Multiobjective Optimisation of Flotation Variables Using Controlled-NSGA-II and Paretosearch

Abstract

Finding the optimum operating points for the maximisation of flotation recovery and concentrate grade can be a very difficult task, owing to the inverse relationship that exists between these two key performance indicators. For this reason, techniques that can accurately find the trade-off are critical for flotation process optimisation. This work extracted well-assessed Gaussian process predictive functions as objective functions for a comparative multiobjective optimisation study using the paretosearch algorithm (PA) and the controlled elitist non-dominated sorting genetic algorithm (controlled-NSGA-II). The main aim was the concomitant maximisation of the copper recovery and the concentrate grade. Comparison of the two applied techniques revealed that the PA discovered the best set of the pareto-optimal solution for both the recovery (93.4%) and concentrate-grade (17.4 wt.%) maximisation.

1. Introduction

Flotation modelling has received much attention since the early 1930s and can be classified into different categories based on its characteristics and usage. For instance Gharai and Venugopal [1] classified flotation models into macro-scale and micro-scale based on the overall process and subprocesses occurring in a flotation system. According to their classification, the micro-scale looks at the chemical and physical correlation among subprocesses in a flotation cell, while the macro-scale is the simplification and combination of a number of micro-scale models for the prediction of the overall behaviour of a flotation cell or even banks [2]. Another classification performed by Quintanilla, Neethling [3] distinguished flotation models into models for simulation purposes and models for control purposes. Despite the numerous models developed under the various classifications highlighted above, froth flotation modelling has always been a difficult task due to the complex nonlinear dynamic nature of the various process variables. The complex nonlinear relationships existing among the flotation variables have presented significant reliability and utilisation challenges for traditional first-principle models (e.g., first-order kinetic approach), which may involve some assumptions [4,5,6,7,8].

With the inability of conventional models like empirical or phenomenological models to fully capture these complex nonlinear relationships, attention has been drawn to modern day machine learning (supervised and unsupervised) algorithms in modelling froth flotation outcomes. Machine learning algorithms, if properly trained with a high-quality data set, have the ability to capture variable relationships for predicting the outcome of a future unseen data set. Commonly applied machine learning algorithms in the modelling of the metallurgical performance of froth flotation include fuzzy logic, support vector machine, decision trees, principal component regression, random forest, and artificial neural network [9,10,11,12,13,14,15]. In recent times, application of a probabilistic Gaussian process regression (GPR) algorithm has acquired much attention over most nonlinear modelling techniques [16]. The successful application of the GPR algorithm in the field of particle separation includes recent work where the flotation recovery was predicted from model-selected flotation variables [17,18]. Another success story of the GPR algorithm in the field of minerals engineering was its application by Patel, Gorai [19] in predicting the iron ore grade from image analysis. The main advantage of the GPR algorithm is its ability to include prior knowledge of the noise in the measured data in the model [20]. Previous publications have compared different machine learning model performances; hence, this will not be repeated here [21,22,23].

Once a machine learning algorithm is well-trained with good prediction accuracy, its predictive function can be used for process optimisation [24]. In machine learning process optimisation, optimal operating points of process variables are ascertained for peak performance after modelling the key performance indicators with other process variables [25,26]. Over the years, several optimisation algorithms including particle swarm optimisation, ant colony algorithm, gravitational search algorithm, evolutionary algorithm, genetic algorithm, simulated annealing, surrogate optimisation, and tabu search have been developed, with regard to recent emerging advanced computational techniques [27]. While some optimisation problems may just involve one goal, some real-life situations like froth flotation require more than one goal to be achieved simultaneously, and therefore, there is a need for multiobjective optimisation (MO) techniques. MO is an enterprising area of research, which has gained much attention in many scientific and engineering fields [28]. MO allows decisionmakers to plan and make decisions with multiple conflicting or sometimes nonconflicting goals.

