Discussions
Although in this example the competitive shortcomings of the OR 9000 can be iden-tified with all the three analysis approaches. Slightly changing the performance scores of OR 9000may induce different results.
The Keeney and Lilien [74]’s measurable value analysis requires ranges for all the evaluation attributes and mid-values for those ranges, while both Bordely and Kirkwood’s approach and our approach require that a target be specified for each attribute. These three methods require that attribute weights be assessed.
Both Bordley and Kirkwood’s approach and our approach consider that the perfor-mance of these two testers sets targets against which the OR 9000 is judged. However, as J941 and Sentry 50 are competitors of OR 9000, there should be some fuzzy uncer-tainty about the target itself. The main advantage of our approach is that it can capture the fuzzy quantities of target achievement. In addition, it is quite natural to consider interdependence among different targets, which is missed in both Keeney and Lilien’s ap-proach and Bordely and Kirkwood’s apap-proach. Our apap-proach can model the dependence phenomena via fuzzy measure and fuzzy integral.
Chapter 6
Prioritized Multi-Attribute
Target-Oriented Decision Analysis
Abstract: In multi-attribute target-oriented decision model, the importance information associated with different targets is important as some targets are more important than others. In general, the importance information plays a fundamental role in the compar-ison between alternatives by overseeing the tradeoffs between respective satisfactions of different targets. A concept closely related to the importance of targets is the priority of targets. Simply speaking, by saying target T1 has a higher priority than target T2, it indicates that we are not willing to tradeoff satisfaction of target T1 until perhaps we attain some level of satisfaction of target T2.
The main objective of this chapter is to study the prioritized multi-attribute target-oriented decision model, where there exists a prioritization of different targets. To do so, firstly, the ordered weighted averaging (OWA) operator will be used to obtain the satisfaction degree for each priority level. Secondly, we suggest that roughly speaking any t-norm can be used to model the priority relationships between the targets in dif-ferent priority levels. To keep the slight change of priority weight, strict Archimedean t-norms perform better in inducing priority weight. As Hamacher family of t-norms pro-vide a wide class of strict Archimedean t-norms ranging from the product to weakest t-norm, Hamacher t-norms are selected to induce the priority weight for each priority level. Thirdly, considering decision maker (DM)’s requirement toward the higher priority levels, abenchmark based approach is proposed to induce priority weight for each priority level, i.e., “the satisfactions of the higher priority attributes are larger than or equal to the DM’s requirements”. We suggest that the weights of lower priority level should depend on the benchmark achievement of all the higher priority levels.
6.1 Introduction
For a multi-attribute decision analysis (MADA) problem, with N attributes and N targets, a decision maker (DM) is defined to be target oriented if his or her utility for outcome (alternative) depends only on which targets are met by that outcome, where there is a single target for each attribute. As our research is based on the value-focused model [75] and we divide MATODA problems into two main steps: (1) on the target achievement of single attribute, (2) aggregation of partial target achievements into a global value. In this chapter, we assume that the target achievements of different attributes have already been obtained according to the target-oriented decision models proposed in Chapter 3 and Chapter 4. Thus we can simply view the MATODA problems as an aggregation problems, and MATODA can be viewed as a special case of MADA problems, where there exists a target for each attribute. In fact, traditional MADA problems assume that there exists one utility function for each attribute. Target-oriented decision model presumes a target-oriented utility. From now on, we shall use MATODA and MADA interchangeably.
In Chapter 5, several cases of MATODA problems have been introduced and discussed:
(1) general representation; (2) independent case; (3) additive preference case; and (4) non-additive case based on fuzzy measure and fuzzy integral. In all these four cases, the importance information associated with different targets/attributes is important as some targets/attributes are more important than others. In this case, the DM usually associates different importance weights with different targets/attributes. There are several approaches to incorporating and/or assigning weights to different targets/attributes [23, 79,94,101,125,126,141,144]. Typical is some form of weighted arithmetic mean, such as quasi-arithmetic means, weighted arithmetic means, weighted quasi-arithmetic means [23].
These aggregation operations work well in situations in which any differences are viewed as being in conflict because the operator reflects a form of compromise behavior among the various targets/attributes [94,137]. In general, the importance information associated with different attributes plays a fundamental role in the comparison between alternatives by overseeing tradeoffs between respective satisfactions of different targets [148, 150].
A concept closely related to the importance of attributes is the priority of attributes [36, 148]. In practical decision making situations, it is usual for DMs to consider different pri-orities of targets/attributes. A typical example is in the case of buying a car based upon the attributes of safety and cost. Assume that the DM specifies two targets; Tsafety and Tcost. In this case, usually we may not allow compensation between the target achieve-ments of cost and safety. Simply speaking, by saying target Tsafety has a higher priority than targetTcost, it indicates that we are not willing to tradeoff satisfaction of targetTcost until perhaps we attain some level of satisfaction of targetTsafety, This kind of MATODA, so-called prioritized MATODA, will be studied in this chapter.
