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doi: 10.3934/jimo.2021169
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Multi-objective chance-constrained blending optimization of zinc smelter under stochastic uncertainty

Yu Chen 1, , Yonggang Li 1, , Bei Sun 1,2,, , Chunhua Yang 1, and  Hongqiu Zhu 1,

1. 

School of Automation, Central South University, Changsha 410083, China
2. 

The Peng Cheng Laboratory, Shenzhen 518000, China

* Corresponding author: Bei Sun (sunbei@csu.edu.cn)

Received  May 2021 Revised  July 2021 Early access September 2021

Considering the uncertainty of zinc concentrates and the shortage of high-quality ore inventory, a multi-objective chance-constrained programming (MOCCP) is established for blending optimization. Firstly, the distribution characteristics of zinc concentrates are obtained by statistical methods and the normal distribution is truncated according to the actual industrial situation. Secondly, by minimizing the pessimistic value and maximizing the optimistic value of object function, a MOCCP is decomposed into a MiniMin and MaxiMax chance-constrained programming, which is easy to handle. Thirdly, a hybrid intelligent algorithm is presented to obtain the Pareto front. Then, the furnace condition of roasting process is established based on analytic hierarchy process, and a satisfactory solution is selected from Pareto solution according to expert rules. Finally, taking the production data as an example, the effectiveness and feasibility of this method are verified. Compared to traditional blending optimization, recommended model both can ensure that each component meets the needs of production probability, and adjust the confident level of each component. Compared with the distribution without truncation, the optimization results of this method are more in line with the actual situation.

Keywords: Uncertainty, multi-objective chance-constrained programming, blending problem, hybrid intelligent optimization algorithm, analytic hierarchy process.
Mathematics Subject Classification: 60A99, 49M37.
Citation: Yu Chen, Yonggang Li, Bei Sun, Chunhua Yang, Hongqiu Zhu. Multi-objective chance-constrained blending optimization of zinc smelter under stochastic uncertainty. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021169
References:
[1]

A. Chakraborty and M. Chakraborty, Multi criteria genetic algorithm for optimal blending of coal, Opsearch, 49 (2012), 386-399.  doi: 10.1007/s12597-012-0089-y.  Google Scholar

[2]

Y. Chen, Y. Li, B. Sun, Y. Li, H. Zhu and Z. Chen, A chance-constrained programming approach for a zinc hydrometallurgy blending problem under uncertainty, Computers & Chemical Engineering, 140 (2020), 106893.  doi: 10.1016/j.compchemeng.2020.106893.  Google Scholar

[3]

K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: Nsga-ii, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.  doi: 10.1109/4235.996017.  Google Scholar

[4]

P. H. Dos Santos, S. M. Neves, D. O. Sant'Anna, C. H. de Oliveira and H. D. Carvalho, The analytic hierarchy process supporting decision making for sustainable development: An overview of applications, Journal of Cleaner Production, 212 (2019), 119-138.  doi: 10.1016/j.jclepro.2018.11.270.  Google Scholar

[5]

F. D. Fomeni, A multi-objective optimization approach for the blending problem in the tea industry, International Journal of Production Economics, 205 (2018), 179-192.  doi: 10.1016/j.ijpe.2018.08.036.  Google Scholar

[6]

O. P. Hilmola, Role of inventory and assets in shareholder value creation, Expert Systems with Applications: X, 5 (2020), 100027.  doi: 10.1016/j.eswax.2020.100027.  Google Scholar

[7]

N. Hovakimyan, F. Nardi, A. Calise and N. Kim, Adaptive output feedback control of uncertain nonlinear systems using single-hidden-layer neural networks, IEEE Transactions on Neural Networks, 13 (2002), 1420-1431.  doi: 10.1109/TNN.2002.804289.  Google Scholar

[8]

Y. Ito, Representation of functions by superpositions of a step or sigmoid function and their applications to neural network theory, Neural Networks, 4 (1991), 385-394.  doi: 10.1016/0893-6080(91)90075-G.  Google Scholar

[9]

B. Liu and B. Liu, Theory and Practice of Uncertain Programming, volume 239, 2009., Springer. doi: 10.1007/978-3-540-89484-1.  Google Scholar

[10]

