We introduce a new facility layout problem, the so-called T-Row Facility Layout Problem (TRFLP). The TRFLP consists of a set of one-dimensional departments with pairwise transport weights between them and two orthogonal rows which form a T such that departments in different rows cannot overlap. The aim is to find a non-overlapping assignment of the departments to the rows such that the sum of the weighted center-to-center distances measured in rectilinear directions is minimized. The TRFLP is a generalization of the well-known Multi-Bay Facility Layout Problem with three rows (3-BFLP). Both problems, the TRFLP and the 3-BFLP, have wide applications, e.g., factory planning, semiconductor fabrication and arranging rooms in hospitals.

In this work we present a mixed-integer linear programming approach for the TRFLP and the 3-BFLP based on an extension of the well-known betweenness variables which now can be equal to one if the corresponding departments lie in different rows. One advantage of our formulation is the calculation of inter-row distances without big-M-type constraints. We provide cutting planes exploiting the crossroad structure in the layout, and hence T-row (3-Bay) instances with up to 18 (17) departments are solved to optimality in less than 7 h. The best known approach for the 3-BFLP is clearly outperformed. Additionally, tight lower bounds for larger instances are calculated to evaluate our heuristically determined layouts.

我们引入了一个新的设施布局问题，即所谓的 T-Row 设施布局问题 (TRFLP)。TRFLP 由一组具有成对运输权重的一维部门和两个正交行组成，这些行形成一个 T，使得不同行中的部门不能重叠。目的是找到部门到行的非重叠分配，使得在直线方向上测量的加权中心到中心距离的总和最小化。TRFLP 是著名的三排多槽设施布局问题 (3-BFLP) 的泛化。TRFLP 和 3-BFLP 这两个问题都有广泛的应用，例如工厂规划、半导体制造和医院房间布置。

在这项工作中，我们为 TRFLP 和 3-BFLP 提出了一种混合整数线性规划方法，该方法基于众所周知的中介变量的扩展，如果相应的部门位于不同的行，则该变量现在可以等于 1。我们公式的一个优点是计算行间距离而没有大 M 类型的约束。我们提供了利用布局中十字路口结构的切割平面，因此具有多达 18 (17) 个部门的 T 排 (3-Bay) 实例在不到 7 小时的时间内得到优化。3-BFLP 最著名的方法显然表现出色。此外，计算较大实例的严格下限以评估我们启发式确定的布局。