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An interactive algorithm for multicriteria programming
Compuf. & Ops Res. Vol. 7. pp. 81-87 Pergamon Press Ltd., 1980. Printed in Great Britain
AN INTERACTIVE ALGORITHM FOR MULTICRITERIA PROGRAMMING-t E. U. CHOOS and D. R. ATKINS University of British Columbia, Vancouver, B.C., Canada Abstract-An interactive algorithm for general multicriteria programming is proposed. Thedecisionmaker merely selects his most preferred point amongst presented alternatives. The algorithm ensures that the alternatives generated in each iteration are evenly distributed over the desiredneighbourhood of the efficient frontier. The algorithm was motivated by the need for solutions to the multiple linear fractional problem and is shown to have a particularly simple and easy to implement stmcture in this case. A discussion is included of why this problem is important and of how it arises in practice, particularlyin financial problems.
INTRODUCTIONAND MOTIVATION The use of interactive algorithms for multicriteria optimization has been proposed by several
authors[lA]. By gaining some partial information about the decision maker’s preferences at even an early stage these interactive algorithms avoid the enumeration of all efficient points[79, 10, 11-131. The purpose of such interactive algorithms is to present to the decision maker, in a series of meetings, a choice of efficient alternatives which are in some sense representative of all those available. Over this series of meetings he is to explore his preferences amongst presented alternatives and finally choose a most preferred one which he is prepared to admit as satisfactory. Many existing such algorithms suffer from two important inadequacies. Firstly, they do not extend naturally to the case when some at least of the criteria are nonlinear functions and secondly they often do not give a large measure of control to ensure the representative nature of the trial efficient points generated. We have stated that these inadequacies are “important” and this claim will be substantiated in the next paragraph in order to illustrate that we are not simply undertaking a minor extension to existing models. The work reported in this paper has not been stimulated by the idea of non-linear criteria in general but has been forced on us by the need to cope with one type of non-linearity in particular, that is where the criteria in question are linear fractionals or ratios. The need has arisen in two practical contexts already and is likely to be a problem of general concern. The first context is with financial planning; people involved with the raising and allocating of funds often have great trouble with single criterion linear programs, which somehow fail to capture the multifaceted nature of the alternatives open to them. On the other hand multi criteria optimization which presents them with a range of efficient alternatives seems to be more in keeping with their normal decision-making style of exploring several different plans. Unfortunately many of the criteria that they would wish to employ are ratios such as return on capital employed, liquidity, earnings per share, dividend per share, earnings cover, dividend cover, etc. The same situation is encountered with medium range production planning but here the ratios in question are productivity ratios, ratios of seasonal to full-time labour, capacity utilization, etc. This problem, which we shall term the Multicriteria Linear Fractional Problem, (MLFP), is not a trivial extension of the linear case as can be seen in the following illustrative “toy” example.
“max” f, = 1 + :,I+ x2 and f2= 22 1+x, ‘IThiswork was partiallysupportedby NRC grant A-4743. SEng U. Choo is a Ph.D. candidate in the Faculty of Commerceand Business Administrationat the University of British Columbia. Canada. He has oublished articles in Journal of OptimizationTheory and Applications, Canadian _. MathematicalBulletinand Nante Maihematica. Werek Atkins is Associate Professor of ManagementScience in the Faculty of Commerce and Business Administrationat U.B.C. He has published articles in the Journalof the U.K. OperationalResearch Society, Journalof Finance, Omegaand the EngineeringEconomist. 81
E. U. CHOOand D. R. ATKINS
82
such that
The efficient region is clearly seen to be as hatched in Fig. 1.
Fig. 1.
Even in this exceptionally simple example the potential problems can be seen, the efficient region is not piecewise linear and the image of the feasible decision space is not convex. Thus methods that proceed by maximizing the weighted sum of criteria, which is itself non-linear of course, will give only the extreme points and will miss potentially interesting efficient points between. This paper presents a tractable method of approaching this type of problem. In the next section an informal description of the algorithm is presented, the tedious details being relegated to an Appendix. A further section illustrates how this would operate on our “toy” example and a final section discusses some particular aspects of the implementation as well as the specialization to the all linear criteria case. AN INFORMAL
DESCRIPTION
OF THE ALGORITHM
In order to use an optimization procedure to generate efficient points we need a surrogate maximand. The familiar form of the weighted sum of criteria we have seen to be useless both because it is highly non-linear and, anyway, will not give the efficient frontier in the non-convex case. An alternative form and hence an alternative characterization of the (weakly) efficient set is given by the Tchebycheff norm which is the maximum weighted deviation from some “ideal” point u*. min max&(uy - fi(x)) for feasible x or min Z s.t. 2 2 pi(u) -fi(x)) for feasible
X.
