Manufacturing Applications of the One-dimensional Cutting Stock Problem as a Team Project

This paper explains the beneficial and practical impact of operations research in two real manufacturing settings. Two manufacturing examples used in student projects were (1) cutting rails (80‘ or 40‘) to manufacture railroad frogs of many sizes and (2) cutting round metal rolls (12‘ to 20‘) to meet customer demands for various lengths of cuts. Student teams in Engineering of Manufacturing Processes and Operations Research courses wrote computer programs. The program first identified all possible patterns that can be cut out of a given stock length. Next, the program created a mathematical model (a text file) as an output. This text file was used as an input for the optimizer software LINGO. When compared to the manual solutions obtained by foremen in two settings, student teams with no prior experience were able to match the manual solution of the foremen in small problems and improve the manual solution by up to 30 % in large problems. After finishing the project, each team wrote a technical team report to document the experience they gained in manufacturing and mathematical modeling. Student assessment was based on student team reports (knowledge gained) and individual team interviews, exit surveys, and the end of semester course evaluations (students’ attitudes). The project outcomes include improved understanding of production-related concepts such as remnant minimization in manufacturing, as well as enthusiasm for operations research and its applications in manufacturing. Introduction Kolb’s experiential learning cycle/spiral [1 3] is often used as a powerfull metacognitive method in many engineering programs. Namely, a learner gains knowledge by answering four questions (Why?, What?, How? and What if?) in succession. A set of activities is associated with each question. This cycle of questions and activities is repeated for deeper learning regardless of the prefered learning style (type) of the learner. Laboratory experiments and other experiential learning activities [4-6] are well recognized parts of Kolb’s learning cycle. Creating products is the primary function of any manufacturing establishment. Product realizationbased learning seems to be a natural model for learning manufacturing engineering [7]. The product realization-based learning can be understood as a part of the project-based learning (PBL) pedagogy which is well accepted in education [8, 9]. PBL is also emphasized as one of the priority educational methods in manufacturing engineering [10] and industrial engineering education [11]. PBL pedagogy is already successfully implemented in some manufacturing processes courses [12, 13]. Students’ experiences described here are based on product-realization learning concepts. In addition, some additional PBL pedagogy strategies and teamwork are implemented. In operations research (OR), the one-dimensional cutting stock [20-29] problem describes the case of cutting standard length stock material into various specified sizes while minimizing the material wasted. Unusable length is called remnant or drop in manufacturing that involves metal works. Solution to this computationally complex optimization problem can be used in many manufacturing applications. To solve it, the problem can be formulated as an integer linear model first, and then solved using a common optimizer software. Since the problem is known to have multiple optimal solutions in some cases, binary variables can be added to identify all optimal solutions, but this is often not necessary. U.S. Customary units are used throughout this paper because both manufacturers exemplified in this work operate using the U.S. Customary units in almost all aspects of their daily operations. All machinists and majority of the management in these facilities are not only unfamiliar with the metric system, but are also outright against it. Curricular Context Applications of operations research in manufacturing can be implemented in either operations research or manufacturing courses. In our case, there are two courses that can benefit from this work: MAE 495 which is an elective senior level course for mechanical engineering majors [14] and EN 471, Operations Research, which is a regular one-semester, three-credits, junior-level course in an industrial engineering program [15]. The Manufacturing Processes course for senior mechanical engineering students was taught from industrial engineering and operations research perspective, similarly to the Operations Research course in an industrial engineering program. Concepts of manufacturing economics and optimization were emphasized. Optimization examples included one-dimensional cutting-stock problem as a project topic. The described experience deals with about 80 students per semester, where students work in teams of three to four students per team. From the four pillars of manufacturing engineering (a. materials and manufacturing processes, b. product, tooling, and assembly engineering, c. manufacturing systems and operations, and d. manufacturing competitiveness [16],”) this work addresses two of them (c. and d.). Educational Goals, Activities, and Outcomes Educational goals of this project include improved understanding of production-related concepts such as remnant minimization in manufacturing, as well as increased enthusiasm for operations research and its applications in manufacturing. Students’ experience consists of several activities: observation of real metal cutting operations, realizing costs due to final length of stock material, programming and running OR-based calculations to minimize remnants, and succesfully competing with experienced foremen’s manual solutions. Several learning outcomes originate from project educational goals and project activities like increased appreciation for manufacturing in general, increased appreciation for OR by experiencing a “real life” manufacturing problem, an understanding of the role of analytical approaches to engineering problem solving, development of written communication skills through writing of technical reports, and development of teamwork skills. These outcomes are closely related to ABET-EAC Criterion 3, a-k student learning outcomes [17], specifically outcome a an ability to apply knowledge of mathematics, science and engineering, outcome g an ability to communicate effectively, and outcome k an ability to use the techniques, skills, and modern engineering tools necessary for engineering practice. Practical Experience This paper reports student experience with one-dimensional cutting stock problem using two very different manufacturing applications. The Facility 1 cuts different sizes of parts from a rail stock to manufacture rail frogs. The Facility 2 cuts different sizes and cross sections of cylindrical parts to manufacture different products. While students did not get an actual metal cutting experience in this project, they observed cutting operations at both facilities. The Mathematical Model Figure 1 shows the simplest version of the classic one-dimensional cutting stock model using m cuts and n patterns. Input variable Di represents the known demand for each cut size I (1 to m). Input matrix aij (m x n) represents number of cut size of type i that can be obtained from pattern j. Output variable Xj represents the number of stocks that should be cut according to pattern j (1 to n).