486 European Journal of Operational Research 76 (1994) North-Holland

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1 486 European Journal of Operational Research 76 (1994) North-Holland Theory and Methodology TOPSIS for MODM Young-Jou Lai, Ting-Yun Liu and Ching-Lai Hwang Department of Industrial Engineering, Kansas State University, Manhattan, KS 66506, USA Received November 1990; revised February 1992 Abstract: In this study, we extend TOPSIS to solve a multiple objective decision making problem. The principle of compromise (of TOPSIS) for multiple criteria decision making is that the chosen solution should have the shortest distance from the positive ideal solution as well as the longest distance from the negative ideal solution. Thus, we reduce a k-dimensional objective space to a two-dimensional objective space by a first-order compromise procedure. We then use membership functions of fuzzy set theory to represent the satisfaction level for both criteria. We obtain a single-objective programming problem by using the max-min operator for the second-order compromise operation. To illustrate the procedure, the Bow River Valley water quality management problem is solved by use of TOPSIS. Keywords: Multiple objective decision making; Positive ideal solution; Negative ideal solution; Fuzzy sets; Membership functions I. Introduction Decision making is the process of selecting a possible course of action from all of the available alternatives. In almost all such problems the multiplicity of criteria for judging the alternatives is pervasive. That is, for many such problems, the decision maker wants to attain more than one objective or goal in selecting the course of action while satisfying the constraints dictated by environment, processes, and resources. Mathematically, these problems can be represented as: max [fl(x), f2(x)... f~(x)] (la) s.t. x~x={xlgi(x)<o,i=l,2... m} (lb) where x is an n-dimensional decision variable vector. The problem consists of n decision variables, m constraints, and k objectives. Any or all of the functions may be nonlinear. In the literature, this problem is often referred to as a vector maximization problem. Because of incommensurability and the conflicting nature of the multiple criteria, the problem becomes complex, but interesting. Kuhn and Tucker published one of the earliest considerations of multiple objectives in In the last two decades a large variety of techniques have been published. The most recent books and monographs are Keeney and Raiffa [13], Hwang and Masud [10], Zeleny [27], Goicoechea, Hansen and Duckstein [7], Yu [25], Steuer [21], Zimmermann [29], and Seo and Sakawa [20]. The recent survey papers include Hwang, Paidy, Yoon and Masud [11], Lieberman [18], White [22] and Correspondence to: Prof. Y.-J. Lai, Department of Industrial Engineering, Kansas State University, Manhattan, KS 66506, USA /94/$ Elsevier Science B.V, All rights reserved SSDI (92)

2 Y-J. Lai et al / TOPSIS for MODM 487 many others. According to the information needed, Hwang and Masud have systematically classified the MODM techniques into four classes: a) no articulation of preference information; b) a priori articulation of preference information; c) progressive articulation of preference information or interactive methods; and d) a posteriori articulation of preference information or nondominated solution generation methods. In this study we will provide a new approach, TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) for MODM, which belongs to the first class (no articulation of preference information), to solve (1). TOPSIS was first developed by Hwang and Yoon [12] for solving a multiple attribute decision making problem. It is based upon the principle that the chosen alternative should have the shortest distance from the Positive Ideal Solution (PIS) and the farthest from the Negative Ideal Solution (NIS). The single criterion of the shortest distance from the given goal or the PIS may be not enough to satisfy decision makers. In practice, we might like to have a decision which not only makes as much profit as possible, but also avoids as much risk as possible. A similar concept has also been pointed out by Zeleny [27] and Hall [9]. Furthermore, to choose the positive and negative ideal solutions as reference points has long been accepted. For example, Christians have their heaven (PIS) and hell (NIS). In the following section, we will discuss the family of dp-distance and its normalization. The TOPSIS approach is presented in Section 3. By use of TOPSIS, we will solve the Bow River Valley water quality management problem in Section 4. We will also discuss the TOPSIS solutions in Section 5. Finally, concluding remarks and future studies will be given in Section dp-distance and normalization To define (a) reference point(s) is usually the first step solving Multiple Criteria Decision Making (MCDM) problems. Global criterion method, goal programming, fuzzy programming and interactive approaches all need initial reference points in order to obtain (a) compromise solution(s). Among the various concepts regarding reference points, the concept of the ideal system is, in our point of view, the most important one. For example, Thomas Moore's Utopia and Christians' heaven have been widely accepted as ideal systems for a long time. With a given reference point, the MCDM problems can then be solved by locating the alternative(s) or decision(s) which are the closest to the reference point (or the ideal point). Thus, the problem becomes how to measure the distance to the reference point. Goal programming measures this distance by using the weighted sum of absolute distances from given goals. The global criteria method measures this distance by using Minkowski's Lp metric. The Lp metric defines the distance between two points, f and f* (the reference point), in k-dimensional space as dp = (f~* -f~)p, where p >_ 1. (2) t= One physical property of dp measurement is well known: when p increases, distance dp decreases, i.e., d~ >_ d 2 > >_ d~ and greater emphasis is given to the largest deviation in forming the total. Specifically, p = 1 implies an equal importance (weights) for all these deviations, while p = 2 implies that these deviations are weighted proportionately with the largest deviation having the largest weight. Ultimately, for p = ~, the largest deviation completely dominates the distance determination, i.e., d~ = max{[l* -L I}. t Distances p = 1, 2, and m are especially operationally important: d x (the Manhattan distance) and d 2 (the Euclidean distance) are the longest and the shortest distances in the geometrical sense; d= (the Tchebycheff distance) is the shortest distance in the numerical sense. Unfortunately, because of the incommensurability among objectives, it is impossible to directly use the above distance family. Thus, we need to normalize the distance family to remove the effects of the