The point u* is chosen to be slightly larger than the maximum of each criterion individually ensuring that no UT is in fact attainable. The weights pi are chosen a prior’ and more will be said about these shortly. The isoquants for this norm are the “corners” lying along the line through u* with direction (l/p,, l/&, . . . ,1/p*) as shown in Fig. 2. The Tchebycheff norm minimization chooses the “corner” closest to u* and still in contact with the feasible region. This final point of contact will be (weakly) efficient. In fact every choice of positive wieghts gives a (weakly) efficient point and every such point comes from such a choice of weights. This gives the characterization we need in the non-convex case because, loosely speaking, the “corner” can probe into the non-convex regions, as shown in Fig. 2. This characterization is developed rigorously in Appendix 1. Although this characterization is used directly in the first
An interactive algorithm for multicriteria programming
83
fz
Fig. 2.
phase of the algorithm the second phase could have been developed without its aid. Nevertheless the choice of the mechanics of the algorithm were directly motivated by this characterization and we believe the logic of the method is best seen in this light. The algorithm proceeds in two phases, and they will be described separately. Phase I This phase does not involve the decision maker and is really a means of getting started. From the point U* a search direction is chosen by selecting the pi which keeps the vertex of the “comer” in the central area of the feasible region. As shown in Appendix 2 a slightly modified form of the Tchebycheff norm minimization reduces in the case of the MLFP to a linear program with a single parameter. We thus use any convenient univariate search method over this parameter to find a point as close to efficient as we like. The “closeness” here is not critical and even a rough approximation will be quite sufficient. This point will be taken as the “best” point for starting phase two. In subsequent iterations though note that the “best” point will be (weakly) efficient although here it is only approximately so. Phase II This is the principal phase and intimately involves the decision maker. The search direction and hence the choice of pi, is that from the “ideal” point to the “best” point, which is then extended on to include several more points. Thus in Fig. 3, y” is the “best” point and y’, with k = 1 up to some maximum k = m are the extra points. gk is the distance between y” and yk. Then taking each criterion in turn, say f, first, we maximize f, subject to all other criteria being at least equal to their value at y”, then y’, all the way to y”. This will give the sequence of (weakly) efficient points y”, y”, . . . y’” for criterion 1 and y20,y*r, etc. for criterion 2 and so on for all the criteria. Now each of these maximizations is of a linear fractional subject to linear constraints so the Charnes and Cooper [ 131transformation can be used to transform the problem into a linear program. These (weakly) efficient solutions are then presented to the decision maker who is asked to select the one that best fits his needs. The selected point now becomes the new
Fig. 3.
E.U.Cmo andD.R.ATKINS
84
“best” point and Phase II is repeated but now with a substantially reduced choice of the distance g, in order that the search space be focused on efficient points of most interest. The choice of m is really the only ad hoc choice to be made and this is unlikely to be difficult in any real context. The iteration ceases when a satisfactory efficient point is reached.
DEMONSTRATIONOFTHEALGORITHMONTHE“TOY"EXAMPLE
The example is
‘hlax” f1= s.t.
1
+x:‘+ x*
x,+x2=zl,
f2 = &
I
X1,X*~O.
We may take u* = (0.6, 1.2) and the Phase I problem is then: min 2 s.t.
z 2 0.6( 1 + x1+ x2) - xi - 0.6 w(I+ zi + xz)B 22 1.2(1+xi)-xz1axl+x2andx,,x230,
1.2w(l +xi)B z>O.
Where i = dl.8. After a few iterations of Step 2 of Phase I we could take w = 2/3B, xl = 7118,x2 = lo/l8 with function values f, = l/5 and f2 = 2/5, clearly a non-efficient point as it is dominated by the efficient point fi = l/5, f2 = 3/7. Phase II now begins with l/B, = 0.4, l/B2 = 0.8 and, arbitrarily m = 2. The consequent sequence of efficient points resulting from Phase II will be
f, 0.0 0.1 0.2 3/14 l/3 0.5 0.2 0.0 k2100 12 1.25 1.5 gk 1.5 1.25 1 1
f2 1.0 213 317 0.4
These are presented to the decision maker who suppose chooses (0.1, 2/3) as the best point. Phase II will be repeated but this time with l/B, = 0.6-0.1, l/B2 = 1.2-2/3 and with 0, the reduction factor of the search direction length, now equal to 2. Notice that from the second iteration onwards the “best” point at the end of the previous iteration will be in the list of (weakly) efficient points for the next.
