A Fuzzy Evolutionary Approach with Taguchi Parameter Setting for the Set Covering Problem

Jingpeng Li and Raymond S K Kwan School of Computing, University of Leeds, Leeds LS2 9JT, UK

Email: li@comp.leeds.ac.uk, R.S.Kwan@leeds.ac.uk

Abstract - The set covering problem has a very wide area of applications, and scheduling is one of its most important applications. At CEC2001, a fuzzy simulated evolution algorithm was reported for a set covering problem in bus and rail driver scheduling. This paper reports on the generalisation of the approach to the class of set covering, which is basically to cover the rows of a zero-one matrix with a subset of columns at minimal cost. The simulated evolution method iteratively transforms a single solution treating its selected columns as a population. At each iteration a portion of the population is probabilistically, biased towards selecting the ‘weaker’ columns, discarded; and the broken solution is then reconstructed by means of a greedy heuristic. The columns a r e evaluated under several criteria formulated based on fuzzy set theory. The set of weights associated with the evaluation criteria were previously either fixed in advance o r calibrated by means of a simplified genetic algorithm. In this paper, we consider some discrete levels of values instead of a continuous range from 0 to 1 for each weight. We have also included two other parameters driving the evolutionary process as additional factors having some discrete levels of values. Taguchi’s orthogonal experimental design is applied. This has the effect of comprehensively evaluating the combinations of factors, although only a small fraction of the possible combinations is explicitly experimented. Better results have been obtained, and comparisons with those previously reported a r e discussed.

I. INTRODUCTION

Set Covering Problems (SCPs) are difficult zero-one optimization problems, which have been proven to be NP- complete [l]. Scheduling is a major application of the SCP. Other application areas are for example resource allocation, pattern recognition, and machine learning.

The SCP is to cover the rows of an m-row, n-column, zero- one matrix (q) with a subset of columns at minimal cost. Defining xi =1 if c o l m j (with an associated cost cj >O) is in the solution and xj =O otherwise, the SCP is to

n

Minimize c jx j (1)

(2)

(3 1

j = I

n

Subject to: ~ u v x j 2 1 , &I={ 1,2,. . ., m}

xi = 0 or 1, j ~ & { 1,2,.. ., n} j=1

Constraint (2) ensures that each row is covered by at least one column, and (3) requires that whole columns be used. In the case that all the cost coefficients cj are equal, the problem is called a unicost SCP, and the objective is to minimize the number of columns.

Integer Linear Programming (ILP) is the traditional approach for the SCP. The relaxed Linear Programming (LP) for the SCP, Le. ignoring the integer constraint, is first solved.

Then, the branch-and-bound tree-search procedure is used to obtain the integer solution. For large size SCPs, this method often has computational difficulty in getting an integer solution, although the relaxed LP can usually be solved relatively quickly in polynomial time [2].

In recent years, some heuristic techniques for the SCP are aimed at getting near-optimal solutions within reasonable time. Beasley [3] gave a Lagrangian heuristic and reported better results than those of some other heuristics. Sen [4] discussed the performance of a simulated annealing algorithm and a simple Genetic Algorithm (GA) on SCP, but few experimental results were given. Huang et al. [5] presented a GA for the unicost SCP, in which a new penalty function to handle the constraints and a mutation operator to accelerate the convergence were proposed, and found better solutions for two large test problems. Beasley and Chu [6] presented a GA-based heuristic with several modifications to the basic genetic procedures, and produced high-quality solutions on a large set of randomly generated problems. The sizes of the test problems in all the above papers were up to 1,000 rows and 10,000 columns.

In this paper, we present a new fuzzy evolutionary approach for solving even larger SCP problems. The approach involves a number of parameters and evaluation weights, the combinations of which are efficiently experimented using Taguchi’s orthogonal arrays [7]-[SI. Data sets for the bus and rail driver scheduling problem [9], a real world SCP, is used to demonstrate this approach.

The driver scheduling problem is normally solved in two phases. The first phase generates a very large number of legal potential shifts. The second phase is an SCP, which selects from the very large pool of potential shifts the cheapest subset to form the solution schedule. At CEC2001, a fuzzy Simulated Evolution (SE) algorithm was reported for the second phase [ 101. It iteratively transforms a single schedule treating its component shifts as a population. At each iteration a portion of the population is probabilistically, biased towards selecting the ‘weaker’ shifts, discarded. The broken schedule is then repaired by means of a greedy heuristic. The shifts are evaluated under several criteria formulated based on fuzzy set theory.

