# A Compromise Programming Application to Support Forest Industrial Plantation Decision-Makers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{®}

**,**a Forest Management Decision Support System, developed by following the multi-criteria decision theory, was used to process the case study. Expressly, the hypothesis that mill managers initially have, that older ages rotation could improve mill production, was not confirmed. Moreover, mill managers lean towards changes in the short term, while the case study shows that changes in rotation size and genetic material take time, and decisions have to be made involving both interests: forest and mill managers.

## 1. Introduction

## 2. Material and Methods

#### 2.1. Case Study

^{3}/ha-year, averaging 35.3 m

^{3}/ha-year [27].

^{3}of non-coniferous pulpwood [28]. The industrial short fiber average consumption was 4.34 m

^{3}of wood to produce one ton of pulp. The literature mentions productivities from 3.5 to 5.1 m

^{3}per ton of dry pulp [13]. To sustain their high level of productivity, Brazilian pulp mills’ main concerns regarding wood quality are (i) wood density [13], (ii) size uniformity of the chips [29], and (iii) chemical composition [30].

^{3}, and the annual nominal interest rate is 7% [32].

^{3}). The other two GenMat2 and GenMat1 classify clones into mid-density (500 kg/m

^{3}) and high density (580 kg/m

^{3}), respectively. The productivity varies from 42 to 52 m

^{3}/ha-year at seven years old. In the business-as-usual scenario, harvesting ages are limited to only two alternatives: six and seven years old.

^{3}of wood had entered into the mill, and (ii) how much pulp had been produced in dry tons of pulp.

^{3}) group and GenMat1 (580 kg/m

^{3}) density are equivalent to E. grandis × E. urophilla hybrids and E.urophilla, respectively. These species are widely used in genetic improvement programs in Brazil to boost volume productivity [34,35].

#### 2.2. Brief Description of Romero^{®}

^{®}

#### 2.2.1. Forest Harvest Schedule Formulation

_{ur}is an area of a unit u managed according to a prescription r, where prescription refers to a sequence of interventions that can happen within the planning horizon. In the Model II formulation, the decision variable is an area of a unit u where one complete forest cycle occurs. Romero

**encompasses an extension of Model II formulation where the decision variable is an area of a unit**

^{®}**u**in which a management intervention changes the previous course of the forest [47]. The formulation proposed by Nobre [47] is described in Appendix A.

#### 2.2.2. Multicriteria Concepts Embedded in Romero^{®}

^{®}’s model formulation. They are implemented according to the following descriptions.

- (a)
**Attributes.**These values are calculated independently from any stakeholder’s choice. In our case study, a “Production” accounting variable could be written using the expression, $\sum _{i=1}^{I}{\displaystyle \sum _{u=1}^{U}{\displaystyle \sum _{k=1}^{K}{V}_{iupk}}}}{X}_{iupk$ which is the total production within the horizon. It is called xTotalProd.- (b)
**Objectives.**They are all available to be selected by the user. If the user selects the attribute xTotalProd to form a maximization objective, then the model creates statements as follows:$$\begin{array}{l}Max\hspace{0.17em}\hspace{0.17em}Z\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{p=1}^{P}xTotalPro{d}_{p}}\\ subject\hspace{0.17em}to:\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Initial\hspace{0.17em}Area)...\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Conservation\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}Area)...\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Production)\hspace{0.17em}{\displaystyle \sum _{i=1}^{I}{\displaystyle \sum _{u=1}^{U}{\displaystyle \sum _{k=1}^{K}{V}_{iupk}{X}_{iupk}}}}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}xTotalPro{d}_{p}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall \hspace{0.17em}p\hspace{0.17em},\hspace{0.17em}p\hspace{0.17em}=\hspace{0.17em}1,2,....P\end{array}$$- (c)
**Targets and Goals.**In statement (1), in the line “Goal”, the value vMin_{p}is a target; decisionmakers want at least vMin_{p}of production in period p. In Romero’s implementation, targets are parameters that are saved in a database and ready to be updated.^{®}$$\begin{array}{l}Production)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{i=1}^{I}{\displaystyle \sum _{u=1}^{U}{\displaystyle \sum _{k=1}^{K}{V}_{iupk}{X}_{iupk}}}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}xTotalPro{d}_{p}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall \hspace{0.17em}p\hspace{0.17em},\hspace{0.17em}p\hspace{0.17em}=\hspace{0.17em}1,2,....P\\ Goal)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}xTotalPro{d}_{p}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\ge \hspace{0.17em}\hspace{0.17em}vMi{n}_{p}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall \hspace{0.17em}p\hspace{0.17em},\hspace{0.17em}p\hspace{0.17em}=\hspace{0.17em}1,2,....P\\ \end{array}$$calculates the deviations but does not minimize them.^{®}- (d)
**Criteria.**In statements (1) and (2) the Production criterion means that the attribute xTotalProd is used to calculate the objective MaxTotalProd.

