A genetic-algorithm-based optimal scheduling system for full-filled tanks in the processing of starting materials for alumina production.

Yang, Chunhua ; Gui, Weihua ; Kong, Lingshuang 等

INTRODUCTION

In the blending process of starting materials for sintering in the production of alumina, the bauxite, the raw limestone, the anthracite, the alkali, the returned lye and waste liquid are mixed together in certain percentages to form raw slurry. The quality of raw slurry directly influences the quality of sintered clinker and the final products (Zhou and Chen, 2004). Nevertheless, there exists a great composition fluctuation in the processing of raw slurry due to the instability of mine sources, the uncertainty of the composition of returned lye and waste liquid, and large time-delay of composition analysis (Yang et al., 2005). In order to obtain the qualified raw slurry for sintering, a batch of full-filled tanks were employed, in which the raw slurry is mixed together to meet technological requirements. The mixing process is called the scheduling of full-filled tanks.

At present, in most alumina smelteries, the operators have to select the number of full-filled tanks based on their individual experience and to determine an appropriate combination by repeatedly computing the average quality indices of raw slurry in each batch of full-filled tanks selected. It is too difficult for manual scheduling methods to find an optimal combination due to the limitation of individual experience and the complexity to consider all technological requirements synthetically. Thus two-times scheduling of full-filled tanks is necessary to obtain qualified raw slurry, where the first scheduling is to obtain raw slurry with more steady quality, and the second scheduling is required to further improve the quality of raw slurry. Such two-times scheduling increases the computation burden of operators and brings much complexity of production process, which leads to energy waste and limits throughput of raw slurry as well. Therefore, it is imperative to find an efficient approach to implement optimal scheduling of full-filled tanks so as to relief the computation burden of operators and to simplify the production process.

In the scheduling process, the relationships of quality indices between the raw slurry mixed by scheduling and the raw slurry from selected tanks are non-linear. At the same time, some technological requirements have to be considered as inequality constraints to ensure high product quality and normal production. If digital 1 and 0 are used to denote one certain tank is chosen or not, the problem to get an optimal scheduling scheme is formulated as a non-linear 0-1 combinatorial optimization problem subject to technological constraints. It is theoretically solvable by the enumeration algorithm (thereafter referred to as EA), which is a traditional approach to solve 0-1 integer combination programming problem. However, with the increasing scale of problems, their computation time increases exponentially. Genetic algorithms (thereafter referred to as GAs) are global probabilistic search algorithms based on the mechanisms of natural selection and "survival of the fittest" from natural evolution, which can find an optimal solution faster than the conventional algorithms due to its ability to direct the search towards relatively "prospective" regions in the search space. GAs have been applied to many fields with good results, such as job-shop scheduling and combinatorial optimization, etc. (Cheng et al., 1996; Alexandre and de Vasconcelos, 2002; Kacem, 2003). In the other hand, it is convenient to represent 0-1 variables by binary encoded strings. Therefore, GAs are good candidates for solving the 0-1 combinatorial optimization problems.

In order to maintain diversity in the population and to sustain the convergence capacity of the basic genetic algorithm (thereafter referred to as BGA), a number of improved strategies for genetic operators have been developed by adapting probabilities of crossover and mutation (Davis, 1989; Srinivas and Patnaik, 1994). In this article, based on the specific industrial background of alumina production, an improved GA is proposed to solve the optimal scheduling problem of full-filled tanks. Firstly, an optimization model for scheduling of full-filled tanks is developed based on material balance principle and expert experience, whose objective is to minimize the error between average quality indices of raw slurry in the selected batch of full-filled tanks and the expected quality indices on condition that all technological requirements are met. Then the proposed genetic algorithm (thereafter referred to as IGA) is used to solve the optimal scheduling problem, where the intervention strategy is introduced into the random process of population initialization to obtain a well-proportioned initial population and the difference between the fitness value of the best solution and the average fitness value of better solutions and the difference between the fitness value of the best solution and the average fitness value of the current population are together used as the yardstick for detecting the convergence of the GA, in which the better solution is defined to be those individuals whose fitness values are less than or equal to the average fitness value of the current population. The probabilities of crossover and mutation (thereafter referred to as [p.sub.c] and [p.sub.m], respectively) are adaptively changed depending on the two differences. The IGA-based optimization system has been applied to the processing of starting materials for alumina production. The application results show that the quality of mixed raw slurry can meet the technological requirements only with the first scheduling, which not only enhances the quality of the raw slurry ready for sintering, but also simplify the blending process by removing the second scheduling process.

The rest of the article is organized as follows. Optimal Scheduling Model of Full-Filled Tanks Section builds an optimal scheduling model of full-filled tanks for the blending process of alumina production with sintering. In Optimal Scheduling Based on Improved Genetic Algorithm Section, an improved genetic algorithm is proposed for solving the optimal problem. Industrial Applications Section describes an industrial application and the conclusions are summarized in Conclusions Section.