This paper reports on new improvements in the fuzzy SE approach, and its generalization from driver scheduling to the class of SCP:

For fuzzy evaluation, the criteria used are now generalized for the SCP without any domain specific knowledge in their formulation.

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In SE’s Evaluation step, a new evaluation function has been designed to replace the former function that was specialized for the driver scheduling problem. In SE’s Construction step, the hnction for assessing which shifts to be used for repairing the broken schedule is now different from that in the Evaluation step for discarding shifts.

0 Taguchi’s experimental design is utilized to reliably set the seven parameters in our proposed algorithm. This method uses orthogonal arrays to perform an initial study of the wide range of parameter space, with a small number of experiments.

We shall first describe the method of fuzzy evaluation for the SCP. A fuzzy SE algorithm is then presented. Taguchi’s orthogonal experimental design for parameter settings will be briefly introduced. Comparative results using large-scale real world problems are given, which are followed by some conclusions.

11. FUZZY EVALUATION FOR SET COVERING

Based on the characteristics of the SCP, we have abstracted five critical factors for assessing the ‘goodness’ of each column j (~EJ). Since the criteria bear some uncertainty, fuzzy set theory [ l l ] is used to introduce the concept of a structural coefficient, which gives column j a quantitative value f i ( j ) ~ [0,1] according to its structural state. The fitter the structure for column j, the largerfi(j) is.

A. Construction of the Factor Set The main factors concerning the column’s structure are the

number of rows it covers (u,), its cost (uz), the ratio of its number of covered rows to its cost (uj), and the average coverage number (number of columns covering the row) of all the rows covered by this column (u4).

Since the relaxed LP for the SCP can be solved relatively easily, the fractional cover in the relaxed LP solution, if the column is included in the solution, is used as the fifth factor

1) Factor ul and u3

The goodness of a column j ( j ~ J) generally increases with the number of rows it covers, denoted as pi, and the ratio of

Pj to its cost cj. A similar formulation of the membership

functions p jk ( ~ E J and k l , 3) for these two factors are therefore used and defined as:

m

where x j l = pi = ; i=l

bj, = maximum number of rows; cj l = minimum number of rows;

bi3 = maximum ratio;

cj3 = minimum ratio.

2 ) Factor u2 and 244

The goodness of a column j UEJ) generally decreases with its cost cj and the average coverage number of all the rows covered by column j, denoted as ai. A similar formulation

of the membership functions p j k ( j ~ J and k-2, 4) for these two factors are therefore used and defined as:

b . - x .

bjk -‘jk

p . Jk = u , V j ~ J ,

where x j 2 = c j ;

bj2 = maximum cost;

cj2 = minimum cost;

( 5 )

x j 4 =aj = T ( a , x g a v ) / T a v , j E J ; i=l j = l i=l

b j 4 = maximum average coverage number;

c j 4 = minimum average coverage number.

3 ) Factor u5 In the relaxed LP solution, the values of the column

variables lie between 0 and 1. These values provide some useful information about the significance of some of the columns identified in the relaxed solution. Empirical studies using cases of driver scheduling have shown that the closer to a value of 1, the higher the chance the column will be included in the integer solution [ 121. Hence the membership function pis ( j ~ J ) for the fractional cover factor is defined as follows:

J 5 lo ( x , 5 - d

p . = e , if column j is in the fractional cover; (6) , otherwise.

Let pis =1 when xis = bj, , and pis =0.01 when

where xis = fractional value of column j in the relaxed LP x . = c . 15 1 5 ’

solution; b = maximum value in fractional cover;

cis = minimum value in fractional cover.

