draft - university of toronto t-space · 114 et al. (2003) proposed to add clique constraints to...

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A new mixed-integer programming model for spatial forest planning

Journal: Canadian Journal of Forest Research

Manuscript ID cjfr-2019-0152.R2

Manuscript Type: Article

Date Submitted by the Author: 24-Aug-2019

Complete List of Authors: Gharbi, Chourouk; Université Laval Faculté des sciences et de génie, Génie mécaniqueRönnqvist, Mikael; Université Laval, Département de génie mécaniqueBeaudoin, Daniel; Universit� LavalCarle, Marc-Andre ; Universite TELUQ, École des sciences de l'administration

Keyword: Mixed integer programming, forest management, adjacency, area restriction model, unit restriction model

Is the invited manuscript for consideration in a Special

Issue? :Not applicable (regular submission)

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Canadian Journal of Forest Research

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1 Title: A new mixed-integer programming model for spatial forest planning

2 Author 1: Chourouk Gharbi

3 Title: Ph.D. candidate

4 Affiliation: FSG - Département de génie mécanique, Université Laval

5 Bureau: Pavillon Adrien-Pouliot, bureau 3501

6 Telephone: 418 208-8528

7 E-mail: [email protected]

8 Author 2: Mikael Rönnqvist

9 Title: Professeur titulaire

10 Affiliation: FSG - Département de génie mécanique, Université Laval

11 Bureau: Pavillon Adrien-Pouliot, bureau 3345

12 Telephone: 418 656-2131, poste 6838


14 Author 3: Daniel Beaudoin

15 Title: Professeur agrégé

16 Affiliation: FFGG - Département des sciences du bois et de la forêt, Université Laval

17 Bureau: Pavillon Abitibi-Price, bureau 2103

18 Telephone: 418 656-2131, poste 8485


20 Author 4: Marc-André Carle

21 Title: Professeur

22 Affiliation: École des sciences de l'administration, Université TÉLUQ

23 Telephone: 418 657-2262, poste 5250


25

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26 Abstract

27 The unit restriction model and the area restriction model are the two main approaches to

28 dealing with adjacency in forest harvest planning. In this paper, we present a new mixed-

29 integer programming (MIP) formulation that can be classified as an area restriction

30 approach and a unit restriction approach. We need to generate a feasible cluster to

31 formulate the model. However, unlike other approaches, there is no need to generate

32 specific model constraints representing computationally burdensome clusters for large

33 cases. We describe and analyze our approach by comparing it with the most efficient

34 approaches presented in the literature. Comparisons are made from a modeling and a

35 computational point of view. Results showed that the proposed model was competitive

36 with regard to modeling complexity and size of formulation. Furthermore, it is easy to

37 implement in standard modeling software.

38 Keywords: Mixed integer programming, forest management, adjacency, area restriction

39 model, unit restriction model.

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40 1 Introduction

41 Aside from an interest in harvesting planning, forest management researchers and

42 practitioners have an interest in other concerns such as connectivity restriction, wildlife

43 habitat protection and landscape conservation, and size distribution. These concerns have

44 led to new spatial forest management rules to control the spatial development of the forest

45 landscape and forest management activities (Baskent & Keles 2005). Spatial forest

46 planning problems, including adjacency restrictions, are often developed to address these

47 spatial patterns. Two stands are considered adjacent when they share a common boundary.

48 The unit restriction model (URM) and area restriction model (ARM) are the two main

49 approaches to deal with adjacency in harvest scheduling models (Murray 1999). In the

50 URM approach, the boundaries of each potential cutting block are predefined;

51 simultaneous harvesting is prohibited in two adjacent units (Baskent & Keles 2005).

52 However, ARM models allow simultaneous harvesting of adjacent units, provided their

53 combined area does not exceed the maximum allowable cut size (Murray 1999). In this

54 approach, harvest block boundaries are not predefined; instead, they are defined through

55 models that determine all the potential harvesting blocks that satisfy a maximum allowable

56 cut (Goycoolea et al. 2009; Baskent & Keles 2005). A so-called green-up period is used to

57 determine how much time is needed before blocks are no longer considered adjacent and

58 suitable for harvesting.

59 Many researchers used integer programming or mixed-integer programming models to

60 model spatial forest planning problems. Decision variables usually correspond to

61 harvesting blocks at a particular period of time under environmental and operational

62 constraints. Objective function may be maximizing profit or net present value, minimizing

63 cost or alternatives. These forest management planning problems are hard and usually

64 require a long problem-solving with exact approaches and large cases. In addition, in

65 practical planning for large forest units, the proposed models cannot be used directly due

66 to the large size of the problems. Formulating and solving ARM models is significantly

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67 more difficult than formulating and solving URM models (Baskent & Keles 2005). But

68 unlike URM models, ARM models are more flexible and generate more possibilities of

69 harvesting plans with higher values (Kašpar et al. 2016).

70 Several authors have addressed spatial modeling with exact methods for solving spatial

71 forest scheduling, such as branch and bound (Crowe et al. 2003; Rebain & MacDill 2003;

72 Constantino et al. 2008), column generation (Barahona et al. 1992; Weintraub et al. 1994),

73 and cutting plane (Könny & Tóth 2013). Exact approaches particularly present

74 computational difficulties for large problems (Goycoolea et al. 2005). To circumvent these

75 drawbacks, other authors proposed heuristics for addressing spatial modeling such as

76 simulating annealing (Öhman & Eriksson 2002; Borges et al. 2014), tabu search (Richards

77 & Gunn 2003; Caro et al. 2003), and genetic algorithm (Moore et al. 2000; Venema et al.

78 2005). Researchers have demonstrated that these methods cannot necessarily reach

79 optimality when solving large problems. These methods, however, can find feasible

80 solutions in reasonable time.

81 Three exact mixed-integer programming formulations were proposed for modeling ARM

82 problems. The first formulation was proposed by McDill et al. (2002), called path

83 formulation. Based on knapsack-type cover inequalities, this formulation requires the

84 enumeration of clusters called minimally infeasible clusters. Their combined area just

85 exceeds the maximum allowable cut size.

86 The second exact formulation proposed was the generalized management unit approach. In

87 this approach, McDill et al. (2002) used sets of units, known as feasible clusters, whose

88 total area does not exceed the maximum allowable cut size (Goycoolea et al. 2009). This

89 formulation was eventually improved by using maximal clique adjacency constraints; It is

90 referred to as the maximal clique-cluster (MCC) formulation. Note that a maximal clique

91 cannot be a subset of another clique.

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92 The third exact formulation was called Area Restriction with Stand-Clear-Cut variables

93 (ARMSC) (Constantino et al. 2008). Unlike the path and MCC formulations, the ARMSC

94 does not need a priori numeration of sets because every management unit can be assigned

95 to an empty clear-cut set in each planning period. Stands within a clear-cut set could be

96 adjacent. The maximum allowable cut restriction is imposed for each clear-cut set and

97 adjacency constraints are imposed in the model to prevent clear-cut overlapping

98 (Constantino et al. 2008). Goycoolea et al. (2009) found the model was not competitive

99 with respect to the quality of the solution given within a time limit.

100 Yoshimoto & Asante (2018) recently proposed an exact formulation called the MF_Model

101 I formulation that does not require a priori numeration of sets. The formulation integrates

102 the unit aggregation to form feasible clusters and the spatial restrictions in one model.

