Model-Free Quality Optimization Strategy for a Batch Process with

Article pubs.acs.org/IECR

Model-Free Quality Optimization Strategy for a Batch Process with Short Cycle Time and Low Operational Cost Shengqiang Zhu, Yi Yang, Bo Yang, Zhijiang Shao, and Xi Chen* State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China ABSTRACT: Batch processes are an important factor in modern industries. Quality optimization of batch processes to determine the proper settings of key process variables is critical to successful production. The traditional method for quality optimization is model-based optimization (MBO), which is a challenging task in many cases. The accuracy of the model can deteriorate, particularly when process conditions are frequently changed, a situation that commonly occurs in some batch processes. In this study, a systematic, model-free optimization (MFO) strategy is proposed for a batch process with short cycle time and low operational cost. Instead of developing a model to correlate the relationship between the process and its quality variables, MFO optimizes the process on the basis of online experimentation rather than function evaluation. A direct search algorithm is proposed by integration of simplex search and the MFO strategy. An adaptive experiment number (AEN) strategy is also presented to enhance the search method. To demonstrate the feasibility of the MFO method, it is used in the quality optimization of injection molding, a typical batch process with short cycle time and low operational cost. A comparison of the use of the MFO method with the MBO method whose models were developed with pure quadratic regression and Kriging methods is also presented. Experimental results demonstrate the efficient performance of the proposed method. experimental costs.11 In addition, model mismatch can be serious, which can in turn lead to optimization failure. Building an accurate quality model involves tremendous effort, so a novel model-free optimization method (MFO) was proposed by Kong et al.12 for batch processes with short cycle time and low operational cost. Instead of developing a quality model, the current study uses an online experimental approach to evaluate quality responses. The approach reduces model construction costs and avoids model mismatch. It is integrated with gradient-based optimization algorithms, such as simultaneous perturbation stochastic approximation, as well as direct search algorithms, such as simplex search, as key components. The proposed method is successfully applied to the injection molding process to control the part weight, part dimensions, and focal length of the molded product. Quality optimization of batch processes with MFO is challenging. The target value of the optimization problem is unknown, unlike in previous quality control cases in which the termination criterion is set as the quality measurement that reaches a preset target within a specific tolerance. Reasonable criteria should be established in consideration of the characteristics of these types of batch processes and optimization algorithms. Additionally, MFO requires repetitive experiments in each iterate to reduce measurement noise and process disturbance, but these result in increased experimental cost and decreased optimization efficiency. A strategy to balance quality estimation precision with experimental cost is therefore needed. In this study, the simplex-MFO method for quality optimization is developed by integration of simplex search with the MFO

1. INTRODUCTION Batch processes are widely used in many chemical processes, such as food, polymer, and pharmaceutical production. ANSI/ISA 88.011 defines a batch process as “a process that leads to the production of finite quantities of material by subjecting quantities of input materials to an ordered set of processing activities over a finite period of time using one or more pieces of equipment.” Quality optimization of batch processes involves searching for the optimal process conditions to meet optimal product quality requirements. With the rapid development in computer and optimization technology, research on the quality optimization of batch processes has attracted considerable interest in industries and academia.2,3 Model-based optimization (MBO) is widely used in the quality optimization of batch processes. The development of a first-principle model is a common research direction in this area. For example, Kamal4,5 developed first-principle models and aimed to achieve optimal operation in injection molding, a typical batch process. However, the approach requires a profound understanding of the process under investigation. In addition, the operating conditions of some batch processes can frequently and drastically change, so developing a first-principle model that is applicable to wide ranges is infeasible. For example, an injection molding process can frequently change its material or mold, so developing a first-principle model that applies to materials of different features and molds with different shapes is difficult. Empirical models are therefore often used. Techniques such as artificial neural network,6 partial least-squares,7 principal component regression,8 support vector machine,9 and the Kriging model10 are commonly used to develop data-driven models. Batch processes are characterized by their time-varying characteristic, strongly nonlinearity, and susceptibility to noise and disturbance. Optimization methods based on empirical models normally require a large amount of data, which lead to high © 2014 American Chemical Society

Received: Revised: Accepted: Published: 16384

April 26, 2014 September 21, 2014 September 30, 2014 September 30, 2014 dx.doi.org/10.1021/ie5017199 | Ind. Eng. Chem. Res. 2014, 53, 16384−16396

Industrial & Engineering Chemistry Research

Article

strategy. An approach that incorporates adaptive experiment number (AEN) is put forward to balance quality estimation precision with experimental cost. Experimental verification is conducted in industrial injection molding.

