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The temperature of aluminum alloy work-pieces in the aging furnace directly affects the quality of aluminum alloy products. Since the temperature of aluminum alloy work-pieces cannot be measured directly, a temperature prediction model based on improved case-based reasoning (CBR) method is established to realize the online measurement of the work-pieces temperature. More specifically, the model is constructed by an advanced case-based reasoning method in which a state transition algorithm (STA) is firstly used to optimize the weights of feature attributes. In other words, STA is utilized to find the suitable attribute weights of the CBR model that can improve the accuracy of the case retrieval process. Finally, the CBR model based on STA (STCBR) was applied to predict the temperature of aluminum alloy work-pieces in the aging furnace. The results of the experiments indicated that the developed model can realize high-accuracy prediction of work-pieces temperature and it has good application prospects in the industrial field.

The Aging Furnace (AF) is an important equipment for the thermal treatment of aluminum alloy work-pieces to enhance their comprehensive performance of anticorrosion property and mechanical properties, such as hardness and ultimate tensile strength [

Thus, this paper introduces a novel weight allocation method that is based on STA to improve the traditional CBR system, and the improved CBR system is called STCBR. To test and verify the effectiveness of the developed method, STCBR is applied in temperature prediction of aluminum alloy work-pieces in AF. The experimental results show that STCBR can realize high-accuracy prediction of work-pieces temperature and has strong robustness.

STCBR model involves five processes: case representation, case retrieval, case reuse, case revise and case retain.

Typically, a case model consists of two parts: feature attributes and solution attributes. Feature attributes are the mathematical description of problems, and solution attributes are solutions to problems. Therefore, the general case model of a source case can be represented by C a s e i as Equation (1):

C a s e i = { f i 1 , f i 2 , ⋯ , f i j , ⋯ , f i n , s i 1 , s i 2 , ⋯ , s i j , ⋯ , s i m } (1)

where n is the number of feature attributes, m is the number of feature attributes, f i j and s i j represent the value of feature attributes and solution attributes of the ith source case.

The purpose of case retrieval is to retrieve one or more cases with the maximum similarity to the new target case from the case base, by calculating the similarity between them. At present, the k nearest neighbor algorithm (KNN) is often used for case retrieval in the CBR system [

At the first step, it is needed to calculate the local similarity s i m i j between

the target case and the source case in the case base, as is shown in Equation (2):

s i m i j = 1 − | f i j − f ′ j | max ( f i j , f ′ j ) (2)

where f i j is the value of the jth feature of the ith source case, and f ′ j is the value of the jth feature attribute of the target case. It is noteworthy that the local similarity describes the similarity degree of the same feature attribute between the target case and the source case. The number of feature attributes is n, there are thus n local similarities needed to be calculated by Equation (2).

After all of the local similarities between the ith source case and the target case are available, the global similarity is introduced at the second step by Equation (3):

s i m i = ∑ j = 1 n w j ⋅ s i m i j (3)

where w j is the weight of the jth feature, and n is the number of feature attributes.

At the third step, sort the cases by similarity based on KNN, and then the cases with maximum global similarity are chosen for case revise and reuse. The maximum global similarity is defined as Equation (4):

s i m max = max ( s i m k ) , k = 1 , 2 , ⋯ , n u m (4)

where num is the number of cases in the case base.

In similarity calculation, the weights of feature attributes have a great impact on retrieved results and the accuracy of problem solving. Thus, it is important to optimize the weights of feature attributes to improve the quality of the CBR system. The optimization problem of weight allocation can be expressed as Equation (5):

{ max F = f ( ω 1 , ω 2 , ⋯ , ω n ) s . t . g ( ω ) = 1 − ∑ j = 1 n ω j = 0 0 ≤ ω j ≤ 1 , j = 1 , 2 , ⋯ , n (5)

where ω = [ ω 1 , ω 2 , ⋯ , ω n ] ∈ R n is the weight vector; f ( ⋅ ) is a fitness function to estimate the prediction accuracy of CBR for the training case set base on the weight vector ω . The fitness function is defined as Equation (6):

f ( ω ) = ∑ l = 1 q hit l (6)

where q is the size of the training case set, and hit j represents that if the lth training case is matched successfully; if it is, hit l is 1, otherwise hit l is 0.

