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Uncertainty in land allocation decisions for Vancouver Island : an application of fuzzy multiple objective… Ells, Arlene Kathryn 1995

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UNCERTAINTY IN LAND ALLOCATION DECISIONS FOR VANCOUVER ISLAND: AN APPLICATION OF FUZZY MULTIPLE OBJECTIVE PROGRAMMING by ARLENE KATHRYN ELLS B.Sc. (Agr.), Nova Scotia Agricultural College, 1993 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Agricultural Economics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1995 ©Arlene Kathryn Ells, 1995 In presenting this thesis In partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree1 that permission for extensive copying of this thesis "for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date <f^lu 14 DE-6 (2/88) ABSTRACT An increasing amount of weight is placed on non-timber values when management or allocation plans for woodlands are considered. In addition, the public is becoming more involved in the planning process. Both trends are evident on Vancouver Island, particularly in the recently completed work of the Commission on Resources and Environment (CORE), a multiple interest group process. A great deal of uncertainty is associated with decision-making situations of this type, due both to the complexity of the problem and the necessity to reconcile conflicting and poorly defined objectives. This study explored the use of fuzzy set theory as a tool for incorporating uncertainty into models used to inform decision-making processes. Two types of uncertainty were considered; vagueness in the identification and ranking of objectives and imprecision in the specification of problem parameters. Fuzzy set theory was used to construct a fuzzy multiple objective linear program (FMOLP) to capture the vagueness of objectives for the land-use allocation problem on Vancouver Island. This model was then extended through the use of possibility theory to a fuzzy possibilistic multiple objective linear program (FPMOLP), allowing consideration of the problem with imprecise parameters. Results were compared to those from a more traditional goal programming approach. Results from the FMOLP provided a better solution than did the goal programming approach when measured by the criteria of the degree of satisfaction of the problem's objectives. The FMOLP provided a higher degree of satisfaction to the majority of objectives considered, and yielded an allocation scheme where more of the land base was in multiple use and protected areas than did the goal program. Results from the FPMOLP were interpreted as a sensitivity analysis of the model with respect to uncertainty of parameter specification. It was demonstrated that the solution was sensitive to the level of parameter uncertainty considere. The methodology employed allowed only for general insights into this problem, proving a poor tool for this type of analysis. i i i TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF FIGURES v LIST OF TABLES vi CHAPTER 1 INTRODUCTION 1 1.1 Background Information 1 1.2 Research Objectives 4 1.3 Thesis Outline 4 CHAPTER 2 FUZZY SET THEORY AND DECISION MODELLING 5 2.1 Uncertainty 5 2.2 Fuzzy Set Theory 8 2.3 Fuzzy Multiple Objective Decision Making 14 CHAPTER 3 A DECISION MODEL FOR LAND USE ON VANCOUVER ISLAND 22 3.1 Net Social Well Being or Economic Efficiency 25 3.2 Forest Sector Employment 32 3.3 Government Revenue 34 3.4 Expansion of Wilderness Protection 35 3.5 Long-Run Sustained Yield 35 3.6 Objective Target Values 36 CHAPTER 4 EMPIRICAL RESULTS 39 4.1 Fuzzy MODM 39 4.2 Fuzzy Possibilistic MODM 46 CHAPTER 5 SUMMARY AND CONCLUSIONS 52 5.1 Methodology 52 5.2 Summary of Results 53 5.3 Restrictions and Further Research 54 REFERENCES 58 iv LIST OF FIGURES Figure 1: Two Forms of Fuzzy Sets 11 Figure 2: Illustration of Fuzzy Decision 16 Figure 3 : Possibility Distribution for (m^ Py) 19 V LIST OF TABLES Table 1: Specification of Fuzzy Objectives 37 Table 2: Simulation Results for Fuzzy MODM: Land Allocation 41 Table 3: Simulation Results for Fuzzy MODM: Monetary Benefits 42 Table 4: Simulation Results for Fuzzy MODM: Non-Monetary Benefits 43 Table 5: Simulation Results for Fuzzy MODM: Degree of Objective Satisfaction 44 Table 6: Simulation Results for Fuzzy Possibilistic MODM: Land Allocation 48 Table 7: Simulation Results for Fuzzy Possibilistic MODM: Non-Monetary Benefits 49 Table 8: Simulation Results for Fuzzy Possibilistic MODM: Monetary Benefits 49 Table 9: Simulation Results for Fuzzy Possibilistic MODM: Degree of Objective Satisfaction 49 v i CHAPTER 1 INTRODUCTION 1.1 Background Information The Government of British Columbia introduced a number of policy initiatives in the past several years that affect the how the province's land base is utilized. These include: (1) the 1991 Protected Areas Strategy (PAS), a strategy aimed at increasing the proportion of the province's preserved wilderness area from about 6.5% of the land base to 12%; (2) the 1992 Timber Supply Review, a re-examination of the forest inventory and determination of sustainable regional harvests to prevent a potential downfall in future timber supply; (3) the 1994 Forest Practices Code, legislation to protect environmental and ecological properties of the working forest; (4) the 1994 Forest Renewal Plan, a plan to capture a greater share of potential forest rents, which are then used to employ workers displaced by planned harvest reductions in new silviculture initiatives; (5) a Forest Land Reserve (created in 1994) that prevents private forest land owners from putting forest land into competing uses (including agriculture); and (6) the 1992 Commission on Resources and Environment (CORE) process, charged with finding a consensus on land-use issues and providing recommendations as to land use allocation. Implementation of PAS targets has subsequently defaulted to the CORE process. Each of these policies affects the size of the available commercial harvest. Each of these policies offers constraints to the way that forest industries may conduct their business in the forests. This points out the growing recognition that the forests provide more to the people of B.C. than simply an industrial input. While the value of the forest as a source of timber is not in doubt, it is no longer the preeminent resource value. This 1 is forcibly brought home in the Vancouver Island CORE process, where the mandate includes a consensus decision on what the Island's woodlands are to provide. The focus of this study is on land-use decisions for forested land on Vancouver Island. The CORE process consists of seeking consensus among different interest groups, each with its own objectives. The principal means for presenting alternative scenarios to the participants in the CORE process has been multiple accounts analysis. The multiple accounts framework is used to overcome the difficulty of including all items in a social cost-benefit (economic efficiency) analysis. Some items, such as regional employment, are simply incommensurable with measures of social welfare, while other items are difficult to measure, such as preservation and biodiversity values. In its standard form, multiple accounts analysis requires that economic efficiency, or net social well being (measured in monetary terms), be the account against which all other accounts are to be compared (van Kooten 1995a; U.S. Water Resources Council 1983). A decision to protect wilderness or logging jobs needs to be judged in terms of its impact on economic efficiency. However, this does not appear to be the philosophy of the CORE process. Rather, the approach employed by CORE treats all accounts as equal, and does not require any comparison against the efficiency standard. Therefore, traditional cost-benefit analysis does not provide information in the form required by those involved in the decision making process. In this case, multiple objective decision making (MODM) is the more appropriate tool of analysis. Other characteristics of the decision regarding land use also indicate that the traditional cost-benefit analysis is not an appropriate tool in this case. • The objectives of society are ill-defined and so one is unable to develop a 2 consensus objective function. • The values that society attaches to various forest activities (such as recreation or preservation of biodiversity) are imprecise at best, or simply unknown. • Land-use and/or silvicultural decisions pertain to an uncertain and distant future. It is not clear that the decisions made today will yield the anticipated results—trees planted today may be unsuited for tomorrow's climate, while land set aside as wilderness into perpetuity may be logged as a result of changing values and a need to produce more wood products. • The effects of silviculture and other forest management decisions are uncertain, both from a biological and socioeconomic perspective. »• There is uncertainty about forest tenures, the macro economy, future product prices, and the ability of, or need for, governments to reduce deficits/debts. These characteristics all embody some type of uncertainty regarding the land-use problem, a state of affairs that cost-benefit analysis is not well-equipped to deal with. These uncertainties arise from a lack of good and complete information regarding both current and future events, as well as from our inability to clearly define and relate relevant objectives. A tool for dealing with these types of uncertainty is fuzzy set theory. This study demonstrates the use of fuzzy set theory in the context of a multiple objective decision-making model for land-use on Vancouver Island. Vancouver Island was selected because it is a region where land-use conflicts are intense and the recent C O R E (1994) land-use recommendations (subsequently adopted by government) have been controversial. 3 1.2 Research Objectives The purpose of the current research is to capture the uncertainty inherent in describing the socioeconomic impacts of land-use and forest management decisions on Vancouver Island. The major objectives of this study are: 1) to develop a fuzzy multiple objective decision making model (FMODM) that incorporates uncertainties both in objective specification and parameter values; 2) to contrast the FMODM approach to a traditional multiple objective approach where uncertainty is not considered; and 3) to compare the results obtained from the two models in application to the land-use problem on Vancouver Island. 1.3 Thesis Outline In Chapter 2, an overview of uncertainty as it relates to decision-making is provided, followed by a description of fuzzy set and possibility theory. A theoretical model for the consideration of uncertainty in decision support models is defined. The derivation of the empirical models and required parameters are presented Chapter 3. Results of the empirical models are discussed in Chapter 4. Finally, in Chapter 5 conclusions, limitations, and suggestions for further research are provided. 4 CHAPTER 2 FUZZY SET THEORY AND DECISION MODELLING Two types of information are required in the decision making process. The first defines the set of feasible solutions available to the decision maker while the second type is the criteria for identifying the preferred or "best" solution. As described in Chapter 1, uncertainty exists with respect to both types when making decisions regarding land-use allocation on Vancouver Island. Uncertainty may exist in several forms, and fuzzy set theory is particularly useful in allowing formal consideration of vagueness and ignorance. 2.1 Uncertainty Current literature concerned with modelling uncertainty provides a wide range of definitions both for the concept of uncertainty itself as well as the various types of uncertainty that may be addressed. The extent of the divergence is demonstrated by the fact that some typologies define uncertainty as a subset of ignorance while others consider ignorance as a subset of uncertainty (Krause and Clark 1993). In what follows, it is the latter conceptualization that is employed. According to Kruse etal. (1991), uncertainty has to do with the degree of belief or faith in the validity of a particular proposition or datum. Uncertainty arises from many sources including error, lack of judgement, imprecision, unreliability, variability, vagueness, ignorance and ambiguity. The theory of fuzzy sets is most widely used in dealing with vagueness and ignorance, although one often sees fuzzy sets recommended as a means for dealing with ambiguity. 