# A Mathematical Model for Reduction of Trim Loss in Cutting Reels at a Make-to-Order Paper Mill
## 2 Literature Review
[32] presented two bi-level mathematical programming models to represent non-exact two-stage, two-dimensional residual bin-packing problem. [PERSON] and [PERSON] [33] proposed a method to control stock and to calculate the near optimal amount of usable leftover. Instead of randomly generated, usable leftovers were used from previous orders in the next order and trim loss in sequential orders was reduced and was explained with the help of computational outcomes. [PERSON] et al. [34] proposed diving heuristic based on column generation and used dynamic programming to solve pricing problem. Most recently, [PERSON] et al. [35] proposed a mathematical model which utilized column generation method and integer solution finding strategies to solve one-dimensional cutting stock problem with useable leftover. One of the most important industrial sectors in the world is the paper industry [36]. This industry is placed third after the chemical and metal industries in terms of water consumption [37]. Due to high consumption of water, the paper industry results in high generation of wastewater [36]. Generally, a paper mill produces effluents of volume ranging between 1.5 to 60 m\({}^{3}\) per ton of paper produced [37]. Additionally, when 1 m\({}^{3}\) of wastewater is discharged from a paper mill, it contains 0.9 m\({}^{3}\) of potentially reusable water and generates a reusable sludge of mass 0.7 kg [38]. The treatment of effluents consequently produces a large amount of sludge ranging between 40 to 50 kg of dry sludge per ton of paper produced, of which 30% is biological sludge and 70% is primary sludge [36; 39]. Usually the primary and secondary sludge originates from sedimentation and biological treatment, respectively [40]. Due to recent increasing restrictive environmental legislation, different challenges regarding management of waste have been faced by the paper industry. Without adequate treatment, industrial wastewater has an adverse effect on the aquatic environment. Land applications is one of the most commonly practiced methods for sludge management [41]. However, unplanned landfilling is also commonly practiced within paper mills which leads to negative outcomes regarding air, water, and soil [42]. Nowadays, people are facing a serious problem of environmental pollution and one of the challenges is its protection. The paper industry is one of the important industrial sectors but it causes an adverse impact on the environment as it discharges wastewater containing toxic pollutants [43; 44]. [PERSON] et al. [45] presented the data regarding waste generation from few European paper mills and this is shown in Table 1.  
In a paper mill, waste paper is produced during the cutting process of reels. The waste paper is reprocessed to make new reels. During reprocessing, chlorine and chlorine dioxide are used for bleaching of pulp and chlorinated organic substances are discharged to the environment [46; 47]. Cutting optimization minimizes the production of waste paper and less paper is reprocessed. As a result, minimum effluents are discharged to the environment from the paper mill. In this paper, we present a new mathematical model for reduction of trim loss in cutting reels at a make-to-order paper mill. The main differences from other models proposed in the literature are the consideration of cutting master reels of flexible widths and stocked over-produced and leftovers reels to fill a customer demand. Master reels are normally produced of standard size, which on cutting into smaller reels to fill an order results in trim losses while the production of master reels of flexible widths mostly results in no trim loss and hence no reprocessing of paper. The cutting of master reels into smaller reels results sometime in over-produced reels and leftover reels, which are stocked to meet future demands. Our mathematical model considers these over-produced reels and leftover reels in stock to meet a demand with minimum or no trim loss.