Abstract

Effective computational methods are important for practitioners and researchersworking in strategic underground mine planning. We consider a class of prob-lems that can be modeled as a variant of the resource-constrained project schedul-ing problem with optional activities (or jobs) and generalized precedence con-straints; the objective maximizes net present value. We provide a computationalreview of math programming and constraint programming, describe and implementnovel problem-size reductions, and develop linear programming- and constraintprogramming-based hybrid techniques to significantly expand the number of com-putationally tractable problem instances. We analyze the efficacy of the methodsapplied to instances both from the literature and derived from real-world mines.These practical, large-scale planning problems cannot even be processed using stan-dard optimization approaches. However, our reduction and hybrid strategies allowus to solve them to within less than 5% of optimality in ten minutes and to withinless than 3.5% of optimality within a day.

1

Hybrid Optimization Strategies forResource-Constrained Project Scheduling

Problems in Underground Mining

Alessandro Hill1, Andrea Brickey2, Marcos Goycoolea3, andAlexandra Newman4

1California Polytechnic State University2South Dakota School of Mines

3Universidad Adolfo Ibanez4Colorado School of Mines

April 16, 2019

1 Introduction and Literature Review1

Production scheduling problems seek to optimally sequence jobs (or activities) over2

time in such a way as to comply with precedence and resource-capacity constraints.3

The models we study assume the form of a resource-constrained project scheduling4

problem (RCPSP) in which each activity is associated with a duration, a profit, and5

a lag (or delay) between the activity’s completion and the start of a subsequent6

activity. The duration is constant, regardless of when an activity is executed, and7

is directly related to the effort required to execute the corresponding activity, which8

consumes a certain amount of one or more given, limited resources, e.g., haulage9

or mill capacity. Precedence relationships arise from geometric and geotechnical10

rules specific to the mining method(s) being employed to construct access to and11

extract ore from the deposit. The goal is to determine start (or end) times for the12

activities such that net present value or some similar utility function is maximized.13

Constraints consist of (i) precedence between selected pairs of activities subject14

to lags, and (ii) per-time-period limits on the amount of resources that can be15

consumed by the activities.16

In strategic underground mine planning, a long-term, e.g., multiple-decade,17

schedule manages the extraction of ore during the lifetime of a mine. To this end,18

the ore body of interest containing economic concentrations of, for instance, cop-19

per and/or gold, is subdivided into smaller, notional, three-dimensional rectangular20

units (constituting a block model), for which extraction times are determined. Not21

all activities must be completed over the given time horizon (6), counter to what22

many project scheduling problems require. In addition to activities that are related23

to extraction (e.g., stoping or hauling), those corresponding to backfilling are nec-24

essary to guarantee stability within the mine. Moreover, the installation of infras-25

tructure and ventilation components significantly complicates underground mine26

2

planning. In this context, mine managers are increasingly relying on computer-27

based decision support, yielding optimization models that are difficult to solve,28

especially for large instances.29

1.1 Related Work30

Resource-constrained project scheduling problems have been studied in parallel,31

but separately – in mine planning, and, more generally, without a specific applica-32

tion. In the former case, the focus is on determining open pit mine schedules in33

which each activity, in this case, the extraction of a block, requires a single time34

period to complete and there are precedence restrictions but no time lags between35

activities. This special case arises because blocks are usually homogeneous in their36

extraction requirements. The more general manufacturing case possesses hetero-37

geneous activities whose durations and lags (or delays) vary. The general RCPSP38

was first introduced in (27) and in (38) and has been receiving notable attention in39

the literature since the late 1990’s. Its variants have applications in logistics, man-40

ufacturing, service operations, and transportation. An overview of existing models41

can be found in (23) and the latest methodological advances are given in (41). The42

most common objective in these applications is to minimize project makespan, i.e.,43

the time required to perform all activities. Instances with just 90 activities given in44

the PSPlib (30) are notoriously difficult to solve. For example, at the time of this45

writing, some of these larger instances cannot be solved at all, while others require46

hours to obtain a solution within approximately 10% of optimality.47

In (44), (45), and (32), scheduling heuristics for the RCPSP with discounted48

cash flows are proposed, while an exact algorithm is suggested in (46). (13) present49

an optimization-based heuristic for an RCPSP in an open-pit mine planning setting.50