3 488 Y.-J. Lai et al. / TOPSIS for MODM f2 c su -c -c Figure 1. Indifference distance curves with different d p incommensurability. Yu and Zeleny [24] and Zeleny [26] normalized the distance family of (2) by using the reference point. The distance family then becomes (k )l/p dp= ~_, [(ft*-ft)/ft*]", wherep>l. (3) t=l We have seen that the amount of dp decreases when parameter p increases. Another property of d, can be revealed by considering indifference (distance) curves which display the relationship between the distance function and human choice behavior. Any point on the same indifference curve has equal distance from the given reference point. The loci of points c units from the reference point of a two-dimensional space with p = 1, 2, and oo are given in Figure 1. From the figure, we find that the length of locus sp (s I < s2 < s~o) increases when dp decreases (d 1 > d 2 > > d ). More points can be contained in the locus with the larger p; or more points are expressed by the same distance with the larger p. Thus, Yoon [23] pointed out that the distance function becomes less specific or less credible as parameter p increases. This is consistent with measurement systems. A measurement system is more credible when it has more classes. For example, suppose we use only integer numbers in measuring temperatures. The Fahrenheit scale can then be said to be more credible than the Centigrade scale because the former divides the temperature range between freezing and boiling into 180 degrees (classes) while the latter divides this range into 100 degrees (classes). A temperature expressed by 20 C can be further divided into 67 F, 68 F and 69 F. Thus the more credible measuring system has smaller class size. The previous indifference curves contain only an homogeneous amount of distance and their length can be viewed as a class size. Thus, the credibility of a distance function dp decreases when parameter p increases. 3. TOPSIS for MODM To obtain a compromise solution for (1), the global criteria method uses the distance family of (3) with the ideal solution being the reference point. The problem becomes how to solve the following auxiliary problem: k )lfp mindp= x~x E [(ft*--ft)/ft*] t=l p

