Mathematical Modeling in the Library

Jon Warwick
London South Bank University

How do you connect a control engineer with an academic librarian via an economist?  The popular view would probably be that the engineer works at the traditionally ‘hard’ end of the engineering discipline solving problems related to structure and design and using tools fashioned from mathematics and physics.  The librarian works in a social environment, dealing with people and trying to run a system in which books are loaned to the public and (hopefully!) returned.  To return to the original question, how do you connect mathematics and the study of physical processes with the need to run a social system as efficiently as possible in which the behaviour of people needs to be understood?  Strangely, one answer to this is further connected to the problems I encounter when trying to have an early morning hot shower every day.  The main problem is that when I start my shower I am rudely awakened by the fact that the water is cold.  I adjust the heat control and nothing happens so I turn it up more and then get drenched in scalding hot water so I yank the setting down to cold again.  This process continues until I eventually get the temperature about right.  There are two processes in operation here.  The first is a process of (my) expectations adjustment in that I assumed the water system would behave as I expected and be warm immediately and then had to adjust my expectations when it did not.  The second is the heat adjustment process which is a feedback mechanism.  Adjusting the heat control warms the water which I then sense and can make further adjustments.  Unfortunately, this feedback mechanism contains a significant delay so that when I adjust the heat control I expect the water to warm but it does not as it takes longer than I expect for the water to mix to the right temperature so rather than adjust my expectation (again) I turn the heat more which, of course, turns out to be too much.

This early morning experience illustrates the notions of expectation and feedback which we all experience on a regular basis every day of our lives.  Control engineers know a great deal about feedback systems and how to model them from the humble thermostat to the massive dampers used to prevent oscillation in bridges and buildings and when we add the notion of expectation modelling into the mix we begin to get a powerful set of tools that can be used investigate a huge range of systems, processes and effects across the spectrum of science, the arts and humanities.  What has this all got to do with a librarian?  Well, in this article we will see how some simple ideas from system dynamics and expectations theory can be used to help the librarian understand and optimise the library system and even predict how users will behave under certain circumstances.

First, we will take a look at expectations theory.  In many situations, the behaviour of a system can be significantly altered by the behaviour of actors within the system. For example, a new road planned to relieve pressure on an existing route finds itself swamped by traffic as drivers change route to take advantage of the new road.  In an academic library, students will rush to take advantage of a recommended text book newly stocked on the shelves.  But these actions are themselves governed by peoples expectations of their success – if a student thinks that everyone will be trying to borrow the new book then he or she may well not bother and if they all think this then the new book is not used.  So actions are governed by expectation of system performance which is itself governed by actions of actors within the system.  A circular problem!  This is not a new problem to economists, however, where future economic activity depends on current expectations.  For example, wage demands could lead to lower company profits which would negatively impact the ability to pay higher wages etc.  Economists suggest that a way forward is the use of rational expectations (see for example Begg, 1982) where an actor’s behaviour is a function of expectations (predictions) and ‘rational’ means that the expectations are exactly those that can be obtained from a model of the system.  So, if we know from the model how an actor’s expectations affect behaviour then together with his/her expectations of system performance this will determine behaviour.  Mathematically, if an actor takes action A then system performance will be s(A) and if the actor expects performance s then behaviour will be R[s].  So the action taken must satisfy A = R[s(A)].  This framework clearly implies feedback so we now need to add the ideas of the control engineer to produce a viable model linking action and system performance.

The founding father of the application of the principles of control engineering and feedback to managed systems was Jay Forrester (see Forrester, 1961) and a number of books have been written on the topic of system dynamics as the field has now become known (a good example is Coyle, 1996).  We will examine the idea of feedback and how it might assist the academic librarian in understanding the library system better.  Let us suppose that a group of students has been recommended to read a selection of books within their subject area.  Research in this area (Warwick, 1984) has shown that if a student tries to borrow a particular book from the library and it is not available then three actions are open to the student:  place a reservation for the book, return at a later date to try and borrow it, or give up using the library copies and perhaps buy the book.  Under the rational expectations model the probability of these actions occurring will depend on the benefit that the student expects to receive from each action – the greater the benefit then the greater the chance of that action.  We can use utility theory (Morgenstern and von Neuman, 1944) to measure the benefit derived from each action (Warwick, 1994) and this will depend, inter alia, on the type of book, how strongly it was recommended, how long it will be useful and so on.  Now, drawing on ideas from system dynamics we can construct an influence diagram showing the feedback mechanisms at play in the library system.  An example is shown in Figure 1 below exploring book issues and reservation choices only:

