Received 20 September 2005; received in revised form 24 January 2006; accepted 24 January 2006 Available online 6 March 2006

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1 Int. J. Miner. Process. 79 (06) Simultaneous optimization of the performance of flotation circuits and their simplification using the jumping gene adaptations of genetic algorithm-ii: More complex problems Chandan Guria a, Mohan Varma b, Surya P. Mehrotra c,1, Santosh K. Gupta d, a Department of Polymer Engineering, Birla Institute of Technology, Ranchi, Mesra , India b Department of Space Engineering and Rocketry, Birla Institute of Technology, Ranchi, Mesra , India c National Metallurgical Laboratory, Jamshedpur , India d Department of Chemical Engineering, Indian Institute of Technology, Kanpur 8 016, India Received September 05; received in revised form 24 January 06; accepted 24 January 06 Available online 6 March 06 Abstract The binary-coded elitist non-dominated sorting genetic algorithm with the modified jumping gene operator (NSGA-II-mJG) is used to obtain global optimal solutions of flotation circuits. Several single-objective and multi-objective optimization problems are solved using the interconnecting cell linkage parameters (fraction flow rates) and the mean cell residence times as the decision variables. In the single-objective problem, the overall recovery of the concentrate stream is maximized for a desired grade of the concentrate. Two two-objective optimization problems are then solved. In one, the number of non-linking streams and the overall recovery of the concentrate are maximized simultaneously. This gives several simple circuits in a systematic manner with only marginally lower recoveries. In the other two-objective optimization problem, the overall recovery of the concentrate is maximized while the total cell volume is minimized. A three-objective problem (maximization of the overall recovery of the concentrate, maximization of the number of non-linking streams and minimization of the total cell volume) is then solved. All the problems constrain the grade of the product to lie at a fixed value. Finally, a complex and computationally intensive four-objective optimization problem is solved. The solution of several practical optimization problems in this study helps develop useful insights into the optimal solutions. 06 Published by Elsevier B.V. Keywords: froth flotation; mineral processing; circuit/network optimization; global optimal; jumping gene; genetic algorithm; multi-objective optimization 1. Introduction Corresponding author. Tel.: , ; fax: addresses: guria@iitk.ac.in (C. Guria), varma_mohan@hotmail.com (M. Varma), spm26@sify.com (S.P. Mehrotra), skgupta@iitk.ac.in (S.K. Gupta). 1 On leave from: Department of Metallurgical Engineering, Indian Institute of Technology, Kanpur 8 016, India. Optimal design of flotation plants depends on the quality of the raw ores and the desired final products and involves (i) the selection of appropriate flotation circuits, (ii) the sizing of the banks as well as the cell units, and (iii) fixing of the operating parameters. Optimal selection of flotation circuits means finding the /$ - see front matter 06 Published by Elsevier B.V. doi:.16/j.minpro

2 150 C. Guria et al. / Int. J. Miner. Process. 79 (06) interconnections of several banks of flotation cells with one or more recycle streams such that one obtains the best possible performance. Significant progress has been made in the optimization of flotation networks. These are summarized by Mehrotra (1988), Kapur and Mehrotra (1989), Yingling (1993) and Loveday and Brouckaert (1995). Guria et al. (05b; Part I of this study) have provided a recent and detailed review of work in this area, as well as a reasonably complete set of references. These details are, therefore, not repeated here. They, themselves, have developed a newer and more robust algorithm and then used it to carry out the multi-objective optimization of flotation circuits. This is an adaptation of the artificial-intelligence based genetic algorithm (GA; Holland, 1975; Goldberg, 1989; Deb, 01; Coello Coello et al., 02). They were able to obtain not only the optimal operating conditions, but were also able to simplify the circuits (having fewer connecting streams) in a systematic manner using multiple-objectives. They used the jumping gene (JG) adaptations of the elitist non-dominated sorting genetic algorithm (NSGA-II; Deb, 01). These include NSGA-II-JG (Kasat and Gupta, 03) and NSGA-II-mJG (Guria et al., 05a,b). Of these, NSGA-II-mJG was found to work best for circuitoriented problems since it converged most rapidly to the global optimal solutions. This is because the optimal solutions may have some decision variables lying exactly at their lower or upper bounds (e.g., mass fractions equal to zero or unity) and this can be achieved most efficiently in NSGA-II-mJG. Guria et al. (05a) obtained optimal solutions that were superior to those obtained using traditional techniques when applied to the relatively simple two-species, two-cell single-objective optimization problem (Mehrotra and Kapur, 1974; maximizing the overall recovery of the concentrate while ensuring a grade of 75% for the concentrate, for a prescribed flotation cell volume of m 3 ). This was because the earlier techniques converged to local optima. The earlier paper of Guria et al. (05b) focused primarily on the technique and only a few problems were solved to illustrate its applicability. As a result of this, much insight could not be developed for the optimal solutions of flotation circuits. Nor were the effects of varying the different input (given) variables too clear. In this study, NSGA-II-mJG is used to optimize several additional and more complex single and multi-objective flotation problems of practical interest, so that these problems are overcome. Pareto optimal solutions are obtained for the more meaningful multi-objective optimization problems, from which a decision maker can select a preferred solution, using his/her intuition and experience. 2. Flotation model Mehrotra and Padmanabhan (1990) have reviewed the kinetics of flotation. A simple macroscopic (lumpedparameter) froth flotation model with distributed δf1 β41 β31 β21 β11 δ11 δf2 δf3 δf4 β 40 β β 43 β30 42 β44 β β 32 β33 β34 β β22 β23 β24 C2 C3 β β β C1 δ12 δ13 δ14 T1 T2 T3 T4 δ21 δ δ22 δ23 δ24 δ31 δ32 δ δ33 δ34 δ41 δ42 δ 43 δ30 δ44 δ40 Fig. 1. Generalized configuration for a four-cell circuit.

