The 1-dimensional Cutting Stock Problem (1-CSP) seeks to fulfill with a minimum trim loss the demand of parts of various lengths by cutting them out from a given stock size. In the Pattern Minimization Problem (1-PMP) one wants an optimal 1-CSP solution that uses a minimum number of distinct cutting patterns. We discuss two formulations of 1-PMP: (1) introduced by Vanderbeck, and (2) obtained b...

We introduce a generalized cake-cutting problem in which we seek to divide multiple cakes so that two players may get their most-preferred piece selections: a choice of one piece from each cake, allowing for the possibility of linked preferences over the cakes. For two players, we show that disjoint envy-free piece selections may not exist for two cakes cut into two pieces each, and they may no...

Cake cutting is a fundamental model in fair division; it represents the problem of fairly allocating a heterogeneous divisible good among agents with different preferences. The central criteria of fairness are proportionality and envy-freeness, and many of the existing protocols are designed to guarantee proportional or envy-free allocations, when the participating agents follow the protocol. H...

The cake-cutting problem refers to the issue of dividing a cake into pieces and distributing them to players who have different value measures related to the cake, and who feel that their portions should be “fair.” The fairness criterion specifies that in situations where n is the number of players, each player should receive his/her portion with at least 1/n of the cake value in his/her measur...

This paper proposes a cake-cutting protocol using cryptography when the cake is a heterogeneous good that is represented by an interval on a real line. Although the Dubins-Spanier moving-knife protocol with one knife achieves simple fairness and truthfulness, all players must execute the protocol synchronously. Thus, the protocol cannot be executed on asynchronous networks such as the Internet....

This paper considers the one-dimensional cutting stock problem with divisible items, which is a new problem in the cutting stock literature. The problem exists in steel industries. In the new problem, each item can be divided into smaller pieces, then they can be recombined again by welding. The objective is to minimize both the trim loss and the number of the welds. We present a mathematical m...

We propose a formulation of the stochastic cutting stock problem as discounted infinite-horizon Markov decision process. At each epoch, given current inventory items, an agent chooses in which patterns to cut objects anticipation unknown demand. An optimal solution corresponds policy that associates state with and minimizes expected total cost. Since exact algorithms scale exponentially state-s...

The constrained 2-D cutting stock problem is an irregular problem with dynamic data structures, highly variable amounts of computation per task, and unpredictable amounts and patterns of communication. This paper describes the design and implementation of a parallel solution to this problem on a cluster of workstations and a distributed memory multicomputer. The key element of our parallel solu...

We address a 1-dimensional cutting stock problem where, in addition to trimloss minimization, we require that the set of cutting patterns forming the solution can be sequenced so that the number of stacks of parts maintained open throughout the process never exceeds a given s. For this problem, we propose a new integer linear programming formulation whose constraints grow quadratically with the...