Fixed Charge Problem

Consider a production planning problem with N products such that the th product requires a fixed production or set-up cost Kj, independent of the amount produced, and a variable cost Cy per unit, proportional to the quantity produced. Assume that every unit of product requires imits of resource i and there are M resources. Given that the product , whose sales potential is dj, sells for p per unit and no more than units of resource i are available i = 1,2. M), the problem is to determine the optimal product mix that maximizes the net profit. 

Formulation The total cost of production (fixed plus variable) is a nonlinear function of the quantity produced. But, with the help of binary (0-1) integer variables, the problem can be formulated as an integer linear program. 

Let the binary integer variable 8y denote the decision to produce or not to produce product . 

Note that Xj can be positive only when 8y = 1, in which case its production is limited by dj and the fixed production cost is included in the objective function. 

3 Constraint with Multiple Right-Hand-Side Constants 

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Many problems in force field investigations arise from the calculation of Coulomb interactions with fixed charges, thereby neglecting possible mutual polarization. With that obvious drawback in mind, Ulrich Sternberg developed the COSMOS (Computer Simulation of Molecular Structures) force field [30], which extends a classical molecular mechanics force field by serai-empirical charge calculation based on bond polarization theory [31, 32]. This approach has the advantage that the atomic charges depend on the three-dimensional structure of the molecule. Parts of the functional form of COSMOS were taken from the PIMM force field of Lindner et al., which combines self-consistent field theory for r-orbitals ( nr-SCF) with molecular mechanics [33, 34]. 

Consider the following simplest prototype problem for stationary electrodiffusion of a univalent symmetric electrolyte through a bipolar ion-exchange membrane with an antisymmetric piecewise constant fixed charge density XN(x). 

Complete neglect of the Coulomb interaction between charges fixed on the network is also a problem. It has been argued [33] that the contribution from fixed charges to the osmotic pressure plays an important role in determining the swelling equilibrium of gels. However, we will not discuss these theoretical problems further because the main concern of the present article is to clarify experimentally the nature of the first-order transition rather than to improve theoretical treatment. 

Mow, V. C., Ateshian, G. A., Lai, W. M., and Gu, W. Y. (1998). Effects of fixed charges on the stress-relaxation behavior of hydrated soft tissues in a confined compression problem. International Journal of Solids Structures, 35 4945-4962. 

For other models of flow of electrolytes through porous media the reader is referred to [2], [5], [6]. To take into account FCD (fixed charge density) one has to impose additional condition on the interface T (w) and the electroneutrality condition. A challenging problem is to use homogenisation methods for the case of finitely deformable skeleton, even hyperelastic. The permeability would then necessarily depend on strains. Such a dependence (nonlinear) is important even for small strain, cf. [7]. It is also important to include ion channels [8]. 

One s first reaction is to reject the adequacy of casting the problem as a linear one, but, as Grossmann and Santibanez show, the use of discrete (zero/one) decisions allow one to include to a very good approximation many of the nonlinearities. For example, a zero/one variable can be associated with the existence or non-existence of a unit. In the cost function that discrete variable can cause one to add in a fixed charge for the unit only if it exists. Also one can define a continuous "flow11 variable for the unit which can be forced to zero if the unit does not exist by the linear constraint ... 

Note also that neither CHEEP nor PDQC assigns identical charges to the two hydrogens that are equivalent by symmetry. The differences are small, on the order of 0.001 e, but nonnegligible. This effect, which had been noted previously in fitting multipoles to potentials, was treated by aligning the major symmetry axis of the molecule with one of the Cartesian axes, a measure that obscures, but does not fix, the problem. 

Note that only constraints corresponding to Boolean variables that are true are imposed. Also, fixed charges y, are only applied to these terms. Assuming that K subproblems (NLPD) are solved in which sets of linearizations / =1,...,AT are generated for subsets of disjunction terms L(ik) = / Vik = true), one can define the following disjunctive OA master problem ... 

After all, we have formulated mathematically the problem as a mixed-integer programming problem stated below. The objective function is composed of the total transportation cost between every facility, the total production cost at plant, and the total operational cost at each facility, the total holding cost at DC over the planning horizon, and the total fixed-charge for the open DCs. 

Nevertheless, these methods are mostly applied with fixed charges (even if these are chosen in a sophisticated way) and with pairwise additivity approximation as well as with the neglect of nuclear quantum effects. Suggestions for polarizable models appeared in literature mainly for water [23], The quality of potential parameterization varies from system to system and from quantity to quantity, raising the question of transferability. Spontaneous events like reactions cannot appear in simulations unless the event is included in the parameterization. Despite these problems, it is possible to reproduce important quantities as structural, thermodynamic and transport properties with traditional MD (MC) mainly due to the condition of the nanosecond time scale and the large system size in which the simulation takes place [24],... 

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