3 C. Guria et al. / Int. J. Miner. Process. 79 (06) Table 1 Details of the flotation problems studied for different grades of feed Problem no Number of cells, m Constraints: grade of product, G d,% composition, 15: : 65 (Good grade) 4: 16: 80 (Intermediate grade) 4: 2: 94 (Poor grade) x j (val:mid:gang), % w 1 :w 2 :w 3 0.8:0.3: :0.3: :0.3:0.05 Grade of feed, % n=3 (Valuable, Middling and Gangue); M F =11.34 kg min 1 ; K val =1.0 min 1 ; K mid =0.1 min 1 ; K gang =0.01 min 1 ;0 λ i (min) ; 0 β, δ and δ F 1; d t =3500 kg/m 3 ; ρ w =00 kg/m 3 ; ρ c =850 kg/m 3 ; d p =35%. flotation rate constants (Mehrotra and Kapur, 1974; Dey et al., 1989) is used for optimization, as in our previous paper (Guria et al., 05b). Assuming perfectly mixed continuous flotation cells with constant average flotation rate constants of the individual species, one can obtain expressions for the overall recovery (M c ) and the grade (G) of the final concentrate stream (mixture from all cells) in terms of the flow rate (M F ) of the finely divided solid feed (ore) to the flotation circuit, the feed composition (x j ), the flotation rate constants (K j ), the mean individual cell residence times (λ i ), the fractional flows (β's and δ's) in the interconnecting cell-linkage streams, and the total number of cells (m). M c and G are defined as M c ¼ and G ¼ X m X n C j;i b i0 i¼1 j¼1 X n M F;j j¼1 X m X n i¼1 j¼1 X m X n i¼1 j¼1 C j;i b i0 w j C j;i b i0 ð1þ ð2þ In Eqs. (1) and (2), C ji, β i0, M Fi and w j are the mass flow rate of the jth species in the concentrate from the ith cell, the fraction of the concentrate stream leaving the circuit from the ith cell, the feed flow rate to the ith cell, and the mass fraction of the valuable mineral associated with the jth species locked in the concentrate stream, respectively. The grade of any single stream can be written using Eq. (2) as P n j¼1 y j w j = P n j¼1 y j ¼ P nj¼1 y j w j, where y j is the mass fraction of the jth species in the stream. The values of C ji (and T ji, the flow rate of the jth species in the tailings from the ith cell) for a given problem can be solved by using the model equations (Mehrotra and Kapur, 1974; Dey et al., 1989) given in paper-i (Guria et al., 05b). The volume of the individual flotation cells is given by X n 1 V i ¼ k i T ji d þ ð0 d pþ þ k X i n C ji ð3þ d t d p q w d c j¼1 Here, d t, d p, d c and ρ w are the density (kg/m 3 ) of the tailing, the mass percent solids in the tailing, the density Table 2 Values of the best set of computational parameters for the SOO problem Problem no. l substr p c p m p jump For all the problems: N max,gen =,000; N p =50; l chrom =32; p 11 1 =0.9; Random seed= ; H 1 = 8. j¼1

4 152 C. Guria et al. / Int. J. Miner. Process. 79 (06) of the concentrate and of pure water, respectively. The first term in Eq. (3) represents the volume occupied by the tailings, while the second term corresponds to the volume occupied by the concentrate. The latter is less than about % and was neglected in Paper 1 of this study. 3. Formulation of the optimization problems A general flotation circuit (involving four cells) is shown in Fig. 1. We consider three species (n = 3), namely, the valuable (val), middling (mid) and the gangue (gang), to be present in the raw ground ore. We start with the solution of a simple, single-objective optimization (SOO) problem, where the overall recovery (M c ) of the concentrate is maximized for a specified (desired) value (G d ) of the grade. The mathematical description of this problem is given in Appendix A (Eqs. (A1.a) (A1.f )). Penalty functions, as described by Guria et al. (05b), are used for handling the constraints. In general, one would also like to get the most simplified flotation circuit for ease of control and operation, but the solution of this SOO problem does not address this point. For this, one needs to formulate a multi-objective optimization (MOO) problem where one of the objectives is to maximize the number (NLS) of non-linking streams (missing streams that have no flow). Problem MOO-A in the Appendix considers two-objective functions: the overall recovery (M c )as well as the number of non-linking streams are maximized simultaneously while constraining the grade to lie at a specified value (G d ). A detailed mathematical description of this MOO-A problem is given in Appendix A (Eqs. (A2.a) (A2.c)). It is to be noted that the maximum number of non-linking streams depends on the number of flotation cells (NLS max for 2, 3 and 4 cells are 9, and 35, respectively). In these two problems, the cell volumes (Eq. (3)) are calculated using the optimal values of the mean cell residence times (decision variables) and the flow rates of the tailings and the concentrate, and there is no control on their values. This could lead to excessively high hold-up volumes. One may be interested, alternatively, Table 3 Optimal solutions for the SOO problems Problem/ case no. Grade of feed, % Grade of product (G d ), % Number of cells (m) Optimal recovery (M c ) Mean cell residence times (λ), min Cell 1 Cell 2 Cell 3 Cell Total volume (V), m 3

