Literature review

There are four main topics that are covered in this paper's proposed model: (1) fleet design, (2) LCC analysis, (3) reliability, and (4) overcapacity. The main objective of this study is to identify which vehicles are actually needed for company operations in order to ensure good fleet design. In one sense, this topic is a well-studied subject and there is a great amount of literature regarding techniques, models, and solutions for different industries and particular problems within this field of the study.

In this study specifically, the work of Coelho et al. (Reference Coelho, Grasas, Ramalhinho, Coelho, Souza and Cruz2016) is of interest because the authors base their analysis on a heterogeneous fleet and propose a model for optimal routing in a vehicle routing problem (VRP) variant, all in order to minimize transportation cost. Because they worked with a heterogeneous fleet, they used a heuristic that they called GILS-VND. This heuristic combines iterated local search (ILS), the greedy randomized adaptive search procedure (GRASP), and the variable neighborhood descent (VND) procedure. Compared with other similar studies, this proposal achieved better economic and technical results. Another study from Coelho et al. (Reference Coelho, Souza, Coelho, Guimarães, Lust and Cruz2012) also suggested that fleet design in a mining operation is a computationally demanding task, thereby supporting the use of heuristics to develop solutions to real-world applications. Finally, King et al. (Reference King, Goycoolea and Newman2017) applied an appropriated branch-and-bound technique to solve the linear programming problem behind the transition between underground and above ground operations. Their technique included rounding heuristics and the authors applied their proposed methodology to a real metal extraction operation. It was a very large problem (50,000 variables and over 1.5 million constraints), but the framework managed to efficiently solve the linear programming problem.

In the case of LCC analysis, the study by Johannknecht et al. (Reference Johannknecht, Gatzen and Lachmayer2016a) combined fleet size, availability, and LCC for gas and oil services. This paper is interesting because it provides a model based on cost efficiency, where fleet size depends on availability. There is an important difference between this study and the proposed model in this study, which is the fact that this study does not uniformly characterize the fleet and also considers overcapacity. However, this study successfully integrated the LCC in order to create a follow-up cost analysis that considered the fleet size. Li and Epureanu (Reference Li and Epureanu2018) optimized the use of a military fleet using the LCC approach. Their model was able to diminish costs related to operation with minor effects on fleet readiness. This was also demonstrated in a real-life military implementation and achieved excellent results.

Reliability is the third important component. In the study by Johannknecht et al. (Reference Johannknecht, Gatzen, Hahn and Lachmayer2016bb), a thorough cost analysis considering availability and LCC for oil and gas operations was made. The authors proposed a framework that considered cost from the early concept phase all the way through to deployment. They acknowledged that this kind of consideration was needed to realistically model the expendables that a company must consider when acquiring a particular vehicle or tool, and then empirically demonstrate its efficiency. Finally, the work of Klosterhalfen et al. (Reference Klosterhalfen, Kallrath and Fischer2014) proposed a model that optimally determined the structure and fleet size for a chemical company while also considering availability constraints and a predefined number of vehicle types. Through this approach, a considerable reduction (120 vehicles) was made to fleet size without compromising company operations.

Finally, overcapacity is another key component to address in our proposed model for fleet design. However, to the best of our knowledge, there are no previously published studies that address this topic as it is presented here. There are, however, studies that have defined and explained the methodology (such as Vestergaard et al., Reference Vestergaard, Squires and Kirkley2003; Kjærsgaard, Reference Kjærsgaard2010), but it has not yet been directly applied to fleet design. Thus, this study will contribute to the existing literature in this important field of study, which is particularly crucial for mining operations.

Fleet sizing with overcapacity

In order to model fleet size while also considering overcapacity, it is necessary to understand this objective and the justification behind it. Overcapacity is a long-term investment strategy aimed at ensuring better operational results through fleet availability and inefficiency cost reduction. For example, transportation costs in large mines are around 50% or 60% of total operational costs (Chaowasakoo et al., Reference Chaowasakoo, Seppälä, Koivo and Zhou2017). Moreover, a system with overcapacity minimizes the failure effect and considers the possibility of replacing an unavailable element with a standby element or increasing the idle capacity of other elements. From a capital expenditure (CAPEX) point of view, fleet sizing with overcapacity increases the required investment but reduces OPEX due to higher fleet availability (Amado Reference Amado2013).

