Basic Feasibility

The arrangement and distribution of anodes in gravel and activated charcoal filters is different. Cathodic protection of activated charcoal filters is basically feasible but requires a large number of electrodes and high protection current densities that are twice those for gravel bed filters, so that an electrically insulating layer can be deposited on the steel wall. 

Result 2 is true in two dimensions as can be seen from the example of relations (7.1), whose constraints have been rewritten in equation form in (7.9). The (xx, x2) coordinates of the vertex at xx = 0, x2 = 1 are given by the (xx, x2) coordinates of the basic feasible solution... 

The optimal vertex corresponds to the basic feasible solution... 

Comparing this definition with the previous one (x is a vertex if and only if it is a basic feasible solution), if x is a basic feasible solution, then n — m nonbasic bounds are active, plus m equalities, so... 

Results 1 and 2 imply that, in searching for an optimal solution, we need only consider vertices, hence only basic feasible solutions. Because a basic feasible solution has m basic variables, an upper bound to the number of basic feasible solutions is the number of ways m variables can be selected from a group of n variables, which is... 

To explain the method, it is necessary to know how to go from one basic feasible solution (BFS) to another, how to identify an optimal BFS, and how to find a better BFS from a BFS that is not optimal. We consider these questions in the following two sections. The notation and approach used is that of Dantzig (1998). 

The simplex method is a two-phase procedure for finding an optimal solution to LP problems. Phase 1 finds an initial basic feasible solution if one exists or gives the information that one does not exist (in which case the constraints are inconsistent and the problem has no solution). Phase 2 uses this solution as a starting point and either (1) finds a minimizing solution or (2) yields the information that the minimum is unbounded (i.e., —oo). Both phases use the simplex algorithm described here. 

If the problem is in feasible canonical form, we have a vertex directly at hand, represented by the basic feasible solution (7.13). But the form provides even more valuable information. By merely glancing at the numbers cpj = m + 1,..., w, you can tell if this extreme point is optimal and, if not, you can move to a better one. Consider first the optimality test, given by the following result. 

We show how the canonical form can be used to improve a nonoptimal basic feasible solution. 

The circled term will be explained soon. The basic feasible solution is... 

For the original basic feasible solution, one reduced cost is negative, namely... 

Substituting this value into Equations (7.21) and (7.22) yields a new basic feasible solution with lower cost ... 

This gives the basic feasible solution (7.24), as predicted. It also indicates that the present solution although better, is still not optimal, because c2, the coefficient of 2 in the/equation, is —1. Thus we can again obtain a better solution by increasing 2 while keeping all other nonbasic variables at zero. From Equation (7.25), the current basic variables are then related to 2 by... 

In the original system (7.18), if the constant on the right-hand side of the second equation had been zero, that is, if the basic feasible solution had been degenerate, then Jtj would have been related to x3 by... 

Let us now formalize the procedures of the previous section. If at least one cj < 0, then, at least if we assume nondegeneracy (all bt > 0), it is always possible to construct, by pivoting, another basic feasible solution with lower cost. If more than one Cj < 0, the variable xs to be increased can be the one with the most negative cy, that is, the one whose relative cost factor is... 

The procedure of the previous section provides a means of going from one basic feasible solution to one whose/is at least equal to the previous/(as can occur, in the degenerate case) or lower, if there is no degeneracy. This procedure is repeated until (1) the optimality test of relations (7.15) is passed or (2) information is provided that the solution is unbounded, leading to the main convergence result. 

Because the number of basic feasible solutions is finite, the algorithm can fail to terminate only if a basic feasible solution is repeated. Such repetition implies that the same value of/is also repeated. Under nondegeneracy, however, each value of /is lower than the previous, so no repetition can occur, and the algorithm is finite. 

If, at some iteration, the basic feasible solution is degenerate, the possibility exists that/can remain constant for some number of subsequent iterations. It is then possible for a given set of basic variables to be repeated. An endless loop is then set up, the optimum is never attained, and the simplex algorithm is said to have cycled. Examples of cycling have been constructed [see Dantzig (1998), Chapter 10]. 

The simplex algorithm requires a basic feasible solution as a starting point. Such a starting point is not always easy to find and, in fact, none exists if the constraints are inconsistent. Phase 1 of the simplex method finds an initial basic feasible solution or yields the information that none exists. Phase 2 then proceeds from this starting... 

Solution. Here no phase 1 is needed because an initial basic feasible solution is obvious. To apply directly the results of the previous sections, we rephrase the problem as... 

Is there a basic feasible solution to the problem Answer yes or no, and explain. 

An LP problem has been converted to standard canonical form by the addition of slack variables and has a basic feasible solution (with xx = x2 = 0) as shown in the following set of equations ... 

The first reported attempts of what was then called "absolute or total asymmetric synthesis" with chiral solid catalysts used nature (naturally ) both as a model and as a challenge. Hypotheses of the origin of chirality on earth and early ideas on the nature of enzymes strongly influenced this period [15]. Two directions were tried First, chiral solids such as quartz and natural fibres were used as supports for metallic catalysts and second, existing heterogeneous catalysts were modified by the addition of naturally occuring chiral molecules. Both approaches were successful and even if the optical yields were, with few exceptions, very low or not even determined quantitatively the basic feasibility of heterogeneous enantioselective catalysis was established. 

Suppose it is desired to minimize a cost function c when the nonnegative system variables x are constrained by inequalities. Take any basic feasible solution and consider how c is altered by a change Ax/ in the variable x. There are two effects to be considered. First there is the direct cost Acy of the material Axy, which is given by... 

Synthesis of pure hydrogen peroxide using solid polymer electrolytes (SPE) could eliminate the need to separate the product from liquid electrolytes (basic or acidic). Designs of the (SPE) fuel cell type of reactor could be investigated for such a process. Tatapudi and Fenton [71, 80] demonstrated the basic feasibility of this process (with or without concurrent anodic ozone evolution). However, new cathode materials and... 

A demonstration unit sponsored by the Environemtal Protection Agency (EPA) to produce RDF at St. Louis proved the basic feasibility of mechanical separation processes, transport and storage techniques, and combustion of fluff RDF to replace 5 to 27 percent of the pulverized coal used in suspension-fired utility boilers. However, the refinement of equipment components and the technical and economic optimization of the basic technology still require a great deal of work. 

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