The conventional approach for solving MO problems has been the use of scalarisation techniques, converting MO to single-objective problems. With these single objective-based techniques, only a single solution is found, leaving multiobjective problems partially solved and incomplete. Several multiobjective evolutionary algorithms (MOEAs) such as the non-dominated sorting genetic algorithm (NSGA) address this problem; however, it suffers from a lack of elitism, high computational complexity, and the requirement for a sharing-parameter ${\sigma }_{share}$ specification. As a result, an improved, elitist-NSGA (NSGA-II) was proposed and shown to perform better than other elitist MOEAs (e.g., pareto archive evolution strategy (PAES) and strength pareto evolution algorithm (SPEA)) in finding diverse solutions and in converging close to the true (actual) pareto-optimal set. These results encourage the application of NSGA-II to more complex real-life MO problems [29,30]. Irrespective of the numerous applications of NSGA-II, it had the issue of premature convergence to a suboptimal solution set, and as such, the proposal of the controlled-NSGA-II comes with controlled elitism for better convergence performance. Another powerful MO technique is the paretosearch algorithm (PA). PA was applied by Gantayet and Dheer [31] in finding the optimal planning strategy for a radial distribution network. Mišo, Branko [32] also applied PA for the multiobjective calibration in a gas–metal arc welding process simulation [33].

In this work, a comparative MO study has been carried out on various rougher flotation variables using the PA and controlled-NSGA-II following a Gaussian process prediction of both copper recovery and the copper concentrate grade. The work aimed at finding the optimum operating points of various rougher flotation variables for the maximisation of both the copper recovery and the concentrate copper grade. PA and controlled-NSGA-II have not been applied in any flotation variables optimisation study previously. A paucity of information exists on the use of the predictive functions of a GPR model as the objective functions for the multiobjective optimisation of industrial flotation variables. Another unique thing about this work is that effort has been made to select the best pareto-optimal solution, a difficult decision to make in MO problems.

2. Research Methodology

This section specifically captures the data collection and pre-processing, predictive model development and assessment, theoretical overview of MO algorithms, and finally, optimisation modelling. A detailed theoretical overview of the GPR algorithm has been presented elsewhere and will not be repeated here [17,20]. MATLAB R2020b (64-bit version) was used for the various algorithm implementations.

2.1. Data Collection and Pre-Processing

A total of 80,000 data observations were collected from the copper mineral (e.g., chalcopyrite and bornite) flotation circuit, specifically the rougher circuit. The extracted data include variables such as ore feed grade, mill throughput, hydrocyclone overflow particle percent passing the 75 µm size, xanthate dosage (tank 1 and 4), frother dosage (tank 1 and 4), airflow rate (tank 1, 2, 3, 4, and 5), and froth depth (tank 1, 2, and 4). These flotation variables are the main variables that are known to enhance both the rougher copper recovery and grade and whose optimisation was of much interest to the metallurgist. All the variables are monitored directly and independently except for the rougher copper recovery, which is estimated from onstream analyser (OSA) readings from rougher feed, cleaner concentrate, and scavenger tails using Equation (1). Furthermore, Ehrig, McPhie [34] have provided more details about the deposit mineralisation. Details of the rougher flotation circuit under consideration have been previously presented. The data preprocessing are consistent with the previous publications, including domain knowledge for outlier removal and data standardisation [17].

$$\mathrm{Recovery}{R}_i=\left({\displaystyle \frac{c_i}{f_i}}\right)\left({\displaystyle \frac{f_{i-t_i}}{c_{i-t_i}}}\right)\times 100\%,i=\mathrm{1,2},3,\dots$$

(1)

,

where

$c_i=$ $i$th concentrate grade;

$f_i=$ $i$th feed grade;

$t_i=$ $i$th tails grade.

The essence of the standardisation is to reduce the impact of the variables with high variance and of different scales or magnitude on the model prediction outcomes. For confidentiality reasons, only standardised results are presented in this work. Equation (2) shows the equation for standardisation. Figure 1 is a visualisation of the variation in the output variables and some input variables.

$$Z_i={\displaystyle \frac{a_i-{\overline{a}}_i}{{\sigma }_a}},i=\mathrm{1,2},3,\dots$$

(2)

,

where

$Z_i=$ transformed observation;

$a_i=$ observation of variable $a$;

${\overline{a}}_i=$ mean of variable $a$;

${\sigma }_i=$ standard deviation of variable $a$.