103 However, it generates large model sizes. Adjacency is modeled using a pairwise method.

104 This formulation was proposed and compared in previous works (McDill et al. 2002;

105 Goycoolea et al. 2005). The results showed that it generated many adjacency constraints.

106 Compared to the path formulation, the MF_Model I did not show better performance. But

107 the MF_Model I was better than the path formulation when the two- and three-year green-

108 up periods were used with a ten-year planning horizon. For other cases, the path

109 formulation is better than the MF_Model I formulation.

110 When it comes to computational time and memory, solving exact formulations for spatial

111 forest planning using ARM approaches is difficult. In most instances, optimal solutions are

112 not reached; it sometimes takes too long to find feasible solutions for practical planning

113 purposes. Many researchers have tried to improve these formulations. For example, Crowe

114 et al. (2003) proposed to add clique constraints to the path formulation to decrease solution

115 time. This formulation did not outperform the original version of the path formulation but

116 provided better LP bound and cut off some fractional solutions. Tóth et al. (2012) were

117 also interested in the path formulation. They proposed to strengthen it by substituting the

118 cover inequalities for others known as extended cover inequalities. In so doing, they proved

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119 that proposed constraints are stronger than original ones. However, they noted that

120 computational time associated with this strengthening was considerable. Tóth et al. (2013)

121 also showed that removing adjacency constraints from the model and considering them as

122 lazy constraints had a significant improvement in solution times. Their methods consist in

123 finding a potential solution without using adjacency constraints, iteratively adding them

124 when they are not respected until the desired gap or the optimal solution is reached. Vielma

125 et al. (2007) replaced strict volume constraints with elastic volume constraints in the MCC

126 formulation to reduce their negative effect on the resolution. They showed an improvement

127 in the time allocated to find feasible solutions for almost all problems. However, their

128 approaches still have some shortcomings, especially in controlling strict volume constraint

129 violation and the quality of the feasible solution found.

130 All exact formulations result in large problems that generally increase exponentially with

131 the number of stands. Most formulations involve a significant preprocessing compilation

132 to enumerate sets. To overcome these shortcomings and deal with adjacency restrictions,

133 we present a new formulation that we call “Full Adjacent Unit (FAU)”. It is a new MIP

134 formulation that can be used as a URM approach and an ARM approach. The FAU/URM-

135 based formulation does not need any a priori enumeration. The FAU/ARM-based

136 formulation requires a priori enumeration of feasible clusters. For its part, the MCC

137 formulation needs a priori enumeration of feasible clusters and maximal cliques, while the

138 path formulation needs a priori enumeration of minimally infeasible clusters. Restriction

139 on adjacency is formulated through a new set of constraints called “full adjacent units”

140 constraints. Only one “full adjacent units” constraint per stand is needed to prohibit

141 simultaneous, adjacent stand harvesting, making the number of constraints linear to the

142 number of stands, hence dramatically reducing the problem size compared to the other

143 formulations proposed in the literature. Our objective here is to present and compare the

144 performance of the proposed FAU formulation to the path and MCC formulations

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145 presented earlier. In addition, we will apply the model on some large cases from forest

146 planning units from Quebec.

147 In Section 2 of this paper, we will present the path and the MCC formulations, followed

148 by the FAU formulation in Section 3. Section 4 describes the experimentations and presents

149 the results. We will end the paper with a few concluding remarks in Section 5.

150 2 Existing models

151 2.1 Notations

152 We will use the notations below for the three model formulations.

Notation Details

P Set of stands

T Set of planning periods

R Set of feasible clusters

pR Set of feasible clusters containing stand p

C Set of maximal cliques

pN Set of units that are adjacent to stand p

p pB card N Number of stands adjacent to stand p

u A spatial unit that could be a stand, a predefined set of

stands or a feasible cluster

pU Set of units containing stand . Note that could p pU

be equal to if represents the stand .{ }p u p

M Set of minimally infeasible clusters

,p tn Net present value of stand in period p t

tv Available volume in period .t

pa Area of stand p

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pg Initial age of stand p

m Number of years in period

s Maximum allowable cut

tu Ratio of upper bound of harvesting volume in period t

tl Ratio of lower bound of harvesting volume in period t

G Average age G

153

154 Figure 1 illustrates a hypothetical forest that includes a set of 11 stands. For simplicity, { }P

155 assume that the maximum allowable cut and that the area of each stand is 30S ha p

156 . Two stands are adjacent if they share a common line. In graph theory, cliques 10pa ha

157 are sets of nodes that are mutually adjacent or connected to each other by arcs. A maximal

158 clique is not a subset of any other clique. Therefore, belongs to while {2, 3} and 2,3,4 C

159 {2, 4} are separate because they are contained in . Additionally, the set of stands 2,3,4

160 adjacent to stand 1 is . The cluster is part of the set of feasible 1 2,3,6,9,10N 1,2,9 R

161 clusters as it is a contiguous cluster with a total area less than or equal to 30 ha. The cluster

162 is part of the set M of minimally infeasible clusters. A minimally infeasible 1,2,6,9

163 cluster’s area just exceeds the maximum allowable cut size. It is a contiguous and infeasible

164 set which becomes feasible or not contiguous if any unit is removed from the set

165 (Goycoolea et al. 2009).

166 Solving spatial forest planning consists of selecting stands to be harvested in each p

167 period of the planning horizon to maximize the net present value while respecting t ,p tn

168 spatial requirements presented in Table 1.

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169 2.2 The path formulation

170 The path formulation was proposed by McDill et al. (2002). This formulation functions by

171 constructing all possible paths or minimally infeasible clusters (Goycoolea et al. 2009).

172 Adjacency constraints are imposed for each path. To ensure a feasible solution,

173 simultaneous harvesting of all stands composing each path (minimally infeasible cluster)

174 is prohibited. The path formulation is described in further detail in Appendix A.

175 2.3 The MCC formulation

176 The MCC formulation defines all feasible clusters and the enumeration of all maximal

177 cliques. The adjacency constraints are satisfied by harvesting at most one feasible cluster

178 among all feasible clusters that intersect each maximal clique. The model is presented

179 further in Appendix B.

180 3 Proposed model (Full adjacent unit formulation)

181 We propose a new approach to formulate the adjacency restriction that can be efficiently

182 applied for URM and ARM approaches. This formulation is simpler than the others

183 discussed above even when applied as an ARM approach, as it is unnecessary to enumerate

184 sophisticated sets of stands. In this formulation, only the generation of feasible clusters is

185 required when the ARM approach is used. This stand clustering method is easy to

186 implement and represents an important advantage per the easiness of modeling the direct

187 cost saving or the net present value (McDill et al. 2002). Although the proposed model can

188 be used as an ARM approach, the adjacency constraints is satisfied mainly based on the

189 adjacency relation between stands, since the adjacency restriction for one stand depends

190 only on its adjacent stands. Yoshimoto & Brodie (1994) concluded that using adjacency

191 relations to model adjacency constraints did not require a modeling or a computational

192 effort such as many other approaches and algorithms. In their work, they compared many

193 methods to model adjacency and used an adjacency matrix called “the original adjacency

194 matrix.” However, in this model, we use adjacency relations between stands without any

195 matrix notation. We define each stand and each time period , one adjacency constraint t

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196 that prohibits the simultaneous harvesting of this stand and its adjacent stands when they

197 are not contained in the same harvesting unit. The adjacency constraint, referred to as the

198 “full adjacent unit constraint,” is easy to model. Below we present the URM application

199 and the ARM application of the FAU formulation.

