Risky Business--How to Optimize a Hydrocarbon Processing Plant
It's All About the Trade-Offs
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By Pierre R. Latour
Pierre R. Latour is president, Clifftent, Inc., firstname.lastname@example.org
Decisions in life involve optimizing risky tradeoffs. Let's see if you recognize and appreciate risky tradeoff problems and opportunities, even if you can't solve them exactly.
Football punter – A football punter, standing at the middle of the 50-yard line, wishes to make a "coffin corner" kick; that is, to kick the ball out of bounds as close to the goal line as possible, but not into the end zone. If successful, the opposing team will get the ball at the yard line where it goes out of bounds; if unsuccessful, it will get the ball at the 20-yard line. Assume the ball travels in a straight line, has enough velocity to cross the side line or goal line, and there is no run-back. If the kicker's launch angle has a normal distribution with a standard deviation σ of 7.5º from where he aims, at what sideline marker should the punter aim so that the expected value of the opposing team's starting position is minimized? What is that expected value (in yards from their goal line)?
A football field, not counting the end zones, is 300 feet by 160 feet. --Steven E Bradley, AZ A'77, Brain Ticklers, The Bent of Tau Beta Pi, Fall 2010. (See answer at end of article.)
Car driver speed limit – There is more or less incentive to drive fast to the posted limit (or beyond) depending on the consequences of being late. And there is more or less penalty for exceeding the limit depending on weather, location and likelihood of a ticket. Further there is some uncertainty about the speedometer reading, traffic radar gun reading and fuzz buster effectiveness. Each driver continuously optimizes his value proposition related to speed, depending on his perception of the shape of his value tradeoff tent and the size of the mishap cliff somewhere beyond the limit. Since 10 mph is too slow and 100 mph is too fast, an optimum exists in between. Goldilocks was correct; not too fast; not too slow; just right.
Offshore oil wells – Down-hole drill mud pressure must be somewhat higher than the expected reservoir pressure to prevent blowouts like the Deepwater Horizon in the Gulf of Mexico. But the lower the pressure, the less the mud supply tank nitrogen blanket compression costs. An important risky profit tradeoff must be optimized for the pressure controller setpoint.
Home thermostat – The winter comfort value increases with temperature setpoint until about 75 ºF and vanishes somewhere above that. Furnace fuel cost increases faster than linearly with temperature until the furnace is at maximum capacity. So a smooth value (comfort – cost) tradeoff hill exists for every thermostat, even one for Earth1.
Crude pipeline temperature – An Alaskan crude pipeline has heating elements for temperature control. Heating crude oil reduces its viscosity, allowing higher pumping rates and profit. But heating costs and corrosion rates increase with temperature, and equipment wear and tear increases can overshadow throughput gains. The profit vs. temperature setpoint tradeoff exists, whether one knows it or not.
Refinery vacuum distillation temperature – Vacuum unit heater coil outlet temperature increases will upgrade lower-value residue to gas oil. But if the temperature is too high, typically 780 ºF, residue entrainment from higher vapor velocity quickly contaminates the gas oil, decreasing its value, and furnace or heat recovery limits may be exceeded, causing severe penalties. Temperature must be set just right, depending on oil rate, molecular weight, pressure, steam, equipment capabilities and economics.
While everyone makes tradeoff determinations like these daily, based on experience, training, emotion and judgment, a rigorous mathematical optimization, using accurate inputs, will undoubtedly give superior results and improve learning and decision-making. This is a fundamental tenant of science, mathematics and engineering.
AGO Pour Point Trade-Off Picture
Consider how a refiner should set atmospheric gas oil (AGO) pour point setpoint illustrated in Figure 1.
Red tent – A typical risky tradeoff is shown for the atmospheric gas oil pour point2. The top red curve with peak at 30 ºC is the steady-state profit tent with peak profit = 100 cents per barrel barrel. This red curve of perfection can only be achieved with σ = 0, perfect control.
Blue hill – The blue frequency distribution data is centered at 28 ºC, with σ = 1.25 ºC.
Pink hill – When the product of the red and blue curves is integrated, the risky expected value profit, $85.27, is on the pink hill at 28 ºC. Moving the blue distribution left and right and reintegrating it generates the pink hill with a top at 28.36 ºC worth $86.00.
Green hill – If σ is cut in half to 0.625 ºC, the green distribution profit at 28.36 ºC increases from $86.00 to $92.95 on the brown hill.