2. MODEL-FREE QUALITY OPTIMIZATION The quality of a batch process product is determined by key operating conditions. Quality optimization of batch processes is the process of searching for the optimal settings for key variables. The optimization can be expressed as min Y = f (X ) + e(X ) X

s. t. g (X ) ≤ 0 LB ≤ X ≤ UB,

X = [x1 , ···, xn]T

(1)

where Y is the quality index to be optimized, X is an n-dimensional vector of the process variables with the lower and upper bounds as LB and UB, respectively, f(X) is a function denoting the relationship between quality and the process variables, e(X) is the total effect of the uncertainty function, and g(X) denotes the inequality constraint. An equality constraint can be converted into an inequality constraint, so the equality constraint is excluded from this general expression. Traditionally, the MBO method starts at the modeling task by building the functions f and g. An optimization algorithm is then run to seek the optimal solution. For a batch process with frequent and dramatic changes in process conditions, the modeling task can be difficult and time-consuming. In this study, an MFO strategy is developed for a type of batch process with short cycle time and low operational cost. 2.1. MFO Strategy. MFO suggests online experimental measurements instead of building a quality model to evaluate quality responses. Figures 1 and 2 compare the flowcharts of

Figure 2. Flowchart of MFO.

MBO and MFO. In the MBO structure, the model is the key to quality optimization. The optimization algorithm is run on the basis of the model to seek the optimal solution. After the optimal solution is derived and implemented, deviation from the real process can be observed; the model is then corrected with new data, and the optimization process is repeated. By comparison, the MFO strategy avoids complicated quality modeling by replacing it with online experiments and direct measurement. MFO requires strong operability and speed features and may therefore not be applicable to all batch processes. However, it can be useful for batch processes with short cycle time and low operational cost. To reduce experiment costs, an efficient MFO algorithm is used to speed up convergence. As the core of MFO, the efficiency improvement and effective implementation of the MFO algorithm significantly affect quality optimization. The MFO algorithm must possess the following characteristics. First, it should accept online experiment values as the feedback information on MFO, whose implementation is not based on the process model. Second, it must be robust enough to deal with noise, which can disturb online experiments. Third, it should rapidly and accurately converge with the optimal solution. Appropriate optimization algorithms must therefore be proposed and integrated into MFO. 2.2. Simplex-MFO Algorithm Development. Optimization algorithms can be categorized into two groups: gradientbased algorithms and gradient-free algorithms. Normally, gradient-based algorithms offer fast convergence, but in some cases, the gradient may not be easy to derive. In the MFO scheme, function evaluation is replaced by direct quality measurement. Although the gradient can still be estimated with the use of a difference operation, accuracy deteriorates as a result of noise and disturbance in the process operation and quality measurement. All of the aforementioned factors affect the efficiency of gradient-based algorithms. Therefore, gradient-free algorithms are considered for the MFO in this project. Simplex search,13 a typical gradient-free algorithm, is integrated into MFO. A strategy with AEN is also proposed to deal with noise and disturbance. The integrated algorithm is referred to as the simplex-MFO in this study. Figure 3 shows the flowchart of the proposed algorithm, whose details are as follows: 2.2.1. Simplex-MFO Initialization. The parameters of the simplex algorithm are set in this stage. The initial guess of the

Figure 1. Flowchart of MBO. 16385

dx.doi.org/10.1021/ie5017199 | Ind. Eng. Chem. Res. 2014, 53, 16384−16396


Article

Figure 3. Flowchart of simplex search based on MFO.

