An overview on method of cyclic shifts for the construction of experimental designs useful in business and commerce

Ahmed, Rashid; Tahir, M. H.; Rajab, Muhammad; Daniyal, Muhammad Article An overview on method of cyclic shifts for the construction of experimental designs useful in business and commerce Pakistan Journal of Commerce and Social Sciences (PJCSS) Provided in Cooperation with: Johar Education Society, Pakistan (JESPK) Suggested Citation: Ahmed, Rashid; Tahir, M. H.; Rajab, Muhammad; Daniyal, Muhammad (2020) : An overview on method of cyclic shifts for the construction of experimental designs useful in business and commerce, Pakistan Journal of Commerce and Social Sciences (PJCSS), ISSN 2309-8619, Johar Education Society, Pakistan (JESPK), Lahore, Vol. 14, Iss. 1, pp. 49-62 This Version is available at: https://hdl.handle.net/10419/216864 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Pakistan Journal of Commerce and Social Sciences 2020, Vol. 14 (1), 49-62 Pak J Commer Soc Sci An Overview on Method of Cyclic Shifts for the Construction of Experimental Designs Useful in Business and Commerce Rashid Ahmed (Corresponding author) Department of Statistics, The Islamia University of Bahawalpur, Pakistan Email: [email protected] M. H. Tahir Department of Statistics, The Islamia University of Bahawalpur, Pakistan Email: [email protected] Muhammad Rajab Department of Statistics, The Islamia University of Bahawalpur, Pakistan Email: [email protected] Muhammad Daniyal Department of Statistics, The Islamia University of Bahawalpur, Pakistan Email: [email protected] Abstract The competitive nature of business today requires the knowledge to be able to manipulate process levels in a predictable fashion and to reduce process variation (quality control). This goal can be achieved through the use of experimentation to identify and confirm the effects of the process variables. Therefore, design of experiments should be an integral part of process management. Experimental designs such as treatment balanced, neighbor balanced and balanced repeated measurements designs, have been extensively used in different fields, especially in business and commerce. Different methods have been used for their construction but method of cyclic shifts is the easiest one. This paper provides the overview, how this method has been used for the construction of the useful designs. Keywords: cyclic shifts; experimental designs; competitive nature of business; polygonal designs; process management; quality control. 1. Introduction Method of cyclic shifts (MOCS) has been widely used to construct different designs. This method was developed by Iqbal (1991). In this method, we can study some standard characteristics and properties of the specific design only through the set(s) of shifts. This method is quite simple to construct several types of important designs. Some of these are balanced incomplete block designs (BIBDs), partially BIBDs, polygonal designs (PDs), neighbor balanced designs (NBDs) and repeated measurements designs (RMDs). In this Method of Cyclic Shifts for the Construction of Experimental Designs 50 article, an overview is provided that how the MOCS is used to construct several types of important designs. This method consists of Rule I & Rule II. According to Sanders et al. (2002), the competitive nature of business today requires the knowledge to be able to manipulate process levels in a predictable fashion and to reduce process variation. The need of this knowledge (the continuous improvement of processes) must be quicker than ever before. This goal requires the ongoing use of