Design of Experiments: Useful Orthogonal Arrays for Number of Experiments from 4 to 16

Abstract

A methodology for the design of an experiment is proposed in order to find as many schemes as possible with the maximum number of factors with different levels for the smallest number of experimental runs. An algorithm was developed and homemade software was implemented. The abilities in generation of the largest groups of orthogonal arrays were analyzed for experimental runs of 4, 6, 8, 9, 10, 12, 14, 15, and 16. The results show that the proposed method permits the construction of the largest groups of orthogonal arrays with the maximum number of factors.

Introduction

Manufacturing process optimizations are powerful methods that provide simulation scenarios that yield the desired outcome [1]. The optimization techniques could contain metaheuristic procedures and/or classical optimization methods [1,2] that involve setting a series of parameters in order to obtain:

• Maximum return on budgets
• Most effective configuration of machines
• Most effective allocation of raw materials
• Optimal workforce allocations to minimize labor and total time

Although the design of experiments concept was introduced by Fisher in the early 1920s [3], the most research on this topic was carried out in the academic environment [4]. One year later, Fisher [5] demonstrated the usefulness of his concept in agricultural experiments; he analyzed the optimum water, rain, sunshine, fertilizer, and soil conditions needed to produce the best crop. Taguchi [6] went further with the design of experiment concept by introducing his approach in 1986. According to the nature of the problem, the Taguchi approach divides optimization problems in two categories, using a log function of desired output as objective functions for optimization (called Signal-to-Noise ratios):

• Static problems (there are several control factors that directly decide the desired value of the output):

  • Smaller-the-Better approach is used when:

    ▪

    The ideal value for all undesirable characteristics is zero

    ▪

    The ideal value is finite and its maximum or minimum value is defined

  • Larger-the-Better
  • Nominal-the-Best approach is used when a specified value is most desired and neither a smaller nor a larger value is desirable.
• Dynamic problems (there is a signal input that directly decides the output):

  ○

  Sensitivity of the slope: the slope should be at the specified value (usually 1) when the output is:

  ▪

  An undesired characteristic (it can be treated as Smaller-the-Better)

  ▪

  A desirable characteristic (it can be treated as Larger-the-Better)

  ○

  Linearity (Larger-the-Better): is used when the dynamic characteristics are required to have direct proportionality between the input and output.

A triad could better characterize the aim of manufacturing process optimization: best quality – less failures – higher productivity. Factorial analysis can be used in order to find the best values for parameters implied in the manufacturing process [7]. Opposite to full factorial analysis, the Taguchi method reduces the number of experimental runs to a reasonable one, in terms of cost and time, by using orthogonal arrays [8].

The Taguchi method is used whenever the settings of interest parameters are necessary, not only for manufacturing processes. Therefore, the Taguchi approach is used in many domains such as: environmental sciences [9,10], agricultural sciences [11], physics [12], chemistry [13], statistics [14], management and business [15], medicine [16].

Choosing the proper orthogonal arrays suitable for the problem of interest is the main difficulty of the Taguchi’s approach. The available literature identified the use of the orthogonal arrays summarized in Table 1.

The literature reported many orthogonal arrays; however, a full scheme that includes all the possibilities of orthogonal arrays, even for a small number of experimental runs, could not be found yet [17]. Starting from this observation the aim of the present study was to generate the largest groups of orthogonal arrays for number of experimental runs from four to sixteen, with the maximum number of factors by using a series of homemade software.

Table 1. Design of experiments: reported scheme

Experimental runs	Scheme	Reference
4	23	[18]
8	27	[19,20,21,22,23]
	25	[24]
	23	[25,26,27]
9	34	[28,29,30,31,32,33]
	33	[34]
16	45	[35,36,37]
	215	[38]
18	21×37	[39,40,41]
	22×36	[42]
	37	[43]
	21×36	[44]
	21×34	[45,46]
	21×33	[47]
27	33	[48]

Table 1. Design of experiments: reported scheme

Experimental runs	Scheme	Reference
4	23	[18]
8	27	[19,20,21,22,23]
	25	[24]
	23	[25,26,27]
9	34	[28,29,30,31,32,33]
	33	[34]
16	45	[35,36,37]
	215	[38]
18	21×37	[39,40,41]
	22×36	[42]
	37	[43]
	21×36	[44]
	21×34	[45,46]
	21×33	[47]
27	33	[48]

Method

Note that searching for as many as possible orthogonal arrays with the highest number of factors possible for the smallest number of experimental runs is not a trivial task. Table 2 presents an example of 12 experimental runs, in which adding of a new orthogonal array (D) to the existent ones (A, B, and C) is not possible, although the maximum number of factors with two levels for which the orthogonal arrays could be obtained is equal with eleven. The information regarding the maximum number of factors with two levels can be checked with Statistica software, Experimental design - Taguchi robust design experiments [49].