L. Ou, G. Luo, A. Ray, C. Li, H. Hu, S. Kelley and S. Park, Understanding the impacts of biomass blending on the uncertainty of hydrolyzed sugar yield from a stochastic perspective, ACS Sustainable Chemistry & Engineering, 6 (2018), 10851-10860.  doi: 10.1021/acssuschemeng.8b02150.  Google Scholar

[11]

Ü. S. Sakallı and Ö. F. Baykoç, Can the uncertainty in brass casting blending problem be managed? a probability/possibility transformation approach, Computers & Industrial Engineering, 61 (2011), 729-738.  doi: 10.1016/j.cie.2011.05.004.  Google Scholar

[12]

Ü. S. Sakallı and Ö. F. Baykoç, An optimization approach for brass casting blending problem under aletory and epistemic uncertainties, International Journal of Production Economics, 133 (2011), 708-718.  doi: 10.1016/j.ijpe.2011.05.022.  Google Scholar

[13]

Ü. S. Sakallı and Ö. F. Baykoç, Strong guidance on mitigating the effects of uncertainties in the brass casting blending problem: A hybrid optimization approach, Journal of the Operational Research Society, 64 (2013), 562-576.  doi: 10.1057/jors.2012.50.  Google Scholar

[14]

M. Savic, D. Nikolic, I. Mihajlovic, Z. Zivkovic, B. Bojanov and P. Djordjevic, Multi-criteria decision support system for optimal blending process in zinc production, Mineral Processing and Extractive Metallurgy Review, 36 (2015), 267-280.  doi: 10.1080/08827508.2014.962135.  Google Scholar

[15]

K. L. Schultz, D. C. Juran and J. W. Boudreau, The effects of low inventory on the development of productivity norms, Management Science, 45 (1999), 1664-1678.  doi: 10.1287/mnsc.45.12.1664.  Google Scholar

[16]

H. A. Taha, Operations Research an Introduction, The Macmillan Co., New York; Collier-Macmillian Ltd., London, 1971.  Google Scholar

[17]

O. S. Vaidya and S. Kumar, Analytic hierarchy process: An overview of applications, European Journal of Operational Research, 169 (2006), 1-29.  doi: 10.1016/j.ejor.2004.04.028.  Google Scholar

[18]

Y. Yang, P. Vayanos and P. I. Barton, Chance-constrained optimization for refinery blend planning under uncertainty, Industrial & Engineering Chemistry Research, 56 (2017), 12139-12150.  doi: 10.1021/acs.iecr.7b02434.  Google Scholar

show all references

References:
[1]

A. Chakraborty and M. Chakraborty, Multi criteria genetic algorithm for optimal blending of coal, Opsearch, 49 (2012), 386-399.  doi: 10.1007/s12597-012-0089-y.  Google Scholar

[2]

Y. Chen, Y. Li, B. Sun, Y. Li, H. Zhu and Z. Chen, A chance-constrained programming approach for a zinc hydrometallurgy blending problem under uncertainty, Computers & Chemical Engineering, 140 (2020), 106893.  doi: 10.1016/j.compchemeng.2020.106893.  Google Scholar

[3]

K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: Nsga-ii, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.  doi: 10.1109/4235.996017.  Google Scholar

[4]

P. H. Dos Santos, S. M. Neves, D. O. Sant'Anna, C. H. de Oliveira and H. D. Carvalho, The analytic hierarchy process supporting decision making for sustainable development: An overview of applications, Journal of Cleaner Production, 212 (2019), 119-138.  doi: 10.1016/j.jclepro.2018.11.270.  Google Scholar

[5]

F. D. Fomeni, A multi-objective optimization approach for the blending problem in the tea industry, International Journal of Production Economics, 205 (2018), 179-192.  doi: 10.1016/j.ijpe.2018.08.036.  Google Scholar

[6]

O. P. Hilmola, Role of inventory and assets in shareholder value creation, Expert Systems with Applications: X, 5 (2020), 100027.  doi: 10.1016/j.eswax.2020.100027.  Google Scholar

[7]

N. Hovakimyan, F. Nardi, A. Calise and N. Kim, Adaptive output feedback control of uncertain nonlinear systems using single-hidden-layer neural networks, IEEE Transactions on Neural Networks, 13 (2002), 1420-1431.  doi: 10.1109/TNN.2002.804289.  Google Scholar

