# Optimization Means Higher Profits

## One way or another, every manufacturer must solve a basic optimization problem—how to produce the most valuable product at the least possible cost. Particularly in high volume, low margin industries like petrochemicals and consumer products, profits depend on squeezing every bit of finished product out of the available raw materials and saving every dime on labor and material costs.

01/01/1999

 KEY WORDS Software for control Optimization Linear Programming Modeling Control software Hierarchical control

One way or another, every manufacturer must solve a basic optimization problem-how to produce the most valuable product at the least possible cost. Particularly in high volume, low margin industries like petrochemicals and consumer products, profits depend on squeezing every bit of finished product out of the available raw materials and saving every dime on labor and material costs.

The industrial automation industry owes its existence to the search for greater profitability. The earliest loop controllers were designed to maintain all flow rates, pressures, and temperatures in a plant at levels required to maximize the quality of the end product. Higher quality products meant higher sale prices and higher profits. Today, integrated control systems are capable of not only maintaining individual process variables at their setpoints, but choosing the setpoints that optimize profitability of the entire process.

## Optimization example

Profit optimization can be a complicated problem involving thousands of income and expense variables. Any number of solutions is possible, but only one can be truly optimal and finding it can involve considerable mathematical effort. Consider instead a simpler optimization problem that every telephone user faces-getting the best deal on long distance phone service.

Suppose there are just three rate structures available for long distance phone calls. Plan A offers a flat fee of 10 cents per minute for all calls all week. Plan B charges \$1 for the first 20 minutes of each call plus 10 cents per minute thereafter. Plan C charges \$6.25 for unlimited weekend calling and 10 cents per minute for weekday calls. These are the conditions of the problem that implicitly define its solution.

So which is the better deal? That depends on the objectives and constraints of the problem as well as its conditions. Objectives define exactly what constitutes a 'good deal.' Constraints define which of the many possible solutions can even be considered. Solutions that don't fit within problem constraints are ruled out no matter how optimal they may appear on paper.

In this example, the objective for most callers would be to minimize each month's phone bill. Others might need the maximum amount of phone time that a fixed payment can buy each month. Smart callers will account for constraints such as the times they are home to use their phones and how long they're willing to allow each call to go on.

The 'best' deal will be different for callers with different objectives and different constraints. For a caller who talks a lot but only on weekends, plan C is the optimal solution to the problem. Conversely, plan A is optimal for a caller who makes a lot of short calls on weekdays and none on weekends. Plan B is best for weekday callers that average at least 10 minutes per call.

## Complications

Now consider the same problem from the point of view of the administrator for a large company's telephone system. Every caller in the company will have different objectives and different constraints, not all of which can be met simultaneously. This is a multivariable optimization problem. The overall optimal solution may or may not minimize the phone bill for any individual caller, but the company's total phone bill can be minimized by judiciously balancing all the competing requirements.

The problem becomes even more complicated in cases where long distance rates change during the day. This is a real-time optimization problem since the optimal solution depends on conditions that vary from minute to minute. Real-time optimization problems are particularly difficult to solve if the conditions of the problem vary randomly.

At the top of the complexity scale are multivariable, real-time, nonlinear optimization problems. Nonlinear problems involve conditions that can not be expressed with simple algebraic expressions like the phone rates. Nonlinear problems often have thousands of solutions that each appear optimal when compared to nearby solutions. It is often difficult to determine which of these local solutions gives the overall or globally optimal solution without actually checking each one.

## Solutions

Fortunately, there are solutions for all these problems and several software packages that can help find them. Some apply specifically to industrial control problems, others are more general purpose. OptimizePro from Visual Solutions (Westford, Mass.) is one example of the former. It can be used to determine optimal parameters for a closed loop controller such as the PID constants required to give minimal settling time plus minimal overshoot. OptimizePro starts with an initial guess supplied by the user and searches for the best solution that meets all of the constraints of the problem.

Among more popular general purpose optimization software packages are Matlab and Mathematica from the MathWorks (Natick, Mass.) and Wolfram Research (Champaign, Il.), respectively. MATLAB's Optimization Toolbox and Mathematica's Global Optimization toolset offer precoded algorithms for solving the most complex optimization problems. If a user can detail the conditions, objectives, and constraints of a problem in mathematical terms, either package can solve it numerically.

For example, suppose there are two phone users-Jack and Jill-at the Crown Hill Company. System administrator M. Goose knows that their calls each cost 10 cents a minute and that Jack always talks at least twice as much as Jill every month. If the total phone budget is limited to \$45 per month and Jill spends at least 100 minutes on the phone each month, what is the least that M. Goose can expect to pay for long distance service?

The diagram shows the answer. The objective, conditions, and constraints have all been reduced to algebraic expressions that the optimization software can handle. The user has also selected a procedure for solving the problem. There are several options, but this is a classic case for linear programming . A linear programming algorithm is essentially a trial-and-error search routine. It systematically selects certain solutions that barely satisfy the constraints, then identifies the one that best meets the problem's overall objective.

