Numerical Modelling of Transportation Cost Minimization in Surface Mining

Numerical Modelling of Transportation Cost Minimization in Surface Mining

Gamal M. A. Mahran1,2

1Deanship of Graduate Studies, King Abdulaziz University, Jeddah 21589, Saudi Arabia.

2Mining and Petroleum Department, Faculty of Engineering, Al-Azhar University, Qena 83513, Egypt.

Abstract:- This study aims to minimize ore transportation cost in surface mining by numerical modelling. Minimization of ore transportation cost is very important in case of multiple locations of ore deposit such as phosphate ore of Aljalamid, in Saudi Arabia, the ore is found in three locations. The present numerical model aims to determine the best position of processing plant. The suggested mathematical based on the factors such as number of main locations, reserves in each location, transportation costs in different locations, and transportation distance from each ore deposit location to required processing plant position. Computer programming was used to solve the considered mathematical model. The main results of this study are determination of processing plant location, Contour map of additional cost in the area around the optimum location, and the Presented model is very useful in similar deposits.

Keywords:- Numerical Modelling, Simulation, Transportation Cost, Cost Minimization, Optimum Processing Plant position.

1. INTRODUCTON

   Cost of transportation represents one of the main items of mining projects. The factors affecting the cost of transportation in mining are transportation distance, ore tonnage, and transportation method. Many transportation models were suggested such as Larwood and Benson Model [1], Anderson Model [2], Bechtel Model [3], and Zimmerman Model [4]. These models tried to minimize the cost transportation [5-10]. Fjellstrom [11] presented a model to estimate the transportation cost of ore to crusher and waste to backfilling rooms in Renstrom mine. Wegener [12] presented many contributions for development of transportation models. Brazil et al. [13] suggested a model to optimize the transportation costs of underground mining roads. This model focused on the underground mine networks. Two models were investigated by Dharma and Ahmad [14]. These models presented for application in two iron ore mines. These models were applied to optimize transportation cost. The numerical results were agreed with real-world situation. Shephard [15] suggested a transportation model based on transported quantity, distance, shipment delay, transport technology, and route. Mahran et al. [16]. Suggested two mathematical models to obtain the optimum location of the processing plant in Bahariya iron ore mines. These models based on ore reserves and transportation distance but it did not take into consideration the transportation costs of different locations in El-Gedida, Ghorabi, Nasser and El-Harra iron ore deposits. Also the gravity center calculations are not accurate due to uniformity ore thickness assumption and method of calculation. Inwood and Keay [17] investigated relationship between trade costs and trade volumes by using modern different tools such as evidence on effective transport costs of iron ore trade. linear programming was used by [18, 19] to minimize cost of transportation in mining. Chen et al. [20] proposed a model based on classical transportation problem and transport path.

   Ahmed et al. [21] presented a study to minimize cost of transportation by Linear Programming Problem. Joshi [22] suggested a method to decrease cost of transportation. The method based on linear programming and was solved by using four different methods. Ahmad [23] presented a method that named Best Candidates Method (BCM) to solve optimization cases. The objective of his method to find optimal solution. Novikov et al. [24] considered an integer model to minimize cost of production of iron ore raw materials and its transportation during mining and ore processing. Saderova et al [25] presented a mathematical model of the production of raw material from a mining area to started point of next technological process. They suggested two methods for modelling. The first methods was based on traffic modelling with mathematical equations. The second method based on computer simulation using ExtendSim8. The previous transportation models were based on ore reserves and transportation distance but it did not take into consideration the transportation costs of different locations. Also the gravity center calculations are not accurate due to uniformity ore thickness assumption and method of calculation. The present model will avoid the shortcomings of the previous models to obtain optimum location of processing plant. The current model applied in Aljalamid area of Saudi Arabia, the ore located in three locations that are Fish, Southern and Western. In this case, selection of optimum location of processing plant represents a real challenge. The suggested model studied factors such as: number of locations, reserves in each location, transportation costs of one ton for one unit distance in different locations, and the distance from the gravity center of each location to optimum location of the processing plant.