OPTIMAL SCHEDULING MODEL OF FULL-FILLED TANKS

Description of the Blending Process

The blending process of raw slurry is shown in Figure 1. Several kinds of raw material, including bauxite, limestone, anthracite, alkali, returned lye, and waste liquid, are firstly put to ball mills with appropriate percentages and ground wetly to form raw slurry. The raw slurry is then sent into tank A1 to AN. A significant fluctuation of raw slurry composition exists because of the instability of mine supplying, the composition uncertainty of returned lye and waste liquid, and large time-delay of composition analysis. In order to obtain the slurry with expected composition, two-times scheduling of full-filled tanks is adopted to find appropriate combinations of full-filled tanks selected, and three batches of tanks are prepared to store raw slurry. The first batch of tanks from tank A1 to AN stores the raw slurry from ball mills. In the first scheduling process, the raw slurry from full-filled tanks selected among tank A1 to AN is mixed to form more steady raw slurry, and then to be pumped into the second batch of tanks from tank B1 to BM. In the second scheduling process, the raw slurry from full-filled tanks selected among tank B1 to BM is mixed again to form qualified raw slurry following the technological requirements. Then the qualified raw slurry is pumped into the third batch of tanks from tank K1 to KL ready for sintering. At last it is sent to clinker kiln to be sintered as clinker.

In the scheduling process, the considered technological requirements are mainly about three quality indices of raw slurry, including the mol ratio of [Na.sub.2]O to [Al.sub.2] [O.sub.3] and [Fe.sub.2][O.sub.3], the mol ratio of CaO to Si[O.sub.2], and the mass ratio of [Al.sub.2][O.sub.3] to Si[O.sub.2] (thereafter referred to as [N/R], [C/S], and A/S, respectively) in raw slurry. The three quality indices are calculated respectively by

[FIGURE 1 OMITTED]

Equations (1) to (3):

[N/R] = a x N/A + b x F (1)

[C/S] = c x C/S (2)

A/S = A/S (3)

The assay compositions of raw slurry, that is, A,C,F,N,S, are mass percentages. In order to directly utilize the assay compositions to calculate the ratios of [N/R] and [C/S], Equations (1) and (2) transform the mol ratio to the mass ratio according to the relationship between mol and mass of corresponding composition, in which the transformation coefficients a, b, and c are 1.645, 0.6375, and 1.071, respectively.

Optimization Model for Full-Filled Tanks Scheduling

In the process of full-filled tanks scheduling, the essential requirement is that the average quality indices of selected tanks should approximate the expected values as close as possible. Thus the objective of the optimal scheduling is to minimize the square errors between them as Equation (4).

min[[[f.sub.[N/R]](X), [f.sub.[C/S]](X), [f.sub.A/S](X)].sup.T] (4)

In order to stabilize the production process, the average quality indices of the raw slurry in tanks remained have to be in an acceptable range. So it is necessary to guarantee that the average quality indices of the raw slurry in the remaining tanks meet certain technological requirements, which are treated as constraints with lower and upper bounds as Equation (5).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

In addition, the number of the selected tanks must be in a suitable range to keep the steady-state production. If the number of selected tanks is too large, more tanks will be occupied for mixing the raw slurry. Not enough empty tanks remained to store the raw slurry from ball mills, which may lead to interruption of the continuous production. If the number of selected tanks is too small, not all quality indices of mixed raw slurry from the selected tanks are satisfied with the expected quality indices, which means that the qualified raw slurry cannot be timely pumped into clinker kiln. Thus, the number of selected tanks has to subject to the following constraint.

[N.sup.min] [less than or equal to] N(X) [less than or equal to] [N.sup.max] (6)

The n-dimensional vector X = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [member of] [R.sup.N] denotes a scheduling scheme supposing that the number of full-filled tanks is n, where [x.sub.i] is either 1 or 0: [x.sub.i] = 1 represents the ith tank is selected and [x.sub.i] = 0 means that the ith tank is not selected. Thus, combining the indices Formulas (1) to (3), [f.sub.[N/R]](X), [f.sub.[C/S]](X), and [f.sub.A/S](X) in the objective Function (4) can be formulated as follows.