2.2. Model Development and Evaluation

For the GPR predictive model development, a portion of the pre-processed data set (30,000 observations) was sampled randomly from 74,633 observations for model training and validation owing to the expensive computational time accompanying a large data set. The required size of the sampled data set was ascertained from an empirical study, which revealed the saturation of the model performance after 30,000 observations. Furthermore, the sampling was repeated until the sample and population statistics (mean, median, minimum, and maximum values) difference were below 5%. Following this, the sampled data set was randomly partitioned into 80% training data (24,000 observations) and 20% validation data (6000 observations), based on the holdout cross-validation technique [35]. It must be noted that the partitioning ratio does not follow a particular rule, but generally, the training data should completely capture the population data. The remaining portion of the pre-processed data set (44,663 observations) served as the testing data set. In the GPR algorithm, the main hyperparameter that affects the performance of the algorithm is the covariance function. In this regard, this work utilised four covariance functions (rational quadratic, squared exponential, exponential, and matern 3/2) in the modelling of each output variable. The mathematical expressions of these covariance functions are shown in Equations (3)–(6).

i. Squared exponential covariance function

$$k\left(x_i,x_j\right)={\sigma }_f^2\mathrm{e}\mathrm{x}\mathrm{p}\left[{\displaystyle \frac{{-d}^2}{{2l}^2}}\right]$$

(3)

;

ii. Rational quadratic covariance function

$$k\left(x_i,x_j\right)={\sigma }_f^2\mathrm{e}\mathrm{x}\mathrm{p}{\left[1+{\displaystyle \frac{d^2}{{2\alpha l}^2}}\right]}^{-\alpha }$$

(4)

;

iii. Exponential covariance function

$$k\left(x_i,x_j\right)={\sigma }_f^2\mathrm{e}\mathrm{x}\mathrm{p}\left[{\displaystyle \frac{-d}{l}}\right]$$

(5)

;

iv. Matern 3/2 covariance function

$$k\left(x_i,x_j\right)={\sigma }_f^2\left(1+{\displaystyle \frac{\sqrt{3d}}{l}}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{\sqrt{3d}}{l}}\right)$$

(6)

;

where the Euclidean distance between $x_{i}and{x}_j,d=\sqrt{\left({x_i-x_j)}^{\prime }\right(x_i-x_j)}$, ${\sigma }_f^2$ is the signal variance of the function, $\alpha$ is the shape parameter for the rational quadratic covariance function, and $l$ is the length scale. In previous research using the GPR model on the input variables, the percent particle size passing 75 µm variable contributed more than the other variables [17].

Having prior knowledge about the outcome (recovery and grade) of a flotation process will continue to be of great interest to the process engineer. The accuracy of the various developed GPR models were assessed by the mean absolute percentage error (MAPE), the correlation coefficient ($r$), the relative root mean square error (RRMSE), and the relative error correction (REC). The various mathematical notations are presented in Equations (7)–(10). For a good performance under the following indicators, the models’ $r$ values should approach 1, and the MAPE and RRMSE values should be close to zero with REC as close to 100% as possible.

$$r={\displaystyle \frac{\sum _{i=1}^n{(y}_i-\overline{y})-(\widehat{y_i}-\overline{\widehat{y}})}{\sum _{i=1}^n{(y}_i-\overline{y}{)}^2\times \sqrt{\sum _{i=1}^n({\widehat{y}}_i-\overline{\widehat{y}}{)}^2}}}$$

(7)

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=\left({\displaystyle \frac{1}{n}}\right){\sum }_{i=1}^n{\displaystyle \frac{|y_i-{\widehat{y}}_i|}{y_i}}\times 100\%$$

(8)

$$\mathrm{R}\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}={\displaystyle \frac{\sqrt{\left(\frac{1}{n}\right)\sum _{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2}}{\frac{1}{n}\sum _{i=1}^n\widehat{y_i}}}$$

(9)

$$\mathrm{R}\mathrm{E}\mathrm{C}=100-\left({\displaystyle \frac{100}{n}}{\sum }_{i=1}^n{\displaystyle \frac{|y_i-{\widehat{y}}_i|}{y_i}}\right)$$

(10)

,

where

$y_i=$ $i$th true copper recovery or copper-concentrate grade value;

$\overline{y}=$ mean of true copper recovery or copper-concentrate grade values;

${\widehat{y}}_i=$ $i$th predicted copper recovery or copper-concentrate grade value;

$\overline{\widehat{y}}=$ mean of predicted copper recovery or copper-concentrate grade value;

$n=$ total number of observations.

