A Stochastic Optimization Approach to Mine Truck Allocation

Transcription

1 A Stochastic Optimization Approach to Mine Truck Allocation CHUNG H. TA 1, JAMES V. KRESTA 2, J. FRASER FORBES* 3, and HORACIO J. MARQUEZ 4 In the mining industry, truck assignment is an important and complex process and an optimal truck allocation can result in significant savings. In this paper, a truck allocation model is formulated using chance-constraine stochastic optimization approach that can accommodate uncertain parameters such as truckload and cycle time. A real-time hauling framework, which consists of the chance-constrained optimization model and a model updater, is developed to compensate for changes in the uncertain key operating parameters. The use of the model updater helps the truck allocation system to adapt to random operational changes. The effectiveness of the chance-constrained approach in dealing with uncertain process parameters, when coupled with model updating, is shown to be a viable implementation framework in the dispatching operation. Keywords: stochastic optimization; truck allocation; chance-constrained; oilsand mining 1. Introduction In the open-pit mining industry, trucks and shovels are the mining technology of choice; however, it is widely recognized that the operation of these trucks and shovels contributes significantly to the overall operation cost. Fortunately, there exist many opportunities to reduce the cost of the truck and shovel operation. Operationally, the main challenges have been to effectively deploy truck resource and to maintain a steady, reliable supply of ore material at the highest possible efficiency. Most mining companies develop and implement the truck dispatching system as an integrated multi-stage system. The first stage involves an allocation of truck resource given a production requirement. These truck solutions are then implemented using a real-time dispatching system, with or without the interaction of the truck dispatcher. The existence of many uncertain operational elements makes accurate allocation more complicated since 1 Syncrude Canada Ltd., Research, Avenue, Edmonton, Alberta, T6N-1H4, Canada 2 Syncrude Canada Ltd., Research, Avenue, Edmonton, Alberta, T6N-1H4, Canada 3 Corresponding author: Dr. J Fraser Forbes, Chemical and Materials Engineering, University of Alberta, Edmonton, T6G-2G6, Phone: (780) , Fax: (780) , fraser.forbes@ualberta.ca 4 Electrical and Computer Engineering, University of Alberta, Electrical and Computer Engineering, Research Facility, Edmonton, Alberta, T6G-2V4, Canada

2 the allocation model is based on historical performance and is implemented without explicitly acknowledging uncertainty. A variety of truck allocation-dispatching approaches have been adopted at different mining operations. In the past, almost all mine operations relied on heuristic rules to assign truck resources. While a heuristic algorithm can still be used successfully in small mining operations, most large mining companies employ either mathematical programming or the combination of mathematical programming and heuristic rule methods in the implementation of the truck dispatching system. Efforts at improving truck allocation and dispatching have been investigated in both academia and industry, and these efforts have met with various degrees of success. The most common approach to the allocation and dispatch problem is to solve the allocation problem via mathematical programming techniques and the dispatching problem via heuristics ([3], [4], [6], [7], [8]). Elbrond and Soumis [3] used a combination of mixed integer, non-linear programming in working with short-range plan and operational plan. Truck dispatching was performed based the concept of minimizing the variance of waiting time of trucks and shovels. The evaluation of the new truck dispatching system based on this concept was reported by Soumis et al. [6] at the Mount Wright Mine. Temeng et al. [7] took a goal programming approach, in which they solved the truck assignment problem as a transportation model that minimizes the total waiting time of shovels and trucks. Xi et al. [7] worked on a multi-step dispatch model, which consists of two linear programming steps to optimize the initial truck flow and implemented a real-time dispatch based on the concept of a minimizing ratio variance rule. Various heuristic rules were used in the dispatching system. Lizotte et al. [4] discussed many rules: maximization of truck utilization, maximization of shovel utilization, and match factor. These authors showed that, depending on mine operation, using a combination of these rules generates a positive dispatching result. While it is clearly recognized that the allocation and real-time dispatch are separate tasks, the division of these tasks and the transition between them has not been clearly defined. The success of these research efforts and of the industrial implementation has been limited for three reasons: 1. allocation is often based on historical information and average values at the beginning of shift rather than on currently observed mining information, which yields an approach more analogous

