Additional equivalent damping ratio (EDR) plays a crucial role in the performance evaluation and design methodology of conventional passive dampers. The inclusion of inerter elements has been extensively demonstrated to significantly enhance the control efficiency of these passive dampers. Recently, negative stiffness (NS) elements have been found potential to enhance the control performance of inerter-based dampers (IBDs). However, little research has investigated the enhancement effect of NS elements on the additional EDR of IBDs. Meanwhile, the existing design methodology for IBDs is response-based, posing challenges in ensuring optimal control efficiency and fully utilizing the essential features of inerter and NS elements. In this study, the tuned negative stiffness inerter damper (TNSID), which integrates the NS element into one of the classical IBDs, namely the tuned inerter damper (TID), is investigated in terms of EDR for seismic application. This study makes two primary contributions. Firstly, it proposes an optimization design methodology aimed at achieving optimal control performance and efficiency of the TNSID simultaneously. Secondly, it unveils the enhancement law for the control performance and efficiency of TNSID. To do this, the analytical formulas for additional EDR and EDR enhancement (EDRE) factors are established as indicators of control performance and efficiency, respectively. Closed-form expressions are derived to optimize the TNSID design parameters. The comparative studies are conducted on both single and multi-degree-of-freedom primary structures, utilizing closed-form expressions, frequency transfer functions, and time history analysis. Results demonstrate that the incorporation of the NS element and the proposed optimization design methodology can further enhance both the control performance and efficiency of TNSID, surpassing those achieved by TID and classical fixed-point optimization theory.