A fuzzy multicriteria model for comparing energy projects

Enqy Vol. 12. No. 7. pp. 599-613, Printed in Great Butam

A FUZZY

1987

0360-5442/87 Pergamon

MULTICRITERIA ENERGY MITALI University

DE

of Waterloo,

MODEL FOR COMPARING PROJECTS

and KEITH Waterloo,

(Ruceit&

$3.00 + 0 00 Journals Ltd

W.

Ontario

30 April

HIPEL NZL 3Gl. Canada

1986)

fuzzy set approach to multicriteria modelling IS presented for selecting the best alternative solution to a large-scale engineering project. The problems of making expert recommendations, amidst conflicting views of different people and institutions. is handled by this technique which incorporates the viewpoints of different interest groups. both quantitative and qualitative. The efficacy of the methodology is demonstrated by applying it to the real world problem of selecting the best site for the Bay of Fundy Tidal Power Project where emphasis is placed on SOCLOeconomic aspects. The direct impacts ofdeveloping tidal power energy on good living, transportation, psychological and political factors are studied by using fuzzy binary relations to model preferences. Abstract-A

1. INTRODUCTION

When designing a large-scale engineering project and especially an energy-generation project, its benefits and detriments on modern society should be kept in proper perspective. Usually, there is an array of possible alternative solutions to any design problem. To assist in ascertaining which alternative solution to ultimately select, benefit-cost analysis has been traditionally employed. However, these projects bring about environmental and social impacts which are often overlooked in the economic analyses because they cannot be measured in terms of monetary units. In large-scale engineering projects, usually intangible factors such as aesthetics, loss of life, environmental quality, and social concerns must be considered in addition to quantitative factors which may include monetary costs and benefits, and the amount of energy generated. Another problem of large-scale engineering projects is that often two or more decision-making parties are involved with a given project and the viewpoints of all the interest groups must be properly incorporated into any type of analysis. The purpose of this paper is to develop a comprehensive decision-making procedure for ascertaining the more preferable solutions to a large-scale energy-development project and to demonstrate its usefulness by applying it to a case study. In the proposed methodology, fuzzy set approaches are used within the framework of multicriteria decision making (MCDM). This MCDM methodology is particularly useful when the objectives are fuzzy in nature, non-commensurable, and of varying degrees of importance. Essentially, this fuzzy set theoretic approach for multicriteria modelling constitutes a method to evaluate large scale energy projects, where there is a need to compare distinct alternatives using both quantitative and qualitative criteria and to accommodate the viewpoints of the different interest groups. Furthermore, approaches are proposed for executing sensitivity analyses in fuzzy MCDM. To demonstrate the efficacy of the fuzzy multicriteria concepts presented, the methodology is employed to determine the most reasonable solution to the problem of site selection of the Bay of Fundy Tidal Power Project, in the Maritime Provinces of Canada. Because the existing study of this problem considered only economic factors,’ inherent advantages of the proposed methodology are clearly portrayed.

2.

FUZZY

MULTICRITERIA

MODELLING

Most of the traditional tools for formal modelling, reasoning and computing are dichotomous and precise in character. In fact, real world systems are very often uncertain and vague in a number of ways. This type of uncertainty (stochastic character) has long 599

MITALI DE and KEITH W. HIPEL

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been handled appropriately by probability theory and statistics. In contrast, vagueness concerning the description of the semantic means of events, phenomena or statements themselves is termed fuzziness, which is studied in a formal framework of fuzzy mathematics. Fuzzy set theory was first initiated by Zadeh’ and followed by many other authors.3-19 Fuzzy set theory provides a powerful tool to handle both quantitative and non-quantitative information. Dejinition

of a fuzzy

set

Let X be a classical set of objects called the universe, whose generic elements are denoted by x. Membership in a classical subset, Yof X, is often viewed as a characteristic function from X to [0, l] such that

444 = :, i

if

if

XE Y, xt+ Y.

where [0, 1) is called the valuation set. If the valuation set is allowed to be the real interval [0,11, Y is called a fuzzy set.2 The grade of membership of x in Y is denoted by f,(x) and the valuation set can be called the membership space M. The set Y is thus a subset of X with no sharp boundary and Y can be completely characterized by the set of ordered pairs

