Abstract
This article utilizes the Normalized Sylvester-Hadamard Matrices of size 2 k x2 k and their associated saturated orthogonal arrays OA(2 k , 2 k - 1, 2, 2) topropose analgorithmbased on factor projection (Backward/Forward) for the construction of three systematic run-after-run2 n-(n-k) fractional factorial designs: (i) minimum cost trend free 2 n-(n-k) designsof resolution III (2 k-1 ≤n≤2 k – 1 – k)by backward factor deletion (ii) minimum cost trend free 2 n-(n-k) designsof resolution III (k+1≤n≤ 2 k-1 – 2+k ) by forward factor addition (iii) minimum costtrend free 2 n-(n-k) designsof resolution IV (2 k-2 ≤n≤2 k-1 -2) ,where each 2 n-(n-k) design is economic minimizing the number of factor level changes between the 2 k successive runs and allows for the estimation of all factor main effects unbiased by the linear time trend,which might be present in the 2 k sequentially generated responses. The article gives for each 2 n-(n-k) design: (i) the defining contrast displaying the design’s alias structure(ii) the k independent generators for sequencingthe design’s 2 n-(n-k) runs by the Generalized Fold over Scheme and (ii) the minimum total cost of factor level changes between the 2 n-(n-k) runs of the design. Proposed designs compete well with existing systematic2 n-(n-k) designs (of either resolution) in minimizing the experimental costandin securing factors’ resistance to the non-negligible time trend. Keywords : Sequential fractional factorial experimentation; Time trend free systematic run orders; Generalizedfoldover scheme for sequencing experimentalruns; The total cost of factor level changes between successive runs; The Normalized Sylvester –Hadamard Matrices; Orthogonal Arrays and factor projection; Design resolution and the alias structure. DOI: 10.7176/JEP/11-25-05 Publication date: September 30 th 2020
Highlights
Experiments are carried out in all fields:industrial, educational, agricultural, medical, etc.,where experimentation has led to many innovations and discoveries
Systematic fractional 2n-kfactorial experiments suffer from the problem that factor effects may be adversely affected by a time trend which might be present among responses of the successive runs
There are a total of 2n-k! run orders to carry out fractional 2n-kfactorial experiments run after run but not all these run orders are resistant to the time trend nor economic
Summary
Experiments are carried out in all fields:industrial, educational, agricultural, medical, etc.,where experimentation has led to many innovations and discoveries. Reference [6] utilized the standard order of the full 2k experiment in (1.1) laying out all main effects Ai (i =1,2,...,k) and their interaction columns in increasing number of level changes [ from 1 up to (2k-1) ] constructed two types of 2n-(nk)designs:minimum cost 2n-(n-k) designs of resolution III (2k-1≤n≤2k-1)and minimum cost 2n-(n-k) designs of resolution IV (2k-2≤n≤2k-1)but regardless of factors’ time trend resistance.neither the defining relations nor the GFS generator sets were reported nor the minimal total cost of factor level changes wascomputed. [22] represented experimental runs of regular 2k-pdesigns as graph vertices applied Travelling Salesman Algorithm to locate graph paths ( i.e. run orders) of minimal distance without regard to factors’ time trend resistance.These minimallysequenced 2k-pdesigns( 4 ≤k≤15 and 1 ≤p≤11) cannot be sequenced by the GFS approach ,since many of theserun orders do not start with the null treatment (1)=(000...0000.Defining contrasts were provided but neither the factors’pattern of level changes nor the total cost of factor level changes werereported.
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