(36) propose such a heuristic to solve an underground mine planning problem with51

a complex set of precedence constraints, inter alia, while (29) implement reformula-52

tions and a decomposition approach to yield solutions to the problem of scheduling53

both open pit and underground operations subject to a transition between the54

two at some (non-predetermined) stage during the mine plan. For comprehen-55

sive reviews of RCPSP variants and corresponding methodologies, we refer to (23)56

and (41).57

1.2 Contribution and Paper Organization58

We evaluate and extend state-of-the-art optimization approaches to the RCPSP, rel-59

evant in underground mine planning models with real-world datasets. We show how60

the integration of constraint programming (CP) and mathematical programming-61

based approaches can be used to obtain practical schedules and tight bounds on62

the corresponding optimal objective function value. Our main contributions are:63

(i) we describe reduction techniques that enable us to optimize large-scale problems64

in practice; (ii) we develop constraint programming-based hybrid scheduling tech-65

niques and assess their efficacy on industry data; and, (iii) we use these techniques66

to solve real-world mining production-scheduling instances in an operationally fea-67

sible amount of time.68

The remainder of this paper is structured as follows: Section 2 provides rele-69

vant definitions and notation. Section 3 discusses preprocessing strategies to reduce70

problem instance size. Section 4 explains the algorithms we use to solve the opti-71

mization problems. Section 5 demonstrates our strategies on a variety of industry72

datasets, and Section 6 concludes.73

3

2 Problem Formulation and Current Solution74

Approaches75

General resource-constrained production scheduling problems seek a minimum-76

makespan sequence of activities such that it contains the entire set of activities.77

By contrast, the RCPSP variant considered in underground mine planning maxi-78

mizes net present value, or discounted cashflows, but also allows optional activities.79

Both the general and the underground-mining-specific variant possess sequencing80

(or precedence) constraints between the activities. We next provide a mathematical81

formulation.82

2.1 Notation and Problem Definition83

Let T be a discrete time horizon, T = {0, 1, ..., T}. We are given a set of activities84

(i.e., jobs) J and a set of resources R available in each time period, i.e., so-called85

“renewable resources” in the scheduling literature. The accomplishment of activity86

(job) j ∈ J is associated with a monetary value pj ∈ R, a duration dj ∈ Z+,87

and a per-period resource utilization ujr ∈ R+ ∀ r ∈ R. Each resource r ∈ R88

has a constant availability Ur ∈ R+ per time period. Furthermore, we are given89

an activity-on-node network represented by an acyclic digraph H with node set90

J corresponding to the set of activities, and arc set A, representing generalized91

precedence relationships (see Figure 1, left), which require that if arc (i, j) ∈ A,92

and if activity j is scheduled then activity i must be scheduled as well. Each arc93

(i, j) ∈ A is associated with a lag lij ∈ Z+0 which is used to model the amount of94

time that must lapse between the end of activity i and the start of activity j. In95

other words, if activity j starts at time t, then activity i has to be finished at time96

t− lij or before.97

We write i ≺ j if there exists a directed path from node i to node j in H.98

Furthermore, if i ≺ j, let d(i, j) represent the length of the longest path from i to j99

(as measured in cumulative lags). For i ≺ j, we say that i is a predecessor of j, and100

j is a successor of i, respectively. A predecessor (successor) i is direct with respect101

to j if (i, j) ∈ A ((j, i) ∈ A).102

A feasible schedule S for the RCPSP is a set of scheduled activities J S ⊆ J103

and activity start times tSj ∈ T ∀j ∈ J S (or activity end times tSj ∈ T ∀j ∈ J S)104

such that:105

(i) if activity j is scheduled, then each direct predecessor i is scheduled and ends106

at least lij time units before the start time of j, and107

(ii) the consumption of each resource in each time period does not exceed its108

availability.109

For a given discount factor δ ∈ (0, 1], the net present value of schedule S is defined110

as the sum of the discounted profits of the scheduled activities:∑j∈J S

e−(1+δ)tSj pj .111

Our variant of the RCPSP accepts the empty schedule as feasible, and corre-112

sponds to PS|prec|∑CFj β

Cj according to the classification scheme in (7). This113

problem is known to be NP-hard (4).114

The following time-indexed formulation for the RCPSP, called a by-formulationor step-formulation, is known to yield the best linear programming-based bounds (2).We define the following variables representing the time by which each activity ends,

4

Figure 1: The activity-on-node network for an RCPSP literature instance (j301-1from (30)) with 30 activities (left) and an optimal schedule in Gantt chart representation(right).

i.e., the activity ends in time t or before:

yjt =

{1, if activity j ends by time t,

0, otherwise

Then, the RCPSP can be written as follows:

