Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 610307 | https://doi.org/10.1155/2015/610307

Svetlana Strbac Savic, Jasmina Nedeljkovic Ostojic, Zoran Gligoric, Cedomir Cvijovic, Snezana Aleksandrovic, "Operational Efficiency Forecasting Model of an Existing Underground Mine Using Grey System Theory and Stochastic Diffusion Processes", Mathematical Problems in Engineering, vol. 2015, Article ID 610307, 18 pages, 2015. https://doi.org/10.1155/2015/610307

Operational Efficiency Forecasting Model of an Existing Underground Mine Using Grey System Theory and Stochastic Diffusion Processes

Academic Editor: Paolo Maria Mariano
Received03 Aug 2015
Revised28 Oct 2015
Accepted29 Oct 2015
Published22 Dec 2015

Abstract

Forecasting the operational efficiency of an existing underground mine plays an important role in strategic planning of production. Degree of Operating Leverage (DOL) is used to express the operational efficiency of production. The forecasting model should be able to involve common time horizon, taking the characteristics of the input variables that directly affect the value of DOL. Changes in the magnitude of any input variable change the value of DOL. To establish the relationship describing the way of changing we applied multivariable grey modeling. Established time sequence multivariable response formula is also used to forecast the future values of operating leverage. Operational efficiency of production is often associated with diverse sources of uncertainties. Incorporation of these uncertainties into multivariable forecasting model enables mining company to survive in today’s competitive environment. Simulation of mean reversion process and geometric Brownian motion is used to describe the stochastic diffusion nature of metal price, as a key element of revenues, and production costs, respectively. By simulating a forecasting model, we imitate its action in order to measure its response to different inputs. The final result of simulation process is the expected value of DOL for every year of defined time horizon.

1. Introduction

Efficiency is a prerequisite for the survival of every mining company, especially in high competitive market environment such as mineral resource industry. Efficiency signifies company’s ability to meet its short or long-term goals. Operational efficiency is defined as the ratio between the input required to run production process and the output gained from the production. In the context of mining business, operational efficiency refers to the length of time until mineral assets are transformed to money. Peak operational efficiency occurs when the right combination of mineral deposit characteristics, human resources, mining technology, and mineral processing come together to optimize mining performance.

Operational efficiency is related to finding the very best way of mining to produce a mineral asset. It enables management of the mining company to increase productivity, increase profitability, improve competitiveness, use freed-up capacity, and enable company to grow or increase its future market value.

Mining company’s management uses different strategies to reach their defined goals. One of the most important elements of a company’s management operations is to forecast what goals are realistic and capability estimation of the company in order to achieve them. Planners try to forecast the behavior of the input variables of production environment and arrive at desirable states. They create their strategies on realistic targets drawn from these forecasts. A forecast is based on past and current production indicators, that is, business numbers. In many cases, the forecast might be burdened by some inaccuracies, so it would be a mistake to base a budget on that. It is very important to emphasize the forecasting acts serve as a basis for further planning.

The forecasting process is performed in a specific environment. If we take into consideration the fact that the environment is changed over time then it is obvious that the forecasts and targets are changed as well. Management must be able to describe environment changes in order to strategically link the forecasting and planning functions, improving the performance of both.

Large capital intensive projects, such as those in the mineral resource industry, are often associated with diverse sources of both endogenous and exogenous uncertainties. These uncertainties can greatly influence the operational efficiency. Having the ability to plan for these uncertainties is increasingly recognized as critical to long-term mining company success. In the mining industry in particular, the relationships between input variables that are controllable and those that are not and the physical and economic outcomes are complex and often nonlinear. Operational efficiency forecasting of mine in today’s environment is much complex than it was just a few years ago. There are typically many variables, which are directly or indirectly associated with the forecasting process.

There is a considerable literature dedicated to the operational efficiency measurement. It includes many approaches, which take into account various aspects of the problem.

