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

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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 and 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.
[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.

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

	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

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.

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

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 $ 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	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

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

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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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.

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.

American Institute of Mathematical Sciences

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

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