(F) min∑j∈J

∑t∈T

e−δtpj(yjt − yj,t−1) (1a)

yjt ≤yj,t+1 j ∈ J , t ∈ T (1b)∑j∈J

ujr∑

t<t′≤t+dj

(yjt′ − yj,t′−1) ≤Ur r ∈ R, t ∈ T (1c)

yj,t+dj+lij ≤yit (i, j) ∈ A, t ∈ T (1d)

yjt ∈{0, 1} j ∈ J , t ∈ T (1e)

The objective given in (1a) maximizes the net present value using discount rate115

δ. Without loss of generality, we consider end times when discounting profits.116

Variable linking inequalities (1b) ensure that if an activity ends by time t, it con-117

sequently ends by all the subsequent time periods. Resource-feasibility is enforced118

through knapsack inequalities (1c) in each period for each resource. Disaggregated119

generalized precedence inequalities (1d) ensure that the sequence and lag informa-120

tion given in A are appropriately considered; these collapse to ordinary precedence121

inequalities without lags if lij = 0. The bounds obtained by formulation (F) can122

be replicated with a strengthened “at” formulation, also called pulse-formulation123

(12, 17).124

2.2 Current Solution Approaches125

To solve the continuous relaxation of formulation (F), which in and of itself can be126

difficult, it is possible to use a decomposition algorithm that reduces the problem127

to that of solving a sequence of maximum-closure or minimum-cut problems. In128

fact, this follows directly from the “by” formulation, since it suffices to dualize the129

resource capacity constraints to obtain such a problem. (27), in the context of min-130

ing, was the first to attempt such an approach with Dantzig-Wolfe decomposition.131

Shortly after, (18), in the context of scheduling, proposed a Lagrangian relaxation132

decomposition approach. A number of similar approaches include (14), (12) and133

(34) in which the resource constraints are dualized.134

5

It is possible to strengthen the linear programming relaxation by using pre-135

processing and cutting planes and via a reformulation of the constraints. (43), and,136

independently, (5) and (19), strengthen the time-indexed formulation by incorpo-137

rating bounds on the earliest and latest possible time an activity can be started.138

For the RCPSP with release dates and due dates, pre-processing has been used139

to further strengthen the formulation through constraint programming techniques140

(see (15)). (12) and (5) develop and apply cutting planes based on clique and cover141

inequalities. In the context of mining, or more generally, when it is not necessary142

to schedule all activities, it is possible to excise activities from consideration (8).143

Given a solution to the linear programming relaxation of formulation (F), it is144

possible to use a list-scheduling heuristic to derive high-quality solutions for RCPSP.145

(37) and (22) propose using an α-points procedure that creates a topological or-146

dering of the activities, and schedules them as early as possible using said ordering147

as a priority list. Ties are broken using the time period after which α percent148

of an activity is scheduled in the linear programming relaxation; this time period149

corresponds to the one in which the activity is executed in the integer-feasible solu-150

tion. Their methods not only perform well in practice but also provide theoretical151

performance guarantees. (34) employ this type of algorithm in challenging com-152

putational tests with excellent results. Independently, and in a mining context,153

(11) develop the TopoSort heuristic, which also relies on the linear-programming154

relaxation solution as a starting point but breaks ties regarding when activities are155

scheduled with the first time period in which a fraction of an activity is scheduled156

as the time at which the activity is executed in the integer-feasible solution. For157

a survey, see (42). Properly branching can significantly improve performance, e.g.,158

as in (16) under criteria such as a “cut-set dominance” rule. However, little work159

with branching rules has been done in the context of mining.160

Local search heuristics that employ optimization as a sub-routine have proven161

effective, e.g., sliding time window heuristics (13) and local search heuristics (1, 11)162

which fix the values, or relax the integrality, of some of the variables in the formu-163

lation; both the relaxed and integer variables are optimized over, with (some subset164

of) the values of the integer variables being permanently fixed, and (some subset165

of) the relaxed variables being made integer in the next iteration; this procedure166

continues until an integer-feasible solution has been established. We refer to (23)167

and (41) for a survey on heuristic solution approaches for RCPSP variants.168

3 Reduction Techniques169

Model instance size significantly affects solvability because it is inextricably in-170

tertwined with the number of variables and constraints in (F). To this end, we171

introduce a variety of techniques to reduce the number of activities |J | and arcs172

|A|.173

3.1 Earliest Activity Start Times174

For each activity j ∈ J , we can compute a lower bound es(j) on the earliest time175

it would be possible to start its execution, as follows:176

es(j) = max{d(i, j) : i ∈ J : i ≺ j}.