4 f2 f2* Y.-J. Lai et al. / TOPSIS for MODM 1 ~ -/I' r' 489 f2" f" I I! fl" Figure 2. Two compromise solutions based on PIS (f*) and NIS (f*) f fl or min dp = x~x k )l/p ]~ [(ft(x*) -ft(x))/ft(x*)] p t=l (4) where x * is the (positive) ideal solution and p = 1, 2,..., ~. The value chosen for p reflects the way of achieving a compromise by minimizing the weighted sum of the deviations of criteria from their respective reference points (ideal solution). Boychuk and Ovchinnikov [2] have suggested p = 1 which is consistent with the argument (p = 1 is the most credible situation) in the previous section. In (4), we only consider one criterion of the least distance from the (positive) ideal solution. However, just a Christians have their heaven and hell, and investors have profit and risk, we would like to have PIS and NIS. The best alternatives or decisions are those that have the shortest distance from PIS and the farthest from NIS. The best investments are those that make as-much-as-possible profit and avoid as-much-as-possible risk. Sometimes the chosen alternative or decision which has the minimum Euclidean distance from PIS has a shorter distance to NIS than some other alternatives. That is, the compromise solution based on PIS is not identical to that which is based on NIS (as indicated in Figure 2). Thus we should consider both criteria simultaneously. Since the solutions A 1 (based on PIS) and A 2 (based on NIS) are usually different, we have to resolve them. Based upon the above arguments, Hwang and Yoon [12] developed TOPSIS to solve multiple attribute decision making problems. In this study, we further extended the concept of TOPSIS to develop a methodology for solving multiple objective decision making problems. Hwang and Yoon used both PIS (f*) and NIS (f-) to normalize the distance family and obtain t = 1 ft* - i"7, where p > 1. (5) Suppose we have the following MODM problem: max/min [ fa(x), f2(x),...,fk(x)] s.t. x~x={xjgi(x){>, =, <}0, i=1,2... m} (6a) (6b)

5 490 Y.-J. Lai et al. / TOPSIS for MODM where fj(x) - Benefit objective for maximization, j ~ J. f,(x) = Cost objective for minimization, i ~ I. In order to use the distance family of (5) to resolve (6), we must first find PIS (f*) and NIS (f-) which are: f* : ( xrnax (or min)fs(x) (or fi(x)), Vj (and i)} (7a) and f-= {min(or max)fs(x ) (or f~(x)), Vj (and i)} (Tb) wherej~jandi~i.f*={ * * f2, f2... fk*} (f-- {fi-..., fk})) is a set of individual positive (negative) ideal solutions, and is a point solution in the k-dimensional objective-functional space. Using the PIS and the NIS, we obtain the following distance functions from them, respectively: {- [fj*-d(x)] p - [L(x)-fi*]"} dp's= )_2 w? / ~,_---~ + z; w;i 7=---~ j~j [ j j i~l [ Ji.#i (8a) and _pdnis---- ~ 2~ Wj / ~--T "]- 2" W] / 7-- [jej L J J ie[ i_ Ji Ji 1/p (8b) where wt, t = 1, 2... k, are the relative importance (weights) of objectives, and p = 1, 2... oo. In order to obtain a compromise solution, we transfer (6) (with the original k incommensurable and conflict objectives) into the following bi-objective problem with two commensurable (but often conflicting) objectives: min d Pis(x) max dpnts(x) s.t. x ~X (9a) (9b) (9c) where p= 1, 2... pp. Since these two objectives are usually conflicting to each other, we cannot simultaneously obtain their individual optima. Each objective achieves only a 'PORTION' of their optima. The term 'PORTION' is essentially fuzzy. Thus we can use membership functions to represent the fuzzy term 'PORTION'. Assume that the membership functions (/Zl(X) and/z2(x)) of two objective functions are linear between dp* and dp' which are: (d~ m)* = min dpis(x) and the solution is x p, xex (d~ m)* = max d~is(x) and the solution is x N, x~x (dpis) =dp PIS (x N) and (d~'s) -dp- NIS(x p).

6 Y.-J. Lai et al. / TOPSIS for MODM 491 lal(x), v.2 (x) i I I i I i -max-min. I solution i /\ ', 0 dp(x) (dpp *s )* (dp NIs ) ' (dppis) ' (dp~* s )* Figure 3. The membership functions of t~l(x) and 1~2(x) (The horizon expresses the n-dimensional space of x.) Then, based on the preference concept, we assign a larger degree to the one with shorter distance from the PIS for/zl(x) and assign a larger degree to the one with farther distance from NIS for tz2(x). Thus, as shown in Figure 3, /z~(x) and /z2(x) can be obtained as the following (see also Lai and Hwang [15]): ].ti(x) = 1 dpis(x! -(dppls)* 1--(dpPiS ) (dpis)* 0 (d.)-d. (., -- NIS ~ NIS p ) if PIS dp (x)<(d~'s) *, t * if (dp ms) >_ dpls(x) ~ (d PIs), if 4, PIS (x) > if dpym(x) > (d~ Is)*, if (d~'s)' < djis( x ) _< (dpn's) *, r if dn's(x)< (dnis).! Now, we can resolve (9) by using the max-min operation which is proposed by Bellman and Zadeh [1] and applied by Zimmermann [28]. The satisfying decision, x *, may be obtained by solving the following problem: ~D(X*) = max{min(~l(x), ~2(x))}. x~x (10) Finally, if a = min(/xl,/z2), we will have the following equivalent model of (10): max a (lla) s.t. /d,l(x) >_a and tz2(x ) >a, (llb) x ~X, where a is the satisfactory level for both criteria of the shortest distance from the PIS and the farthest distance from the NIS. The compromise solution of (11) will be in the dark line as shown in Figure 4, when p = 2. (llc)

