A Genetic Algorithm-Based Approach for Optimizing Short-term Production Schedules of Multi-mine Mineral Value Chains

Abstract

This article presents a customized genetic algorithms (GA) with new crossover and mutation operators to solve the short-term production scheduling problem of multi-mine mineral value chains (MVC). The preceding problem consists of determining the extraction sequence and destination allocation of blocks from all the mines collaboratively while closely meeting the quality and quantity requirements of the processing units subject to relevant technical and operational constraints. The short-term production scheduling is carried out at shorter scales wherein the operations are modeled in a great detail with large number of constraints. This makes the industry-scale instances of the problem computationally intractable for standard mixed-integer programming (MIP) solvers. Thus, a GA-based heuristic approach is developed to obtain near-optimal solutions to the large-scale instances of the problem in a reasonable amount of computational time. Computational experiments show that the developed GA-based method is a promising way to handle industry-scale instances of the problem. Moreover, the sensitivity analysis on various parameter combinations of crossover and mutation operators indicates that the customized global mutation operator, when used in combination with the customized crossover operators, took on average 12.5% less time than the customized local mutation operator to converge to a solution.

Appendix. Mathematical model

1.1 Indices and sets

m :

is a mine; m = 1, 2,..., M

p :

is a processing unit; p = 1, 2,..., P

s :

is a stockpile; s = 1, 2,..., S

w :

is a waste dump; w = 1, 2,..., W

d :

is a destination; d = 1, 2,..., D (where D = P ∪ S ∪ W)

t :

is a time period; t = 1, 2,..., T

e :

is a metal element; e = 1, 2,..., E

i _m :

is a block of mine m; i = 1, 2,..., I_m

1.2 Parameters

I _m :

number of blocks in mine m

ver_pre_i,m ::

set of vertical predecessors of block i of mine m

hor_pre_i,m ::

set of horizontal predecessors of block i of mine m

nadj_i,m ::

number of adjacent blocks of block i of mine m

adj_i,m ::

set of adjacent blocks of block i of mine m

pen_smooth ::

penalty cost associated with non-smooth mining

$M{C_{m}^{t}}:$ :

mining capacity of mine m in period t

grade_e,i,m ::

percentage of element e in block i of mine m

tonnage_i,m ::

tonnage of block i of mine m

$tar\_pro{d_{p}^{t}}:$ :

target tonnage requirement for processing plant p in period t

$tol\_prod_{p}^{t,+}:$ :

positive tolerance allowed from tonnage targets of plant p in period t (in %)

$tol\_prod_{p}^{t,-}:$ :

negative tolerance allowed from tonnage targets of plant p in period t (in %)

$min\_pro{d_{p}^{t}}:$ :

minimum tonnage requirement for plant p in period t

$max\_pro{d_{p}^{t}}:$ :

maximum tonnage requirement for plant p in period t

$tar\_grade_{e,p}^{t}:$ :

target grade requirement of element e at plant p in period t

$tol\textunderscore grade_{e,p}^{t,+}:$ :

percentage of positive tolerance allowed for element e at plant p in period t

$tol\textunderscore grade_{e,p}^{t,-}:$ :

percentage of negative tolerance allowed for element e at plant p in period t

$min\textunderscore grade_{e,d}^{t}:$ :

minimum acceptable grade of element e at destination d in period t

$max\_grade_{e,d}^{t}:$ :

maximum acceptable grade of element e at destination d in period t

$pen\_prod_{p}^{t,+}:$ :

penalty cost for positive deviation from target tonnage requirements for plant p in period t

$pen\_prod_{p}^{t,-}:$ :

penalty cost for negative deviation from target tonnage requirements for plant p in period t

$pen\_grade_{e,p}^{t,+}:$ :

penalty cost for positive deviation of element e from its target grade requirements for plant p in period t

$\mathit {pen}\_{\mathit {grade}}_{e,p}^{t,-}:$ :

penalty cost for negative deviation of element e from its target grade requirements for plant p in period t

$in{v_{s}^{t}}:$ :

inventory of stockpile s at start of period t

cap_s ::

maximum capacity of stockpile s

rehandling_cost_s,p ::

cost associated with reclaiming materials from stock s to plant p

pen_MC_m ::

penalty cost associated with deviation from maximum possible mining capacity for mine m

1.3 Decision variables

$x_{i,m,d}^{t}:$ :

Binary variable which equals 1, if block i of mine m is sent to destination d in period t, 0 otherwise

$y_{s,p}^{t}:$ :