Consider N as the total equipment to be evaluated. Let x _i be defined as follows:

$$x_i = \left\{ {\matrix{ 1 \hfill & {{\rm if}\,{\rm equipment}\,i\,{\rm is}\,{\rm purchased/maintained}} \hfill \cr 0 \hfill & {{\rm if} \,{\rm equipment}\,i \,{\rm is\; not\;}} \hfill \cr} \;} \right.\quad \forall \; i\; \epsilon \; \lcub {1\comma \, \ldots\comma \, N} \rcub.$$

If the analysis is before the beginning of the project, x _i = 1 means that the vehicle would be bought. If an operating fleet is being evaluated, it would mean that the vehicle would be maintained and not sold.

All equipment in the system have two possible operating states: available (with the value 1) and not available (with the value 0). Thus, the first step is to obtain all the possible combinations of operating states. Because operating states are binary, the total number of operating states in the system is J, defined as follows:

(1)$$J = 2^{\sum\nolimits_{i = 1}^N {x_i}}. $$

Consider a combination of system operating states J and the corresponding binary variables used to visualize the operative status of each equipment item are defined as follows:

(2)$$\delta _{ij} = \left\{ {\matrix{ 1 \hfill & {{\rm if}\,{\rm equipment}\,i\,{\rm is}\,{\rm operative}\,{\rm in}\,{\rm state}\,j} \hfill \cr 0 \hfill & {{\rm if}\,{\rm equipment}\,i\,{\rm is}\,{\rm inoperative} \,{\rm in}\,{\rm state}\,j{\rm \;}} \hfill \cr} \;} \right.\quad \forall \; i\; \epsilon \; \lcub {1\comma \, \ldots\comma \, N} \rcub \comma \,\; \forall \; j\; \epsilon \; \lcub {1\comma \, \ldots\comma \, J} \rcub. $$

The available capacity of each combination of operative states can be obtained from the matrix δ:

(3)$$C_j^A = \sum\limits_{i = 1}^N {\delta _{ij}\cdot C_i^{{\rm max}}} \; \quad \forall \; j\; \epsilon \; \lcub {1\comma \, \ldots\comma \, J} \rcub. $$

Given the operative state j of the system, the ratio between the operational capacity of the corresponding equipment $C_{ij}^O$ and its maximum capacity ${\rm Cap}_i^{{\rm max}}$ will be the same for all equipment. To determine the final impact of each piece of equipment, it is necessary to identify the operative states j of these systems, so that the available capacity of the equipment is less than or equal to the required system capacity (otherwise, the impact will be defined as 0 because the system already meets the capacity requirement).

Therefore, the ω _j binary variable is defined as follows:

(4)$$\omega _j = \left\{ {\matrix{ 1 \hfill & {C_j^A \ge C_S^R} \hfill \cr 0 \hfill & \sim \hfill \cr} \;} \right.\quad \forall \; j\; \epsilon \; \lcub {1\comma \, \ldots\comma \, J} \rcub. $$

The operative capacity $C_{ij}^O$ of the equipment, therefore, will be given in the following expression:

(5)$$C_{ij}^O = \displaystyle{{\lpar {\omega_j\cdot C_S^R + \lpar {1-\omega_j} \rpar \cdot C_j^A} \rpar \cdot \delta _{ij}\cdot {\rm Cap}_i^{{\rm max}}} \over {{\rm Cap}_i^{{\rm max}} + \sum\nolimits_{k = 1\, \wedge \,k\ne i}^N {\delta _{kj}\cdot {\rm Cap}_k^{{\rm max}}}}} \quad \forall \; i\; \epsilon \; \lcub {1\comma \, \ldots\comma \, N} \rcub \comma \,\forall \; j\; \epsilon \; \lcub {1\comma \, \ldots\comma \, J} \rcub. $$

When calculating impacts, that is, $C_S^R \gt C_j^A$, the operational state of the system is considered to be the state of reference for all equipment. The aim is to determine the loss of capacity that unavailable equipment exerts on the system, depending on the operational capacity that the system would have if such an equipment were operating. Hence, the formula used to determine the impact will be given in the following equation:

(6)$$I_{ij} = \left\{ {\matrix{ {\displaystyle{{\lpar {C_S^R -C_j^A} \rpar \cdot \lpar {1-\delta_{ij}} \rpar \cdot C_{i1}^O} \over {C_S^R \cdot \sum\nolimits_{k = 1}^N {\lpar {1-\delta_{kj}} \rpar \cdot C_{k1}^O}}}} \hfill & {C_S^R \gt C_j^A} \hfill \cr 0 \hfill & \sim \hfill \cr}} \right.\quad \forall \; i\; \epsilon \; \lcub {2\comma \, \ldots\comma \, N} \rcub \comma \,\; \forall \; j\; \epsilon \; \lcub {2\comma \, \ldots\comma \, J} \rcub. $$

Considering an operating state of the system, the required capacity will always be satisfied. Therefore, the probability based on the availability of each operating state of the system is defined as follows:

(7)$$P_j^A = \prod\limits_{i = 1}^N {A_i^{\delta _{ij}} \cdot {\lpar {1-A_j} \rpar }^{\lpar {1-\delta_{ij}} \rpar }} \quad j\; \epsilon \; \lcub {1\comma \, \ldots\comma \, J} \rcub. $$

The weighted impact of the equipment, considering all possible combinations for the operating state of the system, is given as follows:

(8)$$I_i^\omega = \sum\limits_{\,j = 1}^J {P_j^A \cdot I_{ij}} \quad \forall \; i\; \epsilon \; \lcub {1\comma \, \ldots\comma \, N} \rcub. $$

Finally, the availability of load sharing systems and overcapacity is calculated as follows:

(9)$$A_{{\rm sist}} = \sum\limits_{i = 1}^N {A_i\cdot I_i^\omega}. $$

Once the calculation of every component is ready, the related cost of having (or not having) a specific piece of equipment must be calculated. The first cost to calculate is the operational cost, which is defined in the following equation:

(10)$$C_{{\rm op}_{ij}} = \left\{ {\matrix{ {\displaystyle{{\delta_{ij}\cdot C_j^A} \over \Delta}} \hfill & {C_S^A \gt C_j^R} \hfill \cr {\delta_{ij}\cdot C_j^A} \hfill & \sim \hfill \cr}} \right.\quad \forall \; i\; \epsilon \; \lcub {2\comma \, \ldots\comma \, N} \rcub \comma \,\; \forall \; j\; \epsilon \; \lcub {2\comma \, \ldots\comma \, J} \rcub.$$

where Δ is a scaling factor, introduced to diminish the impact of overcapacity.

With this, we can calculate the operational cost of all scenarios as defined in the following equation:

(11)$$C_{{\rm op}} = \sum\limits_{i = 2}^N {\sum\limits_{\,j = 1}^J {C_{{\rm op}_{ij}}\cdot P_j^A \cdot {\rm FA}}}. $$

Therefore, considering (9), we can calculate the inefficiency cost, as shown in the following equation:

(12)$$C_{{\rm in}} = \lpar {1-A_{{\rm sist}}} \rpar \cdot C_{{\rm inef}}\cdot {\rm FA}\cdot h. $$

where C _inef is the inefficiency cost of the fleet per hour due to unavailability, FA is the actualization factor, and h is the annual operational time.

Finally, we used the following cost-related objective function to minimize the global cost:

(13)$$\mathop {\min} \limits_x \sum\limits_{i = 1}^N {x_iiv_i + C_{{\rm in}} + C_{{\rm op}}}. $$

where x _i is the decision of purchasing/maintaining the equipment i (1 if it is purchased/maintained and 0 if not) and iv _i is the investment/opportunity cost related to equipment x _i.

It should be noted that, in the case of fleets in operation, availability indicators, operating costs, and maintenance costs are much more distorted given their historical values. Therefore, the mathematical model can be summarized as follows:

Mathematical model

$$\mathop {\min} \limits_x \sum\limits_{i = 1}^n {x_iiv_i + C_{{\rm in}} + C_{{\rm op}}}$$

$${\rm s}{\rm. t.}\quad Q_{{\rm sist}} \le Q_{{\rm req}}$$

$$x\; \epsilon \; \lcub {0\comma \,1} \rcub$$

with

Proposed solution algorithm

To solve the proposed model, two alternatives were explored. First, a deterministic approach was taken, solving the model (when possible) with a full calculation of every possible state. Second, a metaheuristic approach was taken, solving the model for all proposed cases in an approximated (but near-optimal) calculation.

