# Modeling and Optimization of Cement Raw Materials Blending Process

Authors: Xianhong Li; Haibin Yu; Mingzhe Yuan
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper focuses on modelling and solving the ingredient ratio optimization problem in cement raw material blending process. A general nonlinear time-varying (G-NLTV) model is established for cement raw material blending process via considering chemical composition, feed flow fluctuation, and various craft and production constraints. Different objective functions are presented to acquire optimal ingredient ratios under various production requirements. The ingredient ratio optimization problem is transformed into discrete-time single objective or multiple objectives rolling nonlinear constraint optimization problem. A framework of grid interior point method is presented to solve the rolling nonlinear constraint optimization problem. Based on MATLAB-GUI platform, the corresponding ingredient ratio software is devised to obtain optimal ingredient ratio. Finally, several numerical examples are presented to study and solve ingredient ratio optimization problems.


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## Body

## 1. Introduction

Cement is a widely used construction material in the world. Cement production will experience several procedures which include raw materials blending process and burning process, cement clinker grinding process, and packaging process. Cement raw material and cement clinkers mainly contain four oxides: calcium oxide or lime (CaO), silica (SiO2), alumina (Al2O3), and iron oxide (Fe2O3). The cement clinkers quality is evaluated by the above four oxides. Hence, ingredient ratio of cement raw material will affect the quality and property of cement clinker significantly. Optimal ingredient ratio will promote and stabilize cement quality and production craft. Therefore, cement raw materials should be reasonably mixed. Hence, it is a significant problem to obtain optimal ingredient ratio.Many publications have studied various cement processes in cement production. In [4], under different ball charge filling ratios, ball sizes, and residence time, a continuous ball mill is studied for optimizing cement raw material grinding process. In [5], an adaptive control framework is presented for raw material blending process, and corresponding optimal control structure is discussed too. In [6, 7], control strategies are presented for cement raw material blending process by the least square methods, neural network methods, and adaptive neural-fuzzy inference methods. In [8–10], model identification and advanced control problems have been discussed, through considering chemical composition variations disturbance, and model predictive controller is used to calculate optimal raw material feed ratio. In [11], a time-varying Kalman filter is proposed to recursive estimate oxide composition of cement raw material via X-ray analysis. In [12], a T-S fuzzy controller is proposed to improve real-time performances in blending process. In [13], the feeder, ball mill, and homogenizing silo are seen as a whole system, input and output data are used to analyze blending process. In [1, 14, 15], fuzzy neural network with particle swarm optimization (FNN-PSO) methods and artificial neural network (ANN) are applied to establish and optimize cement raw material blending process. In [2, 16–24], algebraic methods, least square methods, neural network methods, linear program methods, and empirical methods are used to compute or obtain optimal ingredient ratios in cement raw material blending process. In [3, 25–28], new original raw materials and instruments are introduced in blending process. In [29–33], cement production problems are discussed.This does not give much attention to modeling and obtaining optimal ingredient ratio in blending process. In this paper, ingredient ratio optimization problem is analyzed for cement raw material blending process under various conditions. A G-NLTV model is established for cement raw material blending process. The ingredient ratio optimization problem can be equivalently transformed into convex problems. A framework of grid interior point method is proposed to solve ingredient ratio optimization problem. A software is developed to solve the ingredient ratio optimization problem through MATLAB-GUI. This paper is arranged as follows: raw material blending process and critical cement craft parameters are introduced in Section2; G-NLTV model of raw material blending process under various circumstances is established in Section 3; the grid interior point method framework and cement ingredient software are presented in Section 4; numerical examples in blending process are presented in Section 5; paper contents are concluded in Section 6.


## 2. Raw Material Blending Process and Critical Cement Craft  Parameters

Cement production process could be roughly divided into three stages. The first stage is to make cement raw material, which contains raw material blending process and grinding process. The second stage and third stage are to burn the raw material and grind cement clinkers respectively. The cement raw material blending process is an important link because the blending process will affect the cement clinker quality and critical cement craft parameters, thus the blending process finally affects the cement quality. Figure1 demonstrates cement raw material blending process and its control system. Cement original materials are usually the limestone, steel slag, shale, sandstone, clay, and correct material. The original cement materials should be blended in a reasonable proportion, and then original cement materials are transported into the ball mill which grinds original cement materials into certain sizes. The classifier selects suitable size of original cement material which is transported to the cement kiln for burning.Figure 1
The cement raw material blending process and its control system.The quality of cement raw material and cement clinkers are evaluated by the cement lime saturation factor (LSF), silicate ratio (SR), and aluminum-oxide ratio (AOR). LSF, SR, and AOR are directly determined by the lime, silica, alumina, and iron oxide which are contained in cement raw material. The LSF, SR, and AOR are critical cement craft parameters, thus ingredient ratio determines critical cement crafts parameters. Likewise, critical cement craft parameters are also used to assess the blending process. In cement production, the LSF, SR, and AOR must be controlled or stabilized in reasonable range. Critical cement craft parameters are not stabilized, so it cannot produce high qualified cement. The X-ray analyzer in Figure1 is used to analyze chemical compositions of the original cement material or raw material, then X-ray analyzer can feedback LSF, SR, and AOR in fixed sample time. The LSF, SR, and AOR can be affected by many uncertain factors such as composition fluctuation, and material feeding flow. Table 1 shows the chemical composition of original cement materials. Chemical composition is the time-varying function.  The symbols μj=μj(t), ηj=ηj(t),…, ωj=ωj(t), and φj=φj(t) represent chemical composition of original cement material-j. In Table 1, R2O represents total chemical composition of sodium oxide (Na2O) and potassium (K2O).Table 1
The chemical composition of cement original material.
Composition of the original cement material
SiO2  (%)
Al2O3  (%)
Fe2O3  (%)
CaO  (%)
MgO  (%)
R2O  (%)
SO3  (%)
TiO2  (%)
Cl  (%)
Impurity  (%)
Loss  (%)
Cement material type
Material-1
μ
1
η
1
ρ
1
γ
1
τ
1
r
1
s
1
λ
1
π
1
ω
1
φ
1
Material-2
μ
2
η
2
ρ
2
γ
2
τ
2
r
2
s
2
λ
2
π
2
ω
2
φ
2
…
…
…
…
…
…
…
…
…
…
…
…
Material-i
μ
i
η
i
ρ
i
γ
i
τ
i
r
i
s
i
λ
i
π
i
ω
i
φ
i
…
…
…
…
…
…
…
…
…
…
…
…
Material-n
μ
n
η
n
ρ
n
γ
n
τ
n
r
n
s
n
λ
n
π
n
ω
n
φ
n
Description
Active ingredients in cement
Harmful ingredients in cementWhy chemical composition is the time-varying function? Original cement materials are obtained from nature mine, thus chemical composition is time-varying function. Composition fluctuation is inevitable and it may contain randomness. With economic development, resource consumption is expanding and the resources are consuming. Therefore, original cement materials with stable chemical composition become more and more difficult to find. From the perspective of protecting environment, cement production needs to use parts of waste and sludge, therefore original cement materials composition fluctuation will be enlarged in the long run.To some extent, modelling and optimization of the cement raw material blending process becomes more important and challenge. Because of different original cement material type, different chemical composition, and different requirements on critical cement craft parameters, ingredient ratio should be more scientific and reasonable in blending process. Therefore, ingredient ratio should adapt to the chemical composition fluctuation and guarantee critical cement craft parameters in permissible scope.