Y= C{x,fy(X)}/X~W. Here, fy(x) is the degree to which x satisfies the condition specified by Yor the truthfulness of the statement x is I: The fuzzy set differs from the conventional set as follows. The conventional set may be represented by a characteristic function, which takes on the values of either 0 or 1, meaning not belonging to and belonging to a given set, respectively. In a fuzzy set, the characteristic function is replaced by the membership function which takes on values in the whole interval [0, 11. The values of the membership function express the degree to which a particular element belongs to the fuzzy set. In the decision context, this concept may be used with X as the set of alternatives and Y as some objectives or criteria that can be expressed as a fuzzy subset of X where fr(x) would be a measure of satisfaction that alternative x gives to criteria Y Therefore, in MCDM problems if A is the set of alternatives and C is the set of evaluation criteria, C can be expressed as a fuzzy set of A, where the membership function f,(a) would measure the satisfaction-that alternative a gives to criteria c, and where CE CUE A. Another important concept in fuzzy set theory is that of fuzzy binary relations, which permit the modelling of situations where interactions between elements are more or less strong. Fuzzy binary relations can be considered as generalizations of ordinary binary relations. For a long time, devision making has been concerned with a single evaluation criterion, such as profit, cost or rate of return. However, due to the increasing complexity of decision making, it has become more and more difficult to see the world in a unidimensional manner by using a single criterion to judge what is perceived. In reality, the overall performance of an alternative is not dependent on a single criterion only, but on a variety of criteria at the same time. Decision making can be defined to be the process of selecting a possible course of action from all the available alternatives, in order to meet a set of objectives in the most efficient way, within the given constraints. Basically, three major approaches have been developed in MCDM. The first approach to multiple criteria decision making consists of using utility theory to obtain a function, which is used to compare different alternatives.20.21 The second procedure uses mathematical programming methods to simultaneously or sequentially optimize several objective functions, in order to obtain a solution. The third approach of solving MCDM problems is founded on the concept of ordering where the alternatives

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are classified and ordered according to some preference profile. These and a number of other scattered methods using various approaches are discussed in Refs. 22-25. When alternative solutions are evaluated against a range of criteria, often there are nonquantitative or intangible criteria. For instance, criteria related to aesthetics, good living, emotions, health and safety effects, risk, pollution and other social impacts of large scale engineering projects are usually qualitative in nature and furtunately fuzzy set theory can handle these kinds of criteria in MCDM. Different approaches of fuzzy multicriteria modelling are found in the literature (Refs. 26-30). The fuzzy multicriteria model proposed here incorporates the best features of other approaches into a single comprehensive methodology. For instance, the evaluation matrix is based on the research of Hipel” while the modelling and aggregation of preferences is based on the ELECTRE methods.12.13.28 In the field of energy, a range of decision analysis and systems engineering tools have been employed for enhancing the decision making process. 31 -35 The energy study presented in this paper constitutes one of the first published applications of fuzzy multicriterial modelling to an energy problem. An earlier version of this research was presented by De and Hipe136 at an international conference. To model a MCDM problem effectively, the comprehensive methodology outlined in Fig. 1 is developed in this paper. This methodology is explained with the help of the Bay of Fundy Tidal Power case study.

3. BAY

OF

FUNDY

TIDAL

POWER

GENERATION

CASE

STUDY

3.1. Problem detection As is shown in Fig. 1, a study is initiated by detection of a problem and its related needs. The present and future energy demands of the Maritime Provinces in Canada are similar to those in many other parts of the world. In any modern civilization, there is a remarkable dependence upon geologically stored sources of energy, both for survival at present population levels and for ensuring the existence of various pleasures and comforts. Expanding technology provides both the incentive and the means for exploiting stored energy reserves. Adequate supplies of cheap energy came to be more or less taken for granted, until events of the 1970s and early 1980s led to a vastly altered perception of the energy future. Renewable energy resources gain great importance as a means for ensuring dependable energy supplies and stabilization of costs caused urgent concerns. The recent economic pressures and the dwindling reserves of traditional non-renewable energy resources stressed the need of renewable resources. Such concerns are more emphasized in the Maritime Provinces of Canada, because of the relative scarcity of indigenous energy resources when compared to other parts of Canada. In 1977, statistics revealed that, of the primary energy demand in the Maritime provinces, consisting of the provinces of Nova Scotia, Prince Edward Island and New Brunswick, 40% came from renewable hydro-electric power, 12% from coal and the balance of 48% was supplied by oil. These figures include hydro-electric energy imported from Quebec to New Brunswick but exclude thermal inputs related to exports from New Brunswick to New England. Nova Scotia’s electrical utility production depended on oil to an even greater extent than the Maritimes as a whole, i.e. 62% in contrast with 48%. There seemed to be a relatively small potential for the development of indigenous energy resources to reduce oil dependency. This need in the Maritime provinces instigated an interest in the unharnessed potential of hundreds of billions of k Whr per year in the Bay of Fundy tides. The large tidal ranges found in the upper reaches of the Bay of Fundy (Fig. 2), with its unique physiographic characteristics, offered optimistic possibilities for harnessing at least a portion of this energy for a pollution-free and relatively inflation-free contribution to the growing energy needs in the region. With the aim of reassessing the possibility of Fundy tidal power, the Bay of Fundy Tidal Power Review Board was created in February 1972 by agreement among the governments of Canada, Nova Scotia and New Brunswick. The board conducted the study of the project under designated task areas such as the, Bay of