5 C. Guria et al. / Int. J. Miner. Process. 79 (06) to maximize the recovery and minimize the total cell volume (V), for a desired value (G d ) of the grade. No requirement is made on the number of non-linking streams. The MOO-B problem in the Appendix does precisely this. Eqs. (A3.a) (A3.c) describe this problem mathematically. If one also wants to obtain the most simplified circuit simultaneously, one needs to solve a three-objective optimization problem. The MOO-C problem (Eqs. (A4.a) (A4.d)) in the Appendix maximizes the overall recovery (M c ), minimizes the total cell volume (V) and maximizes the number of non-linking streams (NLS), simultaneously while achieving a desired value (G d ) of the grade. The MOO-D problem (Eqs. (A5.a) (A5.e)) is then solved. This involves four objectives: maximization of the overall recovery (M c ), the number of non-linking streams (NLS), and the grade (G) and minimization of the total cell volume (V). This represents the most complex of the multi-objective optimization problems. decreases. Similarly, the total volume, V, of the cells reduces with decreasing values of m (and lower values of the recovery). Increasing the grade (G d ) of the product reduces the overall recovery (counteracting influence). These observations are consistent with those observed in industrial practice, but are quantified here. The two-objective MOO-A problem described in the Appendix is solved next to explore how simplified circuits can be obtained, though at a cost of reduced recovery. The best values of the computational parameters are given in Table 4 for the twenty eight problems. Fig. 2 shows the Pareto optimal solutions for Problems 1 9 (the remaining plots are not being provided for reasons of brevity, but can be supplied on request). Symbols, as for example, 3C_G35_ F1565_w831, indicate three cells, G d =35.0%, feed with x j =15%, % and 65%, and w j =0.8, 0.3 and 0.1. It may be noted that NLS can have only integral values, 4. Results and discussion Several standard preliminary tests (Kasat and Gupta, 03; Guria et al., 05a) are made on the NSGA-IImJG code so as to ensure that it is free of errors. The code is then used to solve the optimization problems for several values of the input variables (givens) so as to see their effect on the solutions. One important input variable is the quality (or grade) of the feed. We consider three different feeds having good (24.5%), intermediate (12.0%) and poor (8.5%) grades. Again, for any given quality of feed, the first four optimization problems (SOO, MOO-A to MOO-C) are solved for three different values of G d, and for each of these possibilities, three different values of m are considered. Thus, twenty seven problems are solved for each of the first four optimization problems. One additional problem is solved only for the MOO-A optimization problem for a poor grade feed (8.5%) with very high quality of product (G d = 75.0%). Table 1 gives the details of the different cases studied, including the values of w j. This potpourri of solutions enables us to develop considerable insights into the optimal solutions. The last problem, MOO-D, is solved for the feed of intermediate quality for only one case, viz., m=4. The SOO problem is first solved using NSGA-IImJG. The best set of values of the several computational parameters (that gives the highest values of the recovery) is obtained by trial and error. These are given in Table 2. The optimal solutions for this problem are given Table 3. It is observed that for a given feed and G d, the optimal recoveries reduce as the number of cells Table 4 Values of the best set of computational parameters for the MOO-A problems Problem/case no. l substr p c p m p jump For all the problems: N max,gen =,000; N p =50; l chrom =32; p 11 1 =0.9; Random seed= ; H 1 = 8.

6 154 C. Guria et al. / Int. J. Miner. Process. 79 (06) and so there are jumps in the Pareto solutions. Interestingly, it is observed (for all the 28 cases) that NLS falls sharply after some stage, for only a marginal increase in the recovery. This allows one to operate the circuits with reasonably high recoveries, while still using simple circuits. A possible preferred solution from among these Pareto solutions could be those with the simplest circuit (highest value of NLS), without focusing on the recovery (which will, obviously, be the lowest, unless several solutions exist with the same value of NLS, in which case the point with the highest recovery is selected). This point is indicated by a closed square in each of the nine Pareto sets in Fig. 2. Fig. 3 shows the circuits corresponding to these selected points for Problems 1 6. The circuits for the remaining problems can be supplied on request. The mean residence times of the individual cells are given inside the boxes representing them, while the grades of the individual streams are written in parentheses next to them. The corresponding overall recovery (M c ) and the Number of non-linking streams A 4C_G35_F1565_w831 Number of non-linking streams C_G35_F1565_w831 B Number of non-linking streams C_G35_F1565_w831 C Number of non-linking streams Recovery % C_G55_F1565_w831 D Number of non-linking streams Recovery % Recovery % E Problems 1-3 3C_G55_F1565_w831 Number of non-linking streams F 2C_G55_F1565_w831 Number of non-linking streams Recovery % 38 4C_G75_F1565_w G Number of non-linking streams Recovery % Problems C _G75_F1565_w H Number of non-linking streams Recovery % 2C_G75_F1565_w I Recovery % Recovery % Recovery % Problems 7-9 Fig. 2. Pareto optimal solutions for the MOO-A problem for Cases 1 9 intable 1.

