**Appendix B. Proofs of (Non-)Associativity of Fusion Rules**

**Proposition A1.** *The quantified fusion rule as formalised in (7) is not associative, i.e., fu*(*I*1, *I*2, *I*3) = *fu*(*fu*(*I*1, *I*2), *I*3) *with Ii* = *πi.*

**Proof by Counterexample.** Assume three possibility distributions defined by key points (*x*, *π*(*x*)) as follows: *π*<sup>1</sup> = ((0.1, 0),(0.2, 1),(0.6, 1),(0.75, 0)), *π*<sup>2</sup> = ((0.2, 0), (0.3, 1), (0.5, 1), (0.55, 0)), and *π*<sup>3</sup> = ((0.35, 0),(0.7, 1),(0.8, 1),(0.9, 0)). With *j* = 2, fusion results using (7) *<sup>π</sup>*(*f u*) <sup>1</sup> <sup>=</sup> *fu*(*I*1, *<sup>I</sup>*2, *<sup>I</sup>*3) and *<sup>π</sup>*(*f u*) <sup>2</sup> = *fu*(*fu*(*I*1, *I*2), *I*3) are clearly different as shown in Figure A1. This example proves that (7) is not associative.

**Figure A1.** Three possibility distributions fused by the quantified fusion rule (7). Plot (**a**) shows *<sup>π</sup>*(*f u*) <sup>1</sup> <sup>=</sup> *fu*(*I*1, *<sup>I</sup>*2, *<sup>I</sup>*3). Plot (**b**) shows *<sup>π</sup>*(*f u*) <sup>2</sup> = *fu*(*fu*(*I*1, *I*2), *I*3).

**Proposition A2.** *The majority-opinion-guided possibilistic fusion rule (14) is not associative, i.e., fu*(*I*1, *I*2, *I*3) = *fu*(*fu*(*I*1, *I*2), *I*3) *with Ii* = *πi.*

**Proof by Counterexample.** Assume three possibility distributions defined by key points (*x*, *π*(*x*)) as follows: *π*<sup>1</sup> = ((0.1, 0),(0.2, 1),(0.6, 1),(0.75, 0)), *π*<sup>2</sup> = ((0.2, 0), (0.3, 1), (0.5, 1), (0.55, 0)), and *π*<sup>3</sup> = ((0.35, 0),(0.7, 1),(0.8, 1),(0.9, 0)). With *rel* = (1, 1, 1), **v** = (1, 1, 1), **w***<sup>p</sup>* = (0, 0, 1), and **w***<sup>m</sup>* = ( <sup>1</sup> 3 , 1 3 , 1 <sup>3</sup> ), fusion results using (14) *<sup>π</sup>*(*f u*) <sup>1</sup> = *fu*(*I*1, *I*2, *I*3) and *π*(*f u*) <sup>2</sup> = *fu*(*fu*(*I*1, *I*2), *I*3) are clearly different as shown in Figure A2. This example proves that (14) is not associative.

**Figure A2.** Three possibility distributions fused by the majority-opinion-guided possibilistic fusion rule (14). Plot (**a**) shows *<sup>π</sup>*(*f u*) <sup>1</sup> <sup>=</sup> *fu*(*I*1, *<sup>I</sup>*2, *<sup>I</sup>*3). Plot (**b**) shows *<sup>π</sup>*(*f u*) <sup>2</sup> = *fu*(*fu*(*I*1, *I*2), *I*3).

### **References**