experimentation to identify and confirm the effects of the process variables. Therefore, designed experiments should be viewed as an integral part of process management. Whereas the statistical design of experiments (DOE) is a valuable tool for rapidly accumulating this process knowledge. In industrial experimentation, blocks are frequently a period of time when extraneous variables (variables not explicitly manipulated in the experiment) can reasonably be expected to remain constant while an experiment takes place. Periods of time (blocks) for subsequent experiments are selected so that while variables not explicitly manipulated can reasonably be expected to remain constant during the execution of a DOE, some might have changed between experiments (blocks). Polygonal designs are useful for survey and marketing. 2. Construction of BIBDs and PBIBDs BIBDs and PBIBDs have the most significant role in experimental designs to compare each pair of treatments with equal or almost equal precision. Such experimental designs provide the assurance that treatments can be compared with same precision. Using method of cyclic shifts, Yasmin et al. (2015) and Jamil et al. (2017) presented BIBD and PBIBD respectively. 2.1 How to Obtain a BIBD and PBIBD Using Rule I. MOCS is described in this Section only for BIBDs and PBIBDs. Let Sj = [qj1, qj2, …, qj(k-1)], 1 ≤ qji ≤ v-1. A design will be BIBD if each of 1, 2, …, v-1 appears λ times in Sj*. Sj* = [qj1, qj2, …, qj(k-1), (qj1+qj2), (qj2+ qj3),…, (qj(k-2)+qj(k-1)), (qj1+qj2+ qj3), (qj2+qj3+qj4), …, (qj(k-3)+qj(k-2)+qj(k-1)), …, (qj1+qj2+…+ qj(k-1)), and complement of each element], here v-qi is complement of qi. It will be PBIBD (two associate) if  has two values λ2 = λ1+1. Example 2.1: S = [3, 1, 2, 2, 1, 1, 1] provide BIBD for v = 15, k = 8 in 15 blocks through MOCS (Rule I). Ahmed et al. 51 Here S* = [3,1,2,2,1,1,1,4,3,4,3,2,2,6,5,5,4,3,8,6,6,5,9,7,7,10,8,11,12,14,13,13,14,14,14,11, 12,11,12,13,13,9,10,10,11,12,7,9,9,10,6,8,8,5,7,4]. Here each element from 1, 2, 3, …, 14 appears 4 times. Hence it is BIBD with λ = 4. Blocks 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 3 4 5 6 7 8 9 10 11 12 13 14 0 1 2 4 5 6 7 8 9 10 11 12 13 14 0 1 2 3 6 7 8 9 10 11 12 13 14 0 1 2 3 4 5 8 9 10 11 12 13 14 0 1 2 3 4 5 6 7 9 10 11 12 13 14 0 1 2 3 4 5 6 7 8 10 11 12 13 14 0 1 2 3 4 5 6 7 8 9 11 12 13 14 0 1 2 3 4 5 6 7 8 9 10 2.2 How to Obtain a BIBD and PBIBD Using Rule II Let Sj = [qj1, qj2, …, qj(k-2)]t, 1≤ qji ≤ v-2. A design will be BIBD if each of 1, 2, …, v-2 appears λ times in Sj*. Where Sj* = [qj1, qj2, …, qj(k-2), (qj1+qj2), (qj2+qj3),…, (qj(k-3)+qj(k-2)), (qj1+qj2+qj3), (qj2+qj3+qj4), …, (qj(k-4)+qj(k-3)+qj(k-2)), …, (qj1+qj2+…+ qj(k-2))], here, v-1-qi is complement of qi. If λ has two values λ2 = λ1+1 then it is PBIB design with 2-association scheme. Example 2.2: BIBD is constructed from the sets of shifts [1,2,3] and [1,4]t for v = 8, k = 4 in 14 blocks through Rule II. Here S* = [1,2,3,3,5,6,1,4,5,6,5,4,4,2,1,6,3,2] Here each element from 1, 2, 3, …, 6 appears 3 times. Hence it is BIBD with λ = 3. [ Blocks 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 1 2 3 4 5 6 0 1 2 3 4 5 6 1 2 3 4 5 6 0 1 2 3 4 5 6 0 3 4 5 6 0 1 2 5 6 0 1 2 3 4 6 0 1 2 3 4 5 7 7 7 7 7 7 7 Method of Cyclic Shifts for the Construction of Experimental Designs 52 3. Construction of Polygonal Designs PDs are important in survey sampling. Using MOCS, Tahir et al. (2011) constructed PDs for k = 3 with λ = 1, 2, 3, 4, 6, 12 and α = 2, Intizar et al. (2016) constructed PDs in blocks of two different sizes 4 and 2. Following is the procedure to obtain PDs. 3.1 How to Obtain Polygonal Designs Using Rule I Let Sj = [qj1, qj2, …, qj(k-1)], where 1 ≤ qji ≤ v-1.  