Many studies have attempted to include different priorities of attributes into MADA problems in the literature. Generally speaking, approaches to prioritized MADA can be classified into two categories according to our knowledge. Approaches belonging to the first class aim to use non-monotonic intersection operator [58, 142] and triangular norms (t-norms) to model the priority relationships among attributes. For example, Yager [143] uses the non-monotonic intersection operator to deal with MADA problems and presents a type of attribute, called second order attribute. Yager [145] uses the weighted conjunction of fuzzy sets and fuzzy modeling to develop the operators in fuzzy
information structures. Chen and Chen [30] extend the non-monotonic intersection op-erator to present a prioritized multi-attribute fuzzy decision making problems based on the similarity measure of generalized fuzzy numbers. Luo et al [95] give five methods to construct the priority operators that are used for calculating the global degree of satis-faction of a prioritized fuzzy constraint problem based on Dubois et al [39]. The second class of approaches tend to use weighted aggregation operators to model the prioritized MADA. For example, Yager [148] shows that the prioritization of attributes can be mod-eled by using importance weights in which the weights associated with the lower priority attributes are related to the satisfaction of the higher priority attributes. Moreover, they provide some models that allow for the formalization of these prioritized MADA prob-lems using both the Bellman-Zadeh paradigm [12] for MADA and the ordered weighted averaging (OWA) operator. To develop this concept further, Yager [150] proposes a prior-itized averaging/scoring aggregation operator with a strict/weak priority order by means of the product t-norm. Furthermore, taking DM’s requirements into account, Wang and Chen [31,136] suggest that the weights of the lower priority attributes depend onwhether each alternative satisfies the requirements of all the higher priority attributes or not.
In this study, we focus on the second class of prioritized MADA, i.e., priority weighted MADA [31,136, 148, 150]. Although previous research has greatly advanced the priority weighted MADA, there are still some limitations and drawbacks in previous works.
1. Firstly, in prioritized MADA we will have a prioritization of attributes. Attributes in the same priority level should allow different tradeoffs. However, as we shall see in Section 6.3, Yager’s method [148,150] does not preserve this property.
2. Secondly, as suggested by Yager [148, 150], the product triangular norm is used to induce the priority weight for each priority level. However, as there are many types of t-norms available, can any t-norm be used to induce the priority weight? If so, which type of t-norms are better?
3. Thirdly, DM(s) may have a requirement toward the higher priority levels. The method of inclusion of DM’s requirements into satisfaction function proposed by Wang and Chen [31,136] will be too strict for DM to make decision under prioritized environments. In addition, due to the vagueness or impreciseness of knowledge, it is difficult for DMs to estimate their requirements with precision.
Motivated by the above observations, the objective of this paper is to propose a prioritized aggregation operator to overcome the limitations and drawbacks of previous works [31,136,148,150]. Toward this end, firstly, similar with Yager [148,150] and Wang and Chen [31,136], the OWA operator will also be used to obtain the degree of satisfaction for each priority level. To preserve the tradeoffs among the attributes in the same priority level, the degree of satisfaction for each priority level is viewed as a pseudo attribute.
Secondly, we suggest that roughly speaking any t-norm can be used to model the priority relationships between the attributes in different priority levels. To keep the slight change of priority weight, strict Archimedean t-norms perform better in inducing priority weight.
As Hamacher family of t-norms provide a wide class of strict Archimedean t-norms rang-ing from the product to weakest t-norm [110], Hamacher t-norms are selected to induce the priority weight for each priority level. Thirdly, considering DM’s requirement toward the higher priority levels, a benchmark based approach is proposed to induce priority weight for each priority level, i.e., “the satisfactions of the higher priority attributes are
larger than or equal to the DM’s requirements”. We suggest that the weights of lower pri-ority level should depend on the benchmark achievement of all the higher pripri-ority levels.
In particular, Lukasiewicz implication is utilized to compute benchmark achievement for crisp requirements. In case of fuzzy uncertain requirements, as target-oriented decision analysis [17,18] lies in the philosophical root of Simon’s bounded rationality [120] as well as represents the S-shaped value function [70], fuzzy target-oriented decision analysis [62]
is utilized to obtain the benchmark achievement.
The rest of this chapter is organized as follows. In Section 6.2 we recall some basic knowledge of aggregation operators in MADA problems operator and t-norms. In Sec-tion 6.3 we propose a prioritized weighted aggregation operator based on OWA operator and t-norms, we also compare our method with Yager’s prioritized aggregation opera-tor [148, 150]. In Section 6.4, we propose a benchmark based approach to induce the priority weight for each priority level by taking DM’s requirement toward higher priority levels in account. Considering the uncertainties of DM’s requirements, crisp and fuzzy uncertain benchmarks are studied. Comparative analysis with [31, 136] are also given to show the effectiveness and advantages of our proposed approach. Finally, we provide some concluding remarks and future work in Section 6.5.