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Axiomatic Design  265


             Hartley (1928) introduced a logarithmic measure of information in
           the context of communication theory. Hartley, and later Shannon
           (1948), introduced their measure for the purpose of measuring infor-
           mation in terms of uncertainty. Hartley’s information measure is
           essentially the logarithm of the cardinality or source alphabet size (see
           Definition 8.1), while Shannon formulated his measure in terms of
           probability theory. Both measures are information measures, and
           hence are measures of complexity. However, Shannon called his mea-
           sure “entropy” because of the similarity to the mathematical form of
           that used in statistical mechanics.
             Hartley’s information measure (Hartley 1928) can be used to explore
           the concepts of information and uncertainty in a mathematical frame-
           work. Let X be a finite set with a cardinality |X|   n. For example, X
           can be the different processes in a transactional DFSS project, a flow-
           chart, or a set of DPs in a product DFSS project. A sequence can be
           generated from set X by successive selection of its elements. Once a
           selection is made, all possible elements that might have been chosen
           are eliminated except for one. Before a selection is made, ambiguity is
           experienced. The level of ambiguity is proportional to the number of
           alternatives available. Once a selection is made, no ambiguity sus-
           tains. Thus, the amount of information obtained can be defined as the
           amount of ambiguity eliminated.
             Hartley’s information measure I is given by I   log 2 N (bits) where
           N   n and s is the sequence of selection. The conclusion is that the
                 s
           amount of uncertainty needed to resolve a situation, or the amount of
           complexity to be reduced in a design problem is equivalent to the
           potential information involved. A reduction of information of  I bits
           represents a reduction in complexity or uncertainty of  I bits.

           Definition 8.1. A source of information is an ordered pair     (X,P)
           where X   {x 1 , x 2 ,…,x n } is a finite set, known as a source alphabet, and
           P is a probability distribution on X. We denote the probability of x i by p i .
             The elements of set X provide specific representations in a certain
           context. For example, it may represent the set of all possible tolerance
           intervals of a set of DPs. The association of set X with probabilities
           suggests the consideration of a random variable as a source of infor-
           mation. It conveys information about the variability of its behavior
           around some central tendency. Suppose that we select at random an
           arbitrary element of X,say, x i with probability p i . Before the sampling
           occurs, there is a certain amount of uncertainty associated with the
           outcome. However, an equivalent amount of information is gained
           about the source after sampling, and therefore uncertainty and infor-
           mation are related. If X   {x 1 }, then there is no uncertainty and no
           information gained. At the other extreme, maximum uncertainty
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