5 Ambiguity may be defined as the property of possessing several distinct but plausible and reasonable interpretations (Cox 1994). The term "hot" is considered ambiguous until the context for its use has been defined, be it temperature, spiciness or trendiness to name but a few. The failure to define clearly the context associated with a particular event or concept is not a source of uncertainty amenable to formalization through the use of fuzzy set concepts. The recommendation of fuzzy set theory for use in this situation arises from confusion regarding the terms ambiguity and vagueness. Vagueness has perhaps the widest range of interpretations advanced for any of the various types of uncertainty; in fact, completely contradictory definitions can be found. However, there is a consensus within the literature on fuzzy sets. Vagueness is said to occur when an object is completely known but its classification is in doubt because the concept is poorly defined (Barret and'Pattanik 1989). Alternatively, vagueness refers to the lack of clear-cut boundaries for the set of objects to which the symbol or meaning is applied (Fedrizzi 1987). Consider classifying a person as "young", "middle-aged", or "old". You may know their age exactly, but at what age is one to be described as old versus middle-aged. These age categories are poorly defined, they are vague. The concept of vagueness has direct application to decision making in identification of the "optimal" or "best" solution. The characterization of the best solution may be incomplete or unknowable, and thus the relation of any one solution to that ideal is unclear. The solution is known; the description of the best solution is not. This approach is clearly distinct from that of probability theory. Probability deals with the quantification of an uncertain event while fuzzy set theory deals with the quantification of the uncertainty of the description of the event. Put another way, probability deals with 6 unknown elements of fixed and well known sets while fuzzy set theory deals with fixed and known elements of ill-defined sets (Dubois and Prade 1993). Fuzzy set theory also has application in incorporating uncertainty due to ignorance or incomplete information, a condition often encountered in the process of identifying decision alternatives. Ignorance is simply a lack of knowledge regarding the event or object under consideration. In the absence of complete knowledge, the degree of certainty in the value or definition of the object is less than absolute. A related type of uncertainty is found in expert opinions. This form of input to a decision making process contains the uncertainty of the subjectivity of the expert providing the information. Both of these types can be formalized using the concept of a possibility distribution. Fuzzy set theory provides the mathematical methods for describing possibility distributions, although possibility theory itself predates the theory of fuzzy sets. As is the case for probability theory, several interpretations of possibility theory exist. One approach is to consider possibility as a measure of partial membership in a series of crisp subsets, the degree of membership reflecting the possibility that the element belongs in a particular subset (Klir and Folger 1988). Continuing with the example of classifying a person on the basis of age, consider classifying them as being in their twenties, thirties, forties or fifties. This is a series of crisp sets. The highest degree of belief or possibility is assigned to the set representing our best guess for the age of that person. The second highest belief or possibility value is assigned to the set representing our second best guess. As more information regarding the individual is available, more of our belief becomes centred on one of the sets and the value it receives is increased. This process continues until one set is accorded full belief value, we know for certain their age lies 7 within one of the sets. The mechanism for describing this belief assigned across crisp sets is referred to as a fuzzy measure (Klir and Folger 1988). A fuzzy measure is conceptually different from a fuzzy set where the problem is to determine degree of membership in a particular set. A fuzzy set is used to describe the vague concepts with age such as "old" or "young", and to what degree a given individual may belong to one of these fuzzy sets. The use of a fuzzy measure to describe possibility distributions is also conceptually different from probability theory where the focus is on assigning belief value to each of individual elements contained within a single, well-defined set. Another view of possibility is that, although a degree of belief may be assigned to a set as a whole, the assignment of that degree of belief among the individual elements of the set requires more knowledge than is present. In this situation, a possibility measure describes the ordering of the elements within the set in terms of their relative likelihood of occurrence, a preference relation on the possible elements of the set (Dubois and Prade 1993). To say a person is about 30 years old is to implicitly define a set of ages that contains the number 30. Possible ages that would fit the definition of about 30 may lie in the range of 27 to 33. We consider that they are most likely to be 30, less likely to be either 29 or 31, less likely again to be 28 or 32, etc. We can order the ages based on relative possibility, but we cannot assign a cardinal ranking to the possibilities. 2.2 Fuzzy Set Theory Fuzzy set theory and the fuzzy logic that it supports was initially proposed by L. A. Zadeh in 1965. His concern was with the ambiguity and vagueness of natural 8 language, and the attendant inability to convey precise information via linguistic means. Since the early 1980s, interest in fuzzy set theory and associated reasoning systems has grown rapidly as evidenced by the growth in related literature. A major research focus is in the field of artificial intelligence but applications for the social sciences are also being advanced. Decision support modelling is one such area of application as is the process of making decisions itself. Fuzzy sets and membership functions An element x of X is assigned to an ordinary (crisp) set A via the characteristic function uA, such that: uA(x) =1 i fxeA. (1) uA(x) = 0 otherwise. The element has either full membership (uA(x) = 1) or no membership (uA(x) = 0) in the mm set A. The valuation set for the function is the pair of points {0,1}. A fuzzy set A is also described by a characteristic function, the difference being that the function now maps over the closed interval [0,1]. Formally, a fuzzy set A of the universal set X is defined by its membership function M A : X ^ [ 0 , 1 ] , (2) which assigns to each element xeX a real number uA(x) in the interval [0,1], where the mm value of at x represents the grade or degree of membership of x in A (Sakawa, 1993). As an example, consider the set of young people, Y. It is clear that children belong to this set, they have a degree of membership equal to 1. As we consider progressively older people, the description as young becomes less apt. Is a twenty year old young? 9 A twenty-five year old? To capture the uncertainty surrounding their membership in the set of young people, they are assigned a partial degree of membership, something less than one. Membership in the fuzzy set Y may be described as some function of age: As age increases, the degree to which the individual is considered a member of the set of young people decreases. This is an example of a one-sided fuzzy set. Membership in this set approaches zero as age increases, but the membership level never reaches zero. Other functional forms defien set membership over a finite interval, everything lying outside the defined interval is accorded a membership value of 0. It is this latter type of fuzzy set that is used to describe the uncertainty surrounding satisfactory or acceptable objective values. As an example of another form of fuzzy set, return to the discussion of "middle-aged" and "old". As a person ages, they pass from being young through middle-aged to old. Thus, membership of a person in the set of middle-aged people, M, may be described as a function of their age. where a,b,c e A, the set of all ages, and a < b <c. The membership functions used to describe age are linear in this example, and the fuzzy set takes the shape of a triangle. This is an example of a two-sided fuzzy set. If (b - a) = (c - b), then the fuzzy set is said to be symmetric, t h e one- and the two-sided form of a fuzzy set are illustrated in Figure 1. It should be noted that a fuzzy set is described by its membership function and a uY(age) = (1+(0.04*age)2)-1 uM(age) = 0 MM(age) = (age - a)/(b - a) MM(age) = 1 Mw(age) = 1 - (age - b)/(c - b) uM(age) = 0 age > a a > age > b age = b b > age > c age > c. 10 membership function describes a fuzzy set. In this regard the two concepts are interchangeable and discussion of fuzzy sets is often simplified by omitting the explicit reference to the membership function. Figure 1 : Two Forms of Fuzzy Sets H(age) ftr(age) 1 \ 1 0 XJ a b c age Preceding definitions have employed the concept of normalized fuzzy sets. A fuzzy set A, defined over a finite interval, is said to be normal if there exists a n x e X such that uA(x) = 1. It is not necessary that there be a defining element for the fuzzy set, one that would naturally be assigned the membership value of 1. A subnormal fuzzy set is normalized by dividing uA(x) by its height (greatest membership value). Set theoretic operations are defined for fuzzy sets. Among these are the concepts of containment, complement, intersection and union. A fuzzy set A is contained in the mm mm fuzzy set B (A is a subset of B), if and only if the membership function of A is less than or equal to that of B everywhere on X: 11 A c B ~uA(x) <. uB(x) for all x e X. (3) The complement of A (written as A) is defined as: M * M = 1 - M A M (4) The intersection of the fuzzy sets A and B is defined as: A n B - u ( A n S ) = min{uA(x), uB(x)} for all x e X, (5) and the union as: A u B «=• U(AuB) = max{uA(x), uB(x)} for all x e X. (6) Hence, the intersection An B is the largest fuzzy set contained in both A and B, and the mm mm mm mm union A u B is the smallest fuzzy set containing both A and B. These three definitions of set operations are those originally advanced by Zadeh (see Klir and Folger 1988). Subsequent work has seen the advancement of several different definitions for these operations but all include (4)-(6) as special cases. This set of operations provides the basis for consideration of possibility through fuzzy set theory. Another concept required for model building with fuzzy sets is that of the alpha-mm level set. The a-level set A,, is simply that subset of A for which the degree of membership exceeds the level a, and is itself a crisp set (an element either meets the required level of a or it does not). A a is an upper level set of A. The use of alpha level sets provides a means of Aa = {x| uA(x) * a}, a e[0,1]. (7) 12 transferring information from a fuzzy set into a crisp form. Defining an a-level set is referred to as taking an a-cut, cutting off that portion of the fuzzy set whose members do not have the required membership or possibility value. Alpha level sets are also used in defining a convex fuzzy set. A fuzzy set A in X is said to be convex if and only if its a-level sets are convex. Alternatively, A is a convex fuzzy set if and only if MA ^ + (1-A) x2) * min (MxJ.vM) V e X and A e [0,1]. (8) Fuzzy numbers and possibility functions A fuzzy set that describes the situation where a value is "approximately m" or "about n" is said to be a fuzzy number. A fuzzy number is a convex normalized fuzzy set of the real number line. In appearance it resembles the fuzzy set described above for middle age. A standard form of fuzzy number that allows for computational efficiency is that of the L-R fuzzy number. A fuzzy L-R number M is fully characterized by three parameters where m is the mean value of M and a and 3 are the left and right spreads, respectively. It is defined as _(—) x * m a>0 /?