Q. 4 (Confounding, 25 pts). You are performing a single replication of a 23 full
ID: 378948 • Letter: Q
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Q. 4 (Confounding, 25 pts). You are performing a single replication of a 23 full facto- rial experiment with factors A, B,C. As it takes hours to test a single treatment, you decide to complete the experiment in four days by confounding the interaction effects ABC and AB a. (5) Propose the definitions of the blocks and provide a detailed plan of which treat ments to test each day. b. (5) How is ABC confounded with the four days? c. (5) How is AB confounded with the four days? d. (5) Is there any other effect that is confounded? How'? e. (5) Is this a good design? Explain.Explanation / Answer
a) Dual problem is following
Min W = 40y1 + 50y2
2y1 + y2 >= 5
y1 + 2y2 >= 3
y1, y2 >= 0
b) Objective value of both the primal and the dual is same, i.e. 110.
Substituting the solution values of dual problem i.e. y1 and y2 in the constraints equations, we see that there is no slack. 2*7/3 + 1/3 = 5 , and 7/3 + 2*1/3 = 3
Therefore, primal will also not have any surplus. We utilize this information to solve the primal. .
5x1 + 3x2 = 110 ---(1)
2x1 + x2 = 40 ---(2)
x1 + 2x2 = 50 ---(3)
Multiply equation (2) by and subtracting equation (1) from it, we get 6x1 - 5x1 = 40*3 - 110 , or x1 = 10
Substitute this value in equation (3), we get x2 = 20
Therefore, solution of primal is x1 = 10, x2 = 20
d) The meaning of dual variable values to the primal problem is that a unit increase in the RHS of constraint 1 of the primal will lead to an increase in the objective value by 7/3 and similarly, a unit increase in the RHS of constraint 2 of the primal will lead to an increase in the objective value by 1/3
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