3 to an off-line scheduling function than a real time optimization function. Further, the methods used are deterministic; whereas, the environment is stochastic. This is known and the end-users compensate by allocating additional resources to accommodate for this uncertainty. 2. the transition from the allocation solution to the implementation in a dispatch algorithm is difficult and often requires human intervention. 3. during a shift, the mine experiences both minor and major upsets. The dispatch algorithm should manage the minor upsets; however, the major upsets violate the initial allocation and require intervention. This paper does not provide a complete solution to this difficult problem. Rather, it concentrates on truck allocation only and investigates possible solutions to two of these hurdles: using stochastic programming to improve the initial truck allocation reallocation of trucks based on observed mine operation. The truck allocation model in this paper is developed for illustration purposes based on two uncertain parameters, truck cycle time and truckload. This assumption in no way limits the applicability of the approach to problems with a larger set of uncertain parameters. The allocation solution is then implemented in the real-time hauling framework, where simulated operational data, such as truck cycle times and truckloads, are used to determine the distribution characteristic for these uncertain quantities. The distribution information is then used as feedback data for subsequent truck allocation calculations. The work will be presented in three main sections. The first section provides a brief description of a stochastic programming approach for truck allocation. The second section develops a specific truck allocation problem, which is solved in the third section. This yields a real-time allocation process for a time horizon with a data feedback loop, which allows the model to adapt to changes in the mine by updating the mean and variance of the uncertain parameters. The last section presents the results of two scenarios and concludes with discussion of the effectiveness of the chance-constrained programming approach and the real-time hauling framework.

4 2. Truck Allocation Problem The fundamental objective of truck allocation is to support the overall mine objective of maximizing profit over some operating window; however, the truck allocation objective is rarely expressed in this way. It is normally expressed in physical terms that may be relate but not usually directly to the economic objectives. For this reason, there are many objectives that can be used for a truck allocation problem depending on the physical mine configuration, operating conditions, the economic or management goals, and the company culture. For the problem presented in this work the objective function is to minimize the operating and capital cost of ore delivery, expressed as minimizing the truck resources needed to meet a production constraint. While this formulation is specific to the Syncrude operation (Figure 1), switching between different formulations can be easily accomplished. Figure 1 - Ore Handling Operation with Trucks and Shovels The reason for this objective in the Syncrude operation is that the mine and process plant (Extraction) are closely coupled. In the mine operation, it is essential to maintain a steady feed to Extraction and this production is constrained within narrow bands by Extraction demand and the size of the surge pile. The surge pile plays a key role in supplying the ore stream to the Extraction plant. It acts as the buffer that bridges the two distinct processes, the discrete delivery of the ore to the plant and the continuous process

5 of slurrying the feed stream of the ore to the Extraction plant. While the limited size of the surge creates operational challenges, the capital and long-range operational savings justify this strategy. While ore blending is an important element in the overall operation it is handled as part of the mine development by controlling the shovel production throughput. The critical constraint for the truck allocation and dispatch is steady rate of production to Extraction. If this constraint is violate severe economic penalties are incurred. This constraint can be guaranteed with sufficient resources; however, economic penalties are incurred and are related to the operating and capital costs of the extra haul trucks. An important operational objective is to satisfy this constraint with minimum resources. A simple version of a linear truck allocation model is, Minimize TruckResou rce = K( X ( s, Subject to V o s d g [ V ] (Truck Units) (1) + H V V (2) Truck s d g Extraction o Min V 60 Truck = Lo ( s, X ( x, g τ ( s, ) (Tonnes/Hr) (3) Lo ( s, X ( s, CShovel ( s) d g τ ( s, o 60 (Tonnes/Hr) (4) s d ( X ( s, R (5) X ( x, 0 (6) where indices s, g correspond to shovel, dump, and truck type respectively; K( is the cost coefficient 5 of truck type g, based on the unit of 240-Ton trucks, i.e. K( = {1,1.33,1.5}; X ( s, 0 represents the number of trucks of type g, assigned to shovel s and dump d; L o ( s, (Tonnes) is the truck haul capacity for truck type g travelling between shovel s and dump d; ( s, d τ (minutes) are the ore truck o, cycle times; V o represents the initial surge volume and V Truck and VExtraction correspond to the hourly 5 For example: given three truck types including 240-Ton, 320-Ton, and 360-Ton trucks: K =, K = 1.33, K 1.5 (It is assumed that trucks with larger capacity correspond to higher cost.) =