(—) x * m p > o P and written as (m,a,3)LR. Fuzzy numbers of the L-R type can be subjected to a variety of manipulations. Given the fuzzy numbers M=(m,a,P)LR and N=(n,Y,6)LRl the basic L-R fuzzy operators, modified for symmetric possibility functions (where a=p and Y=6), are as follows (Sakawa 1993, pp.26-30): 13 Addition: (m,a)Syme (n,Y)Sym = (m+n, a+Y)Sym (10) Subtraction: (m,a)Syme (n,Y)sym = (m-n, a+Y)Sym (11) Multiplication: (m,a) S y m® (n,Y)Sym = (mn, na+mY)Sym, iff m,n > 0 (12) Scalar multiplication: k ® (m,a)Sym = (km,ka)Sym (13) The concept of a fuzzy number can be used to model possibility distributions. When uncertainty surrounds the true value of a parameter or coefficient, there exists a range of values or crisp subsets of the real numberline that may contain the true value. Within that range, values may be identified as more or less possible. A fuzzy number can be used to describe a continuous possibility distributions for this imprecise parameter. Given a fuzzy number M=(m,a)Sym, the possibility (n) that M takes on a particular value, a, is given by the membership function uM(a); that is, n[M=a] = uM(a). For computational efficiency, possibility distributions are commonly normalized so that the most possible value is defined as having a possibility of 1. A fuzzy model with vague preferences and imprecise coefficients can be written as a crisp LP, as illustrated in the next section. The focus will be on a decision maker seeking to maximize satisfaction of a number of different vaguely defined objectives, subject to imprecise biophysical and socioeconomic constraints. 2.3 Fuzzy Multiple Objective Decision Making One approach of MODM is to establish a specific numeric goal for each of several objectives, formulate an objective function for each, and then seek a solution, through linear programming methods, that minimizes the deviations of the objectives from their 14 respective goals. Priorities are assigned to the objectives, and the model is solved in an iterative fashion such that the highest priority objective(s) are optimized first. Solution values obtained for the first ranked objectives become constraints in solving the model for the second ranked objective(s), and so on. This process is referred to as preemptive priority goal programming (see Lee, Moore and Taylor 1985). Unlike this classical form of goal programming, fuzzy programming allows simultaneous consideration of all objectives and constraints without a requirement for a ranking or weighting system. It is the philosophy of how the decision alternatives are to be measured and ranked that distinguishes the FMOLP from the more traditional approach. As Ignizio (1983) points out, the measurement of the "goodness" of a solution is entirely a matter of philosophy. Fuzzy multiple objective linear programming In the FMOLP model, we are concerned with the uncertainty surrounding the definition of satisfactory solution values for each of the objective functions. Although a precise value for each objective is provided by the model, it is unclear as to how well that value represents the concept of a fully satisfied objective. The term satisfactory solution is vaguely defined; it is a fuzzy set. Thus, a goal or constraint, G(x) or C(x) respectively, may be completely satisfied by choice of the solution vector x (uG(x) = 1 or uc(x) = 1), completely unsatisfied (MG(X) = 0 or uc(x) = 0), or experience some degree of satisfaction (0 < uG(x),uc(x) k 1)- Crisp goals and constraints are accommodated in this framework by defining a crisp set as a specialized case of a fuzzy set. The decision space for the model, uD > is the fuzzy set defined by the intersection of the fuzzy goal and the fuzzy constraint, and is characterized as 15 uD(x) = min (uG(x),uc(x)). The decison space defined is illustrated in Figure 2. It follows, then, that in order to maximize the degree of satisfaction of the goals and constraints, that the objective function for the fuzzy linear programming model is maximize uD(x) = maximize min (uG(x),uc(x)). xeX xeX Figure 2: Illustration of Fuzzy Decision This maxmin operator is but one of several ways to represent the decision. It has the advantage that it is simple and linear, but it may fail to capture the true decision making process. There is an implicit assumption in the use of maxmin that all goals and constraints are weighted equally. This operator also fails to consider tradeoffs available between the various goals and constraints—it is non-compensatory. The solution is obtained where the minimum membership value has been maximized, or the lowest level 16 of satisfaction has been raised as high as possible. Given its limitations, in the absence of good evidence to argue for another decision operator, the maxmin approach is used in what follows. The decision making model can now be written as a crisp linear programming (LP) model. Suppose that the original FMODM is as follows: find x s.t. AjX^bj i = 1,2,...,m (14) x * 0, where m is the number of goals and constraints, Aj refers to the matrix of crisp parameter mm values, bj refers to the vector of constraint values, and * refers to the presence of fuzzy objective or constraint sets. Then the crisp representation of the fuzzy MODM (14) can be written as: max A s.t. u,(x)-A*0, i = 1,2,...,m (15) A e [0, 1] ,and x * 0. where A =uD(x). The use of fuzzy objectives will be illustrated with an example. An objective of the land-use decision model to be developed below will be to preserve wilderness by setting land aside as protected areas. The question is: how much land should be protected? According to the Protected Areas Strategy (PAS), 12% of the land base of B.C. should be protected. Since "undershooting" of this goal will be politically sensitive, it can be argued that 12% serves to define the lower limit of acceptable objective values—any solution that provides a lower percentage of the land base as wilderness will be 17 unacceptable and have a membership value of 0. On the other hand, there are many people who would argue that more land should be set aside. Claims up to 28% have been put forward. If we adopt 28% as a perfectly satisfactory level of forest protection, then the membership function describing the fuzzy set of acceptable or satisfactory solutions is as follows: Uj(x) = 1, if Ape* 28% Mi(x) = 1 + (AjX-12)/(28-12) if 12% < Ax < 28% (16) Mi(x) = 0, if Ape ^  12%. If the solution to the optimization problem allocates 20% of the land base to protected areas, Uj(x) will equal 0.50. Note the assumption of a linear form for the membership function. It is not necessary that membership functions be defined solely in terms of linear functions in order to allow inclusion in the LP form. Other functional forms (piecewise linear, exponential, hyperbolic) can be transformed into linear forms for use in this type of model. Fuzzy possibilistic programming Assuming linear membership functions, the fuzzy model of the preceding section may be re-written as Max A s.t. Ax-bi -Pj( A -1) * 0, (17) X e [0,1], and x ;> 0. where p, refers to the interval over which membership in the set of satisfactiry solutions lies between zero and one (the term (28-12) in (16)). Now consider the situation where mm the elements of the matrix A are not precisely known. The j-th element of A,, ai}, is described by a possibility distribution. Furthermore, assume that these possibility 18 distributions are triangular and symmetric, allowing iy to be written as the fuzzy number (mu. 3y)sym w i t n t n e possibility distribution: rr(ai n(a, n ( a i n(a. = 0 = 1 + (a, - m,)/p, = 1 = 1 - (my - a,)/p„ = 0 ; m, - Pij < a , < my ; ay = my ; my < ay < my + p, ; m, + Py <, ay. This possibility function is depicted in Figure 3. Figure 3 : Possibility Distribution for (miJy Pu) it(a) 1 A it fa) 0 1 ^ mn + pff a To capture the effect of uncertainty in the model parameters, we employ the concept of the alpha cut. This allows the definition of a crisp parameter value derived from the characteristics of the underlying possibility distribution, and permits the use of a standard LP format. The imprecise nature of the technical coefficients is incorporated into model (17) to give the following structure: 19 Max A s.t. [V (1-a )P l ]x-b, -p l (A-1)*0 > (18) A e [0,1], and x * 0 (Lai and Hwang 1994). Model (18) allows for each element in the parameter matrix to be adjusted to mm reflect the level of possibility being considered. Each element, a^ , is transformed to take the value where n(ag) = a, the possibility of the value a^  is alpha. When a = 1, this fuzzy possibilistic MODM (FPMODM) formulation is identical to the fuzzy MODM discussed previously; only the most possible values are considered (a=1). As the level of possibility decreases, parameter values are moved away from the centre value, to values lying below on the defined interval. The solution is now derived using parameter values considered as less possible. This case reflects the situation where the parameter considered most possible are greater than the true parameter values. The model in (18) sets all imprecise parameters to the same level of possibility, one distinct point in the range of possible solutions. An alternate form of this FPMODM model is: Max A s.t. [A + O-aJPiJx-bj-pjfA-l) * 0, — (19) A e [0,1], and x ;> 0. Model (19) uses less possible and higher values for the parameters. This represents the situation where the most possible values lie below the true values. Considered jointly, these two models provide an upper and lower bound for possible solutions by considering the two extreme points of the possibility distributions defined by fifo,) = a. As more of the uncertainty regarding the information in the model 20 is incorporated (the possibility level is decreased), the length of the interval separating these bounds increases. This interval identifies the range of possible solutions. Models (18) and (19) are restrictive in that they consider only the endpoints of this interval, the extreme cases. By comparing the change to the solution with respect to a change in model possibility level, one can perform a type of sensitivity analysis with respect to uncertainty in the definition of parameters. This maxmin approach will have a solution in which it will be impossible to increase the membership value of one objective without reducing the membership value of another. Hence, the resulting solution is Pareto efficient. As illustrated in this chapter, a fuzzy model with vague preferences and imprecise coefficients can be written as a crisp LP, solvable using standard algorithms. Next, this approach is applied to land-use allocation on Vancouver Island. 