Partial solution

By 1960, the elements of modem membrane science had been developed, but membranes were used in only a few laboratory and smaU, specialized industrial appHcations. No significant membrane industry existed, and total annual sales of membranes for aU appHcations probably did not exceed 10 million in 1990 doUars. Membranes suffered from four problems that prohibited their widespread use as a separation process they were too unreHable, too slow, too unselective, and too expensive. Partial solutions to each of these problems have been developed since the 1960s, and in the 1990s membrane-based separation processes are commonplace. 

Another serious problem in heat exchangers is corrosion. Severe corrosion can and does occur in tubing and very often with common fluids such as water. Proper material selection based on a full analysis of the operating fluids, velocities and temperatures is mandatory. Very often, heavier gauge tubing is specified to offset the effects of corrosion, but this is only a partial solution. This should be followed by proper start-up, operating and shut-down procedures. 

We next consider Scheme XIV, for whieh, in the preceding subseetion, a partial solution was obtained. The seheme is rewritten here for eonvenienee. 

Work on mutant cultures provided the first practical, although partial, solution to the dehydration phenomenon. A mutant strain whose parent produced chlortetracycline (2) was... 

A partial solution to the problem of producing sharp peaks at low elution temperatures is to add a small amount of a higher-boiling co-solvent to the main solvent. As suggested by Grob and Muller (23, 24), butoxyethanol can be used as a suitable cosolvent for aqueous mixtures in such cases. 

Twenty-five grams (0.212 mole) of 3-sulfolene, and 15.0 g (0.153 mole) of pulverized maleic anhydride are added to a dry 250-ml flask fitted with a condenser. Boiling chips and 10 ml of dry xylene are added. The mixture is swirled gently for a few minutes to effect partial solution, then gently heated until an even boil is established. During the first 5-10 minutes, the reaction is appreciably exothermic, and care must be exercised to avoid overheating. 

In addition to the careful selection of structural metals, the cathodic protection of water-wetted parts may also be specified. For most boiler plant systems, however, because of the tortuous and extended waterside surfaces involved, the use of cathodic protection is only a partial solution to controlling corrosion and should never be the sole secondary protocol. Rather, cathodic protection functions well when employed as part of a more comprehensive program that includes appropriate internal chemical treatments. 

Twenty-two grams (0.45 mole) of 70% hydrogen peroxide (Note 1) is added dropwise with efficient agitation to a slurry or partial solution of 36.6 g. (0.30 mole) of benzoic acid (Note 2) in 86.5 g. (0.90 mole) of methanesulfonic acid (Note 3) in a 500-ml. tail-form beaker. The reaction temperature is maintained at 25-30° by means of an ice-water bath. The reaction is exothermic during the hydrogen peroxide addition, which requires approximately 30 minutes. During this period the benzoic acid completely dissolves. 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

Another problem with all the dwell time methods is that sinee a speeifie diameter tool is often used to do the polishing, the finished surfaee has a "roughness" of a spatial frequency associated with the tool diameter. This rather eo-herent roughness ean produce diffraction artifacts in the image produeed by the telescope. A partial solution to this problem is to use several tool sizes and do the figuring in stages rather than all at onee. 

A partial solution to this dilemma could be that a large proportion of the protein-rich foods (meat, eggs) consumed by these people came from animals that were themselves fed a C4 diet. We know that dogs typically share the same diet as humans (Katzenberg 1989 Cannon et al. 1999) and are important components of the diet in some sites (eg., Cuello Hammond 1991 van der Merwe et al, this volume). It is unlikely that all the meat consumed by Maya peoples was derived from pure C4 consumers, however, as we have evidence for at least some C3-based animal bones that are presumed to be waste from food preparation. This should a subject of future study to test for the degree of domestication (and consequent feeding on maize) of meat-supplying animals such as turkeys. 

Having defined the process of branching, we must now formally define the mechanisms for controlling the expansion of the subsets. The basic intuition behind each of these mechanisms is that we can measure the quality not just of a single feasible solution, but of the entire subset represented by the partial solution string. 

In our scheduling example, a partial solution represents all the possible completions of a partial schedule. To estimate the quality of the partial solution, we could assume the best scenario, and assign it a lower-bound value based on the lowest possible value of its makespan. There are many ways to estimate the makespan for example, we could simply ignore the remainder of the batches, and report the makespan of the current partial schedule, clearly a lower bound on the final makespan. Stated formally, the lower-bound function g(jr) satisfies the following requirements (Ibaraki, 1978) ... 

