Possibility schema for interdisciplinary forest management evaluation and decision-making

Transcription

1 Possibility schema for interdisciplinary forest management evaluation and decision-making by Martin Herbert Kijazi 1 ABSTRACT Interdisciplinary planning and evaluation of forest management is necessary for sustainable forest management (SFM) schemes involving multiple values of multi-stakeholders. Often, multi-objective forest-planning and evaluation encounter complexity and uncertainty due to inexactness i.e., fuzziness, ambiguity, imprecision and variability of spatial behaviours of ecological and human systems. This paper develops the possibility schema from fuzzy sets and theory of possibility for representation and evaluation of inexact spatial concepts, configurations, and processes, associated with forest ecosystem and stakeholder values. A hypothetical case of interdisciplinary research utilizing criteria and indicators of SFM is used to illustrate the utility of the proposed possibility schema in interdisciplinary forest decisionmaking. The schema can be used for ex-ante appraisal and ex-post evaluation of forest programs. It can also be used for integration of interdisciplinary forest knowledge, including ecological and socio-economic models of SFM. Key words: decision-making, fuzzy sets, inexactness, interdisciplinary evaluation, multiple values, possibility theory, sustainable forest management, uncertainty RÉSUMÉ La planification interdisciplinaire et l évaluation de l aménagement forestier sont nécessaires dans le cas des schémas d aménagement forestier durable (AFD) qui comprennent les multiples valeurs des nombreux intervenants. Souvent, la planification et l évaluation des nombreux objectifs forestiers font face à une complexité et à une incertitude découlant de «l inexactitude», par ex., la confusion, l ambiguïté, l imprécision et la variabilité, des comportements spatiaux des systèmes humains. Cet article élabore un schéma de possibilité des conditions de confusion et de la théorie de la possibilité traitant de la représentation et de l évaluation des concepts spatiaux, des configurations et des processus inexacts, associés à l écosystème forestier et aux valeurs des intervenants. Un exemple hypothétique de recherche interdisciplinaire utilisant les critères et les indicateurs d AFD est utilisé pour illustrer le schéma de possibilité proposé en matière de prise de décision interdisciplinaire en foresterie. Le schéma peut être utilisé pour l intégration des connaissances forestières interdisciplinaires, comprenant également les modèles écologiques et socio-économiques d AFD. Mots clés : prise de décision, conditions de confusion, inexactitude, évaluation interdisciplinaire, valeurs multiples, théorie de la possibilité, aménagement forestier durable, incertitude Introduction Incorporations of interdisciplinary knowledge and multiple values of multi-stakeholders, in multiple-decision units, are common features of sustainable forest management (SFM) planning and evaluation. However, such features increase uncertainty in forest decision-making. Nilsson et al. (2004) have observed widespread uncertainty in manage- Martin Herbert Kijazi ment for the multiple objectives; managers cannot be certain whether all data accurately reflect the current state of all forest values, how the forest will change, or how management practices will affect the forest. According to Leung (1983) complexity of human systems, 1 PhD (forestry) candidate, Faculty of Forestry, University of Toronto, 33 Willcocks Street, Toronto, Ontario, Canada M5S 3B3. tini.herbert@utoronto.ca insufficient or inexact information, and ambiguous cognitive and decision-making significantly increase uncertainty. The same is true of ecological systems. Imprecision observed in ecological systems has led to a shift from conventional linear and single equilibrium, in favour of non-linear and multipleequilibria views of ecosystems, and the need for adaptive environmental management (Holling 1978). Hence, managers must continuously adapt practices, and scientists must constantly adapt experiments, to evolving ecosystems (Gunderson et al. 1995). Traditionally, analysts equated uncertainty with randomness and dealt with it using probability models. But a significant proportion of uncertainty is caused by inexactness 2 of behaviours of social and ecological systems. In SFM exactness in representing, analyzing, and forecasting spatial and temporal (ecological and social) behaviours can be difficult to achieve due to incomplete information and inexact cognition and evaluation. For example SFM requires mainte- 2 The term inexactness represents fuzziness, ambiguity, imprecision and variability of behaviors. MAY/JUNE 2005, VOL. 81, No. 3 THE FORESTRY CHRONICLE 375