SOMEFURTHERCOMMENTS
Although the principle employed in the method could clearly be extended to the more general nonlinear case, the particularly suitable form of the phase two iterations for the MLFP case is that they revert to a sequence of linear programs. This sequence can be made to run very economically by the judicious use of the basis “save and restore” options and the “row masking” facilities available with most large LP systems. In the very special case when all the criteria are in fact linear the method is still not without merit. Phase I is now simply a linear program or the method proposed in [14] can be used and Phase II is a linear program with right hand side parametrics. Even in this special case the algorithm retains its most attractive feature of controlling the dispersion at trial efficient points. This is in marked contrast to methods which maximize linear weighted combinations of the criteria for which the dispersion of trial points can be quite irregular and of course confined to vertices of the efficient region only.
85
An interactive algorithm for multicriteria programming REFERENCES
1. R. Benayoun, J. de Mintgolfier, J. Tergny and 0. Laritchev, Linear programming with multiple objective functions: Step Method (STEM). Mathemoricol Programming 1, 366-375(1971). 2. J. S. Dyer, Interactive goal programming. Mgmr. Sci. 19(l), 62-70 (1972).
3. A. M. Geoffrion, J. S. Dyer and A. Feinberg, An interactive approach for multi-criterion optimization, with an application to the operation of an academic department.Mgmr Sci. 19(4),357-363(1972). 4. R. E. Steuer, An interactive multiple objective linear programming procedure. TIMS Studies in the Management Sci. 6, 225-239(1977). 5. R. E. Steuer and A. T. Schuler, An interactive multiple-objective linear programming approach to problem in forest management. Op. Res. 26(2), 254-269(1978). 6. S. Zoints and T. Wallenius, An interactive programming method for solving the multiple criteria problem. Mgmt Sci. 22 (6) 652-663 (1976). I. J. G. Ecker, N. S. Hegner and I. A. Kouada, Generating all maximal efficient faces for multiple objective linear Programs. Paper presented at TIMSIORSA Conf., New York (1978). 8. J. P. Evans and R. E. Steuer, A revised simplex method for linear multiple objective programs Mathematical Programming 5, 54-72 (1973).
9. T. Gal, A general method for determining the set of all efficient solutions to a linear vector maximum problem. European i Op. Rex 1, 307-322 (1977).
_
10. H. Isermann. The enumeration of the set of all efficient solutions for a linear multiole oroaram. 00. Res. . obiective Quart. 28,71 l-725 (1977). 11. P. L. Yu, A class of solutions for group decisions problems. Mgmt Sci. 19(g),936-946 (1973). 12. P. L. Yu and M. Zelenv. The set of all nondominated solutions in linear cases and a multicriteria simolex method. J. .I
Mathematical Analysis-and Application 49,430+8
.
(1975).
13. M. Zeleny, Compromise Programming. in Multiple Criteria Decision Making, pp. 262-301,J. L. Cochrane and M. Zeleny (Eds.). University of South Carolina Press, Columbia, SC. (1973). 14. A. Charnes and W. W. Cooper, Programming with linear fractional functionals. Naval Res. Logistics Quart. 9,181-l% (1%2).
15. M. E. Posner and C. T. Wu, Contributions to linear max-min programming. Paper presented at TIMSIORSA Conf., University of Wisconsin-Milwaukee, Los Angeles (1978). 16. V. J. Bowman, On the relationship of the Tchebycheff norm and the efficient frontier of multiple-criteria objectives. in Mtdtipte Criteria Decision Making, pp. 76-85, Jouy-en Josas (1975). 11 M. Zeleny, The theory of the displaced ideal, Multiple Criteria Decision Making: Kyoto 1975,pp.15l-205, Springer-Verlag, New York (1975). I,.
APPENDIX
l-CHARACTERIZATION
OF EFFICIENCY
BY TCHEBYCHEFF
NORM
(1) The multicriteria programming problem can be stated as follows:
(MP)“max” F(x) = f(x) = (f,(x), ss.
. ,f&))
XES
where S is a compact subset of R” and f ,, . ,fL are continuous real functions. It is obvious that there exist z’, . . , z’-in S such that fi(zi) = maxf&)
,i = 1,.
uf=fi(z’)+e
,i=l,...,L
xc.5
, L.
(1)
Let (2)
where e is a small positive number. The exact magnitude of t is not important as long as c is small relative to fi(z’), NOTATION For any two vectors y’ and y2, the notation 5 , s and < have the following meaning: y’ 5 y2iffyi’
for every i
y’ G y’iffyt c y? for every i and y’ # y*
y ’ < yziffy! < y? for every i. Definition. Let x’, x*E S. We say that xi is dominated (respectively strictly dominated) by x2 iff F(x’) 6 F(x’) (respectively F(x’) < F(x*)). Defnition. A feasible point x in S is said to be efficient (resp. weakly efficient) iff x is not dominated (resp. strictly dominated) by any other feasible points. For any vector /3 > 0, the generalized Tchebycheff norm of a vector y is defined by
IIYIIB = maxSilnl. I
E. U. CHCZG and D. R. ATKINS
86
Let (P/3) denote the following program min lIu* - F(x)ll~
(PP)
x E s.
