**Author details**

Greanious Alfred Mavondo1 \*, Blessing Nkazimulo Mkhwanazi2 , Joy Mavondo3 , Wisdom Peresuh4 and Obadiah Moyo5

1 National University of Science and Technology, Faculty of Medicine, Pathology Department, Bulawayo, Zimbabwe

2 University of KwaZulu Natal (UKZN), Pietermaritzburg, South Africa

3 Imagegate Diagnostics (PL), Bulawayo, Zimbabwe

4 Labnet Laboratories (P/L), Bulawayo, Zimbabwe

5 Chitungwiza General Hospital, Chitungwiza, Zimbabwe

\*Address all correspondence to<EMAIL_ADDRESS>
© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**1.3 Caving zones**

**Figure 3** depicts the conceptual model of caving consisting of five major zones/regions. The conceptual model was based on the analysis of data collected at Northparkes Mines' E26 block cave, Australia [6] and consists of the following zones:

Caved zone (mobilized zone): This region consists of broken ore blocks that have fallen from the cave back. The material in the caved zone provides support to the cave walls. This is the bottom-most region close to drawbells.

Airgap: Extraction of broken ore creates a void volume inside the cave, and this region is called the airgap. During continuous caving, the height of the airgap formed is a function of the extraction rate of the material from the caved zone.

Zone of discontinuous deformation (yield zone): This region no longer supports the overlying rock mass and adheres to large-scale displacements of rock. Seismogenic zone: An active seismic front occurs due to slip on joints and brittle failure of rock, mainly due to changing stress conditions caused by the progress of the cave.

Surrounding rock mass: Elastic deformation occurs in the rock mass ahead of the seismic front and surrounding the cave.

An interesting point to note is that caved zone is the highest porosity region in the cave (if the airgap is not considered). Over time, as the cave evolves and progresses, the caved zone reaches the economic ore boundary, defined as "fully developed" in terms of ore production. The cave is then termed as a "fully developed" or "mature cave." A mature cave consists of different porosity zones consisting of broken ore and waste (based on the degree of dilution) of various sizes in the caved area with zero airgap. The particles'size changes from finer to coarser as we move from caved zone to the surrounding rock mass zone, as shown in **Figure 3**.


**2. Preliminaries**

In this part, we review some basic concepts of LIFS, PLTS and PULTS, and point out the main disadvantage of these fuzzy sets.

#### *2.1. PLTS and PULTS*

For convenience, all the LST mentioned in this article are represented by *S* = {*sα*|*α* ∈ [0, 2*τ*]} except for special explanations. In order to present the probability distribution information of the HFLTS, Pang et al. [14] proposed PLTS.

**Definition 1** ([14])**.** *Let S* = {*sα*|*α* ∈ [0, 2*τ*]} *be a continuous LTS, then a PLTS is defined as*

$$L(p) = \{ L^{(k)}(p^{(k)}) | L^{(k)} \in \mathcal{S}, p^{(k)} \ge 0, k = 1, 2, \dots, \#L(p), \sum\_{k=1}^{\#L(p)} p^{(k)} \le 1 \},\tag{1}$$

*where L*(*k*)(*p*(*k*)) *is the linguistic term L*(*k*) *associated with the probability p*(*k*)*, and* #*L*(*p*) *is the number of all different linguistic terms in L*(*p*)*. To further reflect the hesitation of DMs, Lin et al. [15] expanded PLTS into PULTS:*

$$\mathcal{S}(p) = \{ \langle [L^k, \mathcal{U}^k], p^k \rangle | p^k \ge 0, k = 1, 2 \cdots, \#\mathcal{S}(p), \sum\_{k=1}^{\#\mathcal{S}(p)} p^k \le 1 \},\tag{2}$$

*where* [*Lk*, *Uk*], *pk represents the uncertain linguistic variable* [*Lk*, *Uk*] *associated with its probability pk. Lk*, *Uk* ∈ *S are the linguistic terms, Lk* ≤ *Uk, and* #*S*(*p*) *is the cardinality of S*(*p*)*.*