A two-dimensional cutting stock problem can be defined as follows: A set of rectangular stock sheets of p different types is available. For each type of sheet Sp we know its length Lp and width Wp. From these sheets we have to cut smaller rectangular pieces of length li and width wi, i = 1, ...,m in order to satisfy a given demand for di pieces of each type. The objective is to minimize the tot...

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 paper provides an introduction to one-dimensional cutting stock problems and solution procedures. The first problem considered requires that both trim loss and pattern changes be controlled. Both linear programming and sequential heuristic procedures are discussed along with the ways they can be used jointly to generate the best possible solutions to this type of problem. Two other importa...

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, the authors present a hybrid genetic approach for the two-dimensional rectangular guillotine oriented cutting-stock problem. In this method, the genetic algorithm is used to select a set of cutting patterns while the linear programming model permits one to create the lengths to produce with each cutting pattern to fulfil the customer orders with minimal production cost. The effec...