## 3. General Dynamic Model of Blending Process

The blending process is to produce qualified cement raw material. In cement raw material blending process, it is a key task to stabilize critical cement craft parameters LSF, SR, and AOR in permissible scope. In practice, formulas in [26, 34] are used to calculate LSF, SR, and AOR as follows:
(3.1)α=(Mγ-1.65Mη-0.35Mρ)(2.8Mμ),or:α=Mγ(2.8Mμ+1.18Mη+0.65Mρ),β=Mμ(Mη+Mρ),Ω=MηMρ,
where α is the LSF, β is the SR, and Ω is the AOR. Without losing generality, it assumes that there has n-type the original cement materials in blending process. The mass of CaO, SiO2, Al2O3, and Fe2O3 in cement raw material can be acquired as
(3.2)Mγ=γ1M1+⋯+γnMn=∑j=1nγjMj,Mμ=μ1M1+⋯+μnMn=∑j=1nμjMj,Mη=η1M1+⋯+ηnMn=∑j=1nηjMj,Mρ=ρ1M1+⋯+ρnMn=∑j=1nρjMj.
LSF, SR, and AOR are affected by the original cement materials mass or mass percentage. Obviously, LSF, SR, and AOR are affected by composition fluctuation. Equation (3.2) is equivalently expressed as
(3.3)MγM=(∑j=1nγjMj)M,MμM=(∑j=1nμjMj)MMηM=(∑j=1nηjMj)M,MρM=(∑j=1nρjMj)MM=M1+M2+⋯+Mn-1+Mn=∑j=1nMj,
where M is total mass of original cement material. Variables are normalized, and (3.3) is further expressed as
(3.4)mγ=∑j=1nγjxj=(γTx),mμ=∑j=1nμjxj=(μTx),mη=∑j=1nηjxj=(ηTx),mρ=∑j=1nρjxj=(ρTx),∑j=1nxj=1,xj=(MjM),x=(x1,…,xn)T,
where  xj(j=1,…,n) is the mass percentage of original cement material-j, x=(x1,…,xn)T is ingredient ratio (mass percentage vector), and γ, μ, η, and ρ are mass percentage vector of CaO, SiO2, Al2O3, and Fe2O3 for cement raw material, respectively. The ingredient ratio x is usually expressed by the percentage form, and mγ, mμ, mη, mργ, μ, η, and ρ are obtained as
(3.5)mγ=MγM,γ=(γ1,…,γn)T,mμ=MμM,μ=(μ1,…,μn)T,mη=MηM,η=(η1,…,ηn)T,mρ=MρM,ρ=(ρ1,…,ρn)T.
In practice, each type of original cement material will possess a certain proportion, thus mass percentage xj will yield
(3.6)εj≤xj≤1,0≤εj≤1(j=1,2,…,n),
where εj(j=1,2,…,n) is minimum mass percentage of original cement material-j. Minimum mass percentage εj is decided by cement production crafts. In cement production, the mass percentage mγ of cement raw material should be limited in permissible scope. Otherwise, cement will lose its inherent nature property as
(3.7)mγ-≤mγ≤mγ+,mγ+=δr0+Δr0,mγ-=δr0-Δr0,
where δγ0 is the expected mass percentage of CaO, Δγ0 is the maximum fluctuation scope, and mγ- and mγ+ are the lower bounded and upper bounded, respectively. δγ0 and Δγ0 are determined by cement production crafts. In actual cement production, critical cement craft parameters LSF, SR, and AOR should be stabilized in permissible scope as follows:
(3.8)α0-≤α≤α0+,β0-≤β≤β0+,Ω0-≤Ω≤Ω0+,
where α0-, β0-, and Ω0- are the minimum lower bounded of LSF, SR, and AOR respectively, and α0+, β0+, and Ω0+ are the maximum upper bounded of LSF, SR, and AOR respectively. Cement raw material is burned in the kiln, to guarantee the quality of the cement clinker, and burning loss and impurity ratio should be limited in allowable range. If raw material has too much impurity, it will affect the clinker quality. So, they can not exceed certain scope and will yield relationships as
(3.9)mφ=MφM,Mφ=φ1M1+⋯+φnMn,mω=MωM,Mω=ω1M1+⋯+ωnMn,mφ=φ1x1+⋯+φnxn=∑j=1nφjxj=φTx≤δφ0,mω=ω1x1+⋯+ωnxn=∑j=1nωjxj=ωTx≤δω0,
where δφ0 and δω0 are the maximum permission loss ratio and impurity ratio, respectively, and φ and ω are loss and impurity percentage vector, respectively. To restrict harmful ingredients and protect environment, harmful ingredients in cement raw material should be reduced as far as possible. In [29–33], it shows that too much harmful ingredients such as magnesium oxide, sodium oxide, trioxide, and potassium will affect burning process and cause cement kiln plug and crust. Harmful ingredients will affect cement clinkers quality and property. Therefore, toxic ingredients in cement raw material should be limited as follows:
(3.10)mτ=MτM,Mτ=τ1M1+⋯+τnMn,mr=MrM,Mr=r1M1+⋯+rnMn,ms=MsM,Ms=s1M1+⋯+snMn,mλ=MλM,Mλ=λ1M1+⋯+λnMn,mπ=MπM,Mπ=π1M1+⋯+πnMn,(3.11)mτ=τ1x1+⋯+τnxn=∑j=1nτjxj=τTx≤δτ0,mr=r1x1+⋯+rnxn=∑j=1nrjxj=rTx≤δr0,ms=s1x1+⋯+snxn=∑j=1nsjxj=sTx≤δs0,mλ=λ1x1+⋯+λnxn=∑j=1nλjxj=λTx≤δλ0,mπ=π1x1+⋯+πnxn=∑j=1nπjxj=πTx≤δπ0,
where δτ0, δr0, δs0, δλ0,  and  δπ0 are the permissible maximum mass percentage of MgO, R2O, SO3, TiO2, and Cl in cement raw material, respectively, and τ, r, s, λ, and  π are composition mass percentage vector of MgO, R2O, SO3, TiO2, and Cl, respectively.In cement production, the cement kiln can be divided into wet kiln and dry kiln. In [30–33], it shows that the cement raw material with high sulphur-alkali ratio (SAR) will cause some problems in dry kiln. Therefore, it is necessary to control the SAR for preventing cement kiln plug and crust. The cement raw material with small SAR will increase the flammability and improve the cement clinkers quality. Some formulas are presented to calculate the SAR for cement raw material. The world famous cement manufacturers such as KHD Humboldt Company, F.L.Smidth Company, and F.C.B Company in [31] propose their formulas to calculate SAR; in practice, any of the following formulas can be used to compute SAR:
(3.12)KHD Humboldt(δθ0=0.7~1.0):θ=Ms(0.85Mr1+1.29Mr2-1.119Mπ)≤δθ0,F.C.B(δθ0=0.3~1.2):θ=Ms(0.85Mr1+1.29Mr2)≤δθ0,F.L.Smidth(δθ0=0.3%):θ=Ms-(0.85Mr1+0.645Mr2)≤δθ0,
where θ is the SAR, Mr1 and Mr2 are the mass or mass percentage of K2O and Na2O, respectively, and δθ0 is the permissible maximum percentage. The Mr1 and Mr2 have the implicit relationships: Mr=Mr1+Mr2, Mr1=ξMr2. ξ is mass ratio between K2O and Na2O. The SAR is limited in permissible scope, which will reduce the environmental pollution. Strictly speaking, the blending process does not include the cement ball mill grinding process. Before cement raw materials are transported into the cement burning kiln, cement raw material blending process is considered as a whole process, thus the grinding process could be seen as part of blending process. For integrity and generality, we consider that the cement raw material blending process includes ball mill grinding process. Then, the mass balance equation of active ingredients SiO2  in ball mill could be obtained as follows:
(3.13)d(Qmμ)dt=Finput-Foutput⟺∑j=1n(μjxjdQdt+Qμjdxjdt+Qxjdμjdt)=Qinput×∑j=1nμjxj-Q×∑j=1nkμ,jμjxj,Finput=Qinputmμ,Foutput=kQmμ+Q×∑j=1nνμ,jμjxjmμ=∑j=1nμjxj,kμ,j=k+νμ,j,(j=1,2,…,n),
where Q is original cement material output flow in ball mill, Qinput is original cement material feed flow, Finput is SiO2 mass in feed flow, Foutput is SiO2 mass in output flow, and kμ,j is the SiO2 ouput mass coefficient of original cement material-j. In (3.13), it assumes that output mass is proportional to the material flow in ball mill and mass composition percentage. Likewise, the A12O3, Fe2O3, and CaO mass balance equation of active ingredients in ball mill will be obtained as follows:
(3.14)∑j=1n(ηjxjdQdt+Qηjdxjdt+Qxjdηjdt)=Qinput×∑j=1nηjxj-Q×∑j=1nkη,jηjxj,{kη,j=k+νη,j,(j=1,2,…,n)},∑j=1n(ρjxjdQdt+Qρjdxjdt+Qxjdρjdt)=Qinput×∑j=1nρjxj-Q×∑j=1nkρ,jρjxj,{kρ,j=k+νρ,j,(j=1,2,…,n)},∑j=1n(γjxjdQdt+Qγjdxjdt+Qxjdγjdt)=Qinput×∑j=1nγjxj-Q×∑j=1nkγ,jγjxj,{kη,j=k+νγ,j,(j=1,2,…,n)}.Therefore, the MgO, R2O, SO3, TiO2, and Cl mass balance equation of harmful ingredients in ball mill could be obtained as follows:
(3.15)∑j=1n(τjxjdQdt+Qτjdxjdt+Qxjdτjdt)=Qinput×∑j=1nτjxj-Q×∑j=1nkτ,jτjxj,{kτ,j=k+ντ,j,(j=1,2,…,n)},∑j=1n(rjxjdQdt+Qrjdxjdt+Qxjdrjdt)=Qinput×∑j=1nrjxj-Q×∑j=1nkr,jrjxj,{kr,j=k+νr,j,(j=1,2,…,n)},∑j=1n(sjxjdQdt+Qsjdxjdt+Qxjdsjdt)=Qinput×∑j=1nsjxj-Q×∑j=1nks,jsjxj,{ks,j=k+νs,j,(j=1,2,…,n)},∑j=1n(λjxjdQdt+Qλjdxjdt+Qxjdλjdt)=Qinput×∑j=1nλjxj-Q×∑j=1nkλ,jλjxj,{kλ,j=k+νλ,j,(j=1,2,…,n)},∑j=1n(πjxjdQdt+Qπjdxjdt+Qxjdπjdt)=Qinput×∑j=1nπjxj-Q×∑j=1nkπ,jπjxj,{kπ,j=k+νπ,j,(j=1,2,…,n)}.
The impurity and loss mass balance equation of ball mill in blending process could be also obtained as follows:
(3.16)∑j=1n(ωjxjdQdt+Qωjdxjdt+Qxjdωjdt)=Qinput×∑j=1nωjxj-Q×∑j=1nkω,jωjxj,{kω,j=k+νω,j,(j=1,2,…,n)},(3.17)∑j=1n(φjxjdQdt+Qφjdxjdt+Qxjdφjdt)=Qinput×∑j=1nφjxj-Q×∑j=1nkφ,jφjxj,{kφ,j=k+νφ,j,(j=1,2,…,n)}.In order to obtain the general nonlinear time-varying dynamic optimization model, we needs to select suitable optimization objective function. In practice, many factors should be considered such as original cement material cost, grind ability, and the error between the actual critical craft and desired critical craft. To reduce the cement cost, an optimal ingredient ratio should be pursued to reduce the original cement material cost. Thus, original cement material cost function is acquired as(3.18)minJ1=min(∑j=1nxjCj)=min(CTx),
where Cj (¥/ton) is the cost of original cement material-j, and J1 is the cost function. To improve the grind-ability, it can pursue an optimal ingredient ratio to reduce the electrical power consumption. Thus, the power consumption function is acquired as
(3.19)minJ2=min(∑j=1nxjPj)=min(PTx),
where Pj (Kwh/ton) is bond grinding power index of original cement material-j, and J2 is power consumption function. Pj represents the grind ability of original cement material-j and also can reflect the ball mill power consumption. To reduce critical cement craft error, it can pursue an optimal ingredient ratio to reduce LSF, SR, and AOR error. Hence, the critical cement craft error function J3 is obtained as follows:
(3.20)minJ3=min{w1(Δα)2+w2(Δβ)2+w3(ΔΩ)2},Δα=α-αd0,Δβ=β-βd0,ΔΩ=Ω-Ωd0,
where wj  (j=1,2,3) is the weight of LSF error, SR error, and AOR error, Δα, Δβ, and ΔΩ are the error of LSF, SR, and AOR, and αd0, βd0, and Ωd0 are the expected LSF, SR, and AOR. Based on the cement production requirements, various objective functions are obtained. Finally, G-NLTV dynamic optimization models of cement raw material blending process are obtained as
(3.21)Model.1:minJ1=min(CTx)Model.2:minJ2=min(PTx)Model.3:minJ3=min(w1(Δα)2+w2(Δβ)2+w3(ΔΩ)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(s.t)(3.1)–(3.12),(3.13)–(3.17),
where Ψ1, Ψ2, and Ψ3 are the function weight. The G-NLTV dynamic optimization model includes the single objective and multiple objectives optimization model. All the optimization models contain algebraic constraints and dynamic constraints.


