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Multiobjective Assessment of Investment Plans by Use of Utility Theory
Copyright © IF AC Control Science and Technology (8th Triennial World Congress) Kyoto. Japan. 1981
MULTI OBJECTIVE ASSESSMENT OF INVESTMENT PLANS BY USE OF UTILITY THEORY Y. Nishikawa*,
J. Nomura**, T. Inaki**, J. Hashizume** and K. Sawada**
·Department of Electrical Engineering, Kyoto University, Yoshida, Kyoto 606, Japan ·*Research and Development Laboratory, Matsushita Electric Works, Ltd., Kadoma, Osaka 571, Japan
Abstract. The paper discusses a multiobjective assessment procedure as applied to the assessment of investment plans in a production firm. There are many conventional methods for assessing the investments. Among those methods, the rat e of return method and the payout period method are used as the ones with th e criterion defined from the economical viewpoint. In addition to the economical viewpoint, we have introduced a n ew criterion defined from the technical viewpoint. The total assessment formula is derived considering buth the economical and th e technical terms by use of a multiattribute utility function. A real problem uf assessing th e investments has s ucc ess full y been d ealt with by the present procedure. Keywurds. Investment ; utility theory
1.
management systems;
I NTRO DUCTION
multiobjective assessment;
systems analysis;
turn, the payout period and the technical level of new equipment. Generally, the rate of return and the payout period can readily be defined and formu· lated quantitatively. But, the technical level of equipment is difficult to quantify. Hence , it is graded into ten classes where each class is defined by the technical level in comparison with the equipments of other production firms.
For a production firm, it is one of the important yet di fficult problems to assess a nd decide invest ment plans, because they are necessary and indispensable for a stabilization and magnificatiun of the production firm or for an increase of profit. Once th e investment plans are executed, they give a significant effect upon the productiun quantities , the quality of goods, th e unit costs of materials, the cost of dynamic force, etc. After beginning the performance, a change of the plan will be impossible practically and will ensue a lot of loss es. So far, there have been developed many methods for assessment of investm en ts. Among them , the rate of return method, MAP I (Machinery and Allied Products Insti tute) method [1J and the payout period method assess the investm ents from the economical point of view. Further , there are many reports about simulation models which relate the number of laborers, the rat es of operation, the production quantities, the sales quantities, etc. All of the above methods, however, assess the investments only from the economical viewpoint and the simulation models are effective for assessing the investments under uncertainties of economical situation. Thus, it is not easy to make a decision on the real problem, without technical considerations, by using these assessment methods or simulation models . In this paper, we have defined the criterion from the technical viewpoint in addition to the economical viewpoint, and then have derived a total assessment function by use of multiattribute utility functions[2], As an application, a real problem of annual investments has successfully been dealt with by the present procedure. The first part of the paper defines three criteria (objectives) of the assessment, i .e., the rate of re-
1539
In the second part, the multiattribute utility function associated with th e three objectives is constructed by using the utility theory . Based on answers to questionnaires put to the utficer in charge, individual utility functions are assessed and parameters in the multiattribute utility function are also evaluated. In the third part, a real problem is investigated. The problem is to assess the four hundred and forty-one kinds of investments. The consequences of the decision with use of the present procedure are compared with those of a heuristic or an intuitional decision in the pas t. The comparison proves the effectiveness of the present procedure.
2.
CRITERIA OF THE I0IVESTMENTS
Selecting criteria for the assessment is the first im portant step in our whole procedure. The criteria to our problem are selected by discussions with several administrators, the accountant general and the vice-president of the firm. Fundamental criteria finally selected are (1) economical and technical assessment, (2) top management policy and industrial policy assessment, and (3) assessment of future circumstances. Figure 1 illustrates these three factors affecting the total assessment. For making a decision definitely, quantification of these criteria is needed. Considerations for the quantification are detailed in what follows.
Y. Nishikawa et al.
1540
top management policy and industrial policy
economical technical
given a series of inquiries of the form, "Is the Jth criteriun more useful than the ith criterion' If ves, how much is the degree'" The replied degree a" ( i, J = 1, 2, ., 6; i '" J) of each person are ar· ranged in a matrix such as shown in Fig. 2.
~J
·t ~ ;
I circumstances
~ __: _l-:~ _-+i_-+:_-j ,1 : 2 ; !---;-~ I 2 ! 2 Fig. 1.
Factors in the assessment of investments.
5 6
2. 1 Economical Criteria There have been developed many methods which as· sess the investment from the economical point of view. The outlines of typical methods are as fol · lows. a) Minimum·cost rul e By comparing the annual cost of the present equip· ments with that of the new equipment, the replace· ment investment is judged. !vIA!'I is a revised method of the minimum · cost rule. This method only estimates the cost saving and it is indifferent to the goal rate of production firm. Hence, it is difficult to decide whether investment is appropriate or not. b) Rate of return By examining ratio of the net profit to the amount of invested money, the investment is judged. For a production firm, this method is better than the other methods from the viewpoint of profit, but there are a lot of uncertainties about economical da · ta In the future. c) Payout period By examining the payout period, the investment is judged. This method is useful to judge the safety of investment and floating capital, but it is not useful to judge whether the investment is profitable or not. d) Capital turnover By examining the capital turnover, the investment is judged. This method is useful to compare the investment plans if they are for the development of new products and for the ltlcrease of productive power, but it is meaningless if they are for ration alization of the production There are many methods for economical estimation of the investment. However, as mentioned above, they are similar more or less [ 3 J . For a real problem of assessing the investment , we have to de· fine and formulate the economical criteria more ef· fectively. We have had some discussions about the economical criteria with the persons in charge (accountants, administrators). In order to get a per· suasive conclusion out of the discussions, we have made use of the method of IWSM ( Interpreti ve Weighted Structural Modeling) [4J. At the outset of this step, we selected six criteria which had been used rather independently in different divisions. They are 1) minimum · cost rule, 2) rate of return, 3) payout period, 4) capital turnover, 5) MAPI, and 5) revised !vIAP!. The person in charge are
Fig. 2.
2
1
Individual matrix obtained from the replies .
In the matrix, the numeral 3 means that the Jth criterion is much more useful than the ith criterion, the numeral 2 means that the Jth criterion is more useful than the ith criterion, and the numeral 1 means that the Jth criterion is a little more useful than the ith criterion. From this matrix, the digraph is obtained by IWSM and is shown in Fig. 3. After representing the digraph of each person , we have had a discussion about the digraphs and con· cept of the economical criteria. Finally we have de· fined the economical criteria as folluws: Rate of 1, = where 1 P
return (P/!) x I00
I, [% J, (1)
investment amount [yenJ , annual av e rage of profit [yenJ.
Payout period ....
I,
[ yearJ,
1, ~ I/(P+D),
(2 )
where D: annual depreciation [yen]. Many formulas to the rate of return and the payout period have been propused so far, but the simplest formula in each method is adopted here. Since there are a lot of uncertainties in the future fore· cast of economical situations, it may be meaningless to use complicated formulas.
1.0
on 'J)
"
c
2
'"
0. 5
'fi
:l
"> :;:
"
::>:::
0
Fig. 3.
Relative usefulness of the economical criteria ( by an accountant).
Multiobjective Assessment of Investment Plans 2. 2 Technical Criterion
X, real·valued functions u(x) and u(y) can be as· signed in such a way that , for all x and y in X,
New equipment; have direct inilu e nce upon the pro· duction technique, the labor requirement, the unit costs of materials, the cost for dynamic force, ete. They have also indirect iniluence upon the infurma· tion oi production technique and the activity uf hu · 11lan resources (researche rs. e ngint'ers). ete
Hence
it is important to assess the investment not only from the economical viewpoint but also from the technical viewpoint. As a first step of technical assessment, an investment plan is graded into ten classes by comparing the new equipment with simi· lar ones in other production firms.
x«y if and only if u(x) ~ u(y).
The function u ( x) is a utility function which trans· forms the preference structure of a person into a corresponding numerical utility structure [5 J . Let X,_ denote X1xX, x ... XX, _IXX,q X... x X rn , and X,, _ de · note XIXX, X ... xX,_I XX"I X"' X X, _I XX,.I " "· XXm. For m : 3. if for some X" X, x X, is preferentially inde · pendent of X,, _ for all j"" i, and X, is utility inde· pendent of X, _ for all I , then we have either u(x) = i:k,u,(x;) ,
The classes are as follows. Al class: The new equipment is unprecedented in the world and it is possible to apply for an i nternat ional patent. A2 class The new equipment is unpr ecedented inside the cuuntry and it can be buasted in the world. Bl class There is no similar equipment to the new equipment in the country and it is possible to apply for a domestic patent. B2 class Ther e are quit e few similar equipments to the new eq ui pment in the country and it can be boasted in the country. Cl class Ther e are similar equipments tu the new equipment in the country but it has many better functions than the similar equipments of other produc· tion firms. C2 class Ther e are many similar equipments to the new equipment in the country and it has almost equivalent functions to the similar equipments of other production firms. Dl class The new equipment has a few worse functions than the similar ones of other production firms . 02 class The new equipment has worse func· tions than the similar ones of other production firms. The new equipment gets behind in El class its functions compared with the simi· lar ones of other production firms. E2 class The new equipment gets behind a lot in its functions compared with the similar ones of other production firms.
i\lUL TIATTRIBUTE UTILITY FUNCTION
The assessment is the preference representation of a decision maker , and its p'ossibility means that the decison maker has the preference structure to the objectives Ii the preference structure of the deci· sion maker is repr ese nted explicitly, the quantitative analysis will be made possible to the assessment procedure. 1-1. L. Keene\' proposed the multiattribute utility functions in 1~74 [ 2J. In his description, the preference of the decision maker is represented explicitly if the utility independence and the preferential independence are confirmed. Let the set X be a product set of m attributes ( cri· teria) X, : X = XI x X, x ... x X m' If X and y are ele· ments in X and x is not preferred to y , a prefer· ence relation is donoted as x"'i,'y. Assuming that4, on X is connected (i e., x4y or y«x) and transi · tive (i.e., if x"'i,' y and y..( Z, then x -< z), and that there is a countable subset of X that is «-de nse in
(3)
/ :.: \
or
3.
1541
u(x) = \,ljl C1+kk,u;
(X,J J-l) /k.
(4)
where u and all u; are scaled from zero to one, k, are scaling constants such that O< k, < 1, and k > -l is a nonzero constant. Equation (3) is called the additive utility function and (4) is the multiplicative utility function [ 6J. By use of the multiattribute utility function for the economical and technical criteria, the quantit:lIive analysis can be done for assessing the investment.
4.
CONSTRUCTION OF ASSESSMENT FORMULA
Let f,' and f,' denote the least and the most desira· ble va!ues of attribute f, ,respectively. Table 1 shows the ranges of the three attributes decided by discussions among the person in charge, where f, denotes the technical criterion. Allowable Ranges of f;
TABLE f,
f;'
f,*
200 Cb
fl
0
f,
8 years
0.5 year
f,
E2 class
Al class
0·'
~o
4. 1 Verifying Mutual Utility Independence and the Additive Structure [ 7 J , [8J Because it is very important to verify independence conditions and the additive structure, we have a short dialogue with the decision maker for verification. First, we ask him about the certainty equivalent for the lottery which is shown in Fig. 4. Figure 4 is the 50 · 50 lottery [ 3 J to the attribute fl where attributes f, and f, are fixed as f, = 0.5, f, = AI. His answer is thirty ·e ight per cent. Further, we put some other questions to him by changing the values of f, and f,. As a result of those questions, we know that the value of certainty equivalent is always thirty · eight per cent and it is indif· ferent to the values of f, and f,· In the same manner, we ask him about the certainty equivalents for the 50-50 lottery to the attribute f, and to f,. Due to those investigations, we are able to verify the condition of mutual utility independence. To verify the additive structure, we ask the decision maker whether the lotteries to the attributes fl and f, are indifferent or not. Figure 5 shows the lotteries. His answer is that the lottery Ll is preferred to the lottery L2. This result violates the condition for additive independence, and then the multiattribute utility function is not of an additive form.