[f.sub.[N/R]](X) = [(a x [[summation].sup.n.sub.1] [x.sub.i][v.sub.i][N.sub.i]/[[summation].sup.n.sub.1] [x.sub.i][v.sub.i][A.sub.i] + b x [[summation].sup.n.sub.1] [x.sub.i][v.sub.i][F.sub.i] - [S.sub.[N/R]]).sup.2] (7)

[f.sub.[C/S]](X) = [(c x [[summation].sup.n.sub.1] [x.sub.i] [v.sub.i][C.sub.i]/[[summation].sup.n.sub.1] [x.sub.i] [v.sub.i][S.sub.i] - [S.sub.[C/S]]).sup.2] (8)

[f.sub.A/S](X) = [([[summation].sup.n.sub.1] [x.sub.i] [v.sub.i] [A.sub.i]/[[summation].sup.n.sub.1][x.sub.i][v.sub.i][S.sub.i] - [S.sub.A/S]).sup.2] (9)

Similarly, [R.sub.[N/R]](X), [R.sub.[C/S]](X), and [R.sub.A/S] (X) in Equation (5) and N(X) in Equation (6) can be formulated as:

[R.sub.[N/R]](X) = a x [[summation].sup.n.sub.1] (1 - [x.sub.i])[v.sub.i][N.sub.i]/[[summation].sup.n.sub.1] (1 - [x.sub.i])[v.sub.i][A.sub.i] + b x [[summation].sup.n.sub.1] (1 - [x.sub.i])[v.sub.i][F.sub.i] (10)

[R.sub.[C/S]](X) = c x [[summation].sup.n.sub.1] (1 - [x.sub.i])[v.sub.i][C.sub.i]/[[summation].sup.n.sub.1] (1 - [x.sub.i])[v.sub.i][S.sub.i] (11)

[R.sub.A/S](X) = [[summation].sup.n.sub.1] (1 - [x.sub.i]) [v.sub.i][A.sub.i]/[[summation].sup.n.sub.1] (1 - [x.sub.i])[v.sub.i][S.sub.i] (12)

N(X) = [n.summation over (i=1)] [x.sub.i] (13)

The raw slurry in each tank is nearly in the same density, so the variable has been eliminated in Equation (7) to (12). Combining Equation (4) to (13), the optimization model of full-filled tanks scheduling is obtained. Obviously, it is a typical multi-objective non-linear combinatorial optimization problem with inequality constraints.

OPTIMAL SCHEDULING BASED ON IMPROVED GENETIC ALGORITHM

Multi-Objective Function Transformation

To solve the optimization problem with lower computation cost, a well-known fitness function in the form of "sum of weights" is introduced. The idea is to associate each objective function with a weighting coefficient standing for the importance grade of its corresponding objective function, and to minimize the weighted sum of sub-objectives, such that a multi-objective problem is turned into a single objective problem (Alexandre and de Vasconcelos, 2002; Yang et al., 2002; Chen et al., 2004; Ramzan and Witt, 2006). It is described as:

F(X) = [[omega].sub.1][f.sub.[N/R]](X) + [[omega].sub.2] [f.sub.[C/S]](X) + [[omega].sub.3][f.sub.A/S](X) (14)

[[omega].sub.1] + [[omega].sub.2] + [[omega].sub.3] = 1 (15)

where F(X) denotes the normalized objective function related to [f.sub.[N/R]](X), [f.sub.[C/S]](X), and [f.sub.A/S](X), [[omega].sub.1], [[omega].sub.2], and [[omega].sub.3] are adjusted according to the importance of [f.sub.[N/R]](X), [f.sub.[C/S]](X), and [f.sub.A/S](X) taking into consideration of actual production requirements.

Fitness Function With Penalty

The penalty strategy plays a very important role for the optimization problem with constraints, especially when the optimal solution lies on or approximates the boundary of the feasible search places. So the inequality constraints in Equation (5) and (6) are handled by using the penalty terms to define the fitness value of an individual. Considering that the optimization objective is to solve the minimum problem in Equation (14), the fitness function with penalty strategy is constructed as follows:

Z(X) = F(X) + [[lambda].sub.1][([R.sub.[N/R]] (X) - [R.sup.lim.sub.[N/R]].sup.2] + [[lambda].sub.2][([R.sub.[C/S]](X) - [R.sup.lim.sub.[C/S]].sup.2] + [[lambda].sub.3] [([R.sub.A/S] (X) - [R.sup.lim.sub.A/S]).sup.2] + [[lambda].sub.4][(N(X) - [N.sup.lim]).sup.2] (16)

where, the penalty factors [[lambda].sub.i] (i=1,2,3,4) take enough large values so as to satisfy the constraints, [R.sup.lim.sub.[N/R]], [R.sup.lim.sub.[C/S]], [R.sup.lim.sub.A/S] defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Encoding of Chromosome

The code in the improved genetic algorithm is expressed by a natural binary code, which is denoted as a chromosome. And each gene on the chromosome represents a variable. The chromosome length depends on the number of variables in the problem considered, which denotes the sum of full-filled tanks. In this case, the optimal individual is the optimal combination and there is no need for decoding. For example,

Chromsome : [[1001 ... 1100].sub.n]

The value of the ith in the binary string denotes the state of the ith tank: selected or not, 1 represents "selected" and 0 represents "not." And the total number of bit "1" is the sum of full-filled tanks that are selected for mixing.