## 4. Analysis of Ingredient Ratio Optimization Problem and Grid Interior Point Framework

The object functionsJ1, J2, and J3 in dynamic optimization models are the convex functions. The Ψ1J1+Ψ2J2,Ψ1J1+Ψ2J3,Ψ1J2+Ψ2J3, and the Ψ1J1+Ψ2J2+Ψ3J3 are also the convex functions. As known, the convex optimization problems have good convergent properties. The optimization problems are the convex optimization problem which is determined by their objective function and constraints. We need to check the constraints of optimization problems shown in Table 2. The constraints (3.1)–(3.12) are algebraic constraints and constraints (3.13)–(3.17) are dynamic constraints. The algebraic constraints and dynamic constraints construct the feasible regions of the optimization problem. The feasible region of constraint (3.12) and constraint (3.8) are obtained as
(4.1)Fθ={x∣θ=Ms(0.85Mr1+1.29Mr2-1.119Mπ)≤δθ0},Fα,β,Ω={x∣α0-≤α≤α0+,β0-≤β≤β0+,Ω0-≤Ω≤Ω0+},
where Fθ and Fα,β,Ω are the feasible regions constructed by constraints (3.12) and (3.8), respectively. SAR θ is equivalently expressed as
(4.2)θ=(Ms/M)((0.85Mr1+1.29Mr2-1.119Mπ)/M)⟺θ=ms(0.85mr1+1.29mr2-1.119mπ)⟺θ=ms((0.85ξ+1.29)mr/(1+ξ)-1.119mπ).
Then, feasible region Fθ can be equivalently written as
(4.3)Fθ={x∣ms((0.85ξ+1.29)mr/(1+ξ)-1.119mπ)≤δθ0}⟺Fθ={x∣(1+ξ)ms≤((0.85ξ+1.29)mr-1.119(1+ξ)mπ)δθ0}⟺Fθ={x∣(1+ξ)ms+1.119(1+ξ)δθ0mπ-(0.85ξ+1.29)δθ0mr≤0}.
Likewise, critical cement craft parameters α, β, and Ω can be equivalently expressed as
(4.4)α=((Mγ-1.65Mη-0.35Mρ)/M)((2.8Mμ)/M)⟺α=(mγ-1.65mη-0.35mρ)(2.8mμ)β=(Mμ/M)((Mη+Mρ)/M)⟺β=mμ(mη+mρ)Ω=(Mη/M)(Mρ/M)⟺Ω=mηmρ.
Then, feasible region Fα,β,Ω can be equivalently written as
(4.5)Fα,β,Ω={x∣(mγ-1.65mη-0.35mρ)(2.8mμ)≥α0-,(mγ-1.65mη-0.35mρ)(2.8mμ)≤α0+,mμ(mη+mρ)≥β0-,mμ(mη+mρ)≤β0+,mηmρ≥Ω0-,mηmρ≤Ω0+}⟺Fα,β,Ω={x∣mγ-1.65mη-0.35mρ-2.8α0-mμ≥0,mγ-1.65mη-0.35mρ-2.8α0+mμ≤0,mμ-β0-(mη+mρ)≥0,mμ-β0+(mη+mρ)≤0,mη-Ω0-mρ≥0,mη-mρΩ0+≤0}⟺Fα,β,Ω={x∣mγ-1.65mη-0.35mρ-2.8α0-mμ≥0,mμ-β0-(mη+mρ)≥0,mη-Ω0-mρ≥0}∩{x∣mγ-1.65mη-0.35mρ-2.8α0+mμ≤0,mμ-β0+(mη+mρ)≤0,mη-mρΩ0+≤0}.
In the previous section, we know that the ms, mπ, mr, mγ, mη, mρ, and mμ are the linear functions of the ingredient ratio (original cement materials mass percentage vector) x=(x1,x2,…,xn)T. Therefore, feasible region Fθ and Fα,β,Ω are the convex or semiconvex region. Constraints (3.8) and (3.12) are nonlinear algebraic constraints, but their feasible regions are also convex or semiconvex region. Hence, feasible regions constructed by constraints (3.1)–(3.12) are obtained as
(4.6)F={x∣Fθ∩Fα,β,Ω∩Fo.e}∈Π,
where F is the feasible region constructed by constraints (3.1)–(3.12), Fo.e is the feasible region constructed by constraints (3.1)–(3.10), and Π is the convex and semiconvex regions set. The constraints (3.1)–(3.10) are the linear algebraic constraints. Hence, the feasible region Fo.e constructed by constraints (3.1)–(3.10) is the convex or semiconvex. Therefore, feasible regions constructed by constraints (3.1)–(3.12) belong to convex or semiconvex region.Table 2
G-NLTV dynamic optimization models of the cement raw material blending process.
Optimization models
Optimization objective functions
Constraints
Single-objectiveoptimization
Model.1
min
J
1
=
min
(
C
T
x
)
(3.1)–(3.12), (3.13)–(3.17)
Model.2
min
J
2
=
min
(
P
T
x
)
(3.1)–(3.12), (3.13)–(3.17)
Model.3
min
J
3
=
min
{
w
1
(
Δ
α
)
2
+
w
2
(
Δ
β
)
2
+
w
3
(
Δ
Ω
)
2
}
(3.1)–(3.12), (3.13)–(3.17)
(Notes:Δα=α-αd0,Δβ=β-βd0,ΔΩ=Ω-Ωd0)
Multiple-objectiveoptimization
Model.4
min
(
J
1
,
J
2
)
=
min{Ψ1J1+Ψ2J2}
(3.1)–(3.12), (3.13)–(3.17)
Model.5
min
(
J
1
,
J
3
)
=
min{Ψ1J1+Ψ2J3}
(3.1)–(3.12), (3.13)–(3.17)
Model.6
min
(
J
2
,
J
3
)
=
min{Ψ1J2+Ψ2J3}
(3.1)–(3.12), (3.13)–(3.17)
Model.7
min
(
J
1
,
J
2
,
J
3
)
=
min{Ψ1J1+Ψ2J2+Ψ3J3}
(3.1)–(3.12), (3.13)–(3.17)The constraints (3.13)–(3.17) are the time-varying differential equation constraints in dynamic model. The constraints (3.13)–(3.17) can be equally written as the following vector form:
(4.7)dQdtμTx+QxTdμdt+QμTdxdt=QinputμTx-QμTΛμx,Λμ=diag(kμ,1,…,kμ,n),dQdtηTx+QxTdηdt+QηTdxdt=QinputηTx-QηTΛηx,Λη=diag(kη,1,…,kη,n),dQdtρTx+QxTdρdt+QρTdxdt=QinputρTx-QρTΛρx,Λρ=diag(kρ,1,…,kρ,n),dQdtγTx+QxTdγdt+QγTdxdt=QinputγTx-QγTΛγx,Λγ=diag(kγ,1,…,kγ,n),dQdtτTx+QxTdτdt+QτTdxdt=QinputτTx-QτTΛτx,Λτ=diag(kτ,1,…,kτ,n),dQdtrTx+QxTdrdt+QrTdxdt=QinputrTx-QrTΛrx,Λr=diag(kr,1,…,kr,n),dQdtsTx+QxTdsdt+QsTdxdt=QinputsTx-QsTΛsx,Λs=diag(ks,1,…,ks,n),dQdtλTx+QxTdλdt+QλTdxdt=QinputλTx-QλTΛλx,Λλ=diag(kλ,1,…,kλ,n),dQdtπTx+QxTdπdt+QπTdxdt=QinputπTx-QπTΛπx,Λπ=diag(kπ,1,…,kπ,n),dQdtωTx+QxTdωdt+QωTdxdt=QinputωTx-QωTΛωx,Λω=diag(kω,1,…,kω,n),dQdtφTx+QxTdφdt+QφTdxdt=QinputφTx-QφTΛφx,Λφ=diag(kφ,1,…,kφ,n).
The constraints (3.13)–(3.17) in the dynamic optimization model reveal that fluctuations of the cement material flow and chemical composition will have important effects on cement raw material ingredient ratio. The derivative of ingredient ratio is affected by the chemical composition and cement material flow. In practical cement production, chemical composition is analyzed and updated by the X-ray analyzer in fixed sampling period which may be quarter hour, half hour, one hour, and even longer. Therefore, it is hard to accurately solve dynamic optimization problem (3.21) because chemical composition and cement material flow could not be continuously and accurately obtained. To simplify the dynamic model, it is assumed that the derivative of feed flow Q and ingredient ratio x are minor, and they can be ignored. Then, the constraint (4.7) can be equivalently expressed as
(4.8)QxTdμdt≅QinputμTx-QμTΛμx,QxTdηdt≅QinputηTx-QηTΛηx,QxTdρdt≅QinputρTx-QρTΛρx,QxTdγdt≅QinputγTx-QγTΛγx,QxTdτdt≅QinputτTx-QτTΛτx,QxTdrdt≅QinputrTx-QrTΛrx,QxTdsdt≅QinputsTx-QsTΛsx,QxTdλdt≅QinputλTx-QλTΛλx,QxTdπdt≅QinputπTx-QπTΛπx,QxTdωdt≅QinputωTx-QωTΛωx,QxTdφdt≅QinputφTx-QφTΛφx,(dQdt≈0,dxdt≈0).