2.3. Optimisation Modelling

Problem Formulation

Using the predictive functions of the best performing copper recovery and concentrate grade models as the objective functions, the multiobjective optimisation of various rougher flotation variables that maximises both the copper recovery and concentrate grade were carried out using the controlled-NSGA-II and PA. The problem formulation is shown in Equation (11). A theoretical overview of the pareto search algorithm and the controlled-NSGA-II has been previously reported; hence, it is not repeated here [36,37] Custódio, Madeira [38].

$$\left\{\begin{array}{c}Maximise\left(Copperrecovery\right)\\ Maximise\left(Copperconcentrategrade\right)\end{array}\right.$$

(11)

such that

1.6 w.t%$\le feed grade\le$ 2.6 w.t%;

78$.0\mathrm{\%}\le Feed particle size\le$ 84.0%;

350.0 t/h $\le Throughput\le$ 900.0 t/h;

37.0 mL/min $\le Xanthate to tank cell 1\le$ 190.5 mL/min;

27.8 mL/min $\le Xanthate to tank cell 4\le$ 142.9 mL/min;

28.1 mL/min $\le Frother to tank cells 1 and 4\le$180.7 mL/min;

900.0 m³/h $\le Airflow to tank cell 1\le$ 1250.0 m³/h;

1100.0 m³/h $\le Airflow to tank cells 2,3,4 and 5\le$ 1250.0 m³/h;

75.0 mm $\le Frothdepthoftankcell1,2,3,4and5\le$ 150.0 mm.

These boundary constraints were developed based on the domain knowledge of the process conditions.

3. Results and Discussion

In this section, results from the various techniques are presented and discussed. In Section 3.1, discussion of the different GPR models from different covariance functions is presented. Section 3.2 details the comparative results of the MO algorithms.

3.1. Predictive Models’ Performance

In the prediction of the copper recovery and concentrate grade, the GPR models with four covariance functions were developed assuming the two output variables were independent of each other. The prediction of each output variable was evaluated by considering the predicted and actual values. This evaluation was accurately carried out using $r$, MAPE, RRMSE, and REC criteria. The summary results of the developed models are shown in

Table 1. Summary of the GPR models’ performance for copper recovery prediction.

Training Data Set
GPR Covariance Function	$\mathit{r}$	MAPE (%)	RRMSE	REC (%)
Squared exponential	0.95	0.04	0.03	99.80
Rational quadratic	0.99	0.02	0.01	99.98
Exponential	0.97	0.02	0.02	99.96
Matern 3/2	0.95	0.03	0.04	99.78
Validation Data Set
GPR Covariance Function	$\mathit{r}$	MAPE (%)	RRMSE	REC (%)
Squared exponential	0.94	0.29	0.46	83.83
Rational quadratic	0.96	0.24	0.38	85.47
Exponential	0.96	0.28	0.42	84.28
Matern 3/2	0.95	0.30	0.46	83.80
Testing Data Set
GPR Covariance Function	$\mathit{r}$	MAPE (%)	RRMSE	REC (%)
Squared exponential	0.95	0.33	0.42	84.10
Rational quadratic	0.97	0.26	0.37	85.39
Exponential	0.96	0.30	0.44	83.98
Matern 3/2	0.97	0.33	0.43	83.97

.

Considering Table 1, the GPR model from the rational quadratic covariance function had a relatively better performance than all the other models and was therefore considered to be the best model for the prediction of rougher copper recovery. Correlation coefficient values of 0.99, 0.96, and 0.97 were observed, respectively, for the training data, validation data, and testing data when the rational quadratic covariance function was used. These values show a clear indication of the strong relationship between the actual and predicted recovery values under the rational quadratic covariance function.

