Cornput.. Envhn. and U&an SysWms, Vol. 14, pp. 167-I 70, 1990 0196-9716/90 $3.00 + .oo Printed in the USA. All tights resewed. Copyright 0 1990 Pergamon Press plc

COMPUTER APPLICATION CONSIDERATIONS

PARALLEL MICROCOMPUTING FOR DATA ENVELOPMENT ANALYSIS

Fred Phillips

IC2 Ins tit&e University of Texas at Austin

Ronald G. Parsons

Donoho Design Group, Inc.

Andrew Donoho

Donoho Design Group, Inc.

ABSTRACT. Data envelopment Analysis (DEA) is an eminently parallelizable process. The multiple linear programming (LP) problems that must be solved in a DEA application need not be solved in any particular or&r, and each LP can jit within a single “transputer.” An experiment using DEA on a newly available Levco Translink (2.5 A4FLOPS) board for the Macintosh personal computer is reported.

In the 1960s and early ‘7Os, operations research (OR) problems involving repeated linear programming (LP) runs took all night to execute on a dedicated mainframe. In the 198Os, the earliest personnel computer (PC) data envelopment analysis (DEA) code required several hours to run on an 8088 machine, for a problem of the size described in this study. The experiment described here brings the solution time for such a problem down to only a few seconds.

Although the Intel 80386 chips used in high-end IBM-compatible microcomputers may

show more raw numeric computing power than the Macintosh (Mac) circuitry, we have made the Mac the primary platform of our research for several reasons. First, the numeric work in which the Intel chips excel is only one part of a project that may include exploratory research, experimental design, data collection, data analysis, project tracking, correspondence, and preparation of reports and presentations. The near-universal cut-and-paste feature of Macintosh applications makes integrating these steps far more efficient for researchers and

more instructive for students, washing out the Intel chip’s slight edge in computation speed. Mac operating systems and application software have been “upwardly compatible,” eliminating the nonproductive time and expense of purchasing and learning new or updated software.

Requests for reprints shoukl be sent to Fred Phillips, IC2 Institute, University of Texas at Austin, 2815 San

Gabriel, Austin, Texas 78705.

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168 E Phillips, R. G. Parsons, and A. Donoho

Also, the graphics capabilities of the Motorola 68000 family of chips lead to attractive user interfaces. These interfaces accelerate the diffusion of new computer techniques, and this is an important consideration in applications-oriented research. Finally, parallel processing hard-

ware and good software drivers were first available for the Macintosh.

TRANSPUTERS

The transputer is a 20 MHz 32-bit RISC processor with 4 Kbytes of high-speed static RAM and a 1.5 MFlop 64-bit floating point processor. Made by the British company Inmos, it is based on Hoare’s Communicating Sequential Processes model of parallel computing (Davidson, 1988; Hoare, 1978). This model is characterized by (1) no data sharing; (2) data transfer by communication; and (3) synchronous communication. These features ensure that there is no bottleneck in accessing shared memory, that processes may reside on different com- puters, and that communication can act to synchronize the programs being executed on the dif- ferent machines.

UPool: Coordinating Parallel Processors for the Macintosh

UPool is a development tool for parallel-processing applications on Mac computers using the TransLink II NuBus boards manufactured by Levco Corporation of San Diego, California. These add-on boards contain varying numbers of transputers (currently up to four) and acceler- ate the Mac by lo-100 times for scientific and engineering calculations. Up001 facilitates mar- rying such parallel-processing systems to MacApp, Apple Computer’s object-oriented pro- gramming environment, decreasing the time and expense required to develop parallel-process- ing applications on the Mac.

The UPool package consists of a set of routines written in Object Pascal which are designed to allocate and schedule processors (in this case, transputers) under MacApp. The burden of allocating, scheduling, executing, and freeing processes is handled completely by UPool in an optimized manner. The MPW (Macintosh Programmer Workshop) Pascal Units contain object definitions and executable code written for use with MacApp 2.0 and MPW 3.0. The objects provide methods which may be used as-is or overridden to customize UPool for particular applications. The package also provides C source code for the transputer processes. The user adds transputer application code in an included file.

UPool’s designer (Donoho) was the author of the popular MacSpin package for visual data analysis. The present authors have combined this Exploratory Data Analysis (EDA) technique with DEA to the mutual benefit of both techniques. EDA has helped us to understand product markets and their geographic variations, to avoid statistical fallacies, and to screen input/output variables prior to executing DEA. It then helps us understand the DEA output and prepare pre- sentations and reports. In the other direction, DEA augments established EDA tools by intro- ducing the efficient/inefficient characterization of points whenever the set of variables can be partitioned sensibly into “inputs” and “outputs.”