Determinist model

This methodology analyzes and calculates the total cost of all possible scenarios for each combination. The number of calculations needs to achieve an optimal fleet is 2^J. The number of calculations and the way data is managed prevent a solution for fleets with a high number of vehicles.

As shown in Algorithm 1, each step is explained and justified.

Metaheuristic approaches

There are 2^J possible scenarios that need to be checked in order to find the optimal solution with the entire model being used in each case. Due to this a high number of possibilities, two metaheuristics were tested to find the optimal solution, namely genetic algorithm (GA) and ant colony optimization (ACO). The reason why we chose this type of algorithm is because model complexity scales quickly in relation to the number of trucks that are currently owned by a company. This means that, in real-life applications, a deterministic approach will be very inefficient, costing the company time and money. International mining companies can have fleets as large as 400 trucks that they then seek to optimize (see the Results section for links regarding this fleet size). This makes computation time, even if there is an ample time in which to run the model, an important factor to consider.

GA is an evolutionary algorithm (EA) that was first presented by Holland and Reitman (Reference Holland and Reitman1977). It is based on a Darwinian approach of the survival of the fittest, an evolutionary process that identifies the best candidate to solve the problem. Through the evolution, genetic operators, such as crossover and mutation, are applied in addition to a selective process that preserves the best individuals over generations. The application to the current problem is presented and described below. ACO, introduced by Dorigo and Gambardella (Reference Dorigo and Gambardella1997), is a population-based algorithm that mimics the behavior of ants and how they communicate with each other in order to find an optimal path to a specific goal. Specifically, each ant follows a path that leads to a solution of the problem, and ants leave pheromones along the best path (the shortest) that are stored. This allows the next generation to know what paths are better. After enough generations have passed, the ants should, in theory, obtain the optimal solution or a near optimal solution.

The GA algorithm pseudocode is shown in Algorithm 2.

The ACO pseudocode is shown in Algorithm 3.

Specifically, we follow the proposed algorithm by Xiong et al. (Reference Xiong, Wang and Yan2006) for the GA approach and Hristakeva and Shrestha (Reference Hristakeva and Shrestha2004) for the ACO approach. The specifications for both algorithms can be seen in Table 1.

Table 1. Hyperparameter settings for ACO and GA

The first column shows the main hyperparameters of the GA. Generations refer to the maximum number of iterations that the algorithm can perform to find the solution. Stall refers to the maximum number of generations have to pass consecutively and, if an error does not improve, the algorithm stops. Population size refers to the number of genes that are candidates to solve the problem. Crossover rate and mutation rate are the percentages at which these two operators are applied. The second column contains the main hyperparameters of the ACO. Maximum iterations refer to the maximum times that the algorithm can be run to solve the problem. The number of ants are the population of ants that search for the optimal solution. Alpha and beta refer to the relative importance of track and visibility, respectively, and rho is the durability of the track.

Simulation experiment

The proposed model was tested in a simulation using different fleet sizes and applied these two metaheuristic approaches to solve it. Table 2 presents a technical description of the trucks used for this simulation, including capacity, availability, and opportunity cost.

Table 2. Fleet technical description

The information in this table is organized as follows: Availability and capacity are unique parameters measuring confidence that are specific to each truck. Opportunity cost refers to how much the truck cost, meaning that, in the optimal solution, the truck not selected is actually a source of positive revenue for the organization.

The computational environment used was Matlab 2018a, Xubuntu 16.04, two Intel Xeon E5-2630 processors, and 12 GB of ram.

The simulation experiment was developed in an incremental fashion. Five scenarios were tested considering fleets of size 10, 12, 15, 18, and 20, in which the GA, ACO and complete enumeration approaches were selected to solve the problem. The results for each approach were then compared. In all simulations, the required capacity was set at 70% of total capacity, inefficiency cost at $110 (USD/h), 10% annual interest rate, and 10 years for an evaluation period. As expected, the complete enumeration (deterministic) found the global optimal solution, so the two metaheuristics were tested against this approach. Additional experiments consisting of 22, 25, and 30 trucks, for which the deterministic approach take over a day to compute, were also performed. The results of all experiments can be seen in Tables 3–5 for the deterministic, ACO, and GA approaches, respectively.