Briciu et al. [1] applied the concept of Cost-Volume-Profit analysis in monitoring and measuring the performance of companies in the mining industry in Romania. Zhao et al. [2] evaluated operating efficiency of Chinese Coal Mining Companies by Malmqusit Productivity Index. Factor analysis was employed by Li et al. to evaluate performance of coal mine companies [3]. Four types of empirical tests have been performed (parametric mean differences, nonparametric Wilcoxon rank sum test, static regression panel estimation, and dynamic regression panel estimation) to estimate managerial and operational efficiency of privatized mining companies in Jordan [4]. A stochastic frontier analysis method was used to estimate profit efficiency in the South African mining sector [5]. The analytic hierarchy process (AHP) methodology was selected for ranking the efficiency of selected platinum mining methods [6]. Improving the efficiency of truck/shovel operations can increase the total operational efficiency of mine. To make the truck and shovel fleet more optimum on the basis of productivity improvements and associated costs reduction, it was essential to calculate the best match factors and truck/shovels assignments (or fleet size) for the system [7].

This paper investigates the relationship between ore production rate, fixed costs, revenues, production costs, working days, and degree of use of production capacity, as input variables, and Degree of Operating Leverage (DOL) as indicator of operational efficiency of an existing underground mine. Multivariable grey model is used to establish the time sequence response formula of DOL. This formula enables us to predict the future values of DOL time series based only on a set of the most recent data. To decrease uncertainty of future values of the operational efficiency we use expert’s knowledge and simulation processes to find future values of input variables affecting them. Estimation of future revenues is based on the application of mean reversion process, normal and uniform distribution. Geometric Brownian motion is used to define future values of production costs. Values of ore production rate, fixed costs, working days, and degree of use of production capacity are estimated by expert’s knowledge.

By simulating a forecasting system, we imitate its action in order to measure its response (output) to different inputs. The simulation allows analysts to describe the uncertainty of variables that influence the value of DOL by different time depending scenarios. The first objective of the use of simulation in the forecasting is to determine the distribution of the DOL from the variables that affect his performance, which results in the average or expected value of DOL for every year of defined time horizon.

2. Grey Information Systems and Multivariable Grey Model

The grey theory established by Dr. Deng includes grey relational analysis, grey modeling, prediction, and decision making of a system in which the model is unsure or the information is incomplete [8]. It provides an efficient solution to the uncertainty, multi-input, and discrete data problem.

Grey models predict the future values of a time series based only on a set of the most recent data depending on the window size of the predictor. It is assumed that all data values to be used in grey models are positive, and the sampling frequency of the time series is fixed. From the simplest point of view, grey models can be viewed as curve fitting approaches [9].

In grey system theory, multivariable grey model () denotes a grey model, where is the order of the differential equation and is the number of variables. The is defined as follows [10]: where and are determined coefficients, is the major sequence factor, is the influencing sequence factors, and is the time sequence variable.

In this paper we are applying [1115]. First we separate the data into two sequences: major sequence factor (it masters the system behaviors) and influencing sequence factors (it influences the system behaviors).

The major sequence factor is represented as follows: The influence sequence factors are Assume the original series of data with equal time samples is represented as , . The first-order Accumulating Generation Operator (AGO) of is defined as , where for . This operator is used to smooth the randomness of the data and to weaken the tendency of variation. The average generation of adjacent sample sequence of is as follows: , where , for .

The first-order grey differential equation of the is According to the , the constructed AGO sequence is The constructed AGO sequence can be represented by the following matrix form: Factors of the matrix are obtained by using the least square method as follows: whereFinally, the can be expressed as follows: The time sequence response formula, that is, the solution of (9), isAccording to (10), we can obtain . The fitting forecast value of can be obtained by the inverse AGO as follows:The predicted values when can be used to check the adequacy of the model and predicted value when can be used as the forecast value for the data series.

The relative percentage error (RPE()) and the average relative percentage error (ARPE()) are as follows:Grey relational (relevance) analysis is part of grey system theory that gives us information about the relevance degree of each input variable in the system. It is utilized to ascertain the primary factors that are needed to make a superiority comparison in the system [11]. The fundamental principle is to recognize the relevance degree among many factors, according to the similarity levels of the geometrical patterns of sequence curves. We can judge whether they are close or not according to the similar degree of sequence curve shape. The more similar the curve is, the higher the correlation degree between relative series is, and vice versa [11].

Set the major sequence factor variable to be analyzed, , , as the dependent variable, and the influencing sequence factors, , , as the independent variables. Because various variables have different physical significances, that is, dimensions, it is difficult to obtain accurate conclusion. To eliminate dimensions from analysis, equalization method is used, and formula is The grey relational coefficient is calculated as follows [16]:where refers to the discriminating coefficient and ; .

