Scheduling problems are one of the most researched topics in the field of operational research. Scheduling problem models have evolved and branched because of the wide range of products, standards, and customer requirements. Recently, the partial job shop scheduling problem, which is a general model of shop scheduling problems, has become a new scheduling problem. Operations in this model are partially ordered, and the order varies for each job. Several problems studied independently in the literature, such as the group shop, mixed shop, job shop, and open shop scheduling problems, are considered special cases of the partial shop scheduling model. Because ant algorithms are known in the literature as effective tools for solving combinatorial optimization problems, this study proposed an ant colony (AC) algorithm for solving partial shop problems with an objective function to minimize makespan. The AC method was examined and evaluated on the renowned “Taillerd” benchmark instances. It was then compared with the hybrid scatter search (HSS) and iterated tabu search (ITS) methods. The AC algorithm’s average deviation for 80 instances ranged between 0% and 1.78%. The AC algorithm outperforms the HSS and ITS methods, according to the computational findings; where the average percentage relative deviation for AC is 0.66%, compared with 0.99% for ITS and 10.14% for HSS.

The ant system (AS) and scheduling problem are well-known concepts in literature. Ant algorithms have been known to be an effective tool for solving combinatorial optimization problems. Elitist AS (EAS), rank-based AS (RAS), ant colony system (ACS), and max-min AS (MMAS) are the variants of the AS algorithm; they are triggered by the different ways of updating the pheromone trail τ, computing the visibility η, and/or other parameters in the basic AS model. The main contribution of this article is twofold. First, the basic AS and its controlled parameters are presented, the key variants of the ant algorithms are explained, and major changes of each variant from the basic model are tracked. Second, sixty papers are collected between 2015 and 2020 based on a search strategy for tracking the implementation of different AS variants in solving scheduling problems. Numerous findings based on a statistical analysis of the collected papers are reported and discussed. This study will allow the researcher to understand the essence of the ant algorithm, recognize the fundamental differences in its five systems, and determine how each of them can be implemented. Tracking a sample of articles that apply an ant algorithm for a specific case study gives researchers new ideas on how to adjust the original model to fit their problem.