Continuous variable which represents the amount of material reclaimed from stockpile s and sent to plant p in period t

$z_{i,m}^{t}:$ :

Integer variable which gives the number of adjacent blocks of block i of mine m not mined in the same period as block i

$dev\textunderscore prod_{p}^{t,+}:$ :

Continuous variable which represents the amount of surplus production from the upper limit of tonnage target for plant p in period t

$dev\textunderscore prod_{p}^{t,-}:$ :

Continuous variable which represents the amount of shortage in production from the lower limit of tonnage target for plant p in period t

$dev\textunderscore grade_{e,p}^{t,+}:$ :

Continuous variable which represents the deviation from the upper limit of grade target of element e for plant p in period t

$dev\textunderscore grade_{e,p}^{t,-}:$ :

Continuous variable which represents the deviation from the lower limit of grade target of element e for plant p in period t

$dev\textunderscore M{C_{m}^{t}}:$ :

Continuous variable which represents the deviation from the maximum possible mining capacity of mine m in period t

1.4 Objective function

The objective function (Eq. (2)) minimizes the deviations from the quality and quantity targets of the processing plants and mines by penalizing these deviations. It also minimizes the frequent movement of excavators as it is a non-productive operation. Finally, the objective function also minimizes the amount of material rehandled from the stockpiles to the processing plants.

Minimize =

$$ \begin{array}{@{}rcl@{}} &&{\sum}_{t=1}^{T} {\sum}_{p=1}^{P}{\sum}_{e=1}^{E} (pen\textunderscore grade_{e,p}^{t,+}\cdot dev\textunderscore grade_{e,p}^{t,+}\\&&+pen\textunderscore grade_{e,p}^{t,-}\cdot dev\textunderscore grade_{e,p}^{t,-})+\\ &&{\sum}_{t=1}^{T} {\sum}_{p=1}^{P} (pen\textunderscore prod_{p}^{t,+}\cdot dev\textunderscore prod_{p}^{t,+}\\&&+pen\textunderscore prod_{p}^{t,-}\cdot dev\textunderscore prod_{p}^{t,-})+\\&&{\sum}_{t=1}^{T} {\sum}_{m=1}^{M} (pen\textunderscore MC_{m}\cdot dev\textunderscore M{C_{m}^{t}}) \\&&+{\sum}_{t=1}^{T} {\sum}_{m=1}^{M} {\sum}_{i=1}^{I^{m}} (pen\textunderscore smooth\cdot z_{i,m}^{t})+\\&& {\sum}_{t=1}^{T} {\sum}_{p=1}^{P} {\sum}_{s=1}^{S} (rehandling\textunderscore cost_{s,p}\cdot y_{s,p}^{t}) \end{array} $$

(2)

1.5 Constraints

The objective function is minimized subject to the constraints which have been outlined in Eqs. (3)–(19). Equation (3) shows the reserve constraint which ensures that a block from a particular mine cannot be mined more than once. In this model, partial mining of blocks is not allowed and thus a block is either completely mined or not mined at all.

$$ {\sum}_{t=1}^{T} {\sum}_{d=1}^{D} x_{i,m,d}^{t} \!\leq\! 1 \quad \forall m = {1,2,...,M}; \forall i={1,2,...,I_{m}} $$

(3)

Equations (4) and (5) ensure that the maximum possible mining capacity is met for each mine of the MVC in each time period. Any deviations from the former is calculated by the decision variable $dev\textunderscore MC_m^t$, which is penalized in the objective function.

$$ \begin{aligned} {\sum}_{d=1}^{D} (x_{i,m,d}^{t}\cdot tonnage_{i,m}) \leq M{C_{m}^{t}} \\ \forall m={1,2,...,M};\forall t={1,2,...,T} \end{aligned} $$

(4)

$$ \begin{aligned} &{\sum}_{d=1}^{D} (x_{i,m,d}^{t}\cdot tonnage_{i,m}) + dev\textunderscore M{C_{m}^{t}}\\ &\geq M{C_{m}^{t}} \quad \forall m={1,2,...,M};\\ &\forall t={1,2,...,T} \end{aligned} $$

(5)

Equation (6) describes precedence relationship between the blocks in vertical direction. The vertical predecessors are defined for each block and this constraint ensures that a block can only be mined in a particular period if all its overlying predecessor blocks have been completely mined by that period.