2. THEORETICAL CONSIDERATION

   1. General

      Choice of optimum location of ore processing factory depends upon the total transportation cost. Transportation cost parameters is dependent mainly upon the reserves of each ore deposit area, the distances of ore transportation from each ore deposit area to the optimum location, and transportation costs per ton in different locations.

   2. Mathematical model

      The suggested model depends on the minimum sum of transportation cost from ore deposit locations to optimum location of processing plant. Ore deposit scattered into different locations (n). Each location has an ore reserves of (Qi) which is considered to be concentrated at a defined point that is the center of gravity having coordinates (xi, yi, zi) the cost of transported 1 ton for 1 kilometer distance from each ore deposit location to optimum location (Ci). All of the ore reserves are to be transported to a location where the mineral processing plant is to be construct so that the transportation cost should be minimized. Hence, this issue can be mathematically expressed as in equations from 1 to 6. The suggested model based on the minimum sum of transportation cost of different ore deposits to optimum location of processing plant as shown in equation (1)

      n

      n

      Q C D Minimum

      (1)

      i i i

      i 1

      Where:

      n: Number of ore deposit locations.

      i: to denote the ith ore deposit location.

      Qi: Reserves of ore deposit location of location number (i) in (tons).

      Ci : Transportation cost of one ton for one unit distance for each path or transportation method in (U.S$/ton. m).

      Di: Distance between different ore locations and processing plant optimum location in (m) x, y, z: The coordinates of processing plant optimum location.

      xi, yi, zi : The coordinates of the gravity center of different ore deposit locations. The total transportation cost (US$) of ore deposit is:

      n

      S QiCi Di

      i1

      (2)

      The distance (Di) between two points by using its coordinates can be mathematically = (x x )2 ( y y )2 (z z )2

      Hence, S from equation (2) may be mathematically expressed as follows:

      n

      i i i

      S Q C (x x )2 ( y y )2 (z z )2

      i i i i i

      i 1

      (3)

      For this sum to be minimum, the partial differentiation in regards to x, y and z should equal zero, which means that the following conditions have to be satisfied:

      s n Q C (x x ) / (x x )2 ( y y )2 (z z )2 0

      i i

      i

      i

      i i

      i i

      i

      i

      i i

      (4)

      x i 1

      s n Q C ( y y ) / (x x )2 ( y y )2 (z z )2 0

      i i

      i

      i

      i i

      i i

      i

      i

      i i

      (5)

      y i1

      s n Q C (z z ) / (x x )2 ( y y )2 (z z )2 0

      i i

      i

      i

      i i

      i i

      i

      i

      i i

      (6)

      z i1

      Where:

      S : Total cost of transportation for all ore deposit locations (US$).

      s , s , s : The partial differentials of equation (3)

      x y z

   3. Calculation of ore reserves and center of gravity of ore deposit

   The objective of gravity center determination is to accumulate the quantity of ore (reserves) of each area or location at a point that has coordinates (xi, yi, zi). Gravity center means an imaginary point in a body of matter where, for convenience in certain calculations, the total weight of the body may be thought to be concentrated [26]. The center of gravity of ore deposit can be determined as follows.

   • Firstly, the ore deposit map or area is divided into suitable number of triangles. Each triangle consists of three boreholes as shown in figure (1).

     Figure 1. Ore deposit area is divided into triangles.

   • Secondly, Determination of ore reserves and center of gravity of each triangle. Gravity center of a triangle is the intersection point of its medians [27] as shown in figure (2). The center of gravity divides each of the medians in the ratio 2:1, which is to say it is located of the perpendicular distance between each side and the opposing point. Its Cartesian coordinates are the means of the coordinates of the three vertices. That is, if the three vertices are a (xa , ya ) , b (xb , yb ) , and c (xc , yc ) , then the x and y coordinates of the center of gravity x, y and can be calculated by equation (7).

