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A visual reference direction approach to solving discrete multiple criteria problems
152
European Journal of Operational Research 34 (1988) 152-159 North-Holland
Theory and Methodology
A visual reference direction approach to solving discrete multiple criteria problems Pekka K O R H O N E N
Helsinki School of Economics, Runeberginkatu 14-16, 00100 Helsinki, Finland
Abstract: In this paper, we propose a new visual interactive method for solving discrete multiple criteria problems. The method is based on the use of a reference direction, which is determined by the aspiration levels for the criteria specified by the decision maker. The reference direction is projected onto the set of efficient alternatives. A subset found in this way is presented to a decision maker in a visual form using computer graphics. He can choose any efficient alternatives he pleases. We need not make any assumptions about the properties of the utility function. The method has been implemented on an I B M / P C 1 microcomputer. The name of the program is VIMDA (a Visual Interactive Method for Discrete Alternatives). Keywords: Multiple criteria, computer graphics, interactive, discrete
1. Introduction Quite a few approaches have been developed to solve discrete multiple criteria problems: the traditional multiattribute utility theory (see, e.g. Farquhar, 1984; Keeney and Raiffa, 1976), the analytic hierarchy process (Saaty, 1980), the outranking method (Roubens, 1982; Roy, 1973), the fuzzy-set approach (Yager, 1980), interactive programming (Korhonen, Wallenius and Zionts, 1984; Marcotte and Soland, 1986), hierarchical interactive approaches (Korhonen, 1986) and some others (Hinloopen, Nijkamp and Rietveld, 1983). Most of these methods are based on restrictive assumptions concerning the decision maker's preference structure or they are developed to solve problems of special structure: e.g. many (qualitaThis work was partly supported by a grant from Foundation for EconomicEducation. Received August 1986; revisedFebruary 1987
tive) criteria and few alternatives (e.g. Korhonen, 1986; Roy, 1973; Roubens, 1982; Saaty, 1980) or few quantitative criteria and many alternatives (e.g. Korhonen, Wallenius and Zionts, 1984; Marcotte and Soland, 1986). The methods of Korhonen, Wallenius and Zionts (1984) and Marcotte and Soland (1986) are based on pairwise comparisons of attainable alternatives and require sometimes a lot of comparisons for identifying an optimal solution. We propose a new interactive method to solve discrete deterministic multiple criteria problems with few criteria and many alternatives. Our aim is to develop the method that is easy to use and understand, permits the decision maker to examine any efficient alternatives he wishes and makes no restrictive assumptions concerning the underlying utility function of the decision maker. In particular, we need not assume that the utility function remains unchanged during the interactive process. We assume the criteria to be quantitative, but the approach may also be used with ordinal
0377-2217/88/$3.50 O 1988, Elsevier Science Publishers B.V. (North-Holland)
P. Korhonen / Visual reference direction approach for solving discrete multiple criteria problems
criteria. Interactive use of computer graphics also plays a central role in the proposed method. The paper consists of four sections and an appendix. In the next section we present some preliminary considerations. Our approach with an illustrative example is given in Section 3. The final section consists of a discussion. In the appendix we prove two lemmas and present a detailed description of our procedure to find a subset for the decision maker's evaluation.
153
alternative in row i. Thus each decision alternative is a point in the criterion space R P. Assume that the D M wishes to maximize each of the p criteria, then the problem is 'max'xi.
(2.1)
iGl
2. Preliminary considerations
Because the feasible set is discrete and finite, it is easy to eliminate all dominated alternatives, i.e. the alternatives x~, i ~ I, for which there exists a x k, k ~ I , such that xij~
2.1. Basic definitions and notation
2.2. Choosing a subset
First we introduce some terms used in this paper. By the term criterion we refer to the concept which represents a magnitude that is of special interest to the decision maker (DM). It is the basis for evaluation. A criterion may be qualitative or quantitative. A criterion is quantitative when its magnitude can be described on a numerical scale and a preference (utility) function is a (strictly) monotonic function over this scale. A criterion is qualitative, if the decision maker can only express ordinal preferences by stating which of a pair of alternatives he prefers more. An aspiration level is a desired or acceptable level of a criterion. A criterion and its aspiration level is termed a goal. For example, if the D M wants to attain a profit level of at least $1000 he has established a goal, where $1000 is his aspiration level corresponding to his particular goal. We use the term reference goal as a synonym for goal. A reference direction is any direction starting from some alternative and describing a preferable change in the criterion space. The term achievement function is a function of the criteria that maps any given point in the criterion space onto the set of efficient points. The concept of efficiency and an achievement function are reasonable only for quantitative criteria. From hence on we deal only with quantitative criteria. Now we introduce some notation to describe the problem. We assume that there is a single decision maker, a set of n (n > 0) deterministic decision alternatives and p criteria ( p > 1), which define an n × p decision matrix X whose elements are denoted by xij, i ~ I = {1, 2 . . . . . n} and j ~ J = { 1 , 2 . . . . . p}. We use x i or i to refer to the
We use a reference direction and an achievement function to generate a subset of the efficient alternatives. A reference direction reflects the desire of the decision maker to improve the values of the criteria. We use as a reference direction the vector from the current alternative to the point defined by the decision maker's aspiration levels. An achievement function finds out attainable and efficient alternatives that bear some relation to the decision maker's aspirations. There are several methods to specify a reference direction. The use of marginal rates of substitution for estimating the gradient of the utility function (Geoffrion, Dyer and Feinberg, 1972) or the technique used in the Boundary Point Ranking method (Hemming, 1976) are two examples of possible techniques. We prefer to use a simple and convenient alternative. The reference direction is projected onto the set of efficient points by using an achievement (scalarizing) function as suggested by Wierzbicki (1980) in his reference point approach. Once the decision maker has specified his aspiration levels for the criteria, we find an efficient solution that minimizes the value of the achievement function. When we apply the achievement function to the reference direction, instead of one point we obtain a set of efficient solutions. When a given direction is projected onto the set of efficient alternatives, the D M does not need to make comparisons between inefficient (or infeasible) solutions. Korhonen and Laakso (1986) have used the same idea in their visual interactive method for solving multiple objective linear programming problems and a similar technique have