optimized variables is given as X0. Each dimension of the initial guess is normalized into the same range [0, 100]. A sequential perturbation method is then used to construct the initial simplex. Each component of X0 is perturbed with a random change ratio of no more than 50% to generate the vertex of the initial simplex. The perturbed vectors and original point construct the initial simplex with n+1 vertices, denoted as a point set V0 = {v1, v2, ..., vn+1} Experiments are conducted to obtain the function evaluation values in each vertex. The vertices in V0 are sorted according to their function evaluations: f (v1) ≤ f (v2) ≤ ... ≤ f (vn + 1)

such as reflection, expansion, outside or inside contraction, and shrinkage, to search for the optimum. The procedure is performed as follows: (a). Reflection. Reflection is first conducted on the initial simplex. A reflection is calculated as vr = (1 + α)vm − αvn + 1

(3)

where i=1

vm = (∑ vi)/n

(2)

n

2.2.2. Simplex Operation. As the kernel of quality optimization, simplex-MFO updates the iterates through strategies,

(4)

vm is the geometrical center of all vertices except the worst one and α is the reflection coefficient. 16386



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If f (vr) ≤ f(v1), which means that the objective function evaluation of the reflection point is better than the best one in the initial simplex, then the expansion operation in step (b) is performed. Else, if f (v1) ≤ f(vr) ≤ f(vn), which means that the objective function evaluation of the reflection point is better than the next-to-worst one but worse than the best one in the initial simplex, then the reflection point is accepted and replaces the worst point to form the new simplex. Else, outside or inside contraction is considered, as shown in step (c). (b). Expansion. The expansion operation is calculated as ve = (1 − γ )vm + γvr

measurement of the ith experiment, and yk̅ is the estimated quality response. However, this treatment increases experiment cost and decreases the efficiency of MFO. To address this problem, a strategy that incorporates AEN is proposed in this study. The key to this strategy is to balance the quality estimation precision and the experimental cost. Two aspects are involved in this strategy. First, the repeated experiment number relates to the feasibility of the iterative solution. For a constrained optimization, the initial guess and the first few iterates may not be feasible solutions, or the constraints are not satisfied. In this stage, the objective function is far from the optimal value with the constraint penalty. Few requirements exist for the quality precision of such infeasible solutions. Therefore, we can skip the repetition of the experiment. However, if the iteration enters the feasible region, we perform additional experiments to obtain a good measurement. Therefore, checking the solution feasibility as follows can vary the number of experiments:

(5)

where γ is the expansion coefficient. If f(ve) ≤ f(vr), then ve replaces vn+1. Otherwise, vr replaces vn+1. (c). Outside or Inside Contraction. If f(vn) ≤ f(vr) ≤ f(vn+1), then the following formula defines the outside contraction operation as voc = (1 − β)vm + βvr

(6)

⎧ ⎪ 1, g (X ) > ξ Nexp = ⎨ ⎪ ⎩ m , g (X ) ≤ ξ

where β is the contraction coefficient. Either vr or voc, whichever has the lower function value, is then accepted and replaces vn+1. Else, if f(vr) ≥ f(vn+1), then the inside contraction is implemented as vic = (1 − β)vm + βvn + 1

where g(X) is the inequality constraints in eq 1, ξ is an additional margin set for the constraint, and m is the repeated experiment time when the feasible regions are entered and whose value is further determined as follows: After an iterate enters the feasible region, the second aspect is activated with the simplex operations considered. Different operations are available in the simplex search, including reflection, expansion, outside contraction, inside contraction, and shrinkage. These operations make different contributions to the optimization process according to each iterative process and result, and they are therefore treated with their different, repeated, respective times. We denote mr, me, moc, mic, and ms as the different repeated times for the reflection, expansion, outside contraction, inside contraction, and shrinkage operations, respectively. The variable m in eq 11 is therefore assigned different values for different simplex operations. Among these operating points, the reflection point is the most critical one because it decides the direction of the following search. However, in terms of the degree of improvement toward the optimal point, the reflection and expansion points make the largest contributions. Therefore, the reflection and expansion points of quality estimation precision should be higher than those of other iterative points. The following rule is set for the proposed simplex-MFO algorithm in consideration of the contributions of the different operations:

(7)

If the function evaluation of the inside contraction point is better than the worst one, that is, f(vic) ≤ f(vn+1), then vic replaces vn+1. If no improvement in performance is observed, then the shrinkage operation is applied, and step (d) is performed. d). Shrinkage. The shrinkage operation is expressed as vi = (1 − δ)v1 + δvi ,

i = 2, ..., n + 1

(8)