The aggregated value of the grey relational coefficient is calculated as follows: The higher is, the bigger impact the th variable has on the variable to be analyzed; on the contrary, the lower is, the smaller impact the th element has on the variable to be analyzed. By this way, we can see how contribution of the input variables to output variable is changed over time.

Finally, the management of the company obtains the aggregated information vector which sublimates the following important indicators:(i)The forecasting model.(ii)The relative percentage error of forecasting model.(iii)The average relative percentage error of forecasting model.(iv)The grey relational coefficient between inputs and output for the th time point.(v)The aggregated grey relational coefficient between inputs and output.

3. Forecasting Model

3.1. Concept of the Model

Overall operational efficiency measures have enjoyed a great deal of interest among planners analyzing the performance of an underground mine. Suppose the production of mine in period can be represented by mining technology set (MTS): where is the input mining technology vector and is the output vector which represents the indicators obtained from production. The MTS is the space of the feasible combinations of input-output vectors in period .

The input vector of the MTS is composed of many variables but they can be divided into three main subspaces: characteristics of mineral deposit, underground mining methods, and mineral processing methods. The output vector is usually composed of the following variables: realized production rate and quality of produced mineral assets.

Operational efficiency is usually expressed by some kind of economic indicator. According to required type of operational efficiency, MTS can be transformed and joined to economic set (ES). This transformation means that some variables of and are used to calculate some variables of ES.

To express the operational efficiency of an existing underground mine we apply the concept of Degree of Operating Leverage (DOL). The DOL is the leverage ratio that sums up the effects of an amount of operating leverage on the company’s earnings (revenues) before interests and taxes (EBIT).

The DOL during the period can be calculated with following formula (Weygandt et al.) [17]: where REV is revenues (USD), PC is production costs (USD), and FC is fixed costs (USD).

Operating leverage refers to the amount of operating fixed costs in the cost structure. DOL is the ratio of the relative change in sales. Implications are related to amplifying the changes in sales volume into larger changes in EBIT. Even in a perfectly competitive market, price plays a role in determining the DOL magnitude. Assume that a firm is currently operating with a positive profit. When market equilibrium price rises, the denominator in (17) rises reducing the DOL with no change in output, operating fixed cost, or unit variable cost. Consequently, breakeven output falls since the contribution margin is larger. DOL can vary due to changes in any of the variables appearing in (17). These variables include management determined choices (operating fixed cost and output levels); market determined parameters (price in a competitive market as time passes); and economic and engineering realities (unit variable costs, given operating fixed cost increases due to new capital acquisition) [18].

The higher value of DOL indicates that the business of mining company is exposed to the greater risk.

The input vector that is used to evaluate operational efficiency is represented as the union of three following sets:is a subset of the input mining technology variables.is a subset of the output mining technology variables.is a set of the external variables.

According to our problem, we create the vector space of input variables as follows: where is ore production rate (t/year), is fixed costs (USD), is revenues (USD), is production costs (USD), is working days (days/year), and is degree of use of production capacity (%).

Let be a sequence of observed DOL values on an annual basis, that is, the major sequence factor. Equation (9) will be used for expressing the forecast model of DOL of an existing underground mine as follows: The time sequence response formula of DOL isThe fitting forecast value of can be obtained by the inverse AGO as follows:For we can make forecasts of DOL as follows:where is future time horizon.

The forecasting model should be able to involve a common time horizon, taking the characteristics of the input variables that directly affect the value of DOL. In this section, the proposed method of forecasting of DOL under multiple uncertainty and nonuniformity of input variables is outlined. The method is based mainly on simulating multiple realizations of the uncertain variables and making forecasts using value expectations.

According to the valuation function defined by (22), the expected value of DOL over future time is defined as follows: Before we start to create forecasting process, it is necessary to divide the set of input variables into two subsets; , where denotes the set composed of variables defined by experts, while denotes the set composed of variables defined by some kind of stochastic law. In our case these subsets areValues of , , , and are of deterministic nature while the others are stochastic.

3.2. Volatility of the Input Variables
3.2.1. Volatility of Revenues

Mining companies not having smelting facilities realize their revenues by selling their metal concentrates as a final product. Estimating revenues is, indeed, a difficult and risky activity. Annual mine revenue is calculated by multiplying the number of units produced and sold during the year by the sales price per unit. The annual mine revenue () is calculated according to the following equation: where is the annual production of ore (), is the value of the metal concentrate, is the grade of ore mined (%), is the mill recovery rate (%), is the metal content of the concentrate (%), and is the number of metal concentrates derived from the ore ( for polymetallic deposits).