6 rate of ore that goes in and out of the surge; (s) denotes the capacity of shovel s (Tonnes/Hr) and C Shovel Dw represents the required hourly amount of waste material to be hauled; R( is the available number of trucks of truck type g; H is the number of hours and is used to limit the time period (Constraint (2)). This initial truck model involves both a linear objective function and a number of linear constraints. The objective is formulated as a function of truck resources used to haul ore material. The goal is to minimize truck resources required to satisfy production constraints. Constraints (2), (4), and (5) are key linear constraints in the model. Constraint (2) places a requirement on the level of the surge while Constraint (4) aims to limit trucks deployed at the shovel, based on the shovel throughput capacities. Constraint (5) puts an upper limit in the available truck resource. The ore blending constraint is omitted from the model in this study for simplicity (such a constraint can be added to the model easily as a minor extension of the current study). Mathematical expressions used in the objective function and various constraints all depend on averaged values. These values include truckload and truck cycle time, which are in turn, a function of many other varying quantities such as travel times (both empty and full), loading time, dump time, waiting/queueing time at the shovels. A solution to this linear model is determined based on these averaged values, and thus may not represent the best solution. One can effectively overcome this model limitation by working with a stochastic truck allocation model, especially when characteristics of these uncertain parameters are known. 3. Probabilistic Truck Allocation The advantage of using stochastic programming over conventional linear programming is that the uncertainty is explicitly incorporated into the numerical solution rather than heuristically accommodated by increasing the available resources beyond the optimal solution, as is the current practice in LP 6 -based approaches. If average values of uncertain or varying parameters are used in the constraint equations, this is equivalent to a stochastic solution where the constraints will be met only 50% of the time; however, 6 Linear Programming

7 with stochastic programming the confidence of meeting the constraints can be explicitly specified (e.g. 95%) Stochastic Optimization As one of the two mainstream stochastic programming (SP) methods, the recourse-based stochastic optimization is formulated on the basis of consistently meeting the constraint that contains uncertainty with the aid of some recourse action. A commonly used example is of allocation/planning for farm planting [1], where the recourse is buying shortfalls at a premium. A recourse-based solution is deemed inappropriate in mining operations involving continuous ore feed. Moreover, recourse-based algorithms demand heavy computer resources in determining the first-stage solution since the technique involves solving the optimization problem over a large number of realizations [1]. The other mainstream technique, chance-constrained optimization, is used in various fields ranging from economics to engineering. Charnes et al. [2] first introduced this chance-constrained problem, which involves probabilistic constraints, while working on the problem of scheduling heating oil production. The probabilistic constraints can be individual and joint. Prekopa [5] showed that, in a limited number of situations, some techniques could be used to convert joint-probabilistic constraints into individual-probabilistic constraints. The truck allocation work in this paper is investigated as an individual, probabilistic constraint problem. The crucial step in solving a chance-constrained problem is to convert the constraint into a deterministic form. For a constraint i, such that, { ix b i } α i Prob a, ( x 0, 0 α 1) (7) i where x are the decision variables, α i is the degree of confidence that the inequality constraint is to be met. Constraint (7) can be interpreted as forcing the constraint a x b to be satisfied at some specified i i level of α i, given that elements of the coefficient vector a i and/or the parameter b i are uncertain or may vary in some random fashion. An equivalent deterministic inequality can easily be derived using the probability theory. In this work, we assume that uncertainty is embedded in the left-hand-side coefficient vector a i and that the uncertain parameters vary according to a Gaussian distribution with known characteristics. The equivalent deterministic constraint is

