**Ziggy MacDonald**- University of Leicester

Figure One: Linear Program Input Box

Having entered the problem formulation the user clicks the 'Preprocess' button which reveals the problem in simplex form via another Applet Window, shown in Figure Two, which reveals the initial basic feasible solution.

Figure Two: Simplex Formulation of the Problem

At this point the user can then choose how to proceed to the optimal solution. The 'Step' method allows the user to get involved with the solution generating process by determining the entering variable. At this point, I have to confess that I became just a little confused as the display did not appear to update between steps, although the message window did give some indication of what was going on. What is important, however, is that this is a clear indication of what is possible on the web and it suggests to me that as Java scripting develops, the sophistication of this type of material will greatly improve.

The other main site (WORMS) has put considerable effort in exploring the capabilities of Java scripting with its experimental JAVA section. This appears to be their 'work-in-progress' space (note) and an attempt has been made to reveal the efforts required to get Java to do the things that we want from the web. The real stuff of interest, however, is not available directly from WORMS (as far as I can tell), as it appear that the more advanced developments of this technology are available from "Simplex Place", a site written by the same author (Moshe Sniedovich) which lives on the same server. The end result is perhaps unrivalled on the web. In addition to a detailed explanation of the theory of the simplex method (written in a readable and lively style), the site provides an interactive simplex tool created with JavaScript which is embedded in the web page, removing the possible confusion of Applet windows floating around. It's a similar animal to the OTC's impressive Simplex Java Applet, but it lives in a far richer teaching environment and has more in common with the traditional methodological approach of textbooks. This is the essence of my particular attraction to this site. The user can discover the simplex method from the first principles of gaussian elimination, through the determination of entering variable (referred to as the Greedy Rule!) and leaving variable to the optimality test, and at each stage of the learning process there is a Java tool for practising the methods (with OTC's Simplex Applet, the instructional material is separate from the interactive tool). Having worked through this support material the user can then proceed to interact directly with each tableau on the road to a final solution. This is illustrated in figure Three.

Figure Three: Interactive Tableau

The student makes use of the tool in Figure Three as though it were a traditional simplex tableau (unlike that provided by OTC). In terms of learning, the student can use this tool to check all the calculations for each set of row operations required to pivot each tableau. What is particularly promising about this site is the author's obvious desire to keep on improving the tool. Promised features include a facility to describe the order of the problem and the ability to deal with 'greater than or equal to' constraints explicitly (i.e. by utilising the Big M tableau method). If this occurs by the time I teach the subject next academic session I will certainly be recommending this site to my students (in addition to Excel Solver!).

Figure Four: On-Line Graphical Solution

In Figure Four one can see the graphical solution to the 'Diet Problem', previously discussed in MacDonald (1996) and Estelles et al (1996), as created with this interactive tool. Not only can students enter the constraints etc. via a spread-sheet like interface and observe the corresponding graphical representation as it changes, the application is written such that one can directly alter the constraints by clicking and dragging the graphical view. As the graphical view is manipulated, the problem definition given above it is automatically resolved. Although the on-line version is perhaps a little sensitive (you have to remember to press enter after each change you make in the problem definition) this is clearly the way forward for interactive web-based teaching. Not only would I recommend this site to anyone charged with introducing students to linear programming, I will certainly be referring my third year business economics to this page.

Other miscellaneous resources include the Imperial College Management School OR-Library, maintained by John Beasley. This library contains data-sets for a variety of OR type problems from assignment problems to vehicle routing problems. The LP section contains twenty seven data files, but beyond this the material is of limited value. Perhaps of greater use is the Mathematical Programming Glossary, maintained by Harvey Greenberg. The site contains material contributed by a variety of sources (including a lot of LP-related material) and is excellent for brief but detailed information on all aspects of mathematical programming. Finally there is the standard LP FAQ (Frequently Asked Questions). The OTC maintains these for LP and they provide a useful resource for students wishing to gain an introductory insight into the world of LP.

MacDonald, Z., 1995, "Teaching Linear Programming using Microsoft Excel Solver", *Computers in Higher Education Economic Review*, 9 (3) 7-10

MacDonald, Z., 1996, "Economic Optimisation: An Excel Alternative to Estelles et al's GAMS Approach.", *Computers in Higher Education Economic Review*, 10 (3) 2-5

MPS format is the de facto standard ASCII medium among most of the commercial LP codes. MPS input format was originally introduced by IBM to express linear and integer programs in a standard way. The main feature of MPS format is that
it is a fixed column format (as opposed to entering the model as equations), and everything (variables, rows, etc.) gets a name, thus care must be taken that all information is placed in the correct columns.

Department of Economics

University of Leicester

University Road

Leicester. UK

LE1 7RH