Fuzzy Model for Selection of Underground Mine Development System in a Bauxite Deposit

In this paper, a fuzzy programming model, incorporating fuzzy measures of costs and ore reserves, is developed to evaluate different design alternatives in the context of the selection of the underground mine development system. The bauxite deposit is usually mined using the sublevel mining method. This method extracts the ore via sublevels, which are developed in the ore body at regular vertical spacing. In such an environment, we consider the development system as a weighted network interconnecting all sublevels with surface breakout point using the minimum cost of development and haulages. Selection of the optimal development system is based on the application of Convex Index and composite rank. The uncertainties related to the future states of transportation costs are modeled with a special stochastic process, the Geometric Brownian Motion. The results indicate that this model can be applied for solving underground mine development problems.


Introduction
The investment environment associated with the mining industry is unique when compared with the environment encountered by typical manufacturing industries. Some of the characteristics of mining which are often proclaimed as unique are as follows: capital intensity, long preproduction periods, high risk and nonrenewable resource [1]. Mines' development system investments provide a good example of irreversible investment. Such investment requires careful analysis because, once the investment takes place, it cannot be recouped without significant loss a value. Obviously, the selection of an underground mine development system belongs to strategic planning. Tahernejad et al. [2] emphasize that a lack of scientific planning, poor management and a lack of clear strategies are the most important problems of Iran's dimensional stone mines.
The problem discussed in this paper is 'Kostari' mine, a small-scale open-pit bauxite mine located in Bosnia-Herzegovina. Production of bauxite from the open-pit mine is approaching the end. The management of the company has estimated that the remaining reserves of bauxite can only be mined by an underground method. The question of which underground mine development system is suitable for accessing and exploiting a deposit is one that mine engineers and planners are faced with, when investigating the most efficient production system. Basically, there are three main development systems to gain access to an ore body: vertical shaft, decline (ramp) and adit. These three systems can be mutually combined, and in that, the number of potential alternatives is increased. Generally speaking, the process of selection of an underground mine development system encompasses the identification, evaluation and selection among alternatives.
To solve the problem of selecting a suitable development system, we consider an underground mine development system as a network interconnecting all access points with surface breakout point, using the minimum cost of development and haulage. The major task is to design the lowest costfeasible development system, respecting all operational constraints.
In our case, conditions prevailing on the surface and deposit are not suitable to apply adit as the development system. Shaft, Decline and Shaft-Decline development systems are identified as potential alternatives for the evaluation process. From each sublevel access point, we span the ore transportation network to the surface in three alternative directions. Each direction corresponds to one development system. Every section of the network is weighted by an adequate fuzzy cost function, which combines the cost needed to build up the section and the cost of ore transportation along it. The values of this function are changed over the project time, using a stochastic process, Geometric Brownian Motion, to simulate them. A set of potential alternatives is preferenceranked, according to the ascending order of Convex Index. By evaluating the networks spanned from each sublevel, it can be seen how depth, sublevel ore reserves and fixed production rate affect the efficiency of the development system.

Fuzzy Sets Theory
In order to deal with vagueness of human thought, Zadeh [8] first introduced the fuzzy set theory. This theory was oriented to the rationality of uncertainty, owing to imprecision or vagueness. A fuzzy set is a class of objects with a continuum of grades of membership. The role of fuzzy sets is significant when applied to complex phenomena not easily described by traditional mathematical methods, especially when the goal is to find a good approximate solution [9]. Modeling using fuzzy sets has proved to be an effective way of formulating decision problems, where the information available is subjective and imprecise [10].

Linguistic Variable
A linguistic variable is a variable whose values are words or sentences in a natural or artificial language [11]. As an illustration, age is a linguistic variable if its values are assumed to be fuzzy variables, labeled young, not young, very young, not very young, etc., rather than the numbers 0, 1, 2, 3… [12].

