# A Mathematical Model for Reduction of Trim Loss in Cutting Reels at a Make-to-Order Paper Mill
## 5 Results and Discussion  
The marketing department of the company provided the data used in the computational results and it corresponds to real problems solved at the paper mill as discussed earlier. The algorithm was implemented using both FORTRAN and MATLAB programming languages on a computer and the results were obtained with a Core i7 and 1.99 GHz in reasonable computational time. The paper mill managers contacted our institute to improve their system. The issues which the mill managers were facing were: spending more time on calculating the feasible pattern generation, finding a width of master reels that can result in minimum trim loss, decision about provision of stocked reels to fill an order, etc. It was a very interesting study, and also it was a convenient sample as the industry was close to our institute and it was easy to collect data and share findings. It was also easy for us to trace the results of implementation. For the purpose of comparative analysis and to validate the solution procedure adopted, two cases were considered. Case 1 was master reels with adjustable widths and Case 2 was master reels with fixed widths. The number of patterns of master reels decreased when more stocked reels were selected to meet demand and vice versa. Therefore, for longer widths of master reels in Case 2, the number of patterns of master reels remained equal to that of Case 1, but for shorter widths, the number decreased gradually as represented in Figure 2(a). The same was anticipated for stocked reels in both cases, but the number increased gently for shorter widths of master reels in Case 2, as can be observed in Figure 2(b). With greater width of the master reel, a lesser number of master reels were used to fill an order and vice versa. Hence, in Case 1, reels of different widths of master reels were selected for cutting, which resulted in slightly higher number of master reels used than only few master reels of fixed width, while in Case 2, the number of master reels selected for cutting increased with decrease in the width of master reels as shown in Figure 2(c). In cutting stocked reels to meet demand, initially the number of reels selected for cutting remained same in both cases, but then increased with decrease in the widths of master reels in Case 2, as can be seen in Figure 2(d). Increasing the area of master reels for cutting resulted in a decrease in the area of stocked reels and conversely decreasing the area of master reels resulted in an increase in the area of stocked reels. Thus, there is an inverse relationship between master reels and stocked reels in Case 2 and the total width of master reels selected for cutting in Case 1 was higher than some results of Case 2 as one can see in Figure 2(e), while the total width of stocked reels remained same in both cases but then diverged for smaller widths of Case 2 as compared in Figure 2(f). As already known, a solution is considered best when both master reels and stocked reels are cut to meet demand with no or minimum trim loss. Here, the proposed model resulted in no trim loss in cutting master reels and stocked reels. As depicted in Figure 2(g), in Case 1, the trim loss in cutting master reels and stocked reels was zero while in Case 2, the loss was much higher than Case 1 as seen in Figure 2(h). The optimal solution in Case 1 produced few over-produced reels and no trim loss. Therefore, the value of the objective function in Case 1 was less than all other values of objective function in Case 2 as presented in Figure 2(i). Figure 3: _Cont_. In Case 2, there are trim losses in cutting master reels and stocked reels and as trimming requires an additional knife, thus the number of required knives at winder in Case 1 was less than Case 2 as given in Figure 3j. The paper mill has validated the proposed model and is in use currently. The method finds a better solution and considerable economical, operational, and environmental benefits are proclaimed. The model significantly reduces trim loss, thereby improves productivity.