Mine production scheduling within capacity constraints

JC Najor and PC Hagan

The University of New South Wales (UNSW), Sydney

 

The conventional approach used in the determination of material movement rate for truck and shovel fleets is normally based on trucking capacity with little or no allowance made for interaction with other mining processes. This often leads to over-estimation of fleet capacity resulting in mine schedules that often cannot be achieved. An alternate approach to scheduling that accounts for queuing theory and constraints on fleet capacity due to downstream processes such as the primary crusher was investigated for a large open cut mine.

Truck productivity and required operating hours were calculated using the capacity constrained approach and compared to the conventional method of calculation. The capacity constrained model tended to predict lower levels of fleet productivity than the conventional approach depending on travel time and loading time with differences of up to 40%. When the calculated fleet capacity factors were applied to an actual mine schedule covering a 2.5 year period, the difference in total material movement represented a shortfall of nearly 8%.

 

 

Introduction

The core mining process of most large-scale open cut mining operations is centred on materials handling; a process often involving a combination of excavators/shovels and trucks moving broken rock from one or more working areas to either the primary crusher for processing, stockpile or waste rock dump. Aside from processing of ore, materials handling has an enormous leverage on the economic performance of a mining project because of its impact on both capital cost and operating cost, requiring a large investment (and often re-investment every five to ten years depending on the type of equipment) on the truck and shovel fleet and significant resources in terms of people, consumables and fuel to operate and maintain the equipment.

At many of the large open cut iron ore mines in the Pilbara region of Western Australia, many tens of millions of tonnes are moved each year each requiring upwards of twenty trucks each costing over $AUD2million together with the costs of operators and maintainers.

Significant financial savings can accrue to a project through efficient management of the fleet by maximising use of the equipment and minimising the resources necessary to operate and support the fleet. Significant potential savings can also be accrued by ensuring that fleet size is matched to targets for material movement. To achieve this requires good planning practises aided by reliable modelling techniques.

The conventional approach to determination of fleet size is often based on trucking capacities which has not always produced a reliable estimate.

A different approach to determining fleet size was investigated at one Pilbara mine involving a qualitative and quantitative assessment in relation to production planning. The model incorporates constraints based on equipment capacity and aims to account for the impact of downstream plant capacity. The model is based on queuing theory which accounts for the stochastic or time-variable behaviour of truck-shovel systems (Kappas & Yegulalp, 1991).

In particular the project investigated differences in truck productivity and operating hours for a given schedule as determined by the conventional approach compared to a capacity constrained model.

METHODOLOGY

The capacity constrained model is based on a spreadsheet in which a series of truck productivity matrices were generated for each of three material types handled in the mine, these being high grade crusher feed (HG-CF), low grade crusher feed (LG-CF), and other material (OM). The latter comprising waste material and low grade material destined for stockpile.

The matrices indicate how truck productivity is affected by cycle time. The matrices also account for the crusher feed rate in the case of crusher feed material and lowest cost per tonne for other material. Typical truck productivity profiles were generated for each material type. A calculation of truck productivities was also made using the conventional approach. The current medium-term schedule for the mine was then used to quantify the changes in terms of operating hours required for a given schedule.

Figure 1. Schematic of a queuing system.

(after Carmichael, 1987).

Incorporation of Queuing Theory

Queuing theory was incorporated in the capacity constrained model to account for the time lost as a result of waiting by units in a truck/shovel system.

In most truck/shovel systems, trucks do not normally arrive at the shovel to be “serviced” in a predictable manner, nor does it take exactly the same time for the shovel to service each truck. The interaction between the randomness of inter-arrival times of trucks and the shovel service time results in either trucks to queue at the shovel, or the shovel being idle while waiting for a truck to arrive (Elbrond, 1990). Figure 1 shows a schematic of a typical queuing system with a truck and shovel combination.

 

The Queue Formula

The general equation describing the probability of number of trucks queuing at a shovel under steady states conditions can be expressed as:

                       1)

where P(n) = probability n trucks are at the shovel i.e. one truck being serviced while (n – 1) are waiting, P(0) = probability that the shovel is idle; M is number of trucks servicing a shovel; Ts is truck cycle time (delayed) in min; T1 is shovel load time (delayed) in min; m is service rate that is the rate the shovel can load the trucks if it does not have to wait for them to arrive which equals 60/T1, l is arrival rate which equals 60/Ts and r = l/m.