200 3.1 Full adjacent unit constraint formulation in URM approach.

201 Considering: as a binary variable that is equal to 1 if unit is harvested in period utX u t

202 and 0 otherwise. The general structure of the FAU constraint is:

'

''''

1 1; ,p p

pp

ut u tu U u Up

u Up N

X X p P t TB

(1)

203

204 In relation to the URM, harvesting blocks are predefined. Consequently, a stand is always

205 contained in one harvesting unit. When the potential harvesting blocks are not predefined,

206 stands represent the decision variables. In this case, the FAU constraint becomes:

''

1 1; ,p

pt p tp Np

X X p P t TB

(2)

207 Referring to Figure 1, suppose that each stand has an area less than or equal to the maximal

208 allowable cut. With , if the stand 6 is harvested in period , no other stand 6 1,3,7,9N t

209 from can be harvested in the same period. Otherwise, all stands adjacent to stand 6 6N

210 could be simultaneously harvested if they are not mutually adjacent. The adjacency

211 constraint for the stand 6 in period can thus be expressed as:t

6 1 3 7 91 14t t t t tX X X X X

(3)

212

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213 3.2 Full adjacent unit constraint formulation in ARM approach

214 Under the ARM approach, potential harvesting units are the feasible clusters (i.e., all

215 possible stand combinations that form a contiguous set with a total area that does not

216 exceed the maximal allowable cut). The FAU constraint is then:

'

''''

1 1; ,p p

pp

rt r tr R r Rp

r Rp N

X X p P t TB

(4)

217 Where:

218 is a binary variable that is equal to 1 if cluster is harvested in period , and 0 rtX r t

219 otherwise.

220 Again, by referring to Figure 1, suppose that each stand area is equal to 10 and the maximal

221 allowable cut is equal to 20 ha. The adjacency constraint for stand 6 becomes:

14 ( ) 14 13 19 20 21 1 2 3 5 6 8 10 11 12 14 22 23 26 27 28 29X X X X X X X X X X X X X X X X X X X X Xt t t t t t t t t t t t t t t t t t t t t (5)

222 Such that:

223 is the set of the feasible clusters containing stand 6.6 4 13 19 20 21{ , , , , }R C C C C C

224 is the set of the feasible clusters 1 2 3 5 6 8 10 11 12 14 22 23 26 27 28 29{ , , , , , , , , , , , , , , , }C C C C C C C C C C C C C C C C

225 containing stands adjacent to stand 6.

226 Clearly, some stands within are mutually adjacent, hence they could not be 6 \ 6N

227 harvested at the same time. This restriction is not supported by the adjacency constraints

228 for stand 6. However, it will be enforced when adjacency constraints for all stands are

229 developed.

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230 3.3 The full adjacent unit formulation

231 The general structure of the FAU model is as follows:

ut utt T u U

Max n X (6)

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

1utt T u U

X

u U (7)

'

''''

1 1p p

pp

ut u tu U u Up

u Up N

X XB

,p P t T (8)

( 1) ( 1)t ut ut u t u t t ut utu U u U u U

l v X v X u v X

1..( 1)t nt (9)

p

p p p ut pp P t T p Pu U

a g ntm tm g X G a

(10)

0,1utX ,u U t T (11)

232 The objective function (6) maximizes the net present value of the harvested units.

233 Constraint set (7) guarantees that each stand is harvested once at the most. Constraint set

234 (8) is the “Full adjacent units” constraints used to account for adjacency. Constraint set (9)

235 enforces sustainable yield constraints, while constraint set (10) imposes the average ending

236 age of the stands. Finally, constraint set (11) states the binary requirement for variables.

237 We refer to this formulation as the Full Adjacent Units formulation (FAU1).

238 3.4 Strengthening formulation

239 In the FAU1 presented above, reflects the number of stands affected by the harvesting pB

240 of stand . could be large when the forest is tight and the number of adjacent stands p pB

241 for each stand becomes larger. It is possible to reduce the feasible region by reducing the

242 value of , making the formulation tighter. In Figure 1, consider only stand 7 for which pB

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243 . Sets are the maximal cliques that contain stand 7 3,6,8,9N 3,6,7 , 6,7,9 , 7,8

244 7. The harvesting of stand 7 will affect three sets of stands: Even if stand 3,6 , 6,9 , 8 .

245 7 is not harvested, stands 3 and 6 could never be harvested together because of their

246 adjacency. Hence, if , the adjacency constraint for stand 7 remains true. To 7 3B

247 generalize the idea, the FAU1 formulation could become tighter if were equal to the pB

248 number of maximal cliques containing stand rather than the number of stands adjacent p

249 to . Thus, the adjacency constraints becomes: p

'

'''''

1 1; ,p p

p

p

ut u tu U u Up

u Up N

X X p P t TB

(8’)

Such that: 'p =card(maximalcliques containing )B p

250 This formulation is referred to the FAU2.

251 If stand 7 is not harvested, stands that could be concurrently harvested are , or{3,9} {8,9}

252 . Thus, the maximum number of stands that could be harvested simultaneously is {6,8}

253 exactly 2 that is smaller than the number of maximal cliques containing stand 7. Note that

254 if the adjacency constraint is still valid. Generally speaking, if represents the '7 2,B '

pB

255 maximum number of stands adjacent to and the stands are not mutually adjacent, the p

256 FAU2 formulation could become stronger. The adjacency constraints become:

'

''''

1 1; ,''

p p

p

p

ut u tu U u Up

u Up N

X X p P t TB

(8’’)

Such that: ''p i 1..

; 1..; 1.. ; 1..

=max(card(Q));Q={p }i pi p j

i mp N i mP N i m j m

B

257 The resulting formulation is identified as FAU3.

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258 The proposed model is promising due to the linear relationship between the number of

259 constraints and the number of stands. The number of constraints will neither be affected by

260 the number of stands nor by the approach used to formulate the URM or ARM problem.

261 However, in other formulations, such as the path formulation, the number of constraints

262 grows exponentially when the number of stands or the maximum allowable cut increases

263 (Goycoolea et al. 2009). However, if, for example, the number of units included in a cluster

264 can be limited to five or six, the overall number is not too large, improving chances of

265 becoming computationally tractable. This is like other successful applications, such as

266 vehicle routing problem or scheduling, where set partitioning or set covering models are

267 combined with a priori enumeration of feasible routes or schedules. Even if the number of

268 constraints has a linear behaviour in some other models, the proposed model is expected to

269 yield the smallest number of constraints.

270 4 Case studies and experiments

271 4.1 Settings

272 Tests were conducted to evaluate:

273 The preprocessing effort in terms of implementation and computation time. Special

274 attention is paid to the most complicated type of clusters—the maximal clique.

275 The size of different formulations. We focus on the number of rows and columns

276 generated by the ARM-based approaches. Spatial attention is paid to the number of

277 constraints.