A Parallel Computing Experiment With DEA

All tests were run on a Mac IIcx with a 68030 chip and a 68882 math coprocessor. We used the double precision option of an off-the-shelf simplex code for linear programming (Press et al., 1988) written in C. We wrote a simple DEA driver, also in C, to generate matrices and to call the simplex routine. Allocating tasks to the transputers was left to UPool (McLemore, Parsons & Donoho, 1989; Donoho, 1989).

We used the original CCR (Chames, Cooper & Rhodes, 1978) engineering ratio efficiency

Parallel Microcomputing 169

model to solve a problem with 54 decision making units (DMUs) and seven input/output vari- ables.

The first run was compiled and executed as if on a 68000 chip (the chip around which the original Macintosh was built), with numerical calculations done via SANE (Standard Apple Numerics Environment). Run time was 5 minutes, 24 seconds. The second run was compiled and executed as if on a 68020 chip, with numerical calculations done via direct calls to the

68882. Run time decreased sharply, to 14.6 seconds. The third run used the 68030 plus two transputers. Run time was between 5.6 and 5.7 seconds. A fourth run was made using four transputers, and the run time was 5.0 seconds. When we sacrificed numerical precision (going from double precision arithmetic to single precision arithmetic), the run time was reduced to 4.0 seconds.

IMPLICATIONS

Further research is needed to benchmark DEA models other than the CCR ratio model, and to incorporate the extended efficiency diagnostics that have characterized more recent theoreti- cal and applied work in DEA. The communication burden between the central processor and the transputers should further be investigated; this burden will put a bound on how fast solution times decrease relative to the number of transputers used. In DEA applications, the transputers do not have to communicate with each other. At present, we are beginning to see communica- tions bottlenecks, because roughly four kilobytes of data are pushed to each transputer for each DMU. However, rather than communicate the entire simplex matrix to the transputer for every DMU, a more sophisticated experiment would communicate only those (two) rows that need to be changed. This would reduce the communication bandwidth requirement by a factor of ten.

With four transputers per Levco board and five slots available in the Mac II, up to 20 trans- puters may be supported by a single Mac. Future products with eight transputers per board would double this, and of course distributed computing is possible under the Hoare model, uti-

lizing very large numbers of communicating transputers. The behavior of DEA in these mas- sively parallel environments is unknown.

One of our hypotheses had been that a user interface based on a matrix of scatter-plots - with plotted DMUs changing color according to their efficiency value as each is computed - would be the appropriate one. It is now evident that the computer we have assembled solves the DEA too fast for such an interface to be useful, as the human eye cannot make sense of 54 dots (replicated in n plots!) changing colors within a span of five seconds or less. Design work on an appropriate user interface continues.

This work opens the possibility of real-time efficiency analysis (with user interfaces that may be, for example, descendants of MacSpin) for fast, ongoing multifunction processes that can be monitored by instrumentation. Examples include police and emergency medical service (EMS) dispatching, evaluation of integrated circuit designs, etc. Related possibilities include interactive visual data analysis in the “foreground” while DEA or expert statistical systems are simultaneously computing in the “background.”

REFERENCES

Charnes, A., Cooper, W.W., & Rhodes, E. (1978, November). Measuring the efficiency of decision making units. European Journal of Operadonal Research, 2,6.

Davidson, C. (1988, January). Transputers and how to feed them. APDALq, 5-7. Donoho, A.W. (1989, April). UPOOL - Parallel processing on the Mac. Frameworks: The Journal of MacApp

Object-Oriented Program Development, 3, I. Hoare, C.A.R. Communicating sequential processes. Communs ACM, 21,8,666. The Mathematical Programming Society. (1988). The 13th International Symposium on Mathematical Programming,

170 E Phillips, R. G. Parsons, andA. Donoho

Tokyo. McLemore, C., Parsons, R.G.. & Donoho, A.W. (1988). WOOL: MacApp objectsfor multiprocessing. Austin, Texas:

Donoho Publishing Group. Press, W.H., Flanneay, BP.. Teukolski, LA., & Vetterling, W.T. (1988). Numerical recipes in C: The art of scientific

computing. Cambridge: Cambridge University Press.