The first major component of the mine revenue calculation is annual production of concentrate. One of the key variables associated with annual concentrate production is the tonnage of ore mined. Annual ore tonnage is derived from the mining project schedule and is denoted .

The second key variable associated with determining the annual production of saleable units is the grade of the ore mined. The concept of the ore grade () is defined as the ratio of useful mass of metal to the total mass of ore and its critical value fluctuates over space and can be estimated by normal probability density function. The use of normal distribution is based on geostatistical methods developed for evaluation of grades in mineral deposits. The deposit is sampled by means of drill-holes. Each sample gives us the information about mass of metal (grade) and we can create histogram of grades values. According to the obtained histogram adequate probability distribution function is selected. In most cases of mineral deposits normal distribution is used as representative function: At first sight, it seems that does not change its value over time, but if we take into consideration the mining front advances over time through different parts of deposit, then it is also time depending. For simplicity, we adopt the values of which belong to the same interval, defined by , over future time horizon.

Most ores require beneficiation before saleable product can be produced. The resulting milling losses must be estimated and appropriate recovery percentages established. These recoveries are commonly estimated from a metallurgical testing program. Percentage recovery is the third basic variable which must be estimated to arrive at a final estimate of the annual production of saleable units extracted from the mine and is denoted . A specific stochastic behavior that is used to quantify uncertainty related to mill recovery rate () is uniform probability density function. Perhaps it is virtually certain that mill recovery will lie in interval but it is possible to achieve it by mineral processing control. For example, flotation is the most widely used method for the concentration of fine grained minerals. It takes advantage of the different physicochemical surface properties of minerals in particular their wettability, which can be a natural property or artificially changed by chemical reagents. By altering the hydrophobic (water repelling) or hydrophilic (water attracting) conditions of their surfaces, mineral particles suspended in water can be induced to adhere to air bubbles passing through a flotation cell or to remain in the pulp. The air bubbles pass to the upper surface of the pulp and form a froth, which, together with the attached hydrophobic minerals, can be removed. The tailings, containing the hydrophilic minerals, can be removed from the bottom of the cell. According to that, within interval no value is more likely than any other. This gives rise to a uniform distribution: It seems that does not change its value over time, but if we take into consideration the fact that the flotation is influenced by many parameters and fact that it is very difficult to keep them constantly over time, then it is also time depending. For simplicity, we adopt the values of which belong to the same interval but takes the stochastic values for every year of the time horizon.

The second major component of the mine revenue calculation is unit sales price or unit metal concentrate sales price (). It directly depends on mineral asset price, metal content of the concentrate, and metal recovery rate. Estimating future mineral prices is an exercise for which a high error of estimation invariably exists. The characteristically long preproduction periods of mining projects mean that their success will be determined by mineral prices five to ten years in the future.

The market risks related to mineral asset price () are modeled with a special stochastic process, a mean reversion process. The mean reversion process has economic logic; for example, although the commodity prices have sensible short-term oscillations, they tend to revert back to a “normal” long-term equilibrium level. The past values of the changes in this uncertain factor help predict the future. We will use a model where the metal spot price is assumed to follow the stochastic process [19]:Let ; applying Ito’s lemma allows the characterization of the log price by an Ornstein-Uhlenbeck stochastic mean reverting process: with where is the long-run equilibrium metal price, measures the speed of mean reversion to the long-run mean log price , is an increment to a standard Brownian motion, and refers to the price volatility rate. The metal price adjustment mechanism is accounted for by market forces. The correct discrete-time format for the continuous-time process of mean reversion is the stationary first-order autoregressive process [20], so the sample path simulation equation for is performed by using exact discrete-time expression:where is the fixed time interval from time to and is the normally distributed random variable.

By substituting (31) to , we have exact discrete-time equation for , given by In order to estimate the parameters of the mean reversion process, we run the following regression: where and . Hence, if we regress observation against , we can obtain estimates of and . is the standard deviation obtained from the regression. The speed of mean reversion () is the negative of the slope, while the long-run equilibrium is the intercept estimate of that regression divided by the speed of mean reversion.

