Modeling and Optimization of Cement Raw Materials Blending Process

Table 7

Optimization models and results for cement materials in Table 4.

	Optimization Models
	Model.1: 𝐽1=24𝑥1+25𝑥2+50𝑥3+30𝑥4+28.7𝑥5
	Model.2: 𝐽2=12.45𝑥1+12.10𝑥2+18.98𝑥3+14.70𝑥4+15.66𝑥5
	Model.3: 𝐽3=𝑤1(0.96−𝛼)^2+𝑤2(1.90−𝛽)^2+𝑤3(1.25−Ω)^2
	(𝛼𝑑0=0.96,𝛽𝑑0=1.90,Ω𝑑0=1.25,𝑤1=0.5,𝑤2=0.3,𝑤3=0.2)
	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)–(3.5)
(1)	𝑀𝜇=8.52𝑥1+62.74𝑥2+7.92𝑥3+3.15𝑥4+44.77𝑥5
	𝑀𝜂=1.23𝑥1+17.94𝑥2+50.27𝑥3+21.3𝑥4+26.04𝑥5
	𝑀𝜌=1.31𝑥1+4.06𝑥2+13.01𝑥3+38.55𝑥4+4.49𝑥5
	𝑀𝛾=46.05𝑥1+2.40𝑥2+2.94𝑥3+5.17𝑥4+8.42𝑥5
	𝑥1+𝑥2+𝑥3+𝑥4+𝑥5=1.0, 𝑥1≥0, 𝑥2≥0, 𝑥3≥𝜀, 𝑥4≥0, 𝑥5≥𝜀, (𝜀=0.0001)
(2)	𝛼=(𝑀𝛾−1.65𝑀𝜂−0.35𝑀𝜌)/(2.8𝑀𝜇)
	𝛽=𝑀𝜇/(𝑀𝜂+𝑀𝜌), Ω=𝑀𝜂/𝑀𝜌
(3)	𝑀𝜑=40.09𝑥1+7.99𝑥2+24.74𝑥3+30.25𝑥4≤39.00
	𝑀𝜏=2.49𝑥1+0.94𝑥2+0.79𝑥3+1.53𝑥4+1.67𝑥5≤3.00
	𝑀𝑠=0.02𝑥1+0.64𝑥2+0.14𝑥3+0.05𝑥4+0.95𝑥5≤0.8
	𝑀𝑟=0.28𝑥1+3.25𝑥2+0.19𝑥3+0.62𝑥5≤0.9
	𝑀𝑟1=0.21𝑥1+3.25𝑥2+0.19𝑥3+0.62𝑥5≤0.8,𝑀𝑟2=0.07𝑥1≤0.1
	𝑀𝜋=0.0243𝑥1+0.09𝑥2+0.25𝑥3+0.013𝑥4+0.043𝑥5≤0.2
(4)	𝜃=𝑀𝑠/(0.85𝑀𝑟1+1.29𝑀𝑟2−1.119𝑀𝜋)≤0.7
(5)	0.94≤𝛼≤0.98;1.80≤𝛽≤2.00;1.15≤Ω≤1.35
Models	Optimal ingredient ratio	Notes
Model.1	𝑥^∗1=88.257%;𝑥^∗2=7.503%;𝑥^∗3𝑥=0.010%;∗4=3.731%;𝑥^∗5=0.499%;𝐽^∗1=24.32497;
Model.2	𝑥^∗1=87.565%;𝑥^∗2=8.480%;𝑥^∗3𝑥=0.040%;∗4=3.905%;𝑥^∗5=0.010%;𝐽^∗2=12.51115;
Model.3	𝑥^∗1=87.805%;𝑥^∗2=7.791%;𝑥^∗3𝑥=0.878%;∗4=3.516%;𝑥^∗5=0.010%;𝐽^∗3=0.000;
Model.4	𝑥^∗1=87.555%;𝑥^∗2=8.414%;𝑥^∗3𝑥=0.010%;∗4=3.912%;𝑥^∗5=0.109%;𝐽^∗1+𝐽^∗2=36.83931;	Ψ1=Ψ2𝐽=1∗1=24.32659
Model.5	𝑥^∗1=88.257%;𝑥^∗2=7.503%;𝑥^∗3𝑥=0.010%;∗4=3.731%;𝑥^∗5=0.499%;𝐽^∗1+𝐽^∗3=24.33017;	Ψ1=Ψ2𝐽=1∗1=24.32497
Model.6	𝑥^∗1=87.565%;𝑥^∗2=8.480%;𝑥^∗3𝑥=0.040%;∗4=3.905%;𝑥^∗5=0.010%;𝐽^∗2+𝐽^∗3=12.51635;	Ψ1=Ψ2𝐽=1∗2=12.51115
Model.7	𝑥^∗1=87.555%;𝑥^∗2=8.414%;𝑥^∗3𝑥=0.010%;∗4=3.912%;𝑥^∗5=0.109%;𝐽^∗1+𝐽^∗2+𝐽^∗3=36.84451.	Ψ1=Ψ2=Ψ3𝐽=1∗1𝐽=24.32659∗2=12.51273

Where 𝑥₁, 𝑥₂, 𝑥₃, 𝑥₄, and 𝑥₅ are ingredient ratio of the limestone, clay, iron, correction materials, and coal ash, respectively.