Set-packing problem

Reverse auction. Combinatorial auctions are also proposed for procurement problems in markets with one buyer and multiple sellers. The reverse combinatorial auction is formulated as a set covering problem rather than a set packing problem. An interesting (and complicating) issue that arises in this setting is that there are various business rules that are used to constrain the choice of winners. These business rules appear as side constraints in the winner determination problem. The winner determination problem with no side constraints can be written as ... 

The first constraint ensures that overlapping sets of goods are never assigned. The second ensures that no bidder receives more than one subset. Call this formulation CAPl. Problem CAP as formulated here is an instance of what is known as the set-packing problem. ... 

It is natural to think that since CAP is an instance of the set packing problem it must be NP-hard. One must be careful in drawing such a conclusion, since it depends on how the input to an instance of CAP is encoded. Suppose one takes the number of bids as a measure of the size of the input and this number is exponential in M. Any algorithm for CAP that is polynomial in the number of bids but exponential in the number of items would, formally, be polynomial in the input size but impractical for large M In particular, an instance of CAP in which bids were submitted on every possible combination of items... 

In any case, the upshot is that effective solution procedures for the CAP must rely on two things. The first is that the number of distinct bids is not large. The second is that the underlying set packing problem can be solved reasonably quickly. For a list of solvable instances of the set packing problem and their relevance to CAP see de Vries and Vohra [26]. 

For this reason the extensive literature on the set packing problem and its relatives is relevant for computational purposes. 

It is found that the viscosity of a paste made from a fixed polymer/plasticiser ratio depends to a great extent on the particle size and size distribution. In essence, in order to obtain a low-viscosity paste, the less the amount of plasticiser required to fill the voids between particles the better. Any additional plasticiser present is then available to act as a lubricant for the particles, facilitating their general mobility in suspension. Thus in general a paste polymer in which the pastes have a wide particle size distribution (but within the limit set by problems of plasticiser absorption and settling out, so that particles pack efficiently, will... 

Preliminaries for a Set of Problems Dealing with a Single Packed Bed Reactor... 

Another set of problems was studied in connection with the PEIs 179,180 [136]. It was found that all these PEIs do not show an LC-phase,but form a smectic crystalline solid state with upright mesogens (smectic-E like). However, the PEIs can in principle adopt two different kinds of chain packing and layer structures as illustrated in Fig. 29. The first is that the mesogens pack in an antipar-... 

This is a set packing formulation and is NP-hard [89]. There are special cases under which the structure of this problem simplifies and allows for polynomial time solutions. Many special cases arise out of constraints that reduce the constraint matrix to be totally unimodular [35]. A common example is the case where adjacent plots of land are being sold and bidders might want multiple plots but they need to be adjacent. However, real world problems will often not satisfy the fairly severe restrictions that provide a totally unimodular constraint matrix. Moreover, if the bidding language is not expressive then this can interact with the incentive properties of an auction because a bidder is not able to express her true valuation, even if that would be her equilibrium strategy. We wait until Section 4 for an extensive discussion of the interaction between computational constraints and incentives. 

Figure 12. Comparison of Coulomb and Tammes angles, r( ) Ihe optimum angular diameter of a spherical cap in the Tammes packing problem. SXN) is the minimum angular separation between adjacent charges in the spherical Coulomb problem. The graph shows the relative difference between the two sets of angles, Eq. (4.13).

These problems can be associated with the fact that various pore size SEC packings, either in a bank of individual pore size columns or in a mixed gel column set, respond differently in this extremely polar solvent. In addition the... 

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

Equipment instructions must be properly and promptly filed. A separate file should be set up for each piece of equipment and should include the smallest note packed with it. A lack of such files is an invitation to future problems. In one instance, a strip chart recorder that had not been used for several years seemed just right for a procedure but needed some minor repairs and parts. After a couple of hours of diligent search, the instruction book was finally found tucked away in a drawer. It could easily have been missed. In other less fortunate cases, it took time-consuming correspondence with the manufacturers to produce information that should have been immediately available in the file cabinet. 

The ZOE cement is easy to mix and a greater amount of powder can be incorporated into this cement (5 1 by mass) than any other, where even 4 1 by mass is unusual. Because the ZOE cement is sensitive to moisture it can be formulated to have a long working time under normal room conditions (23 °C, relative humidity 50 %) and a rapid set once placed in the warm and moist conditions of the mouth. This is a considerable clinical advantage, making it convenient to use. The cement can be used in a war pack for use on the battlefield. Nevertheless, sensitiveness to humidity can give rise to problems in use under tropical conditions. 

So far, Santos has been able to express the relation between a set of coefficients af, aj J 6 / describing a vector field and the overall curvature of the stream lines of this vector field. Based on the curvature field, they constructed the measure E of the curvature distribution in the simulation box. Provided that the homogeneous curvature field of curvature c0 is the one that minimizes E, the problem of packing has been recast as a minimization problem. However, the lack of information about the gradient of the error function to be minimized does not facilitate the search. Fortunately, appropriate computer simulation schemes for similar minimization problems have been proposed in the literature [105-109]. 

Figure 11(A) shows a principle sketch of a SEC set-up. The eluent (solvent) is pumped at a constant flow rate through the system. A small amount of polymer solution (typically 200 pL) is injected manually or with an autosampler. The main part comprises a set of columns (typically 3-4 columns+pre-column) typically packed with microporous styrene-divinylbenzene, porous glass, or silica. In the routine analytical laboratory it is especially useful to have a pre-column in order to collect impurities that might be present in the samples. If many different samples are to be analyzed, it is necessary to check the reliability of the columns frequently to avoid artefacts due to residues from previous samples still held on the column. In order to avoid problems, samples should be purified before they...

Absorption columns can be modeled in a plate-to-plate fashion (even if it is a packed bed) or as a packed bed. The former model is a set of nonlinear algebraic equations, and the latter model is an ordinary differential equation. Since streams enter at both ends, the differential equation is a two-point boundary value problem, and numerical methods are used (see Numerical Solution of Ordinary Differential Equations as Initial-Value Problems ). 

Optimization of a distributed parameter system can be posed in various ways. An example is a packed, tubular reactor with radial diffusion. Assume a single reversible reaction takes place. To set up the problem as a nonlinear programming problem, write the appropriate balances (constraints) including initial and boundary conditions using the following notation ... 

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