$$ \begin{aligned} &{\sum}_{d=1}^{D} x_{imd}^{t} \leq {\sum}_{r=1}^{t} {\sum}_{d=1}^{D} x_{jmd}^{r} \quad \\&\forall m={1,2,...,M};\forall t={1,2,...,T};\\ &(i,j \in {1,2,...,I_{m}}; j \in ver\textunderscore pre_{i,m}) \end{aligned} $$

(6)

Equation (7) maintains intra-bench precedence relationship between the blocks. This constraint is similar to constraint (6) and it ensures that a block can be mined via side cut in a period only if all its horizontal predecessors have been completely mined by that period. This constraint helps in ensuring that a minimum mining width is available for an excavator to extract blocks via side cut. It also encourages mining of spatially connected blocks in the same period to limit unnecessary shovel moves.

$$ \begin{aligned} & {\sum}_{d=1}^{D} x_{imd}^{t} \leq{\sum}_{r=1}^{t} {\sum}_{d=1}^{D} x_{jmd}^{r} \quad \\&\forall m={1,2,...,M};\forall t={1,2,...,T};\\ &(i,j \in {1,2,...,I_{m}}; j \in hor\textunderscore pre_{i,m}) \end{aligned} $$

(7)

Equation (8) ensures smooth and continuous mining with minimum movement of excavating equipment from one part of the mine to other. To ensure this, adjacent blocks have been defined for each block i of mine m. The number of adjacent blocks k of block i not mined in the same period as block i is stored in the decision variable $z_{i,m}^t$, which is penalized in the objective function.

$$ \begin{aligned} &{\sum}_{d=1}^{D} (x_{i,m,d}^{t}\cdot nadj_{i,m})- {\sum}_{k=1}^{nadj_{i,m}} {\sum}_{d=1}^{D} x_{k,m,d}^{t} - z_{i,m}^{t} \leq 0 \\& \forall m={1,2,...,M};\\ &\forall t={1,2,...,T}; (i,k \in {1,2,...,I_{m}}; k \in adj_{i,m}) \end{aligned} $$

(8)

Equations (9) and (10) ensure that the tonnage targets are closely met for all the processing plants of the MVC in each period. Any deviations from the upper and lower limits of tonnage targets are calculated by the variables $dev\textunderscore prod_p^{t,+}$ and $dev\textunderscore prod_p^{t,-}$ respectively in each period and are minimized in the objective function.

$$ \begin{aligned} &{\sum}_{m=1}^{M} {\sum}_{i=1}^{I_{m}} (x_{i,m,p}^{t}\cdot tonnage_{i,m}) + {\sum}_{s=1}^{S} y_{s,p}^{t} \\&+dev\textunderscore prod_{p}^{t,-}\ \geq min\textunderscore pro{d_{p}^{t}} \quad \\&\forall p={1,2,...,P}; \forall t={1,2,...,T} \end{aligned} $$

(9)

$$ \begin{aligned} &{\sum}_{m=1}^{M} {\sum}_{i=1}^{I_{m}} (x_{i,m,p}^{t}\cdot tonnage_{i,m}) + {\sum}_{s=1}^{S} y_{s,p}^{t} \\&-dev\textunderscore prod_{p}^{t,+} \leq max\textunderscore pro{d_{p}^{t}} \quad \\&\forall p={1,2,...,P}; \forall t={1,2,...,T} \end{aligned} $$

(10)

Equations (11) and (12) ensure that the material fed to the processes have average grades within acceptable grade range for all the elements in each period. Any deviations from the upper and lower limits of grade requirements for the elements are penalized in the objective function.

$$ \begin{aligned} &{\sum}_{m=1}^{M} {\sum}_{i=1}^{I_{m}} (x_{i,m,p}^{t}\cdot \mathit{tonnage}_{i,m}\cdot\! (\mathit{grade}_{e,i,m} - \mathit{min}\textunderscore \mathit{grade}_{e,p}^{t})) +\\& {\sum}_{s=1}^{S} (y_{s,p}^{t}\cdot grade_{e,s} - min\textunderscore grade_{e,p}^{t})+dev\textunderscore grade_{e,p}^{t,-} \geq 0 \quad \\&\forall e={1,2,...,E};\forall p={1,2,...,P}; \forall t={1,2,...,T} \end{aligned} $$

