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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

 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:

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##### References:
Blending process
Semi-underground warehouse
Block diagram of multi-objective chance-constrained blending optimization
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)
Optimization idea
Multi-objective hybrid intelligent optimization algorithm
Roasting process
Calciner temperature and feed rate (the upper is the calcination temperature and the lower is the feed amount of the thrower)
Evaluation mechanism of working condition
Pareto front at different probability levels
Pareto front at different probability levels
Distribution of blending results under different distributions
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
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
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
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.
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.
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.
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
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
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
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
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
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
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
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
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
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
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
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