St.
The proof of the following theorem can be found in Bowman[lS]. Theorem 1. Let x* be a feasible point in S. If x* is efficient, then x* solves (&I) for some /3 > 0. In general, the inverse of Theorem I is not true. For weakly efficient points, we have the following characterization. Theorem 2. Let x* be a feasible point in S. Then x* is weakly efficient iff x* solves (Pfl) for some /3 > 0. Proof:
(e)
Suppose x* solves (Pp) for some /3 > 0. If x* is strictly dominated by x’ in S, then F(x*) < F(x’) and thus Pi(uf-fi(x*))>Bi(ur-fi(x’))
,i=l,...,L.
It follows that II@* - F(x*)lle ’ IIn*-
w$?.
This contradicts the hypothesis that x* solves (P/I). Hence x* is weakly efficient. (3). Suppose x* is weakly efficient. Set i = 1,.
pi = (UT-ii(x’,
, 12.
Then p > 0 and )lu*- F(x*)([~= I. Assume that x* does not solve (P/3). Let x” be an optimal solution of (Pfl). Then 1/u*$ll;X&(U?
-fi(X”))
JUT-fJx”)
<
FwJll, < u*- F(x*)lla
I ,i=l,...,L
< F(x’?.
Thus x* is not weakly efficient. This is a contradiction. Hence x* solves (Pfi). It follows from Theorem 2 that the multiparametric program (P@) generates all weakly efficient solutions of (MP). The problem (Pp) seeks the feasible solution which is “nearest”, in the sense of generalized Tchebycheff norm (1.[Ia,to the ideal point U* hot attainable) where pi gives the relative importance of the deviation of fi value from uf. It is difficujt and not practical to compute the weight vector fi that gives the optimal solution. The ideal point u* chosen is different from that in [I I, 171in order to have fi > 0. APPENDIX 2-DETAILS OFTHE ALGORITHM-PHASE If the maximum of each criterion separately occurs at xi, then define
I
1 = 147- minfi(xj) Pi
i
and let B be the normalization factor where -;2 = zj$ Define the problem (QBw) by (Q@v) min I S.t.&=Pi(Uf
I
I
-f!(X))-
WB
i=l,...,L
where
ci=x +ri fdx)=diTx + ti
and diTx+ti>O
for i=l,...,L.
Then (Qpw) is a linear program with a single parameter IV,and if z(p) is the optimal value of (P/3) (see Appendix l), then z(/3)= inf{w 20:h(w) = 0) where h(w) is the optimal value of (QBw). It is clear that h(w) is a nonnegative nonincreasing function of w over [O,m]and that there exists a w’ satisfying h(w’) = 0. We now have an univariate search problem over [0, w’]for which the following procedure will be sufficient to solve to whatever accuracy is required.
An interactive algorithm for multicriteria programming Step 0 Step 1
87
Put the interval [L, H] = [0, w’] H+L w= 2 evaluate h(w)
Step 2 If h(w) 2 0 put L = w. If h(w) = 0 put H = w and stop when [L, H] is small enough and h(w) = 0. Otherwise return to Step 1.
Let the point obtained be I? and
Also initially set B = 0 and D = $ - 3.
Phase II We have the “best” point y” either from Phase I or a previous iteration of Phase II. Set l/pi = uf - y: and let B be the normalization factor defined as in Phase I. We update the scalar 19,that acts to contract the search direction and concentrate our attention on smaller parts of the efficient surface, by 0 = 0 f 1.Ifwe have chosen m inspection points then define, gb=k.$fork=O,l
,...,
tn.
For each criterion i we now solve max 0x) s.t.
fi(x)3uf---(gkt3)
forallj
;
Ax 5 b. This is solved for each k successively until the maximand stops increasing. For each i and k the optimal solution vector xi’ gives the criteria vector
If there are no feasible solutions other than y’, we repeat Phase II, otherwise the decision maker is asked to choose the most preferred yp from the y’. If this is satisfactory we stop, otherwise we repeat Phase II with y” = yp. and 3 = [lu* - yql. For the case of multiple linear fractional problem the sequence of maximizations takes on the particularly simple form of a sequence of linear programs. We can see this by using the Chames and Cooper [I41 transformation x’ = nx, for real n > 0. y” is obtained by solving the following linear program. max
CiTxx’ t
riTj
s.t. Ax’ - bq 5 0 cjrx’
trpj 2 uf(~jTx'ttj7))-~djTx'tljq)(gkt C)for allj Bi di’x’ t
till = 1 n 2 0.
Because of the non-convex nature of the problem in general and also because of the unknown nature of the decision makers implicit utility function, this procedure cannot guarantee convergence to anything other than a local optimal. It is the purpose of the wide coverage of the efficient region initially with the algorithm to try and minimize any serious consequences of this.