(11)

$$ \begin{aligned} &{\sum}_{m=1}^{M} {\sum}_{i=1}^{I_{m}} (x_{i,m,p}^{t}\!\cdot \mathit{tonnage}_{i,m}\cdot\! (\mathit{grade}_{e,i,m} - \mathit{max}\textunderscore \mathit{grade}_{e,p}^{t})) +\\ &{\sum}_{s=1}^{S} (y_{s,p}^{t}\cdot \mathit{grade}_{e,s}-\mathit{max}\textunderscore \mathit{grade}_{e,p}^{t})-dev\textunderscore grade_{e,p}^{t,+} \leq 0 \quad \\&\forall e={1,2,...,E};\forall p={1,2,...,P}; \forall t={1,2,...,T} \end{aligned} $$

(12)

It should be noted that the grade_e,s, grade of element e in material reclaimed from stock s, changes as soon as a batch of new material are stocked in it. This dynamic changing of stockpiles grade make constraints (11) and (12) non-linear.

Equation (13) ensures that the inventory of a stockpile cannot exceed its maximum capacity at any point. Equations (14) and (15) deal with the flow of materials from the stockpiles to the processing plants in the MVC. These constraints ensure that the amount of material sent from a stockpile s to all the processes in period t cannot exceed the stockpile’s inventory at the end of period (t − 1). It has been assumed in this model that the materials sent to stockpile s in period t cannot be reclaimed from it in the same period.

$$ \begin{aligned} &in{v_{s}^{0}}+ {\sum}_{m=1}^{M} {\sum}_{i=1}^{I_{m}} {\sum}_{r=1}^{t} (x_{i,m,s}^{r}\cdot tonnage_{i,m})\\&- {\sum}_{p=1}^{P} {\sum}_{r=1}^{t} y_{s,p}^{r} \leq cap_{s} \quad \\&\forall s={1,2,...,S};\forall t={1,2,...,T} \end{aligned} $$

(13)

$$ \begin{aligned} {\sum}_{p=1}^{P} y_{s,p}^{1} \leq in{v_{s}^{0}} \quad \forall s={1,2,...,S}; t=1 \end{aligned} $$

(14)

$$ \begin{aligned} &{\sum}_{p=1}^{P} y_{s,p}^{t} \leq in{v_{s}^{0}} +{\sum}_{m=1}^{M} {\sum}_{i=1}^{I_{m}} {\sum}_{r=1}^{(t-1)} (x_{i,m,s}^{r}\cdot tonnage_{i,m})\\&-{\sum}_{p=1}^{P} {\sum}_{r=1}^{(t-1)} y_{s,p}^{r} \quad \\&\forall s={1,2,...,S}; t={2,3,...,T} \end{aligned} $$

(15)

Equations (16) and (17) ensure that the average grade of element e in the materials sent to stockpile s is within its acceptable grade range. Only the batches of materials with grades within some acceptable range could be sent to the processes or stocks.

$$ \begin{aligned} &{\sum}_{m=1}^{M} {\sum}_{i=1}^{I_{m}} (x_{i,m,s}^{t}\cdot tonnage_{i,m}\\&\cdot (grade_{e,i,m}-min\textunderscore grade_{e,s})) \geq 0 \quad \\&\forall e={1,2,...,E};\forall s={1,2,...,S}; \forall t={1,2,...,T} \end{aligned} $$

(16)

$$ \begin{aligned} &{\sum}_{m=1}^{M} {\sum}_{i=1}^{I_{m}} (x_{i,m,s}^{t}\cdot tonnage_{i,m}\\&\cdot (grade_{e,i,m}-max\textunderscore grade_{e,s})) \leq 0 \quad \\&\forall e={1,2,...,E};\forall s={1,2,...,S}; \forall t={1,2,...,T} \end{aligned} $$

(17)

Equations (18) and (19) define limits on average grade of element e in the materials that have to be sent to the waste dumps.

$$ \begin{aligned} &{\sum}_{m=1}^{M} {\sum}_{i=1}^{I_{m}} (x_{i,m,w}^{t}\cdot tonnage_{i,m}\\&\cdot (grade_{e,i,m}-min\textunderscore grade_{e,w})) \geq 0 \quad \\&\forall e={1,2,...,E};\forall w={1,2,...,W}; \forall t={1,2,...,T} \end{aligned} $$

(18)

$$ \begin{aligned} &{\sum}_{m=1}^{M} {\sum}_{i=1}^{I_{m}} (x_{i,m,w}^{t}\cdot tonnage_{i,m}\\&\cdot (grade_{e,i,m}-max\textunderscore grade_{e,w})) \leq 0 \quad \\&\forall e={1,2,...,E};\forall w={1,2,...,W}; \forall t={1,2,...,T} \end{aligned} $$

(19)