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Applications of Interactive Multi-Objective Programming Methods: a Survey
Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993
APPLICATIONS OF INTERACTIVE MULTI-OBJECTIVE PROGRAMMING METHODS: A SURVEY H. Nakayama Department of Applied Mathematics, KOfliJn University, 8·9·1 Okarrwto, JligashiNJda, Kobe 658, Japan
Abstrad: Many practical problems often have several objectives conflicting with each other, and we need to make a balanced decision from the total viewpoint. For th ese problems, the traditional mathematical programming is not valid, and instead the multi-object ive programming have been developed. Among them, the aspiration level approach has been widely recognized to be effective in many practical fields. As one of the aspiration level approach, the author developed the satisficing trade-off method . And, he has been applying the method to several kinds of practical problems for these ten years. In this paper, a survey on applications of the method to practical problems will be given showing noteworthy points in real application.
Key Words: multi-objective programming, applications of mult.i -objective programming, satisficing trade-off method, aspiration level approach
1
Introduction
Inte ractive multi-object ive programming search a solution in an int.eractive way with OM while eliciting information on his/he r value judgment. Along this line, several methods have been developed remarkably for the last about fifteen years: Among th e m, the aspirat.ion level approach is now recognized very effective in practi ce, b ecause (i) it does not require any consistency of OM 's judgment , (ii) aspiration levels reflect the wish of OM ve ry well , and (iii) aspiration levels play the role of probe bette r than the we ight for objective functions. This will be reviewed briefly in the followings.
Multi-objective programming problems are formulated as follows: (MOP) Minimize
f(x) == (11 (x), h(x), ... , fr(x))
over x E X. The constraint set X may be given by
j
= 1, ... ,m.,
and/or a subset of Rn itself. For the problem (MOP), we define Pare to solutions as follows :
x
Definition 1.1 A solution is said Pareto optimal, if there is no better solution x E X other than namely, if
f(x)
i
f(x)
for any
x,
x -::f.
2
x EX.
Why is the Weighting Method Ineffective?
In multi-objective programming problems , th e final decision is made on the basis of the value judgm e nt of OM. He nce it is important how we elicit the value judgme nt of OM. In many practical cases, th e vector objective fun ct ion is scalarized in such a manne r that the valu e judgme nt of OM call be incorporated.
In general, there may be many Pare to solutions. The final decision is made among them taking the total balance over all criteria into account. This is a problem of value judgment of decision maker (in abbreviation, OM) . The totally balancing over criteria is usually called trade-off. It should be noted that there are very many criteria, say, over one hundred in some practical problems such as erection management of cable stayed bridge, and camera lens design. Therefore, it is very important to develop effective methods for helping OM to tradeoff easily even in problems with very many cri teria.
The most well known scalarization technique is the linear weighted sum:
LW;!i(X).
(1)
i=1
The valuejudgment of OM is reflected by the weight.
407
normalized in this way is WI = W2 = W3 = 1/3 and the renewed weight by the same normalization is w~ = 10/13, w~ = 2/13, = 1/13. We can observe that w~ is less than W2. Now increase the normalized weight W2 to be greater than 1/3. To this end, set the un normalized weight Wl = 10, W2 = 7 and W3 1. With this new weight , we have a solution (1 - 10/.JI50,1 - 7/.JI50, 1 - 1/.JI50). In spite that the normalized weight W2" = 7/18 is greater than the original one (= 1/3) , the obtained solution is still worse than the previous one.
Although this type of scalarization is widely used in many practical problems, there is a serious drawback in it . Namely, it can not provide a solution among sunken parts of Pareto surface due to "duality gap" for nonconvex cases. Even for convex cases, for example, in linear cases, even if we want to get a point in the middle of line segment between two vertices, we merely get a vertex of Pareto surface, as long as the well known simplex method is used. This implies that depending on the structure of problem, the linear weighted sum can not necessarily provide a solution as DM desires.
w;
=
As is read ily seen in the above example, it is usually very difficult to adjust the weight in order to obtain a solution as DM wants. Therefore, it seems much better to take th e aspiration level of DM rather than the weight as th e prob e. Interactive multi-obj ective programming techniques based on aspiration levels have been develop ed so that the drawbacks of the traditional goal programming may be overcome. In the following section, we shall discuss the satisficing trade-off method developed by the author (Nakayama 1984) as one of them.
In the traditional goal programming (CharnesCooper, 1961), some kind of metric function from the goal is used as the one representing the preference of DM . For example, the following IS well known :
r
r
(
~ Wi I/;( x) -
)
lip
It IP
(2)
The preference of DM is reflected by the weight Wi, the value of p, and the value of the goal It. If the value of p is chosen appropriately, a Pareto solution among a sunken part of Pare to surface can be obtained by minimizing the function (2). However, it is usually difficult to pre-determine appropriate values of them . Moreover, the solution minimizing (2) can not be better than the goal even though the goal is underestimated .
3
Satisficing 'frade-off Method
In the aspiration level approach, the aspiration is modified as follows : level at the k-th iteration
t
(3)
r,
Here, the operator P selects the Pare to solution nearest in some sense to the given aspiration level The operator T is the trade-off operator which if DM does changes the k-th aspiration level not compromise with the shown solution PCl).
t.
