An emergency supply policy for an inventory replenishment model with returns and partial backorders

Abstract This paper studies a continuous-review inventory replenishment model with a limited storage capacity S in an uncertain environment. We assume that the demands and returns follow independent Poisson processes. We further assume a ra1079 shelf life, a random lead time, and early loss. The storage is managed according to the base-stock ( S , s ) policy for $$s&lt;S,S&gt;0.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> In case of overstock, each returned item exceeding S is transferred to a foreign facility. If during the lead time a demand reaches zero stock, we consider two alternatives: either allow partial backordering up to $$L_{B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>B</mml:mi> </mml:msub> </mml:math> items, beyond which the unsatisfied demand is lost, or call for an immediate and costly emergency supply up to level $$0&lt;Q_{B}^{e}\le S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:msubsup> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mi>e</mml:mi> </mml:msubsup> <mml:mo>≤</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> . Our objective is to study how the thresholds s , S , $$L_{B},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>B</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and $$Q_{B}^{e}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mi>e</mml:mi> </mml:msubsup> </mml:math> are impacted by the system’s parameters, such as returns, demands, and costs. Using a Markovian framework, we derive the steady-state probabilities for the inventory level, and construct closed-form expressions for the average cost functions. Then, we numerically investigate the impact of the different parameters on the best policy and on the threshold levels. We compare the two alternatives and identify situations in which calling for an emergency supply is economically profitable.