If each of 2, …, v-2 appears λ times in S* except 1 and v-1 which do not appear then this design is CPD with α = 1.  If each of 3, …, v-3 appears λ times in S* except 1, 2, v-1 and v-2 which do not appear then this design is CPD with α = 2.  If each of α+1, …, v-(α+1) appears λ times in S* except 1, 2,…, α, v-1, v-2,…, v-α which do not appear then this design is CPD with joint distance α. Here, S* contains (i) each shift of S, (ii) sum (mod v) of each of two, three, …, k-1 consecutive shifts, and (iii) complement of each element in (i) & (ii). Here, v-qi is complement of qi. For a circular polygonal design with α = 1, the concurrence matrix is: For a circular polygonal design with joint distance the concurrence matrix is Example 3.1: A circular polygonal design for v = 11, k1 = 3, k2 = 2, λ = 1 & α = 1 is constructed using [2, 3]+[4]. The required design is: Ahmed et al. 53 Blocks 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 0 1 5 6 7 8 9 10 0 1 2 3 4 Blocks 12 13 14 15 16 17 18 19 20 21 22 0 1 2 3 4 5 6 7 8 9 10 4 5 6 7 8 9 10 0 1 2 3 3.2 How to Obtain Polygonal Designs Using Rule II Let Sj = [qj1, qj2, …, qj(k-2)]t, where 1 ≤ qji ≤ v-1.  If each of 2, …, v-3 appears λ times in S* except 1 and v-2 which do not appear then this design is CPD with α = 1.  If each of 3, …, v-4 appears λ times in S* except 1, 2, v-2 and v-3 which do not appear then this design is CPD with α = 2.  If each of α+1, …, v-1-(α+1) appears λ times in S* except 1, 2,…, α, v-2,…, v-1-α which do not appear then this design is CPD with joint distance α. Example 3.2: [2,3]+[4]t provide PD for v = 8, k = 3, λ = 2 and α = 1. Here S* = [2,3,5,4,5,4,2,3] Here each element from 2, 3, …, 5 appears exactly twice. Hence it is polygonal design with λ = 2 and α = 1. Blocks 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 1 2 3 4 5 6 0 1 2 3 4 5 6 2 3 4 5 6 0 1 4 5 6 0 1 2 3 5 6 0 1 2 3 4 7 7 7 7 7 7 7 4. Construction of Balanced and Strongly Balanced RMDs RMDs have application in many branches such as business, commerce, agriculture, food science, animal husbandry, biology, education, psychology, and pharmacology. In the following literature, MOCS is used to construct RMDs. Method of Cyclic Shifts for the Construction of Experimental Designs 54 Iqbal and Jones (1994), Iqbal and Tahir (2009), Iqbal et al. (2010), Bashir et al. (2018), Rajab et al. (2018), Rasheed et al. (2018), Ahmed et al. (2018), Khan et al. (2019), Daniyal et al. (2019), Rasheed et al. (2019), and Ahmed et al. (2019) constructed CBRMDs & CSBRMDs in periods of equal and unequal sizes. Jabeen et al. (2019), Nazeer et al. (2019), Jabeen et al. (2019) constructed minimal CSPBRMDs in periods of equal and two different sizes. Hussain et al. (2020) constructed CWBRMDs in two different periods sizes. Abdullah et al. (2019) constructed such designs in non-circular periods. Following is the to obtain the BRMDs. 4.1 How to Obtain BRMDs in Circular Periods using Rule I Let S = [ , , …, ], where 1 ≤ qi ≤ v-1. If each of 1, 2, …, v-1 appears λʹ in S* then it be CBRMD in periods of size p, where S* = [ , , …, , v - ( + + …+ ) mod v]. Design will be non-binary if sum of any two, three, …, or (p-1) consecutive elements of S is 0 (mod v), otherwise binary. Example 4.1: Set of shifts [1,2,8]+[3,4,6] produce non-binary CBRMD for v = 9 and p = 4. Here S* = [2,1,8,7,3,4,6,5] Here each element from 1,2,3,…,8 appears once. Therefore, it is minimal CBRMDs with λʹ = 1. Periods Subjects 1 2 3 4 5 6 7 8 9 1 02 13 24 35 46 57 68 70 81 2 10 21 32 43 54 65 76 