In addition, one of the most serious drawbacks in the goal programming is that they consider that a desirable solution can be obtained by adjusting the weight . It should be noted that there is no positive co-relation between the weight Wi and the value I(x) for the resulting solution x. Example 2.1 Let YI = II(x),Y2 = h(x) and Y3 = !a(x), and let the feasible region in the objective space be given by
Suppose that the goal is (YI' ]12, Y3) = (0,0, O) .The solution minimizing the metric function (2) with p = 1 and WI = W2 = W3 = 1 is (YI, Y2, Y3) = (11/J3, 1-1/J3, 1-1/J3) . Now suppose that DM wants to decrease the value of It more and that of h a little more, and hence modify the weight into w~ = 10, w~ = 2, w; = 1. The solution corresponding to the new weight is (1 - 10/ViOS, 1 2/ViOS, 1 - I/ViOS) . Note that the value of h is worse than before in spite that DM wants to improve it and hence increase the weight of h up to twice. Someone might think that this is due to the normalization of weight. Therefore, we normalize the weight by WI +W2+W3 1. The original weight
t
Of course, since P(t) is a Pare to solution, there exists no feasible solution which makes all crite-k ria better than P(J ), and thus DM has to tradeoff among criteria if he wants to improve some of criteria. Based on this trade-off, a new aspiraSimilar protion level is decided as To cess is continued until DM obtains an agreeable solution. This id ea is implemented in DIDASS (Grauer et al. 1984) and the satisficing trade-off method (Nakayama 1984). While DIDASS leaves the trade-off to t.he heuristi cs of DM , the satisficing trad e-off met hod provides a device based on the sensitivity analysis.
pct).
3.1
On The Opearatioll P
The operation which gives a Pare to solution pCl) nearest to yk is performed by some auxiliary scalar optimization. It has been shown in SawaragiNakayama-Tanino (1985) that the only one scalarization technique, which provides any Pare to solution regardless of the structure of problem , is of the
=
408
Tchebyshev norm typ e. However, the scalarization function of Tchebyshev norm type yields not only a Pare to solution but also a weak Pare to solution. Since weak Pareto solutions have a possibility that there may be another solution which improves a critera while others being fixed, they are not. necessarily "efficient" as a solution in decision making. In order to exclude weak Pareto solutions, the following one of the augmented Tchebyshev type can be used: max W;fi(X)
l
+ Q' ~ W;fi(X). ~
they are. Let the index set of each class be denoted by
J~, J~ , respectively. Clearly, ]~+1 < J;(xk) for all i E I}. Usually, for i E I~, we set ]~+1 I;(x k ) . For i E I~ , DM has to agree to increase the value -k+1 of l i . It should be noted that an appropriate sacrifice of Ij for j E J~ is needed for getting the improvement of l i for i E Ij.
=
A utomatic Trade-off
(1)
It is of course possible for DM to answer new aspiration levels of all objective functions. In practical probelms, however , we often encounter cases with very many objective functions, for which DM tends to get tired with answering new aspiration levels for all objective functions. Therefore, it is more practical in problems with very many objective functions for DM to answer only his/her improvement rather than both improvement and relaxation. At this stage, we can use the assignment of sacrifice for /j (j E JR) which is automatically set in the equal proportion to AiWi, namely, by
i=l
where Q' is usually set a sufficiently small positive number , say 10- 6 . Here , the weight Wi is given as follows: Let It be an ideal value which is usually given in such a way that It < Min{J;(x)1 x E X}, and let I.i be a nadir value which is usually given by
I.i
=
max li(Xj)
l~j~r
(5)
where XJ~
= arg xEX min/J(x).
Jj,
(6)
( 10)
For this circumstance, we set
= =---
(7)
w:=~-~
(8)
Wi
li - It
wh ere N is the number of eleme nts of the set JR, and A is the Lagrange multiplier associated with the constraints in Problem [Q]. The reason why (l0) is available is that (A1W1, ... ,Arwr) is the normal vector of the tangent hyperplane of the Pareto surface under appropriate conditions (Nakayama 1991-b).
or
I.i - It
The minimization of (4) with (7) or (8) is Ilsually performed by solving the following equivalent optimization problem, because the original one is not smooth:
(Q)
Minimize
z
I3y doing this, in cases that there are a large number of criteria, the burden of DM can be decreased so much. Of course, if DM does not agree with this quota D.lj laid down automatically, he/she can modify it in a manual way.
+ Q'L~=l W;fi(X)
subject to
Exact Trade-off
Wi(J;(X) - ]i) < z
(9)
In linear or quadratic cases, we can evaluate the exact t.rade-off in an extend ed form of the automatic trade-off st.ated above. This implies that we can calculate exactly the amount of relaxation such that the new aspiration level is on the Pare to surface (N akayama, 1992) . The main idea in it is that the parametric optimization technique is used in stead of th e simple sensitivity analysis. Using this technique, a new Pareto solution can be obtained without solving the auxiliary sclarized optimization problem again. This implies that we can obtain the new solution very quickly. Therefore, using some graphic presentation as computer outputs, DM can see the trade-off among critera in a dynamic way, e.g . as an animation. This makes D t\I's judgement easier .
x E X.
