Linking mathematics with applications: The comparative assessment process

Mathematics and Computers in Simula& North-Holland

33 (1992) 469-476

469

Bob Anderssen CSIRO Divi’ 11oi Mathematics and Statistics, GPO E s 1965, Car.berra, ACT 2601

1. Introduction Like any human activity such as exercise, music or writing, simulation means different things to different people. Here, ecur interest in simulation is as an aid to decision making in the sense of exploring the extert and structure of feasible solutions. Such solutions arc determined by the mathematical formulatiort defining the underlying model of the problem requiring the decision. We therefore view it as a comparative assessment process, since the final choice will be made on the basis of comparisons of the vari-us alternatives. As will be explained and motivated in so&medetail below, the comparative process can be viewed as having the following overall structure of BUILD and represents

+ SOLVE

a supplementation

IDENTIFY

-+ DISPLAY

4 AMEND

-+ COMPARE,

to the general

problem-solving

process of

--t FORMULATE

in that it is a. more specific implementation SOLVE

-+.MO’D&C-+

SOLVE

+ INTERPRET,

assessment

(1)

(2)

for the + INTERPRET

(3)

steps than is usuaily discussed. In fact, this is one way of defining simulnfion, by viewing it as an implementation of the two final steps in the generai problem-solving process. Sinulnfzon is then seen to complement and supplement problem-solving which can be viewed as defining tile full process from the moment identification begins. Simulation thereby becomes the continuing step has yielded an appropriate process of exploration and comparison once the formulation mathematical model which describes the relationship between available data (constraints) and the iaformation required for decision making purposes. links In terms of the title of this paper, there are two ways in which mathematics In terms of the problem-solving with applications within the framework of problem-solving. formalism of (Z), they are:

0378~754/92/$05.OO Q 1992 - Elsevier Science Publishers B.V. Allrights

r~servd

B. Anderssen I Linking mathematics with applications

470

s the IDER’TI~‘Y -+ FORhl(/‘LATE mathematical model for its solution;

step which links t,he application and

with a relevrrnt

a the SOLVE -+ INTERPRET step which links the model ba.ck to the application relating possible solutions back to the data and constraints.

It is the latter

by

type of linking which is the focus of this paper.

It is interesting to reflect that most problems arise in one problems, and therefore simulation, as defined above as comparative as a t.ype of trial-and-error methodology for the solution of such underdetermined. This interpretation of simulation will be pursued below.

way or another as inverse assessment, can be viewed problems when they are in some detail in Section 2

Other frameworks have been proposed and analysed for the problem-solving process. They include the IDEAL Problem Solver of Bransford and Stein [5] and various strategies for hlathetnatical Modelling (e.g. Klamkin [7], and some of the references cited in the General Supplementary References in that book). In one way or another, they reduce to one or other of tlte above canonical processes. M’hen constructing a simulation system for some specific problem, the first step is to define the goals and then to look for a suitable implementaion which achieves these goals. Brcausc we are interested in the construction of computer systems which are user.friendly and specific, at1 ol,viJus framework for the choice of such goals is usability desz$n (Gould et nl. [G]j. In this way, the above comparative assessment interpretation of sitnulation yields a modus operandi for the implementation of usability design. This is the point of view which is pursued in this paper, and discussed in some detail in Section 3. The way in which the comparisons are made depends explicitly on thca nature of the decision making under consideration. It can vary greatly from one situation to the next even though tile same basic mathematical problem is being solved. In some sitrations, such as market research, the process is quite ad hoc and subjective. In others, such as the design of lakes for pollution control, specific well-defined quantitative measures (eg. flow and residence time patterns) are use to choose between alternatives. Because the comparisons are what distinguishes the decision making in one problem from the next. their choice i; problem-context specific and therefore will not be pursued in great detail in this paper. In order to illustrate the above points and the related discussion below, we shall consider a specific example in Section 4; namely, the NESSIE system developed by menrbcrs of tlte SSiPO Uivision of Xlathematics and Statistics in the Yarralumla Laboratory in Canberra.

3. fb?&rsscn I Linking mathematics with applications

2. Simulation

and Comparative

471

Assessment

Wc now turn to an examination of the role of simulation mined problems, since they typify situations where the available more than one solution. It leads naturally to motivation for the tion of the compa.rative assessment process as well as for the role problem-solving.

in thr solution of undrrdcterdata a.nd constraints support. above mcnt,ionctI nlotlularizaof comparative assessment in