Every vertex, except the best one, is replaced with a new one according to the equation. (e). Bound Activation. If bounds for the optimized variables exist, then the simplex operation may leave the bounded area. The following strategy is proposed to lead the search back into the feasible region. Suppose the bounded region is D. If the operation point v̂k ϵ D, then vk = v̂k. Otherwise, a rule to choose the nearest point within the feasible region by minimizing the Euclidean distance from the current point is activated:

vk = arg min||v − vk̂ || vϵD

(9)

(f). New Simplex. After the worst point is replaced through reflection, expansion, or contraction operations, or after the points other than the best one are updated through shrinkage operations, a new simplex is constructed. This process is repeated until the termination criteria are satisfied. 2.2.3. Experiments with AEN. Unlike traditional optimization, MFO relies on real experiments to perform function evaluation. The actual process is always accompanied by noise and disturbance, so the measurement deviates from the true value. One common way to deal with noise is to repeat the experiment and measurement and then use the average value to approximate the true value: yk̅ =

1 TpTm

mr > me > moc > m ic > ms ≥ 1

i=1 j=1 Tm

(12)

Figure 3 shows that following each simplex operation is a module for experiments with AEN to evaluate the function at the new point. Equations 11 and 12 are combined to determine the number of experiments at the tested point and thus save on experiment costs while maintaining balanced precision. 2.2.4. Termination Criteria. Quality control problems involve a preset quality, and MFO can stop when this product quality is met. In quality optimization problems, however, the optimal solution is initially unknown. Traditional gradient-based judgment is also difficult because of noise in the MFO measurement. The process of the simplex search iterates by replacing the worse point with a better one in the simplex. The simplex search contains improvements in two directions, namely, the response values and the simplex shape. The response values are connected

∑ ∑ ykij Tp

(11)

(10)

where Tp and Tm are the repeated experiment and measurement number, respectively, yijk represents the quality response in the jth 16387



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factors make the injection molding process a good example to demonstrate the MFO strategy. The experiments were conducted on an HTL68/JD reciprocating screw injection molding machine. A commercially available injection-molding grade of polystyrene (PS), type GPPS 666D, was used in the experiments. The refractive index of the material was 1.59. Figure 4 shows that the plastic optical

with function evaluations through experiments, whereas the simplex shape is determined by the coordinates of the iterative points. The simplex search terminates only when no changes are observed in either direction. Therefore, 2-fold termination criteria are adopted in the simplex-MFO method. Criterion 1 involves the response value precision in the simplex: ε1 =

1 n+1

j=1

∑ Yj n+1

(13)

where Yj is the function evaluation values of the jth vertex among n + 1 vertices in the simplex. If ε1 becomes small, then all the vertices have very close response values. Criterion 2 evaluates the simplex shape by calculating the longest distance among all the vertices in the simplex as follows: ε2 =

max

||Xi − Xj||

i , j = 1, ···, n + 1 i≠j

(14)

where Xi,Xj are the coordinates of the ith and jth vertices, respectively. If ε2 becomes small, then all the vertices are very close. Finally, the process is terminated if both criteria are consecutively satisfied for a certain time. K is set as the counter variable, and the termination criteria are set as (ε1 ≤ ε1̅ ) ∩ (ε2 ≤ ε2̅ ) ∩ (K ≥ K max )

(15)

Figure 4. Shape of the plastic optical lens used in the test.

where ε1̅ and ε2̅ are the preset tolerances and Kmax is the preset constant for the counter.

lenses were produced with a lens-shaped mold. The designed focal length, f tg, was 80 mm. Two quality indices, the part weight and the focal length, were used in the optimization. The weight was measured offline with a precision electronic balance. The resolution of the balance was 0.001 g. The focal length was measured offline with a focimeter, which had a resolution of 0.01 diopter (D). A diopter is a unit of measurement of the optical power of a lens or curved mirror, which is equal to the reciprocal of the focal length measured in meters. The relationship of the focal length measured by millimeter f (mm) and by diopter d (m−1) is 1 f = 103 (16) d