8 T 1 ( 1 α i ) i aix + x Cx F b, where C is the covariance matrix of the random vector, for example the truck cycle times T and the truckloads L; 1 F is the inverse of the cumulative Gaussian distribution. If α 0. 5, then i 1 ( 1 ) 0 F α and the equivalent deterministic constraint is rewritten as i 2 T [ ] x Cx ( a x b ) ( 1 ) i i F α i. This form is more convenient because this constraint form has continuous derivatives of every order with respect to all variables. The resulting deterministic constraint is a quadratic, but deterministic, and can be solved easily using available techniques. In many other stochastic optimization problems, random components can also appear in the objective function. Depending on the problem context, the objective function can be based on maximizing the benefit or minimizing the incurred cost using the expected value of the random variables. Alternatively, the objective can also be to minimize the variance of the deviation of a random quantity or the probability that the desired level of a quantity is not met. Resolution to these types of stochastic optimization problems is beyond the scope of this article and thus omitted. The truck allocation problem in this article is limited to the stochastic optimization problem with individual probabilistic constraint and the objective function based on the mean values of the random parameters Chance-Constrained Truck Allocation Problem The truck allocation problem is formulated as a stochastic linear optimization model with the decision variables being the number and type of trucks that are allocated to the ore shovels. A chance-constrained model similar to the one presented earlier is shown as follows, Minimize TruckResou rce = K( X ( s, Subject to Prob s d g { V + H[ V V ] V } α (Truck Units) (8) o Truck Extraction Min (9) V 60 Truck = Lo ( s, X ( x, g τ ( s, ) s d g o (Tonnes/Hr) (10)

9 Lo ( s, X ( s, CShovel ( s) d g τ ( s, o 60 (Tonnes/Hr) (11) s d ( X ( s, R (12) X ( x, 0 (13) The only difference between the two models is in Constraint (9), which is a probabilistic constraint. Here, α represents the confidence level at which Constraint (9) must be satisfied. Meeting the production constraint is the key operational constraint and Constraint (9) expresses the production demand in a probabilistic manner. As previously discusse the surge pile, though small in size, is vital in providing a steady stream of ore material to the Extraction plant. Constraint (9) ensures that, at the end of the H-hour long perio the surge volume remains equal or larger than a specified level with 95% confidence. The decision variables of the optimization problem are integer, as they represent the actual number of trucks to be deployed. Mathematically, this constraint causes the problem to be more difficult to solve than problems with continuous variables. The stochastic linear allocation model can be easily converted into a quadratic deterministic model, which can be solved directly with a mixed-integer nonlinear solver. While a commercial solver such as DICOPT 7 can be used to solve for the truck solution, it can take a long time for problem convergence. An alternate approach, which involves solving two sub-models, is chosen for the work in this paper. The first sub-model is a probabilistic chance-constrained optimization model and must be converted into a non-linear deterministic model before it can be solved with available non-linear techniques. The model can be mathematically presented as follows, Minimize TruckResou rce( 1) = K( X ( s, Subject to s d g 240-Ton Truck Units (14) 7 DICOPT (DIscrete and Continuous OPTimizer) was developed by J. Viswanathan and Ignacio E. Grossman at the Engineering Design Research Center (EDRC) at Carnegie Mellon University.