In one way or another, decision making is connected with examining altctrnativc>s ratI1cl than determining specific well-defined outcomes (though the latter invariably plays a s(~coiItlar~ role), and therefore is characterized by situations where tte available data ant1 constraints support more than one solution (usually, infinitely many). For example, when buying grorc+s. it is not a matter of calculating the cost of a number of given items, but of determining which items to buy in order to stay within budget while looking after the diet, and sustcnanc(~ of the family. The mathematical models which characterize such situations arr stiiti to 1)~ undrrdefermined. They are often classified as inverse or improperly posed problcms (though this classification includes other types of problems), in order to distinguish them from for.c~nrrl problems which is the name given to mathematical models for which the solution is fornlally a specific well-defined (properly posed) outcome. Hence, in the above example, c;ilculal.ing the cost of the groceries is the forward problem. Consequently, the solution of such problems reduces to Pnmpling and hypothc~s;zing a.bout the extent and structure of the non-uniqueness supported by thr available data and constraints. This process commences only when the relationship brtwc%rn the data and tllc. properties of interest have been formulated. Simulation in the form of trial-and-rrror and rrlatctl exploratory methods, such as Monte Carlo Inversion, is often used. The task of choosing F~OIII among the various alternatives which the simulation identifies falls to the decision maker. This leads naturally to the need to COMPARE, and helice to the conclusion that comparative of simulation assessment is the appropriate f:xmework in which to interpret the application to such situations. It is the structure of such exploratory methods which yields natural mot.ivation for the in such methods are (cf Rndrrsscn and modularizatien of equation (1). Th e k ey components Seneta [3], [4]): (i) select (BUILD

or AMEND)

a possible so!ution

from the class of admissablc

rc>alistic

solutions; (:i) for a given admissable solution, compute (SOLVE) the propcrtjcs which corresponds to t,he information required for decision making purposes (usually a. forward problrm); (iii) accept satisfy the relevant

as possible solutions only the OIICS for which thr (DISPLAY) data and constraints up to predetermined levels;

(ib.) COMPARE

the cocsistent

properties

of the accepted

solutions.

proprltirs

B. Anderssen I Linking mathematicswith applicarions

472

In this way, independent motivation is obtained for the implementatior, ative assessment process via the modularization of equation (1).

3. Usability

computer

of the compar-

Design

When one builds a commercial product, even when it is software system, the competing constraints include (i) identifying (ii) establishing (iii) protecting

a market

or a user-friendly

need;

a market edge; intellectual

property;

(iv) focussing

on the needs of the poteutial

(v) producing

quality;

(vi) meeting

deadlincz.

clients;

The first two constraints aim to ensure that one has a suitable product to develop and a potential market for its sale. The last two relate to the implementation of the development. The third is connected 14th how the overall development is managed. From the point of view of ensuring that the success of t.he product in the marketplace is maximized, it is the focusse’ng on the needs of ihe poleztia: clients which determines how the product (user-friendly system) is designed. In software engineering, such focusing is well recognized as the central activity in product design once the market. need and edge have been identified. is mentioned in the introduction, a mod :s operandi has been prcposed for the management of such focussing. It goes under the name of usability design (Gould et al. [6]). It therefore follows from the discvlssfo:. of the prel,iaus sect&s that comparative assessm.ent represents one way of responding to user needs in the sense uf al!owing them to focus on the comparative decision making associated with their problem, and consequently represents a possible framework in which product design and development can respond to the goals and aims of usability design. Since the

BUILD

+ SOLVE

+ DISPLAY

4 AMEND

-+ COMPA_RE

process and modularization

can be viewed as one way of implementing comparative assessment, it becomes, in the light of the above discussion, a practical framework in which to do product design and development in accordance with usability design principles. i’i’e now turn to explore some consequences

of usability

design for the implementa&n

of

B. Anderssen I Linking mathematics with clpplications

et al. [6] and others on usability

the above modularization. As is clear from the work of Gould design, the key design criteria (usability measures) are: (a!) easy to understand

473

and learn;

(b) easy and quick to use; (c) focusses on the information (d) facilitates (e) builds

comparative

the confidence

(fj explicitly (g) results

friendly available

required

for decision

making

purposes;

assessment; of the decision

(generates more-or-less

maker in addressing

enthusiasm

the task at hand;

in the user);

in real-time

on (colour)

hard c3py.

Succinct justification for these design criteria is encapsulated in the recognition that a computer system is user-friendly only when its use by the client leads rapidly to the conclusion that it supports the specific decision making process for which it was designed. If, as has already been of exploration and comparison ical model, then a framework for accommodating the above usability measures). The key modularization, are:

1. The client (BUILD).

must

explained above, simulation is viewed as the continuing process once the formulation step has yielded an appropriate mathematis needed for its implementation that yields a natural structure sets of constraints (competing commercial constraints and the components in such a framework, when it is based on the above

be able to specify the explicit

nature

of his problem

quickly

and easily

2. In order to ensure that the clien: is able to concentrate fully on the relevant decision making responsibilities within the context in which the probiem arose, matters connected wit,h how the problem is solved must be controlled by the designers and buildeis of the system (SOLVE). This need to differentiate clearly between decision-making and proble,rlsolving h_.; been discussed in some detail in Anderssen [I]. 3. The results PLAY). 4. Modifications (AMEND).