[8]

Y. Ito, Representation of functions by superpositions of a step or sigmoid function and their applications to neural network theory, Neural Networks, 4 (1991), 385-394.  doi: 10.1016/0893-6080(91)90075-G.  Google Scholar

[9]

B. Liu and B. Liu, Theory and Practice of Uncertain Programming, volume 239, 2009., Springer. doi: 10.1007/978-3-540-89484-1.  Google Scholar

[10]

L. Ou, G. Luo, A. Ray, C. Li, H. Hu, S. Kelley and S. Park, Understanding the impacts of biomass blending on the uncertainty of hydrolyzed sugar yield from a stochastic perspective, ACS Sustainable Chemistry & Engineering, 6 (2018), 10851-10860.  doi: 10.1021/acssuschemeng.8b02150.  Google Scholar

[11]

Ü. S. Sakallı and Ö. F. Baykoç, Can the uncertainty in brass casting blending problem be managed? a probability/possibility transformation approach, Computers & Industrial Engineering, 61 (2011), 729-738.  doi: 10.1016/j.cie.2011.05.004.  Google Scholar

[12]

Ü. S. Sakallı and Ö. F. Baykoç, An optimization approach for brass casting blending problem under aletory and epistemic uncertainties, International Journal of Production Economics, 133 (2011), 708-718.  doi: 10.1016/j.ijpe.2011.05.022.  Google Scholar

[13]

Ü. S. Sakallı and Ö. F. Baykoç, Strong guidance on mitigating the effects of uncertainties in the brass casting blending problem: A hybrid optimization approach, Journal of the Operational Research Society, 64 (2013), 562-576.  doi: 10.1057/jors.2012.50.  Google Scholar

[14]

M. Savic, D. Nikolic, I. Mihajlovic, Z. Zivkovic, B. Bojanov and P. Djordjevic, Multi-criteria decision support system for optimal blending process in zinc production, Mineral Processing and Extractive Metallurgy Review, 36 (2015), 267-280.  doi: 10.1080/08827508.2014.962135.  Google Scholar

[15]

K. L. Schultz, D. C. Juran and J. W. Boudreau, The effects of low inventory on the development of productivity norms, Management Science, 45 (1999), 1664-1678.  doi: 10.1287/mnsc.45.12.1664.  Google Scholar

[16]

H. A. Taha, Operations Research an Introduction, The Macmillan Co., New York; Collier-Macmillian Ltd., London, 1971.  Google Scholar

[17]

O. S. Vaidya and S. Kumar, Analytic hierarchy process: An overview of applications, European Journal of Operational Research, 169 (2006), 1-29.  doi: 10.1016/j.ejor.2004.04.028.  Google Scholar

[18]

Y. Yang, P. Vayanos and P. I. Barton, Chance-constrained optimization for refinery blend planning under uncertainty, Industrial & Engineering Chemistry Research, 56 (2017), 12139-12150.  doi: 10.1021/acs.iecr.7b02434.  Google Scholar