21 CHAPTER 3 A DECISION MODEL FOR LAND USE ON VANCOUVER ISLAND Vancouver Island consists of nearly 3.35 million hectares (ha), of which 2.4 million ha or 72.5% is publicly owned and 10.2% is officially designated as wilderness. The Vancouver Island CORE report (CORE 1994) recommends that this land base be allocated among the following five land-use categories: (1) protected area , (2) regionally-significant land, (3) multi-resource use area, (4) cultivation use areas , and (5) settlements. While CORE specifies how land may be used in each of these designations, what will be allowed in practice is only vaguely known. During deliberations, the Vancouver Island CORE employed the categories "high-intensity resource use", "integrated resource use", "low-intensity resource use", "protected areas" and "settlement" (van Kooten 1995b). It is not clear how these categories relate to the previous ones. In addition, subsequent CORE reports for other regions of the province have made use of this latter set of categories in defining land use, and prescriptions for appropriate levels of resource use and development have been advanced. Therefore, the latter categories are used in the current analysis. The Vancouver Island CORE identifies 2,210,255 ha of publicly owned land on the Island, and classifies it according to timber production potential. It is this area under direct government control that is the focus of this analysis. Hence, "settlement" lands and other private lands are ignored. As discussed earlier, objectives for land use decisions are vague. The 1989 Parksville Old-Growth Workshop identified a number of goals that are considered to reflect the general public's expectations regarding forestland use in the province (B.C. 22 Ministry of Forests 1990). Eight goals can be identified; in no particular order, they are as follows: 1. achieve the highest possible revenue from timber harvest; 2. maximize the benefits citizens obtain from forest recreation activities; 3. obtain the greatest nonuse benefits from forests, as measured in monetary terms; 4. maintain forest employment; 5. be sure that harvests do not exceed long-run, sustained yield (LRSY); 6. collect substantial direct revenues from the forest industry since forest lands are publicly owned; 7. seek to maximize the contribution of the forest sector to provincial GDP (in absolute as opposed to relative terms), because higher levels of GDP lead to greater government revenue; and 8. expand wilderness protection. Vague terminology renders each of these objectives fuzzy. It follows, then, that the values for each of the goals cannot be precisely known. This vagueness regarding desired objective values can be modeled through the specification of fuzzy objectives. Use of the maxmin operator allows the problem to be solved without ranking the objectives, allowing for uncertainty in this area as well. Uncertainty is present in the lack of knowledge regarding parameter values for use in modelling this land-use decision. Land-use decisions require policy makers to take account of long-term conditions and effects about which there is little categorical information. Defining the parameters as imprecise coefficients allows explicit 23 consideration of this form of uncertainty. For convenience, the planning period for this model is set at 100 years and will consider one entire rotation of the working forest. While this rotation length is somewhat greater than required based on mean annual increment (MAI) considerations, given the increasing weight of non-timber considerations influencing harvest age, it is not unreasonably long. The model is also based purely on consideration of second-growth forests. This is patently not the case on Vancouver Island where a significant amount of high-value old growth remains. However, the model as formulated is static, yielding a once-and-for-all land allocation. This does not allow consideration of the transition from partial reliance on old growth to complete reliance on second and subsequent rotations for the generation of timber-related values as well as a substantial portion of non-timber values. This complete reliance will be the case on Vancouver Island once the transition is completed so this model portrays the future of the Island industry. Three types of MODM models for land-use decisions on Vancouver Island are compared to illustrate the usefulness of fuzzy MODM. The first is traditional, deterministic, multiple goal programming. The second is fuzzy multiple objective decision making (FMODM), which incorporates the fuzziness in identifying target values, as explained in Chapter 2 above. Finally, possibilistic fuzzy multiple objective decision making (FPMODM), with both fuzzy constraints and imprecise parameters, is considered. The first step is specification of the parameters, keeping in mind the objectives of the problem. 24 •r 3.1 Net Social Well Being or Economic Efficiency The first three goals identified above are concerned with social well being, as measured by consumer and producer surpluses. Summing the values in these three categories provides a measure of net social well being or economic efficiency in the terminology of cost-benefit analysis. A question arises as to how these three goals are best modeled—separately or as a combined total. Observation of the CORE process leads one to conclude that decision makers indeed view these as separate and distinct objectives in the decision-making process. Also, while the third and eighth goals appear to be the same, this is not the case. Forest lands could be kept out of timber production even though they are not officially designated for protection. Logging benefits Logging benefits per hectare are calculated as the difference between price and cost per cubic meter (m3) of delivered wood. Two attributes of the site dictate harvested wood volume, namely, site quality and the harvest intensity of the managed site. Site quality is characterized as good, medium or poor based on the commercial volume of timber currently on the site, the species on the site, and the site's ability to grow a future stand of trees. On Vancouver Island, the Crown land consists of 223,842 hectares (ha) of very high productivity area (referred to as good sites in this model), 891,016 ha of high to moderate productivity (medium sites), 771,288 ha of low to poor productivity (poor sites), and 324,359 ha of non-productive or unclassified land (CORE 1994). There is substantial variability among sites, even in the same category, in terms of their ability to provide logging benefits. The average harvest volume per ha given the age of a stand, by species and site class, for the B.C. coastal region, is as specified in 25 the FOREST6.0 model (Phelps et al. 1990a, 1990b). The variability within site classes is taken into account through considering the age of harvest. The harvest age for the model will be 100 years, but the volume harvested will vary from that available at 90 years of age to that available at 110 years. The volume available for harvest will also vary by the management intensity of the forest site. The category "high intensity management" is taken to mean that forestry is a priority and that intensive levels of silviculture will be practiced. Under "integrated management" it is assumed that basic silviculture will be performed and some allowance for non-timber values will be made. Land designated for "low intensity" management receives no silvicultural investment but is available to provide harvest volume from the naturally regenerated stock. Of course, "protected" area will see no timber harvest. Taking the influence of both site quality and management intensity into consideration, nine variables are required to fully describe the possible combinations of site class and management intensity. Intervals for the fuzzy variables representing harvested wood volume can be defined by taking the maximum and minimum volumes available for each site class under each management regime at both 90 and 110 years of age. All species are accorded equal weight so the lower bound becomes the lowest yield figure across age class, management and species, while the upper bound is the highest such value. Therefore, on good sites under high intensity, harvestable volume varies from 870 m3/ha to 1510 m3/ha; similarly, harvestable volumes vary from 859 to 1404 m3/ha on the same sites with integrated management. Assuming a symmetrical triangular distribution of the form (my, p(j) for these fuzzy variables, the centre value, my, is the arithmetic mean of the upper and lower values, and the possibility distributions 26 describing the fuzzy harvest parameters are: High intensity use, good site (1190 m3/ha, 320m3/ha) High intensity use, medium site ( 861 m3/ha, 184 m3/ha ) High intensity use, poor site ( 499 m3/ha, 167 m3/ha) Integrated use, good site (1131.5 m3/ha, 272.5 m3/ha) Integrated use, medium site .( 749.5 m3/ha, 176.5 m3/ha) Integrated use, poor site ( 379.5 m3/ha, 179.0 m3/ha) Low intensity use, good site (868 m3/ha, 192 m3/ha) Low intensity use, medium site (525 m3/ha, 127 m3/ha) Low intensity use, poor site (288.5 m3/ha, 107.5 m3/ha ) The next step is to ascertain the value of the harvest. Wood prices are available from the input information for the FOREST6.0 simulation model in a form similar to that of harvest volume information. Possibility functions for price are calculated in the same manner as for wood volume. These distributions are then scaled upwards to reflect the average wood price implied for the Coast region in 1992, based on aggregate industry income statement information reported by Price Waterhouse (1993). Using the value of sales and transfers of logs, and removing the value of logs purchased for resale, an average price of $70.71 /m3 is derived for logs harvested on the coast of B.C. Symmetric possibility functions for the fuzzy price variables prices are: High intensity use, good site ($81.08 /m3, $16.31 /m3) High intensity use, medium site ($69.48 lm3, $17.75 lm3) High intensity use, poor site ($64.12 lm3, $14.38 lm3) Integrated use, good site ($72.39 lm3, $11.70 lm3) Integrated use, medium site ($69.07 lm3, $12.97 lm3) Integrated use, poor site ($64.58 lm3, $14.79 lm3) Low intensity use, good site ($72.39 lm3, $12.32 lm3) Low intensity use, medium site ($69.02 lm3, $12.66 lm3) Low intensity use, poor site ($63.79 lm3, $14.51 lm3) 27 Calculation of delivered wood costs follows the methodology outlined above, with the exception that costs vary with management intensity but are constant across site qualities. Two cost values are reported for the B.C. coastal region, one for low cost and another for high cost sites. The possibility distributions are based on the mean of the two cost figures and are scaled to reflect an average cost of delivered wood after stumpage fees, rents and royalties of $65.13/m3 (Price Waterhouse 1993). The resulting symmetric possibility distributions for delivered wood costs are as follows: High intensity use ($61.06 /m3, $5.50 /m3) Integrated use ($65.07 /m3, $5.21 /m3) Low intensity use ($65.32 /m3, $4.57 /m3) These costs do not include costs of silviculture. The Ministry of Forests (1992) provides average cost data for silvicultural activity in 1992. Basic silviculture was applied at a cost of $21.20 /ha, while incremental silviculture represented an added expense of $20.00/ha. These costs are added in the appropriate management categories. Net logging benefits are calculated as the difference of total revenue per hectare and total costs per hectare. Using the definitions of fuzzy addition, subtraction and multiplication provided in equations (10) - (12), net logging benefits per hectare per year are: High intensity use, good site ($197 /m3, $324 /m3) High intensity use, medium site ( $31 /m3, $164 /m3) High intensity use, poor site (-$26 /m3, $104 /m3) Integrated use, good site , ( $62 /m3, $211 /m3) Integrated use, medium site ( $9/m3, $143 An3) Integrated use, poor site (-$23 /m3, $75 /m3) Low intensity use, good site ($61 /m3, $160 lm3) Low intensity use, medium site ( $19 /m3, $95 /m3) Low intensity use, poor site ( -$4 /m3, $53 /m3) 28 The negative value associated with harvest of poor sites is consistent with a competitive industry: most of the current harvest is obtained from the better quality sites, and there is no margin to allow harvest of the inferior site class. The spreads associated with each of the logging benefits parameters are large in comparison to the centre and identify the possibility of net negative returns to harvest. Again, this is not inconsistent with current forest industry conditions, where logging operations are considered to be cost centres and operated such that costs of harvest are minimized. This is an implicit recognition that negative benefits are generated at some subset of sites. Recreation values Timber benefits are but one of the values derived from the Island's land base. In terms of social well-being, one should also consider the value provided through recreational access and use, as well as passive-use (or nonuse) values. Ministry of Forests estimates place the recreation plus option value