Towards the Numerical Solution of a Large and Difficult MILP Problem of Closed Loop Layout

This paper contributes to two different areas: (1) layout problems and (2) integer programming. (1) One important class of layout problems arises when rectangles are to be placed on the plane. The rectangles represent machines with their close environments or small production cells. There is a flow of the semi-finished products among the rectangles. The problem is to place the rectangles without overlapping such that the total transportation, i.e. the sum of the products of the flows and distances, is minimal. A subcase of this general problem is the closed loop layout problem where the track of the transportation is a rectangle as well. The rectangular layout problem has a set of benchmark problems starting with 4 cells and for each even number there is a problem up to 18 cells. To the best knowledge of the author the optimal solution is known only for 4 and 6 cells. This paper provides the optimal solution of the 8 cell problem and suggest a very good feasible solution for the 10 cell problem. The latter is supposed to be optimal. (2) The closed loop layout problem is modeled by a mixed 0-1 programming problem. It is a large scale problem, however recent leading solvers can solve even larger problems. In spite of this fact, it is too difficult even for the top solvers. For even 8 or 10 cells, the model cannot be solved in a single run. It means that the structure of the problem is one which makes it extremely difficult. A possible numerical solution can be that the basic problem is cut into several subproblems and the subproblems are solved, more precisely fathomed. This procedure can be characterized as follows: it is a branch and bound structure above the single-run branch and bound procedure of the optimizer. It is described in the paper. It was strong enough to solve exactly the 8 cell problem. The 10 cell problem is still under the procedure. The procedure uncovered some special properties of the integer programming solution methods and suggests some principles to obey in the solution of large scale MILP problems.