Because H is acyclic, this value can be computed in linear time using a topological177

sorting of the nodes, and allows us to eliminate variables. Moreover, if j ∈ J is178

6

such that es(j) + dj > T , then activity j can be removed, because it is impossible179

to complete it during the given time horizon. To translate these early start times180

to the variables in (F) that represent when an activity ends, we offset them by the181

duration of activity j, dj182

3.2 Weighted Transitive Reduction183

We generalize the notion of the ultimate pit, or envelope of blocks which are eco-184

nomical to remove in the context of open pit mine planning, to eliminate variables.185

For a background on this technique, see (8) and (21).186

Consider the following relaxation of formulation (F), which can be obtained byeliminating inequalities (1c):

(F) min∑j∈J

∑t∈T

e−δtpj(yjt − yj,t−1) (2a)

yjt ≤yj,t+1 j ∈ J , t ∈ T (2b)

yj,t+dj+lij ≤yit (i, j) ∈ A, t ∈ T (2c)

yjt ∈{0, 1} j ∈ J , t ∈ T (2d)

Observe that formulation (F) takes the form of a maximum closure problem. That187

is, by renaming the indices, objective function, and precedences, it can be written188

as:189

max{cw : wi ≤ wj ∀(i, j) ∈ A, wi ∈ {0, 1}}. (3)

If we relax all knapsack constraints related to resource consumption, we obtain190

an integral optimal solution (9). Moreover, the resulting optimization problem191

reduces to a maximum closure, or minimum cut problem (33, 34), which (i) can be192

efficiently solved in polynomial time, and (ii) admits a unique minimal and optimal193

solution w1, in the sense that all other optimal solutions w∗ satisfy w1 ≤ w∗. For194

background on the maximum closure problem and a description of an algorithm195

that effectively solves it in practice, see (20), (10), and (24).196

Proposition. Let y1 be an optimal and minimal solution of (1a)-(1e), and y2197

any optimal solution of (2a)− (2d). For i = 1, 2, let J i = {j ∈ J : yijT = 1}. If T198

is sufficiently large, then, J 1 ⊆ J 2.199

Proof. Suppose not, and consider the non-empty set J∆ = J 1 \ J 2.200

For each j ∈ J , and t ≥ 1, define:201

wkj,t+k =

{y1jt if j ∈ J∆ and t+ k ∈ T

0 otherwise.

Observe that since J∆ is non-empty, the optimality and minimality of y1 implies202

that, for all k ≥ 0:203 ∑j∈J

∑t≥1

e−δtpj(wkjt − wkj,t−1) > 0.

Given that T is assumed to be large, we can also assume there exists k such that204

y2 +wk is feasible for (2a)− (2d). This, however, contradicts the optimality of y2,205

and so we conclude that J∆ must be empty. �206

This proposition implies that we can compute y2 and use it to eliminate each207

activity j ∈ J \ J 2 from the problem. Computing y2 is tractable, as mentioned208

above.209

7

3.3 Transitivity Closure210

Consider an arc (i, j) ∈ A. If d(i, j) > lij , then arc (i, j) can be removed from A211

without affecting the set of feasible solutions to the problem. All such redundant212

arcs can be identified efficiently with complexity O(|J ||A|), as follows:213

214

1. Compute a topological ordering of the nodes J , which can be accomplished215

in O(|J |+ |A|), as first shown by (28).216

2. For each activity i ∈ J , let A+(i) be the set of all arcs with tail i. Test the217

redundancy of all arcs (i, j) ∈ A+(i) as follows: By traversing the set of nodes218

in J in topological order, starting from i, compute the distance d(i, j) for all219

j � i. This can be done in O(|A|). Then, compare each distance d(i, j) to lij220

to determine redundancy, which can be done in O(|J ||A|).221

222

We refer to the resulting precedence graph as the weighted transitive reduction of223

the original precedence graph.224

3.4 Sub-Makespan Arcs225

Consider two activities i, j ∈ J such that i ≺ j. Assume that these two activities226

are such that they cannot be processed simultaneously due to the fact that together227

they consume more resources than is available in any time period. Assume that for228

some r ∈ R, uir + ujr > Ur. Then, if d(i, j) < di, add arc (i, j) with lag lij = di.229

That is, add a lag forcing j to start after i is finished, i.e., (i, j) with lij = di. Note230

that if arc (i, j) already exists, redefine lij = max(lij ,di) instead.231

This idea can be generalized as follows. Consider the set of activities J ij =232

{k ∈ J : i ≺ k ≺ j}. Solve the minimum makespan problem (considering resource233

limitations) in the subgraph induced by J ij , and let mij be the optimal value.234