1542
Y. Nishikawa
<= 05
CE-
05
f, = 200 o Q
f, = 05year
f, : A
f,
= 0
class
I
a~.
The value of parameter k can be calculated by th e following equation which is derived from Equation (4) with f, =f: for all I:
I+k =( l+kk,)(l+kk,)(l+kk,)
00
~ f, = 200'b, f,: Al class Lottery LI
~f' = O'b' ~f' = Ooo,
E2 class
Fig. 5.
\
f, = 20000, f,: E2 class f, = 0.5 year
Lotteri"s to the attributes
I.
and
f,.
\
0.8
f,: Al class
Lottery L2 : ~
05
(5)
1.0 0.9
f, :
(k :.-. -J).
Table 2 shows the values of k, and k, evaluated from the responses of the vice·president.
50·50 lottery to the attribute f,.
Fig. 4.
et
f\
0.7 ~
0.6
'0 .':-
0.5
:::>
I
! i
I
"" ""
0.4 03
0.5 I
2
I
I
I
I
""" I
~
-1
3
:
:,
!
0.1
For each individual utility function u, (f,), we put the normalization condition as
!
! ! ,
\
o
u, U,o) = O.O,
I
I
0.2
4. 2 Construction of Multiattribute Utility Function
i
i
5
-..........
6
..............
7
8
f, (year) Fig. 7.
Utility function for the payout period.
u, U:) = 1.0. The values of U i for fi between f,o and f,' have been decided by questionnaires to the decision maker, i.e. , th e vice · president of the firm. Figures 6 through 8 show the individual utility func· tions obtained from his responses . The scaling cOnstants k i are evaluated by using a lott ery shown in Fig. 9. In Fig. 9, f i - implies the set of all attributes oth~r than f i . If, at a certain value p' of the probability p, the lottery and the certainty equivalent become indifferent to eac h other, then we have the fo llowing equalities:
I. 0
r-----r-,---,-----,---,----,----,---,---Q
0.9 f----t-
+-------t-__+-+-----t-+-----1r'------j
O. 8 t__-t-_+__-t--+---+--+__~>---t---j 0.7t__-+----+---+__--+---t__-+----+---+__~ 0.6r-~--_+--_+__--+---t---t___+--_+__~
~
O. 5 r---+---+--+---~-_+-_+__-t--t---1
.':-
0.4~__+--_+--__+--+---t-----tf____+--__+~
'0
0.3~__+-__+-+-_<>--_+-+---f---__+-~
P'u(f,', f/, f,')+(I-P')u(f,o, f,o, f,o)
0.2r-~--_+--_+__--+---t-~t___+--_+__~
= uCf,*, fio- ), and uCf,o, f,o, f,o) =o, uCf,', f,', f,')=I, °E2 El
D2
DI
uU,', f,'_) = [(!+kk ,J-I J/k=k, . Hence Fig. 8.
1.0 0.9
"3
0.6 0.5
.':- 0.4
.-
:::>
0.3 0.2
oI
/
o o
/
/
/
/
/
I
!
!
I-P
i I I i j I I
!
i
Fig. 9.
Evaluating the scaling constants.
I
I
I TABLE 2
100
150
200
f,[%)
Fig. 6.
- - - - - - - - f,o , f,o, f,o
Certainty L.....-=-____ ----'__equivalent ----'----__----'-----'______~ fi, , f.'-
I
50
A2 Al
P .__--------- f,·. f;. f:
Lottery
i I I
82131
Utility function for the te chnic al level of equipment.
,
V
Cl
class)
J-I--f-- I-
V
0. 7
c...:::
/y
I
0.8 r--'
C2
f, [
Utility function for the rate of return.
Values of k, and k
k,
0.30
k,
0.30
k,
0.35
k
0.16
Multiobjective Assessment of Investment Plans 5.
1543
Figure 12 shows the percentage of the plans put In each rank.
EXAMPLE
A real problem considered here is from the assess· ment of investment plans of the production firm in 1980. There are four hundred and forty·one invest· ment plans.
5. 1 Transformation of the Assessment Formula Before applying the assessment formula to the real problem, we have had discussions among the persons in charge about the problems. Their opinions to be noted are such as follows: (1) Since the persons in charge (accountants, ad· ministrators) are not familiar to the compli· cated mathematical presentation, it is important to define the assessment formula as simple as possible. (2) If the assessment formula is not simple, the top· managers will not be able to understand its meaning, then it will not be applicable to the real problems.
Copy
Fig. 10.
1·0 system.
As a result of the discussions, we have decided to simplify the assessment formula. We substitute the values of k, .and k into (4) giving
u = 0.30u,(/, )+0.30U, (/,) +0.35 u, (/,) 1.0
+0.0 15u, (/,) u, (/, )+O.017U, (/, )u, (I,)
DIO Cl)
+0.017u, (/,) u,(/,) + 0.001 u, (/, )u, (I, )u, (/,).
::l
(6 )
..0
Equation (6) consists of the linear terms and the nonlinear terms, and the coefficients of the nonline· ar terms are relatively small. Because of this, we have defined a simplified assessment formula as fol· lows:
u = 0.30u, (/, )+0.30u,(/, )+0.40u,(/,).
r-------------------~~--~~D5------~
(7)
D,
,
0; u
c 0
u
Cl)
D,OO D
This procedure may seem meaningless from the result of non-additive independence, but this is quite useful for the practical application. The above formula is used for setting a kind of standard for the assessment.
I
D71 11
"" 0. 5
D,
o
1 1
I I 1 1
D,
Cl)
The coefficients of Ui (I,) are evaluated again by the questionnaires put to the decision maker.
I
-- -- - - - - ------D,?T
<;;;
E 0
I
:
DOl D,
_ Y
...
0
-5
1
'0 i:'
I 1 1 1
~
;::J
D,: operational di visi7n 1IX
0
0.5
1.0
X
5. 2 Result of Assessment According to the purpose of investments, the investment plans are classified into ten categories. They are for the new products, for the increase of production power, for the rationalization, for the research and development, for the welfare, etc. Among these categories, the investment plans for the research and development, for the welfare of the employee and for the safety are paid no regard at present. The total percentage of these is twentyseven . The investment plans from fourteen operational divisions are handled by using a computer. The input and output system is shown in Fig. 10. Figure II shows the configuration of the operational divisions on the x, y·plane where x·axis indicates the utility of technical attribute u,(/,) and y-axis indi· cates the utility of economical attribute [u,(/,)+ u,(/,) J/ 2. The investment plans are classified into three ranks according to the value of u, i.e., A rank B rank Crank
7.0 s. u, 5.0 s. u < 7.0, and U< 5.0.
Utility of the technical attribute Fig. 11. Configuration of the investment plans by divisions.
not
A
rank 50 %
17 %
Fig. 12. Ratio of the A, Band Cranks.
1544 b.
Y. Nishikawa et aZ .
CO:--iC LUS!001
\\'e have derived a furmu la for the assessment of ill\·es tm .:n t plans bv use uf the multiattribute utilit v functiun. As a res ult uf rea l applic'ation, we have fo und t hat thc' propused assess ment formula wurks ,;u ", el!. The main find ings are s uch as follow s. ( 1) (:: )
The decisiun uf the tup ·managers can b e mad e murt' r"adilv and qui ckl y th an before because uf th e definite assessment cr it eria. Since the assessmen t standard is defined clea r Iv. evervone can giv e eas il y his ideas or ad v iccs on th e in ves tm ents.
( 3)
,\s a ll the in ves tm en t plans are represented on the x. y ·plane ( x: th e utility of technical at trib ute, )': the utilit y of eco nomi ca l attribut e), it is easy to see th e re lativ e utiliti es uf those p lans. :-'lureove r, th e principa l int en ti on of each uperati una l d ivis iun can eas ily be un ders tuod.
REFERE0ICES [ l J Terb orgh, G.
L:2 .1 L:lj [ -1 J
L5]
[ 6] [ 7J
D y nami c i:"lu ipment Po licy, :-'IcGraw· Hill , ;\Iew York , 19-19 Kecne y, R. L. i\ lulti ple Ut il ity Functiuns , Operations Re seach, Vol. 22, pp. L~·3-1, 19 7-1. Scheub le, P. A. How to Figure Eq uipm ent Replacement, llClT'1'arc/ Bu s iness Rev ieu', Sep tem b er -October , 1955. :\'is hik awa, Y . and 01omu ra, J. Interpreti ve Weigh ted Structural Modelin g ( lWSM) and Its App li ca tions to a Schedu ling and a Budget Al location, '!1;\/S :!.Jth int allationa l ,Heeting, 1979 . Fishburn, P . Utility Theo ry. Management Sciellce, Vo!. 14, pp . :135 -378, 1968. Keeney, R. L. an d Raiffa, H. Decis ion wit h Multiple Objectives, ] ohn-Wil ey, New York , 1976. Keene y, R. L. The Art of Assessing Multi attribu te Util it y FU ;lc tiun s, Organizationa l Be havior and H uman Performance, Vo!. 19, pp . 267 -3 10 , 1977.
L 8]
Raiffa , H. Dec is ion Analysis, Addison-\Vesley, Re ad ing, i\ lass., 1961).
AC K'JUWLEDGME N T
The authors wish to express th eir hearty th an ks to :-'lr. K. Kobayashi, Vice -pres ident of Matsushita Elc-ctric Works, Ltd ., for his excel lent advices and cuupe ratiuns in doin g thi s work.
Discussion to Pap e r 50.4 Y. Y. Haimes (USA) : The use of systems analysis and multiobjective optimization would be limited if aspects of risks and uncertainties were not addressed. Could you share with us your experience with the Vice President (as a decision - maker) in terms of using explicit considerations of risks and un ce rtainties in the analysis? Risk and un certainty associated with finances, market f o rces, etc.? Y. Nishikawa (Japan): Up to n ow we have not explicitly formulated elements directly rela ted to uncertainties. Of course there are va rious kinds of uncertainties in our problems, e . g . variation of future demand, future economic circumstance, etc. \,e have treated these uncertainties through the investigation of the past data and the estimation for the future together with decision-makers. Due to t h is kind of discussion and investigation, the DMs can get brain models based on which I wi ll determine their utility function, then all the uncertain elements are included in t he final expression of the utility functions. A. Straszak (Poland): Traditionally, multiobjective decision problems arose in the public sector. To what extent could this situation take place in the private sector? Could you comment on the Japanese style of management with relation to multiobjective techniques? Y. Nishikawa (JapanA: As a matter of course, the final goal or objective of a private organization is the increase of its profit, although it sometimes aims to contribute to social welfare or something like that. In that sense, the objective of optimization in a private-sector problem may seem just to be one. But, in reality, its objective has many
aspects such as short-term profit and longterm prosperity, and encouragemen t of its social credit, etc. Then, a model of its ac tivity has to take various factors and then objectives into consideration, as the complete identification of the final single objective is almost impossible not only for systems analysts but also for a decisionmaker himself. There are, without doubt, some salient features in the way of decision - making in Japanese organ izations , th ough clear discussion at this place is quite difficult. In order to deal with them, we have contacted with several decision-makers at several hierarchies, first beginning with lower-ranked decision-makers and then shifting to higherranked ones. By so doing, a hierarchical structure in decision procedure could be dealt with in our analysis, more or less. This also answers, to some extent, Professor Okada's question. N. Okada (Japan): I suspect that there may be a hierarch ical sequence or process in which decisions are made. Couldn't it be the case that a top level decision is made which is passed on to another level in which the next decision is made, and so on? I'n other words, isn't it likely that prio rity is given to one criterion, rather than the rest of the criterion? Do your research results suggest that this is the case? Or does your model explicitly incorporate these kind of institutional dimensions? Y. Nishikawa (Japan): Hierarchy in the structure of objective functions and hierarchy in the decision procedure are basically important matters. In our study, considerations of these points are included in screening and structuring the objective function through the discussion between the decisionmakers and the system analysts, mainly helped by our method of IWSM.
MULTI OBJECTIVE ASSESSMENT OF INVESTMENT PLANS BY USE OF UTILITY THEORY Y. Nishikawa*,
J. Nomura**, T. Inaki**, J. Hashizume** and K. Sawada**
·Department of Electrical Engineering, Kyoto University, Yoshida, Kyoto 606, Japan ·*Research and Development Laboratory, Matsushita Electric Works, Ltd., Kadoma, Osaka 571, Japan
Abstract. The paper discusses a multiobjective assessment procedure as applied to the assessment of investment plans in a production firm. There are many conventional methods for assessing the investments. Among those methods, the rat e of return method and the payout period method are used as the ones with th e criterion defined from the economical viewpoint. In addition to the economical viewpoint, we have introduced a n ew criterion defined from the technical viewpoint. The total assessment formula is derived considering buth the economical and th e technical terms by use of a multiattribute utility function. A real problem uf assessing th e investments has s ucc ess full y been d ealt with by the present procedure. Keywurds. Investment ; utility theory
1.
management systems;
I NTRO DUCTION
multiobjective assessment;
systems analysis;
turn, the payout period and the technical level of new equipment. Generally, the rate of return and the payout period can readily be defined and formu· lated quantitatively. But, the technical level of equipment is difficult to quantify. Hence , it is graded into ten classes where each class is defined by the technical level in comparison with the equipments of other production firms.
For a production firm, it is one of the important yet di fficult problems to assess a nd decide invest ment plans, because they are necessary and indispensable for a stabilization and magnificatiun of the production firm or for an increase of profit. Once th e investment plans are executed, they give a significant effect upon the productiun quantities , the quality of goods, th e unit costs of materials, the cost of dynamic force, etc. After beginning the performance, a change of the plan will be impossible practically and will ensue a lot of loss es. So far, there have been developed many methods for assessment of investm en ts. Among them , the rate of return method, MAP I (Machinery and Allied Products Insti tute) method [1J and the payout period method assess the investm ents from the economical point of view. Further , there are many reports about simulation models which relate the number of laborers, the rat es of operation, the production quantities, the sales quantities, etc. All of the above methods, however, assess the investments only from the economical viewpoint and the simulation models are effective for assessing the investments under uncertainties of economical situation. Thus, it is not easy to make a decision on the real problem, without technical considerations, by using these assessment methods or simulation models . In this paper, we have defined the criterion from the technical viewpoint in addition to the economical viewpoint, and then have derived a total assessment function by use of multiattribute utility functions[2], As an application, a real problem of annual investments has successfully been dealt with by the present procedure. The first part of the paper defines three criteria (objectives) of the assessment, i .e., the rate of re-
1539
In the second part, the multiattribute utility function associated with th e three objectives is constructed by using the utility theory . Based on answers to questionnaires put to the utficer in charge, individual utility functions are assessed and parameters in the multiattribute utility function are also evaluated. In the third part, a real problem is investigated. The problem is to assess the four hundred and forty-one kinds of investments. The consequences of the decision with use of the present procedure are compared with those of a heuristic or an intuitional decision in the pas t. The comparison proves the effectiveness of the present procedure.
2.
CRITERIA OF THE I0IVESTMENTS
Selecting criteria for the assessment is the first im portant step in our whole procedure. The criteria to our problem are selected by discussions with several administrators, the accountant general and the vice-president of the firm. Fundamental criteria finally selected are (1) economical and technical assessment, (2) top management policy and industrial policy assessment, and (3) assessment of future circumstances. Figure 1 illustrates these three factors affecting the total assessment. For making a decision definitely, quantification of these criteria is needed. Considerations for the quantification are detailed in what follows.
Y. Nishikawa et al.
1540
top management policy and industrial policy
economical technical
given a series of inquiries of the form, "Is the Jth criteriun more useful than the ith criterion' If ves, how much is the degree'" The replied degree a" ( i, J = 1, 2, ., 6; i '" J) of each person are ar· ranged in a matrix such as shown in Fig. 2.
~J
·t ~ ;
I circumstances
~ __: _l-:~ _-+i_-+:_-j ,1 : 2 ; !---;-~ I 2 ! 2 Fig. 1.
Factors in the assessment of investments.
5 6
2. 1 Economical Criteria There have been developed many methods which as· sess the investment from the economical point of view. The outlines of typical methods are as fol · lows. a) Minimum·cost rul e By comparing the annual cost of the present equip· ments with that of the new equipment, the replace· ment investment is judged. !vIA!'I is a revised method of the minimum · cost rule. This method only estimates the cost saving and it is indifferent to the goal rate of production firm. Hence, it is difficult to decide whether investment is appropriate or not. b) Rate of return By examining ratio of the net profit to the amount of invested money, the investment is judged. For a production firm, this method is better than the other methods from the viewpoint of profit, but there are a lot of uncertainties about economical da · ta In the future. c) Payout period By examining the payout period, the investment is judged. This method is useful to judge the safety of investment and floating capital, but it is not useful to judge whether the investment is profitable or not. d) Capital turnover By examining the capital turnover, the investment is judged. This method is useful to compare the investment plans if they are for the development of new products and for the ltlcrease of productive power, but it is meaningless if they are for ration alization of the production There are many methods for economical estimation of the investment. However, as mentioned above, they are similar more or less [ 3 J . For a real problem of assessing the investment , we have to de· fine and formulate the economical criteria more ef· fectively. We have had some discussions about the economical criteria with the persons in charge (accountants, administrators). In order to get a per· suasive conclusion out of the discussions, we have made use of the method of IWSM ( Interpreti ve Weighted Structural Modeling) [4J. At the outset of this step, we selected six criteria which had been used rather independently in different divisions. They are 1) minimum · cost rule, 2) rate of return, 3) payout period, 4) capital turnover, 5) MAPI, and 5) revised !vIAP!. The person in charge are
Fig. 2.
2
1
Individual matrix obtained from the replies .