for the Vancouver forest region at $111.11 million per year (B.C. Ministry of Forests 1991). The Vancouver forest region represents a land base of 3.4 million hectares. Dividing total recreational value (use plus its option value) for the Vancouver forest by 3.4 million gives an average recreational value for the area of $33/ha/yr. This value has been obtained with current management regimes and the current level of protected area. Legislated management regime for B.C. is best described by the integrated management regime used in this study. It is probable that the public will not value land under intensive management as highly for recreational purposes as that under integrated management since recreational opportunities will be curtailed by the intensive timber focus of the area. On the other hand, land under the low intensity regime can be expected to offer little in the way of increased recreational 29 opportunities as compared to the integrated management areas, the same activities are being pursued and logging ongoing. Protected areas, with potentially more stringent guidelines as to appropriate recreational activities, will provide less in the way of recreational value than the low and integrated management intensity regimes. In short, integrated and low management areas are assumed to provide the same level of recreational benefits while high intensity and protected areas both provide somewhat less. There is little to indicate what the magnitude of these differences might be. Therefore, it is assumed that there is a 40% reduction in recreational value for land allocated to protected areas and a 50% reduction for land allocated to high intensity resource use. Centres for the possibility distributions are scaled to preserve the gross average of $33/ha/yr, with spreads arbitrarily set at $10/ha/yr for all classes. Symmetric possibility distributions for recreational benefit parameters are as follows: High intensity use ($21.94/ha, $10.00/ha) Integrated use ($43.86/ha, $10.00/ha) Low intensity use ($43.86/ha, $10.00/ha) Protected areas ($26.32/ha, $10.00/ha) Passive-use values The major economic benefits of wilderness protection, biodiversity guidelines, and other similar policies are the passive-use benefits. Void et al (1994) conducted a survey to determine the values that B.C. residents place on wilderness protection in the province. The Province's Protected Areas Strategy (B.C. Ministry of Forests 1992) was the motivating factor in designing the survey. The mean maximum annual willingness to pay (WTP) for a doubling and tripling of wilderness areas were $136 and $168 per household, respectively. In the survey, the authors identified a base level of protected 30 land area in B.C. of 5% of the total. A number of assumptions need to be made in order to derive an average annual value for passive-use for Vancouver Island. First, it is assumed that the number of households on Vancouver Island corresponds to the to the size of the labour force; CORE (1994) reports a labour force of 295,230. For the Island, protection levels of 5%, 10% and 15% of the land base represent areas of 165,000 ha, 330,000 ha, and 495,000 ha, respectively. Prior to CORE recommendations, about 341,000 ha on Vancouver Island (10% of total) were identified as protected areas. Using data from Void et al. (1994) and the aforementioned assumptions, each household on the Island is prepared to pay $32/yr (the difference of $168 and $132) to increase the amount of protected area to 495,000 ha (15% of the total) from the current 10% level. This corresponds to an average annual payment of $26.69/ha of protected area, a level of payment assumed to hold over the entire range of possible protected area allocations. This calculation involves only the resident population of Vancouver Island and does not consider the possibility that others would also be willing to pay to preserve wilderness on the Island. In this respect, the figure of $29.69/ha represents an estimate of a minimum value for passive-use benefits. At the extreme, the state of the forest may be considered a global resource and thus the entire world population would have to be considered when assigning value. Having derived a passive-use value for protected areas, it is necessary to do so for the remaining land allocation categories. Clearly the value of passive-use attributes falls with increasing management intensity, but there is little information for quantifying this relationship. The assumption is that the provision of passive-use value from land under low intensity management is reduced to one-half that of protected areas and that 31 integrated management area provides one-fourth the passive-use value of protected area. Land under high intensity management is assumed to provide nothing to the passive-use objective. Spreads for these fuzzy numbers are set to allow the range of possible values to begin at 0 and extend to twice the hypothesized value. These hypothesized possibility distributions for passive-use benefits are summarized as follows: High intensity use ($0 /ha,$0 /ha) Integrated use ( $6.67 /ha, $6.67 /ha) Low intensity use ($13.35 /ha, $13.35 /ha) Protected areas ($26.69 /ha, $26.69 /ha) 3.2 Forest Sector Employment The maintenance of direct forest employment is another of the goals identified by the Old-Growth Strategy Workshop. Achievement of this goal has the effect of stabilizing regional economies and minimizing political and social costs associated with unemployment. Forest related employment may be generated both by the forest industry and the forest-related tourism and recreation industry. Price Waterhouse (1993) reports 1.18 jobs/1,000 m3 of wood harvested for the coastal industry. It would appear that some of the jobs associated with the Island harvest are located in mainland mills. For this reason the employment estimate is reduced slightly to 1.16 jobs/Tj00 m3. The spread for this fuzzy number is set at 0.07, consistent with the magnitude of the variance of job levels as reported by Statistics Canada (in COFI 1992) for the past decade. As forest industry related jobs are directly a function of wood harvest levels, the multiplication of this possibility distribution by those for harvest volumes yields symmetric possibility distributions for direct timber jobs: 32 High intensity use, good site High intensity use, medium site High intensity use, poor site (0.0138 jobs/ha, (0.0100 jobs/ha, (0.0058 jobs/ha, 0.0045 jobs/ha) 0.0027 jobs/ha) 0.0029 jobs/ha) Integrated use, good site Integrated use, medium site Integrated use, poor site (0.0131 jobs/ha, (0.0087 jobs/ha, (0.0044 jobs/ha, 0.0040 jobs/ha) 0.0026 jobs/ha) 0.0023 jobs/ha) Low intensity use, good site Low intensity use, medium site Low intensity use, poor site (0.0101 jobs/ha, (0.0061 jobs/ha, (0.0033 jobs/ha, 0.0028 jobs/ha) 0.0018 jobs/ha) 0.0015 jobs/ha) There is little information about the relationship between employment and other uses of the forest. There are about 88,400 direct jobs in tourism throughout B.C.; dividing by the public land area of the province gives about 0.001 jobs per hectare. However, tourism is defined in a way that is unrelated to recreation; rather, it refers to any spending by persons outside of their normal shopping region (Cavanagh and McDougall 1992). Hence, the number of tourist jobs to be created by forest-based recreation and other activities will be substantially lower. Evidence from the Kamloops Resource Management Area Plan suggests that there are about 0.0001 direct tourist jobs/ha. A study by Clayton Resources Lt. and Robinson Consulting & Associates Ltd. (undated) indicates that the Valhalla wilderness area resulted in the generation of 0.0003 tourist-related jobs/ha (both direct and indirect) (see also Matas 1993). Given the paucity of information concerning the relationship between tourism and recreation jobs and land use, the following possibility distributions are assumed: High intensity use (0.0002 jobs/ha, 0.00015 jobs/ha) Integrated use (0.0002 jobs/ha, 0.00015 jobs/ha) Low intensity use (0.0003 jobs/ha, 0.00025 jobs/ha) Protected area (0.0003 jobs/ha, 0.00025 jobs/ha) 33 3.3 Government Revenue The sixth and seventh goals identified at the Parksville workshop have to do with government revenues. Maintaining government revenue means that the government will not have to increase borrowing or increase taxes in order to maintain current provision of services. In this analysis, d/recf revenues to the provincial government do not include employee taxes paid as a result of indirect and induced employment, and revenues accruing to the federal government are ignored. In 1992, the provincial government and municipalities received $5.27/m3 of harvest, while the province collected $9.05/m3 in stumpage fees for a total revenue of $14.32/m3 (Price Waterhouse 1993). It is assumed that revenues can vary by as much as $5 /m3. Therefore, a possibility function for direct government revenues would be: ($14.32/m3, $5.007m3). Indirect revenues are examined by looking at the contribution of forestry to provincial GDP. Forestry accounts for a substantial proportion of provincial GDP, indicating a high dependence on forest operations. A reduction in GDP signals lower federal and provincial government revenues, and an increase in expenditures on unemployment insurance and welfare. By dividing the forestry component of GDP by the annual timber harvest, it turns out that each cubic metre of harvest contributes about $70 directly to GDP. An interval of $20 /m 3 is chosen so that the possibility function for indirect government revenue is assumed to be: ($70/m3, $10/m3). The distributions describing the contribution to government revenue are each combined with the fuzzy sets describing harvest volumes to obtain the following fuzzy parameters: 34 direct revenue indirect revenue High intensity use, good site High intensity use, medium site High intensity use, poor site Integrated use, good site Integrated use, medium site Integrated use. poor site Low intensity use, good site Low intensity use, medium site Low intensity use, poor site ($170/ha, $105/ha) ($123/ha, $69/ha) ( $71/ha, $49/ha) ($162/ha, $96/ha) ($107/ha, $63/ha) ( $54/ha, $45/ha) ( $124/ha, $71/ha) ( $75/ha, $44/ha) ( $41/ha, $30/ha) ($833/ha, $343/ha) ($603/ha, $215/ha) ($349/ha, $167/ha) ( $792/ha, $304/ha) ($525/ha, $199/ha) ($266/ha, $163/ha) ($608/ha, $221/ha) ($368/ha, $141/ha) ( $202/ha, $104/ha) 3.4 Expansion of Wilderness Protection Currently, about 10.2% or 341,000 ha of Vancouver Island is protected, representing 15.5% of the Crown land. PAS objectives require that 12% of the land be protected, although some argue that, given the importance of coastal versus interior forests, more than 12% should be preserved. Estimates of appropriate values range as high as 20% or more. Regardless of the level of protection, there is nothing uncertain about the contribution of a hectare of land towards this objective. It is a crisp parameter—one hectare of land allocated to protected area provides one hectare of protected area. 3.5 Long-Run Sustained Yield The final objective identified is that of maintaining long-run, sustained yield (LRSY) harvest. This objective is often invoked to ensure that benefits from the timber resource are maintained over time. By design, this model solves for an even flow of benefits over the rotation period. Hence, LRSY is a redundant objective and not included here. 35 3.6 Objective Target Values The objectives of the model are all modeled as fuzzy greater than constraints. Thus, the degree of satisfaction increases as the value of the objective function increases. The value deemed to be the lowest possible to generate any satisfaction of the objective defines the lower limit of the constraint interval (bj - dj). The value deemed to be the lowest value at which complete satisfaction of the objective is attained defines the upper limit of the interval (bj). Two approaches can be used to determine the upper and lower values. The levels may be provided by a decision-maker or an expert in the area, relying on a subjective understanding of both the limits inherent in the system as well as what would constitute a satisfactory level of achievement. A second approach is to define the upper and lower bounds as the maximum and minimum levels that the system can provide when each objective is considered in isolation. This is an objective approach to defining the fuzzy constraints and is appropriate when there is little information available regarding the problem, preventing initial specification of unrealistic objectives. In practice, it would be preferable to use the second method to set initial parameters, and then use feedback information from users to refine the objective intervals, an incorporation of new information on values or preference structure. The specification of satisfactory levels of achievement for this model employs both approaches, without the benefit of any interactive procedure. The initial upper and lower bounds are provided in Table 1. 