Fuzzy Numbers
A fuzzy numberM is a convex normalized fuzzy setM of the real line R [12]: -it exists such that one x 0 ∈ R with μM (x 0 ) = 1 (x 0 is called mean value ofM) -μM (x)is piecewise continuous. Triangular fuzzy number can be defined as a triplet (a,b,c). The membership function is defined as [13]:

Problem Formulation
Considering a directed network (in the context of the orehaulage direction) that is composed of a finite set of nodes and a set of directed arcs, we denote each arc by an order pair (i, j), where i and j are different nodes, respectively. The arc length is the distance needed to traverse (i, j) from node i to j. It is denoted by l(i, j). Formally, the problem is to find the fuzzy least cost path from the origin node (access point) to the destination node (surface breakout point). Figure 1 shows an example of the development network with several possible development paths, and Fig. 2 presents an adequate directed graph of the possible ore-haulage paths for the production area (sublevel) 1-2. The fuzzy least cost path problem can be formulated as the following form. The objective functioñ    Fig. 2 The graph of the ore-haulage system for the production area 1-2 has to be minimized, subject to: x i j ∈ {0, 1} ∀(i,j) ∈ A ( 4 ) wheref i -the triangular fuzzy cost function in node i is added to the outgoing edge from node i, but not to an incoming edge;f i j -the triangular fuzzy cost function between node i and node j (for i =1, 2,…,n and j = 1, 2, . . . , n); x i j -the decision variable defines binary variables x i j , where x i j = 1 if the form (i to j) is on the path and x i j = 0 otherwise;C D A i -total cost of building (driving) of development forms that are used only for the purpose of accessing production area; ore haulage will not be done along them ($);C D H i -total cost of building (driving) of development form that is used for the purpose of accessing production area, and to haul the ore along it ($);C H i j (t)-the cost of ore haulage along the development form, which changes over project time ($).
The objective function, defined by Eq. (2), refers only to one sublevel. Let G(P) be a graph of P, where P = { p 1 , p 2 ,…, p y } is a set of fuzzy possible paths from sublevel to surface (see Fig. 2). The solution of Eq. (2) is defined by the least cost path p = min( p 1 , p 2 ,…,p y ) having the minimum Convex Index. The path p is the most suitable path, and we assign rank 1 to it. The rest of the paths are ranked further, in ascending order of their Convex Index values, and we assign values 2, 3,. . .,y to them, respectively. It means there is one rank order of given paths p 1 , p 2 ,…,p y (for example, p 3 , p 1 , p 2 ; with assigned values p 3 →1, p 1 →2, p 2 →3) for one sublevel. Suppose there are k sublevels that should be mined during the planned period. If we take into consideration the previous assumption, then there are k rank orders of given paths, one for each sublevel.
According to above discussion, our problem can be represented as an alternatives, attributes, evaluations ( A,X ,E) model. We consider: a finite set of alternatives, i.e., development systems A( p 1 , p 2 ,…, p y ); a finite set of attributes, i.e., assigned values X (x 1 ,x 2 ,…,x k ), according to solution of Eq.
(2); and a set of evaluations of alternatives with respect to attributes.
The main goal of our model formulation was to determine the development system that should be used to access all k sublevels, and to haul ore from them to the surface. The solution is based on the composite ranking of given alternatives. The optimal development system for a given ore deposit is selected by the following equation: The complete procedure will be described in Sect. 3.3.