In addition to the elimination of partial solutions on the basis of their lower-bound values, we can provide two mechanisms that operate directly on pairs of partial solutions. These two mechanisms are based on dominance and equivalence conditions. The utility of these conditions comes from the fact that we need not have found a feasible solution to use them, and that the lower-bound values of the eliminated solutions do not have to be higher than the objective function value of the optimal solution. This is particularly important in scheduling problems where one may have a large number of equivalent schedules due to the use of equipment with identical processing characteristics, and many batches with equivalent demands on the available resources. 

The intuitive notion behind a dominance condition, D, is that by comparing certain properties of partial solutions x and y, we will be able to determine that for every solution to the problem y(y) we will be able to find a solution to Yix) which has a better objective function value (Ibaraki, 1977). In the flowshop scheduling problem several dominance conditions, sometimes called elimination criteria, have been developed (Baker, 1975 Szwarc, 1971). We will state only the simplest ... 

Definition. A dominance relation, D, is a partial ordering of the partial solutions of the discrete decision processes in X, which satisfies the following three properties for any partial solutions, x and y. 

Definition. An equivalence condition, EQ, between two partial solutions, r and y is a binary relationship, x.EQ.y, which has the following properties ... 

Let us briefly discuss the theoretical results providing the basis for the improved efficiency of branch-and-bound algorithms. Let F = [x g(.x) lower-bound test. Then, the set L, defined by L =Fr X%, contains all the partial solutions, which can be terminated only by an equivalence relation. Recall that, by definition, no node in X% can be terminated by a dominance rule. 

In addition to having to assign state variables to the strings of the DDF, we also have to assign properties to the alphabet symbols. In our flowshop example, the alphabet symbols can be interpreted as batches to be executed with a series of processing times. Thus, if we use the notation, (jc), to denote the state of partial solution, x, then... 

If we are successful, then we have verified that for the conditions prevailing in the example, the partial solutions, x and y would indeed be equivalent, as far as Condition-a is concerned. We now move to Condi-tion-b, which ensures that the equivalent node will play an equivalent role in the enumeration as the one that was eliminated. Thus, as we examine the children of x, y, regardless of whether they are members of the feasible set, we would verify that their lower-bound values were equal, and that if we had any existing dominance, or equivalence conditions that the equivalent descendant of x, i.e., xu, participates in the same relationships as does yu. 

In these cases there is no well defined notion of a looser constraint, the choice is then either to force those variables to be equal in x and y, or to find some path from their value to a constraint on another inter- or intrasituational variable and thus be able to show that their values in jc, y should obey some ordering based on these other constraints. This topic is the subject of current research, but is not limiting in the flowshop example, since no such constraints exist. Lastly, it is not enough to assert conditions on the state variables in x and y, since we have made no reference to the discrete space of alternatives that the two solutions admit. Our definition of equivalence and dominance constrains us to have the same set of possible completions. For equivalence relationships the previous statement requires that the partial solutions, x and y, contain the same set of alphabet symbols, and for dominance relations the symbols of JC have to be equal to, or a subset of those of y. Thus our sufficient theory can be informally stated as follows ... 

The analysis of the branching structure turns the preceding deduction process around. We have all the facts available to us at the end of the solution synthesis, i.e., at the end of solving a particular problem. Our task is to select and connect subsets of those facts to prove new results that are useful for deriving new control information. In essence, we have to turn facts about solutions and partial solutions at lower levels in the tree, into constraints on the properties of states and alphabet interpretations higher in the tree. 

The goal of the reasoning is to prove that two partial solutions are equivalent to one another, or that one dominates the other. To do this, we will start with conditions that contain... 

For example, consider trying to prove that one solution, x, will dominate another solution, y, if they both have scheduled the same batches, but the end-times of each machine are earlier in the partial schedule (partial solution), X, than in y. The general theory will not be couched in these terms, but more abstractly, in terms of the properties of binary operators. 

This explicit inclusion of operationality allows us to declare explicitly some facts about the pair (x, y), such as their state variable values, as operational. This will not permit the explanation to stop at other partial solutions, whose states have not been declared operational thus we will use this approach. 

This is high-level description of the explanation process can be illustrated in the flowshop example. To instantiate the target concept, we use the partial solution strings that represent x and y call them a-, [Pg.319]

If we assume that the variables are ordered from the first unit to the last, we will begin by examining the start-times on the first unit. The start-times are not operational, but the end-times on the units for partial solution X, y are. The start-times of the first unit in the next state, which are equal to the end-times of the first unit in jc and y are analyzed by using one of the less-equal implications. The end-times, which are operational, are then compared. 

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