must relate to the information

of the original

5. The overall structure making (COMPARE).

must

problem

facilitate

required

for decision

must be easily derived

the comparative

making

purposes

from the original

assessment

process

(DIS-

problem

for decision

This type of modularization not only yields a useful framework in which to construct a user-friendly system such as NESSIE. It also yields, as the above discussion indicates, a natural

B. Anderssen ! Linking mathemtics with applicat’ms

474

framework in which to satisfy the above sets of constraints as t,hcy impinge on the spwific TJ ,is aspect will be pursued in some detail brlow WIIW SIWlI< problem under examination. is discussed. M’ithin the overall problem-solving process into which the design and construction of user-friendly systems fit, the identification stage accommodates the matters connected with marketing (ie (i) and (ii)), the formulate stage addresses matters connected wit.11ease of understanding and use (ie (a) and (b)), th e solve stage tackles matters connected with c!uality and stage mops up mattrrs connected intellectual property (ie (iii) and (v)), and the interpretation with clients’ needs and information used for decision making (ie (iv) and (c)j, In conclusion, we pinpoint, for the design of user-friendly systems, two ma.jor advarltagcls associated with the use of the above modularization within the framework of usability tlr+n. They are: 1. For the client, responsibilities.

it allows attention

to be focussed

on the decision

making

process

a.nd

2. Since the SOLVE activity is not of direct interest to the client (as long as it correctly evaluates the consequences for the types of situation which the client, const~cis using BUILD), the SOLVE module can remain under the direct conLro1 of the owns. lo. 2;;:; way, the essential (mathematical) intellectual property of the syster2 can be pndmic~(I by only allowing the client to have “remote” access to SGLVZ

4.

NESSIE

NESSIE was constructed specifically to assist environmental specialists and local govcrnment personnel with designing artificial lakes for pollution contra! and other purpose: [Andcrsscn et al. [Z]). The requirements of usability design force the development of the (soft.waIi,) prodilct to be focused on the client’s needs as the to;] priority. T1.e use of comparative assessment as outlined above as the basis for the implementation yielded a system where the client is able to BUILD an explicit representation of a possible lake; DISPLAY the flow and residence time patterns for that lake as determined by SOLVE; am AMEND the initial lake dcb,ign so that the flow and residence time patterns for different designs determined by SOLVE can bc COMPARED utilizing DISPLAY. In this way, the overall comparative assessment process reduces to exploring and compari!rg the flow and residence time patterns for difrerent lake designs. JJowever, the way in which the Information ir- t,hese patterns is lltilized depencls 011 the type of decision making under consideration. For exan,pie, the rcsidcncc time patt.crIls l>laJ the more important role when considerations of overall water ouality arc paramount. Implicitly, straints:

for us, the strategy

in designing

NESSIE

was gr:idcd

by the following

con-

3. Anderssen I Linking mathematics with applications

o the client is a decision the client * a product

475

maker who does riot wish to be bogged down by unnecessary

detaii;

wishes to s4ve a specific problem; is something

particular

and specialized

which must be highly focussed.

Without

at the time being fully aware of its existence, we implicitly followed the ?r.qnbi[i!y et al. [6]. For us, t.he overriding mot,ivation was the need to construct a mathematical product. In this way, we arrived at the above BUILD etc modularization for the construcion of NESSIE. design process of Gould

Acknowledgem~mt The support gratefully acknowledged.

and encouragement

of my colleague

(Dr.)

John

hloon~y

is

References [l] R. S. Andersser

7 Mathematics

in action,

Search 22 (1991), 203 - 205.

[2] R. S. Anderssen, C. R. Dietrich and P. Green, Designing artificial lakes a- pollution devices, Mathematics and Cumputers in Simulation 32 (1990), 77 - 82. [3] Ii. S. Arderssen and E. Seneta, A simple statistical estimation proccdurc inversion in geophysics, Pure and Appfied Geophysics 91 (I 971), 5-18.

for Monte

control

Carlo

[4] R. S. Artderssen

inversion (19’2j,

and E. Seneta, A simple statistical estimation procedure for Monte Carlo in geophysics II: efficiency and Hempel’s paradox, PzLre and Applied Groph!j.qic.+ 96 5-14.

/5] J. D. Bransford 1981.

and B. S. Stein,

The IDEAL

Problem

Solver,

W. H. Freeman,

16’ .J. D. Gould, S. J. Boies, and C. T,ewis, Making usable, useful, productivity computer applications, Communications cf the A CM 34 (1991), 75 - 85. [7] hl. S. Klamkin

(Editor),

Mathematic~f

Modelling,

SIAM, Philadelphia,

1987.

New York.

- enhancing