On a Network

This paper deals with a problem of assigning n persons to n nodes (jobs) in a given network, whose arc lengths are dependent upon the assignment, so that the length of a critical path in the network may be minimized. The problem is first formulated as an integer programming probem, and then a branch-and-bound algorithm is proposed. The algorithm makes use of lower bounds obtained from the generated LP (linear programming) problems. Each LP problem is efficiently solved by reducing them to a series of rather small LP problems with two types of auxiliary problems : the ordinary assignment problems and the critical path problems. 1 . I ntroducti on The ordinary assignment problem assigns n persons to n jobs so that the sum of the assignment costs may be minimized. A variant of this assignment problem will be discussed in this paper. Consider the assignment of n persons (machines) to n jobs, the execution of which must satisfy certain precedence relations represented as a network. The objective to be minimized is the time of completing all jobs. It is formally defined as follows. Let G = [ V, A 1 be an acyclic network with a set of vertices V = { v 0' v 1' v2'···, vn+l } and a set of (directed) arcs A ={ al' a2, ... , at } c V,·V. o Each vertex represents a job and each arc a precedence relacion between jobs ; arc at = ( v., v. ) implies that the execution of v. cannot be initiated prior ~ J J to the completion of Vi' Vo is the dummy job representing the start; hence no arc is incident to VO' Vn+l is the dummy job representing the completion of all jobs ; hence no arc is incident from V n + l . We assume that all (directed) paths start from Vo and end at Vn+l' Let M = { m l' m 2' ... , mn } be the set of n persons (machines) to be assigned to jobs, and let dZa(t) be the length of at E A when mk is assigned 70 © 1976 The Operations Research Society of Japan An Assignment Problem on a Network to Va(t) where Va(t) is the vertex in V from which the arc at is incident, i.e., at = ( Va(t)' Vj ) for some Vj . The length of. at = ( Va(t)' Vj ) may be considered as the execution time of Va(t) plus the setup time of Vj , that generally depends on both va(t) and Vj . For simplicity d~o (lengths of arcs from Vo) are considered to be O. Our problem P is now defined as follows : Find an assignment of n elements in M to n jobs in V (i.e., one-to-one mapping M -+V ) that minimizes 71 the length of the critical path (longest path) from Vo to Vn+l. As a special case, this is equivalent to the ordinary assignment problem if G = [ V, A ] is given by V = { VO' v 1'···, vn+l } and A = {( vo' v 1 ), (V 1' v 2 ), ••• , (Vn , Vn+l )}. The above problem naturally arises in optimal scheduling of a set of n jobs, on which a technological ordering described by a PERT-like network G is imposed, if the selection of a person (or a machine) to execute each job is permitted. For a given G, the problem asks to find an assignment of n persons with different ability, experience, education and 'so forth (or n machines of different performance) in such a way that minimizes the total completion time of all n jobs. Note that the differences in persons (or machines) are repret sented solely by the lengths dka(t) of arc at· As an example, consider that G represents the process of painting a house (or a toy, an automobile, etc.) in n colours. Each node represents to paint one given colour. The preceding relation specified by G shows the order in which n colours are painted; e.g., black is painted after white, etc. The t length dka(t) of at =(Va(t) , Vj ) is the sum of the time of painting colour Va(t) and the time of waiting until it dries enough to start painting the next colour v j . Then the above problem is to assign n spray guns of varied performance to n nodes so that the entire process may be completed in the minimum time. (It is natural to assume that one spray gun is prepared for each colour . ) The dynamic assignment models discussed in Charnes et al.[l] is also similar to ours in that an assignment on an n-stage network is considered, though they use somewhat different constraints and objective function. t In case the length dka(t) of at =(Va(t)' Vj ) is dependent on Va(t) only, it is sometimes more convenient to use the network in which each arc (rather than node) represents a job, as in the ordinary PERT network. It would be obvious that the following theory can be easily modified to this model ; hence the discussion of this model will be omitted. Copyright © by ORSJ. Unauthorized reproduction of this article is prohibited. 72 T. lbaraki. H. Ishii and H. Mine 2. Formulation as an JP Problem The above assignment problem P is formulated as an IP (integer programming) problem in this section. An algorithm based on the branch-and-bound principle will then be developed in the subsequent sections. Branch-and-bound algorithms have attained certain success for other variants of the assignment problem, such as the quadratic assignment problem [4)[6) and the multidimensional assignment problem [10]. Let the path-arc ~~trix of G be given by IT = ( nst ), where if atE A is on the s-th path of G otherwise, .' " 1,2, ... , So ( 1,2, ... , to ( the number of paths in G ) the number of arcs ). Introduce 0-1 variables x ki which specify an assignment in the following manner :