A theoretical minimal solution for heuristics: the case of the spatial harvest timber problem. (English) Zbl 1511.91091
Summary: Heuristic methods are widely used to address the spatial optimization of timber harvests at the forest level. These methods have been shown to solve the timber harvest problem in a timely and computationally efficient manner. However, solutions provided by any heuristic are often suboptimal, and inquiries often arise regarding the quality of those solutions. One way to assess the quality of the solutions is to compare them against a minimum solution estimated using probability functions associated with the theoretical distribution of the solution space. A thorough theoretical framework is proposed to estimate the parameters of the probability distribution of the solution space based on the noncentral chi-square (\(\chi^2\)) distribution as an underlying distribution for the instances. A case study using the Lincoln Tract dataset suggests that the best objective function out of six thousand solutions was 0.26, whereas the theoretical minimal solution was \(2 \times 10^{-6}\).
MSC:
| 91B76 | Environmental economics (natural resource models, harvesting, pollution, etc.) |
| 90C27 | Combinatorial optimization |
| 90C59 | Approximation methods and heuristics in mathematical programming |
Keywords:
statistical optimum estimation techniques (SOET); statistical bounds; extreme value theory; noncentral chi-square \(( \chi^2)\); forestryReferences:
| [1] | András, S.; Baricz, A., Properties of the probability density function of the non-central chi-squared distribution, J. Math. Anal. Appl., 346, 2, 395-402 (2008) · Zbl 1145.60010 |
| [2] | Bettinger, P., Heuristic algorithm teaching tool (HATT) version 2.01. U.S. copyright registration number TXu 1-897-176 (2013), Warnell School of Forestry and Natural Resources, University of Georgia, Athens, GA, USA |
| [3] | Bettinger, P.; Boston, K.; Kim, Y. H.; Zhu, J., Landscape-level optimization using tabu search and stand density-related forest management prescriptions, European J. Oper. Res., 176, 2, 1265-1282 (2007) · Zbl 1102.90017 |
| [4] | Bettinger, P.; Demirci, M.; Boston, K., Search revision within s-metaheuristics: impacts illustrated with a forest planning problem, Silva Fennica, 49, 2, 1232-1252 (2015) |
| [5] | Bettinger, P.; Graetz, D.; Boston, K.; Sessions, J.; Chung, W., Eight heuristic planning techniques applied to three increasingly difficult wildlife planning problems, Silva Fennica, 36, 2, 561-584 (2002) |
| [6] | Bettinger, P.; Sessions, J.; Boston, K., A review of the status and use of validation procedures for heuristics used in forest planning, Math. Comput. For. Nat.-Res. Sci., 1, 1, 26-37 (2009) |
| [7] | Boender, C. G.E.; Rinnooy Kan, A. H.G.; Timmer, G. T.; Stougie, L., A stochastic method for global optimization, Math. Program., 22, 125-140 (1982) · Zbl 0525.90076 |
| [8] | Boston, K.; Bettinger, P., An analysis of Monte Carlo integer programming, simulated annealing, and tabu search heuristics for solving spatial harvest scheduling problems, For. Sci., 45, 2, 292-301 (1999) |
| [9] | Brandeau, M. L.; Chiu, S. S., Sequential location and allocation: worst case performance and statistical estimation, Locat. Sci., 1, 4, 289-298 (1993) · Zbl 0926.90053 |
| [10] | Busby, G. M.; Binkley, C. S.; Chudy, R. P., Constructing optimal global timberland investment portfolios, For. Policy Econ., 111, December 2019, Article 102083 pp. (2020) |
| [11] | Carling, K.; Meng, X., Confidence in heuristic solutions?, J. Global Optim., 63, 2, 381-399 (2015) · Zbl 1332.90221 |
| [12] | Carling, K.; Meng, X., On statistical bounds of heuristic solutions to location problems, J. Comb. Optim., 31, 4, 1518-1549 (2016) · Zbl 1347.90069 |