7 C. Guria et al. / Int. J. Miner. Process. 79 (06) total cell volume (V) are also mentioned. It is known that the cells in a flotation circuit are referred to as rougher (s), cleaner(s), re-cleaner(s), and scavenger(s) cells, depending on their structural and operational parameters. The feed is normally introduced in a rougher cell where crude separation is effected. The concentrate is then re-floated in one or more cleaner/re-cleaner cells to improve its grade. Similarly, the tailings stream from the rougher stage is re-floated in a scavenger cell so as to extract, as far as possible, the residual mineral from the gangue before it leaves the circuit. In Fig. 3, the cells are indicated as rougher (R), cleaner (C) or scavenger (S) cells. The following general observations can be made from the results (of all the 28 problems): 1. There is only a single feed entry (always to the rougher cell) to the circuit, irrespective of the values of x j, G d, w j and m. 2. For given values of x j, G d, and w j, the total cell volume decreases with a decrease in the number of cells at the cost of decreased M c. β 30(41.33) β (19.48) β 43(12.77) (24.50) 3(R) (17.79) β 12(11.22) 2(S 1) (16.) δ 41(.30) 1(S 3) (17.33) 4(S 2) (18.32) δ 32(12.78) δ 24(.99) M c = % Cell volume = m 3 Problem 1 δ (.13) δ 44[0.46] β 30(41.70) β (19.76) β 23(12.21) (24.50) 3(R) (15.48) 1(S 1) (17.02) 2(S 2) (14.06) δ 31(12.79) δ 12(11.01) M c = % Cell volume = m 3 δ 22[0.76] Problem 2 δ (.23) β (42.95) β (18.17) (24.50) 1(R) (19.98) 2(S) (16.76) δ 12(12.98) M c = % Cell volume = m 3 Problem 3 δ 22[0.44] δ (.81) Fig. 3. Circuits corresponding to the points indicated in Fig. 2 (MOO-A problem; Eq. (A2)) for Problems 1 6 (Table 1).

8 156 C. Guria et al. / Int. J. Miner. Process. 79 (06) β (55.0) β 24(.38) β 41(39.53) β 31(25.89) (24.50) 4(R) (16.03) 1(C) (3.99) (S 1) (8.78) (S 2) (9.69) δ 43(15.37) δ 14(27.01) δ 32(13.08) M c = % Cell volume = m 3 δ (11.79) Problem 4 β 23(41.06) β (55.0) β 21(22.96) (24.50) 2(R) (14.71) 3(S) (17.19) 1(C) (9.48) δ 23(14.73) δ 30(11.98) M c = % Cell volume = m 3 δ 12(25.) Problem 5 β (55.0) β 21(41.03) (24.50) 2(R) (19.99) 1(C) (16.25) δ 12(.64) M c = % Cell volume = m 3 δ (13.18) Problem 6 Fig. 3 (continued). 3. For a given number of cells and the quality of feed, an increase in G d leads to a decrease in the overall recovery, with the total cell volume decreasing. We now analyze the simplified circuits shown in Fig. 3 further. When starting with a good grade feed and when the required product (concentrate) grade is only 35.0% (e.g., G d =35.0%), the optimal simplified circuit configuration for the cases of 4-, 3- and 2-cells (Fig. 3, Problems 1 3) does not include cleaner cells for further improvement of the product quality (Dey et al., 1989). For the same grade of feed when the required G d is 55.0% (Fig. 3, Problems 4 6), the optimal simplified circuit configurations for the 4-cell and the 3-cell cases involve a combination of rougher, cleaner and scavenger (R C S) cells, i.e., (R C S S) for the 4-cell and (R C S) for the 3-cell configuration, while only a rougher cleaner (R C) combination is used (and not R S) for the 2-cell case. Meloy (1983) and Dey et al. (1989) also suggest the use of R C S circuits for maximum separation for intermediate values of G d. For a still higher concentrate grade (e.g., G d =75.0%) and starting with a good grade feed, most of the circuits (Problems 7 9, not shown in Fig. 3) are found to comprise of