3.2
On The Operation T
In cases DM is not satisfied wit.h th e solution for p(fk), he/she is requested to answer his/h er new aspiration level ]k+1. Let xk denote the Pare to solution obtained by projection P(f\ and classify the objective functions into the following three groups: (i) the class of criteria which are to be improved more , (ii) the class of criteria which may be relaxed, (iii) the class of criteria which are acceptable as
409
( a) car navigation system
3.3
Interchange of Objectives and Constraints
4. planning
In the formulation of the auxiliary scalarized optimization prolem [Q], change the right hand side of the equation (9) into f3iz, namely
(a) scheduling of string selection in steel manufact uring (U eno et al. 1(90)
(11 )
(b) long term planning of atomic power plants
=
As is read ily seen, if f3i 1, then the function J; is considered to be an objective function, for which the aspiration leve l is not necessarily attained, but the level of fi should be better as much as possible. On th e other hand, if f3i = 0, then J; is considered to be a const raint function , for which the aspiration level should be guaranteed. In many practical problems, there is almost no cases in whi ch we consider th e role of objective and constraint fixed from the beginning, but usually we want to interchange them depending on th e situation. Using the formula (11), this can be done very easily. In add ition , if the value of f3i is set in the middle of 0 and 1, J; can play a role in the middle of objective and constraint which is neith er a complete objective nor a complete constraint (Kamenoi et at. 19(2). This is also very effective in many practical problems.
7i
In the following , some of examples are introduced briefly.
4.1
7i
4
Feed Formation
Stock farms in Japan are modernized recently. Above all, the feed ing system in some farms is fully controled by computer: Each cow has its own place to eat which has a locked gate. And each cow has a key 011 her neck, which can open the corresponding gate only. Every day, on the basis of ingredient analysis of milk and/or of the growth situation of cow, the appro priate blending ratio of materials from several view points should be made. There are about 20-30 kinds of raw materials for feed in cow farms such as corn, cereals, fish meal, etc.
Applications
About 10 criteria are usually taken into account for feed form at ion of cow:
Interact ive multi-objective programming methods have been applied to a wide range of practical problems. Good examples in engineering applications can be seen in Eshenauer et al. (1990). The author himself has applied to several real problems:
• cost • neutrition - protein - TDN - cel lulose
1. blending
- calc ium
(a) feed (Nakayama et at. 19(2)
- magn esIUm - cic.
(b) plastic materials (Nakayama et at. 1(86)
• stock amount of mater ials • etc .
(c) cement production (Nakayama 1991-a)
This feeding problem is well known as the diet problem from the beginnillg of the history of mathemati cal progra mming, which can be formulated as the traditional linear programming. It is very easy to consider it as multi-obj ective linear programming. The author develop ed a software for feed formation in an inteactive way between designer and computer using the satisficing trade-off method. So far , this software has been used in a stock farm in some university and feed companies, and brought good res ults (Nakayama et al. 1(92) .
(d) portfolio (Nakayama 1(89)
2. design (a) camera lens (b) erect ion management of cab le-stayed bridge (Furukawa et al. 1(86)
3. route search
410
4.2
Erection Management of Cahle Stayed Bridge
shim-adjustment. This operation is very easy for th e designer, and the visual information of t.rad eoff among criteria is user-friendly. The software was used for real bridge construction , say, Tokiwa Great Bridge (Ube City) and Karasuo Harp Bridge (Kita-Kyusyu City) in 1992.
In erection of cable stayed bridge, the following criteria are considered for accuracy control (Furukawa et al. 1986): 1. 11.
residual error in each cable tension, residual error in camber at each node,
Ill.
amount of shim adjustment for each cable,
IV.
number of cables to be adjust.ed.
Since the change of cable rigidity is small enough to be neglected with respect to shim adjustment., both the residual error in each cable tension and that in each camber are linear functions of amount of shim adjustment. Let us define n as the number of cable in use, flTi (i = 1, ... , n) as t.he difference between the designed tension values and the measured ones, and Xik as the tension change of i-th cable caused from the change of the k-th cable length by a unit. The residual error in cable tension caused by the shim adjustment is given by
shim
(mm) tensio n r - - - - , , - - -- - - - - ; r r - U - (ton) camber f - - - - . . " - ------,,.---llit-
(mm)
Fig.l A phase of the proposed erection management system
n
Pi
= IflTi -
I:>ikfl1kl
(i=l, ... ,n) 4.3
k=l
Let m be the number of nodes, fl::j (j = 1, ... , m) the difference between the designed camber values and the measured ones, and Yjk the camber change of j-th node caused from t.he change of t.he k-th cable length by a unit. Then the residual error in the camber caused by the shim adjustments of flll' ... ,flln is given by
Recently, the t.raffic control is actively promot.ed from both sides of roads and cars. From t.he side of roads, e.g., the traffic flow is controlcd by adjusting the signal time, or signs which tell drivers information calling their attention, and so on. On the other hand, from the side of cars, many car manufacturing companies are trying t.o equip some high grade cars with navigat.ion systems. which tell drivers their locat.ions where t.hey are by use of satellites. In addition to this map informat.ion , they would like to give an information about the shortest route by their navigation systems. In the following, we shall report a trial along this line with some experimental results.
n
qj
= IflZj -
L Yjkfl1k I
(j=l, ... ,m)
k=l
In addition, the amount of shim adjustment can be treated as objective functions of 7'i
= Ifllil
Multi-objective Route Search
ill Car Navigation Systems
(i=l, ... ,n)
And the upper and lower bounds of shim adjustment inherent in t.he structure of the cable anchorage are as follows;
The well known Dijkstra method is applied for finding th e "shortest" rout.e. However. t.he original Dijkstra method searches for an opt.imal rout.e in an area spreading over as th e shape of circular disk with the origin of the start.ing point.. Since, in many practical cases, this method takes "ve n t.rivially inefficient paths int.o account, we limit our concern to an area of ellipsoid with focuses of the starting point and the des tination.