2 nance of healthy 3 ecosystems; but healthy represents inexact condition a level of this attribute rather than its presence or absence. Given imprecision, uncertainty, subjectivity and ignorance so common in SFM the application of conventional, Boolean (bivalent) logic, which assumes perfect information and precise cognition and decision-making, has serious limitations in the analysis and representation of spatial information and evaluation of SFM schemes. Instead, fuzzy logic (herein, logic of inexactness) based on fuzzy sets theory (Zadeh 1965, Kaufmann 1975), and theory of possibility (Zadeh 1978) may be more suited for such analysis. The objective of this paper is to develop a possibility schema for interdisciplinary planning and evaluation of SFM using fuzzy logic and theory of possibility. The first section reviews uncertainty in forest management. The second section describes fuzzy nature of forest phenomena and events. The third section deals with fuzzy conceptualization and representation of forest phenomena and events. The fourth section develops the possibility schema for SFM. The merits and utility of the proposed schema are presented in the discussion section, prior to conclusions. Uncertainty in Forest Management Uncertainty means inability to assemble information, which quantitatively and qualitatively prescribe or predict deterministically and numerically a system, its behaviour, or other characteristics (Zimmerman 2000). The forest sector is subject to much uncertainty. Nilsson et al. (2004) and Kangas and Kangas (2004) have identified several sources of such uncertainty including: (1) the need to integrate socio-economic and ecological issues and preferences of multi-stakeholders; (2) limited knowledge of biological patterns, processes and responses to forest management; (3) impact of natural disturbances fires, insects, and diseases; (4) unclear political constraints and policy specifications, future prices of products and costs of management options, and technological innovations; (5) long-term planning horizon; and (6) large spatial units. Conventionally, probability theory was used to deal with uncertainty in decision-making, given the strong statistical and timber-centered tradition in forestry (Kangas and Kangas 2004). However, probability theory can not deal with non-probabilistic uncertainty arising from inexactness of spatial behaviour of eco-systems and disturbance events, and that of social behaviour, including cognitive behaviour and value judgments about forest ecosystems and values. Nonconventional approaches that can deal with such uncertainty include: 1) fuzzy set theory (Zadeh 1965) for dealing with uncertainty due to ambiguity of concepts, 2) possibility theory (Zadeh 1978) and 3) evidence theory (Dempster 1967a, b; Schafer 1976). The two later theories deal with subjective beliefs and expert judgments, as well as partial information and pure ignorance (Kangas and Kangas 2004). A brief treatment of fuzzy nature of forest phenomena is presented next. Fuzzy Nature of Forest Phenomena and Events Fuzziness is different from randomness. According to Bellman and Zadeh (1970) randomness involves uncertainty about the membership or non-membership of an object in a non-fuzzy class, while fuzziness involves the gradual transition from non-membership to membership of an object to a fuzzy class; a member of a subset belonging to the universe of a discourse for a given phenomenon is assigned a value representing its grade of membership with values ranging from 1 (highest grade of membership) to zero (non-member). A statement such as the probability that a forest fire will burn forest A is 0.8, is a probabilistic statement about the uncertainty of the occurrence of the non-fuzzy event forest fire. In contrast, suppose that 120 ha of forest are burnt; a statement such as the grade of membership of 120 ha burnt, in the class,severeforest fire, is 0.4 is a classification statement 4 concerned with the membership of 120 ha burnt in the fuzzy class severe ofa non-fuzzy event forest fire. Similarly we can speak of a grade of membership of a fuzzy phenomenon, biodiversity (which is normally not amenable to binary classification diverse or not diverse ). Instead this fuzzy phenomenon is, linguistically, characterized with fuzzy concepts such as highly diverse (or high species richness). Although uncertainty may be due to randomness or fuzziness, it may also be a result of both; e.g., the probability of having big forest fires in Alberta concerns the probability of the random occurrence of the fuzzy event big forest fire. A combination of probability and fuzzy sets theory (Zadeh 1978) may be necessary to analyze this type of phenomenon. Conceptualizing and Representing Inexactness in Forestry In forests, gradual variation of most spatial phenomena (site productivity, biodiversity, suitable wildlife habitat, etc.) makes it difficult or impossible to establish a non-fuzzy boundary and allow dichotomous regionalization. Fuzzy logic is suited to more natural depiction of such spatial concepts and phenomena given that a spatial unit may belong to a region or phenomenon only to a certain degree. In the context of fuzzy set theory, the concept large (for a clearcut) can be depicted as a fuzzy subset defined by a membership function, µ large,which maps the universe of discourse, clearcut in hectares, to a membership set M A [0, 1]. The grade of membership, µ large (A), represents the degree of belonging of an element A in the universe of discourse U to the concept large,with µ large (A) = 0 and µ large (A) = 1 representing complete non-membership and complete membership respectively. The degree of belonging increases from zero to one as the value of A increases (Fig. 1). The basic assumption is that a small variation in A should have at most a small effect on the degree of belonging of A to large.thus, the concept large can be viewed as a continuous function, which imposes a fuzzy restriction on the value of the base variable, clearcut size. Such characterization allows statements of intermediate truth, by not assuming exactness of concepts. 3 Henceforth, words in italics represent fuzzy concepts and phenomena. 4 That is, a statement which can be used for ex ante or ex post classification of the severity, or area-wise extent, of occurrence of forest fires. 376 MAI/JUIN 2005, VOL. 81, N o. 3 THE FORESTRY CHRONICLE