P. Korhonen / Visual reference direction approach for solving discrete multiple criteria problems
154
also been suggested by Winkels and Meika (1984) for projecting the estimated gradient of an unknown utility function onto the efficient frontier. However, in the discrete case the projection problem is more difficult to solve than in multiple objective linear programming. There are also several ways to specify an achievement function. The characterizations of achievement functions are considered in more detail by Wierzbicki (1986). We use the following simple form for an achievement function: f(g,
xi, w) = max (gj - x i j ) / w j,
(2.2)
jEJ
where i E I , g ~ R p is a given reference goal vector called a reference point, and w ~ R e is a given weighting vector. If criterion j is maximized wj > 0, otherwise wj < 0. Since we have assumed that all criteria are to be maximized, wj > 0 in our considerations. By minimizing f ( g , x,, w), i ~ 1, for given g and w we find an efficient solution Xk, k ~ I (Wierzbicki, 1980, 1986). Given an initial solution x h, h ~ I, a reference point g and a weighting vector w we can define a parametrized achievement function F for the reference direction d = g - x h as follows: F ( t , d, x h, xi, w ) = f ( x
h + td, xi, w ) ,
(2.3)
where t >_ 0 and i ~ I. By solving the parametric programming problem m i n F ( t , d, xh, x i, w ) , i~l
for all t>_
0,
(2.4)
we obtain a set of decision alternatives Xm, FFI~. M C I, as a solution. The index set M is assumed to be an ordered set M=(m
1, m 2 . . . . . m k ) ,
k~
2.3. Visual interaction
The alternatives in set M are presented for evaluation to the D M both numerically and graphically. The values of the criteria of the alternatives belonging to M are plotted on the screen (on the y-axis). The current alternative m 1 is plotted on the left and the last alternative m k on the right. The values of the criteria of the subsequent alternatives are connected by a line. Different line patterns or colors can be used for different criteria. A sample graphical display is shown in Figure 1, where the cursor (a vertical line) points to the fourth alternative. The numerical values of the criteria for the alternative are shown on the top line of the display. As the cursor is moved to the left or to the right the numerical values of the criteria corresponding to the alternative are displayed. Geoffrion et al. suggested the use of graphics as a possible method for solving the step-size problem as early as 1972, and e.g. Winkels and Meika (1984) have further used computer graphics in connection with the G D F method. However, their approach is not interactive in the same sense as ours.
For each criterion the scale is chosen so that the m a x i m u m value is at the top and the minimum value is at the bottom of the display. This kind of presentation gives the decision maker holistic and exact information on available alternatives in a very convenient way. It is possible for the decision maker to choose the most preferred alternative from among the alternatives presented on this display.
P_r
473
__ H a
65
£!.
!,6
.................. M.~,......£.3.~. ............
(2.5)
where for m j, m h ~ M , j < h , there exists a t* such that X m j is the solution of (2.4), when t = t*, and x,,, is not a solution for any t < t*. The set M consists of the indices of the alternatives to be presented to the DM. In appendix, we prove two lemmas and describe a procedure to find the set M. The problem is solved in two parts. First we determine the explicit form of the parameterized achievement functions: vi(t ) = F(t, d, xh, xi, w), for all i ~ I, and then use these functions in problem (2.4).
~....,....,.., ~
Go
.........~,'-"
left
and
-~ t o
x~ight
(End
=
exit)
Figure 1. A graphical representation of a subset of alternatives
P. Korhonen / Visualreferencedirection approachfor solvingdiscretemultiple criteriaproblems
3. Development of the approach
Table 1 The types of criteria are specified
The proposed method is a modification of the approach of Korhonen and Laakso (1986) to solve general multiple criteria problems. In the discrete case an efficient curve is replaced by a set of efficient alternatives. This subset is presented in a visual form to the decision maker. At each iteration the decision maker is asked to choose the most preferred alternative from the subset. This alternative becomes the starting solution for the next iteration. The decision maker specifies new aspiration levels for the criteria. This information is used to determine a new reference direction. The process is repeated until he is unable to find a better solution than the current one. If it is desired to check the optimality of the final point, we have to make some assumptions concerning the utility function. If it is assumed to be quasiconcave, we can use the method by Korhonen, Wallenius, and Zionts (1984) to eliminate alternatives on the basis of the decision maker's sequential choices until one alternative remains. The approach is implemented on an I B M / P C 1 microcomputer under name VIMDA (a Visual Interactive Method for Discrete Alternatives). The dimensions of the problem in VIMDA are: n = 500 and p = 10. We illustrate the steps of the approach in terms of a numerical example using the steps of program VIMDA. Suppose the owner of a laundry wants to choose a washing machine. In addition to price, he wants to consider washing time and consumption of electricity and water as the most important criteria. He has collected data on these four criteria for thirty-three different types of machines. These data are given in Zeleny (1982, pp. 210-211). We refer to alternatives by indices. Step O. Find an initial (efficient) solution. Ask the decision maker to specify which criteria are to be maximized and which are to minimized (see Table 1). Define weights wj as follows:
VIMDA The types of criteria (max/min):
(rJabs (maxi(rj*x,j)) wj= f [rj
if m a x i ( ~ * xlj) 4=0, if maxi( ~ * xij ) = O,
where § is 1 if criterion j is maximized, and - 1 if it is minimized. (Other choices are also possible.)
Criteria Price Wash.time El.consumption Water consumption
155
Types MIN MIN MIN MIN
Find an initial alternative by optimizing one of the criterion. Here we optimize the last criterion. In our example all criterion are to be minimized and alternative 2, b = (425, 80, 1.5, 110), has the 'best' water consumption (see Table 2). Step 1. Find a reference direction. The values of the current solution and the maxim u m and minimum values of each criterion are displayed in Table 2. The decision maker is asked to give aspiration levels for the criteria. Usually, at the first iteration, the decision maker gives ideal values as aspiration levels for criteria. In our example these are (395, 50, 1.4, 110). The vector from the current alternative to the point defined by the decision maker's aspiration levels is used as a reference direction. For our problem d = (395 - 425, 5 0 - 80, 1 . 4 - 1 . 5 , 1 1 0 - 110) =(-30,
-30,-0.1,0).
Step 2. Find a subset of efficient solutions. In this step we determine the ordered index set M, as described in Section 2. In our example M = {2, 21, 29, 7, 8, 11, 5, 26}. Step 3. Find the most preferred solution from among set M.