9 C. Guria et al. / Int. J. Miner. Process. 79 (06) essentially rougher cleaner scavenger (R C S) combinations (Dey et al., 1989), except for the 2-cell problem, which consists of only an R C combination. However, when the starting feed is of poor grade (see Table 1, Problem 28) but the required concentrate grade is high (e.g., G d =75.0%), it is noted that the circuit configuration is shifted from R C S StoR C C Sin a 4-cell circuit (Fig. 4). Addition of cleaner cells in the flotation circuit reflects the difficulty in achieving higher separations (grades). Some additional details of the Pareto solutions for the MOO-A problem for the three feeds are summarized in Table 5. Here, the column for the minimum (best) M c refers to solutions corresponding to the points indicated in Fig. 2 (highest attainable recovery with the simplest circuit), while the entry for the maximum M c refers to the right-most points in Fig. 2. Table 5 shows that for the good-grade feed, as the complexity of the flotation circuits increases (i.e., lower values of NLS), the overall recovery increases, albeit slightly, but the total cell volume increases by a reasonable amount. Such multiobjective optimization problems can, thus, help plant personnel in selecting the best (preferred) designs/ operating conditions. Table 5 suggests that for the intermediate-quality feed (grade of feed=12.0%), it is not possible to obtain very high quality (G d ) product with reasonable values of the recovery. Hence, the values of G d have to be lowered. For this feed and with G d =25%, it is found (detailed results not shown) that all simplified circuits consist of rougher scavenger (R S) cells with no recycle (Dey et al., 1989). When G d is taken as 35.0%, the optimal circuits are combinations of rougher cleaner scavenger (R C S) cells for the 4- and 3-cell cases, while for 2-cell circuits, there is no cleaner cell. When G d is taken at the still higher value of 55.0%, the 4- cell circuit consists of a rougher cleaner scavenger (R C S) configuration with more cleaner cells, whereas for the 3-cell problem there is no scavenger cell. The presence of more cleaner cells reflects the difficulty of separation for feeds of lower grade. For the 2-cell β 34(54.75) β 13(27.25) β 40(75.0) β 23(38.74) (24.50) 3(R) (8.24) δ 32(21.12) 2(S 1) (3.97) δ 21(17.51) 1(S 2) (9.90) 4(C) (0.40) δ (14.50) δ 43 (50.74) M c = % Cell volume = m 3 Problem 7 β 24(11.99) β 30(75.0) (8.50) 1(R) (4.45) β 12(38.57) 2(C 1) (13.45) β 23(61.63) 3(C 2) (2.60) 4(S) (16.91) δ 14(6.56) δ 21(15.00) δ 32(46.01) δ 40(5.46) M c = % Cell volume = m 3 Problem 28 Fig. 4. Circuit configurations for product grade of 75.0% starting with good grade (24.5%) and poor grade (8.5%) feeds, respectively.

10 158 C. Guria et al. / Int. J. Miner. Process. 79 (06) Table 5 Results of the MOO-A problem for feeds having grades of 24.5%, 12.0% and 8.5% Prob. no Grade of feed (%) Product grade (%) No. of cells Minimum (best) M c Maximum M c Recovery (%) Cell volume (m 3 ) NLS Recovery (%) Cell volume (m 3 ) NLS problem, the circuit is an R S combination with a selfrecycle stream from the concentrate from the scavenging cell, indicating a non-trivial circuit. The presence of the self-recycle stream helps increase the separation, and so enables the attainment of G d. Qualitatively similar results are observed as the grade of the feed becomes even poorer (e.g., for grade = 8.50%). The most simplified circuits (not shown) are found to comprise of R C S configurations for G d = 25% for the 4- and 3-cell cases, and R S circuits for the 2-cell case. It is found that as G d increases, the number of cleaner cells increases for the 4-cell problem. Results for the MOO-B problem (minimum V with maximum M c for a specified G d ) are shown in Fig. 5, for a feed having an intermediate grade of 12.0%. The values of the best computational parameters for these nine problems are given in Table 6. Interestingly, after some value of M c, further marginal increases require considerable increases in the total volume of the cells. This information is extremely useful to a decision maker, since he/she can select an appropriate point on the Pareto set. Results of the MOO-B problem are summarized in Table 7. Here, the maximum value of M c corresponds to the highest obtainable recovery (in Fig. 5) while the minimum (best) M c corresponds to the recovery where the slope of the Pareto set in Fig. 5 changes sharply (a good operating point). All the circuits for this problem are quite complex (several streams). The three-objective problem, MOO-C, is solved for a feed having an intermediate grade of 12.0%. The values of the best computational parameters are also given in Table 6 for the intermediate quality feed and for Problems 18. The Pareto optimal surfaces for three problems (Problems 12) are shown in Fig. 6 (the complete set can be supplied on request). It is observed (for all nine cases) that the Pareto surface shoots up (high V) at some stage (at low NLS). This is associated with only marginal increases in M c. This is similar to what happens for the 2-objective Problems,

