain the solution for the mass flow rates of the jth species from the ith cell in the tailings by solving eq A.10. This may be written as

Tj,i ) φj,i(MF,j, δ, β, R)

(A.13)

MF,j ) MFxj

(A.14)

with

When the Tj,i values are known, one can obtain the mass flow rates of the jth species in the concentrate stream from the ith cell, using eq A.7:

If

Rj,i ≡

1 1 + Kjλi

(A.4)

the mass flow rates of the jth species from the ith cell in the concentrate and tailings streams are given by2

(A.12)

Cj,i ) φj,i

(

)

1 - Rj,i Rj,i

(A.15)

The overall recovery (Rc) and the grade (G) can now be written for the generalized flotation network: m n

Cj,i ) Fj,i(1 - Rj,i)

(A.5) Rc )

and

Tj,i ) Fj,iRj,i

∑ ∑Cj,iβi0 i)1 j)1

(A.16)

n

MF,j ∑ j)1

(A.6)

Therefore,

m n

Cj,i ) Tj,i

(

)

1 - Rj,i Rj,i

(A.7)

Substituting for Cj,i in eq A.3 gives m

Fj,i ) MF,jδFi +

∑ k)1

Tj,k

[( ) 1 - Rj,k Rj,k

βki + δki

]

m

∑ Tj,k k)1

[( ) 1 - Rj,k Rj,k

]

βki + δki

(A.17)

m n

∑ ∑Cj,iβi0 i)1 j)1 (A.8)

On further substituting this expression in eq A.6, we obtain2

Tj,i ) MF,jδFiRj,i + Rj,i

G)

∑ ∑Cj,iβi0wj i)1 j)1

where wj is the mass fraction of the valuable mineral in the jth species in the concentrate stream (which is assumed to be the same for all the cells). Knowing the flow rates of the tailings (from eq A.9) and the flow rates of the concentrates (from eq A.15), one can obtain the volumes of the individual flotation cells:2

(A.9)

with i ) 1, 2,..., m and j ) 1, 2,..., n. Equation A.9 comprises of a set of n × m linear algebraic equations that can be solved numerically for the several Tj,i, in terms of the linkage and design parameters.

n

V i ) λi

[

1

Tj,i + ∑ d j)1

]

(100 - dp) d p Fw

(A.18)

where d, dp, and Fw are the densities (kg/m3) of the

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Figure 14. Overall flowchart of NSGA-II-mJG (similar to that of NSGA-II or NSGA-II-JG).

solids, the mass-percent solids in the tailing, and the density of pure water, respectively. Appendix B. Details of the Binary-Coded Elitist Nondominated Sorting Genetic Algorithm with the Modified Jumping Gene Operator (NSGA-II-mJG) The binary-coded algorithms NSGA-II and NSGA-IIJG have been described in considerable detail in refs 23, 26, 28, and 29 and are not described here. The overall flowchart of these two algorithms is similar to that for NSGA-II-mJG, as shown in Figure 14. Here, only a description of the mJG operation (to be performed on the Np daughter chromosomes in box D of Figure 14) is provided. The mJG operation (Figure 15) proceeds as follows. First, one starts with box D in Figure 14 (Np daughter chromosomes). Then, (1) A chromosome is chosen (sequentially) from box D. (2) A random number (r) is generated. If r e pmJG (a parameter that is provided as an input to the code), this chromosome is selected for the mJG operation. If not, go to Step 1 and select the next chromosome in box D (until all of the chromosomes in box D have been checked). If r e pmJG, generate another random number r and select the decision variable (from among N decision variables in the chromosome) that is to be changed (if i/N e r e (i + 1)/N, select the (i + 1)th decision variable). (3) Yet another random number r is generated. If r e p11...1 (another parameter), all of the binaries associated

Figure 15. Detailed flowchart of the mJG operation.

with the selected decision variable in the chromosome are replaced by ones. If r g p11...1, all such binaries are replaced by zeros. These steps are repeated until all the chromosomes in box D have been processed. The procedure in Figure 14 then is continued. Literature Cited (1) Gaudin, A. M. Flotation; McGraw-Hill: New York, 1957. (2) Mehrotra, S. P.; Kapur, P. C. Optimal-Suboptimal Synthesis and Design of Flotation Circuits. Sep. Sci. 1974, 9, 167. (3) Karnopp, D. Random Search Techniques for Optimization Problems. Automatica 1963, 1, 111.