3. APPLICATION OF INJECTION MOLDING Injection molding is a typical batch process with short cycle time and low cost. It is a major plastic processing technique to convert thermoplastic into different types of products with the use of a reciprocating screw injection molding machine. The entire process consists of four sequential stages: filling, packingholding, cooling, and plastication. In the filling stage, the screw moves forward and pushes the melt through the runner, gate, and finally, into the cavity. During this period, the cavity pressure gradually increases until the cavity is completely filled, at which point the pressure rapidly increases, and the process then switches to the packing-holding stage. During the packingholding stage, additional materials are packed into the cavity at a high pressure to compensate for the material shrinkage that results from cooling. This process continues until the gate freezes off. Subsequently, the melt can no longer be pushed into the mold. The cooling stage continues, which is accompanied by a gradual reduction in cavity pressure, until the solidification is sufficient to eject the molded part from the mold without damage. Concurrent with the cooling of the polymer in the mold, the polymer in the barrel undergoes plastication to prepare the melt for the next cycle. The processing cost involves the use of plastic material and electricity, which are all quite cheap for each cycle. Moreover, the time cost for a cycle, typically less than 1 min, is short. Similar to those of other batch processes, the optimal settings of key process variables based on precise closedloop control14,15 and effective monitoring16 are essential to obtain quality products. Meanwhile, the rapidly varying market requires a flexible manufacturing system with which molds and materials can be frequently changed, and this condition often results in difficulties in model development. All of these

The objective of the quality optimization of the injectionmolded plastic lenses is to determine the proper setting for key variables on the basis of the following considerations: keeping the focal length of the plastic lenses at their targets within a proper tolerance while minimizing the weight of the lenses to save material. The mathematical formulation of the above quality optimization can be expressed as ̃ X) min J = W( X

͠ X ) − FLtg || ≤ FL tol , s. t. ||FL( X = [x1 , ···, xn]T , xiL ≤ xi ≤ xiU ,

i = 1, ···, n

(17)

where X represents the process conditions with a dimension of n, xLi and xUi denote the lower and upper bounds of xi, respectively, and xi denotes the ith component of X. Then, 16388



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⎧ ̃ ⎪ W(X ) = W(X ) + eW (X ) ⎨ ⎪ ͠ X ) = FL(X ) + eFL(X ) ⎩ FL(

problem shown in eqs 17 and 19, several primary parameters were first set. The product weight ranged from 7.4 to 7.9 g, which was further normalized within 7 and 8 g. The tolerance of the focal length was set to 70 and 90 mm, and the normalized interval was set to 50 and 110 mm. The penalty parameter M in eq 19 was set to 100. For the termination criteria, the tolerance of criterion 1, ε1̅ , was set to 0.1, and that of criterion 2, ε2̅ , was set to 10%, and Kmax = 3. 4.1. Experiments Based on Simplex. The simplex method was used without the AEN strategy as the first test to prove the effectiveness of MFO. With the existence of noise and disturbance considered, the quality measurement at each iterate was first conducted by consistent repetition of the experiments for five times. The results are presented as follows. Without loss of generality, a random point was selected as the initial point: X0 = [105 105 45 70 90]T. Figure 5 depicts the iterative result of the simplex method. Figure 5a shows the trajectory of all test points, where the first six points formed the original simplex. In the initial simplex, the best point was from run 4 and the worst point was from run 6. The simplex strategies used in eq 3 indicate that the seventh reflection point (run 7) was obtained, and its result was evaluated with an online experiment. The experimental result shows that the reflection point was better than the best point of the initial simplex; therefore, the expansion strategy was implemented in eq 5 to obtain the expansion point (run 8). After a comparison of the results of the reflection and expansion points, the reflection point was accepted as the new best point and was used to replace the worst point in the initial simplex. Therefore, a new simplex was formed. This process was repeated until the termination criteria were satisfied. Figure 5a indicates that a total of 31 runs were made before MFO converged. Five experiments were repeated at each point, so a total of 155 batch experiments were conducted. Some test points might have increased the objective function, such as points 12 and 14 in Figure 5a. Figure 5b shows the simplex evolution results, which recorded the best results during the iteration. The simplex was changed 14 times until the process converged. Figure 5c shows the weight trajectory of MFO. The part weight was decreased from 7.850 to 7.801 g. The focal length trajectory, as shown in Figure 5d, was decreased from its original value to less than 70 mm to obtain a feasible result within the constraints. The value of termination criterion 1 calculated with eq 13 is represented in Figure 5e. After the seventh iteration, termination criterion 1 went into the preset tolerance of 0.1. Meanwhile, termination criterion 2 in eq 14 did not meet the requirement until the 12th iteration. After this, termination criterion 2 went into the preset tolerance of 0.1. To further test the proposed method, another randomly generated point, X0 = [108 116 48 67 41]T, was selected as a different initial guess to demonstrate the robustness of the MFO method with repeated experiments. The other settings of test 2 were the same as those in test 1. Figure 6 shows the iterative results. A comparison of Figures 6d and 5d shows that the initial guess was on the other side of the feasible region. After construction of the initial complex with the first six runs, MFO allowed the search to enter the feasible region. The part weight, as shown in Figure 6c, was first increased from 7.486 to 7.826 g and then decreased to 7.800 g at the optimal solution. Finally, a total of 11 iterations and 26 run tests were performed before test 2 was terminated. Five repeated experiments were required at each point, so a total of 130 batch experiments were conducted for this test.