7 492 Y.-J. Lai et al / TOPSIS for MODM f2 f2" - ( d2 PIs )* -~,,.~ \ I- f d2eis ( fl, f2 ) =D f2" f" I i I [ ft ft- fl* Figure 4. The compromise solution existing in the dark line, when p = 2 For the special case of p = ~, we will use the following problem (instead of (9)): min max S.t. d PiS d NIS W,[(fi(x)-f,*)/(f7--fi*)] < dpis, Wj[(fj(x)--%7)/(fj*--fj-)] _>d NIS, wi[(f7 --fi(x))/(f]---f/*)] >d NIs, X ~X, (12a) (12b) (12c) where d P~s and d~ Is are not real distances, but the largest and smallest components of the k-dimensional distance functions, respectively. Similarly, we can compute the membership functions and obtain (11). In order to clearly demonstrate the TOPSIS approach, let us discuss the following numerical example of the Bow River Valley water quality management problem. (12d) (12e) (12f) (12g) 4. Bow River Valley water quality management problem The Bow River Valley water quality management problem discussed here is based on a hypothetical case developed by Dorfman and Jacoby [5] and modified by Monarchi, Kisiel and Duckstein [19] and Hwang and Masud [10]. It is concerned with the pollution problems of a river basin, the Bow River Valley, whose main features are shown in Figure 5. Industrial pollution originates form the Pierce-Hall Cannery, located near the head of the valley, and municipal waste pollution comes from two outlets located at Bowville and Plympton. A state park is located between the cities, and the lower end of the valley is a part of the state boundary line. The Bow Valley Water Pollution Control Commission is made up of representatives from all three waste dischargers and members of the state and federal government. The commission is responsible for maintaining river water quality by setting the waste reduction requirements at the three sources of pollution, but it must act with an awareness of the effect of any additional effluent treatment costs on the economy of the valley.

8 E-Z Laietal./TOPS~rMODM 493 Bow River (4.75) Bowville IPopulation 250,000 0 i0 JPierece-Hall Cannery (6.75) 50 Robin State Park (2.0) (5.1) Plympton Population 200, I00 STATE LINE (i.0) (Values in parentheses at the sides are current DO levels in milligrams per liter.) Figure 5. Main features of the Bow River Valley [19] The commission is concerned with the following six objectives: the DO (Dissolved Oxygen) levels at Bowville, Robin State Park, and Plympton; the percent return on investment at the Pierce-Hall Cannery; and the addition to the tax rate for Bowville and Plympton. The first problem of the commission is to determine a waste treatment policy which satisfies the constraint (g3(x)) that the DO level at the state line is above 3.5 mg/l. The treatment levels of waste discharges at the Pierce-Hall Cannery, Bowville and Plympton (Xl, x 2 and x3, respectively) are the decision variables and are restricted between 0.3 and 1.0. To reduce its complexity, the original problem formulation has been slightly modified without losing its original meaning. Both objectives corresponding to DO levels at Bowville and Plympton are changed to the constraints (gl(x) and g2(x)) of DO > 6.0. The modified problem is given below: max fx(x) = (x 1-0.3) (x 2-0.3) (w 1-0.3) (w 2-0.3) (mg/l DO at state park) (13a) max f2(x) = {[59/(1.09-XlZ)] - 59} (% earning for cannery) (13b) min f3(x) {[532/(1.09-x~)] - 532} ($/$1000 additional tax at Bowville) (13c) min.1:4(x) {[450/( x2)] - 450} (13d) ($/$1000 additional tax at Plympton) s.t. gl(x) = (Xl- 0.3 ) >6, (13e) g2(x) = (x, - 0.3) (x 2-0.3) (w I ) (w ) > 6, g3(x) = (X ) (x ) (x ) (w, ) (w ) (w ) >_ 3.5, (13f) 0.3<X i<l.0, i=l,2and3, (13g) where w i = 0.39/(1.39 -x}), i = 1, 2 and 3. Let the above constraints be denoted by x ex.

