18 Elements of Multicriteria Decision Making Abdollah Eskandari and Ferenc Szidarovszky University of Arizona, Tucson, Arizona Abbas Ghassemi New Mexico State University, Las Cruces, New Mexico 1 INTRODUCTION Our life is filled with situations when decisions have to be made. Buying a car, selecting a wife, choosing the best route to work, etc., show such situations. In engineering design we always face decisions. The consequences of our deci- sions are usually complex; they can by measured and evaluated only by several criteria. In environmental engineering these criteria include water and air quality, and their effects on people, livestock, plants, and wildlife, among others. There- fore all of these measures should by incorporated into the decision process, that is, we have to take multiple criteria into account. This chapter will give a brief introduction into modeling and solving such engineering decision problems. 2 MODELING DECISION PROBLEMS Every decision-making problem is based on choice. We need to have options to select from. Options, or in other words, decision alternatives, may be the types of Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. cars available on the market, or the different technology variants to treat a region or to build a water treatment plant. The first step in modeling decision problems is to see which set of alternatives we may choose from. This set can be finite or infinite. If the number of decision alternatives is finite, then the decision problem is called discrete. For example, if finitely many technical variants are available for an engineering project, then the decision problem is discrete. In this case we have to prepare a list of the alternatives. If the number of alternatives is infinite, then no such list is possible. In such cases decision variables have to be introduced, and the decision alternatives are identified by the different values of the decision variables. If x denotes the capacity of a wastewater treatment plant, then the different values of this variable represent the different variants for the capacity. The decision variables are usually collected in the compo- nents of a decision vector. If x 1 , x 2 , . . . , x m are the decision variables, then x =(x 1, x 2 , . . . , x m ) is the decision vector. If the decision alternatives are repre- sented by the different continuous values of a decision vector, then the decision problem is called continuous. Sometimes discrete problems with a large number of alternatives are approximated by continuous models. The second step of the modeling process is known as the feasibility check. There might be many different reasons why some alternatives should be dropped from the list. For example, our budget poses a constraint on which cars could be purchased. In the case of a discrete problem, the infeasible alternatives have to be dropped from the list. The result of the feasibility check is therefore a reduced list of alternatives. In the case of a continuous problem, we are unable to check all options one by one, since there are usually too many of them or we might even have infinitely many possibilities. In such cases feasibility has to be expressed by certain constraints, which model the additional requirements that determine feasibility. As an example, assume that a function g o (x 1 , . . . , x m ) represents the cost of a project, and we cannot spend more than an amount of Q o dollars. Then this budgetary constraint can be formulated as the inequality g o (x 1 , . . . , x m ) < = Q o (1) In most practical problems, several similar constraints should be satisfied for feasibility. These constraints are either < = , or > = , or = types. Without restricting generality, we may assume mathematically that all constraints are given by inequalities of > = type. If an inequality is defined by a < = relation, then it must have the form g 1 (x 1 , . . . , x m ) < = Q 1 (2) which is equivalent to the relation −g 1 (x 1 , . . . , x m ) > = −Q 1 (3) Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. which is a > = type of inequality. If a constraint is given by an equality, then it can be rewritten as a pair of > = type inequalities in the following way. Assume that g 2 (x 1 , . . . , x m )=Q 2 (4) in a constraint of equality type. It is equivalent to the pair of inequalities g 2 (x 1 , . . . , x m ) < = Q 2 g 2 (x 1 , . . . , x m ) > = Q 2 (5) By multiplying the first inequality by (–1) we have the pair −g 2 (x 1 , . . . , x m ) > = −Q 2 g 2 (x 1 , . . . , x m ) > = Q 2 (6) Therefore in our further discussions we will always assume that the feasible alternative set of continuous decision problems is given by a system of > = inequalities g 1 (x 1 , . . . , x m ) > = Q 1 g 2 (x 1 , . . . , x m ) > = Q 2 . . . (7) g n (x 1 , . . . , x m ) > = Q n This inequality system can be formulated in a more simple