10 60 VTruck = Lo, τ ( s, Prob V Truck d g o ( s, X ( s, d { V + H[ VTruck VExtraction ] VMin } α CShovel () s m C () s 0 o (16) Tonnes/H (17) VTruck Shovel, where m 1 Tonnes/H (18) s d ( (15) X ( s, R (19) X ( s, 0 (20) This first sub-model is almost identical to the general chance-constrained model presented earlier (e.g. Expression (8) to (13)) with the exception of Constraint (18), which was added to prevent solutions with zero trucks being assigned to the shovels. Minimum ore throughput from each working shovel must be maintained because mining ore material from the ore pit is also a part of the production requirements. The input coefficient m is used to specify the minimum amount of ore that needs to be mined by the shovels. The optimal solution to the first-sub model contains continuous numbers of trucks, X(s,, of truck type g to be allocated to between shovel s and dump d. The solution process is not complete without obtaining a practical number of trucks to be deployed. The second sub-model serves to carry out the task of determining the ultimate discrete truck solution. Its model is a mixed integer linear optimization model and its mathematical format is shown below Minimize TruckResou rce 2) = K( Y ( s, ( 240-Ton Truck s d g Units Subject to K ( Y ( s, TruckResource(1) 240-Ton Truck s d g Units L o ( s, Y ( s, CShovel ( s) d g τ ( s, o 60 L o ( s, Y ( s, m CShovel ( s) d g s d τ ( s, o 60 ( (21) (22) Tonnes/H (23) Tonnes/H (24) Y ( s, R (25) ( ) i = 1: Y i ( x, 0, i = 2,3... (26) Y ( i 1) ( s, 1 Y ( i) ( x, Y ( i 1) ( s, g, + 1 Y ( s, 0 (27)

11 The linkage between the two models is based on the truck resource quantity, TruckResource(1), which corresponds to the optimal, but fractional, number of trucks found from solving the first submodel, is generally known to be a theoretical optimal truck resource. The objective of the second submodel is to minimize the deployed truck resource, TruckResource(2), that corresponds to a discrete truck solution denoted by Y(s,. The optimal objective value found from solving the first sub-model, TruckResource(1), helps define the lower bound of the objective function value of the second sub-model (Constraint (22)). Constraints (23) to (25) are duplicates of Constraints (17) to (19) in the first sub-model with the exception of the number of trucks being discrete. Initial conditions (Constraint (26)) are added to ensure realistic transition of truck solution between two consecutive periods. This constraint requires that previous truck allocation be taken into account as input parameters. This constraint helps ensure a gradual change transition of truck solutions from one time period to the next. By breaking the original model into two sub-models, we can manage to reduce the complexity in the original problem and obtain a faster convergence to the problem solution Real-Time Allocation In daily mine planning, trucks are allocated and this allocation plan is communicated to the dispatcher, who will then make the final decision on individual truck assignment; however, this means of operation demands a highly experienced dispatcher to ensure a successful operation. The objective of this research is to support the dispatchers decision making using an optimization-based technique. Unlike various past studies where real-time data is used only during the dispatching stage, this study incorporates operational real-time data at the allocation stage (Figure 2). Truck allocation and truck dispatching are two distinct tasks, but they occur sequentially within the allocation period. Initially, trucks are assigned to the hauling operation at the start of the time period. The hauling operation depends on the real-time truck dispatch system to ensure the most efficient hauling operation. The truck dispatch system may work at the minute level, while the allocation happens less frequently. Since the truck allocation process also depends on many varying parameters, such as the truck cycle times and the truckloads, realtime allocation should enhance the efficiency of the hauling operation.

12 Indee in the event of the breakdown of a shovel or a crusher at the dump, the mine environment is completely different, requiring a reallocation of the truck resource to maintain the same level of efficiency. Acting as a useful decision support tool, a real-time allocation can quickly provide the new optimal allocation solution to the dispatcher. Furthermore, truck reallocation can also be triggered manually or automatically when some pre-programmed conditions are met. An automatic allocation advisor can alert the dispatcher of the mining condition that can lead to a bigger future problem or a situation where savings can be made. Realizations of uncertain parameters Planner Input Optimizer Allocation Truck Solution X i Plant (Mine Operation) Updater Statistical Information on Truckloa Cycle Time Past-period characteristic data Figure 2 - Optimization and Plant Simulation with Update The process starts with the collection of mine data in past time period. The mine data is fed to the Updater module, which serves to modify and prepare the input data for the optimization model. In this study, the updater module was designed as a simple low-pass filter, whose main job is to dampen abrupt changes in the probability distribution characteristics for the truck cycle times and truckloads. The main idea of the updater is to allow the system to adapt to changing conditions in the mine environment, while at the same time eliminating high frequency variation that must be either handled at the dispatch level or ignored. Using the plant input, adjusted by the updater and with manual input from the dispatcher, the optimizer module, with embedded chance-constrained allocation model as previously presente generates a truck allocation solution X i that is to be implemented in the next time period. Figure 3 provides details on how the updater determines the new distribution characteristic of the truck cycle times and truckload. Distribution information R t, gathered during the mine operation is used in conjunction with that corresponding to the current solution P t, to derive the updated characteristics, which

