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.