The Cost Functions
The general fuzzy cost function for development form that is used for access and ore haulage is: where l = the total length of the development form (m); Q = the estimated quantity of ore to be transported along the development form (t),Q =P r ·R.P r = estimated reserves (t);R = reserve recovery ratio (%);δ = the unit cost of building of a development form ($/m);c(t) = the unit cost of ore haulage ($/tm); λ = number of sublevels that will be accessed from the specified development form, or number of sublevels that will use it as a haulage path or both (λ = 1, 2, . . ., k). We first analyze the cost of vertical development form, called 'shaft'. The length of the vertical shaft is the difference in height between the top and the base of the shaft. In the development network, we treat the shaft as a vertical line segment with variable cost of the form: whereã s andb s are operational parameters associated with the hoist costs, n s is the number of parts of the shaft between loading point and surface. For example, for the (1-2), (2), (3), (4), (5) ore-haulage path (see Fig. 2), n s takes a value of 2, and for the (1-2), (1-1), (4), (5) ore-haulage path, n s takes a value of 1. The shaft cost function for the second ore-haulage path is different from the shaft cost function for the first ore-haulage path, and its value is represented by a 'bold curve' between nodes (4) and (5). The cost function of the horizontal development form, called the 'drive', is: When we define the cost function of the declined development form called the 'ramp', the physical constraint related to transportation path gradient has to be involved. This operational constraint shows that each ramp must have a maximum allowable absolute gradient of r , where r depends on the type of mining transportation equipment, such as haulage trucks. Let (β), connecting these two points, is equal to absolute value of the slope from A to B. The cost function of the ramp is: The cost function of the ore-pass is a little bit different from the general cost function because there is no second term related to haulage cost. Gravitation is used as a way to transport ore. The cost function of the ore-pass is: where α is the gradient of the ore-pass. The ore reserve quantities to be excavated and transported depend directly on the way of stoping, called the 'mining method'. Efficiency of the mining method can be expressed by the reserve recovery ratio. Fuzzy linguistic variables used to describe recovery ratio are as follows: very low (VL), low (L), medium (M), high (H) and very high (VH). The next step transforms the fuzzy linguistic variables to triangular fuzzy numbers, as shown in Table 1.
To estimate an adequate value for the reserve recovery ratio, it is necessary to get opinions of experts dealing with underground mining methods. Suppose we have p experts and each of them has given their opinion. The final value is expressed by an aggregated fuzzy number obtained by averaging the fuzzy opinions of the experts. Table 2 presents the expert estimation process.
The concept of the triangular fuzzy numbers is also applied to the estimation of the unit cost of construction of development form. For example, the cost of a shaft-sinking operation depends directly on rock mass properties. Generally, it is very hard to predict with certainty what may happen when sinking at depth. Accordingly, we cannot define the cost of sinking as a crisp value; we use triangular fuzzy numberδ(δ 1 , δ 2 , δ 3 ) to define it.
The uncertainties related to the future states of costs of transportation are modeled using a special stochastic process, Geometric Brownian Motion. Certain stochastic processes are functions of a Brownian motion process and have many applications in finance, engineering and the sciences. Some special processes are solutions of Itô-Doob-type stochastic differential equations [14]. We applied a continuous time process using the Itô-Doob-type stochastic differential equation to describe movement of unit costs of transportation in this study: where μ is the drift and σ is the volatility, W t is a normalized Brownian motion. In order to estimate the parameters of the Brownian motion process (μ,σ ), we run the following regression: The main objective of using simulation in the selection of the development system is to determine the distribution of the unit costs of transportation for every year of the project. In this way, we obtain the sequence of probability density functions of unit costs, where T is a total project time. Sequence of obtained pdf i of unit costs can be transformed into a sequence of triangular fuzzy numbers of unit costs, c i ∼TFN i , i = 1, 2, .., T , i.e, c 1 ∼pdf 1 → c 1 ∼TFN 1 ; c 2 ∼pdf 2 → c 2 ∼TFN 2 ;…; c i ∼pdf i → c i ∼TFN i . The way of transformation is based on the following fact; the support of the membership function and the pdf are the same, and the point with higher probability (likelihood) has the higher possibility. For more details, see [15]. The uncertainty in the parameter is modeled by triangular fuzzy number with the membership function, which has the support of η − 2σ < X < η + 2σ, set up for around 95 % confidence interval of normal distribution function. If we take into consideration that the triangular fuzzy number is defined as a triplet (a,b,c), then a and c are lower bound and upper bound obtained from lower and upper bound of 5 % of the distribution, and the most promising value b is equal to mean value of the distribution [16].
The production plan can be defined approximately, as follows: wherẽ t φ−1 -point when the previous sublevel is mined and φ-th sublevel is started to be mined (year); t φ−1 +t s -point when the φ-th sublevel is mined and φ +1 sublevel is started to be mined (year); t c -construction period (year); Y p -yearly production rate (t/year);t s -time of mining of φ-th sublevel (year); ktotal number of sublevels.
According to the defined production plan, unit costs of transportation related to the φ-th sublevel are expressed as follows: where E(t φ )-de-fuzzified value of fuzzy triangular numbert φ = (a, b, c) It is obtained according to the following equation: (a + b + c)/3. In this way, the interval of time within each sublevel to be mined is transformed into a crisp interval time; i.e.: Unit costs of transportation related to mining of the φ-th sublevel are calculated as follows: wherec[E(t φ−1 )]-unit costs of transportation in the year when mining of the φ-th sublevel is started;c[E(t φ−1 +t s )]unit costs of transportation in the year when mining of the φ-th sublevel is over.

