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

Yu Chen; Yonggang Li; Bei Sun; Chunhua Yang; Hongqiu Zhu

doi:10.3934/jimo.2021169

Article Contents

2022, Volume 18, Issue 6: 4491-4510. Doi: 10.3934/jimo.2021169

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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

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

Received: April 30, 2021

Revised: June 30, 2021

Early access: September 2021

Published: September 2022

Abstract Full Text(HTML) Figure(12) / Table(17) Related Papers Cited by

Abstract

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:

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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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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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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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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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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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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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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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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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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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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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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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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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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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Table 14. Main component parameters of Zinc concentrates

Ore bin	Zn (%)					Pb (%)					SiO2 (%)			AP RMB/t
Ore bin	$ u $	$ \sigma $	$ a $	$ b $	Dis	$ u $	$ \sigma $	$ a $	$ b $	Dis	$ u $	$ \sigma $	Dis	AP RMB/t
#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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Table 15. Mean value of each component

Ore bin	Zn (%)	Pb (%)	SiO2 (%)	AP RMB/t
Ore bin	E	E	E	AP RMB/t
#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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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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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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