| [13] | Cascio, A.; Clutter, M., Risk and required return assessments of equity timberland investments in the United States, For. Prod. J., 58, 10, 61-70 (2008) |
| [14] | Caulfield, J. P.; Newman, D. H., Dealing with timberland investment risk: theory versus practice for institutional owners, J. For. Econ., 5, 253-268 (1999) |
| [15] | Cohen, J. D., Noncentral chi-square: Some observations on recurrence, Amer. Statist., 42, 2, 120-122 (1988) |
| [16] | Conde, E., A minimum expected regret model for the shortest path problem with solution-dependent probability distributions, Comput. Oper. Res., 77, 11-19 (2017) · Zbl 1391.90509 |
| [17] | Cooke, P., Statistical inference for bounds of random variables, Biometrika, 66, 2, 367-374 (1979) · Zbl 0406.62011 |
| [18] | Cubbage, F.; Kanieski, B.; Rubilar, R.; Bussoni, A.; Olmos, V. M.; Balmelli, G.; Donagh, P. M.; Lord, R.; Hernández, C.; Zhang, P.; Huang, J.; Korhonen, J.; Yao, R.; Hall, P.; Del La Torre, R.; Diaz-Balteiro, L.; Carrero, O.; Monges, E.; Thu, H. T.T.; Frey, G.; Howard, M.; Chavet, M.; Mochan, S.; Hoeflich, V. A.; Chudy, R.; Maass, D.; Chizmar, S.; Abt, R., Global timber investments, 2005 to 2017, For. Policy Econ., 112, Article 102082 pp. (2020) |
| [19] | Dannenbring, D., Procedures for estimating optimal solution values for large combinatorial problems, Manage. Sci., 12, 1273-1283 (1977) · Zbl 0377.90051 |
| [20] | Derigs, U., Using confidence limits for the global optimum in combinatorial optimization, Oper. Res., 33, 5, 1024-1049 (1985) · Zbl 0587.90070 |
| [21] | Dueck, G.; Scheuer, T., Threshold accepting: A general purpose optimization algorithm appearing superior to simulated annealing, J. Comput. Phys., 90, 1, 161-175 (1990) · Zbl 0707.65039 |
| [22] | Epstein, R.; Weintraub, A.; Sapunar, P.; Nieto, E.; Sessions, J. B.; Sessions, J.; Bustamante, F.; Musante, H., A combinatorial heuristic approach for solving real-size machinery location and road design problems in forestry planning, Oper. Res., 54, 6, 1017-1027 (2006) · Zbl 1167.90575 |
| [23] | Fisher, R.; Tippett, L., Limiting forms of the frequency distribution of the largest or smallest member of a sample, Camb. Philos. Soc., 24, 180-190 (1928) · JFM 54.0560.05 |
| [24] | Giddings, A. P.; Rardin, R. L.; Uzsoy, R., Statistical optimum estimation techniques for combinatorial optimization problems: A review and critique, J. Heuristics, 20, 3, 329-358 (2014) |
| [25] | Global, C., Timber Primer: Timberland Asset Class OverviewTech. Rep (2017), Campbell Globall LLC: Campbell Globall LLC Portland, OR |
| [26] | Glover, F., Tabu search, part i, ORSA J. Comput., 1, 190-206 (1989) · Zbl 0753.90054 |
| [27] | Glover, F., Tabu search, part II, ORSA J. Comput.1, 2, 4-32 (1990) · Zbl 0771.90084 |
| [28] | González-González, J. M.; Vázquez-Méndez, M. E.; Diéguez-Aranda, U., A note on the regularity of a new metric for measuring even-flow in forest planning, European J. Oper. Res., 282, 3, 1101-1106 (2020) · Zbl 1431.91278 |
| [29] | Gumbel, E., Statistics of Extremes (1958), Columbia University Press: Columbia University Press New York, NY, USA · Zbl 0086.34401 |
| [30] | Holland, J., Adaptation in Natural and Artificial Systems: an Introductory Analysis with Application to Biology (1975), The University of Michigan Press: The University of Michigan Press Ann Harbor, MI, USA · Zbl 0317.68006 |
| [31] | Jin, X.; Pukkala, T.; Li, F., Fine-tuning heuristic methods for combinatorial optimization in forest planning, Eur. J For. Res., 135, 4, 765-779 (2016) |
| [32] | Kettani, H., 2006. Contributions to the theory of the non-central chi-square distribution. In: International Conference on Scientific Computing. CSC 2006, Las Vegas, NV, p. 7. |
