The combination of dependence-based interval-valued evidential reasoning approach with balanced scorecard for performance assessment

Abstract

The evidential reasoning (ER) approach can model multiple attribute decision analysis problems with both quantitative and qualitative attributes under the uncertain environment. In real situations, however, giving precise (crisp) assessments for alternatives is often too restrictive and difficult for experts, due to the incompleteness or the lack of information, knowledge and data. And there may be dependence among attributes which is difficult to mathematically model or ill known. To deal with these situations, this paper develops a dependence-based interval-valued ER (shortly called DIER) approach based on Denoeux’s cautious conjunctive rule which is an operator to combine belief functions from dependent sources. A pair of nonlinear optimization problems considering the relative weights of attributes is constructed based on the cautious rule to aggregate dependent attributes. Furthermore, due to the dependence of balanced scorecard (BSC) perspectives, the combination of DIER approach with BSC, called the DIER-BSC, is formed to implement the performance assessment under the uncertain environment. Finally, the performance assessment of the sensors department in a manufacturing company, which provides oxygen supplying and cooling devices for aviation, is demonstrated as an example to verify the validity and usefulness of DIER-BSC.

Introduction

As a practical method of modeling multiple attribute decision analysis (MADA) problems with both quantitative and qualitative attributes under the uncertain environment, the evidential reasoning (ER) approach has been developed in the 1990s and in recent years (Chin et al., 2009, Guo et al., 2007, Wang et al., 2006, Yang and Sen, 1994, Yang and Sen, 1997, Yang and Singh, 1994, Yang and Xu, 2002a, Yang and Xu, 2002b).

However, due to the incompleteness or the lack of information, knowledge and data, experts or decision makers may often feel too restrictive and difficult in some situations to give precise (crisp) assessments which results in partial or total ignorance. To deal with these situations, experts are encouraged to give interval-valued assessments which are interval-valued belief structures (IBSs) (Denoeux, 1999, Lee and Zhu, 1992, Wang et al., 2006, Wang et al., 2007, Yager, 2001). Correspondingly, the ER approach is extended as the interval-valued ER (shortly called IER) approach (Wang, Yang, Xu, & Chin, 2006).

Although the IER approach can effectively deal with conflicting attributes by considering the relative weights of attributes, it is incapable of handling dependent attributes. That is to say, it cannot avoid that some common parts on different attributes are counted twice, which is naturally difficult to describe, as demonstrated by Dempster (1967).

When the interaction between items of evidence can be modeled mathematically (e.g., Dubois and Prade, 1986, Smets, 1986, Wu et al., 1996), Dempster’s rule or the conjunctive rule of transferable belief model can be extended to apply in the IER approach. However, due to the fact that the interaction between items of evidence is usually ill known, or almost impossible to describe in practice, sometimes it cannot be modeled mathematically (Denoeux, 2008).

An operator proposed by Denoeux, called the cautious conjunctive rule, provides a feasible basis to implement the aggregation of dependent attributes in the IER approach, whether the interaction between interval-valued assessments can be modeled mathematically or not. The operator is commutative, associative and idempotent simultaneously, unlike other existing operators. And it is suitable for ubiquitous cases instead of some special cases, such as simple belief functions (the simple belief function has at most two focal elements; and if it has two, the frame is one of them) and separable belief functions (Denoeux, 2008).

Additionally, although Kallel and Hegarat-Mascle extended the cautious conjunctive rule by parameterizing the non-distinctness with their generalized discounting in order to estimate the actual correlation between two belief functions (Kallel & Hegarat-Mascle, 2009), the extended rule is not suitable to be applied in the IER approach. First, it has no associativity property except the parameters {0, 1}. Second, precisely parameterizing the non-distinctness among attributes is not easy for the decision maker. Third, as they stated, their current work is not enough to practically handle three non-distinct sources and solve the conflict between sources.

Aiming at the MADA problems with dependent attributes under interval uncertainties, this paper extends the IER approach to a new dependence-based interval-valued ER (shortly called DIER) approach. A pair of nonlinear optimization problems based on the cautious conjunctive rule considering the relative weights of attributes is constructed to aggregate dependent attributes.

The consideration of the relative weights of attributes in the optimization problems certainly transforms the assessments on attributes to non-dogmatic ones (the frame is a focal element), which guarantees the prerequisite of applying the cautious conjunctive rule. The difference between the consideration of the relative weights of attributes and the discounting process is explained by introducing the ER nonlinear models for aggregating interval-valued assessments in Section 2.4.

To implement the performance assessment under the interval-uncertain and dependent environment, this paper combines the DIER approach with the balanced scorecard (BSC), which is a well performance assessment framework developed by Kaplan and Norton, 1992, Kaplan and Norton, 1993, Kaplan and Norton, 1996a, Kaplan and Norton, 1996b to provide a holistic consideration of financial and non-financial measures. The combination forms the DIER-BSC to effectively handle the intrinsic dependence among BSC perspectives. Its detailed necessity and process will be discussed in Section 4.

The rest of this paper is organized as follows. Section 2 reviews the relevant concepts. Section 3 introduces the DIER approach. Section 4 presents the necessity, modeling and procedure of DIER-BSC. The performance assessment of the sensors department in a manufacturing company, which provides oxygen supplying and cooling devices for aviation, is implemented as an example to demonstrate the validity and usefulness of DIER-BSC in Section 5. And this paper is concluded in Section 6.