What is this “Pareto Frontier”?
Part of my research concerns the usage of what is called “multi-objective optimization” to making chemical processes function more efficiently. An important concept, which I have pondered in other areas since learning about it, is the Pareto frontier.
First let’s examine a single-objective optimization problem of interest to a chemical engineer. Suppose we are manufacturing refrigerators, and we want to minimize the production cost of a fridge. We can create a big function for the price of a single refrigerator, input all of the important constraints on the optimization, and then use MATLAB or gPROMS or CPLEX to solve the problem. Okay, great, we have built a cheap fridge. But there a whole host of other objectives that are also of great interest to the consumer, and despite higher costs, may compel the consumer to buy a competitor’s more expensive fridge than ours. What other kinds of objectives might be favorable for us to optimize over? Well, I can brainstorm a few:
- Minimize power consumption
- Maximize expected lifetime of operation
- Minimize environmental impact
- Maximize volume (for a big fridge)
- Minimize volume (for a compact fridge)
- Minimize noisiness/vibrations
- Minimize purchase cost
(For perspective, this is a comparatively small list of objectives; some lens design problems in the field of optics can have several hundred competing objectives describing lens quality.)
In general, it is impossible to simultaneously extremize all of these objectives with a single refrigerator design. This is where multiobjective optimization comes into play. In multiobjective optimization, we search for what is called the Pareto frontier, which is a multiobjective analog to an optimum for a single objective. We search for the set of designs for which there is no “strictly better” (or, “dominated”) design available. For example, let’s consider a case of two objectives, of minimizing power consumption and minimizing purchase cost. Look at these possible designs:
- $800 capital cost, 40 kwh power consumption
- $500, 80 kwh
- $1000 capital cost, 30 kwh
- $900, 60 kwh
The red design is completely inferior to the first design, since the refrigerator costs more AND uses more power. The other two refrigerators cannot be said to be inferior, since they either cost more and use less power, or cost less and use more power.
A Pareto frontier is the set of what is called the “non-dominated set” of refrigerator designs. Graphically, it would look something like this:
We see the expensive fridges use less power, and vice versa. Inferior designs lie above and the right of the green curve.
The Pareto frontier is a useful concept for describing all sorts of phenomena described by competing objectives. Businesses, which are in direct competition with other businesses for similar products, can make money by carving out a niche elsewhere on the Pareto frontier that their competitors have not entered yet. For example, McDonald’s offers very cheap burgers of disputable quality. Five Guys however, offers much better quality food… for more money. Serious money can be made by breakthroughs which expand the boundary of the Pareto frontier – that is, by producing Five Guys-quality food for the same price as McDonald’s. Expanding the Pareto frontier however (for the burger joint example), would require research and development into identifying and refining production processes, employee training, cooking equipment design, ingredient quality, farming practices, and meat-raising practices.
The Pareto Frontier and My Views on Dating
The Pareto frontier neatly summarizes the human practice of dating and intimate relationships as well. This is a realm where the number of competing objectives and constraints explodes. As a male, here are some things I look for in a female, and a list of constraints:
- Maximize beauty
- Maximize classiness
- Maximize intelligence
- Good conversationalist
- Maximum sex drive
- Maximize mental health
- Would make a good mother
- No felony convictions or dishonorable discharges from the military. (number of felonies/DD’s is an integer constraint)
- A cut-off on debt-to-income ratio (no high credit card balances and no high student loan balances; this is a continuous constraint)
- No drug involvement (boolean yes-no)
- No children (integer constraint, with a remote exception allowed for a widow who’s husband died in an accident or in the Iraq/Afghanistan wars).
- Hard constraints on classiness: no tramp stamps, no sexual body piercings, no tattooed breasts, no neck or facial tattoos, no facial piercings except for small gem nose piercing (no nose rings) (a set of boolean yes-no constraints)
- Is employed or in school (boolean)
- Is not obese (percentage body fat is a continuous variable)
- No serious mental health problems (set of booleans)
- No violent tendencies (semi-fuzzy, though the number of convictions for violent crimes would be quantitative)
- No serious expensive health problems, like diabetes (booleans)
- Is between the ages of 26 and 32 (integer)
- Is not married (boolean)
- Zero or one divorces (integer)
- No sexually transmitted diseases (set of booleans for each possible STD)
- Is a Christian (boolean)
It is interesting to note that the objectives here are fuzzy and difficult to quantify, but the constraints are almost entirely quantitative in nature (either integers, boolean true-false, or continuous variables). It is also interesting to note that these constraints are entirely related to personal behavior on the part of the female. The reader might complain about my list of objectives, but the actual objectives are not what affects the problem most strongly. The constraints affect the problem far more than the objectives do, since this is what carves out my “feasible set” of women. When feasibility is satisfied, then I give that female consideration. The saying that there is “plenty of fish in the sea” does not ring true, since while there are approximately 150 million females in America, the number of feasible dating partners plummets when put through the sieve above. A very small percentage of women satisfy these constraints. Perhaps I am too picky, but I see little compelling reason to involve myself with women that don’t act right.
Most males my age have completely flip-flopped the constraints I have above with the objectives. I am attempting to find the most physically attractive female I can under constraints related to her behavior. For other guys, it is the complete opposite. They will attempt to “maximize the quality of her behavior” under hard constraints on physical beauty, such as:
- Hair color
- Minimum breast size
- Minimum height
- Certain ethnicity/race
- Physically attractive body type, e.g. nice legs, flat stomach, thin figure
- Athletic/gym-going habits
It is no wonder that males my age get into so much trouble with females. Eventually, guys figure out to do the optimization in the reverse manner. But how much damage they do to themselves before they figure it out depends on their own intelligence, stubbornness, and capability of getting their manhood under control.