On the Lambert W function: EOQ applications and pedagogical considerations

The Lambert W function dates back to Euler’s time and despite offering solutions to many operations management problems it is still relatively unknown. This may be due to the fact that it is only incorporated into specialist mathematical software and is not generally available in common spreadsheet applications. This obscurity is rather unfortunate as it is relatively easy to use when the significance and mechanics of the function are known. In order to illustrate the use of the Lambert W function we consider two Economic Order Quantity (EOQ) scenarios: an EOQ model with perishable inventory; and a Net Present Value (NPV) analysis of an EOQ problem with trim loss. Both scenarios are motivated by real world situations. Via these two examples we reflect upon the pedagogical aspects of using the Lambert W function, specifically at a postgraduate level, and provide guidance on the manipulation of equations containing exponentials. We present a Lambert W function ‘look-up’ table for classroom use (which we suggest is no more difficult to use than the standard normal table popular in operations management texts) and a Microsoft Excel ‘Add-In’ for self-study and practical use. We also illustrate the use of the Laplace transform to conduct NPV analyses of the EOQ model, and demonstrate the close relation between the Laplace transform and the Lambert W function. It is hoped that is paper will accelerate the resurgence of the use of the Lambert W function in our field, as it provides exact analytical solutions to many problems that currently are viewed as not having explicit solutions.