The original cement materials chemical composition will fluctuate with time. To solve the optimization problem (3.21), dynamic optimization models should be transformed into discrete form. Thus, dynamic constraint (4.8) in optimization model can be transformed into the following discrete forms:
(4.9)μ(k)=fμ(μ(k-1),x),η(k)=fη(η(k-1),x),ρ(k)=fρ(ρ(k-1),x),γ(k)=fγ(γ(k-1),x),τ(k)=fτ(τ(k-1),x),r(k)=fr(r(k-1),x),s(k)=fs(s(k-1),x),λ(k)=fλ(τ(k-1),x),π(k)=fπ(π(k-1),x),ω(k)=fω(ω(k-1),x),φ(k)=fφ(φ(k-1),x).
It is noted μ(k)=μ(kTs),…,φ(k)=φ(kTs), and x=x(k) in (4.9), and Ts is the sampling period. Differential equation is transformed into difference equation. Constraints (3.1)–(3.12) in dynamic optimization model are transformed into the following discrete forms:
(4.10)h(μ(k),…,φ(k),x)=0g(μ(k),…,φ(k),x)≤0⟺h(fμ(μ(k-1),x),…,fφ(φ(k-1),x),x)=0g(fμ(μ(k-1),x),…,fφ(φ(k-1),x),x)≤0,
where h(·) and g(·) are the discrete equality and inequality constraint vectors, respectively. Hence, the continuous time dynamic model is transformed into the following discrete time form:
(4.11)Model.1:minJ1=min(CTx)Model.2:minJ2=min(PTx)Model.3:minJ3(k,x)=min(w1(Δα(k,x))2+w2(Δβ(k,x))2+w3(ΔΩ(k,x))2)Δα(k,x)=Γα(γ(k),η(k),ρ(k),μ(k),x)Δβ(k,x)=Γβ(μ(k),η(k),ρ(k),x)ΔΩ(k,x)=ΓΩ(η(k),ρ(k),x)Model.4:min(J1,J2)=min{Ψ1J1+Ψ2J2}Model.5:min(J1,J3(k,x))=min{Ψ1J1+Ψ2J3(k,x)}Model.6:min(J2,J3(k,x))=min{Ψ1J2+Ψ2J3(k,x)}Model.7:min(J1,J2,J3(k,x))=min{Ψ1J1+Ψ2J2+Ψ3J3(k,x)}s.t:(4.9)-(4.10).It should be noted that (i) the continuous time dynamic optimization model is transformed into discrete time rolling optimization model; (ii) chemical composition and cement material flow cannot be obtained in a continuous and accurate way, thus it is necessary to transform the continuous model into the discrete model; (iii) it is difficult and complex to directly solve the continuous-time dynamic ingredient ratio model; (iv) the dynamic model of discrete time form is equivalent to a static optimization problem in a specific sampling time. Without losing the generality, the discrete time model can be expressed as the general form in a specific sampling period as follows:(4.12)minf(x)s.t:h(x)=0,g(x)≤0,a≤x≤b,
where f(x):Rn→R, h(x):Rn→Rm, and g(x):Rn→Rq are the smooth and differentiable functions, x is the decision variable, and n, m, and q denote the number of the decision variables, equality constraints, and inequality constraints, respectively. The discrete model is seen as a general linear or nonlinear static optimization problem in certain sampling period. In recent years, various optimization algorithms and optimization toolboxes or softwares in [35–53] are developed to solve optimization problems. The optimization methods in [35–53] such as the Newton methods, conjugate gradient methods, steepest descent methods, interior point methods, trust region methods, quadratic programming (QP) methods, successive linear programming (SLP) methods, sequential quadratic programming (SQP) methods, genetic algorithms, and particle swarm algorithms are well established to solve constraint optimization problems. Based on interior point methods in [43–51], a framework of grid interior point method is presented for dynamic cement ingredient ratio optimization problem. The optimization problem (4.12) could be transformed into following form:
(4.13)minf(x)-υ∑i=1qlnδis.t:h(x)=0,gε(x)+δ=0gε(x)=(g(x)T,(a-x)T,(x-b)T)T,
where υ>0 is the barrier parameter, the slack vector δ=(δ1,δ2,…,δq)T>0 is set to be positive, and gε(x) is an expanded inequality constraint. It introduces the Lagrange multipliers y and z for barrier problem (4.13) as follows:
(4.14)L(x,y,z,δ)=f(x)-υ∑i=1qlnδi+yT(gε(x)+δ)+zTh(x),
where L(x,y,z,δ) is Lagrange function, y=(y1,y2,…,yq)T and  z=(z1,z2,…,zm)T are Lagrange multipliers for constraints gε(x)+δ and h(x), respectively. Based on Karush-Kuhn-Tucker (KKT) optimality conditions [41–43], optimality conditions for optimization problem (4.13) can be expressed as
(4.15)∇xL(x,y,z,δ)=∇f(x)+(∇gε(x))Ty+(∇h(x))Tz=0,∇δL(x,y,z,δ)=-υSδ-1e+y=0⟺-υe+SδYe=0,∇yL(x,y,z,δ)=gε(x)+δ=0,∇zL(x,y,z,δ)=h(x)=0,
where Sδ=diag(δ1,δ2,…,δq) is the diagonal matrix and its elements are the components of the vector δ, e is a vector of all ones, ∇h(x) and ∇gε(x) are the Jacobian matrices of the vectors h(x) and gε(x), respectively, ∇f(x) is the grand of function f(x), and Y=diag(y1,y2,…,yq) is a diagonal matrix and its elements are the components of vector y. The system (4.15) is KKT optimal condition of optimization problem (4.13). When in the search procedure, δ and y should remain positive (δ>0,y>0). To obtain the iteration direction, we can make the point (x+Δx,δ+Δδ,z+Δz,y+Δy) satisfy the KKT conditions (4.15), then the following system will be obtained:
(4.16)∇f(x+Δx)+(∇gε(x+Δx))T(y+Δy)+(∇h(x+Δx))T(z+Δz)=0-υe+(Sδ+ΔSδ)(Y+ΔY)e=0gε(x+Δx)+(δ+Δδ)=0,h(x+Δx)=0⇒(4.17)(∇2f(x)+∇2gε(x)Ty+∇2h(x)Tz)Δx+∇gε(x)TΔy+∇h(x)TΔz+(∇f(x)+∇gε(x)Ty+∇h(x)Tz)=0,-υe+SδYe+SδΔy+YΔδ=0,gε(x)+∇gε(x)Δx+δ+Δδ=0,∇h(x)Δx+h(x)=0.
The system (4.17) is obtained by ignoring the higher order incremental system (4.16), and replacing nonlinear terms with linear approximation, system (4.17) is written in the following matrix form:
(4.18)(H(x,y,z)0∇h(x)T∇gε(x)T0Sδ-1Y0I∇h(x)000∇gε(x)I00)(ΔxΔδΔyΔz)=(-∇f(x)-∇gε(x)Ty-∇h(x)TzυSδ-1e-y-h(x)-gε(x)-δ)(4.19)H(x,y,z)=∇2f(x)+∇2gε(x)Ty+∇2h(x)Tz,
where H(x,y,z) is the Hessian matrix in system. Finally, the new iterate direction is obtained via solving the system (4.18), which is the essential process of the interior point method. Thus, the new iteration point can be obtained in the following iteration:
(4.20)(x,δ,z,y)⟵(x,δ,z,y)+ζ1(Δx,Δδ,Δz,Δy),
where ζ1 is the step size. Choosing the step size ζ1 holds the δ, y>0 in search process. In this paper, a framework of grid interior point method is presented for optimization problem of ingredient ratio in raw material blending process. The grid interior point method framework is depicted as follows.