11 C. Guria et al. / Int. J. Miner. Process. 79 (06) C_G25_F41680_w C_G25_F41680_w C_G25_F41680_w8305 Cell volume, V (m 3 ) Cell volume, V (m 3 ) Cell volume, V (m 3 ) Cell volume, V (m 3 ) Recovery, M c (%) 3.0 4C_G35_F41680_w Recovery, M c (%) 1.0 4C_G55_F41680_w8305 Cell volume, V (m 3 ) Recovery, M c (%) Recovery, M c (%) Problems C_G35_F41680_w8305 2C_G35_F41680_w Recovery, M c (%) Recovery, M c (%) Problems C_G55_F41680_w8305 2C_G55_F41680_w8305 Cell volume, V (m 3 ) Cell volume, V (m 3 ) Cell volume, V (m 3 ) Cell volume, V (m 3 ) Recovery, M c (%) Recovery, M c (%) Recovery, M c (%) Problems Fig. 5. Pareto optimal frontiers (MOO-B; Eq. (A3)) for intermediate grade feed (12.0%). G d =25.0%, 35.0% and 55.0% and m=4, 3, 2. MOO-A and MOO-B (the qualitative similarity of Figs. 5 and 6 is to be noted). The preferred solution can be selected quite easily from the Pareto surfaces. The arrows on the three Pareto surfaces in Fig. 6 indicate a good compromise among the most simplified circuit, with the maximum recovery and the minimum cell volume. The corresponding simplified circuits are shown in Fig. 7 for the three cases of Fig. 6. Here, for low values of G d of 25.0%, the circuit is mostly an R C S combination (except the 2-cell problem, where the circuit is an R S combination). Similar circuit configurations are also obtained (not shown) for the other two higher values of G d. Though, both the MOO-A and MOO-C problems give similar circuit configurations, MOO-C gives lower recoveries and cell volumes as compared to MOO-A. In order to see the variation of the

12 160 C. Guria et al. / Int. J. Miner. Process. 79 (06) Table 6 Values of the best computational parameters for the MOO-B and MOO-C problems for feed having a grade of 12.0% Problem l substr MOO-B MOO-C no. p c p jump p m p jump For all the problems: N max,gen =,000; N p =50; l chrom =32; p 11 1 =0.9; Random seed=0.1234; H 1 = 8 ; p m =0.01 (MOO-B); p c =0.98 (MOO-C). recovery with the number of non-linking streams and the total cell volume, we (somewhat arbitrarily) locate two points in Fig. 6 Problem, the first point corresponds to the minimum value of NLS (most complex circuit), while the second corresponds to the maximum value of V. The circuits for these two points are shown in Fig. 8. Higher total volumes give higher recoveries (with intermediate complexities of the circuit), while the most complex circuits lead to lower volumes and recoveries. After solving the relatively simple two-objective optimization problems and building up our insights as well as computational skills, we solved a very general, four-objective problem, involving all four of the important objectives, Problem MOO-D (for m = 4). Fig. 9 shows the results of this problem with the best set of parameters (that give the smoothest results: N max,gen = 00; N p =50; l chrom =32; l substr =1536; p c =0.978; p m =0.001; p jump =0.01; p 11 1 =0.90; Random seed= ). The results cannot be shown graphically in an easy form (as in the two-objective problems), and so another approach is taken to present them. We rearrange the results so that M c increases continuously with the chromosome number and plot each of the objectives as a function of the (new) chromosome number. The results are shown in Fig. 9. This is a more complex problem, and much higher scatter is observed, particularly in the plots of NLS (a discrete variable) and V. Still, some trends can be observed. The grade and the volume worsen almost continuously as M c improves (hence the results comprise a Pareto set). What is interesting is that the total volume does not increase sharply at some stage, and so there is no obvious choice of a preferred solution, as in the earlier two-objective problems. One could possibly select a preferred solution from among this set of Pareto solutions. Table 7 Results of the MOO-B problem for feed having a grade of 12.0% Prob. no Product grade, G d (%) No. of cells Minimum (best) M c a Maximum M c Recovery (%) Cell volume (m 3 ) NLS Recovery (%) Cell volume (m 3 ) NLS a At point of change in slope in Fig. 5.

13 C. Guria et al. / Int. J. Miner. Process. 79 (06) C-G25-F41680-w8305 Total cell volume, m No. of non-linking streams Problem Recovery, (M c) % C-G25-F41680-w Total cell volume, m No. of non-linking streams Problem Recovery, (M c) % C-G25-F41680-w8305 Total cell volume, m No. of non-linking streams Problem Recovery, (M c) % Fig. 6. Pareto optimal surface (MOO-C; Eq. (A4)) for intermediate grade feed (12.0%) and G d =25.0% (m=4, 3 and 2). Chromosome no. 28 corresponds to a simple circuit (with NLS=35) with a reasonably high value of M c of 38.56%, a reasonable grade of 22.54%, and a total volume, which is not too large, of 1.85m 3. The corresponding circuit/detailed solution is given in Fig.. These results illustrate the power of the technique