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2633 (4) Gurin, L. S. Random Search in the Presence of Noise. Eng. Cybern. (Engl. Transl.) 1966, 2, 252. (5) Lee, E. S. Random Search Technique. In Quasilinearization and Invariant Imbedding; Mathematics in Science and Engineering, Vol. 41; Academic Press: New York, 1968; p 95. (6) Umeda, T.; Hirai, A.; Ichikawa, A. Synthesis of Optimal Processing System by an Integrated Approach. Chem. Eng. Sci. 1972, 27, 795. (7) Umeda, T.; Ichikawa, A. A. Modified Complex Method for Optimization. Ind. Eng. Chem., Process Des. Dev. 1971, 10, 229. (8) Green, J. C. A. The Optimization of Flotation Networks. Int. J. Miner. Process. 1984, 13, 83. (9) Reuter, M. A.; van Deventer, J. S. The Use of Linear Programming in the Optimal Design of Flotation Circuits Incorporating Regrinding Mills. Int. J Miner. Process. 1990, 28, 15. (10) Luss, R.; Jaakola, T. H. I. Optimization by Direct Search and Systematic Reduction of the Size of Search Region. AIChE J. 1973, 19 (9), 760. (11) Dey, A.; Kapur, P. C.; Mehrotra, S. P. A Search Strategy for Optimization of Flotation Circuits. Int. J. Miner. Process. 1989, 26, 73. (12) Yingling, J. C. Parameter and Configuration Optimization of Flotation Circuits, I. A Review of Prior Work. Int. J. Miner. Process. 1993, 38, 21. (13) Mehrotra, S. P. Design of Optimal CircuitssA Review. Min. Metall. Process. J. 1988, 5, 142. (14) Yingling, J. C. Parameter and Configuration Optimization of Flotation Circuits. II. A New Approach. Int. J. Miner. Process. 1993, 38, 41. (15) Loveday, B. K.; Brouckaert, C. J. An Analysis of Flotation Circuit Design Principles. Chem. Eng. J. 1995, 59, 15. (16) Grassi, F.; Schena, G. D. A General Expression for the Global Recovery of a Chain of Reactors. Int. J. Miner. Process. 1995, 43, 137. (17) Schena, G. D.; Villeneuve, J.; Noel, Y. A Method for Financially Efficient Design of Cell-Based Flotation Circuits. Int. J. Miner. Process. 1995, 46, 1. (18) Schena, G. D.; Zanin, M.; Chiarandini, A. Procedures for the Automatic Design of Flotation Networks. Int. J. Miner. Process. 1997, 52, 137. (19) Abu-Ali, M. H.; Abdel Sabour, S. A. Optimizing the Design of Flotation Circuits: An Economic Approach. Miner. Eng. 2003, 16, 55. (20) Holland, J. H. Adaptation in Natural and Artificial Systems; University of Michigan Press: Ann Arbor, MI, 1975. (21) Goldberg, D. E. Genetic Algorithms in Search, Optimization and Machine Learning; Addison-Wesley: Reading, MA, 1989. (22) Deb, K. Optimization for Engineering Design: Algorithms and Examples; Prentice Hall of India: New Delhi, 1995. (23) Deb, K. Multi-objective Optimization Using Evolutionary Algorithms. Wiley: Chichester, U.K., 2001. (24) Coello Coello, C. A.; van Veldhuizen, D. A.; Lamont, G. B. Evolutionary Algorithms for Solving Multi-Objective Problems; Kluwer: New York, 2002. (25) Srinivas, N.; Deb, K. Multiobjective Optimization Using Nondominated Sorting in Genetic Algorithms. Evol. Comput. 1994, 2, 221. (26) Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Trans. Evol. Comput. 2002, 6, 182. (27) Bhaskar, V.; Gupta, S. K.; Ray, A. K. Applications of Multiobjective Optimization in Chemical Engineering. Rev. Chem. Eng. 2000, 16, 1.

