Determining the Minimum Number of Warehouses and their Space-Size for Storing Compatible Items

We present an exact procedure for determining the smallest number of necessary warehouses and their space-size for storing compatible items. The required floor space housing of every item is known. The method developed here refers to store compatible items in the same warehouse in order to diminish the maximum necessary space-size of every warehouse and consequently to the determination of the minimum number of needed warehouses. The problem is formulated in the context of graph theory. Compatible items stored in the same warehouse are the elements of a color class of a specific coloring of a weighted conflict graph G = (V, E, W), where the vertices of V represent the items to be stored and all couples of non-compatible items define the edge set E. The elements of W are the numbers assigned to the vertices of V that express the required storing space of every corresponding item. That is the problem is reduced to find a coloring of G that correspond to an optimal solution.