174 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Panel member 1 A 2 A 3 A 4 B 5 B 6 C 7 C P B P C P B P C P B P C P A P C P A P C P B P A P B P A A A A B B B B P B P B P B P A P A P A P A three samples two sam ties Now, if N(Y) is the number of times perfume Y is the most perfumed choice, and the group decision is made on the basis of the highest value of N(Y), with some further criterion for decision in the case of a tie, in the above case with three samples we have, N(A)= 3, N(B) =2. N(C)--2 Accordingly perfume A is selected out of the set A, B and C. If, however, only two samples are considered by the panel, say A and B, we see from the second table above which is obtained by deleting perfume C from the first table, N(A) = 3, N(B) = 4 and so B would be chosen from the set A, B. In short, even though each member of a panel is coherent, it is possible for a group decision procedure to be incoherent. As a further example of the kind of problem that can arise in group decision-making, consider the following three panel member, three product problem Panel 1 A P B P C member 2 B P C P A 3 C P A P B Let the mechanism for producing a panel choice be as follows: each indi- vidual ranks the alternatives as shown: for each pair (X, Y), the number N(X,Y), being the number of times perfume X is preferred to Y, is deter- mined: the alternatives for the panel are then ranked on the basis of the numbers N(X, Y). We find that N(A, B) = 2 N(B, C) = 2 V(C, A) = 2 N(B, A) = 1 N(C, B) = 1 N(A, C) = 1

DECISION ANALYSIS 175 If the first pair discussed is A and B, B is discarded in fayour of A. In the next comparison between A and C, A is discarded and perfume C selected. Now, had the first perfumes compared been B and C or A and C, the choice would have been A and B respectively. Thus we have a panel of coherent members with a decision rule for obtaining a group choice which has the undesirable quality of being dependent upon the order of consideration of the samples. It is now evident the process of obtaining a group choice needs to be transitive, or coherent, if any useful information is to result. What is required is a 'Welfare function, W', whose input data are the coherent preferences of the panel members, and whose output are the unique panel choice. This function is, by definition, a coherent function. We return for the moment to the question of recognizing a good choice S* from S t. It is evident the choice S* is a function of both the specific panel used and the welfare function--or panel choice procedure. It is arguable that by having a sufficiently large panel, the individual preferences of panel members will in some sense be evened out and two panels of similar com- position and sufficiently large size would be expected to give the same choice given a specific welfare function. This assumption is directly measurable. If, as is often assumed, a sufficiently large panel will produce a choice which is more or less independent of the individual panel members, the choice S* is directly determined by the welfare function used. This being so, the main choice the soap manufacturer has to make is the selection of a welfare function W. Presumably he will value highly any welfare function he believes reflects the preference of the market at large. How can this selection of welfare function be made? Suppose the manufacturer has, say, two possible welfare functions, Wx and W•, and is contemplating perfuming a product. Following the usual procedure, tenders S i are invited and a panel test to determine preferences duly conducted. Since the market as such might object to the particular product in question being perfumed, the manufacturer may well append a non-perfumed product S o to those submitted to the panel. Now, if both welfare functions give the same choice, that is So) = So) = s* the perfume choice S* is clear. If this is not the case, the welfare function to choose is ideally that which will lead to the greatest increased profit for the manufacturer over a specific period. An analysis of past occasions in which