As the sum of the probabilities must equal unity then

                              2)

If we define service time and arrival rate in terms of the number of trucks servicing a shovel then:

                                          3)

whereby

                 4)

Production Rate of the System

The production rate of the truck/shovel system can be calculated as:

P  =  m E Ct                             5)

where E is the proportion or fraction of time the shovel takes to load trucks which equals 1 - P(0) and Ct is truck payload in tonnes.

As the number of trucks working on the shovel, M, increases then the fraction of time spent by the shovel loading trucks, E, will approach unity as time spent waiting for the arrival of trucks will approach zero. Therefore at the limit, lim P = Ct, that is the production rate of the system is limited by the truck payload.

 

 

Table 1.

Queuing theory model.

Input Parameters

 

Load + Spot Time (min)

2.1

Travel + Dump Time (min)

6.0

Crusher Feed Rate (t/h)

2100

Truck Payload (t)

220

Efficiency

1.00

Cost Inputs

 

Truck Op. Cost ($/h)

150

Shovel Op. Cost ($/h)

250

 

Truck Fleet
Size

Shovel Prod.
(t/oph)

Truck Prod.
(t/oph)

Shovel Cost
($/t)

Truck Cost
($/t)

Total Cost
($/t)

0

0

0

0

0

0

1

1650

1650

$0.15

$0.09

$0.24

2

3106

1553

$0.08

$0.10

$0.18

3

4315

1438

$0.06

$0.10

$0.16

4

5240

1310

$0.05

$0.11

$0.16

5

5874

1175

$0.04

$0.13

$0.17

6

6256

1043

$0.04

$0.14

$0.18

7

6456

922

$0.04

$0.16

$0.20

8

6546

818

$0.04

$0.18

$0.22

9

6582

731

$0.04

$0.21

$0.24

10

6595

659

$0.04

$0.23

$0.27

11

6599

600

$0.04

$0.25

$0.29

12

6600

550

$0.04

$0.27

$0.31

13

6600

508

$0.04

$0.30

$0.33

14

6600

471

$0.04

$0.32

$0.36

15

6600

440

$0.04

$0.34

$0.38

16

6600

412

$0.04

$0.36

$0.40

17

6600

388

$0.04

$0.39

$0.42

18

6600

367

$0.04

$0.41

$0.45

19

6600

347

$0.04

$0.43

$0.47

20

6600

330

$0.04

$0.45

$0.49

 

Other Queue Characteristics

Queuing theory can also provide other information about queue characteristics (Bertinshaw, 2004). These include:

·      the expected length of the waiting line, Ew

·      the expected number being serviced and waiting, En

·      the expected time a truck must wait before being serviced, Et

the expected time in the system, Ey

whereby

6)

      7)

               8)

9)

 

Sample Spreadsheet

These queuing formulae can be translated into a spreadsheet model. An advantage of this approach is that it provides an ability to observe the affect of fleet size on production and cost. An example of a spreadsheet is given in Table 1 which shows the input parameters used to develop the model, i.e. load & spot time, travel & dump time, truck payload, and truck/shovel costs. The table also shows the amount of production for a truck/shovel system as determined by queuing calculations and cost per tonne of material moved. This data can then be used to determine the optimum number of trucks for that system.

 

Figure 2. Productivity curves for
the capacity constrained model.

Figures 2 and 3 were generated using the production and cost data provided in Table 1. Figure 2 shows that shovel productivity increases and truck productivity decreases with fleet size. The graph also indicates a relatively large number of trucks are required for a shovel to reach the theoretical capacity for this system i.e. eight or nine trucks.

Figure 3 shows the corresponding cost curves indicating loading costs decrease and haulage costs increase with fleet size. Based on the lowest cost per tonne, the graph suggests that the optimum fleet size for this system is either 3 or 4 trucks.

RESULTS

High Grade Crusher Feed Material (HG-CF)

Figure 4 shows the differences in truck productivity profiles between the capacity constrained and the conventional approach in the case of HG-CF material for three levels of travel time. The graph indicates that truck productivities calculated using the capacity constrained model are nearly always less than those calculated using the conventional approach irrespective of load and travel time. Hence the capacity constrained model provides a more conservative estimate of potential mining capacity. Conversely the conventional approach tends to over estimate mining capacity.

Figure 3. Cost curves for
the capacity constrained model.

The graph also highlights differences between truck productivities tends to increase with load times and decreases with travel time. This would suggest for example that as a pit gets deeper or new pits are brought on-line further from the crusher then the difference in estimated truck productivities will become less significant.