Table 2. Design of experiment: 12 experimental runs

Experimental runs	Factor(levels)
	A(2)	B(2)	C(2)	D(2)
1	0	0	0	-
2	0	0	0	-
3	0	0	0	-
4	0	1	1	-
5	0	1	1	-
6	0	1	1	-
7	1	0	1	-
8	1	0	1	-
9	1	0	1	-
10	1	1	0	-
11	1	1	0	-
12	1	1	0	-

Table 2. Design of experiment: 12 experimental runs

Experimental runs	Factor(levels)
	A(2)	B(2)	C(2)	D(2)
1	0	0	0	-
2	0	0	0	-
3	0	0	0	-
4	0	1	1	-
5	0	1	1	-
6	0	1	1	-
7	1	0	1	-
8	1	0	1	-
9	1	0	1	-
10	1	1	0	-
11	1	1	0	-
12	1	1	0	-

As already mentioned, the objective of the research was to obtain the largest set of orthogonal arrays by using a series of homemade software. Note that there was no rule about generating these sets, as proven by the previous example from Table 2.

An application was designed for generating orthogonal arrays from a list of factor levels (level array) using a recursive function (recurs), a class (liste) for storing current array (lst instance of liste), and an initialization function (fill_first_oa):

      class liste{
       function __construct(b,n){
        this->b=b;//b: base of numeration
        this->n=n;//n: number of experiments (n%b=0)
        this->m=n/b; //number of repetitions
        for(s=0,i=0;i<b;i++){
         this->d[i]=this->m;//clusters initialization
         s+=i*this->m;
        }
        this->s=s;//sum of elements
        for(i=0;i<n;i++){
         this->v[i]=0;//elements initialization
        }
       }
      }
      function recurs(&ar,&o_a_a,$it,&is_OA){
       if(it>=ar->n) return;//nothing to recourse
       if(check_empty(ar)){
       if(check_orto(ar,o_a_a)){
         is_OA=TRUE;
         return;
        }//ar is OA with o_a_a
       }else{
        for(i=1;i<ar->b;i++){//0 is the default
         $ar->v[it]=i;//try with i
         $ar->d[i]--;
         recurs(ar,o_a_a,it+1,is_OA);
         if(is_OA) return;
         ar->d[i]++;//try with 0
         ar->v[it]=0;
         recurs(ar,o_a_a,it+1,is_OA);
         if(is_OA) return;
        }
       }
      }
      function fill_first_oa(lvn,expn,&o_a_a_var){
       for(k=0,i=0;i<expn;i++){
        o_a_a_var[0][i]=k++;
        k%=lvn;
       }//fill like 0,1,2,0,1,2 (lvn=3)
      }
      //main program for Orthogonal Arrays (OA)
       ...
       fill_first(levels[0],expn,o_a_a);// first OA
       for(i=1;i<n;i++){//n: number of planned OAs
        ...
        lst = new liste(levels[i],expn);
        ...
        recurs(lst,o_a_a,0,stop);//stop: no more OAs;
        ...
        //display intermediary OAs (o_a_a)
        ...
       }
	  

Another application was designed for generating the orthogonal arrays having the same number of levels (levels) for a given number of experimental runs (expn) starting with a list of already found orthogonal arrays (orto_list, which is an array of arrays), using a recursive function (rec), and an orthogonal testing function (ort, which also add the new OA to the list on successful):

      function rec(&a,b,na,va,nb,vb,pa,pb,bufb,nbufb,
         &orto_list,&norto_list){//output data
       if(va==0){//all combinations were exhausted
        for(i=0;i<na;i++){
         c[bufb[i]]=(i-i%nb)/nb;
        }//it’s time to check our OA
        nc=(na/nb-1)*na/2;
        ort(c,na,nc,orto_list,norto_list);
       }else //do recursion for all remained combinations
        if(vb==0){
        j=0;
        for(i=0;i<va;i++){
         if(a[i]==b[j]){
          bufb[]=b[$j];
          nbufb++;
          if(j<nb-1)j++;
         }else{
          newa[]=a[i];
         }
        }
        rec(newa,array(),na,va-nb,nb,nb,0,0,bufb,nbufb,
           orto_list,norto_list);
       }else{
        for(i=pa;i<va-vb+1;i++){
         b[pb]=a[i];
         rec1(a,b,na,va,nb,vb-1,i+1,pb+1,bufb,nbufb,
           orto_list,norto_list);
        }
       }
      }
	  