Therefore, [a = b j 5

(7)

B. Fuzzy Evaluation Based on fuzzy mathematics, fuzzy evaluation considers

various criteria influencing the structure of a column, and sets up a mathematical model to evaluate the efficiency of this

I

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column. Therefore, for column j (~EJ), the formulation of its structural coefficient J;(j) by the method of fuzzy evaluation is:

5

f1 ( j ) = ( w k ' P j k 1 3 . (8) k = l

Where wk denotes the corresponding weights for criteria uk ( k l , 2, 3, 4, 5 ) . They all satisfy the normalizing condition

wk = 1 and wk 2 0. If the k-th criterion were dominant in

assessing the set structure, its weight should have a high value.

From the analysis above it can be seen that the main task of the above fuzzy evaluation model is to find a suitable weight distribution among the fizzy membership functions. These five weights, along with two other parameters given later, could be determined by Taguchi's orthogonal experimental design, which is described in section IV.

5

k=l

111. A FUZZY SIMULATED EVOLUTION ALGORITHM

The Simulated Evolution (SE) algorithm is a general optimisation technique originally proposed by Kling and Banerjee [13] for the placement problem. It combines the feature of iterative improvement and constructive perturbation to avoid getting stuck at poor local optima.

In this paper, a fuzzy SE algorithm is applied to mimic generations of evolution on a single solution. It executes a sequence of Construction, Evaluation, Selection and Mutation steps in a loop until a user specified parameter (e.g. cpu-time, or the solution cost) is reached or no improvement has been achieved for a number of iterations. Throughout the evolution, the currently best solution is retained and finally retumed as the final solution.

A. Construction The Construction step takes a partial solution as the input,

and produces a complete solution as the output. All the existing column assignments in the partial solution remain unaffected. Therefore, the Construction step is to assign columns to all the uncovered rows to complete a partial solution. Note that the partial solution in the first iteration of the loop is set to be empty.

Each of the remaining unassigned rows i has a coverage list of length L, , i.e. containing Li possible columns that can cover it. The greed-based constructor assumes that the desirability of adding columnj(jEJ ) into the partial solution generally increases with its function value F'( j ) , which can be formulated as:

F ' ( j ) = f , ( j ) x C a , , VjE J . (9) id'

Where 50') is the structural coefficient defined as formula (8), and Z'is the set of rows to be covered. However, to introduce diversification, we randomly select one of the candidates, not necessarily the top candidate, from a Restricted Candidate List (RCL) consisting of columns with Y

largest function values F ' ( j ) . From empirical studies we find that r I 4 achieves better solutions.

Let J ' = { 1,2,. . . , t} the set of columns in a partial solution, and S, = {ila, = 1,ie I } the set of rows covered by column j ,

the steps to generate a complete solution are: Step 1 Set z ' = z - ~ ( s , . : ~ ' E J * ) .

Step 2 If Z'= q3 then stop: J ' i s a complete solution and

C(J ' ) = c(c , . : j' E J ' ) . Otherwise locate a row

t'E I' having L,, = min(L; : i 'E Z') , and then randomly select a c o h n n Sk within RCL from the coverage list of row t' . Proceed to step 3.

Step 3 Add k to J' , set 1'= I f - S , , and retum to step 2. Before the Construction, some rows may already be over-

covered, i.e. covered more than once by the existing columns in the partial solution. The other columns added by the Construction step are each chosen to cover at least one currently uncovered row, but they increase the amount of over-cover as well. Thus some columns might become redundant later, causing all their rows covered by other columns. It should be pointed out that, in the next Selection, these redundant columns will be removed automatically because of their zero goodness.

B. Evaluation In this step, goodness of the individual column in a

complete cover J' is computed. The evaluation function F ( j * ) for column j * ( j * E 1') should be normalized.

Besides the structural coefficient J ; ( j ' ) , another normalized

function to reflect the coverage status for column j' should

be combined. Hence the overall evaluation function F ( j ' )

consists of two parts: structural coefficient f , ( j ' ) E [0,1] and

over-cover penalty f, ( j ' ) E [OJ] , which can be formulated as:

F ( j ' ) = fi ( j' ) x f , ( j ' ), Vj* E J * . (10) The ratio of the non-overlapped number of rows to total

number of rows in j * ( j * E J ' ) is also regarded as an

important criterion, which can be formulated as f 2 ( j * ) below:

[ o , C a,. > 1; where bii. = j * E J '

' (1, otherwise.