     C.G. (x, y) a b c [ xa xb xc , ya yb yc ]

     (7)

     3 3 3

     Figure 2. Gravity center determination of a triangle.

   • Thirdly, Determination of Z coordinate of gravity center of a triangle. Z coordinate of gravity center of a triangle can be determined for the point, which was determined in the second step. Z coordinate of gravity center of the triangle can be calculated by equation (8).

   z z t

   Where:

   o.b

   • tb

   2

   (8)

   z- : z coordinate of the gravity center of the triangle.

   z : z coordinate of ground surface at ( x, y ) coordinates of C.G. to.b : thickness of overburden at ( x, y ) coordinates of C.G.

   tb : Average thickness of the bed of required triangle.

   • Fourthly, Calculation of gravity center of ore deposit that consists of a number of triangles as shown in figure (3) by using the following equations (9,10,11).

   n

   n

   x

   i1

   Ri xi

   n

   n

   Ri

   i1

   Figure 3. Ore deposit consists of a number of triangles.

   (9)

   n

   y Ri yi i1

   n

   z Ri zi

   n

   n

   Ri

   i1 (10)

   n

   n

   Ri

   i1

   i1

   (11)

   Where:

   x = x coordinate of the center of gravity of the ore deposit. y = y coordinate of the center of gravity of the ore deposit. z = z coordinate of the center of gravity of the ore deposit.

   x i = x coordinate of the center of gravity of the i th triangle of ore deposit.

   yi = y coordinate of the center of gravity of the i th triangle of ore deposit.

   zi = x coordinate of the center of gravity of the i th triangle of ore deposit.

   i

   i

   R = The reserves of the i th triangle of ore deposit.

   n = Number of triangles of ore deposit.

3. RESULTS AND DISCUSSIONS

The suggested mathematical model was applied in Aljalamid phosphate ore deposit, in the northern part of Saudi Arabia, the ore is spread on three different locations that are Fish area, Southern area and Western area.

1. The center of gravity of ore deposit.

   Data of boreholes were obtained from Maaden Company for Aljalamid deposit. These data represent three different locations. Determination of gravity center coordinates of ore deposit consists of two steps as follows: First, calculation of the gravity center of each triangle of ore deposit using equations (7, 8). Second, application of the moment method (Equations 9,10,11). The ore reserves and gravity center of different locations of Aljalamid phosphate ore deposits are shown in Table 1.

   Table 1. Reserves and coordinates of gravity center of different locations of Aljalamid phos-phate ore deposit

   Ore deposit locations	Reserves (Tons)	C.G. Coordinates
   		x (m)	y(m)	z (m)
   Fish Area location (L1)	146,130,862	291095	245985	756
   Southern Area location (L2)	117,299,884	290119	191354	712
   Western Area location (L3)	109,819,355	242008	216954	723

2. Optimum Processing Plant Location

   The optimum location of processing plant requires the number of ore deposit locations(n), reserves (Qi) and center of gravity for

   each location (xi,yi,zi). The Equations (4, 5&6) of model can be used in the form of 12, 13 and 14 as follows.

   s (Q C (x x ) (x x )2 ( y y )2 (z z )2 ) (Q C (x x )

   (x x )2 ( y y )2 (z z )2 )

   x 1 1 1 1

   1 1 2 2 2 1 1 1

   n n

   n n

   2 2 2

   2 2 2

   …………. (Qn Cn (x xn ) (x x n ) ( y y ) (z z ) ) 0………………………………………………..(12)

   s (Q C ( y y ) (x x )2 ( y y )2 (z z )2 ) (Q C ( y y )

   (x x )2 ( y y )2 (z z )2 )

   y 1 1 1 1

   1 1 2 2 2 1 1 1

   n n

   n n

   2 2 2

   2 2 2

   ………… (Qn Cn ( y yn ) (x x n ) ( y y ) (z z ) ) 0………………………………………………..(13)

   s (Q C (z z ) (x x )2 ( y y )2 (z z )2 ) (Q C (z z )