and compact form by introducing the vectors g(x)=        g 1 (x 1 , . . . , x m ) g 2 (x 1 , . . . , x m ) . . . g n (x 1 , . . . , x m )        and Q =        Q 1 Q 2 . . . Q n        (8) Then system (7) can be rewritten as g(x) > = Q (9) We will use the notation X for the set of the feasible decision alternatives of continuous problems. By using the above notation we may write X = {x|x ∈ R m , g(x) > = Q} (10) where R m denotes the set of all m-element real vectors. The above discussion shows that the result of the feasibility check of continuous problems is the Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. formulation of the system of constraint (7) or (9), and the definition of the feasible set (10). It is worthwhile to mention that even discrete problems can be demon- strated in this way. If there are r alternatives to select from, then X can be defined as the discrete set X = {1,2, . . . , r}. The third step of modeling decision problems is the formulation of the evaluation criteria. Each criterion evaluates the goodness of the different alterna- tives from a specific point of view. If only one decision maker is present, then the criteria reflect his or her complicated evaluation system. In the case of multiple decision makers, the evaluation systems of all decision makers are combined and summarized in the criteria. In the first case a suitable alternative is determined that gives an acceptable trade-off among the criteria, and in the second case an appropriate compromise solution is determined that is acceptable by all parties involved in the decision-making process. Some criteria are easy to quantify. For example, cost, profit, volume, capacity, etc., have specific values for each decision alternative. Some other criteria might not be easily quantifiable. For example, the esthetic consequences of a forestry treatment technology can be judged only subjectively. Subjective judgments can be made verbally, such as very good, good, fair, bad, unacceptable, or in any other similar way. These subjective measures can be then quantified. For example, on a [0, 100] scale, the above evaluation categories can be identified by 100, 75, 50, 25, and 0, respectively. In the case of a discrete problem, a payoff matrix is hence constructed, where the rows correspond to the decision alterna- tives, the columns correspond to the evaluation criteria, and the (i, j) element a ij of this matrix is the evaluation of alternative i with respect to criterion j as shown in Figure 1. After the payoff matrix is constructed, the decision problem is considered to be mathematically well defined. In the case of a continuous problem, let ƒ 1 , ƒ 2 , . . . , ƒ s denote the evaluation criteria as earlier, let x be the decision vector, and assume that the feasibility of the decision vector is given by the system of inequalities (7) or (9). Then the decision problem is mathematically well defined by the multicriteria optimization problem Maximize ƒ j (x 1 , . . . , x m )(j = 1, 2, . . . , s) (11) Subject to g l (x 1 , x 2 , . . . , x m ) > = Q l (l = 1, 2, . . . , n) or, using vector notation, Maximize ƒ(x) (12 ) Subject to g(x) > = Q where Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. ƒ(x)=        ƒ 1 (x 1 , . . . , x m ) ƒ 2 (x 1 , . . . , x m ) . . . ƒ s (x 1 , . . . , x m )        In most applications the problem formulation is itself a complicated, lengthy project, and needs a continuous interaction with all parties involved and interested in the decision problem and its solution. The process should start with identifying decision makers, or in other words, the stakeholders. Next we have to find out what they can decide on in order to find the set of decision alternatives. There are very often regulations, requirements, public opinion, etc., which influence and restrict choices. These restrictions determine the set of feasible alternatives. The formulation of the evaluation criteria is also itself a complicated process. In every engineering project we wish to accomplish certain goals, such as cleaning wastewater, providing certain products to the public, etc. At the same time, we want to avoid certain negative effects such as worsening air quality, water quality, paying too much for the project, etc. After the goals and the negative effects have been identified verbally, we have to find those quantities which can be used to characterize the goodness of the different alternatives with respect to the goals and additional consequences. These quantities should measure how the goals are satisfied if the different alternatives are selected and carried out, how certain it is that the project will be successful under the different alternatives, and how severe the additional negative consequences will be. These measurable or at least subjectively quantifiable quantities are usually selected as Criteria Alternatives 1 2 3 … s 1 2 3  r 11 a 12 a 13 a … 1 s a 21 a 22 a 23 a … 2 s a 31 a 32 a 33 a … 3 s a    1 r a 2 r a 3 r a … rs a FIGURE 1 Illustration of a payoff matrix. Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. the evaluation criteria. Without restricting generality, we may assume that all criteria are maximized. If smaller values are better for a criterion, then by multiplying this criterion by (–1) we obtain a maximizing criterion. If a target value has to be reached in the case of a certain criterion, then the distance between the actual and target values has to be minimized, or the negative of this distance has to be maximized. In the final problem formulation in both the discrete and continuous cases, we will always assume therefore that all criteria are to be maximized. In the decision science literature problem (11) or (12) and the one given in Figure 1 are called multiple–criteria decision problems, or multiobjective pro- gramming problems. In the next section the solution of such problems will be defined and examined, and in the later parts of this chapter we will give the fundamentals of the most popular solution algorithms. A comprehensive summary and additional details are presented in Ref. 1. 3 MULTIOBJECTIVE PROGRAMMING As a simple illustration, consider the problem of designing a wastewater treatment plant. Assume that there are two design alternatives which cost $2,000,000 and $2,500,000, respectively. Assume that the capacities are 1500 m 3 /day and 2000 m 3 /day, respectively. It is also assumed that both versions result in the same output quality. Since lower cost but higher capacity is better, the first criterion has to be transformed into a maximizing criterion by multiplying it by (–1). The resulting payoff matrix is given in Table 1. Notice that this problem is a discrete multiobjective programming problem. As a simple continuous example, consider the following problem. Assume that a combination of three alternative technologies can be used in a wastewater treatment plant. Let x 1 ,x 2 denote the proportion (in percent) of the application of technology variants 1 and 2; then 1 − x 1 − x 2 is the proportion of the third technology. Therefore we have two decision variables, x 1 and x 2 , and they have to satisfy the conditions x 1 > = 0, x 2 > = 0, and x 1 + x 2 < = 1. The feasible set is illustrated in Figure 2. TABLE 1 Payoff Matrix for Wastewater Treatment Plan Alternatives/Criteria –Cost (–$) Capacity (m 3 /day) 1 –2,000,000 1,500 2 –2,500,000 2,000 Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. Assume that there are two major pollutants that have to be removed from the wastewater, and the three technology variants remove 3, 2, 1 mg/m 3 , respec- tively, of the first kind of the pollutant, and 2, 3, 1 mg/m 3 of the second kind of pollutant. Then the total removed amount of the two pollutants can be obtained as [3x 1 + 2x 2 + 1(1 − x 1 − x 2)] V and [2x 1 + 3x 2 + 1(1 − x 1 − x 2 )]V, where V is the total amount of treated wastewater. Maximizing those objectives is equivalent to maximizing 2x 1 + x 2 and x 1 + 2x 2 . Therefore we have the following multiobjec- tive programming problem: Maximize 2x 1 + x 2 and x 1 + 2x 2 Subject to x 1 > = 0, x 2 > = 0 (13) x 1 + x 2 < = 1 Notice that (13) is a continuous problem with two decision variables and two objective functions. In many decision problems we are also interested in knowing the set of all feasible objective values. For each x ∈ X, the vector ƒ(x)=(ƒ 1 (x 1 , . . . , x m ), ƒ 2 (x 1 , . . . , x m ), . . . , ƒ s (x 1 , . . . , x m )) is called the objective vector at x, and the set of all feasible objective vectors, H = {u|u = ƒ(x) with some x ∈ X} is called the objective space. If there are s objectives, then H ⊂ R s . In our discrete problem we have two objectives, therefore the objective space is two-dimensional. Since there are only two alternatives, the objective space has only two points: (–2,000,000, 1,500) and (–2,500,000, 2,000). In every 2 x (0,1) 1 x (1,0) X FIGURE 2 Illustration of the feasible set for the continuous example. Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. discrete problem, every point of the objective space corresponds to a decision alternative, but different alternatives may have the same point, if the objective function values are identical. For our discrete example, Figure 3 shows these points when the first objective is given on the horizontal axis and the second objective is identified by the vertical axis. Point A1 shows the simultaneous objective function values for the first alternative, and point A2 shows the same values for the second alternative. The question that arises now is the choice among the two alternatives, which is the same as selecting the “better” point among A1 and A2. If cost is our only concern, then point A1 (that is, alternative 1) must