(i=l , ... ,n).
Fig.l shows one phase of erection management system of cable stayed bridge using the satisficing trade-off method. The residual error of each criterion and the amount of shim adjust.ment are represented by graphs . The aspiration level is inputted by a mouse on the graph. After solving the auxiliary min-max problem, the Parcto solution according to the aspiration level is represented by a graph in a similar fashion. This procedure is continued until the designer can obt.ain an appropriate
Now, the question is "vVhat is the s/lOr/csl?". So far, almost of all cars equipped wi th navigation systems consider only the physical dist.ance. This is not appropriat.e for many practical c
terms of a physical distance . On the other hand, expert drivers may not care about the easiness to drive, but want to arrive at the destination in the shortest time. Of course, the shortest time does not mean the shortest distance. According to the above consideration , we make a hierachy of criteria for evaluating the optimal route as in Fig.2.
[4] Grauer, M., Lewandowski , A. and Wierzbicki A.P. (1984), DIDASS Theory, Implementation and Experiences, in M. Grauer and A.P. Wierzbicki (eds.) Interactive Decision Analysis, Proceedings of an International Workshop on Interactive Decision A nalysis and Interpretative Computer Intelligence, Springer: 22-30. [5] Kamenoi, T ., Sugimoto, H. and Nakayama, H. (1992) , Studies on Revised Satisficing Trade-off Method for Structure Design, Journal of the Japan Soc. of Civil Engineers, 441/1-18, 117-126 (in Japanese)
distance limited velocity time
Optimal Route
jam time for right turn
[6] Nakayama, H. (1984), Proposal of Satisficing Trade-off Method for Multi- objective Programming, Transact. SICE, 20: 29-35 (in Japanese)
easiness for right turn
[7] Nakayama, H. (1989), An Interactive Support System for Bond Trading, in A. G. Lockett and G. Islei (eds.) Improving Decision Making in Organizations, Springer, 325-333
cost
[8] Nakayama, H. (1991-a), Satisficing Trade-off Method for Problems with Multiple Linear Fractional Objectives and its Applications, in A. Lewandowski and V. Volkovich (eds .) Multiobjective Problems of Maht ematical Programming, Springer, 42-50.
condition of path
Fig.2
Hierarchy of criteria in multiobjective route search
[9] Nakayama, H. (1991-b), Trade-off Analysis Based Upon Parametric Optimization, in P.Korhonen, A. Lewandowski and J . Wallenius (eds.), Multiple Criteria Decision Support, Springer, 42-52
Now, AHP (Analytic Hierarchy Process) can be applied for determining weight on each criteria. Of course, the weight depends on drivers . For practical use, however, it is better for drivers to input only the weight among time, cost and easiness, because it would be troublesome to input the weights for all criteria. The weights for criteria on the lowest level may be decided by AHP for sampled data through questionaires made in apriori. Through the above procedure, scores corresponding to the preference of drivers are assigned on arcs and nodes on the map . After this, the well known Dijkstra method can be applied for finding the optimal route as stated previously.
[10] Nakayama, H. (1992), Trade-off Analysis using Parametric Optimization Techniques, European J. of Operational Research, GO, 87-98 [11] Nakayama, H., Mitani, K. and Yoshida, F. (1992), A Support System for Multi-objective Feed Formation in Stock Farming, Proc. of Software Conference, Osaka, 17-20 , (in Japanese). [12] Nakayama, H., Nomura, J., Sawada, K. and Nakajima, R. (1986). An Application of Satisficing Trade-off Method to a Blending Problem of Industrial Materials, in G . Fandel et al. (Eds.), Large Scale Modelling and Interactive Decision Analysis. Springer, 303-313. [13] Nakayama, H. and Sawaragi, Y. (1984). Satisficing Trade-off Method for Interactive M ultiobjective Programming Methods, in M. Grauer and A.P. Wierzbicki (eds.) Interactive Decision Analysis, Proceedings of an International Workshop on Interactive Decision A nalysis and Interpretative Computer- Intelligence, Springer: 113-122.
References [1] Charnes, A. and Cooper, W . (1961), Management Models and Industrial Applications of Linear Programming, Vol. 1, John Wiley, New York [2] Eschenauer, H.A., Koski, J . and Osyczka, A. (1990), Multicriteria Design Optimization, Springer
[14] Sawaragi, Y., Nakayama, H. and Tanino, T. (1985). Theory of Multiobjective Optimization, Academic Press
[3] Furukawa, K., Inoue, K . Nakayama, H. and Ishido, K. (1986), Management of Erection for Cable Stayed Bridge using Satisficing Trade-off Method, Journal of the Japan Soc. of Civil Engineers, 374/1-6, 495-502 . (in Japanese)
[I5] Ueno, N., Nakagawa, Y., Tokuyama, H., Nakayama, H. and Tamura, H., (1990) A Multiobjective Planning for String Selection in Steel Manufactureing, Communications of the Operations Resmrch Society of Japan, 35 , 656-661 (in Japanese)
412
APPLICATIONS OF INTERACTIVE MULTI-OBJECTIVE PROGRAMMING METHODS: A SURVEY H. Nakayama Department of Applied Mathematics, KOfliJn University, 8·9·1 Okarrwto, JligashiNJda, Kobe 658, Japan
Abstrad: Many practical problems often have several objectives conflicting with each other, and we need to make a balanced decision from the total viewpoint. For th ese problems, the traditional mathematical programming is not valid, and instead the multi-object ive programming have been developed. Among them, the aspiration level approach has been widely recognized to be effective in many practical fields. As one of the aspiration level approach, the author developed the satisficing trade-off method . And, he has been applying the method to several kinds of practical problems for these ten years. In this paper, a survey on applications of the method to practical problems will be given showing noteworthy points in real application.