3 Fig. 1. An example of membership function of fuzzy subset large for a forest clearcut. Fig. 2. An example of intersection of membership function of fuzzy subsets large for a forest clearcut and severe for forest fire. The intersection is indicated by the thicker curve. Set operations can be applied to the membership functions of fuzzy sets. An intersection of fuzzy sets is defined as minimum (or logical and ) of their membership functions, whereas the union is defined as maximum (or logical or ) of their membership functions (Ells et al. 1997). Symbolically, intersection and union of fuzzy sets X and Y can respectively be represented as [1] X Y= µ X (A) µ Y (A) = min (µ X (A), µ Y (A)) U [2] XUY = µ X (A) V µ Y (A) = max (µ X (A), µ Y (A)) For example, if X is a fuzzy set representing a severe forest fire and Y is a fuzzy set representing a large forest clearcut, the intersection of a severe forest fire and large clearcut (occurrence of both in a spatial unit) can be represented as [3] X Y= µ severe (A) µ large (A) = min (µ severe (A), µ large (A)) U The intersection of severe forest fire and large clearcut is depicted in Fig 2. The grade of membership of each event is represented by a separate curve. At any given point on the horizontal axis, the intersection of the grade of memberships is represented by the curve with smaller grade of membership at that point as indicated by the thickened sections of the two curves. Similar reasoning as the foregoing can be used to develop possibility schema for SFM evaluation and decisionmaking as presented in the next section. Possibility Schema for SFM Using Criteria and Indicators: An Interdisciplinary Approach From theoretical approaches formulated for conceptualization of inexact spatial concepts (Leung 1983), a conceptualization of inexact concepts and phenomena characterizing SFM as fuzzy subsets, of a universe of discourse namely SFM, having imprecise spatial connotations is proposed. Consider an example of a forest community, where (for simplicity) SFM is defined by only two criteria, site productivity (for timber) and biodiversity. Such a community may be characterized by a linguistic proposition, community B is productive and diverse where diversity is measured by species richness, and site productivity is measured by site index. The above linguistic proposition can be defined by a possibility distribution function, utilizing basic set operations (e.g., Min procedure proposed by Bellman and Zadeh 1970): [4] Poss (species richness = a, site index = b) = min (µ diverse (a), µ productive (b)) The possibility distribution function above states that, if the species richness is a and site index is b, then the possibility of a forest community being both diverse in species and productive for timber equals the minimum of its grade of membership of being diverse and that of being productive. The following discussion extends this logic into an integrative and interdisciplinary schema for SFM decision-making. MAY/JUNE 2005, VOL. 81, No. 3 THE FORESTRY CHRONICLE 377

4 The approach to Equation 4 above can be used to integrate interdisciplinary knowledge from measurement of various criteria and indicators, C&I, of SFM into a common denominator for the purpose of decision-making, while taking into account uncertainty. Suppose SFM is defined by the following six Canadian Council of Forest Ministers (CCFM 2000) criteria: (1) Conservation of biological diversity; (2) Ecosystem condition and productivity; (3) Soil and water conservation; (4) Global ecological cycles; (5) Multiple benefits; and (6) Society s responsibility. Suppose elements and indicators of these criteria are measured appropriately to produce the following respective indices for each criterion: (1) biodiversity index; (2) eco-health index; (3) conservation index; (4) global-health index; (5) multiple-benefits index; and (6) societal index. The indices above can be used to construct fuzzy classes, characterizing SFM as a universe of discourse. For example, using the eco-health index, the concept healthy in the measurement of ecosystem condition and productivity can be depicted as a fuzzy subset defined by a membership function, µ healthy,which maps the universe of discourse, ecosystem condition and productivity, to a membership set M A [0, 1]. The grade of membership µ healthy (A) represents the degree of belonging of an element A (that is assessed) in the universe of discourse U to the concept healthy,with µ healthy (A) = 0 representing complete non-membership and µ healthy (A) = 1 