In this thesis, we examine two topics in the area of educational measurement. The first topic studies how to best design two achievement tests with common items such that a population mean-ability growth is measured as precisely as possible. The second examines how to calibrate newly developed test items optimally. These topics are two optimal design problems in achievement testing. Paper I consist of a simulation study where different item difficulty allocations are compared regarding the precision of mean ability growth when controlling for estimation method and item difficulty span. We take a more theoretical approach on how to allocate the item difficulties in Paper II. We use particle swarm optimization on a multi-objective weighted sum to determine an exact design of the two tests with common items. The outcome relies on asymptotic results of the test information function. The general conclusion of both papers is that we should allocate the common items in the middle of the difficulty span, with the two separate test items on different sides. When we decrease the difference in mean ability between the groups, the ranges of the common and test items coincide more.

In the second part, we examine how to apply an existing optimal calibration method and algorithm using data from the Swedish Scholastic Aptitude Test (SweSAT). We further develop it to consider uncertainty in the examinees' ability estimates. Paper III compares the optimal calibration method with random allocation of items to examinees in a simulation study using different measures. In most cases, the optimal design method estimates the calibration items more efficiently. Also, we can identify for what kind of items the method works worse.

The method applied in Paper III assumes that the estimated abilities are the true ones. In Paper IV, we further develop the method to handle uncertainty in the ability estimates which are based on an operational test. We examine the asymptotic result and compare it to the case of known abilities. The optimal design using estimates approaches the optimal design assuming true abilities for increasing information from the operational test.