   (x x )2 ( y y )2 (z z )2 )

   z 1 1 1 1

   1 1 2 2 2 1 1 1

   n n

   n n

   2 2 2

   2 2 2

   ………… (Qn Cn (z zn ) (x x n ) ( y y ) (z z ) ) 0…………………………………………………(14)

   The unknowns in equations 12, 13 and 14 are x, y and z coordinates of the optimum location of the processing plant. These equations represent the mathematical model. Computer program was developed to solve these equations.

3. Model validation

   Validation step of model is one of main important steps of modelling. Validation based on the values of equations 12, 13 and 14 equal zero by using the input data of different locations and the calculated result of optimum location. Model v alidation can be summarized as follows:

   1. Specify number of main locations of ore deposit (n)

   2. Specify ore reserves of each location Qi

   3. Specify gravity center coordinates for each location (xi,yi,zi)

   4. Specify cost of transportation in different locations or paths to optimum location.

   5. Input the above data in steps from 1 to 4.

   6. Run the developed program to obtain optimum location coordinates of processing plant.

   7. Calculate the values of the equations.12, 13 & 14 using the coordinates of optimum location resulted from step 6, and input values of steps from 1 to 4.

   8. If the calculated vales of equations (12, 13, and 14) in step 7 were zero or within the permissible errors it means the model is valid.

   9. The above steps were repeated for 10 varied cases with different program inputs (variables), n, Q, C and center of gravity coordinates.

      Table 2 shows the obtained results for 10 validation cases together with their corresponding errors. As an example, the output result of case no. 1 is shown in Figure 4 and Figure 5. Where Figure 4 shows a snapshot of the program with the input variables and Figure 5 shows a snapshot of program final results of the optimum location for case No.1.

      Figure 4 A screenshot of the program after completion of ariables enteredfor case No. 1.

      Figure 5 A screenshot of the program after the completion of program run for case No.1

      Published by : http://www.ijert.org

      International Journal of Engineering Research & Technology (IJERT)

      ISSN: 2278-0181

      Vol. 10 Issue 10, October-2021

      Table 2. Validation Table

      			First Location				Second Location				Third Location			Fourth Location if any				Processing Plant Optimum Location			Eq.1 2	Eq.1 3	Eq.1 4
      Case No.	No. of Locatio	Q ton	X m	Y m	Z m	Q ton	X m	Y m	Z m	Q ton	X m	Y m	Z m	Q ton	X m	Y m	Z m	X m	Y m	Z m
      1	3	10000	1000	1000	100	15000	3000	2000	200	20000	2000	3000	300	_	_	_	_	2034.29257	2895.39495	289.53949	0.0	0.0	0.0
      2	4	5000	500	500	50	8000	1000	1500	70	7000	2000	1000	60	9000	2500	500	80	1997.62097	998.13330	60.12598	0.0	0.0	0.0
      3	3	60000	800	700	90	75000	1100	950	110	90000	1300	800	115	_	_	_	_	1126.63943	885.40152	109.49180	0.0	0.0	0.0
      4	4	75000	1200	1000	150	80000	1500	1800	140	9000	2000	1600	175	68000	2500	900	130	1580.09806	1368.02748	143.46178	0.0	0.0	0.0
      5	3	150000	29400	25000	1100	165000	33500	39000	1140	173000	38600	28200	1210	_	_	_	_	35624.5718 5	29567.3155 8	1172.7006 2	0.0	0.0	0.0
      6	4	258000	15300	11230	276	195800	17200	13394	245	188600	18100	12850	264	210000	19100	10960	239	17591.2146 2	12530.0822 3	256.78175	0.0	0.0	0.0
      7	3	310700	36970	11230	320	344270	30900	15610	339	442870	41830	17330	361	_	_	_	_	37298.5825 5	14357.9684 1	338.66142	0.0	0.0	0.0
      6	4	570280	55370	61700	1230	390900	68300	81370	1270	410380	54800	73230	1290	87100	83120	95419	1320	56407.4765 0	71864.6469 9	1276.3955 7	0.0	0.0	0.0
      9	3	51200	52300	41100	360	63400	59420	70900	345	69500	65310	45600	355	_	_	_	_	63703.6133 5	46721.6731 4	354.86492	0.0	0.0	0.0
      10	4	95800	53120	71225	317	83400	57340	85340	335	76800	77600	86460	350	79200	81300	70250	345	63945.4206 2	79182.7847 7	334.47552	0.0	0.0	0.0