If every row in j ' has been covered by one or more other

columns in J' as well, then f, ( j ' ) = 0 ; conversely if none

of the rows in j ' is over-covered, f2 ( j ' ) = 1 .

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C. Selection In this step it will be determined whether a column

j ' ( j * E J') is retained for the next generation, or discarded and placed in a queue for the new allocation. This is done by comparing its goodness F ( j * ) to (p, - k , ) , where p, is a random number generated for each generation in the range [0, 13, and k, is a value smaller than 1.0. If F ( j * ) > (p, -k,) then j * will survive in its present position; otherwise j' will be removed from the current evolutionary solution. The rows it covers, except those also covered by other columns in the solution, are then released for the next Construction. By using this Selection process, column j' with larger goodness

F ( j ' ) has higher probability of survival in the current solution.

The purpose of subtracting ksE [0,1] from p, is to improve the SE's convergent capability. Without it, in the case of p, close to 1, nearly all the columns will be removed from the solution, which is obviously undesirable. A suitable setting of the selection valve k, is important to the algorithm's performance.

D. Mutation To escape from local minima in the solution space,

capabilities for uphill moves must be incorporated. This is carried out in the Mutation step by probabilistically discarding even some superior components of the solution. Therefore, following the Selection step, each retained column j * (j* ~f ) has a chance to be mutated, i.e. randomly discarded from the partial solution at a given rate ofp,, and releases its covered rows, except those also covered by other retained columns, for the next generation. The mutation ratep, should be much smaller than the selection rate to guarantee convergence. Like the selection valve k, , p m is also an influencing parameter in the SE.

1 2 3 4 5 6 I 8 9

IV. TAGUCHI METHOD FOR PARAMETER DESIGN

Seven parameters are investigated in our proposed algorithm, namely weight wk for criteria u k ( k l , 2, 3 ,4 , 5) in the fuzzy evaluation model, selection valve k, in the Selection step, and mutation rate p , in the Mutation step. These parameters will influence the SE's performance greatly, and are difficult to determine. Common approaches for parameter design lead either to a very long time span for trying out all combinations, or to a premature termination of the design process with results far from optimal in most cases.

A GA was presented in our earlier work [14] to calibrate the fuzzy weight distribution, and some good solutions were obtained as by products. However, the weights so obtained may not be always good for the SE even though the results were rather satisfactory in some cases [ 151. The first reason is that GA and SE are evolutionary algorithms with very different mechanisms, therefore a good weight distribution under GA may not be always suitable for SE. The second reason is due to the different construction methods: the one

A B C D

1 1 1 1 1 2 2 2 1 3 3 3 2 1 2 3 2 2 3 1 2 3 1 2 3 I 3 2 3 2 1 3 3 3 2 1

used in GA is deterministic rather than randomized as is in SE.

For full experimentation on the fuzzy SE algorithm, the first six parameters at five value levels each would require 15,625 (56) possible experimental evaluations, and each experiment is quite time-consuming. The time to conduct such a detailed search for the optimal solution is prohibitive. Naturally, we would like to reduce the number of experiments to a practical point, and still reach a near optimal solution. For the problem of choosing an appropriate parameter configuration, Taguchi's Orthogonal Experimental Design (OED) provides a solution.

A. Basic Concepts The OED for parameter design provides a systematic and

efficient approach to determine near optimal parameter settings. The objective is to select the best combination of control factors (parameters) so that the product or process is most robust with respect to noise factors. The OED applies orthogonal arrays from experimental design theory to study a large number of variables with a small number of experiments, significantly reducing the number of experimental configurations. Moreover, the conclusions drawn from small-scale experiments are valid over the entire experimental region spanned by the control factors and their settings.

Orthogonal arrays are a special set of Latin squares [16], constructed by Taguchi to lay out the experimental design. By using the table, the required experimental situations are defined. Consider a 3-level and 4-factor orthogonal array shown in Table 1 below:

In this array, the columns are mutually orthogonal. That is, for any pair of columns, all combinations of factor levels occur, and occur an equal number of times. Here there are four factors A, B, C, and D, each at three levels. This array is designated by the symbol L9, with the 9 indicating nine rows, or configurations, to be tested. Specific test characteristics of each experiment are identified in the associated row of the table, and the number of columns represents the maximum number of factors that can be studied using that array. Thus L9 (34) means that nine experiments are to be carried out to study four variables at three levels. Note that this design reduces 81 (34) configurations to 9 experiments.

