The authors consider the following ‘symmetric’ generalization of the well-known asset-selling problem. Assume that there are two activities, each of which must be carried out a given number of times. An example is the disposal of i and j containers of toxic waste of two different kinds. Exactly one of the activities is performed per time period. The random rewards (possibly negative) for performing the different tasks are i.i.d. and have a known two-dimensional joint distribution. In addition, there are fixed costs c, c1 and c2 (possibly negative) per period as long as activities of both kinds, or only of the first or only of the second kind must be performed. Based on a realization of the two random rewards, each period a choice must be made which of the two activities to perform such that the expected total reward becomes maximal. The authors show for arbitrary discount factor the existence of an optimal control limit policy with recursively computable critical numbers dij. Some explicitly solvable cases are presented. Under appropriate assumptions the dij’s are monotone in i and/or j which allows, despite a 4-dimensional state space, to represent the optimal policy by a family of 2-dimensional monotone decision regions. And then, the limits of the dij’s can be used to identify ‘large’ sets of states where the optimal decision can be found without numerical computations. The basic tool is the integrated distribution function of the difference of the two rewards. The paper generalizes many of the known results on asset-selling problems without recall.