36 Table 1: Specification of Fuzzy Objectives Lower Bound Upper Bound Spread Logging ($'000,000) 'B V" 1 / » Recreation ($'000,000) Passive-use ($'000,000) Direct employment Qobs) 48.1 49.9 8.6 13,345 139.2 640.5 341,000 72.0 23.9 90.1 40.2 60.0 53.4 15,700 2,355 174.0 34.8 854.0 213.5 660,000 319,000 Direct revenue ($'000,000) Indirect revenue ($'000,000) Protected area ('000 ha) For logging benefits, the level for complete satisfaction is set as the maximum available from the model if only logging benefits are considered. The minimum represents the amount generated from a working forest of 700,000 hectares, a scenario rejected by CORE as providing insufficient returns. Recreation and passive-use benefit intervals are defined by the maxima and minima available from the system (see Table 1). The matter of employment levels is a highly political one. It is unlikely that an acceptable minimum level is the lowest that the system could provide. The assumption here is that the current level provided by the forest industry is fully satisfactory, although it will be difficult to maintain current employment in the future as technological developments lead to a decreasing number of jobs per unit of harvest. Jobs related to recreation are also considered satisfactory at current levels, although, in actual fact, it would probably be less than satisfactory if current levels were simply maintained. However, recreation contributes only a very small number of jobs in comparison to those derived from timber harvest; requiring an increase in this component has little impact in the model. The lower level for job provision is arbitrarily set at 15% below the current 37 level, on the assumption that it would be politically unwise for government to allow employment levels to drop below this figure (Table 1). Maximal values for both direct and indirect revenue are determined by the ability of the system to generate revenues and timber-related GDP. Lower bounds again reflect the political nature of these objectives. It is assumed that a decrease of more than 20% in direct revenues, or of 25% in indirect revenues (i.e., forestry's contribution to GDP), would be politically unsatisfactory. Values for these fuzzy objectives are also provided in Table 1. The final objective is that of wilderness expansion. Any increase in protected areas will most likely come from Crown land. A doubling of protected area on the Island would mean that 660,000 ha would be removed from consideration for the working forest, or about 30% of total Crown land. It is assumed that protecting almost a third of the public land on Vancouver Island would prove sufficient to allow all PAS objectives to be met; thus, the decision maker is assumed to be completely satisfied at that level of wilderness protection. The lower level for the fuzzy objective is defined as the current area under protection, a level below current legislated requirements and thus considered unacceptable. The data are summarized in Table 1. 38 CHAPTER 4 EMPIRICAL RESULTS Two formulations of a crisp MODM (in the form of a preemptive priority goal programming model), a fuzzy MODM and a possibilistic fuzzy MODM were constructed in EXCEL5.0. The results of the crisp formulations are compared to the fuzzy MODM. The fuzzy model is then expanded to include consideration of imprecise coefficients, with the results compared to the basic fuzzy model. 4.1 Fuzzy MODM The crisp MODM formulations are all based on a single preemptive priority goal programming model, adapted from van Kooten (1995b). Priority rankings are consistent with the original and are as follows: 1. Net social well-being 2. Employment 3. Direct government revenue 4. Indirect government revenue 5. Expansion of wilderness protection. The net social well being objective function can be formulated either as the sum of the relevant economic efficiency objectives (logging, recreation and passive-use) or as three distinct objectives, each receiving the same, number one priority. In the first situation, the target value is defined as the sum of the independent targets, with the objective being to minimize the positive or negative deviation from this "compound" target. This approach is labelled GOAL1. However, GOAL1 fails to provide the 39 legislated level of protected area, and, therefore, a second model is specified (GOAL2); it incorporates as a constraint the minimum PAS objective of 12%. When the net social benefit objectives were treated as separate objectives of the same priority ranking, the solution was entirely predicated upon the initial iteration of the model. Remaining objectives had no weight in determining land allocation. Due to this failure to consider all lower ranked objectives, the results of this second method of modelling the net social benefit objective are not reported in this paper. Target values for the (crisp) goal programs were defined as the midpoint of the intervals identified for the fuzzy objectives in the fuzzy model (Table 1). The approach is to minimize the deviation from the target value regardless of whether that deviation is positive or negative. To allow unrestricted positive deviations would cause the model to maximize net social well-being with a land allocation that left little slack available to incorporate the remaining objectives in subsequent iterations. The fuzzy MOLP (FUZZY) considers all objectives as independent and equal in terms of priority. There is little difference between GOAL1 and GOAL2 in the land allocation scheme identified (Table 2). With the addition of the constraint regarding protected areas, sufficient area is moved out of the poor site category to allow the constraint to be met. This illustrates one of the deficiencies of the general approach. There is no provision for the inclusion of better quality sites into protected areas even though this would be required if protection of a representative sample of ecosystems was to occur. In fact, the model construction is such that it prejudices against such an outcome. In all cases, protected area is increased only at the expense of poor quality sites. 40 Table 2: Simulation Results for Fuzzy MODM: Land Allocation (hectares) Site Quality Management Model Intensity GOAL1 GOAL2 FUZZY High 223,842 223,842 209,848 Good Integrated 0 0 13,994 Low 0 0 0 High 462,013 462,796 363,747 Medium Integrated 0 0 527,269 Low 429,003 428,220 0 High 0 0 0 Poor Integrated 0 0 0 Low 771,288 694,647 658,671 Protected 324,359 401,000 436,976 Total Allocated 2,210,505 2,210,505 2,210,505 The goal programming formulations concentrate good quality sites into high intensity management regimes, and divide medium sites relatively equally between high and low management categories. All poor sites are allocated to the low intensity system. In contrast, the FUZZY model places the larger proportion of medium quality forest land under integrated management as well as a small amount of good quality area. Total area assigned to the high management regime is less, and protected area is greater (a solution obtained without constraints regarding allocation). The effects of this differential in allocation schemes are evident in the levels attained for each of the objective functions. Both GOAL1 and GOAL2 provide a higher level of total social benefits than does the FUZZY model (Table 3). Most of this difference is found in the level of logging benefits. This is surprising as less area is 41 under intensive forest management in the FUZZY scheme. It should be noted that the interpretation of logging benefits as a social benefit is a bit fuzzy in and of itself. Strictly speaking, they can be looked upon as producer surplus, rent accruing to the logging industry. However, in B.C., logging operations are generally treated as a cost centre by the forest industry. Thus, the objective is to minimize costs as opposed to producing any profit or surplus. So, while logging benefits (as defined in this model) are a social benefit, their importance relative to the other sources of social well-being is unclear; also, it is not clear whether this is the best approach for incorporating producer surplus into the equation. This is the only area where the goal program provides a significantly higher return than does the fuzzy formulation, and the FUZZY model returns the highest level of total monetary "benefits". This advantage lies in the assignment of medium quality land to the integrated category as opposed to the lower yielding, low management category. Table 3: Simulation Results for Fuzzy MODM: Monetary Benefits (millions of dollars) Benefits Model GOAL1 GOAL2 FUZZY Logging Recreation Passive Use Direct Revenue Indirect Revenue $ 63.4 $ 76.2 $ 24.7 $ 159.2 $ 778.3 $ 63.8 $ 74.9 $ 25.7 $ 156.1 $ 763.0 $ 55.3 $ 76.7 $ 24.0 $ 166.7 $814.8 Turning attention to the non-monetary benefits (Table 4), it is clear that the FUZZY 42 model provides higher objective values than do the crisp MODMs. In spite of the fact that it allocates less land to high intensity management, the FUZZY model provides the greatest annual harvest volume, 11.6 million m3. Although this result appears at odds with that of lower logging benefits, these are consistent results. The harvest volume in the FUZZY model is generated under a less intensive system with attendant price penalties, and, on average, each cubic meter is worth less. As the forest industry provides the bulk of the forest-related jobs, it is no surprise that the highest level of employment is generated by the model that provides the greatest harvest volume. Superior levels of harvest volume and employment are provided by the model that also provides the superior amount of protected area, a result contrary to what might be expected. It is interesting that all three models provide an annual harvest that is in excess of the current LRSY of about 11.0 million m3. This is largely due to the application of intensive silvicultural practices to a substantial portion of the Crown land base, a situation contrary to current conditions. Table 4: Simulation Results for Fuzzy MODM: Non-Monetary Benefits Model G O A L ! GOAL2 FUZZY Direct Employment Gobs) 13,493 13,239 14,054 Protected Area (hectares) Harvest Volume 324,359 401,000 436,976 11.1 10.9 11.6 ('000,000 m3) 43 As described in Chapter 2, there is a membership function (14) associated with each of the objective functions that indicates the degree of satisfaction achieved foe that objective. By assumption, the value of the membership function increases linearly with an increase in the objective function's value. It is the degree of membership of the objective function value in the set of satisfactory values that is reported in Table 5. As one might expect given the results presented above, the degree of satisfaction observed is greatest under the FUZZY solution for all objectives other than logging and passive-use benefits. Table 5: Simulation Results for Fuzzy MODM: Degree of Objective Satisfaction Model GOAL1 GOAL2 FUZZY Logging 0.64 0.66 0.30* Recreation 0.65 0.62 0.67 Passive Use 0.31 0.33 0.30* Direct Revenue 0.58 0.49 0.79 Indirect Revenue 0.65 0.57 0.82 Employment 0.06 0.00 0.30* Protected Area 0.00 0.19 0.30* indicates minimum satisfaction level for FMODM model The target values for the