According to STA, a solution to an optimization problem is regarded as a state and the process of updating current solution is regarded as a state transition [

{ x ( k + 1 ) = A k x ( k ) + B k u ( k ) y ( k + 1 ) = f ( x ( k + 1 ) ) (7)

where x ( k ) ∈ R n stands for a state corresponding to a current solution to the optimization problem; A k ∈ R n × n and B k ∈ R n × m are state transition matrices with appropriate dimensions, which are usually regarded as transformation operators for the optimization algorithm; u ( k ) is a function of x ( k ) and historical states; and f ( x ( k + 1 ) ) is the fitness function.

There are four special state transformation operators which are designed to solve the continuous optimization problems:

(1) Rotation transformation:

x k + 1 = x k + α 1 n ‖ x k ‖ 2 R r x k (8)

where α is a positive constant called the rotation factor; R r ∈ R n × n is a random matrix with its entries belonging to the range of [ − 1 , 1 ] ;and ‖ ⋅ ‖ 2 is the 2-norm of a vector.

(2) Translation transformation:

x k + 1 = x k + β R t x k − x k − 1 ‖ x k − x k − 1 ‖ 2 (9)

where β is a positive constant called the translation factor; R t ∈ R n × n is a random variable with its components in the range of [0,1]. The translation transformation will be performed only when a better solution is found.

(3) Expansion transformation:

x k + 1 = x k + γ R e x k (10)

where γ is a positive constant called the expansion factor; R e ∈ R n × n is a random diagonal matrix with its entries obeying Gaussian distribution.

(4) Axesion transformation:

x k + 1 = x k + δ R a x k (11)

where δ is a positive constant called the axesion factor; R a ∈ R n × n is a random diagonal matrix with its elements obeying Gaussian distribution and only one random index having a nonzero value.

The optimization flowchart of the feature weights using STA is shown in

Phase 1 (parameters initialization). Set the values of SE (the search enforce-

ment, which means the times of the transformation), α , β , γ , δ (operation factor), f c (a constant coefficient used for lessening the α ), and K (times of iteration). Set k = 0 and generate initial solution x ( k ) randomly.

Phase 2 (state transition operation). Perform state transition operations for x ( k ) . The specific operations for x ( k ) are given as follows:

best ← Expansion ( x ( k ) , S E , β , γ )

best ← Rotation ( x ( k ) , S E , α , β )

best ← Axesion ( x ( k ) , S E , β , δ )

Next, the operator will be described in detail, taking the expansion operator as an example:

(a) Make S E copies of x ( k ) and carry out an expansion operation for each copy state by Equation (10); the result is newstates 0 = { x 1 0 ( k ) , x 2 0 ( k ) , ⋯ , x S E 0 ( k ) } .

(b) Perform CBR process for the training data set, and then calculate the fitness value of each state of newstates by Equation (5) and make a ranking of the states according to the fitness value in descending order; and assign the state which is sorted as 1 to best .

(c) x ( k − 1 ) ← x ( k ) , x ( k ) ← b e s t ; make S E copies of x ( k ) and carry out a translation operation for each copy state by Equation (9); the result is newstates 1 = { x 1 1 ( k ) , x 2 1 ( k ) , ⋯ , x S E 1 ( k ) } .

(d) Perform CBR process for the training data set, and then calculate the fitness value of each state of newstates 1 by Equation (5) and make a ranking of the states according to the fitness value in descending order; and assign the state which is sorted as 1 to best .

Phase 3 (iteration or termination). If k < K , then k = k + 1 , α = α / f c , and go to step 2. Otherwise, assign best as the optimal weight vector ω * .

Target and source cases are generally impossible to be exactly the same, thus it is significant to study how to make an appropriate adjustment for retrieved cases to achieve accurate results [

s t = ∑ i = 1 m s i m i ⋅ s i ∑ i = 1 m s i m i (12)

where m is the number of cases with the maximum similarity.

To verify the validity of the improved CBR proposed in this paper, some experiments are conducted based on actual production data which were collected from an aging furnace in Southwest Aluminum Co. Ltd., China. The detailed location of the work-piece thermocouples and the working room thermocouples is shown in

thermocouples, Z1 and Z2, are installed in the Zone 1 and Zone 2 of the work-piece. Based on the production data drawn from real factory, six feature attributes are chosen from data records directly, which are working room temperature of Area 1 f_{1}, working room temperature of Area 2 f_{2}, the duration of heating period f_{3}, the duration of holding period f_{4}, the set temperature f_{5} and alloy state f_{6}, respectively. Work-piece temperature of Zone 1 s_{1} and Work-piece temperature of Zone 2 s_{2} are chosen as solution attributes.