Table 2 Specification of aspiration levels VIMDA The specification of aspiration levels for criteria The name of a current solution: 2 Criteria Lower Upper Current bounds bounds values Price 395 595 425 Wash.time 50 80 80 El.consumption 1.4 1.8 1.5 Water consumption 110 140 110
Aspiration levels 395 50 1.4 110
156
P. Korhonen / Visual reference direction approach for solving discrete multiple criteria problems
The alternatives corresponding to the index set M are presented to the decision maker in a visual form. He is asked to indicate his most preferred alternative. The values of the criteria are plotted on the screen using distinct colors (red, yellow and green) and line patterns for the criteria. The cursor can be moved from alternative to alternative and the corresponding numerical values of criteria are displayed simultaneously. The scale for each criterion j is chosen in such a way that maxixij is at the top of the screen and minix~j is at the bottom. If an improved solution is found or the decision maker is willing to consider other directions, return to step 1, otherwise stop. Figure 1 is a sample display at the first iteration. Let us assume that alternative 7 is the most preferred choice of the decision maker from this set. The current alternative for the next iteration is alternative 7. Let us assume that the aspiration levels for the next iteration are (450, 60, 1.5, 130). Now the set M = (7, 8, 19, 11, 18, 26}. The new display is described in Figure 2. If we assume that the washing time is one of the most important criteria, the decision maker can choose alternative 19, b = (543, 57, 1.6, 120), as the most preferred alternative from this set. If the aspiration levels of the decision maker for criteria are the same as at the previous iteration the set M = {19, 14, 8, 7, 6, 3, 27, 32}. If he still prefers the current solution 19 to the others he may be willing to stop. In our approach we are not primarily interested to prove the optimality of the final solution. We prefer the satisfaction of the decision maker to proving the optimality, mathematically. If it is desired to check the optimality of the final point, we have to make some assumptions about the utility function. If the utility function is assumed to be quasiconcave, we can use the method by Korhonen, Wallenius, and Zionts (1984) for eliminating alternatives on the basis of the decision maker's sequential choices. We can utilize all preference information cumulated so far during the process. If more than one alternative remain we can restart the search process with the remaining alternatives. We terminate the process when only one alternative remains--which is optimum.
P~
543
Na
57
E1.!.,6
...............N . , . . . . 1 . 2 0 .........
I
~f--¸7 i.- -¸ ........
/,J
"z
::7........
""-....
t/
J i..... '..... ~-
to
left
i
and
"+
_i
to
]
,
--'------'~--~----____ x~ight
(Edd
ex
Figure 2. The display at the second iteration
When people used our approach, we noticed an interesting fact. Some of them were making cycles during the search process, i.e., either they chose the same alternative many times although not at subsequent iterations or they chose an alternative at some stage even it was shown but not chosen earlier. It implies that their preference structure were not transitive. In the other words, we do not have proper information about their utility function. Therefore we have not implemented the optimality check in program VIMDA.
4. Discussion In this paper our main purpose was to develop an interactive decision aid for the decision maker to evaluate alternatives close to his aspiration levels. In our opinion, a full benefit from an interactive approach can be obtained by interactive utilization of computer graphics; visual representation enables the decision maker to evaluate a large set of possible alternatives simultaneously. To avoid irrelevant information only nondominated solutions are presented to the decision maker for evaluation. Besides the use of visual aids, our approach has three more desirable features. First, the decision maker is free to examine any set of the efficient frontier he pleases at any moment, i.e. he is neither confined to evaluate solutions with some special properties, nor is his freedom limited by his earlier choices during the search process. Second, we need no specific assumptions about the underlying utility function of the decision maker during the
P. Korhonen / Visual reference direction approach for solving discrete multiple criteria problems
interactive process. The utility function can even be assumed to be changing due to learning and 'changes of mind' during the search process. If we are interested in proving the optimality of the final solution, then we have to make some assumptions about the utility function. Third, the decision maker is asked to give very simple information at each iteration. He has only to specify his aspiration levels for criteria and then evaluate reasonable alternatives by using an informative representation. We have developed our approach assuming quantitative criteria. However, the approach can be used even with ordinal criteria, i.e. when the decision maker is only able to rank alternatives according to each criterion. He can express his aspiration levels in terms of ordinal numbers and the approach tries to find the alternatives that reflect his aspirations. Although, the choice of the subset of efficient alternatives depends on the weights used, it primarily depends on the aspiration levels specified by the decision maker. By setting weights in a reasonable way we can help the decision maker to find the alternatives he likes. Preliminary experiments indicate that the decision makers find the approach easy to use and understand. They also seem to like colors, figures and spreadsheet interface. Especially, the use of colors appears to enhance the attractiveness of the method.
Appendix. A procedure to find the set M In this present a indices of maker for
appendix we prove two lemmas and procedure to find the set M of the the alternatives given to the decision evaluation.
Lemma 1. The functions vfit), i ~ I, are convex. Proof. Let o ~ [0, 1]. Using the definition of v~(t):
v i ( t ) = max (Xhj + tdj - x i j ) / w j , j~J
we obtain that
o v , ( t , ) + (1 - o ) v i ( t 2 ) = m a x o ( x h j + t,dj - x i j ) / w j jGJ
(A.1)
+
max(1 rEJ
157
O ) ( X h r "1- t 2 d r - X i r ) / W r
>~ max (Xh, + (Off + (1 -k~J
o)t2)dk -
x,k)/w,
O , ( a t I "~- (~ -- o ) t 2 ) ,
which proves the convexity of v,(t).
Lemma 2. The functions v i(t), i ~ 1 and t >__O, are piecewise linear and are of the form
Vi(t ) = b i r + t c i r
when t E [ t / r , tar+l,,),
where r ~ [ 1 , 2 . . . . . ni] , ni<~p, and ti(n,+l) = ~ . Moreover, b~k < bij and c~k > c,j, if k > j . Proof. For each t we can write
vi( t ) = max (Xhj + tdj - x,j) / w j jGJ
= (Xhr-
Xir)/WrJt - tdr/w,
= b,(t) + tc,(t), where bi(t ) = (Xh, -- X i , ) / W , and ci(t ) = d r / w r. Consider t 1 > 0, t 2 > 0 and t 2 > tl, and assume that ci(tl)4= c,(t2). By the definition of v,(t) we have
vi( tl) = bi( t,) + t,c,( t,) >~ b,(t2) + t,ci( t2) , v i ( t : ) = bi( t2) + t2c,( t2) >I bi( t,) + t2ci( t,), Hence it follows that
bi(t,)-bi(t2)+tx[c,(t,)-G(t2)
] >10,
(A.2a)
bi(t2)-bi(tl)+t2[ci(t2)-ci(t,)
] >10.
