Modeling and Optimization of Cement Raw Materials Blending Process

Table 8

Optimization models and results for cement materials in Table 5.

	Optimization Models
	Model.1: 𝐽1=18𝑥1+25𝑥2+48𝑥3+9𝑥4
	Model.2: 𝐽2=11.24𝑥1+12.50𝑥2+19.86𝑥3+13.80𝑥4
	Model.3: 𝐽3=𝑤1(1.02−𝛼)^2+𝑤2(1.80−𝛽)^2+𝑤3(1.10−Ω)^2
	(𝑤1=0.5,𝑤2=0.3,𝑤3=0.2,𝛼𝑑0=1.02,𝛽𝑑0=1.80,Ω𝑑0=1.10)
	Model.4: min(𝐽1,𝐽2)=min(Ψ1𝐽1+Ψ2𝐽2)
	Model.5: min(𝐽1,𝐽3)=min(Ψ1𝐽1+Ψ2𝐽3)
	Model.6: min(𝐽2,𝐽3)=min(Ψ1𝐽2+Ψ2𝐽3)
	Model.7: min(𝐽1,𝐽2,𝐽3)=min(Ψ1𝐽1+Ψ2𝐽2+Ψ3𝐽3)
	Subject to (3.1)–(3.4)
(1)	𝑀𝜇=1.02𝑥1+69.56𝑥2+11.05𝑥3+56.39𝑥4
	𝑀𝜂=1.29𝑥1+16.42𝑥2+2.22𝑥3+22.77𝑥4
	𝑀𝜌=3.35𝑥2+77.85𝑥3+10.18𝑥4
	𝑀𝛾=69.26𝑥1+2.45𝑥3+1.13𝑥4
	𝑥1+𝑥2+𝑥3+𝑥4+𝑥5=1.00, 𝑥1≥0,𝑥2≥0,𝑥3≥0,𝑥4≥0
(2)	𝛼=(𝑀𝛾−1.65𝑀𝜂−0.35𝑀𝜌)/(2.8𝑀𝜇)
	𝛽=𝑀𝜇/(𝑀𝜂+𝑀𝜌), Ω=𝑀𝜂/𝑀𝜌
(3)	𝑀𝜑=24.65𝑥1+5.83𝑥2+1.06𝑥3≤30.00
	𝑀𝜏=2.71𝑥3+2.16𝑥4≤1.50
(4)	1.00≤𝛼≤1.04,1.70≤𝛽≤1.90,0.95≤Ω≤1.25
Models	Optimal ingredient ratio	Notes
Model.1	𝑥^∗1=75.007%;𝑥^∗2=14.973%;𝑥^∗3𝑥=3.620%;∗4=6.400%;𝐽^∗1=19.55801;
Model.2	𝑥^∗1=76.090%;𝑥^∗2=19.530%;𝑥^∗3𝑥=3.624%;∗4=0.755%;𝐽^∗2=11.81783;
Model.3	𝑥^∗1=75.442%;𝑥^∗2=19.623%;𝑥^∗3𝑥=4.257%;∗4=0.678%;𝐽^∗3=0.00000;
Model.4	𝑥^∗1=75.654%;𝑥^∗2=14.798%;𝑥^∗3𝑥=3.562%;∗4=5.985%;𝐽^∗1+𝐽^∗2=31.45261;	Ψ1=Ψ2𝐽=1.0∗1=19.56587
Model.5	𝑥^∗1=75.007%;𝑥^∗2=14.973%;𝑥^∗3𝑥=3.620%;∗4=6.400%;𝐽^∗1+𝐽^∗3=19.56571;	Ψ1=Ψ2𝐽=1.0∗1=19.55801
Model.6	𝑥^∗1=76.090%;𝑥^∗2=19.530%;𝑥^∗3𝑥=3.624%;∗4=0.755%;𝐽^∗2+𝐽^∗3=11.82553;	Ψ1=Ψ2𝐽=1.0∗2=11.81783
Model.7	𝑥^∗1=75.654%;𝑥^∗2=14.798%;𝑥^∗3𝑥=3.562%;∗4=5.985%;𝐽^∗1+𝐽^∗2+𝐽^∗3=31.46031.	Ψ1=Ψ2=Ψ3𝐽=1.0∗1𝐽=19.56587∗2=11.88674

Where 𝑥₁, 𝑥₂, 𝑥₃, and 𝑥₄ are ingredient ratio of the carbide slag, clay, sulfuric acid residue, and cinder, respectively.