## 1 Introduction
Besides, some research considering the cost of the inerter systems has been carried out as well. The cost of the inerter system is mainly related to its control force and parameter requirements. In [6], the authors pointed out that due to the inter-story installation scheme, to obtain an identical structural control performance with the tuned mass damper, the apparent mass and damping coefficient requirement of the TID needed to be enlarged, consequently causing the force exerted from the TID on the structure to increase. Considering the potentially increased cost of the inerter system, [PERSON] et al. [7] proposed a trade-off solution. They developed a dual-objective design approach for the structure equipped with a different inerter system, giving Pareto optimality between the structural control performance and the control force of the inerter system. [PERSON] et al. [30] confirmed that two inerter systems, instead of a single one, installed at the bottom story could reduce the size and control force of the device, thus facilitating the manufacture and improving the structural control performance at the same time. However, using the Pareto optimality to reduce the control force of the inerter system is based on sacrificing the control performance. Utilizing multiple inerter systems, being equipped at each story or concentrated at the bottom layer, to meet various optimization goals, requires many quantities and types of inerter systems, which will still increase the overall cost. Indeed, improving the utilization efficiency of the inerter system is an effective way to reduce its cost. For improving the utilization efficiency of the inerter system, instead of utilizing the structural shear deformation between consecutive layers, different installation schemes considering the vibration characteristics of the structure have been studied. [PERSON] et al. [31] proposed outrigger TVMDs utilizing the bending deformation instead of the shear deformation of high-rise structures. [PERSON] and [PERSON] [32] confirmed the high performance of the practical application of the TVMDs installed across three layers using a chevron steel brace in a 37-story building. Additionally, proposing to connect the inerter to a toggle brace, [PERSON] et al. [33] were the first to use the support with an amplification effect to enhance the efficiency of the inerter. However, the outrigger TVMD is only suitable for outrigger structures. The chevron steel brace and toggle brace are inevitably bulky due to buckling issues. Besides, it is inconvenient to adjust the displacement transition ratio of the chevron steel brace, which results in different sizes of inerter which are needed to achieve an optimal distribution of the inerter systems. In contrast, the cable-bracing system has advantages in terms of being lightweight due to its tension-only and adjustability characteristics. [PERSON] et al. [34] proposed a cable-bracing inerter system (CBIS), supporting the inerter system with a simple cable-bracing system, and illustrated its advantages in terms of installation convenience and end constraint release. [PERSON] and [PERSON] [35; 36] studied a damped cable system, utilizing prestressed cables to connect fluid viscous spring-dampers and the structure. Utilizing pretensioned rods combined with levers, [PERSON] et al. [37] developed interlayer bracings and outer bracings to connect the magneto-rheological dampers between layers and between the ground and a specific layer, respectively. [PERSON] [38] combined the pretensioned cables with a seesaw member and proposed a cable-bracing seesaw system to support a pair of fluid dampers. The cable-bracing seesaw system connecting the grounded dampers directly to the top story exhibited an excellent damping performance. Making full use of the characteristics of the CBIS and maximizing its utilization efficiency, in this paper, a cross-layer installed CBIS (CICBIS), able to cross more layers and suitable for shear-type MDOF structures, is proposed.