The Generalized Assignment Problem (GAP) is the problem of finding the minimal cost assignment of jobs to machines such that each job is assigned to exactly one machine, subject to capacity restrictions on the machines. We propose a class of greedy algorithms for the GAP. A family of weight functions is defined to measure a pseudo-cost of assigning a job to a machine. This weight function in turn is used to measure the desirability of assigning each job to each of the machines. The greedy algorithm then schedules jobs according to a decreasing order of desirability. A relationship with the partial solution given by the LP-relaxation of the GAP is found, and we derive conditions under which the algorithm is asymptotically optimal in a probabilistic sense.
Romeijn, H. E., & Romero Morales, D. (2000). A Class of Greedy Algorithms for the Generalized Assignment Problem. Discrete Applied Mathematics, 103(1-3), 209–235. https://doi.org/10.1016/S0166-218X(99)00224-3