Academic literature on the topic 'Symmetrical fractional factorials'

In this paper, we develop theorems on finite and infinite summation formulas by utilizing the q and (q,h) anti-difference operators, and also we extend these core theorems to q(α) and (q,h)α difference operators. Several integer order theorems based on q and q(α) difference operator have been published, which gave us the idea to derive the fractional order anti-difference equations for q and q(α) difference operators. In order to develop the fractional order anti-difference equations for q and q(α) difference operators, we construct a function known as the quantum geometric and alpha-quantum geometric function, which behaves as the class of geometric series. We can use this function to convert an infinite summation to a limited summation. Using this concept and by the gamma function, we derive the fractional order anti-difference equations for q and q(α) difference operators for polynomials, polynomial factorials, and logarithmic functions that provide solutions for symmetric difference operator. We provide appropriate examples to support our results. In addition, we extend these concepts to the (q,h) and (q,h)α difference operators, and we derive several integer and fractional order theorems that give solutions for the mixed symmetric difference operator. Finally, we plot the diagrams to analyze the (q,h) and (q,h)α difference operators for verification.

Stated choice experiments are increasingly becoming popular due to their ability to optimize information gain with limited resources. Many designs have been developed for the selection of various attributes and their levels to form choice sets. One such design is blocked fractional factorial design (BFFD). Stated choice experiments for symmetric attributes of 4 choice sets of sizes 2 and 4 and 8 choice sets of sizes 2 and 4 were developed using BFFDs. Generators for the stated choice set of sizes 2 and 4 with resolution three, four, and five were developed. The alias structures and confounding effects for the designs were derived, as well as their clear effects if any for estimation. The A -efficiency was used to compute the efficiencies of the proposed designs since it has better statistical properties. The computed efficiencies for the proposed designs reveal that 4 choice sets of size 4 designs are more efficient. Finally, a practical application of the proposed method was carried out for four choice sets of size 4 using 2 V 5 − 1 − 2 design with attributes and levels of service quality in public transport.