Population Initialization With Intervention Strategy

As mentioned above, in order to keep the steady-state production, the number of selected tanks should be in the range from [N.sup.min] to [N.sup.max]. In the practical scheduling process, it is limited between 3 and 8. If the initial population can meet the requirement and be distributed uniformly, it would be fast to find an optimal combination needed by the production technology.

However, it is impossible to make the initial population to meet the technological requirement only by the random function. So the intervention strategy is adopted to affect the random process of population initialization. By the strategy, the total number of "1" in each chromosome of initial population is restricted to the range from 3 to 8. But, when the number of full-filled tanks is more than 10, more than 90% of chromosomes in initial population are the combination of 7 or 8 tanks because of the influence of the longer coding and the random of the initial function, which results in the uneven distribution of initial population. In order to obtain the well-proportioned initial population, the total number of "1" in some chromosomes is constrained to between 3 and 6 by the further intervention. The intervention strategy is automatically executed in the implementation program of IGA.

Improved Genetic Operators

The probabilities of crossover and mutation have great effects on the GA's performance such as the speed and probability to global convergence (Deb and Beyer, 2004). In order to improve these performances, Srinivas and Patnaik (1994) used the difference between the fitness value of the best solution and the average fitness value of the current population to detect the convergence of the GA. However, if the better solutions have the same or closer fitness values, they will be copied to the next generation with larger probability, which may result in premature convergence of the GA to a local optimum. In order to overcome the problem, it is also essential to identify whether the GA is converging to a local optimum by observing the average fitness value of the better solutions in relation to the fitness value of the best solution. So, the two differences are together adopted to detect the convergence of the GA in our article. The chosen expressions for [p.sub.c] and [p.sub.m] are in the form of:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

[[bar.f].sub.total] is the average fitness value of the current population, [[bar.f].sub.better] is the average fitness of the better solutions whose fitness values are less than or equal to [[bar.f].sub.total]. In order to constrain [p.sub.c] and [p.sub.m] to the typical ranges 0.0-1.0 and 0.0-0.5, respectively, set [P.sub.c0] = 0.5, [P.sub.c1] = [P.sub.c2] = 1.0, [P.sub.m0] = 0.25, [P.sub.m1] = [P.sub.m2] = 0.5.

The improved adaptive algorithm assures that [p.sub.c] and [p.sub.m] for some of better solutions vary inversely with [f.sub.min] - [[bar.f].sub.better], and for the other better solutions they vary inversely with [f.sub.min] - [[bar.f].sub.total]. The inferior solutions, the fitness values of which are larger than [[bar.f].sub.total], are disrupted with maximum probabilities [p.sub.c] and [p.sub.m].

Evolution Strategy

If there is no interference in GA evolution progress of raw slurry optimal scheduling system, it is possible for the individual to become an inferior one after employing crossover and mutation, especially when the number of the selected tanks in the individual is more than 8.

Adopting the interfering strategy, those individuals that have the first ten smallest fitness values of the current generation are directly copied to the next generation without applying any genetic operators and the last 20 inferior ones are reproduced by the initial function, which guarantees that the better individuals are preserved in future generations.

Implementation of Optimal Scheduling of Full-Filled Tanks

The implementation steps of IGA are as follows:

Step 1: Randomly generate an initial population with the natural binary code by utilizing a random function. The intervention strategy is performed to firstly assure the total number of "1" in each individual is constrained to between 3 and 8. When the number of full-filled tanks is more than 10, the total number of "1" in some chromosomes is constrained to between 3 and 6 to make the individuals distribute uniformly in the range from 3 to 8.

Step 2: Compute the fitness values of all individuals using (16) by applying the penalty function to the operating cost of each solution.

Step 3: Selection: arranging the individuals according to their fitness values and choosing the first ten better individuals to get into the next generation directly and the last twenty inferior ones reproduced by the initial function.

Step 4: Crossover: randomly choosing pairs of individuals from the current population with probability [p.sub.c] in (21). That is, firstly assign a random value to a temporary parameter p by a random function, if p is less than the crossover probability [p.sub.c], the current individual and next individual are selected to be crossover pairs. Then randomly deciding a single crossover bit and exchanging the bit at the point.

Step 5: Mutation: randomly changing the value of each bit of the solution with probability [p.sub.m] in (22) to produce new individuals. That is, firstly assign a random value to a temporary parameter p by a random function, if p is less than the crossover probability [p.sub.m], the current bit of the individual is changed.

Step 6: Check the termination condition. If the evolving process satisfies the convergence condition or reaches the maximum generation predetermined, the evolution terminates and returns the best solution in current population; else loop to Step (2).

[FIGURE 2 OMITTED]

The flow chart of the IGA procedure is shown in Figure 2.

The algorithm can effectively guarantee the diversity of population, however, it cannot always find the optimal solution due to the limitation of the evolution generation. So, the IGA is more suitable to satisfactorily solve the optimization problems which are predominant in industry process.