Brown hill – When the product of the red and green curves is integrated, the risky expected value profit is on the brown hill, with the new top at 28.81 ºC worth $94.45. If σ returns to 1.25, the blue distribution again, profit at 28.81 ºC drops from the brown hill, $94.45, to the pink hill, $85.00, so one should back off to 28.36 ºC to regain $86.00.
Profit meters – These risky profit hills for each key performance indicator (KPI) and controlled variable (CV) are the profit meter, showing where you stand relative to the hilltop and red peak of perfection if you got your inputs right. Just point to your current position on the hill3.
If the brown hill is correct, any input data error will mislead to a different best mean and a corresponding loss down the brown hill. This proves incorrect data always leads to measurable financial losses. This method quantifies that loss. Honesty is indeed the best policy.
All you have to do is find the right hill and stay on top. This is how to manage risk. It optimizes setpoint setting and quantifies the financial value of improved dynamic performance, reduced variance. This also quantifies the mantra: It is better to play it on the safe side. The hilltop obeys the law of diminishing returns.
This one-independent-variable-at-a time optimization theory guarantees a unique solution, so anyone who properly deploys it will perform better on average than those with equivalent talent and skill who do not.
The greatest gain from reduced variance comes at the hilltop where the curvature is greatest; the gain diminishes as the mean is further removed from the top and the clifftent function approaches linear, so deviations cancel. This is why control engineers should focus on aligning setpoints to economics first for easy profit, before working on reducing variance, even where gains are greatest. To do this, they must know the clifftent for each CV.
There is an optimum percentage of the 30 ºC limit violations and an optimum margin limit – hilltop mean, nσ. (The optimum nσ is never 6σ; it depends on the tradeoff situation.)
Finally, the conventional process control benefit claim procedure is incorrect on three counts. First, it assumes the starting mean is best; this is wrong. Strike one. Then it assumes reduction in variance about that mean makes no money; this is wrong. Strike two. Finally, it assumes that with better control, the new mean is moved to the new best value; this is wrong. Strike three. The conventional procedure is arbitrary and excludes the very real variance reduction benefit that has remained intangible and hidden4-6 for so long.
How the Hydrocarbon Processing Industry Operates
Operating chemical manufacturing businesses is a pillar of chemical engineering, after product and plant development. The way the hydrocarbon processing industry (HPI) operates its plants can be described in a straightforward way.
Purpose – What is the purpose of operating an oil refinery or the HPI? Maximum production? Optimum yields? On spec products with no quality giveaway? Minimum energy? Minimum inventory? Minimum maintenance? Maximum reliability? Perfect safety? Zero emissions? Obey the law? Customer satisfaction? Job security? Job satisfaction? Shareholder value? Can all these goals be unified?
A useful objective for optimizing risky HPI operating tradeoffs is maximizing expected value profit over appropriate periods4-15. The universal HPI performance measure is expected value profit rate, not maximum production, not minimum energy, not minimum inventory, not zero mishaps.
Situation – How do you operate an oil refinery2, 6, 13, 14 or petrochemical7 plant? HPI operation consists of setting important operating conditions such as production rate, product quality, efficiency and inventory. "Important" means the settings are economically significant. These targets, limits and specs are directly related to control system setpoints and constraints for flows, temperatures, pressures and levels. The settings should be set right, and the plant should be held at them. If plant constraints are set too conservatively, the allowable operating domain is too restricted, and profitability suffers. If constraints are set too liberally, the allowable domain exceeds feasibility, and ugly things happen.
HPI operations are filled with tradeoffs, risky tradeoffs and penalty pitfall cliffs. As in all decisions of life, HPI operations require optimizing risky tradeoffs6. The only thing operators can influence is the mean and variance of CVs and KPIs. Even major equipment changes only provide new feasible setpoints. Every significant CV/KPI has a risky tradeoff associated with it. Every CV/KPI has a profit meter3, 6 associated with it.
Profit tradeoffs are shaped like a tent, invariably concave downward, with a peak at the proper limit. Many CV profit functions have a discontinuity near the peak, a cliff. Exceeding limits can have serious penalties. HPI operations are full of cliffs2, 6. Do not fall off big ones very often. But maximum profit results from tight control near the cliff edge.
