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tinctions that applied to every situation, even though with less and less relevance.

The third item on the list, representation of problems, also raises a question of the locus of power (rather than of specific mechanisms related to ill-structured problems). At a global level we talk of the representation of a problem in a mathematical model, presumably a translation from its representation in some other global form, such as natural language. These changes of the basic representational system are clearly of great importance to problem solving. It seems, however, that most problems, both well structured and ill structured, are solved without such shifts. Thus the discovery of particularly apt or powerful global representations does not lie at the heart of the handling of ill-structured problems.

More to the point might be the possibility that only special representations can hold ill-structured problems. Natural language or visual imagery might be candidates. To handle ill-structured problems is to be able to work in such a representation. There is no direct evidence to support this, except the general observations that human beings have (all) such representations, and that we do not have good descriptions of them.

More narrowly, we often talk about a change in representation of a problem, even when both representations are expressed in the same language or imagery. Thus we said that Fig. 10.12 contained two representations of the LP problem, the original and the one for the simplex method. Such transformations of a problem occur frequently. For example, to discuss the application of heuristic search to inverting matrices we had to recast the problem as one of getting from the matrix A to I, rather than of getting from the initial data (A, I, AX = I) to X. Only after this step was the application of the method possible. A suspicion arises that changes of representation at this level-symbolic manipulation into equivalent but more useful form-might constitute a substantial part of problem solving. Whether such manipulations play any special role in handling ill-structured problems is harder to see. In any event, current research in artificial intelligence attempts to incorporate this type of problem solving simply as manipulations in another, more symbolic problem space. The spaces used by theorem provers, such as LT, are relevant to handling such changes.

Method identification, the next item, concerns how a problem statement of a method comes to be identificd with a new problem, so that each of the terms in the problem statement has its appropriate referent in the problem as originally given. Clearly, some process performs this identification, and we know from casual experience that it often requires an

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exercise of intellect. How difficult it is for the LP novice to "see" a new problem as an LP problem, and how easy for an old hand!

Conceivably this identification process could play a critical role in dealing with ill-structured problems. Much of the structuring of a problem takes place in creating the identification. Now it might be that methods still play the role assigned to them by our hypotheses, but even so it is not possible to instruct a computer to handle ill-structured problems, because it cannot handle the identification properly. Faced with an appropriate environment, given the method and told that it was the applicable one, the computer still could not proceed to solve the problem. Thus, though our hypotheses would be correct, the attempt to give them substance by describing methods would be misplaced and futile.

Little information exists about the processes of identification in situations relevant to this issue. When the situation is already formalized, matching is clearly appropriate. But we are concerned precisely with identification from a unformalized environment to the problem statement of a method. No substantial programs exist that perform such a task. Pattern recognition programs, although clearly designed to work in "natural" environments, have never been explored in an appropriately integrated situation. Perhaps the first significant clues will come out of the work, mentioned at the beginning of this chapter and still in its early stages, on how a machine can use a hand and eye in coordination. Although the problems that such a device faces seem far removed from management science problems, all the essentials of method identification are there in embryo. (Given that one has a method for picking up blocks, how does one identify how to apply this to the real world, seen through a moving television eye?)

An additional speculation is possible. The problem of identification is to find a mapping of the elements in the original representation (say, external) into the new representation (dictated by the problem statement of the method to be applied). Hence there are methods for the solution to this, just as for any other problem. These methods will be like those we have exhibited. (Note, however, that pattern recognition methods would be included.) The construction of functions in the induction method may provide some clues about how this mapping might be found. As long as the ultimate set of contacts with the external representation (represented in these identification methods as generates and tests) were rather elementary, such a reduction would indeed answer the issue raised and leave our hypotheses relevant.

An important aspect of problem solving is the acquisition of new information, the next item on the list. This occurs at almost every step, of course, but most of the time it is directed at a highly specific goal; for instance, in method identification, which is a major occasion for assimilating information, acquisition is directed by the problem statement. In contrast, we are concerned here with the acquisition of information to be used at some later time in unforeseen ways. The process of education provides numerous examples of such accumulation.

For an ill-structured problem one general strategy is to gather additional information, without asking much about its relevance until obtained and examined. Clearly, in the viewpoint adopted here, a problem may change from ill structured to well structured under such a strategy, if information is picked up that makes a strong method applicable.

The difficulty posed for our hypotheses by information acquisition is not in assimilating it to our picture of methods. It is plausible to assume that there are methods for acquisition and even that some of them might be familiar; for example, browsing through a scientific journal as generateand-test. The difficulty is that information acquisition could easily play a central role in handling ill-structured problems but that this depends on the specific content of its methods. If so, then without an explicit description of these methods our hypotheses cannot claim to be relevant. These methods might not formalize easily, so that ill-structured problems would remain solely the domain of human problem solvers. The schemes whereby information is stored away yet seems available almost instantly --as in the recognition of faces or odd relevant facts--are possibly aspects of acquisition methods that may be hard to explicate.