278 The quality of solutions given by the solver within a time limit of 10,000 seconds.

279 MCC and FAU models will be used for URM and ARM tests because they are URM-based

280 and ARM-based approaches. Because the path formulation is purely an ARM-based

281 approach, it will be used only for ARM tests. We conducted many tests using three small

282 real forests available on the Internet (Integrated Forest Management). The first forest is in

283 Northern California (El Dorado). The other forests are in Canada: Shulkell, Nova Scotia,

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284 and NBCL in New Brunswick. We chose these forests because of their use in other studies

285 that discussed the same problem, hence ensuring a more reliable comparison of model

286 performance. These cases are referred later to “small cases,” used for size and ARM

287 solution quality tests. Eight real forests situated in Quebec’s 04251 unit management are

288 also used for size, ARM and URM solution quality tests. Forests in these cases are referred

289 later to “large cases” (denoted form 04251_1 to 04251_8). To test the size of various

290 formulations, we generated eight square forests (denoted from s1 to s8). These hypothetical

291 forests were not used for testing the quality of the solutions.

292 Parameters used to test the models were the same as those used in Goycoolea et al. (2009).

293 The net present values were calculated with a 3% discount rate between each planning

294 period. The sustainable yield ratios are respectively fixed at and . We 1,15tu 0,85tl

295 used an average ending age of 40. The maximum clear-cut size restriction for El Dorado,

296 NBCL and Shulkell are 48.56 ha, 32.37 ha and 16.19 ha, respectively. The maximum clear-

297 cut size restriction for the other real cases is 40 ha.

298 The ARM runs used small cases, while URM runs were done on an Intel Xeon processor

299 at 3.07 GHz frequency with 8 GB of random-access memory using Windows 7. ARM runs

300 using large cases were done on an Intel Xeon processor at 2.6 GHz frequency with 43 GB

301 of random-access memory using Windows 7. All programs were written in AMPL. Default

302 settings were used for all integer and linear programming solutions in CPLEX 12.5 (except

303 for the optimality gap, set at ). A maximum time limit of 10,000 seconds was set on the 510

304 optimizer, excluding the time required for preprocessing the three formulations and the

305 time required by AMPL to generate the model. Note that these steps were fast.

306 4.2 Implementation and cluster generation

307 To implement the various formulations discussed in this paper, many clusters need to be

308 generated (minimally infeasible clusters for the path formulation, feasible clusters for the

309 MCC, and the FAU formulations and maximal cliques for the MCC formulation). The

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310 algorithm to construct feasible clusters, minimally infeasible clusters and maximal cliques

311 is explicitly detailed in Appendix C.

312 The size of various clusters for experimenting with each real forest is presented in Table 2.

313 Generating feasible clusters, including minimally feasible clusters, was quick in all cases.

314 With respect to El Dorado, an approximate duration of five minutes was needed to generate

315 minimally infeasible clusters. However, more than 10 hours was needed to generate this

316 cluster type for the smallest Quebec case. This cluster type was not generated for all large

317 cases. For small cases, the time to generate maximal cliques was short, through the time

318 increased with large cases. Table 3 shows the time needed for generating maximal cliques

319 with large cases. It reached nearly eight hours for 04251_8.

320 We had remarked that the number of these different clusters was different in some works

321 treating the same problem with the same instances. Table 4 gives some examples showing

322 the different values found for maximal cliques, feasible clusters and minimally infeasible

323 clusters in the case of El Dorado. ‘N.O’ indicates the set type not used in the cited work.

324 We did not exclude any stands in the cluster construction. Removing some stands from the

325 model could alter its solvability and change the number of different sets and number of

326 adjacency constraints. But some works, such as Goycoolea et al. (2009), stated that “stands

327 having an area larger than the maximum imposed limit were ignored.” Nevertheless, in the

328 algorithm of enumerating sets, the reader does not find any indication about the inclusion

329 of these stands in clusters. Despite this restriction, the numbers of clusters were almost

330 same as ours in their work, and the same number of stands and the same maximal allowable

331 cuts were used. Because of these inconsistencies and the time needed to generate some

332 clusters, it would be better to avoid using sophisticated sets in the formulations.

333 4.3 Problem size comparison

334 In this section, we evaluate and compare the size of formulations in number of variables

335 and number of constraints with ARM-based models. Empirically, in the simplex method,

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336 the number of iterations and the number of computations in a pivot step are bounded by

337 polynomial functions of the number of variables and the number of constraints. But in some

338 cases, the number of simplex iterations could be bounded by an exponential function in the

339 number of constraints, making the number of constraints more meaningful than the number

340 of variables (Bixby 2002). Therefore, we will focus on the number of constraints in our

341 analysis. Different types of real and hypothetical forests are used, too.

342 The first experiment compared the size of various model formulations using the three real

343 forests when The dimension of sets are presented in Table 2 and formulation sizes 3.T

344 are presented in Table 5. As was shown in Goycoolea et al. (2009), the number of feasible

345 clusters is 15–29 per stand; in the small real cases, the number of minimally infeasible

346 clusters is 8–16 per stand. However, these numbers depend on the ratio of stand area and

347 maximal allowable cut in forest, not on formulation. Moreover, the number of maximal

348 cliques containing each stand is between one and two. This number depends on the forest’s

349 adjacency structure; consequently, it could be bigger for some forests. The path

350 formulation generates the smallest number of variables, while the FAU formulations give

351 the smallest number of constraints. That said, the constraint numbers generated by the path

352 formulation is by far larger than those generated by the FAU and the MCC models.

353 Goycoolea et al. (2009) demonstrated that the constraint number generated with the path

354 formulation usually increased in an exponential curve. Only the constraint number

355 variation with the MCC and the FAU models are presented in Figure 2. Even if the number

356 of constraints grows linearly with the two models, the FAU still has the smallest number

357 of constraints.

358 In the previous El Dorado example, the MCC and FAU formulations approximately exhibit

359 the same behaviour. In a third experiment, we tested the size of these two formulations on

360 other instances having different configurations than El Dorado. Because of the simple

361 adjacency structure of square forests, it is easy to compute the number of maximal cliques

362 and the number of constraints in the clique cluster formulation without running the model.

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363 Table 6 presents the number of constraints in the MCC and FAU formulations for the eight

364 square forests when the number of time periods is three. The number of stands for these

365 instances ranges between 16 and 121 (Table 6). Figure 3 shows how the number of

366 constraints grows when the number of stands increases in a square forest. The two models

367 exhibit linear behaviour. However, the FAU formulation results in smaller problems

368 compared to MCC formulation.

369 For other cases, the behaviour of the number of constraints could change for the MCC

370 formulation depending on forest structure. But the MCC formulation is still linear with the

371 proposed formulation. These experiments demonstrate that the MCC formulation size is

372 sensitive to the forest structure and size. The FAU model, for its part, usually results in an

373 easy-to-compute number of constraints. This number follows a linear function of the stands

374 number. The MCC formulation can result in a large number of constraints in small cases.

375 Thus, the FAU model presents more advantages per size.

376 The second experiment compared the size of the three formulations when , using 3T

377 different values of maximum clear-cut in the El Dorado case (Table 7). The aim is to test

378 the effect of the maximum clear-cut size on the formulations. Because the MCC and FAU

379 formulations use the same decision variables, it is expected that the number of columns

380 would be the same. As Table 5 and Table 7 illustrate, the number of columns is by far

381 larger than the number of rows in these two formulations. Such is not the case for path

382 formulation, as increasing the maximum allowable cut generates more feasible clusters,

383 thereby increasing the number of variables. In practice, this number is still limited, for the

384 maximum allowable cut does not have any effect on the number of constraints.