### 4.1. Grid Interior Point Method Framework

The following steps are considered.Step 1.
The feasible regionF={x∣h(x)=0,g(x)≤0,a≤x≤b} is divided into N small pieces of feasible region without any intersection (F=UFi), and Fi={x∣h(x)=0,g(x)≤0,a+(i-1)×Θ≤x≤a+i×Θ}, and Θ is the interval length (Θ=(b-a)/N;i=1,2,…,N).Step 2.
Fori=1:N,each small feasible region will do the following steps.Step 3.
Choose an initial iteration point(x(0,i),δ(0,i),z(0,i),y(0,i)) in the feasible region set Fi={x∣h(x)=0,g(x)≤0,a+(i-1)×Θ≤x≤a+i×Θ}, and the δ(0)>0, y(0)>0, k=0.Step 4.
Constructing current iterate, we have the current iterate valuex(k,i), δ(k,i), z(k,i) and y(k,i) of the primal variable x, the slack variable δ, and the multipliers y and z, respectively.Step 5.
Calculate the Hessian matrixH(x,y,z) of the Lagrange system L(x,y,z,δ), and the Jacobian matrix ∇h(x) and ∇g(x) are of the vectors h(x) and g(x) in the current iterate (x(k,i),δ(k,i),z(k,i),y(k,i)).Step 6.
Solve the linear system (4.18) and construct the iterate direction (Δx,Δδ,Δz,Δy). Solve the linear matrix equation (4.18), and then we can obtain the primal solution Δx, multipliers solution Δz, Δy, and also the slack variable solution Δδ.Step 7.
Choosing the step sizeζ1 holds the δ, y>0 in the search process, ζ1∈[0,1]. Update the iterate values: (x(k+1,i),δ(k+1,i),z(k+1,i),y(k+1,i))←(x(k,i),δ(k,i),z(k,i),y(k,i))+ζ1(Δx,Δδ,Δz,Δy),k←k+1.Step 8.
Check the ending conditions for regionFi. If it is not satisfied, go to Step 5, else the minimum fmin,i of feasible region Fi is obtained, i←i+1, go to Step 3.Step 9.
Compare the minimumfmin,i of feasible region Fi, output the minimum fmin=min{fmin,i(i=1,2,…,N)}, end.Based on the grid interior point method framework, the algorithm structure diagram of cement raw material blending process is shown in Figure2. In this paper, we develop the ingredient ratio software for cement raw material blending process based on the MATLAB-GUI and grid interior point method. The ingredient ratio software interface is shown in Figures 3 and 4. The ingredient ratio software has strong features which include single objective optimization model, multiple objectives optimization model, and robust ingredient ratio. The software achieves ingredient ratio for four, five, and six types of original cement materials, of course the software can be further improved to achieve ingredient ratio for more types of original cement materials. In practice, it does not exceed eight types of original cement material.Figure 2
Optimization algorithm structure diagram of cement raw material blending process.Ingredient ratio software for cement raw material blending process.
(a)
(b)
(c)
(d)Figure 4
Optimization results of ingredient ratio software for cement raw material blending process.


