This paper develops and analyzes a time-dependent optimal stopping problem and its application to the decision making process concerning organ transplants. Offers (organs for transplant) appear at jump times of a Poisson process. The values of the offers are i.i.d. random variables with a known distribution function. These values express the degree of histocompatibility between the donor and the recipient. The sequence of offers is independent of the jump times of the Poisson process. The decision about acceptance or rejection must be made at the time of appearance of the offer. When the offer is accepted, the decision process terminates and a reward is gained. The reward depends on the value of the selected offer, a discount function and a cost function related to aversion to risk. Otherwise, the decision process is continued until a random time T. After this time the reward is equal to zero. The aim of the decision maker is to maximize the expected mean reward. A first-order differential equation for the optimal mean reward is obtained.