13 is required for the new allocation period t+1. The information used in the updater includes the mean and standard deviation of the truck cycle times and truckloads. R t P t+1 Updater Pt + ( λ ) R t [ σ µ ] P = λ 1 t+ 1 P, R = L L where 0 λ 1, σ τ µ τ P t Figure 3 - The Updater Module The coefficient λ serves to control the frequencies that will be passed to the allocation problem. When λ is close to 1, the updater only allows very long-term trends in changes in the mining operation to be passed to the optimizer; whereas, when λ is small the updater will pass short-term variation to the optimizer Case Studies A simplified hauling configuration of 2 ore shovels and 1 crusher was simulated as an illustration of the proposed real-time allocation approach. Trucks travel on a fixed route connecting the two ore shovels to the common crusher, with the objective of satisfying the production requirement with a minimum truck resource. A custom discrete event simulator was developed to carry out the simulation of the mine operation. Data collected from the operation showed that many of the uncertain parameters are normally distributed with specified means and standard deviations. The varying parameters include truck speeds, shovel load times, truck dump times, and truckloads. These normally distributed parameters ultimately contribute to uncertain truck cycle times and truckloads. The simulator simulates truck queues at the shovel and dump locations. Trucks will go into a queuing state when the shovel is busy and similarly, loaded trucks will have to queue up at the dump location or go into a wait state when the surge pile is full.

14 In the case study, the allocation period was chosen to be 3 hours (H = 3). This should be long enough to collect sufficient data to be statistically meaningful, but at the same time, must be short enough to allow the optimization system to be responsive to a low surge pile. Further, the system was programmed to trigger a new allocation process before the end of the 3-hour period when the surge level is below a specified threshold. This feature is critical when process upsets occur. The simulation study was conducted over 2 scenarios to demonstrate the effect of the chanceconstrained optimization technique in the real-time truck allocation problem. The first scenario assumes a normal operating condition. The second scenario involves an unexpected reduction in mean truck speeds, which could be caused by poor weather. Plots of the variation of the surge level (Tonnes) over a 30-hour time horizon are used to show the performance of the ore hauling operation when truck allocation with feedback is integrated into the process. Important data shown in each time period (3-hour lon includes the total truck resource (units of 240-Ton truck), the number of trucks to be deployed of specific truck type, the expected mean ore rate and the actual mean ore rate realized in the (simulated) operation. Scenario 1 The first scenario simulates operation with no upsets (Figure 4). The truck allocation solutions appear conservative as the surge level remains above the desired level for most of the time. This result is due two factors: 1) the integer restriction on the truck allocation means that constraints will be exceeded since a fractional solution would exactly satisfy constraints; 2) the probabilistic constraint ensures that the surge level is above the desired level at the end of the perio 95% of the time. The truck allocation process stewards to the surge level, not the ore rate, which is in agreement with the probabilistic constraint (Constraint (9)). When the surge level is higher than the desired level, less aggressive truck solutions are generated since there is an abundance ore supply in the surge. In these cases, the resulting ore rate is lower than the extraction ore rate (refer to Calc. Rate in Figure 4). The only exception is in period 8 where the both Calc. Rate and Simu. Rate are above 6000 Tonnes/H. A higher truck resource is allocated in this period to return the surge to a comfortable level.