Figure 1.  Blending process
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Figure 2.  Semi-underground warehouse
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Figure 3.  Block diagram of multi-objective chance-constrained blending optimization
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Figure 4.  Statistical diagram and distribution fitting diagram of each component($ a $: Truncated normal distribution for high-purity ore; $ b $: Lognormal distribution for high-purity ore; $ c $: Lognormal distribution for High-quality ore)
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Figure 5.  Optimization idea
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Figure 6.  Multi-objective hybrid intelligent optimization algorithm
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Figure 7.  Roasting process
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Figure 8.  Calciner temperature and feed rate (the upper is the calcination temperature and the lower is the feed amount of the thrower)
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Figure 9.  Evaluation mechanism of working condition
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Figure 10.  Pareto front at different probability levels
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Figure 11.  Pareto front at different probability levels
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Figure 12.  Distribution of blending results under different distributions
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Table 1.  Classification methods
#5 High-silicon ore #4 High-lead ore #3 Low-purity ore #2 High-purity ore #1 High-quality ore
Zn% < 44 44 < Zn < 47 >47
Pb% >1.8 < 1.8 < 1.8 < 1.8
SiO2% >3 < 3 < 3 < 3
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#5 High-silicon ore #4 High-lead ore #3 Low-purity ore #2 High-purity ore #1 High-quality ore
Zn% < 44 44 < Zn < 47 >47
Pb% >1.8 < 1.8 < 1.8 < 1.8
SiO2% >3 < 3 < 3 < 3
Table 2.  Common composition range of zinc concentrates
Zn(%) Fe(%) SiO2(%) Pb(%) Sb(%) Ge(%) Co(%)
Min 41.46 2.93 1.25 6.71 0.013 0.0027 0.00125
Max 55.37 17.2 7.65 0.72 1.21 0.0025 0.006
Requirement 47> < 12 < 3 < 1.8 < 0.1 < 0.006 < 0.004
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Zn(%) Fe(%) SiO2(%) Pb(%) Sb(%) Ge(%) Co(%)
Min 41.46 2.93 1.25 6.71 0.013 0.0027 0.00125
Max 55.37 17.2 7.65 0.72 1.21 0.0025 0.006
Requirement 47> < 12 < 3 < 1.8 < 0.1 < 0.006 < 0.004
Table 3.  Test valve of zinc concentrate from one supplier
Date Suppliers Material Zn% Pb% SiO2%
2020/9/7 Company of A Zinc concentrates 49.32 1.79 2.47
2020/9/7 Company of A Zinc concentrates 49.50 1.64 2.69
2020/9/7 Company of A Zinc concentrates 49.33 1.78 2.51
2020/9/7 Company of A Zinc concentrates 49.51 1.71 2.69
2020/9/7 Company of A Zinc concentrates 49.27 1.30 3.64
2020/9/7 Company of A Zinc concentrates 44.59 1.35 3.71
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Date Suppliers Material Zn% Pb% SiO2%
2020/9/7 Company of A Zinc concentrates 49.32 1.79 2.47
2020/9/7 Company of A Zinc concentrates 49.50 1.64 2.69
2020/9/7 Company of A Zinc concentrates 49.33 1.78 2.51
2020/9/7 Company of A Zinc concentrates 49.51 1.71 2.69
2020/9/7 Company of A Zinc concentrates 49.27 1.30 3.64
2020/9/7 Company of A Zinc concentrates 44.59 1.35 3.71
Table 4.  Model parameters description