Key Words: multi-objective programming, applications of mult.i -objective programming, satisficing trade-off method, aspiration level approach
1
Introduction
Inte ractive multi-object ive programming search a solution in an int.eractive way with OM while eliciting information on his/he r value judgment. Along this line, several methods have been developed remarkably for the last about fifteen years: Among th e m, the aspirat.ion level approach is now recognized very effective in practi ce, b ecause (i) it does not require any consistency of OM 's judgment , (ii) aspiration levels reflect the wish of OM ve ry well , and (iii) aspiration levels play the role of probe bette r than the we ight for objective functions. This will be reviewed briefly in the followings.
Multi-objective programming problems are formulated as follows: (MOP) Minimize
f(x) == (11 (x), h(x), ... , fr(x))
over x E X. The constraint set X may be given by
j
= 1, ... ,m.,
and/or a subset of Rn itself. For the problem (MOP), we define Pare to solutions as follows :
x
Definition 1.1 A solution is said Pareto optimal, if there is no better solution x E X other than namely, if
f(x)
i
f(x)
for any
x,
x -::f.
2
x EX.
Why is the Weighting Method Ineffective?
In multi-objective programming problems , th e final decision is made on the basis of the value judgm e nt of OM. He nce it is important how we elicit the value judgme nt of OM. In many practical cases, th e vector objective fun ct ion is scalarized in such a manne r that the valu e judgme nt of OM call be incorporated.
In general, there may be many Pare to solutions. The final decision is made among them taking the total balance over all criteria into account. This is a problem of value judgment of decision maker (in abbreviation, OM) . The totally balancing over criteria is usually called trade-off. It should be noted that there are very many criteria, say, over one hundred in some practical problems such as erection management of cable stayed bridge, and camera lens design. Therefore, it is very important to develop effective methods for helping OM to tradeoff easily even in problems with very many cri teria.
The most well known scalarization technique is the linear weighted sum:
LW;!i(X).
(1)
i=1
The valuejudgment of OM is reflected by the weight.
407
normalized in this way is WI = W2 = W3 = 1/3 and the renewed weight by the same normalization is w~ = 10/13, w~ = 2/13, = 1/13. We can observe that w~ is less than W2. Now increase the normalized weight W2 to be greater than 1/3. To this end, set the un normalized weight Wl = 10, W2 = 7 and W3 1. With this new weight , we have a solution (1 - 10/.JI50,1 - 7/.JI50, 1 - 1/.JI50). In spite that the normalized weight W2" = 7/18 is greater than the original one (= 1/3) , the obtained solution is still worse than the previous one.
Although this type of scalarization is widely used in many practical problems, there is a serious drawback in it . Namely, it can not provide a solution among sunken parts of Pareto surface due to "duality gap" for nonconvex cases. Even for convex cases, for example, in linear cases, even if we want to get a point in the middle of line segment between two vertices, we merely get a vertex of Pareto surface, as long as the well known simplex method is used. This implies that depending on the structure of problem, the linear weighted sum can not necessarily provide a solution as DM desires.
w;
=
As is read ily seen in the above example, it is usually very difficult to adjust the weight in order to obtain a solution as DM wants. Therefore, it seems much better to take th e aspiration level of DM rather than the weight as th e prob e. Interactive multi-obj ective programming techniques based on aspiration levels have been develop ed so that the drawbacks of the traditional goal programming may be overcome. In the following section, we shall discuss the satisficing trade-off method developed by the author (Nakayama 1984) as one of them.
In the traditional goal programming (CharnesCooper, 1961), some kind of metric function from the goal is used as the one representing the preference of DM . For example, the following IS well known :
r
r
(
~ Wi I/;( x) -
)
lip
It IP
(2)
The preference of DM is reflected by the weight Wi, the value of p, and the value of the goal It. If the value of p is chosen appropriately, a Pareto solution among a sunken part of Pare to surface can be obtained by minimizing the function (2). However, it is usually difficult to pre-determine appropriate values of them . Moreover, the solution minimizing (2) can not be better than the goal even though the goal is underestimated .
3
Satisficing 'frade-off Method
In the aspiration level approach, the aspiration is modified as follows : level at the k-th iteration
t
(3)
r,
Here, the operator P selects the Pare to solution nearest in some sense to the given aspiration level The operator T is the trade-off operator which if DM does changes the k-th aspiration level not compromise with the shown solution PCl).
t.