For the experiment, there are 2230 groups of data in total, and 103 groups are selected as testing data, and the remaining 2127 groups as case base and training data for feature weights allocation. The testing data comes from 2 batches of different products as shown in

In order to validate the performance of the developed CBR model, three different CBR models were established based on the same case base. The CBR model based on STA proposed in this paper is referred to as STCBR. Furthermore, the traditional method with equal weight as EWCBR, and the genetic algorithm as GACBR:

(1) The setting parameters of STCBR [

(2) In GACBR [

(3) EWCBR allocates average weights for each attributes.

Batch | Parameters | ||||
---|---|---|---|---|---|

Alloy state | Aging temperature of 1^{st} aging period (˚C) | Holding time of 1^{st} aging period (minute) | Aging temperature of 2^{ed} aging period (˚C) | Holding time of 2^{ed} aging period (minute) | |

#1 | 7075-T6 | 140 ± 1 | 300 | - | - |

#2 | 7050-T6 | 121 ± 1 | 210 | 177 ± 1 | 210 |

CBR model | Weight of each feature | |||||
---|---|---|---|---|---|---|

f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | |

STCBR | 0.2500 | 0.1786 | 0.2143 | 0.0357 | 0.0714 | 0.2500 |

GACBR | 0.0833 | 0.2917 | 0.2083 | 0.1250 | 0.1667 | 0.1250 |

EWCBR | 0.1667 | 0.1667 | 0.1667 | 0.1667 | 0.1667 | 0.1667 |

different feature attribute on reasoning results. For GACBR and STCBR, the weights allocation methods are more reasonable and credible because they are performed based on the information of data rather than experiences of experts. For GACBR model, the weight of working room temperature of Area 2 and duration of rising period are assigned as 0.2917 and 0.2083, which are much larger than those of the other features. It is presented that working room temperature of Area 2 and the duration of rising period play more important roles in predicting the temperature of work-pieces for GACBR model. However, for STCBR model, the alloy state and working room temperature of Area 1 are of most correlation with the temperature of work-pieces.

closer the predicted values are to the 45 measured values, the higher the method’s precision is. From

The absolute errors of different prediction models are shown in

To analyze the error in a numerical way, two indices Mean-SE (mean absolute error) and Max-SE (maximum absolute error) are used to evaluate the performance of each model as shown in

CBR model | Mean absolute error (˚C) | Maximum absolute error (˚C) | ||
---|---|---|---|---|

7075-T6 | 7050-T6 | 7075-T6 | 7050-T6 | |

STCBR | 0.5642 | 0.6158 | 3.2512 | 3.1438 |

GACBR | 0.6226 | 0.6898 | 5.600 | 5.7353 |

EWCBR | 0.9920 | 1.2873 | 9.9477 | 6.7332 |

from disturbances of environment or insufficiency of cases.

Due to the significant role of online prediction of the work-pieces temperature for aging furnace, an improved CBR method was introduced in establishing the temperature prediction model. In order to promote the performance of problem solving of the CBR model, a novel optimization approach STA which combines

CBR model | Advantages | Disadvantages | Application |
---|---|---|---|

EWCBR | Simple weights allocation method and low modeling complexity. | Low prediction accuracy, lower reliability and robustness | The system which has rich expert experience and mechanism knowledge |

GACBR | High prediction accuacy | Longer time spend of weights allocation, lower reliability and robustness. | The system which has certain expert experience with little mechanism knowledge |

STCBR | High prediction accuacy and strong robustness | Longer time spend of weights allocation | The system which has certain expert experience with little mechanism knowledge |

global search with local search to avoid the local minima is proposed to optimize feature weights. Then the established model was verified by the practical production data in AF, and the experiment results show the advantages of the STCBR model, which effectively promotes the prediction accuracy of traditional CBR in general and has practical value to apply to the aging furnace in industry.

Future research may be focused on enriching the existing STA models and improving the case revise strategy to realize better prediction performance.

This research was supported by National Natural Science Foundation of China (61174132) and Doctoral Fund of Ministry of Education of China (20130162110067). These financial contributions are gratefully acknowledged.

Zhu, Q., Shen, L., He, J.J. and Gui, W.H. (2017) Temperature Prediction of Aluminum Alloy Work-Pieces in Aging Furnaces Based on Improved Case-Based Reasoning. International Journal of Nonferrous Metallurgy, 6, 47-59. https://doi.org/10.4236/ijnm.2017.64004