We consider off-line vector assignment problems. The goal is to assign input vectors to machines so that a given target function is minimized. The target function usually gives some measure of the quality of the distribution of input vectors among machines. In a previous study (Epstein and Tassa, J. Algorithms 48, 360-384 (2003)) we dealt with this problem where the cost function is symmetric; namely, all machines are identical. The dynamic programming techniques that we used in our previous study do not extend to asymmetric settings. Here, we deal with an asymmetric setting where the cost functions per machine may be different for different machines. Using graph-based techniques, we design a polynomial time approximation scheme for a wide class of asymmetric target functions. Other than a significant extension of the class of cost functions to which our current scheme applies, the present PTAS is much simpler the previous one. It should be noted that asymmetric cost functions appear very naturally in the so called line-up problem that motivated our study.