172 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Very Profit favourable ml Favourable Unfavourable Z Rop(X Zj)p(Zj) j (1) (2) Z• Za X X Xx 250 000 60 000 - 100 000 x• o o o p(Z) 0.4 0.3 0.3 (1) pur I z) 0.8 0.8 0.2 (2) p(x I z) 0.2 0.2 0.8 88 400 - 400 0 '0' (1) For X i = X, X = Xx, I Rep(X I Z•)p(Z.•) is given by 250 000 x (0.8) x (0.4) + 60 000 x (0.8) x (0.3) - 100000 x (0.2) x (0.3) = 88 400 (2) and for Xx, X by 250000 x (0.2) x (0.4) + 60 000 x (0.2) x (0.3) - 100 000 x (0.8) x (0.3): - 400. From expression (5) we now have for the expected value of Xx and X2 using the imperfect survey information EV(XO = oe88 400 + oe0, EV(X•) = oe0. Thus the value of perfect information is given by oe88 400 - oe88 000 (expected value without information) = oe400, and the (2) decision procedure is Xx if X obtains, otherways X•.. Evidently the manager would be ill advised to pay more than oe400 for this particular survey since the expected value of his decision will only increase by this amount. It is a matter of common sense that a manager with a low opinion as to the marketability of a product is unwilling to pay much for further information. The above process provides us with a numerical value for this information. A slightly more general expression for the value of imperfect information here is Zoemax ZR, P(XIZ•)p(Zj) } - max ZR•jp(Z•) z•. i j j j which represents the difference between the expected value with imperfect information and that without it. This, the Baysian approach to a decision problem, uses all the infor- mation available in a logical and consistent manner to analyse the decision situation. When evaluating a numerical measure of the expected conse- quences of a particular choice Xi, use is made of both the manager's priors

DECISION ANALYSIS 173 and the sample result X. This seems reasonable since the information X is a function of the number of people surveyed, the format of the survey, etc., whereas the manager's priors p(Z) are based upon a different source of information, namely his experience and knowledge to date. Here the broadest possible data base is used for the decision. For further reading on this topic, the reader is recommended to look at (4) and (5). SUBJECTIVE TESTS AND INFORMATION In the previous section are discussed the use of information which was essentially an objective measure of a specific quantity such as market share. We discuss in this section the value and use of tests and information which are subjective in nature. Consider the situation of, say, a soap manufacturer who wishes to per- fume his product. His procedure might be to compile a perfume specification and duly invite say six perfumers to submit a sample as tender. Prestonably the perfume specification indicated any price limit the manufacturer wished to impose. If all six perfumers submit tenders, the manufacturer needs to choose between six perfumes, Sx, S2, S3, S4, $5, So for his product. It appears a common method employed to make this choice is to use a consumer panel. Here a team of 30 or so individuals are requested to give their opinion and preference on the samples Si, and a particular one $* selected on the basis of this information. Such a process has the attraction of being simple to operate and also of avoiding any real decision on the part of the manufacturer in a highly subjective area. Panel tests An obvious question here is how one would recognize a good choice of $* and also what is expected of it? Before we discuss these points further, it will be useful to consider some possible problems associated with a panel test of the type described above. We shall assume each and every member of the panel to be a coherent individual capable of giving an informed opinion. Being coherent individuals, their preferences are transitive in that if S•PSj and SjPS•, then S•PS•, where P denotes preferred to. Suppose three firms A, B and C have submitted perfume tenders, and the preference orderings of a seven man panel is as follows: