E L S E V I E R European Journal of Operational Research 83 (1995) 245-246

EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH

L e t t e r t o t h e E d i t o r

A comment on Mode s, modelling and modellers: an application to risk analysis" by B. Wahlstr6m

European Journal of Operational Research 75 (1994) 477-487

W . W . C o o p e r

The University of Texas at Austin, College of Business Administration, Management Department, CBA 4.02, Austin, TX 78712, USA

Accepted November 1994

As noted in our references, the above article appears in a special issue of EJOR on risk man- agement. It seems to me to suffer~ from deficien- cies which start with the following definition of "model" as taken by Wahlstr6m from Collins English Dictionary (1986):

" A simplified representat ion or description of a system or complex entity, especially one designed to facilitate calcula- tions and predictions."

A better definition is the following, as taken from Kohler's Dictionary for Accountants (6th edition, Prentice-Hall, 1983):

Model: Any system of relations used to represent another system of relations. Examples: A set of algebraic relations used to represent the graphs portrayed in a chart such as a breakeven chart. A set of blueprints used to guide the construction of a house or a piece of machinery. A chart or system of accounts with accompanying portrayals of flows that can be used to repre- sent the transactions conducted by an enterprise. Note that in all cases the relation is symmetric since, for instance, the algebraic relations may be modeled by the corresponding geometric figures or the blueprints may be drawn to conform to an already constructed house. For many purposes, one may single out one set of relations to evaluate the other. This is then said to be an evaluation of the model. Such an evalua- tion may be descriptive as when, for instance, the blueprints are checked against the house that was actually built or when the accounts are examined to see whether they portray the transactions that occurred. The evaluation may also be nor-

mative as when the house resulting from the blueprints is evaluated for its s tructure or esthetic qualities relative to what was specified in the blueprints or when the accounting system is evaluated for its efficacy in use. The former are somet imes said to be descriptive models and the latter are referred to as normative models.

One inadequacy of Wahlstr6m's definition arises from its omission of the control aspect of modeling, as in the above example :o f a set blueprints - to which the finally built structure must conform! Such control features often form a part of the function to be performed by OR models which are used to originate a plan and to evaluate subsequent actions. This use of models is also found in common parlance in forms such as "models of conduct", "model laws", etc.

Another deficiency in Wahlstr6m's article arises from his confounding two things in his statement that "Risks are usually defined as a function of probability and costs". The usages in insurance are bet ter in distinguishing "risk" as the "chance of occurrence of an undesirable event" and "severity" as the "level (e.g., of dam- age) that can materialize". The following quota- tion from page 387 of Brockett et al. (1992) can clarify what is involved,

"Af te r the 1953 f lood the Dutch Parl iament required all dikes to be built so that there would be a chance of 1 : 10,000

0377-2217/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD! 0 3 7 7 - 2 2 1 7 ( 9 4 ) 0 0 3 2 9 - 7

246 W.W. Cooper ~European Journal of Operational Research 83 (1995) 245-246

or less that the sea would exceed the level of the dike. This condition can be represented by the following simple chance constraint: Pr(x > L) > a, where Pr means probability, L, the sea level, is a random variable and x, the level of the dike, is a decision variable. If ce = 9,999/10,000 then Pr(x < L) < 1 - a = 1/10,000 satisfies the risk condition imposed by the Dutch Government (i.e., the risk of occurrence of this undesirable event does not exceed 1 : 10,000). Having identified (1 - a) as the measure of risk, we next direct attention to the "levels" at which undesirable events may occur - such as a level of one inch or a level of one foot, etc., by which the sea may exceed the height of the dike. To provide for such differences, we could use L as another random variable to represent the sea level and relate it to L via L = aL + d, where a and d are prescribed constants.

Note the t radeoff tha t this last expression per- mits be tween " level" and "r isk" - a t radeoff which is concea led from view by Wahl s t r6m in a m a n n e r ana logous to the way in which his defini- t ion of mode l conceals the "cont ro l func t ion" of model l ing.

Final ly I t u rn to his s t a t ement (p. 480) that " T h e most essent ial concept in the model l ing process is causality." Here again control (possibly involving r edundanc ies ) is concea led beh ind an emphasis on simplicity in a search for ease of unde r s t and ing . Wahl s t r6m thereby omits the im- po r t an t dis t inct ion be tween necessary and suffi- c ient condi t ions, of causali ty as emphas ized by Ackoff (1974, p. 16) with the following example:

" A n acorn is necessary bu t no t sufficient for an oak." Many other condi t ions are also requi red for an oak to mater ia l ize and, in the presence of uncer ta inty , it may be desirable to add extra condi t ions to help ensure desired outcomes - as is of ten done in s imula t ing large and complex systems despite an accompanying increase in model complexity tha t may conceal one or more necessary condi t ions for " the cause" to be ident i- fied.

Wahls t r6m's article evident ly falls seriously short of supplying the guidance that its title sug- gests for model l ing uses in opera t ional research and rela ted approaches to risk analysis.

References

Ackoff, R. (1974), Redesigning the Future: A Systems Approach to Societal Problems, Wiley, New York. Wiley & Sons, Inc., 1974).

Brockett, P.L., Charnes, A., Cooper, W.W., Kwon K.H., and Ruefli, T.W. (1992), "A chance constrained programming approach to empirical analyses of mutual fund investment strategies", Decision Sciences 23, 385-408.

Cooper, W.W., and Ijiri, Y. (eds.) (1983), Kohler's Dictionary for Accountants 6th edn., Prentice-Hall, Englewood Cliffs, NJ.

Wahlstr/Sm, B. (1994), "Models, modelling and modeUers: an application to risk analysis", European Journal of Opera- tional Research 75, 477-487.