The machine loading problem

Abstract

The Machine Loading Problem is a special case of the general Linear Programming Problem and may be described mathematically as follows: Subject to the following constraints all x. @ 0 ij find values X;{ which minimize the functionZE where, for th~specific problem of the opti~ll ~ cijxij' allocatlon of products to machines, m = number of available machine types n ~ number of distinct products M.. total production time available on machine i I (during a basic time period) P.. total production requirement for product j aij_-amount of time required to produce one unit of product j on machine i bij. 1 in this particular application ci~=~ cost of producing one unit of product j on machine i xi~=j amount of product j to be produced on machine i 28-1-2o Origin of the Geometric Patterns in the Machine Loading Problem The mathematical form of the ~achine l~ading Problem differs from that of the Transportation problem only t~rough the presence of coefficients aij which are not all = io This minor generalization has a drastic effect: given a '~basis'~ m~eo~ a distribution~ in table forms let us draw all possible horizontal and vertical lines interconnecting basis elements~ No longer (as was the case in the Trans~ portation Problem) do we obtain a connected branched chain (topolo~-ically~ a "tree"), but rather a number of disconnected pieces~ each piece consisting of a closed loop (or an entry in the special "slack" column) with attached branched side-chains (trees)° A simple case is illustrated in Figure i (see next page)° From the point of View of the full Linear Programming matrix of coefficients, this corresponds to the following mathematical equivalent: a set of linearly independent basis vectors is expressed iu terms of auxiliary vectors~ ~ich themselves are no longer independent but are now linearly dependent° it then becomes necessary to solve simultaneous sets of equations dbr~ ing each iteration. As a consequence of the radically altered topo!o-gy, existing programs for the Transportation Problem cannot be simply adapted to the more general Machine Losding version. A new approach was required for the present program, which, during the process of solution, guides the 704 to detect and trace all loops and trees in systematic fashion. Introduction of a new and removal of an existing basis elements which occurs in each iteration of the algorithm; will alter the topological connectivity of loops and attached chains° For example, row …