Technical Note—Error Noted in “Order-Based Cost Optimization in Assemble-to-Order Systems” by Lu and Song (2005)

AbstractDiscrete convexity, which extends submodularity to integer vectors, has been used in the economics and management literature to characterize the behavior of optimal policies. One of its variants, called L♮-convexity, has enabled recent advances in various operations management systems. In a paper published by Operations Research in 2005, an assemble-to-order inventory system was shown to have the L♮-convexity property, which was used to motivate an efficient algorithm. In a technical note, “Error Noted in ‘Order-Based Cost Optimization in Assemble-to-Order Systems’ by Lu and Song (2005)” by Bolandnazar, Huh, McCormick, and Murota, the authors show that the proof in that paper is incorrect and L♮-convexity may not hold. Despite this error, the authors credit Lu and Song for introducing this useful concept to the operations management community.One of the main results of “Order-Based Cost Optimization in Assemble-to-Order Systems” [Lu Y, Song J-S (2005) Order-based cost optimization in assemble-to-order systems. Oper. Res. 53(1):151–169] is proposition 1(c), which states that the cost function of an assemble-to-order inventory system satisfies a discrete convexity property called L♮-convexity. We construct a counterexample showing that this result is incorrect, and hence their proposed steepest decent algorithm may not work.