be our choice. If capacity is our only concern, then point A2 with alternative 2 is our choice. The dilemma we face here is the conflict between the two objectives, since higher capacity can be obtained by selecting A2 but then the cost becomes worse. If both objectives are important to us, then neither of the alternatives is better than the other and hence either alternative seems to be a reasonable choice. With decision science terminology we might say that no alternative is dominated by another alternative and therefore both alternatives are nondominated. In the economic literature, nondominated alternatives are called Pareto optimal. Let us define these terms mathematically. In a discrete problem, let p and q be two alternatives, and let ƒ(p) and ƒ(q) denote the corresponding objective vectors. We say that alternatives p and q are equivalent if ƒ(p)=ƒ(q). Here the equality of vectors is defined by component-wise equality. That is, two alternatives are equivalent to each other if they result in the same values in all objective functions. Similarly, we say that alternative p dominates alternative q, if ƒ(p) > = ƒ(q) and there is a strict inequality in at least one of the objectives. Here the notation ƒ(p) > = ƒ(q) means that each Objective 2 A2 2,000 A1 1,500 -2,500,000 -2,000,000 FIGURE 3 Objective space for a discrete problem. Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. component of vector ƒ(p) is greater than or equal to the corresponding component of ƒ(q). In our example, ƒ(1) = (–2,000,000, 1,500) and ƒ(2) = (–2,500,000, 2,000) and none of the alternatives dominates the other one. If no other alternatives dominate an alternative p, then it is called nondominated. In our example, therefore, both alternatives 1 and 2 are nondominated. In the case of continuous problems, the objective space can be constructed by representing the set H = {u|u =(u 1 , . . . , u s )|u j = ƒ j (x) with some x ∈ X for j = 1,2, . . . , s} in the s-dimensional space. In our continuous example u 1 = 2x 1 + x 2 and u 2 = x 1 + 2x 2 From these equations we see that x 1 = 2u 2 − u 1 3 and x 2 = 2u 1 − u 2 3 The constraints of problem (13) have now the form 2u 2 − u 1 3 > = 0 2u 1 − u 2 3 > = 0 and 2u 2 − u 1 3 + 2u 1 − u 2 3 > = 1 These inequalities can be simplified as u 2 > = u 1 2 u 2 < = 2u 1 u 1 + u 2 < = 3 (14) The set of feasible payoff vectors (u 1 ,u 2 ) satisfying these conditions are illustrated in Figure 4. The vertices A, B, C of the triangle can be obtained as the intercepts of the lines u 2 = u 1 /2 and u 2 = 2u 1 , u 2 = 2u 1 and u 1 + u 2 = 3, u 2 = u 1 /2 and u 1 + u 2 = 3, respectively. Simple calculation shows that A = (0, 0), B = (1, 2) and C = (2, 1). We will next show that the nondominated objective vectors form the linear segment connecting points B and C including the endpoints. All points between A and B including point A are dominated by the objective vector B, since both coordinates of B are larger. Similarly, all points between A and C are dominated by C. Let now u be a point inside the triangle. The continuation of the linear segment connecting points A and u intercepts the linear segment between B and C in a point D. Since the slope of the line connecting A and u is positive (between 1 ⁄ 2 and 2), the objective vector D dominates u. Hence u is dominated. If D is any point of the linear segment connecting B and C (including the endpoints), then it is impossible to increase any objective value without worsening the value of the other objective function. Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. Thus, the points of the linear segment between B and C are the nondominated objective vectors. Any nondominated point and the corresponding decision alternative is a reasonable choice. The nondominated decision alternatives can be obtained in the following way: X o = {x|x ∈ X, ƒ(x)∈H o } (15) where H o is the set of nondominated objective vectors. In the case of our example, the decision vector x =(x 1 ,x 2 ) is nondominated if and only if 1 < = 2x 1 + x 2 < = 2 and (2x 1 + x 2) +(x 1 + 2x 2 )= 3 These relations can be rewritten as x 1 + x 2 = 1 and 0 < = x 1 < = 1 (16) and the set of the nondominated decision vectors is the linear segment connecting the points (1, 0) and (0, 1) in Figure 2. Hence any point of this linear segment is a reasonable selection. These nondominated alternatives do not use the third technology variants. In the case of our discrete example both alternatives are nondominated, and in the case of the continuous example there are infinitely many nondominated alternatives. Notice that the different nondominated solutions give different objective function values, contrary to single-objective optimization problems, where all alternative optimal solutions give the same optimal objective function value. In other words, in single-objective optimization problems the different optimal solutions are equivalent to each other, but in the presence of multiple FIGURE 4 Objective space for a continuous problem. Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. [...]... objectives The knowledge of the status quo point or a nadir and the suggested direction of improvement or the specification of an ideally best point and the required direction of relaxation makes the use of direction-based methods possible Distance-based methods are used when the directions of improvement or relaxation are replaced by distance functions This diversity of types of preference information... vector of relaxation B ideal point S2 H S1 A direction vector of improvement u1 FIGURE 8 Example of dominated solutions of direction-based method Copyright 2002 by Marcel Dekker, Inc All Rights Reserved In the case of the discrete problem of Table 1, let ƒ∗, , ƒ∗ denote the 1 s coordinates of the ideally best point The objective vector of alternative i is given by the ith row of the payoff matrix:... best for any of the decision makers, but it is a solution that can be accepted by all of them This kind of methodology is known as social choice A comprehensive summary of social choice procedures can be found in Refs 2 and 3 This section introduces the fundamentals of this methodology The data, or mathematical model, consists of the sets of alternatives and criteria, the importance weights of the different... feasible and therefore the optimal value of t is t∗ = 1, and the corresponding solution is x∗ = 1 and x∗ = 0 1 2 There is no guarantee that the solution obtained by either variant of this method is nondominated Such examples are shown in Figure 8, where A denotes the nadir and B is the ideal point The directions for improvement and relaxation are shown as well The solutions are the points S1 and S2;... 3.5 Distance-Based Methods Similarly to direction-based methods, two major distance-based methods are applied in practical cases The more frequently used variant is based on the knowledge of an “ideally best” point and a distance function of s-dimensional vectors The solution that gives the closest objective vector to the ideal point is selected as the solution Depending on the selected type of distance,... discrete example of Table 1 assume that the lowest cost is 0 and the largest acceptable cost is 3,000,000 Then the worst and best values for the first objective are ƒ1∗ = −3,000,000 and ƒ∗ = 0 The normalized elements of the first column of 1 the payoff matrix become − a11 = a11 − ƒ1∗ ƒ∗ − ƒ1∗ 1 = −2,000,000 + 3,000,000 1 = 0 + 3,000,000 3 = −2,500,000 + 3,000,000 1 = 0 + 3,000,000 6 and − a21 = a21... may chose the weights c1 = 2⁄3 and c2 = 1⁄3 Therefore the weighted average of the normalized objectives becomes 2⁄3(x1 + 1⁄2x2) + 1⁄3(1⁄2x1 + x2) = 5⁄6x1 + 4⁄6x2 and the optimal solution is x∗ = 1 and x∗ = 0 1 2 3.4 Direction-Based Methods Two direction-based methods are used in practical applications The more frequently applied variant is based on the knowledge of a point of the objective space which... which is the sum of the weights of all criteria which prefer alternative i1 against i2 Similarly, N(i2,i1) = ∑ {wj|ai j 2 < ai1j} (54) j and alternative i1 is considered better than i2 if and only if N(i1,i2) > N(i2,i1) (55) Consider again the example of Table 4, and assume that comparisons are made in the order 1, 2, 3, 4 We first compare alternatives 1 and 2 by computing N(1,2) = 0.4 and N(2,1) = 0.3... and N (4, 3) = 0.3 + 0.2 = 0.5 Therefore alternatives 3 and 4 are equivalent, but alternative 2 is better than 4 Hence alternative 4 is dropped and both alternatives 2 and 3 are considered as the social choice solution 5 CONFLICT RESOLUTION In this section we assume that there are s decision makers, and each has a well-defined objective function on the set of all alternatives Let X denote the set of. .. of the different objective functions, and a vector defining the direction of relaxation Similarly to problem (30), this concept can be modeled as the following optimization problem: Minimize t Subject to u∗ − tv ∈ H (31) where u∗ is the ideally best point, and vector v gives the direction of relaxation In the case of the continuous example (13), assume that u∗ = (3,3) and v = (2,3) The definition of . removed from the wastewater, and the three technology variants remove 3, 2, 1 mg/m 3 , respec- tively, of the first kind of the pollutant, and 2, 3, 1 mg/m 3 of the second kind of pollutant. Then. multiobjective pro- gramming problems. In the next section the solution of such problems will be defined and examined, and in the later parts of this chapter we will give the fundamentals of the most. result of the feasibility check of continuous problems is the Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. formulation of the system of constraint (7) or (9), and the definition of