For error statistics, the MAPE and RRMSE were considered in this work. The trained GPR model developed with rational quadratic covariance function had MAPE values of 0.02%, 0.24%, and 0.26% when rougher copper recovery was predicted using the training, validation, and testing data sets, respectively. On average, these recorded MAPE values indicate that GPR model developed with the rational quadratic covariance function was distant by 0.02%, 0.24%, and 0.26% from the actual training, validation, and testing recovery values, respectively. With regard to the MAPE, the GPR model developed with the rational quadratic covariance function was able to explain 99.98%, 99.76%, and 99.74% of the total useful recovery information in the training, validation, and testing data, respectively. The low MAPE values of the rational quadratic covariance function shows its higher prediction accuracy than all the other investigated covariance functions in predicting rougher copper recovery.

In terms of the RRMSE, the results of the various GPR models shown in Table 1 indicate that all the models had a very satisfactory rougher copper recovery prediction. However, the rational quadratic covariance function had the best performance obtaining RRMSE values of 0.01 (training data), 0.38 (validation data), and 0.37 (testing data).

In general, the REC reveals the percent closeness of the data to the best fit line. From Table 1, the GPR model developed with the rational quadratic covariance function recorded the highest REC values of 99.98% (training data), 85.47% (validation data), and 85.39% (testing data), confirming their close association with their actual copper recovery values. This close association obtained by the GPR model developed with the rational quadratic covariance function is visualised in Figure 2. Comparing these results to previous work, as highlighted in Section 2.1, it can be seen that similar results were obtained even though different year ranges of the data set were considered in each case.

The prediction of the concentrate grade was also determined utilising the same GPR covariance functions highlighted above. The performances of the various GPR models are shown in

Table 2. Summary of the models’ performance for the prediction of the rougher copper concentrate grade.

Training Data Set
GPR Covariance Function	$\mathit{r}$	MAPE (%)	RRMSE	REC (%)
Squared exponential	0.96	0.02	0.02	99.95
Rational quadratic	0.98	0.02	0.01	99.97
Exponential	0.97	0.02	0.01	99.97
Matern 3/2	0.99	0.01	0.01	99.99
Validation Data Set
GPR Covariance Function	$\mathit{r}$	MAPE (%)	RRMSE	REC (%)
Squared exponential	0.97	0.19	0.29	91.81
Rational quadratic	0.98	0.18	0.27	91.86
Exponential	0.97	0.19	0.28	91.82
Matern 3/2	0.98	0.17	0.25	92.92
Testing Data Set
GPR Covariance Function	$\mathit{r}$	MAPE (%)	RRMSE	REC (%)
Squared exponential	0.97	0.23	0.28	92.20
Rational quadratic	0.97	0.21	0.29	91.95
Exponential	0.98	0.19	0.28	92.56
Matern 3/2	0.99	0.16	0.26	92.99

. It is evident that the development of the GPR model using the matern 3/2 covariance function marginally outperformed the other investigated models making it the best model in the prediction of the rougher concentrate copper grade. Referring to Table 2, the $r$ values > 0.97, the MAPE values < 0.18%, the RRMSE < 0.27, and the REC values > 92% were recorded for the matern 3/2 covariance model. The covariance functions’ copper recovery and concentrate grade predictive performance differences are due to the fact that during the transformation of the data set into higher dimensional space for the unique discrimination of the labels, the transformations conducted by these covariance functions (rational quadratic for rougher copper recovery and Matern 3/2 for rougher copper concentrate grade) had a better higher dimensional space correlation structure between the output variables and input variables as compared to the other covariance functions investigated in this work.

Furthermore, comparing the results in Table 1 and Table 2 generally shows that the overall concentrate grade prediction accuracy is marginally better than that of the rougher copper recovery. This is expected because as compared to the rougher copper recovery, the rougher copper concentrate grade had a lower range of values giving it more instances per label and therefore an easier characterisation. Just like the recovery prediction, Figure 3 shows the closeness between actual and predicted concentrate grade under the GPR matern 3/2 covariance model.