**controls attributes, objectives, targets, and goals. It always calculates many other attributes like in the Production line of the statement (2). Depending on the decision-makers choices, the attributes will be used in goals or objectives.**

^{®}^{1}, f

^{2}, …, f

^{n}, and they are subject to a set of constraints represented by a vector of functions Q(x) (statement (3)).

^{1}and f

^{2}of statement (3) are the mill production and the forest profits, respectively. The function Q(x) will be the entire Model formulation proposed by Nobre [47]. To do so, Romero

**implements the two following steps.**

^{®}**calculation process.**

^{®}**shows the payoff matrix and the Pareto frontier as described above and saves the entire MOLP solution of each point along the curve; however, the decision process needs to go further. The next phase is to select a small set of solutions that could best fit a specific group of decision-makers along the Pareto frontier. One of the most productive ways to accomplish this task was proposed in 1973 by Yu and Zeleny under the name of Compromising Programming [48,49].**

^{®}#### 2.2.3. Compromise Programming (CP)

_{1}) and Min(L

_{∞}), which are also described in Appendix B.

**calculates both distances to the ideal point, with a minimum of L**

^{®}_{1}and a minimum of L

_{∞}[53]. That is to say, the compromise set is a closed interval of the Pareto frontier, and Romero

**shows the distances Min(L**

^{®}_{1}) and Min(L

_{∞}) along the Pareto frontier, later shown graphically in the results section.

_{1}) and Min(L

_{∞}) along the Pareto frontier, Romero

**implemented the following sequence of steps:**

^{®}- The user inputs how many points he wants to calculate in the Pareto-efficient set. The user should set as many points as possible depending on the time each run (to obtain one point) takes.
- Romero
^{®}- o
- distributes those points between the best and worst values of each attribute,
- o
- runs the MOLP model to calculate the efficient point using the constraint methodology,
- o
- saves the results (attribute values, decision variables, and deviation variables) of all points into the database.

- The user updates the decision-makers’ weights for each criterion according to their preferences.
- Romero
:^{®}- o
- calculates L
_{1}and L_{∞}for each efficient point, - o
- finds the minimum L
_{1}and minimum L_{∞}using as SQL to implement and find a proxy of the linear programming models of the statements (A8) and (A9) in Appendix B.

#### 2.2.4. Romero^{®} Architecture

^{®}

**architecture and embedded mathematical programming. Most importantly, this section focuses on how the system guarantees the interactive MCDM principles.**

^{®}**comprises the general components shown in Figure 6.**

^{®}**prepares the results to calculate the payoff matrix and automatically sets the parameters to the Pareto frontier. Two steps are needed to use the constraint method: (1) get the range from minimum and maximum values of all attributes and (2) use these range values as constraints. Romero**

^{®}**can recreate a solution when decision-makers choose any point along the Pareto frontier because it saves all decision variable results. To calculate the two points Min(L**

^{®}_{1}) and Min(L

_{∞}), Romero

**runs a minimization SQL query from results stored in the database**

^{®}**.**

#### 2.3. Romero’s^{®} Application to the Case Study

^{®}

^{®}was configured to evaluate the consequences of harvesting the eucalyptus plantations at older ages.

**,**allowing genetic material changes after clear-cuts, as shown in Figure 7. The transition types cl1, cl2, and cl3, represent clear-cuts; the cl1 clear-cuts and re-establishes the plantation with GenMat1, cl2 with GenMat2, and cl3 with GenMat3. After a cl1 intervention, it will be alternatively possible to have interventions of all three types (cl1, cl2, and cl3). Figure 7 shows the ina transition type, which represents the initial forest status. After an intervention of the ina type, interventions cl1, cl2, or cl3 may follow; ina does not change the value of any forest attribute.