(A.2b)
By adding inequalities (A.2a) and (A.2b) we obtain
It 2 - t , ] [ c , ( t 2 ) - c,(t,)] >/0.
(A.3)
Since t 2 > t I and ci(t2) 4= c,(tl) , inequality (A.3) holds iff c , ( t 2 ) > c~(tl). Inequality (A.2a) is nonnegative only if bi(tl) > bi(t2). If ci(t2) = G(tl), then from inequalities (A.2a) and (A.2b) it follows that bi(t2) = bi(ta). Because b i(t) and c~(t) are monotonic functions with at most p different values, there are n~ different t values; t,j, j = 1, 2 . . . . . n,, n ~ < p , where bi(t ) and ci(t ) change their values. By writing b,j = bi(tij ) and c~y = c(t,j) we can complete the proof. To illustrative our problem we use a simple example. Let us assume we have a 5 × 4 decision
158
P. Korhonen / Visual reference direction approach for solving discrete multiple criteria problems
Phase 1: Preliminary computations
alternative matrix
X=
1 0 4 2 9
3 7 5 8 5
8 3 2 1 2
1] 4 6 • 4 3
Find the index vectors m i referring to the rank orders of the elements of the vectors Yi = W(Xh -Xi), for all i, i = 1, 2 . . . . . n. Let the element mix refer to the largest element of the vector Yi and mip refer to the smallest element.
Let alternative 3: (4, 5, 2, 6) be our current alternative and assume that the D M has specified the vector (5, 6, 7, 5) as his aspiration level vector. T h e reference direction is thus (5, 6, 7, 5 ) (4, 5, 2, 6) = (1, 1, 5, - 1). Using formula (A.1) we can n o w find function vi(t ) for each alternative i, i = 1, 2 . . . . . 5. These functions are described in Fig. 3. T o describe the procedure to find the set M assume that the current alternative is x h and the reference point specified by the D M is q. The reference direction is d, d = q - x h. D e n o t e by W a diagonal matrix with reciprocal values of the weight vector w along the main diagonal: W = diag(1/w~, 1/w2,...,
1/Wp)
F r o m Figure 3, we can see that the set M, in our example, consists of three alternatives: (3, 2, 1) in this order. All these alternatives give a minim u m value for a parametrized achievement function (2.3) with some p a r a m e t e r value t. N o t e that we have not picked alternative 5 into the set M, because it does not give a better value for an achievement function than alternative 3, which is already included in the set M. T h e procedure consists of three phases.
-...~
..... •
//.
/
/
Phase 2: Compute the functions vi, i = 1, 2, . . . . n S t e p 2 . 1 . Set j : = l , n i : = 0 , s : = j , and t : = 0 . Step 2.2. Set bij :=Yimis , Cij : = din, , t~j:= t, and n i : = n i -]- 1. Step 2.3. Find r, r 6 (s, p], such that dml r > dm,,
and
Zr =
m i n . >, ( z, },
where z u = (Yimis -- Yimi, ) / ( d ' i r d m,s ) Step 2.4. If such r exists then s : = r , j : = j + l , t := z., and go to 2.2; otherwise, stop. Functions vi(t ) are found as follows: vi(t ) := bij + tcij, when t ~ [tij, ti(j+u), for j < ni, and vi(t ) := bi. ' + tci.,, when -
-
t >~ tin /
Phase 3: F i n d the index set M Step 3.1. Find the first alternative r. Define K := { l, 2 . . . . . n } and find r as vr (0) = br, (0) = m i n i vi (0) = m i n ibi~ (0). Set M . ' = ( r } and ki.'= 1, for all i ~ K . Step 3.2. D e t e r m i n e the range of the current linear part of Or(t ) . If kr>~n r then set T : = ~ ; otherwise, set T : = tr(kr+l ). Set S T : = [trk, T]. Step 3.3. F o r each i, i ~ K, find the maximal u~ for which vi(t ) > v~(t), for all t ~ [0, ui] nST. A: If crk" - c~k' ~< 0 then set u~ := T; otherwise, c o m p u t e Z := (bik ' -- ark,)//(Crk ~ -- Cik,)
3 t ~
Figure 3. The parametrized achievement functions for each alternative
If z >/T, set u i := T, and if z < ti(k,+l), set u i := z; otherwise set k i .'= k i + 1 and return to A. Step 3.4. Redefine k,, for each i ~ K, and set K. Set k i := m a x ( k i I tik, <~ Ui }" If k i = n i then set K : = K N {i}. If K = ~ then stop.
P. Korhonen / Visual reference direction approach for solving discrete multiple criteria problems
Step 3.5. Update M. Find uj = If % = T , otherwise and go to
m a x ~ x U i. set k r : = k r + l and go to 3.2; set r .'=j and M := M tA { j ) 3.2.
Acknowledgements The author wishes to thank prof. Subhash Narula for many helpful suggestions and comments.