Then, if d(i, j) < mij , add arc (i, j) with lag lij = mij . Note that if arc (i, j)235

already exists, redefine lij = mij , instead of adding a redundant arc.236

3.5 Elimination of Trivial Activities237

We can eliminate each zero-profit activity j ∈ J (i.e., pj = 0) if ujr = 0 for all238

r ∈ R. These types of activities are commonly added by mine planning schedulers239

as markers. When eliminating an activity j ∈ J , it is important to add an arc240

connecting each direct predecessor of j to each direct successor of j with the corre-241

sponding lags. That is, for every pair of arcs (i, j) and (j, k) in A, add an arc (i, k)242

with lag lij + ljk.243

4 Hybrid Optimization Algorithm244

In this section, we describe a hybrid algorithm combining both linear programming245

(LP) and constraint programming (CP). We execute the preprocessing steps de-246

scribed in Section 3 until no further reductions can be made. Then, we solve the247

LP to obtain an upper bound on the optimal objective function value; addition-248

ally, through a rounding technique on the resulting, fractional solution, we obtain249

an initial integer-feasible solution. CP is then used to improve upon the starting250

solution. We next explain the algorithm in more detail.251

8

1. Solving the LP relaxation252

It is well known that solving the LP relaxation of problem (1a) - (1e) can253

be difficult due to problem instance size. In fact, time-indexed formulations254

of the RCPSP grow linearly with the number of time periods, of which real-255

world problems tend to possess copious amounts, easily in the thousands.256

Decomposition algorithms address this difficulty; see, for example, (12), who257

use Lagrangian relaxation for the makespan version of RCPSP. In order to258

solve the LP relaxation, we use the decomposition algorithm proposed by259

(35), which is an extended version of the Bienstock-Zuckerberg algorithm260

(3) incorporating auxiliary variables and algorithmic speed-ups; however, we261

update the version in that reference to IBM ILOG CPLEX, Version 12.8 (26)262

for optimizing our master problems. Solving the LP relaxation provides an263

upper bound on the objective function of maximizing net present value, and264

can be used to estimate the quality of schedules obtained in later steps.265

2. Rounding the LP relaxation solution266

To obtain an integer-feasible solution to problem (1a) - (1e) we use a gen-267

eralized version of the TopoSort heuristic, which was originally proposed by268

(11) in the context of scheduling problems in open pit mine planning. The269

generalized TopoSort heuristic is described in Algorithm 1, and differs from270

its open pit variant by considering arc lags. Note that as long as coefficients271

ljt are non-negative for all j ∈ J and t ∈ T , the TopoSort algorithm is guar-272

anteed to produce a feasible solution. After running the TopoSort algorithm,273

we run the α-points procedure for one hundred different values of parameter274

α, evenly discretized between zero and one (37, 22).275

We call the combination of 1. and 2. the mathematical programming method-276

ology.277

3. Using constraint programming278

CP has been shown in a number of recent studies to be the most effective tech-279

nique for solving small and difficult instances of the RCPSP for both makespan280

minimization and net present value maximization; see, for example, (40, 39).281

Little has been published, however, on the effectiveness of CP for larger in-282

stances of the RCPSP. In the context of this study, we note that CP can be283

used both as an alternative means of generating an initial feasible solution to284

an instance of RCPSP and as a means to improve a solution produced by some285

other method. We use the IBM ILOG CP Optimizer Version 12.8 (ILOG CP)286

on an appropriate CP model (31) for both of these purposes, which iterates287

between using constraint programming and other techniques to compute a se-288

quence of improving upper and lower bounds to the RCPSP until optimality289

can be proven, or until a solution with a desired gap is found. Although the290

solver is effective at generating and improving the quality of feasible solutions,291

it is usually very slow to generate upper bounds (via a technique that is not292

entirely transparent to the user) stronger than those from the LP relaxation.293

Since ILOG CP does not allow the user to import externally computed upper294

bounds, we configure it to stop each time it generates a new and improving295

feasible solution. We then compare the quality of this solution against the up-296

per bound provided by the LP solution. If the gap is below a pre-established297

value, we stop. Otherwise, ILOG CP continues attempting to improve the298

solution, or attains the desired gap with its own computed upper bounds.299

9

Algorithm 1: The TopoSort Heuristic.

Input: An instance of RCPSP, as defined by (1a)-(1e), and a feasible solution y∗ of

the LP relaxation.

Output: A feasible solution y of the RCPSP instance, satisfying the integrality

constraint.