- regulated wood flow,
- genetic diversification to mitigate risk exposure to plagues and diseases, and
- production sustainability.

**parameters and constraints. After setting the parameters and acceptable prescriptions for future management, it is possible to define the prime conflicts accurately.**

^{®}**(Table 4), was chosen to represent forest managers’ aspirations. At the same time, mill managers’ aspirations are represented by MaxMillPrd (Maximize Mill Production). MaxNPV and MaxMillPrd are determined using accounting xTotalNPV and xTotalMillPrd**

^{®}©**.**

_{i}, which refers to the area in which the intervention i will occur

**.**Romero

^{®}was used to set up the parameters, handle parameter combinations, run different scenarios, and report the results.

_{i}. Likewise, the following list of the constraints defines the feasibility, which depends on the same decision variables x

_{i}.

## 3. Results

**could support the multi-stakeholders simulated teams, starting by defining the minimum and maximum ages that foresters would be allowed to harvest.**

^{®}**to run. The simulation finishes when Romero**

^{®}**shows the results window. Table 6 summarizes the running information of Romero**

^{®}**after each scenario.**

^{®}**had to deal with, the more time-consuming the data processing became. The most time-consuming step is the concrete model building routine when Pyomo builds an abstract model. Naturally, the larger the model, the longer it took to build it.**

^{®}**will cycle through solving, changing the values of a few parameters defining the next objective, and solving again. These latter steps run faster than the model building steps (6–7 s for the smallest problem and 18–19 s for the biggest problem).**

^{®}**to facilitate the search for consensus between stakeholders. The third criterion, maximize final stock, was included to add more substance to the analysis. It considers the consequences of regulating the annual pulpwood demand and maximizing the standing wood stock at the end of the planning horizon. NPV and pulp parameters were only calculated and left unconstrained.**

^{®}_{1}to the industrial managers’ best value. Likewise, b and d are the distances from L∞ and L

_{1}to forest managers’ best value. The sum of the distances from point L∞ to the Optimum point is a + b

**.**The sum of the distances from point L

_{1}to the Optimum point is c + d. Herein, these distances (a, b, c, and d) from the Optimum point are called losses.

_{1}is the point where the sum of both losses is at its minimum along the Pareto frontier (line segments c + d). In other words, L

_{1}is the point where stakeholders lose less together. On the other hand, L∞ is the minimum of the max between them, which corresponds to line segment a in Figure 12. At L∞, there are no noteworthy losses for any of the stakeholders.

_{1}. The columns “Best Value” and “Worst Value” repeat the optimum point’s values. The “Interval” column refers to the difference between the best and the worse values. The Distance column presents the distances between the point on the Pareto front and the line that crosses the optimum point. These distances are calculated according to statement (A2).

_{1}. Point 10 of Table 8, which is located between L

_{∞}and L

_{1}(Figure 12) can be considered a good compromise in this case study. Even if the decision-makers agree to select point 10, they might need a more in-depth understanding of this particular solution. Therefore, point 10 should be examined to determine if it meets the needs of the decision-maker.

## 4. Discussion

**can support large-scale multi stakeholders’ decision processes. Any harvest-schedule-based problem involving diverse objectives can be addressed by the approach when parties involved in the negotiation are willing to seek consensus interactively driven by the compromise programming principle.**

^{®}**can guide to a compromise solution where stakeholders assume responsibilities regarding the decisions’ impact as prescribed by Bruña-García and Marey-Pérez [4] and other authors [20,21].**

^{®}**contribute to filling the gap on the use of MCDM techniques into FMDSS capable of supporting participatory processes.**

^{®}## 5. Conclusions

**’s support was essential to consider and inspect all aspects of the solution; it was an information-based decision-making process as preconized by literature.**

^{®}^{®}and its embedded models and methods, when applied to an industrial forest plantation case involving interactions among groups with diverse interests, demonstrated that incorporating MDCM techniques in a participatory process could improve the decision-making process and the quality of the decisions. Expressly, the hypothesis proposed in the study case that longer rotations could improve mill production, as expected by the mill managers, could not be confirmed. Nevertheless, Romero

