Tool Carousel Design via Integer Programming L R Foulds* Department of Management Systems University of Waikato New Zealand lfoulds@waikato.ac.nz J M Wilson Business School Loughborough University Great Britain j.m.wilson@lboro.ac.uk Abstract We describe a branch and bound algorithm for an assignment problem subject to a special set of side constraints. The problem has application in the design of tool carousels for certain flexible manufacturing systems. The resulting model represents a special case of the restricted facilities layout problem in which it is forbidden to locate any facility in certain zones. The bounds for the algorithm are generated by relaxing the side constraints and using the Hungarian method to solve the resulting assignment problem. Partitioning in a manner similar to subtour elimination for the travelling salesman problem leads to encouraging computational results. Keywords: algorithms; assignment; heuristics; integer programming; side constraints 1. Introduction For a flexible manufacturing system tools will frequently be located in a carousel as shown in Figure 1. The carousel has a given number of pockets, of equal area, located around its perimeter into which tools are to be located. We now make three assumptions: • the pockets are all of unit area and the tools are of various (integral) areas; • it is forbidden to locate any tools in certain pockets (denoted in Figure 1 by hatched areas) for technical reasons; • the total area of the set of tools to be located is no greater than the number of available pockets. An adjacency rating is known for each pair of tools, representing the desirability that the pair be located in adjacent pockets in the carousel. The problem we will consider is: assign all tools to the pockets (with each tool occupying a continuous sequence of pockets equal to its area) so that the sum of the adjacency ratings of tools is a maximum. Other models of tool carousel design for flexible manufacturing systems have been discussed by Wilson [9] and Foulds and Wilson [4]. It should be noted that if there are no forbidden zones in the problem it reduces to the travelling salesman problem (TSP) whereas if it has exactly one forbidden zone it reduces to the TSP path problem (i.e. one where it is not necessary to return to the starting city). Hence the problem is NP-hard (see for instance Garey and Johnson [5]). The problem to be discussed in this paper can be formulated as an assignment problem with a particular set of side constraints. The contribution of this paper is: to show that the problem is a special case of the restricted facilities layout problem with forbidden location zones and to report on computational experience in solving the model by a branch and bound algorithm. In the next section we introduce a model of the tool carousel design problem. 2. A Model