      IJERTV10IS100099

      www.ijert.org

      (This work is licensed under a Creative Commons Attribution 4.0 International License.)

      207

4. Optimum processing plant location of Aljalamid phosphate ore.

The mathematical model was used to calculate the optimum location of the processing plant. The required data to apply the mathematical model are the reserves, coordinates of the gravity centers and transportation cost of one ton for unit distance of the different Aljalamid phosphate ore deposits. These data are given in Table 1. According to the output of the computer program as shown in figure 6, the coordinates of the optimum location of the processing plant are approximately (278842, 223149, 735) in the east, north, and elevation directions, respectively. The obtained optimum location of processing plant related to the different ore deposit locations of Aljalamid phosphate ore is shown in figure 7.

Figure 6 A Screanshot of the optimum location results achieved after the completion of pogram run.

Figure 7 Map shows the optimum location of processing plant related to the different ore deposit locations of Aljalamid phosphate ore.

1. Effect of processing plant location deviation from optimum

   The current presented results showed the optimum location of mineral processing plant in an ideally theoretical case. Due to any reason whatsoever, it may be impossible to construct the processing plant at the determined optimum location. Now, it is of importance to investigate the transportation of ore to any location somehow around the optimum location. Off course this deviation from the optimum plant location will increase transportation cost. Total transportation cost of the ore can be determined by the equation (3) as follows:

   n

   S Q * C * (x x )2 ( y y )2 (z z )2

   i i i i i

   i 1

   The additional total transportation cost percentage due to deviation of processing plant location from the optimum plant location can be calculated from equation (15).

   Additional Cost % ( S SO )*100

   SO

   (15)

   S: is total transportation cost of the ore for the different locations to the processing plant location. (Equation 3).

   So: is the total ore transportation cost to the optimum processing plant location, It can be calculated from Equation 3, when x, y and z coordinates refer to optimum processing plant location i.e So is a special case of S when transportation of ore is going to be to the optimum mineral processing location.

   Figure 8 shows a contour map of the additional cost percentage compared to the minimum for different selected mineral processing plant locations. The figure shows that choosing one of three locations at Aljalamid (Fish area L1, Southern area L2 & Western area L3) may be increase the transportation cost to 8 %, 22% and 28 % respectively compared with the optimum location. Also, the figure shows that there may be more than 28 % increase in the ore transportation cost due to an incorrect selection of the processing plant location.

   Figure 8 Contour map of additional cost % related deviation of processing plant location from optimum. (L1= Fish Area, L2 =Southern Area, and L3 refers for Western Area)

   4. CONCLUSIONS From the obtained results, the following conclusions can be made:

   • The phosphate ore in Aljalamid is found in 3 main locations having ore reserves of 146, 117, and 109 million tons respectively.

   • Centers of gravity for three locations of ore deposit were determined.

   • The presented model satisfies minimum total cost of ore transportation from all ore deposit locations to processing plant location.

   • Both of the mathematical model and Computer program to solve the model were validated.

   • The processing plant optimum location were obtained.

   • Deviation of processing plant location from optimum increases the transportation cost of ore deposit.

   • The presented model will be useful in similar ores.

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