Chore division, introduced by Gardner in 1970s [10], is the problem of fairly dividing a chore among n different agents. In particular, in an envy-free chore division, we would like to divide a negatively valued heterogeneous object among a number of agents who have different valuations for different parts of the object, such that no agent envies another agent. It is the dual variant of the cel...

Utilization of metalworking fluids in the technological process of metalworking brings its advantages and disadvantages, mainly associated with the contamination of the working air through cutting fluid mist. The primary aim of this article is to approximate the mechanism of aerosols generation which contaminates the working air and to draw the attention to the problem of quantification of gene...

In the classical Gilmore-Gomory model for the one-dimensional cutting stock problem (1D-CSP) we have to deal implicitly with a huge number of variables representing all feasible patterns. The advantage is a strong relaxation and absence of symmetries. To reduce the number of variables, we can restrict them to subpatterns, i.e., partial patterns which are combined to produce whole patterns. Each...

In this paper the polyhedron of the cutting stock problem is investigated with respect to facet-de ning inequalities. For some classes of valid inequalities this property is proved.

We give a unified geometric approach to some theorems about primitive elements and palindromes in free groups of rank 2. The geometric treatment gives new proofs of the theorems.

The present paper reports an intelligent computer-aided nesting (CAN) system for optimal nesting of two-dimensional parts, especially parts with complicated shapes, with the objective of effectively improving the utilization ratio of sheet materials. This paper also systemically reviews the nesting algorithms that were developed to perform various nesting tasks, and attacks the irregular part n...

We investigate the two-stage guillotine two-dimensional cutting stock problem. This problem commonly arises in the industry when small rectangular items need to be cut out of large stock sheets. We propose an integer programming formulation that extends the well-known Gilmore and Gomory model by explicitly considering solutions that are obtained by both slitting some stock sheets down their wid...

We propose an online form of the cake cutting problem. This models situations where agents 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 fairness properties that online cake cutting procedures can posse...

We present a variable neighborhood search (VNS) for the 3-staged 2-dimensional cutting stock problem employing “ruin-and-recreate”-based very large neighborhood search in which parts of the incumbent solution are destroyed and rebuilt using construction heuristics and dynamic programming. Experimental results show that for instances where the sizes of the elements are not too small compared to ...

This paper looks at different rules in dissolving a common value partnership when one partner holds proprietary information. In winner’s bid auction (WBA) and loser’s bid auction (LBA), there exists unique mixed strategy equilibrium. “Payoff equivalence” is established in the sense that partners’ expected payoffs are the same under the two auction formats. The informed partner benefits from an ...