## 4.1. Grid Interior Point Method Framework

The following steps are considered.Step 1.
The feasible regionF={x∣h(x)=0,g(x)≤0,a≤x≤b} is divided into N small pieces of feasible region without any intersection (F=UFi), and Fi={x∣h(x)=0,g(x)≤0,a+(i-1)×Θ≤x≤a+i×Θ}, and Θ is the interval length (Θ=(b-a)/N;i=1,2,…,N).Step 2.
Fori=1:N,each small feasible region will do the following steps.Step 3.
Choose an initial iteration point(x(0,i),δ(0,i),z(0,i),y(0,i)) in the feasible region set Fi={x∣h(x)=0,g(x)≤0,a+(i-1)×Θ≤x≤a+i×Θ}, and the δ(0)>0, y(0)>0, k=0.Step 4.
Constructing current iterate, we have the current iterate valuex(k,i), δ(k,i), z(k,i) and y(k,i) of the primal variable x, the slack variable δ, and the multipliers y and z, respectively.Step 5.
Calculate the Hessian matrixH(x,y,z) of the Lagrange system L(x,y,z,δ), and the Jacobian matrix ∇h(x) and ∇g(x) are of the vectors h(x) and g(x) in the current iterate (x(k,i),δ(k,i),z(k,i),y(k,i)).Step 6.
Solve the linear system (4.18) and construct the iterate direction (Δx,Δδ,Δz,Δy). Solve the linear matrix equation (4.18), and then we can obtain the primal solution Δx, multipliers solution Δz, Δy, and also the slack variable solution Δδ.Step 7.
Choosing the step sizeζ1 holds the δ, y>0 in the search process, ζ1∈[0,1]. Update the iterate values: (x(k+1,i),δ(k+1,i),z(k+1,i),y(k+1,i))←(x(k,i),δ(k,i),z(k,i),y(k,i))+ζ1(Δx,Δδ,Δz,Δy),k←k+1.Step 8.
Check the ending conditions for regionFi. If it is not satisfied, go to Step 5, else the minimum fmin,i of feasible region Fi is obtained, i←i+1, go to Step 3.Step 9.
Compare the minimumfmin,i of feasible region Fi, output the minimum fmin=min{fmin,i(i=1,2,…,N)}, end.Based on the grid interior point method framework, the algorithm structure diagram of cement raw material blending process is shown in Figure2. In this paper, we develop the ingredient ratio software for cement raw material blending process based on the MATLAB-GUI and grid interior point method. The ingredient ratio software interface is shown in Figures 3 and 4. The ingredient ratio software has strong features which include single objective optimization model, multiple objectives optimization model, and robust ingredient ratio. The software achieves ingredient ratio for four, five, and six types of original cement materials, of course the software can be further improved to achieve ingredient ratio for more types of original cement materials. In practice, it does not exceed eight types of original cement material.Figure 2
Optimization algorithm structure diagram of cement raw material blending process.Ingredient ratio software for cement raw material blending process.
(a)
(b)
(c)
(d)Figure 4
Optimization results of ingredient ratio software for cement raw material blending process.


