حساسیت تصمیم گیری با پارامترهای ابزار تجارت کردن غیر دقیق با استفاده از ابزار خطی مرزی

In earlier work we have developed methods for analysing decision problems based on multi-attribute utility hierarchies, structured by mutual utility independence, which are not precisely specified due to unwillingness or inability of an individual or group to agree on precise values for the trade-offs between the various attributes. Our analysis is based on whatever limited collection of preferences we may assert between attribute collections. In this paper we introduce three methods to assess the robustness of our selected decision. Two of these use the metric generated by the weights of the boundary linear utility. The third applies directly to the space of trade-off parameters.

In two earlier papers we have developed a methodology for decision analysis with multi-attribute utilities which does not require the specification of precise trade-offs between different risks. Multi-attribute utilities may be imprecisely specified, due to an unwillingness or inability on the part of a client to specify fixed risk trade-offs or because of disagreement within a group with responsibility for the decision. In [3] we introduced our approach to constructing imprecise multi-attribute utility hierarchies. We described the structure which we use, which is based on a utility hierarchy with utility independence at each node, and explained the notion of imprecise utility trade-offs for such a hierarchy, based on limited collections of stated preferences between outcomes. This leads to a set R of possible trade-off specifications h. In this context we used a concept of Pareto optimality to reduce the set of alternatives. These methods and some associated theory are summarised in Section 2 of this paper. We are particularly concerned with problems where the number of alternatives among which we must choose is large. Many real decision problems, for example in experimental design, have very large spaces of possible choices. Relaxing the requirement for precise trade-off specification reduces our ability to eliminate choices by dominance and can leave us with a large class of choices, none of which is dominated by any other over the whole range of possible trade-offs allowed by the imprecise specification. We are therefore faced with the need for methods to reduce the decision space which are tractable even when the decision space is very large and there is a complicated multi-attribute utility structure to consider. Such methods should respect the range R of trade-offs and favour choices which perform well, in some sense, compared to others over the whole of R. In [4] we described ways to reduce the class of alternatives that we must consider, by eliminating choices which are ‘‘e-dominated” and combining choices which are ‘‘e-equivalent”. We explored the effects of different values of e and of different parts of the hierarchy to see when and why choices are eliminated.To choose a single choice d* from our reduced list, we can use the boundary linear utility approach described in [3], or choose the choice which is the last to be eliminated as we increase the value of our e criterion as described in [4]. We can then find the set D* of choices which are ‘‘almost equivalent” to d* and perhaps use secondary considerations to choose among them. We review boundary linear utility in Section 3 of this paper. In Section 4 we describe methods, based on the boundary linear utility, for exploring the sensitivity of possible choices to variation in the utility trade-offs. This helps us to find a decision which, as far as possible, is a good choice over the whole range of possible trade-offs. The practical implementation of our approach is illustrated throughout by an example concerning the introduction of a new course module at a university, which we first described in [4].

In [3,4] and this paper we have described an approach to multi-attribute decision analysis where the trade-offs between attributes are not precisely specified. Imposing the condition of utility independence makes the dimensionality of the tradeoff specification finite and allows us to work in terms of ranges for trade-off parameters. However, by imposing this condition only at the nodes of a utility hierarchy we can relax the requirement for mutual utility independence between all attributes. In our earlier papers we discussed how to reduce the number of alternatives for consideration and how to make a robust choice. In this paper we have considered the examination of sensitivity of our choice, in particular using the boundary linear utility. Our procedure may be summarised briefly as follows. Having identified the attributes of interest, the forms of marginal utilities and the possible choices and elicited beliefs about the values, we address the multi-attribute utility. 1. We construct a hierarchy combining utilities at nodes where the assumption of mutual utility independence between the parents can be made. Nodes where just two utilities are combined are binary (3). Nodes where more than two utilities are combined may be additive (2) or multiplicative (4). In each case the scaling preserves the (0,1) range and the interpretation. 2. We elicit ranges for the trade-off parameters at each node. See Section 5 of [3]. 3. We eliminate choices which are dominated. To do this it is only necessary to consider the vertices of the feasible region R. See Section 6 of [3]. We also eliminate choices which are equivalent to retained choices, retaining just one member of each equivalence class. 4. When there is still a large number of choices remaining we eliminate more using the concepts of almost-preference as described in [4]. Choices which are almost-dominated are eliminated and only one member of each almost-equivalence class is retained. This requires the choice of a small utility tolerance, e. 5. As described in Section 5 of [4], we explore the hierarchy to identify which nodes are influential in the elimination of choices and to examine sensitivity of the selection to the imposition of the elicited trade-off ranges. 6. We determine a candidate choice d* either (a) by using a boundary linear utility as described in Section 7 of [3] or (b) by selecting the last remaining choice as we increase the tolerance e, as described in Section 5 of [4]. 7. As described in Section 4 of this paper, we examine the robustness of our choice d* in comparison to other retained alternatives. We have proposed three procedures to do this. The first two are based on the metric induced by the coefficients of the boundary linear utility. The third uses distances in the space of the trade-off parameters themselves. In each case we wish to see whether d* ‘‘does well” in comparison to other choices over a large part of the feasible region of trade-offs. Volume sensitivity. We examine sensitivity to the choice of boundary linear utility coefficients k, by finding the proportion of the volume of possible k vectors where the difference in Uk between d* and another alternative is at least e. Distance in k-space. A choice of k may have been used to select d*. We examine how far this choice would have to be changed before another choice would be made. Sensitivity in the h-metric. A choice of k also determines a central h value, h0. Our choice d* is preferred to all others over some region around h0. We examine how far we can expand this region before we reach a point where some other choice would be preferred. 8. Finally we can return to the discarded members of the almost-equivalence class of d* and consider whether we wish to change to one of these, perhaps using secondary criteria. In this paper we have described and illustrated the use of three sensitivity measures. Only one, volume sensitivity, gave any computational challenge and, in this case, we have shown how this may be overcome using an efficient Monte Carlo algorithm.The example illustrated the use of our methods. We gained a better understanding of the issues which are important in making our choice and greater confidence in our selection of d1. We saw that d2 posed the most important challenge to the choice of d1 and identified node V as the main basis for this challenge. We believe that, in many difficult decision problems where a range of trade-off specifications must be considered, our methods could lead to the selection of a choice which is, in practical terms, close to optimal everywhere in the range.