15 Truck allocation throughout the time horizon shows a realistic transition between any two periods. The difference of the truck resources in two consecutive periods is not more than 1 unit of 240-ton truck. Such allocation solutions are more realistic as they closely reflect the actual operation in the truck dispatching office. The surge behavior in the first period is unrealistic due to initialization. According to the specific simulation algorithm, haul trucks all start in the RunEmpty state; therefore, it takes some time for the first ore load to reach the crusher, resulting in a short period of quick depletion of the surge. Ore delivery caught up quickly, helping the surge recover from the previous unrealistic depletion. Further, the allocated truck resource appears higher than neede causing the rise of the surge level in the first period. This effect also lingers into the second period. If this is a critical issue the simulation initial values can be modified. Scenario 2 The conservatism in the truck solution is a blessing for an operation with upsets. Figure 5 shows that, when trucks are driven with much slower speeds, truck cycle times become longer; the ore in the surge is depleted in the third period. Since the reallocation does not occur until the end of the perio high inventory of ore in the surge helps offset the reduced ore supply due to the change in the operating conditions. The system begins to adapt to the change of condition in the subsequent periods, allocating more trucks to the hauling task to maintain the required level of the ore in the surge. The recovering attempt begins to occur in period 4, when a 320-Ton truck is added to the hauling fleet (total truck resource is increased by 1.33 units from period 3). In subsequent periods, more trucks are allocated to help raise the surge level to the expected level. The surge build-up is slow and gradual because of the conservative update rule (50% of the effect of the change in truck cycle time is accounted for in the feedback rule). Truck allocation solutions from period 4 onward correspond to the Calc. Rate consistently above the 6000 Tonnes/Hour ore rate to Extraction. These allocation solutions demonstrate the ability of the process to adapt to changes in the operating condition. However, in the current instance of the simulation, it takes almost 7 time periods for the surge to recover to the desired level. For mine dispatchers with low

16 risk tolerance, it is possible to raise the desired surge level to a higher level, for example 9000 Tonnes instead of 7000 Tonnes. But bringing the surge level close to the maximum level may lead to inefficient truck use, as trucks are more likely to wait at the dump when the surge becomes full. In both cases, truck allocation appears to favor the smaller truck size, allocating the 240-ton trucks before the 320-ton trucks or even using 360-ton trucks. This behavior is influenced by the relative difference between the cost coefficients, which were assigned based on the loading capacities. Alternatively, the cost of renting trucks could be used. 4. Conclusions Stochastic optimization, specifically the chance-constrained metho can be successfully applied in a truck allocation problem. This technique makes use of the probability distribution information of the uncertain parameters to quantify and incorporate the effect within the truck allocation problem. The chance-constrained technique requires reasonable computation and is simple to implement, which makes it a suitable technique to be used for the truck allocation problem. This allocation problem is best solved within a real-time framework, providing the dispatcher with the truck solutions that they can either use directly or compare with heuristic-based solutions. The study shows that the implementation of the truck allocation using current operating data allows adaptation to disturbances in the mine, freeing the dispatcher from the complex task of gathering and assessing all the information to make new allocation decision. Real-time allocation does not preclude the implementation of the real-time dispatching system, which is critical in dealing with up-to-the-minute dispatching decisions. The model used in this work is simplistic and can be improved by including 1) an ore-blending constraint, and 2) a constraint to ensure a smooth transition of truck solutions between periods. 5. References 1. Birge J. R. and Louveaux F., Introduction to Stochastic Programming, Springer, Charnes A. and Cooper W. W., Cost horizons and certainty equivalents: An Approach to stochastic programming of heating oil, Management Science 4, pp , 1958.

17 3. Elbrond J. and Soumis F., Towards integrated production planning and truck dispatching in open pit mines, International Journal of Surface Mining, pp. 1-6, Lizotte Y., Bonates E. and Leclerc A., Analysis of truck dispatching with dynamic heuristic procedures, Off- Highway Haulage in Surface Mines, Golosinski & Srajer (eds), Balkema, Rotterdam, pp , Prekopa A., Stochastic Programming, Kluwer Academic Publishers, Soumis F., Ethier J. and Elbrond J., Evaluation of the new truck dispatching in the Mount Wright Mine, 21 APCOM Proceeding, pp , Temeng V. A., Francis O. O. and Frendewey, Jr., J. O., Real-time truck dispatching using a transportation algorithm, International Journal of Surface Mining, Reclamation and Environment 11, pp , Xi Y. and Yegulalp T. M., Optimum dispatching algorithm for Anshan open-pit mine, APCOM Proceedings 24, pp , 1994.