goal programs were the mid-points of the fuzzy objective intervals, corresponding to a degree of satisfaction of 0.50. GOAL1 summed logging, recreation and passive-use benefits into a single objective; its "optimal" land allocation generated a positive deviation for the goals of logging and passive-use benefits, while undersupplying land categories generating recreation and option benefits. The FUZZY 44 model provides an objective satisfaction level greater than 0.5 only for recreational values. The FUZZY model clearly provided for greater satisfaction of objectives in all remaining cases. The failure of GOAL1 to meet even minimum requirements for protected area has already been addressed. GOAL2 fails to provide any degree of satisfaction for the employment objective. In no case was an objective completely satisfied (Uj(x)=1). This upper level of satisfaction was most often defined as the maximum obtainable from the system rather than as a direct social or economic goal. Thus, one would not expect total satisfaction of any one objective when all objectives are considered. Focusing on the FUZZY model as described in Table 5, the minimum degree of satisfaction is attained for four of the seven objectives. Given that the model provides a Pareto efficient solution, the interpretation is that it is impossible to increase the satisfaction level for any one of these four without compromising that of at least one of the other three. The standoff is between logging benefits and employment on the one hand, and passive use values and protected areas on the other. This situation reflects the reality of the conflicts identified in the Vancouver Island land use debate. All of the results discussed in this section are a function of the intervals defined for the fuzzy constraints of the FMODM model and are sensitive to change in the bounds of those intervals. Also, the comparison of fuzzy and non-fuzzy solutions is conditioned on the use of the midpoints of the fuzzy constraint intervals as the precise goals for the crisp models. However, with this midpoint as the goal value, the crisp models were unable to provide for all objectives without the use of minimum value constraints. To raise the goal levels of the higher priority objectives would simply exacerbate this 45 situation. To lower the goal levels may help to achieve some satisfaction in all areas, but those satisfaction levels for the higher priority objectives would be lower than in the current solution. In either case, the overall advantage would continue to lie with the FMODM model. The FMODM model, given the objectives as specified, provides a more satisfactory solution to the land allocation question than does the traditional LP approach. The minimum objective satisfaction levels are higher and the majority of objectives are better satisfied. 4.2 Fuzzy Possibilistic MODM As discussed in Chapter 3, there is a great deal of uncertainty surrounding the precision of the parameters in the model, and symmetric possibility distributions are used to model this uncertainty. By lowering the value of a (the FMODM has an implicit a value of 1), the effect of this uncertainty upon optimal land allocation can be explored. At any value of a<1 there are two solutions to consider. The first is from model (18), where parameter values take on a less-likely and lower value (the LOWER results); the second if from (19), generating a solution based on parameter values of the same possibility but higher value (the UPPER model results). The results from the two models are provided in Tables 6 though 9. Before moving to a detailed discussion of results, there are two points regarding the interpretation of the possibility distributions worth considering. The first is that, although each individual possibility distribution is assumed symmetric, this does not presuppose symmetry in solution sets. The response in any one parameter value to a change in possibility level is strictly a function of the spread (Py) defined for that number. 46 The spread completely defines the slope of the linear possibility distribution and thus the rate of change in value. It is the net result of all such independent movements that determines the ultimate solution. The second point regards the interpretation of movement along the possibility distribution. Examining the effects of parameter values lying below those judged most likely is not synonymous with taking a more pessimistic approach nor with any sort of risk assessment. In the first case, a lower value for a cost variable would have to be considered a more optimistic outlook, while a lower value for a price variable would be considered a more pessimistic approach. The methodology used in this paper adjusts all parameter values in the same direction and to the same degree; statements regarding a more desirable scenario as defined by possibility levels are not appropriate. In the second case, risk is a concept defined in terms of probability theory and utility functions (Keeney and Raiffa, 1976). This interpretation is clearly not supported by this fuzzy methodology. The most obvious result obtained from the variation of the possibility level is in the asymmetry of the feasible solution space. While solutions may be obtained for any value of alpha using the UPPER model, feasible solutions do not exist below the possibility level of 0.92 for the LOWER model. Parameter values become insufficient to provide any satisfaction of at least one of the objectives; in this case, the limiting objective is timber benefits. As noted previously, the FPMODM model with a=1 is identical to the FMODM model (FUZZY) discussed in the previous section. An unexpected result is that the LOWER model, a = 0.95, provides for over 10% more protected area than do any of the other scenarios considered. The minimum 47 amount is provided by the FUZZY model. The rationale for this is that the LOWER model concentrates the good and medium quality sites into the high intensity management category, a shift of over 400,000 ha from the integrated management allocation level of the FUZZY model. This occurs in response to the lower estimation of both wood yield and wood value. This causes a large reduction in passive-use benefits as the high intensity management category does not contribute to this objective. The shortfall is replaced by the allocation of poor quality area, with it's negative logging value, into the protected area category (Table 6). Harvest volume is increased under this LOWER scenario and job numbers fall slightly (Table 7). Monetary benefits are also slightly lower with the largest change observed in recreation benefits; logging benefits are virtually unchanged (Table 8). The same objectives remain binding as in the FUZZY model with the exception of protected areas (Table 9). Table 6: Simulation Results for Fuzzy Possibilistic MODM: Land Allocation Site Quality Management Intensity LOWER FUZZY UPPER a =0.95 a=1 a =0.95 a =0.9 a =0.80 High 223,842 209,848 136,697 69,680 0 Good Integrated 0 13,994 87,145 154,162 223,842 Low 0 0 0 0 0 High 753,259 363,747 330,668 300,593 213,323 Medium Integrated 137,757 527,269 560,328 590,423 677,693 Low 0 0 0 0 0 High 0 0 0 0 0 Poor Integrated 0 0 0 0 0 Low 543,453 658,671 645,434 632,048 605,625 3rotected 552,194 436,976 450,213 463,599 490,022 Total Allocated 2,210,505 2,210,505 2,210,505 2,210,505 2,210,505 48 Table 7: Simulation Results for Fuzzy Possibilistic MODM: Non-Monetary Benefits LOWER FUZZY UPPER a =0.95 a =1 a =0.95 a =0.9 a =0.80 Direct Employment 13,948 14,054 14,151 14,250 14,445 Gobs) Protected Area 552,194 436,976 450,213 463,499 490,022 (hectares) Harvest Volume 11.7 11.6 11.5 11.4 11.2 ('000,000 m3) Table 8: Simulation Results for Fuzzy Possibilistic MODM: Monetary Benefits LOWER FUZZY UPPER Benefits g =0.95 g=1 g =0.95 g =0.9 g =0.80 Logging $54.2 $55.3 $56.3 $57.3 $66.1 Recreation $64.7 $76.7 $79.9 $82.9 $88.1 Passive Use $21.7 $24.0 $26.2 $28.4 $32.6 Direct Revenue $163.2 $166.7 $170.0 $173.3 $179.7 Indirect Revenue $806.3 $814.8 $822.7 $830.6 $846.0 Table 9: Simulation Results for Fuzzy Possibilistic MODM: Degree of Objective Satisfaction LOWER FUZZY UPPER g =0.95 g =1 g =0.95 g =0.9 g =0.80 Logging 0.26* 0.30* 0.34* 0.38* 0.75 Recreation 0.37 0.67 0.75 0.82 0.95 Passive Use 0.26* 0.30* 0.34* 0.38* 0.47* Direct Revenue 0.69 0.79 0.88 0.98 1.00 Indirect Revenue 0.78 0.82 0.85 0.89 0.96 Employment 0.26* 0.30* 0.34* 0.38* 0.47* Protected Area 0.66 0.30* 0.34* 0.38* 0.47* * indicates minimum satisfaction level in FPMODM model 49 The results obtained from the UPPER model as alpha is decreased are as expected. All parameters of the model increase in value as a decreases, resulting in a higher provision of benefits from each hectare of land considered. Increasing yields and wood values allow less area to be allocated to the high intensity regime and more to integrated management. The result is an increase in both recreation and passive-use benefits. Harvest volume declines and job provision increases, evidence of the less possible higher per ha yield estimates and a greater number of jobs per unit of harvest. The relationships between objective values remain relatively constant; those objectives binding in the FUZZY model are also those binding |n the UPPER model for alpha greater than 0.9. In the a=0.8 scenario, logging benefits have increased greatly relative to other objective values, and they are no longer binding on the solution. The solution provided by the FUZZY model is sensitive to overestimation of the true parameter values. If values are realized at a generally lower level than those judged most likely, a large shift in resource allocation is required to obtain the best solution as judged by the maximization of minimum objective satisfaction. Results of the UPPER model seem to indicate that we have little to fear from the general underestimation of parameter values. If true values are realized at some level above those judged most likely, the error in planning has simply been that more land was allocated to high intensity use than was required, and protected area levels could have been higher without jeopardizing other objectives. This is not necessarily the case. Silvicultural costs in this model are only the direct operating costs of applying specific silvicultural treatments. No account is made of any capital cost associated with the ability to carry out such treatments, or of the administrative and bureaucratic costs surrounding the business of 50 doing silviculture in B.C. As noted earlier, results from models (18) and (19) can be used to gain some understanding about the sensitivity of the solution to the uncertainty in parameter definition. However, it is a crude approach for considering imprecise parameters and offers only a few generalized insights to this land-use allocation question. 51 CHAPTER 5 SUMMARY AND CONCLUSIONS An increasing amount of weight is placed on non-timber values when management or allocation plans for woodlands are considered. In addition, the public is becoming more involved in the planning process. Both trends are evident on Vancouver Island, particularly in the recently completed work of the Commission on Resources and Environment (CORE), a multiple interest group process. The mandate of the Vancouver Island CORE was to obtain a consensus decision on land-use allocation for the Island, where a substantial portion of the land base is publicly-owned forest land. The philosophy of this CORE process did not appear consistent with that of traditional cost-benefit analyses in that a single economic efficiency account did not set the measurement standard for evaluating options. In addition, cost-benefit analysis does not provide a means for considering the uncertainty associated with the identification of objectives and information relevant to the CORE process. 