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In the era of globalization, institutions face increasing challenges in effectively managing their resources to achieve diverse and differentiated objectives. The unbalanced multi-objective assignment problem reflects these challenges, as the distribution of limited resources requires achieving a balance between objectives such as minimizing time, cost, etc. In this study, an application case in the industrial field of rolling process was investigated, where the objective of the assignment problem is to minimize total assignment. In order to tackle this problem, the multi-objective problem was addressed by converting it into a single-objective problem through the weighted sum method by assigning a weight to each objective. A new heuristic method was then applied to solve the unbalanced single-objective assignment problem where the machines were loaded with more than one job. The Hungarian method was also used to solve the problem in order to assess the proposed method, and the outcomes were then compared. The proposed method proved efficient as it reduced idle time by 28%, increased machine efficiency by 5% and reduced costs by an average of 15%.
In the era of globalization, institutions face increasing challenges in effectively managing their resources to achieve diverse and differentiated objectives. The unbalanced multi-objective assignment problem reflects these challenges, as the distribution of limited resources requires achieving a balance between objectives such as minimizing time, cost, etc. In this study, an application case in the industrial field of rolling process was investigated, where the objective of the assignment problem is to minimize total assignment. In order to tackle this problem, the multi-objective problem was addressed by converting it into a single-objective problem through the weighted sum method by assigning a weight to each objective. A new heuristic method was then applied to solve the unbalanced single-objective assignment problem where the machines were loaded with more than one job. The Hungarian method was also used to solve the problem in order to assess the proposed method, and the outcomes were then compared. The proposed method proved efficient as it reduced idle time by 28%, increased machine efficiency by 5% and reduced costs by an average of 15%.
Let α \alpha be a set of n n elements and δ \delta be a nonnegative integer. A δ \delta -partition of α \alpha is a set of pairwise disjoint nonempty subsets of α \alpha such that the union of the subsets is equal to α \alpha and every subset has a size greater than δ \delta . We formulate an algorithm for computing all δ \delta -partitions of a given n n -element set and show that the algorithm runs in O ( n ) {\mathcal{O}}\left(n) space and O ( n ) {\mathcal{O}}\left(n) delay time between any two successive outputs of δ \delta -partitions of the given set. An application of the notion of δ \delta -partitions is illustrated in the following scheduling problem. Suppose a factory has n n machines and m ≤ n m\le n jobs to complete daily. Every job can be accomplished by operating at least δ + 1 \delta +1 machines. A machine cannot work on multiple jobs simultaneously. According to a utilization policy of the factory’s management, no machine is allowed to be idle, so all machines should be running on some job. Find a daily schedule of the factory’s machines satisfying all the mentioned constraints. Let α \alpha be the set of the factory’s machines. Then, an α \alpha ’s δ \delta -partition with m m subsets is a legal schedule if every subset (in the δ \delta -partition) includes exclusively δ + 1 \delta +1 or more machines that run on the same job.
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