The orthogonal array facilitates the experimental design process by assigning factors to the appropriate columns. For this L9 (34) array, factors A, B, C, and D are arbitrarily

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assigned to columns 1, 2, 3, and 4 respectively. The experimenter may use different designators for the columns, but the nine trial-runs will cover all combinations, independent of column definition. In this way, the orthogonal array also ensures that factors influencing the quality of solutions are properly investigated and controlled during the initial design stage.

B. The OED Approach Our goal is to determine the best setting for each parameter

so that the solution cost is minimized. As a first step towards our goal, the initial levels of the seven control factors in our algorithm can be arbitrarily chosen. Without any pre- knowledge about the influence of the weights wk (kl, 2,3,4, 5) on the algorithm, it is reasonable to set the levels of wk evenly over the full applicable range [0, 13. However the range for the selection valve k, and the mutation rate p , should be much narrower, since according to our former experience, the SE usually yields better solutions with k, E [0.20,0.30] andp,E [0.05,0.06] respectively.

Due to the minor role of the mutation step in SE, the seventh factor ofp, is relatively less important than the other six. To maintain a balance between necessary precision and number of experiments, we define factors wk ( k l , 2, 3, 4, 5) and k, to be 5 levels, andp, to be 2 levels (shown in Table 2). These seven factors are assigned to the Lso (2' x 56) orthogonal array, a segment of which is shown in Table 3. This is a economical and efficient design for dealing with these seven factors using only 50, rather than 3 1,250 (2l x 56) experimental trials.

I Data I Rows I Columns I D;;?

By studying the main effects of each factor, the general trends of the influencing factors can be characterised. The characteristics can be controlled, such that a lower, or a higher, value in a particular factor produces the preferred result. Thus, the levels of influencing factors to produce the best results can be predicted.

Since the main purpose of this paper is to test the suitability of the proposed approach for the SCP, we only perform an initial investigation about the wide range of parameter settings, and use OED to find a suitable range of the control factors. Therefore, we simply choose the parameter configuration from Table 3 that leads to the best results, and skip the follow-on process of ANOVA and firther experiments.

Best-known solutions Cover I Cost I Elapsed

V. COMPUTATIONAL RESULTS

The algorithm presented in this paper was coded in Borland C++, and run on a Pentium I1 333 MHz machine with 196 megabyte RAM using the Windows 98 operating system. To test the SE, eight real-world large size SCPs originating from the public transport industry are solved. Problem instances prefixed by B are bus problems, and T are train problems. Details of these test problems, including number of rows, number of columns, and density (percentage of ones in the aii matrix), are given in Table 4.

Control factors

1. Weight wI

TABLE 4. DETAILS OF THE TEST PROBLEMS AND RELATED BEST-KNOWN SOLUTIONS

(Note: Results of cases marked by asterisks are obtained by the hybrid GA, while others are obtained by the specialized set covering ILP.)

Levels 1 2 3 4 5

0.1 0.3 0.5 0.7 0.9

TABLE 2. CONTROL FACTORS AND THEIR LEVELS

2. Weight w2 3. Weight w3 4. Weight w4 5. Weight ws 6. Selection valve k., 7. Mutation rate pm

0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.22 0.24 0.26 0.28 0.30 0.05 0.06 -

TABLE 3. SEGMENT OF L50(2' x 56 ) ORTHOGONAL ARRAY (The values in parenthesis represent the factor levels)

- Test run

1 2 3 4 5 6 7 8 9

10

-

...

I . w , 2. w2 0.1(1) 0.1(1) 0.1(1) 0.3(2) 0.1(1) OS(3) 0.1(1) 0.7(4) 0.1(1) 0.9(5) 0.3(2) 0.1(1) 0.3(2) 0.3(2) 0.3(2) 0.5(3) 0.3(2) 0.7(4) 0.3(2) 0.9(5)

Control factors 3.w3 4. wq 5. w5 0.1(1) 0.1(1) 0.1(1) 0.3(2) 0.3(2) 0.3(2) o . sb j os i3 j o.+j 0.7(4) 0.7(4) 0.7(4) 0.9(5) 0.9(5) 0.9(5) 0.3(2) OS(3) 0.7(4) OS(3) 0.7(4) 0.9(5) 0.7(4) 0.9(5) 0.1(1) 0.9(5) 0.1(1) 0.3(2) 0.1(1) 0.3(2) 0.5(3)

6. k,, 0.22( 1) 0.24(2) 0.26(3) 0.28(4) 0.30(5) 0.30(5)

0.24(2) 0.26(3) 0.28(4)

0.22(1)

... ... ... ... ... ...