Table 3. Results from the deterministic approach

This table shows the results of solving the problem by using the complete enumeration approach. Best cost refers to the total cost of the trucks selected. Time refers to how much time it takes for the algorithm/approach to find the solution. # Trucks In refer to how many trucks must be considered in the optimal solution of the problem. Trucks Left refer to which trucks, specifically, are left out of the solution.

Table 4. Results from the ACO approach

This table shows the results of solving the problem by using the ACO approach. Best cost refers to the total cost of the trucks selected. Time refers to how much time it takes for the algorithm/approach to find the solution. # Trucks In refer to how many trucks must be considered in the optimal solution of the problem. Trucks Left refer to which trucks, specifically, are left out of the solution.

Table 5. Results from the GA approach

This table shows the results of solving the problem by using the GA approach. Best cost refers to the total cost of the trucks selected. Time refers to how much time it takes for the algorithm/approach to find the solution. # Trucks In refer to how many trucks must be considered in the optimal solution of the problem. Trucks Left refer to which trucks, specifically, are left out of the solution.

The simulations that used small fleet sizes were developed in order to determine the relationship between the actual global minimum and the minimums found by the metaheuristic approaches. We can see that, in general, both metaheuristics managed to find the global minimum but under different conditions. ACO found the exact global minimum for small fleet sizes, that is, 10, 12, and 15 trucks, but only found a near optimal solution for 18 (0.34% difference) and 20 trucks (1.75% difference). On the other hand, GA did not behave well for small fleet sizes, resulting in considerable differences with the optimal case (2.43% for 10 trucks and 0.34% for 12 trucks). However, GA interestingly find the exact global minimum for 15 and 18. Yet, it also failed to determine the actual global minimum for 20 trucks, having a difference of 1.07%. This is a very important result because it shows that this kind of algorithm can actually find the optimal solution for particular problems, opening up the possibility of applying this method to a real-world application with excellent results.

Considering the above, the time consumption analysis was also worth performing, especially given the fact that the actual global minimum can be reached through metaheuristic methods. For very small cases, as expected, the deterministic approach was faster than the metaheuristic methods. This is not a surprising result because, for small problems, the algorithm takes more time in preparing their configurations than actually finding the solution. Specifically, for the first two simulations, there was a big difference in time (×5 for ACO and ×3 approximately for GA). Yet, in the simulation of 15 trucks, ACO managed to reduce time consumption by around 38%, and GA was slightly higher (3.63% higher). When the 18 trucks simulation was performed, time was reduced in both approaches by a seventh of the time that it took the deterministic approach to complete the same problem. Significantly, GA managed to actually find the global minimum, which is a promising result for larger applications. In the case of 20 trucks, these findings made this even more evident, given that the deterministic approach took almost 2 days to find the solution, while the metaheuristic methods took just seconds.

It is important to notice the actual trucks that were selected in each case. For the fleet size of 10 trucks, only ACO found the actual global minimum, which was to select all the trucks. In contrast, GA dropped the 8th truck, finding a higher cost function. For the 12 trucks, both ACO and GA kept 11 of the 12 trucks, but again only ACO actually dropped the correct one (the third one) and GA dropped the 10th, again finding a higher cost function.

An interesting result happened in the 15 fleet size simulations. In that case, both ACO and GA found the global minimum and, therefore, dropped the correct truck, the third one. Until this simulation, we had noticed that ACO behaved better than GA in solving the problem, both in terms of cost function as well as identification of the actual solution and time consumption.

However, for higher fleet sizes, GA achieved better results, finding the global minimum (and therefore, selecting the correct trucks to drop) for 18. Both algorithms failed to find the optimal minimum for 20 trucks, but ACO selected the same amount of trucks in the fleet (17 trucks), but made different choices than GA, which selected 18 of the 20 trucks. Yet, GA actually reached a solution close to the actual minimum, regardless of selecting one more truck. When taking time into consideration, we believe that this result could be due to the configurations regarding the number of ants, maximum iterations, and alpha setting of the ACO algorithm. This could be a topic for further research as finding the best hyperparameters for this algorithm will not be considered in this work. Thus, it was relatively expected that ACO would take less time but fail to find the optimal solution.