Questions – How do you optimize tradeoffs? Are risks involved? How do you account for risk? Is safety involved? Do economics matter? Does the maintenance state of the equipment matter? Is there any established method for doing it right? Any best practice? Any universal decision method? Or is it up to organization, empiricism, engineering judgment, culture and management experience? Is an IT provider a profit or cost center? If the financial value of IT services, products and solutions cannot be quantified, they should be a cost center, based on the faith theory,8 experience or judgment. If they can quantify their financial value, they can become a profit center.
IT Problems – Is there a standard decision process for gathering pertinent information and setting setpoints? Does the value of data, information, measurement and control depend upon what you do with it? If so, is there a standard method for doing something with it? Deciding what to do to set setpoints? If not, then it will not be possible to predict the value of data, information, measurement, control or unclear decisions.
A major mantra for IT since its inception is decision support, providing appropriate and timely information so that managers and operators can make better decisions to operate the HPI. The value of data, instruments, computers, models, algorithms, software, services and information depends on what you do with it, obviously. If using something makes more money and not using it makes less, it should be credited for adding value. However, there has never been a rigorous method for best practice decision making for setting operating condition means2, 6, 7 and, hence, no way to quantify the contribution of improved information and equipment from those decisions or even a method to define precisely what information is needed for those operating decisions. Clifftent eliminates this technological impediment for IT2-13. It finally provides a rigorous operating decision making method that shows the value lost when incorrect data is used.
Business – In the early 1990s it became apparent the instrumentation, process control and IT businesses selling products, services, models, technology and projects were failing to identify, capture and sustain significant profits for the HPI and themselves8,9.
Chemical engineering has not provided a comprehensive method for aligning process control system setpoints to economics3, 6. Information technology hasn't either. Process control reduces variance, but its value could not be properly quantified until Clifftent integrated it with optimum mean setting in 19966
Consequences – It is now clear that current decision-making practice
- Cannot be assured that it is operating at its best,
- Cannot measure operating performance relative to its base case or perfection,
- Cannot reconcile disconnected performance measures,
- Cannot learn to improve operation well from imprecise decision methods,
- Cannot quantify that any deviations from optimum operation loses money2, 7
- Cannot quantify the value of reduced variance and improved dynamic performance of systems6, 8
- Cannot properly determine the value of data, information, components, models, controllers or solutions,
- Cannot manage risk consistently
How should the HPI operate?
The HPI should operate as it always has, but determine operating setpoints with a rigorous mathematical method, using accurate data, with a unified purpose; that is, Clifftent6
Solution – Shoring up the foundation of and repairing the disconnected contributions to HPI operations, the HPI must adopt best operating practices. This means defining the decisions to be made and the proper method for determining them. Then determine the required input data, establish procedures for assuring input accuracy and ensure faithful deployment and use. Analyze past performance for sustained improvement. Decision support requirements are clear and value measurable.
Re-optimize risky tradeoffs rigorously whenever any change occurs. Adopt a mathematically sound method with correct inputs for a proper unified purpose. Since 1996, Clifftent6 has offered a rigorous method for setting CV means. It provides a way to take into account risk factors; i.e., accounting for the chances of violating constraints versus the consequence of violating them. It is built on the steady state profit as a function of the controlled variable mean, which is always a tent- shaped trade-off, sometimes with a discontinuous cliff at the constraint peak. It is based on the discovery that every CV has a financial tradeoff, not too high and not too low, just right. An optimum always exists. Of course these business 101 truisms are well-known. That is why Clifftent fits so naturally into solution businesses offering profit performance.
Method – Here is how to optimize risky tradeoffs.
- Establish a unified objective such as expected value profit rate over the near term.
- Specify a base case CV mean, usually its current value.
- Determine the location and size of any profit cliff in the neighborhood of the limit and maximum theoretical profit at the limit.
- Set process model requirements for credits as each CV approaches its limit and the damage when it is exceeded.
- Specify economic gain sensitivity to approach the limit and loss when exceeding the limit.
- Determine the steady-state profit rate vs. CV/KPI mean. This is the financial sensitivity function. It is always shaped like a tradeoff tent, often with a discontinuous cliff in the neighborhood of a limit value.
- Determine the uncertainty in each CV/KPI; forecast its near term statistical distribution (some are not Gaussian) and variance. This is a primary duty of operators, before resetting setpoints.
- Calculate the expected value profit hill function vs. CV mean and locate the maximum hilltop CV and corresponding profit. Calculate best move size, expected gain and new expected percentage of spec violation.