14 162 C. Guria et al. / Int. J. Miner. Process. 79 (06) β (13.57) β 30 (30.47) β 40 (50.) 1(S 2 /C) (6.75) β 13 (.01) 2(S 3 ) (14.37) 3(S 1 ) (5.05) 4(R) (0.81) (12.0) δ 12(7.45) δ 31 (8.90) δ (6.19) δ 43 (.56) Mc = % Cell volume = m 3 Problem β 12 (57.21) β (70.49) β 30 (24.46) (12.0) 1(R) (0.08) 2(C) (7.52) 3(S) (19.68) δ 13 (11.81) δ 21 (31.59) δ 30 (7.04) Mc = % Cell volume = m 3 Problem 11 β (18.35) β (25.84) 2(S) (2.79) 1(R) (17.40) (12.0) δ (6.67) Mc = % Cell volume = 0.44 m 3 δ 12 (7.19) Problem 12 Fig. 7. Simplified circuits for the points corresponding to the arrows in Fig. 6. used and the range of interesting and important problems that can be solved by GA. 5. Conclusions The elitist, binary-coded, non-dominated sorting genetic algorithm with the modified jumping gene operator, NSGA-II-mJG, is a powerful AI-based technique for obtaining global optimal solutions for flotation circuits. A few single-, two-, three- and fourobjective optimization problems have been solved for different values of the input variables. Optimal solutions and Pareto sets are obtained for single and multiobjective optimization problems, respectively. The results obtained are consistent with intuitive expectations, but quantify them. Multi-objective optimization problems in which one objective is to maximize the number of non-linking streams give solutions having the most simplified circuits, even though they are associated with (marginally) lower values of the overall recovery. The Pareto solutions provide a set of several equally good (non-dominated) solutions to a decision

15 C. Guria et al. / Int. J. Miner. Process. 79 (06) (a) δ F1[0.35] δf2[0.03] δf4 β40[0.37] β42[0.23] β43[0.36] β 30 β 44 β31[0.19] β β δ δ12[0.08] δ13[0.02] δ14[0.02] δ21[0.13] δ[0.69] δ 22 δ23[0.25] δ24[0.55] δ31[0.36] δ32[0.03] δ[0.04] δ33[0.13] δ 34 δ41[0.30] δ42[0.21] δ30[0.29] δ 44 M c = % Cell volume = 0.29 m 3 NLS = 13 (minimum) δ40[0.37] (b) δ F1 β 31[0.14] β 30[0.28] β 21 β β 32 β δ12 δ 13[0.35] δ 21[0.06] δ22[0.47] δ 23 δ 14[0.35] δ 31[0.08] δ 33[0.30] δ 34 M c = % Cell volume = 3.36 m 3 NLS = 25 δ 30[0.08] δ 44 δ 40[0.] Fig. 8. Optimal circuits at the (a) lowest value of NLS and (b) the largest value of V, for the Pareto surface in Fig. 6, Problem. maker from which one can select the preferred solution, using intuition/experience. Symbols C ji d c d p d t d w mass flow rate of the jth species in the concentrate from the ith cell, kg min 1 density of the concentrate, kg m 3 mass percent solids (loading) in the tailings density of the tailings, kg m 3 density of water, kg m 3 f i ith objective function G d grade of the (overall) concentrate stream, percent H i penalty parameters (Guria et al., 05b) K i specific flotation rate constant of the ith species (same in each cell), min 1 l chrom total number of binaries in a chromosome l substr number of binaries characterizing each decision variable m number of cells in the circuit

16 164 C. Guria et al. / Int. J. Miner. Process. 79 (06) Recovery, Mc (%) Total cell volume, V, m 3 0 4C_F41680_w Chromosome number 5 4C_F41680_w Grade, G (%) No of non-linking streams (NLS) 40 4C_F41680_w Chromosome number 0 4C_F41680_w Chromosome number Chromosome number Fig. 9. Pareto optimal solutions for the MOO-D problem (Eq. (A5)) for intermediate grade feed (12.0%), m=4. M C M F M F,j overall recovery of the concentrate stream, percent mass flow rate of total solids in the fresh feed, kg min 1 mass flow rate of the jth species in the feed, kg min 1 n number of species in the feed n number of decision variables N max,gen maximum number of generations N p number of chromosomes in the population crossover probability p c p jump jumping gene probability p m mutation probability p probability T ji mass flow rate of the jth species in the tailings stream from the ith cell, kg min 1 V i volume of the ith cell, m 3 V total volume of m cells (¼ P m i¼1 V i ), m 3 w j mass fraction of the valuable mineral in the jth species (same for all cells) x j mass fraction of the jth species in the feed mass fraction of the jth species in a stream y j β 41(7.88) (12.0) 1(R) (.0) β 21(11.62) β (22.54) δ 12(7.08) 2(S1) (.0) β 32(6.22) δ 23(5.78) 3(C) (.0) δ 30(5.13) 4(S 2) δ 43(5.32) Recovery = % Cell volume = 1.85 m 3 Grade = % NLS = 35 Fig.. Simplified circuit for chromosome no. 28 in Fig. 9.

