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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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
B. Liu and B. Liu, Theory and Practice of Uncertain Programming, volume 239, 2009., Springer.
doi: 10.1007/978-3-540-89484-1. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
H. A. Taha, Operations Research an Introduction, The Macmillan Co., New York; Collier-Macmillian Ltd., London, 1971. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
B. Liu and B. Liu, Theory and Practice of Uncertain Programming, volume 239, 2009., Springer.
doi: 10.1007/978-3-540-89484-1. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
H. A. Taha, Operations Research an Introduction, The Macmillan Co., New York; Collier-Macmillian Ltd., London, 1971. |
[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. |
[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. |












#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 |
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 |
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 |
Model Parameters | explanatory notes |
i | i = 1, 2, 3, 4, 5; |
price per ton of zinc concentrate i; | |
Maximum allowable of y, | |
amount of blending (t); | |
allowance of mixed zinc concentrates; | |
minimum demand for mixed zinc concentrates; | |
allowance of raw material i, and | |
minimum demand for raw material i. | |
Random variables | |
zinc content percentage of raw material i; | |
y content percentage of raw material i. | |
Decision variables | |
amount of zinc concentrates i. |
Model Parameters | explanatory notes |
i | i = 1, 2, 3, 4, 5; |
price per ton of zinc concentrate i; | |
Maximum allowable of y, | |
amount of blending (t); | |
allowance of mixed zinc concentrates; | |
minimum demand for mixed zinc concentrates; | |
allowance of raw material i, and | |
minimum demand for raw material i. | |
Random variables | |
zinc content percentage of raw material i; | |
y content percentage of raw material i. | |
Decision variables | |
amount of zinc concentrates i. |
Algorithm 1. |
Step 1: Use the uniform distribution to create decision variable |
Step 2: Use the Monte Carlo method to produce |
Step 3: Take |
Step 4: Select the |
Algorithm 1. |
Step 1: Use the uniform distribution to create decision variable |
Step 2: Use the Monte Carlo method to produce |
Step 3: Take |
Step 4: Select the |
Algorithm 2. |
Step 1: Use the uniform distribution to create decision variable |
Step 2: Use the Monte Carlo method to generate |
Step 3: Get number |
Step 4: Estimate the probability |
Algorithm 2. |
Step 1: Use the uniform distribution to create decision variable |
Step 2: Use the Monte Carlo method to generate |
Step 3: Get number |
Step 4: Estimate the probability |
Criterion | Excellent | Good | Generally | Bad | Worst |
SZR |
96 |
94 |
92 |
90 |
|
RN |
0.5% |
1% |
2% |
4% |
|
Zn% |
50 |
48 |
47 |
46 |
|
Pb% |
1 |
1.5 |
1.8 |
2.0 |
|
SiO2% |
1.5 |
2.0 |
3.0 |
3.5 |
Criterion | Excellent | Good | Generally | Bad | Worst |
SZR |
96 |
94 |
92 |
90 |
|
RN |
0.5% |
1% |
2% |
4% |
|
Zn% |
50 |
48 |
47 |
46 |
|
Pb% |
1 |
1.5 |
