Who determines the valuation of shares that want to be sold back to a private company?

We have a small family private company that consists mainly of vacation properties (unlikely to be sold immediately) and some cash liquidity. A person with 27% of the shares is looking to sell them back to the company.

I see that there are various methods to value these shares. If we looked at the total value of the assets (property + cash), it would appear that the shares are worth much more than just cash alone. My question is, who has the authority to decide the method of valuation (board? directors?) and then whether the seller is obligated to accept that price per share?

Small Business Financing
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If the family company has a written shareholders agreement, that should outline how the company should ultimately be valued at any given time, as well as the method by which an existing shareholder can sell his or her shares back to the company. If no such language exists, or language that speaks to whether the board or directors of the company have the power to rule on these matters, the majority shareholders should try to come up with a methodology for how to value the company that the seller of shares agrees with. If an agreement can not be reached as to how to value the company, then there are a number of third party organizations that can come in and give you a fair market value after reviewing your books. The downside to this is that these companies are often pricey.