| [33] | Los, M.; Lardinois, C., Combinatorial programming, statistical optimization and the optimal transportation network problem, Transp. Res. B, 16, 2, 89-124 (1982) |
| [34] | Martins, I.; Ye, M.; Constantino, M.; da Conceição Fonseca, M.; Cadima, J., Modeling target volume flows in forest harvest scheduling subject to maximum area restrictions, Top, 22, 1, 343-362 (2014) · Zbl 1298.90049 |
| [35] | McRoberts, K. L., A search model for evaluating combinatorially explosive problems, Oper. Res., 19, 6, 1331-1349 (1971) · Zbl 0228.90021 |
| [36] | Mei, B., Investment returns of US commercial timberland: Insights into index construction methods and results, Can. J. Forest Res., 47, 2, 226-233 (2017) |
| [37] | Messina, V.; Bosetti, V., Integrating stochastic programming and decision tree techniques in land conversion problems, Ann. Oper. Res., 142, 1, 243-258 (2006) · Zbl 1101.91063 |
| [38] | Murray, A. T., Spatial restrictions in harvest scheduling, For. Sci., 45, 1, 45-52 (1999) |
| [39] | Nydick, R. L.; Weiss, H. J., An analytical evaluation of optimal solution value estimation procedures, Nav. Res. Logist., 41, 2, 189-202 (1994) · Zbl 0798.90134 |
| [40] | Pickands, J. I., Statistical inference using extreme order statistics, Ann. Statist., 3, 1, 119-131 (1975) · Zbl 0312.62038 |
| [41] | Plà-Aragonés, L. M.; Vizvári, B.; Kristensen, A. R., Methods and applications in natural resources management, Ann. Oper. Res., 219, 1, 1-3 (2014) · Zbl 1301.00058 |
| [42] | Reiter, S.; Sherman, G., Discrete optimizing, J. Soc. Ind. Appl. Math., 13, 3, 864-889 (1965) · Zbl 0163.41201 |
| [43] | Rencher, A. C.; Schaalje, G. B., Linear Models in Statistics, 672 (2008), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. Hoboken, New Jersey, USA · Zbl 1136.62045 |
| [44] | Restrepo, H.; Zhang, W.; Mei, B., The time-varying role of timberland in long-term, mixed-asset portfolios under the mean conditional value-at-risk framework, For. Policy Econ., 113, Article 102136 pp. (2020) |
| [45] | Robson, A. D.S.; Whitlock, J. H., Estimation of a truncation point, Biometrika, 51, 1, 33-39 (1964) · Zbl 0163.40503 |
| [46] | Roise, J. P., Multicriteria nonlinear programming for optimal spatial allocation of stands, For. Sci., 36, 3, 487-501 (1990) |
| [47] | State of Oregon, J. P., Oregon Administrative Rules, Oregon Revised Statutes. ORS 527.740 Harvest Type 3 Limitations (2021), Oregon Secretary of State: Oregon Secretary of State Salem, OR |
| [48] | van der Watt, P., A note on estimation of bounds of random variables, Biometrika, 67, 3, 712-714 (1980) |
| [49] | Wackerly, D. D.; Mendenhall III, W.; Sheaffer, R. L., Mathematical Statistics with Applications, 912 (2008), Thomson Learning, Inc.: Thomson Learning, Inc. Belmont, CA, USA |
| [50] | Wan, Y.; Clutter, M. L.; Mei, B.; Siry, J. P., Assessing the role of U.S. timberland assets in a mixed portfolio under the mean-conditional value at risk framework, For. Policy Econ., 50, 118-126 (2015) |
| [51] | Wan, Y.; Clutter, M. L.; Siry, J. P.; Mei, B., Assessing the impact of macroeconomic news on the U.S. forest products industry portfolio across business cycles: 1963-2010, For. Policy Econ., 28, 15-22 (2013) |
| [52] | Washburn, C. L.; Binkley, C. S., Do forest assets hedge inflation?, Land Econom., 69, 3, 215-224 (1993) |
| [53] | Wrds, C. L., Center for Research in Security Prices Files on Stocks Indexes and TreasuryTech. Rep (2018), Wharton Research Data Services |
| [54] | Zhu, J., Spatial Harvest Scheduling in the Southeastern United States: Estimating the Impact on Landowners of Different Sizes of Spatial Configuration of Ownership, 154 (2006), University of Georgia, Athens, GA, USA, (Ph.D. thesis) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