References Farquhar, P. (1984), "Utility assessment methods", Management Science 30 (11) 1283-1300. Geoffrion, A., Dyer, J., and Feinberg, A. (1972), "An interactive approach for multi-criterion optimization, with an application to the operation of an academic department", Management Science 19 (4) 357-368. Hemming, T. (1978), Multiple Objective Decision Making under Certainty, The Economic Research Institute at the Stockholm School of Economics, Stockholm. Hinloopen, E., Nijkamp, P., and Rietveld, P. (1983), "The regime method: A new multicriteria technique", in: P. Hansen (ed.), Essays and Surveys on Multiple Criteria Decision Making, Springer, 146-155. Keeney, R., and Raiffa, H. (1976), Decisions with Multiple Objectives, Wiley, New York. Korhonen, P., Wallenius, J., and Zionts, S. (1984), "Solving the
159
discrete multiple criteria problem using convex cones". Management Science 30 (11) 1336-1345. Korhonen, P. (1986), "A hierarchical interactive method for ranking alternatives with multiple qualitative criteria", European Journal of Operational Research 24 (2) 265-276. Korhonen, P., and Laakso, J. (1986), "A visual interactive method for solving the multiple criteria problem", European Journal of Operational Research 24 (2) 277-287. Marcotte, O., and Soland, R. (1986). "An interactive branchand-bound algorithm for multiple criteria optimization", Management Science 32 (1) 61-75. Roubens, M. (1982), "Preference relations on actions and criteria in multicriteria decision making", European Journal of Operational Research 10 (1) 51-55. Roy, B. (1973), "How outranking relation helps multiple criteria decision making", in: J. Cochrane and M. Zeleny (eds.), Multiple Criteria Decision Making, University of South Carolina Press, Columbia, SC, 179-201. Saaty, T. (1980), The Analytic Hierarchy Process, McGraw-Hill, New York. Wierzbicki, A. (1980), "The use of reference objectives in multiobjective optimization", in: G. Fandel and T. Gal (eds.), Multiple Criteria Decision Making, Theory and Application, Springer, New York. Wierzbicki, A. (1986), " O n the completeness and constructiveness of parametric characterizations to vector optimization problems", OR Spektrum 8 73-87. Winkels, H.-M., and Meika, M. (1984), "An integration of efficiency projections into the Geoffrion-approach for multi-objective linear programming", European Journal of Operational Research 16 (1) 113-127. Yager, R. (1980), "A new methodology for ordinal multiobjective decisions based on fuzzy sets", Decision Sciences 12. Zeleny, M. (1982), Multiple Criteria Decision Making, McGraw-Hill, New York.
European Journal of Operational Research 34 (1988) 152-159 North-Holland
Theory and Methodology
A visual reference direction approach to solving discrete multiple criteria problems Pekka K O R H O N E N
Helsinki School of Economics, Runeberginkatu 14-16, 00100 Helsinki, Finland
Abstract: In this paper, we propose a new visual interactive method for solving discrete multiple criteria problems. The method is based on the use of a reference direction, which is determined by the aspiration levels for the criteria specified by the decision maker. The reference direction is projected onto the set of efficient alternatives. A subset found in this way is presented to a decision maker in a visual form using computer graphics. He can choose any efficient alternatives he pleases. We need not make any assumptions about the properties of the utility function. The method has been implemented on an I B M / P C 1 microcomputer. The name of the program is VIMDA (a Visual Interactive Method for Discrete Alternatives). Keywords: Multiple criteria, computer graphics, interactive, discrete
1. Introduction Quite a few approaches have been developed to solve discrete multiple criteria problems: the traditional multiattribute utility theory (see, e.g. Farquhar, 1984; Keeney and Raiffa, 1976), the analytic hierarchy process (Saaty, 1980), the outranking method (Roubens, 1982; Roy, 1973), the fuzzy-set approach (Yager, 1980), interactive programming (Korhonen, Wallenius and Zionts, 1984; Marcotte and Soland, 1986), hierarchical interactive approaches (Korhonen, 1986) and some others (Hinloopen, Nijkamp and Rietveld, 1983). Most of these methods are based on restrictive assumptions concerning the decision maker's preference structure or they are developed to solve problems of special structure: e.g. many (qualitaThis work was partly supported by a grant from Foundation for EconomicEducation. Received August 1986; revisedFebruary 1987
tive) criteria and few alternatives (e.g. Korhonen, 1986; Roy, 1973; Roubens, 1982; Saaty, 1980) or few quantitative criteria and many alternatives (e.g. Korhonen, Wallenius and Zionts, 1984; Marcotte and Soland, 1986). The methods of Korhonen, Wallenius and Zionts (1984) and Marcotte and Soland (1986) are based on pairwise comparisons of attainable alternatives and require sometimes a lot of comparisons for identifying an optimal solution. We propose a new interactive method to solve discrete deterministic multiple criteria problems with few criteria and many alternatives. Our aim is to develop the method that is easy to use and understand, permits the decision maker to examine any efficient alternatives he wishes and makes no restrictive assumptions concerning the underlying utility function of the decision maker. In particular, we need not assume that the utility function remains unchanged during the interactive process. We assume the criteria to be quantitative, but the approach may also be used with ordinal
0377-2217/88/$3.50 O 1988, Elsevier Science Publishers B.V. (North-Holland)
P. Korhonen / Visual reference direction approach for solving discrete multiple criteria problems
criteria. Interactive use of computer graphics also plays a central role in the proposed method. The paper consists of four sections and an appendix. In the next section we present some preliminary considerations. Our approach with an illustrative example is given in Section 3. The final section consists of a discussion. In the appendix we prove two lemmas and present a detailed description of our procedure to find a subset for the decision maker's evaluation.
153
alternative in row i. Thus each decision alternative is a point in the criterion space R P. Assume that the D M wishes to maximize each of the p criteria, then the problem is 'max'xi.