385 4.4 Solutions quality comparison

386 We considered three planning horizons for periods 1, 3 and 5 to solve integer problems for

387 each real instance. Twelve time periods were also considered for the El Dorado case. For

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388 each problem, we used different variants detailed in Table 8. Only variant 1 (Only harvest

389 once and adjacency constraints) was considered for large cases tests.

390 For each run, we recorded the value of the best feasible solution found. If the optimal

391 solution was reached, we recorded the time it took to solve the problem. If no provable

392 optimal solution was found, we recorded the optimality gap (in percentage) given by

393 CPLEX which is computed as: . The best node is the best node-best integer

100best integer

Gap

394 current best upper bound found by the solver (ILOG S.A 2003). Results are summarized

395 in tables. The first column represents the objective value (OBV). If optimality was reached,

396 the time required to complete the solution was indicated in parentheses. Otherwise, we

397 indicated the feasible solution obtained within the time limit.

398 4.4.1 The integer programming problem resolution for small cases

399 We used the ARM-based modeling approach for small cases to test different formulations.

400 Results of the ARM resolution problems with small problems are summarized in Table 9.

401 To make reading the table easier, we presented only results with the smallest and the

402 biggest periods (this did not affect the result interpretation), excluding results of FAU1 and

403 FAU2 in Table 9. Note that all results are discussed in the text. With the FAU models, the

404 solver was able to find optimal solutions for several instances. Where optimal solutions

405 were not found, the solver provided solutions within a 1% gap. These observations were

406 reported in the El Dorado case: the MCC formulation reached optimality for different

407 variants with single periods and for variant 1 with the multi periods; the path formulation

408 returned optimal solutions for all variants with single periods and for variant 1 with three

409 periods; FAU1 did not reach optimality for any variants; FAU2 gave optimal solutions only

410 for variant 1 with a single period; and FAU3 gave optimal solutions for all variants with a

411 single period. The maximum gap within the time limit was 1.04% for FAU1 and FAU2 and

412 0.8% for FAU3. The MCC formulation and the path formulation solved all problem variants

413 in the NBCL and Shulkell cases, reaching optimality with the various values of time

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414 periods. Optimal solutions were found with FAU1, FAU2 and FAU3 for NBCL instances

415 when three and five periods were considered for all problem variants. Within a single

416 period, the maximum gap for the different problem variants was 0.24%. With respect to

417 forests in Shulkell, Nova Scotia, the FAU1 formulation gave feasible solutions within the

418 time limit with a maximum gap of 1.05%. FAU2 and FAU3 reached optimality except for

419 variant 1 when three and five periods were considered.

420 For El Dorado and Shulkell, the smallest cases, it would be better to use the MCC

421 formulation, even if the performance of the three formulation is similar. When NBCL’s

422 forest size increased, the path formulation performance decreased. The MCC and the FAU

423 formulation performance was close.

424 4.4.2 The integer programming problem resolution for large cases

425 Large cases were used to test approaches that could be applied in URM and ARM

426 approaches and FAU and MCC formulations.

427 4.4.2.1 URM resolution

428 Results of URM resolution are presented in Table 10. The two formulations found optimal

429 solutions for the single period. The FAU formulation gave feasible solutions with a gap

430 less than 13% when three periods were considered and less than 4% when five periods

431 were considered. With three time periods, the MCC returned a “run out of memory” for

432 04251_4 and 04251_5. For the other cases, feasible solutions were found with a maximum

433 gap of 6%. The same formulation returned a “run out of memory” with five time periods

434 for 04251_5. For the other cases, solutions found were within a maximum gap of 2%.

435 Optimality gaps given within the time limit for MCC formulation were a bit slower than

436 those given with the FAU model, but the MCC formulation could not find solutions with

437 the available memory for three runs.

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438 4.4.2.2 ARM resolution

439 Results of ARM resolution are presented in Table 11. The FAU model did not find optimal

440 solutions. When feasible solutions were found, the recorded gap was less than 17 % for all

441 cases except 04251_3 with five periods. The FAU returned a “run out of memory” for

442 04251_6, 04251_7, 04251_8 with five time periods. The MCC model found optimal

443 solutions for all cases within a single period except 04251_6 and 04251_7, providing

444 feasible solutions with other periods and for all other cases. For 46% of runs, gaps were

445 large and reached 41,955%. For 04251_8 with five periods, the MCC returned a “ran out

446 of memory” status.

447 Problems generated using the ARM approach were bigger than those generated using the

448 URM approach. Memory was not great enough for the two models to find feasible solutions

449 within the time limit. However, results given by the FAU formulation were more

450 convincing because of the huge gaps given by the MCC formulation.

451 4.4.2.3 Computational time

452 When models found optimal solutions for all CPLEX runs, the MCC model was faster.

453 When optimal solutions were found within the time limit, the smallest computational time

454 was given by the MCC, path and FAU. However, when comparing the whole

455 computational time needed to solve the problem considering the preprocessing steps, the

456 FAU outperformed the path and the MCC models. In fact, when large cases were used,

457 time to generate minimally infeasible clusters was huge even when the smallest case was

458 tested. In the case of the MCC model, time to generate maximal cliques was needed in the

459 preprocessing phase when it used as an URM. When used as ARM approach, the

460 preprocessing time should have considered time to generate feasible clusters and maximal

461 cliques. When used as ARM, only the time to generate feasible clusters in the FAU model

462 was needed in the preprocessing step. No preprocessing was needed for the FAU model

463 when applied in the URM. Table 12 shows the total time needed to solve the mono-period

464 URM model with the FAU and MCC models when large cases are used.

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465 5 Concluding remarks

466 In this article, we have proposed a new mixed-integer programming formulation that can

467 be used as a URM and an ARM approach to model a spatial forest management problem

468 called the FAU formulation. Its performance was compared with the two most competitive

469 ARM formulations found in the MCC and path literatures with regard to model size and

470 the quality of obtained solutions. The various ARM model formulations allowed the same

471 feasible region and the same optimal solutions, though they differed in size and

472 performance.

473 The FAU formulation enjoys many advantages. Firstly, it is the simplest to implement

474 because, unlike other formulations, it does not require generating complicated sets. It also

475 enjoys a competitive advantage with respect to the number of constraints, especially when

476 solving large cases, and represents the smallest total computational time. Moreover, FAU

477 favours a robust approach with respect to problem size. Unlike other approaches discussed

478 in this paper, the FAU model is neither sensitive to forest size nor structure. It usually

479 addresses size problems in line with the number of stands. As displayed in the experiments,

480 the FAU model caused fewer problems and forest structure had a significant effect on

481 problem size. This fact had a huge impact on problem size generated by the MCC

482 formulation, leading to many constraints. In terms of the quality of solutions, the FAU

483 model found optimal solutions or feasible solutions with low gaps in most cases.

484 All formulations perform well when tested in real forests. Despite the small size of forests

485 in some case studies, methodologies reviewed in this work resulted in large numbers of

486 sets (feasible clusters, minimally infeasible clusters, cliques), increasing model size.

487 These methodologies caused another inconvenience: Constructing these clusters required

488 an algorithmic effort that could explain inconsistencies in some works solving the same

489 problems with the same instances. Although these are not major inconsistencies, they can

490 result in non-feasible solutions given by the different formulations. Accuracy is crucial to

491 ensure all the variables and constraints are considered. Using the FAU model could avoid

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492 fuzziness resulting from some clusters such as maximal cliques and ensure feasibility.