9 494 Y.-J. Lai et al. / TOPSIS for MODM Table 1 PIS payoff table of (13) fl f2 f3 f4 XI X2 X3 max fl ~ * max f * min f * min f * PIS: f * = (6.7922, , , ). Table 2 NIS payoff table of (13) fl f2 f3 ):4 xl x2 x3 min fl min f max f max f NIS: f- = (4.7797, , , ). To solve this problem, we first obtain PIS and NIS (as shown in Tables 1 and 2) by computing (7). Next, we compute (8) and obtain the following equations: dp P's = {wi [( f,( x))/2.0125] p + w~'[( f2( x))/5.9321] p + w~'[(f3 (x) )/8.6391] + WaP[(f4(x) )/ ] p)i/p, (14a) d~ Is = (wl [(fl(x) )/2.0125] p + w~'[(f2 (x) )/5.9321] p + w~'[( f3( x))/8.6391] p + w~' [ ( f4 ( x))/ ] P/1/p. (14b) In this manner, (9) is obtained. In order to get numerical solutions, let us assume that the importance (or weight) is the same among these four objectives and p = 2. The payoff table of (9) is shown in Table 3, when p = 2. Next we will compute (11): max o~ s.t. [ d~is(x) ]/ > a, [ d~is( x)]/ > a, X EX, where d~'is(x) and dnis(x) are shown in (14), except p = 2 and w t = 0.25, for t = 1, 2, 3 and 4. The Table 3 PIS payoff table of (9), when p = 2 d~ Is an Is /1 fz /3 f4 min d2 PIs * ' max d Nls ' " d* = (0,3092, ) and d~ = (0.5013, ).

10 Y.-J. Lai et al. / TOPSIS for MODM 495 Table 4 Solutions for equal weights among objective functions Problem fl f2 f3 f4 Xl X2 X3 PIS NIS p=l: min dl Pls AR a 48% 90% 66% 97% max d Nls AR 48% 90% 66% 97% p= 2: mind Pm AR 55% 84% 61% 98% max d NIs AR 0% 100% 94% 97% d~' = (0.3092, ), d~ = (0.5013, ) max a AR 29% 100% 78% 97% a* = 66.68% p =oo: mind PIs AR 61% 61% 61% 94% max d Nls AR 61% 61% 61% 94% " AR (Achieved Rate) [ = ( fj - f~ )/( fj* - f~ )' for j G J (f:~ -fi)/(f~ -fi*), for/g/. Table 5 Solutions for unequal weights among objective functions (w I = 0.4, w 2 = 0.3, w 3 = 0.2 and w 4 = 0.1) Problem fl f2 f3 f4 Xl x2 x3 PIS NIS p=l: mind Pm AR a 66% 98% 34% 99% max d~ Is AR 66% 98% 34% 99% p=2: mind PIs , AR 63% 83% 50% 99% max d NIs AR 78% 100% 0% 100% d~ = (0.3348, ) and d~ = (0.4685, ) max a AR 68% 100% 28% 99% a* = 65% p=oo: mind PIs AR 72% 63% 45% 94% max d Nls AR 71% 39% 54% 99% d* = (0.1106, ) and d" = (0.1865, ) max a AR 71% 60% 49% 98% a* = 89% a AR (Achieved Rate) [ = ( fj - f~ )/(fi* - f7 ), for j G J (f7 -fi)/(f7 - f,*), for/g/.

