Solving the multi-criteria: Total completion time, total late work, and maximum earliness problem

Within this research, The problem of scheduling jobs on a single machine is the subject of study to minimize the multi-criteria and multi-objective functions. The first problem, minimizing the multi-criteria, which include Total Completion Time, Total Late Work, and Maximum Earliness Time (∑𝐶 𝑗 , ∑𝑉 𝑗 , 𝐸 𝑚𝑎𝑥 ) , and the second problem, minimizing the multi-objective functions ∑𝐶 𝑗 + ∑𝑉 𝑗 + 𝐸 𝑚𝑎𝑥 are the problems at hand in this paper. In this study, a mathematical model is created to address the research problems, and some rules provide efficient (optimal) solutions to these problems. It has also been proven that each optimal solution for ∑𝐶 𝑗 + ∑𝑉 𝑗 + 𝐸 𝑚𝑎𝑥 is an efficient solution to the problem (∑𝐶 𝑗 , ∑𝑉 𝑗 , 𝐸 𝑚𝑎𝑥 ) . Because these problems are NP-hard problems so it is difficult to determine the efficient (optimal) solution set for these problems so some special cases are shown and proven which find some efficient (optimal) solutions suitable for the discussed problem, and highlight the significance of the Dominance Rule (DR), which can be applied to this problem to enhance efficient solutions.