Comparison of IGA With BGA and EA

Comparison of computation complexity

One of the main requirements in the optimal scheduling of full-filled tanks is to find the optimal combination of selected full-filled tanks as fast as possible so as to keep the continuous production. In general, the allowed computation-time to obtain a combination scheme is constrained within 2 min. To demonstrate the computation performance of IGA, the experiments are performed repeatedly with different number of full-filled tanks 10 times. In all experiments, the population size for the GAs is 100. For the BGA, [p.sub.c] and [p.sub.m] take the static values of 0.6 and 0.001, respectively. For the IGA, [p.sub.c] and [p.sub.m] are determined according to Equations (20) and (21), and the intervention strategy is introduced into the initial process of population. The average consumed time to find the optimal solution is listed in Table 1. Also tabulated is the number of instance where the GAs stuck at a local optimum.

From Table 1, it is clear that EA always finds the optimal solution, but it takes more than 120 s to complete the optimization computation when the number of full-filled tanks is more than 22. Though the BGA can find the optimal solution in less than 1 min, it stuck at a local optimum with greater probability than IGA. The IGA outperforms the BGA both in terms of the speed and probability to global convergence.

Comparison of solution quality

The running data in Table 2 are used to test the quality of IGA solutions. In Table 2, the first column is the tank number which is fully filled with raw slurry, while columns 2-6 represent the mass percentage of calcium oxide (CaO), sodium oxide ([Na.sub.2]O), silicon oxide (Si[O.sub.2]), ferric oxide ([Fe.sub.2][O.sub.3]), alumina ([Al.sub.2][O.sub.3]) of raw slurry in the corresponding tanks. Here, the total number of tanks is 30 and there are 18 full-filled tanks. The optimal scheduling results of IGA and EA are shown in Table 3. According to the technological requirements, the associated parameters in the optimization model are set as follows:

1) The lower and upper bound of selected tank numbers: [N.sup.min]=3, [N.sup.max]=8.

2) The desired indices values: [S.sub.[N/R]] = 0.98, S[C/S] = 2.010, [S.sub.A/S] = 4.80.

3) The minimum and maximum of [N/R] in the left tanks: [R.sup.min.sub.[N/R]] = 0.98, [R.sup.max.sub.[N/R]] = 1.10.

4) The minimum and maximum of [C/S] in the left tanks: [R.sup.min.sub.[C/S]] = 1.950, [R.sup.max.sub.[C/S]] = 2.050.

5) The minimum and maximum of A/S in the left tanks: [R.sup.mix.sub.A/S] = 4.70, [R.sup.max.sub.A/S] = 4.85.

The optimization parameters of IGA are given as follows:

1) The population size = 100.

2) The convergence condition: [square root of Z(X)] [less than or equal to] 0.05, the maximum generation = 200.

3) The weighting coefficients: [[omega].sub.1] = [[omega].sub.2] = [[omega].sub.3] = 1/3.

Table 3 demonstrates all optimal combination schemes in different number of selected tanks. It is shown that the quality of mixed raw slurry related to the IGA is as good as that of EA except the case when the number of selected tanks is 7. But it is noticeable that the solution is not optimal but still satisfactory because its fitness and the average quality index of the full-filled tanks remained are all in the desired range. Therefore, the optimization solutions of the IGA not only assure that the average quality indices of raw slurry in the selected tanks are satisfactory, but also guarantee that the quality indices of raw slurry in each full-filled tank remained are in the desired ranges.

Choice of [[omega].sub.1], [[omega].sub.2], and [[omega].sub.3]

In order to evaluate the effects of [[omega].sub.1], [[omega].sub.2], and [[omega].sub.3] on the solution quality of the IGA, the solution quality of the IGA has been investigated by varying [[omega].sub.1], [[omega].sub.2], and [[omega].sub.3]. The experiment is performed by using the data in Table 2 and the experimental results are listed in Table 4. On analyzing the results, it is noticed that the IGA can find the satisfactory solution when the sub-objectives are given different attention as [[omega].sub.1], [[omega].sub.2], and [[omega].sub.3] is changed. Smaller is the value of [[omega].sub.i], the corresponding average index of selected tanks will be paid less attention. In general, the three sub-objectives in Equation (14) are assigned the same weight to obtain three eligibility indices of mixed raw slurry. However, [[omega].sub.1], [[omega].sub.2], and [[omega].sub.3] take different values when there does not exist the optimal combination which makes three indices of mixed raw slurry meet together the technological requirements with the same attention. In this case, A/S is paid more attention than the other two indices because of its importance, that is, its corresponding weight should take greater value than other quality indices.

