We consider a fundamental integer programming (IP) model for cost–benefit analysis and flood protection through dike building in the Netherlands, due to Zwaneveld and Verweij (2017). Experimental analysis with data for the IJsselmeer shows that the solution of the linear programming relaxation of the IP model is integral. This naturally leads to question whether the polytope associated to the IP is always integral.

In this paper we first give a negative answer to this question by proving the non-integrality of the polytope. Secondly, we establish natural conditions that guarantee the linear programming relaxation of the IP model is integral. We show that these conditions are indeed satisfied by the recent data on flood probabilities, damage and investment costs of IJsselmeer. Finally, we show that the IP model can be solved in polynomial time when the number of dike segments, or the number of feasible barrier heights, are bounded.

由于Zwaneveld和Verweij（2017），我们考虑在荷兰通过堤防建设进行成本效益分析和防洪的基本整数规划（IP）模型。IJsselmeer数据的实验分析表明，IP模型线性规划松弛的解决方案是不可或缺的。这自然会引起质疑，即与IP相关的多位点是否始终是必不可少的。

在本文中，我们首先通过证明多表位的非完整性来对这个问题给出否定的答案。其次，我们建立自然条件，以保证IP模型的线性编程松弛是完整的。我们证明这些条件确实得到了有关艾瑟尔湖的洪水概率，破坏和投资成本的最新数据的满足。最后，我们表明，当堤防段的数量或可行的障碍高度的数量有界时，可以在多项式时间内求解IP模型。