Model Parameters explanatory notes
i i = 1, 2, 3, 4, 5;
$ {{P_i}} $ price per ton of zinc concentrate i;
$ \overline {{T_y}} $ Maximum allowable of y,
$ y=\{SiO2,Pb,Fe,S,Ni,Co,Sb,Ge\}$
$ m $ amount of blending (t);
$ \bar m $ allowance of mixed zinc concentrates;
$ \underline m $ minimum demand for mixed zinc concentrates;
$ {{\rm{X}}_{\max i}} $ allowance of raw material i, and
$ {{\rm{X}}_{\min i}} $ minimum demand for raw material i.
Random variables
$ {\tilde W_i} $ zinc content percentage of raw material i;
$ {{{\tilde T}_{yi}}} $ y content percentage of raw material i.
Decision variables
$ {{X_i}} $ amount of zinc concentrates i.
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Model Parameters explanatory notes
i i = 1, 2, 3, 4, 5;
$ {{P_i}} $ price per ton of zinc concentrate i;
$ \overline {{T_y}} $ Maximum allowable of y,
$ y=\{SiO2,Pb,Fe,S,Ni,Co,Sb,Ge\}$
$ m $ amount of blending (t);
$ \bar m $ allowance of mixed zinc concentrates;
$ \underline m $ minimum demand for mixed zinc concentrates;
$ {{\rm{X}}_{\max i}} $ allowance of raw material i, and
$ {{\rm{X}}_{\min i}} $ minimum demand for raw material i.
Random variables
$ {\tilde W_i} $ zinc content percentage of raw material i;
$ {{{\tilde T}_{yi}}} $ y content percentage of raw material i.
Decision variables
$ {{X_i}} $ amount of zinc concentrates i.
Table 5.  Eq. (12a) acquisition process
Algorithm 1.
Step 1: Use the uniform distribution to create decision variable $ x $;
Step 2: Use the Monte Carlo method to produce $ L $ = 1000 independent random matrices $ {\xi ^{\rm{1}}},{\xi ^{\rm{1}}}, \cdots ,{\xi ^L} $ based on the distribution, and get the sequence $ \left\{ {{J_1},{J_2}, \cdots ,{J_L}} \right\} $;
Step 3: Take $ L' $ as the integer part of $ {\partial _1}L $;
Step 4: Select the $ L' $th element $ {J_{L'}} $ as an estimate of $ {U_a} = \underline J $ in the sequence $ \left\{ {{J_1},{J_2}, \cdots ,{J_L}} \right\} $ based on the law of large numbers.
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Algorithm 1.
Step 1: Use the uniform distribution to create decision variable $ x $;
Step 2: Use the Monte Carlo method to produce $ L $ = 1000 independent random matrices $ {\xi ^{\rm{1}}},{\xi ^{\rm{1}}}, \cdots ,{\xi ^L} $ based on the distribution, and get the sequence $ \left\{ {{J_1},{J_2}, \cdots ,{J_L}} \right\} $;
Step 3: Take $ L' $ as the integer part of $ {\partial _1}L $;
Step 4: Select the $ L' $th element $ {J_{L'}} $ as an estimate of $ {U_a} = \underline J $ in the sequence $ \left\{ {{J_1},{J_2}, \cdots ,{J_L}} \right\} $ based on the law of large numbers.
Table 6.  Eq. (12c) acquisition process
Algorithm 2.
Step 1: Use the uniform distribution to create decision variable $ x $;
Step 2: Use the Monte Carlo method to generate $ L $ = 1000 independent random matrices $ {\xi ^{\rm{1}}},{\xi ^{\rm{1}}}, \cdots ,{\xi ^L} $ according to the distribution, and obtain the sequence $ \left\{ {{g_1},{g_2},{g_3}, \cdots {g_L}} \right\} $ by Eq. (13);
Step 3: Get number $ L' $ that satisfies the inequality $ {g_i} \ge \underline W ,i = 1,2, \cdots ,L $ in sequence $ \left\{ {{g_1},{g_2},{g_3}, \cdots {g_L}} \right\} $;
Step 4: Estimate the probability $ {U_{Zn}} $ based on the frequency $ L'/L $ according to Kolmogorov's law of strong numbers.