MITALI DE and KEITH W. HIPEL

602

Problem

detection/expression

of need

1 Problem

definition

I I

Determination

of evoluatlon

Deflmtion

of

criteria

alternatives

I Evaluation of alternatives with respect to each criterion by means of fuzzy sets

1 Modelling of preferences using fuzzy binary relations

1 Synthesis of information for comparing the alternative solutions by aggregation of preferences

Ordering

of alternatives

t Solutton

of the decision

problem

Fig. 1. Fuzzy MCDM methodology.

Fundy physiography, tidal power plant design, tidal power generation and system studies and environmental aspects and found the project feasible and financially viable.

techniques, market to be economically

3.2 Problem definition The needs are the basis for the formulation of the organizational goal, which is often stated in an abstract, elusive and unclear manner. In the present application, the definition of the project goal is selection of the best possible site for the location of the tidal energy generation project. Although the project goal is defined at this stage, it is not clear as to how such a goal can be measured or achieved. The specific problem investigated in this paper using a fuzzy multicriteria model based upon socio-economic factors is the solution of the most appropriate site for the location of the tidal power generation plant. 3.3 Determination

of evaluation

criteria

The next step in MCDM is to obtain a set of operational objectives by which the project goal may be achieved in terms of the priorities of the decision makers. Formulation of these objectives in explicit and measurable criteria constitutes an important component of the MCDM process. With respect to the major decision makers (namely the government, business enterprise and the public) in the Bay of Fundy Tidal project, four evaluation criteria are considered for the purpose of ranking the possible sites based on the initial assessment of community attitudes obtained from a non-scientific sampling of fifteen key

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projects

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informants. Some important factors to consider when formulating evaluation criteria include relationships between the identified criteria, approximate resource requirements, measurement scales, and organizational structure of the engineering project. The criteria identified by Harvey et ak3’ are as follows. (1) cr. Labour supply in the impact area. The labour supply criterion is a composite of three subfactors: (a). L,, labour pool (the ratio of the number of workers required to the total construction labour force available in the area). (b) L,, construction share (the ratio of the construction workers in the area to the total labour force available in the area). (c) L,, construction labour flows (the number of construction workers working in the area). (2) L’2.Relative economic need in the impact area. The relative need factor is a composite of two subfactors: (a) R,, income growth, measured by the ratio of the average income per tax return in 1974 to average income per tax return in 1964. (b) R,, population growth, measured by the ratio of site area population in 1976 by site area population in 1966. (3) L’~.Supply of key public services. This criteria is composed of two subfactors: (a) S,, Housing, measured by the number of rental units available as a percentage of the total number of housing units in the area. (b) S,, Education/health (S,), community (S,), recreation/business/personal services determined by the number of industry workers per thousand population. (4) cd. Dispersement capability. This criteria refers to the number of centers of population located in the total impact area. This criteria measures dispersement over space. (5) c5. Benefits uersus costs. Benefits and costs of the specified criteria are estimated, with respect to the project, by weighting costs against benefits and the results are used as fifth criteria. 3.4 Assessment

of weights of importance

The assessment of a weight for each criterion is an important problem in MCDM as decision makers must decide, which objectives or criteria are most important. The alternatives in this study are first evaluated where all criteria have the same importance. As part of a sensitivity analysis, weights are assigned according to the relative importance of the criteria and the evaluation is repeated. A recommended method for eliciting criteria weights is the analytic hierarchy process method, which is a hierarchical scaling method proposed by Saaty.3” 3.5 Dejnition