In the matrix, the numeral 3 means that the Jth criterion is much more useful than the ith criterion, the numeral 2 means that the Jth criterion is more useful than the ith criterion, and the numeral 1 means that the Jth criterion is a little more useful than the ith criterion. From this matrix, the digraph is obtained by IWSM and is shown in Fig. 3. After representing the digraph of each person , we have had a discussion about the digraphs and con· cept of the economical criteria. Finally we have de· fined the economical criteria as folluws: Rate of 1, = where 1 P
return (P/!) x I00
I, [% J, (1)
investment amount [yenJ , annual av e rage of profit [yenJ.
Payout period ....
I,
[ yearJ,
1, ~ I/(P+D),
(2 )
where D: annual depreciation [yen]. Many formulas to the rate of return and the payout period have been propused so far, but the simplest formula in each method is adopted here. Since there are a lot of uncertainties in the future fore· cast of economical situations, it may be meaningless to use complicated formulas.
1.0
on 'J)
"
c
2
'"
0. 5
'fi
:l
"> :;:
"
::>:::
0
Fig. 3.
Relative usefulness of the economical criteria ( by an accountant).
Multiobjective Assessment of Investment Plans 2. 2 Technical Criterion
X, real·valued functions u(x) and u(y) can be as· signed in such a way that , for all x and y in X,
New equipment; have direct inilu e nce upon the pro· duction technique, the labor requirement, the unit costs of materials, the cost for dynamic force, ete. They have also indirect iniluence upon the infurma· tion oi production technique and the activity uf hu · 11lan resources (researche rs. e ngint'ers). ete
Hence
it is important to assess the investment not only from the economical viewpoint but also from the technical viewpoint. As a first step of technical assessment, an investment plan is graded into ten classes by comparing the new equipment with simi· lar ones in other production firms.
x«y if and only if u(x) ~ u(y).
The function u ( x) is a utility function which trans· forms the preference structure of a person into a corresponding numerical utility structure [5 J . Let X,_ denote X1xX, x ... XX, _IXX,q X... x X rn , and X,, _ de · note XIXX, X ... xX,_I XX"I X"' X X, _I XX,.I " "· XXm. For m : 3. if for some X" X, x X, is preferentially inde · pendent of X,, _ for all j"" i, and X, is utility inde· pendent of X, _ for all I , then we have either u(x) = i:k,u,(x;) ,
The classes are as follows. Al class: The new equipment is unprecedented in the world and it is possible to apply for an i nternat ional patent. A2 class The new equipment is unpr ecedented inside the cuuntry and it can be buasted in the world. Bl class There is no similar equipment to the new equipment in the country and it is possible to apply for a domestic patent. B2 class Ther e are quit e few similar equipments to the new eq ui pment in the country and it can be boasted in the country. Cl class Ther e are similar equipments tu the new equipment in the country but it has many better functions than the similar equipments of other produc· tion firms. C2 class Ther e are many similar equipments to the new equipment in the country and it has almost equivalent functions to the similar equipments of other production firms. Dl class The new equipment has a few worse functions than the similar ones of other production firms . 02 class The new equipment has worse func· tions than the similar ones of other production firms. The new equipment gets behind in El class its functions compared with the simi· lar ones of other production firms. E2 class The new equipment gets behind a lot in its functions compared with the similar ones of other production firms.
i\lUL TIATTRIBUTE UTILITY FUNCTION
The assessment is the preference representation of a decision maker , and its p'ossibility means that the decison maker has the preference structure to the objectives Ii the preference structure of the deci· sion maker is repr ese nted explicitly, the quantitative analysis will be made possible to the assessment procedure. 1-1. L. Keene\' proposed the multiattribute utility functions in 1~74 [ 2J. In his description, the preference of the decision maker is represented explicitly if the utility independence and the preferential independence are confirmed. Let the set X be a product set of m attributes ( cri· teria) X, : X = XI x X, x ... x X m' If X and y are ele· ments in X and x is not preferred to y , a prefer· ence relation is donoted as x"'i,'y. Assuming that4, on X is connected (i e., x4y or y«x) and transi · tive (i.e., if x"'i,' y and y..( Z, then x -< z), and that there is a countable subset of X that is «-de nse in
(3)
/ :.: \
or
3.
1541
u(x) = \,ljl C1+kk,u;
(X,J J-l) /k.
(4)
where u and all u; are scaled from zero to one, k, are scaling constants such that O< k, < 1, and k > -l is a nonzero constant. Equation (3) is called the additive utility function and (4) is the multiplicative utility function [ 6J. By use of the multiattribute utility function for the economical and technical criteria, the quantit:lIive analysis can be done for assessing the investment.
4.
CONSTRUCTION OF ASSESSMENT FORMULA
Let f,' and f,' denote the least and the most desira· ble va!ues of attribute f, ,respectively. Table 1 shows the ranges of the three attributes decided by discussions among the person in charge, where f, denotes the technical criterion. Allowable Ranges of f;
TABLE f,
f;'
f,*
200 Cb
fl
0
f,
8 years
0.5 year
f,
E2 class
Al class
0·'
~o
4. 1 Verifying Mutual Utility Independence and the Additive Structure [ 7 J , [8J Because it is very important to verify independence conditions and the additive structure, we have a short dialogue with the decision maker for verification. First, we ask him about the certainty equivalent for the lottery which is shown in Fig. 4. Figure 4 is the 50 · 50 lottery [ 3 J to the attribute fl where attributes f, and f, are fixed as f, = 0.5, f, = AI. His answer is thirty ·e ight per cent. Further, we put some other questions to him by changing the values of f, and f,. As a result of those questions, we know that the value of certainty equivalent is always thirty · eight per cent and it is indif· ferent to the values of f, and f,· In the same manner, we ask him about the certainty equivalents for the 50-50 lottery to the attribute f, and to f,. Due to those investigations, we are able to verify the condition of mutual utility independence. To verify the additive structure, we ask the decision maker whether the lotteries to the attributes fl and f, are indifferent or not. Figure 5 shows the lotteries. His answer is that the lottery Ll is preferred to the lottery L2. This result violates the condition for additive independence, and then the multiattribute utility function is not of an additive form.
1542
Y. Nishikawa
<= 05
CE-
05
f, = 200 o Q
f, = 05year
f, : A
f,
= 0
class
I
a~.
The value of parameter k can be calculated by th e following equation which is derived from Equation (4) with f, =f: for all I:
I+k =( l+kk,)(l+kk,)(l+kk,)
00
~ f, = 200'b, f,: Al class Lottery LI
~f' = O'b' ~f' = Ooo,
E2 class
Fig. 5.
\
f, = 20000, f,: E2 class f, = 0.5 year
Lotteri"s to the attributes
I.
and
f,.
\
0.8
f,: Al class
Lottery L2 : ~
05
(5)
1.0 0.9
f, :
(k :.-. -J).
Table 2 shows the values of k, and k, evaluated from the responses of the vice·president.
50·50 lottery to the attribute f,.
Fig. 4.
et
f\
0.7 ~
0.6
'0 .':-
0.5
:::>
I
! i
I
"" ""
0.4 03
0.5 I
2
I
I
I
I
""" I
~
-1
3
:
:,
!
0.1
For each individual utility function u, (f,), we put the normalization condition as
!
! ! ,
\
o
u, U,o) = O.O,
I
I
0.2
4. 2 Construction of Multiattribute Utility Function
i
i
5
-..........
6
..............
7
8
f, (year) Fig. 7.
Utility function for the payout period.
u, U:) = 1.0. The values of U i for fi between f,o and f,' have been decided by questionnaires to the decision maker, i.e. , th e vice · president of the firm. Figures 6 through 8 show the individual utility func· tions obtained from his responses . The scaling cOnstants k i are evaluated by using a lott ery shown in Fig. 9. In Fig. 9, f i - implies the set of all attributes oth~r than f i . If, at a certain value p' of the probability p, the lottery and the certainty equivalent become indifferent to eac h other, then we have the fo llowing equalities:
I. 0
r-----r-,---,-----,---,----,----,---,---Q
0.9 f----t-
+-------t-__+-+-----t-+-----1r'------j
O. 8 t__-t-_+__-t--+---+--+__~>---t---j 0.7t__-+----+---+__--+---t__-+----+---+__~ 0.6r-~--_+--_+__--+---t---t___+--_+__~
~
O. 5 r---+---+--+---~-_+-_+__-t--t---1
.':-
0.4~__+--_+--__+--+---t-----tf____+--__+~
'0
0.3~__+-__+-+-_<>--_+-+---f---__+-~
P'u(f,', f/, f,')+(I-P')u(f,o, f,o, f,o)
0.2r-~--_+--_+__--+---t-~t___+--_+__~
= uCf,*, fio- ), and uCf,o, f,o, f,o) =o, uCf,', f,', f,')=I, °E2 El
D2
DI
uU,', f,'_) = [(!+kk ,J-I J/k=k, . Hence Fig. 8.
1.0 0.9
"3
0.6 0.5
.':- 0.4
.-
:::>
0.3 0.2
oI
/
o o
/
/
/
/
/
I
!
!
I-P
i I I i j I I
!
i
Fig. 9.
Evaluating the scaling constants.
I
I
I TABLE 2
100
150
200
f,[%)
Fig. 6.
- - - - - - - - f,o , f,o, f,o
Certainty L.....-=-____ ----'__equivalent ----'----__----'-----'______~ fi, , f.'-
I
50
A2 Al
P .__--------- f,·. f;. f:
Lottery
i I I
82131
Utility function for the te chnic al level of equipment.
,
V
Cl
class)
J-I--f-- I-
V
0. 7
c...:::
/y
I
0.8 r--'
C2
f, [
Utility function for the rate of return.
Values of k, and k
k,
0.30
k,
0.30
k,
0.35
k
0.16
Multiobjective Assessment of Investment Plans 5.
1543
Figure 12 shows the percentage of the plans put In each rank.
EXAMPLE
A real problem considered here is from the assess· ment of investment plans of the production firm in 1980. There are four hundred and forty·one invest· ment plans.
5. 1 Transformation of the Assessment Formula Before applying the assessment formula to the real problem, we have had discussions among the persons in charge about the problems. Their opinions to be noted are such as follows: (1) Since the persons in charge (accountants, ad· ministrators) are not familiar to the compli· cated mathematical presentation, it is important to define the assessment formula as simple as possible. (2) If the assessment formula is not simple, the top· managers will not be able to understand its meaning, then it will not be applicable to the real problems.
Copy
Fig. 10.
1·0 system.
As a result of the discussions, we have decided to simplify the assessment formula. We substitute the values of k, .and k into (4) giving
u = 0.30u,(/, )+0.30U, (/,) +0.35 u, (/,) 1.0
+0.0 15u, (/,) u, (/, )+O.017U, (/, )u, (I,)
DIO Cl)
+0.017u, (/,) u,(/,) + 0.001 u, (/, )u, (I, )u, (/,).
::l
(6 )
..0
Equation (6) consists of the linear terms and the nonlinear terms, and the coefficients of the nonline· ar terms are relatively small. Because of this, we have defined a simplified assessment formula as fol· lows:
u = 0.30u, (/, )+0.30u,(/, )+0.40u,(/,).
r-------------------~~--~~D5------~
(7)
D,
,
0; u
c 0
u
Cl)
D,OO D
This procedure may seem meaningless from the result of non-additive independence, but this is quite useful for the practical application. The above formula is used for setting a kind of standard for the assessment.
I
D71 11
"" 0. 5
D,
o
1 1
I I 1 1
D,
Cl)
The coefficients of Ui (I,) are evaluated again by the questionnaires put to the decision maker.
I
-- -- - - - - ------D,?T
<;;;
E 0
I
:
DOl D,
_ Y
...