(18)

where W(X) and FL(X) denote the true values of the part weight and ͠ (X) represent the measurefocal length, respectively, W̃ (X) and FL ments of these quality attributes, and eW(X) and eFL(X) denote the measurement errors of weight and focal length, respectively. eFL(X) is relatively large for two reasons. First, the molding process suffers from various disturbances that result in quality variation. Second, the measurement of the focal length requires the part to be properly positioned (so that the laser can pass through the center of the lens), and the operation requires an experienced operator. Therefore, a relatively large tolerance of the focal length constraint was selected for this case study. The target focal length, FLtg, was set to 80 mm, and the tolerance FLtol was set to 10 mm. The penalty function method9 is used to transform a nonlinear constrained problem to an unconstrained problem by addition of penalty items as follows: ̃ X ) + M × Φ2 min J = W( X

s. t. xiL ≤ xi ≤ xiU ,

i = 1, ..., n

X = [x1 , ..., xn]T ⎧ ͠ X ) − FL tg || ≤ FL tol 0, if || FL( ⎪ Φ=⎨ ⎪|| FL( ͠ X ) − FL tg || , if || FL( ͠ X ) − FL tg || > FL tol ⎩ (19)

where M denotes the penalty parameter and Φ is the penalty function for the constraint. The optimized variables include the pressure settings of the first and second injection stages (x1, x2), stroke proportion for the stage division (x3), packing pressure (x4), and packing time (x5). Table 1 shows the descriptions and Table 1. Process Variables for Optimization variable

variable description

bounds

unit

x1 x2 x3 x4 x5

pressure of the 1st injection stage pressure of the 2nd injection stage stroke proportion for the stage division packing pressure packing time

[90−120] [90−120] [30−60] [60−75] [4−15]

bar bar % bar s

Table 2. Fixed Experimental Conditions barrel temperature (°C) 1st zone

2nd zone

3rd zone

4th zone

mold temperature (°C)

cooling time (s)

stroke (mm)

210

200

200

190

60

40

40

bounds of the decision variables. Table 2 shows the other key experimental conditions set, such as the barrel temperatures, hot runner temperatures, cooling time, and injection stroke. The simplex-MFO algorithm was used in the quality optimization as formulated above. The performance of this algorithm is demonstrated in the subsequent section.

4. RESULTS AND DISCUSSION A series of experiments was conducted to prove the effectiveness of MFO for quality optimization. To deal with the optimization 16389



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Figure 5. Results of MFO experiment 1. The trajectories of MFO.

