Mathematical Methods of Operations Research

Ackooij, Wim

doi: 10.1007/s00186-014-0478-5pmid: N/A

The unit commitment problem, aims at computing the production schedule that satisfies the offer-demand equilibrium at minimal cost. Often such problems are considered in a deterministic framework. However uncertainty is present and non-negligible. Robustness of the production schedule is therefore a key question. In this paper, we will investigate this robustness when hydro valleys are made robust against uncertainty on inflows and the global schedule is robust against uncertainty on customer load. Both robustness requirements will be modelled by using bilateral joint chance constraints. Since this is a fairly large model, we will investigate several decomposition procedures and compare these on several typical numerical instances. The latter decomposition procedures are clearly a prerequisite if robust unit commitment is ever to be used in practice. We will show that an efficient decomposition procedure exists and can be used to derive a robust production schedule. The obtained results are illustrated on a convex simplification of a unit commitment problem in order to avoid the use of heuristics. The investigated decomposition approaches can be applied trivially to a non-convex setting, but will need to be followed by appropriate heuristics. How this may work in practice is also illustrated.

Garnaev, Andrey; Kikuta, Kensaku

doi: 10.1007/s00186-014-0479-4pmid: N/A

In this paper we suggest a new class of search games, namely, screening and hiding versus search games. This new class of search games is motivated by the book Search and Screening by Koopman who is the founder of the modern search theory. In our game a hider is not just trying to find the best places to hide resources from a searcher, but also it applies efforts to screen their allocations making it more difficult for the searcher to find them. This hiding and screening strategy of the hider versus search strategy of the searcher makes the plot more real-world relative comparing to widely investigated in literature just hiding versus search games. For the suggested game we found the equilibrium strategies of both players as well as we proved uniqueness of the equilibrium.

Oliveira, Paulo

doi: 10.1007/s00186-014-0480-ypmid: N/A

A strongly polynomial-time algorithm is proposed for the strict homogeneous linear-inequality feasibility problem in the positive orthant, that is, to obtain $$x\in \mathbb {R}^n$$ x ∈ R n , such that $$Ax > 0$$ A x > 0 , $$x> 0$$ x > 0 , for an $$m\times n$$ m × n matrix $$A$$ A , $$m\ge n$$ m ≥ n . This algorithm requires $$O(p)$$ O ( p ) iterations and $$O(m^2(n+p))$$ O ( m 2 ( n + p ) ) arithmetical operations to ensure that the distance between the solution and the iteration is $$10^{-p}$$ 10 - p . No matrix inversion is needed. An extension to the non-homogeneous linear feasibility problem is presented.

Lohmann, Edwin; Borm, Peter; Slikker, Marco

doi: 10.1007/s00186-014-0481-xpmid: N/A

In this paper sequencing situations with Just-in-Time (JiT) arrival are introduced. This new type of one-machine sequencing situations assumes that a job is available to be handled by the machine as soon as its predecessor is finished. A basic predecessor dependent set-up time is incorporated in the model. Sequencing situations with JiT arrival are first analyzed from an operations research perspective: for a subclass an algorithm is provided to obtain an optimal order. Secondly, we analyze the allocation problem of the minimal joint cost from a game theoretic perspective. A corresponding sequencing game is defined followed by an analysis of a context-specific rule that leads to core elements of this game.

Arin, J.; Katsev, I.

doi: 10.1007/s00186-014-0482-9pmid: N/A

We introduce and characterize a new solution concept for TU games: The Surplus Distributor Prenucleolus. The new solution is a lexicographic value although it is not a weighted prenucleolus. The SD-prenucleolus satisfies core stability, strong aggregate monotonicity, null player out property in the class of balanced games and coalitional monotonicity in the class of monotonic games with veto players. We characterize the solution in terms of balanced collections of sets and we provide a simple formula for computing it in the class of monotonic games with veto players.

Göhl, Markus; Borgwardt, Karl

doi: 10.1007/s00186-014-0483-8pmid: N/A

This paper deals with the average-case-analysis of the number of pivot steps required by the simplex method. It generalizes results of Borgwardt (who worked under the assumpution of the rotation-symmetry-model) for the shadow-vertex-algorithm to so-called cylindric distributions. Simultaneously it allows to analyze an extended dimension-by-dimension-algorithm, which solves linear programing problems with arbitrary capacity bounds $$b$$ b in the restrictions $$Ax\le b$$ A x ≤ b , whereas the model used by Borgwardt required strictly positive right hand sides $$b$$ b . These extensions are achieved by solving a problem of stochastic geometry closely related to famous results of Renyi and Sulanke, namely: assume that $$a_1,\ldots ,a_m$$ a 1 , … , a m are uniformly distributed in a cylinder. How many facets of $${{\mathrm{conv}}}(a_1,\ldots ,a_m,0)$$ conv ( a 1 , … , a m , 0 ) will be intersected by a two-dimensional shadow plane along the axis of the cylinder? The consequence of these investigations is that the upper bounds of Borgwardt (under his original model) still apply when we accept distributions with arbitrary right hand sides.