AN INTERACTIVE ALGORITHM FOR MULTICRITERIA PROGRAMMING-t E. U. CHOOS and D. R. ATKINS University of British Columbia, Vancouver, B.C., Canada Abstract-An interactive algorithm for general multicriteria programming is proposed. Thedecisionmaker merely selects his most preferred point amongst presented alternatives. The algorithm ensures that the alternatives generated in each iteration are evenly distributed over the desiredneighbourhood of the efficient frontier. The algorithm was motivated by the need for solutions to the multiple linear fractional problem and is shown to have a particularly simple and easy to implement stmcture in this case. A discussion is included of why this problem is important and of how it arises in practice, particularlyin financial problems.
INTRODUCTIONAND MOTIVATION The use of interactive algorithms for multicriteria optimization has been proposed by several
authors[lA]. By gaining some partial information about the decision maker’s preferences at even an early stage these interactive algorithms avoid the enumeration of all efficient points[79, 10, 11-131. The purpose of such interactive algorithms is to present to the decision maker, in a series of meetings, a choice of efficient alternatives which are in some sense representative of all those available. Over this series of meetings he is to explore his preferences amongst presented alternatives and finally choose a most preferred one which he is prepared to admit as satisfactory. Many existing such algorithms suffer from two important inadequacies. Firstly, they do not extend naturally to the case when some at least of the criteria are nonlinear functions and secondly they often do not give a large measure of control to ensure the representative nature of the trial efficient points generated. We have stated that these inadequacies are “important” and this claim will be substantiated in the next paragraph in order to illustrate that we are not simply undertaking a minor extension to existing models. The work reported in this paper has not been stimulated by the idea of non-linear criteria in general but has been forced on us by the need to cope with one type of non-linearity in particular, that is where the criteria in question are linear fractionals or ratios. The need has arisen in two practical contexts already and is likely to be a problem of general concern. The first context is with financial planning; people involved with the raising and allocating of funds often have great trouble with single criterion linear programs, which somehow fail to capture the multifaceted nature of the alternatives open to them. On the other hand multi criteria optimization which presents them with a range of efficient alternatives seems to be more in keeping with their normal decision-making style of exploring several different plans. Unfortunately many of the criteria that they would wish to employ are ratios such as return on capital employed, liquidity, earnings per share, dividend per share, earnings cover, dividend cover, etc. The same situation is encountered with medium range production planning but here the ratios in question are productivity ratios, ratios of seasonal to full-time labour, capacity utilization, etc. This problem, which we shall term the Multicriteria Linear Fractional Problem, (MLFP), is not a trivial extension of the linear case as can be seen in the following illustrative “toy” example.
“max” f, = 1 + :,I+ x2 and f2= 22 1+x, ‘IThiswork was partiallysupportedby NRC grant A-4743. SEng U. Choo is a Ph.D. candidate in the Faculty of Commerceand Business Administrationat the University of British Columbia. Canada. He has oublished articles in Journal of OptimizationTheory and Applications, Canadian _. MathematicalBulletinand Nante Maihematica. Werek Atkins is Associate Professor of ManagementScience in the Faculty of Commerce and Business Administrationat U.B.C. He has published articles in the Journalof the U.K. OperationalResearch Society, Journalof Finance, Omegaand the EngineeringEconomist. 81
E. U. CHOOand D. R. ATKINS
82
such that
The efficient region is clearly seen to be as hatched in Fig. 1.
Fig. 1.
Even in this exceptionally simple example the potential problems can be seen, the efficient region is not piecewise linear and the image of the feasible decision space is not convex. Thus methods that proceed by maximizing the weighted sum of criteria, which is itself non-linear of course, will give only the extreme points and will miss potentially interesting efficient points between. This paper presents a tractable method of approaching this type of problem. In the next section an informal description of the algorithm is presented, the tedious details being relegated to an Appendix. A further section illustrates how this would operate on our “toy” example and a final section discusses some particular aspects of the implementation as well as the specialization to the all linear criteria case. AN INFORMAL
DESCRIPTION
OF THE ALGORITHM
In order to use an optimization procedure to generate efficient points we need a surrogate maximand. The familiar form of the weighted sum of criteria we have seen to be useless both because it is highly non-linear and, anyway, will not give the efficient frontier in the non-convex case. An alternative form and hence an alternative characterization of the (weakly) efficient set is given by the Tchebycheff norm which is the maximum weighted deviation from some “ideal” point u*. min max&(uy - fi(x)) for feasible x or min Z s.t. 2 2 pi(u) -fi(x)) for feasible
X.