Based on the above discussion, the best performing predictive function in each instance was extracted as the objective function for the flotation variables optimisation that would maximise both the rougher copper recovery and the rougher copper concentrate grade.

3.2. Best Multiobjective Algorithm Selection

The optimisation capability of the PA and controlled-NSGA-II has been evaluated. A total of 70 sets of pareto-optimal solutions were investigated for each algorithm. Out of these, four pareto-optimal solutions were further explored for each algorithm based on a rougher copper recovery constraint of at least 93% at a satisfactory grade. Figure 4 visualises the optimal solutions found by the PA and controlled-NSGA-II.

From Figure 4, it is evident that the optimal solutions from the PA and controlled-NSGA-II are very similar. However, comparing the two algorithms, the pareto-optimal solutions found by the PA lies marginally above that of the controlled-NSGA-II, making it a better technique for this work. As such, only the pareto-optimal solutions found by the PA will be further discussed.

Selection of the Best Solution

The PA optimal solutions are shown in

Table 3. Pareto-optimal solutions found by PA. * Froth depth of tank cells 2 and 4 represent tank cells 3 and 5, respectively.

Variable		Solution 1	Solution 2	Solution 3	Solution 4
Flotation feed grade (wt.%)		2.52	1.60	2.52	2.52
Percent particle passing 75 µm		78.00	80.22	78.00	78.00
Mill throughput (t/h)		600.00	670.56	679.69	679.69
Xanthate dosage in tanks 1 and 4 (ml/min)	1	69.11	66.98	190.50	69.11
	4	29.93	27.80	29.93	29.93
Frother dosage in tanks 1 and 4 (ml/min)	1	40.94	38.81	40.94	40.94
	4	28.10	28.10	28.10	28.10
Air flow rate in tanks 1 to 5 (m3/h)	1	1215.63	1213.49	1215.63	900.00
	2	1195.00	1192.87	1195.00	1195.00
	3	1165.63	1163.49	1165.63	1165.63
	4	1212.50	1210.37	1212.50	1212.50
	5	1100.00	1100.00	1100.00	1100.00
Froth depth in tanks 1 to 5 (mm)	1	150.00	150.00	150.00	150.00
	2/3 *	135.35	133.22	135.35	135.35
	4/5 *	150.00	150.00	150.00	150.00
Copper recovery (%)		93.72	93.84	93.40	93.41
Concentrate copper grade (wt.%)		16.77	15.41	18.16	17.40

. The results from Table 3 clearly indicate that there is an inverse relationship between the rougher copper recovery and the rougher copper concentrate grade. In other words, the conditions that maximise the rougher copper recovery are the same conditions that minimise the rougher copper concentrate grade. Therefore, to select the best set of pareto-optimal solutions, there should be a proper trade-off between the rougher copper recovery and rougher copper concentrate grade, with an emphasis on variables that have a significant economic impact on the process. In this work, emphasis was placed on the throughput, xanthate dosage, feed particle size, and frother dosages.

From Table 3, solutions 1, 3, and 4 had a comparatively coarse grinding of 78% passing 75 µm making them better options over solution 2, with regard to the feed particle size. In terms of the throughput, solution 1 had the least tonnages of material treated per hour followed by solution 2, with solutions 3 and 4 recording the same throughput of almost 680 t/h. At this stage, it can be seen that the best set of solutions lies between solutions 3 and 4, as they have the same feed particle size and throughput. It is worth noting that despite the lower feed grade of solution 2, the recovery was subtly higher than the other solutions. The higher recovery value may be due to the corresponding finer grind, with the low feed grade also contributing to a lower concentrate grade relative to the other solutions [39,40]. With the equivalent performance of solutions 3 and 4, their reagent consumptions (xanthate and frother dosages) were assessed. Again, from Table 3, solutions 3 and 4 had the same reagent consumptions except for the xanthate consumption in tank cell 1, where solution 3 consumed almost thrice the quantity of xanthate of solution 4. With similar rougher copper recovery values of 93.40% and 93.41% attained by solutions 3 and 4, respectively, it can be observed that solution 3 had a higher rougher copper concentrate grade of 18.16 wt.% as compared to solution 4, which had 17.40 wt.%. However, the argument is that the difference in the rougher copper concentrate grade will not be enough to compensate for the xanthate cost, which may have contributed to the higher grade, if the processing plant is to be run with solution 3 [41,42,43,44]. Moreover, the concentrate obtained at the roughing stage will further undergo upgrade during the cleaning stage and that is why emphasis is often placed on the recovery over the concentrate grade during the roughing stage. Based on the above analysis, solution 4 was selected as the best set of pareto-optimal solutions in this work.

