Solution methods for a class of finite-horizon vector-valued Markov decision processes

This paper investigates and develops solution methods for a class of finite-horizon Markov decision processes characterized by additive or multiplicative vector rewards. Two concepts of optimality are treated: (1) optimality in the space of return vectors, whereby a policy is optimal if it delivers a maximal total reward from any initial state; and (2) optimality in the space of return functions, whereby a policy is optimal if its total reward function is maximal among all total reward functions. The paper elucidates the relation between the two concepts, proposes a procedure for utilizing this relation to determine the set of optimal policies under concept (1), and formulates a dynamic programming approach to calculating optimal policies under concept (2). The paper demonstrates that dynamic programming yields all optimal policies under concept (2). The paper’s results are illustrated with numerical experiments and a multi-objective stochastic inventory control problem.