Fractional Factorial Experiment - Assignment Example

Experimental Design and Process Analysis 2 March Fractional Factorial Experiment Sometimes, when undertaking a full factorial experiment, the amount of treatments to be assessed increases. This causes an increase in the cost of the experiment, the time required to perform the experiment and the amount of resources required to fully run the experiment. The runs needed to do the full factorial experiment exhausts the resources available to complete it. Hence, to complete the experiment a fractional factorial experiment is done ensuring a great deal of savings in terms of time, cost and resources. Thus, using fractional factorial is a direct means of saving and it results to better effectiveness and competence of the experimental design (Anthony, 73). In this experiment, a one half fractional experiment was conducted. It was done in k – 1 runs. The experiment was conducted to assess the effect of five factors on a coil spring. The free height of the spring was denoted as y. The goal of the experiment was to obtain a height of eight inches or a height as close to eight inches as possible. The number of factors used was five, while the number of runs was sixteen with three replicates for each run. The experiment was a 2k – 1 design, thus it was a one half fractional factorial experiment. It was denoted as 25 – 1. Based on the evaluation of the cause and effects, the following factors and factor levels were chosen for the experiment. Below is a table of the factors and the factor levels for both high denoted by a plus sign and low depicted by a minus sign. Table 1: Factors and Factor Levels Factor Level + − A Temperature: high (0C) 1027 1004 B Time for heating in seconds 25 23 C Time used in transfer in seconds 12 10 D Time for holding down in seconds 3 2 E Temperature of the smother oil (0C) 66 - 77 54 - 66 Source: From experiment. From the determined factors and factor levels the fractional factorial experiment was conducted. This was because the number of runs needed for a full factorial experiment would have been a lot. Because the resources at hand coupled with the knowledge that the relations would not be of concern, a 25 – 1 fractional factorial was used. The design matrix and the height data are given in the table below. Table 2: Design Matrix and Height Data Factors A B C D E Free Height ӯ si2 lnsi2 − + − + − + − + − + − + − + − + + + − − + + − − + + − − + + − − + + + + − − − − + + + + − − − − − + + − + − − + − + + − + − − + − − − − − − − − + + + + + + + + 7.78 7.78 7.81 8.15 8.18 7.88 7.50 7.56 7.50 7.59 7.56 7.75 7.94 8.00 7.88 7.69 8.09 8.06 7.56 7.62 7.44 7.56 7.81 7.69 7.50 7.25 7.12 7.88 7.88 7.44 7.50 7.56 7.50 7.63 7.75 7.56 7.32 7.44 7.44 7.56 7.69 7.62 7.18 7.18 7.25 7.81 7.50 7.59 7.7900 8.0700 7.5200 7.6333 7.9400 7.9467 7.5400 7.6867 7.2900 7.7333 7.5200 7.6467 7.4000 7.6233 7.2033 7.6333 0.0003 0.0273 0.0012 0.0104 0.0036 0.0496 0.0084 0.0156 0.0373 0.0645 0.0012 0.0092 0.0048 0.0042 0.0016 3.6717 -8.1117 -3.6009 -6.7254 -4.5627 -5.6268 -3.0031 -4.7795 -4.1583 -3.2888 -2.7406 -6.7254 -4.6849 -5.3391 -5.4648 -6.4171 -0.0254 Source: From the experiment The effects of the fractional factorial experiment were then determined. The equation used was: Effect k = Σy+ − Σ y- n+ n- Effect A = ΣyA+ − ΣyA- = 61.973 – 60.2033 = 0.2212125 nA+ nA- 8 8 Effect B = ΣyB+ − ΣyB− = 61.7933 – 52.7497 = 1.13045 nB+ nB− 8 8 Effect C = ΣyC+ − ΣyC− = 61.2033 – 60.9733 = 0.02875 nC+ nC− 8 8 Effect D = ΣyD+ − ΣyD− = 61.5033 – 60.673 = 0.1037875 nD+ nD− 8 8 Effect E = ΣyE+ − ΣyE− = 60.0499 – 62.1267 = -0.2596 nE+ nE− 8 8 Analysing the effect aliasing in the fractional factorial experiment. It was found that in the design matrix, col D = col A × col B ×col C. This meant that: ӯ(D+) – ӯ (D−) = ӯ (ABC+) – ӯ (ABC−) Thus, the design was not able to differentiate D from ABC. Effect D which was the main effect was aliased with the interaction ABC. Hence, D was equal to ABC which meant that I was equal ABCD. I was the column of plus’s which was the characteristic element in the set of multiplications. I is equal to ABCD was the characterizing relation for the 25-1 design experiment. This means that all the effect aliasing relations are: A = BCD, B = ACD, C = ABD, D = ABC, AB = CD, AC = BD, AD = BC, E = ABCDE, AE = BCDE, BE = ACDE, CE = ABDE, DE = ABCE, ABE = CDE, ACE = BDE, BDE = BCE. The main effect is named clear if it is not aliased with other main effects or two factor interactions. It is named strongly clear if it is not aliased with other main effects, two factor interactions or three factor interactions. Thus, it can be approximated on the theory of negligible three – factor and other interactions that are greater. A main effect that is strongly clear can be approximated on the theory of negligible four – factor and other greater interactions (Montgomery, 54). Hence, for the coil spring fractional factorial experiment. The design of this experiment was 25-1 which had I equal to ABCD. The clear effects of this experiment were ABCD, and the strongly clear effects were E, AE, BE, CE and DE. This design had a resolution IV. The location effects that were analysed are shown in table 3: Table 3: Factorial Effects, coil spring experiment Effect ӯ lns2 A 0.221 1.891 B 0.176 0.569 C 0.029 -0.247 D 0.104 0.216 E -0.260 0.280 AE 0.085 -0.589 BE -0.165 0.589 CE 0.054 1.111 DE 0.027 0.129 AB 0.017 -0.002 AC 0.020 0.425 BC -0.035 0.670 ABE 0.010 -1.089 ACE -0.040 -0.432 ADE -0.047 0.854 Source: From Experiment The formula used was ӯ = 7.636 + 0.1106xA + 0.0519xD + 0.0881xB – 0.1298xE + 0.0423xAxE – 0.0827xBxE. With ABCE being equal to negative, positive, positive and negative respectively, substituting and computing resulted in: ӯ = 7.637 + 0.110 (-1) + 0.0519xE + 0.0881 (+1) – 0.1298 (-1) + 0.0423 (-1)(-1) – 0.0827 (+)(-1) ӯ = 7.8683 + 0.0519 xE, which gives, 8.0 = 7.8683 + 0.0519xE, (8.0 – 7.8683) ÷ 0.0519 = xE 2.54 = xE Works Cited Antony, Jiju. Design of Experiments for Engineers and Scientists. Butterworth – Heinemann, 2003. Print Montgomery, Douglas. Design and analysis of experiments. Wiley, 1976. 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