The main program calls the rec function after initializing the identical permutation (a[i]=i). Then, the results are displayed (orto_list):

rec(a,array(),expn,expn,expn/levels,expn/levels,0,0,array(),0,
orto_list,norto_list);

The rec function generates all possible distinct permutations of, for example, a set like:

{0,0,0,1,1,1,2,2,2}

by taking into account that the changing of a pair of positions (from 0 to 9 in our case) is relevant only for the positions that point to different values.

Thus, the complexity of the problem solved by rec function is (b = base, 3 in above example, n = number of experimental runs, 9 in the example above, which give a total number of combinations equal with 1680):

(n,b) ∙ (n-b,b) ∙ (n-2b,b)∙…

Results and Discussion

The programs presented in the previous section were run in order to reach the aim of the research for the following number of experiments: 4, 6, 8, 9, 10, 12, 14, 15, and 16. The largest groups of orthogonal arrays were generated. The results according to the number of factors and associated levels by the number of experimental runs were summarized and presented in Table 3. Note that only the maximum number of factors with associated levels according to the number of experimental runs was reported. The schemes that were included into the reported ones were not displayed; for example, for L₉ only the 3⁵ scheme was reported and not the 3⁴ scheme, as this was obviously included into the 3⁵ scheme. As presented in Table 3, a total number of sixty-three schemes were identified: 2 for L₄, 3 for L₆, 12 for L₈, 4 for L₉, 2 for L₁₀, 18 for L₁₂, 2 for L₁₄, 9 for L₁₅, and 11 for L₁₆.

Note that the following are true (see Table 3):

• Any level of any factor is a number that divides the number of experimental runs. This is the explanation for missing the orthogonal arrays for L₅, L₇, L₁₁, and L₁₃ (the number of the experimental runs could be divided just by themselves);
• In every experimental runs, in at least one case, the highest level of factor is equal with the number of experimental runs;

For the number of experimental runs equal with 4, 8, 12, and 16, the highest number of factors is given by the expression: n_F = n_E – 1 (where n_F = number of factors, n_E = number of experimental runs), as the Hadamard matrix shown [50].

The comparison of the resulted schemes (Table 3) with the table of orthogonal arrays maintained by Sloane [51] (who identified one scheme for L₄, L₆, L₉, L₁₀, L₁₄, and L₁₅, 2 schemes for L₈, four schemes for L₁₂, and seven schemes for L₁₆) indicates that the number is greater and the contribution to the orthogonal arrays database is significant. Furthermore, there are not any schemes reported by Sloane [51] that cannot be identified by using the implemented software (see the Material section).

The analysis of the available literature and software suggests that orthogonal arrays that are reported for the first time were identified.

Table 3. Number of experimental runs (n_exp), maximum number of factors (n_f), levels (L_i) and maximum number of factors for specified levels (∑MF_i)

nexp	nf	L1	∑MF1	L2	∑MF2	L3	∑MF3		nexp	nf	L1	∑MF1	L2	∑MF2	L3	∑MF3
4	3	4	2	2	1				12	5	12	1	2	4
		2	3								6	5
6	3	6	1	3	2						4	2	3	2	2	1
		3	3								4	1	3	2	2	2
		2	1	3	2						3	1	2	4
8	7	4	4	2	3					4	12	4
		4	2	2	5					3	12	1	6	2
		2	7						14	6	7	5	2	1
	6	8	3	4	2	2	1			5	14	1	7	4
		8	1	4	3	2	2		15	8	5	4	3	4
		8	1	4	1	2	4			7	5	5	3	2
		4	6								15	1	3	6
	5	8	3	2	2						5	2	3	5
		8	2	2	3						5	1	3	6
		8	1	4	3	2	1				3	7
		8	1	2	4					6	15	1	5	5
	4	8	4								5	6
9	5	3	5								5	3	3	3
	4	9	4						16	15	2	15
		9	2	3	2					14	4	1	2	13
		9	1	3	3					13	8	1	2	12
10	6	10	1	5	5						4	2	2	11
	3	5	2	2	1					12	16	1	2	11
12	11	2	11							10	16	1	8	1	2	8
	10	4	1	2	9					9	4	9
	9	4	2	2	7						4	3	2	6
	7	4	4	2	3					7	8	7
		4	3	2	4					5	16	5
		3	7								16	2	2	3
	6	4	6						nexp = number of experimental runs
		4	3	3	1	2	2		nf = total number of factors
		4	2	3	1	2	3		Li = levels for associated number of factors i
		3	4	2	2				∑MFi = total number of maximum factors i
		3	3	2	3