1 For each j ∈ J define:

δ(j) = |{i ∈ J : (i, j) ∈ A}|,Sj = {k ∈ J : (j, k) ∈ A},y∗j,0 = 0, y∗j,T+1 = 1,

es(j) = min{t ∈ 1, . . . , T + 1 : y∗jt > 0},

E(j) =

T+1∑t=1

t(y∗jt − y∗j,t−1).

2 y ← 0.

3 while J 6= ∅ do4 Choose ∈ J that solves min{E(j) : δ(j) = 0, es(j) ≤ T} (break ties choosing an

activity with the highest objective value).

5 for t = 1, . . . , T do

6 yt ← 1, ∀t ≥ t.

7 if

(∑j∈J

ujr(yjτ − yj,τ−dj ) ≤ Ur, ∀r ∈ R, τ ∈ T

)then

8 break.

9 else yt ← 0, ∀t ≥ t.

10 if yt = 1 then

11 for k ∈ S do

12 δ(k)← δ(k)− 1.

13 es(k)← max{es(k), t+ ljk}.

14 J ← J \ {}.

15 return y

5 Computational Study300

We conduct our computational experiments using Linux 2.6.32 as the operating301

system and an x86 64 architecture, with four eight-core Intel Xeon E5-2670 pro-302

cessors and 128 GB of RAM. All mining instances follow an underground method303

similar to the one displayed in Figure 2, in which ore is extracted one vertical level304

at a time from the highest to the lowest. Extraction of sections of ore on each level305

represent activities which result in profit (as the recovered ore is processed and306

sold), while development and backfilling activities incur a cost but are necessary to307

access the ore. Labor capacity, as well as equipment availability, limit the mining308

rate and constitute the resource constraints.309

We evaluate our approach by applying it to five groups of instances from different310

mines: A-1 through C-5 are academic (17), and D and E are based on real-world311

10

Figure 2: A sublevel stoping mining operation, consisting of activities such as developmentand drilling, in which ore is extracted from vertical levels (25)

data sets with a five- and ten-year time horizon at daily fidelity, respectively. We312

make slight modifications to the real-world data sets to disguise the data without313

altering the underlying structure. The complexity of the precedence constraints314

can be seen in Figure 3 where (A) highlights the development, e.g., decline, needed315

to access the deposit, (B) the ventilation development required to ensure proper316

airflow for safety and comfort of miners, (C) the production drifts used to access317

the mining areas, and (D) the stopes extracted. Geotechnical requirements may318

create additional precedence between different areas or levels within the mine.319

We conduct two types of numerical experiments: (i) those showing the effec-320

tiveness of the preprocessing techniques from §3 and the corresponding solution321

times and optimality gaps resulting from the implementation of the Bienstock-322

Zuckerberg algorithm and the TopoSort heuristic, i.e., the mathematical program-323

ming approach, to obtain an integer-feasible solution (see Table 1); and (ii) those324

contrasting the implementation of constraint programming alone with the hybrid325

strategies of either upper-bounding techniques only, or upper-bounding techniques326

and the TopoSort heuristic that yields an integer-feasible solution (see Table 2).327

We report preprocessing reductions only in terms of the number of variables as328

there are confounding effects related to the reduction in the number of constraints.329

While some instances benefit very little from the preprocessing techniques that330

eliminate fewer than 1% of the variables in a given problem instance, others see331

as much as a 26% reduction, depending on the geology of the orebody, the mining332

method employed, and how the block model was constructed. Solution times for333

most instances using just the mathematical programming methodology are fewer334

than three minutes, but the most computationally expensive instance requires more335

than 15.5 hours before the algorithm terminates. Correspondingly, most of the336

fast solving instances result in optimality gaps of less than 3%; the more difficult337

instances possess gaps in the 10-20% range.338

11

Figure 3: (A) Each design element is divided into activities (jobs) and each activityis associated with other activities through precedence. For example, in Quadrant A,decline segment 2 can only begin once segment 1 has been completed. (B) To ensurea safe working environment, all working areas need to be ventilated in an undergroundmine. In Quadrant B, the decline is constrained from advancing until its previous sectionhas been connected to the ventilation circuit through the ventilation access and raise. (C)The mine is divided into different levels used to access extraction areas, i.e., stopes, viaproduction drifts that are excavated from the main decline. As depicted in Quadrant C,stope extraction at a given level can only begin once the production drifts have been fullycompleted. (D) In Quadrant D, the stopes on the lower level can only be extracted whenstopes directly above them have been extracted to conform to precedences associatedwith a top-down mining method.