Model of Selection of Development System
The selection of the underground mine development system in a fuzzy environment is the task of finding the shortest path in a fuzzy weighted network. In this paper, we apply an algorithm for the fuzzy shortest path problem based on the Convex Index [17]. To apply this algorithm, it is necessary to introduce the following definitions.

Definition 1
The α-cut interval is obtained as follows for all α ∈ [0, 1]: = (a, b, c), for all α, λ ∈ [0, 1] , where λ is the index of optimism. IfÃ and B are two triangular numbers, then in the Convex Index, we haveÃ <B if CoI (Ã) < CoI (B) According to the above derivation, we can pose our problem for one access point (sublevel) as follows: I nput: for each access point (sublevel), a mine designer creates a directed haulage network of n nodes with fuzzy edge cost functions. Out put : the fuzzy least cost function and corresponding least cost development haulage path.

Definition 2 Convex Index (CoI); LetM be a triangular fuzzy number, then CoI
Algorithm for solving the problem of selection of underground mine development system is composed of the following steps: ), for all possible path cost functions F i (X, t)=(a i (t),b i (t),c i (t)), i =1,2,…,y, using Definition 2. 7. Step7. Determine the actual least cost development haulage path with the minimum CoI i and assign a value of 1 to it. 8. Step8. Make a rank order of the rest of paths according to ascending order of Convex Index, and assign value to each path (2,3,…,y). 9. Step9. As the ore body is dipping, repeat Step13. For each row of E matrix, compute the sum of all the terms in the row. The alternative that corresponds to the row with the minimum sum of the terms is the optimal development system for given ore deposit [see Eq. (7)].

A Numerical Example Statement
To illustrate the proposed procedure, we applied it to a study considering the selection of an underground mine development system needed for the exploitation of a bauxite deposit.
Since the stripping ratio is approaching the planned value, the management of the company is faced with the problem of increasing production costs at an active open-pit mine. The management has decided to start a project of underground mining of the remaining bauxite reserves. Part of the deposit that should be mined is located between level 710 and level 494. Deposit is inclined at an angle of about 70 o . The underground mine should be designed for the capacity of production of 200,000 t/year. According to geological and mining conditions, the sublevel mining method is selected. The height of sublevel is 8 m. Two geological and two mining experts (E p = 4) were consulted to estimate the adequate values of reserve recovery ratio for each sublevel separately. The relevant operational data related to bauxite reserves are shown in Table 3. Data related to costs of construction and transportation are given in Table 4.