Both results indicate that MFO was effective for quality optimization. The average result with the use of repeated experiments decreased the effect of noise or disturbance at the cost of increasing the total number of experiments. Therefore, a dynamic experiment strategy was proposed and tested as follows. 4.2. Experiments Based on Simplex-MFO with AEN. In tests 1 and 2, each experiment was repeated five times to decrease the effect of noise or disturbance. Tests 3 and 4 were further conducted to demonstrate the effectiveness of the AEN strategy. Considering the noise and disturbance in measurement, ξ, the additional margin shown in eq 11, was set to 5 mm to deal with the constraints of focal length. Therefore, when the focal length

was less than 65 mm or exceeded 95 mm, the AEN strategy would set the experiment number to 1 to save in operation cost. Otherwise, the experiment numbers for the simplex operation, as well as those for the shrinkage, inside contraction, outside contraction, expansion, and reflection, were set according to eq 12, where the parameters were set as mr = 5, me = 4, moc = 3, mic = 2, ms = 1. The experiment was first conducted with the same initial simplex as in test 1. Figure 7 shows the detailed results. Among the first six points that formed the initial simplex, all experiment numbers were set to 1, except for run 4 whose focal length was within the dynamic experiment threshold and thus was set to 5. Although these six points were the same as those in test 1 in 16390



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terms of input, the output results were slightly different because of noise and disturbance. Therefore, although the traditional simplex method is a deterministic algorithm, the simplex-MFO proposed in this study has a stochastic feature. The trajectories in Figure 7 gradually deviated from those presented in Figure 5. Eventually, meeting the termination criteria required 16 iterations and 35 run tests. Although the run number increased compared with that in test 1, the total number of experiments was reduced to 80 by application of the AEN strategy for different simplex operations. The total runs decreased by 48.39% compared with those in the experiments that were repeated five times.

Figure 7c shows that the part weight was eventually reduced to 7.797 g. Figure 7d depicts that the focal length was also led to the feasible region. The next experiment was conducted with the use of the initial simplex generated in test 2. Figure 8 presents the results. With the AEN strategy, the simplex-MFO took 17 iterations and 30 runs to terminate. A total of 93 experiments were conducted for the entire optimization process. Compared with that of test 2, the total number of experiments was decreased by 28.46%. The eventual part weight was optimized to 7.758 g, and the focal length was also led in the feasible region. 16391



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4.3. Comparison between MFO and MBO. A comparative study on MFO and MBO was also conducted on the injection molding machine. Two indices were used to evaluate the performance of different quality optimization methods. The first index was the total number of experiments needed for the optimization. This number was set based on two reasons. First, the MFO results show that it can obtain the optimal solution within about 30 runs. Second, Latin hypercube sampling was used for the design of experiment. The experiment has two levels with five factors, so the experiment run number was eventually set to 25=32 to build the model for MBO in this study.

Table 3 lists the results of the comparison. Rows 2 to 5 show the results. The MFO with AEN (MFO3 and MFO4) and that without AEN (MFO1 and MFO2) were compared. The results indicate that the simplex-MFO with AEN strategy effectively maintained a balance between quality estimation precision and experimental cost, as well as provided a good means for the quality optimization of the injection molding process. The findings also indicate that the process did not converge to the same solution for the different tests and was affected by noise and disturbance. However, all the solutions are acceptable for industrial applications. 16392



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omitted because of data limitation (32 in this case). The mathematical form is

To reduce the noise and disturbance, each experiment was repeated five times in each condition. The second index is the optimal result in terms of the objective function and constraints. The MBO and MFO results were then compared in terms of the number of experiments. Two modeling techniques, the polynomial regression model and Kriging model,17 were used in the comparison. Sequential quadratic programming was used for MBO with the use of the function fmincon in Matlab.18 4.3.1. MBO Based on Quadratic Polynomial Regression. Quadratic polynomial regression was first adopted to build the models of weight and focal length. The cross-product terms were

i=1

y = β0 +

j=1

∑ βi xi + ∑ βjjxj2 n

n

(20)

where xi,xj denote the ith and jth inputs of n-dimensional input vector X, respectively. y is the output. β0, βi, and βjj are the coefficients of the model. The 32 experimental and predicted values were compared as shown in Figures 9 and 10. Obvious errors were found at a number of points. The root-mean-square-error of the weight and focal length were 0.087 and 7.0352, respectively. 16393



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Figure 9. Weight model and residual based on quadratic polynomial regression.

Figure 10. Focal length model and residual based on quadratic polynomial regression.

Figure 11. Weight model and validation based on Kriging technique.