11 496 Y.-J. Lai et al. / TOPSIS for MODM compromise solution is shown in Table 4. Furthermore, in Table 4, we also provide the cases of p = 1 and o~. Table 5 presents the solution for the cases of unequal weights (w I = 0.4, WE = 0.3, W 3 = 0.2 and w 4 = 0.1) with p = 1, 2 and oo. 5. Discussion of TOPSIS solutions From the previous example, we can see that the solutions of mind PIS and max dl Nls are the same in both cases of equal weight and unequal weight. Thus, we do not need to solve (9) and (11). Similarly, the solutions of min d PIs and max d NIs are also the same in the case of 'equal weight', but are different in the case of unequal weight. When p = 2, the solutions of min d PIs and max d2 NIs are always different. For those cases of different solutions, we need to solve (9) and (11) in order to obtain a compromise solution which has the shortest distance from the PIS and the longest distance from the NIS. In order to display these properties of TOPSIS solutions, when p = 1, 2 and 0% it is necessary to go back to the objective representation space. For simplicity, let us assume that we have only two objectives which are going to be maximized. For the case of p = 1, we have the following distance functions: where dpis=wl[(f~'-fl(x))/(f,*-fl)] q-w2[(fff --f2(x))/(fff --f2) ] = Cl PIS- {Wlfl(x)/(f~-fl) + [wef2(x)/(fff -f2)]}, dniis=wl[(fl(x) -fl)/(f,*-f;)] +Wz[(fz(X)-f2)/(fff -f2)] = cn's + {[wlf,(x)/(f,* --fl)] + [w2f2(x)/(f~ --f2)]}, C PIS and C NIS are constants. It is obvious that both slopes of the above equations are -[w2(fl*-f{)/wl(fff-f2)], in the objective representation space. The indifference curves of dpis(fl, f2) = C and dlms(fl, f2) = C' are linear and have the same constant MRS (Marginal Rate of Substitution) as shown in Figure 6. Therefore, the solutions of mind P1s and max dl ms are the same whether the weights of the objectives are equal or not. The compromise solution of TOPSIS can be obtained by either minimizing d Pls or maximizing d ms. The satisfactory degree of both criteria is 1 (both criteria are fully satisfied). f2 f2* -- f, [N~dlPIS(fl, f2 )=C f2- f- l l fl- fl* Figure 6. The compromise solution when p = I

12 Y.-J. Lai et al. / TOPSIS for MODM 497 When p = 2, the distance functions become: dp~s= {[w,(f~* -f,(x))/(f( ~ -fl)]2 + [w2(f2* -f(x))/(f2 -f;)]2} '/2 = {[Wl(fl ~ -fl(x))] 2-{- [W2(f~ --f2(x))]2} 1/2, d NIS= {[wl(fl(x ) -fl)//(f~-fl)] 2 + [w2(f2 (X) -f2)//(f~ -f2)]2} 1/2 where W 1 = wl/(f~-fl) and W 2 = w2/(f~-f2). The indifference curves for both problems can be determined to be and dpls(fl, f2) = D= {[Wl(f~ -fl(x))] 2+ [w2(ft -f2(x))]2} 1/2 [Wl(fff -fi(x))]2+ [W2(f~-f2(x))]2=D 2 dnis(fl, f2) w~-d'= {[Wl(fl(x)-fl)] 2+ [W2(f2(x) -f2)]2} 1/2 [Wl( fl( x ) --fl)]2 + [W2(f2( x ) -f~-)]2 =D'2 where D and D' are constants. In general, both curves will not interact at the same point in the nondominated solution region (see also Figure 4). Thus, we need to find a compromise solution in order to satisfy the criteria of the shortest distance from the PIS and the longest distance from the NIS as much as possible. Thus, a TOPSIS solution has the highest possible degree of satisfaction, a*, for both criteria. This is always true when 2 < p < 0o. When p = 0% both criteria lead to the following problems: min d PIS (15a) s.t. Wl[(fl*--fl(x))/(f~'- f~-)] _<d Pro, (15b) Wz[(f[ -f2(x))/(f[ -f2)] < dpis, (15C) X ~ X, (15d) and max d NIS s.t. Wl[(f,(x)-fl)//(f~-fl)] ~-- dnis, w2[(f2(x) -f2)/(f~ -f2-)] -> dnis, x~x. (16a) (16b) (16c) (16d) Equation (15) is actually equivalent to: min s.t. d PIs w,{1- [(fl(x) -f?)/(f~-f~-)]} _<d~ ~s, W2{1 -- [(f2(x) --f2)/( f~ -f2-)] } < dpis, xex,