**led the compromise to an unexpected point where the possibility of harvesting in younger ages allowed an earlier replacement of less efficient genetic material.**

^{®}**was shown capable of promoting interaction among stakeholders while seeking consensual decisions.**

^{®}**approach can be packed as applying rigorous multicriteria principles to a MOLP harvest schedule formulation. It employed an interface that promotes compromise and consensual planning. Furthermore, Romero**

^{®}**demonstrated through the case study application that strategic scenario analysis is still an essential tool to evaluate future impacts of short-term decisions.**

^{®}## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Model Formulation

## Appendix B. Compromise Programming Definitions

_{p}. This text does not intend to go deep into the L

_{p}definitions; however, it is crucial to have a clear understanding of the distance concepts [49] to sanction the use of the two limits of the compromise set: Min(L

_{1}) and Min(L

_{∞}).

_{p}represents the family of distance functions. For each value of p, there is a different distance value L

_{p}for the same pair of points x

^{1}and x

^{2}. A particular case L

_{p}is for p = 2 when the two-dimensional “Euclidean distance” is obtained. A second particular case of L

_{p}is for p = 1, called “Manhattan distance”, which, in a two-dimensional space, is the sum of the cathetus lengths. Therefore, L

_{1}is the largest distance between x

^{1}and x

^{2}, while L

_{2}is the shortest one according to Pythagoras theorem.

_{∞}is called “Chebysev distance” and tends to the value that can be calculated as follows:

_{∞}is given by only one deviation of the set of n deviations: the larger one. If L

_{1}and L

_{∞}are calculated for each point along the Pareto frontier, it is possible to determine two shorter distances, the Min(L

_{1}) and Min(L

_{∞}).

_{p}(statement (A5)) are contained in a “compromise set”, which is a subset of the efficient solutions set (Pareto frontier). In addition, the limits of this subset are given by L

_{1}and L

_{∞}(statements (A6) and (A7)). Two years later, Freimer and Yu demonstrated similar conditions to a more than two-criterion problem. Moreover, according to the traditional economic analysis and the theorems of Yu, the utility functions intercept a Pareto frontier in one of the points of the “compromise set”.

_{1}is a summation of the deviations between each attribute value from this ideal value (statement (A6)). The minimum L

_{1}is the maximum efficiency point, where the best point is achieved considering the value of the attributes. However, this point may be unbalanced, that is to say, in that point; one attribute can have much more value than others. Moreover, if the decision-makers have different preferences that can be translated into weights (from 0 to 1), L

_{1}could be mathematically represented as follows using previous definitions of statements (A3) and (A6):

_{∞}, it is necessary to select the maximum deviation between attribute individual deviations. That is, only the largest deviation of each attribute from its best value is considered. Considering all previous definitions, L

_{∞}could be mathematically represented as follows:

_{1}) is presented by Romero (1993) and can be written as follows, based on statements (A3) and (A8).

_{∞}) a linear programming model can be written as follows:

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Area | 21,400 ha | ||

Management units | 18 | ||

Area of each unit | From 500 to 1900 ha | ||

Allowable harvest ages | 6 and 7 years old | ||

Silvicultural costs | R$9500.00/ha | ||

Pulpwood price | R$50/m^{3} | ||

Discount rate | 7% | ||

Groups | Average Productivity (m ^{3}/ha.year at 7 years old) | Average Wood Density (kg/m ^{3} at 7 years old) | |

Species Eucalyptus spp. | GenMat1 | 42 | 580 |

GenMat2 | 48 | 500 | |

GenMat3 | 52 | 420 |

Scenarios | Allowable Ages |
---|---|

216 | 5, 6, 7 |

217 | 6, 7, 8 |

218 | 6, 7, 8, 9 |

Mill Requirements | Implementation |
---|---|

Regulated wood flow | From period 2 onwards No decrease above 10% No increase above 25% |

Genetic material safety | 60% maximum area for any GenMat type 20% minimum area for any GenMat type |

Production sustainability | Age class control 10% flexibility |

Objectives | Objective Description | Attribute (Accounting Variable) |
---|---|---|