The experimental results show that the real part weight at this point was 7.775 g, a difference of 0.209 g from the model prediction, and the focal length was 63.50 mm, which was completely unacceptable in this case. All the above-mentioned data are also listed in Table 3, where the total experiment number is calculated based on the sum of the modeling and

Offline optimization was further conducted with the proposed model. The optimal solution was considered as X* = [90 90 25.8 60 65]T, with the predicted optimal part weight as 7.566 g and the focal length as 90.00 mm, just on the border of the constraint. The above optimal setting point was then used on the injection molding machine for validation. 16394



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Figure 12. Focal length model and validation based on Kriging technique.

Table 3. Experimental Results Comparison between MFO and MBO test

initial point

total experiment no.

total iteration no.

total run no.

opt X*

opt Y*

model error

MFO1 MFO2 MFO3 MFO4 MBO1 MBO2

[105 105 45 70 90] [108 116 48 67 41] [105 105 45 70 90] [108 116 48 67 41] [108 116 48 67 41] [108 116 48 67 41]

155 130 80 93 165 181

14 11 16 17

31 6 35 30

[107 107 257 67 48] [113 118 266 69 55] [108 108 252 67 56] [96 118 245 60 128] [90 90 25.8 60 65] [118 94 27.6 60 97]

7.801 7.800 7.797 7.758 7.775 7.775

0.209 0.018

shows that Kriging modeling is acceptable for the quality optimization. However, using the Kriging model does not demonstrate any advantage in terms of the total experiment runs compared with those of MFO with AEN. The above experimental results are further summarized in Table 3 for a clear comparison. Rows 2 to 5 present the optimization results of the simplex-MFO, and the last two rows list those of MBO. For MBO, quadratic polynomial regression cannot provide an optimal condition that satisfies the quality requirements. The Kriging technique is more effective than quadratic polynomial regression in this application. The total number of experiments of the simplex-MFO with AEN strategy, as indicated by rows 4 and 5, are significantly less than those of MBO and MFO with a constant number of repetitive experiments.

validation experiments. Therefore, with a comparable experiment number as that of the MFO method, the quadratic polynomial regression did not establish a correlation and was therefore not appropriate for the quality optimization of the injection molding process. 4.3.2. MBO Based on Kriging Method. The Kriging model was used in the injection molding quality optimization. The Matlab toolbox DACE19 was used to construct the Kriging model. The Kriging model was created as follows: y = f (X )T β* + r(X )γ *

(21)

where f(X) β* denotes the regression model determined by the base function f(X) = [f1(X),···,f p(X)] and the regression parameters β*, r(X) represents a stochastic model correlation between different sample points, and γ*denotes the correlation parameter. The linear regression model and the Gaussian correlation model were selected. The same 32 experimental data were used to construct the process model. A total of 16 extra experimental data were used for validation, resulting in the increase of the total experiment number of MBO2 by 16 than the MBO1 as shown in Table 3. Figures 11 and 12 present the model predictions of weight and focal length with the use of the training and validation samples. These models correlated the training samples well because the Kriging model was considered as an unbiased estimation. However, deviations were still found in a few validation samples. Offline optimization was conducted with the proposed model. The optimal solution was X* = [118 94 27.6 60 97]T. The optimal weight was 7.758 g, and the eventual focal length was 75.49 mm. Further experimental validation shows that the real part weight was 7.775 g, which exceeded the model prediction by 0.018 g, and the real focal length was 70.16 mm, which was quite close to the border of the constraint. The finding T

5. CONCLUSION A systematic MFO method was proposed for a type of batch processes with short cycle time and low cost. A simplex-MFO algorithm was developed for an online search of the optimal solution. Strategies of adaptive experiment number and 2-fold termination criteria were proposed for the optimization. To demonstrate the quality optimization, experiments were conducted in the injection molding process to minimize part weight subjected to a specific range of focal length. A comparison between the MFO and MBO methods was also conducted. The results indicate that MFO performed well in the quality optimization of the injection molding process. The proposed method also shows the potential to be extended to other batch processes with rapid production and low-cost features.

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AUTHOR INFORMATION

Corresponding Author

*Tel: +86-571-87953966. E-mail: [email protected]. 16395



Article

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the financial support of 863 Program (No. 2012AA040305) and National Natural Science Foundation of China (Nos. 61374205 and 61374167).

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REFERENCES

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Model-Free Quality Optimization Strategy for a Batch Process with

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