This paper presents an approach for solving a new real problem in Cutting and Packing. At its core is an innovative mixed integer programme model that places irregular pieces and defines guillotine cuts. The two-dimensional irregular shape bin packing problem with guillotine constraints arises in the glass cutting industry, for example, the cutting of glass for conservatories. Almost all cuttin...

This paper addresses a stock-cutting problem with rotation of items and without the guillotine cutting constraint. In order to solve the large-scale problem effectively and efficiently, we propose a simple but fast heuristic algorithm. It is shown that this heuristic outperforms the latest published algorithms for large-scale problem instances. Keywords—Combinatorial optimization, heuristic, la...

This study proposes a deterministic model to solve the two-dimensional cutting stock problem (2DCSP) using a much smaller number of binary variables and thereby reducing the complexity of 2DCSP. Expressing a 2DCSP with m stocks and n cutting rectangles requires 2n2 + n(m + 1) binary variables in the traditional model. In contrast, the proposed model uses n2 + n log2 m binary variables to expres...

This paper discusses cake-cutting protocols when the cake is a heterogeneous good, represented by an interval on the real line. We propose a new desirable property, the meta-envy-freeness of cake-cutting, which has not been formally considered before. Meta-envy-free means there is no envy on role assignments, that is, no party wants to exchange his/her role in the protocol with the one of any o...

This paper discusses cake-cutting protocols when the cake is a heterogeneous good that is represented by an interval in the real line. We propose a new desirable property, the meta-envy-freeness of cake-cutting, which has not been formally considered before. Though envy-freeness was considered to be one of the most important desirable properties, envy-freeness does not prevent envy about role a...

We propose an online form of the cake cutting problem. This models situations where players arrive and depart during the process of dividing a resource. We show that well known fair division procedures like cut-and-choose and the Dubins-Spanier moving knife procedure can be adapted to apply to such online problems. We propose some desirable properties that online cake cutting procedures might p...

Let us imagine that red candles and blue candles are placed on the top of a cake. We want to cut it with a knife into two pieces in such a way that all the red candles are on one of the two pieces and all the blue candles are on the other piece. A question is when we can cut it successfully. The following Kirchberger’s theorem [5] answers this question in a general setting. Let X be a finite su...

In this paper, we address the combined assortment and trim loss minimization problem. In real settings, when different stock lengths are available, a common problem is to select a subset of stock lengths from which to cut the ordered items. In the cutting and packing literature, this problem is known as the assortment problem. Solving the assortment and trim loss minimization problem in a singl...