493 Above all, constructing cluster types such as maximal cliques is time-consuming.

494 In most instances, solving integer problems took a long time, even with small cases. In

495 this paper, we proposed a new mathematical formulation that could be a URM and an ARM

496 approach. When compared with the path and the MCC formulations in small cases, no

497 method consistently outperformed another when solving integer problems. When

498 compared with the MCC formulation in a URM context using large cases, however, the

499 FAU outperformed the MCC. In an ARM context, feasible solutions with small gaps were

500 found for many cases, even if the memory limit was not enough to solve the FAU. The

501 MCC formulation left large gaps that reached nearly 420%. CPLEX spent a lot of time

502 trying to solve the linear programming relaxation problem, but CPLEX could not find an

503 upper bound that addressed huge gaps.

504 Regardless of the selected formulation, it is challenging to use exact approaches to solve

505 ARM models for computational time and memory usage. Using decomposition methods

506 and developing customized heuristics to solve these problems can still be relevant when

507 handling large instances. Alternatively, one could strengthen existing integer programming

508 formulations or develop cutting plane approaches.

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509 References

510 Barahona, F., Weintraub, A. & Epstein, R., 1992. Habitat Dispersion in Forest Planning 511 and the Stable Set Problem. Operations Research, 40(December 2015), pp.S14–S21.

512 Baskent, E.Z. & Keles, S., 2005. Spatial forest planning: A review. Ecological Modelling, 513 188, pp.145–173.

514 Bixby, R.E., 2002. Solving real-world linear programs : A decade and more of progress. 515 Operations Research, 50, pp.3–15.

516 Borges, P., Eid, T. & Bergseng, E., 2014. Applying simulated annealing using different 517 methods for the neighborhood search in forest planning problems. European Journal 518 of Operational Research, 233(3), pp.700–710. Available at: 519 http://dx.doi.org/10.1016/j.ejor.2013.08.039.

520 Caro, F., Constantino, M., Martins, I. & Weintraub, A., 2003. A 2-Opt Tabu Search 521 Procedure for the Multiperiod Forest Harvesting Problem with Adjacency, Greenup, 522 Old Growth, and Even Flow Constraints. Forest Science, 49(5), pp.738–751.

523 Constantino, M., Martins, I. & Borges, J.G., 2008. A New Mixed-Integer Programming 524 Model for Harvest Scheduling Subject to Maximum Area Restrictions. Operations 525 Research, 56(3), pp.542–551.

526 Crowe, K., Nelson, J. & Boyland, M., 2003. Solving the area-restricted harvest-scheduling 527 model using the branch and bound algorithm. Canadian Journal of Forest Research, 528 33(9), pp.1804–1814.

529 Goycoolea, M., Murray, A., Vielma, J.P., & Weintraub, A., 2009. Evaluating approaches 530 for solving the area restriction model in harvest scheduling. Forest Science, 55(2), 531 pp.149–165.

532 Goycoolea, M. Murray, A., Barahona, F., Epstein, R., & Weintraub, A., 2005. Harvest 533 Scheduling Subject to Maximum Area Restrictions: Exploring Exact Approaches. 534 Operations Research, 53(3), pp.490–500.

535 ILOG S.A, 2003. ILOG AMPL CPLEX System Version 9.0 User’s Guide,

536 Kašpar, J., Marušák, R. & Bettinger, P., 2016. Time Efficiency of Selected Types of 537 Adjacency Constraints in Solving Unit Restriction Models. , pp.1–14.

538 Könny, N. & Tóth, S.F., 2013. A cutting plane method for solving harvest scheduling 539 models with area restrictions. European Journal of Operational Research, 228(1), 540 pp.236–248.

541 McDill, M.E., Rebain, S. a. & Braze, J., 2002. Harvest scheduling with area-based 542 adjacency constraints. Forest Science, 48(4), pp.631–642.

543 Moore, C.T., Conroy, M.J. & Boston, K., 2000. Forest management decisions for wildlife 544 objectives: System resolution and optimality. Computers and Electronics in 545 Agriculture, 27(1–3), pp.25–39.

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546 Murray, A.T., 1999. Spatial restrictions in harvest scheduling. Forest Science, 45(1), 547 pp.45–52.

548 Öhman, K. & Eriksson, L.O., 2002. Allowing for spatial consideration in long-term forest 549 planning by linking linear programming with simulated annealing. Forest Ecology 550 and Management, 161, pp.221–230.

551 Rebain, S. & MacDill, M.E., 2003. A Mixed-Integer Formulation of the Minimum Patch 552 Size Problem. Forest Science, 49(4), pp.608–618. Available at: 553 http://www.ingentaconnect.com/content/saf/fs/2003/00000049/00000004/art00012.

554 Richards, E.W. & Gunn, E. a, 2003. Tabu search design for difficult forest management 555 optimization problems. Canadian Journal of Forest Research, 33(6), pp.1126–1133.

556 Tóth, S.F., McDill, M., Könnyu, N., & George, S. 2012. A strengthening procedure for the 557 path formulation of the area-based adjacency problem in harvest scheduling models. 558 Mathematical and Computational Forestry and Natural-Resource Sciences, 4(1), 559 pp.27–49. Available at: http://www.scopus.com/inward/record.url?eid=2-s2.0-560 84860314388&partnerID=tZOtx3y1.

561 Tóth, S.F., McDill, M., Könnyu, N., & George, S. 2013. Testing the use of lazy constraints 562 in solving area-based adjacency formulations of harvest scheduling models. Forest 563 Science, 59(2), pp.157–176.

564 Venema, H.D., Calamai, P.H. & Fieguth, P., 2005. Forest structure optimization using 565 evolutionary programming and landscape ecology metrics. European Journal of 566 Operational Research, 164(2), pp.423–439. Available at: 567 http://linkinghub.elsevier.com/retrieve/pii/S0377221703008877 [Accessed January 568 22, 2015].

569 Vielma, J.P., Murray, A.T., Ryan, D., & Weintraub, A. 2007. Improving computational 570 capabilities for addressing volume constraints in forest harvest scheduling problems. 571 European Journal of Operational Research, 176, pp.1246–1264.

572 Weintraub, A., Barahona, F. & Epstein, R., 1994. A column generation algorithm for 573 solving general forest planning problems with adjacency constraints. Forest science, 574 40(1), pp.142–161. Available at: 575 http://www.ingentaconnect.com/content/saf/fs/1994/00000040/00000001/art00012.

576 Yoshimoto, A. & Asante, P., 2018. A New Optimization Model for Spatially Constrained 577 Harvest Scheduling under Area Restrictions through Maximum Flow Problem. Forest 578 Science, 64, pp.392–406.

579 Yoshimoto, A. & Brodie, J.D., 1994. Comparative analysis of algorithms to gennerate 580 adjacency constraints. Canadian Journal of Forest Research, 24, pp.1277–1288.

581

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Table 1 : Spatial requirements (constraints) considered for mathematical formulations

Constraint Type Details

Only harvest once constraint Each stand should be harvested at least once

during the planning horizon.

Adjacency constraint In URM, two adjacent harvesting blocs should

not be harvested simultaneously. In ARM,

adjacent stands should not be harvested

concurrently during the green-up period while

their combined area exceeds the maximum

allowable cut. In this study, we limit the green-

up period to one period.