(2.1)
iGl
2. Preliminary considerations
Because the feasible set is discrete and finite, it is easy to eliminate all dominated alternatives, i.e. the alternatives x~, i ~ I, for which there exists a x k, k ~ I , such that xij~
2.1. Basic definitions and notation
2.2. Choosing a subset
First we introduce some terms used in this paper. By the term criterion we refer to the concept which represents a magnitude that is of special interest to the decision maker (DM). It is the basis for evaluation. A criterion may be qualitative or quantitative. A criterion is quantitative when its magnitude can be described on a numerical scale and a preference (utility) function is a (strictly) monotonic function over this scale. A criterion is qualitative, if the decision maker can only express ordinal preferences by stating which of a pair of alternatives he prefers more. An aspiration level is a desired or acceptable level of a criterion. A criterion and its aspiration level is termed a goal. For example, if the D M wants to attain a profit level of at least $1000 he has established a goal, where $1000 is his aspiration level corresponding to his particular goal. We use the term reference goal as a synonym for goal. A reference direction is any direction starting from some alternative and describing a preferable change in the criterion space. The term achievement function is a function of the criteria that maps any given point in the criterion space onto the set of efficient points. The concept of efficiency and an achievement function are reasonable only for quantitative criteria. From hence on we deal only with quantitative criteria. Now we introduce some notation to describe the problem. We assume that there is a single decision maker, a set of n (n > 0) deterministic decision alternatives and p criteria ( p > 1), which define an n × p decision matrix X whose elements are denoted by xij, i ~ I = {1, 2 . . . . . n} and j ~ J = { 1 , 2 . . . . . p}. We use x i or i to refer to the
We use a reference direction and an achievement function to generate a subset of the efficient alternatives. A reference direction reflects the desire of the decision maker to improve the values of the criteria. We use as a reference direction the vector from the current alternative to the point defined by the decision maker's aspiration levels. An achievement function finds out attainable and efficient alternatives that bear some relation to the decision maker's aspirations. There are several methods to specify a reference direction. The use of marginal rates of substitution for estimating the gradient of the utility function (Geoffrion, Dyer and Feinberg, 1972) or the technique used in the Boundary Point Ranking method (Hemming, 1976) are two examples of possible techniques. We prefer to use a simple and convenient alternative. The reference direction is projected onto the set of efficient points by using an achievement (scalarizing) function as suggested by Wierzbicki (1980) in his reference point approach. Once the decision maker has specified his aspiration levels for the criteria, we find an efficient solution that minimizes the value of the achievement function. When we apply the achievement function to the reference direction, instead of one point we obtain a set of efficient solutions. When a given direction is projected onto the set of efficient alternatives, the D M does not need to make comparisons between inefficient (or infeasible) solutions. Korhonen and Laakso (1986) have used the same idea in their visual interactive method for solving multiple objective linear programming problems and a similar technique have
P. Korhonen / Visual reference direction approach for solving discrete multiple criteria problems
154
also been suggested by Winkels and Meika (1984) for projecting the estimated gradient of an unknown utility function onto the efficient frontier. However, in the discrete case the projection problem is more difficult to solve than in multiple objective linear programming. There are also several ways to specify an achievement function. The characterizations of achievement functions are considered in more detail by Wierzbicki (1986). We use the following simple form for an achievement function: f(g,
xi, w) = max (gj - x i j ) / w j,
(2.2)
jEJ
where i E I , g ~ R p is a given reference goal vector called a reference point, and w ~ R e is a given weighting vector. If criterion j is maximized wj > 0, otherwise wj < 0. Since we have assumed that all criteria are to be maximized, wj > 0 in our considerations. By minimizing f ( g , x,, w), i ~ 1, for given g and w we find an efficient solution Xk, k ~ I (Wierzbicki, 1980, 1986). Given an initial solution x h, h ~ I, a reference point g and a weighting vector w we can define a parametrized achievement function F for the reference direction d = g - x h as follows: F ( t , d, x h, xi, w ) = f ( x
h + td, xi, w ) ,
(2.3)
where t >_ 0 and i ~ I. By solving the parametric programming problem m i n F ( t , d, xh, x i, w ) , i~l
for all t>_
0,
(2.4)
we obtain a set of decision alternatives Xm, FFI~. M C I, as a solution. The index set M is assumed to be an ordered set M=(m
1, m 2 . . . . . m k ) ,
k~
2.3. Visual interaction
The alternatives in set M are presented for evaluation to the D M both numerically and graphically. The values of the criteria of the alternatives belonging to M are plotted on the screen (on the y-axis). The current alternative m 1 is plotted on the left and the last alternative m k on the right. The values of the criteria of the subsequent alternatives are connected by a line. Different line patterns or colors can be used for different criteria. A sample graphical display is shown in Figure 1, where the cursor (a vertical line) points to the fourth alternative. The numerical values of the criteria for the alternative are shown on the top line of the display. As the cursor is moved to the left or to the right the numerical values of the criteria corresponding to the alternative are displayed. Geoffrion et al. suggested the use of graphics as a possible method for solving the step-size problem as early as 1972, and e.g. Winkels and Meika (1984) have further used computer graphics in connection with the G D F method. However, their approach is not interactive in the same sense as ours.
For each criterion the scale is chosen so that the m a x i m u m value is at the top and the minimum value is at the bottom of the display. This kind of presentation gives the decision maker holistic and exact information on available alternatives in a very convenient way. It is possible for the decision maker to choose the most preferred alternative from among the alternatives presented on this display.
P_r
473
__ H a
65
£!.
!,6
.................. M.~,......£.3.~. ............
(2.5)
where for m j, m h ~ M , j < h , there exists a t* such that X m j is the solution of (2.4), when t = t*, and x,,, is not a solution for any t < t*. The set M consists of the indices of the alternatives to be presented to the DM. In appendix, we prove two lemmas and describe a procedure to find the set M. The problem is solved in two parts. First we determine the explicit form of the parameterized achievement functions: vi(t ) = F(t, d, xh, xi, w), for all i ~ I, and then use these functions in problem (2.4).
~....,....,.., ~
Go
.........~,'-"
left
and
-~ t o
x~ight
(End
=
exit)
Figure 1. A graphical representation of a subset of alternatives
P. Korhonen / Visualreferencedirection approachfor solvingdiscretemultiple criteriaproblems
3. Development of the approach
Table 1 The types of criteria are specified
The proposed method is a modification of the approach of Korhonen and Laakso (1986) to solve general multiple criteria problems. In the discrete case an efficient curve is replaced by a set of efficient alternatives. This subset is presented in a visual form to the decision maker. At each iteration the decision maker is asked to choose the most preferred alternative from the subset. This alternative becomes the starting solution for the next iteration. The decision maker specifies new aspiration levels for the criteria. This information is used to determine a new reference direction. The process is repeated until he is unable to find a better solution than the current one. If it is desired to check the optimality of the final point, we have to make some assumptions concerning the utility function. If it is assumed to be quasiconcave, we can use the method by Korhonen, Wallenius, and Zionts (1984) to eliminate alternatives on the basis of the decision maker's sequential choices until one alternative remains. The approach is implemented on an I B M / P C 1 microcomputer under name VIMDA (a Visual Interactive Method for Discrete Alternatives). The dimensions of the problem in VIMDA are: n = 500 and p = 10. We illustrate the steps of the approach in terms of a numerical example using the steps of program VIMDA. Suppose the owner of a laundry wants to choose a washing machine. In addition to price, he wants to consider washing time and consumption of electricity and water as the most important criteria. He has collected data on these four criteria for thirty-three different types of machines. These data are given in Zeleny (1982, pp. 210-211). We refer to alternatives by indices. Step O. Find an initial (efficient) solution. Ask the decision maker to specify which criteria are to be maximized and which are to minimized (see Table 1). Define weights wj as follows:
VIMDA The types of criteria (max/min):
(rJabs (maxi(rj*x,j)) wj= f [rj
if m a x i ( ~ * xlj) 4=0, if maxi( ~ * xij ) = O,
where § is 1 if criterion j is maximized, and - 1 if it is minimized. (Other choices are also possible.)