This allows us to conclude that, considering the time required for the deterministic approach to complete simulations, the experiment with larger fleet's sizes would be pointless to solve using this approach because of the clear exponential growth that occurs with increases in fleet size. Thus, three more simulations were made that only considered the results of the metaheuristic methods. Their results are shown in Tables 6 and 7.

Table 6. Extra ACO simulations

This table shows additional results obtained by using the ACO approach, specifically for greater fleet sizes, which are not possible to solve by using the complete enumeration approach. Best cost refers to the total cost of the trucks selected. Time refers to how much time it takes for the algorithm/approach to find the solution. # Trucks In refer to how many trucks must be considered in the optimal solution of the problem. Trucks Left refer to which trucks, specifically, are left out of the solution.

Table 7. Extra GA simulations

This table shows additional results obtained by using the GA approach, specifically for greater fleet sizes, which are not possible to solve by using the complete enumeration approach. Best cost refers to the total cost of the trucks selected. Time refers to how much time it takes for the algorithm/approach to find the solution. # Trucks In refer to how many trucks must be considered in the optimal solution of the problem. Trucks Left refer to which trucks, specifically, are left out of the solution.

In this extended simulation, we saw that GA managed to obtain better cost functions than ACO in two of the three experiments. However, ACO found the better solution for the smallest of these simulations (of 22 trucks). The time consumption observation that was previously described was also seen here. Indeed, GA took more time but was able to find better solutions to the problem. A very interesting result was in relation to the actual truck composition that each approach found. GA tended to drop slightly less trucks compared with ACO. Although the difference was just one truck in these simulations, this difference could result in a bigger difference for bigger fleet sizes. An important point to consider is that, in general, better solutions were found when more trucks were selected rather than when more trucks were dropped, an important fact for larger fleets.

The important reduction in time consumption presented an opportunity to apply the proposed model to real mining industry situations, where the companies have to optimize very large fleets. To name a few, two of the biggest Chilean mining operations work with 172 haul trucks (Escondida¹) and 120 trucks (Chuquicamata²). Rio Tinto³ (Australia) works with 400 haul trucks and Grasberg⁴ (Indonesia) works with 170 trucks. Although the simulation made in the study was relatively small compared with the real fleet sizes used today in the mining industry, they allow for a fair comparison between the two models. Now, given what happened with the deterministic approach, the exponential time that was needed to make all the necessary calculations makes it impossible to apply this method to such large fleets. The fairly small amount of time that ACO and GA needed to find near-optimal solutions make them good possibilities that could be effective in real-world applications. In order to better visualize this, Figure 1 showed a log-time graph with the three approaches.

Fig. 1. Time consumption based on fleet size.

This figure is a log–log graph of fleet size and shows how long it takes the algorithm/approach to find the optimal solution for that particular configuration. The deterministic approach can solve a problem only to a certain point, after which it can no longer solve problems. Also, metaheuristic approaches are better suited for larger problems as they (1) can solve them within a reasonable time and (2) for smaller problems, the complete enumeration is faster and is guaranteed to find the optimal solution.

This figure clearly shows the rapid growth of the deterministic approach, even in log transformation. Both ACO and GA had a slower tendency, expressed in the slope of the log-time function. The higher slope for the deterministic approach means that it consumes much more time when there are more trucks in the fleet. This allows us to estimate the time required for every approach in a real-world application. Considering that most companies perform fleet optimization once a year, ACO could solve this problem in less than a year (in 338 days) for a fleet composed of 130 trucks, whereas GA could solve the problem in less than a year (333 days) for a fleet composed of 64 trucks. Considering the companies that were cited early, both approaches could actually solve problems for even larger fleets, depending on the computational environment and algorithm enhancement that could be performed on the approaches presented here. This is open to future research. However, this work has already shown that large fleets can be optimized – while also considering overcapacity, availability, and inefficient cost – in less than a year.