# A Mathematical Model for Reduction of Trim Loss in Cutting Reels at a Make-to-Order Paper Mill
###### Abstract  
One of the main issues in a paper mill is the minimization of trim loss when cutting master reels and stocked reels into reels of smaller required widths. The losses produced in trimming at a paper mill are reprocessed by using different chemicals which contributes to significant discharge of effluent to surface water and causes environmental damage. This paper presents a real-world industrial problem of production planning and cutting optimization of reels at a paper mill and differs from other cutting stock problems by considering production and cutting of master reels of flexible widths and cutting already stocked over-produced and useable leftover reels of smaller widths. The cutting process of reels is performed with a limited number of cutting knives at the winder. The problem is formulated as a linear programming model where the generation of all feasible cutting patterns determines the columns of the constraint matrix. The model is solved optimally using simplex algorithm with the objective of trim loss minimization while satisfying a set of constraints. The solution obtained is rounded in a post-optimization procedure in order to satisfy integer constraints. When tested on data from the paper mill, the results of the proposed model showed a significant reduction in trim loss and outperformed traditional exact approaches. The cutting optimization resulted in minimum losses in paper trimming and a lesser amount of paper is reprocessed to make new reels which reduced the discharge of effluent to the environment. process optimization for waste reduction; combinatorial optimization; integer programming; waste generation 2520201052740.3390/app10155274.390/app10155274. problem is the number of times a feasible pattern associated with master reels, over-produced reels, and useable leftovers are cut to satisfy ordered quantities while considering minimization of waste. The involvement of various operational and technological constraints in the planning process causes the trim loss minimization problem more interesting and difficult. As requested by a paper mill at Peshawar, a system has been developed and described in this paper for the support of its production planning which focuses on the production of master reels and cutting master reels, over-produced reels and leftovers to fill an order. Being part of COOL, which is a broader system, this work intends to support the application of an optimizing plan for the production of paper and stock management. New mathematical models and approaches are developed to solve the constraints that arise in practical situations at a paper mill. Though there are various types of cutting stock problems, a problem which has not been frequently studied and can be included in the broad family of cutting stock problem, consists of considering master reels of flexible widths, over-produced reels of smaller widths and useable leftovers when objects are cut to satisfy demands. In cutting master reels, over-produced reels and leftovers are stocked for future use and are not wasted, therefore, the problem is called cutting stock and useable leftover problem. A one-dimensional approach has been devised and trim loss at the end of the reels is disregarded in the considered problem formulation. The same approach can also be applied in other industries as well, such as steel [1; 2] wood, leather, and plastic industry [3]. A limited number of cutting knives at winder cut the reels into smaller reels of required widths. All feasible patterns are generated and the knives are adjusted in such a way that trim loss is reduced. Knife setting can benefit in reducing pollution load by minimizing trim loss and reducing reprocessing of paper trimming. In the reprocessing and recycling of paper, a harsh chemical called bleach is used for the purpose of paper brightness, which is harmful to both environment and health.