Figure 1:  Part of A Causal Loop Diagram of the Borrowing Process

Using this diagram the librarian can begin to see how feedback affects user expectations and hence behaviour.  In a causal loop diagram, the arrows show links between concepts and the ‘+’ or ‘-‘ sign indicates the direction of the relationship.  A ‘+’ sign indicates a direct relationship and a ‘-‘ sign an inverse relationship.  The ‘D’ written against some links indicates that there might be a significant delay along the link.  In diagrams such as this, we look for feedback loops that might give an indication of system behaviour and these come in two forms:  positive or reinforcing loops in which the overall effect is of increase or decrease without limit, and negative or balancing loops in which the overall effect is of re-establishing stability.  So, for example, in Figure 1 if the rate of demand to the library increases, then this increases library demands yet to be satisfied which increases the rate of book issue and the number of copies on loan.  However, with more copies on loan, the number of copies available on the shelf decreases so the expected utility of borrowing decreases (less likely to find a copy) and so demand will reduce.  This is a balancing loop.  Also the model can be used to highlight conflicting effects.  For example, increasing the loan period will reduce the return rate of books, reducing the number of copies on the shelf hence reducing the utility of borrowing.  But increasing the loan period will also reduce the utility of placing a reservation (you’d have to wait longer) which reduces the number of reservations and this eventually leads to an increased number of books on the shelf and an increased utility from borrowing – but there is likely to be significant delay in this effect.

A model such as this can now enable us to see expectations and feedback in action.  The model allows the estimation of system performance in response to an action to be made in the form of estimating the utility of that action.  The utility derived from that action then can be used to give us the likelihood that the action will be undertaken.  The librarian can assess the effects on user behaviour of changing library policy (perhaps the loan period or the number of copies available) and make decisions accordingly.  The next stage of analysis would be to construct a full system dynamics simulation model of the library system to allow quantification of these ideas but we will not proceed to that stage here.

So the academic librarian benefits from the work of the control engineer and the economist.  There are many other examples in the system dynamics literature of models developed in areas where one would least expect to see engineering principles offering useful insights including the arts (Alessi, 2005), society (Mahon, 1997) and teaching at all levels (Langheim, 2004).  I do suspect, however, that my morning shower tomorrow will follow the usual ritual – some expectations are difficult to shift!

References

Allessi, S. (2005) The Application of System Dynamics Modelling in Elementary and Secondary School Curricular. The University of Iowa,       http://www.c5.cl/ieinvestiga/actas/ribie2000/charlas/alessi.htm, accessed 27th October 2005.

Begg, D. K. H., (1982) The Rational Expectations Revolution in Macroeconomics.  Philip Allan.

Coyle, G. (1996) System Dynamics Modelling. Chapman and Hall.

Forrester, J. (1961) Industrial Dynamics. MIT Press, Cambridge MA.

Langheim, R, (2004) The State of Education:  An Examination of Systems Thinking in the K-12 Environment in the United States. 22nd International Conference of the Systems Dynamics Society, Oxford, England.

Mahon, I. (1997) Simulation of a System Collapse:  the Case of Easter Island. 15th International Conference of the Systems Dynamics Society, Istanbul, Turkey.

Morgenstern, O. and von Neuman, J. (1944) Theory of Games and Economic Behaviour. Princeton University Press.

Warwick, J. (1994) “Measuring the Benefit Derived by Students from Recommended Undergraduate Texts”. Collection Management, 18(3/4), pp. 139-151.

Warwick, J. (1984) An Operational Research Study of Duplication Policy for Undergraduate Recommended Reading. PhD Thesis, CNAA.