Using dynamic programming to determine optimal pipe welding locations

This paper discusses the use of dynamic programming in plant piping for the purpose of determining optimal welding locations where welding costs are minimized. Pipes are constructed and assembled by welding elbows, T-joints, and material pipes together. Partial pipes, called spools, which are small enough to be transported, are welded at the factory; these parts are then assembled and welded at the site to form an entire pipe system. The conditions we have considered here include the following: the unit costs for factory welding and for site welding; the size of the spools, restricted by the transportation mode; some parts of the pipes where welding is not possible and other parts where welding is necessary; and the lengths of material pipes. The functions in which the welding costs are minimized are the following: (x) = miny ∈ Y (x) {C1+G(y, x)+F(y)} (x = 1, 2, 3,··· ,m) G(y,x) = mint ∈ W (y, x) {C2+G(t, x)} (y = x – 1,x – 2,···). Here, x, y, and t represent positions on the pipe; the center line of the pipe is partitioned into meshes, and the boundaries of these meshes are labeled with numbers beginning at O, the initial point, and ending at m, the terminal point. Y(x) is the set of points y such as the part between y and x can form one spool which can be transported. W(y, x) is the set of points t between y and x which can be the factory-welding spot closest to y. F(x) gives the minimum value for the sum of the site welding cost and the factory welding cost from the initial point to x, whereas G(y,x) gives the minimum value for the factory welding cost within the spool from y to x. C1 and C2 are the unit costs for site welding and factory welding, respectively. In order to demonstrate the applicability of the present method, it is applied to several examples and its results are compared with those by the conventional, manual method. It indicates that the new method reduces the cost by 10% and that the welding locations are determined with only about 20% of the work processes required by manual calculation.