Lattice-shifted photonic crystal waveguides (LSPCWs) are effective for generating wide-band on-chip slow light at room temperature. In this study, we integrated the LSPCW with multi-heaters using CMOS-compatible process, and demonstrated electrically-tunable slow light. Seven heaters were placed on each side of the LSPCW with air slots for thermal isolation. On-demand temperature distributions were formed by controlling each heating power independently. When the heating powers were optimized, a clear delay peak of slow light corresponding to the flat photonic band of the LSPCW was observed, which suggests that fabrication errors in the air-hole diameters were less than 5 nm and related band fluctuation was well compensated by the heating. When a linear temperature distribution was added to this condition, the delay was reduced up to 54 ps. When a quadratic distribution was added, the group delay dispersion was generated in the range from -10.2 to 17.5 ps/nm. We applied the tunable delay to the delay scanning in optical correlator. Here, output pulse were compressed to 0.6 ps through self-phase modulation and dispersion compensation in external fibers, and its delay was tuned in range of 17 ps. At a scanning frequency of 100 Hz, which is <10 times faster than that of conventional mechanical delay scanners, pulse lengths of 0.3 - 6 ps were measured with a <95% accuracy. We are also fabricating all on-chip optical correlator consisting of this slow light delay scanner and two-photon-absorption photo-diodes.