- Study best mean change and corresponding profit gain from the suboptimum base to the optimum settings for reasonableness and merit. Specify the specific IT information needed for setpoint decisions, optimization of risky tradeoffs and holding HPI operations at maximum profit rate.
- Alter any input assumptions and resolve for a new optimum mean and corresponding profit change. Implement the change and confirm the results
Uses – Here are important applications.
- Deploy a comprehensive, holistic method for tracking correct information (the truth) for every significant CV/KPI.
- Set operating conditions to maximize risky tradeoff expected value profit.
- Several applications have been solved: low sulfur fuel oil sulfur6, vessel vapor velocity13, alky DIB nC4in top iC414, olefin plant7 C3 in C3=, ACU cuts and FCC regenerator slide valve pressure drop.
- Design and use profit meters for every CV/KPI of interest in operating management dashboards. Align HPI operations to its economics. This is worth >$1/barrel crude for oil refineries2, 8, 12, 13.
- Learn from your history to forecast near-term CV variance to manage risks mathematically.
- Determine the incentive and value of IT with the mathematical proof that acting on correct data always maximizes profit; acting on incorrect data always results in a quantifiable loss. This was first done for AGO pour point2.
- Provide a clear justification tool for components such as instruments, analyzers, laboratories, control algorithms, models, computers, actuators, valves, and alarm shutdown systems, that can specify their performance contribution to CV performance above their competitor or existing components. Control components must reduce CV variance or improve accuracy. IT components must improve tradeoff profit function accuracy.
- Measure financial performance of installed solutions and alarm systems so they can be licensed fairly based on shared-risk and shared-reward business partnerships to maximize expected value profit to both operating company user and technology solution sustainer.
HPI Operations Engineering
Once the proper method for HPI operations is adopted, operations will remain at optimum and performance can be measured.
Discoveries – You cannot properly determine the value of reduced variance without optimizing the mean first. In fact the farther the mean is from its optimum, the less the value of reduced variance because deviations effects cancel out. Its maximum value is about the optimum mean where deviation effects are additive. Setpoint optimization cannot be properly done without accounting for variance risk.
The use and value of control and optimization are not separate; they are fundamentally linked. The value of each component of Clifftent, equipment and information can be determined by their contribution to determining and holding the optimum mean.
Human judgment and experience can now focus on the accuracy, reality, validity and truth of useful input data for near term forecasts; setpoint determination decision-making follows naturally. Operator focus changes from CV setpoint setting to CV variance forecasting. Those with sufficient know-how of their business and corresponding CV clifftents can reap measurable gains with little cost; others must develop that business know-how to capture and sustain higher profit rates8, 9.
Model the physical and financial consequences of exceeding limits and specs like temperature, pressure, level, composition, vapor velocity Process Modeling –13, product quality, distillation flood, compressor surge/load, heater duty and FCC circulation reversals. This is just as important as process modeling the effects for approaching limits like these6. Clifftents shown in Figure 2 are real and essential.
Instruments – Measure the performance and value of instruments, models, APC, IT and CIM solutions on HPI operations. They must provide the correct data more accurately and promptly or reduce CV variance directly.
Multivariable Control – Set CV limits and measure dynamic control financial performance with clifftent. Quantify the financial value of controller tuning.
LP and Online Optimizers – Linear programming optimization is the standard tool for planning and scheduling oil refinery operations since 1970. Their solution is invariably at a constraint set boundary intersection of dependent variables. The LP constraint value inputs should be set by optimizing the risky tradeoff for each6.
On-line, real-time optimizers of basic process models such as olefin plants, catalytic crackers, distillation columns and blenders also find the best constraint set boundary intersection of dependent variables used for setting setpoints. However they don't really determine the best setpoint value; that is an optimizer input for the constraint. So the constraint value inputs to these optimizers should be set by optimizing the risky tradeoff for each16, 17.
If the financial value of IT can be quantified rigorously, if the cost of delivery and sustainability of that performance is substantially less than the value creating a sustaining profit stream, and if that profit is shared equitably between the HPI opco customer and the IT solution provider, then IT has a chance to become a profit center. IT –
All HPI Equipment –The Clifftent risky tradeoff optimization method allows measurement of the value added by any component of a control, alarm or information system. Since it is universal, it can also be used to determine the value of any plant equipment change, since they all have a bearing on the best operating conditions: flows, temperatures, efficiencies, compositions, constraints and economics. Just identify all CVs affected, update their clifftents and sum the value of changing them to the new optimum operating settings.
