# A Mathematical Model for Reduction of Trim Loss in Cutting Reels at a Make-to-Order Paper Mill
## 3 Mathematical Model
#### Decision Variables
Considering all decision variables non-integer in the first phase, which is a rare case, the new objective function and constraints for the second phase are given as:  
\[\textbf{Minimize}:Z=\sum_{a=1}^{l}\sum_{c=1}^{n}X_{ac}+\sum_{b=1}^{m}\sum_{d=1}^ {r}X_{bd} \tag{9}\]  
**Subject to:**  
\[\sum_{a=1}^{l}A_{ai}\sum_{c=1}^{n}X_{ac}+\sum_{b=1}^{m}B_{bi}\sum_{d=1}^{r}X_{bd }\geq S_{i}\ \ \ i=1,2,\ldots,u \tag{10}\]  
\[\sum_{a=1}^{l}A_{ai}\sum_{c=1}^{n}X_{ac}+\sum_{b=1}^{m}B_{bi}\sum_{d=1}^{r}X_{ bd}\geq D_{i}\ \ \ i=1,2,\ldots,u \tag{2}\]0  
The objective function (9) minimizes the summation of all non-integer decision variables resulted in the first phase of the solution. The constraints (10) limit the shortages whereas constraints (11) are binary restrictions on the decision variables. Finally, the decision variables obtained in the second phase are added with pre-decimal parts of the corresponding decision variables in the first phase. After the optimal integer solution is obtained, other measures are calculated by the equations given below. The total area of an order is calculated using Equation (12) while total master reels and stocked reels used are enumerated utilizing Equations (13) and (14), respectively. The gross area of the selected master and stocked reels can be estimated using Equations (15) and (16), respectively. Finally, Equations (17) and (18) evaluate the net area of selected master reels and stocked reels, respectively. The problem falls in the domain of linear programming because the objective as well as all the constraints are linear in the decision variables. The non-linear models can either be converted to linear models using approximation techniques or they can be solved using non-exact/meta-heuristic approaches which requires higher computational requirements. Furthermore, our developed algorithm solves the linear problem in adequate amount of time. \[\sum_{a=1}^{l}A_{ai}\sum_{c=1}^{n}X_{ac}+\sum_{b=1}^{m}B_{bi}\sum_{d=1}^{r}X_{ bd}\geq D_{i}\ \ \ i=1,2,\ldots,u \tag{2}\]1  
\[\sum_{a=1}^{l}A_{ai}\sum_{c=1}^{n}X_{ac}+\sum_{b=1}^{m}B_{bi}\sum_{d=1}^{r}X_{ bd}\geq D_{i}\ \ \ i=1,2,\ldots,u \tag{2}\]2 Shortages and surpluses algorithm. \[\sum_{a=1}^{l}A_{ai}\sum_{c=1}^{n}X_{ac}+\sum_{b=1}^{m}B_{bi}\sum_{d=1}^{r}X_{ bd}\geq D_{i}\ \ \ i=1,2,\ldots,u \tag{2}\]3 \[\sum_{a=1}^{l}A_{ai}\sum_{c=1}^{n}X_{ac}+\sum_{b=1}^{m}B_{bi}\sum_{d=1}^{r}X_{ bd}\geq D_{i}\ \ \ i=1,2,\ldots,u \tag{2}\]4 (14) \[\sum_{a=1}^{l}A_{ai}\sum_{c=1}^{n}X_{ac}+\sum_{b=1}^{m}B_{bi}\sum_{d=1}^{r}X_{ bd}\geq D_{i}\ \ \ i=1,2,\ldots,u \tag{2}\]5 (15) \[\sum_{a=1}^{l}A_{ai}\sum_{c=1}^{n}X_{ac}+\sum_{b=1}^{m}B_{bi}\sum_{d=1}^{r}X_{ bd}\geq D_{i}\ \ \ i=1,2,\ldots,u \tag{2}\]6 (16) \[\sum_{a=1}^{l}A_{ai}\sum_{c=1}^{n}X_{ac}+\sum_{b=1}^{m}B_{bi}\sum_{d=1}^{r}X_{ bd}\geq D_{i}\ \ \ i=1,2,\ldots,u \tag{2}\]7 (17) \[\sum_{a=1}^{l}A_{ai}\sum_{c=1}^{n}X_{ac}+\sum_{b=1}^{m}B_{bi}\sum_{d=1}^{r}X_{ bd}\geq D_{i}\ \ \ i=1,2,\ldots,u \tag{2}\]8  
The trim loss comprises of both reusable and unusable reels. All those reels that cannot be sold are transferred for recycling while those greater than a certain width are considered as reusable reels for the next orders. The benefits of the proposed model for paper mill industry are: it takes less computational time for its solution due to linearity, it is used to solve the trim loss minimization problem for flexible widths of master reels, it is used to consider the cutting of stocked over-produced and leftover reels to fill an order, it considers the limitation of cutting blades at winder and it is used to minimize the total cost in terms of master reels and stocked reels. The constraints of the model are used to meet the demand and provide limited stocked reels to customers. As the model minimizes trim loss, thus the reprocessing of paper trimming is minimized and as a result, the discharge of chlorinated compounds and other toxic organics to the environment are minimized. Additionally, the model finds a better solution and proclaims considerable economical and operational benefits to the industry.