Criteria Price Wash.time El.consumption Water consumption
155
Types MIN MIN MIN MIN
Find an initial alternative by optimizing one of the criterion. Here we optimize the last criterion. In our example all criterion are to be minimized and alternative 2, b = (425, 80, 1.5, 110), has the 'best' water consumption (see Table 2). Step 1. Find a reference direction. The values of the current solution and the maxim u m and minimum values of each criterion are displayed in Table 2. The decision maker is asked to give aspiration levels for the criteria. Usually, at the first iteration, the decision maker gives ideal values as aspiration levels for criteria. In our example these are (395, 50, 1.4, 110). The vector from the current alternative to the point defined by the decision maker's aspiration levels is used as a reference direction. For our problem d = (395 - 425, 5 0 - 80, 1 . 4 - 1 . 5 , 1 1 0 - 110) =(-30,
-30,-0.1,0).
Step 2. Find a subset of efficient solutions. In this step we determine the ordered index set M, as described in Section 2. In our example M = {2, 21, 29, 7, 8, 11, 5, 26}. Step 3. Find the most preferred solution from among set M.
Table 2 Specification of aspiration levels VIMDA The specification of aspiration levels for criteria The name of a current solution: 2 Criteria Lower Upper Current bounds bounds values Price 395 595 425 Wash.time 50 80 80 El.consumption 1.4 1.8 1.5 Water consumption 110 140 110
Aspiration levels 395 50 1.4 110
156
P. Korhonen / Visual reference direction approach for solving discrete multiple criteria problems
The alternatives corresponding to the index set M are presented to the decision maker in a visual form. He is asked to indicate his most preferred alternative. The values of the criteria are plotted on the screen using distinct colors (red, yellow and green) and line patterns for the criteria. The cursor can be moved from alternative to alternative and the corresponding numerical values of criteria are displayed simultaneously. The scale for each criterion j is chosen in such a way that maxixij is at the top of the screen and minix~j is at the bottom. If an improved solution is found or the decision maker is willing to consider other directions, return to step 1, otherwise stop. Figure 1 is a sample display at the first iteration. Let us assume that alternative 7 is the most preferred choice of the decision maker from this set. The current alternative for the next iteration is alternative 7. Let us assume that the aspiration levels for the next iteration are (450, 60, 1.5, 130). Now the set M = (7, 8, 19, 11, 18, 26}. The new display is described in Figure 2. If we assume that the washing time is one of the most important criteria, the decision maker can choose alternative 19, b = (543, 57, 1.6, 120), as the most preferred alternative from this set. If the aspiration levels of the decision maker for criteria are the same as at the previous iteration the set M = {19, 14, 8, 7, 6, 3, 27, 32}. If he still prefers the current solution 19 to the others he may be willing to stop. In our approach we are not primarily interested to prove the optimality of the final solution. We prefer the satisfaction of the decision maker to proving the optimality, mathematically. If it is desired to check the optimality of the final point, we have to make some assumptions about the utility function. If the utility function is assumed to be quasiconcave, we can use the method by Korhonen, Wallenius, and Zionts (1984) for eliminating alternatives on the basis of the decision maker's sequential choices. We can utilize all preference information cumulated so far during the process. If more than one alternative remain we can restart the search process with the remaining alternatives. We terminate the process when only one alternative remains--which is optimum.
P~
543
Na
57
E1.!.,6
...............N . , . . . . 1 . 2 0 .........
I
~f--¸7 i.- -¸ ........
/,J
"z
::7........
""-....
t/
J i..... '..... ~-
to
left
i
and
"+
_i
to
]
,
--'------'~--~----____ x~ight
(Edd
ex
Figure 2. The display at the second iteration
When people used our approach, we noticed an interesting fact. Some of them were making cycles during the search process, i.e., either they chose the same alternative many times although not at subsequent iterations or they chose an alternative at some stage even it was shown but not chosen earlier. It implies that their preference structure were not transitive. In the other words, we do not have proper information about their utility function. Therefore we have not implemented the optimality check in program VIMDA.
4. Discussion In this paper our main purpose was to develop an interactive decision aid for the decision maker to evaluate alternatives close to his aspiration levels. In our opinion, a full benefit from an interactive approach can be obtained by interactive utilization of computer graphics; visual representation enables the decision maker to evaluate a large set of possible alternatives simultaneously. To avoid irrelevant information only nondominated solutions are presented to the decision maker for evaluation. Besides the use of visual aids, our approach has three more desirable features. First, the decision maker is free to examine any set of the efficient frontier he pleases at any moment, i.e. he is neither confined to evaluate solutions with some special properties, nor is his freedom limited by his earlier choices during the search process. Second, we need no specific assumptions about the underlying utility function of the decision maker during the
P. Korhonen / Visual reference direction approach for solving discrete multiple criteria problems
interactive process. The utility function can even be assumed to be changing due to learning and 'changes of mind' during the search process. If we are interested in proving the optimality of the final solution, then we have to make some assumptions about the utility function. Third, the decision maker is asked to give very simple information at each iteration. He has only to specify his aspiration levels for criteria and then evaluate reasonable alternatives by using an informative representation. We have developed our approach assuming quantitative criteria. However, the approach can be used even with ordinal criteria, i.e. when the decision maker is only able to rank alternatives according to each criterion. He can express his aspiration levels in terms of ordinal numbers and the approach tries to find the alternatives that reflect his aspirations. Although, the choice of the subset of efficient alternatives depends on the weights used, it primarily depends on the aspiration levels specified by the decision maker. By setting weights in a reasonable way we can help the decision maker to find the alternatives he likes. Preliminary experiments indicate that the decision makers find the approach easy to use and understand. They also seem to like colors, figures and spreadsheet interface. Especially, the use of colors appears to enhance the attractiveness of the method.
Appendix. A procedure to find the set M In this present a indices of maker for
appendix we prove two lemmas and procedure to find the set M of the the alternatives given to the decision evaluation.
Lemma 1. The functions vfit), i ~ I, are convex. Proof. Let o ~ [0, 1]. Using the definition of v~(t):
v i ( t ) = max (Xhj + tdj - x i j ) / w j , j~J
we obtain that
o v , ( t , ) + (1 - o ) v i ( t 2 ) = m a x o ( x h j + t,dj - x i j ) / w j jGJ
(A.1)
+
max(1 rEJ
157
O ) ( X h r "1- t 2 d r - X i r ) / W r
>~ max (Xh, + (Off + (1 -k~J
o)t2)dk -
x,k)/w,
O , ( a t I "~- (~ -- o ) t 2 ) ,
which proves the convexity of v,(t).