Business – Adopting this best practice assures the HPI it is making the best of the hands it is holding18-23.
Operating companies are more willing to share a percentage of the profit action with solution providers when they see clear profit gains and providers are dedicated to identifying, capturing and sustaining significant benefits for them. The adversarial relationship between both parties hungry for profits matures to a lasting business partnership where both win. This allows solution providers to license based on value-added financial performance9, 10, 12.
Operating companies seeking performance guarantees should operate with Clifftent, show their potential solution partner their base case operation and weaknesses, and solicit the provider's offer for a percentage of the improvement he creates, where it comes from, how he creates it and the commercial risk he assumes. It is conceivable the operating company can secure a risk-free, somewhat variable profit stream with minimum risk, which is the purpose of performance guarantees.
This means if a solution provider can deliver sustained performance2 of about $1.3/barrel crude at a cost of about $0.1/barrel, he can offer an operating company a no-risk variable annuity worth about $1.0/barrel, net. For a 200-kbpd (thousands of barrels per day) oil refinery, that is $72 kk/year or net present value (NPV) (72, 30 years, 10%) = $680 kk. And the solution provider nets NPV = $136 kk. All they have to agree on is how to measure performance and that 100/20 profit split.
Process Control Practice Renewal
The 2010 call for renewal of HPI process control and IT practices24 was built on this standard method for setpoint decision-making. Process control engineers now have the way to prove the value of their work and stay on real hilltops.
Clifftent is not a fad like Quality Is Job 1, ISO9000, Six Sigma, DCS, integration, CIM, APC, automation, real-time optimization and cloud computing, because it is the fundamental mathematical way to operate the HPI by optimizing risky tradeoffs using universal ideas of process behavior, economic sensitivity, statistics and calculus. It unifies setting every CV/KPI setpoint, limit, target and tolerance for all HPI plants to maximize a single, universal performance measure; expected value profit rate. All HPI plants are or should already be operated this way; Clifftent only improves the ad hoc, imprecise methods used everywhere from the beginning. It directly converts know-how into profit.
To maximize his opponents' runback at 100 yards, a perfect σ = 0 punter should always take dead aim at the coffin corner intersection of the goal and side lines, an angle from the middle of the 50 yard line of arc tan 150/80 = 61.93º. But the real σ = 7.5º punter is best advised to play it on the safe side and aim at the six-yard line, an angle of arc tan 132/80 = 58.78º. His expected value runback would average from the 14-yard line. Of course, he should practice to reduce σ <7.5º and then aim closer to the coffin corner.
--Steven E Bradley, AZ A'77, Brain Ticklers, TheBent of Tau Beta Pi, Winter 2011
Optimization theory guarantees any punter that uses this method properly will out-perform, on average, all other punters of equal talent and skill, σ = 7.5º, who do not.
- Latour, P.R., "Engineering Earth's Thermostat with CO2?", Invited Speech, The Industry Forecast Forum, Hydrocarbon Processing, Omni Hotel, Houston, TX, December 3, 2009 and Hydrocarbon Processing, V89, n2, February 2010, pp. 25, 28.
- Latour, P.R., "Align HPI Operations to Economics – Clifftent Optimizes Risky Tradeoffs at Limits," Hydrocarbon Processing, Vol. 87, No.12, December 2008, pp. 103-111
- Latour, P.R., "Why Tune Control Loops? Why Make Control Loops?" Hydrocarbon Processing, Vol. 81, No. 9, September 2002, pp. 11-12.
- Latour, P.R., "Modeling Intangible, Hidden Benefits from Better Product Quality Control," International Conference on Productivity and Quality in the Hydrocarbon Process Industry, Hydrocarbon Processing Magazine/Coopers and Lybrand, Houston, Texas, February 27, 1992.
- Latour, P.R., "Quantify quality control's intangible benefits", Hydrocarbon Processing, V71, n5, May 1992, pp. 61-66.
- Latour, P.R., "Process Control: Clifftent Shows It's More Profitable than Expected," Hydrocarbon Processing, Vol. 75, No. 12, December 1996, pp. 75-80. Republished in Kane, Les, Ed, Advanced Process Control and Information Systems for the Process Industries, Gulf Publishing, Co, 1999, pp. 31-37.