17 C. Guria et al. / Int. J. Miner. Process. 79 (06) Greek letters β ij fraction of concentrate outlet stream from cell i, going to cell j δ ij fraction of tailings outlet stream from cell i, going to cell j δ Fi fraction of fresh feed going to cell i λ i residence time in cell i, min density of water, kg m 3 ρ w Subscripts, superscripts d desired value F feed gang gangue materials L lower bound mid middling materials o stream leaving the circuit (not recycled) U upper bound val valuable materials. Acknowledgement Partial financial support from the Department of Science and Technology, Government of India, New Delhi [through grant III-5(13)/01-ET] is gratefully acknowledged. Bounds on the decision variables: 0Vb; dv1:0 ða1:eþ k i;l Vk i Vk U i ; i ¼ 1; 2; N ; m ða1:fþ A2. Multi-objective optimization (MOO-A) problem maxf 1 ðb; d; d F ; kþum c ða2:aþ maxf 2 ðb; d; d F ; kþunls ða2:bþ subject to Eqs: ða1:bþ ða1:f Þ ða2:cþ A3. MOO-B maxf 1 ðb; d; d F ; kþum c ða3:aþ minf 2 ðb; d; d F ; kþuv ða3:bþ subject to Eqs: ða1:bþ ða1:f Þ ða4:cþ A4. MOO-C maxf 1 ðb; d; d F ; kþum c ða4:aþ Appendix A. Optimization problems solved A1. Single-objective optimization (SOO) problem maxf 1 ðb; d; d F ; kþum c ða1:aþ subject to (s. t.): Constraints: maxf 2 ðb; d; d F ; kþunls minf 3 ðb; d; d F ; kþuv subject to Eqs: ða1:bþ ða1:f Þ A5. MOO-D (m=4) ða4:bþ ða4:cþ ða4:dþ G ¼ G d ða1:bþ maxf 1 ðb; d; d F ; kþum c ða5:aþ Model equations (Guria et al., 05b) ða1:cþ maxf 2 ðb; d; d F ; kþunls ða5:bþ Model constraints 2 (Guria et al., 05b) ða1:dþ minf 3 ðb; d; d F ; kþuv ða5:cþ X m 2 Model constraints (Guria et al., 05b): i¼1 X m i¼1 X m i¼1 d Fi ¼ 1:0 d ki þ d k0 ¼ 1:0; k ¼ 1; 2; N ; m ðbþ b ki þ b k0 ¼ 1:0; k ¼ 1; 2; N ; m ðcþ These represent the fact that for any stream being split, the sum of the flows after the split is the same as that before the split. These are accommodated in the optimization code as described by Guria et al. (05b). ðaþ maxf 4 ðb; d; d F ; kþug ða5:dþ subject to Eqs: ða1:cþ ða1:f Þ ða5:eþ Decision variables: b T u½b k0 ; b ki Š; k ¼ 1; 2; N ; m; i ¼ 1; 2; N ; m d T u½d k0 ; d ki Š; k ¼ 1; 2; N ; m; i ¼ 1; 2; N ; m d T F u½d FiŠ; i ¼ 1; 2; N ; m k T u½k i Š; i ¼ 1; 2; N ; m ða6þ

18 166 C. Guria et al. / Int. J. Miner. Process. 79 (06) M F Input (specified) variables: x j ; K j ; w j m References ðj ¼ 1; 2; N ; nþ ða7þ Coello Coello, C.A., van Veldhuizen, D.A., Lamont, G.B., 02. Evolutionary Algorithms for Solving Multi-Objective Problems. Kluwer, New York. Deb, K., 01. Multi-Objective Optimization Using Evolutionary Algorithms. Wiley, Chichester, UK. Dey, A.K., Kapur, P.C., Mehrotra, S.P., A search strategy for optimization of flotation circuits. Int. J. Miner. Process. 26, Goldberg, D.E., Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading, MA. Guria, C., Verma, M., Mehrotra, S.P., Gupta, S.K., 05a. Multiobjective optimal synthesis and design of froth flotation circuits for mineral processing using adaptation of genetic algorithm. Ind. Eng. Chem. Res. 44, Guria, C., Verma, M., Gupta, S.K., Mehrotra, S.P., 05b. Simultaneous optimization of the performance of flotation circuits and their simplification using the jumping gene adaptations of genetic algorithm. Int. J. Miner. Process. 77, Holland, J.H., Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor, MI. Kapur, P.C., Mehrotra, S.P., Modeling of flotation kinetics and design of optimum flotation circuits. In: Fuerstenau, D.D. (Ed.), Challenges in Mineral Processing. AIME, pp Kasat, R., Gupta, S.K., 03. Multi-objective optimization of an industrial fluidized-bed catalytic cracking unit (FCCU) using genetic algorithm (GA) with jumping gene operator. Comput. Chem. Eng. 27, Loveday, B.K., Brouckaert, C.J., An analysis of flotation circuit design principles. Chem. Eng. J. 59, Mehrotra, S.P., Design of optimal circuits a review. Min. Met. Proc. J. 5, Mehrotra, S.P., Kapur, P.C., Optimal suboptimal synthesis and design of flotation circuits. Sep. Sci. 9, Mehrotra, S.P., Padmanabhan, N.P.H., Flotation kinetics a review. Trans. Indian Inst. Met. 43, Meloy, T.P., Optimization for grade of profit in mineral processing circuits circuit analysis. Int. J. Miner. Process. 11, Yingling, J.C., Parameter and configuration optimization of flotation network, I. A review of prior work. Int. J. Min. Process. 38,