of the alternatives

Following the formal development of criteria, the next stage of MCDM is the formulation of alternative courses of action. Here, an alternative represents one of the feasible choices available to solve the problem. To get a clear picture of the alternatives available for the selection of a site for the project, the location and physiography is briefly discussed. Location and Physiography: As shown in Fig.2, the Bay of Fundy is located between the provinces of New Brunswick and Nova Scotia. The outer portion of the bay is 200 km long from the Gulf of Maine to Cape Chignecto. The width varies from nearly 120km between Yarmouth, Nova Scotia, and Cutler, Maine, to some 45 km at the inner end. This portion of the bay slopes seaward from an average depth of 40m in the vicinity of Cape Chignecto to 130m at its outer limit. The most significant feature of the Bay of Fundy, from the point of view of developing tidal power, is the range of its tides. These tides are among the highest in the world, with spring tide ranges of about 16 m in Minas Basin and 14 m in Chignecto Bay. The magnitude of the spring tides decreases towards the mouth of the Bay of Fundy and is only 9m at Saint John. These large tides result from amplification both within the Bay of Fundy and in the Gulf of Maine. The Fundy tides are not only very large but are also very regular and are a semi-diurnal type. From the point of view of their size and regularity, the Fundy tides are well suited to the development of tidal power. Socio-economic Aspects of Tidal Power Generation: When choosing the Bay of Fundy site, the review board considered aspects such as geology, seismology, climate, winds, ice, waves, tidal currents, ecology and environment. An evaluation of the socio-economic Er:yI.::.F

604

MITALI DE and KEITH W. HIPEL

implications constitutes an essential facet of an overall appraisal of a tidal power development. Primarily, the aim of the study is to identify and examine potential stresses and strains in the social system that may arise as a result of the implementation of the proposed Fundy Tidal Power concepts as three possible sites identified by the committee. For the purpose of analysis in this study, the immediate site area is defined as being within 10 road miles (16 km) of the actual site and the total impact areas are defined as being within a radius of approximately 60 road miles (96 km) of the actual site. The natural geographic features of the three areas, selected for consideration, as possible sites and the existing road networks in both New Brunswick and Nova Scotia, show that the particular impact areas will be dependent upon the choice of a northern or southern starting point for any one of the sites under consideration, due to the different patterns of access implied. Regardless of the size of the projects, this would hold and hence the three sites were subdivided into the following possible categories. These choice of sites formed the alternatives ai, i = 1 . . ...6 to the MCDM problem. (1) UI. B9 South: Tenny Cape, Nova Scotia, (impact area entirely in Nova Scotia) (2) a2. B9 North: Economy Point, Nova Scotia, (impact area entirely in Nova Scotia, except for Sackville, New Brunswick area). (3) u3. A8 South: Boss Point, Nova Scotia, (impact area is evenly divided between New Brunswick and Nova Scotia). (4) cr.+.A8 North: Pecks Point, New Brunswick. New Brunswick, (though A6S and A8N have (5) 4. A6 South: Cape Maringoin, different starting points, their impact areas are the same, being mainly in New Brunswick but extending into Cumberland county and North-west part of Colchester county, Nova Scotia). (6) a6. A6 North: Mary’s Point, New Brunswick (impact area entirely in New Brunswick, except for Amherst Nova Scotia area). The map in Fig. 2 from Harvey,37 graphically illustrates the impact areas and the expected change in location of the impact areas as a direct consequence of choosing one starting point over another.

