A survey of stochastic inventory models with fixed costs: Optimality of ( <i>s</i> , <i>S</i> ) and ( <i>s</i> , <i>S</i> )‐type policies—Continuous‐time case

Fixed costs of ordering items or setting up a production process arise in many real‐life scenarios. In their presence, the most widely used ordering policy in the stochastic inventory literature is the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="">$(s,S)$</mml:annotation> </mml:semantics> </mml:math> policy. Optimality of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="">$(s,S)$</mml:annotation> </mml:semantics> </mml:math> policies and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="">$(s,S)$</mml:annotation> </mml:semantics> </mml:math> ‐type policies have been examined for various inventory models, including those with the inventory level being reviewed in every period or continuously, finite and infinite horizons, discounted‐cost and average‐cost criteria, backlogging and lost‐sales practices, standard and generalized demands and/or costs, deterministic and stochastic lead times, single‐product and multi‐product settings, and coordinated pricing‐inventory decisions. We comprehensively survey the vast literature accumulated over seven decades in two papers. This paper is devoted to continuous‐time models, and the companion paper, also published in this journal issue, reviews discrete‐time models. We go over model specifications, proof techniques, specific results, and limitations of the articles published in the literature. We conclude each paper by providing corresponding suggestions for extensions and directions for future research.