An analysis model based on boundary limits for multi-channel spectrograph with the broad band and high throughput is built. The relationships among performance requirements, initial parameters, project budget, and risk of the multi-channel spectrograph are discussed. The mathematic model can get structural parameters of each sub-system of the multi-channel spectrograph quickly according to the given system requirements. It also provide a valuable method to evaluate the feasibility of the design and cost budget reasonably at the beginning of the project. A multi-channel spectrograph based on volume phase holographic gratings is designed for the 4 m telescope. It has a wavelength range from 350 nm to 1000 nm and resolution of 5000 at the blazed wavelength in each channel. The peak efficiency of the whole spectrograph is over 53%, and the monochromatic enclosed energy at 80% is better than 15 μm with the whole working band, which meets the requirements of system performance.