The point u* is chosen to be slightly larger than the maximum of each criterion individually ensuring that no UT is in fact attainable. The weights pi are chosen a prior’ and more will be said about these shortly. The isoquants for this norm are the “corners” lying along the line through u* with direction (l/p,, l/&, . . . ,1/p*) as shown in Fig. 2. The Tchebycheff norm minimization chooses the “corner” closest to u* and still in contact with the feasible region. This final point of contact will be (weakly) efficient. In fact every choice of positive wieghts gives a (weakly) efficient point and every such point comes from such a choice of weights. This gives the characterization we need in the non-convex case because, loosely speaking, the “corner” can probe into the non-convex regions, as shown in Fig. 2. This characterization is developed rigorously in Appendix 1. Although this characterization is used directly in the first
An interactive algorithm for multicriteria programming
83
fz
Fig. 2.
phase of the algorithm the second phase could have been developed without its aid. Nevertheless the choice of the mechanics of the algorithm were directly motivated by this characterization and we believe the logic of the method is best seen in this light. The algorithm proceeds in two phases, and they will be described separately. Phase I This phase does not involve the decision maker and is really a means of getting started. From the point U* a search direction is chosen by selecting the pi which keeps the vertex of the “comer” in the central area of the feasible region. As shown in Appendix 2 a slightly modified form of the Tchebycheff norm minimization reduces in the case of the MLFP to a linear program with a single parameter. We thus use any convenient univariate search method over this parameter to find a point as close to efficient as we like. The “closeness” here is not critical and even a rough approximation will be quite sufficient. This point will be taken as the “best” point for starting phase two. In subsequent iterations though note that the “best” point will be (weakly) efficient although here it is only approximately so. Phase II This is the principal phase and intimately involves the decision maker. The search direction and hence the choice of pi, is that from the “ideal” point to the “best” point, which is then extended on to include several more points. Thus in Fig. 3, y” is the “best” point and y’, with k = 1 up to some maximum k = m are the extra points. gk is the distance between y” and yk. Then taking each criterion in turn, say f, first, we maximize f, subject to all other criteria being at least equal to their value at y”, then y’, all the way to y”. This will give the sequence of (weakly) efficient points y”, y”, . . . y’” for criterion 1 and y20,y*r, etc. for criterion 2 and so on for all the criteria. Now each of these maximizations is of a linear fractional subject to linear constraints so the Charnes and Cooper [ 131transformation can be used to transform the problem into a linear program. These (weakly) efficient solutions are then presented to the decision maker who is asked to select the one that best fits his needs. The selected point now becomes the new
Fig. 3.
E.U.Cmo andD.R.ATKINS
84
“best” point and Phase II is repeated but now with a substantially reduced choice of the distance g, in order that the search space be focused on efficient points of most interest. The choice of m is really the only ad hoc choice to be made and this is unlikely to be difficult in any real context. The iteration ceases when a satisfactory efficient point is reached.
DEMONSTRATIONOFTHEALGORITHMONTHE“TOY"EXAMPLE
The example is
‘hlax” f1= s.t.
1
+x:‘+ x*
x,+x2=zl,
f2 = &
I
X1,X*~O.
We may take u* = (0.6, 1.2) and the Phase I problem is then: min 2 s.t.
z 2 0.6( 1 + x1+ x2) - xi - 0.6 w(I+ zi + xz)B 22 1.2(1+xi)-xz1axl+x2andx,,x230,
1.2w(l +xi)B z>O.
Where i = dl.8. After a few iterations of Step 2 of Phase I we could take w = 2/3B, xl = 7118,x2 = lo/l8 with function values f, = l/5 and f2 = 2/5, clearly a non-efficient point as it is dominated by the efficient point fi = l/5, f2 = 3/7. Phase II now begins with l/B, = 0.4, l/B2 = 0.8 and, arbitrarily m = 2. The consequent sequence of efficient points resulting from Phase II will be
f, 0.0 0.1 0.2 3/14 l/3 0.5 0.2 0.0 k2100 12 1.25 1.5 gk 1.5 1.25 1 1
f2 1.0 213 317 0.4
These are presented to the decision maker who suppose chooses (0.1, 2/3) as the best point. Phase II will be repeated but this time with l/B, = 0.6-0.1, l/B2 = 1.2-2/3 and with 0, the reduction factor of the search direction length, now equal to 2. Notice that from the second iteration onwards the “best” point at the end of the previous iteration will be in the list of (weakly) efficient points for the next.
SOMEFURTHERCOMMENTS
Although the principle employed in the method could clearly be extended to the more general nonlinear case, the particularly suitable form of the phase two iterations for the MLFP case is that they revert to a sequence of linear programs. This sequence can be made to run very economically by the judicious use of the basis “save and restore” options and the “row masking” facilities available with most large LP systems. In the very special case when all the criteria are in fact linear the method is still not without merit. Phase I is now simply a linear program or the method proposed in [14] can be used and Phase II is a linear program with right hand side parametrics. Even in this special case the algorithm retains its most attractive feature of controlling the dispersion at trial efficient points. This is in marked contrast to methods which maximize linear weighted combinations of the criteria for which the dispersion of trial points can be quite irregular and of course confined to vertices of the efficient region only.