Let denote a price scenario with spot prices , where is determined by (32). Figure 1 presents a sample path of the metal price (e.g., lead) simulated using the above equation.

In the purpose of simulation, we apply a metal price scenario for the time interval, , with increment .

The unit value of metal concentrate sales price can be expressed as follows: where is the metal recovery rate (%). The value of is calculated in different ways with respect to type of metal.

3.2.2. Volatility of Production Costs

Production costs (PC) are incurred directly in the production process. These costs include the ore and waste development of individual stopes, the actual stoping activities, the mine services providing logistical support to the miners, and the milling and processing of the ore at the plant. The uncertainties related to the future states of unit production costs are modeled with a special stochastic process, the geometric Brownian motion. Certain stochastic processes are functions of a Brownian motion process and these have many applications in finance, engineering, and the sciences. Some special processes are solutions of Itô-Doob type stochastic differential equations (Ladde, Sambandham) [21].

In this model, we apply a continuous time process using the Itô-Doob type stochastic differential equation to describe movement of unit production costs. A general stochastic differential equation takes the following form: Here, , is a Brownian motion, and ; this is the cost process.

is called the geometric Brownian motion, which is solution of the following linear Itô-Doob type stochastic differential equation: where is the drift, is the volatility, and is normalized Brownian motion.

Using the Itô-Doob formula applied to , we can solve this equation: Equation (37) describes an operating cost scenario with spot costs .

Let denote a cost scenario with spot costs , where is determined by (37). Figure 2 presents a sample paths () of the unit production cost simulated using (37) times.

In the purpose of CO simulation, we apply a cost scenario for the time interval, , with increment .

The annual production costs are expressed as follows:

3.3. Variables Based on Expert’s Knowledge

To estimate an adequate value of these variables, it is necessary to get opinions of experts dealing with them. Suppose we have experts and each of them has given his/her opinion. The final value is expressed by averaging the opinions of experts. Table 1 presents the expert estimation process.


ExpertValue

x1,1x1,2

x2,1x2,2

x5,1x5,2

x6,1x6,2

3.4. DOL Forecasting Model Based on Expert’s Knowledge and Simulation

The model is developed on the basis of expert’s knowledge and simulation of DOL changing over defined time horizon and takes into account the variability of input parameters. By simulating a forecasting system, we imitate its action in order to measure its response (output) to different inputs. The advantage of simulating a system is the possibility of replicating its evolution as many times as necessary in independent conditions. The simulation allows analysts to describe the uncertainty of variables that influence the value of DOL by different time depending scenarios. The values of DOL are usually forecasted on an annual basis.

The first objective of the use of simulation in the forecasting is to determine the distribution of the DOL from the variables that affect his performance, which results in the average or expected value of DOL for every year of defined time horizon. The relation between uncertain variables affecting the value of DOL is described by (22).

For each simulation, the input values and DOL result represent one possible state of nature. Simulated values of DOL are obtained by performing the following calculations:where denotes the number of simulations.

Space of simulation for and , where the first six rows concern the evolution path of the input variables while the last one concerns evolution path of the DOL, can be represented as follows:Once the simulations , , have been obtained, they can be used to estimate the distribution of the DOL for every year of the defined time horizon. The expected values of the DOL can be expressed by the following vector space: According to (41), the management of the company obtains the forecasting information vector of DOL: This vector concerns the information of the possible future states of the Degree of Operating Leverage over defined time horizon.

Space of simulation for and , where the first seven rows concern the evolution paths of the DOL and input variables while the rest concern evolution paths of the grey relational coefficient, can be represented as follows: Once the simulations , , have been obtained, they can be used to estimate the distribution of the for every year of the defined time horizon. The expected values of the can be expressed by the following vector space: The expected values of the aggregated grey relational coefficient are calculated as follows: According to (45), the management of the company obtains the forecasting information vector of the aggregated grey relational coefficient:

4. Application of the Procedure

4.1. A Numerical Example Statement

A management of an operating underground zinc mine is trying to forecast the future states of operational efficiency for five years ahead. For this problem, the input parameters that are required for the forecasting are given in Tables 2 and 3. In order to define values of production rate, fixed costs, working days, and degree of use of production capacity, three experts are included into process of estimation. Note that the situation is hypothetical and numbers used are to permit calculation. Values of the input parameters are similar to the real environment of mining.