Lemma 2. The functions v i(t), i ~ 1 and t >__O, are piecewise linear and are of the form
Vi(t ) = b i r + t c i r
when t E [ t / r , tar+l,,),
where r ~ [ 1 , 2 . . . . . ni] , ni<~p, and ti(n,+l) = ~ . Moreover, b~k < bij and c~k > c,j, if k > j . Proof. For each t we can write
vi( t ) = max (Xhj + tdj - x,j) / w j jGJ
= (Xhr-
Xir)/WrJt - tdr/w,
= b,(t) + tc,(t), where bi(t ) = (Xh, -- X i , ) / W , and ci(t ) = d r / w r. Consider t 1 > 0, t 2 > 0 and t 2 > tl, and assume that ci(tl)4= c,(t2). By the definition of v,(t) we have
vi( tl) = bi( t,) + t,c,( t,) >~ b,(t2) + t,ci( t2) , v i ( t : ) = bi( t2) + t2c,( t2) >I bi( t,) + t2ci( t,), Hence it follows that
bi(t,)-bi(t2)+tx[c,(t,)-G(t2)
] >10,
(A.2a)
bi(t2)-bi(tl)+t2[ci(t2)-ci(t,)
] >10.
(A.2b)
By adding inequalities (A.2a) and (A.2b) we obtain
It 2 - t , ] [ c , ( t 2 ) - c,(t,)] >/0.
(A.3)
Since t 2 > t I and ci(t2) 4= c,(tl) , inequality (A.3) holds iff c , ( t 2 ) > c~(tl). Inequality (A.2a) is nonnegative only if bi(tl) > bi(t2). If ci(t2) = G(tl), then from inequalities (A.2a) and (A.2b) it follows that bi(t2) = bi(ta). Because b i(t) and c~(t) are monotonic functions with at most p different values, there are n~ different t values; t,j, j = 1, 2 . . . . . n,, n ~ < p , where bi(t ) and ci(t ) change their values. By writing b,j = bi(tij ) and c~y = c(t,j) we can complete the proof. To illustrative our problem we use a simple example. Let us assume we have a 5 × 4 decision
158
P. Korhonen / Visual reference direction approach for solving discrete multiple criteria problems
Phase 1: Preliminary computations
alternative matrix
X=
1 0 4 2 9
3 7 5 8 5
8 3 2 1 2
1] 4 6 • 4 3
Find the index vectors m i referring to the rank orders of the elements of the vectors Yi = W(Xh -Xi), for all i, i = 1, 2 . . . . . n. Let the element mix refer to the largest element of the vector Yi and mip refer to the smallest element.
Let alternative 3: (4, 5, 2, 6) be our current alternative and assume that the D M has specified the vector (5, 6, 7, 5) as his aspiration level vector. T h e reference direction is thus (5, 6, 7, 5 ) (4, 5, 2, 6) = (1, 1, 5, - 1). Using formula (A.1) we can n o w find function vi(t ) for each alternative i, i = 1, 2 . . . . . 5. These functions are described in Fig. 3. T o describe the procedure to find the set M assume that the current alternative is x h and the reference point specified by the D M is q. The reference direction is d, d = q - x h. D e n o t e by W a diagonal matrix with reciprocal values of the weight vector w along the main diagonal: W = diag(1/w~, 1/w2,...,
1/Wp)
F r o m Figure 3, we can see that the set M, in our example, consists of three alternatives: (3, 2, 1) in this order. All these alternatives give a minim u m value for a parametrized achievement function (2.3) with some p a r a m e t e r value t. N o t e that we have not picked alternative 5 into the set M, because it does not give a better value for an achievement function than alternative 3, which is already included in the set M. T h e procedure consists of three phases.
-...~
..... •
//.
/
/
Phase 2: Compute the functions vi, i = 1, 2, . . . . n S t e p 2 . 1 . Set j : = l , n i : = 0 , s : = j , and t : = 0 . Step 2.2. Set bij :=Yimis , Cij : = din, , t~j:= t, and n i : = n i -]- 1. Step 2.3. Find r, r 6 (s, p], such that dml r > dm,,
and
Zr =
m i n . >, ( z, },
where z u = (Yimis -- Yimi, ) / ( d ' i r d m,s ) Step 2.4. If such r exists then s : = r , j : = j + l , t := z., and go to 2.2; otherwise, stop. Functions vi(t ) are found as follows: vi(t ) := bij + tcij, when t ~ [tij, ti(j+u), for j < ni, and vi(t ) := bi. ' + tci.,, when -
-
t >~ tin /
Phase 3: F i n d the index set M Step 3.1. Find the first alternative r. Define K := { l, 2 . . . . . n } and find r as vr (0) = br, (0) = m i n i vi (0) = m i n ibi~ (0). Set M . ' = ( r } and ki.'= 1, for all i ~ K . Step 3.2. D e t e r m i n e the range of the current linear part of Or(t ) . If kr>~n r then set T : = ~ ; otherwise, set T : = tr(kr+l ). Set S T : = [trk, T]. Step 3.3. F o r each i, i ~ K, find the maximal u~ for which vi(t ) > v~(t), for all t ~ [0, ui] nST. A: If crk" - c~k' ~< 0 then set u~ := T; otherwise, c o m p u t e Z := (bik ' -- ark,)//(Crk ~ -- Cik,)
3 t ~
Figure 3. The parametrized achievement functions for each alternative
If z >/T, set u i := T, and if z < ti(k,+l), set u i := z; otherwise set k i .'= k i + 1 and return to A. Step 3.4. Redefine k,, for each i ~ K, and set K. Set k i := m a x ( k i I tik, <~ Ui }" If k i = n i then set K : = K N {i}. If K = ~ then stop.
P. Korhonen / Visual reference direction approach for solving discrete multiple criteria problems
Step 3.5. Update M. Find uj = If % = T , otherwise and go to
m a x ~ x U i. set k r : = k r + l and go to 3.2; set r .'=j and M := M tA { j ) 3.2.
Acknowledgements The author wishes to thank prof. Subhash Narula for many helpful suggestions and comments.
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