Numerical Example Solution
Three development systems have been evaluated: Decline development system; Shaft development system; and Shaft-Decline development system.
Decl i ne devel opment s yst em: main decline starts from level 768 and ends on level 718. The length of the main decline is 452 m and the gradient is 1:9. First sublevel access decline starts from level 718 and ends on the first mining level 710. The ore is removed from stope using Load Haul Dump vehicles. The ore is then dumped into a mine truck to be hauled to the surface via main decline. Construction period ist c = (1 1, 2 1, 5) year. There is no need to interrupt the production when lower levels are developed for mining.

Sha f t devel opment s yst em
: the shaft is sunk from level 768 to level 662. On level 718, the horizontal drive is constructed from the shaft to the deposit. Access decline is constructed from the horizontal drive to sublevel 710, and further via sublevels 702, 694, 686, 678 and 670, to the main transport level 662. Ore-pass connects sublevels 710, 702, 694, 686, 678 and 670 with the main transport level 662. Main transport level 662 connects ore-pass with the loading point located near the shaft. The ore is removed from the stope using Load Haul Dump vehicles and dumped down an orepass, where it falls to the main transport level 662. The ore is then dumped into a mine truck and transported to the shaft, to be hoisted up to the surface in skips and emptied into bins at the surface. Construction period ist c = (1, 5 1, 7 2) year.  Step7 and Step8. Rank order of the haulage paths according to ascending order of Convex Index; see Table  11.
Step9. Rank order of the haulage paths for all sublevels; see Table 12.
Step11. S,A v diagram according to data from Table 12 (see Fig. 5).

E =
Haulage paths, from sublevel S-710 to the surface, are weighted by corresponding fuzzy cost functions. Since they are represented in fuzzy format, a ranking procedure based on fuzzy number comparison needs to be applied. Comparison procedure was carried out in Step 5 and 6, where α-cut interval and Convex Index of fuzzy number were used as a basis for comparison. We assigned value of 1 to the haulage path p 2 , value of 2 to p 3 , and value of 3 to p 1 , according to ascending order of Convex Index; Step 7 and 8. The procedures of comparison and assignation were repeated for the rest sublevels (S-702,…,S-494), and S,CoI i and S,A v diagrams were created on the basis of obtained outcomes; Steps 9-11. The number of columns of matrix E is equal to the total number of sublevels, while the number of rows corresponds to the number of underground mine development systems which we evaluated. Elements of matrix E are equal to numerical outcomes (ranks) of the assignation procedure; Step 12.
Obtained composite rank order indicates the following rank of underground mine development systems: Shaft-Decline development system; Decline development system; and Shaft development system.
For comparison purposes, Fig. 6 shows the changeover points between proposed underground mine development systems, with respect to mine depth and sublevel ore reserves.        results more reliable. Network modeling gives the opportunity for the underground mine to be presented almost as a real physical model.

Conclusion
The selection of an underground mine development system is classified as a strategic decision-making process, which has the most influence on the future of a mine. A major design task is to determine the mine development system needed to provide access to each sublevel and provide haulage paths to transport excavated ore to the surface. We consider the development system as a network interconnecting all sublevels with surface breakout point, using the minimum costs of development and haulage. The selection model takes into account costs related to the construction of the development system, quantities of ore to be transported, lengths of transportation paths and unit costs of ore transportation. In essence, the selection of the underground mine development system corresponds to the selection of any other investment proposal. If we take into consideration this fact, then the network model enables a decision maker, who is not familiar with underground mining, to view this problem in a completely understandable way. The relevance of the proposed model is supported by the fact that the results were obtained under uncertainty. One of the main advantages of the proposed methodology is the quantification of uncertainties by fuzzification of the input data. In this way, we included risks into the process of strategic decision making, such as the selection of underground mine development system. Future scope of the work may consider to the extension of the model in the sense of a multi-criteria decision-making process, where composite rank is retained as one criterion and new criteria are added. New criteria can include indicators such as equipment reliability, complexity of construction and complexity of transportation route.