## 5. Numerical Results for Blending Process

In production, many field operating engineers will give an ingredient ratio of original cement materials based on critical cement crafts and their experiences. In this paper, a G-NLTV model and ingredient ratio software are shown to provide optimal ingredient ratios for cement raw material blending process under different production requirements. Three numerical examples are shown to depict the proposed method. It does not consider the differential or difference equation constraint because output mass coefficient and flow of original cement materials are unknown. Tables3–5 in the Appendix display only original cement materials chemical composition in a specific sampling period, wherein the chemical composition in Table 3 [1] is used to produce cement raw materials by a cement enterprise in Shan Dong province of China.Table 3
Chemical compositions of cement original materials in certain sampling period [1].
Material type
SiO2 (%)
Al2O (%)
Fe2O (%)
CaO(%)
Loss(%)
Impurity(%)
Power(Kwh/ton)
Cost(¥/ton)
Limestone
4.50
0.99
0.24
45.00
40.56
2.91
12.45
25.00
Sandstone
65.00
5.76
1.61
0.52
2.62
8.90
12.94
15.00
Steel slag
17.50
6.90
29.00
31.49
0.30
13.45
19.89
68.00
Shale
45.31
23.30
6.10
8.63
10.34
6.32
28.60
20.00
Coal ash
59.26
24.55
8.07
3.73
8.32
6.07
28.60
20.00Table 4
Chemical compositions of cement original materials in certain sampling period [2].
Material type
Loss(%)
SiO2 (%)
Al2O3  (%)
Fe2O3  (%)
CaO(%)
MgO(%)
SO3  (%)
K2O (%)
Na2O (%)
Cl(%)
Limestone
40.09
8.52
1.23
1.31
46.05
2.49
0.02
0.21
0.07
0.0243
Clay
7.99
62.74
17.94
4.06
2.40
0.94
0.64
3.25
0.00
0.09
Iron
24.74
7.92
50.27
13.01
2.94
0.79
0.14
0.19
0.00
0.25
Correction
30.25
3.15
21.30
38.55
5.17
1.53
0.05
0.00
0.00
0.013
Coal ash
0.00
44.77
26.04
4.49
8.42
1.67
0.95
0.62
0.00
0.043
Assuming the cost and bond power index for the cement material in Table4 are 24.00 ¥/ton, 25.00 ¥/ton, 50.00 ¥/ton, 30.00 ¥/ton, 28.70 ¥/ton, 12.45 Kwh/ton, 12.10 Kwh/ton, 18.98 Kwh/ton, 14.70 Kwh/ton, and 15.66 Kwh/ton, respectively.Table 5
Chemical compositions of cement original materials in certain sampling period [3].
Material type
Loss (%)
SiO2 (%)
Al2O3 (%)
Fe2O3 (%)
CaO (%)
MgO (%)
Carbide slag
24.65
1.02
1.29
0.00
69.26
0.00
Clay
5.83
69.56
16.42
3.35
0.00
0.00
Sulfuric acid residue
1.06
11.05
2.22
77.85
2.45
2.71
Cinder
0.00
56.39
22.77
10.18
1.13
2.16
Assuming the cost and bond power index for the cement material in Table5 are 18.00 ¥/ton, 25.00 ¥/ton, 48.00 ¥/ton, 9.00 ¥/ton, 11.24 Kwh/ton, 12.50 Kwh/ton, 19.86 Kwh/ton, and 13.80 Kwh/ton, respectively.There are five types of original cement material in Table3, and they are the limestone, sandstone, steel slag, shale, and coal ash. The steel slag is the most expensive material, the sandstone is the cheapest material, the limestone has the best grind ability, and the shale has the poorest grind ability. The optimization models (discrete time) and optimal ingredient ratios under different production requirements are presented in Table 6 and Figure 5. Model.1 has the smallest cost with the optimal ingredient ratio x1=84.003%, x2=7.687%, x3=3.203%, x4=0.010%, and x5=5.097%. Model.2 has the smallest power consumption with the optimal ingredient ratio x1=84.145%, x2=8.021%, x3=3.795%, x4=0.010%, and x5=4.029%. Model.3 has the smallest critical cement craft deviation with optimal ingredient ratio x1=84.046%, x2=7.335%, x3=3.587%, x4=0.010%, and x5=5.021%. Model.4, Model.5, Model.6, and Model.7 are the multiple objectives optimization model which could be equivalently transformed into single objective optimization model via introducing weight Ψ1, Ψ2, and Ψ3. Model.4 makes balance between material cost and power consumption with optimal ingredient ratio x1=84.658%, x2=7.349%, x3=3.122%, x4=0.010%, and x5=4.681%. Model.5, Model.6, and Model.7 have the same optimal ingredient ratio with Model.1, Model.2, and Model.4, respectively because the objective function J3 is far less than the objective function J1 and J2. In addition, the weight of objective function J3 is not far larger than the weight of objective function J1 and J2, therefore they have the same optimal ingredient ratio.Table 6
Optimization models and results for cement materials in Table3.
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
(
w
1
=
0.5
,
w
2
=
0.3
,
w
3
=
0.2
,
α
d
0
=
1.00
,
β
d
0
=
2.70
,
Ω
d
0
=
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
)
M
μ
=
4.5
x
1
+
65
x
2
+
17.5
x
3
+
45.31
x
4
+
59.26
x
5
M
η
=
0.99
x
1
+
5.76
x
2
+
6.9
x
3
+
23.3
x
4
+
24.55
x
5
M
ρ
=
0.24
x
1
+
1.61
x
2
+
29.0
x
3
+
6.1
x
4
+
8.07
x
5
M
γ
=
45.0
x
1
+
0.52
x
2
+
31.49
x
3
+
8.63
x
4
+
3.73
x
5
x
1
+
x
2
+
x
3
+
x
4
+
x
5
=
1.00
x
1
≥
0
,
x
2
≥
0
,
x
3
≥
0
,
x
4
≥
ε
,
x
5
≥
0, (ε=0.0001)
(
2
)
α
=
(
M
γ
-
1.65
M
η
-
0.35
M
ρ
)
/
(
2.8
M
μ
)
β
=
M
μ
/
(
M
η
+
M
ρ
)
,
Ω
=
M
η
/
M
ρ
(
3
)
M
φ
=
40.56
x
1
+
2.62
x
2
+
0.3
x
3
+
10.34
x
4
+
8.32
x
5
≤
38.00
M
ω
=
2.91
x
1
+
8.9
x
2
+
13.45
x
3
+
6.32
x
4
+
6.07
x
5
≤
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
3
*
=
3.203
%
;
x
4
*
=
0.010
%
;
x
5
*
=
5.097
%
;
J
1
*
=
25.35308
;
Model.2
x
1
*
=
84.145
%
;
x
2
*
=
8.021
%
;
x
3
*
=
3.795
%
;
x
4
*
=
0.010
%
;
x
5
*
=
4.029
%
;
J
2
*
=
13.42398
;
Model.3
x
1
*
=
84.046
%
;
x
2
*
=
7.335
%
;
x
3
*
=
3.587
%
;
x
4
*
=
0.010
%
;
x
5
*
=
5.021
%
;
J
3
*
=
0.000
;
Model.4
x
1
*
=
84.658
%
;
x
2
*
=
7.349
%
;
x
3
*
=
3.122
%
;
x
4
*
=
0.010
%
;
x
5
*
=
4.681
%
;
J
1
*
+
J
2
*
=
38.86875
;
Ψ
1
=
Ψ
2
=
1.0
J
1
*
=
25.36388
Model.5
x
1
*
=
84.003
%
;
x
2
*
=
7.687
%
;
x
3
*
=
3.203
%
;
x
4
*
=
0.010
%
;
x
5
*
=
5.097
%
;
J
1
*
+
J
3
*
=
25.35828
;
Ψ
1
=
Ψ
2
=
1.0
J
1
*
=
25.35308
Model.6
x
1
*
=
84.145
%
;
x
2
*
=
8.021
%
;
x
3
*
=
3.795
%
;
x
4
*
=
0.010
%
;
x
5
*
=
4.029
%
;
J
2
*
+
J
3
*
=
13.42918
;
Ψ
1
=
Ψ
2
=
1.0
J
2
*
=
13.42398
Model.7
x
1
*
=
84.658
%
;
x
2
*
=
7.349
%
;
x
3
*
=
3.122
%
;
x
4
*
=
0.010
%
;
x
5
*
=
4.681
%
;
J
1
*
+
J
2
*
+
J
3
*
=
38.87395
.
Ψ
1
=
Ψ
2
=
Ψ
3
=
1.0
J
1
*
=
25.36388
J
2
*
=
13.50487
Wherex1, x2, x3, x4, and x5 are ingredient ratio of the limestone, sandstone, steel slag, shale, and coal ash, respectively.Figure 5
Optimal ingredient ratio for cement materials in Table3.There are five types of original cement materials in Table4, and they are the limestone, clay, iron, correction, and coal ash. The iron is the most expensive material, the limestone is the cheapest material, the clay has the best grind ability, and the iron has the poorest grind ability. The optimization models (discrete time) and optimal ingredient ratios under different production requirements are presented in Table 7 and Figure 6. Model.1 has the smallest material cost with the optimal ingredient ratio x1=88.257%, x2=7.503%, x3=0.010%, x4=3.731%, and x5=0.499%. Model.2 has the smallest power consumption with the optimal ingredient ratio x1=87.565%, x2=8.480%, x3=0.040%, x4=3.905%, and x5=0.010%. Model.3 has the smallest critical cement craft deviation with optimal ingredient ratio x1=87.805%, x2=7.791%, x3=0.878%, x4=3.516%, and x5=0.010%. Model.4 makes balance between material cost and power consumption with optimal ingredient ratio x1=87.555%, x2=8.414%, x3=0.010%, x4=3.912%, and x5=0.109%. Model.5, Model.6, and Model.7 have the same optimal ingredient ratio with Model.1, Model.2, and Model.4, respectively.Table 7