In addition, one of the most serious drawbacks in the goal programming is that they consider that a desirable solution can be obtained by adjusting the weight . It should be noted that there is no positive co-relation between the weight Wi and the value I(x) for the resulting solution x. Example 2.1 Let YI = II(x),Y2 = h(x) and Y3 = !a(x), and let the feasible region in the objective space be given by
Suppose that the goal is (YI' ]12, Y3) = (0,0, O) .The solution minimizing the metric function (2) with p = 1 and WI = W2 = W3 = 1 is (YI, Y2, Y3) = (11/J3, 1-1/J3, 1-1/J3) . Now suppose that DM wants to decrease the value of It more and that of h a little more, and hence modify the weight into w~ = 10, w~ = 2, w; = 1. The solution corresponding to the new weight is (1 - 10/ViOS, 1 2/ViOS, 1 - I/ViOS) . Note that the value of h is worse than before in spite that DM wants to improve it and hence increase the weight of h up to twice. Someone might think that this is due to the normalization of weight. Therefore, we normalize the weight by WI +W2+W3 1. The original weight
t
Of course, since P(t) is a Pare to solution, there exists no feasible solution which makes all crite-k ria better than P(J ), and thus DM has to tradeoff among criteria if he wants to improve some of criteria. Based on this trade-off, a new aspiraSimilar protion level is decided as To cess is continued until DM obtains an agreeable solution. This id ea is implemented in DIDASS (Grauer et al. 1984) and the satisficing trade-off method (Nakayama 1984). While DIDASS leaves the trade-off to t.he heuristi cs of DM , the satisficing trad e-off met hod provides a device based on the sensitivity analysis.
pct).
3.1
On The Opearatioll P
The operation which gives a Pare to solution pCl) nearest to yk is performed by some auxiliary scalar optimization. It has been shown in SawaragiNakayama-Tanino (1985) that the only one scalarization technique, which provides any Pare to solution regardless of the structure of problem , is of the
=
408
Tchebyshev norm typ e. However, the scalarization function of Tchebyshev norm type yields not only a Pare to solution but also a weak Pare to solution. Since weak Pareto solutions have a possibility that there may be another solution which improves a critera while others being fixed, they are not. necessarily "efficient" as a solution in decision making. In order to exclude weak Pareto solutions, the following one of the augmented Tchebyshev type can be used: max W;fi(X)
l
+ Q' ~ W;fi(X). ~
they are. Let the index set of each class be denoted by
J~, J~ , respectively. Clearly, ]~+1 < J;(xk) for all i E I}. Usually, for i E I~, we set ]~+1 I;(x k ) . For i E I~ , DM has to agree to increase the value -k+1 of l i . It should be noted that an appropriate sacrifice of Ij for j E J~ is needed for getting the improvement of l i for i E Ij.
=
A utomatic Trade-off
(1)
It is of course possible for DM to answer new aspiration levels of all objective functions. In practical probelms, however , we often encounter cases with very many objective functions, for which DM tends to get tired with answering new aspiration levels for all objective functions. Therefore, it is more practical in problems with very many objective functions for DM to answer only his/her improvement rather than both improvement and relaxation. At this stage, we can use the assignment of sacrifice for /j (j E JR) which is automatically set in the equal proportion to AiWi, namely, by
i=l
where Q' is usually set a sufficiently small positive number , say 10- 6 . Here , the weight Wi is given as follows: Let It be an ideal value which is usually given in such a way that It < Min{J;(x)1 x E X}, and let I.i be a nadir value which is usually given by
I.i
=
max li(Xj)
l~j~r
(5)
where XJ~
= arg xEX min/J(x).
Jj,
(6)
( 10)
For this circumstance, we set
= =---
(7)
w:=~-~
(8)
Wi
li - It
wh ere N is the number of eleme nts of the set JR, and A is the Lagrange multiplier associated with the constraints in Problem [Q]. The reason why (l0) is available is that (A1W1, ... ,Arwr) is the normal vector of the tangent hyperplane of the Pareto surface under appropriate conditions (Nakayama 1991-b).
or
I.i - It
The minimization of (4) with (7) or (8) is Ilsually performed by solving the following equivalent optimization problem, because the original one is not smooth:
(Q)
Minimize
z
I3y doing this, in cases that there are a large number of criteria, the burden of DM can be decreased so much. Of course, if DM does not agree with this quota D.lj laid down automatically, he/she can modify it in a manual way.
+ Q'L~=l W;fi(X)
subject to
Exact Trade-off
Wi(J;(X) - ]i) < z
(9)
In linear or quadratic cases, we can evaluate the exact t.rade-off in an extend ed form of the automatic trade-off st.ated above. This implies that we can calculate exactly the amount of relaxation such that the new aspiration level is on the Pare to surface (N akayama, 1992) . The main idea in it is that the parametric optimization technique is used in stead of th e simple sensitivity analysis. Using this technique, a new Pareto solution can be obtained without solving the auxiliary sclarized optimization problem again. This implies that we can obtain the new solution very quickly. Therefore, using some graphic presentation as computer outputs, DM can see the trade-off among critera in a dynamic way, e.g . as an animation. This makes D t\I's judgement easier .
x E X.
