The teacher assignment problem: A special case of the fixed charge transportation problem

A basic model of the task of assigning classes to professors, in such a way that the average number of distinct subjects assigned to each professor is minimized, is formulated as a mixed integer program. It turns out that the problem is a special case of the fixed charge transportation problem which in some cases corresponds to finding a basic solution of a transportation problem which is as degenerate as possible. We present an equivalent alternative formulation of the problem which makes it easy to prove that it is NP-hard in the strong sense and an exact branch and bound algorithm for its solution based on this alternative formulation is outlined. Computational experiments with data from a concrete problem, concludes the paper. 9 1997 Elsevier Science B.V. Ke)~vords: Branch and bound; Fixed charge transportation problem; 0-1 integer programming; Complexity 1. I n t r o d u c t i o n In the classical transportation model, the cost of transportation from producer i to consumer j is proportional to the number of units xq which are transported. However, in many practical situations, the actual cost of transporting xq units from i to j is the sum of a fixed (starting) cost of serving consumer j from producer i and a cost proportional to the number of units transported. 9 Corresponding author. E-mail: [email protected]. I E-mail: [email protected] In other words the cost is given by the following formula i, j ) = [ f i j + cijxij i f x i j > 0 cost( 0 if xij = 0 This modification of the transportation problem is called the fixed charge transportation problem. If all cij's are zero it is called the pure fixed charge transportation problem. The model treated in this paper is a special case of the pure fixed charge transportation problem in which all f q ' s are equal to one. But the algorithm we present is based on a new equivalent formulation of the problem. This important special case of the 0377-2217/97/$17.00 9 1997 Elsevier Science B.V. All rights reserved. PH S0377-2217(96)00082-3 464 T.tL thdtberg. D.M. Cardoso / European Journal of Operational Research I01 (1997) 463-473 fixed charge transportation problem originated from a practical problem of assigning subjects to teachers and for this reason we have maintained the terminology of this particular practical problem when describing the model. However many other types of practical problems, including the problem of minimizing the total number of order-selection operations in a warehouse (see [7]), can also be stated within this model. Several special purpose algorithms for solving the fixed charge transportation problem and the more general fixed charge network flow problem have been proposed, including strong cutting plane techniques [6,8] and branch and bound algorithms [1]. However the special cost structure of our model allows us to give an alternative formulation of the problem which leads to a more direct approach to its solution. The pure fixed charge transportation problem has not been studied much in the past, but see [4] for a recent approach based on constraint generation procedures for a set covering reformulation. In Section 2 the Teacher Assignment Problem is formally stated as a mixed integer linear program. In Section 3 we give an alternative combinatorial formulation which establishes the base of which an exact algorithm is developped in Section 4. Finally, in Section 5, the results of the computational experiments with the implemented algorithm are presented. 2. The teacher assignment problem In this section we will introduce a basic model of the problem of assigning subjects to professors. The objective of the assignment will be to minimize the average number of distinct subjects taught by each professor. Let there be given a set, 1, o f m subjects and a set, J, of n professors. It is obvious that any valid assignment must satisfy Exij=hi, i~ l (1) j ~ J Exij"Fsj=bj, j E J (2) iEI where x~j > 0 is the number of hours of subject i taught by professor j, Iq is the total number of hours to be lectured of subject i (the product of the subjects weekly hours and the number of classes needed for the subject), and b/ is the maximum number of hours that professor j can teach. For convenience we have introduced slack variables sj>O ( j ~ J ) in Eq. (2) representing the unused teaching capacity of professor j. We will not impose any additional restrictions on the assignment. In particular there will be no minimum number of hours that each professor should teach. To count the number of distinct subjects we introduce zero-one variables dij indicating whether professor j is teaching subject i or not. 1 i f x i j > 0 dij = 0 if xii = 0 The complete basic model then becomes: The teacher assignment problem (TAP). minimize f = E E dij (3) j ~ J iEI subject to ~ x i j = h i, i ~ l (4) j ~ J