[FIGURE 3 OMITTED]

INDUSTRIAL APPLICATIONS

An IGA-based optimization system is developed for the blending process of alumina production, which consists of an optimization computer, six real-time monitoring computers, and three distributed controllers. The optimization computer exchanges real-time information with real-time monitoring computers by an Ethernet network, the main function of which is to implement the optimization computation. The monitoring computers are used to control and monitor the whole production process in real-time and to communicate with the distributed controllers by DH+ Network. The distributed controllers, including 36 closed-loops, are constructed to implement the real-time control of feed rate of raw materials for 6 ball mills.

The application software is developed by VC++ 6.0 on the optimization computer, which is mainly composed of an expert optimization module, an optimal scheduling module, database and user interface. The expert optimization module is used to obtain the optimal mixing percentages of raw materials, and the optimal scheduling module is to find an optimal combination of full-filled tanks. The database holds the measurement data and the quality requirements of raw slurry and to store the optimization results. The user interface is used to configure optimization parameters and to display and to print data, graphs, optimization results, etc.

The developed optimization system has been running in Zhongzhou Branch of China alumina industry since November 2005. To investigate the application advantages of IGA in long term, 60 groups of production data were continuously collected in October 2006, where each group of data includes the average [N/R], [C/S], A/S of selected tanks by IGA, the three average indices of the left tanks and the desired indices that will slightly change with the production condition. At the same time, in order to compare the two approaches of intelligent scheduling and manual scheduling, 60 groups of data of mixed raw slurry based on manual scheduling were also collected from the operation log in October 2004 before the developed system started running. These data are used to illustrate the application results of IGA-based optimization system shown in Figure 3. The results show that three average indices of mixed raw slurry based on IGA (unmarked solid lines on left panel) are very close to the desired indices (circle lines), and the maximum error of [N/R], [C/S], and A/S is 0.02, 0.03, and 0.09, respectively, which improves the quality of mixed raw slurry apparently. From the right panel, it is shown that the average indices of the left tanks by IGA are kept in the required ranges, which guarantees the succession of the whole production process.

Since the optimization system was put into service in the alumina smeltery, the eligibility rate of [N/R], [C/S], and A/S of mixed raw slurry is raised to 99%, 96%, and 94% only with the first scheduling. The blending process is simplified from two-times scheduling to one-time scheduling, and the first two batches of tanks, including tank A1 to AN and tank B1 to BM, can be used to store raw slurry, which would increase enormously the throughput of raw slurry. Furthermore owing to the less composition fluctuation of mixed raw slurry, the eligibility rate of clinker is increased a lot, whose eligibility rate of [N/R], [C/S], A/S increases by 0.34%, 0.46%, 5.95%, respectively, such that contributes to stabilizing the subsequent process of alumina production.

CONCLUSIONS

In this article, an optimal model of full-filled tanks scheduling has been presented for the blending process of alumina production by combining material balance principle with expert experiences, and an improved genetic algorithm with penalty strategy has been proposed to solve the optimization problem by considering the multi-objective function, non-linearity and multiple inequality constraints existed in the optimal model. The improvements increase the convergence speed of the GA and effectively prevented the GA from getting stuck at a local minimum, and enhanced the performance of the global convergence, which made the optimal scheduling of full-filled tanks completed in within 1 min of the computation-time constraint.

This IGA-based optimization system was applied to an alumina smeltery in China. The application results showed that the optimal scheduling scheme met the technological requirements and reduced the composition fluctuation of raw slurry. The mixed raw slurry is qualified for sintering only with the first scheduling, which simplifies the blending process by removing the second scheduling process, and also increases enormously the throughput of raw slurry. Owing to the less composition fluctuation of mixed raw slurry, the eligibility rate of clinker is increased a lot, which contributes to stabilizing the subsequent process of alumina production.

ACKNOWLEDGEMENTS

The authors acknowledge the financial support from the National Natural Science Foundation (grant number: 60574030, 60634020), 973 Program (grant number: 2002CB312200), NCET Program of China (grant number: 04-0751), and 863 Program (grant number: 2006AA04Z181).