Optimization models and results for cement materials in Table4.
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
(
α
d
0
=
0.96
,
β
d
0
=
1.90
,
Ω
d
0
=
1.25
,
w
1
=
0.5
,
w
2
=
0.3
,
w
3
=
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
M
η
=
1.23
x
1
+
17.94
x
2
+
50.27
x
3
+
21.3
x
4
+
26.04
x
5
M
ρ
=
1.31
x
1
+
4.06
x
2
+
13.01
x
3
+
38.55
x
4
+
4.49
x
5
M
γ
=
46.05
x
1
+
2.40
x
2
+
2.94
x
3
+
5.17
x
4
+
8.42
x
5
x
1
+
x
2
+
x
3
+
x
4
+
x
5
=
1.0, x1≥0,  x2≥0, x3≥ε, x4≥0, x5≥ε, (ε=0.0001)
(
2
)
α
=
(
M
γ
-
1.65
M
η
-
0.35
M
ρ
)
/
(
2.8
M
μ
)
β
=
M
μ
/
(
M
η
+
M
ρ
), Ω=Mη/Mρ
(
3
)
M
φ
=
40.09
x
1
+
7.99
x
2
+
24.74
x
3
+
30.25
x
4
≤
39.00
M
τ
=
2.49
x
1
+
0.94
x
2
+
0.79
x
3
+
1.53
x
4
+
1.67
x
5
≤
3.00
M
s
=
0.02
x
1
+
0.64
x
2
+
0.14
x
3
+
0.05
x
4
+
0.95
x
5
≤
0.8
M
r
=
0.28
x
1
+
3.25
x
2
+
0.19
x
3
+
0.62
x
5
≤
0.9
M
r
1
=
0.21
x
1
+
3.25
x
2
+
0.19
x
3
+
0.62
x
5
≤
0.8
,
M
r
2
=
0.07
x
1
≤
0.1
M
π
=
0.0243
x
1
+
0.09
x
2
+
0.25
x
3
+
0.013
x
4
+
0.043
x
5
≤
0.2
(
4
)
θ
=
M
s
/
(
0.85
M
r
1
+
1.29
M
r
2
-
1.119
M
π
)
≤
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
3
*
=
0.010
%
;
x
4
*
=
3.731
%
;
x
5
*
=
0.499
%
;
J
1
*
=
24.32497
;
Model.2
x
1
*
=
87.565
%
;
x
2
*
=
8.480
%
;
x
3
*
=
0.040
%
;
x
4
*
=
3.905
%
;
x
5
*
=
0.010
%
;
J
2
*
=
12.51115
;
Model.3
x
1
*
=
87.805
%
;
x
2
*
=
7.791
%
;
x
3
*
=
0.878
%
;
x
4
*
=
3.516
%
;
x
5
*
=
0.010
%
;
J
3
*
=
0.000
;
Model.4
x
1
*
=
87.555
%
;
x
2
*
=
8.414
%
;
x
3
*
=
0.010
%
;
x
4
*
=
3.912
%
;
x
5
*
=
0.109
%
;
J
1
*
+
J
2
*
=
36.83931
;
Ψ
1
=
Ψ
2
=
1
J
1
*
=
24.32659
Model.5
x
1
*
=
88.257
%
;
x
2
*
=
7.503
%
;
x
3
*
=
0.010
%
;
x
4
*
=
3.731
%
;
x
5
*
=
0.499
%
;
J
1
*
+
J
3
*
=
24.33017
;
Ψ
1
=
Ψ
2
=
1
J
1
*
=
24.32497
Model.6
x
1
*
=
87.565
%
;
x
2
*
=
8.480
%
;
x
3
*
=
0.040
%
;
x
4
*
=
3.905
%
;
x
5
*
=
0.010
%
;
J
2
*
+
J
3
*
=
12.51635
;
Ψ
1
=
Ψ
2
=
1
J
2
*
=
12.51115
Model.7
x
1
*
=
87.555
%
;
x
2
*
=
8.414
%
;
x
3
*
=
0.010
%
;
x
4
*
=
3.912
%
;
x
5
*
=
0.109
%
;
J
1
*
+
J
2
*
+
J
3
*
=
36.84451
.
Ψ
1
=
Ψ
2
=
Ψ
3
=
1
J
1
*
=
24.32659
J
2
*
=
12.51273
Wherex1, x2, x3, x4, and x5 are ingredient ratio of the limestone, clay, iron, correction materials, and coal ash, respectively.Figure 6
Optimal ingredient ratio for cement materials in Table4.There are four types of original cement materials in Table5, and they are the carbide slag, clay, sulfuric acid residue, and cinder. The sulfuric acid residue is the most expensive material, the cinder is the cheapest material, the carbide slag has the best grind ability, and the sulfuric acid residue has the poorest grind ability. The optimization models (discrete time) and optimal ingredient ratios under different production requirements are presented in Table 8 and Figure 7. Model.1 has the smallest material cost with the optimal ingredient ration x1=75.007%, x2=14.973%, x3=3.620%, and x4=6.400%. Model.2 has the smallest power consumption with the optimal ingredient ratio x1=76.090%, x2=19.530%, x3=3.624%, and x4=0.755%. Model.3 has the smallest critical cement craft deviation with optimal ingredient ratio x1=75.442%, x2=19.623%, x3=4.257%, and x4=0.678%. Model.4 makes balance between material cost and power consumption with optimal ingredient ratio x1=75.654%, x2=14.798%, x3=3.562%, and x4=5.985%. Model.5, Model.6, and Model.7 have the same optimal ingredient ratio with Model.1, Model.2, and Model.4, respectively.Table 8
Optimization models and results for cement materials in Table5.
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
(
w
1
=
0.5
,
w
2
=
0.3
,
w
3
=
0.2
,
α
d
0
=
1.02
,
β
d
0
=
1.80
,
Ω
d
0
=
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
M
η
=
1.29
x
1
+
16.42
x
2
+
2.22
x
3
+
22.77
x
4
M
ρ
=
3.35
x
2
+
77.85
x
3
+
10.18
x
4
M
γ
=
69.26
x
1
+
2.45
x
3
+
1.13
x
4
x
1
+
x
2
+
x
3
+
x
4
+
x
5
=
1.00, x1≥0,x2≥0,x3≥0,x4≥0
(
2
)
α
=
(
M
γ
-
1.65
M
η
-
0.35
M
ρ
)
/
(
2.8
M
μ
)
β
=
M
μ
/
(
M
η
+
M
ρ
), Ω=Mη/Mρ
(
3
)
M
φ
=
24.65
x
1
+
5.83
x
2
+
1.06
x
3
≤
30.00
M
τ
=
2.71
x
3
+
2.16
x
4
≤
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
3
*
=
3.620
%
;
x
4
*
=
6.400
%
;
J
1
*
=
19.55801
;
Model.2
x
1
*
=
76.090
%
;
x
2
*
=
19.530
%
;
x
3
*
=
3.624
%
;
x
4
*
=
0.755
%
;
J
2
*
=
11.81783
;
Model.3
x
1
*
=
75.442
%
;
x
2
*
=
19.623
%
;
x
3
*
=
4.257
%
;
x
4
*
=
0.678
%
;
J
3
*
=
0.00000
;
Model.4
x
1
*
=
75.654
%
;
x
2
*
=
14.798
%
;
x
3
*
=
3.562
%
;
x
4
*
=
5.985
%
;
J
1
*
+
J
2
*
=
31.45261
;
Ψ
1
=
Ψ
2
=
1.0
J
1
*
=
19.56587
Model.5
x
1
*
=
75.007
%
;
x
2
*
=
14.973
%
;
x
3
*
=
3.620
%
;
x
4
*
=
6.400
%
;
J
1
*
+
J
3
*
=
19.56571
;
Ψ
1
=
Ψ
2
=
1.0
J
1
*
=
19.55801
Model.6
x
1
*
=
76.090
%
;
x
2
*
=
19.530
%
;
x
3
*
=
3.624
%
;
x
4
*
=
0.755
%
;
J
2
*
+
J
3
*
=
11.82553
;
Ψ
1
=
Ψ
2
=
1.0
J
2
*
=
11.81783
Model.7
x
1
*
=
75.654
%
;
x
2
*
=
14.798
%
;
x
3
*
=
3.562
%
;
x
4
*
=
5.985
%
;
J
1
*
+
J
2
*
+
J
3
*
=
31.46031
.
Ψ
1
=
Ψ
2
=
Ψ
3
=
1.0
J
1
*
=
19.56587
J
2
*
=
11.88674
Wherex1, x2, x3, and x4 are ingredient ratio of the carbide slag, clay, sulfuric acid residue, and cinder, respectively.Figure 7
Optimal ingredient ratio for cement materials in Table5.Overall, numerical examples are presented to demonstrate various optimization problems in the blending process. The dynamic optimal ingredient ratio could be obtained in the blending process and can help to promote the cement quality if raw material chemical composition is updated with time.


## 6. Conclusions

This paper focuses on modelling and solving the ingredient ratio optimization problem in cement raw material blending process. A general nonlinear time-varying (G-NLTV) model of the raw material blending process is established by taking raw material chemical composition fluctuations, feed flow fluctuations, various craft constraints, and various production requirements into account. Various objective functions are presented to obtain optimal ingredient ratios under different cement production requirements. To simplify G-NLTV model and solve the optimization problem with conveniences, the optimal ingredient ratio problem is transformed into discrete time single objective rolling or multiple objectives rolling optimization problem. A framework of grid interior point method is proposed to solve the rolling optimization problem. Based on MATLAB-GUI, the corresponding ingredient ratio software is developed to achieve the optimal ingredient ratio for the cement blending process. Finally, numerical examples are shown to study and solve the ingredient ratio optimization problems in cement raw material blending process.



---

*Source: 392197-2012-12-10.xml*