Sustainable yield constraint For each period, the amount of timber

harvested should be ranged between an upper

bound and a lower bound. The upper bound is

equal to a ratio times the volume harvested tu

in the previous period. The lower bound is

equal to a ratio times the volume harvested tl

in the previous period. This constraint satisfies

the non-declining even-flow of timber

requirement.

Average ending age constraint At the end of the planning horizon, the average

forest age should be at least equal to an average

age.

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Table 2: Characteristics of real forests

Instance StandsTotal

area

Minimum

stand

area

Average

stand

area

Maximum

stand area

Maximum

clear-cut

size

Feasible

clusters

Minimally

infeasible

cluster

Maximal

cliques

------------------------------(ha)-------------------------------

El

Dorado1363 21147.03 4.05 15.52 47.09 48.56 21411 20627 2033

NBCL 5881 60393.10 0.99 10.27 99.95 32.37 94912 51781 5948

Shulkell 1039 4498.75 0.13 4.33 112.36 16.19 29630 12875 1092

04251_1 10282 1 7.96 44 40 218888 - 12394

04251_2 20736 1 6.81 44 40 322646 - 24134

04251_3 31514 1 7.03 124 40 447800 - 36593

04251_4 42374 1 7.52 124 40 630125 - 50791

04251_5 52819 1 7.54 124 40 804687 - 63492

04251_6 63275 1 7.55 124 40 1052838 - 76616

04251_7 73782 1 8.04 124 40 1235682 - 90579

04251_8 83826 1 8.13 124 40 1561763 - 104144

Table 3: CPU to generate maximal clique for large cases

Case Stands Time (min)

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04251_1 10282 5.55

04251_2 20736 14.12

04251_3 31514 48.59

04251_4 42374 111.49

04251_5 52819 173.49

04251_6 63275 254.19

04251_7 73782 293.28

04251_8 83826 464.51

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Table 4: Different values of clusters in different work

Maximal

cliques

Feasible

clusters

Minimally

infeasible

clusters

Tóth et al. (2012) 2033 21411 20629

Constantino et al. (2008) 2033 20047 20630

Goycoolea et al. (2009) 2041 N.O N.O

This work 2033 21411 20627

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Table 5: Size of the different formulations when T=3

MCC Path FAU

Instances Rows Columns Rows Columns Rows Columns

El Dorado 7462 64236 63244 4089 5452 64236

NBCL 23370 284739 141224 17643 23169 284739

Shulkell 4295 88893 39668 3117 4136 88893

04251_1 47464 218888 - - 41128 218888

04251_2 93138 322646 - - 82944 322646

04251_3 141293 447800 - - 126056 447800

04251_4 194747 630125 - - 169496 630125

04251_5 243295 804687 - - 211276 804687

04251_6 293123 1052838 - - 253100 1052838

04251_7 345519 1235682 - - 295128 1235682

04251_8 396258 1561763 - - 335304 1561763

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Table 6: Constraints number generated with the FAU and the MCC formulation with square forests

Number of Constraints

Example MCC FAU

s1 82 64

s2 136 100

s3 204 144

s4 286 196

s5 382 256

s6 492 324

s7 616 400

s8 754 484

Table 7: Impact of the maximal allowable cut on the formulations size when T=3

S=48.56 S=53.42 S=58.27

-----------------------------------------------ha----------------------------------------------

El Dorado Rows Columns Rows Columns Rows Columns

MCC 7462 64236 7467 102696 7467 166596

Path 101007 4089 160797 4089 261189 4089

FAU 5457 64236 5457 102696 5457 166596

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Table 8: Different problem variants

Variant Details

Variant 1 Only harvest once constraints

Adjacency constraints



Volume constraints



Volume constraints

Ending age constraints

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Table 9: Integer problem resolution for small cases

MCC Path FAU3

Case Period VariantOBV

(Opt time)

( s)Gap % OBV

(Opt time)

( s)Gap % OBV

(Opt time)

( s)Gap %

T=1 variant1 2154060 (2.714) - 2154060 (113.64) - 2154060 (137.53) -

T=1 variant2 2154060 (2.714) - 2154060 (113.35) - 2154060 (137.74) -

El DoradoT=1 variant3 2154060 (4.55) - 2154060 (114.20) - 2154060 (236.09) -

T=12 variant1 3195578.87 (82.69) - 3194555.46 - 0.1 3194380 - 0.1

T=12 variant2 2753226.78 - 0.7 2761863.3 - 0.39 2760580 - 0.4

T=12 variant3 2755661.74 - 0.61 2754289 - 0.66 2749140 - 0.8

T=1 variant1 2,3E+08 (2.74) - 2,3E+08 (1.04) - 2,3E+08 (14.44) -

T=1 variant2 2,3E+08 (2.69) - 2,3E+08 (1.06) - 2,3E+08 (14.64) -

NBCLT=1 variant3 2,3E+08 (7.02) - 2,3E+08 (1.09) - 2,3E+08 (73.24) -

T=5 variant1 3,23E+08 (79.27) - 3,23E+08 (30.85) - 3,23E+08 (425.58) -

T=5 variant2 3,12E+08 (1538.76) - 3,12E+08 (41.94) - 3,12E+08 (747.30) -

T=5 variant3 3,12E+08 (1717.21) - 3,12E+08 (64.95) - 3,12E+08 (1049.36) -

T=1 variant1 2227174 (0.88) - 2227174 (3.400) - 2227170 (5.38) -

T=1 variant2 2227174 (0.85) - 2227174 (3.13) - 2227170 (5.33) -

ShulkellT=1 variant3 2227174 (2.18) - 2227174 (3.05) - 2227170 (10.20) -

T=5 variant1 9823166 (54.58) - 9823166 (1941.59) - 9822100 - 0.1

T=5 variant2 7465390 (8273.84) - 7465400 (18879.3) - 7464940 (320.69) -

T=5 variant3 7465389 (3761.31) - 7465411 (18746.3) - 7464850 (339.58) -

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Table 10: Integer problem resolution with URM-based approaches

FAU1 MCC

Case PeriodOBV

(Opt time)

( s)Gap % OBV

(Opt time)

( s)Gap %

04251_1 T=1 51360400 (4.165) - 51360400 (0.92) -

04251_1 T=3 100559894 - 1 100941000 - 0.5

04251_1 T=5 105118163 - 0.1 105171000 - 0.08

04251_2 T=1 90708200 (8.096) - 90708200 (2.62) -

04251_2 T=3 174938394 - 2 176157000 - 0.9

04251_2 T=5 182753982 - 0.6 183740000 - 0.07

04251_3 T=1 137147738 (14.555) - 137147738 (5.31) -

04251_3 T=3 265401769 - 2 267777000 - 0.7

04251_3 T=5 277883378 - 0.5 278798000 - 0.1

04251_4 T=1 199024000 (22.44) - 199024000 (9.34) -

04251_4 T=3 381021791 - 5 “out of memory” - -

04251_4 T=5 414011894 - 0.4 414652000 - 0.1

04251_5 T=1 257782000 (32.32) - 257782000 (15.97) -

04251_5 T=3 496132873 - 6 “out of memory” - -

04251_5 T=5 540579705 - 0.5 541999000 - 0.2

04251_6 T=1 314188105 (40.9) - 314188105 (11.94) -

04251_6 T=3 621140649 - 4 618188000 - 3

04251_6 T=5 654345835 - 1 663745000 - 0.4

04251_7 T=1 390574324 (101.1) - 390574324 (17.5) -

04251_7 T=3 725092142 - 13 783773000 - 2

04251_7 T=5 822388697 - 2 836555000 - 0.4

04251_8 T=1 459334000 (139) - 459334000 (42.8) -

04251_8 T=3 864506889 - 13 898824000 - 6

04251_8 T=5 954276385 - 4 “out of memory” - -

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Table 11: Integer problem resolution with ARM-based approaches