1.8 |
2.0 |
|
SiO2% |
1.5 |
2.0 |
3.0 |
3.5 |
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 |
Target | Criterion | Importance of index | Excellent | Good | General | Poor | Worst |
Total score u | SZR |
9 | 9 | 7 | 5 | 3 | 1 |
RN |
7 | 9 | 7 | 6 | 4 | 2 | |
Zn% |
4 | 9 | 8 | 6 | 3 | 1 | |
Pb% |
5 | 9 | 7 | 6 | 2 | 1 | |
SiO2% |
3 | 9 | 8 | 5 | 2 | 1 |
Target | Criterion | Importance of index | Excellent | Good | General | Poor | Worst |
Total score u | SZR |
9 | 9 | 7 | 5 | 3 | 1 |
RN |
7 | 9 | 7 | 6 | 4 | 2 | |
Zn% |
4 | 9 | 8 | 6 | 3 | 1 | |
Pb% |
5 | 9 | 7 | 6 | 2 | 1 | |
SiO2% |
3 | 9 | 8 | 5 | 2 | 1 |
SZR |
RN |
Zn% |
Pb% |
SiO2% |
|
SZR |
1 | 9/7 | 9/4 | 9/5 | 3 |
RN |
7/9 | 1 | 7/4 | 7/5 | 7/3 |
Zn% |
4/9 | 4/7 | 1 | 4/5 | 4/3 |
Pb% |
5/9 | 5/7 | 5/4 | 1 | 5/3 |
SiO2% |
1/3 | 3/7 | 3/4 | 3/5 | 1 |
SZR |
RN |
Zn% |
Pb% |
SiO2% |
|
SZR |
1 | 9/7 | 9/4 | 9/5 | 3 |
RN |
7/9 | 1 | 7/4 | 7/5 | 7/3 |
Zn% |
4/9 | 4/7 | 1 | 4/5 | 4/3 |
Pb% |
5/9 | 5/7 | 5/4 | 1 | 5/3 |
SiO2% |
1/3 | 3/7 | 3/4 | 3/5 | 1 |
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 |
Target | Criterion | Weights | Excellent | Good | General | Poor | Worst |
Total score u | SZR |
0.3103 | 0.36 | 0.28 | 0.2 | 0.12 | 0.04 |
RN |
0.2414 | 0.3214 | 0.25 | 0.2143 | 0.1429 | 0.0714 | |
Zn% |
0.1379 | 0.3333 | 0.2963 | 0.2222 | 0.1111 | 0.037 | |
Pb% |
0.1724 | 0.36 | 0.28 | 0.24 | 0.08 | 0.04 | |
SiO2% |
0.1034 | 0.36 | 0.32 | 0.2 | 0.08 | 0.04 |
Target | Criterion | Weights | Excellent | Good | General | Poor | Worst |
Total score u | SZR |
0.3103 | 0.36 | 0.28 | 0.2 | 0.12 | 0.04 |
RN |
0.2414 | 0.3214 | 0.25 | 0.2143 | 0.1429 | 0.0714 | |
Zn% |
0.1379 | 0.3333 | 0.2963 | 0.2222 | 0.1111 | 0.037 | |
Pb% |
0.1724 | 0.36 | 0.28 | 0.24 | 0.08 | 0.04 | |
SiO2% |
0.1034 | 0.36 | 0.32 | 0.2 | 0.08 | 0.04 |
Operation of the system | Range of |
Proportion |
Excellent | u |
0.5 |
Good | 0.9 |
0.6 |
General | 0.8 |
0.8 |
Poor | 0.7 |
1 |
Operation of the system | Range of |
Proportion |
Excellent | u |
0.5 |
Good | 0.9 |
0.6 |
General | 0.8 |
0.8 |
Poor | 0.7 |
1 |
Ore bin | Zn (%) | Pb (%) | SiO2 (%) | AP RMB/t | ||||||||||
Dis | Dis | Dis | ||||||||||||
#1 | 1.01* | 0.594 | – | – | LN | 1.021 | 0.38 | 0 | 1.8 | N | 0.21 | 0.35 | LN | 14701 |
#2 | 0.122 |
0.351 | – | – | LN | 1.24 | 0.3 | 0 | 1.8 | N | 0.19 | 0.38 | LN | 13236 |
#3 | 0.103 |
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 | ||||||||||
Dis | Dis | Dis | ||||||||||||
#1 | 1.01* | 0.594 | – | – | LN | 1.021 | 0.38 | 0 | 1.8 | N | 0.21 | 0.35 | LN | 14701 |
#2 | 0.122 |
0.351 | – | – | LN | 1.24 | 0.3 | 0 | 1.8 | N | 0.19 | 0.38 | LN | 13236 |
#3 | 0.103 |
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 |
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 |
Zn% | SiO2% | Pb% | |||||||
Min | 47 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 280 |
Max | 55 | 3 | 1.8 | 20 | 970 | 840 | 350 | 300 | 300 |
Zn% | SiO2% | Pb% | |||||||
Min | 47 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 280 |
Max | 55 | 3 | 1.8 | 20 | 970 | 840 | 350 | 300 | 300 |
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 |
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