# A Mathematical Model for Reduction of Trim Loss in Cutting Reels at a Make-to-Order Paper Mill
## 2 Literature Review
So, trim loss minimization and provision of stocked reels are important aspects in solving cutting stock problems which are considered in this model. Other contributions of this paper are the consideration of various operational and technological constraints in the production and cutting process. Operational constraints include the combination of reels of identical weight of paper per unit width and combination of reels of identical internal and external diameters. Additionally, to optimize the use of available machinery at the paper mill, the minimum and maximum widths are imposed to the cutting patterns. Operational constraints include edge trim loss at the two ends, maximum diameter of the master reels, and the limitation of cutting blades at winder. The limited cutting blades has been considered in the mathematical model and feasible patterns are generated according to the number of available cutting blades. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{5}{c}{**Paper Mill**} \\ \cline{2-5}  & **SCA** & **Norske Skog** & **Stora Enso** & **Holmen** \\ \hline Mill production (millions of tonnes) & 9.9 & 4.8 & 15.1 & 2.3 \\ Total waste generated (kg/ton product) & 163 & 163 & 155 (dry) & 160 \\ Recovered waste (kg/ton product) & 115 & 138 & - & 136 \\ Waste sent to landfill (kg/ton product) & 47 & 16 & 22 & 23 (wet) \\ Hazardous waste (kg/ton product) & 0.3 & 1.5 & 0.3 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Generation of waste from few European pulp and paper mills. The linear programming model with the objective of trim loss minimization is solved through simplex algorithm which normally results in non-integer solution. The constraints of the model are regarding meeting the customer demand, provision of stocked reels, and limitation on cutting blades at winder. A non-integer solution is rounded by a new post-optimization procedure which is relatively easy and takes less computational time. The minimization of trim loss and provision of stocked reels results in reduced reprocessing of paper. In the reprocessing of paper, harsh chemicals are used for different purposes. These chemicals are discharged in wastewater to the environment which are harmful and can increase the risk of different types of cancers and also affect aquatic life badly. The paper mill has confirmed that by implementing the proposed method, significant improvements in terms of trim loss minimization and reduction in discharge of effluents into the environment has been achieved and it outperformed other traditional approaches. However, the trim loss minimization problem has been analyzed in various contexts in literature. Our study proposes a new mathematical model for the minimization of trim loss in cutting master reels of flexible widths into smaller reels, provision of stocked over-produced and leftover reels to fill an order, limitation of cutting blades in the feasible pattern generation, consideration of various operational and technological constraints, rounding the non-integer solution by a new post-optimization procedure and the minimization of chlorinated compounds by minimizing the reprocessing of paper trimming to protect environment. The next section contains the mathematical model for the considered problem.