FAU1 MCC

Case PeriodOBV

(Opt

time)

(s)

Gap % OBV

(Opt

time)

(s)

Gap %

04251_1 T=1 68950462 - 12 70598745 (38.51) -

04251_1 T=3 95078477.4 - 1.8 78613300 - 38580

04251_1 T=5 93727249.8 - 3 95864500 - 0.1

04251_2 T=1 117239409 - 8 118915000 (38.4) -

04251_2 T=3 154186000 - 1.7 155666000 - 0.1

04251_2 T=5 154853000 - 1.4 154893000 - 0.6

04251_3 T=1 177618000 - 8 180119000 (55.4) -

04251_3 T=3 234708000 - 2 239430000 - 0.08

04251_3 T=5 213913000 - 39919 239085000 - 0.2

04251_4 T=1 261756000 - 9 265204000 (214.12) 0

04251_4 T=3 358656000 - 3 327306000 - 25780

04251_4 T=5 358072648 - 4 327662000 - 37615

04251_5 T=1 341825743 - 11 347904000 (106) -

04251_5 T=3 473757684 - 4 433009000 - 26100

04251_5 T=5 473803607 - 4 433308000 - 38096

04251_6 T=1 421085511 - 13 430466000 - 9973

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04251_6 T=3 585966062 - 4 536530000 - 28691

04251_6 T=5“out of

memory”-

-535765000 -

41955.4

04251_7 T=1 528151540 - 14 540376000 (211.05) -

04251_7 T=3 745382423 - 6.4 690071000 - 26690

04251_7 T=5“out of

memory”-

-690605466 -

38950

04251_8 T=1 627837259 - 17 645794184 275.13 0

04251_8 T=3 830779598 - 15 830787050 - 29336.2

04251_8 T=5“out of

memory”-

-“out of memory”

-

Table 12: Total time to solve mono-period URM problem

Case Stands FAU_time(s) MCC_time(s)

04251_1 10282.00 4.16 333.92

04251_2 20736.00 8.10 849.82

04251_3 31514.00 14.56 2920.71

04251_4 42374.00 22.44 6698.74

04251_5 52819.00 32.32 10425.37

04251_6 63275.00 40.90 15292.30

04251_7 73782.00 101.10 17697.90

04251_8 83826.00 139.00 28009.60

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Figure 1: Hypothetical forest example with 11 units

0 20000 40000 60000 80000 1000000

50000

100000

150000

200000

250000

300000

350000

400000

450000

MCC FAU

Num

ber

of c

onst

rain

ts

Number of stands

Figure 2: Number of constraints generated with the FAU and the MCC models in real cases

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0 20 40 60 80 100 120 1400

100

200

300

400

500

600

700

800

MCC FAU

Number of stands

Num

ber

of c

onst

rain

ts

Figure 3: Number of constraints generated with the FAU and the MCC models with square forest

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Table 1 : Spatial requirements (constraints) considered for mathematical formulations ..26

Table 2: Characteristics of real forests ..............................................................................27

Table 3: CPU to generate maximal clique for large cases.................................................27

Table 4: Different values of clusters in different work......................................................29

Table 5: Size of the different formulations when T=3 ......................................................30

Table 6: Constraints number generated with the FAU and the MCC formulation with square

forests.................................................................................................................................31

Table 7: Impact of the maximal allowable cut on the formulations size when T=3 ........31

Table 8: Different problem variants ..................................................................................32

Table 9: Integer problem resolution for small cases..........................................................33

Table 10: Integer problem resolution with URM-based approaches.................................34

Table 11: Integer problem resolution with ARM-based approaches.................................35

Table 12: Total time to solve mono-period URM problem...............................................36

Figure 1:Hypothetical forest example with 11 units..........................................................38

Figure 2: Number of constraints generated with the FAU and the MCC models in real cases

...........................................................................................................................................38

Figure 3: Number of constraints generated with the FAU and the MCC models with square

forest ..................................................................................................................................39

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Appendix A: Path formulation

pt ptt T p P

Max n X (A1)


1ptt T

X

p P (A2)

1ptp I

X I

,I M t T (A3)

( 1) ( 1)t pt pt p t p t t pt pt

p P p P p Pl v X v X u v X

1..( 1)t nt (A4)

_

( ( ) )p p p pt pp P t T p P

a g ntm tm g X G a

(A5)

0,1ptX ,p P t T (A6)

Where: is a binary variable which is equal to one if stand is harvested at period ptX p t

and zero otherwise.

The objective function (A1) consists of maximizing the net present value. Equation (A2)

ensures that each stand is harvested at least once. Equation (A3) imposes that in each period

at least one stand in each minimally infeasible cluster of the set M is not harvested.

Equation (A4) enforces sustainable yield constraints, while equation (A5) is the average

ending age constraints. Finally, equation (A6) states the binary requirement for the

variables.

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Appendix B: Maximal Clique-Cluster formulation

rt rtt T r R

Max n X (B1)


1p

rtt T r R

X

∀𝑝є𝑃 (B2)

1rtr RR C

X

,K C t T (B3)

( 1) ( 1)t rt rt r t r t t rt rtr R r R r R

l v X v X U v X

1..( 1)t nt (B4)

_

( ( ) )p p p rt pp P t T r R p P

a g ntm tm g X G a

(B5)

0,1rtX ,r R t T (B6)

Where:

: A binary variable equal to 1 if cluster is harvested in period , and 0 otherwise.rtX r t

The objective function (B1) consists in maximizing the net present value. Equation (B2)

guarantees that each stand is harvested at least once. Equation (B3) ensures for each period

that if a cluster is harvested then no adjacent clusters or clusters with the same stands may

be harvested. Equation (B4) enforces sustainable yield constraints, while equation (B5)

imposes the average ending age constraints. Finally, equation (B6) states the binary

requirement for the variables.

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Appendix C: Pseudo code for cluster generation (feasible, minimally infeasible and

maximal cliques)

p

1 p

1

1

1

begini=0e=0j=0h=0for p P doif a s then

i=i+1k=i{p}is feasiblecluster C k .

elsee=e+1{p}is minimally infeasiblecluster M e

end ifwhile k i

for p C kfor p N

if {p } C k is feasible thenk=k+1C k ={p } C k-1

elseif {p } C k is minimal

1

1 p

1

1

1

ly infeasiblecluster then

M e ={p } C iend if

end forend fork=k+1

end whilej=j+1h=j{p}is a cliqueQ h .while h j

for p Q hfor p N

if {p } Q h is clique then n=n+1

{p } Q h is a non maximalclique W[n]else

h=h+1Q h ={p }

Q h-1end if

end forend forh=h+1end while

end for

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It is obvious that maximal clique sets are contained in \Z WQ

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draft - university of toronto t-space · 114 et al. (2003) proposed to add clique constraints to...

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