## 2 Theory  
### Design of Split-Plots  
In split-plot designs, there is at least one factor which is hard to change, meaning that this factor imposes a restriction on complete randomization of the experimental runs. The hard-to-change factors are called WP factors [1]. For instance, one of the most hard-to-change factors at industrial scale is the temperature of big furnaces. Complete randomization requires the furnace to be turned on, reach the desired temperature, operate at this temperature for the desired amount of time, and then be turned off and allowed to cool down. The process should be repeated as many times as the experimental runs are, with only one sample at a time being treated, and the order of the temperatures tested should be completely random. However, it is obvious that this procedure is extremely time-consuming. What would make more sense at industrial scale would be to treat simultaneously all the samples that are to be treated at the same temperature. This way of performing the experiment would impose a restriction on complete randomization of the factor temperature. In other words, temperature is a hard-to-change factor (thus, a WP factor) and, because of that, its levels cannot be changed completely randomly.  
It must be clarified though, that the order in which the different levels of the factor (i.e., different temperatures) will be tested in the latter case must still be random. To illustrate this, assume that eight runs should be performed; half of them should be treated at 500 \({}^{\circ}\)C and the rest at 800 \({}^{\circ}\)C. In a split-plot experiment, all four runs that are to be performed at one of the temperatures will be performed first. After the treatment of these four runs is completed, the furnace should be turned off and allowed to cool down. Then, the process will be repeated for the rest four runs in the same way. In other words, the furnace will be switched on and off only twice in the split-plot design (compared to the eight times that would be required for a CRD). Whether the temperature of 500 \({}^{\circ}\)C or 800 \({}^{\circ}\)C will be tested first is something that will be decided randomly. The two different levels of the factor temperature constitute two different WPs.  
Now let us assume that the four runs which were treated under the same temperature had to stay in the furnace for different treatment times, e.g., 1 h, 2 h, 3 h, and 4 h. The treatment time is an easy-to-change factor, since the only requirement is to remove the samples from the furnace at the indicated times. Complete randomization of the easy-to-change factors is possible and these factors are called SP factors. Since there are four runs performed under the same temperature conditions simultaneously, we say that we have four SP within each WP. In our example, it should be randomly decided which position in the furnace each of the four samples treated under the same temperature (i.e., the four SP) will occupy. To illustrate this, let us assume that it was randomly decided that the runs treated at 800 \({}^{\circ}\)C would be performed first. Then, the position of each of the four samples in the furnace would be decided randomly. After these runs are finished, the runs at 500 \({}^{\circ}\)C would follow; however, a new randomization should then be applied to these sub-plots (i.e., the previous randomization order is not used again).