162 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS that this possible restriction exists whether or not decision-analysis is used, and represents an initial and subjective identification of a 'sub-decision' on the part of the decision-maker. Starr (2) refers to the resulting problem as decision-making under ambiguity. A second difficulty can arise when evaluating the results Rij. This matrix can sometimes be compiled by direct observation, but more likely costly experimentation or an operational research type study is required. White (3) defines the process of evaluating R o given environment Zi and strategy X i as the object study. Having determined Ro, there is still the problem of deciding an order of preference between results. Any specific result R e could be viewed in a multitude of ways. Consider, for example, the decision concerned with the selection of a marketing policy for a product. The results of various policies could be measured by r• -- value of sales in the first 6 months, rs -- value of sales in the first 2 years, ra = total net profit gained over first year, r• -- expenditure on advertising over first 6 months. Clearly there are numerous other measures one might be concerned with here such as possible counter measures by competitors, etc. If the result R is restricted to the above four measures, we see it has the form of a four-dimen- sional vector R(rx, r2, rs, r•) with different strategies and environments giving rise to different values of rx. Suppose the two possible strategies Xx, X2 (under a specific environment) give rise to X 1 -- R(5 000, 23 000, 2 000, 1 400) X• -• R(8 000, 19 000, 1 900, 1 850) which result is preferred ? Before this question is answered it is necessary for the decision-maker to have a clear idea as to his objectives in making the decision. Perhaps the objective is to maximize profits. If so, over what period of time? Should a period of time be specified: presumably the decision- maker would wish to select that policy which maximizes his profits over this period, subject to his company being in a viable trading position for the subsequent period of time. The true objective is seldom simple in reality and will likely be reviewed within a dynamic context. There are some techniques available to assist with this preference selection procedure. Suppose in the above marketing problem the decision-

DECISION ANALYSIS 1(53 maker decided that the order of paramount importance and concern was rs, r4, r•. and r•, and a fractional increase in rs was preferable to any increase in r4, r•. and rx: that a fractional increase in r• was preferable to any extent of increase in r2 and rx: and that an increase in r•. was superior to any increase in rx. Then the order rs, r•, r2 and r• is a lexicographic ordering (3) of the out- come parameters for the decision-maker. His choice of R would be that with the best value of rs. Should two or more policies have the same best value of rs, the choice is made between them on the basis of the next most important variable r•, and so on. Alternatively, it might be possible to determine a value function V, whose argument is an outcome R s, with the property V(Rs) V(R,) if and only if R s is preferred to R, (3). Clearly the decision function is itself dependent upon the decision-maker. The existence of such a decision function reduces the selection procedure to a mathematical problem, namely find R* where V(R)* = max V(Rt). Yet another approach to the problem exists using utility theory (4). Here each outcome is given a single numerical value (utility) which reflects its desirability as measured by the decision maker. The most desirable out- come is that with the greatest utility. The final difficulty we shall mention here is that due to the uncertainty introduced through the unknown environment Z. According to the type of uncertainty, decision problems are divided into those of certainty, uncer- tainty and risk. In what follows, we will assume the result R can always be determined, and furthermore, that it is a scalar quantity such as cash or time. This being so, most of the difficulties mentioned in this section will not apply. Decisions under certainty In this case an outcome R is known for certain once the alternatives are specified. Any uncontrolled events are assumed known or irrelevant to the decision. Here either the environment is unique or the decision is indepen- dent of the environment. Thus the result R is a function of X only, i.e. R = R(X,).