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The subject is a deterministic continuous-time continuous-state inventory control model. Stock is replenished by ordering from one of a number of suppliers incurring a different cost per item and a different set-up cost. Taking the cost of procurement into account, the objective is to minimize the total discounted cost over an infinite planning horizon. The size of the order that is to be placed and the supplier with which it is to be placed are to be decided. Earlier studies of the problem have relied substantially on the assumption that the set-up cost of every supplier is strictly positive. Removing this restriction calls for a significant modification of the adopted approach. This is realized in the present study. It is shown that there is a stable unique optimal policy of a type that encompasses (s, S) and generalized (s, S) policies. Conditions that are necessary and sufficient for it to reduce to each of these types are established. The case of two suppliers is studied in detail, properties of the solution are investigated, numerical examples illustrating various aspects are included, and the connection with antecedent results is assessed.

A deterministic continuous-time continuous-state inventory model is studied. In the absence of intervention, the level of stock evolves by a process governed by a differential equation. The inventory level is monitored continuously, and can be adjusted upwards at any time. The decision maker can order from several suppliers, each of which charges a different ordering and purchasing cost. The problem of selecting the supplier and the size of the order to minimize the total inventory cost over an infinite planning horizon is formulated as the solution of a quasi-variational inequality (QVI). It is shown that the QVI has a unique solution. This corresponds to a generalized (s, S) policy under amenable conditions, which have been characterized in an earlier work by the present authors. Under the complementary conditions a new type of optimal control policy emerges. This leads to the concept of a hyper-generalized (s, S) policy. The theory behind a policy of this type is exposed.

This paper is concerned with determining the optimal inventory policy for an infinitehorizon deterministic continuous-time continuous-state inventory model, where, in the absence of intervention, changes in inventory level are governed by a differential evolution equation. The decision maker has the option of ordering from several suppliers, each of which entails differing ordering and purchasing costs. The objective is to select the supplier and the size of the order that minimizes the discounted cost over an infinite planning horizon. The optimal policy is formulated as the solution of a quasi-variational inequality. It is shown that there are three possibilities regarding its solvability: it has a unique solution that corresponds to an (s, S) policy; it does not admit a solution corresponding to an (s, S) policy but does have a unique solution that corresponds to a generalized (s, S) policy; or, it does not admit a solution corresponding to an (s, S) policy or a generalized (s, S) policy. A necessary and sufficient condition for each possibility is obtained. Examples illustrate their occurrence.

This paper concerns a general formulation for a longstanding problem. This is the problem of determining a replenishment schedule that minimises the total cost of stocking and holding an inventory in a deterministic finite-horizon inventory model. The formulation permits the treatment of seemingly unrelated models in a single framework. These include classical lot-size models, batching models, repair models, recovery models and others. Admissible control policies are restricted to a partition of some closed interval on the real line. The solution of a mixed integer nonlinear programming problem (MINLP) delivers the optimal partition. It is shown that the MINLP possesses an optimal solution under very mild conditions. The theory of submodular functions on a lattice is the key to handling the integer variable. This theory permits the recovery and generalisations of earlier results on the interleaving property of optimal partitions and a convexity property of the value of the objective function. Past intractable inventory models with demand driven by a general differential equation and in the absence and in the presence of shortages and inflation are solved. This generalises a number of existing results in the literature.