ParameterObserved period
Year
12345

DOL1.2191.4211.3331.6761.437
Production rate (t/year)10000095000100000100000100000
Fixed costs (USD)12000001500000110000014000001600000
Revenues (USD)
 Ore grade (%)4.113.964.043.823.87
 Mill recovery rate (%)78.5278.0181.0579.0281.56
 Metal content of the concentrate (%)53.853.853.853.853.8
 Metal recovery rate (%) 

8585858585
 Zinc metal price (USD/t)21602195195019102160
 Unit value of metal concentrate sales price (USD/t) 
18361866165816241836
 Revenues (USD)110658841077387710137834914527910823137
Production costs (USD)
 Unit production costs (USD/t)43.8760.1157.4056.7655.64
 Production costs (USD)43870005714000574000056760005564000
Working days (day/year)330310340300320
Degree of use of production capacity (%)9085938288


ParameterValue
Year
678910

Production rate (t/year)10000010500097000100000110000
Fixed costs (USD)16000001800000150000017000002000000
Revenues (USD)
 Ore grade (%)-normal distributionMin. 3.45; medium 4.06; max. 4.68; volatility 0.205
 Mill recovery rate (%)-uniform distributionMin. 77; medium 78.5; max. 80; volatility 0.866
 Metal content of the concentrate (%)53.8
 Metal recovery rate (%)85
 Zinc metal price (USD/t)-mean reversion process Spot value 2113; equilibrium metal price 2277; speed of
mean reversion 0.9221; price volatility rate 0.2734
Production costs (USD)
 Unit production costs (USD/t)-geometric Brownian motionSpot value 65; drift 0.02382; cost volatility 0.09351
Working days (day/year)330330340340340
Degree of use of production capacity (%)8690928593
Number of simulations500

4.2. A Numerical Example Solution

Step 1 (forecasting model). Transformations of the original data sequences by AGO are as follows: Calculation of the average generation of adjacent sample sequence of is as follows: Construction of the matrix and matrix is as follows: The constructed AGO sequence of our problem is represented by the following matrix form: According to we obtain the following factors of the matrix . Values of the determined coefficients are ; ; ; ; ; ; .

The AGO time sequence response formula of DOL isAGO and inverse AGO values of the time sequence response formula of DOL are represented by Table 4.


ErrorYear
12345

DOL-AGO values2.2923.6425.2916.546
DOL-inverse AGO values1.2191.0731.3501.6491.255

The adequacy of the obtained time sequence response formula is obtained by using (12) and summarized in Table 5.


ErrorYear
12345

DOL-observed values1.2191.4211.3331.6761.437
DOL-fitted values1.2191.0731.3501.6491.255
RPE (%)024.471.281.6312.70
ARPE (%)10.02

Step 2 (grey relevance analysis). Applying (13) we obtain the following results of equalization (see Tables 6 and 7).


DOL(k)1.2191.4211.3341.6771.437

X1(k)10000095000100000100000100000

X2(k)12000001500000110000014000001600000

X3(k)110658841077387610137834914527810823136

X4(k)43870005714000574000056760005564000

X5(k)330310340300320

X6(k)9085938288

Sum1675330518083273170782681632166218087545


DOL′(k)7.28E − 087.86E − 087.81E − 081.03E − 077.95E − 08

0.0059690.0052530.0058550.0061270.005529

0.0716280.082950.0644090.0857760.088459

0.6605190.5957920.593610.5603150.598375

0.2618590.3159830.33610.3477590.307615

1.97E − 051.71E − 051.99E − 051.84E − 051.77E − 05

5.37E − 064.7E − 065.45E − 065.02E − 064.87E − 06

Absolute values of are represented by Table 8.


0.0059690.0052530.0058550.0061270.005529

0.0716280.082950.0644090.0857750.088459

0.6605190.5957920.593610.5603150.598375

0.2618590.3159830.33610.3477590.307615

1.96E − 051.71E − 051.98E − 051.83E − 051.76E − 05

5.30E − 064.62E − 065.37E − 064.92E − 064.79E − 06

Extreme values of are represented by Table 9.


Min.Max.

0.0052530.006127
0.0644090.088459
0.5603150.660519
0.2618590.347759
1.71E − 051.98E − 05
4.62E − 065.37E − 06

Grey relational (relevance) analysis of is represented by Table 10.


DOL()/