تجزیه و تحلیل منافع، فرصت ها، هزینه ها، و ریسک ها (BOCR) با AHP-ANP : اعتبارسنجی انتقادی

This paper shows that the usual multiplicative synthesis of alternative priorities for benefits, opportunities, costs and risks, obtained from separate Analytic Hierarchy or Network models, can be ambiguous. The ratio of benefit and opportunity priorities to cost and risk priorities can be misleading when assessing the profitability of a project. The same holds for their additive synthesis. Both types of synthesis have been advocated in AHP/ANP literature. A quotient of these priorities with weights as coefficients, not powers, will however produce sound results, provided that the four separate models are properly related to each other by weights that make the priorities on the four factors commensurate and are obtained from magnitude comparisons. Similarly, additive synthesis with properly weighted factor priorities based on relative magnitudes will produce sound results, although use of reciprocal values of costs and risks, as often advocated in the literature, is not recommended; negative costs and risks priorities should be used instead.

A decision on whether or not to undertake a project usually requires investigating the positives (benefits) and negatives (costs) of that project and an attempt to express those in monetary terms such as dollars. If that project has a benefit/cost ratio >1, its benefits outweigh its costs. If there are several projects to choose among, those projects will usually be ordered according to their respective benefit/cost ratios. The ones with a ratio >1 are the attractive ones exceeding the break-even point, and the one with the highest ratio among these gives the highest return on money spent and would therefore very likely be the one to be chosen. The problem is that often benefits and costs are difficult to express in monetary terms, especially when some of the benefits or costs are intangible, such as “improved accuracy” or “learning efforts”. The Analytic Hierarchy Process (AHP) developed by Saaty [1] has been advocated as an approach that not only can deal with both tangibles and intangibles but also helps organize all aspects involved in a hierarchic structure where the benefit or cost aspects act as criteria and the projects as alternatives. Usually, we have separate hierarchies: one costs hierarchy and one benefits hierarchy. One has to pairwise compare the importance of cost criteria in the cost hierarchy, and the same with respect to the benefit criteria in the separate benefits hierarchy. These processes produce relative criteria weights expressed on a derived ratio scale, usually normalized to the unity sum for each family of criteria in each hierarchy. The alternative projects are pairwise compared with respect to each criterion on the lowest level of each hierarchy; their derived priorities are expressed on a ratio scale as well, again usually normalized to the unity sum per criterion. Synthesis of the alternative priorities and the criteria weights using a weighted sum produces composite alternative priorities for each hierarchy. For each alternative, its composite benefit priority is then divided by its composite cost priority. The resulting ratio value serves as a means to rank the alternatives and choose the best one, i.e. the alternative with the highest benefit/cost-priority ratio. Examples of benefit/cost analysis using the AHP were published in Saaty [1] and [2]. Benefit/cost analysis with the AHP has been criticized though. The main criticism is that different hierarchies produce priorities on different derived ratio scales which are usually not commensurate. The quotient of two ratio scales is again a ratio scale but has lost its clear relationship with the individual scales. The result may therefore appear meaningful as a measure of profitability, whereas it is in fact not: benefit/cost-priority ratios may be larger than unity when in fact the monetary costs exceed the monetary benefits. This is demonstrated in Wedley et al. [3]. Wedley et al. [4] reviewed previous literature which proposed procedures for proper benefit/cost analysis with the AHP, including suggestions by Saaty [2, page 151] to produce more meaningful ratios. Wedley et al. [3] suggested a formal magnitude adjustment procedure that converts the benefit and cost hierarchies to a common unit thus assuring that resulting benefit/cost ratios do have the desirable property of correctly indicating the break-even point. The questioning procedures they proposed are however cognitively difficult. Wedley et al. [4] proposed using linking pin methods, thereby potentially easing the questions for relating benefits to costs. In the AHP literature thus far, only risks (R) have been added to the B/C ratio, see for example Saaty [2, 164–166]. For each alternative a B/(C∗R) ratio is computed based on priorities that are obtained from three different hierarchies: a benefits, a costs, and a risks hierarchy. More recent publications by Saaty [5] and Saaty and Ozdemir [6] showed that a fourth factor, opportunities (O), can be added to the analysis. This allows a full BOCR analysis using a (B∗O)/(C∗R) ratio where positives not only include benefits but opportunities as well, and negatives not only costs but also risks. A full BOCR analysis is in some ways similar to a SWOT analysis, where not only the strong points (S) of a firm but also its (external) opportunities (O) are taken into account such as good chances of entering a new market and other favourable situations. Opportunities in BOCR analysis usually catch expectations about positive spin-off, future profits and revenue of future positive developments, whereas benefits represent current revenue or those profits from positive developments one is relatively certain of. Likewise, a firm’s weak (W) points may not tell the whole story of negative aspects in SWOT analysis; external threats (T) concerning competition or unfavourable developments in society must be dealt with as well. Risks in BOCR analysis are supposed to catch the expected consequences of future negative developments, whereas costs represent (current) losses and efforts and consequences of negative developments one is relatively certain of. BOCR analysis enables therefore a potentially richer analysis than a mere BC analysis, although many of the aspects that define the factors and their relationships are usually difficult to specify and quantify. Recently, the Analytic Network Process with supporting software was developed by Saaty [5], enabling one to model systems with feedback and dependence. Especially in the context of the ANP, many applications of BOCR analysis are offered where a network model with one or more networks for each of the four BOCR factors is set up instead of a hierarchy. This allows for modelling interrelationships between the elements defining each of the four overall factors. Not only B/C analysis but also B/(C∗R) analysis has been criticized, for example by Millet and Wedley [7]. These authors argue that the product of costs and risks is not meaningful or justified, with the added argument that differences in relative importance are not accounted for. Similar arguments may hold for BOCR analysis as well. Furthermore, it has been argued (Wedley [8]) that opportunity and risk priorities could be regarded as probabilities. Higher opportunity and risk priorities would then represent greater opportunity or risk likelihoods, and their products with benefit and cost priorities, respectively, would produce expected benefit and expected cost values. This paper does not enter into this discussion, although it does, for other reasons that will become clear later on, criticise the quotient of products as proposed by Saaty [5]. This paper will not question a BOCR analysis per se, nor will it address the way in which composite alternative priorities on each of the four merit factors (B, O, C, R) are computed, be it using AHP-based or ANP-based models. It is rather the computational method for getting meaningful synthesized results that is the paper’s subject. It therefore investigates the way in which the composite priorities of alternatives on each of the four factors (B, O, C, R) are synthesized into a final BOCR value (Fig. 1).It shows, supported by a validation example using fictitious monetary values, that one cannot be certain that any of the multiplicative or additive expressions for synthesis proposed by Saaty [5] and Saaty and Ozdemir [6] produce the correct ordering of alternatives, or give a reliable indication of profitability, except in special cases. It argues that this is due to the incommensurability of the composite priorities on the four factors. In the context of B/(C∗R) analysis, Millet and Wedley [7] touched upon this issue without elaborating it. The current paper suggests improved ways of synthesis that do not show the deficiencies mentioned above. The remainder of this paper is organized as follows. First, the reference example is introduced, with some additional assumptions. Then the results using the formulas for synthesis proposed in Saaty [5] and Saaty and Ozdemir [6] and implemented in the SuperDecisions [9] software are shown and critically discussed, after which the improved approach is presented in several steps. The paper concludes with a summary of the merits of the improved approach, and attempts to draw a generally valid conclusion.