The last three items on the list name things that can be constructed by a problem solver and that affect his subsequent problem-solving behavior. Executive construction occurs because the gross shape of a particular task may have to be reflected in the top-level structure of the procedure that solves it. The induction method, with the three separate induction tasks mentioned, provides an example. Each requires a separate executive structure, and we could not give a single unified procedure to handle them all. Yet each uses the same fundamental method. Relative to our hypotheses, construction seems only to provide additional loci for problem-solving power. This item could become important if it were shown that solutions are not obtained to ill-structured problems without some construction activity.

The extended discussion of the parts of the problem-solving process other than methods, and the ways in which they might either refute or nullify our two hypotheses, stems from a conviction that the major weakness of thesc hypotheses is the substantial incompleteness of our knowledge about problem solving. They have been created in response to partial evidence, and it scems unlikely that they will emerge unscathed as some of these other parts become better known.

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6.2. Measures of Informational Demands

Throughout the chapter we have talked as if adding information to a problem statement leads to a decrease in generality and an increase in power. Figure 10.3 is the baldest form of this assertion. At the most general level it seems plausible enough. Here one crudely identifies the number of conditions in the problem statement with the size of the space being searched: as it gets smaller, so the problem solver must grow more powerful. At a finer level of analysis, however, this assertion seems often violated, and in significant ways; for example, a linear programming problem is changed into an integer programming problem by the addition of the constraint that the variables {x} range over the positive integers rather than the positive reals. But this makes the problem harder, not easier. Of course, it may be that existing methods of integer programming are simply inefficient compared to what they could be. This position seems tenuous, at best. It is preferable, I think, to take as a major difficulty with these hypotheses that they are built on foundations of sand.

6.3. Vague Information

It is a major deficiency of these hypotheses (and of this chapter) that they do not come to grips directly with the nature of vague information. Typically, an ill-structured problem is full of vague information. This might almost be taken as a definition of such a problem. except that the term vague is itself vague.

All extant ideas for dealing with vagueness have one concept in common: they locate the vagueness in the referent of a quite definite (hence un-vague) expression. To have a probability is to have an indefinite event, but a quite definite probability. To have a subset is to have a quite definite expression (the name or description of the subset) which is used to refer to an indefinite, or vague, element. Finally, the constructs of this chapter are similarly definite. The problem solver has a definite problem statement, and all the vagueness exists in the indefinite set of problems that can be identified with the problem statement.8

• Reitman's proposals, although we have not described them here, have the same definite character (17]. So also does the proposal by Zadeh for "fuzzy" sets (24).

The difficulty with this picture is that, when a human problem solver has a problem he calls ill structured, he does not seem to have definite

ressions which refer to his vague information. Rather he has nothing definite at all. As an external observer we might forin a definite expression describing the range (or probability distribution) of information that the subject has, but this “meta" expression is not what the subject has that is this information.

It seems to me that the notion of vague information is at the core of the feeling that ill-structured problems are essentially different from well-structured ones. Definite processes must deal with definite things, say, definite expressions. Vague information is not definite in any way. This chapter implies a position on vague information; namely, that there are quite definite expressions in the problem solver (his problem statement). This is a far cry from a theory that explains the different varieties of vague information that a problem solver has. Without such explanations the question of what is an ill-structured problem will remain only half answered.

7. CONCLUSION

The items just discussed—other aspects of problem solving, the measurement of power and generality, and the concept of vagueness—do not exhaust the difficulties or deficiencies of the proposed hypotheses. But they are enough to indicate their highly tentative nature. Almost surely the two hypotheses will be substantially modified and qualified (probably even compromised) with additional knowledge. Even so, there are excellent reasons for putting them forth in bold form.

The general nature of problems and of methods is no longer a quasiphilosophic enterprise, carried on in the relaxed interstices between the development of particular mathematical models and theorems. The development of the computer has initiated the study of information processing, and these highly general schema that we call methods and problemsolving strategies are part of its proper object of study. The nature of generality in problem solving and of ill-structuredness in problems is also part of computer science, and little is known about either. The assertion of some definite hypotheses in crystallized form has the virtue of focusing on these topics as worthy of serious, technical concern.

These two hypotheses (to the extent that they hold true) also have some general implications for the proper study of management science. They say that the field need not be viewed as a collection of isolated mathematical gems, whose application is an art and which is largely

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