• Previous Article
    Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction
  • JIMO Home
  • This Issue
  • Next Article
    Portfolio optimization for jump-diffusion risky assets with regime switching: A time-consistent approach
doi: 10.3934/jimo.2021169
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Multi-objective chance-constrained blending optimization of zinc smelter under stochastic uncertainty

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.

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. ChenY. LiB. SunY. LiH. 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. DebA. PratapS. 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 SantosS. M. NevesD. O. Sant'AnnaC. 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. HovakimyanF. NardiA. 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. OuG. LuoA. RayC. LiH. HuS. 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. SavicD. NikolicI. MihajlovicZ. ZivkovicB. 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. SchultzD. 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. YangP. 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. ChenY. LiB. SunY. LiH. 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. DebA. PratapS. 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 SantosS. M. NevesD. O. Sant'AnnaC. 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. HovakimyanF. NardiA. 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. OuG. LuoA. RayC. LiH. HuS. 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. SavicD. NikolicI. MihajlovicZ. ZivkovicB. 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. SchultzD. 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. YangP. 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
Figure 2.  Semi-underground warehouse
Figure 3.  Block diagram of multi-objective chance-constrained blending optimization
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)
Figure 5.  Optimization idea
Figure 6.  Multi-objective hybrid intelligent optimization algorithm
Figure 7.  Roasting process
Figure 8.  Calciner temperature and feed rate (the upper is the calcination temperature and the lower is the feed amount of the thrower)
Figure 9.  Evaluation mechanism of working condition
Figure 10.  Pareto front at different probability levels
Figure 11.  Pareto front at different probability levels
Figure 12.  Distribution of blending results under different distributions
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
#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
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
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.
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.
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.
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
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
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
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
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
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
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
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
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
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
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
Pb 0.6 0.8 0.8
SiO2 0.6 0.8 0.95
Colour Blue Red Green
[1]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099

[2]

Tone-Yau Huang, Tamaki Tanaka. Optimality and duality for complex multi-objective programming. Numerical Algebra, Control & Optimization, 2022, 12 (1) : 121-134. doi: 10.3934/naco.2021055

[3]

Liwei Zhang, Jihong Zhang, Yule Zhang. Second-order optimality conditions for cone constrained multi-objective optimization. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1041-1054. doi: 10.3934/jimo.2017089

[4]

Azam Moradi, Jafar Razmi, Reza Babazadeh, Ali Sabbaghnia. An integrated Principal Component Analysis and multi-objective mathematical programming approach to agile supply chain network design under uncertainty. Journal of Industrial & Management Optimization, 2019, 15 (2) : 855-879. doi: 10.3934/jimo.2018074

[5]

Min Zhang, Gang Li. Multi-objective optimization algorithm based on improved particle swarm in cloud computing environment. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1413-1426. doi: 10.3934/dcdss.2019097

[6]

Yuan-mei Xia, Xin-min Yang, Ke-quan Zhao. A combined scalarization method for multi-objective optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2669-2683. doi: 10.3934/jimo.2020088

[7]

Henri Bonnel, Ngoc Sang Pham. Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions. Journal of Industrial & Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789

[8]

Xia Zhao, Jianping Dou. Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1263-1288. doi: 10.3934/jimo.2018095

[9]

Tien-Fu Liang, Hung-Wen Cheng. Multi-objective aggregate production planning decisions using two-phase fuzzy goal programming method. Journal of Industrial & Management Optimization, 2011, 7 (2) : 365-383. doi: 10.3934/jimo.2011.7.365

[10]

Ya Liu, Zhaojin Li. Dynamic-programming-based heuristic for multi-objective operating theater planning. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020145

[11]

Editorial Office. RETRACTION: Peng Zhang, Chance-constrained multiperiod mean absolute deviation uncertain portfolio selection. Journal of Industrial & Management Optimization, 2019, 15 (2) : 537-564. doi: 10.3934/jimo.2018056

[12]

Han Yang, Jia Yue, Nan-jing Huang. Multi-objective robust cross-market mixed portfolio optimization under hierarchical risk integration. Journal of Industrial & Management Optimization, 2020, 16 (2) : 759-775. doi: 10.3934/jimo.2018177

[13]

Shoufeng Ji, Jinhuan Tang, Minghe Sun, Rongjuan Luo. Multi-objective optimization for a combined location-routing-inventory system considering carbon-capped differences. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021051

[14]

Qiang Long, Xue Wu, Changzhi Wu. Non-dominated sorting methods for multi-objective optimization: Review and numerical comparison. Journal of Industrial & Management Optimization, 2021, 17 (2) : 1001-1023. doi: 10.3934/jimo.2020009

[15]

Danthai Thongphiew, Vira Chankong, Fang-Fang Yin, Q. Jackie Wu. An on-line adaptive radiation therapy system for intensity modulated radiation therapy: An application of multi-objective optimization. Journal of Industrial & Management Optimization, 2008, 4 (3) : 453-475. doi: 10.3934/jimo.2008.4.453

[16]

Xiliang Sun, Wanjie Hu, Xiaolong Xue, Jianjun Dong. Multi-objective optimization model for planning metro-based underground logistics system network: Nanjing case study. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021179

[17]

Wendai Lv, Siping Ji. Atmospheric environmental quality assessment method based on analytic hierarchy process. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 941-955. doi: 10.3934/dcdss.2019063

[18]

Namsu Ahn, Soochan Kim. Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021037

[19]

Nguyen Thi Toan. Generalized Clarke epiderivatives of the extremum multifunction to a multi-objective parametric discrete optimal control problem. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021088

[20]

Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial & Management Optimization, 2021, 17 (2) : 695-709. doi: 10.3934/jimo.2019130

2020 Impact Factor: 1.801

Metrics

  • PDF downloads (89)
  • HTML views (112)
  • Cited by (0)

[Back to Top]