0
-5
1
'0 i:'
I 1 1 1
~
;::J
D,: operational di visi7n 1IX
0
0.5
1.0
X
5. 2 Result of Assessment According to the purpose of investments, the investment plans are classified into ten categories. They are for the new products, for the increase of production power, for the rationalization, for the research and development, for the welfare, etc. Among these categories, the investment plans for the research and development, for the welfare of the employee and for the safety are paid no regard at present. The total percentage of these is twentyseven . The investment plans from fourteen operational divisions are handled by using a computer. The input and output system is shown in Fig. 10. Figure II shows the configuration of the operational divisions on the x, y·plane where x·axis indicates the utility of technical attribute u,(/,) and y-axis indi· cates the utility of economical attribute [u,(/,)+ u,(/,) J/ 2. The investment plans are classified into three ranks according to the value of u, i.e., A rank B rank Crank
7.0 s. u, 5.0 s. u < 7.0, and U< 5.0.
Utility of the technical attribute Fig. 11. Configuration of the investment plans by divisions.
not
A
rank 50 %
17 %
Fig. 12. Ratio of the A, Band Cranks.
1544 b.
Y. Nishikawa et aZ .
CO:--iC LUS!001
\\'e have derived a furmu la for the assessment of ill\·es tm .:n t plans bv use uf the multiattribute utilit v functiun. As a res ult uf rea l applic'ation, we have fo und t hat thc' propused assess ment formula wurks ,;u ", el!. The main find ings are s uch as follow s. ( 1) (:: )
The decisiun uf the tup ·managers can b e mad e murt' r"adilv and qui ckl y th an before because uf th e definite assessment cr it eria. Since the assessmen t standard is defined clea r Iv. evervone can giv e eas il y his ideas or ad v iccs on th e in ves tm ents.
( 3)
,\s a ll the in ves tm en t plans are represented on the x. y ·plane ( x: th e utility of technical at trib ute, )': the utilit y of eco nomi ca l attribut e), it is easy to see th e re lativ e utiliti es uf those p lans. :-'lureove r, th e principa l int en ti on of each uperati una l d ivis iun can eas ily be un ders tuod.
REFERE0ICES [ l J Terb orgh, G.
L:2 .1 L:lj [ -1 J
L5]
[ 6] [ 7J
D y nami c i:"lu ipment Po licy, :-'IcGraw· Hill , ;\Iew York , 19-19 Kecne y, R. L. i\ lulti ple Ut il ity Functiuns , Operations Re seach, Vol. 22, pp. L~·3-1, 19 7-1. Scheub le, P. A. How to Figure Eq uipm ent Replacement, llClT'1'arc/ Bu s iness Rev ieu', Sep tem b er -October , 1955. :\'is hik awa, Y . and 01omu ra, J. Interpreti ve Weigh ted Structural Modelin g ( lWSM) and Its App li ca tions to a Schedu ling and a Budget Al location, '!1;\/S :!.Jth int allationa l ,Heeting, 1979 . Fishburn, P . Utility Theo ry. Management Sciellce, Vo!. 14, pp . :135 -378, 1968. Keeney, R. L. an d Raiffa, H. Decis ion wit h Multiple Objectives, ] ohn-Wil ey, New York , 1976. Keene y, R. L. The Art of Assessing Multi attribu te Util it y FU ;lc tiun s, Organizationa l Be havior and H uman Performance, Vo!. 19, pp . 267 -3 10 , 1977.
L 8]
Raiffa , H. Dec is ion Analysis, Addison-\Vesley, Re ad ing, i\ lass., 1961).
AC K'JUWLEDGME N T
The authors wish to express th eir hearty th an ks to :-'lr. K. Kobayashi, Vice -pres ident of Matsushita Elc-ctric Works, Ltd ., for his excel lent advices and cuupe ratiuns in doin g thi s work.
Discussion to Pap e r 50.4 Y. Y. Haimes (USA) : The use of systems analysis and multiobjective optimization would be limited if aspects of risks and uncertainties were not addressed. Could you share with us your experience with the Vice President (as a decision - maker) in terms of using explicit considerations of risks and un ce rtainties in the analysis? Risk and un certainty associated with finances, market f o rces, etc.? Y. Nishikawa (Japan): Up to n ow we have not explicitly formulated elements directly rela ted to uncertainties. Of course there are va rious kinds of uncertainties in our problems, e . g . variation of future demand, future economic circumstance, etc. \,e have treated these uncertainties through the investigation of the past data and the estimation for the future together with decision-makers. Due to t h is kind of discussion and investigation, the DMs can get brain models based on which I wi ll determine their utility function, then all the uncertain elements are included in t he final expression of the utility functions. A. Straszak (Poland): Traditionally, multiobjective decision problems arose in the public sector. To what extent could this situation take place in the private sector? Could you comment on the Japanese style of management with relation to multiobjective techniques? Y. Nishikawa (JapanA: As a matter of course, the final goal or objective of a private organization is the increase of its profit, although it sometimes aims to contribute to social welfare or something like that. In that sense, the objective of optimization in a private-sector problem may seem just to be one. But, in reality, its objective has many
aspects such as short-term profit and longterm prosperity, and encouragemen t of its social credit, etc. Then, a model of its ac tivity has to take various factors and then objectives into consideration, as the complete identification of the final single objective is almost impossible not only for systems analysts but also for a decisionmaker himself. There are, without doubt, some salient features in the way of decision - making in Japanese organ izations , th ough clear discussion at this place is quite difficult. In order to deal with them, we have contacted with several decision-makers at several hierarchies, first beginning with lower-ranked decision-makers and then shifting to higherranked ones. By so doing, a hierarchical structure in decision procedure could be dealt with in our analysis, more or less. This also answers, to some extent, Professor Okada's question. N. Okada (Japan): I suspect that there may be a hierarch ical sequence or process in which decisions are made. Couldn't it be the case that a top level decision is made which is passed on to another level in which the next decision is made, and so on? I'n other words, isn't it likely that prio rity is given to one criterion, rather than the rest of the criterion? Do your research results suggest that this is the case? Or does your model explicitly incorporate these kind of institutional dimensions? Y. Nishikawa (Japan): Hierarchy in the structure of objective functions and hierarchy in the decision procedure are basically important matters. In our study, considerations of these points are included in screening and structuring the objective function through the discussion between the decisionmakers and the system analysts, mainly helped by our method of IWSM.