In this example, the answer is simple - M. Goose's total phone bill will be minimized if both Jack and Jill talk as little as possible. Note that this solution lies at the intersection of two constraints. This is a typical result for a constrained optimization problem. If the theoretical optimum can't be met exactly, the closest solution that barely meets the most restrictive constraints will have to do.

Honeywell's Profit Suite technology incorporates a layered approach to control and optimization. At the first level, Profit Controller provides localcontrol and optimization. Each Profit Controller provides multivariable model predictive control and dynaminc optimization for some portion of the process using Honeywell's RMPCT control technology. For large scale optimization and dynamically coordinates the underlying Proft Controllers. At both levels the optimal dynamic path to reach the optimal targets is determined on a minute-to-minute basis to account for interactions and control/constraint interactions. At the top level, ProfitMax uses a first-principles nonlinear model of the entire process to compute optimization targets and process gain and economic information which may then be downloaded to Profit Optimizer and Profit Controller.

## Real problems

This is exactly what happens in a typical processing plant, whether the end product is petroleum, chemicals, or dog food. Capacity and safety limits generally prevent the production process from being run flat out at its absolute maximum rate. An acceptable operating point must be found that meets all constraints and maximizes profitability as much as possible.

That's why linear programming is an essential tool in all of the most popular optimization software designed specifically for process control applications. Connoisseur from The Foxboro Company (Foxboro, Mass.), for example, uses a linear programming algorithm to search for optimal setpoints for all the loops in the process control system. A process mathematical model provides the conditions of the problem by specifying how each process variable depends on all the others.

Once the optimal setpoints have been determined, Connoisseur can automatically download them to a distributed control system (DCS) or a stand-alone loop controller for implementation. This is an example of hierarchical control . Each individual controller attempts to meet just its own individual objectives as defined by its setpoint. Setpoints that optimize overall objectives of the control system are chosen at a higher level in the control hierarchy. All interactions among the individual loops are accounted for in the choice of setpoints.

Hierarchical control systems with plant-wide optimization capabilities are also available from Honeywell Hi-Spec Solutions (Phoenix, Ariz.), Intelligent Optimization (Houston, Tex.), ADERSA (Verrieres-le-Buisson, France), and AspenTech (Houston, Tex.). Honeywell implements a different hierarchical control strategy with its Profit Suite technology (diagram).

## Better models

Several of these vendors also offer real-time optimization software. For example, RT-OPT from AspenTech monitors plant behavior of the plant and updates its process models accordingly. This accounts for any changes in the relationships among the process variables that may have occurred over time as a result of fouling, equipment wear, etc. With a more accurate, up-to-date process model, RT-OPT can safely push the plant to operate very close to its constraints. A less accurate model would require a more conservative operating point with a wider safety margin.

The quest for model accuracy has been an on-going concern for vendors and users of process optimization software. Sometimes a simple set of equations based on first principles such as the laws of thermodynamics and physics will suffice to represent behavior process. In larger plants, however, processes are much more complicated and the theoretical principles that govern behavior of the process are not entirely evident.

In such cases, an empirical model is often used to define conditions of the optimization problem. An empirical model involves equations that are fit to the process data numerically rather than analytically. The parameters of these equations may or may not have any physical significance, but if chosen correctly, the equations can still show how one process variable depends on all others. Process Perfecter from Pavilion Technologies (Austin, Tex.) and NeuCOP Optimizer from the NeuralWare division of AspenTech (Pittsburgh, Pa.) both use an artificial neural networks and historical process data to model and optimize a process empirically.

## Applications

It's no coincidence that most vendors mentioned so far in this article have offices, if not headquarters, in Texas. The commercialization of online optimization and control technology has been driven largely by the economic needs of the petrochemical industry. A refinery that produces 189,000 liters gallons of fuel per hour will net \$12,000 per day by improving their profitability by just one penny per gallon. ROMeo from SIMSCI (Brea, Calif.) and HYSYS Real Time Optimization (HYSYS.RTO+) from Hyprotech (Calgary, Alberta, Canada) are two more software packages designed to solve multivariable process optimization problems encountered in the refining and petrochemical industries.

GE Continental Control (Houston, Tex.) focuses its MVC optimization technology on chemical facilities in particular. Chemical processes tend to be more nonlinear than refining and require more sophisticated real-time modeling and optimization techniques. Other industries currently benefiting from optimization software include aerospace, mining, and food processing. The Foresight Optimizing Controller from PSE Optima (Salt Lake City, Ut.) is used for global optimization in all of these.

Optimization tools are constantly finding new applications in industries as diverse as polymers production, power generation, and pulp & paper. Improved optimization techniques are under development as well. Who knows, someone may eventually figure out how to find the best deal for long distance calls!

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