indicating complete membership. Other concepts characterizing other criteria i.e., diverse, conserved, maintained, balanced, and participatory for criteria 1, 3, 4, 5 and 6 respectively can be conceptualized similarly. Suppose that (for brevity of presentation), SFM is qualified by three criteria corresponding with fuzzy classes diverse in species and ecosystems, balanced in utilization and contribution of multiple values, and participatory in society s responsibility based on interdisciplinary experts and other stakeholders evaluation of measurements (indices) that characterize SFM. Sustainability level of a given forest region B under a given management scheme, and these three criteria, may then be defined by a possibility distribution function, using the Min operator 5 : [5] Poss (biodiversity index = a, multiple-benefits index = b, societal index = c) = min (µ diverse (a), µ balanced (b), µ participatory (c)) The possibility distribution function above states that, if biodiversity index = a, multiple-benefits index = b, and societal index = c, then the possibility for sustainability of a forest scheme i.e., possibility of being diverse in species and ecosystems, balanced in utilization and contribution of multiple benefits, and participatory in decision-making, equals the minimum of its grade of membership of being diverse, being balanced, and being participatory. The decision-maker s goal can be to find a schema which maximizes minimum grade of membership 6 ; this has been seen before in Maness 5 Other aggregation criteria such as average and weighted average can also be used depending on the preference. Min operation gives equal weight to all criteria. Otherwise weighted average could be used. 6 Alternatively the goal can be to find a schema that equals or surpasses a pre-specified threshold. and Farrell (2004) whereby a multi-objective optimization model was created for forest development planning for an integrated forest products company located in the East Kootenay area of British Columbia, Canada. The mathematical model uses a fuzzy MAXMIN approach, where each indicator represents an objective in the model. In the current case, SFM can then be characterized as an aggregate, interdisciplinary, and multi-stakeholder (fuzzy logic-based) decision-making represented by a membership function µ Decision (SFM). Common aggregation criteria such as Min (Bellman and Zadeh 1970) or average (Dubois and Prade 1988), can be used to aggregate membership function values for all six criteria under alternative forest management schemes to generate the aggregated decision function µ Decision (SFM). The optimum value/decision, µ Decision Optimal (SFM) involves maximizing the membership function µ Decision (SFM). Thus adding fuzzy subsets healthy, conserved, and balanced for criteria 2, 3 and 4 respectively to Equation 5 we obtain: [6] µ Decision (SFM) = min (µ diverse (a), µ balanced (b), µ participatory (c), µ maintained (d), µ healthy (e), µ conserved (f)) [7] µ Decision Optimal (SFM) = max (µ Decision (SFM)) The aggregate decision function above assumes that all criteria are of equal weight. Otherwise, the criteria would be weighted 7 accordingly such that [8] µ Decision (SFM) weighted = ( 1 µ diverse (a), 2 µ balanced (b), 3 µ participatory (c), 4 µ maintained (d), 5 µ healthy (e), 6 µ conserved (f)), Where 1, 2., n are corresponding weights which add up to 1 and n = number of fuzzy classes (six in this case.) A hypothetical example of the application of possibility schema in interdisciplinary SFM decision-making is provided in Table 1. (For a practical example of the approach see, for example, Maness and Farrell 2004). The approach in the current hypothetical example makes use of the CCFM C&I of SFM and three alternative forest management schemes in which the membership values for each criterion can be obtained based on the performance of each alternative scheme with respect to C&I. Suppose Scheme 1 concerns continuing with status quo forest management with the following membership function values, in brackets, corresponding to Biodiversity (0.4), Ecosystem condition and productivity (0.8), Soil and water conservation (0.6), Global ecological cycles (0.6), Multiple benefits (0.7) and Society s responsibility (0.6); Scheme 2 involves marginal increase in protected areas (wildlife reserves, riparian zones, sensitive sites, etc.) leading to increased membership value for Biodiversity conservation (to 0.5), Soil and water conservation (to 0.7), and Global ecological cycles (to 0.7), while reducing the membership value for multiple benefits (to 0.7) due to restrictions in use of protected areas); and Scheme 3 involves marginal increase in intensification of forest management (intensive 7 If the targets are set appropriately then this takes the place of weights and they are not needed. 378 MAI/JUIN 2005, VOL. 81, N o. 3 THE FORESTRY CHRONICLE