In this paper benefits–opportunities–costs–risks (BOCR) analysis using AHP/ANP methodology was addressed. The computation of the composite priorities for the alternatives on each of the four BOCR factors may be based on hierarchy (AHP) or network (ANP) models, but is not the subject of this paper. It is rather the final synthesis of these four types of composite priorities that was investigated, using a multiplicative expression or two variants of an additive expression as advocated in the AHP/ANP literature. In order to be able to draw conclusions as to the validity of these types of synthesis, monetary equivalents of priorities were used in several scenarios. In reality, these monetary values are not known; they were merely used to investigate whether or not synthesis of priorities reproduces monetary results, assuming perfect consistency. The analysis in this paper suggests that it is crucial to express priorities on benefits, opportunities, costs and risks in commensurate terms. Otherwise, the results are meaningless or even deceiving (by incorrectly suggesting profitability of alternatives) and contradictory (showing rank reversals with results from more or less equivalent synthesis expressions), leading to bad decisions. This paper argues that although any set of weights reflecting relative importance can express priorities on a common priority scale, it is only a weighting scheme based on relative B, O, C and R magnitudes that will not only create commensurate priorities but also serve the purpose of validation (close approximation or even exact representation of “true” values) and allow a sound profitability analysis, equivalent to monetary break-even analysis even when the BOCR factors are intangible. If the choice must be made from a given set of alternatives without reconsidering that set, then the most sensible thing to do perhaps is to choose the one that has the highest priority. Nevertheless, if that priority would also be a reliable indicator of profitability, one may thereby be prompted to indeed try to improve an alternative or search for a better one if the indication should be unfavourable. This paper also argues that in a return on investment ratio oriented analysis, a synthesis expression should be used that is a quotient of positives (B and O) to negatives (C and R) with the rescaling weights as coefficients, not powers. The revised synthesis expression comes in two variants: one where rescaled (and thus commensurate) benefit and opportunity priorities are added in the nominator (and the same with the rescaled cost and risk priorities in the denominator), and one where they are multiplied. Multiplication produces a new unit that has no intuitive interpretation whereas addition has the advantage of keeping the original unit in both the nominator and denominator and should be recommended for this reason. Using a priority synthesis expression where negatives appear as reciprocals is not recommended even if they are made commensurate by weights that take the magnitude of reciprocal values into account. Further, in a net value oriented analysis, an additive synthesis expression should be used where rescaled cost and risk priorities are subtracted from rescaled benefits and opportunities priorities. Finally, if additional weights based on personal values are to be used to take account of feelings of relative importance of the four factors, this paper suggests that they should be applied to the rescaled priorities. Similar questions such as the ones proposed by Wedley et al. [3] regarding BC analysis enable BOCR priorities to achieve the desirable attributes of regular BOCR analysis. The disadvantage, however, is the cognitive difficulty to compare aggregate (or average) BOCR factors. Potentially simpler techniques for achieving commensurability were proposed by Wedley et al. [4] using linking pin technology. Since the unit of a derived ratio scale is arbitrary, a proportional transformation of the scale can put the unit in any hierarchy node. Furthermore, the link between hierarchies does not have to be at their topmost node (Schoner et al. [15]) but across a common alternative or well-chosen specific sub-factors. Before linking, one merely has to identify a node that becomes the unit of measurement for each hierarchy. Other nodes are then expressed in terms of that unit. Comparing abstract totalities seems mentally more demanding than comparing single nodes. However, how this should be done when having networks for the BOCR factors rather than hierarchies remains open for further investigation. A very recent suggestion was put forward by Saaty [16] and seems to be along the lines of the ideas above. It is based on the comparison of the ratings of one alternative across the four factors, which seems one of the ways in which their relative magnitudes may be captured provided that the other alternatives are expressed in terms relative to the referent alternative.