Table 3. Number of experimental runs (n_exp), maximum number of factors (n_f), levels (L_i) and maximum number of factors for specified levels (∑MF_i)

nexp	nf	L1	∑MF1	L2	∑MF2	L3	∑MF3		nexp	nf	L1	∑MF1	L2	∑MF2	L3	∑MF3
4	3	4	2	2	1				12	5	12	1	2	4
		2	3								6	5
6	3	6	1	3	2						4	2	3	2	2	1
		3	3								4	1	3	2	2	2
		2	1	3	2						3	1	2	4
8	7	4	4	2	3					4	12	4
		4	2	2	5					3	12	1	6	2
		2	7						14	6	7	5	2	1
	6	8	3	4	2	2	1			5	14	1	7	4
		8	1	4	3	2	2		15	8	5	4	3	4
		8	1	4	1	2	4			7	5	5	3	2
		4	6								15	1	3	6
	5	8	3	2	2						5	2	3	5
		8	2	2	3						5	1	3	6
		8	1	4	3	2	1				3	7
		8	1	2	4					6	15	1	5	5
	4	8	4								5	6
9	5	3	5								5	3	3	3
	4	9	4						16	15	2	15
		9	2	3	2					14	4	1	2	13
		9	1	3	3					13	8	1	2	12
10	6	10	1	5	5						4	2	2	11
	3	5	2	2	1					12	16	1	2	11
12	11	2	11							10	16	1	8	1	2	8
	10	4	1	2	9					9	4	9
	9	4	2	2	7						4	3	2	6
	7	4	4	2	3					7	8	7
		4	3	2	4					5	16	5
		3	7								16	2	2	3
	6	4	6						nexp = number of experimental runs
		4	3	3	1	2	2		nf = total number of factors
		4	2	3	1	2	3		Li = levels for associated number of factors i
		3	4	2	2				∑MFi = total number of maximum factors i
		3	3	2	3

One observation that resulted from the investigated cases refers to the modality of constructing orthogonal arrays. The orthogonal arrays could be classified as fixed-level (all factors have the same number of levels) and mixed-level (the factors have different levels). The linear programming can be used for fixed-level orthogonal arrays construction [52]. The mixed-level orthogonal arrays can be constructed using the expansive replacement method [17] or the “mixed spreads" approach [53]. The analysis of the orthogonal arrays obtained by using the developed programs revealed that a new factor could be constructed by the linear combination of two existing factors. Thus, a factor that is independent from all the other orthogonal arrays factors results ((10)_x · A + B, (100)_x · A + (10)_x · B + C, where x = number of levels, A, B, and C = elements of vector as modulo x values (from zero to n_E – 1, n_E = number of experimental runs)).

Let us take for example the L₈ (2⁷) orthogonal array (see Table 4).

Table 4. L₈ (2⁷) orthogonal array

Experimental run	Factor(Levels)
	A(2)	B(2)	C(2)	D(2)	E(2)	F(2)	G(2)
1	0	1	1	1	0	0	0
2	1	1	1	0	1	1	0
3	0	1	0	0	1	0	1
4	1	1	0	1	0	1	1
5	0	0	1	0	0	1	1
6	1	0	1	1	1	0	1
7	0	0	0	1	1	1	0
8	1	0	0	0	0	0	0

Table 4. L₈ (2⁷) orthogonal array

Experimental run	Factor(Levels)
	A(2)	B(2)	C(2)	D(2)	E(2)	F(2)	G(2)
1	0	1	1	1	0	0	0
2	1	1	1	0	1	1	0
3	0	1	0	0	1	0	1
4	1	1	0	1	0	1	1
5	0	0	1	0	0	1	1
6	1	0	1	1	1	0	1
7	0	0	0	1	1	1	0
8	1	0	0	0	0	0	0

The L₈ (4¹×2⁵) orthogonal array is obtained (see Table 5) by the linear combination of the two-level factors A and B (Table 4).