5.1 Methodology This study explored the use of fuzzy set theory as a means of incorporating uncertainty due to vagueness and imprecision into models used to inform decision-making processes. Vagueness is found in the identification and ranking of objectives, and compromises the ability to distinguish preferred decision alternatives. This vagueness was modelled through the use of a Fuzzy Multiple Objective Linear Program (FMOLP), where objective values are assigned a degree of membership in the fuzzy set of satisfactory solution values. The fuzzy set methodology was applied to the land-use 52 problem on Vancouver Island and results were compared to those obtained from a more traditional approach, namely, preemptive priority goal programming. Imprecision or ignorance is encountered in the specification of parameters to a problem. Possibility theory was used to extend the FMOLP to include these two types of uncertainty in land allocation modelling. Results of this Fuzzy Possibilistic Multiple Objective Linear Program (FPMOLP) were interpreted as a type of sensitivity analysis with respect to uncertainty in the problem description. 5.2 Summary of Results The FMOLP yielded a solution that provided some degree of satisfaction for all objectives, while the traditional model failed to provide any degree of satisfaction for at least one of the objectives without the use of appropriate constraints. For the majority of objectives considered, the degree of satisfaction provided by the FMOLP model exceeded that of the crisp model. Land-use allocation under the two approaches was distinctly different. The FMOLP solution allocated about 25% of the land base to integrated timber management, while the traditional model concentrated land into the extreme categories of low (natural regeneration and growth) and intensive timber management intensity. The area assigned to the protected category was greatest in the FMOLP solution, as were the number of direct jobs provided and the annual harvest volume. This "better" solution was obtained without the necessity of specifying precise value for objectives, and without an explicit ranking or weighting of the objective functions. The FMOLP solution also clearly identified those objectives that were in direct conflict with each other, and thus the areas where compromise is required if satisfaction levels are to be increased. 53 The FPMOLP model was solved assuming a number of different possibility levels for parameter value realization, and results were compared against those of the FMOLP model (where all parameters take their most possible value). In general terms, the model as specified was sensitive to the possibility of lower realizations of parameter values. Feasible solutions existed only when parameter values assigned a possibility level of 92% or more were considered. The "best" solution obtainable at the 95% possibility level (all parameters are realized at a value that is judged to have a 95% possibility of occurring) yielded a land allocation scheme where nearly one-half of the available land base was assigned to the high timber management regime. This allocation scheme also provided the highest level of protected area of the cases considered. All other objectives were less well satisfied than in the base FMOLP model. As parameter value realizations greater than those judged most likely were considered, changes to the FPMOLP solution were less dramatic and more intuitively consistent. For the most part, higher parameter values in this model represent a more favourable situation. This is reflected in the increasing objective function values and associated satisfaction levels as higher, but less likely, parameter values are considered. Allocation schemes show increasing areas of land in integrated timber management and protected areas with a corresponding decrease in land assigned to both the high and low management categories. 5.3 Restrictions and Further Research In comparing of the FMOLP model with the traditional MODM approach, both can be considered as only the first step in an exploration of the problem at hand. Trade-off 54 information is available form the dual side of the problem, and this, in conjunction with a solution in the form presented in this paper, can be used to re-specify and refine the intervals used to define the degree of objective satisfaction. Implicitly, this respecification will introduce new information regarding values and preferences of the interest groups; values and preferences that have been shaped and clarified by the information supplied in the initial formulation. This re-specified model yields a solution conditioned on the fact that additional information regarding the objectives was available and uncertainty had been reduced. This gradual refinement of objectives may facilitate reaching a consensus agreement. It would seem that the fuzzy approach is more consistent with this type of interaction, no a priori assignment of priority or desired value is required, and the model explicitly includes room for compromise positions in the form of the intervals specified for the objectives. The methodology chosen for the inclusion of parameter uncertainty (the FPMOLP) proved to be simplistic and restrictive, allowing little in the way of specific insights into the problem. Although it highlighted the potential danger of being overly optimistic regarding the model coefficients, the results of working with underestimated values are misleading. The danger is in assuming that the effects of this underestimation are benign; resources are still being improperly allocated with attendant costs and inefficiencies. The true value of this particular FPMOLP formulation may be as an initial exposure to explicitly considering ignorance in model specification. Further research is required both in regards to the theoretical problem and the empirical problem. Theoretically, work is required on the role of possibility theory in modelling uncertainty. Alternative models to the one advanced in this paper are 5 5 available; effort needs to be expended in understanding how they differ from this approach and the advantages they may offer. This general area is currently the focus of much debate and controversy, and new insights and methodologies are being continually advanced. Failure of this model to deal convincingly with parameter uncertainty should not prejudice against further investigation in this area. Empirically, the reflection of reality offered by the model is blurred by the static nature of the model. Development of a dynamic model would allow foreseeable changes in objectives as well as in technical and technological parameters to be included. Also, the dynamic nature of the forest resource could be explicitly considered, particularly the transition from the resources available from old growth to those available in subsequent rotations. As mentioned earlier, a major shortcoming is the failure to allow the movement of higher quality land into protected areas, clearly at odds with the philosophy and intent of the Protected Areas Strategy. The use of dollar values in describing the non-timber flows derived from the forest is inconsistent with the philosophy that abandons the requirement for accounts to be evaluated against a dollar valued standard. The FMOLP model allows for objectives to be defined and measured in units that are more natural and intuitive; recreational value may be better represented by units reflecting the degree or level of use (e.g., user-days), while enumeration and measurement of various attributes of wilderness areas may provide for a more realistic description of the benefits they provide. Two major assumptions in the model specification were those of the validity of the maxmin operator in describing the decison process and the validity of the choice of a linear function to describe the membership functions. No empirical testing was undertaken for either assumption. 56 Further refinement of both the FMOLP and the FPMOLP model could offer insight into the requirement for silvicultural investment in the forest of Vancouver Island. Solutions obtained in this research indicate a clear requirement for investment in intensive silviculture in order to best meet the objectives that were described. 57 REFERENCES Barret, C.B. and P.K. Pattanik, 1989. "Fuzzy Sets, Preference and Choice: Some Conceptual Issues", Bulletin of Economic Research 41: 229-253. B.C. Ministry of Forests. 1990. Towards An Old-growth Strategy: Executive Summary of Old-growth Workshop Recommendations. Summary of the Parksville workshop of November 3-5, 1989. Victoria, B.C. B.C. Ministry of Forests, 1991. Outdoor Recreation Survey 1989/90. How British Columbians Use and Value their Public Forest Lands for Recreation. Recreation Branch Technical Report 1991-1. Victoria: Queen's Printer for British Columbia. Cavanagh, Carl and Ruth McDougall, 1992. "Estimating Tourism GDP, 1981-1991". In British Columbia Economic Accounts 1982-1991. Victoria: B.C. Ministry of Finance and Corporate Relations. Clayton Resources Lt. and Robinson Consulting & Associates Ltd., undated. Economic Impacts of Land Allocation for Wilderness Purposes: A Retrospective Analysis of the Valhalla Parkin British Columbia. Study prepared for the B.C. Forest Industry Land Use Task Force. Vancouver. Cox, E., 1994. The Fuzzy Systems Handbook. Cambridge, MA: Academic Press. Commission on Resources and Environment, 1994. Vancouver Island Land Use Plan. Volume 1. Victoria: Government of British Columbia. February. 260pp. Maps. Council of Forest Industries of British Columbia, 1992. British Columbia Forest Industry Statistical Tables. Vancouver, B.C. Dubois, D. and H. Prade, 1993. "Fuzzy Sets and Probability: Misunderstandings, Bridges and Gaps", Proceedings of IEEE International Conference on Fuzzy Systems. Volume 2. Fedrizzi, M., 1987. "Introduction to Fuzzy Sets and Possibility Theory". In Optimization Models using Fuzzy Sets and Possibility Theory edited by J. Kacprzyk and SAOrlovski. Dordecht, The Netherlands: D. Reidel Publishing Co. Ignizio, J.P., 1983. "Generalized Goal Programming", Computers and Operations Research 10: 277-289. Keeney, RL . and H. Raiffa, 1971. Decisions with Multiple Objectives: Preferences and Value Tradeoffs. New York: John Wiley and Sons. Klir, G. and T. Folger, 1988. Fuzzy Sets, Uncertainty, and Information. Englewood Cliffs, New Jersey: Prentice Hall. 58 Krause, P. and D. Clark, 1993. Representing Uncertian Knowledge: An Artificial Intelligence Approach. Dordrecht, The Netherlands: Kluwer Academic Publishers. Kruse, R., E. Schwecke and J. Heinsohn, 1991. Uncertainty and Vagueness in Knowledge Based Systems. Heidelberg: Springer-Verlag. Lai, Y-J. and C'L. Hwang, 1994. Fuzzy Multiple Objective Decision Making. Berlin: Springer-Verlag. Lee, S.M., L.J. Moore and B.W. Taylor III, 1985. Management Science. 2nd Edition. Boston: Allyn and Bacon. Matas, Robert, 1993. "In the beginning there was Moresby", Globe and Mail Saturday, Nov. 13. p.D3. Phelps, S.E., W.A. Thompson, T.M. Webb, T.M., P.J. McNamee, D. Tait and C.J. Walters, 1990a. British Columbia Silviculture Planning Model: Structure and Design. Unpublished Report. Victoria: B.C. Ministry of Forests. Phelps, S.E., W.A. Thompson, T.M. Webb, T.M., P.J. McNamee, D. Tait and C.J. Walters, 1990b. British Columbia Silviculture Planning Model: User Manual. Unpublished Report. Victoria: B.C. Ministry of Forests. Price Waterhouse, 1993. The Forest Industry in British Columbia, 1992. Report Tables. Vancouver: Price Waterhouse. Sakawa, M., 1993. Fuzzy Sets and Interactive Multiobjective Optimization. New York: Plenum Press. U.S. Water Resources Council, 1983. Economic and Environmental Principles and Guidelines for Water and Related Land Resources Implementation Studies. Washington DC: Mimeograph, March 10. 137pp. van Kooten, G.C., 1995a. "Economics of Protecting Wilderness Areas and Old-Growth Timber in British Columbia", The Forestry Chronicle 71(Jan/Feb): 52-58. van Kooten, G.C., 1995b. "Modelling Public Forest Land Use Tradeoffs on Vancouver Island", Journal of Forest Economics 2: In press. Void, T., B. Dyck, M. Stone, R. Reid and T. Murray, 1994. Wilderness Issues in British Columbia: Preliminary Results of a 1993 Province-wide Survey of British Columbia Households. Victoria: BC Forest Service, BC Parks and BC Environment, mimeograph. 30pp. App. 59 

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