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.
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 Section 2; 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. Figure 1 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.
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 Figure 1 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 , , , and represent chemical composition of original cement material-. In Table 1, R2O represents total chemical composition of sodium oxide (Na2O) and potassium (K2O).
Why 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: where is the LSF, is the SR, and is the AOR. Without losing generality, it assumes that there has -type the original cement materials in blending process. The mass of CaO, SiO2, Al2O3, and Fe2O3 in cement raw material can be acquired as 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 where is total mass of original cement material. Variables are normalized, and (3.3) is further expressed as where is the mass percentage of original cement material-, is ingredient ratio (mass percentage vector), and , , , and are mass percentage vector of CaO, SiO2, Al2O3, and Fe2O3 for cement raw material, respectively. The ingredient ratio is usually expressed by the percentage form, and , , , , , , and are obtained as In practice, each type of original cement material will possess a certain proportion, thus mass percentage will yield where is minimum mass percentage of original cement material-. Minimum mass percentage is decided by cement production crafts. In cement production, the mass percentage of cement raw material should be limited in permissible scope. Otherwise, cement will lose its inherent nature property as where is the expected mass percentage of CaO, is the maximum fluctuation scope, and and are the lower bounded and upper bounded, respectively. and 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: where , , and are the minimum lower bounded of LSF, SR, and AOR respectively, and , , and 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 where and are the maximum permission loss ratio and impurity ratio, respectively, and and 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: where , , , , and are the permissible maximum mass percentage of MgO, R2O, SO3, TiO2, and Cl in cement raw material, respectively, and , , , , and 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: where is the SAR, and are the mass or mass percentage of K2O and Na2O, respectively, and is the permissible maximum percentage. The and have the implicit relationships: , . 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: where is original cement material output flow in ball mill, is original cement material feed flow, is SiO2 mass in feed flow, is SiO2 mass in output flow, and is the SiO2 ouput mass coefficient of original cement material-. 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:
Therefore, the MgO, R2O, SO3, TiO2, and Cl mass balance equation of harmful ingredients in ball mill could be obtained as follows: The impurity and loss mass balance equation of ball mill in blending process could be also obtained as follows:
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 where (¥/ton) is the cost of original cement material-, and 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 where (Kwh/ton) is bond grinding power index of original cement material-, and is power consumption function. represents the grind ability of original cement material- 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 is obtained as follows: where () is the weight of LSF error, SR error, and AOR error, , , and are the error of LSF, SR, and AOR, and , , and 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 where , , and 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 functions , , and in dynamic optimization models are the convex functions. The , and the 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 where and are the feasible regions constructed by constraints (3.12) and (3.8), respectively. SAR is equivalently expressed as Then, feasible region can be equivalently written as Likewise, critical cement craft parameters , , and can be equivalently expressed as Then, feasible region can be equivalently written as In the previous section, we know that the , , , , , , and are the linear functions of the ingredient ratio (original cement materials mass percentage vector) . Therefore, feasible region and 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 where is the feasible region constructed by constraints (3.1)–(3.12), is the feasible region constructed by constraints (3.1)–(3.10), and is the convex and semiconvex regions set. The constraints (3.1)–(3.10) are the linear algebraic constraints. Hence, the feasible region 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.
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: 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 and ingredient ratio are minor, and they can be ignored. Then, the constraint (4.7) can be equivalently expressed as 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: It is noted , and in (4.9), and 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: where and are the discrete equality and inequality constraint vectors, respectively. Hence, the continuous time dynamic model is transformed into the following discrete time form:
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: where , , and are the smooth and differentiable functions, is the decision variable, and , , and 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: where is the barrier parameter, the slack vector is set to be positive, and is an expanded inequality constraint. It introduces the Lagrange multipliers and for barrier problem (4.13) as follows: where is Lagrange function, and are Lagrange multipliers for constraints and , respectively. Based on Karush-Kuhn-Tucker (KKT) optimality conditions [41–43], optimality conditions for optimization problem (4.13) can be expressed as where is the diagonal matrix and its elements are the components of the vector , is a vector of all ones, and are the Jacobian matrices of the vectors and , respectively, is the grand of function , and is a diagonal matrix and its elements are the components of vector . The system (4.15) is KKT optimal condition of optimization problem (4.13). When in the search procedure, and should remain positive . To obtain the iteration direction, we can make the point satisfy the KKT conditions (4.15), then the following system will be obtained: 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: where 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: where is the step size. Choosing the step size holds the , 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 region is divided into small pieces of feasible region without any intersection , and , and is the interval length .