18 Surge Volume (Tonnes) vs. Time Ore rate to Extraction: 6000 Tonnes/Hr Main Constraint: Prob { InitSurge + Hours * ( SuppliedRate - OreRate ) >= DesiredSurge} >= 95 % Surge Vol Rate to Extraction Surge (Tonnes) Tot. Res. # Trucks [9,0,0] 8.00 [8,0,0] 7.33 [6,1,0] 7.00 [7,0,0] 7.00 [7,0,0] 7.33 [6,1,0] 7.33 [6,1,0] 8.00 [8,0,0] 7.00 [7,0,0] 7.33 [6,1,0] Calc. Rate Simu. Rate :00 10:00 13:00 16:00 19:00 22:00 01:00 04:00 07:00 10:00 Time Total truck resource is measured in units of 240-T payloa Truck Types used: [240-T,320-T,360-T] Figure 4 - Surge Level vs. Time (Scenario 1, CCP, 3 Truck Types)

19 Surge Volume (Tonnes) vs. Time Ore rate to Extraction: 6000 Tonnes/Hr Main Constraint: Prob { InitSurge + Hours * ( SuppliedRate - OreRate ) >= DesiredSurge} >= 95 % Surge Vol Rate to Extraction Surge (Tonnes) Tot. Res. # Trucks 2000 Calc. Rate Simu. Rate 0 Speed reduction starts from Period [9,0,0] [7,1,0] [7,0,0] [7,1,0] [8,1,0] [10,0,0] [10,0,0 [11,0,0] [10,0,0] [10,0,0] :00 10:00 13:00 16:00 19:00 22:00 01:00 04:00 07:00 10:00 Time Total truck resource is measured in units of 240-T payloa Truck Types used: [240-T,320-T,360-T] Figure 5 - Surge Level vs. Time (Scenario 2, CCP, 3 Truck Types

20 6. Biographies Chung H. Ta works as a research engineer at the Research department in Syncrude Canada Ltd. He obtained his bachelor degrees in Electrical Engineering and Computer Science from the University of Saskatchewan in 1988 and his M.Sc. degree from the University of Alberta in His research interest is in the area of operations research and decision support system in a production environment. Dr. James V. Kresta graduated from McMaster University with a Ph.D. on Application of Multivariate Statistics in Chemical Engineering in For the last 12 years Dr. Kresta has been working for Syncrude Research, working on forensic data analysis, process control and optimization. For the past 5 years he has lead a series of projects to develop Integrated Decision Support for the Mine and Bitumen Production areas of Syncrude. J. Fraser Forbes is a Professor in the Department of Chemical & Materials Engineering at the University of Alberta, Canada. He has over 20 years of experience in control/systems engineering as a researcher, educator and practitioner, and has worked in the steel, food products, forest products, and petrochemical industries. His current research interests include the use of optimization techniques in the design and synthesis of industrial automation systems. Horacio J. Marquez received the B. Sc. degree from the Instituto Tecnologico de Buenos Aires (Argentina), and the M.Sc.E and Ph.D. degrees in electrical engineering from the University of New Brunswick, Fredericton, Canada, in 1987, 1990 and 1993, respectively. From 1993 to 1996 he held visiting appointments at the Royal Roads Military College, and the University of Victoria,Victoria, British Columbia. Since 1996 he has been with the Department of Electrical and Computer Engineering, University of Alberta, where he is currently a Professor and Department Chair. Dr. Marquez is the Author of Nonlinear Control Systems: Analysis and Design, (Wiley, 2003). He received the 2003/2004 University of Alberta McCalla Research Professorship. His current research interests include nonlinear dynamical systems and control, nonlinear observer design, robust control, and applications.