# A Mathematical Model for Reduction of Trim Loss in Cutting Reels at a Make-to-Order Paper Mill
## 2 Literature Review  
Since there is very rich literature on trim loss minimization problems, therefore, studies that are most relevant to our current work have been reviewed in this section. In literature, various types of cutting stock problems have been investigated [7; 8; 9; 10]. Effective algorithms for solving cutting stock problems are beneficial to reduce production cost and improve material utilization [11]. [PERSON] et al. [12] presented reel and sheet cutting at a paper mill. To satisfy customer orders, the reels were first cut into smaller widths which were then cut into finished reels or sheets. The proposed solution procedure comprises of three stages. At the first stage, all feasible cutting patterns were enumerated. The second stage solved the linear programming model and a rounding procedure was used in the third stage for obtaining integer solution. [PERSON] et al. [13] discussed the roll assortment optimization in a paper mill. The production of large reels on cyclical basis was considered and the reels produced were then cut into smaller size. The problem was formulated as a binary nonlinear programming model and two solution methods were presented. The first method was column generation and fast pricing heuristic based on branch-and-price algorithm while the second was marginal cost heuristic. [PERSON] et al. [14] considered a two-stage extension of one-dimensional cutting stock problem and an exact simultaneous column-and-row generation algorithm was presented for its solution. The pricing sub-problem was a knapsack problem and three algorithms were proposed for the problem including branch-and-price. The performances of the three algorithms were compared by conducting extensive computational experiments which revealed that a hybrid algorithm which integrated the other two algorithms was performing best out of all the three algorithms. For a bi-objective one-dimensional trim loss problem, [PERSON] et al. [15] presented a new model and an efficient solution algorithm. The two objectives of the model were: the minimization of trim loss and the minimization of the number of partially-cut large objects. A two-stage least-loss algorithm was used to determine the patterns which minimized the trim loss and near-optimum solution was produced. Computational tests confirmed the efficiency and near-optimality of the proposed algorithm. In the context of furniture production, [PERSON] et al. [16] studied the integrated lot sizing and cutting stock problem. A mathematical model was proposed and tested that captured the production process of the selected factory. A column-generation solution method was proposed to solve the problem and an intensive computational study was conducted using real data. It has been shown that the integrated approach performed well both in terms of reduction in total cost of raw as well as inventory costs of small pieces. [PERSON] et al. [17] presented an integer linear programming model for the scheduling and one-dimensional cutting stock problem with heterogeneous orders. A novel metaheuristic algorithm was proposed based on a fix-and-optimize strategy hybridized with a random local search. The performance of the proposed metaheuristic was compared with IBM CPLEX by carrying out extensive computational experiments which showed that the proposed metaheuristic performed better than CPLEX in larger instances while in small instances, CPLEX showed a marginal advantage. In steel industries, [PERSON] et al. [18] studied the one-dimensional cutting stock problem with divisible items. The main objectives were to minimize both the number of welds used to recombine small pieces and the trim loss. A nonlinear integer formulation was first developed for the NP-hard problem and to fulfill the needs of the company, a heuristic based on dynamic programming was also designed. The computational experiments revealed that the algorithm performed well on real life instances.