3.2
On The Operation T
In cases DM is not satisfied wit.h th e solution for p(fk), he/she is requested to answer his/h er new aspiration level ]k+1. Let xk denote the Pare to solution obtained by projection P(f\ and classify the objective functions into the following three groups: (i) the class of criteria which are to be improved more , (ii) the class of criteria which may be relaxed, (iii) the class of criteria which are acceptable as
409
( a) car navigation system
3.3
Interchange of Objectives and Constraints
4. planning
In the formulation of the auxiliary scalarized optimization prolem [Q], change the right hand side of the equation (9) into f3iz, namely
(a) scheduling of string selection in steel manufact uring (U eno et al. 1(90)
(11 )
(b) long term planning of atomic power plants
=
As is read ily seen, if f3i 1, then the function J; is considered to be an objective function, for which the aspiration leve l is not necessarily attained, but the level of fi should be better as much as possible. On th e other hand, if f3i = 0, then J; is considered to be a const raint function , for which the aspiration level should be guaranteed. In many practical problems, there is almost no cases in whi ch we consider th e role of objective and constraint fixed from the beginning, but usually we want to interchange them depending on th e situation. Using the formula (11), this can be done very easily. In add ition , if the value of f3i is set in the middle of 0 and 1, J; can play a role in the middle of objective and constraint which is neith er a complete objective nor a complete constraint (Kamenoi et at. 19(2). This is also very effective in many practical problems.
7i
In the following , some of examples are introduced briefly.
4.1
7i
4
Feed Formation
Stock farms in Japan are modernized recently. Above all, the feed ing system in some farms is fully controled by computer: Each cow has its own place to eat which has a locked gate. And each cow has a key 011 her neck, which can open the corresponding gate only. Every day, on the basis of ingredient analysis of milk and/or of the growth situation of cow, the appro priate blending ratio of materials from several view points should be made. There are about 20-30 kinds of raw materials for feed in cow farms such as corn, cereals, fish meal, etc.
Applications
About 10 criteria are usually taken into account for feed form at ion of cow:
Interact ive multi-objective programming methods have been applied to a wide range of practical problems. Good examples in engineering applications can be seen in Eshenauer et al. (1990). The author himself has applied to several real problems:
• cost • neutrition - protein - TDN - cel lulose
1. blending
- calc ium
(a) feed (Nakayama et at. 19(2)
- magn esIUm - cic.
(b) plastic materials (Nakayama et at. 1(86)
• stock amount of mater ials • etc .
(c) cement production (Nakayama 1991-a)
This feeding problem is well known as the diet problem from the beginnillg of the history of mathemati cal progra mming, which can be formulated as the traditional linear programming. It is very easy to consider it as multi-obj ective linear programming. The author develop ed a software for feed formation in an inteactive way between designer and computer using the satisficing trade-off method. So far , this software has been used in a stock farm in some university and feed companies, and brought good res ults (Nakayama et al. 1(92) .
(d) portfolio (Nakayama 1(89)
2. design (a) camera lens (b) erect ion management of cab le-stayed bridge (Furukawa et al. 1(86)
3. route search
410
4.2
Erection Management of Cahle Stayed Bridge
shim-adjustment. This operation is very easy for th e designer, and the visual information of t.rad eoff among criteria is user-friendly. The software was used for real bridge construction , say, Tokiwa Great Bridge (Ube City) and Karasuo Harp Bridge (Kita-Kyusyu City) in 1992.
In erection of cable stayed bridge, the following criteria are considered for accuracy control (Furukawa et al. 1986): 1. 11.
residual error in each cable tension, residual error in camber at each node,
Ill.
amount of shim adjustment for each cable,
IV.
number of cables to be adjust.ed.
Since the change of cable rigidity is small enough to be neglected with respect to shim adjustment., both the residual error in each cable tension and that in each camber are linear functions of amount of shim adjustment. Let us define n as the number of cable in use, flTi (i = 1, ... , n) as t.he difference between the designed tension values and the measured ones, and Xik as the tension change of i-th cable caused from the change of the k-th cable length by a unit. The residual error in cable tension caused by the shim adjustment is given by
shim
(mm) tensio n r - - - - , , - - -- - - - - ; r r - U - (ton) camber f - - - - . . " - ------,,.---llit-
(mm)
Fig.l A phase of the proposed erection management system
n
Pi
= IflTi -
I:>ikfl1kl
(i=l, ... ,n) 4.3
k=l
Let m be the number of nodes, fl::j (j = 1, ... , m) the difference between the designed camber values and the measured ones, and Yjk the camber change of j-th node caused from t.he change of t.he k-th cable length by a unit. Then the residual error in the camber caused by the shim adjustments of flll' ... ,flln is given by
Recently, the t.raffic control is actively promot.ed from both sides of roads and cars. From t.he side of roads, e.g., the traffic flow is controlcd by adjusting the signal time, or signs which tell drivers information calling their attention, and so on. On the other hand, from the side of cars, many car manufacturing companies are trying t.o equip some high grade cars with navigat.ion systems. which tell drivers their locat.ions where t.hey are by use of satellites. In addition to this map informat.ion , they would like to give an information about the shortest route by their navigation systems. In the following, we shall report a trial along this line with some experimental results.
n
qj
= IflZj -
L Yjkfl1k I
(j=l, ... ,m)
k=l
In addition, the amount of shim adjustment can be treated as objective functions of 7'i
= Ifllil
Multi-objective Route Search
ill Car Navigation Systems
(i=l, ... ,n)
And the upper and lower bounds of shim adjustment inherent in t.he structure of the cable anchorage are as follows;
The well known Dijkstra method is applied for finding th e "shortest" rout.e. However. t.he original Dijkstra method searches for an opt.imal rout.e in an area spreading over as th e shape of circular disk with the origin of the start.ing point.. Since, in many practical cases, this method takes "ve n t.rivially inefficient paths int.o account, we limit our concern to an area of ellipsoid with focuses of the starting point and the des tination.