(28) Kasat, R. B.; Kunzru, D.; Saraf, D. N.; Gupta, S. K. Multiobjective Optimization of Industrial FCC Units Using Elitist Nondominated Sorting Genetic Algorithm. Ind. Eng. Chem. Res. 2002, 41, 4765. (29) Nandasana, A. D.; Ray, A. K.; Gupta, S. K. Dynamic Model of an Industrial Steam Reformer and Its Use for Multiobjective Optimization. Ind. Eng. Chem. Res. 2003, 42, 4028. (30) Kasat, R. B.; Gupta, S. K. Multi-objective Optimization of an Industrial Fluidized-Bed Catalytic Cracking Unit (FCCU) Using Genetic Algorithm (GA) with the Jumping Genes Operator. Comput. Chem. Eng. 2003, 27, 1785. (31) Fichera, M. A.; Chudacek, M. W. Batch Cell Flotation ModelssA Review. Miner. Eng. 1992, 5, 41. (32) Kapur, P. C.; Mehrotra, S. P. Modelling of Flotation Kinetics and Design of Optimum Flotation Circuits. In Challenges in Mineral Processing; Fuerstenau, D. W., Ed.; AIME: New York, 1989; p 300. (33) Imaizumi, T.; Inoue, T. Kinetic Consideration of Froth Flotation. In Mineral Processing, Proceedings of the 6th International Mineral Processing Congress, Cannes; Roberts, A., Ed.; Pergamon: New York, 1963; p 581. (34) Inoue, T.; Imaizumi, T. Some Aspects of Flotation Kinetics. In 8th International Mineral Processing Congress, Leningrad; Pergamon: New York, 1968; p 8. (35) Woodburn, E. T.; Loveday, B. K. The Effect of Variable Residence Time on the Performance of a Flotation System. J. S. Afr. Inst. Min. Metall. 1965, 65, 612; Discussion, 1966, 66, 649. (36) Loveday, B. K. Analysis of Froth Flotation Kinetics. Trans. Inst. Miner. Met. 1966, 75, C219. (37) Harris, C. C.; Chakravarti, A. Semi-Batch Froth Flotation Kinetics: Species Distribution Analysis. Trans. Soc. Min. Eng., AIME 1970, 247, 162. (38) Kapur, P. C.; Mehrotra, S. P. A Phenomenological Model for Flotation Kinetics. Trans. Inst. Miner. Met. 1973, 82, C229. (39) Jowett, A. Investigation of Residence Time of Fluid in Froth Flotation Cells. Br. Chem. Eng. 1961, 6, 254. (40) Niemi, A.; Paakkinen, U. Simulation and Control of Froth Flotation Circuits. Automation 1969, 5, 551. (41) Arbiter, N.; Harris, C. C. Flotation Kinetics. In Froth Flotation, 50th Anniversary; Fuerstenau, D. W., Ed.; AIME: New York, 1962; p 215. (42) Ray, A. K.; Gupta, S. K., Mathematical Methods in Chemical and Environmental Engineering, 2nd Ed.; Thomson Learning: Singapore, 2004. (43) Guria, C.; Verma, M.; Gupta, S. K.; Mehrotra, S. P. Multiobjective Optimization of Froth Circuits for Mineral Processing using the Jumping Gene Adaptations of Genetic Algorithm. Submitted to Int. J. Miner. Process., 2004. (44) Chankong, V.; Haimes, Y. Y. Multiobjective Decision Making: Theory and Methodology; Elsevier: New York, 1983. (45) Tarafder, A.; Rangaiah, G. P.; Ray, A. K. Multiobjective Optimization of an Industrial Styrene Monomer Manufacturing Process. Chem. Eng. Sci., in press. (46) Sareen, R.; Gupta, S. K. Multiobjective Optimization of an Industrial Semibatch Nylon 6 Reactor. J. Appl. Polym. Sci. 1995, 58, 2357.

Received for review April 13, 2004 Revised manuscript received August 26, 2004 Accepted September 9, 2004 IE049706I