NOMENCLATURE

[N/R] one of quality index of the raw slurry,
 mol ratio of [Na.sub.2]O to both of [Al.sub.2]
 [O.sub.3] and [Fe.sub.2][O.sub.3]
[C/S] one of quality index of the raw slurry, mol
 ratio of CaO to Si[O.sub.2]
A/S one of quality index of the raw slurry, mass
 ratio of [Al.sub.2][O.sub.3] to Si[O.sub.2]
a,b,c transformation coefficients in Equations
 (1) to (3)
A total mass percentage of [Al.sub.2][O.sub.3]
 in raw slurry
[A.sub.i] mass percentage of [Al.sub.2][O.sub.3] in the
 ith tank
BGA basic genetic algorithm
C total mass percentage of CaO in raw slurry
[C.sub.i] mass percentage of CaO in the ith tank
EA enumeration algorithm
[f.sub.[N/R]](X) the square error between the average [N/R]
 of the selected tanks and the desired value
[f.sub.[C/S]](X) the square error between the average [C/S]
 of the selected tanks and the desired value
[f.sub.A/S](X) the square error between the average A/S of
 the selected tanks and the desired value
f fitness of the mutation individual
f' smaller fitness in the two crossover
 individuals
[[bar.f].sub.better] average fitness of the better individuals of
 current population
[f.sub.min] smallest fitness of current population,
 fitness of best individual
[[bar.f].sub.total] average fitness of current population
F(X) objective function
F total mass percentage of [Fe.sub.2][O.sub.3]
 in raw slurry
[F.sub.i] mass percentage of [Fe.sub.2][O.sub.3] in
 the ith tank
GAs genetic algorithms
IGA improved genetic algorithm
N total mass percentage of [Na.sub.2]O in raw
 slurry
[N.sub.i] mass percentage of [Na.sub.2]O in the ith tank
N(X) number of the selected tanks
[N.sup.max] allowed maximum number of selected tanks
[N.sup.min] allowed minimum number of selected tanks
[p.sub.c] crossover probability
[p.sub.m] mutation probability
[R.sub.[N/R]](X) average [N/R] of the raw slurry in the
 left tanks
[R.sub.[C/S]](X) average [C/S] of the raw slurry in the
 left tanks
[R.sub.A/S](X) average A/S of the raw slurry in the
 left tanks
[R.sup.max.sub.[C/S]] allowed maximum of average [C/S] of raw
 slurry in left tanks
[R.sup.min.sub.[C/S]] allowed minimum of average [C/S] of raw
 slurry in left tanks
[R.sup.max.sub.[N/R]] allowed maximum of average [N/R] of raw
 slurry in left tanks
[R.sup.min.sub.[N/R]] allowed minimum of average [N/R] of raw
 slurry in left tanks
[R.sup.max.sub.A/S] allowed maximum of average A/S of raw
 slurry in left tanks
[R.sup.lim.sub.A/S] allowed minimum of average A/S of raw
 slurry in left tanks
S total mass percentage of Si[O.sub.2] in
 raw slurry
[S.sub.i] mass percentage of Si[O.sub.2] in the
 ith tank
[S.sub.[C/S]] desired value of [C/S]
[S.sub.[N/R]] desired value of [N/R]
[S.sub.A/S] desired value of A/S
[v.sub.i] volume of the ith tank
X vector of control variables, [x.sub.i]
Z(X) fitness function

Greek Symbols

[[omega].sub.i] weighting factors

[[lambda].sub.i] penalty factors

Manuscript received July 21, 2007; revised manuscript received December 1, 2007; accepted for publication December 1, 2007.

REFERENCES

Alexandre H. F. and J. A. de Vasconcelos, "Multiobjective Genetic Algorithms Applied to Solve Optimization Problems," IEEE Trans. Magn. 38, 1133-1136 (2002).

Chen, X. F., W. H. Gui, Y. L. Wang and L. H. Cen, "Multi-Step Optimal Control of Complex Process: A Genetic Programming Strategy and its Application," Eng. Appl. Artif. Intell. 17, 491-500 (2004).

Cheng, R., M. Gen and Y. Tsujimura, "A Tutorial Survey of Job-Shop Scheduling Problems Using Genetic Algorithms Representation," Comput. Ind. Eng. 30, 983-997 (1996).

Davis, L., "Adapting Operator Probabilities in Genetic Algorithm," Proc. of Third International Conference on Genetic Algorithms, 61-69 (1989).

Deb K. and H. G. Beyer, "Self-Adaptive Genetic Algorithms with Simulated Binary Crossover," Evol. Comput. 9, 137-221 (2004).

Kacem, I., "Genetic Algorithm for the Flexible Job-Shop Scheduling Problem," Proc. of IEEE International Conference on Systems, Man and Cybernetics, 3464-3469 (2003).

Ramzan N. and W. Witt, "Multi-Objective Optimization in Distillation Unit: A Case Study," Can. J. Chem. Eng. 84, 604-613 (2006).

Srinivas M. and L. M. Patnaik, "Adaptive Probabilities of Crossover and Mutation in Genetic Algorithms," IEEE Trans. Syst. Man Cybern. 24, 656-667 (1994).

Yang, C. H., G. Deconinck, W. H. Gui and Y. G. Li, "An Optimal Power-Dispatching System Using Neural Networks for the Electrochemical Process of Zinc Depending on Varying Prices of Electricity," IEEE Trans. Neural Netw. 13, 229-236 (2002).

Yang, C. H., X. G. Duan, Y. L. Wang and W. H. Gui, "Blending Expert System for Raw Mix Slurry in Production of Alumina With Sintering Process," J. Cent. S. Univ. (Nat. Sci. Ed.) 36, 648-652 (2005).