13 _ 498 Y.-J. Lai et al. / TOPSIS for MODM or max s.t. _deis d PIS Wl[(f,(x) --f{)/(f?--fl)] > Wl---oo, w2[(f2(x)--f2)/(f~'--f2)] x~x, >_w2-d PIS, Thus, if w 1 = w 2 = 0.5, we may set d~ Is = d PIS, d PIS = d NIS. We obtain: max d NIS (17a) s.t. w,[(f,(x) -f{)/(f* -fl)] > dnis, (17'0) W2[(f2(x) --f2)/(f2* --f2)] - u~.anis, (17c) x ~ X which has the same solution as (16). That is, when the objectives are equal important a TOPSIS solution can be obtained by solving either (15) or (16), and its satisfactory level is equal to 1 (both criteria are fully satisfied). If w I # w 2, (17) will not be obtained and we will have different solutions for both criteria. Thus, we must solve (11) to obtain a TOPSIS solution with satisfactory degree a*. (17d) 6. Concluding remarks and future studies While pervasive approaches such as global criterion methods, goal programming, fuzzy programming and interactive methods only consider the single criterion of the shortest distance from goal(s) or the PIS, TOPSIS provides a broader principle of compromise for solving multiple criteria decision making problems. The principle of the shortest distance from the PIS and the longest distance from the NIS can be compared to the principle of eastern and western religions; i.e., leading human beings to heaven and away from hell. This is also confirmed by the following statements by Zeleny [27]: "Do humans strive to be as close as possible to the ideal or as far away as possible from the anti-ideal? Our answer - both". TOPSIS for MODM first proposes a methodology by using two distance criteria to evaluate multiple objectives. It transfers k objectives (criteria), which are conflicting and non-commensurable, into two objectives (the shortest distance from the PIS and the longest distance from the NIS), which are commensurable and most of the time conflicting. This process can be considered as a first-order compromise procedure. After that, we can solve the bi-objective problem by balancing the satisfactions for each new criterion and obtain TOPSIS's compromise solution by a second-order compromise, the max-min operator is then considered a suitable one to resolve the conflict between the new criteria (the shortest distance from the PIS and the longest distance from the NIS). For our bi-objective problem, the compromise solution will exist at the point where the satisfactory levels of both criteria are the same. This harmonic result is important, especially from the point of view of group decision making. However, the max-min operator is not compensatory between these two criteria. That is, we cannot increase the satisfactory level of the shortest distance from the PIS by decreasing the satisfactory level of the longest distance from the NIS. Therefore, one may like to use some other compensatory operators to resolve this conflict. Dubois and Prade [6], Zimmermann [29] and Chen and Hwang [3] have provided wide discussions and survies of current compensatory operators. As to membership functions, simple linear functions are assumed to be given in this study. This may not satisfy real-world decision problems. Fortunately, there are various membership functions discussed in Zimmermann [29], Seo and Sakawa [20] and Dombi [4]. In future studies, applying compensatory operators and various membership functions to solve multiple objective programming problems should be emphasized.

