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A Fixed Rate Production Problem with Poisson Demand and Lost Sales Penalties
Author(s) -
Faaland Bruce,
McKay Mark,
Schmitt Thomas
Publication year - 2019
Publication title -
production and operations management
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.279
H-Index - 110
eISSN - 1937-5956
pISSN - 1059-1478
DOI - 10.1111/poms.12931
Subject(s) - poisson distribution , lost sales , queue , mathematical optimization , computer science , inventory control , production (economics) , function (biology) , holding cost , lead time , safety stock , mathematical economics , mathematics , operations research , economics , operations management , statistics , microeconomics , supply chain , law , biology , programming language , evolutionary biology , political science
We solve a variation of a classic make‐to‐stock inventory problem introduced by Gavish and Graves. A machine is dedicated to a single product whose demand follows a stationary Poisson distribution. When the machine is on, items are produced one at a time at a fixed rate and placed into finished‐goods inventory until they are sold. In addition, there is an expense for setting up the machine to begin a production run. Our departure from Gavish and Graves involves the handling of unsatisfied demand. Gavish and Graves assumed it is backordered, while we assume it is lost, with a unit penalty for each lost sale. We obtain an optimal solution, which involves a produce‐up‐to policy, and prove that the expected time‐average cost function, which we derive explicitly, is quasi‐convex separately in both the produce‐up‐to inventory level Q and the trigger level R that signals a setup for production. Our search over the ( Q , R ) array begins by finding Q 0 , the minimizing value of Q for R = 0. Total computation to solve the overall problem, measured in arithmetic operations, is quadratic in Q 0 . At most 3 Q 0 cost function evaluations are required. In addition, we derive closed‐form expressions for the objective function of two related problems: one involving make‐to‐order production and another for control of an N ‐policy M / D /1 finite queue. Finally, we explore the possibility of solving the lost sales problem by applying the Gavish and Graves algorithm for the backorder problem.