3.6 Evaluation of the alternatives It is at this stage of the MCDM process that the experience and knowledge of the decision maker become important since different points of view are explained, discussed and utilized in the detailed evaluation of the different solutions. Because alternatives may not be evaluated at the same performance level, imprecision in predicting the values of the alternatives justifies the use of evaluations based on the theory of fuzzy sets. Fuzzy sets provide a systematic method for simultaneously incorporating both quantitative and qualitative information into the decision process without obscuring the given information. In evaluating any finite set of n alternatives A = (a,, u2,. . . . . a,), against a set of m criteria C = (c,, c2,. . . . . cm), a matrix of evaluations can be formed (Table 1). In this evaluation matrix, the intersection of row ci and column aj reopresents an evaluation fi(aj), which is a fuzzy set indicating how well alternativej satisfies the ith criteria. The evaluation is carried out by using the idea of a membership function defined over the range [0, 1] against some qualitative scale where 0 represents least importance and 1 denotes most importance. Table 1 presents a general picture of an evaluation matrix thus formed for a single decision maker. In addition, if there are k interest groups then for the kth group or position, each element ft in the evaluation matrix fk measures the value of criteria cj for alternative aj. Therefore, in terms of fuzzy set theory, for the kth interest group, jth alternative and ith criterion a fuzzy set is defined as the set of ordered pairs given by [aj, f:(aj)] where membership function f’ for the kth group and ilh factor, maps the alternative aj into the interval [O, l] and the value of the membership function is now fki(aj).18 The use of a membership function to numerically represent the degree to which an element belongs to the set is the central concept of fuzzy set theory. Using this concept, the evaluation matrix of this case

Comparing

energy

projects

45'

,

Fig. 2.Total impact

area for the Bay of Fundy

Project.

Table

matrix

maker

1. Evaluation

for a single decision

ALTERNATIVES a_

C

=2

R I T E R 1

A

c

’

a,

a4

I

a

n

MITALI DE and KEITH W. HIPEL

606

study is constructed as shown in Table 2. Each entry value in the evaluation matrix lies between 0 and 1, indicating the satisfaction provided by the corresponding alternative to the respective criterion. Technical information, such as costs, are evaluated based on an incremental scale. The cheapest value brings highest satisfaction (equal to l), while the most expensive value is assigned the lowest satisfaction (equal to 0). Qualitative information, such as good living, is evaluated with respect to a numerical scale. The scale used to indicate numerical data considers very adequate as 1.0, adequate as 0.75, regular as 0.50, fairly adequate as 0.25 and completely inadequate as 0. 3.7 Modelling of preferences When a decision maker compares a pair of alternatives, there arises those special cases when the decision maker has definite preferences with respect to criteria. However, preference question on most pairs is not answered by a simple yes or no and ends up being less definite, vague or ambiguous. Thus, it is appropriate to model preference relations over a set of alternatives using fuzzy set theory. Preferences are commonly modelled as binary relations resulting from pairwise comparisons. The binary relations are assumed to have some minimal properties so that the alternatives can be consistently ordered. The fuzzy binary relations can be considered as a generalization of the ordinary binary relations in the sense that each ordered pair is allowed a grade of membership in the interval [0, I], instead of only the points (0, l}, as in the case of ordinary binary relations. Fuzzy preference relations have the advantage of allowing different degrees and various strengths of preference to be reflected in the preference model. To model the preference of two alternatives a and be A, classical decision theory introduces two preference situations which are strict preference and indifference. Because these two situations are assumed to be transitive they may lead to unrealistic conclusions.16 Transitivity of preferences requires that if outcome x is preferred to y and y is preferred to z, then x must be preferred to z. Preferences are intransitive when the first two conditions hold, but in the third condition z is preferred to x. This situation can be improved by introducing an indifference threshold 4. A preference threshold p(p > q) may also be incorporated into the modelling of preferences to represent a weak preference when the bounds of indifference and strict preference are difficult to define. Both thresholds lie inside the interval [0, l]. This concept of preference relations is used extensively in the ELECTRE methods 12,13,16.28 In decision theory, when constructing preference relations, different forms of criteria has to be classified as well. Criteria are classified as true criteria, criteria with linear preference, quasi-criteria, pseudo-criteria, and apparent pseudo-criteria.‘2.13*28 Therefore, in a MCDM problem, the decision maker has to choose the criteria type, as well as the required thresholds, to suit the problem at hand. For the purpose of this case study, pseudo-criteria (linear preference with indifference threshold), which is the most general of all possible types of criteria, is used. To give a more precise definition, consider fi(a) and f,(b) as the corresponding evaluation of alternatives a and b respectively, for a particular evaluation criteria i. Then, the fuzzy preference relation aPb such that 0 < aPb < 1, will be represented by

aPB =

0

if fi(a) G fi(b),

g(z)

if fi(a) > fi(b).

where g(z) = g[fi(a) - f:(b)]. Then a case of pseudo-criteria is where p and 4 represent preference threshold and indifference threshold respectively, and

g(z) =

0,

ocz