The present paper covers in a comprehensive manner the formation of the Cu-based Cu-Zr-Al bulk metallic glasses (BMGs). Composition optimization for BMG formation is realized by using an “e/a-variant criterion”. This criterion is incorporated into the ternary phase diagram by a straight composition line, which is defined by linking a specific binary composition to the third element. There e/a-variant composition lines are constructed: (Cu9Zr4)1-xAl, (Cu61.8Zr38.2)1-xAlx and (Cu56Zr44)1-xAlx, where Cu9Zr4, Cu61.8Zr38.2 and Cu56Zr44 are specific cluster compositions in the Cu-rich Cu-Zr binary system. No Cu-based BMGs are obtained in the composition line (Cu9Zr4)1-xAlx using our suction casting techniques, while BMGs are obtained within an e/a span from 1.24 to 1.3 and from 1.28 to 1.36 respectively along the other two lines (Cu61.8Zr38.2)1-xAlx and (Cu56Zr44)1-xAlx. Thermal analysis results indicate that the BMGs on every composition line manifest increased thermal stability and glass forming ability (GFA) with increasing e/a ratios. The maximum appears in Cu58.1Zr35.9Al6 with the e/a value of 1.3, which belongs to the (Cu61.8Zr38.2)1-xAlx series. The characteristic thermal parameters of this BMG are Tg = 760K, Tg/Tm = 0.659 and Tg/Tl = 0.648, which are all superior to those reported for the known Cu55Zr40Al5 BMG.