MaxNPV | Maximize Net Present Value | xTotalNPV |

MaxMillPrd | Maximize Mill Production | xTotalMillPrd |

MaxStck | Maximize Final Stock | xTotalStock |

Scenario | Ages (Years) | Highest NPV (M R$) | Lowest NPV (M R$) | Forest NPV Reduction | Highest Pulp Production (K t) | Lowest Pulp Production (K t) | Mill Production Reduction |
---|---|---|---|---|---|---|---|

216 | 5, 6, 7 | 262,293 | 224,246 | 16.97% | 5149 | 4845 | 6.29% |

217 | 6, 7, 8 | 266,445 | 232,677 | 14.51% | 5022 | 4671 | 7.51% |

218 | 6, 7, 8, 9 | 266,612 | 232,677 | 14.58% | 5022 | 4656 | 7.85% |

Scenarios | ||||
---|---|---|---|---|

216 | 217 | 218 | ||

number of constraints | 53,304 | 23,280 | 38,049 | |

number of variables | 76,588 | 33,145 | 55,096 | |

time consumption (min:s) | ||||

iGen | 00:01.0 | 00:01.0 | 00:02.0 | |

Mem structure optimization | 00:01.0 | 00:00.0 | 00:00.0 | |

Building abstract model | 00:01.0 | 00:00.0 | 00:00.0 | |

Building concrete model | 21:21.0 | 03:44.0 | 12:33.0 | |

Solving 1st criteria and save results | MaxNPV | 00:19.0 | 00:07.0 | 00:11.0 |

Solving 2nd criteria and save results | MaxStck | 00:18.0 | 00:06.0 | 00:12.0 |

Solving 3rd criteria and save results | MaxMillPrd | 00:19.0 | 00:07.0 | 00:12.0 |

Criteria | Clear-Cut Area (ha) | Production (K m^{3}) | Productivity (m^{3}/ha) | %Max Stock | Forest NPV (M R$) | Pulp Production (K t) | Final Stock (K m^{3}) |
---|---|---|---|---|---|---|---|

MaxNPV | 82,992 | 19,708 | 237 | 88.42% | 266,445 | 4671 | 624 |

MaxMillPrd | 84,622 | 18,856 | 223 | 91.79% | 232,677 | 5022 | 648 |

MaxStck | 81,695 | 18,335 | 224 | 239,414 | 4801 | 706 |

Point | Criterion | Value | Best Value | Worst Value | Interval | Distance | Sum | Max | ||
---|---|---|---|---|---|---|---|---|---|---|

(M R$) | ||||||||||

9 | MaxMillPrd | 4879 | 5022 | 4671 | 350 | a | 0.4071 | |||

9 | MaxNPV | 254,004 | 266,445 | 232,677 | 33,767 | b | 0.3684 | 0.7755 | 0.4071 | L_{∞} |

10 | MaxMillPrd | 4900 | 5022 | 4671 | 350 | 0.3479 | ||||

10 | MaxNPV | 252,227 | 266,445 | 232,677 | 33,767 | 0.4210 | 0.7690 | 0.4210 | ||

11 | MaxMillPrd | 4919 | 5,022 | 4671 | 350 | c | 0.2929 | |||

11 | MaxNPV | 250,449 | 266,445 | 232,677 | 33,767 | d | 0.4736 | 0.7666 | 0.4736 | L_{1} |

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**MDPI and ACS Style**

Nobre, S.R.; Diaz-Balteiro, L.; Rodriguez, L.C.E. A Compromise Programming Application to Support Forest Industrial Plantation Decision-Makers. *Forests* **2021**, *12*, 1481.
https://doi.org/10.3390/f12111481

**AMA Style**

Nobre SR, Diaz-Balteiro L, Rodriguez LCE. A Compromise Programming Application to Support Forest Industrial Plantation Decision-Makers. *Forests*. 2021; 12(11):1481.
https://doi.org/10.3390/f12111481

**Chicago/Turabian Style**

Nobre, Silvana Ribeiro, Luis Diaz-Balteiro, and Luiz Carlos Estraviz Rodriguez. 2021. "A Compromise Programming Application to Support Forest Industrial Plantation Decision-Makers" *Forests* 12, no. 11: 1481.
https://doi.org/10.3390/f12111481