14 Y.-J. Lai et al. / TOPSIS for MODM 499 Finally, it should be noted that to reach a single compromise decision is not the end of a decision process. Lewandowski and Wierzbicki [16] pointed out: "One is related to the fact that decisions are concerned with future events and have dynamic consequences... Reaching a single decision is a process, possibly with many phases and recourses and with a role of learning during this process... Separate decision processes are embedded in a longer learning process of the decision maker to become a master expert, with its much more complicated dynamics". Thus, an interactive decision support system (see [8,14,17]) with integration-oriented adaptation and learning features is very helpful in solving practical decision problems. Based on TOPSIS for MODM, we are further developing an expert decision-making support system involving various membership functions and operators and fuzzy constraints, to meet dynamic decision environments. References [1] Bellman, R.E., and Zadeh, L.A., "Decision-making in a fuzzy environment", Management Science B 17 (1970) [2] Boychuk, L.M., and Ovchinnikov, V.O., "Principle methods for solution of multicriterion optimization problems (survey)", Soviet Automatic Control 6 (1973) 1-4. [3] Chen, S.J., and Hwang, C.L., Fuzzy Multiple Attribute Decision Making, Springer-Verlag, Heidelberg, [4] Dombi, J., "Membership function as an evaluation", Fuzzy Sets and Systems 35 (1990) [5] Dorfman, F., and Jacoby, H., "A model for public decision illustrated by a water pollution policy problem", in: The Analysis and Evaluation of Public Expenditures: The PPB System", Joint Economic Committee, 91st Congress, 1st Session, 1969, [6] Dubois, D., and Prade, H., "A review of fuzzy set aggregation connectives", Information Science 36 (1985) [7] Goicoechea, A., Hansen, D.R., and Duckstein, L., Multiobjective Decision Analysis with Engineering and Business Applications, Wiley, New York, [8] Grauer, M., Lewandowski, A., and Wierzbicki, A.P., "DIDASS - Theory, implementation and experiences", in: M. Grauer and A.P. Wierzbicki (eds.), Interactive Decision Analysis, Lecture Notes in Economics and Mathematical Systems, No. 229, Springer-Verlag, Heidelberg, 1983, [9] Hall, III, A.D., Metasystems Methodology: A New Synthesis and Unification, Pergamon, Oxford, [10] Hwang, C.L., and Masud, A.S.M., Multiple Objective Decision Making: Methods and Applications, Springer-Verlag, Heidelberg, [11] Hwang, C.L., Paidy, S.R., Yoon, K., and Masud, A.S.M., "Mathematical programming with multiple objectives: A tutorial", Computers & Operations Research 7 (1980) [12] Hwang, C.L., and Yoon, K., Multiple Attribute Decision Making: Methods and Applications, Springer-Verlag, Heidelberg, [13] Keeney, R.L., and Raiffa, H., Decision with Multiple Objectives." Preferences and Value Tradeoffs, Wiley, New York, [14] Lai, Y.J., and Hwang, C.L., "Interactive fuzzy linear programming", Fuzzy Sets and Systems 45 (1992) [15] Lai, Y.J., and Hwang, C.L., "A new approach to some possibilistic linear programming problems", Fuzzy Sets and Systems 49 (1992) [16] Lewandowski, A., and Wierzbicki, A.P., "Decision support systems using reference point optimization", in: A. Lewandowski and A.P. Wierzbicki (eds.), Aspiration Based Decision Support Systems - Theory, Software and Applications, Lecture Notes in Economics and Mathematical Systems, No. 331, Springer-Verlag, Heidelberg, 1989, [17] Lewandowski, A., Kreglewski, T., Rogowski, T., and Wierzbicki, A.P., "Decision support systems of DIDAS family (dynamic interactive decision analysis & support)", in: A. Lewandowski and A.P. Wierzbicki (eds.), Aspiration Based Decision Support Systems - Theory, Software and Applications, Lecture Notes in Economics and Mathematical Systems, No. 331, Springer-Verlag, Heidelberg, 1989, [18] Lieberman, E.R., "Soviet multi-objective programming methods: An overview", in: A. Lewandowski and V. Volkovich (eds.), Multiobjective Problems of Mathematical Programming, Lecture Notes in Economics and Mathematical Systems, No. 351, Springer-Verlag, Heidelberg, 1991, [19] Monarchi, D.E., Kisiel, C.C., and Duckstein, L., "Interactive multi-objective programming in water resources: A case study", Water Resources Research 9 (1973) [20] Seo, F., and Sakawa, M., Multiple Criteria Decision Analysis in Regional Planning - Concepts, Methods and Applications, D. Reidel, Dordrecht, [21] Steuer, R.E., Multiple Criteria Optimization Theory, Computation, and Applications, Wiley, New York, [22] White, D.J., "A bibliography on the applications of mathematical programming multiple-objective methods", Journal of the Operational Research Society 41 (1990) [23] Yoon, K., "A reconciliation among discrete compromise solutions", Journal of the Operational Research Society 38 (1987)

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