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Note: the eye
indicates that input representation
is not under the control the inputting process.
Figure 10.1. General schema of a problem solver.
ized program or plan for behavior that manipulates the internal representation in an attempt to solve the problem. For the type of problem solvers we have in mind-business men, analysts, etc.-there exist many relatively independent methods, so that the total behavior of the problem
solver is made up as an iterative cycle in which methods are selected on the basis of current information (in the internal representation) and tried with consequent modification of the internal representation, and a new method is selected.
Let us stay with this view of problem solving for a while, even though it de-emphasizes some important aspects, such as the initial determination of an internal representation, its possible change, and the search for or construction of new methods (by other methods) in the course of problem solving. What Fig. 10.1 does emphasize is the method-the discrete package of information that guides behavior in an attempt at problem solution. It prompts us to inquire about its anatomy.
1.1. The Anatomy of a Method
Let us examine some method in management science. The simplex method of linear programming will serve admirably. It is well known, important, and undoubtedly a method. It also works on well-structured problems, but we will take due account of this later. The basic linear programming problem is as follows.
Given: a set of positive real variables
Figure 10.2 shows the standard data organization used in the simplex method and gives the procedure to be followed. We have left out the procedure for getting an initial solution, that is, we assume that the tableau of Fig. 10.2 is initially filled in. Likewise, we have ignored the detection of degeneracy and the procedure for handling it.
There are three parts to the simplex method. The first is the statement of the problem. This determines the entities you must be able to identify in a situation and what you must know about their properties and mutual interrelationships. Unless you can find a set of nonnegative numerical quantities, subject to linear constraints and having a linear objective function to be maximized, you do not have a LP problem, and
the method cannot help you. The second part of the method is the procedure that delivers the solution to the problem. It makes use only of information available in the problem statement. Indeed, these two are coupled together as night and day. Any information in the problem statement which is known not to enter into the procedure can be discarded, thereby making the problem just that much more general.
The third part of the method is the proof or justification that the procedure in fact delivers the solution to the problem (or delivers it within certain specified limits). The existence of this justification has several consequences. One, already noted, is the complete adaptation of means to ends of the shaping of the problem statement so that it is as general as possible with respect to the procedure. Another consequence is a toleration of apparent meaninglessness in the procedure. It makes no difference that there seems to be neither rhyme nor reason to the steps of the method in Fig. 10.2. Careful analysis reveals that they are in fact just those steps necessary to the attainment of the solution. This feature is characteristic of mathematical and computational methods generally and sometimes is even viewed as a hallmark.
An additional part of the simplex method is a rationale that can be used to make the method understandable. The one usually used for the simplex is geometrical, with each constraint being a (potential) boundary plane of the space of feasible solutions. Then the simplex procedure is akin to climbing from vertex to vertex until the maximal one is reached. This fourth part is less essential than the other three.
The first three parts seem to be characteristic of all methods. Certainly, examples can be multiplied endlessly. The quadratic formula provides another clear one:
Find x such that ax2 + bx + c = 0.
compute x = b/2a±1⁄41⁄2a √b2 — 4ac.
(substitute formula in ax2 + bx + c and show by algebraic manipulation that 0 results).
In each case a justification is required (and forthcoming) that establishes the relation of method to problem statement. As we move toward more empirical methods, the precision of both the problem statement and the procedure declines, and concurrently the precision of the justification; in fact, justification and plausible rationale merge into one.
1.2. Generality and Power
We need to distinguish the generality of a method from its power. A method lays claim via its problem statement to being applicable to a certain set of problems, namely, to all those for which the problem statement applies. The generality of a method is determined by how large the set of problems is. Even without a well-defined domain of all problems, or any suitable measure on the sets of problems, it is still often possible to compare two problem statements and judge one to be more inclusive than another, hence one method more general than the other.
A method that is applicable only to locating warehouses is less general than one that is applicable to problems involving the location of all physical resources. But nothing interesting can be said about the relative generality of a specific method for inventory decisions versus one for production scheduling.
Within the claimed domain of a method we can inquire after its ability to deliver solutions: the higher this is, the more powerful the method. At least three somewhat independent dimensions exist along which this ability can be measured. First, the method may or may not solve every problem in the domain; and we may loosely summarize this by talking of the probability of solution. Second, there may exist a dimension of quality in the solution, such as how close an optimizing method gets to the peak. Then methods can differ on the quality of their solutions. (To obtain a simple characteristic for this requires some summarization over the applicable domain, but this feature need not concern us here.) Third, the method may be able to use varying amounts of resources. Then, judgments of probability of solution and of quality are relative to the amount of resources used. Usually the resource will be time, but it can also be amount of computation, amount of memory space, number of dollars to acquire information, and so on. For example, most iterative methods for solving systems of equations do not terminate in a finite number of iterations, but produce better solutions if run longer; the rate of convergence becomes a significant aspect of the power of such methods.
In these terms the simplex method would seem to rank as one of limited generality but high power. The restrictions to linearity, both in the constraints and the objective function, and to a situation describable by a set of real numbers, all constrict generality. But the simplex method is an algorithm within its domain and guarantees delivery of the complete solution. It is not the least general method, as is indicated by the transportation problem with its more specialized assumptions; nor is it the most powerful method for its domain, since it can be augmented with additional schemes that obtain solutions more expeditiously.
Evidently there is an inverse relationship between the generality of a method and its power. Each added condition in the problem statement is one more item that can be exploited in finding the solution, hence in increasing the power. If one takes a method, such as the simplex method, and generalizes the problem statement, the procedure no longer solves every problem in the wider domain, but only a subset of these. Thus the power diminishes. The relationship is not one-one, but more like a limiting relationship in which the amount of information in the problem statement puts bounds on how powerful the method can be. This re