- Latour, P.R., "Align Olefin Operations to Economics – Clifftent Optimizes Setpoints," Presented at 2007 Spring AIChE Meeting Ethylene Producers Conference, Houston, Texas, April 23, 2007. Published in Conference Proceedings CD.
- Latour, P.R., "Demise and Keys to the Rise of Process Control," Hydrocarbon Processing, Vol.85, No. 3, March 2006, pp. 71-80 and Letters to Editor, Process Control, Hydrocarbon Processing, Vol. 85, No, 6, June 2006, p. 42.
- Latour, P.R., "Does the HPI Do Its CIM Business Right?" Hydrocarbon Processing, Vol. 76, No. 7, July 1997, pp. 15-16 and "Optimize the $19-Billion CIMfuels Profit Split," Vol. 77, No. 6, June 1998, pp. 17-18.
- Latour, P.R., "Decisions about Risk Reduction," Letter to Editor, Hydrocarbon Processing, Vol. 80, No. 6, June 2001, p. 39.
- Latour, P.R., "Quantifying Financial Values," Hydrocarbon Processing, Vol. 80, No. 7, July 2001, pp. 13-14.
- Latour, P.R., "Why Invest in Process Control?" Control, Vol. XV, No, 5, May 2002, pp. 41-46.
- Latour, P.R., "Set Vapor Velocity Setpoints Properly," Hydrocarbon Processing, Vol. 85, No. 10, October 2006, pp. 51-56.
- Latour, P.R., "Align Alkylation Separation to Economics," Hydrocarbon Processing, Vol. 88, No.1, January 2009, p. 98.
- Latour, P.R., "APC for Min Maintenance or Max Profit? Part 1," Hydrocarbon Processing, Vol. 88, No.10, October 2009, p. 15, and "Part 2", Vol. 88, No. 11, November 2009, p. 13.
- Friedman, Y.Z. & Latour, P.R., "Dr. Pierre Latour's Views on APC", Hydrocarbon Processing, Vol. 84, No. 11, November 2005, pp. 17-18.
- Friedman, Y.Z. (& Martin, G.D., Latour, P.R.), "APC Survey", Exchange of Letters to the Editor, Hydrocarbon Processing, Vol. 85, No. 10, October 2006, pp. 45-46 and Vol. 85, No,11, November 2006, pp. 45-52.
- Latour, P.R., "Role of RIS/APC for Manufacturing RFG/LSD," National Petroleum Refiners Association 1994 Annual Meeting, San Antonio, Texas, March 21, 1994.
- Latour, P.R., "Mission: Plan to Use RIS/APC for Manufacturing RFG/LSD," Fuel Reformulation, Vol. 4, No. 4, July/August 1994, p. 20.
- Latour, P.R., "Benefits of Modern Refinery Information Systems for Manufacturing Cleaner Fuels," API Reformulated Fuels Conference, American Energy Week 1995, G. R. Brown Convention Center, Houston, Texas, January 31, 1995.
- Latour, P.R., "CIMFUELS," bi-monthly contributing editorial, FUEL Reformulation, September 1995 - February 1998.
- Latour, P.R., "Why Invest in Process Control?," Control, Vol. XV, No. 5, May 2002, pp. 41-46.
- McMahon, T.K. & Latour, P.R., "Clifftent for Process Optimization", Control, Vol. 17, No.12, December 2004, p. 66.
- Latour, P.R., "Process Control Practice Renewal 2010", Hydrocarbon Processing, Vol. 89, No. 4, April 2010, p. 94; "Process Control Practice Renewal 2010―Purpose," Vol. 89, No. 8, August 2010, p. 13; "Process Control Practice Renewal 2010―Performance," Vol. 89, No.10, October 2010, p. 15; "Process Control Practice Renewal―Consequences," Vol. 89, No. 12, December 2010, p. 86. "Process Control Practice Renewal―Select CVs", Vol. 90, No. 1, January 2011, p. 90.
- Randall Bartlett, "Thinking Like an Economist: A Guide to Rational Decision-Making," The Teaching Company, http://www.teach12.com/tgc/courses/course_detail.aspx?cid=5511
- Michael A Roberto, "The Art of Critical Decision-Making", The Teaching Company, http://www.teach12.com/tgc/courses/course_detail.aspx?cid=5932
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