On the Lambert W function: EOQ applications and pedagogical considerations
On the Lambert W function: EOQ applications and pedagogical considerations
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发表时间：
2010
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通讯作者：
S. Disney;R. D. Warburton
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作者：
S. Disney;R. D. Warburton
兰伯特 W 函数可以追溯到欧拉时代，尽管它为许多运营管理问题提供了解决方案，但它仍然相对不为人所知。这可能是因为它仅被纳入专业数学软件中，并且在常见的电子表格应用程序中通常不可用。这种模糊性是相当不幸的，因为当函数的意义和机制已知时，它相对容易使用。为了说明 Lambert W 函数的用法，我们考虑两种经济订单数量 (EOQ) 场景：具有易腐烂库存的 EOQ 模型；以及带有调整损失的 EOQ 问题的净现值 (NPV) 分析。这两种场景都是由现实世界的情况激发的。通过这两个例子，我们反思了使用兰伯特 W 函数的教学方面，特别是在研究生水平，并为包含指数的方程的操作提供了指导。我们提供了供课堂使用的 Lambert W 函数“查找”表（我们认为该表的使用并不比运营管理文本中流行的标准普通表更难）和供自学和实际使用的 Microsoft Excel“插件”。我们还说明了如何使用拉普拉斯变换对 EOQ 模型进行 NPV 分析，并证明了拉普拉斯变换与 Lambert W 函数之间的密切关系。希望这篇论文能够加速 Lambert W 函数在我们领域的使用复兴，因为它为许多目前被认为没有明确解决方案的问题提供了精确的分析解决方案。
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.