Zhou Z. K. and H. W. Chen, "New Research on Burden Calculation for Raw Mix Slurry in Production of Alumina With Sintering Process," World Nonferrous Met. 41-45 (2004).

Chunhua Yang, Weihua Gui, Lingshuang Kong * and Xiaoli Wang School of Information Science and Engineering, Central South University, Changsha 410083, China

* Author to whom correspondence may be addressed. E-mail addresses: lshkong@mail.csu.edu.cn; ychh@mail.csu.edu.cn

Table 1. Comparison of computation complexity of IGA, BGA, and EA

The number of Consumed time (s)
full-filled tanks IGA BGA EA

10 0.2 0.2 0.3
16 0.9 3.8 6.5
18 1.6 8.3 20.7
22 4.3 18.5 >120
28 10.6 30.6 >120
30 11.2 50.2 >120

The number of Stuck times
full-filled tanks IGA BGA EA

10 0 0 0
16 0 3 0
18 1 2 0
22 0 5 0
28 0 3 0
30 1 7 0

Table 2. Mass percentages of slurry composition in full-filled tanks

Full-filled tank CaO [Na.sub.2]0 Si[0.sub.2]
number/mass
percentages

A6 11.00 18.73 5.22
A7 10.20 17.62 5.84
A8 11.28 17.02 5.77
A10 10.60 17.18 5.45
A11 10.38 16.92 5.80
A12 10.90 16.74 5.39
A13 10.50 18.43 5.05
A14 10.80 16.76 6.14
A15 10.10 19.63 5.12
A16 11.00 17.23 6.04
A17 11.35 18.25 5.36
A20 10.65 17.98 5.56
A21 10.75 17.91 5.94
A22 10.85 17.36 6.11
A24 11.00 16.83 6.11
A25 11.02 17.11 6.18
A27 10.10 17.53 5.77
A28 10.20 17.21 5.63

Full-filled tank [Fe.sub.2] [Al.sub.2]
number/mass [0.sub.3] [0.sub.3]
percentages

A6 3.25 25.93
A7 3.34 27.07
A8 3.36 27.75
A10 3.19 28.26
A11 3.41 27.23
A12 3.49 27.97
A13 3.22 26.84
A14 3.44 27.91
A15 3.26 26.42
A16 3.31 27.69
A17 3.21 25.91
A20 3.41 27.23
A21 3.36 26.95
A22 3.22 27.02
A24 3.38 28.00
A25 3.21 27.55
A27 3.42 27.46
A28 3.16 26.18

Table 3. Comparison of optimization results of IGA, BGA, and EA

Number of Algorithm Solution Average indices of
selected selected tanks
tanks ([N/R], [C/R], A/S)

3 IGA A11, A13, A25 0.98, 2.004, 4.79
 EA

4 IGA A12, A14, A15, 0.98, 2.014, 4.80
 A24
 EA

5 IGA A6, A7, A10, 0.98, 2.010, 4.80
 A11, A16
 EA

6 IGA A12, A15, A16, 0.99, 2.018, 4.74
 A20, A22, A27
 EA A8, A12, A15, 0.98, 2.010, 4.80
 A16, A22, A27

7 IGA A12, A13, A16, 0.98, 2.010, 4.80
 A20, A21, A25,
 A27
 EA

8 IGA A7, A11, A12, A13, 0.98, 2.010, 4.80
 A14, A17, A21, A27
 EA

Number of Algorithm [square root Average indices of
selected of Z(X)] left tanks ([N/R],
tanks [C/R1, A/S)

3 IGA 0.01 0.99, 2.015, 4.76
 EA

4 IGA 0.004 0.99, 2.014, 4.76
 EA

5 IGA 0 0.99, 2.015, 4.75
 EA

6 IGA 0.02 0.99, 2.018, 4.74

 EA 0 0.99, 2.015, 4.75

7 IGA 0 1.00, 2.016, 4.75

 EA

8 IGA 0 0.99, 12.017, 4.74

 EA

Table 4. Effect of [[omega].sub.l], [[omega].sub.2], and [[omega].sub.3]
on the solution quality of the IGA

 The solution Average indices of
 selected tanks ([N/R],
 [C/R], A/S)

1:1:1 A6, A7, A10, A11, A16 0.98, 2.010, 4.80
2:2:1 A12, A20, A21, A22 0.98, 2.010, 4.75
2:1:2 A12, A14, A15, A20, A25, A28 0.98, 2.004, 4.80
1:2:2 A12, A13, A16, A20, A21, 1.00, 2.010, 4.80
 A25, A27

 [square root of Z(X)] Average indices of
 left tanks ([N/R],
 [C/R1, A/S)

1:1:1 0 0.99, 2.015, 4.75
2:2:1 0.05 0.99, 2.015, 4.77
2:1:2 0.006 0.99, 2.018, 4.75
1:2:2 0.02 0.98, 2.016, 4.74