Therefore, we seek to reduce both the solution times and the gaps by introducing339

constraint programming. In and of itself, constraint programming performs signifi-340

cantly more poorly on all instances (third through fifth columns of Table 2) than the341

mathematical programming approach (second column of Table 2, which reiterates342

the last column of Table 1), even if the constraint programming algorithm is given343

more time. This is also the case when the code associated with the mathematical344

programming approach (whose solution times are almost completely attributable345

to the time required to solve the linear-programming relaxation) is slow. We at-346

tribute this performance to the structure of some of the instances which are not347

very constrained; as such, the constraint programming techniques cannot propa-348

gate the domain nearly as effectively. By contrast, the mathematical programming349

approach generates faster solutions with smaller optimality gaps due to its ability350

to exploit the precedence-constrained knapsack structure and the efficacy of the351

TopoSort heuristic.352

Providing the constraint programming solver with an upper bound produces353

tighter gaps more quickly; for example, after six hours of run time, gaps that av-354

12

erage approximately 37% with CP alone decrease to under 10% with the bound.355

In the majority of our instances, the linear-programming bound that the math-356

ematical programming approach provides is good (within 3% of the best integer357

solution found overall), suggesting that such a bound is helpful information for the358

constraint-programming solver. However, the hybrid strategy of providing the con-359

straint programming solver both with the integer-feasible solution from TopoSort360

as a lower bound and with the linear programming relaxation solution from the361

Bienstock-Zuckerberg algorithm as an upper bound unequivocally result in the best362

performance. Within ten minutes of solution time, the average gap across all in-363

stances reduces to less than 5%. The gap is further tightened to less than 3.5% after364

24 hours. Because the TopoSort heuristic is terminated when an integer-feasible so-365

lution is found, longer solution time resulting from the mathematical programming366

approach does not result in an improved optimality gap. By contrast, integer-367

feasible solutions generated by the mathematical programming approach can serve368

as initial-feasible solutions to the ILOG CP solver, and reduce the gap by an ad-369

ditional 50%, on average, even with a total of six hours of runtime (see Table 2).370

In some cases, constraint programming does not improve over the solution from371

the mathematical programming approach, but these cases generally correspond to372

those for which the mathematical programming approach determines a good solu-373

tion. For the more difficult cases, in particular, those for which the gap is between374

10 and 20% at the termination of the mathematical programming run, constraint375

programming reduces the gap, sometimes by an order of magnitude.376

While it is apparent that a hybrid strategy consisting of intertwining constraint377

programming with the mathematical programming approach provides the provably378

best solutions in the shortest amount of time for the instances we test, increased379

run time (beyond the 10-minute – and certainly six-hour – limit) has limited ability380

in most cases to significantly further reduce the gap. This implies that the con-381

sequences of determining good-quality solutions early in the search procedure are382

paramount to finding a good solution overall.383

13

Tab

le1:

Pro

ble

mC

har

acte

rist

ics,

Pre

pro

cess

ing

Eff

ecti

venes

s,an

dSol

uti

onT

ime

and

Qual

ity

usi

ng

just

the

Mat

hem

atic

alP

rogr

amm

ing

Met

hodol

ogy

Pro

blem

Chara

cteristics

Pre

pro

cessing

Resu

lts

#of

#of

#of

tim

e#

ofre

new

able

#of

#of

pre

ced

ence

Vari

ab

les

BZ

solv

eG

ap

aft

erIn

stance

acti

vit

ies

pre

ced

ence

sp

erio

ds

reso

urc

esva

riab

les

con

stra

ints

elim

inate

d(%

)ti

me

(sec

.)T

op

oS

ort

(%)

A-1

1,59

838

,016

602

87,2

6025

8,87

918.4

12.9

71.0

0A

-21,

881

65,6

2160

210

1,62

131

4,16

122.1

72.5

31.3

4A

-31,

944

67,8

9860

210

4,63

232

3,89

423.5

41.8

61.5

6A

-41,

961

94,5

0860

210

8,76

637

5,17

921.3

95.4

72.7

6A

-52,

140

71,5

8360

211

8,22

940

9,22

524.2

66.0

62.5

3A

-62,

453

111,

065

602

135,

204

481,

108

26.3

58.3

62.5

5B

-11,

157

104,

754

302

24,4

1997

,189

0.5

40.8

34.0

5B

-21,

160

103,

624

302

24,7

1499

,745

0.3

70.7

42.8

9B

-31,

176

110,

801

302

25,3

3910

4,34

70.1

80.9

32.3

9C

-11,

920

1,95

690

798

,865

136,

322

23.0

044.3

813.3

0C

-23,

322

3,78

918

08

336,

178

513,

065

19.8

893.0

311.6

4C

-33,

322

3,78

918

07

336,

178

535,

818

16.1

2152.2

613.5

7C

-45,

937

8,04

436

57

1,20

4,12

32,

121,

401

10.8

13,3

93.4

311.5

0C

-510

,083

15,2

4873

07

4,18

8,78

77,

954,

316

6.0

339,5

32.1

814.6

9D

7,55

312

,874

1,82

68

6,66

8,93

412

,679

,465

17.6

911,9

84.6

28.7

0E

10,0

9411

,703

3,65

34

20,9

82,7

1736

,238

,496

13.3

556,0

21.0

620.6

0

14

Tab

le2:

Gap

Impro

vem

ents

thro

ugh

Con

stra

int

Pro

gram

min

gw

her

eU

BD

eriv

esth

rough

the

BZ

Alg

orit

hm

and

LB

thro

ugh

Top

oSor

t

CP

Alo

ne

CP

with

UB

CP

with

UB

and

LB

Instance

TopoG

ap

Gap

at

Gap

at

Gap

at

Gap

at

Gap

at

Gap

at

Gap

at

Gap

at

Gap

at

(%)

600

sec.(%

)21,600

sec.(%

)86,400

(%)

600

sec.(%

)21,600

sec.(%

)86,400

(%)

600

sec.(%

)21,600

sec.(%

)86,400

(%)

A-1

1.00

21.65

20.99

20.99

1.10

1.00

1.00

1.00

1.00

1.00

A-2

1.34

23.67

23.07

22.99

1.72

1.00

1.00

1.13

1.00

1.00

A-3

1.6

26.42

22.85

22.83

1.91

1.00

1.00

1.34

1.00

1.00

A-4

2.80

32.51

29.92

29.79

3.73

2.46

2.28

2.01

1.55

1.54

A-5

2.53

40.46

38.20

38.00

3.46

2.19

1.88

2.03

1.67

1.59

A-6

2.60

42.00

41.03

41.00

3.00

1.40

1.34

2.10

1.40

1.40

B-1

4.05

72.26

72.11

72.06

15.44

15.00

14.82

2.44

2.10

2.02

B-2

2.89

71.03

70.80

70.80

14.66

13.89

13.85

2.00

2.00

2.00

B-3

2.40

67.60

67.41

67.41

7.78

7.60

7.60

2.40

2.40

2.40

C-1

13.30

16.35

4.84

4.10

12.00

9.53

8.82

1.00

1.00

1.00

C-2

11.64

25.89

23.77

23.29

21.37

19.12

18.61

7.34

5.92

5.89

C-3

13.57

23.25

21.24

21.10

13.72

11.47

11.31

7.62

5.36

5.06

C-4

11.50

57.78

53.50

52.93

27.87

20.58

19.58

9.56

6.85

6.12

C-5

14.69

58.28

55.38

54.83

27.14

22.08

21.12

13.14

10.80

9.94

D8.70

16.15

13.51

13.23

8.94

6.07

5.78

7.60

5.88

5.88

E20.60

36.96

35.73

35.28

13.28

11.65

11.02

8.44

7.88

7.69

Lin

earAvera

ge(%

)7.20

39.52

37.15

36.91

11.07

9.13

8.81

4.45

3.61

3.47

15

6 Conclusion384

Many strategic underground mine planning models possess a similar structure, that385

of a resource-constrained precedence problem, which we exploit in our mathemat-386

ical programming approach. Constraint-programming techniques can also be used387

to help find good, feasible solutions. Using both, combined with problem-size re-388

duction methods, we can significantly improve solution quality while simultaneously389

reducing solution times for real-life mining instances. Specifically, for the instances390

we test, with preprocessing and the hybrid strategy, the instances can be solved391

to within less than 5% of optimality in ten minutes and to within less than 3.5%392

of optimality within a day. These instances would otherwise be intractable with a393

state-of-the-art general-purpose solver such as CPLEX employing either its math-394

programming or constraint-programming algorithm. Even an algorithm specially395

tailored to provide fast solutions to the resource-constrained project scheduling396

problem fails to close the gap below 1% for six of the 17 instances; adding constraint397

programming to the quiver of algorithmic tactics results in an effective technique398

on all instances we test.399

Acknowledgements400

We acknowledge funding from Alford Mining Systems in partial support of this401

project, and the Center for Mathematical Modeling (Basal) of the University of402

Chile. We thank Akshay Chowdu of the South Dakota School of Mining and Tech-403

nology for his help in creating some of the instances and Levente Sipeki for per-404

forming preliminary computation.405

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