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Algorithm 2.
Step 1: Use the uniform distribution to create decision variable $ x $;
Step 2: Use the Monte Carlo method to generate $ L $ = 1000 independent random matrices $ {\xi ^{\rm{1}}},{\xi ^{\rm{1}}}, \cdots ,{\xi ^L} $ according to the distribution, and obtain the sequence $ \left\{ {{g_1},{g_2},{g_3}, \cdots {g_L}} \right\} $ by Eq. (13);
Step 3: Get number $ L' $ that satisfies the inequality $ {g_i} \ge \underline W ,i = 1,2, \cdots ,L $ in sequence $ \left\{ {{g_1},{g_2},{g_3}, \cdots {g_L}} \right\} $;
Step 4: Estimate the probability $ {U_{Zn}} $ based on the frequency $ L'/L $ according to Kolmogorov's law of strong numbers.
Table 7.  The qualitative assessment method
Criterion Excellent Good Generally Bad Worst
SZR $ x1 $/% $ x1 $$> $96 96$> $$ x1 $$> $94 94$> $$ x1 $$> $92 92$> $$ x1 $$> $90 90$> $$ x1 $
RN $ x2 $ 0.5% $> $$ x2 $ 1%$> $$ x2 $$> $0.5% 2%$> $$ x2 $$> $1% 4%$> $x2$> $2% $ x2 $$> $4%
Zn% $ x3 $ 50$> $$ x3 $$> $48 48$> $$ x3 $$> $47 $ x3 $$> $50 47$> $$ x3 $$> $46 46$> $$ x3 $
Pb% $ x4 $ 1$> $$ x4 $ 1.5$> $$ x4 $$> $1 1.8$> $$ x4 $$> $1.5 2.0$> $$ x4 $$> $1.8 $ x4 $$> $2.0
SiO2% $ x5 $ 1.5$> $$ x5 $ 2.0$> $$ x5 $$> $1.5 3.0$> $$ x5 $$> $2.0 3.5$> $$ x5 $$> $3.0 $ x5 $$> $3.5
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Criterion Excellent Good Generally Bad Worst
SZR $ x1 $/% $ x1 $$> $96 96$> $$ x1 $$> $94 94$> $$ x1 $$> $92 92$> $$ x1 $$> $90 90$> $$ x1 $
RN $ x2 $ 0.5% $> $$ x2 $ 1%$> $$ x2 $$> $0.5% 2%$> $$ x2 $$> $1% 4%$> $x2$> $2% $ x2 $$> $4%
Zn% $ x3 $ 50$> $$ x3 $$> $48 48$> $$ x3 $$> $47 $ x3 $$> $50 47$> $$ x3 $$> $46 46$> $$ x3 $
Pb% $ x4 $ 1$> $$ x4 $ 1.5$> $$ x4 $$> $1 1.8$> $$ x4 $$> $1.5 2.0$> $$ x4 $$> $1.8 $ x4 $$> $2.0
SiO2% $ x5 $ 1.5$> $$ x5 $ 2.0$> $$ x5 $$> $1.5 3.0$> $$ x5 $$> $2.0 3.5$> $$ x5 $$> $3.0 $ x5 $$> $3.5
Table 8.  Scale of importance
Number Explanation
1 Equally important
3 Slightly important
5 Strongly important
7 Very strongly important
9 Absolutely important
2, 4, 6, 8 Intermediate value
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Number Explanation
1 Equally important
3 Slightly important
5 Strongly important
7 Very strongly important
9 Absolutely important
2, 4, 6, 8 Intermediate value
Table 9.  The score of the criteria and the five levels
Target Criterion Importance of index Excellent Good General Poor Worst
Total score u SZR $ x1 $ 9 9 7 5 3 1
RN $ x2 $ 7 9 7 6 4 2
Zn% $ x3 $ 4 9 8 6 3 1
Pb% $ x4 $ 5 9 7 6 2 1
SiO2% $ x5 $ 3 9 8 5 2 1
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Target Criterion Importance of index Excellent Good General Poor Worst
Total score u SZR $ x1 $ 9 9 7 5 3 1
RN $ x2 $ 7 9 7 6 4 2
Zn% $ x3 $ 4 9 8 6 3 1
Pb% $ x4 $ 5 9 7 6 2 1
SiO2% $ x5 $ 3 9 8 5 2 1
Table 10.  Pairwise comparison matrix of criterion levels
SZR $ x1 $ RN $ x2 $ Zn% $ x3 $ Pb% $ x4 $ SiO2% $ x5 $
SZR $ x1 $ 1 9/7 9/4 9/5 3
RN $ x2 $ 7/9 1 7/4 7/5 7/3
Zn% $ x3 $ 4/9 4/7 1 4/5 4/3
Pb% $ x4 $ 5/9 5/7 5/4 1 5/3
SiO2% $ x5 $ 1/3 3/7 3/4 3/5 1
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SZR $ x1 $ RN $ x2 $ Zn% $ x3 $ Pb% $ x4 $ SiO2% $ x5 $
SZR $ x1 $ 1 9/7 9/4 9/5 3
RN $ x2 $ 7/9 1 7/4 7/5 7/3
Zn% $ x3 $ 4/9 4/7 1 4/5 4/3
Pb% $ x4 $ 5/9 5/7 5/4 1 5/3
SiO2% $ x5 $ 1/3 3/7 3/4 3/5 1
Table 11.  Values of the random consistency index