7. P m 0.05(1) 0.05(1) 0.05(1) 0.05( 1) 0.05(1) 0.05(1) 0.05( 1) 0.05(1) 0.05(1) 0.05(1)

...

A statistical method, the Analysis of Variance (ANOVA), is commonly used to analyse the results of the OED, and to determine how much variation each factor has contributed.

The best-known schedules are usually obtained by the TRACS I1 system [17], which is a commercial system based on ILP with more than 100 person-years devoted in its development. In cases (marked by asterisk) where TRACS I1 has difficulty in finding solutions, results achieved by hybrid GAS incorporating strong domain knowledge are cited [ 181.

To give fair comparison of the computational results, each test problem was run by using the same pseudo random number seed at the beginning of the program. In the OED process, iteration number of the SE was set to be 200 for each experiment to all problems. After the best parameter setting is determined, the program run for final results will be terminated if no improvement has been achieved for 1000 iterations. The benchmark results of the solution cost are compiled in Table 5. Since our proposed algorithm, the hybrid GA, and the branch-and-bound phase of the ILP

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process take the relaxed LP solution as a starting point, the elapsed time for them are compared.

TABLE 5. COMPARATIVE RESULTS

Computational results show that the solutions derived by the SE approach are very close to the previous best-known solutions. The negative percentage deviation indicates the percentage improvement from the previous best-known solution. In terms of cover size, our results are 1.53% larger on average. But in terms of solution cost, our results are 0.31% lower on average: in 6 out of the 8 test problems, the SE-based heuristic has generated better solution values. In T7, our solution cost is worse by 3.38%. However, it should be noted that the cover size of T7 in our solution is smaller by 0.5 1%.

In term of the elapsed time, compared with those of other approaches, it is obvious that our results are much faster in general, particularly for larger cases.

In addition to finding the best solutions, another task for the initial OED is to explore whether there exists a generally good pattern of parameter setting. According to the experiments using Ls0 (2’ x 56) orthogonal array, we find that, in all cases, the best result is produced by the same parameter configuration of (0.3, 0.9, 0.1, 0.3, 0.5, 0.28, 0.06) for wk ( k l , 2, 3, 4, 5), k,, and pm respectively. It shows that our fuzzy SE approach for the SCP is quite robust. Also, even without the process of ANOVA, better results may be achieved by another set of OED (using the Ls0 orthogonal array) with narrower ranges centred on the parameter configuration found.

VI. CONCLUSIONS

In this paper, we have developed a novel fuzzy simulated evolution approach for the non-unicost set covering problem. We first design a function to evaluate, under fuzzified criteria, the structure of each column. This hnction is embedded into the Evaluation step and the Construction step of a proposed evolutionary algorithm, which mimics generations of evolution on a single solution. In each generation an unfit portion of the working solution is removed. The broken solution is then repaired by a greedy algorithm specialized for the SCP.

In our proposed algorithm, there are seven investigated control factors. Using the “change-one-factor-at-a-time” method of experimentation, a prohibitively large number of 3 1,250 experiments need to be carried out. In this paper, the

Ls0 orthogonal array from Taguchi’s experimental design theory is applied to reduce the number of experiments to 50.

Instances from the driver scheduling problem, a real-world SCP in the transportation industry, have been used to test the evolutionary approach. It has demonstrated that in general our approach can produce no-worse solutions much faster than some other approaches. Particularly, this approach is suitable for situations where quick and high quality solutions of large size SCPs are desirable.

Finally, it should be mentioned that the solution quality could be improved further if we use more refined parameter settings. This can be implemented by arranging Taguchi’s orthogonal experimental design within narrower ranges, together with the analysis of variance.

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