(i=l , ... ,n).
Fig.l shows one phase of erection management system of cable stayed bridge using the satisficing trade-off method. The residual error of each criterion and the amount of shim adjust.ment are represented by graphs . The aspiration level is inputted by a mouse on the graph. After solving the auxiliary min-max problem, the Parcto solution according to the aspiration level is represented by a graph in a similar fashion. This procedure is continued until the designer can obt.ain an appropriate
Now, the question is "vVhat is the s/lOr/csl?". So far, almost of all cars equipped wi th navigation systems consider only the physical dist.ance. This is not appropriat.e for many practical c
terms of a physical distance . On the other hand, expert drivers may not care about the easiness to drive, but want to arrive at the destination in the shortest time. Of course, the shortest time does not mean the shortest distance. According to the above consideration , we make a hierachy of criteria for evaluating the optimal route as in Fig.2.
[4] Grauer, M., Lewandowski , A. and Wierzbicki A.P. (1984), DIDASS Theory, Implementation and Experiences, in M. Grauer and A.P. Wierzbicki (eds.) Interactive Decision Analysis, Proceedings of an International Workshop on Interactive Decision A nalysis and Interpretative Computer Intelligence, Springer: 22-30. [5] Kamenoi, T ., Sugimoto, H. and Nakayama, H. (1992) , Studies on Revised Satisficing Trade-off Method for Structure Design, Journal of the Japan Soc. of Civil Engineers, 441/1-18, 117-126 (in Japanese)
distance limited velocity time
Optimal Route
jam time for right turn
[6] Nakayama, H. (1984), Proposal of Satisficing Trade-off Method for Multi- objective Programming, Transact. SICE, 20: 29-35 (in Japanese)
easiness for right turn
[7] Nakayama, H. (1989), An Interactive Support System for Bond Trading, in A. G. Lockett and G. Islei (eds.) Improving Decision Making in Organizations, Springer, 325-333
cost
[8] Nakayama, H. (1991-a), Satisficing Trade-off Method for Problems with Multiple Linear Fractional Objectives and its Applications, in A. Lewandowski and V. Volkovich (eds .) Multiobjective Problems of Maht ematical Programming, Springer, 42-50.
condition of path
Fig.2
Hierarchy of criteria in multiobjective route search
[9] Nakayama, H. (1991-b), Trade-off Analysis Based Upon Parametric Optimization, in P.Korhonen, A. Lewandowski and J . Wallenius (eds.), Multiple Criteria Decision Support, Springer, 42-52
Now, AHP (Analytic Hierarchy Process) can be applied for determining weight on each criteria. Of course, the weight depends on drivers . For practical use, however, it is better for drivers to input only the weight among time, cost and easiness, because it would be troublesome to input the weights for all criteria. The weights for criteria on the lowest level may be decided by AHP for sampled data through questionaires made in apriori. Through the above procedure, scores corresponding to the preference of drivers are assigned on arcs and nodes on the map . After this, the well known Dijkstra method can be applied for finding the optimal route as stated previously.
[10] Nakayama, H. (1992), Trade-off Analysis using Parametric Optimization Techniques, European J. of Operational Research, GO, 87-98 [11] Nakayama, H., Mitani, K. and Yoshida, F. (1992), A Support System for Multi-objective Feed Formation in Stock Farming, Proc. of Software Conference, Osaka, 17-20 , (in Japanese). [12] Nakayama, H., Nomura, J., Sawada, K. and Nakajima, R. (1986). An Application of Satisficing Trade-off Method to a Blending Problem of Industrial Materials, in G . Fandel et al. (Eds.), Large Scale Modelling and Interactive Decision Analysis. Springer, 303-313. [13] Nakayama, H. and Sawaragi, Y. (1984). Satisficing Trade-off Method for Interactive M ultiobjective Programming Methods, in M. Grauer and A.P. Wierzbicki (eds.) Interactive Decision Analysis, Proceedings of an International Workshop on Interactive Decision A nalysis and Interpretative Computer- Intelligence, Springer: 113-122.
References [1] Charnes, A. and Cooper, W . (1961), Management Models and Industrial Applications of Linear Programming, Vol. 1, John Wiley, New York [2] Eschenauer, H.A., Koski, J . and Osyczka, A. (1990), Multicriteria Design Optimization, Springer
[14] Sawaragi, Y., Nakayama, H. and Tanino, T. (1985). Theory of Multiobjective Optimization, Academic Press
[3] Furukawa, K., Inoue, K . Nakayama, H. and Ishido, K. (1986), Management of Erection for Cable Stayed Bridge using Satisficing Trade-off Method, Journal of the Japan Soc. of Civil Engineers, 374/1-6, 495-502 . (in Japanese)
[I5] Ueno, N., Nakagawa, Y., Tokuyama, H., Nakayama, H. and Tamura, H., (1990) A Multiobjective Planning for String Selection in Steel Manufactureing, Communications of the Operations Resmrch Society of Japan, 35 , 656-661 (in Japanese)
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