In as much as the optimum operating values have been determined for the various rougher flotation variables, variables like the throughput, feed grade, and feed particle size often vary simultaneously and are very difficult to maintain at a particular set point in a typical industrial operation. Two of these variables (feed grade and feed particle size) have been further analysed to ascertain the simultaneous impact of a varying feed grade and feed particle size on the copper recovery and concentrate grade. To accomplish this, 50 typical industrial observations of feed grade and feed particle size were generated within their specified constraint bounds shown in Equation (10), having all possible generated combinations of the initial feed grade and particle size observations to yield 2500 observations for the two variables. Now, varying the feed grade and particle size simultaneously, while maintaining the other variables at optimum operating points, the best performing predictive models were used to simulate both the rougher copper recovery and rougher copper concentrate grade. The simulation results are visualised in Figure 5. From Figure 5a, it is evident that an increasing feed grade with a decreasing feed particle size enhances the rougher copper recovery and vice versa. The increase in rougher copper recovery occurs as a result of the larger exposure of the copper mineral surface for reagent contact, with a finer grind. However, this phenomenon is not the same with the rougher copper concentrate grade, as shown in Figure 5b. From Figure 5b, an increasing feed grade with a decreasing feed particle size decreases the rougher copper concentrate grade and vice versa. The rougher copper concentrate grade decreases because fine grinding activates the surfaces of other gangue minerals leading to their floatability and subsequent dilution of the rougher copper concentrate grade. Whereas certain combinations of feed particle size and feed grade in their specified constraint bounds are likely to reduce the predicted copper recovery and concentrate grade, the grade and recovery values within the range of 15.70–17.40 wt.% and 93.36–93.48%, respectively, are still attainable when all the other variables are kept at their optimum points. With much emphasis on recovery during the roughing stage, the results affirm that a near optimal rougher copper recovery at a satisfactory grade is achievable within the specified feed grade and particle size constraint bounds when the other variables are kept at their ascertained optimum points.

4. Conclusions

In this work, Gaussian process regression (GPR) models developed under different covariance functions have been used in copper recovery and concentrate copper grade prediction. Following this, the predictive functions of the best models (covariance type) ascertained during the prediction of each output variable were extracted as the objective functions. Using these two objective functions subjected to some process constraints, the PA and controlled-NSGA-II were used to investigate the optimum variable conditions that would maximise both the copper recovery and concentrate copper grade. The main conclusions are as follows:

(1)

The model assessment showed good copper recovery and concentrate grade prediction from the flotation variables using the GPR model.

(2)

The rational quadratic covariance function worked best for copper recovery prediction, while the matern 3/2 covariance function worked best for the concentrate copper grade prediction.

(3)

The comparison between the PA and controlled-NSGA-II revealed that the PA, although the controlled-NSGA-II also worked very well, finds the best set of pareto-optimal solutions for both the copper recovery and concentrate grade maximisation.

(4)

Selecting a final optimal solution requires domain knowledge. For example, the optimal solution for the rougher flotation stage (focussed on maximising the throughput) may differ from that of the cleaner flotation stage (focus may be on higher product grades).

(5)

Further analysis of the best set of pareto-optimal solutions also revealed that a near optimal rougher copper recovery at a satisfactory grade is still attainable with varying feed grades and particle sizes when all the other flotation variables are kept at their optimum values.

The significance of this work is that the developed models could be used to simulate both copper recovery and concentrate in a mineral processing plant following similar conditions. Moreover, operating the rougher circuit at the suggested optimum points would result in a more stabilised flotation performance.