87 08 3 31 42 53 64 75 86 07 18 20 4 23 34 45 56 67 78 80 01 12 Periods Subjects 10 11 12 13 14 15 16 17 18 1 04 15 26 37 48 50 61 72 83 2 30 41 52 63 74 85 06 17 28 3 73 84 05 16 27 38 40 51 62 4 47 58 60 71 82 03 14 25 36 4.2 How to Obtained BRMDs in Circular Blocks Using Rule II Let S1 = [ , , …, ] and S2 = [ , , …, ]t, where 1 ≤ qij ≤ v-2. If each of 1, 2, …, v-2 appears λʹ in S* then it is CBRMD in periods of size p, where S* = [ , , …, , (v-1)-{( + + …+ ) mod(v-1)}, , , …, ]. Ahmed et al. 55 Example 4.2: Sets of shifts [2,1,6]+[3,4]t provide following non-binary CBRMD for v = 8, k = 4. Here S* = [1,2,6,5,3,4] and each element from 1, 2, 3, …, 6 appears exactly once. Hence it is minimal CBRMDs with λʹ = 1. Periods Subjects 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 02 13 24 35 46 50 61 07 17 27 37 47 57 67 2 20 31 42 53 64 05 16 30 41 52 63 04 15 26 3 32 43 54 65 06 10 21 03 14 25 36 40 51 62 4 23 34 45 56 60 01 12 70 71 72 73 74 75 76 4.3 How to Obtain a CSBRMD Using Rule I Let S1 = [ , , …,q1(p-1)], where 0 ≤ ≤ v-1. If each of 0, 1, 2, …, v-1 appears λʹ S* then it is CSBRMDs in period of size p, where S* = [ , , …, q1(p-1) , v-( + + …+ q1(p-1)) mod v]. It will be minimal if λʹ = 1. Example 4.3: [1,3,2,4,6,5] provides following CSBRMDs for v = 7 and p = 7. Here S* = [1,3,2,4,6,5,0] Here each element from 0,1,2,3,…,6 appears exactly once. Hence it is minimal CSBRMDs with λʹ = 1 using rule I. Periods Subjects 1 2 3 4 5 6 7 1 00 11 22 33 44 55 66 2 10 21 32 43 54 65 06 3 41 52 63 04 15 26 30 4 64 05 16 20 31 42 53 5 36 40 51 62 03 14 25 6 23 34 45 56 60 01 12 7 02 13 24 35 46 50 61 4.4 How to Obtain BRMDs in Linear Periods Using Rule I Let S = [ , , …, ], where 1 ≤ qi ≤ v-1. If each of 1, 2, …, v-1 appears λʹ in S* then it is BRMD in linear periods of size p, where S* = [ , , …, ]. Method of Cyclic Shifts for the Construction of Experimental Designs 56 Example 4.4: Sets of shifts [2, 1, 3,4]+[5, 6, 8, 7] provide following CBRMDs in linear periods for v = 9 in p = 5. Here S* = [2,1,3,4,5,6,8,7] and each element from 1, 2, 3,…, 8 appears once, therefore, it is minimal BRMDs with λʹ = 1 in linear periods, using rule I. Periods Subjects 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 2 20 31 42 53 64 75 86 07 18 50 61 72 83 04 15 26 37 48 3 32 43 54 65 76 87 08 10 21 25 36 47 58 60 71 82 03 14 4 63 74 05 16 27 38 40 51 62 12 23 34 45 56 67 78 80 01 5 26 37 40 51 62 73 04 15 06 81 02 13 24 35 46 57 68 70 4.5 How to Obtain BRMDs in Linear Periods Using Rule II Let Sj = [qj1,qj2, …,qj(p-1)] and Si = [qi1, qi2,…, qi(p-2)]t, where 1≤ qij ≤ v-2. If each of 1, 2,…, v-2 appears λʹ in S* then it is non-linear BRMD in periods of size p, where S*= [qj1, qj2, …, qj(p-1), qi1, qi2,…, qi(p-2)] mod(v-1). Example 4.5: Sets of shifts [1,2,3,4]+[5,6,7]t provide following CBRMDs for v = 9 & p = 5 in linear periods, using Rule II. Here S* = [1,2,3,4,5,6,7] and each element from 0, 1, 2, 3, …, 7 appears exactly once. Hence it is minimal BRMDs in linear blocks with λʹ = 1, using Rule II. Periods Subjects 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 2 10 21 32 43 54 65 76 87 08 50 61 72 83 04 15 26 37 48 3 31 42 53 64 75 86 07 18 20 25 36 47 58 60 71 82 03 14 4 63 74 85 06 17 28 30 41 52 02 13 24 35 46 57 68 70 81 5 16 27 38 40 51 62 73 84 05 80 81 82 83 84 85 86 87 88 4.6 How to Obtain SBRMD in Linear Periods Using Rule I Let S = [ , , …, ], where 0 ≤ qi ≤ v-1. If each of 1, 2, …, v-1 appears λʹ in S* then it is BRMD in linear periods of size p, where S* = [ , , …, ]. Example 4.6: Set of shifts [1,2,3,0]+[4,5,6,7] produce SBRMD for v = 8 & p = 5 in linear periods.