Modeling and Optimization of Cement Raw Materials Blending Process

Table 6

Optimization models and results for cement materials in Table 3.

	Optimization Models
	Model.1: 𝐽1=25𝑥1+15𝑥2+68𝑥3+20𝑥4+20𝑥5
	Model.2: 𝐽2=12.45𝑥1+12.94𝑥2+19.89𝑥3+28.6𝑥4+28.6𝑥5
	Model.3: 𝐽3=𝑤1(1.00−𝛼)^2+𝑤2(2.70−𝛽)^2+𝑤3(1.55−Ω)^2
	(𝑤1=0.5,𝑤2=0.3,𝑤3=0.2,𝛼𝑑0=1.00,𝛽𝑑0=2.70,Ω𝑑0=1.55)
	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=4.5𝑥1+65𝑥2+17.5𝑥3+45.31𝑥4+59.26𝑥5
	𝑀𝜂=0.99𝑥1+5.76𝑥2+6.9𝑥3+23.3𝑥4+24.55𝑥5
	𝑀𝜌=0.24𝑥1+1.61𝑥2+29.0𝑥3+6.1𝑥4+8.07𝑥5
	𝑀𝛾=45.0𝑥1+0.52𝑥2+31.49𝑥3+8.63𝑥4+3.73𝑥5
	𝑥1+𝑥2+𝑥3+𝑥4+𝑥5=1.00
	𝑥1≥0,𝑥2≥0,𝑥3≥0,𝑥4≥𝜀,𝑥5≥0, (𝜀=0.0001)
(2)	𝛼=(𝑀𝛾−1.65𝑀𝜂−0.35𝑀𝜌)/(2.8𝑀𝜇)
	𝛽=𝑀𝜇/(𝑀𝜂+𝑀𝜌),Ω=𝑀𝜂/𝑀𝜌
(3)	𝑀𝜑=40.56𝑥1+2.62𝑥2+0.3𝑥3+10.34𝑥4+8.32𝑥5≤38.00
	𝑀𝜔=2.91𝑥1+8.9𝑥2+13.45𝑥3+6.32𝑥4+6.07𝑥5≤7.00
(4)	0.98≤𝛼≤1.02,2.60≤𝛽≤2.80,1.45≤Ω≤1.65
Models	Optimal ingredient ratio	Notes
Model.1	𝑥^∗1=84.003%;𝑥^∗2=7.687%;𝑥^∗3𝑥=3.203%;∗4=0.010%;𝑥^∗5=5.097%;𝐽^∗1=25.35308;
Model.2	𝑥^∗1=84.145%;𝑥^∗2=8.021%;𝑥^∗3𝑥=3.795%;∗4=0.010%;𝑥^∗5=4.029%;𝐽^∗2=13.42398;
Model.3	𝑥^∗1=84.046%;𝑥^∗2=7.335%;𝑥^∗3𝑥=3.587%;∗4=0.010%;𝑥^∗5=5.021%;𝐽^∗3=0.000;
Model.4	𝑥^∗1=84.658%;𝑥^∗2=7.349%;𝑥^∗3𝑥=3.122%;∗4=0.010%;𝑥^∗5=4.681%;𝐽^∗1+𝐽^∗2=38.86875;	Ψ1=Ψ2𝐽=1.0∗1=25.36388
Model.5	𝑥^∗1=84.003%;𝑥^∗2=7.687%;𝑥^∗3𝑥=3.203%;∗4=0.010%;𝑥^∗5=5.097%;𝐽^∗1+𝐽^∗3=25.35828;	Ψ1=Ψ2𝐽=1.0∗1=25.35308
Model.6	𝑥^∗1=84.145%;𝑥^∗2=8.021%;𝑥^∗3𝑥=3.795%;∗4=0.010%;𝑥^∗5=4.029%;𝐽^∗2+𝐽^∗3=13.42918;	Ψ1=Ψ2𝐽=1.0∗2=13.42398
Model.7	𝑥^∗1=84.658%;𝑥^∗2=7.349%;𝑥^∗3𝑥=3.122%;∗4=0.010%;𝑥^∗5=4.681%;𝐽^∗1+𝐽^∗2+𝐽^∗3=38.87395.	Ψ1=Ψ2=Ψ3𝐽=1.0∗1𝐽=25.36388∗2=13.50487

Where 𝑥₁, 𝑥₂, 𝑥₃, 𝑥₄, and 𝑥₅ are ingredient ratio of the limestone, sandstone, steel slag, shale, and coal ash, respectively.