IIE Transactions (2013) 45, 1332–1344 Copyright C  “IIE” ISSN: 0740-817X print / 1545-8830 online DOI: 10.1080/0740817X.2012.725506 A two-dimensional bin packing problem with size changeable items for the production of wind turbine flanges in the open die forging industry JONGSUNG LEE 1 , BYUNG-IN KIM 1,∗ and ANDREW L. JOHNSON 2 1 Department of Industrial & Management Engineering, Pohang University of Science and Technology (POSTECH), Pohang, Kyungbuk 790-784, Republic of Korea E-mail: bkim@postech.ac.kr 2 Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX 77843, USA Received December 2011 and accepted August 2012 Efficient cutting design is essential to reduce the costs of production in the open die forging industry. This article discusses a slab cutting design problem that occurs when parallel piped items are cut from raw material steel slabs with varying widths and lengths to meet a volume requirement. The problem is modeled as a two-dimensional cutting stock problem or bin packing problem with size-changeable items. Cut loss and guillotine cut constraints are included. A knapsack-based heuristic algorithm is proposed and it is tested by a real-world manufacturer who is cutting steel for wind turbine flanges. The firm generates an annual cost reduction of approximately US $2000 000. Keywords: Bin packing problem, size changeable items, knapsack, open die forging 1. Introduction In 2010, the custom open die forging industry in North America recorded over 1.6 billion dollars in sales (Rothaer- mel, 2011). Open die forging consists of pressing a piece of metal between flat or simple curved dies to form a pre- determined shape and characteristics. The entire process requires seven steps: cutting, heating, forging, heat treat- ment, machining, inspection, and packing. This research is motivated by a real-world wind turbine flange manufac- turer seeking to reduce costs by producing less scrap in the cutting step. The company uses three types of raw mate- rials, steel slab, steel bloom, and round bar, based on the component requirements and the raw material character- istics. When the raw material is a bloom or a round bar, it can be cut to specific lengths, meaning that the cutting design problem described in this article could be consid- ered a typical one-dimensional cutting stock problem or bin packing problem. When the raw material is a steel slab, it can be cut in lengths and widths that vary within a size range while maintaining a specified volume. The exact di- mensions of the cut item are not critical since the item can ∗ Corresponding author be reshaped in the forging step, but the size range is limited by the maximum permissible length of the forging machine used and the given maximum ratio between the length and the width. For example, if a given area of an item is 10 m 2 (assume that the thickness of a slab is fixed), and the given maximum ratio is 1.2, then the lengths can range between 2.89 (= sqrt(10/1.2)) m and 3.46 (= 2.89 × 1.2) m. Numer- ous width, height combinations are possible, such as (2.89, 3.46), (3.00, 3.33), (3.10, 3.23), (3.23, 3.10), (3.33, 3.00), and (3.46, 2.89). To improve productivity and reduce costs, the industry prefers guillotine cuts; i.e., a cutting machine or gas flame that cuts raw materials from edge to edge. Thus, the prob- lem becomes a two-dimensional guillotine cut bin packing problem in which items’ widths and lengths vary within a specific range. We limit our discussion to a steel slab since it is the most prevalent raw material, while noting that our proposed approach can also handle cutting design prob- lems for blooms and round bars. Prior to deploying our proposed algorithm, the manu- facturer assigned a single engineer who selected one slab at a time and then manually generated packing patterns by trial and error using only a simple piece of software to aid with calculations. Each slab necessitated repeated trial and error in order to attain a design with satisfactory levels of scrap. Typically, 10 hours per day were needed to develop 0740-817X C  2013 “IIE” Downloaded by [86.50.136.35] at 05:12 22 August 2013