Table 5. L₈ (4¹×2⁵) orthogonal array

Experimental run	Factor(Levels)
	Y(2)	C(2)	D(2)	E(2)	F(2)	G(2)
1	1	1	1	0	0	0
2	3	1	0	1	1	0
3	1	0	0	1	0	1
4	3	0	1	0	1	1
5	0	1	0	0	1	1
6	2	1	1	1	0	1
7	0	0	1	1	1	0
8	2	0	0	0	0	0

Table 5. L₈ (4¹×2⁵) orthogonal array

Experimental run	Factor(Levels)
	Y(2)	C(2)	D(2)	E(2)	F(2)	G(2)
1	1	1	1	0	0	0
2	3	1	0	1	1	0
3	1	0	0	1	0	1
4	3	0	1	0	1	1
5	0	1	0	0	1	1
6	2	1	1	1	0	1
7	0	0	1	1	1	0
8	2	0	0	0	0	0

The L₈ (4²×2³) orthogonal array presented in Table 6 is obtained by the linear combination of C and D factors (Table 4).

Table 6. L₈ (4²×2³) orthogonal array

Experimental run	Factor(Level)
	Y(4)	Z(4)	E(4)	F(4)	G(4)
1	1	3	0	0	0
2	3	2	1	1	0
3	1	0	1	0	1
4	3	1	0	1	1
5	0	2	0	1	1
6	2	3	1	0	1
7	0	1	1	1	0
8	2	0	0	0	0

Table 6. L₈ (4²×2³) orthogonal array

Experimental run	Factor(Level)
	Y(4)	Z(4)	E(4)	F(4)	G(4)
1	1	3	0	0	0
2	3	2	1	1	0
3	1	0	1	0	1
4	3	1	0	1	1
5	0	2	0	1	1
6	2	3	1	0	1
7	0	1	1	1	0
8	2	0	0	0	0

The L₈ (8¹×2⁴) orthogonal array could also been obtained (see Table 7) by the linear combination of A, B and C factors (see Table 4).

Table 7. L₈ (8¹×2⁴) orthogonal array

Experimental run	Factor(Level)
	Q(8)	D(2)	E(2)	F(2)	G(2)
1	3	1	0	0	0
2	7	0	1	1	0
3	2	0	1	0	1
4	6	1	0	1	1
5	1	0	0	1	1
6	5	1	1	0	1
7	0	1	1	1	0
8	4	0	0	0	0

Table 7. L₈ (8¹×2⁴) orthogonal array

Experimental run	Factor(Level)
	Q(8)	D(2)	E(2)	F(2)	G(2)
1	3	1	0	0	0
2	7	0	1	1	0
3	2	0	1	0	1
4	6	1	0	1	1
5	1	0	0	1	1
6	5	1	1	0	1
7	0	1	1	1	0
8	4	0	0	0	0

Although some new factors with associated levels could be obtained, the linear combination of factors could not retrieve the maximum numbers of possible combinations. This can be observed by looking at the obtained results presented in Table 3. For example, the proposed method identified a number of twelve schemes as the largest groups of orthogonal arrays for L₈:

• Seven factors: 4⁴×2³, 4²×2⁵, 2⁷
• Six factors: 8³×4²×2¹, 8¹×4³×2², 8¹×4¹×2⁴, 4⁶
• Five factors: 8³×2², 8²×2³, 8¹×4³×2, 8¹×2⁴
• Four factors: 8⁴

The aim of the research was reached: the largest groups of orthogonal arrays for the studied number of experimental runs were identified. It is already known that the Taguchi approach can satisfy the needs of optimum process design and can reduce manufacturing costs [4]. The orthogonal arrays used by the Taguchi approach allow the study of the simultaneous effect of several factors efficiently, providing better results in smaller number of experimental runs [1,4]. The orthogonal arrays resulted from the present research have an advantage as compared with known orthogonal arrays: they allow the investigation of a greater number of factors with different levels. Having a clear list of all the largest sets of orthogonal arrays possible for a given number of experimental runs, the design of experiments could be improved and simplified. A more useful optimization of manufacturing design could be obtained by using the greatest number of factors’ levels and the smallest number of experimental runs.

The generation of the largest groups of orthogonal arrays could be further developed. The proposed algorithm could be applied for other desired numbers of experiments. Note that, as the number of experimental runs increases, the time needed for generating the maximum numbers of orthogonal arrays with the highest levels increases too.