5 Table 1. A hypothetical SFM decision-making possibility schema utilizing fuzzy logic and criteria and indicators of SFM Membership Alternative forest Criteria functions management schemes 1 2 a 3 Biodiversity µ diverse (a) Ecosystem condition and productivity µ healthy (b) Soil and waster conservation µ conserved (c), Global ecological cycles µ balanced (d), Multiple benefits µ sustainable (e), Society s responsibility µ participatory (f) Aggregated fuzzy decision (by Min operator) µ Decision (SFM) Optimum decision (By Max operator) µ Decision Optimal (SFM) 0.5 a the optimum or rather best scheme has highest-minimum value mechanization in harvesting and tending, artificial regeneration, fertilization, chemical weed and pest control, etc.) leading to decrease in membership value of Biodiversity (to 0.3), Ecosystem condition and productivity (to ), Soil and water conservation (to 0.5), Global ecological cycles (to 0.3), Multiple benefits (to 0.6) and Society s responsibility (to 0.6). In this example, Scheme 2 is the best scenario of the three because it maximizes the membership function (it has the highest-minimum membership value) aggregated by min operator (see Table 1). Discussion Preceding sections have revealed that human and ecological systems are essentially complex. Holling (2000) has provided an integrative perspective of the complexity and uncertainty of human and natural systems, which asserts that human behaviour and nature s dynamics are linked in an evolving system; the seeming paradox of change and stability inherent in evolving systems is the essence of sustainable futures; there is a need for policies that are dynamic and evolutionary; and that policies must expect results that are inherently uncertain and explicitly address that uncertainty through active probing, monitoring, and response. From this perspective it is then clear that fuzzy logic, and the possibility schema presented in this paper, is of critical importance as it can facilitate both integration of interdisciplinary knowledge and explicitly consider uncertainty (whether it arises from stochastic events or imprecise 8 Assuming increased timber productivity due to intensification is offset by decreased overall biological productivity due to soil erosion and rutting, and elimination of non-timber biomass. human and nature s behaviours) in forest management decision-making. The fuzzy (set) logic and theory has several attractive features: (1) flexibility experiences of different individuals (e.g., members of an interdisciplinary team of experts such as foresters, socio-economists and ecologists and various forest stakeholders such as representatives of environmental groups, forest industries, and forest communities) can be translated into a program for decision making purposes; (2) it can be easily translated into computerized programs such as Geographic Information Systems (GIS) mapping (e.g., Nadeau et al. 2003) (GIS is increasingly used as a decision-making tool in forest management); (3) it has computational and intuitive ease (Kangas and Kangas 2004); (4) it can easily be used in conjunction with possibility theory (Zadeh 1978) and evidence theory (Dempster 1967a, b; Schafer 1976) and classical probability theory; this is particularly important in dealing with subjective beliefs and cognitions, and concepts with fuzzy meanings, which cannot be dealt with by classical probability theory alone (Kangas and Kangas 2004). Another important feature of this logic is that it can be utilized in a wide range of other analytical tools such as multicriteria decision analysis and fuzzy mathematical programming, which have a great potential to solve complex SFM problems involving interdisciplinary research and multistakeholder values. Examples of works in these fields include: timber harvest scheduling in a fuzzy decision environment (Bare and Mendoza 1992), fuzzy goal programming in forestry (Pickens and Hof 1991), a multi-objective evaluation model for sustainable forest management using criteria and indicators (Maness and Farrell 2004) and fuzzy methods for assessing criteria and indicators of sustainable forest management (Mendoza and Prabhu 2004). Conclusions The proposed logic of inexactness and possibility schema provide a promising framework for the conceptualization of imprecise spatial concepts in SFM, and seem to be appropriate for analyzing imprecision of spatial behaviours of human and ecosystems and associated value systems; hence, they provide effective tools for analysis of forest policy, institutions and management planning in situations of inexact information and value-based standards embedded in interdisciplinary expert knowledge and inexact value cognition and judgment of different forest stakeholders. The schema can therefore be used in ex-ante appraisal and ex-post evaluation of forest programs. The schema may also be a foundation for future work in the integration of the existing ecological and socio-economic models of SFM, including the C&I approach and the formulation of new models of uncertainty in spatial analysis and planning in forest management. Acknowledgement I would like to acknowledge the anonymous reviewers of this paper for their highly constructive suggestions. References Bare, B.B. and G.A. Mendoza Timber harvest scheduling in a fuzzy decision environment. Canadian Journal of Forest Research 22(4): Bellman, R. E. and L. A. Zadeh Decision-making in a fuzzy environment. Management Sciences 17: B141 B164. MAY/JUNE 2005, VOL. 81, No. 3 THE FORESTRY CHRONICLE 379

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