Step 2. For ,each small feasible region will do the following steps.
Step 3. Choose an initial iteration point in the feasible region set , and the , , .
Step 4. Constructing current iterate, we have the current iterate value , , and of the primal variable , the slack variable , and the multipliers and , respectively.
Step 5. Calculate the Hessian matrix of the Lagrange system , and the Jacobian matrix and are of the vectors and in the current iterate .
Step 6. Solve the linear system (4.18) and construct the iterate direction . Solve the linear matrix equation (4.18), and then we can obtain the primal solution , multipliers solution , , and also the slack variable solution .
Step 7. Choosing the step size holds the , in the search process, . Update the iterate values: .
Step 8. Check the ending conditions for region . If it is not satisfied, go to Step 5, else the minimum of feasible region is obtained, , go to Step 3.
Step 9. Compare the minimum of feasible region , output the minimum , end.
Based on the grid interior point method framework, the algorithm structure diagram of cement raw material blending process is shown in Figure 2. 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.
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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. Tables 3–5 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.
There are five types of original cement material in Table 3, 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 %, %, %, %, and %. Model.2 has the smallest power consumption with the optimal ingredient ratio %, %, %, %, and %. Model.3 has the smallest critical cement craft deviation with optimal ingredient ratio %, %, %, %, and %. 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 , , and . Model.4 makes balance between material cost and power consumption with optimal ingredient ratio %, %, %, %, and %. 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 is far less than the objective function and . In addition, the weight of objective function is not far larger than the weight of objective function and , therefore they have the same optimal ingredient ratio.
There are five types of original cement materials in Table 4, 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 %, %, %, %, and %. Model.2 has the smallest power consumption with the optimal ingredient ratio %, %, %, %, and %. Model.3 has the smallest critical cement craft deviation with optimal ingredient ratio %, %, %, %, and %. Model.4 makes balance between material cost and power consumption with optimal ingredient ratio %, %, %, %, and %. Model.5, Model.6, and Model.7 have the same optimal ingredient ratio with Model.1, Model.2, and Model.4, respectively.
There are four types of original cement materials in Table 5, 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 %, %, %, and %. Model.2 has the smallest power consumption with the optimal ingredient ratio %, %, %, and %. Model.3 has the smallest critical cement craft deviation with optimal ingredient ratio %, %, %, and %. Model.4 makes balance between material cost and power consumption with optimal ingredient ratio %, %, %, and %. Model.5, Model.6, and Model.7 have the same optimal ingredient ratio with Model.1, Model.2, and Model.4, respectively.
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.
Appendix
See Tables 3, 4, and 5.
Nomenclature
| : | mass percentage of original cement material- |
| : | mass percentage of original cement material- |
| : | mass percentage of original cement material- |
| : | CaO mass percentage of original cement material- |
| : | MgO mass percentage of original cement material- |
| : | mass percentage of original cement material- |
| : | mass percentage of original cement material- |
| : | mass percentage of original cement material- |
| : | Cl mass percentage of original cement material- |
| : | Impurity mass percentage in original cement material- |
| : | Mass loss percentage of original cement material- in the cement kiln burning process |
| : | Original cement material- mass |
| : | total mass of original cement material |
| : | total mass of original cement material |
| : | total mass of original cement material |
| : | CaO total mass of original cement material |
| : | MgO total mass of original cement material |
| : | total mass of original cement material |
| : | total mass of original cement material |
| : | total mass of original cement material |
| : | Cl total mass of original cement material |
| : | Impurity total mass of original cement material |
| : | Total mass loss of original cement material in cement kiln burning process |
| : | total mass percentage of original cement material |
| : | total mass percentage of original cement material |
| : | total mass percentage of original cement material |
| : | CaO total mass percentage of original cement material |
| : | MgO total mass of original cement material |
| : | total mass percentage of original cement material |
| : | total mass percentage of original cement material |
| : | total mass percentage of original cement material |
| : | Cl total mass percentage of original cement material |
| : | Impurity total mass percentage of original cement material |
| : | Total mass loss percentage of original cement material in cement kiln burning process. |
Acknowledgments
This work is partially supported by the research project no. 2009AA04Z 155, no. KGCX2-EW-104, and no. 2010CB 334705. The authors would like to thank reviewers for their helpful suggestions.