There are many possible uses for a reliable, easy to use, general purpose, nonlinear optimization code such as LSGRG2. Those described below represent a sample of applications which have been successfully carried out by LSGRG2 users. Additional details may be obtained from the references, or by sending an e-mail request to Leon Lasdon at email@example.com.
1. Blending of Chemical and Petroleum Products
This class of applications involves blending multiple products from several input stocks in order to minimize costs or to maximize revenues. Constraints include meeting product quality specifications (e.g., on octane number, vapor pressure, etc.) while satisfying product demands and limits on input stock availabilities. The use of nonlinear models permits more accurate representation of how product qualities depend on the qualities and quantities of input stocks. Successful applications of this type are described in more detail in reference 1. Some are used "on line" to control blending operations in real time, while others pertain to short or medium term planning. On line uses permit optimal correction of planned blends when conditions change unexpectedly, e.g., different component qualities or quantities, limited storage capacity, a forced mixing of new product with old, etc. Planning applications permit one to perform tradeoff studies of cost versus product quality specifications, or to examine the effect of new input streams.
Texaco designed an interactive blending system called OMEGA, which contained LSGRG2. It was used on line by blenders in all seven Texaco domestic refineries and in two foreign ones, and is also used in planning studies. OMEGA was credited with direct dollar savings conservatively estimated at $30 million annually, plus many intangible benefits. Reference 7, which describes Texaco's design and use of this system, was a finalist in the competition for 1987's best Management Science application. This reference appeared in a special issue of the journal "Interfaces" in 1988.
2. Engineering Design
LSGRG2 had been used in a wide variety of optimal design applications. These include heat exchanger network design (to minimize cost, see reference 1), design of offshore drilling rigs, and electronic system design (see references 2 and 3). Nonlinear optimization models permit imposing a wide variety of system performance constraints, and either meeting these at minimal cost or maximizing performance subject to cost limitations. Optimizer output permits the user to assess the effect on system cost of relaxing or tightening system specifications. Another major benefit is more efficient use of the designers' time. One optimization run can replace a time consuming trial and error process, aimed at satisfying all design specifications. In addition, GRG2 will find the best design meeting these constraints, or it will inform the designer that the constraints cannot all be satisfied, while producing a design that comes as close as possible to satisfying them.
LSGRG2 was been used by Bell Helicopter Textron and the US Army Applied Technology Laboratory for optimal design of helicopter and aircraft components, and for flight path optimization. Digital Equipment Corporation has studied the optimal "burn-in" of electronic components to achieve prescribed system reliability with LSGRG2, and the Naval Undersea Systems Center, in conjunction with Raytheon, Inc, have used it to design large sonar arrays for submarines. This application is described in reference 8.
3. Optimal Operation of Process Units
Computing power is now very cheap, while energy prices are high and fluctuate widely, and profit margins for most process industry firms are slim. This combination has focused the attention of many companies on how to operate units (or entire plants) in manufacturing processes to minimize costs while maintaining product qualities and meeting demands. Such applications are particularly common in the oil and chemical industries. Since models of process units are usually nonlinear, an efficient, general purpose nonlinear optimizer like LSGRG2 is needed. Applications may involve use of the optimizer on line to adjust process inputs as conditions change, or off line to compute optimal operating strategies under a variety of conditions.
LSGRG2 is by far the most widely used optimizer in this rapidly growing area. Its combination of reliability and speed, and the ease with which it can be embedded in a larger system, have led to its adoption by several firms for all their process optimization applications. It has also been used for these purposes by most of the oil and chemical companies listed earlier in this brochure.
4. Fitting Models to Data
A primary task of scientists and engineers is relating causes to effects. This is often done by conducting a set of experiments to measure inputs and outputs, postulating mathematical relations between them, and determining the parameters in these relations to best fit the measured data. If some coefficients appear nonlinearly, the fit cannot be done by conventional linear regression techniques. LSGRG2 solves such nonlinear regression problems very efficiently, while allowing the user to impose bounds or other constraints on the model coefficients. Most special purpose model fitting routines cannot handle constraints. Further, different fitting criteria can be tested - least squares, least sum of absolute deviations, and minimizing the largest absolute deviation.
One LSGRG2 user built an interactive system around LSGRG2, using it to fit a statistical distribution to failure rate data of a purchased product. The maximum likelihood criterion was used. System output permitted this client to gain better control of an important and costly incoming material.
Dow Chemical Co. embedded LSGRG2 in a system for designing and testing dynamic models called SimuSolv (see reference 9). SimuSolv deals with models composed of differential and algebraic equations, and contains facilities for estimating model parameters, solving, and optimizing. LSGRG2 is used in the estimation and optimization operations.
5. Product pricing under government regulations
Over some periods of time, various form of price controls have been imposed. These often prescribe limits on the (sales weighted) average price of a group of products, rather than on individual product prices. This gives the user the freedom to set these prices (subject to the overall limit) to achieve some goal, e.g., maximal revenues. There may be other restrictions on prices, stemming from marketing considerations. LSGRG2 has been successfully used in one such application. The resulting sophisticated pricing policy has yielded what the user feels is a major advantage over its competitors.
6. Optimal Control
LSGRG2 has been used to solve both discrete and continuous time optimal control problems. For continuous time problems, the time functions representing the independent (control) variables are discretized in some way. The differential or difference equations describing the system are solved within LSGRG2's user-provided model routine. The ability of LSGRG2 to handle constraints, including simple bounds on the controls and/or states, and terminal constraints, distinguish it from many other special purpose systems. One can also "customize" LSGRG2 for such applications by computing the derivatives it needs (using the adjoint equations) in another user-provided subroutine. This can dramatically increase accuracy and reduce run time.