An examination of truck factor profiles defined as the ratio of capacity constrained to conventional approach estimates indicates differences of up to 40% in productivities whenever load times exceeds 3.9 min for HG-CF material. Hence in hard-to-dig areas such as poorly blasted ground where load times are likely to be significantly higher, the differences in estimated truck productivities will be more significant with production rates likely to be over-estimated.

Figure 4. Capacity constrained (CC) vs conventional approach (CA) of
truck productivity for various load times and travel times (min).

Similar results were observed in the analysis for LG-CF and OM.

Medium Term Schedule Analysis

The impact of the two approaches on total operating hours for a given schedule was evaluated using the medium term schedule at the mine having a 2.5 year duration.

Data was sourced from the XPAC scheduling software package on a block-by-block basis in terms of tonnes and travel time for each material type. The total operating hours required for each material type was then calculated based on the corresponding truck productivity matrices for the capacity constrained and conventional models using the formula:

        10)

Considering first the capacity constrained model whereby average truck productivity was calculated as 408 t/oph. Assuming a loading time of 3.3 min then the total operating hours required to mine 43.5 Mt of HG-CF material would be:

Total Operating Hours  =  763 126 t @ 700 t/oph + 2 188 172  t @ 700 t/oph +…+2 236 028  t @ 233 t/oph + 177 683  t @ 233 t/oph  =  118 718 h

Using the conventional approach whereby average truck productivity was calculated as 442 t/oph then the total operating hours required to move the same amount of material would be:

Total Operating Hours  =  763 126  t @ 846 t/oph + 2 188 172  t @ 750 t/oph +…+2 236 028  t @ 246 t/oph + 177 683  t @ 237 t/oph  =  110 119 h

According to the capacity constrained model 118,718 hrs would be required to mine the HG-CF material with an average truck productivity of 408 t/oph while only 110,119 hrs would be required according to the conventional approach having an average truck productivity of 442 t/oph, a difference of 8,599 hrs or nearly 8% more time over the same scheduling period.

Figure 5. Effect of conventional vs capacity constrained approaches
on avg. truck productivity and the resulting difference
in estimated operating hours for various load times.

Figure 5 highlights differences in truck productivity between the two scheduling approaches as a function of load time for HG-CF material and the resulting impact in terms of the difference in estimated operating hours. The graph shows as load time increases, the difference in average truck productivities between the two approaches increases, resulting in a greater difference in the estimated operating hours required for the given mine schedule. For example, with a load time of 2.3 min there was a difference of 5,037 hrs in operating hours; at 3.3 min and 4.3 min the difference increased to 8,599 hrs and 16,500 hrs respectively. Similar trends were seen for LG-CF and OM.

conclusion

The inclusion of capacity constraints into a mine scheduling model that accounted for truck productivity resulted in significantly reduced material movement rates compared to the conventional approach.

Truck productivities based on the capacity constraint model were found to be much lower than those determined using the conventional approach.

Differences in truck productivity were sensitive to components in cycle time calculation; the differences became more pronounced with increased loading time and decreased travel time.

Little difference was observed with short duration load times and longer travel times. Hence for operations with short haulage distances where travel times are consequently low, the capacity constrained approach should be considered. For operations with longer haulage distances, truck productivities will be similar regardless of the approach adopted.

While extra effort is required to account for capacity constraints, the likely result will be schedules that better reflect actual fleet capacity. The capacity constraint model allows different mining scenarios to be evaluated with a focus on sensitivity helping to optimise planning outcomes.

Acknowledgements

The authors would like to thank the support provided by Hamersley Iron Pty Ltd and particularly the contribution and encouragement from Mr Jeremy Sinclair who made possible this project.

References

Bertinshaw, R, 2004. Shovel Truck Course Notes. (Mining & Resource Technology Pty Ltd: Perth).

Carmichael, D, 1987. Engineering queues in construction and mining. (John Wiley & Sons: New York).

Elbrond, J, 1990. Haulage System Analysis: Queuing Theory. Surface Mining. 2nd Edition. (Ed: B A Kennedy), pp 743-748, (Society for Mining, Metallurgy, and Exploration, Inc.: Littleton, Colorado).

Kappas, G and Yegulalp, T M, 1991. An application of closed queuing networks theory in truck-shovel systems, in International Journal of Surface Mining and Reclamation. No. 5, pp. 45-53.