n 1 2 3 4 5 6 7 8 9 10 11
RI 0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.46 1.49 1.52
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n 1 2 3 4 5 6 7 8 9 10 11
RI 0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.46 1.49 1.52
Table 12.  Weight of each index
Target Criterion Weights Excellent Good General Poor Worst
Total score u SZR $ x1 $ 0.3103 0.36 0.28 0.2 0.12 0.04
RN $ x2 $ 0.2414 0.3214 0.25 0.2143 0.1429 0.0714
Zn% $ x3 $ 0.1379 0.3333 0.2963 0.2222 0.1111 0.037
Pb% $ x4 $ 0.1724 0.36 0.28 0.24 0.08 0.04
SiO2% $ x5 $ 0.1034 0.36 0.32 0.2 0.08 0.04
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Target Criterion Weights Excellent Good General Poor Worst
Total score u SZR $ x1 $ 0.3103 0.36 0.28 0.2 0.12 0.04
RN $ x2 $ 0.2414 0.3214 0.25 0.2143 0.1429 0.0714
Zn% $ x3 $ 0.1379 0.3333 0.2963 0.2222 0.1111 0.037
Pb% $ x4 $ 0.1724 0.36 0.28 0.24 0.08 0.04
SiO2% $ x5 $ 0.1034 0.36 0.32 0.2 0.08 0.04
Table 13.  Expert rules
Operation of the system Range of $ u $ Proportion
Excellent u$> $0.9 0.5
Good 0.9$> $u$> $0.8 0.6
General 0.8$> $u$> $0.7 0.8
Poor 0.7$> $u 1
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Operation of the system Range of $ u $ Proportion
Excellent u$> $0.9 0.5
Good 0.9$> $u$> $0.8 0.6
General 0.8$> $u$> $0.7 0.8
Poor 0.7$> $u 1
Table 14.  Main component parameters of Zinc concentrates
Ore bin Zn (%) Pb (%) SiO2 (%) AP RMB/t
$ u $ $ \sigma $ $ a $ $ b $ Dis $ u $ $ \sigma $ $ a $ $ b $ Dis $ u $ $ \sigma $ Dis
#1 1.01* 0.594 – – LN 1.021 0.38 0 1.8 N 0.21 0.35 LN 14701
#2 0.122$ \Box $ 0.351 – – LN 1.24 0.3 0 1.8 N 0.19 0.38 LN 13236
#3 0.103$ \triangle $ 0.201 – – LN 1.133 0.41 0 1.8 N 0.185 0.42 LN 12157
#4 46.12 1.9 40 52 N 0* 0.4 – – LN 0.22 0.4 LN 13368
#5 45.31 1.62 40 52 N 1.17 0.31 0 1.8 N 0.1* 0.55 LN 13128
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Ore bin Zn (%) Pb (%) SiO2 (%) AP RMB/t
$ u $ $ \sigma $ $ a $ $ b $ Dis $ u $ $ \sigma $ $ a $ $ b $ Dis $ u $ $ \sigma $ Dis
#1 1.01* 0.594 – – LN 1.021 0.38 0 1.8 N 0.21 0.35 LN 14701
#2 0.122$ \Box $ 0.351 – – LN 1.24 0.3 0 1.8 N 0.19 0.38 LN 13236
#3 0.103$ \triangle $ 0.201 – – LN 1.133 0.41 0 1.8 N 0.185 0.42 LN 12157
#4 46.12 1.9 40 52 N 0* 0.4 – – LN 0.22 0.4 LN 13368
#5 45.31 1.62 40 52 N 1.17 0.31 0 1.8 N 0.1* 0.55 LN 13128
Table 15.  Mean value of each component
Ore bin Zn (%) Pb (%) SiO2 (%) AP RMB/t
E E E
#1 50.275 1.006 1.302 14701
#2 45.8 1.218 1.305 13236
#3 42.87 1.09 1.314 12157
#4 46.12 2.93 1.32 13368
#5 45.31 1.154 4.259 13128
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Ore bin Zn (%) Pb (%) SiO2 (%) AP RMB/t
E E E
#1 50.275 1.006 1.302 14701
#2 45.8 1.218 1.305 13236
#3 42.87 1.09 1.314 12157
#4 46.12 2.93 1.32 13368
#5 45.31 1.154 4.259 13128
Table 16.  Limitation requirements
Zn% SiO2% Pb% $ x1 $ $ x2 $ $ x3 $ $ x4 $ $ x5 $ $ m $
Min 47 0 0 0 0 0 0 0 280
Max 55 3 1.8 20 970 840 350 300 300
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Zn% SiO2% Pb% $ x1 $ $ x2 $ $ x3 $ $ x4 $ $ x5 $ $ m $
Min 47 0 0 0 0 0 0 0 280
Max 55 3 1.8 20 970 840 350 300 300
Table 17.  Different probability levels
Pb 0.6 0.8 0.8
SiO2 0.6 0.8 0.95
Colour Blue Red Green
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Pb 0.6 0.8 0.8
SiO2 0.6 0.8 0.95
Colour Blue Red Green
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