85
An interactive algorithm for multicriteria programming REFERENCES
1. R. Benayoun, J. de Mintgolfier, J. Tergny and 0. Laritchev, Linear programming with multiple objective functions: Step Method (STEM). Mathemoricol Programming 1, 366-375(1971). 2. J. S. Dyer, Interactive goal programming. Mgmr. Sci. 19(l), 62-70 (1972).
3. A. M. Geoffrion, J. S. Dyer and A. Feinberg, An interactive approach for multi-criterion optimization, with an application to the operation of an academic department.Mgmr Sci. 19(4),357-363(1972). 4. R. E. Steuer, An interactive multiple objective linear programming procedure. TIMS Studies in the Management Sci. 6, 225-239(1977). 5. R. E. Steuer and A. T. Schuler, An interactive multiple-objective linear programming approach to problem in forest management. Op. Res. 26(2), 254-269(1978). 6. S. Zoints and T. Wallenius, An interactive programming method for solving the multiple criteria problem. Mgmt Sci. 22 (6) 652-663 (1976). I. J. G. Ecker, N. S. Hegner and I. A. Kouada, Generating all maximal efficient faces for multiple objective linear Programs. Paper presented at TIMSIORSA Conf., New York (1978). 8. J. P. Evans and R. E. Steuer, A revised simplex method for linear multiple objective programs Mathematical Programming 5, 54-72 (1973).
9. T. Gal, A general method for determining the set of all efficient solutions to a linear vector maximum problem. European i Op. Rex 1, 307-322 (1977).
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10. H. Isermann. The enumeration of the set of all efficient solutions for a linear multiole oroaram. 00. Res. . obiective Quart. 28,71 l-725 (1977). 11. P. L. Yu, A class of solutions for group decisions problems. Mgmt Sci. 19(g),936-946 (1973). 12. P. L. Yu and M. Zelenv. The set of all nondominated solutions in linear cases and a multicriteria simolex method. J. .I
Mathematical Analysis-and Application 49,430+8
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(1975).
13. M. Zeleny, Compromise Programming. in Multiple Criteria Decision Making, pp. 262-301,J. L. Cochrane and M. Zeleny (Eds.). University of South Carolina Press, Columbia, SC. (1973). 14. A. Charnes and W. W. Cooper, Programming with linear fractional functionals. Naval Res. Logistics Quart. 9,181-l% (1%2).
15. M. E. Posner and C. T. Wu, Contributions to linear max-min programming. Paper presented at TIMSIORSA Conf., University of Wisconsin-Milwaukee, Los Angeles (1978). 16. V. J. Bowman, On the relationship of the Tchebycheff norm and the efficient frontier of multiple-criteria objectives. in Mtdtipte Criteria Decision Making, pp. 76-85, Jouy-en Josas (1975). 11 M. Zeleny, The theory of the displaced ideal, Multiple Criteria Decision Making: Kyoto 1975,pp.15l-205, Springer-Verlag, New York (1975). I,.
APPENDIX
l-CHARACTERIZATION
OF EFFICIENCY
BY TCHEBYCHEFF
NORM
(1) The multicriteria programming problem can be stated as follows:
(MP)“max” F(x) = f(x) = (f,(x), ss.
. ,f&))
XES
where S is a compact subset of R” and f ,, . ,fL are continuous real functions. It is obvious that there exist z’, . . , z’-in S such that fi(zi) = maxf&)
,i = 1,.
uf=fi(z’)+e
,i=l,...,L
xc.5
, L.
(1)
Let (2)
where e is a small positive number. The exact magnitude of t is not important as long as c is small relative to fi(z’), NOTATION For any two vectors y’ and y2, the notation 5 , s and < have the following meaning: y’ 5 y2iffyi’
for every i
y’ G y’iffyt c y? for every i and y’ # y*
y ’ < yziffy! < y? for every i. Definition. Let x’, x*E S. We say that xi is dominated (respectively strictly dominated) by x2 iff F(x’) 6 F(x’) (respectively F(x’) < F(x*)). Defnition. A feasible point x in S is said to be efficient (resp. weakly efficient) iff x is not dominated (resp. strictly dominated) by any other feasible points. For any vector /3 > 0, the generalized Tchebycheff norm of a vector y is defined by
IIYIIB = maxSilnl. I
E. U. CHCZG and D. R. ATKINS
86
Let (P/3) denote the following program min lIu* - F(x)ll~
(PP)
x E s.
St.
The proof of the following theorem can be found in Bowman[lS]. Theorem 1. Let x* be a feasible point in S. If x* is efficient, then x* solves (&I) for some /3 > 0. In general, the inverse of Theorem I is not true. For weakly efficient points, we have the following characterization. Theorem 2. Let x* be a feasible point in S. Then x* is weakly efficient iff x* solves (Pfl) for some /3 > 0. Proof:
(e)
Suppose x* solves (Pp) for some /3 > 0. If x* is strictly dominated by x’ in S, then F(x*) < F(x’) and thus Pi(uf-fi(x*))>Bi(ur-fi(x’))
,i=l,...,L.
It follows that II@* - F(x*)lle ’ IIn*-
w$?.
This contradicts the hypothesis that x* solves (P/I). Hence x* is weakly efficient. (3). Suppose x* is weakly efficient. Set i = 1,.
pi = (UT-ii(x’,
, 12.
Then p > 0 and )lu*- F(x*)([~= I. Assume that x* does not solve (P/3). Let x” be an optimal solution of (Pfl). Then 1/u*$ll;X&(U?
-fi(X”))
JUT-fJx”)
<
FwJll, < u*- F(x*)lla
I ,i=l,...,L
< F(x’?.
Thus x* is not weakly efficient. This is a contradiction. Hence x* solves (Pfi). It follows from Theorem 2 that the multiparametric program (P@) generates all weakly efficient solutions of (MP). The problem (Pp) seeks the feasible solution which is “nearest”, in the sense of generalized Tchebycheff norm (1.[Ia,to the ideal point U* hot attainable) where pi gives the relative importance of the deviation of fi value from uf. It is difficujt and not practical to compute the weight vector fi that gives the optimal solution. The ideal point u* chosen is different from that in [I I, 171in order to have fi > 0. APPENDIX 2-DETAILS OFTHE ALGORITHM-PHASE If the maximum of each criterion separately occurs at xi, then define
I
1 = 147- minfi(xj) Pi
i
and let B be the normalization factor where -;2 = zj$ Define the problem (QBw) by (Q@v) min I S.t.&=Pi(Uf
I
I
-f!(X))-
WB
i=l,...,L
where
ci=x +ri fdx)=diTx + ti
and diTx+ti>O
for i=l,...,L.
Then (Qpw) is a linear program with a single parameter IV,and if z(p) is the optimal value of (P/3) (see Appendix l), then z(/3)= inf{w 20:h(w) = 0) where h(w) is the optimal value of (QBw). It is clear that h(w) is a nonnegative nonincreasing function of w over [O,m]and that there exists a w’ satisfying h(w’) = 0. We now have an univariate search problem over [0, w’]for which the following procedure will be sufficient to solve to whatever accuracy is required.
An interactive algorithm for multicriteria programming Step 0 Step 1
87
Put the interval [L, H] = [0, w’] H+L w= 2 evaluate h(w)
Step 2 If h(w) 2 0 put L = w. If h(w) = 0 put H = w and stop when [L, H] is small enough and h(w) = 0. Otherwise return to Step 1.
Let the point obtained be I? and
Also initially set B = 0 and D = $ - 3.
Phase II We have the “best” point y” either from Phase I or a previous iteration of Phase II. Set l/pi = uf - y: and let B be the normalization factor defined as in Phase I. We update the scalar 19,that acts to contract the search direction and concentrate our attention on smaller parts of the efficient surface, by 0 = 0 f 1.Ifwe have chosen m inspection points then define, gb=k.$fork=O,l
,...,
tn.
For each criterion i we now solve max 0x) s.t.
fi(x)3uf---(gkt3)
forallj
;
Ax 5 b. This is solved for each k successively until the maximand stops increasing. For each i and k the optimal solution vector xi’ gives the criteria vector
If there are no feasible solutions other than y’, we repeat Phase II, otherwise the decision maker is asked to choose the most preferred yp from the y’. If this is satisfactory we stop, otherwise we repeat Phase II with y” = yp. and 3 = [lu* - yql. For the case of multiple linear fractional problem the sequence of maximizations takes on the particularly simple form of a sequence of linear programs. We can see this by using the Chames and Cooper [I41 transformation x’ = nx, for real n > 0. y” is obtained by solving the following linear program. max
CiTxx’ t
riTj
s.t. Ax’ - bq 5 0 cjrx’
trpj 2 uf(~jTx'ttj7))-~djTx'tljq)(gkt C)for allj Bi di’x’ t
till = 1 n 2 0.
Because of the non-convex nature of the problem in general and also because of the unknown nature of the decision makers implicit utility function, this procedure cannot guarantee convergence to anything other than a local optimal. It is the purpose of the wide coverage of the efficient region initially with the algorithm to try and minimize any serious consequences of this.